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+{"text":"\n\\section{Introduction}\n\nThe inherent randomness of the wireless medium can be utilized for extracting a shared secret, since wireless channels exhibit the feature of \\emph{reciprocity}. This approach is referred to as channel-reciprocity based key generation (CRKG). The underlying assumption is that an eavesdropper (Eve) cannot obtain the same channel state, and thus cannot compute the key. The general feasibility of the approach has been reported by several early works in the literature~\\cite{src:li2006securing,src:wilson2007channel},\nwhich have been extended by subsequent studies related to practical key agreement~\\cite{src:liu2012exploiting,src:pierrot2013}. In particular, there have been some works that deal with the removal of temporal correlation, by methods like principal component analysis (PCA)~\\cite{src:chen2011secret}, beamforming~\\cite{src:madiseh2012applying} or linear prediction~\\cite{src:mcguire2014bounds}.   \n\nThroughout the paper, we use \\textit{cross-correlation}, \\textit{mutual information}, and \\emph{secret-key rates} as performance metric. The theoretical foundation of secret-key rates has been established by Maurer~\\cite{src:maurer1993} and Ahlswede et al.~\\cite{src:ahlswede1993}. They coined the information-theoretic \\emph{source-type model}, where Alice, Bob and Eve have access to a jointly random source, and derived bounds on the secret-key \\emph{capacity}. Their result is used in a large body of research, especially for Gaussian channels, e.g., reference~\\cite{src:wallace2009key} for a multi-observation model or \\cite{src:wilson2007channel} for the application to UWB channels. \n\nHowever, some of the popular beliefs regarding the capabilities of the eavesdropper have to be challenged. Many previous works, e.g.,~\\cite{DBLP:conf\/mobicom\/MathurTMYR08sh,DBLP:conf\/mobicom\/JanaPCKPK09sh}, have relied on the assumption that the channel of Alice-to-Bob gets uncorrelated to that of Eve, as long as Eve is positioned more than half a wavelength away from Alice and Bob, commonly referred to as Jake's model~\\cite[Chapter 3.2.1]{src:goldsmith2005wireless}. In the literature, this is usually referred to as \\emph{spatial decorrelation}~\\cite{src:zhang2016key}.  A study~\\cite{src:pierrot2013} has questioned this assumption by practical evaluation. Recently, a comprehensive study~\\cite{src:he2016toward} has shown that for many popular correlation models of scattering environments, the eavesdropper might obtain largely correlated observations, especially if Eve is located within the line-of-sight beam of Alice and Bob. \n\nIn this work, we intend  and shed more light on the threats for CRGK from passive eavesdropping. \nAs a consequence, we extend the work of~\\cite{src:he2016toward} by providing more elaborated practical measurements. We quantify the leakage of Alice and Bob in relation to Eve with respect to the distance, especially for low ranges that introduce near-field effects. The measurement setup is designed with the objective to generate \\emph{reproducible} results, such that we can justify\n\\emph{stationary} random processes. This is a fundamental necessity in order to obtain meaningful results, which has sometimes been overlooked in previous work. The cross-correlation and achievable secret-key rate serve as the performance metrics that indicate the common randomness available to Alice and Bob, and likewise, the information loss to Eve. We evaluate the metrics for the original data and the processed versions after down-sampling or decorrelation. The results demonstrate that the close physical presence of Eve in the communication setting significantly changes the channel statistics. This phenomenon is so far not covered by conventional channel models for CRKG.\n\nSection~\\ref{sec:systemmodel} introduces the system model and elaborates on both the processing of the measured data and the performance metrics on security. The measurement setup is described in section~\\ref{sec:measurements}. The evaluation and results of the measurement campaign are presented in section~\\ref{sec:evaluation}. Finally, section~\\ref{sec:conclusion} concludes the paper. \n\n\\section{System model} \n\\label{sec:systemmodel}\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[\nblock\/.style={draw, drop shadow, fill=white, rectangle, minimum height=0.75cm, minimum width=3.5em},\npublicch\/.style={draw, drop shadow, fill=white, rectangle, rounded corners,minimum height=1.5em, minimum width=2.5em},\n]\n\\node[block] (alice) at (0,0) {Alice};\t\n\\node[block] (eve) at ($(alice)+(2.5,-1.25)$) {Eve};\n\\node[block] (bob) at ($(alice)+(5,0)$) {Bob};\n\n\\draw[thick,->] ($(alice)+(1,0.2)$) -- node[above] {$h_{ab,k}$} ($(bob)+(-1,0.2)$);\n\\draw[thick,->] ($(bob)+(-1,-0.1)$) -- node[below] {$h_{ba,k}$} ($(alice)+(1,-0.1)$);\n\\draw[thick,->,dashed] (alice) -- node[below,pos=0.5] {$h_{ae,k}$} (eve);\n\\draw[thick,->,dashed] (bob) -- node[below,pos=0.3] {$h_{be,k}$} (eve);\n\\end{tikzpicture}\n\\caption{Overview of the system model.} \\label{fig:systemmodel}\n\\end{figure}\nAs depicted in Figure~\\ref{fig:systemmodel}, we consider Alice, Bob and Eve measuring the channel $h_{ab,k}\\in\\mathbb{R}$, $h_{ba,k}\\in\\mathbb{R}$, $h_{ae,k}\\in\\mathbb{R}$ and $h_{be,k}\\in\\mathbb{R}$, which represent the state of Alice-to-Bob, Bob-to-Alice, Alice-to-Eve and Bob-to-Eve channels, respectively, and $k$ denotes a discrete time instant. We model these variables as joint stationary and ergodic random processes. In general, Eve gets two channel states $\\left(h_{ae,k},h_{be,k}\\right)$, however, in this study we focus on $h_{ae,k}$ only. In the following, we use the labels $x_k:=h_{ba,k}$ for Alice, $y_k:=h_{ab,k}$ for Bob, and $z_k:=h_{ae,k}$ for Eve. Furthermore, we define the vector process\n$\\mybold{v}_k:=\\left(x_{k}, y_{k}, z_{k}\\right)^T$.\n\n\\subsection{Processing}\nFor different $k$, the random vectors $\\mybold{v}_k$ are likely to exhibit correlation in time, since the wireless channel is varying only slowly in indoor environments. In order to remove the temporal dependencies, we perform two alternative options of processing, namely either downsampling or decorrelation. We show both options for $x_{k}$ only, since we have the same processing for $y_{k}$ and $z_{k}$. \n\n\\subsubsection{Downsampling}\nIf we keep only every $N_m$th variable of the process $x_{k}$, we effectively\ndownsample by factor $N_m$ and obtain\n\\begin{align}\n\\label{eq:donwsampled}\nx_{k}^{\\text{ds}}=x_{kN_m}.\n\\end{align} \nThe generated $x^{\\text{ds}}_k$ can be assumed independent under the condition that the process does not exhibit any dependence after an interval of $N_m$ variables. Subsequently, we assume that the $\\mybold{v}^{\\text{ds}}_k=\\left(x_{k}^{\\text{ds}},y_{k}^{\\text{ds}},z_{k}^{\\text{ds}}\\right)^T$ are identically and independently distributed (i.i.d.) for different $k$. \n\n\\subsubsection{Decorrelation}\n We need to provide an estimator for the autocorrelation function \n\\begin{align}\n\\label{eq:autocorrest}\n\\hat{r}_{xx}[l] = \\frac{1}{N-l} \\sum_{i=0}^{N-l-1}x_{i}x_{i+l}.\n\\end{align}\nThis estimator is unbiased if the process is correlation-ergodic. The linear forward predictor for $x_{k}$ of order $N_m$ is given by\n\\begin{align}\n\\label{eq:predictor}\n\\hat{x}_{k} = \\sum_{i=1}^{N_m} a_i x_{k-i},\n\\end{align}\nwhere $a_i\\in\\mathbb{R}$ are parameter coefficients, which can be computed by Levinson-Durbin recursion based on Yule-Walker equations~\\cite{src:vaidyanathan2007the}. We define \n\\begin{align}\n\\label{eq:decorr}\nx_{k}^{\\text{de}} = x_{k}-\\hat{x}_{k}\n\\end{align}\nas \\emph{innovation sequence}, which is orthogonal to past $h_{ab,k-i}$ for $i>0$. However, orthogonal (or uncorrelated if zero-mean) variables do not necessarily imply independence, especially not \\emph{joint} independence of $\\mybold{v}^{\\text{de}}_k=\\left(x_{k}^{\\text{de}},y_{k}^{\\text{de}},z_{k}^{\\text{de}}\\right)^T$ for different $k$. Decorrelation is practically more relevant than downsampling (even if no i.i.d. can be achieved), since the information loss is significantly lower.\n\n\\subsection{Performance metrics}\nThroughout the paper, we use (1) the Pearson correlation and (2) secret-key rates as performance metrics for security.\n\n\\subsubsection{Pearson correlation}\nThe Pearson correlation provides a measure of linear dependence between two data series. The values span between $-1$ and $1$, where $1$ refers to absolute correlation, $0$ to no correlation, and $-1$ to perfect inverse correlation. It is a wide-used metric for secrecy of practical secret-key generation~\\cite{src:he2016toward}. Given a finite collection of $N$ pairs $\\left(x_{k},y_{k}\\right)$ from the process, we use the estimator\n\\begin{align}\n\\label{eq:pearson}\n\\rho_{xy}=\\frac{\\sum\\limits_{i=0}^{N-1}\\left(x_{i}-\\bar{x}\\right)\\left(y_{i}-\\bar{y}\\right)}{\\sqrt{\\sum\\limits_{i=0}^{N-1}\\left(x_{i}-\\bar{x}\\right)^2}\\sqrt{\\sum\\limits_{i=0}^{N-1}\\left(y_{i}-\\bar{y}\\right)^2}},\n\\end{align}\nwhere $\\bar{x}=\\frac{1}{N}\\sum_{j=0}^{N-1}x_{j}$ and $\\bar{y}=\\frac{1}{N}\\sum_{j=0}^{N-1}y_{j}$ are the sample means.\n\n\n\n\\subsubsection{Secret-key rate} \n\\label{sec:sk}\nWe introduce the information-theoretic secret-key rate and use the downsampled process~\\eqref{eq:donwsampled}.\nRecall that the $\\mybold{v}^{\\text{ds}}_k$ are i.i.d. We characterize $\\mybold{v}^{\\text{ds}}_k$ by the joint probability density function $f_{\\mybold{v}^{\\text{ds}}_k}$. We apply a lower bound on secret-key capacity based on the source-type model, under the following conditions:\n\\begin{enumerate}\n\\item The joint probability density function $f_{\\mybold{v}^{\\text{ds}}_k}$ is known a priori at all terminals.\n\\item Alice and Bob exchange messages over an authenticated, public channel with unlimited communication capacity.\n\\item Eve remains passive at all times.\n\\end{enumerate}\nSubsequently, the asymptotic bound is given by~\\cite{src:ahlswede1993}\n\\begin{align}\n\\label{eq:sklower}\nC_{\\text{sk}} &\\geq \\mui\\left(x_{k}^{\\text{ds}};y_{k}^{\\text{ds}}\\right) \\notag\\\\\n& \\qquad-\\min\\left[ \\mui\\left(x_{k}^{\\text{ds}};z_{k}^{\\text{ds}}\\right), \\mui\\left(y_{k}^{\\text{ds}};z_{k}^{\\text{ds}}\\right) \\right]=:R_{\\text{sk}}\n\\end{align}\nfor each $k$, since the process is stationary. Since the actual probability distributions are unknown in practice, we evaluate the lower bound~\\eqref{eq:sklower} by estimations, based on a finite number of measured samples. We utilize a $k$-nearest neighbor estimator (NNE) for the mutual information, which is based on the idea and implementation of~\\cite{src:kraskov2004}. Mutual information is a function of joint and marginal probability densities. For  a measure of the joint density, the estimator computes the distance between a tuple of samples and its $k$th-next neighbor. A similar approach is provided for the marginal densities. To best of our knowledge, the reliability of the NNE has not been studied systematically. However, results in~\\cite{src:kraskov2004} indicate that at least for multivariate Gaussian variables, the estimation error is very low if $N>10^4$ samples are used for the estimation.\n\nNote that the bound~\\eqref{eq:sklower} could have been defined with the original $\\mybold{v}_k$ or the decorrelated processes~\\eqref{eq:decorr}, such that less information is discarded than in case of downsampling. However, in order to obtain an accurate estimation of~\\eqref{eq:sklower}, we require i.i.d. samples for the two following reasons: \n\\begin{enumerate}\n\\item The bound~\\eqref{eq:sklower} has been derived under the assumption of an unlimited number of i.i.d. observations from a random source. Therefore, a value of $R_{\\text{sk}}$ measured in bits per observation, is meaningful only if the time series is i.i.d. as well.\n\\item The NNE of~\\cite{src:kraskov2004} requires i.i.d. samples, since it relies on Khinchin's theorem~\\cite[p. 277]{src:papoulis2002probability}. If the time series of samples exhibits some dependence in time, the estimator might induce an undesired bias.\n\\end{enumerate}\n \nTherefore, if we apply the process $\\mybold{v}_k$ or its decorrelated modification~\\eqref{eq:decorr}, we have an approximation of the lower bound $R_{\\text{sk}}$~\\eqref{eq:sklower} only. While approximating the common information of Alice and Bob is a rather \"safe\" option, we need to be cautious regarding Eve. In order to minimize the risk of underestimating Eve, we verify our results obtained from $\\mybold{v}_k$ or the decorrelated version~\\eqref{eq:decorr} by comparing them with the downsampling approach, since it provides a more accurate description of the information leakage to Eve. Unfortunately, by removing samples from the estimation, the NNE gets more biased.\n\n\n\n\n\\begin{figure}[h]\n\t\t\\centering\n\\includegraphics[width=0.5\\textwidth]{figures\/raumplan\/Raumplan_Setup2.pdf}\n  \t\t\\caption{The testbed includes several experimental setups for performance evaluation as well as for security analysis. \n  \t\tAlice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.}\n        \\label{fig:setup}\n\\end{figure}\n\n\\section{Measurements}\n\\label{sec:measurements}\nThe testbed is applied at the premises of our research group, which is an office area in a university building. \nAlice is positioned at a predestined access point position. Bob and Eve are mounted on an automated antenna positioning setup, which is located at several predestined \"end-device\" positions (cf.\\ Figure~\\ref{fig:setup}). For this, we choose positions which are representative for security-related IoT devices, such as doorknobs (keyless entry systems), window frames (perimeter fence intrusion sensor), and wall (motion detectors) positions. Due to a lack of space, in this version of the paper we restrict  ourselves to a description of one representative realization of all experiments. We will also provide a full version of the paper with results of $23$ further positionings in the building.\n\n\\begin{table}\n\\caption{Parameters of the measurement setup}\n\\begin{center}\n\\begin{tabular}{|l | c | l| }\n\t\\hline\n\tParameter &  Variable & Value \\\\\n\t\\hline\\hline\n\tSampling interval & $T_s$    & $100$ msec  \\\\\\hline\n\tProbing duration & $T_p$    & $<5$ msec  \\\\\\hline\n\tStep size & $\\Delta_d$    & $5$ mm  \\\\\\hline\n\tAccuracy of step size & $\\hat{\\Delta}_d$    & $\\pm 0.05$ mm  \\\\\\hline\n\tGeometrical distance Bob-Eve & $\\Delta_{BE}$    & $[0,30]$ cm  \\\\\\hline\t\n\tGeometrical distance Alice-Bob & $\\Delta_{AB}$    & $5$ m  \\\\\\hline\t\n\tSamples per step & $N$    & $3\\cdot 10^5$   \\\\\\hline\n\\end{tabular}\n\\end{center}\n\n\\label{tab:parameters}\n\\end{table}\n\nWe perform mobile, long-time narrow-band channel measurements on $2.4$~GHz (wavelength $12.5$~cm). The data exchange protocol is implemented on three Raspberry Pi $2$ platforms (credit-card sized computer). \nAll devices are equipped with a CC$2531$ USB enabled IEEE $802.15.4$ communication interface\\footnote{http:\/\/www.ti.com\/tool\/cc2531emk}. The CC$2531$ is a true SoC solution for IEEE $802.15.4$ applications, that is compatible to network layer standards for resource-constrained devices: ZigBee, WirelessHART, and 6LoWPAN. The platform is equipped with proprietary PCB antennas, i.e., \\textit{Meandered Inverted-F antenna} (MIFA), with the size of $5\\times 12$~mm. These antennas provide good performance with a small form factor. The platform and antenna design are widely used in commercial products and suited for systems where ultra-low-power consumption is required. \n\nIn order to establish common channel probing, Alice periodically sends data frames to Bob and waits for acknowledgments. Eve also receives these request-response pairs. When receiving a probe, all three devices extract Received Signal Strength Indicators (RSSI) values and, thus, can measure a channel-dependent sequence over time. For evaluation of the channel measurements, we store and process the realizations of  $\\mybold{v}_k:=\\left(x_k, y_k, z_k\\right)^T$, locally on a monitoring laptop\n\n\nTable~\\ref{tab:parameters} lists the relevant parameters of our measurement setup. We obtain a complete realization of $\\mybold{v}_k$ on every sampling interval $T_s=100$~msec. The protocol ensures that Alice, Bob, and Eve can probe the channel within a probing duration $T_p<5$~msec. We want to analyze the joint statistical properties of the samples with respect to the position of Eve in the scene. As a consequence, we apply an automated antenna positioning system, which is constructed from a low-reflective material, cf. Figure~\\ref{fig:setup}. \nIt moves the antenna of Eve on a linear guide towards the fixed antenna of Bob in step size $\\Delta_d=5$~mm with accuracy $\\hat{\\Delta}_d=\\pm 0.05$~mm. The variable distance $\\Delta_{BE}$ ranges from $0$ to $30$~cm in order to provide $60$ different locations. Alice's antenna is placed orthogonal to the linear guiding at a fixed distance $\\Delta_{AB}=5$~m. For each position of Eve's antenna, we record at least $N$ samples.  \n\nAlice and Bob extract the common randomness $x_k$ and $y_k$ from a time-varying channel. Since we aim for meaningful and reproducible results, we have to create an environment which provides the joint stationarity to the random process. \nTherefore, with a distance of $10$~cm to Alice's antenna, we deploy  a curtain of $30 \\times 30$~cm aluminum strips that continuously rotates at $ \\approx 0.1$~rotations per second, cf. Figure~\\ref{fig:setup}. However, the rotation itself inserts a deterministic component into the channel. The evolution of the self-dependence of channel gains  --- we show exemplary $x^{\\text{ds}}_k$ --- is illustrated in Figure~\\ref{fig:MI_cyclic}. It shows that the mutual information decays rapidly and vanishes after four samples, corresponding to approximately $400$~ms. However, due to the continuously rotating curtain of aluminum strips, we discover strong stochastical dependencies after $96$ samples, corresponding to approximately $9.6$~s. Therefore, we adapt a random source (Unix file \\texttt{\/dev\/urandom}) to the motor controller and program the instrument to rotate with random speed between $0.240$ rad\/s and $1$~rad\/s in random direction and with random interval lengths $0^\\circ, 1^\\circ, \\ldots 60^\\circ$ (uniformly distributed).\n Figure~\\ref{fig:MI_cyclic} shows that no strong stochastical dependencies are given anymore.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[%\nwidth=7cm,\nheight=4cm,\nscale only axis,\nxmin=0,\nxmax=120,\nxlabel={$l$},\nymin=0,\nymax=0.6,\nylabel={$I(x_0;x_l)$ [bits\/observation]},\nlegend style={draw=black,fill=white,legend cell align=left,at={(0.7,0.9)}}\n]\n\\addplot [color=blue,solid,very thick,dashed]\n  table[]{figures\/muiVsDelayAliceOld-1.tsv};\n\\addlegendentry{Continuous rotation}; \n\\addplot [color=red,solid,very thick]\n  table[]{figures\/muiVsDelayAliceNew-1.tsv};\n\\addlegendentry{Random rotation}; \n\\end{axis}\n\\end{tikzpicture}%\n\t\\caption{Self-dependence of channel gains with respect to time delay. Setup is equipped with aluminum strips of either continuous or random rotation.}\n\t\\label{fig:MI_cyclic}\n\\end{figure}\n\n\n\\section{Evaluation and Results}\n\\label{sec:evaluation}\nWe now use the experimental measurements to evaluate and compare the results of the Pearson correlation~\\eqref{eq:pearson}, mutual information, as well as the achievable bound of the secret-key capacity~\\eqref{eq:sklower}, as a function of attacker's distance $\\Delta_{BE}$ to Bob. We interpret the original measurements as realizations of $\\mybold{v}_k$. In addition, we have the decorrelated and downsampled outcomes, denoted by the processes $\\mybold{v}^{\\text{de}}_k$ and $\\mybold{v}^{\\text{ds}}_k$, respectively. The decorrelated samples are obtained by a linear prediction of order $N_m = 30$. To generate the i.i.d. random vectors $\\mybold{v}^{\\text{ds}}_k$ we downsample $\\mybold{v}_k$ by the factor $N_m = 30$. In subsubsection~\\ref{sec:sk}, we have already outlined the necessity of i.i.d. random vectors to obtain accurate estimations. This is not given for $\\mybold{v}_k$ and $\\mybold{v}^{\\text{de}}_k$, however, they provide valid approximations, as the results indicate later on.\nWe present three~Figures~\\ref{fig:original}, \\ref{fig:DS}, \\ref{fig:decorr} with three Subfigures a)-c) each, which are arranged in a 3x3 matrix on the next page. The \\emph{rows} denote the Figures as follows.\n\\begin{enumerate}\n\\item Fig.~\\ref{fig:original} illustrates the results for the \\emph{original} process $\\mybold{v}_k$.\n\\item Fig.~\\ref{fig:DS} shows the results for the \\emph{downsampled} process $\\mybold{v}^{\\text{ds}}_k$ of~\\eqref{eq:donwsampled}.\n\\item Fig.~\\ref{fig:decorr} depicts the results for the \\emph{decorrelated} process $\\mybold{v}^{\\text{de}}_k$ of~\\eqref{eq:decorr}.\n\\end{enumerate}\nThe \\emph{columns} constitute Subfigures as follows. For convenience, we introduce generic labels $X\\in\\left\\lbrace x_k,x^{\\text{de}}_k,x^{\\text{ds}}_k \\right\\rbrace$ for Alice, $Y\\in\\left\\lbrace y_k,y^{\\text{de}}_k,y^{\\text{ds}}_k \\right\\rbrace$ for Bob and $Z\\in\\left\\lbrace z_k,z^{\\text{de}}_k,z^{\\text{ds}}_k \\right\\rbrace$ for Eve.\n\\begin{enumerate}\n\\item Subfigures a) show the Pearson correlation~\\eqref{eq:pearson} vs. geometrical distance $\\Delta_{BE}$ between the three pairs (Alice$\\leftrightarrow$Bob $\\rho_{XZ}$, Alice$\\leftrightarrow$Eve $\\rho_{XY}$, Bob$\\leftrightarrow$Eve $\\rho_{YZ}$).\n\\item Subfigures b) zoom into the correlation $\\rho_{XY}$ of Alice$\\leftrightarrow$Bob.\n\\item Subfigures c) depict the three mutual information results ($I(X;Y)$, $I(X;Z)$, $I(Y;Z)$) and the secret-key rate $R_{\\text{sk}}$ of ~\\eqref{eq:sklower} vs. geometrical distance $\\Delta_{BE}$.\n\\end{enumerate}\n\nMost of the practical key generation schemes use downsampling or decorrelation on the original observations $\\mybold{v}_k$. We introduce the Figs. \\ref{fig:original}, \\ref{fig:DS} and \\ref{fig:decorr} in order to analyze whether downsampling and decorrelation obscure certain features of the channel that are important for the security evaluation of the system. We start with a comparison of the cross-correlation behavior between Alice and Bob, as well as to a potential attacker. By comparing Figure~\\ref{fig:original} (a-b) and Figure~\\ref{fig:DS} (a-b) we see that no significant differences in $\\rho_{XY}$ and $\\rho_{XZ}$ occur after downsampling. (Further, $\\rho_{XZ}$ and $\\rho_{YZ}$ are almost identical due to channel reciprocity between Alice and Bob.) The high similarity is due to the fact that even the process $\\mybold{v}_k$ does not exhibit much dependency in time, as already hinted in Figure~\\ref{fig:MI_cyclic}. As a consequence, the results obtained for $\\mybold{v}_k$ expose a valid approximation of the cross-correlation. As it can be seen from Figure~\\ref{fig:DS}, in case of downsampling the results are more noisy, since much fewer samples are available for the estimations.\n\n\n\\begin{figure*}[htp!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:original}\n\\end{figure*}\n\nAfter decorrelation, the results (see Figure~\\ref{fig:decorr}) show that (unlike in case of downsampling) the correlation decreases on average by $\\approx 0.05$, which can have a significant negative impact on the performance of a potential quantization scheme, cf.~\\cite[Figure 3]{WiComSec-Phy-QuantAna}. \nFurthermore, the difference between the minimum and maximum value significantly decreases. Whereas in the original (and downsampled) signal the difference is $0.995-0.98=0.015$, the difference is $0.97-0.89=0.08$ for the decorrelated signal. \nThis probably stems from errors of the autocorrelation estimate~\\eqref{eq:autocorrest}, which is necessary for the linear forward prediction. Another reason might be the Pearson correlation where single outliers (e.g., strong peaks) significantly influence the result. Analyzing the impact of decorrelation techniques on the reciprocity and security in detail is left for future work.\n\n\n\\begin{figure*}[htp!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:DS}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:decorr}\n\\end{figure*}\n\n\n\n\n\n\nBy analyzing the attacker's opportunity, we observe a wavelength dependent behavior of the correlation between $z_k$ and $x_k$ (or $y_k$), as illustrated in Subfigures a). The following findings hold for all three processes: $\\mybold{v}_k$, $\\mybold{v}^{\\text{ds}}_k$, $\\mybold{v}^{\\text{de}}_k$. The correlation vs. distance function $\\rho_{XZ}$ (and $\\rho_{YZ}$) looks similar to the channel diversity function known from Jake's model~\\cite{src:goldsmith2005wireless}, which is a zero-order Bessel function\\footnote{A zero-order Bessel function is expected for the cross-correlation behavior of two receivers if uniformly distributed scatterers are given. According to Jake's model the first zero correlation is given after $\\approx 0.4\\lambda$, where $\\lambda$ is the wavelength of the carrier~\\cite{src:goldsmith2005wireless,DBLP:books\/daglib\/0025266}.} (cf. Figure~\\ref{fig:bessel}). However, the highest correlation is not at distance $\\Delta_{BE} =0$, where the correlation is only $0.2$. The highest cross-correlation is given at a distance of $\\Delta_{BE} \\approx 12.5$~cm, which is the wavelength of the $2.4$~GHz carrier. The first correlation of zero is given at a distance of $4$~cm. \n\n\\begin{figure}[htp]\n\t\t\\centering\n\\includegraphics[width=0.375\\textwidth]{figures\/bessel\/bessel.pdf}\n  \t\t\\caption{Bessel function versus distance.}\n        \\label{fig:bessel}\n\\end{figure}\n\n\n\n\n\nNote that the cross-correlation behavior of $x_k$ to $y_k$ is not independent of Eve's antenna position. Figure~\\ref{fig:original}(b) illustrates the correlation behavior in detail. The correlation has an \"oscillating\" behavior with a wavelength of approximately $11$~cm, whereby at a distance of $5$~cm the curve decreases rapidly to the lowest level of $\\approx 0.98$. The reason for that might be the non-perfect uniformly distributed scatterers in the environment, which are the basis of Jake's model. \nThe oscillating behavior in Alice's and Bob's original observation is also given in the downsampled and decorrelated versions, cf. Figure~\\ref{fig:DS}(b) and Figure~\\ref{fig:decorr}(b).\nThis behavior is contradictory to theoretical approaches based on Jake's Doppler spectrum~\\cite{DBLP:books\/daglib\/0025266}. The reason might be because the narrow band fading models do not include coupling and near field effects between both antennas for the spatial evaluation of autocorrelation, cross-correlation, and power spectral density (cf. \\cite[Chapter 3.2]{src:goldsmith2005wireless}). \n\n\nThe boundary $B$ between the near field zone and the far field zone can usually be determined by the following relationship: $B\\geq \\frac{2D^2}{\\lambda}$, where $D$ is the largest antenna size~\\cite{DBLP:journals\/comnet\/DlugoszT10}. We estimated the size of our antenna to be $6$~cm. Therefore, the boundary is $\\approx 5.7$~cm. \nAnalyzing near field boundaries in detail is left for future work. \n\n\n\n\nCompared to the cross-correlation behavior between the i.i.d. samples   $x^{\\text{ds}}_k$ and $y^{\\text{ds}}_k$ (after downsampling), both mutual information $I(X;Y)$ and $R_{sk}$ have very similar oscillating behavior, shown in Subfigures c). The (minimum, maximum) values of the correlation are ($0.980$, $0.995$) and the ones of the mutual information are ($2.1$, $2.75$). \nBy analyzing Eve's observation, we see only a slight similarity between the mutual information $I(X;Z)$ (and $I(Y;Z)$) to the correlation behavior of her observation $\\rho_{XZ}$ (and $\\rho_{YZ}$). The similarity can be found by comparing the maximum absolute values. For instance, the highest correlation occurs at $10$~cm with a value of $0.5$, and corresponds to the highest mutual information of $0.5$~bits per sample. \nHowever, the Bessel-like behavior is not evident. Notably is the fact that the attackers observation $z_k$ does not significantly impact $R_{sk}$. Our results show that $R_{sk}$ is mainly dependent on $x_k$ and $y_k$. \nHowever, Eve's antenna affects Alice's and Bob's observation and, therefore, affects $R_{sk}$. Table~\\ref{tab:results} summarizes our results.\n\n\n\\begin{table}\n\\caption{Averaged results of our experiment.}\n\\begin{center}\n\\begin{tabular}{|l | c | c | c | }\n    \\hline\n    & $\\mybold{v}_k$ &  $\\mybold{v}^{\\text{ds}}_k$ &\n$\\mybold{v}^{\\text{de}}_k$ \\\\\n    \\hline\\hline\n    $\\rho_{x_k,y_k}$ & $\\approx 0.99$  & $\\approx 0.99$  & $0.94$  \\\\\\hline\n    $\\rho_{y_k,z_k}$ & $\\approx 0.09$  & $\\approx 0.09$  & $ \\approx\n0.07$  \\\\\\hline\n    $I(X;Y)$         & $\\approx 2.92$  & $\\approx 2.89$  & $\\approx \n2.31$  \\\\\\hline\n    $I(Y;Z)$         & $\\approx 0.26$  & $\\approx 0.27$  & $0.10$  \\\\\\hline\n    $R_{\\text{sk}}$ & $\\approx 2.67$  & $\\approx 2.63$  & $\\approx 2.22$  \\\\\\hline\n\n\\end{tabular}\n\\end{center}\n\n\\label{tab:results}\n\\end{table}\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, we have provided an important pillar to bridge the gap between theory and practice-oriented approaches for CRKG. Our experimental study helps to provide a better understanding of channel statistics in wireless environments for security applications. \nWe present reproducible results based on a relevant environment which justifies the joint stationarity of a random process.  \nWe show results of cross-correlation, mutual information and secret-key rates, which are dependent on attacker's (or third device's) position. \nAs a result, we discovered that the \\textit{observer effect} occurs, which most probably originates from near field distortions. \nWe believe the effect needs to be considered in the future. Common channel models like Jake's model for channel diversity need to be extended in order to be valid for key generation setups. Furthermore, it might be pertinent, for instance, to detect the proximity of Eve. Basing on our results two bidirectionally communicating nodes might recognize a third device, its relative position, and its motion in the proximity. Further studies might use complex-valued channel profiles to analyze  third party positioning based and motion based influences.\n\n\n\n\n\\section{Full Measurement}\n\\begin{figure*}\n\t\t\\centering\n\\includegraphics[width=0.75\\textwidth]{figures\/raumplan\/Raumplan_Setup.pdf}\n  \t\t\\caption{The testbed includes several experimental setups for performance evaluation as well as for security analysis. \tAlice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.}\n        \\label{fig:setup_ur}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_original_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_ds_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_decorr_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_original_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_ds_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_decorr_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_original_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_ds_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_decorr_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_original_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_ds_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_decorr_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_original_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_ds_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_decorr_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_original_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_ds_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_decorr_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_original_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_ds_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_decorr_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_original_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_ds_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_decorr_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_original_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_ds_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_decorr_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_original_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_ds_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_decorr_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_original_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_ds_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_decorr_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_original_11}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_ds_11}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_decorr_11}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_original_12}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_ds_12}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_decorr_12}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_original_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_ds_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_decorr_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_original_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_ds_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_decorr_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_original_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_ds_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_decorr_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_original_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_ds_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_decorr_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_original_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_ds_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_decorr_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_original_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_ds_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_decorr_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_original_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_ds_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_decorr_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_original_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_ds_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_decorr_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_original_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_ds_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_decorr_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_original_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_ds_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_decorr_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_original_23}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_ds_23}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_decorr_23}\n\\end{figure*}\n\n\n\\end{appendices}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHigher-order derivative terms play important roles in the several contexts, e.g., inflation models, modified gravity, renormalization of gravity, and so on. From a phenomenological and theoretical viewpoint, their embeddings into supersymmetry (SUSY) or supergravity (SUGRA) are also interesting. In particular, there exist many non-renormalizable terms in SUGRA and it is quite natural to consider the extension including higher-order derivative terms and the effects of them on cosmology and particle phenomenology. The higher-order derivative terms of a chiral superfield in 4D SUSY or SUGRA and their cosmological applications have been investigated so far, e.g., in Refs.~\\cite{Khoury:2010gb,Khoury:2011da,Baumann:2011nm,Farakos:2012je,Koehn:2012ar,Farakos:2012qu,Koehn:2012te,Farakos:2013zya,Gwyn:2014wna,Aoki:2014pna,Aoki:2015eba,Ciupke:2015msa,Bielleman:2016grv}. \n\n\nThe Dirac-Born-Infeld (DBI) action \\cite{Born:1934gh,Dirac:1962iy} includes such higher-order derivative terms. It was first proposed as a nonlinear generalization of Maxwell theory. The DBI action is also motivated by string theory, which is a promising candidate for a unified theory including gravity. In the context of string theory, an effective action of D-brane is described by a DBI-type action, which consists of Maxwell terms $F_{\\mu \\nu }$ as well as the ones of scalar fields $\\partial _{\\mu }\\phi^i\\partial _{\\nu }\\phi^jg_{ij}$ and a two-form $B_{\\mu \\nu }$ in general, \n\\begin{align}\nS_{\\rm{DBI}}=\\int d^Dx \\sqrt{-g}\\left( 1-\\sqrt{{\\rm{det}}(g_{\\mu \\nu }+ \\partial _{\\mu }\\phi ^i\\partial _{\\nu }\\phi ^jg_{ij}+B_{\\mu \\nu }+F_{\\mu \\nu })} \\right) . \\label{string DBI}\n\\end{align}\n\nSUSY Dp-brane actions in $D$ dimension are also important for the effective theory of superstring. With a component formalism, such actions have also been discussed in many literature. For example, in Refs.~\\cite{Aganagic:1996pe,Aganagic:1996nn}, the authors construct SUSY Dp-brane actions with local kappa symmetry based on a component formalism in 10 dimensional spacetime. In a similar way, the p-brane action in various dimensions has also been discussed in Ref.~\\cite{Bergshoeff:2013pia}. In Refs.~\\cite{Howe:1996mx,Cederwall:1996pv,Cederwall:1996ri,Bergshoeff:1996tu}, the SUSY Dp-brane in SUGRA background is constructed by considering the background super-vielbein on the brane and couplings between them. \n\nAn approach based on superfields is useful for constructing a manifestly SUSY invariant action and generalizing it. Within the formalism, such 4D $\\mathcal{N}=1$ SUSY extensions of the DBI action have been known partially. The DBI action of a vector superfield, which corresponds to the case with $\\phi^i=B_{\\mu \\nu }=0$ in Eq. $\\eqref{string DBI}$, is constructed in Refs.~\\cite{Cecotti:1986gb,Bagger:1996wp,Rocek:1997hi,Kuzenko:2002vk,Kuzenko:2005wh}. In particular, in Refs.~\\cite{Bagger:1996wp,Rocek:1997hi}, it is shown that such an action appears from the partial breaking of 4D $\\mathcal{N}=2$ SUSY. Its SUGRA embedding has also been discussed in Refs.~\\cite{Cecotti:1986gb,Kuzenko:2002vk,Kuzenko:2005wh,Abe:2015nxa}. Its application to inflation models has been investigated in Ref.~\\cite{Abe:2015fha}. Furthermore, in global SUSY,  multiple $U(1)$~\\cite{Ferrara:2014oka,Ferrara:2014nwa} and massive~\\cite{Ferrara:2015ixa}  extensions of the DBI action have been discussed. In particular, for the case with multiple U(1) vector multiplets, linear actions ~\\cite{Andrianopoli:2014mia}, general conditions for partial SUSY breaking ~\\cite{Andrianopoli:2015wqa,Andrianopoli:2015rpa}, and c-maps ~\\cite{Andrianopoli:2016eub} have also been discussed.\n \nFor the DBI action of scalar fields, which corresponds to the case with $F_{\\mu \\nu }=B_{\\mu \\nu }=0$ in Eq. $\\eqref{string DBI}$, its SUSY extension has been done via partially broken $\\mathcal{N}=2$ SUSY theory, where the Goldstino multiplet is an $\\mathcal{N}=1$ real linear superfield \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}. However, there has never been the SUGRA extension of the DBI action of a real linear superfield. In this paper, we discuss the embedding of the DBI action of a real linear superfield into SUGRA. The action of a chiral superfield can be found in Ref.~\\cite{Koehn:2012ar}. In general, it is known that the action with a chiral superfield can be rewritten in terms of the one with a real linear superfield, and vice versa (via linear-chiral duality \\cite{Siegel:1979ai}). Therefore, our action, which will be discussed in this paper, would be equivalent to that derived in Ref.~\\cite{Koehn:2012ar} through the duality transformation. We will discuss this point and the differences between their result and ours.\n\nIn Refs. \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}, the DBI action of a real linear multiplet is realized with a chiral multiplet, which is constrained by a specific $\\mathcal{N}=1$ SUSY constraint. We will investigate the corresponding constraint which is a key for the construction of DBI action, in SUGRA. To achieve this, we use a formulation based on conformal SUGRA \\cite{Kaku:1978nz,Kugo:1982cu,Kugo:1983mv}\\footnote{We will use the superconformal tensor calculus~\\cite{Kaku:1978nz,Kugo:1982cu,Kugo:1983mv}. See also another formulation, conformal superspace~\\cite{Butter:2009cp,Kugo:2016zzf}.}, where  one can treat off-shell SUGRA with different sets of auxiliary fields in a unified manner. Because of the restrictions on the SUGRA embedding of the ${\\cal N}=1$ constraint, we will find that the DBI action of a real linear superfield can be realized only in the so-called new minimal formulation of SUGRA. Furthermore, we will extend the DBI action to the matter coupled version of it.\n\nThe remaining parts of this paper are organized as follows. First, we will briefly review the SUSY DBI action of a real linear superfield in Sec.~\\ref{review}. There, we will find that the constraint imposed between a chiral and real linear superfield is important for the construction. Then, we will extend the constraint to that in conformal SUGRA in Sec.~\\ref{extension}. After a short review of conformal SUGRA, we will also review the concept of the {\\it{u-associated}} derivative which is crucial for the superconformal extension. Using this {\\it{u-associated}} derivative, we will complete the embedding and find that the constraint can be consistently realized in the new minimal SUGRA. With the constraint, we will construct the corresponding action in the new minimal SUGRA, and write down the bosonic component action in Sec.~\\ref{Component action}. The linear -chiral duality and the matter coupled extension will be also discussed there. Finally, we will discuss the correspondence and differences between results in related works and ours in Sec.~\\ref{discussion}, and summarize this paper in Sec.~\\ref{summary}. In Appendix.~\\ref{explicit}, the explicit components of the multiplet including the {\\it{u-associated}} derivative are shown.\n\nIn this paper, we use the unit $M_P=1$ where $M_P=2.4\\times 10^{18}$ GeV is the reduced Planck mass, and follow the conventions of \\cite{Wess:1992cp} in Sec.~\\ref{review} and of \\cite{Freedman:2012zz} in other parts. $a,b\\cdots$ denote Minkowski indices and $\\mu,\\nu\\cdots$ denote curved indices.\n\n\\section{Review of DBI action in global SUSY}\\label{review}\nIn this section, we briefly review the DBI action of a real linear superfield in global SUSY \\cite{Bagger:1997pi}.  We use a chiral superfield $X$ and a real linear superfield $L$ which satisfy the conditions,\n\\begin{align}\n\\bar{D}_{\\dot{\\alpha }} X=0, \\ \\ \\ D^2L=\\bar{D}^2L=0,\n\\end{align}\nwhere $D_{\\alpha }$ and $\\bar{D}_{\\dot{\\alpha }}$ are a SUSY spinor derivative and its complex conjugate.\nTo construct the DBI action for $L$, we consider the following constraint between $X$ and $L$,\n\\begin{align}\nX-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L=0 , \\label{global constraint}\n\\end{align}\nwhere $\\bar{X}$ is a complex conjugate of $X$ \\footnote{In Ref.~\\cite{Bagger:1997pi}, the constraint $\\eqref{global constraint}$ has been obtained from the tensor multiplet in $\\mathcal{N}=2$ SUSY through partial breaking of it. Here, we do not discuss its origin and we just use the constraint as a guideline to obtain the DBI action. In Sec.~\\ref{discussion}, we will briefly comment on the relation between the partial breaking of ${\\cal N}=2$ SUSY and our construction.}. The equation. $\\eqref{global constraint}$ can be solved with respect to $X$ and we obtain\n\\begin{align}\nX=\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L +\\frac{1}{2}\\bar{D}^2\\Biggl[ \\frac{D^{\\alpha }LD_{\\alpha }L\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L }{1-\\frac{1}{2}A+\\sqrt{1-A+\\frac{1}{4}B^2}}\\Biggr],\\label{solution for global constraint}\n\\end{align}\nwhere\n\\begin{align}\nA\\equiv \\frac{1}{2}\\{ D^2( \\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L)+{\\rm{h.c.}}  \\}, \\ \\ \\ B\\equiv \\frac{1}{2}\\{ D^2( \\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L)-{\\rm{h.c.}}  \\}.\n\\end{align}\nUsing this solution $\\eqref{solution for global constraint}$, we can construct the SUSY DBI action as \n\\begin{align}\n\\mathcal{L}=\\int d^2 \\theta X(L) +{\\rm{h.c.}}. \\label{global DBI}\n\\end{align}\nOne can check that the bosonic part of the Lagrangian $\\eqref{global DBI}$ produces,\n\\begin{align}\n\\mathcal{L}_B=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2} ,\\label{component form of global DBI}\n\\end{align}\nwhere $C$ and $B_a$ are a real scalar and a constrained vector satisfying $\\partial ^a B_a=0$, in the real linear superfield, and we use the notation $B\\cdot \\partial C \\equiv B^a \\partial _aC $. It is known that, through the linear-chiral duality, Eq. $\\eqref{component form of global DBI}$ produces the DBI action of a complex scalar, which can be interpreted as the 4D effective D3-brane action. \nWe call Eq. $\\eqref{component form of global DBI}$ the DBI action of a real linear superfield in this paper.\n\nIt is worth noting that Eq. $\\eqref{solution for global constraint}$ satisfies the nilpotency condition, i.e., $X^2=0$, due to the Grassmann property of the SUSY spinor derivative, $\\bar{D}_{\\dot{\\alpha }}$. This reflects the underlying Volkov-Akulov SUSY \\cite{Volkov:1972jx,Rocek:1978nb}. Instead of writing the action like Eq. $\\eqref{global DBI}$, we can also rewrite the same system imposing the constraint $\\eqref{global constraint}$ by a chiral superfield Lagrange multiplier $\\Lambda $,\n\\begin{align}\n\\mathcal{L}=\\int d^2 \\theta \\biggl[ X +\\Lambda \\left( X-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L\\right) +\\tilde{\\Lambda }X^2\\biggr]  +{\\rm{h.c.}}. \\label{global DBI with constraint}\n\\end{align}\nHere we have introduced another Lagrange multiplier $\\tilde{\\Lambda }$, which ensures the nilpotency of $X$. Indeed, we need not require this condition in the Lagrangian since $X$ satisfies $X^2=0$ after integrating out $\\Lambda$ first and solving $X$ with respect to $L$, but the condition is still consistent and makes the calculation simple as far as we focus on the bosonic part of the action, as we will see in the following section.\n\\section{Extension to 4D $\\mathcal{N}=1$ conformal SUGRA}\\label{extension}\nIn this section, we generalize the SUSY DBI action $\\eqref{global DBI with constraint}$ discussed in Sec.~\\ref{review} to that in SUGRA.  \n\n\\subsection{Review of conformal SUGRA} \\label{conformal SUGRA}\nTo construct the action in SUGRA, we use conformal SUGRA formulation. Then, let us briefly review the basics of the conformal SUGRA before proceeding to the specific construction of the DBI action. \n\nIn this formulation, there are extra gauge symmetries such as dilatation, $U(1)_A$ symmetry, S-SUSY and conformal boost in addition to translation, Lorentz transformation and SUSY. The commutation and anti-commutation relations are governed by the superconformal algebra and its representation $\\Phi$ called a superconformal multiplet has the following components, \n\\begin{align}\n\\Phi =\\{ \\mathcal{C}, \\mathcal{Z},\\mathcal{H},\\mathcal{K},\\mathcal{B}_a,\\Lambda ,\\mathcal{D}\\} ,\\label{general multiplet}\n\\end{align}\nwhere $\\mathcal{Z}$ and $\\Lambda $ are spinors; $\\mathcal{B}_a$ is a vector; the others are complex scalars.  We also denote the superconformal multiplet $\\Phi$ by its first component $\\mathcal{C}$,\n\\begin{align}\n\\Phi =\\langle \\mathcal{C} \\rangle ,\n\\end{align}\nwhere $\\langle  ...\\rangle $ represents the superconformal multiplet which has $\\mathcal{C}$ as the first component. $\\mathcal{C}$ must be invariant under the transformations of S-SUSY and conformal boost in order for $\\Phi=\\langle \\mathcal{C}\\rangle$ to be a superconformal multiplet \\cite{Kugo:1983mv}.\n \nA superconformal multiplet is characterized by the charge $(w,n)$ under dilatation and $U(1)_A$ symmetry called the Weyl weight and the chiral weight, respectively. For example, a chiral multiplet $X$ has $(w,w)$, in order to satisfy\n\\begin{align}\n\\bar{\\mathcal{D}}_{\\dot{\\alpha}}X=0, \\label{dbar}\n\\end{align}\nwhere $\\bar{\\mathcal{D}}_{\\dot{\\alpha}}$ is a spinor derivative \\cite{Kugo:1983mv}. For a real linear multiplet $L$ defined by, \n\\begin{align}\n\\Sigma L =\\bar{\\Sigma } L=0,\n\\end{align}\nwhere $\\Sigma $ ($\\bar{\\Sigma }$) is a (anti-) chiral projection operator, the values of each weight are determined as $(w,n)=(2,0)$. We will discuss these operators, $\\mathcal{D}_\\alpha$ and $\\Sigma $, more precisely in the following subsections.\n\nThe chiral multiplet consists of the following components, $\\{z,P_L\\chi,F\\}$, where $z$ and $F$ are complex scalars and $P_L\\chi$ is a chiral spinor; $P_L=(1+\\gamma_5)\/2$ is a left-handed projection operator. It is embedded into a general superconformal multiplet $\\eqref{general multiplet}$ as\n\\begin{align}\n\\{ z,-\\sqrt{2}iP_L\\chi , -F,iF,iD_az,0,0  \\} , \\label{embedding chiral}\n\\end{align}\nwhere $D_a$ is a superconformal covariant derivative. On the other hand, a real linear multiplet has components, $\\{C,Z,B_a\\}$, where $C$ is a real scalar, $Z$ is a Majorana spinor and $B_a$ is a constrained vector which satisfies $D^aB_a=0$. A real linear multiplet is embedded into a general superconformal multiplet $\\eqref{general multiplet}$ as\n\\begin{align}\n\\{ C,Z,0,0,B_a,-\\slash{D}Z,-\\Box C\\}, \\label{embedding linear}\n\\end{align}\nwhere $\\slash{D} \\equiv \\gamma ^a D_a$.\n\nFor later convenience, we also introduce a multiplication rule for superconformal multiplets. For a function of multiplets $f(\\mathcal{C}^I)$, where $I$ classifies different multiplets, we have\n\\begin{align}\n\\nonumber \\langle f({\\cal C}^I)\\rangle=\\biggl[ &f,f_I\\mathcal{Z}^I, f_I\\mathcal{H}^I-\\frac{1}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\mathcal{Z}^I, f_I\\mathcal{K}^I+\\frac{i}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\gamma _5\\mathcal{Z}^I, f_I\\mathcal{B}_a^I-\\frac{i}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\gamma_a \\gamma _5\\mathcal{Z}^I,\\\\\n\\nonumber &f_I\\Lambda^I-\\frac{i}{2}\\gamma _5\\left( \\mathcal{K}^I-\\slash{\\mathcal{B}}^I-i\\gamma _5\\slash{D}\\mathcal{C}^I+i\\gamma _5\\mathcal{H}^I\\right) f_{IJ}\\mathcal{Z}^J-\\frac{1}{4}\\left( \\bar{\\mathcal{Z}}^J\\mathcal{Z}^I\\right) \\mathcal{Z}^Kf_{IJK},\\\\\n\\nonumber &f_I\\mathcal{D}^ I+\\frac{1}{2}f_{IJ}\\left( \\mathcal{K}^I\\mathcal{K}^J+\\mathcal{H}^I\\mathcal{H}^J-\\mathcal{B}^{aI}\\mathcal{B}_a^J-D_a\\mathcal{C}^ID^a\\mathcal{C}^J-2\\bar{\\mathcal{Z}}^J\\Lambda^I-\\bar{\\mathcal{Z}}^J\\slash{D}\\mathcal{Z}^I\\right)\\\\\n&-\\frac{1}{4}f_{IJK}\\bar{\\mathcal{Z}}^J(\\mathcal{H}^K-i\\gamma _5\\mathcal{K}^K-i\\slash{\\mathcal{B}}^K\\gamma _5)\\mathcal{Z}^I+\\frac{1}{16}f_{IJKL}(\\bar{\\mathcal{Z}}^J\\mathcal{Z}^I)(\\bar{\\mathcal{Z}}^K\\mathcal{Z}^L) \\biggr] , \\label{formulaF}\n\\end{align}\nwhere $f_{IJ\\cdots }$ is $\\partial f\/\\partial \\mathcal{C}^I\\partial\\mathcal{C}^J\\cdots$ and $\\bar{\\mathcal{Z}}\\equiv \\mathcal{Z}^T\\hat{C}$ ($\\hat{C}$ is a charge conjugation matrix).\n\nWe also need action formulas to construct a superconformal action. For a chiral multiplet $X=\\{z,P_L\\chi,F\\}$ with its weight $(3,3)$, there exists the so-called F-term formula~\\cite{Kugo:1982cu}, \n\\begin{align}\n[X]_F=\\int d ^{4}x\\sqrt{-g}{\\rm{Re}} \\biggl[ F+\\frac{1}{\\sqrt{2}}  \\bar {\\psi }_{\\mu}\\gamma ^{\\mu}P _{L}\\chi +\\frac{1}{2}z\\bar {\\psi }_{\\mu}\\gamma ^{\\mu \\nu}P _{R}\\psi _{\\nu} \\biggr] , \\label{Fformula}\n\\end{align}\nwhere $\\psi _{\\mu}$ is a gravitino.\nFor a real multiplet $\\phi =\\{C,Z,H,K,B_a,\\Lambda,D\\}$ with its weight $(2,0)$, we can apply the following D-term formula~\\cite{Kugo:1982cu},\n\\begin{align}\n\\nonumber  [\\phi ]_D= \\int d^{4}x\\sqrt{-g}\\biggl[ &D-\\frac{1}{2}i\\bar{\\psi}\\cdot \\gamma \\gamma _{5}\\lambda  -\\frac{1}{3}CR+\\frac{1}{3}(C\\bar{\\psi }_{\\mu}\\gamma ^{\\mu \\rho \\sigma }-i\\bar{Z }\\gamma ^{\\rho \\sigma }\\gamma _{5})D_{\\rho }\\psi _{\\sigma }\\\\\n    &+\\frac{1}{4} \\varepsilon ^{abcd}\\bar {\\psi }_{a}\\gamma _{b}\\psi _{c}\\left(B _{d}-\\frac{1}{2}\\bar{\\psi }_{d}Z \\right)\\biggr] . \\label{Dformula}\n\\end{align}\nHere, all the components of $\\phi$ are real (Majorana).\n\nUsing these superconformal multiplets, the multiplication rule $\\eqref{formulaF}$, and the action formulas $\\eqref{Fformula}$ and $\\eqref{Dformula}$, we can construct superconformal invariant actions. Finally, we fix some parts of the extra gauge symmetries by imposing the condition to one of the superconformal multiplets $\\Phi_0$ called a compensator multiplet, and obtain the $\\rm{Poincar\\acute{e}}$ SUGRA action.\n\n\n\n\\subsection{{\\it{u-associated}} derivative} \\label{u-associated derivative}\nNow, we have prepared the tool for constructing the DBI action in SUGRA. Within the conformal SUGRA formulation, we will discuss a constraint corresponding to that in global SUSY, \n\\begin{align}\nX-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L=0 , \\label{global constraint 2}\n\\end{align}\nin the following. However, it seems to be a nontrivial task to extend the term including SUSY spinor derivatives,\n\\begin{align}\n\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L \\label{global derivative}\n\\end{align}\n to that in conformal SUGRA. \n \nTo treat the term $\\eqref{global derivative}$ in conformal SUGRA, we need the spinor derivative defined as a superconformal operation. In Ref.~\\cite{Kugo:1983mv}, it is pointed out that the spinor derivative in conformal SUGRA, $\\mathcal{D}_{\\alpha }$ ($\\bar{\\mathcal{D}}_{\\dot{\\alpha }}$), cannot be defined on a superconformal multiplet $\\Phi$ unless $\\Phi$ satisfies a specific weight condition, $w=-n$ ($w =n$). This is because $\\mathcal{D}_{\\alpha }\\Phi $ ($\\bar{\\mathcal{D}}_{\\dot{\\alpha }}\\Phi$) is not generically a superconformal multiplet, i.e., the first component of it is S-SUSY and conformal boost inert only when $w=-n$ ($w=n$) is satisfied. \nThen, it is obvious that we cannot define $\\bar{\\cal D}_{\\dot{\\alpha}}L$ as a superconformal multiplet since $L$ has the weight with $(2,0)$.\n\nHowever, the authors in Ref.~\\cite{Kugo:1983mv} also proposed an improved spinor derivative operation, which can be defined on any supermultiplet. They introduced another multiplet, ${\\bf{u}}$, called a {\\it{u-associated}} multiplet, \n\\begin{align}\n{\\bf{u}} =\\{ \\mathcal{C}_u, \\mathcal{Z}_u,\\mathcal{H}_u,\\mathcal{K}_u,\\mathcal{B}_{au},\\Lambda _u,\\mathcal{D}_u \\} ,\n\\end{align}\nin order to force the first component of ${\\cal D}_\\alpha \\Phi$ to be invariant under S-SUSY and conformal boost. To be specific, they defined the {\\it{u-associated}} spinor derivative as \n\\begin{align}\n\\mathcal{D}^{({\\bf{u}})}_{\\alpha }\\Phi =\\langle (P_L\\mathcal{Z})_{\\alpha }+i(n+w)\\lambda _{\\alpha }\\mathcal{C}\\rangle ,\\ \\ \\ \\lambda _{\\alpha } \\equiv \\frac{i(P_L\\mathcal{Z}_u)_{\\alpha }}{(w_u+n_u)\\mathcal{C}_u}, \\label{def u}\n\\end{align}\nwhere $w_u$ and $n_u$ are the Weyl and chiral weight of a {\\it{u-associated}} multiplets, respectively. Unless $w_u+n_u=0$, we can choose any multiplet as the {\\it{u-associated}} multiplet. \nThen, we can define the spinor derivative for an arbitrary superconformal multiplet by this {\\it{u-associated}} spinor derivative.\n\nFor our purpose, we need the {\\it{u-associated}} spinor derivative acting on a real linear multiplet, $\\mathcal{D}^{({\\bf{u}})}_{\\alpha }L $. More generally, we can consider \n\\begin{align}\n\\mathcal{D}^{({\\bf{u}}_1)}_{\\alpha }({\\bf{u}}_2L), \\label{u-derivative}\n\\end{align}\nwhere ${\\bf{u}}_1$ is a {\\it{u-associated}} multiplet and ${\\bf{u}}_2$ is an additional multiplet. These multiplets must satisfy ${\\bf{u}}_1\\neq {\\bf{u}}_2$, since $\\mathcal{D}^{({\\bf{u}})}_{\\alpha }{\\bf{u}}$ is identically zero obviously from the definition $\\eqref{def u}$.\\footnote{As we will discuss, we choose ${\\bf{u}}_1$ and ${\\bf{u}}_2$ as compensators, which become some parts of the gravity multiplet after superconformal gauge fixings. In the global SUSY expression $\\eqref{global derivative}$, all the fields in the gravitational multiplet decouple from it. Therefore, it is natural to consider a possibility that a compensator appears as in Eq. $\\eqref{u-derivative}$.} Using this {\\it{u-associated}} spinor derivative, Eq. $\\eqref{global derivative}$ can be generalized to the one in conformal SUGRA as\n\\begin{align}\n\\frac{1}{{\\bf{u}}_3}\\bar{\\mathcal{D}}^{({\\bf{u}}_1)}(\\bar{\\bf{u}}_2L)\\bar{\\mathcal{D}}^{({\\bf{u}}_1)}(\\bar{\\bf{u}}_2L) , \\label{u-derivative part}\n\\end{align}\nwhere we have introduced a new multiplet ${\\bf{u}}_3$\\footnote{We will refer all of ${\\bf {u}}_i$ as {\\it u-associated} multiplets.} for generality and omitted the spinor index, $\\dot{\\alpha }$, and we have also defined the conjugate of a {\\it{u-associated}} derivative as $\\bar{\\mathcal{D}}_{\\dot{\\alpha }}^{\\bf{u}} \\Phi = (\\mathcal{D}_{\\alpha }^{\\bf{u}}(\\Phi )^*)^*$ following Ref. \\cite{Kugo:1983mv}.\n\nLet us comment on the weight of the multiplet $\\eqref{u-derivative part}$. The operator $\\bar{\\mathcal{D}}^{({\\bf{u}})}_{\\dot{\\alpha} }$ has the weight $(1\/2,3\/2)$, then the total weight of Eq. $\\eqref{u-derivative part}$ is $(2w_2-w_3+5,2n_2-n_3+3)$, where $w_i$ and $n_i$ with $i=1,2,3$ are the Weyl and chiral weights of ${\\bf{u}}_i$, respectively. \n\nFurthermore, Eq. $\\eqref{global constraint 2}$ is a ``chiral\" constraint since the first and second term in Eq. $\\eqref{global constraint 2}$ are chiral multiplets. Then, we require a condition that the multiplet $\\eqref{u-derivative part}$ is a chiral multiplet, that is, \n\\begin{align}\n\\bar{\\mathcal{D}}\\biggl[ \\frac{1}{{\\bf{u_3}}}\\bar{\\mathcal{D}}^{({\\bf{u_1}})}(\\bar{\\bf{u_2}}L)\\bar{\\mathcal{D}}^{({\\bf{u_1}})}(\\bar{\\bf{u_2}}L)  \\biggr]  =0 . \\label{condition for chiral}\n\\end{align}\nTo apply $\\bar{\\mathcal{D}}$ for Eq. $\\eqref{u-derivative part}$, the Weyl and chiral weight of Eq. $\\eqref{u-derivative part}$ must satisfy $w=n$ as mentioned before,\n\\begin{align}\n2w_2-w_3+5=2n_2-n_3+3. \\label{weight condition between w3n3 and w2n2}\n\\end{align}\nThe condition $\\eqref{condition for chiral}$ implies that \n\\begin{align}\nP_R \\mathcal{Z}'=0, \\label{condition for chiral2}\n\\end{align}\nwhere $P_R= (1-\\gamma _5)\/2$ is a right-handed projection operator and $\\mathcal{Z}'$ is the second component of the multiplet $\\eqref{u-derivative part}$. The equation $\\eqref{condition for chiral2}$ can be written explicitly as\n\\begin{align}\n\\nonumber &\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{\\mathcal{Z}}_2^c\\biggl[P_R\\tilde{Z}+k   P_R\\tilde{\\mathcal{Z}}_1^c- P_R\\tilde{\\mathcal{Z}}_3\\biggr] \n+\\bar{\\tilde{Z}}P_R\\tilde{Z}\\biggl[P_R\\tilde{\\mathcal{Z}}_2^c+k P_R\\tilde{\\mathcal{Z}}_1^c- P_R\\tilde{\\mathcal{Z}}_3\\biggr] \\\\\n\\nonumber &-k  \\bar{\\tilde{\\mathcal{Z}}}_1^cP_R\\tilde{\\mathcal{Z}}_1^c\\biggl[\\left( 1-2k \\right) \\left( P_R\\tilde{Z}+P_R\\tilde{\\mathcal{Z}}_2^c\\right) +P_R\\tilde{\\mathcal{Z}}_3\\biggr] \\\\\n\\nonumber &-2k \\biggl[\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{\\mathcal{Z}}_1^c \\left( 2P_R\\tilde{Z}-P_R\\tilde{\\mathcal{Z}}_3\\right)+\\bar{\\tilde{Z}}P_R\\tilde{\\mathcal{Z}}_1^c\\left( 2P_R\\tilde{\\mathcal{Z}}_2^c-P_R\\tilde{\\mathcal{Z}}_3\\right) \\biggr] \\\\\n&-2i\\biggl[i\\tilde{\\mathcal{H}}_2^*+\\tilde{\\mathcal{K}}_2^*-k \\left( i\\tilde{\\mathcal{H}}_1^*+\\tilde{\\mathcal{K}}_1^*\\right) \\biggr] \\biggl[P_R\\tilde{\\mathcal{Z}}_2^c+P_R\\tilde{Z}-k P_R\\tilde{\\mathcal{Z}}_1^c\\biggr] \n-2\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{Z}P_R\\tilde{\\mathcal{Z}}_3=0,\\label{explicit condotion for chiral} \n\\end{align}\nwhere \n\\begin{align}\n&{\\bf{u}}_i =\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} ,\\ \\ \\ (i=1,2,3),\\\\\n&\\tilde{Z}\\equiv \\frac{1}{C}Z, \\ \\ \\ \\tilde{\\mathcal{Z}}_i\\equiv \\frac{1}{C_i}\\mathcal{Z}_i, \\ \\ \\ \\tilde{\\mathcal{H}}_i(\\tilde{\\mathcal{K}}_i)\\equiv \\frac{1}{C_i}\\mathcal{H}_i(\\mathcal{K}_i), \\label{tilde}\\\\\n&k \\equiv \\frac{w_2+n_2+2}{w_1+n_1},\n\\end{align}\nand $``c\"$ denotes the charge conjugation for spinors. \n\nAs a summary, we find that the superconformal realization of Eq. $\\eqref{global derivative}$ is the multiplet $\\eqref{u-derivative part}$ satisfying the conditions $\\eqref{weight condition between w3n3 and w2n2}$ and $\\eqref{explicit condotion for chiral}$. \n\n\\subsection{Old minimal versus New minimal}\nWe have found, in the previous subsection ~\\ref{u-associated derivative}, the conditions for extending Eq. $\\eqref{global derivative}$ to that in conformal SUGRA. Here, we will choose a conformal compensator $\\Phi_0$ as {\\it{u-associated}} multiplets, ${\\bf{u}}_i$. Then, we have two choices of compensators; one of them is a chiral compensator $S_0$ realizing the old minimal SUGRA and the other is a real linear compensator $L_0$ realizing the new minimal SUGRA.\\footnote{We do not discuss the case of the   non-minimal formulation which is realized with a complex linear compensator.}  \n\nNow, we will examine what forms of ${{\\bf{u}}_i}$ with both compensators are allowed. Let us start from the old minimal SUGRA realized with a chiral compensator,\n\\begin{align}\nS_0=\\{ z_0,-\\sqrt{2}iP_L\\chi_0 , -F_0,iF_0,iD_az_0,0,0  \\},\n\\end{align}\nwith its weight $(1,1)$. Here we assume that the multiplets ${\\bf{u}}_i$ take the following form \n\\begin{align}\n{\\bf{u}}_i=S_0^{p_i}\\bar{S}_0^{q_i},\\ \\ \\ (i=1,2,3), \\label{s0s0}\n\\end{align}\nwhere $p_i$ and $q_i$ are the power of $S_0$ and $\\bar{S}_0$, and satisfy $p_1\\neq 0$ since $w_1+n_1=(p_1+q_1)+(p_1-q_1)=2p_1$ must be nonzero by a definition of the {\\it{u-associated}} multiplet. Here we have to stress that Eq. $\\eqref{s0s0}$ is the most general form except for the case including derivative operators on a compensator,\\footnote{For example, $S_0\\Sigma \\bar{S}_0$ could be considered.} which might produce higher-derivative terms of gravity. Using Eq. $\\eqref{embedding chiral}$ and the multiplication rule $\\eqref{formulaF}$, the components of the multiplet in Eq. $\\eqref{s0s0}$ are written as \n\\begin{align}\n\\nonumber &\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} \\\\\n\\nonumber  &= \\{ z_0^{p_i}z_0^{*q_i},\\sqrt{2}iz_0^{p_i-1}z_0^{*q_i-1}(q_iz_0P_R\\chi_0 -p_iz_0^*P_L\\chi_0 ),\\\\\n\\nonumber &z_0^{p_i-2}z_0^{*q_i-2}\\left( -q_iz_0^2z_0^*F_0^*-p_iz_0z_0^{*2}F_0+\\frac{1}{2}q_i(q_i-1)z_0^2\\bar{\\chi}_0P_R\\chi _0+\\frac{1}{2}p_i(p_i-1)z_0^{*2}\\bar{\\chi}_0P_L\\chi _0\\right) ,\\\\\n\\nonumber &z_0^{p_i-2}z_0^{*q_i-2}\\left( -iq_iz_0^2z_0^*F_0^*+ip_iz_0z_0^{*2}F_0+\\frac{i}{2}q_i(q_i-1)z_0^2\\bar{\\chi}_0P_R\\chi _0-\\frac{i}{2}p_i(p_i-1)z_0^{*2}\\bar{\\chi}_0P_L\\chi _0\\right) , \\\\\n&...,...,...\\}, \\label{explicit s0s0}\n\\end{align}\nwhere we have omitted the components, $\\mathcal{B}_{ai},\\Lambda _i$ and $\\mathcal{D}_i $, which are not necessary to evaluate  Eq. $\\eqref{explicit condotion for chiral}$.  \nOne finds that Eq. $\\eqref{explicit condotion for chiral}$ cannot be satisfied by Eq. $\\eqref{s0s0}$ by the following reason: Terms including ${\\cal H}_i$ and ${\\cal K}_i$ must vanish by themselves since any other terms cannot cancel them. After substituting Eq. $\\eqref{explicit s0s0}$ into such a part, we obtain\n\\begin{align}\n\\nonumber &i\\mathcal{\\tilde{H}}_2^*+\\tilde{\\mathcal{K}}_2^*-k \\left( i\\tilde{\\mathcal{H}}_1^*+\\tilde{\\mathcal{K}}_1^*\\right) =2iF_0^*z_0^{*-1}+i\\bar{\\chi}_0P_R\\chi _0z_0^{*-2}(p_2^2-p_2p_1-p_1+1).\n\\end{align}\nApparently, the first term cannot be eliminated no matter how we choose the parameters $p_i$ and $q_i$, and the other terms in Eq. $\\eqref{explicit condotion for chiral}$ cannot eliminate it because they do not contain $F_0^*$. \nTherefore, we find that Eq. $\\eqref{s0s0}$ cannot be a solution of Eq. $\\eqref{explicit condotion for chiral}$. This means that Eq. $\\eqref{u-derivative part}$ cannot be realized as a chiral constraint in the old minimal SUGRA.\n\nNext, we examine the case in the new minimal SUGRA with a real linear compensator\n\\begin{align}\nL_0=\\{ C_0,Z_0,0,0,B_{0a},-\\slash{D}Z_0,-\\Box C_0\\}\n\\end{align}\nwith its weight $(2,0)$. In the same way as the old minimal case, we assume the general form of ${\\bf{u}}_i$ as\n\\begin{align}\n{\\bf{u}}_i=L_0^{r_i},\\ \\ \\ (i=1,2,3), \\label{l0}\n\\end{align}\nwhose components are \n\\begin{align}\n\\nonumber &\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} \\\\\n &= \\{ C_0^{r_i}, r_iC_0^{r_i-1}Z_0,-\\frac{1}{4}r_i(r_i-1)C_0^{r_i-2}\\bar{Z}_0Z_0, \\frac{i}{4}r_i(r_i-1)C_0^{r_i-2}\\bar{Z}_0\\gamma _5Z_0,...,...,...\\} . \\label{explicit l0}\n\\end{align}\nHere we have used Eq. $\\eqref{embedding linear}$ and Eq. $\\eqref{formulaF}$. Then, after substituting Eq. $\\eqref{explicit l0}$ into Eq. $\\eqref{explicit condotion for chiral}$ with the Fierz rearrangement, Eq. $\\eqref{explicit condotion for chiral}$ is summarized as\n\\begin{align}\n(2r_2-r_3+1)\\left\\{ CP_RZ\\bar{Z}_0P_RZ_0+C_0P_RZ_0\\bar{Z}P_RZ\\right\\}=0. \\label{L0 condition}\n\\end{align}\nTo satisfy Eq. $\\eqref{L0 condition}$, the coefficient must be zero,\n\\begin{align}\n2r_2-r_3+1=0.\\label{L0 condition2}\n\\end{align}\nThen, we find that the chiral condition $\\eqref{explicit condotion for chiral}$ is satisfied as long as the {\\it{u-associated}} multiplets follow the condition $\\eqref{L0 condition2}$.\n\nNoting that $w_i=2r_i$ and $n_i=0$ in the ansatz $\\eqref{l0}$, the weight condition $\\eqref{weight condition between w3n3 and w2n2}$ which the chiral multiplet should obey is now reduced to\n\\begin{align}\n2r_2-r_3+1=0. \\label{L0 condition22}\n\\end{align}\nThis is nothing but Eq. $\\eqref{L0 condition2}$ which is satisfied automatically.\n\nTherefore, we conclude that one can make a multiplet in Eq. $\\eqref{u-derivative part}$ a chiral one with the real linear compensator if Eq. $\\eqref{L0 condition2}$ is satisfied. Here and hereafter, we focus on the case of the new minimal SUGRA with $r_1=r_3=1$ and $r_2=0$ for simplicity. In this case, the multiplet in Eq. $\\eqref{u-derivative part}$ becomes\n\\begin{align}\n\\frac{1}{L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L . \\label{Lo derivative part}\n\\end{align}\nWe present the components of this chiral multiplet $\\eqref{Lo derivative part}$ explicitly in Appendix A.\n\n\\subsection{Embedding the constraint into conformal SUGRA} \\label{embedding}\nLet us consider the remaining terms, $X$ and $X\\bar{D}^2\\bar{X}$ in Eq. $\\eqref{global constraint 2}$. For $X$, we just regard it as a superconformal chiral multiplet with the weight $(w,w)$. In order to extend the second one, $X\\bar{D}^2\\bar{X}$, to a superconformal multiplet, we replace it with $X\\Sigma \\bar{X}$, where $\\Sigma $ is a chiral projection operator in conformal SUGRA. However, $\\Sigma $ cannot always be applied for any multiplet $\\Phi$ in the same way as the spinor derivative $\\mathcal{D}$. It can be applied only when $\\Phi$ satisfies the following weight condition, \n\\begin{align}\nw_{\\Phi }=n_{\\Phi }+2. \\label{condition for projection}\n\\end{align}\nTherefore, we compensate the weight of $\\bar{X}$, which has the weight $(w,-w)$, by the real linear compensator multiplet $L_0^s$, where $s$ is the power of $L_0$, \n\\begin{align}\nX\\Sigma \\left( \\frac{1}{L_0^s}\\bar{X} \\right) . \\label{compensated second term}\n\\end{align}\nHere, the term, $\\frac{1}{L_0^s}\\bar{X}$, has the weight $(-2s+w,-w)$. According to Eq. $\\eqref{condition for projection}$, $s$ must satisfy the condition,\n\\begin{align}\ns=w-1. \\label{weight condition of v1}\n\\end{align}\nTaking into account this condition and the fact that $\\Sigma $ raises the weight by (1,3), Eq. $\\eqref{compensated second term}$ has the weight $(3,3)$, which is correct  for a chiral multiplet. Since the total weight of Eq. $\\eqref{compensated second term}$ must be the same as the first term $X$, the value of $w$ is determined as \n\\begin{align}\nw=3. \\label{weight condition of v2}\n\\end{align}\nThen, we find $s=2$ from Eq.  $\\eqref{weight condition of v1}$, and Eq. $\\eqref{compensated second term}$ becomes  \n\\begin{align}\nX\\Sigma \\left( \\frac{1}{L_0^2}\\bar{X} \\right) . \\label{compensated second term2}\n\\end{align}\nFinally, the weight of the multiplet in Eq. $\\eqref{u-derivative part}$ with that in Eq. $\\eqref{l0}$ is $(3,3)$ as long as Eq. $\\eqref{L0 condition22}$ is satisfied, then Eq. $\\eqref{Lo derivative part}$ is automatically satisfied. \n\nTherefore, we find the complete embedding of a global SUSY expression $\\eqref{global constraint 2}$,\n\\begin{align}\nX+\\frac{1}{2}X\\Sigma \\left( \\frac{1}{L_0^2}\\bar{X} \\right) +\\frac{1}{4L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L =0,\\label{local constraint}\n\\end{align}\nwhere $X$ is a chiral multiplet with $(3,3)$, $L$ is a real linear multiplet with $(2,0)$, and $L_0$ is a real linear compensator with $(2,0)$.\n\n\n  \n\n\n\\section{Component action}\\label{Component action}\nIn this section, we derive the DBI action based on the constraint $\\eqref{local constraint}$ in the new minimal SUGRA.\n\n\\subsection{Minimal action} \\label{Minimal action}\nWe first consider the minimal extension of the action $\\eqref{component form of global DBI}$. The action corresponding to Eq. $\\eqref{global DBI with constraint}$ is expected to be\n\\begin{align}\nS=&[2X]_F+\\Biggl[2\\Lambda \\left\\{ X+\\frac{1}{2}X\\Sigma \\left(  \\frac{\\bar{X}}{L_0^2}\\right) +\\frac{1}{4L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L \\right\\} \\Biggr] _F+[\\tilde{\\Lambda }X^2]_F +\\Biggl[\\frac{3}{2}L_0V_R\\Biggr] _D, \\label{SL0}\n\\end{align}\nwhere $V_R\\equiv \\log \\frac{L_0}{S\\bar{S}}$, $S$ is a chiral multiplet with $(1,1)$, and we have assigned the weights of the Lagrange multiplier chiral multiplet $\\Lambda$ to $(0,0)$ and also $\\tilde{\\Lambda }$ to $(-3,-3)$ in such a way that the total weight is equal to $(3,3)$.\nThe last term in Eq. $\\eqref{SL0}$ is responsible for the kinetic term of the gravitational multiplet. Note that this term is invariant under the transformation $S\\to Se^{i\\Theta}$ where $\\Theta$ is a chiral multiplet with the weight $(0,0)$ since $[L_0(\\Theta+\\bar{\\Theta})]_D\\equiv0$ by the nature of a real linear multiplet. Due to this additional gauge invariance, we have gauge degrees of freedom other than superconformal ones. After imposing the gauge fixing condition for this additional gauge symmetry as $S=\\{1,0,0\\}$, the bosonic part of $\\eqref{SL0}$ is given by\n\\begin{align}\n\\nonumber S_B=& \\int d^4 x \\sqrt{-g}\\Biggl[ \\Biggl( F_X (1+\\Lambda )-\\frac{|F_X|^2\\Lambda }{C_0^2}-\\frac{\\Lambda}{4C_0}(B_a-i\\hat{D}_aC)^2\\\\\n\\nonumber &+\\frac{C\\Lambda}{2C_0^2}(B_a-i\\hat{D}_aC)(B_0^a-i\\hat{D}^aC_0)\n-\\frac{C^2\\Lambda}{4C_0^3}(B_{0a}-i\\hat{D}_aC_0)^2 +{\\rm{h.c.}}\\Biggr) \\\\\n&-\\frac{3}{2}\\hat{\\Box} C_0\\log C_0-\\frac{3}{2}\\hat{\\Box} C_0-\\frac{3}{4C_0}(B_0\\cdot B_0+\\hat{D}C_0\\cdot \\hat{D}C_0) \n+3A\\cdot B_0 \\Biggr] , \\label{SBfull}\n\\end{align}\nwhere $\\Lambda$ and $F_X$ are a scalar component of the chiral multiplet $\\Lambda$ and an auxiliary field of $X$, and $\\hat{D}_{\\mu}$ is a superconformal covariant derivative only including bosonic fields, for example,\n\\begin{align}\n\\hat{D}_{\\mu}C=\\partial_{\\mu}C-2b_{\\mu}C,\n\\end{align}\nwhere $b_{\\mu}$ is the gauge field of dilatation. The third term in Eq. $\\eqref{SL0}$, $\\tilde{\\Lambda }X^2$, imposes the nilpotency condition for $X$. Thanks to this, we can drop the scalar component of the chiral multiplet $X$ since the first scalar component can be represented as a fermion bilinear after solving $X^2=0$. That is why, we have inserted this term into the action from the beginning. \nIntegrating out the gauge field of $U(1)_A$ symmetry $A_\\mu$, we obtain    \n\\begin{align}\nB_{0a}=0.\n\\end{align}\nTo eliminate the dilatation symmetry and conformal boost symmetry, we impose the following $D$-gauge and $K$-gauge conditions, \n\\begin{align}\nC_0=1,\\ \\ \\ \\ b_{\\mu}=0.\n\\end{align}\nThese conditions simplify the action $\\eqref{SBfull}$, which becomes\n\\begin{align}\n\\nonumber S_B=\\int d^4 x \\sqrt{-g}\\Biggl[ &\\frac{1}{2}R+\\Bigl( F_X (1+\\Lambda ) -|F_X|^2\\Lambda \\\\\n&-\\frac{\\Lambda }{4}(B\\cdot B-2iB\\cdot \\partial C-\\partial C\\cdot \\partial C)+{\\rm{h.c.}}\\Bigr) \\Biggr] . \\label{FX}\n\\end{align}\nThen, eliminating the auxiliary field $F_X$ leads to\n\\begin{align}\nS_B=\\int d^4 x \\sqrt{-g}\\Biggl[ &\\frac{1}{2}R+\\frac{1}{2\\lambda }\\Bigl(  (\\lambda +1)^2+\\chi ^2\\Bigr) -\\frac{1}{2}(B\\cdot B-\\partial C\\cdot \\partial C)\\lambda -B\\cdot \\partial C \\chi \\Biggr] , \\label{FXX}\n\\end{align}\nwhere $\\lambda={\\rm Re}\\Lambda$ and $\\chi={\\rm Im}\\Lambda$.\nFinally, we obtain the following conditions from the E.O.Ms for $\\lambda $ and $\\chi $,\n\\begin{align}\n&\\frac{\\chi }{\\lambda }=B\\cdot \\partial C ,\\\\\n&\\frac{1}{\\lambda ^2}=1-(B\\cdot \\partial C)^2-B\\cdot B+\\partial C\\cdot \\partial C .\n\\end{align}\nSubstituting them into the action $\\eqref{FXX}$, we obtain the on-shell DBI action of a real linear multiplet,\n\\begin{align}\n S_B=\\int d^4 x \\sqrt{-g}\\Biggl[ \\frac{1}{2}R+1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}\\Biggr] .  \\label{DBI}\n\\end{align}\nThis is almost the same form as Eq. $\\eqref{component form of global DBI}$ except for that our action $\\eqref{DBI}$ is formulated in curved background.\n\nBefore closing this subsection, let us discuss the linear-chiral duality. It is known that the action of a real linear multiplet can be rewritten in terms of that of a chiral multiplet. However, in the case with the action including derivative terms such as Eq. $\\eqref{SL0}$, it is nontrivial to take this duality transformation in a manifestly SUSY way.\\footnote{In global SUSY, the dual action has been obtained at the level of superfield in Ref.~\\cite{GonzalezRey:1998kh}.} Then, we focus only on the bosonic part $\\eqref{DBI}$ and discuss this duality at the component level of bosonic part.\n\nWe start from the following Lagrangian which is the relevant part in the action $\\eqref{DBI}$,\n\\begin{align}\n\\mathcal{L}=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}. \\label{Ori}\n\\end{align}\nTo rewrite this Lagrangian $\\eqref{Ori}$ in terms of the complex scalar of a chiral multiplet, we first relax the constraint on the vector field $B_a$. We impose it by the E.O.M for a scalar field $\\ell$, that is, we use \n\\begin{align}\n\\mathcal{L}=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}+B\\cdot \\partial \\ell ,\\label{LL}\n\\end{align}\nwhere $B_a$ is now an unconstrained vector. The Lagrangian $\\eqref{LL}$ is equivalent to the original one $\\eqref{Ori}$ since the variation with respect to $\\ell$ leads to the constraint, $\\partial _a B^a=0$.\nInstead of $\\ell$, varying with respect to $B_a$ gives\n\\begin{align}\n\\partial ^a\\ell + (\\partial ^a C B\\cdot \\partial C+B^a) \\{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2\\} ^{-1\/2} =0. \\label{du}\n\\end{align}\nOur task is now to solve this equation $\\eqref{du}$ with respect to $B_a$.\nBy taking scalar products of Eq. $\\eqref{du}$ with $B_a, \\partial _a C$ and $\\partial _a \\ell $, we obtain three independent equations and can solve them with respect to $B^2$, $B\\cdot\\partial C$, and $B\\cdot \\partial \\ell$. The solutions are\n\\begin{align}\n&B^2=\\frac{(\\partial \\ell) ^2(1+(\\partial C)^2)^2-(\\partial C\\cdot \\partial \\ell )^2(2+(\\partial C)^2)}{Y^2},\\\\\n&B\\cdot \\partial C=-\\frac{\\partial C\\cdot \\partial \\ell }{Y},\\\\\n&B\\cdot \\partial \\ell =\\frac{-(\\partial \\ell) ^2(1+(\\partial C)^2)+(\\partial C\\cdot \\partial \\ell )^2}{Y},\n\\end{align}\nwhere\n\\begin{align}\nY \\equiv \\{ (1+(\\partial C)^2)(1+(\\partial \\ell) ^2)-(\\partial C\\cdot \\partial \\ell )^2\\}^{1\/2}.\n\\end{align}\nSubstituting these solutions into the Lagrangian $\\eqref{LL}$, we obtain the dual action,\n\\begin{align}\n\\nonumber \\mathcal{L}&=1-\\sqrt{1+(\\partial C)^2+(\\partial \\ell) ^2+(\\partial C)^2(\\partial \\ell) ^2-(\\partial C\\cdot \\partial \\ell )^2}\\\\\n&=1-\\sqrt{1+\\partial \\phi \\cdot \\partial \\bar{ \\phi }-\\frac{1}{4}(\\partial \\phi)^2 (\\partial \\bar{ \\phi })^2+\\frac{1}{4}(\\partial \\phi \\cdot \\partial \\bar{ \\phi })^2} ,\\label{LC}\n\\end{align}\nwhere we have defined a complex scalar $\\phi =\\ell + iC$. The Lagrangian $\\eqref{LC}$ can be written as the DBI form\n\\begin{align}\n\\mathcal{L}=1-\\sqrt{{\\rm{det}} \\left( g_{ab}+\\frac{1}{2}\\partial_a \\phi \\partial_b \\bar{\\phi} \\right)}.\\label{LC2}\n\\end{align}\nThis Lagrangian $\\eqref{LC2}$ agrees with the one constructed in Ref.~\\cite{Koehn:2012ar} using a chiral multiplet directly. \n  \n\n\\subsection{Matter coupled extension} \\label{matter}\nFinally, we discuss the matter coupled DBI action given by\n\\begin{align}\nS=&[2f(\\Phi^{I})X]_F+\\left[2\\Lambda \\left\\{ X+\\frac{1}{2}X\\Sigma\\left(\\frac{\\bar{X}}{M(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})}\\right)+\\frac{1}{4L_0}\\bar{\\cal D}^{(L_0)}L\\bar{\\cal D}^{(L_0)}L\\right\\}\\right]_F\\nonumber\\\\\n&+[{\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})]_D+[\\tilde{\\Lambda}X^2]_F\\label{mDBI},\n\\end{align}\nwhere $\\Phi^I$ ($\\bar{\\Phi}^{\\bar{J}}$) is a (anti-) chiral matter multiplet; $f(\\Phi)$ is a holomorphic function of $\\Phi^I$ with $(0,0)$; $M(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$ and ${\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$ are real functions of $\\Phi^I,\\bar{\\Phi}^{\\bar{J}}$ and  $L_0$ with $(4,0)$ and $(2,0)$, respectively. Note that we have omitted superpotential term $[W(\\Phi^I)]_F$, where $W(\\Phi^I)$ is a holomorphic function of $\\Phi^I$ with the weight $(w,n)=(3,3)$, since the term is irrelevant to the following discussion. Taking into account the nilpotency condition on $X$, the bosonic component of the action~(\\ref{mDBI}) is given by\n\\begin{align}\nS_B=&\\int d^4x \\sqrt{-g}\\Biggl[ \\Biggl( F_X(f+\\Lambda) -\\frac{\\Lambda |F_X|^2}{M}-\\frac{\\Lambda}{4C_0}(B_a-i\\hat{D}_aC)^2 \\nonumber\\\\\n&+\\frac{C\\Lambda}{2C_0^2}(B_a-i\\hat{D}_aC)(B_0^a-i\\hat{D}^aC_0)-\\frac{C^2\\Lambda}{4C_0^3}(B_0^a-i\\hat{D}^aC_0)^2+{\\rm h.c.}\\Biggr)+{\\cal L}_m\\Biggr],\n\\end{align}\nwhere\n\\begin{align}\n{\\cal L}_m=&-\\frac{1}{3}({\\cal F}-{\\cal F}_{C_0}C_0)R(b)+\\frac{1}{2}{\\cal F}_{C_0C_0}(\\hat{D}C_0\\cdot \\hat{D}C_0-B_0\\cdot B_0)\\nonumber\\\\\n&+2{\\cal F}_{I\\bar{J}}(F^I\\bar{F}^{\\bar{J}}-\\hat{D}\\Phi^I \\cdot \\hat{D}\\bar{\\Phi}^{\\bar{J}})+\\left(-i{\\cal F}_{C_0I}B_0\\cdot \\hat{D}\\Phi^I+{\\rm h.c.}\\right)\\label{defLm}.\n\\end{align}\nIn the above expression, $\\Phi^I$ ($\\bar{\\Phi}^{\\bar{J}}$) and $F^I$ ($\\bar{F}^{\\bar{J}}$) represent the scalar and auxiliary components of the (anti-) chiral matter multiplet, and subscripts denote the derivative with respect to the corresponding scalar. $R(b)$ becomes a Ricci scalar when $b_{\\mu}=0$ is imposed as the $K$-gauge condition.\n\nBefore setting superconformal gauge conditions, we integrate out the auxiliary field $F_X$ and the Lagrange multiplier $\\Lambda$. We can easily solve the E.O.M for $F_X$ and obtain\n\\begin{align}\nS_B=&\\int d^4x \\sqrt{-g}\\Biggl[\\frac{M}{2\\lambda}\\left\\{(\\lambda+p)^2+(\\chi+q)^2\\right\\}-\\frac{\\lambda}{2C_0}(B\\cdot B-\\hat{D}C\\cdot \\hat{D}C)\\nonumber\\\\\n&-\\frac{\\chi}{C_0}B \\cdot \\hat{D}C+\\frac{C\\lambda}{C_0^2}(B_0\\cdot B-\\hat{D}C_0\\cdot \\hat{D}C)+\\frac{C\\chi}{C_0^2}(B_0\\cdot \\hat{D}C+B\\cdot \\hat{D}C_0)\\nonumber\\\\\n&-\\frac{C^2\\lambda}{2C_0^3}(B_0 \\cdot B_0-\\hat{D}C_0\\cdot \\hat{D}C_0)-\\frac{C^2\\chi}{C_0^3}B_0\\cdot \\hat{D}C_0+{\\cal L}_m\\Biggr],\\label{mDBI2}\n\\end{align}\nwhere $\\lambda={\\rm Re}\\Lambda$, $\\chi={\\rm Im}\\Lambda$, $p={\\rm Re}f$, and $q={\\rm Im}f$. Note that, at this stage, the matter Lagrangian ${\\cal L}_m$ is not affected by the DBI sector. Next, we eliminate $\\lambda$ and $\\chi$ by using their E.O.Ms, which are given by\n\\begin{align}\n&-\\frac{M}{2\\lambda^2}\\left\\{(\\lambda+p)^2+(\\chi+q)^2\\right\\}+\\frac{M}{\\lambda}(\\lambda+p)+\\mathcal{A}=0,\\\\\n&\\frac{M}{\\lambda}(\\chi+q)+\\mathcal{B}=0,\n\\end{align}\nwhere\n\\begin{align}\n&\\mathcal{A}\\equiv -\\frac{1}{2C_0}(B\\cdot B-\\hat{D}C\\cdot \\hat{D}C)+\\frac{C}{C_0^2}(B_0\\cdot B-\\hat{D}C_0\\cdot \\hat{D}C)-\\frac{C^2}{2C_0^3}(B_0\\cdot B_0-\\hat{D}C_0\\cdot \\hat{D}C_0),\\\\\n&\\mathcal{B}\\equiv -\\frac{1}{C_0}B\\cdot \\hat{D}C+\\frac{C}{C_0^2}(B_0\\cdot \\hat{D}C+B\\cdot \\hat{D}C_0)-\\frac{C^2}{C_0^3}B_0\\cdot \\hat{D}C_0.\n\\end{align}\nSolutions for them are\n\\begin{align}\n&\\lambda|_{\\rm sol}^{-1}=\\frac{1}{p}\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}},\\\\\n&\\chi|_{\\rm sol}=-q-\\frac{\\lambda|_{\\rm sol}}{M}\\mathcal{B}.\n\\end{align} \nSubstituting the above solutions into the action~(\\ref{mDBI2}), we obtain a relatively simple form\n\\begin{align}\nS_B=\\int d^4x\\sqrt{-g}\\left[Mp\\left(1-\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}}\\right)-q\\mathcal{B}+{\\cal L}_m\\right].\\label{mDBI3}\n\\end{align}\n\nThe remaining issue is the elimination of auxiliary fields $B_0^a$ and $A_a$. However, it is difficult to do it because of the presence of nonlinear terms of $B_0^a$ contained in the first term in Eq.~(\\ref{mDBI3}). In addition, ${\\cal L}_m$ has $A_aA^a$ as well as mixing terms between $B_0^a$ and $A_a$ in general cases. Therefore, integration of those auxiliary fields is technically difficult and we cannot obtain the complete on-shell action.\\footnote{The general matter coupled system in the new minimal SUGRA not including higher-order derivative terms can be found in Ref.~\\cite{Ferrara:1983dh}.}\n\nAlthough a general case is difficult to complete the remaining task, we can continue our discussion for the following special case. Let us consider the following choice of ${\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$,\n\\begin{align}\n{\\cal F}=L_0\\log \\left(\\frac{L_0G(\\Phi^i,\\bar{\\Phi}^{\\bar{j}})}{S\\bar{S}}\\right), \\label{special form}\n\\end{align}\nwhere $\\Phi^i$ is a matter chiral multiplet with its weight $(0,0)$, $G(\\Phi^i,\\bar{\\Phi}^{\\bar{j}})$ is a real function of $\\Phi^i$ and $\\bar{\\Phi}^{\\bar{j}}$, and $S$ is a chiral multiplet with $(1,1)$. This action is also invariant under the transformation $S\\to Se^{i\\Theta}$ in the same way as the last term in Eq. $\\eqref{SL0}$, which characterizes the new minimal SUGRA.\n\nWe use the $D$-gauge condition to make the Ricci scalar term canonical. From Eq.~(\\ref{defLm}), we can find an appropriate $D$-gauge choice~\\cite{Ferrara:1983dh}\n\\begin{align}\n{\\cal F}-{\\cal F}_{C_0}C_0=-\\frac{3}{2}.\n\\end{align}\nAs the choice of the additional gauge, we set ${\\cal F}_{C_0}=0$~\\cite{Ferrara:1983dh}. Then, we can solve these gauge conditions with respect to $C_0$ and $S$ and obtain\n\\begin{align}\nS\\bar{S}=&\\frac{3}{2}eG,\\\\\nC_0=&\\frac{3}{2}.\n\\end{align} \nUsing the $K$-gauge, we also set a condition $b_\\mu=0$.\n\nUnder these conditions, ${\\cal L}_m$ becomes\n\\begin{align}\n{\\cal L}_m=&\\frac{1}{2}R+2{\\cal F}_{i\\bar{j}}(F^i\\bar{F}^{\\bar{j}}-\\partial_a\\Phi^i\\partial^a\\bar{\\Phi}^{\\bar{j}})-\\frac{1}{2}B_0^aB_{0a}\\nonumber\\\\\n&+(-i{\\cal F}_{C_0i}B_0^a\\partial_a \\Phi^i+{\\rm h.c.})+(iB_0^a\\partial_a\\log S+{\\rm h.c.})+2B_0^aA_a,\n\\end{align}\nwhere $A_a$ is the $U(1)_A$ gauge field mentioned above. We find that the E.O.M for $A_a$ gives a constraint $B_0^a=0$ and the difficulty due to the nonlinear term of $B_0^a$ is circumvented in this case. This result is irrelevant to other parts of the action (\\ref{mDBI3}) since they do not contain terms of $A_a$. $F^i$ can be eliminated by their E.O.Ms, and we finally obtain the following on-shell action,\n\\begin{align}\nS_B=\\int d^4x\\sqrt{-g}\\left[Mp\\left(1-\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}}\\right)-q\\mathcal{B}+\\frac{1}{2}R-2{\\cal F}_{i\\bar{j}}\\partial_a\\Phi^i\\partial^a\\bar{\\Phi}^{\\bar{j}}\\right], \\label{special matter}\n\\end{align}\nwith\n\\begin{align}\n\\mathcal{A}=\\frac{1}{3}(\\partial C\\cdot \\partial C-B\\cdot B),\\ \\ \\ \\mathcal{B}=-\\frac{2}{3}B\\cdot \\partial C.\n\\end{align}\nHere, the real function $M$ should be understood as $M|_{C_0=3\/2}$.\nNote that, in this case, we cannot add superpotential terms of $\\Phi^i$ by the following reason: To obtain the constraint $B_0^a=0$, we assumed that only $S$ has the weight $(w,n)=(1,1)$ and a special form of ${\\cal F}$ giving ${\\cal F}_{S\\bar{S}}=0$, otherwise such a constraint does not appear. For the superconformal invariance, the superpotential $W$ should have $(3,3)$. From the weight condition, a possible form is $W=S^3g(\\Phi^i)$ but this term is forbidden by the symmetry under $S\\to Se^{i\\Theta}$ which the D-term part $[{\\cal F}]_D$ has. Therefore, we cannot add any superpotential terms of matter multiplets.\n\\section{Relation between our results and other works}\\label{discussion}\nHere, we comment on the differences between ours and the results in Ref.~\\cite{Koehn:2012ar}, in which the DBI action of a chiral multiplet is constructed in the old minimal SUGRA. As we mentioned before, the DBI action of a real linear multiplet can be rewritten in terms of a chiral multiplet through the linear-chiral duality and the whole action of a chiral multiplet is obtained in global SUSY in terms of superfield \\cite{GonzalezRey:1998kh}. The authors of Ref.~\\cite{Koehn:2012ar} embedded the dual chiral multiplet action into the old minimal SUGRA. On the other hand, our starting point is the action of a real linear multiplet, more precisely, the constraint $\\eqref{global constraint}$ imposed upon it. This constraint has its origin in the tensor multiplet of $\\mathcal{N}=2$ SUSY \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}. Indeed, in global SUSY case, the real linear multiplet corresponds to a Goldstino multiplet for the broken SUSY. From such a viewpoint, our construction is important since it makes the connection with the partial breaking of $\\mathcal{N}=2$ SUSY much clearer . \n\nAlthough the ways of construction are different, our action would realize their result. Indeed, at the bosonic component level, we have found the correspondence between the result in Ref.~\\cite{Koehn:2012ar} and ours. However, we also found that the action cannot be realized in the old minimal SUGRA when we do not consider the case including higher-derivative terms of a chiral compensator, which may contradict the result of Ref.~\\cite{Koehn:2012ar}. Unlike the DBI action of a real linear multiplet, that of a vector multiplet can be constructed in both of the old and new minimal SUGRA \\cite{Abe:2015nxa}. The difference originates from the necessity of {\\it u-associated} derivatives in the DBI action of a real linear multiplet. For a vector superfield case, we can construct the DBI action only with the chiral projection operator $\\Sigma$, which does not require {\\it u-associated} multiplet to make the operand superfield a primary superfield~\\cite{Kugo:1983mv,Butter:2009cp,Kugo:2016zzf}. It is interesting to explore these reasons and we expect that the direct derivation of the constraint $\\eqref{global constraint}$ and also DBI action from $\\mathcal{N}=2$ SUGRA are necessary to understand this issue, which would be our future work \\footnote{For the DBI action of a vector multiplet, such attempts have been recently discussed \\cite{Kuzenko:2015rfx}. There, the partial breaking of ${\\cal N}=2$ SUSY in some ${\\cal N}=1$ SUSY background has been discussed.}.\n\n\n\n\n\\section{Summary}\\label{summary}\nIn this paper, we have discussed superconformal generalization of a DBI action of a real linear superfield known in global SUSY. \n\nTo achieve this, we have focused on the constraint $\\eqref{global constraint}$ between a chiral multiplet and a real linear multiplet, which comes from the partial breaking of 4D $\\mathcal{N}=2$ SUSY \\cite{Bagger:1997pi}. However, it is a nontrivial task to embed this constraint into conformal SUGRA due to the existence of the SUSY spinor derivative, which in general, cannot be applied for arbitrary multiplets in conformal SUGRA. Instead of using an original spinor derivative, we have adopted the {\\it{u-associated}} spinor derivative, proposed in Ref.~\\cite{Kugo:1983mv}. We obtained the condition $\\eqref{weight condition between w3n3 and w2n2}$ and $\\eqref{explicit condotion for chiral}$ by requiring that the corresponding constraint $\\eqref{u-derivative part}$ in conformal SUGRA becomes a chiral constraint. Surprisingly, we have found that these conditions can be realized only in the new minimal formulation of SUGRA when we choose the general power function of compensator as the {\\it{u-associated}} multiplet. Then, we have derived the condition $\\eqref{L0 condition2}$ which {\\it{u-associated}} multiplets must satisfy.\n\nAfter embedding the constraint into the new minimal SUGRA, we have shown the component action which is formulated in curved spacetime. We have also discussed the linear-chiral duality at the level of bosonic components and rewritten the action from a complex scalar field of a chiral multiplet. Finally, we have constructed the action where matter multiplets are directly coupled to the DBI sector. Due to the appearance of nonlinear terms for vector field $B_{0a}$, we have restricted the discussion to the special form of matter function $\\eqref{special form}$ and derived the bosonic action $\\eqref{special matter}$.  \n\nIn this paper, we have shown that the DBI action of a real linear multiplet cannot be realized in the old minimal SUGRA as a naive embedding of the constraint $\\eqref{global constraint}$, which may contradict the result of Ref.~\\cite{Koehn:2012ar}. The duality relation between the old and new minimal SUGRA \\cite{Ferrara:1983dh} is generically not obvious when there exist higher-derivative terms. For example, the non-minimal coupling of gravity is realized only in new minimal SUGRA \\cite{Farakos:2012je} as in the case of the DBI action we discussed here. Such an issue may be revealed with the help of deep understanding of SUGRA system with higher-order derivative terms.  \n\nTo investigate our model further, we need the direct derivation of the constraint from $\\mathcal{N}=2$ SUGRA. And also, the remaining part in Eq. $\\eqref{string DBI}$, i.e., a term including $B_{\\mu \\nu }$, and possible combinations of the Maxwell, scalar and 2-form parts have not been constructed. We leave them for future work.\n\n\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank Taichiro Kugo for helpful discussions and comments. YY would like thank also to Hiroyuki Abe and Yutaka Sakamura for useful discussion and collaboration in the related work. The work of YY is supported by JSPS\nResearch Fellowships for Young Scientists No. 26-4236 in Japan.\n\\begin{appendix}\n\\section{The components of {\\it{u-associated}} spinor derivative multiplet}\\label{explicit}\nHere we show the explicit component form of \n\\begin{align}\n\\frac{1}{L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L. \\label{Lo derivative part2}\n\\end{align}\nAs we have seen in Sec.~\\ref{extension}, Eq. $\\eqref{Lo derivative part2}$ is a chiral multiplet with weight $(3,3)$. The components of this multiplet, $\\{ z',P_L\\chi' , F'\\}$, are  \n\\begin{align}\nz'&=\\frac{C^2}{C_0} \\left(\\bar{\\tilde{Z}}-\\bar{\\tilde{Z}}_0  \\right) P_R\\left(   \\tilde{Z}-\\tilde{Z}_0  \\right),\\\\\n\\nonumber P_L\\chi' &=\\frac{\\sqrt{2}C^2}{C_0}P_L\\biggl[ \\left(  \\tilde{\\slash {B}}-i\\slash{D}\\tilde{C}-\\tilde{\\slash {B}}_0+i\\slash{D}\\tilde{C}_0\\right) \\left( \\tilde{Z}-\\tilde{Z}_0 \\right) -\\frac{3i}{2}\\tilde{Z}_0\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\\\\n&-\\frac{i}{2}\\tilde{Z}_0\\bar{\\tilde{Z}}P_R\\tilde{Z}+\\frac{i}{4}\\gamma ^a\\tilde{Z}_0\\bar{\\tilde{Z}}\\gamma _a\\gamma _5 \\tilde{Z}+i\\tilde{Z}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0-\\frac{i}{2}\\gamma ^a\\tilde{Z}\\bar{\\tilde{Z}}_0\\gamma _a\\gamma _5 \\tilde{Z}_0\\biggr] ,\\\\\n\\nonumber F'&=\\frac{C^2}{C_0}\\biggl[ -\\left( \\tilde{B}_a-i D_a\\tilde{C}\\right) ^2 +2\\left( \\tilde{B}_a-i D_a\\tilde{C} \\right) \\left( \\tilde{B}^a-i D^a\\tilde{C} \\right) -\\left( \\tilde{B}_{0a}-i D_{0a}\\tilde{C}\\right) ^2\\\\\n\\nonumber &+i\\bar{\\tilde{Z}}_0\\gamma _5\\left(  \\tilde{\\slash {B}}-i\\slash{D}\\tilde{C} \\right)\\left( \\tilde{Z}-\\tilde{Z}_0  \\right)+\\frac{i}{2}\\bar{\\tilde{Z}} \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}\\\\\n\\nonumber &-2i\\bar{\\tilde{Z}} \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}_0+\\frac{3i}{2}\\bar{\\tilde{Z}}_0 \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}_0\\\\\n\\nonumber &+2\\left( \\bar{\\tilde{Z}}-\\bar{\\tilde{Z}}_0 \\right)P_R \\slash{D}\\left( \\tilde{Z}-\\tilde{Z}_0 \\right) +\\frac{1}{2}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\bar{\\tilde{Z}}\\tilde{Z}+\\frac{1}{2}\\bar{\\tilde{Z}}P_R\\tilde{Z}\\bar{\\tilde{Z}}_0\\tilde{Z}_0\\\\\n&+2\\bar{\\tilde{Z}}P_R\\tilde{Z}_0\\bar{\\tilde{Z}}\\tilde{Z}_0-3\\bar{\\tilde{Z}}P_R\\tilde{Z}_0\\bar{\\tilde{Z}}_0\\tilde{Z}_0-3\\bar{\\tilde{Z}}\\tilde{Z}_0\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0+\\frac{1}{2}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\bar{\\tilde{Z}}_0\\tilde{Z}_0\\biggr] ,\n\\end{align}\nwhere the fields with $\\tilde{}$ are divided by the first components of the multiplet they belong to, in the same way as Eq. $\\eqref{tilde}$, and the superconformal derivative $D_a$ is understood to act only on the numerator but not on the denominator, e.g., $D^a\\tilde{C} \\equiv D^aC\/C =D^a \\log C$.\n\n\n\\end{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe famous Hasse--Minkowski theorem gives precise conditions for a quadratic form to have a non-trivial zero. For the solubility of a system of two quadratic forms there are still many open questions. In this paper we restrict to the study of smooth intersections of two quadrics in five variables. Such a system describes a geometric object $X \\subseteq \\mathbb P^4$ called a \\textit{del Pezzo surfaces of degree $4$}.\n\nAs in the Hasse--Minkowski theorem, a natural first step is to study the inclusion $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ of the rational points into the \\textit{adelic points}. Since we are considering homogeneous equations we can identify  $X(\\mathbb{A}_\\mathbb{Q})$ with $\\prod_{p \\leq \\infty} X(\\mathbb{Q}_p)$, that is, an adelic solution consists of local solutions in each completion of $\\mathbb{Q}$. We say that a class of varieties satisfies the \\textit{Hasse principle} if for each member of the family the existence of local solutions implies the existence of a global solution. In general the Hasse principle can fail. One such example was the quartic del Pezzo surface\n$$\n\\begin{cases}\nx^2-5y^2=uv;\\\\\nx^2-5z^2=(u+v)(u+2v),\n\\end{cases}\n$$\nby Birch and Swinnerton-Dyer \\cite{BSD}.\n\nThis failure can be explained by the Brauer--Manin obstruction, as introduced by Manin \\cite{ManinICM}; any element $\\mathcal{Q}$ in the Brauer group of $X$ determines an intermediate set\n\\[\nX(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}} \\subseteq X(\\mathbb{A}_\\mathbb{Q})\n\\]\nwhich can be used in some cases to show that $X(\\mathbb{Q})=\\emptyset$ even if there are adelic points on $X$. It has been conjectured by Colliot-Th\\'el\\`ene and Sansuc \\cite{CTSansuc} that for varieties such as del Pezzo surfaces, the Brauer--Manin obstruction is the \\textit{only obstruction to the Hasse principle}, that is, $X(\\mathbb{Q})$ is empty if and only if $X(\\mathbb{A}_{\\mathbb{Q}})^{\\Br} := \\bigcap_{\\mathcal{Q} \\in \\Br X} X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}$ is.\n\nPioneering work of Swinnerton-Dyer shows that for quartic del Pezzo surfaces the Brauer group $\\Br X\/\\Br_0 X$ is isomorphic to $\\left(\\mathbb{Z}\/2\\mathbb{Z} \\right)^i$ for $i \\in\\{0,1,2\\}$ \\cite{SDBrauerGroupCubicSurfaces}. Furthermore, the Brauer group can be explicitly computed from the equations \\cite{BBFL}, \\cite{VAV}.\n\nThe conjecture by Colliot-Th\\'el\\`ene and Sansuc was proven by Wittenberg \\cite{WittenbergBook} for quartic del Pezzo surface with a trivial Brauer group subject to the Schinzel hypothesis and the conjectured finiteness of Tate--Shafarevich groups of elliptic curves. These techniques were then applied by V\u00e1rilly-Alvarado and Viray for certain surfaces with a Brauer group of order $2$.\n\nWe will study the arithmetic of quartic del Pezzo surfaces with a Brauer group of order $4$, using fibration $X \\dashrightarrow \\mathbb P^1$ into curves. First we show that the existence of a commonly studied fibration implies the Hasse principle.\n\n\\begin{thm}[Thm.~\\ref{thm:conicfibrations}]\\label{thm:thm1}\nLet $X$ be a quartic del Pezzo surface with $\\# \\Br X\/\\Br_0 X =4$ over a number field $K$. If $X$ admits a conic fibration $X \\dashrightarrow \\mathbb P^1$ then $X(K)\\neq \\emptyset$.\n\\end{thm}\n\nThe work of V\u00e1rilly-Alvarado and Viray in the case of a smaller Brauer group, however uses maps $X \\dashrightarrow \\mathbb P^1$ obtained by embedding $X$ anticanonically and projecting away from a plane. They ask if a Brauer group of order $4$ can be ``vertical'' with respect to such maps. The second result of this paper is that this is not the case.\n\n\\begin{thm}[Thm.~\\ref{thm:nonewpoints}]\nLet $X \\subseteq \\mathbb P^4$ be an anticanonically embedded quartic del Pezzo surface over a number field $K$ with $\\# \\Br X\/\\Br_0 X =4$ and $X(K)=\\emptyset$. Then $\\Br X$ is not vertical with respect to a map $f \\colon X \\dashrightarrow \\mathbb P^1$ obtained by projecting away from a plane.\n\\end{thm}\n\nThis shows that the techniques of \\cite{CTSSD}, \\cite{WittenbergBook} and \\cite{VAV} cannot be directly applied to prove that the Brauer--Manin obstruction is the only one to the Hasse principle for quartic del Pezzo surfaces with a Brauer group of order $4$.\n\nSurfaces with two independent classes in the Brauer group have rarely shown up in the literature. Jahnel and Schindler \\cite{JS} prove that quartic del Pezzo surfaces with a Brauer group of order $2$ (and even those for which the conjecture is true) are Zariski dense in the moduli space of all quartic del Pezzo surfaces. In contrast, Mitankin and Salgado study a subfamily where infinitely many members have a Brauer group of order $4$, but all of these surfaces admit a conic fibrations and hence have a rational point in the light of Theorem~\\ref{thm:thm1}.\n\nWe will restrict to a different subfamily in which each member has a Brauer group of order $4$, which we will show to be non-empty.\n\n\\begin{thm}[Thm.~\\ref{thm:failureWA}, Thm.~\\ref{thm:curlyABorC}]\\label{thm:intro2}\nLet $X$ be a quartic del Pezzo surface with $\\#\\Br X\/\\Br_0 X =4$ given by a system\n\\[\n\\begin{cases}\ny^2-pz^2 = (A_1u+B_1v)(C_1u+D_1v);\\\\\nz^2-pz^2 = (A_2u+B_2v)(C_2u+D_2v),\n\\end{cases}\n\\]\nwhere $p$ is an odd prime. Then $X$ fails weak approximation and there is at most one element $\\mathcal{Q} \\in \\Br X\/\\Br_0 X$ for which $X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}=\\emptyset$.\n\\end{thm}\n\nNote the contrast with the result that if $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} = \\emptyset$ then there exists an $\\mathcal{Q} \\in \\Br X$ such that $X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}=\\emptyset$ for any quartic del Pezzo surface \\cite[Rem.~2 following Lem.~3.4]{CTPoonen}.\n\nFor every quartic del Pezzo surface $X$ with $\\Br X\/\\Br_0 X$ has order $4$, we can write down two explicit generators $\\mathcal{A}_X$ and $\\mathcal{B}_X$, where $\\mathcal{A}_X$ is uniquely defined and $\\mathcal{B}_X$ is any other non-trivial element. We produce examples in which either generator obstructs the Hasse principle.\n\n\\begin{thm}\nConsider the surfaces\n$$\nY\\colon\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(2u-13v)(u-6v),\n\\end{cases}\n$$\nand\n$$\nS \\colon \\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u+v)(153u+179v).\n\\end{cases}\n$$\n\\begin{enumerate}\n\\item[(a)] The surfaces $Y$ and $S$ are everywhere locally soluble.\n\\item[(b)] The Brauer groups $\\Br Y\/\\Br_0 Y$ and $\\Br S\/\\Br_0 S$ are both isomorphic to $(\\mathbb Z\/2\\mathbb Z)^2$.\n\\item[(c)] We have $Y(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{A}_Y} = \\emptyset$ for $\\mathcal{A}_Y = \\left(13,\\frac{u-6v}v\\right) \\in \\Br Y$, and $S(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{B}_S} = \\emptyset$ for $\\mathcal{B}_S = \\left(13,\\frac{y+z}u\\right) \\in \\Br S$.\n\\end{enumerate}\n\\end{thm}\n\nUsing Theorem~\\ref{thm:intro2} we see that $\\mathcal{A}_Y$ and $\\mathcal{B}_S$ are the unique elements in respectively $\\Br Y$ and $\\Br S$ with these properties.\n\nTo prove for a general quartic del Pezzo surface with a Brauer group of order $4$ that $X(\\mathbb{Q})\\ne\\emptyset$ when neither $\\mathcal A_X$, $\\mathcal B_X$ nor $\\mathcal{A}_X+\\mathcal{B}_X$ obstructs the Hasse principle is still open.\n\n\\subsection*{Acknowledgements}\n\nThis paper is a product of an internship of the second author at IST Austria under the supervision of the first author.\n\n\\section{Preliminaries}\n\nLet us fix our terminology.\n\n\\begin{defi}\nLet $k$ be a field. A $k$-scheme $X$ is called \\textit{nice} if it is proper, geometrically integral and smooth over $k$. A \\textit{surface} will be a nice $2$-dimensional $k$-scheme.\n\\end{defi}\n\nAll varieties under consideration in this paper will be nice $\\mathbb Q$-surfaces. So by the properness we can identify $X(\\mathbb{A}_\\mathbb{Q})$ with $\\prod_{p\\leq \\infty} X(\\mathbb{Q}_p)$ where we write $\\mathbb{Q}_\\infty := \\mathbb R$. We have the diagonal embedding $X(\\mathbb{Q}) \\hookrightarrow X(\\mathbb{A}_\\mathbb{Q})$.\n\nWe will say that a class $\\mathcal S$ of varieties satisfies the \\textit{Hasse principle} when $X(\\mathbb{Q})=\\emptyset$ if and only if $X(\\mathbb{A}_\\mathbb{Q}) = \\emptyset$ for all $X \\in \\mathcal S$. If $X(\\mathbb{Q})$ is dense in $X(\\mathbb{A}_\\mathbb{Q})$ we say that X satisfies \\textit{weak approximation}.\n\n\\subsection*{The Brauer--Manin obstruction}\n\nThe failure of the Hasse principle or weak approximation can be explained using the Brauer group. So let us consider $\\Br X := \\H^2(X_{\\text{\\'et}},\\mathbb{G}_m)$ and its natural subgroup $\\Br_0 X := \\Im(\\Br \\mathbb{Q} \\to \\Br X)$. Since $X$ is nice we have an inclusion $\\Br X \\hookrightarrow \\Br \\kappa(X)$, and we will only be interested in elements $\\mathcal Q \\in \\Br X$ of order $2$. Hence such elements are always represented by a quaternion algebra $(f,g)$ over the function field $\\kappa(X)$ of $X$.\n\nRecall the \\textit{invariant map} $\\inv_v \\mathcal Q \\colon X(\\mathbb{Q}_v) \\to \\mathbb{Q}\/\\mathbb{Z}$ of an element $\\mathcal Q=(f,g) \\in \\Br X[2]$ at a place $v$ \\cite[Thm.~1.5.36]{Poonen}, which are constant for $\\mathcal Q \\in \\Br_0 X$. At a point $P \\in X(\\mathbb{Q})$ for which $f,g\\in \\mathcal O^\\times_{X,P}$ the invariant $\\inv_v \\mathcal Q(P)$ coincides with the Hilbert symbol $(f(P),g(P)) \\in \\{\\pm 1\\}$ after identifying the groups $\\{\\pm 1\\}$ and $\\frac12\\mathbb{Z}\/\\mathbb{Z}$. By abuse of terminology we will say that $\\inv_v\\mathcal Q$ is \\textit{surjective} if $\\# \\Im(\\inv_v \\mathcal Q) = 2$.\n\nFor $\\mathcal Q \\in \\Br X$ we can consider\n\\[\nX(\\mathbb{A}_\\mathbb{Q})^{\\mathcal Q}  := \\{(P_v)_v \\in X(\\mathbb{A}_\\mathbb{Q})\\mid \\sum_{v \\leq \\infty} \\inv_v \\mathcal Q(P_v) =0\\}.\n\\]\nBy global reciprocity we have $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}} \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ or even $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\Br} \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ for $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} := \\bigcap X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}}$.\n\nIf $X(\\mathbb{A}_\\mathbb{Q}) \\ne \\emptyset$, but $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} = \\emptyset$, then the Hasse principle fails for $X$ and we say there is a \\textit{Brauer--Manin obstruction to the Hasse principle}. If $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} \\subsetneq X(\\mathbb{A}_\\mathbb{Q})$, then $X$ cannot satisfy weak approximation, and there is a \\textit{Brauer--Manin obstruction to weak approximation}.\n\nNote that if an invariant map of $\\mathcal Q$ is surjective, then there is a Brauer--Manin obstruction to weak approximation, but not necessarily to the Hasse principle.\n\n\\subsection*{Quartic del Pezzo surfaces}\n\nWe will be interested in a particular type of surface, namely del Pezzo surface of degree $4$. For a general treatise on general del Pezzo surfaces and their arithmetic the reader is referred to \\cite{arithmeticdps}. We will give the following characterisation of quartic del Pezzo surfaces, which we will take as the definition.\n\n\\begin{defi}\nA \\textit{del Pezzo surface of degree $4$} is a surface $X \\subseteq \\mathbb P^4$, hence in particular smooth, given by two quadratic forms $Q=0=\\tilde Q$.\n\\end{defi}\n\nIt is known that the Brauer group of a quartic del Pezzo surface modulo constants is either trivial, of order $2$ or isomorphic to the Klein four-group. Algorithms to determine this isomorphism class and explicit generators are given in both \\cite{BBFL} and \\cite{VAV}. Let us introduce some notation from the latter algorithm to determine if we are dealing with a Brauer group of order $4$.\n\nRepresent the quadratic forms $Q$ and $\\tilde Q$ by symmetric matrices $M$ and $\\tilde M$. For a point $T=(\\kappa \\colon \\lambda) \\in \\mathbb P^1(\\bar{\\mathbb{Q}})$ we define the symmetric matrix $M_T = \\kappa M + \\lambda \\tilde M$, with associated quadratic form $Q_T$ over $\\kappa(T)$. Let $f$ be the discriminant of $M_T$, then $\\mathscr{S}=V(f)\\subseteq\\mathbb{P}^1$ is the locus of the degenerate fibres. For the quadratic forms $Q_T$ of rank $4$ we will need the determinant $\\varepsilon_T$ of its restriction to a linear subspace of codimension $1$ on which it is nondegenerate. The invariant $\\varepsilon_T$ is well defined up to squares.\n\n\\begin{pro}[{\\cite[\\textsection 4.1]{VAV}}]\\label{pro:Brauergrouporder4}\nLet $X \\subseteq \\mathbb P^4$ be a quartic del Pezzo surface. Then $\\#\\Br X\/\\Br_0 X =4$ if and only if there exists an $\\varepsilon\\not\\in\\mathbb{Q}^{\\times, 2}$, and three degree 1 points $T_i \\in \\mathscr{S}$ such that each $Q_{T_i}$ has rank $4$ and satisfies $\\varepsilon\\cdot\\varepsilon_{T_i}\\in\\mathbb{Q}^{\\times, 2}$.\n\\end{pro}\n\n\\section{A Brauer group of order $4$}\n\nFrom now on $X\/\\mathbb Q$ will be a quartic del Pezzo surface anticanonically embedded in $\\mathbb P^4$. Such a surface is the intersection of two quadrics. We will be particularly interested in the $X$ for which the Brauer group modulo constants has order $4$. \n\n\\begin{pro}\\label{pro:gen_sys}\nA quartic del Pezzo surface $X$ over $\\mathbb Q$ satisfies $\\Br X\/\\Br \\mathbb Q \\cong (\\mathbb Z\/2\\mathbb Z)^2$ precisely if it can be given by a system of equations of the form\n\\begin{equation}\\label{gen_sys}\n\\begin{cases}\nd_0y^2-\\varepsilon x^2=a_0u^2+2b_0uv+c_0v^2;\\\\\nd_1z^2-\\varepsilon x^2=a_1u^2+2b_1uv+c_1v^2,\n\\end{cases}\n\\end{equation}\nsuch that\n\\begin{enumerate}\n\\item $\\varepsilon\\not\\in\\mathbb{Q}^{\\times, 2}$,\n\\item $d_0=b_0^2-a_0c_0$ and $d_1=b_1^2-a_1c_1$,\n\\item $\\varepsilon d_0d_1d_2\\in \\mathbb{Q}^{\\times, 2}$ where $d_2:=(b_1-b_0)^2-(a_1-a_0)(c_1-c_0)$, and\n\\item the quadratic forms $a_iu^2+2b_iuv+c_iv^2$ do not have a common projective root.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nProposition~4.2 in \\cite{VAV} and Proposition~\\ref{pro:Brauergrouporder4} yield that after a linear change of variables the matrices of $Q_{T_i}$ have the following form\n$$\nM_{T_0}=\n\\begin{pmatrix}\na_0 & b_0 & 0 & 0 & 0\\\\\nb_0 & c_0 & 0 & 0 & 0\\\\\n0 & 0 & m_0 & 0 & 0\\\\\n0 & 0 & 0 & n_0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix},\\ \nM_{T_1}=\n\\begin{pmatrix}\na_1 & b_1 & 0 & 0 & 0\\\\\nb_1 & c_1 & 0 & 0 & 0\\\\\n0 & 0 & m_1 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & k_1\n\\end{pmatrix}\n\\ \n\\text{ and }\n\\ \nM_{T_2}=\n\\begin{pmatrix}\na_2 & b_2 & 0 & 0 & 0\\\\\nb_2 & c_2 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & n_2 & 0\\\\\n0 & 0 & 0 & 0 & k_2\n\\end{pmatrix}.\n$$\nWe get that\n$$\n\\varepsilon_{T_0}=-d_0m_0 n_0,\\quad\\varepsilon_{T_1}=-d_1m_1 k_1 \\quad \\text{ and } \\quad \\varepsilon_{T_2}=-d_2n_2k_2,\n$$\nwhere $d_i:=b_i^2-a_ic_i$. Since the $M_{T_i}$ are linearly dependent we can assume $M_{T_2}=M_{T_1}-M_{T_0}$. Without loss of generality we can set $m_0=m_1$ equal to $\\varepsilon$. We can still change $n_0$ and $k_1$ independently up to squares, and the condition $\\varepsilon\\cdot\\varepsilon_{T_i}\\in\\mathbb{Q}^{\\times,2}$ allows us to take $n_0=-d_0$ and $k_1=-d_1$. The condition $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ is then automatically satisfied. The last condition is equivalent to the projective scheme defined by \\eqref{gen_sys} is smooth.\n\nFrom Proposition~\\ref{pro:Brauergrouporder4} it is easy to check that any such system defines a surface with a Brauer group of order $4$.\n\\end{proof}\n\nThe algorithm in \\cite{VAV} even allows us to represent the elements of the Brauer group by explicit quaternion algebras. Given the explicit element of the Brauer group we can now discuss whether the Brauer--Manin obstruction to Hasse principle is the only one. For simplicity we restrict to surfaces over the rational numbers, but the results and proofs in this and the last section equally apply to quartic del Pezzo surfaces with a Brauer group of order $4$ over any number fields. \n\nFor quartic del Pezzo surfaces with a large Brauer group which admit conic fibrations there cannot be a Brauer--Manin obstruction to the Hasse principle.\n\n\\begin{thm}\\label{thm:conicfibrations}\nLet $X\/\\mathbb Q$ be a quartic del Pezzo surface with $\\# \\Br X\/\\Br_0 X =4$. If $X$ admits a conic fibration $X \\dashrightarrow \\mathbb P^1$ then $X(\\mathbb Q)\\neq \\emptyset$.\n\\end{thm}\n\n\\begin{proof}\nSince $X$ is a nice variety over a number field we conclude from the Hochschild--Serre spectral sequence \\cite[Cor.~6.7.8]{Poonen} that $\\Br X\/\\Br_0 X \\cong \\H^1(G_{\\mathbb Q}, \\Pic \\bar X)$. We will need to understand the Galois action on the geometric Picard group.\n\nFor a general quartic del Pezzo surface the action of the absolute Galois group $G_{\\mathbb Q}$ on $\\Pic \\bar X$ factors through a subgroup $H \\subseteq W_5$. Here $W_5$ is the finite group Weyl group which acts on $\\Pic \\bar X$, see \\cite[\\textsection 1.5]{arithmeticdps}. Since $\\Pic \\bar X$ is torsion-free we conclude from the inflation-restriction sequence that $\\H^1(G_{\\mathbb Q}, \\Pic \\bar X) \\cong \\H^1(H, \\Pic \\bar X)$. For each of the 196 different subgroups of $W_5$ (up to conjugation) we can compute the first cohomology group directly and we find that only four subgroups yield a Brauer group of order $4$.\n\nAgain using the Hochschild--Serre spectral sequence we see that $\\Pic X \\hookrightarrow (\\Pic \\bar X)^{G_{\\mathbb Q}}$ is injective. For the four relevant Galois actions on $\\Pic \\bar X$ we see that $(\\Pic \\bar X)^{G_{\\mathbb Q}}$ has rank $1$ in three cases, and rank $2$ in one case.\n\nIn the case of rank $1$ we conclude that $\\Pic X$ is free of rank $1$, since it contains the canonical class. If $X$ would admit a conic fibration then the class of the fibre should be a (rational) multiple of the canonical class, and have self-intersection $0$. Clearly a contradiction.\n\nIn the case of rank $2$ we find that the action of Galois permutes two exceptional curves, whose intersection pairing equals $1$. Hence their intersection point is defined over the base field.\n\\end{proof}\n\n\\begin{rema} More precisely, in the latter case there are multiple pairs of such intersecting lines which sum to the same class in $\\Pic X$. One can conclude from this that among the quartic del Pezzo surfaces with a Brauer group of order $4$ the following statements are equivalent.\n\\begin{enumerate}\n\\item $\\rk \\Pic X =2$.\n\\item $X$ contains two conjugate intersecting lines.\n\\item $X$ admits a conic fibration.\n\\end{enumerate}\n\\end{rema}\n\nFor general maps $f\\colon X\\dashrightarrow \\mathbb{P}^1$ the following notion will be very useful. We say that $\\Br X$ is \\textit{vertical} with respect to $f$ if $\\Br X\\subseteq f^*(\\Br \\kappa(\\mathbb{P}^1))$.\n\nFor quartic del Pezzo surfaces with vertical Brauer groups there is a method to prove that $X(\\mathbb{A}_\\mathbb{Q})^{\\Br}\\neq\\emptyset$ \nimplies $X(\\mathbb{Q})\\neq\\emptyset$ (see \\cite{WittenbergBook} and \\cite{CTSSD}). V\u00e1rilly-Alvarado and Viray proved in \\cite{VAV} that the Brauer group is vertical for certain quartic del Pezzo surfaces, and they concluded that the Brauer--Manin obstruction to Hasse principle is the only one for these surface of ``BSD-type''.\n\nNext they considered quartic del Pezzo surface with a Brauer group of order $4$. First they noted that if $X(\\mathbb{Q})\\neq \\emptyset$, then $\\Br X$ is vertical with respect to a specific map. Next they asked if $X(\\mathbb{Q})= \\emptyset$ whether the Brauer group can be vertical, in particular for rational maps $f \\colon X \\dashrightarrow \\mathbb P^1$ obtained from projecting away from a plane. They suggested the following approach.\n\nLet $T_i$ be the three degree 1 points of $\\mathscr{S}$, see Proposition~\\ref{pro:Brauergrouporder4}. For $P_i\\in V(Q_{T_i})(\\mathbb{Q})$, denote $M(P_0,P_1,P_2)$ the matrix whose $(i,j)$th entry is $\\frac{\\partial Q_{T_i}}{\\partial x_j}(P_i)$. Then take a hyperplane $H\\subseteq \\mathbb{P}^4$ which does not pass through the vertices of $V(Q_{T_i})$, and define\n$$\nY^{(X)}=\\{(P_0,P_1,P_2)\\in H^3\\mid \\rnk M(P_0,P_1,P_2)\\le 2 \\text{ and } Q_{T_i}(P_i)=0\\}.\n$$\nThere is an inclusion $X(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})$, which is described in \\cite{VAV}.\n\nIf we can find a rational point on $Y^{(X)}$, then we obtain the verticality of $\\Br X\/\\Br\\mathbb{Q}$ with respect to a certain map $f$. We already know that if $X(\\mathbb{Q})\\neq \\emptyset$, then $Y^{(X)}(\\mathbb{Q})\\neq\\emptyset$. The question is, whether there is a point in $Y^{(X)}(\\mathbb{Q})$ which does not come from a point in $X(\\mathbb{Q})$ (see \\cite{VAV}, Question~6.3). Here we show that the answer is no.\n\n\\begin{defi}\nWe define a point $P=(u \\colon v \\colon x \\colon y \\colon z) \\in \\mathbb P^4(\\mathbb{Q})$ to be equivalent to the points $(u \\colon v \\colon \\pm x \\colon \\pm y \\colon \\pm z)$. This induces an equivalence relation $\\sim$ on $Y^{(X)}(\\mathbb{Q})$.\n\\end{defi}\n\nIn other words, two points $\\mathcal P = (P_0,P_1,P_2)$ and $\\mathcal P' = (P'_0,P'_1,P'_2)$ are equivalent precisely if the $P_i$ and $P'_i$ are the same projective point up to a possible change of sign in the last three coordinates.\n\n\nConsider a point $\\mathcal P \\in Y^{(X)}(\\mathbb{Q})$. Although $\\mathcal P$ need not lie the image of the map $X(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})$, we now show that it is equivalent to a point that does.\n\n\\begin{thm}\\label{thm:nonewpoints}\nThe composite map\n$$\nX(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})\\longrightarrow Y^{(X)}(\\mathbb{Q})\/\\sim\n$$\nis surjective.\n\\end{thm}\n\n\\begin{proof}\nWe will represent $X$ by a system of equations as in \\eqref{gen_sys}. In addition, we can assume without loss of generality $a_0 \\ne 0$ and hence after completing the square that $b_0=0$. This implies that $d_0 = -a_0c_0$ and we conclude that also $c_0 \\ne 0$.\n\nConsider the point $(P_0,P_1,P_2)\\in Y^{(X)}(\\mathbb{Q})$ where $P_i=(u_i:v_i:x_i:y_i:z_i)$. Our goal is to prove that $(P_1,P_2,P_3)\\sim (P,P,P)$ for some $P\\in X(\\mathbb{Q})$. By the definition of $Y^{(X)}$ using $T_0=(1:0)$, $T_1=(0:1)$ and $T_2=(-1:1)$ as in the proof of Proposition~\\ref{pro:gen_sys}, we have\n$$\n\\mathrm{rank\\,}\n\\begin{pmatrix}\n2a_0u_0                & 2c_0v_0                & 2\\varepsilon x_0  & -2d_0y_0  & 0    \\\\\n2a_1u_1+2b_1v_1        & 2b_1u_1+2c_1v_1        & 2\\varepsilon x_1  & 0      & -2d_1z_1\\\\\n2(a_1-a_0)u_2+2b_1v_2  & 2b_1u_2+2(c_1-c_0)v_2  & 0            & 2d_0y_2   & -2d_1z_2\n\\end{pmatrix}\n=2.\n$$\nWrite $\\ell_i \\in\\mathbb{Q}^5$ for the rows of the above matrix. The condition on the rank implies the existence of a non-trivial relation $\\kappa\\ell_0+\\lambda \\ell_1+\\mu \\ell_2=0$. After multiplying the coordinates of $P_0$, $P_1$ and $P_2$ by respectively $\\kappa$, $-\\lambda$ and $\\mu$, we obtain $\\ell_0-\\ell_1+\\ell_2=0$. This implies \n\\begin{equation}\\label{xyz}\nx_0=x_1,\\ y_0=y_2\\ \\text{ and }\\ z_1=z_2,\n\\end{equation} \nand the first two columns give us the relations\n\\begin{equation}\\label{subst}\nu_0=\\frac{1}{a_0}\\Bigl(a_1u_1+b_1v_1-(a_1-a_0)u_2-b_1v_2\\Bigr)\n\\quad \\text{ and } \\quad\nv_0=\\frac{1}{c_0}\\Bigl(b_1u_1+c_1v_1-b_1u_2-(c_1-c_0)v_2\\Bigr).\n\\end{equation}\nCondition $Q_{T_i}(P_i)=0$ gives us\n\\begin{multline}\\label{quadr}\n(a_0u_0^2+c_0v_0^2)-(a_1u_1^2+2b_1u_1v_1+c_1v_1^2)+((a_1-a_0)u_2^2+2b_1u_2v_2+(c_1-c_0)v_2^2)=\\\\\n(d_0y_0^2-\\varepsilon x_0^2)-(d_1z_1^2-\\varepsilon x_1^2)+(d_1z_2^2-d_0y_2^2)=0.\n\\end{multline}\nSubstituting \\eqref{subst} into \\eqref{quadr}, we obtain the following:\n\\begin{multline}\\label{quad_form}\n(a_0^2c_0+b_0^2a_0-a_1a_0c_0)(u_1-u_2)^2+2(a_1b_1c_0+b_1c_1a_0-b_1a_0c_0)(u_1-u_2)(v_1-v_2)+\\\\\n(b_1^2c_0+c_1^2a_0-c_1a_0c_0)(v_1-v_2)^2=0.\n\\end{multline}\nThis is a quadratic form in $u_1-u_2$ and $v_1-v_2$ whose determinant equals \n$$\n-a_0c_0(b_1^2-a_1c_1)(b_1^2-(a_1-a_0)(c_1-c_0))=d_0d_1d_2.\n$$ \nFrom $\\varepsilon\\notin\\mathbb{Q}^{\\times,2}$ and $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ we conclude $d_0d_1d_2\\notin\\mathbb{Q}^{\\times,2}$, so \n\\eqref{quad_form} implies $u_1-u_2=0=v_1-v_2$. Then \\eqref{subst} implies that $u_0=u_1=u_2$ and $v_0=v_1=v_2$.\n\nFinally, conditions \\eqref{xyz} and $Q_{T_i}(P_i)=0$ give us $x_0=x_1=\\pm x_2$, $y_0=y_2=\\pm y_1$ and $z_1=z_2=\\pm z_0$, \nwhich means that $P_i \\in X(\\mathbb{Q})$ and $P_0 \\sim P_1 \\sim P_2$ which proves the proposition.\n\\end{proof}\n\n\nWe deduce that the answer to Question~6.3 in \\cite{VAV} is no. We conclude that one cannot directly apply to machinery of \\cite{CTSSD} and \\cite{WittenbergBook} to prove there are rational points on quartic del Pezzo surfaces with a Brauer group of order $4$.\n\n\\section{A subfamily}\n\nWe will study the arithmetic of quartic del Pezzo surfaces with a Brauer group of order $4$. In \\cite{MS} such surfaces appeared, but those all had a rational point for obvious geometrical reasons; on these surfaces there is a pair of intersecting lines which, as a pair, are fixed by the Galois action. We will restrict to a subfamily for which such behaviour does occur.\n\n\\begin{defi}\nLet $\\mathcal X$ be the set of isomorphism classes of quartic del Pezzo surfaces for which the two quadratic forms $Q_i=a_iu^2+b_iuv+c_iv^2$ split over $\\mathbb Q$, and $\\varepsilon$ is an odd prime number $p$.\n\\end{defi}\n\n\\begin{pro}\\label{pro:sys_subfam}\nAny $X \\in \\mathcal X$ can be written as\n\\begin{equation}\\label{sys}\n\\begin{cases}\ny^2-px^2=Muv;\\\\\nz^2-px^2=(Au+Bv)(Cu+Dv),\n\\end{cases}\n\\end{equation}\nfor $A,B,C,D,M,N\\in\\mathbb{Z}$ which satisfy\n\\begin{enumerate}\n\\item[\\textbf{(C1)}] $(AD+BC-M)^2-4ABCD=pN^2$, and\n\\item[\\textbf{(C2)}] $ NM(AD-BC)\\neq 0$.\n\\end{enumerate}\n\\end{pro}\n\nCondition (C1) is a restatement of $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ and (C2) is equivalent to $X$ being smooth. \n\n\n\\begin{proof}\nIt is an immediate consequence of Proposition \\ref{pro:gen_sys}.\n\\end{proof}\n\nTo study the members of this family which fail the Hasse principle we will first consider the question of local solubility. For places $v$ such that $p\\in\\mathbb{Q}_v^{\\times,2}$ we have $(0\\colon 0\\colon 1\\colon \\sqrt{p}\\colon \\sqrt{p})\\in X(\\mathbb{Q}_v)$, so $X(\\mathbb{Q}_v)\\neq\\emptyset$. The following proposition covers all but two of the other places.\n\n\n\\begin{pro}\\label{pro:loc_sol_subfam}\nConsider $X=X_{p,A,B,C,D,M,N}$. Let $v\\not \\in \\{2,p\\}$ be a place such that $p\\notin\\mathbb{Q}_v^{\\times,2}$ and $v\\nmid N$. Then $X(\\mathbb{Q}_v)\n\\neq\\emptyset$.\n\\end{pro}\n\n\\begin{proof}\nObviously $v\\neq \\infty$, so it corresponds to a prime number $q$. The Chevalley--Warning theorem yields that the system \\eqref{sys} has a non-trivial solution $(u,v,x,y,z)\\in\\mathbb{F}_q^5$. Let $(u_0,v_0,x_0,y_0,z_0)$ be an arbitrary lift to $\\mathbb{Z}_q^5$. We distinguish four cases.\n\n\\noindent\\textbf{Case 1.} $q \\nmid y_0,z_0$. In this case take \n$$\ny_1=\\sqrt{Mu_0v_0+px_0^2}\\in\\mathbb{Z}_q,\\quad z_1=\\sqrt{(Au_0+Bv_0)(Cu_0+Dv_0)+px_0^2}\\in\\mathbb{Z}_q,\n$$ \nthen $(u_0\\colon v_0\\colon x_0\\colon y_1\\colon z_1)\\in X(\\mathbb{Q}_q)$.\n\n\n\n\\noindent\\textbf{Case 2.} $q \\mid y_0$ and $q \\nmid x_0,z_0$. Take \n$$\nx_1=\\sqrt{-\\tfrac{M}{p}u_0v_0}\\in\\mathbb{Z}_q,\\quad z_1=\\sqrt{(Au_0+Bv_0)(Cu_0+Dv_0)+px_1^2}\\in\\mathbb{Z}_q,\n$$ \nthen $(u_0\\colon v_0\\colon x_1\\colon 0\\colon z_1)\\in X(\\mathbb{Q}_q)$.\n\n\\noindent\\textbf{Case 3.} $q \\mid x_0, y_0$ and $q \\nmid z_0$. We deduce that $q \\mid u_0v_0$. If $q \\mid u_0$, take $(0\\colon v_0\\colon 0\\colon 0\\colon \\sqrt{BDv_0^2})\\in X(\\mathbb{Q}_q)$. If on the other hand $q \\mid v_0$, take $(u_0\\colon 0\\colon 0\\colon 0\\colon \\sqrt{ACu_0^2})\\in X(\\mathbb{Q}_q)$.\n\n\\noindent\\textbf{Case 4.} $q \\mid y_0, z_0$. Subtracting the two equations in \\eqref{sys} yields\n$$\nACu^2+(AD+BC-M)uv+BDv^2\\equiv 0\\mod q,\n$$\nwhich can be rewritten to\n$$\n(2ACu+(AD+BC-M)v)^2 \\equiv (AD+BC-M)^2v^2-4ABCDv^2 \\equiv pN^2v^2 \\mod q,\n$$\nand equivalently\n$$\n(2BDv+(AD+BC-M)u)^2 \\equiv pN^2u^2 \\mod q.\n$$\nFrom $\\left(\\frac{p}{q}\\right)=-1$ and $q\\nmid N$ we obtain $u\\equiv v\\equiv 0\\mod q$, and \\eqref{sys} gives us $x\\equiv 0\\mod q$. Contradicting the non-triviality of $(u,v,x,y,z)\\in\\mathbb{F}_q^5$.\n\\end{proof}\n\n\n\n\n\nMost of our examples will satisfy $N=1$ so $X$ will be locally soluble everywhere away from $2$ and $p$.\n\nThe algorithm in \\cite[\\textsection 4.1]{VAV} allows to write down the explicit elements of $\\Br X\/\\Br \\mathbb{Q}$. Choosing $P_{T_0}=(0\\colon 1\\colon 0\\colon 0\\colon 0)\\in Q_{T_0}(\\mathbb{Q})$, $P_{T_1}=(-B\\colon A\\colon 0\\colon 0\\colon 0)\\in Q_{T_1}(\\mathbb{Q})$ and $P_{T_2}=(0\\colon 0\\colon 0\\colon 1\\colon 1)\\in Q_{T_2}(\\mathbb{Q})$, we obtain\n\\begin{equation}\\label{Br_subfam}\n\\Br X\/\\Br \\mathbb{Q}=\\left\\{\\id, \\left(p,\\frac{u}{Au+Bv}\\right), \\left(p,\\frac{z-y}{u}\\right), \\left(p,\\frac{Au+Bv}{z-y}\n\\right)\\right\\}.\n\\end{equation}\n\n\n\\begin{defi}\\label{defi:ABC}\nFor any $X\\in\\mathcal{X}$ given by \\eqref{sys}, we will write $\\mathcal{A}=\\left(p,\\frac{u}{Au+Bv}\\right)$, $\\mathcal{B}=\\left(p,\\frac{z-y}{u}\\right)$ and $\\mathcal{C}=\\left(p,\\frac{Au+Bv}{z-y}\\right)$ and assume the dependency on $X$ to be understood.\n\\end{defi}\n\n\\begin{rema}\nThe above notation actually depends on the representation \\eqref{sys} we chose for the equivalence class $X\\in\\mathcal{X}$. One can see that the change of variables $\\tilde{u}=\\frac{Au+Bv}{AD-BC}$, $\\tilde{v}=\\frac{Cu+Dv}{AD-BC}$, $\\tilde{y}=z$, $\\tilde{z}=y$ provides another representation of $X$ in the form \\eqref{sys}, but with possibly different coefficients, and $\\mathcal{B}$ for the new representation is exactly $\\mathcal{C}$ for the old one. \nHowever, one can show that $\\mathcal{A}$ does not depend on the representation of $X$.\n\n\n\n\\end{rema}\n\n\n\n\\begin{rema}\nOne can see that the classes $\\mathcal{A}$ and $\\mathcal{B}$ are also represented by\n\\begin{equation}\\label{Br_gr_rewr}\n\\mathcal{A}=\\left(p,\\frac{Mv}{Au+Bv}\\right) \\quad \\text{ and } \\quad\n\\mathcal{B}=\\left(p,\\frac{AC(z+y)}{u}\\right).\n\\end{equation}\n\\end{rema}\n\n\n\n\n\n\\section{Simultaneous obstructions to the Hasse principle}\n\nWe use the notation from the previous section. In particular, $X$ is a quartic del Pezzo surface given by explicit equations as in Proposition~\\ref{pro:sys_subfam}. We then see from \\eqref{Br_subfam} and Definition~\\ref{defi:ABC} that\n\\[\n\\Br X\/\\Br \\mathbb Q = \\{1,\\mathcal A, \\mathcal B, \\mathcal C\\}.\n\\]\nWe will concern ourselves with the Brauer--Manin obstruction to weak approximation and the Hasse principle. The first results is on weak approximation.\n\n\\begin{thm}\\label{thm:failureWA}\nAny $X \\in \\mathcal X$ fails weak approximation.\n\\end{thm}\n\nThe second result is about the Hasse principle and seems related to the following fact by Colliot-Th\\'el\\`ene and Poonen \\cite[Rem.~2 following Lem.~3.4]{CTPoonen} on quartic del Pezzo surfaces: suppose that $X(\\mathbb A_\\mathbb{Q})^{\\Br} = \\emptyset$ then there exists an element $\\mathcal Q \\in \\Br X$ such that\n $X(\\mathbb A_\\mathbb{Q})^{\\mathcal Q} = \\emptyset$. However, neither result implies the other.\n\n\\begin{thm}\\label{thm:curlyABorC}\nOnly one of $\\mathcal A$, $\\mathcal B$ and $\\mathcal C$ can give an obstruction to the Hasse principle on $X \\in \\mathcal X$.\n\\end{thm}\n\nTheorem~\\ref{thm:failureWA} and Theorem~\\ref{thm:curlyABorC} are direct consequences of the following proposition.\n\n\\begin{pro}\\label{prop:surjectiveinvariantmap}\nConsider an $X\\in\\mathcal{X}$. Assume $X(\\mathbb{Q}_p)\\neq\\emptyset$, then the invariant map at $p$ of $\\mathcal{A}$, $\\mathcal{B}$ or $\\mathcal{C}$ is surjective.\n\\end{pro}\n\nWe will need the following lemmas.\n\n\\begin{lem}\\label{lem:quadres}\nLet $p\\equiv 1\\mod 4$ be a prime. Consider the set $S(a,b)=\\{a+by\\mid y\\in\\mathbb{F}_p^{\\times,2}\\}$ for $a,b\\in\\mathbb{F}_p^\\times$. Denote $S(a,b)_\\zeta=\\{x\\in S(a,b)\\mid \\left( \\frac xp\\right)=\\zeta\\}$. \n\\begin{enumerate}\n\\item[(a)] If $a,b\\in\\mathbb{F}_p^{\\times,2}$, then $|S(a,b)_0|=1$, $|S(a,b)_1|=\\dfrac{p-5}{4}$ and $|S(a,b)_{-1}|=\\dfrac{p-1}{4}$.\n\\item[(b)] If $a\\in\\mathbb{F}_p^{\\times,2}$, $b\\notin\\mathbb{F}_p^{2}$, then $|S(a,b)_0|=0$, $|S(a,b)_1|=\\dfrac{p-1}{4}$ and $|S(a,b)_{-1}|=\\dfrac{p-1}{4}$.\n\\end{enumerate}\nIn particular, for $a\\in\\mathbb{F}_p^{\\times,2}$ we conclude that $S(a,b)$ contains a non-square.\n\\end{lem}\n\n\\begin{proof}\nBoth parts follow from the identity $\\sum_{x\\in\\mathbb{F}_p} (\\frac{a+bx^2}{p})=-(\\frac{b}{p})$ where $a,b\\in\\mathbb{F}_p^\\times$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:quadres_table}\nLet $p\\equiv 1\\mod 4$ be a prime and fix $a,b,c,d\\in\\mathbb{F}_p^{\\times,2}$. Then there exists an $y_0\\in\\mathbb{F}_p^\\times$ such that either\n\\begin{enumerate}\n\\item[(i)] $a+by_0^2,\\ c+dy_0^2\\in\\mathbb{F}_p^{\\times}\\setminus\\mathbb{F}_p^{\\times,2}$, or\n\\item[(ii)] $a+by_0^2\\in \\mathbb{F}_p^{\\times}\\setminus\\mathbb{F}_p^{\\times,2}$ and $c+dy_0^2=0$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nApply the statement (a) of Lemma~\\ref{lem:quadres} for $S(a,b)$ and $S(c,d)$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:surjectiveinvariantmap}]\nWe will first make a few reductions.\n\nFirst suppose that $p \\mid M, A, C$. Then the system \\eqref{sys} is equivalent to the one with $M$, $A$ and $B$ divided by $p$. Also if $p^2 \\mid M, A, C$ then we obtain an equivalent system by dividing these three coefficients by $p^2$. This shows that we can simply to the cases where\n\\begin{equation}\\label{gcd_cond_1}\n p \\nmid \\gcd(A,C,M) \\text{ and } p \\nmid \\gcd(B,D,M),\n\\end{equation}\nand\n\\begin{equation}\\label{gcd_cond_2}\np^2\\nmid\\gcd(A,B,M) \\text{ and } p^2\\nmid \\gcd(C,D,M).\n\\end{equation}\n\nConsider the case $p \\equiv 3 \\mod 4$. For a point $P=(u\\colon v \\colon x \\colon y \\colon z) \\in X(\\mathbb Q_p)$ we compute the invariant map at $P$ and $P'=(u\\colon v \\colon x \\colon -y \\colon -z)$. We see that\n\\[\n\\inv_p \\mathcal{B}(P) \\ne \\inv_p \\mathcal{B}(P')\n\\]\nand this proves the proposition in this case.\n\nNow we can assume \n\\begin{equation}\\label{p_1_mod4}\np\\equiv 1\\mod 4.\n\\end{equation}\nCondition (C1) implies $(p,ABCD)_p=1$. Consider the case $(p,AC)_p=(p,BD)_p=-1$. For a point $P=(u\\colon v\\colon x\\colon y\\colon z)\\in X(\\mathbb{Q}_p)$, we define $P'=(u\\colon v\\colon x\\colon -y\\colon z)\\in X(\\mathbb{Q}_p)$. One can see that $\\inv_p\\mathcal{B}(P)\\neq\\inv_p\\mathcal{B}(P')$ using \\eqref{Br_gr_rewr}, and this proves the proposition in this case.\n\nSo now we can assume\n\\begin{equation}\\label{hilbert_cond}\n(p,AC)_p=(p,BD)_p=1.\n\\end{equation}\n\nFor equations satisfying (C1) and (C2), and the additional assumptions \\eqref{gcd_cond_1} up to \\eqref{hilbert_cond} we consider different cases depending on the valuation of $M$ and $m = \\max\\{v_p(A),v_p(B),v_p(C),v_p(D)\\}$. Without loss of generality we can assume that $p^m \\mathrel{\\|} A$ and $p^{m+1} \\nmid B,C,D$.\n\nIn the first two cases we show that the invariant map of $\\mathcal{B}$ at $p$ is surjective.\n\\begin{enumerate}\n\\item[\\textbf{Case 1.}] $\\mathrm{v}_p(M)=0$ and $m\\geq 1$\n\\item[\\textbf{Case 2.}] $\\mathrm{v}_p(M)=1$ and $m=1$\n\\end{enumerate}\n In the next two cases we show that the invariant map of $\\mathcal{A}$ at $p$ is surjective.\n\\begin{enumerate}\n\\item[\\textbf{Case 3.}] $\\mathrm{v}_p(M)=0$ and $m=0$\n\\item[\\textbf{Case 4.}] $\\mathrm{v}_p(M)$ is odd and $m=0$\n\\end{enumerate}\nIn the last four cases we can make a change of variables to reduce to one of the previous cases.\n\\begin{enumerate}\n\\item[\\textbf{Case 5.}] $\\mathrm{v}_p(M)=1$ and $m\\geq 2$\n\\item[\\textbf{Case 6.}] $\\mathrm{v}_p(M) \\geq 3$ is odd and $m \\geq 1$\n\\item[\\textbf{Case 7.}] $\\mathrm{v}_p(M) \\geq 2$ is even and $m=0$\n\\item[\\textbf{Case 8.}] $\\mathrm{v}_p(M) \\geq 2$ is even and $m\\geq 1$\n\\end{enumerate}\n\n\\noindent\n\\textbf{Surjectivity for $\\mathcal{B}$}.\nTo prove $\\inv_p\\mathcal B$ is surjective we will find points $P_i=(u_i\\colon v_i\\colon x_i\\colon y_i\\colon z_i)\\in X(\\mathbb{Q}_p)$ such that \n$$\n\\left(\\frac{u_1(z_1-y_1)}{p}\\right)\\left(\\frac{u_2(z_2-y_2)}{p}\\right)=-1,\n$$ \nsince this implies $\\inv_p\\mathcal{B}(P_1)+\\inv_p\\mathcal{B}(P_2)=\\frac{1}{2}$.\n\n\\noindent\\textbf{Case 1.} Since $p \\mid A$, condition (C1) implies that \n$\nBC\\equiv M\\not\\equiv 0\\mod p.\n$\nHence $\\mathrm{v}_p(B)=\\mathrm{v}_p(C)=0$.\n\n\\textbf{Case 1a.} If $\\mathrm{v}_p(D)=0$ then \\eqref{p_1_mod4} and \\eqref{hilbert_cond} imply $-BD\\in\\mathbb{Z}_p^{\\times, 2}$.\nChoose $r\\in\\mathbb{Z}_p^{\\times}$ such that $r\\sqrt{-BD}\\notin\\mathbb{Z}_p^{\\times, 2}$. Consider $y_2=\\frac{1}{2}\\left(\\frac{BD}{r}-r\n\\right)$ and note that\n\\begin{equation*}\n-\\frac{DM}{C}\\equiv -BD\\mod p\n\\text{\\quad and\\quad}\n\\left(\\frac{A}{M}y_2^2+B\\right)\\left(\\frac{C}{M}y_2^2+D\\right)\\equiv y_2^2 + BD\\equiv \\frac{1}{4}\\left(r+\\frac{BD}{r}\\right)^2 \\not\\equiv 0\\mod p.\n\\end{equation*}\nThen take \n$$\nP_1=\\left(-\\frac DC \\colon 1 \\colon 0\\colon \\sqrt{-\\dfrac{DM}C}\\colon 0\\right) \\; \\text{ and } \\; P_2=\\left(\\frac{y_2^2}{M}\\colon 1\\colon 0\\colon y_2\\colon \\sqrt{\\left(\\frac{A}{M}y_2^2+B\\right)\\left(\\frac{C}{M}y_2^2+D\\right)}\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\\textbf{Case 1b.} If $\\mathrm{v}_p(D)\\geq 1$ then take $y_1\\in\\mathbb{Z}_p^{\\times, 2}$ and $y_2\\notin \\mathbb{Z}_p^{\\times, 2}$.\nThen\n\\begin{equation}\n\\left(\\frac{A}{M}y_i^2+B\\right)\\left(\\frac{C}{M}y_i^2+D\\right)\\equiv \\frac{BC}{M}y_i^2\\equiv y_i^2\\mod p,\n\\end{equation}\nso we can choose $z_i\\in\\mathbb{Z}_p$ such that $z_i\\equiv -y_i\\mod p$ and $z_i^2=\\left(\\frac{A}{M}y_i^2+B\\right)\\left(\\frac{C}{M}y_i^2+D\\right)$.\nNow take \n$$\nP_i=\\left(1\\colon \\frac{y_i^2}{M}\\colon 0\\colon y_i\\colon z_i\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\n\n\\noindent\\textbf{Case 2.} In this case we have $p \\mathrel{\\|} M$ and $p \\mathrel{\\|} A$ and $p^2 \\nmid B,C,D$. From \\eqref{gcd_cond_1} we get $\\mathrm{v}_p(C)=0$, so condition (C1) implies $p \\mid B$. From $p^2 \\nmid B$, \\eqref{gcd_cond_1} and (C1) we conclude in order that $\\mathrm{v}_p(B)=1$, $\\mathrm{v}_p(D)=0$ and $\\mathrm{v}_p(N)\\ge 1$.\n\nIntroduce $A=pA'$, $B=pB'$, $M=pM'$ and $N=pN'$. Then (C1) becomes $(A'D+B'C-M')^2-4A'B'CD=pN'^2$ and we conclude $\\mathrm{v}_p(A'D+B'C-M)=0$. \nOur system is equivalent to\n$$\n\\begin{cases}\n-x^2+\\dfrac{y^2}{p}=M'uv;\\\\\n\\dfrac{4A'C}{p}(z^2-y^2)=\\bigl(2A'Cu+(B'C+A'D-M')v\\bigr)^2-pN'^2v^2.\n\\end{cases}\n$$\n\\indent\\textbf{Case 2a.} Suppose that $M'\\cdot\\frac{B'C+A'D-M'}{2A'C}\\in\\mathbb{Z}_p^{\\times 2}$. \nChoose $r_1\\in\\mathbb{Z}_p^{\\times 2}$ and $r_2\\not\\in\\mathbb{Z}_p^{\\times 2}$ and define \n$$\ny_i=\\frac{1}{2}\\left(\\frac{A'C}{r_i}-r_i\\right)N',\\quad z_i=\\frac{1}{2}\\left(\\frac{A'C}{r_i}+r_i\\right)N'\\quad \\text{ and } \\quad x_i=\\sqrt{M'(B'C+A'D-M')\\cdot 2A'C+py_i^2}.\n$$ \nNow we can just take\n$\nP_i=(-(B'C+A'D-M')\\colon 2A'C\\colon x_i\\colon py_i\\colon pz_i)\\in X(\\mathbb{Q}_p).\n$\n\n\\textbf{Case 2b.} Now suppose that $M'\\cdot\\frac{B'C+A'D-M'}{2A'C}\\notin\\mathbb{Z}_p^{\\times 2}$. Let $(u:v:x:y:z)\\in X(\\mathbb{Q}_p)$ be represented by a primitive tuple in $\\mathbb{Z}_p$. Obviously, $\\mathrm{v}_p(y),\\mathrm{v}_p(z)\\ge 1$ so write $y=py_1$ and $z=pz_1$. Then \n$$\n(2A'Cu+(B'C+A'D-M')v)^2-pN'^2v^2=4A'Cp(z_1^2-y_1^2),\n$$ \nhence $u\\equiv -\\frac{B'C+A'D-M'}{2A'C} v\\mod p$. This turns the first equation into $M'\\cdot\\frac{B'C+A'D-M'}{2A'C} \nv^2\\equiv x^2\\mod p$. It is possible only if $u\\equiv v\\equiv x\\equiv 0\\mod p$. This contradicts the assumption $X(\\mathbb{Q}_p) \\ne \\emptyset$.\n\\bigskip\n\n\\noindent\\textbf{Surjectivity for $\\mathcal{A}$.}\nIn Cases 3 and 4 we apply the same technique for proving that $\\inv_p \\mathcal A$ is surjective.\n\n\\noindent\\textbf{Case 3.} By assumption $p \\nmid MABCD$. Since $(p,AC)_p=(p,BD)_p=1$, we have $AC,BD\\in \\mathbb{Z}_p^{\\times, 2}$.\n\n\n\\textbf{Case 3a.} If $ABM\\notin \\mathbb{Z}_p^{\\times, 2}$ then $\\mathcal A$ has different invariants at the points\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\in X(\\mathbb{Q}_p)\\quad \\text{ and } \\quad P_2=(0\\colon 1\\colon 0\\colon 0\\colon \\sqrt{BD})\\in X(\\mathbb{Q}_p).\n$$\nas one can see using \\eqref{Br_gr_rewr}.\n\n\n\\textbf{Case 3b.} Now let $ABM\\in\\mathbb{Z}_p^{\\times, 2}$. It follows from Lemma \\eqref{lem:quadres_table} that there exists a $y_0\\in \\mathbb{Z}_p$ such that $1+\\frac{B}{MA}y_0^2$, $\\frac{C}{A}+\\frac{D}{MA}y_0^2\\not\\in\\mathbb{Z}_p^{\\times, 2}$, or $\\frac{C}{A}+\\frac{D}{MA}y_0^2 \\equiv 0 \\mod p$ and $1+\\frac{B}{MA}y_0^2\\not\\in\\mathbb{Z}_p^{\\times, 2}$. In the first case we consider\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\quad \\text{ and } \\quad\nP_2=\\left(1\\colon \\frac{y_0^2}{M}\\colon 0\\colon y_0\\colon A\\sqrt{\\left(1+\\frac{B}{MA}y_0^2\\right)\\left(\\frac{C}{A}+\\frac{D}{MA}y_0^2\\right)}\\right)\n\\in X(\\mathbb{Q}_p).\n$$\nIn the second case we choose a $y_1\\in\\mathbb{Z}_p$ such that $y_1\\equiv y_0\\mod p$ and $\\frac{C}{A}+\\frac{D}{MA}y_1^2=0$. We then work with\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\quad \\text{ and } \\quad  P_2=\\left(1\\colon \\frac{y_1^2}{M}\\colon 0\\colon y_1\\colon 0\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\\noindent\\textbf{Case 4.} Let $M=p^{2k+1}M'$ with $k\\geq 0$ and $M' \\in \\mathbb Z_p^\\times$. From (C1) we obtain that $AD\\equiv BC\\mod p$. Lemma~\\ref{lem:quadres}(b) for $S\\left(1,\\frac{B}{M'A}\\right)$ implies the existence of an $x_0\\in\\mathbb{Z}_p^\\times$ such that $1-\\frac{B}{M'A}x_0^2\\in\\mathbb{Z}_p^\\times\\setminus\\mathbb{Z}_p^{\\times 2}$. Then \n$$\n1-\\frac{D}{M'C}\nx_0^2\\equiv 1-\\frac{B}{M'A}x_0^2\\mod p,\n$$ \nso we can take \n$$\nP_1=(1:0:0:0:\\sqrt{AC})\\, \\text{ and }\\, P_2=\\left(1:-\\frac{x_0^2}{M'}:p^kx_0:0:\\sqrt{AC\\left(1-\\frac{Bx_0^2}{M'A}\\right)\\left(1-\\frac{Dx_0^2}{M'C}\\right)+p^{2k+1}x_0^2}\\right).\n$$\n\n\\noindent\\textbf{Case 5.} We have $p\\mathrel{\\|} M$ and $p^2 \\mid A$. Conditions (C1) and \\eqref{gcd_cond_1} imply $p\\nmid C$, $p \\mid B$, $p \\nmid D$ and $p \\mid N$. Then \\eqref{sys} is isomorphic to the system with coefficients $(p^{-2}A,p^{-1}B,C,pD,p^{-1}M,p^{-1}N)$ under the morphism $(u\\colon v \\colon x \\colon y \\colon z) \\mapsto (pu\\colon v \\colon x\\colon y \\colon z)$. For the new system we have $\\mathrm{v}_p(M)=0$ and $m\\geq 1$, which was considered in Case 1.\n\n\\noindent\\textbf{Case 6.} From $p^{2k+1} \\mathrel{\\|} M$ and $p\\mid A$ we get, as in the previous case, $p \\mid B,N$ and $p \\nmid C,D$. Considering (C1) modulo $p^{2k+1}$ we get $(AD-BC)^2 \\equiv pN^2 \\mod p^{2k+1}$. From this we conclude that $p^{k+1} \\mid AD-BC$ and $p^k \\mid N$. In particular, $p^2 \\mid A$ if and only if $p^2 \\mid B$. So we conclude $p\\mathrel{\\|} A,B$ from \\eqref{gcd_cond_2}. This implies $2\\mathrm{v}_p(AD-BC+M) = \\mathrm{v}_p(pN^2-4ABCD) = 2$ and hence $\\mathrm{v}_p(AD+BC)=1$. From this we deduce that\n\\[\n2\\mathrm{v}_p(AD-BC)=\\mathrm{v}_p(pN^2 +2(AD+BC)M-M^2) = 2k+2.\n\\]\nHence $p^{k+1} \\mathrel{\\|} AD-BC$.\n\nOur system of equations is equivalent to \\eqref{sys} with the coefficients\n\\[\n(p^{-2k}MD,-p^{-2k-1}MB,-C,p^{-1}A,p^{-2k-1}(AD-BC)^2)\n\\]\nunder the morphism $(u\\colon v \\colon x \\colon y \\colon z) \\mapsto (\\frac{Au+Bv}{AD-BC} \\colon p\\frac{Cu+Dv}{AD-BC} \\colon p^{-k}x \\colon p^{-k}y \\colon p^{-k}z)$. This new system satisfied $\\mathrm{v}_p(M)=1$ and $m=1$, which was discussed in Case 2.\n\n\\noindent\\textbf{Case 7.} Now we know that $p^{2k} \\mathrel{\\|} M$ and $p\\nmid ABCD$. Condition (C1) implies $AD\\equiv BC\\mod p$, so $\\mathrm{v}_p(AD+BC)=0$. From $\\mathrm{v}_p(M)=2k$ and (C1) we obtain that $\\mathrm{v}_p(AD-BC)=k$. Now consider the coefficients\n\\[\n(p^{-2k}MD,-p^{-2k}MB,-C,A,p^{-2k}(AD-BC)^2)\n\\]\nwith variables $(\\frac{Au+Bv}{AD-BC}\\colon \\frac{Cu+Dv}{AD-BC}\\colon p^{-k}x\\colon p^{-k}y\\colon p^{-k}z)$ for the system \\eqref{sys}. This is precisely Case 3.\n\n\\noindent\\textbf{Case 8.} Now we assume $p^{2k} \\mathrel{\\|} M$ and $m=1$. Without loss of generality $p \\mathrel{\\|} A$ and as in the previous cases we can prove $p \\mathrel{\\|} B$, $p\\nmid CD$, $\\mathrm{v}_p(AD+BC)=1$ and $\\mathrm{v}_p(AD-BC)\\ge k+1$. The system \\eqref{sys} with coefficients $(p^{-2k}MD,-p^{-2k-1}MB,-C,p^{-1}A,p^{-2k-1}(AD-BC)^2)$ with variables $(\\frac{Au+Bv}{AD-BC}\\colon p\\frac{Cu+Dv}{AD-BC}\\colon p^{-k}x\\colon p^{-k}y\\colon p^{-k}z)$ is isomorphic to our system. This new system satisfies $\\mathrm{v}_p(M)$ is odd and $m=0$, which is precisely Case~4.\n\\end{proof}\n\n\\section{Explicit examples}\n\nAll known examples of Brauer--Manin obstructions for quartic del Pezzo surfaces occur on surfaces with a Brauer group modulo constants of order $2$. The first such obstruction was described by Birch and Swinnerton-Dyer in \\cite{BSD}. Work by Jahnel and Schindler \\cite{JS} showed that the locus of quartic del Pezzo surfaces with a Brauer group of order $2$ for which the Brauer--Manin obstruction is the only one, is dense in the moduli space of all del Pezzo surfaces of degree $4$.\n\nMitankin and Salgado \\cite{MS} restricted to the subfamily in which a positive proportion has a Brauer group of order $4$. Such surfaces can be written as\n$$\n\\begin{cases}\nx_0x_1-x_2x_3=0;\\\\\na_0x_0^2+a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2=0,\n\\end{cases}\n$$\nwhere $a_0a_1,a_2a_3,-a_0a_2\\in\\mathbb{Q}^{\\times,2}$ and $-a_0a_4(a_0a_1-a_2a_3)\\notin\\mathbb{Q}^{\\times,2}$. The Hasse principle, however, trivially holds for such surfaces, because of the rational point $(1:0:\\sqrt{-a_0\/a_2}:0:0)$.\n\nIn this section we will exhibit families of quartic del Pezzo surfaces with a Brauer group of order $4$ with a Brauer--Manin obstruction to the Hasse principle.\n\n\\subsection{Obstructions coming from $\\mathcal A$}\n\nLet us define our first subfamily. Consider a prime number $p\\equiv 1\\mod 4$ and integers $a,b$ such that $ab=p-1$. The surface $Y_{p,a,b}$ is given by the system\n$$\n\\begin{cases}\ny^2-px^2=uv;\\\\\nz^2-px^2=(au-pv)(u-bv).\n\\end{cases}\n$$\nObviously, $Y_{p,a,b}=X_{p,a,-p,1,-b,1,1}$, and one can check that conditions (C1) and (C2) are satisfied with $N=1$.\n\nFor such surfaces we have $\\mathcal{A}=\\left(p,\\frac{u}{au-pv}\\right)$, $\\mathcal{B}=\\left(p,\\frac{z-y}{u}\\right)$ and $\\mathcal{C}=\\left(p,\\frac{au-pv}{z-y}\\right)$.\n\n\\begin{pro}\\label{pro:ex1_inv_B}\nConsider the surface $Y=Y_{p,a,b}$.\n\\begin{enumerate}\n\\item[(a)] The variety $Y$ is everywhere locally soluble.\n\\item[(b)] The invariant map of $\\inv_p\\mathcal{B} \\colon Y(\\mathbb A_\\mathbb{Q}) \\to \\frac12\\mathbb{Z}\/\\mathbb{Z}$ is surjective.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nLocal solubility at all places $v\\not\\in \\{2,p\\}$ has already been proven in Proposition \\ref{pro:loc_sol_subfam}. Local solubility at $p$ and the surjectivity of the invariant map of $\\mathcal{B}$ at $p$ follow from Case 1 in the proof of Proposition~\\ref{prop:surjectiveinvariantmap}. For local solubility at $2$ one can simply consider all possibilities of $a$, $b$ and $p$ modulo $8$.\n\\end{proof}\n\nSo we might expect an obstruction coming from $\\mathcal{A}$.\n\n\\begin{pro}\\label{pro:ex1_inv_A}\nFor all $(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})$ we have: $\\inv_v \\mathcal{A}(s_v)=0$ for $v\\neq p$, and:\n$$\n\\inv_p \\mathcal{A}(s_p)=\\begin{cases}\n0 & \\text{ if }\\left(\\dfrac{a}{p}\\right)=1;\\\\\n\\dfrac{1}{2} & \\text{ if }\\left(\\dfrac{a}{p}\\right)=-1.\n\\end{cases}\n$$\n\\end{pro}\n\n\\begin{proof}\nOne can prove this by direct computation. However, we will apply results from \\cite{Beffeval} and \\cite{CTS13} to conclude that $\\inv_v$ is constant for $v \\ne p$ and compute it at a single point in $X(\\mathbb{Q}_v)$ to conclude $\\inv_v$ is identically zero.\n\nFinally, consider the case $v=p$. Let $s_p=(u\\colon v\\colon x\\colon y\\colon z)$ with $(u,v,x,y,z)$ a primitive tuple in $\\mathbb{Z}^5_p$. Suppose $p \\mid u$, then immediately $p \\mid y,z$. Denote $u=pu'$,  then \n$$\nu'v\\equiv -x^2\\equiv (au'-v)(pu'-bv)\\equiv u'v+bv^2\\mod p\n$$\nwhich means that $p \\mid v$ and then $p \\mid x$, which contradicts the primitivity of $(u,v,x,y,z)$. Thus, $\n\\mathrm{v}_p(u)=0$, so $\\left(p,\\frac{u}{au-pv}\\right)_p=\\left(\\frac{u(au-pv)}{p}\\right)=\\left(\\frac{a}{p}\\right)$. This proves the proposition \nin the case $v=p$.\n\\end{proof}\n\n\nWe can now compute the Brauer--Manin obstruction coming from $\\mathcal{A}$.\n\n\\begin{thm}\\label{thm_ex1}\nConsider the surface $Y_{p,a,b}$. Note that from $ab=p-1$ and $p\\equiv 1\\mod 4$ we conclude $(\\frac{a}{p})=(\\frac{b}{p})$. \nAlso, $(\\frac{a}{p})=(\\frac{b}{p})=-1$ holds if and only if $p\\equiv 5\\mod 8$, and $a$ and $b$ are both even.\n\\begin{enumerate}\n\\item[(a)] If $(\\frac{a}{p})=(\\frac{b}{p})=-1$ then $Y_{p,a,b}(\\mathbb{Q})=\\emptyset$ and this failure of the Hasse principle is \nexplained by the Brauer--Manin obstruction coming from $\\mathcal{A}$.\n\\item[(b)] Otherwise, we have\n\\[\nY_{p,a,b}(\\mathbb A_\\mathbb{Q})^{\\Br X} \\ne \\emptyset.\n\\]\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nThe first observation comes from the following facts: for $p\\equiv 1\\mod 4$ we have $(\\frac{-1}{p})=1$ and $(\\frac{q}{p})=1$ for odd prime divisors $q$ of $p-1$. Also, $(\\frac{2}{p})=1$ if $p\\equiv 1\\mod 8$ and $(\\frac{2}{p})=-1$ if $p\\equiv 5\\mod 8$.\n\nIf $(\\frac{a}{p})=(\\frac{b}{p})=-1$ then Proposition \\ref{pro:ex1_inv_A} implies that $\\sum_v\\inv_v\\mathcal{A}(s_v)=\\frac{1}{2}$ for all $(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})$, so $Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})^\\mathcal{A}=\\emptyset$.\n\nIf $(\\frac{a}{p})=(\\frac{b}{p})=1$ then Proposition \\ref{pro:ex1_inv_A} and Proposition \\ref{pro:ex1_inv_B} imply that \n$\\sum_v\\inv_v\\mathcal{A}(s_v)=0$ for all ${(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})}$, and invariant maps of $\\mathcal{B}$ and $\\mathcal{C}$ at $p$ are \nsurjective. Thus, $Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})^{\\Br}\\neq\\emptyset$.\n\\end{proof}\n\n\n\n\n\\begin{exa}\nConsider a surface $Y_{13,2,6}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(2u-13v)(u-6v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that $Y_{13,2,6}(\\mathbb{Q})=\\emptyset$.\n\\end{exa}\n\n\\begin{exa}\nConsider a surface $Y_{13,1,12}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u-13v)(u-12v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that there is no Brauer--Manin obstruction for $Y_{13,1,12}$ to the Hasse principle, and it has a trivial rational point $(1:0:0:0:1)\\in Y_{13,1,12}(\\mathbb{Q})$.\n\\end{exa}\n\n\n\\begin{exa}\nConsider a surface $Y_{13,12,1}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(12u-13v)(u-v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that there is no Brauer--Manin obstruction for $Y_{13,12,1}$, and it has a rational point $(1:-3:2:7:16)\\in Y_{13,12,1}(\\mathbb{Q})$ which is actually non-trivial to find.\n\\end{exa}\n\n\\begin{rema}\nNote that we can change $a$ and $b$ up to squares without changing the isomorphism class of the surface. Unless $(a,b)$ equals either $(1,p-1)$ or $(-1,-(p-1))$, up to squares, there does not seem to be an obvious rational point. However, for each explicit surface we studied we were always able to find such a point.\n\nThe machinery from \\cite{CTSSD} and \\cite{WittenbergBook} seems ideally suited to proving such a statement by making $X$ into an elliptic fibration using the projection $f \\colon X \\dashrightarrow \\mathbb P^2$ away from a plane studied in the beginning of this paper. Unfortunately the ``condition (D)'' in those works is not satisfied, which relates to the fact that the Brauer group of $X$ is not vertical.\n\\end{rema}\n\n\\begin{rema}\nAlso, note that the technique described above would not just establish the existence of a rational point, but even prove that these are Zariski dense. We already know these two statements to be equivalent by work of Salberger and Skorobogatov \\cite{SalbergerSkorobogatov91}.\n\\end{rema}\n\n\\subsection{Obstructions coming from $\\mathcal B$}\n\nWe will now consider a second subfamily for which $\\mathcal B$ might give an obstruction.\n\n\\begin{defi}\\label{def_ex_B}\nLet $p$ be an odd prime, and $a$ and $b$ integers such that\n\\begin{enumerate}\n\\item $(a+b-1)^2-4ab=p$,\n\\item $4a\\equiv 4b\\equiv 1\\mod p$,\n\\item $a\\equiv b\\equiv 1\\mod 2$, and\n\\item $a\\equiv 1\\mod 8$.\n\\end{enumerate}\nLet $S_{p,a,b}$ be the surface given by\n\\begin{equation}\\label{sys_ex_B}\n\\begin{cases}\ny^2-px^2=uv;\\\\\nz^2-px^2=(u+v)(au+bv).\n\\end{cases}\n\\end{equation}\n\\end{defi}\n\nSuch triples $(p,a,b)$ do exist, as one can see from considering $p \\equiv 5 \\mod 8$, an integer $t \\equiv 3\\frac{p-1}{4}\\mod 8$ and\n\\[\na_t=t^2p^2-tp-\\frac{p-1}{4} \\quad \\text{ and } \\quad  b_t=t^2p^2+tp-\\frac{p-1}{4}.\n\\]\n\nOne can also check that $S_{p,a,b}=X_{p,1,1,a,b,1,1}$ satisfies conditions (C1) and (C2).\n\n\\begin{pro}\n\\\n\\begin{enumerate}\n\\item[(a)] All $S_{p,a,b}$ are everywhere locally soluble.\n\\item[(b)] For all $S_{p,a,b}$ the invariant map of $\\mathcal{A}$ at $p$ is surjective.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nProposition \\ref{pro:loc_sol_subfam} addresses the local solubility at all places $v\\not\\in\\{2,p\\}$. Case 3 in the proof of Proposition~\\ref{prop:surjectiveinvariantmap} establishes the local solubility at $p$ and the surjectivity of the invariant map of $\\mathcal{A}$ at $p$. The local solubility at $2$ follows from the point $(1:0:0:0:\\sqrt{a})\\in X(\\mathbb{Q}_2)$, since $a\\equiv 1\\mod 8$.\n\\end{proof}\n\nWe will now consider the obstruction coming from $\\mathcal{B}$.\n\n\\begin{thm}\\label{thm_ex_B}\nWe have $S_{p,a,b}(\\mathbb A_\\mathbb{Q})^\\mathcal{B}=\\emptyset$ and in particular $S_{p,a,b}(\\mathbb{Q})=\\emptyset$.\n\\end{thm}\n\n\\begin{proof}\nWe can prove that $\\inv_v$ is identically zero for $v \\ne p$ as we did in the proof of Proposition~\\ref{pro:ex1_inv_A}.\n\nFinally, consider the case $v=p$. Let $s_p=(u:v:x:y:z)$ such that $(u,v,x,y,z)$ is primitive in $\\mathbb{Z}_p$, hence in particular $\\mathrm{v}_p(u)\\mathrm{v}_p(v)=0$. Assume $\\mathrm{v}_p(u)=0$; the other case is similar. The system \\eqref{sys_ex_B} together with the condition $4a\\equiv 4b\\equiv 1\\mod p$ gives us\n$$\nv\\equiv \\frac{y^2}{u}\\mod p\\quad \\text{ and } \\quad z^2\\equiv \\frac{1}{4}(u+v)^2\\mod p.\n$$\nLet us write $z \\equiv \\frac{\\varepsilon}2(u+v) \\mod p$ with $\\varepsilon = \\pm 1$. Then we get\n\\begin{align*}\nz\\pm y \\equiv \\frac{\\varepsilon}2(u+v) \\pm y  & \\equiv \\frac{\\varepsilon}2 \\frac{y^2}u \\pm y + \\frac{\\varepsilon}2u\\\\\n & \\equiv \\frac{\\varepsilon}2u\\left(\\frac{y^2}{u^2} \\pm 2 \\varepsilon \\frac yu + 1 \\right)\\\\\n& \\equiv \\frac{\\varepsilon}2u\\left(\\frac yu \\pm \\varepsilon\\right)^2 \\mod p.\n\\end{align*}\n\nSo at least one of $\\eta = \\frac u{z\\pm y}$ is non-zero modulo $p$ at $s_p$, and we have $(p,\\eta)=-1$ since $(\\frac{2}{p})=-1$. We conclude $\\inv_p\\mathcal{B}(s_p)=\\frac{1}{2}$.\n\\end{proof}\n\n\n\n\n\\begin{exa}\nConsider $S_{13,153,179}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u+v)(153u+179v).\n\\end{cases}\n$$\nTheorem \\ref{thm_ex_B} shows that $S_{13,153,179}(\\mathbb{Q})=\\emptyset$ and hence this surface does not satisfy the Hasse principle.\n\\end{exa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last decade, hybrid organic-inorganic materials with the perovskite structure, most notably CH$_3$NH$_3$Pb(I,Cl)$_3$, have become the most promising light-harvesting active layer for the implementation of high efficiency and low cost solar cells \\cite{Graetzel_2014,Hodes_2013,Xiao_2016,Lee_2012,Heo_2013,You_2014,Zhou_2014,Nanni_2014}. As a matter of fact, the power conversion efficiency of perovskite solar cells (PSCs) has shown a very fast growth, increasing from 3.8\\% \\cite{Kojima_2009} in 2009 to 22.1\\%  \\cite{NREL_eff_chart} in 2016. The ease of fabrication and the high radiative efficiency have made perovskites also extremely attractive for the development of bright LEDs and lasers \\cite{Veldhuis_2016,Sutherland_2016,Palma_2016,Palma_2016_2}.\n\nNevertheless, a major drawback originates from the poor material stability\\cite{Xu_2016,Xiao_2016}: decomposition after exposure to moisture, cracks and defects generated by thermal stresses, crystal phase transition, UV-light exposure represent the principal causes of aging\\cite{Xu_2016,Xiao_2016}.\nMoreover, a high density of trap states, both on surfaces and in grain boundaries \\cite{Xiao_2016,Draguta_2016,Wu_2015,Kim_2014}, are present in polycrystalline perovskites, used for solar cells and light emitters. \nIndeed, even though theoretical predictions show that deep trap states are not generally formed inside perovskites grains, the opposite is observed at the grain boundaries and at the surfaces\\cite{Xiao_2016,Park_2016_book}.\nTherefore, sophisticated passivation strategies are essential for increasing the efficiency of PSCs and light emitters \\cite{Son_2016,Giordano_2016,deQuilettes_2016,Park_2016_book}.\n\nFor CH$_3$NH$_3$PbI$_3$ (MAPI) perovskites, a correlation between the solar cell efficiency and the grain morphology was recently demonstrated \\cite{Shao_2016}. Results of local short-circuit photocurrent, open-circuit voltage and dark drift current within individual grains correlate these quantities to different crystal facets, as a consequence of a facet-dependent density of trap states \\cite{Leblebici_2016} and it was also proven that the structural order of the electron transport layer (ETL) impacts the overall cell performance \\cite{Shao_2016}. Moreover, the nature of grain boundaries was shown to affect the carrier recombination kinetics because of non-radiative pathways that would also play a role in the process of charge separation and collection \\cite{deQuilettes_2016}.\n\n\n\nIt turns out that the possibility of controlling the morphology of the perovskite thin film and the understanding if different realizations of the ETL can modify the morphology of the grains are of the utmost relevance to further improvements of perovskite based solar cells.\n\nRecently, graphene and related two-dimensional materials have been introduced in the device structure in order to improve the charge injection and\/or collection at the electrodes: an enhancement of the power conversion efficiency\\cite{Acik_2016} and a long-term stability\\cite{Agresti_2016} was obtained.\nAs a matter of fact, interfaces between perovskite and transport layers have been recently demonstrated to dramatically affect the charge recombination processes and material instability within the working device.\\cite{Capasso_2016}\nIn fact, when free charges are fast injected from perovskite to the electron transport layer, the perovskite degradation is slowed down and the non-radiative recombination is reduced\\cite{Ahn_2016}. \nIn particular, the insertion of graphene flakes into the mesoporous-TiO$_2$ layer (mTiO$_2$) and of lithium-neutralized graphene oxide (GO-Li) as interlayer at perovskite\/mTiO$_2$ interface showed enhanced conversion efficiency and stability on both small and large area devices by demonstrating the crucial role of graphene interface engineering in perovskite-based devices\\cite{Agresti_2017}. Thus, the influence of mesoscopic-graphene modified substrates onto the perovskite film need to be investigated more in details to finely control the photovoltaic performance of complete devices.\n\nGiven the typical size of the grains (a few hundreds of nanometers), high resolution techniques as TEM (Transmission Electron Microscopy), AFM (Atomic-Force Microscopy), SNOM (Scanning Near-field Optical Microscopy), etc.~are employed to investigate the grain morphology but a difficult task is to correlate physical properties at the nanoscale with the device performance measuring, for instance, the $I$-$V$ curve of the cell\\cite{Leblebici_2016}.\nThus, it would be of  extreme relevance to extract information on the active film morphology with much easier techniques on a length scale of tens\/hundreds of microns and even larger so to assess the homogeneity of the film deposition and the reliability of the synthesis protocol and post-deposition treatments. Moreover, the high spatial resolution analysis can explore a limited thickness of the film, and therefore it appears difficult to get information in the case of a real device.\nPhotoluminescence (PL) spectroscopy is an effective tool to investigate the film quality: in fact, from the comparison of samples with different ETLs, it is possible to extract quantitative information on the carrier capture and transport and, by the spectral shape, identify the crystalline phase of the active layer and evaluate the density of traps\/defects.\nMoreover, given the limited thickness (few hundreds of nm) of the perovskite film in solar cells and given the steep behavior of the absorption coefficient in MAPI\\cite{Loper_2015}, by varying the excitation wavelength in the range 300-700\\,nm, it is possible to probe thicknesses of the film from a hundred of nm to the whole film layer.\n\nIn this paper we aim to establish the effects of different graphene-based\nETLs in sensitized MAPI solar cells. In particular we will study the ETL\neffects on the carrier collection efficiency and on the MAPI morphology along\nthe thickness.\nBy picosecond time-resolved measurements we correlate the carrier recombination dynamics to the crystalline quality of the active material in presence\/absence of the ETL. We will find an increase of the electron collection efficiency up to a factor 3 with respect to standard mTiO$_2$.\nTaking advantage of the absorption coefficient dispersion, we are able to assess the film morphology along the thickness. In fact, by tuning the excitation wavelength, we investigate a thickness range from 150 to 400\\,nm and we can highlight the morphology changes induced by different ETLs.\nOur results will indicate that, when a graphene doped mesoporous TiO$_2$ (G+mTiO$_2$) with the addition of a GO-Li interlayer is used as ETL, the morphology of the MAPI film embedded in the mesoporous layer is frozen in the tetragonal phase, regardless of the temperature. In addition, the defect concentration is about one order of magnitude lower than that found with the other ETLs.\n\n\n\n\n\n\n\n\n\\section{Results and discussion}\n\nFour types of samples were prepared using different combinations of ETLs: mTiO$_2$, G+mTiO$_2$, mTiO$_2$ plus GO-Li interlayer and G+mTiO$_2$ plus GO-Li interlayer. \nThe different sample structures are schematically shown in Fig.\\,\\ref{Fig1} and listed in Tab.\\,\\ref{Tab1}. For clear comparison a sample of MAPI on FTO without ETL (reference sample) was also investigated. \nDetails on the sample preparation are reported in the Experimental Section.\nIt is worth noting that PL analysis was carried out on simple photo-electrodes, lacking of the hole collecting layer and the bottom contact: this allows us to focus only on the electron collection and transport after the electron-hole pairs creation due to the photon absorption. Moreover, PL experiments can be realized illuminating the samples either from the perovskite film side (side A) or the FTO side (side B). Varying the excitation wavelength and the excitation side, we can differently penetrate into the MAPI film and selectively probe spatial regions few tens of nm near the ETL or far from it.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig1.pdf}\n\\caption{Structures of the investigated samples. The ETLs are indicated in red.}\n\\label{Fig1}\n\\end{figure}\n\n\n\\begin{table}[b]\n\\caption{List of the investigated samples and description of their corresponding ETLs.}\n\\label{Tab1}\n\\addtolength{\\tabcolsep}{3.8mm}\n\\begin{tabular*}{\\columnwidth}{l@{\\hspace{22mm}}l}\n\\hline\nSample    & ETL  \\\\\n\\hline\nReference & No ETL       \\\\\nETL 1         & mTiO$_2$   \\\\\nETL 2         & G+mTiO$_2$  \\\\\nETL 3         & mTiO$_2$ plus GO-Li\\\\\nETL 4         & G+mTiO$_2$ plus GO-Li \\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\nIn Fig.\\,\\ref{Fig2} PL decays at room temperature, after excitation with photons of 2.06\\,eV, are compared for the different samples, while in the inset a PL spectrum at room temperature is shown: typical spectra of the tetragonal phase are observed\\cite{Milot_2015,Dar_2016}. \nBy exciting from side A (Fig.\\,\\ref{Fig2}a), the PL decays are identical for all samples and no effect related to the presence of the ETL is detected. \nIn contrast there is a significant difference in the PL decay by exciting from side B depending on the sample, as shown in Fig.\\,\\ref{Fig2}b: faster decays are observed in presence of ETL, especially when graphene and\/or GO-Li are used.\nIt is worth noting that the PL decay of reference sample excited from side B is slower with respect to the decays of side A. This can be attributed to non-radiative states at the surface of the uncovered perovskite film (side A).\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig2.pdf}\n\\caption{PL decay (at the PL peak energy) and PL spectra (in the inset) at room temperature, after ps excitation at 2.06\\,eV with an average intensity of 10\\,W\/cm$^2$. a) From side A. b) From side B.}\n\\label{Fig2}\n\\end{figure}\n\nTo extract information about the efficiency in the carrier injection from perovskite to ETL, we first fitted the decays of Fig.\\,\\ref{Fig2}b with a double exponential\\cite{Son_2016,Bi_2016}, taking into account the laser pulse repetition period \\cite{Warren_2013}. The fitting function $I(t)$ can be written as\n\\begin{equation}\nI(t) = f(t)+g(t)\n\\end{equation}\nwhere $f(t)$ is the original double exponential and $g(t)$ the correction term due to the intrinsic periodic nature of TCSCP measurements\\cite{Warren_2013}:\n\\begin{equation}\nf(t)=\\Theta(t-t_0)[Ce^{-(t-t_0)\/\\tau_1}+ \n(1-C)e^{-(t-t_0)\/\\tau_2}]\n\\end{equation}\n\\begin{equation}\ng(t)=C\\frac{e^{-(t-t_0)\/\\tau_1}}{e^{T\/\\tau_1}-1}+ \n(1-C)\\frac{e^{-(t-t_0)\/\\tau_2}}{e^{T\/\\tau_2}-1} \n\\end{equation}\nIn the previous equations $\\theta$(t) is the Heaviside function, $\\tau_1$ and $\\tau_2$ are the decay time constants, $T$ is the laser pulse repetition period (13.15\\,ns), $C$ is the contribution of $\\tau_1$ exponential to the fit and $t_0$ is a constant. \nThe results obtained by the fitting procedure are reported in Tab.\\,\\ref{Tab2}.\nInserting graphene-based ETLs in the samples reduces $\\tau_1$ from 25\\,ns to 15\\,ns while the reduction of $\\tau_2$ depends on the ETL.\nIn the literature\\cite{Son_2016,Bi_2016} the longer decay constant ($\\tau_1$) is ascribed to the radiative recombination in MAPI while the shorter decay constant can be attributed to the carrier removal from MAPI layer towards the ETL.\n\nTo get rid of the local inhomogeneities in the samples, which can give rise to variation of the TI-PL intensity and considering that the PL time evolution does not depend on the detection spot, we estimate the TI-PL intensity from the PL decay. It turns out that we can express the integrated PL intensity ($I_\\mathrm{PL}$) as\n\\begin{equation}\nI_\\mathrm{PL}=\\eta P,\n\\end{equation}\nwhere $\\eta$ is the radiative efficiency and $P$ is the pump intensity, equals to 10\\,W\/cm$^2$ for all the TR-PL measurements.\nAssuming a unitary radiative efficiency for the reference sample, we can evaluate the change in $\\eta$ for the different ETLs (see Tab.\\,\\ref{Tab2}). Lower $\\eta$ is obtained in case of ETLs 2, 3 and 4: thus the insertion of graphene and\/or GO-Li interlayer improves the electron capture from perovskite to ETL by a factor between two and three with respect to ETL 1.\nWe want to remark that this result does not necessarily imply an increase of the solar cell short circuit current density ($J_\\mathrm{sc}$) of the same amount. In fact hole collection by the hole transport layer\\cite{Pydzinska_2016} and non-radiative recombination in the ETL have to be taken into account. However, as shown in the Supporting Information, a significant increase of $J_\\mathrm{sc}$ is measured for complete cell when ETL 4 is used.\n\n\\begin{table}\n\\caption{Results of the fits of data of Fig.\\,\\ref{Fig2}b: $\\tau_1$ and $\\tau_2$ are the decay time constants, $C$ is the contribution of $\\tau_1$ to the fit and $\\eta$ is the radiative efficiency.}\n\\label{Tab2}\n\\addtolength{\\tabcolsep}{3.8mm}\n\\begin{tabular*}{\\columnwidth}{lcccc}\n\\hline\nSample    & $\\tau_1$ (ns) & $\\tau_2$ (ns) & $C$  &$\\eta$ \\\\\n\\hline\nReference & 25       & --    & 1.00 &1 \\\\\nETL 1         & 25       & 2.05   & 0.43 & 0.48 \\\\\nETL 2         & 15       & 1.99   & 0.36 & 0.27\\\\\nETL 3         & 15       & 1.24   & 0.20 &0.16\\\\\nETL 4        & 15       & 1.30   & 0.34 &0.24\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\nMore insight in the role of the ETL can be obtained by PL spectra at low temperature ($T=11$\\,K). In Fig.\\,\\ref{Fig3} PL spectra, obtained by exciting the samples from side A (Fig.\\,\\ref{Fig3}a) and side B (Fig.\\,\\ref{Fig3}b), are reported.\nThe PL spectra from side A show two peaks. As expected for $T < 150$\\,K in MAPI perovskite \\cite{Milot_2015,Kong_2015}, the peak at about 1.65\\,eV is attributed to the orthorhombic phase of MAPI. But the major contribution to the spectra comes from the other peak, centered at 1.55\\,eV. This emission is likely due to the sum of two contributions: the radiative recombination arising from the residual tetragonal phase at 1.56\\,eV and the radiative recombination from localized states below 1.52\\,eV. In the literature, these localized states are identified with radiative traps\\cite{Wu_2015,Fang_2015} or, recently, to methylammonium-disordered domains in orthorhombic phase of MAPI\\cite{Dar_2016}.\nWe want to remark that, in both interpretations of the low energy side emission as radiative traps or disordered domains, a carrier localization is present. More relevant is the fact that the low energy states are radiative and do not produce a loss of photogenerated carriers.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig3.pdf}\n\\caption{TI-PL spectra at 11\\,K for the various samples after excitation at 2.06\\,eV  with an average intensity of 10\\,W\/cm$^2$. a) From side A. b) From side B.}\n\\label{Fig3}\n\\end{figure}\n\nConsidering that the absorption length of the MAPI film at about 2\\,eV is roughly 200\\,nm \\cite{Loper_2015} and that the thickness of the perovskite layer in our samples is about 350\\,nm (see Fig.\\,S5 in the Supporting Information), we can conclude that the emission, exciting from side A, comes mostly from the MAPI film and that no effect related to the presence of the ETL is detected. \nOn the contrary, the  excitation from side B can reveal the ETL effect on the MAPI. As a matter of fact, relevant differences are observed between the emissions from side B (Fig.\\,\\ref{Fig3}b). First of all we observe a smaller  contribution of the orthorhombic phase for all the samples with respect to the excitation from side A. Moreover, in the case of GO-Li plus G+mTiO$_2$ (ETL 4), we detect a strong reduction of the radiative traps at 1.52\\,eV and the dominance of the emission from the tetragonal phase at 1.56\\,eV.\n\nSuch results suggest that an incomplete phase transition occurs for the perovskite wrapped into the mesoporous ETL layer. In particular, in the case of ETL 4, where the PL lineshape shows a negligible low energy tail and a smaller linewidth with respect to the other samples, we can argue that the crystallization of the MAPI film is very good, as also confirmed by the scanning electron microscopy (SEM) image reported in Fig.\\,S4 of the Supporting Information. However, the interaction of MAPI and ETL inhibits the phase transition, at least for a thickness of about 200\\,nm (see SEM cross section in Fig.\\,S5 of the Supporting Information). In order to confirm and test our hypothesis we performed PL measurements as a function of temperature, excitation wavelength and excitation power.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig4.pdf}\n\\caption{Temperature-dependent measurement on ETL 4 by exciting with photon energy of 2.06\\,eV with an average intensity of 10\\,W\/cm$^2$. The labels D, T and O stand for defects (trap states), tetragonal and orthorhombic, respectively. a) PL spectra from 10 to 300\\,K after excitation on side A. b) PL spectra from 10 to 300\\,K after excitation on side B. c) Position of the PL peaks of the spectra in Fig.\\,\\ref{Fig4}a and Fig.\\,\\ref{Fig4}b as a function of temperature. The excitation side is indicated in brackets. d) FWHM of the PL peaks of Fig.\\,\\ref{Fig4}a and Fig.\\,\\ref{Fig4}b as a function of temperature. The excitation side is indicated in brackets. Red solid line shows the fitting of FWHM through Eq.\\,\\eqref{Eq_Gamma}.}\n\\label{Fig4}\n\\end{figure}\n\nIn Fig.\\,\\ref{Fig4}a we report the emission spectra of ETL 4 exciting from side A, varying the sample temperature from 10 to 300\\,K. As already shown before, at low temperature we observe two bands, one at 1.65\\,eV corresponding to the orthorhombic phase and one at about 1.55 eV corresponding to the sum of the emission from optically active trap states and the emission from a residual tetragonal phase.\nAs expected\\cite{Kong_2015,Fang_2015,Dar_2016,Wright_2016}, increasing the temperature, the orthorhombic phase emission shifts at higher energy (see Fig.\\,\\ref{Fig4}c), showing a monotonic increase of its full width at half maximum (FWHM) (see Fig.\\,\\ref{Fig4}d), and it disappears above 150\\,K, where the phase transition of MAPI from orthorhombic to tetragonal phase occurs.\nBy increasing the temperature, the low energy band shows instead the typical S shape both in the peak emission energy (see Fig.\\,\\ref{Fig4}c) and its FWHM (see Fig.\\,\\ref{Fig4}d). This is an indication of the phase transition from orthorhombic to tetragonal phase, with the concurrent lower contribution of the traps.\nAbove 150\\,K the PL spectrum has only one peak, arising from the tetragonal phase emission, which continues to monotonically blue shift increasing the temperature (see Fig.\\,\\ref{Fig4}c), as expected\\cite{Kong_2015,Fang_2015,Dar_2016,Wright_2016}. \nApart from different relative weights of the emission bands, for all samples we find PL spectra very similar to the one of the sample with ETL 4 when the excitation is performed from side A.\n\nA very similar trend with temperature, as shown in Fig.\\,\\ref{Fig4}a for ETL 4, is found for ETL 1, 2 and 3, irrespective of the excitation side, and this trend is commonly reported in the literature\\cite{Dar_2016}. On the contrary, relevant differences are observed in case of ETL 4 exciting from side B (Fig.\\,\\ref{Fig4}b). By increasing the temperature, the PL spectrum shows a single band, corresponding to the tetragonal phase, with a monotonic increase of the emission energy (see Fig.\\,\\ref{Fig4}c) and the FWHM (see Fig.\\,\\ref{Fig4}d). Such behavior indicates  that the  MAPI film embedded in the mesoporous ETL side remains in the tetragonal phase even down to 10\\,K.\n\nThe FWHM of the PL bands as a function of temperature can be fitted taking into account the temperature-independent inhomogeneous broadening and the interaction between carriers and acoustic and longitudinal optical (LO) phonons\\cite{Dar_2016,Wright_2016}, using the following equation:\n\\begin{equation}\n\\Gamma(T)=\\Gamma_0+\\gamma_{\\mathrm{ac}}T + \\frac{\\gamma_{\\mathrm{LO}}}{e^{E_{\\mathrm{LO}}\/k_\\mathrm{B}T}-1},\n\\label{Eq_Gamma}\n\\end{equation}\nwhere $\\Gamma_0$ is the inhomogeneous broadening, $\\gamma_{\\mathrm{ac}}$ and $\\gamma_\\mathrm{LO}$ are the acoustic and LO phonon-carrier coupling strengths, respectively, and $E_\\mathrm{LO}$ is the LO phonon energy.\nWe fitted the FWHM data extracted from the PL from side A: in particular we considered the orthorhombic phase from 10\\,K to 150\\,K and the tetragonal phase from 150\\,K up to room temperature. If the two set of data can be fitted with a single function, we can conclude that the acoustic and optical phonons causing the PL line broadening have very similar energies for the two phases. The solid line in Fig.\\,\\ref{Fig4}d shows the best fitting curve and a good agreement is reached between the data and the model with the fitting parameters $\\Gamma_0 = (23\\pm1)$\\,meV, $\\gamma_\\mathrm{ac} = (30\\pm5)$\\,\\micro eV\/K, $\\gamma_\\mathrm{LO} = (75\\pm5)$\\,meV, $E_\\mathrm{LO} = (19\\pm1)$\\,meV. These values  well agree with  data in the literature \\cite{Dar_2016,Wright_2016}.\n\nA further evidence proving that the emission band at low temperature in ETL 4 must be attributed to a tetragonal crystalline phase of MAPI with good quality, is given in Fig.\\,\\ref{Fig5}a. We report two spectra, already shown before, from side A (red curve) and side B (black curve), acquired at the same average excitation intensity $I_0=10$\\,W\/cm$^2$.\nDecreasing the excitation intensity by one order of magnitude, the emission from side B changes completely, showing a spectrum (blue curve) similar to the side A (red curve), with the dominant contribution of the traps and a small signal from the orthorhombic phase. This is explained by the fact that, lowering the power density, only the trap energy levels close to the band gap tail are filled.\nIn addition, the PL spectrum of the sample ETL 4, obtained by exciting from side B with an intensity of $I_0\/10$ (blue curve in Fig.\\,\\ref{Fig5}a), is very similar to the spectra of the ETLs 1, 2 and 3 (Fig.\\,\\ref{Fig3}b) by exciting from side B with an higher intensity $I_0$. This means that the trap density in the mesoporous region of MAPI in ETL 4 is lower, of about one order of magnitude, with respect to the other samples. This result agrees with the better crystal quality found by SEM analysis for ETL 4 (see Fig.\\,S4 in the Supporting Information).\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig5.pdf}\n\\caption{a) Normalized PL spectra of ETL 4 at $T=11$\\,K, after excitation at 2.06\\,eV, for different excitation densities and excitation sides (in brackets). $I_0$ corresponds to an average intensity of 10\\,W\/cm$^2$. b) Normalized PL spectra of ETL 4 at $T=11$\\,K for different excitation photon energies and excitation sides (in brackets).}\n\\label{Fig5}\n\\end{figure}\n\nAs stated previously, to confirm that the crystalline nature of the film changes when in contact with the ETL, in particular in presence of GO-Li plus G+mTiO$_2$, we probed the ETL 4 along the thickness exploiting the different absorption coefficient of MAPI varying the excitation photon energies.\nWe spanned the range from 1.73 to 3.1\\,eV, where FTO and cTiO$_2$ have a low and nearly constant absorption.\nThe result of this experiment, performed at 11\\,K, is reported in Fig.\\,\\ref{Fig5}b. Let us focus on side B. At high photon energy excitation, 3.1\\,eV (blue curve) and 2.06\\,eV (orange curve), the MAPI is excited for a few tens of nanometers close to the ETL and, as already observed above, the spectra show only the tetragonal phase (which should not be observed at this temperature since the only stable phase below 150\\,K is the orthorhombic). Decreasing the excitation photon energy down to 1.73\\,eV, the absorption coefficient of MAPI decreases and therefore the sample is excited more uniformly in depth. The resulting spectrum (red curve) shows an increase of the contribution of the orthorhombic phase.\nFor comparison, a spectrum from side A with an excitation of 2.06\\,eV is reported (black curve). The comparison of this spectrum with that at 1.73\\,eV from side B (red curve) shows, apart from a difference in the intensity of the orthorhombic phase, exactly the same contribution from the radiative traps and the tetragonal phase. The observed behavior proves that the crystalline nature of MAPI at low temperature is influenced by the interaction with the ETL, which inhibits the MAPI phase change into the orthorhombic form.\n\nOur results demonstrate a substantial improvement of the active layer morphology of ETL 4 with an efficient carrier capture from the ETL.\nThis is confirmed by a remarkable increase in power conversion efficiency (PCE) for complete devices, obtained by $I$-$V$ characterization (see Fig.\\,S6, S7 and Tab.\\,S1 in the Supporting Information), which is mainly ascribed to an improved short circuit current density ($J_\\mathrm{SC}$).\n\n\n\\section{Conclusions}\n\nWe investigated the effects of different graphene-based ETLs in sensitized MAPI photoelectrodes. In particular we have compared four different samples with the following ETLs: mTiO$_2$, G+mTiO$_2$, mTiO$_2$ plus GO-Li interlayer and G+mTiO$_2$ plus GO-Li interlayer. \nWe have studied the ETL effects on the carrier collection efficiency and on the MAPI morphology and quality along the thickness.\nIn presence of ETL, we found faster PL decays by exciting on FTO side with respect to the MAPI side which is explained by efficient electron removal from MAPI layer due to the ETL. In particular an increase of the electron collection efficiency up to a factor 3 with respect to standard mTiO$_2$ is reported.\n\nMoreover, the MAPI layer embedded in G+mTiO$_2$ plus GO-Li ETL shows a crystalline quality much better than the other samples, with a trap density about one order of magnitude lower.\nExploiting the dispersion of the MAPI absorption coefficient, we could probe the sample along the thickness, finding that the morphology of the MAPI film embedded in the G+mTiO$_2$ plus GO-Li ETL is frozen in the tetragonal phase, regardless of the temperature.\nMoreover, the observed morphology improvement of the MAPI encapsulated in the mTiO$_2$ plus GO-Li layer supports the increased efficiency measured for the complete devices.\n\nFinally, our results show that graphene based ETLs significantly improve both the carrier collection and the crystalline quality of the active material, opening new routes to the development of efficient and stable MAPI solar cells.\n\n\n\\section{Experimental section}\n\n\\textbf{Sample preparation.} \nSolar cells photoelectrodes were prepared on Fluorine-doped Tin Oxide (FTO) conductive glass (Pilkington TEC 8, 8\\,\\ohm\/$\\square$, 25\\,mm${}\\times{}$25\\,mm). \nThe substrates were cleaned in an ultrasonic bath, using three sequential steps: detergent with de-ionized water, acetone and 2-Propanol (10\\,min for each step).\nThe substrates were covered by a compact layer of TiO$_2$ (cTiO$_2$).\nA solution of acetylacetone (2\\,mL), titanium diisopropoxide (3\\,mL) and ethanol (45\\,mL) was deposited onto the FTO substrates by Spray Pyrolysis Deposition at 450\\,\\C. The final thickness of the cTiO$_2$ layer was measured about 50\\,nm by a Dektak Veeco 150 profilometer.\n\nThe mTiO$_2$ layer was obtained starting by an ethanol solution of 18NR-T titania paste (Dyesol) dissolved in pure ethanol (1:5 by weight), stirred overnight. \nThe graphene-doped mTiO$_2$ was obtained by adding sonicated graphene ink (1\\% in vol.),\nprepared by dispersing 5\\,g of graphite flakes (+100 mesh, $\\ge$75\\%, Sigma Aldrich) in 500\\,mL of N-methyl-2-pyrrolidone, NMP (Sigma Aldrich). The initial dispersion was ultrasonicated (VWR) for 6 hours and subsequently ultracentrifuged using a SW32Ti rotor in a Beckman-Coulter Optima XPN ultracentrifuge at 10000\\,rpm ($\\sim$12200 g) for 30 mins at 15\\,$^\\circ$C. After ultracentrifugation, the upper 80\\% supernatant was extracted by pipetting. The concentration of the graphitic flakes is calculated from the OAS (see Fig.\\,S1 in the Supporting Information), giving a concentration of 0.25\\,g L$^{-1}$. The morphology of the flakes, i.e., lateral size and thicknesses are characterized by transmission electron microscopy (TEM) and atomic force microscopy (AFM), respectively (see Fig.\\,S2 in the Supporting Information) giving a lateral size distribution of 150\\,nm and thickness of 1.7\\,nm. Raman spectroscopy data are found in the Supporting Information (see Fig.\\,S3).\nBoth standard mTiO$_2$ and G+mTiO$_2$ dispersions were sonicated 10\\,min prior to be spin-coated in air at 1700\\,rpm for 20\\,s onto the cTiO$_2$ surface, followed by a calcination step at 450\\,\\C\\ for 30\\,min.\n\nThe GO-Li interlayer was realized by spin coating (2000\\,rpm for 10\\,s) 200\\,\\micro L of GO-Li dispersion in ethanol\/H$_2$O (3:1) prepared as reported in Ref.\\,\\onlinecite{Agresti_2016b}. After the deposition, the substrates were annealed at 110\\,\\C\\ for 10\\,min.\n\nThe photo-electrodes were completed by depositing the perovskite active layer in dry conditions (relative humidity less than 30\\%) by a double step method: a lead iodide solution (PbI$_2$ in N,N-dimethylformamide, 1\\,M, heated at 70\\,\\C) was spin coated at 6000\\,rpm for 10\\,s on heated substrates (50\\,\\C) which were then dipped into a CH$_3$NH$_3$I (Dyesol) in anhydrous 2-propanol solution (10\\,mg\/mL) for 15\\,min. Finally, the samples were heated at 80\\,\\C\\ for 20\\,min in air. The MAPI absorbing layer has a typical thickness of 350\\,nm with a perovskite penetration into mesoporous layer of about 200\\,nm.\nThe samples were not encapsulated. \nComplete solar cells were realized with the same structure of the investigated samples to compare the PL results with the current-voltage ($I$-$V$) characteristics.\nIn the case of complete devices doped spiro-OMeTAD (73.5\\,mg\/mL) in chlorobenzene solution doped with tert-butylpyridine (TBP 26.77\\,\\micro L\/mL), lithium bis(trifluoromethanesulfonyl)imide (LiTFSI 16.6\\,\\micro L\/mL), and cobalt(III) complex (FK209 from Lumtec, 7.2\\,\\micro L\/mL) is spin coated (2000\\,rpm for 20\\,s) onto the tested photo-electrodes. The final devices are completed by Au counter-electrode thermal evaporation (100\\,nm). $I$-$V$ characteristics are recorded under AM1.5G solar simulator Solar Constant from KHS at 1000\\,W\/m$^2$ (1\\,sun). Results on the solar cells are shown in the Supporting Information.\n\n\\textbf{Optical Absorption Spectroscopy (OAS).}\nThe OAS of the as-produced inks was carried out in the 300-1000\\,nm range with a Cary Varian 5000i UV-vis-NIR spectrometer. The absorption spectra was acquired using a 1\\,mL quartz glass cuvette. The ink is diluted to 1:7 in NMP. The NMP solvent baseline was subtracted. The concentration of graphitic flakes is determined from the extinction coefficient at 660\\,nm, using $A = \\alpha l c$ where $l$ [m] is the light path length, $c$ [gL$^{-1}$] is the concentration of dispersed graphitic material, and $\\alpha$ [Lg$^{-1}$m$^{-1}$] is the absorption coefficient, with $\\alpha \\sim 1390$\\,Lg$^{-1}$m$^{-1}$ at 660\\,nm.\\cite{Lotya_2009}\n\n\\textbf{Transmission electron microscopy (TEM).}\nThe exfoliated flakes morphology is characterized by using a TEM JOEL JEM 1011, using an acceleration voltage of 100\\,kV. The sample preparation was performed diluting the ink  in NMP (1:10). 20\\,\\micro L of the diluted sample were drop cast on copper grids (200 mesh), and dried in vacuum overnight.  Statistical analyses are fitted with log-normal distributions.\n\n\\textbf{Atomic force microscopy (AFM).}\nThe dispersions are diluted 1:30 in NMP. 100\\,\\micro L of the dilutions are drop-casted onto Si\/SiO$_2$ wafers. AFM images are acquired with Bruker Innova AFM in tapping mode using silicon probes (frequency = 300\\,kHz, spring constant = 40\\,Nm$^{-1}$). Statistical analysis are fitted with log-normal distributions.\n\n\\textbf{Raman spectroscopy.}\nThe graphene inks are drop-cast onto Si\/SiO$_2$ wafers (LDB Technologies Ltd.) and dried under vacuum. Raman measurements are collected with a Renishaw inVia confocal Raman microscope using an excitation line of 514\\,nm with a 100X objective lens, and an incident power of $\\sim 1$\\,mW on the sample. 20 spectra are collected for each sample. Peaks are fitted with Lorentzian functions.\n\n\\textbf{Scanning electron microscopy (SEM).}\nElectrodes are imaged by aim of a field-emission scanning electron microscope FE-SEM (JOEL JSM-7500 FA). The acceleration voltage is set at 5\\,kV. Images are collected using the in-lens sensors (secondary electron in-lens image, SEI) and the secondary electron sensor (lower secondary electron image, LEI). No coating is applied.\n\n\n\\textbf{Photoluminescence spectroscopy (PL).}\nPL experiments were performed, in a quasi back-scattering geometry, keeping the samples in a closed cycle cryostat and the temperature was changed from 10 to 300\\,K. Time integrated photoluminescence (TI-PL) measurements were performed exciting the samples by different mode-locked ps laser sources: a tunable (700--850\\,nm and 350--425\\,nm with the second harmonic generator) Ti-Sapphire laser operating at 81.3\\,MHz repetition rate with 1.2\\,ps pulses and a 4\\,ps Rhodamine 6G dye laser synchronously pumped by the second harmonic of a mode-locked Nd-YAG laser, operating at 76\\,MHz. The PL signal was spectrally dispersed by a 50\\,cm monochromator providing a spectral resolution of 1\\,meV and detected by a microchannel plate photomultiplier.\nTime resolved (TR-PL) measurements were carried out exciting the samples by the ps dye laser operating at 600\\,nm and using time-correlated single photon counting technique (TCSPC) with a temporal resolution of about 60\\,ps.\n\n\n\n\\begin{acknowledgments}\nFB acknowledges funding from the Italian Ministry for Education, University and Research within the Futuro in Ricerca (FIRB) program (project DeLIGHTeD, Protocollo RBFR12RS1W). We warmly acknowledge Franco Bogani for fruitful discussions. This work was partially supported by ENTE CARIFI grant n.\\,2015\/11162 and from the European Union's Horizon 2020 research and innovation programme under grant agreement n. 696656 - GrapheneCore1.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe doping phase diagram near the underdoped region is one of the\nimportant and long-debated issues with high-$T_{c}$ cuprates\n\\cite{LeeRMP06}.\nSince the parent compound is antiferromagnetic (AFM) Mott insulator,\nthe AFM correlation plays a significant role in the emergence of\nsuperconductivity by doping charge carries.\nThe intrinsic proximity of the superconducting (SC) phase with the\nAFM phase is also shared by the phase diagrams of other SC\nmaterials, such as iron pnictides \\cite{StewartRMP11} and heavy\nfermion superconductors \\cite{NicklasPRB07}.\nThe multi-layered cuprate superconductors exhibit the coexistence of\nAFM and SC states at underdoping discovered by nuclear magnetic\nresonance measurements \\cite{MukudaJPSJ08,MukudaJPSJ09}.\nHowever, the AFM phase and the SC phase never coexist in the phase\ndiagram of single-layered cuprates such as\nLa$_{2-x}$Sr$_{x}$CuO$_{4}$ \\cite{KeimerPRB92} and\nBi$_{2}$Sr$_{2}$CuO$_{6+\\delta}$ \\cite{KatoJSSC97}.\nIn particular, Bi$_{2}$(Sr$_{2-x}$La$_{x}$)CuO$_{6+\\delta}$ systems\nshows that the three-dimensional AFM region, separated by the SC\nphase, even survives until a high underdoping level\n\\cite{KawasakiPRL10}.\n\nThe existence of the coexisting state has been found by analytical\nand numerical approaches in Hubbard$-$type models\n\\cite{ReissPRB07,JarrellEPL01,AichhornPRB07,DSPRL05,KobayashiPhysicaC10,CaponePRB06,KancharlaPRB08}\nand $t-J-$type models\n\\cite{InabaPhysicaC96,HimedaPRB99,YamasePRB04,ShihPRB04,ShihLTP05,PathakPRL09,WatanabePhysicaC10}.\nThey seem to contribute the underlying mechanism to the coexisting\nstate observed in multi-layered cuprates.\nHowever, a proper mechanism to explain why these two phases do not\nlike to coexist in single-layered cuprates remains needed.\nInterestingly, some previous studies proposed the spin-bag mechanism\nfor superconductivity since two spin bags would attract each other\nto form a Cooper pair and lower the total energy\n\\cite{SchriefferPRB89,WengPRB90,EderPRB94}.\nAs for doping more holes, therefore, it is necessary to re-examine\nhow the local distortion of the AFM background around holes\ninfluences AFM order and SC order.\n\nOn the other hand, one of the most exciting experimental results is\nthe observation of quantum oscillations in the hole-doped cuprates\nwhich pointed to electron pockets \\cite{LeBoeufNat07,LeBoeufPRB11}.\nIn particular, they proposed that these electron pockets probably\noriginate from the Fermi surface reconstruction caused by the onset\nof a density-wave phase, e.g. the AFM phase.\nUnfortunately the electron-like Fermi pockets have never been found\nin most of hole-doped cuprates using angle-resolved photoemission\nspectroscopy (ARPES) \\cite{YangNat08,MengNat09,LuARCMP12}.\nThus, to comprehend the loss of the electron pocket observed by\nARPES experiments, we inquire to what extent into the electronic\ncorrelations ignored in mean-field calculations.\n\nIn this work, we study Gutzwiller's trial wave functions with the\ncoexistence of AFM order and SC order by means of variational Monte\nCarlo (VMC) method.\nTo improve the trial state, we further consider the off-site\ncorrelations between two electrons by applying suitable Jastrow\ncorrelators.\nSurprisingly, the long-range AFM order is strongly enhanced due to\nthe local ferromagnetic (FM) Jastrow correlation, or precisely local\nAFM distortion, giving rise to the disappearance of the coexisting\nstate near underdoping in the phase diagram.\nBesides, the spin-spin correlation in the non-coexisting state\ntransfers the quasiparticle spectral weight from the antinodal\nelectron pockets to the lower AFM band, and also the nodal hole\npockets can remain until superconductivity occurs.\nTherefore, it is expected that the signal of the electron pockets\naround antinodes cannot be found in many hole-doped compounds by\nusing ARPES.\n\n\\section{Theory}\nLet us begin by the Hamiltonian on a square lattice of size\n$16\\times16$,\n\\begin{eqnarray}\nH=-\\sum_{i,j,\\sigma}t_{ij}\\tilde{c}_{i\\sigma}^{\\dag}\\tilde{c}_{j\\sigma}+J\\sum_{\\langle\ni,j\\rangle}\\left(\\mathbf{S}_{i}\\cdot\\mathbf{S}_{j}-\\frac{1}{4}n_{i}n_{j}\\right),\n\\label{e:equ1}\n\\end{eqnarray}\nwhere the hopping $t_{ij}=t$, $t'$, and $t''$ for sites i and j\nbeing the nearest, second-nearest, and third-nearest neighbors,\nrespectively.\nOther notations are standard. We restrict the electron creation\noperators $\\tilde{c}_{i\\sigma}^{\\dag}$ to the subspace without\ndoubly-occupied sites.\nIn the following, the bare parameters $(t',t'',J)\/t$ in the\nHamiltonian are set to be in the hole-doped regime:\n$(-0.3,0.15,0.3)$.\nIn order to understand how AFM order and SC order compete in\nvariational phase diagram, we choose the mean-field ground state\nincluding both AFM order and SC order (AFSC) as a starting point,\n\\begin{eqnarray}\n|\\Psi_{AFSC}\\rangle=\\prod'_{\\bf{k},s=\\{a,b\\}}\\gamma_{\\bf{k}\\uparrow}^{s}\\gamma_{-\\bf{k}\\downarrow}^{s}|0\\rangle,\\label{e:equ2}\n\\end{eqnarray}\nwhere the prime means the product only includes momenta inside the\nmagnetic zone boundary (MZB).\nNote that $s$ represents the quasiparticle coming from the upper AFM\nband ($s=b$) or the lower AFM band ($s=a$).\nThe Bogoliubov's quasiparticle operators $\\gamma_{\\bf{k}\\sigma}^{s}$\nare defined as\n\\begin{eqnarray}\n\\gamma_{\\bf{k}\\sigma}^{s}=u_{\\bf{k}}^{s}\\hat{s}_{\\bf{k}\\sigma}-\\sigma\nv_{\\bf{k}}^{s}\\hat{s}_{-\\bf{k}\\bar{\\sigma}}^{\\dag}.\\label{e:equ5}\n\\end{eqnarray}\nThe coefficients $u_{\\bf{k}}^{s}$ and $v_{\\bf{k}}^{s}$ are the BCS\ncoherence factor of AFM quasiparticles corresponding to the $s$\nband,\n\\begin{eqnarray}\n(u_{\\bf{k}}^{s})^{2}&=&\\frac{1}{2}\\left(1+\\frac{\\xi_{\\bf{k}}^{s}}{\\sqrt{(\\xi_{\\bf{k}}^{s})^{2}+\\Delta_{\\bf{k}}^{2}}}\\right),\\nonumber\\\\\n(v_{\\bf{k}}^{s})^{2}&=&1-(u_{\\bf{k}}^{s})^{2},\\label{e:equ6}\n\\end{eqnarray}\nwhere the AFM band dispersion\n$\\xi_{\\bf{k}}^{b\/a}=\\epsilon_{\\bf{k}}^{+}\\pm\\sqrt{(\\epsilon_{\\bf{k}}^{-})^{2}+m^{2}}$\nand\n$\\epsilon_{\\bf{k}}^{\\pm}\\equiv\\left(\\varepsilon_{\\bf{k}}\\pm\\varepsilon_{\\bf{k+Q}}\\right)\/2$.\nHere $\\varepsilon_{\\bf{k}}$ is the normal-state dispersion.\n$\\Delta_{\\bf{k}}$($=2\\Delta\\left(\\cos\\bf{k_{x}}-\\cos\\bf{k_{y}}\\right)$)\nis $d$-wave pairing amplitude and $m$ AFM order parameter.\nThe annihilation operators for AFM bands, $\\hat{s}_{\\bf{k}\\sigma}$,\nare given by\n\\begin{eqnarray}\n\\left(\n  \\begin{array}{c}\n    a_{\\bf{k}\\sigma} \\\\\n    b_{\\bf{k}\\sigma} \\\\\n  \\end{array}\n  \\right)=\\left(\n          \\begin{array}{cc}\n            \\alpha_{\\bf{k}} & \\sigma\\beta_{\\bf{k}} \\\\\n            -\\sigma\\beta_{\\bf{k}} & \\alpha_{\\bf{k}} \\\\\n          \\end{array}\n        \\right)\\left(\n                 \\begin{array}{c}\n                   c_{\\bf{k}\\sigma} \\\\\n                   c_{\\bf{k+Q}\\sigma} \\\\\n                 \\end{array}\n               \\right),\\label{e:equ3}\n\\end{eqnarray}\nwith $\\bf{Q}=(\\pi,\\pi)$ and the coefficients\n\\begin{eqnarray}\n\\alpha_{\\bf{k}}^{2}&=&\\frac{1}{2}\\left(1-\\frac{\\epsilon_{\\bf{k}}^{-}}{\\sqrt{(\\epsilon_{\\bf{k}}^{-})^{2}+m^{2}}}\\right),\\nonumber\\\\\n\\beta_{\\bf{k}}^{2}&=&1-\\alpha_{\\bf{k}}^{2}.\\label{e:equ4}\n\\end{eqnarray}\n\nIn order to introduce more correlations in the mean-field wave\nfunction, we first formulate the trial wave function fixing the\nnumber of electrons $\\hat{P}_{N_{e}}$ with on-site Gutzwiller\nprojector\n$\\hat{P}_{G}(=\\prod_{i}\\left(1-\\hat{n}_{i\\uparrow}\\hat{n}_{i\\downarrow}\\right))$\nand charge-charge Jastrow correlator ($\\hat{P}_{J}^{CC}$)\n\\cite{CPC08,CPC12},\n\\begin{eqnarray}\n|\\Psi_{CC}\\rangle=\n\\hat{P}_{N_{e}}\\hat{P}_{G}\\hat{P}_{J}^{CC}|\\Psi_{AFSC}\\rangle.\\label{e:equ7}\n\\end{eqnarray}\nMore importantly, we also consider the correlation between spins by\nusing spin-spin Jastrow correlator ($\\hat{P}_{J}^{SS}$),\n\\begin{eqnarray}\n|\\Psi_{CCSS}\\rangle=\\hat{P}_{J}^{SS}|\\Psi_{CC}\\rangle.\\label{e:equ82}\n\\end{eqnarray}\nThe Jastrow correlator is constructed by classical Boltzmann\noperator, $\\hat{P}_{J}^{i}=e^{\\hat{H}_{i}}$, encoding the intersite\ncorrelations.\nFor the sake of simplicity, $\\hat{H}_{i}$ depicting charge ($i=CC$)\nand spin ($i=SS$) parts are chosen to be diagonal in real-space\nconfiguration.\nThe charge-charge Jastrow correlator describes the short- and\nlong-range correlations between holes in the lattice system.\nThus,\n\\begin{eqnarray}\n\\hat{H}_{CC}=\\sum_{i<j}\\eta_{ij}\\hat{n}_{i}^{h}\\hat{n}_{j}^{h},\\label{e:equ81}\n\\end{eqnarray}\nwith\n$\\eta_{ij}\\equiv\\ln(r_{ij}^{\\alpha}v_{\\gamma}^{\\delta_{j,i+\\gamma}})$.\nHere $r_{ij}$ is the chord length of $|\\vec{r}_{i}-\\vec{r}_{j}|$ and\n$\\hat{n}_{i}^{h}=1-\\sum_{\\sigma}\\hat{n}_{i\\sigma}$.\nWe consider three parameters $v_{\\gamma}$, the nearest ($\\gamma=1$),\nsecond-nearest ($\\gamma=2$) and third-nearest ($\\gamma=3$)\nneighbors, standing for short-range hole-hole repulsion if\n$v_{\\gamma}<1$.\nThe factor $r_{ij}^{\\alpha}$ denotes attractive long-range\n($r_{ij}>1$) and repulsive short-range ($r_{ij}<1$) correlations\nbetween holes if $\\alpha>0$.\n\nA similar formalism to the spin-spin correlation has been considered\nat half-filling \\cite{HuesPRL88}.\nWe further imitate the formalism described above to write down the\nspin-spin Jastrow correlator,\n\\begin{eqnarray}\n\\hat{H}_{SS}=\\sum_{i<j}\\kappa_{ij}\\hat{S}_{i}^{z}\\hat{S}_{j}^{z},\\label{e:equ8}\n\\end{eqnarray}\nwhere\n$\\kappa_{ij}\\equiv\\ln(r_{ij}^{\\beta}w_{\\gamma}^{\\delta_{j,i+\\gamma}})$\nand $\\hat{S}_{i}^{z}$ the spin operator along $z$ direction at site\n$i$.\nThe only difference from the charge counterpart is that the\nsign of $\\hat{S}_{i}^{z}\\hat{S}_{j}^{z}$ determines the type of\nmagnetic correlations.\nIn other words, it will be the FM (AFM) correlation if\n$\\hat{S}_{i}^{z}\\hat{S}_{j}^{z}>0$ ($<0$).\nIn addition to the parameter $\\beta$ controlling the long-range spin\ncorrelations, we consider the other three parameters\n$w_{\\gamma=1,2,3}$ for the neighboring spin-spin correlations.\nFor example of the FM case, the short-range correlation would be\nsuppressed when $w_{\\gamma}<1$.\nOn the other hand, the factor $r_{ij}^{\\beta}$ control the\nlong-range ($r_{ij}>1$) and short-range ($r_{ij}<1$) correlations.\nIn the long-range case of $\\beta<0$, for instance, $r_{ij}^{\\beta}$\nwould decrease the FM correlation but conversely increase the AFM\ncorrelation.\n\nIn addition to the ground state, we also propose a trial wave\nfunction for the low-lying excitation of the Gutzwiller-projected\ncoexisting state simply generated by Gutzwiller projecting the\nmean-field excited state\n\\begin{eqnarray}\n|\\Psi_{AFSC}^{\\bf{k}\\sigma\ns}\\rangle=\\left(\\gamma_{\\bf{k}\\sigma}^{s}\\right)^{\\dag}|\\Psi_{AFSC}\\rangle.\\label{e:equ9}\n\\end{eqnarray}\nHere we have applied the particle-hole transformation\n\\cite{YokoyamaJPSJ88,CPCPRB12} into Eq.(\\ref{e:equ9}) to avoid the\ndivergence from the nodes of the mean-field wave function.\nThe Gutzwiller-projected excited state with both AFM order and SC\norder fixing to $N_{e}-1$ electrons is written as\n\\begin{eqnarray}\n|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle=\n\\hat{P}_{N_{e}-1}\\hat{P}_{G}\\hat{P}_{J}^{CC}\\hat{P}_{J}^{SS}|\\Psi_{AFSC}^{\\bf{k}\\sigma\ns}\\rangle.\\label{e:equ10}\n\\end{eqnarray}\nHence we can compute the excitation energies\n$E_{k}(\\equiv\\langle\\Psi_{\\bf{k}\\sigma}^{s}|H|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle-\\langle\\Psi_{0}|H|\\Psi_{0}\\rangle)$\nfor either upper ($s=b$) or lower ($s=a$) AFM quasiparticles.\nFurthermore, the quasiparticle spectral weight measured from ARPES\ncan be obtained by calculating\n\\begin{eqnarray}\nZ_{\\bf{k}}^{-}\\equiv\\frac{\\left|\\langle\\Psi_{\\bf{k}\\sigma}^{s}|c_{-\\bf{k}\\bar{\\sigma}}|\\Psi_{0}\\rangle\\right|^{2}}{\\langle\\Psi_{\\bf{k}\\sigma}^{s}|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle\\langle\\Psi_{0}|\\Psi_{0}\\rangle}.\\label{e:equ11}\n\\end{eqnarray}\nSome details in the VMC calculation should be noticed.\nThe boundary condition we use is periodic along both directions.\nIn order to achieve a reasonable acceptance ratio, the simulation\nconsists of a combination of one-particle moves and two-particle\nmoves.\nThe variational parameters of the Gutzwiller-projected coexisting\nstate are optimized by using the stochastic reconfiguration method\n\\cite{SorellaPRB01}.\nAll physical quantities are evaluated using the optimized\nparameters.\nWe also take a sufficient number of samples ($=2\\times10^{5}$) to\nreduce the statistical errors, and keep the sampling interval\n($\\sim40$) long enough to ensure statistical independence between\nsamples.\n\n\\section{Results}\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=3.5in,width=3in]{fig1.png}}\\end{center}\n\\caption{(a) Variational phase diagram plotted by staggered\nmagnetization $M_{s}$ (squares) and superconducting order parameter\n$\\Delta_{SC}$ (circles). Filled and empty symbols represent\n$|\\Psi_{CC}\\rangle$ and $|\\Psi_{CCSS}\\rangle$, respectively. (b) The\ndifference of the energy components between $|\\Psi_{CCSS}\\rangle$\nand $|\\Psi_{CC}\\rangle$ as a function of hole doping $\\delta$ in\n$16\\times16$ lattice.}\\label{fig1}\n\\end{figure}\n\nWe first consider the trial state with only the charge-charge\nJastrow correlator to better demonstrate the variational phase\ndiagram.\nThen we further include the spin-spin Jastrow correlator to\nsee how the phase diagram changes.\nOrder parameters shown in the phase diagram are determined by the\nstaggered magnetization\n\\begin{eqnarray}\nM_{s}=\\frac{1}{N}\\sum_{i}\\langle\\hat{S}_{i}^{z}\\rangle\ne^{i\\bf{Q}\\cdot\\bf{R}_{i}}\\label{e:mag}\n\\end{eqnarray}\nand the long-range pair-pair correlation function\n\\begin{eqnarray}\nC_{PP}(R)=\\frac{1}{N}\\sum_{i,\\alpha,\\alpha'}\\lambda_{\\alpha,\\alpha'}\\langle\\Delta_{i,\\alpha}^{\\dag}\\Delta_{i+R,\\alpha'}\\rangle.\\label{e:ppcf}\n\\end{eqnarray}\nThe creation operator\n$\\Delta_{i,\\alpha}^{\\dag}$($\\equiv\\tilde{c}_{i\\uparrow}^{\\dag}\\tilde{c}_{i+\\alpha\\downarrow}^{\\dag}-\\tilde{c}_{i\\downarrow}^{\\dag}\\tilde{c}_{i+\\alpha\\uparrow}^{\\dag}$)\ncreates a singlet on the bond $(i,i+\\alpha)$, $\\alpha=x,y$.\nThe factor $\\lambda_{\\alpha,\\alpha'}$ describes $d$-wave symmetry:\n$\\lambda_{\\alpha,\\alpha'}=1$($-1$) as\n$\\alpha=\\alpha'$($\\alpha\\neq\\alpha'$).\n\nIn Fig.\\ref{fig1}(a), without the spin-spin Jastrow correlators as\nindicated by $|\\Psi_{CC}\\rangle$, there exists a region showing the\ncoexistence of AFM order and SC order within doping\n$\\delta\\lesssim0.125$ in the phase diagram\n\\cite{ShihLTP05,PathakPRL09,WatanabePhysicaC10}, where $M_{s}$ and\n$\\Delta_{SC}$($\\equiv\\sqrt{C_{PP}(R>2)}$) are finite.\nLet us turn to the case with both charge-charge and spin-spin\nJastrow correlators denoted by $|\\Psi_{CCSS}\\rangle$.\nObviously the coexisting region disappears and a clear boundary\nseparating the AFM phase and the SC phase shows up at doping\n$\\delta=0.156$.\nNote that near the boundary the spin-spin Jastrow correlator can\ngreatly improve the ground-state energies from $0.3\\%$ to $0.7\\%$.\nFrom the numerical optimization, we find the spin-spin Jastrow\ncorrelator can provide a conduit to vary the mean-field AFM order in\n$|\\Psi_{AFSC}\\rangle$.\nSurprisingly, the optimized spin-spin Jastrow parameters slightly\ndisplay short-range FM correlations in the AFM background (e.g. at\n$\\delta=0.156$ the spin-spin Jastrow weights $w_{\\gamma}$ for\n$\\gamma=1$, $2$ and $3$ would be increased to $1.12$, $1.02$ and\n$1.01$, respectively).\nThe local FM correlation introduced by the Jastrow factors is\nharmful to the mean-field AFM order.\nTo make them balance, it is inevitable to largely enhance the AFM\nbackground in $|\\Psi_{AFSC}\\rangle$.\nThe surprising competition between AFM order and SC order near the\nphase boundary is mainly due to the hugely enhanced AFM order\nfurther leading to the diminished SC order.\n\nTo further demonstrate the energy competition, we analyze the\ndifference of the energy components in the Hamiltonian between\n$|\\Psi_{CCSS}\\rangle$ and $|\\Psi_{CC}\\rangle$ shown in\nFig.\\ref{fig1}(b).\nOur data clearly show that within $0.04<\\delta<0.2$ the spin Jastrow\ncorrelator helps the trial mean-field state gain much more energy\nfrom the second-nearest-neighbor hopping term.\nOn the other hand, the competing energy primarily comes from the\nspin-spin superexchange interaction.\nFrom real-space point of view, holes prefer to move along diagonal\ndirection in strong AFM background so that the hopping energy from\nthe second nearest neighbors ($t'$) is likely to compete with the\nsuperexchange energy ($J$).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=2.4in,width=3.4in]{fig2.png}}\\end{center}\n\\caption{The difference of the momentum distribution function\nbetween $|\\Psi_{CCSS}\\rangle$ and $|\\Psi_{CC}\\rangle$ for doping (a)\n$\\delta=0.125$, (b) $\\delta=0.156$ and (c) $\\delta=0.188$ plotted in\nthe first Brillouin zone. (d) The next-nearest-neighbor energy\ncomponent, $-4t'\\cos(k_{x})\\cos(k_{y})$. The black diamond is the\nhalf-filled Fermi surface. White (Purple) regions present the\npositive (negative) values. Red lines mean zero.}\\label{fig2}\n\\end{figure}\n\nIn momentum space, it is apparent that the $t'$ energy gain would\ninfluence how the band dispersion evolves from Fermi pocket to Fermi\nsurface as increasing doping.\nIn Fig.\\ref{fig2}(a)-(c), the difference of the momentum\ndistribution function between $|\\Psi_{CCSS}\\rangle$ and\n$|\\Psi_{CC}\\rangle$ shows how electrons distribute in the band\nstructure.\nAt $\\delta=0.125$ (Fig.\\ref{fig2}(a)), obviously electrons in the\nsystem would prefer to stay around \"hot spots\" rather than living\nnear nodes and antinodes, which hole pockets and electron pockets\nseem to be observed as well.\nThe hot spot is defined as the momenta along the MZB that can be\nconnected by ($\\pi,\\pi$) momentum scattering.\nOnce doping is increased to $0.156$ which is the phase boundary\n(Fig.\\ref{fig2}(b)), hole pockets become larger and electron pockets\nslightly shrink.\nNow that electrons like to circle just outside the electron pockets,\nthey attempt to form a large Fermi surface.\nIndeed, as further increasing doping to $0.188$ where the long-range\nAFM order almost disappears (Fig.\\ref{fig2}(c)), a clear Fermi\nsurface in which electrons cluster together can be seen.\nSo far, we also understand the reason why the system gain much\nenergy from $t'$ term since the hot spots are located right at the\npurple region shown in Fig.\\ref{fig2}(d).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=3.7in,width=2.4in]{fig3.png}}\\end{center}\n\\caption{(a) Spin-spin, (b) hole-hole and (c) pair-pair correlation\nfunctions for the optimized state $|\\Psi_{CCSS}\\rangle$\n($|\\Psi_{CC}\\rangle$), denoted by red circle (black square) symbols.\nThere are 40 doped holes in $16\\times16$ lattice\n($\\delta=0.156$).}\\label{fig3}\n\\end{figure}\n\nIn Fig.\\ref{fig3}, we compute the spin-spin, hole-hole and pair-pair\ncorrelation functions (already shown in Eq.(\\ref{e:ppcf})) defined\nas,\n\\begin{eqnarray}\nC_{CC}(\\bf{R})&=&\\frac{1}{N}\\sum_{i}\\langle\\hat{n}_{i}^{h}\\hat{n}_{i+\\bf{R}}^{h}\\rangle,\\\\\nC_{SS}(\\bf{R})&=&\\frac{1}{N}\\sum_{i}\\langle\n\\hat{S}_{i}^{z}\\hat{S}_{i+\\bf{R}}^{z}\\rangle\ne^{i\\bf{Q}\\cdot\\bf{R}}.\\label{e:equ12}\n\\end{eqnarray}\nThe doping density we choose to present is $0.156$.\nFigure \\ref{fig3}(a) illustrates that the spin-spin Jastrow\ncorrelators indirectly induce the stronger AFM background showing a\nconstant tail in the staggered spin-spin correlation function which\nimplies a clear AFM order.\nNote that The enhancement of the AFM order mainly arises from the\nmean-field wave function $|\\Psi_{AFSC}\\rangle$.\nFurthermore, we find in Fig.\\ref{fig3}(b) that the hole-hole\ncorrelation function makes no difference even if including the\nspin-spin Jastrow correlators, except that the short-range part\nbecomes less staggered.\nFor spin and charge, there is no correlation for their long-range\nbehavior.\nFinally, we can also see in Fig.\\ref{fig3}(c) that as considering\n$\\hat{P}_{J}^{SS}$ the pair-pair correlation almost vanishes at\nlarge distances so that the SC properties is not available.\n\nNext, it would be interesting to examine the low-lying\nsingle-particle excitation spectra near the phase boundary.\nIn Fig.\\ref{fig4}, by applying the ansatz (Eq.(\\ref{e:equ10})) to\nthe single-particle excitation, we calculate two quasiparticle band\ndispersions ($s=a,b$) and their corresponding spectral weight for\nremoving one particle defined by Eq.(\\ref{e:equ11}).\nIn order to compare with the excitations with\/without spin-spin\nJastrow correlators $\\hat{P}_{J}^{SS}$, we plot their excitation\nenergy $E_{\\bf{k}}$ along the high symmetric momenta in\nFig.\\ref{fig4}(a).\nIn the case where the trial state only includes the charge-charge\nJastrow factors, its optimized mean-field parameters $\\Delta\\gg m$.\nDue to large $d$-wave BCS pairing contribution, the dispersions thus\nshow convex around the antinodes and almost zero gap between the two\nbands at nodes.\nEspecially, the upper AFM band is beneath the lower AFM band near\nthe antinodal regions, and hence there is a clear signal of electron\npockets arising from the upper AFM band shown in Fig.\\ref{fig4}(c).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=2in,width=3.4in]{fig4.png}}\\end{center}\n\\caption{(a) The quasi-particle excitation dispersion $E_{\\bf{k}}$\nfor different optimized states (denoted in the legend of (c)) along\nhigh symmetric momenta at $\\delta=0.156$. Empty (Filled) symbols\nrepresent the upper (lower) AFM band and squares (circles) the trial\nstate $|\\Psi_{CC}\\rangle$ ($|\\Psi_{CCSS}\\rangle$). Due to much\nsmaller $\\Delta$ than $m$ for the trial state $|\\Psi_{CCSS}\\rangle$,\nwe simply plot the lower AFM band (red circles) below the Fermi\nlevel (pink line) except the nodal regions for clear demonstration.\nThe quasiparticle spectral weight $Z_{\\bf{k}}^{-}$ are obtained from\n(b) the lower AFM band and (c) the upper AFM band.}\\label{fig4}\n\\end{figure}\n\nWhen further considering spin-spin Jastrow correlators, the\noptimized mean-field parameters $m\\gg\\Delta$.\nSuch a huge AFM parameter $m$ gives rise to a typical AFM band\ndispersion and opens a AFM gap between these two bands at nodes, as\nindicated by red circles in Fig.\\ref{fig4}(a).\nInterestingly, Fig.\\ref{fig4}(c) shows that near antinodes the\nquasiparticle spectral weight of the upper AFM band disappear and\ntransfer to almost entire lower AFM band (see Fig.\\ref{fig4}(b)).\nIn particular, a clear hole pocket of the lower AFM band centering\naround $\\bf{Q}\/2$ is also observed in Fig.\\ref{fig4}(b).\nThe Gutzwiller and Jastrow correlators arising from electronic\ncorrelation firmly influence the low-lying quasiparticle excitation\nspectra of the mean-field state $|\\Psi_{AFSC}\\rangle$.\nTherefore, the loss of the electron pockets due to electron\ncorrelations provides a route to figure out why electron pockets\nhave never been found in most of hole-doped cuprates measured by\nARPES.\n\n\\section{Conclusions}\nSumming up, by using VMC approach we have studied the coexisting\nstate with both AFM order and SC order simultaneously underneath the\nGutzwiller's projection and Jastrow correlators.\nWe have thereby re-examined the variational ground-state phase\ndiagram and found that the AFM phase competes with the SC phase as\nfurther considering off-site spin correlations.\nThe reasoning for the competition is that the mean-field AFM order\nis considerably enhanced due to short-range FM correlation\nintroduced by the spin-spin Jastrow factors, further leading to the\nvanished SC order.\nAs well, we have first investigated the Gutzwiller-projected\nquasiparticle excitations of the coexisting state.\nBased on the Gutzwiller ansatz, passing through the boundary between\nAFM and SC phases, we have observed the loss of electron pockets\nnear antinodes coming from the upper AFM band and the occurrence of\nhole pockets near nodes arising from the lower AFM band as long as\nthe spin-spin Jastrow correlators are included.\nTherefore, such a strongly correlated electron system needs to be\ncarefully inspected in the explanation for the low-lying\nquasiparticle excitations observed by ARPES experiments.\n\n\\section{Acknowledgments}\n\\label{Acknowledgment} Greatly thanks S.-M. Huang, W. Ku and T.-K.\nLee for helpful discussions. This work is supported by the\nPostdoctoral Research Abroad Program sponsored by National Science\nCouncil in Taiwan with Grant No. NSC 101-2917-I-564-010 and by CAEP\nand MST. All calculations are performed in the National Center for\nHigh-performance Computing in Taiwan.\n\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}