diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpqzo" "b/data_all_eng_slimpj/shuffled/split2/finalzzpqzo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpqzo" @@ -0,0 +1,5 @@ +{"text":"\\section{Results}\n\\label{sec:results}\n\nOdors are mixtures of odorant molecules that are ligands of olfactory receptors.\nAny odor can be described by a vector~$\\boldsymbol c = (c_1, c_2, \\ldots, c_{{N_{\\rm l}}})$ that specifies the concentrations of all ${N_{\\rm l}}$ possible ligands. During a single sniff, the ligands in the odor~$\\boldsymbol c$ come in contact with the~${N_{\\rm r}}$ different odor receptors.\nIn the simplest case, the sensitivity of receptor~$n$ to ligand~$i$ can be described by a single number~$S_{ni}$ and the total excitation~$e_n$ of receptor~$n$ is given by~\\cite{McGann2005, Lin2006}\n\\begin{align}\n\te_n &= \\sum_i S_{ni} c_i\n\t\\;.\n\t\\label{eqn:excitation}\n\\end{align}\n\nTypical receptors have a non-linear dose-response curve~\\cite{Reisert2009} and the output~$a_n$ is thus a non-linear function of~$e_n$.\nMoreover, receptors are subject to noise~\\cite{Lowe1995}, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, from stochastic binding, which limits the number of distinguishable outputs.\nTo capture both effects, we consider receptors with only two output states, which corresponds to large noise~\\cite{Koulakov2007}.\nIn this case, the activity~$a_n$ of receptor~$n$ is given by\n\\begin{align}\n\ta_n &= \\begin{cases}\n\t\t0 & e_n < 1 \\\\\n\t\t1 & e_n \\ge 1\t\n\t\\end{cases}\n\t\\;,\n\t\\label{eqn:receptor_activity}\n\\end{align}\n\\mbox{i.\\hspace{0.125em}e.}\\@\\xspace, the receptor is active if its excitation~$e_n$ exceeds a threshold.\n\\Eqsref{eqn:excitation}--\\eqref{eqn:receptor_activity} describe the mapping of the odor~$\\boldsymbol c$ to the activity pattern~$\\boldsymbol a=(a_1, a_2, \\ldots, a_{{N_{\\rm r}}})$, where the receptor array is characterized by the sensitivity matrix~$S_{ni}$, see \\figref{fig:schematic}C.\nThis activity pattern is then analyzed by the brain to infer the odor~$\\boldsymbol c$.\nSuch a distributed representation of odors in activity patterns has been compared to compressed sensing \\cite{Stevens2015};\nhere we focus on how this representation can be tuned to match the structure of natural odors. \n\nWe assume that the structure of natural odors in a given environment can be captured by a probability distribution $P_{\\rm env}(\\boldsymbol c)$ from which odors are drawn.\n$P_{\\rm env}(\\boldsymbol c)$ can encode, for example, the fact that some ligands are more common than others or that some ligands are strongly correlated or anti-correlated in their occurrence. \nSince natural odor statistics are hard to measure~\\cite{Wright2005}, we work with a broad class of distributions~$P_{\\rm env}(\\boldsymbol c)$ characterized by a few parameters.\nWe define $p_i$ to be the probability with which ligand~$i$ occurs in a random odor.\nThe correlations between the occurrence of ligands are captured by a covariance matrix~$p_{ij}$.\nWe expect $p_i$ to be small since any given natural odor typically contain tens to hundreds of ligands~\\cite{Knudsen1993, Lin2006}, which is a small subset of all ${N_{\\rm l}} \\gtrsim 2100$ ligands~\\cite{Wright2005}.\nWhen a ligand~$i$ is present, we assume its concentration $c_i$ has mean~$\\mu_i$ and standard deviation~$\\sigma_i$.\nThus, the full natural odor statistics $P_{\\rm env}(\\boldsymbol c)$ are parameterized by $p_i$, $\\mu_i$, and $\\sigma_i$ for all ligands~$i$ and a covariance matrix~$p_{ij}$ in our model.\n\n\\subsection{Optimal receptor arrays}\nAn optimal receptor array must tailor receptor sensitivities $S_{ni}$ so that the odors-to-activity mapping \ngiven by \\Eqsref{eqn:excitation}--\\eqref{eqn:receptor_activity}\ndedicates more activity patterns to more frequent or more important odors as specified by $P_{\\rm env}(\\boldsymbol c)$. In information-theoretic terms, the array must maximize the mutual information $I(\\boldsymbol c, \\boldsymbol a)$~\\cite{Atick1992}. \nIn our model, the mapping from~$\\boldsymbol c$ to~$\\boldsymbol a$ is deterministic and $I$ can be written as the entropy of the output distribution~$P(\\boldsymbol a)$, \n\\begin{align}\n\tI &= -\\sum_{\\boldsymbol a} P(\\boldsymbol a) \\log_2 P(\\boldsymbol a)\n\t\\;,\n\t\\label{eqn:MI_def}\n\\end{align}\nwhere the sum is over all possible activity patterns~$\\boldsymbol a$.\nNote that $P(\\boldsymbol a) = \\int \\text{d} \\boldsymbol c \\, P(\\boldsymbol a | \\boldsymbol c) P_{\\rm env} (\\boldsymbol c)$, where $P(\\boldsymbol a | \\boldsymbol c)$ describes the mapping from $\\boldsymbol c$ to $\\boldsymbol a$.\nConsequently, $I$ depends on $S_{ni}$ and the odor environment $P_{\\rm env}(\\boldsymbol c)$.\nIn fact, $I$ is maximized by sensitivities~$S_{ni}$ that are tailored to $P_{\\rm env}(\\boldsymbol c)$ such that all activity patterns~$\\boldsymbol{a}$ are equally likely \\cite{Atick1992,Laughlin1981}.\n\nThe mutual information~$I$ can be approximated~\\cite{Sessak2009} in terms of the mean activities $\\mean{a_n}$ and the covariance between receptors, $\\cov(a_n, a_m)= \\mean{a_n a_m} - \\mean{a_n} \\mean{a_m}$, encoded by $P(\\boldsymbol a)$,\n\\begin{align}\n\tI &\\approx \n\t- \\!\\sum_n \\bigl[\\mean{a_n}\\log_2 \\mean{a_n} + (1 - \\mean{a_n})\\log_2(1 - \\mean{a_n})\\bigr]\n\t\\notag \\\\ & \\quad\n\t- \\frac{8}{\\ln 2} \\sum_{n 1$, where $e_{\\rm b}$ and $e_{\\rm d}$ are the excitations caused by the~$s_{\\rm b}$ shared and the $s - s_{\\rm b}$ different ligands, respectively.\nApproximating the probability distribution of the excitations as a log-normal distribution, we can calculate the expected distance~$h$, see SI.\n\\figref{fig:mixture_discrimination}B shows that this approximation (solid lines) agrees well with numerical calculations (symbols).\nThe figure also shows that mixtures can only be distinguished well if the concentration of the constituents is in the right range.\nThis is because receptors are barely excited for too small concentrations while they are saturated for large concentrations. \nThe distance~$h$ also strongly depends on the number~$s_{\\rm b}$ of shared ligands between the two mixtures, which has also been shown experimentally~\\cite{Bushdid2014}.\nThe distance vanishes for $s_{\\rm b} = s$, but \\figref{fig:mixture_discrimination}B shows that a single different ligand can be sufficient to distinguish mixtures in the right concentration range (green line).\nThis range increases with the width~$\\lambda$ of the sensitivity distribution, similar to the range over which concentrations can be measured, see \\Eqref{eqn:concentration_range}.\n\n\\begin{figure}[t]\n\t\\centerline{\n\t\t\\includegraphics[width=\\figwidth]{\"Figures\/Fig5\"}\n\t}\n\t\\caption{%\n\t\tThe discriminability of mixtures strongly depends on the concentrations at which odors are presented. \n\t\t(A) Maximal mixture size~$s_{\\rm max}$ (from \\Eqref{eqn:mixture_size_max}) as a function of the ligand concentration~$c$ for different widths~$\\lambda$ of the sensitivity distribution at ${N_{\\rm r}}\/\\eta=300$.\n\t\tDotted lines indicate where $c$ is below the detection threshold for single ligands.\n\t\t(B)~Mean difference~$h$ in the activation pattern of two mixtures of size~$s=10$ as a function of $c$ for different numbers $s_{\\rm b}$ of shared ligands and widths~$\\lambda$.\n\t\tAnalytical results (lines) are compared to numerical simulations (symbols).\n\t\t\\label{fig:mixture_discrimination}\n\t}\n\\end{figure}%\n\n\\subsection{Experimentally measured receptor arrays}\n\nThe response of receptors to individual ligands has been measured experimentally for flies~\\cite{Muench2015} and humans~\\cite{Mainland2015}.\nWe use these published data to estimate the statistics of realistic sensitivity matrices as described in the SI.\n\\figref{fig:sensitivities} shows the histograms of the logarithms of the sensitivities for flies and humans.\nBoth histograms are close to a normal distribution, with similar standard deviations~$\\lambda_{\\rm exp} \\approx 1.1$, which implies log-normally distributed sensitivities.\nUsing a simple binding model between receptors and ligands, $\\lambda_{\\rm exp}$ can also be interpreted as the standard deviation of the interaction energies, see SI.\nConsequently, these interaction energies exhibit a similar variation on the order of one $k_{\\rm B} T$ for both organisms, which could be caused by the biophysical similarity of the receptors.\n\n\\begin{figure}[t]\n\t\\centerline{\n\t\t\\includegraphics[width=\\figwidth]{\"Figures\/Fig6\"}\n\t}\n\t\\caption{%\n\t\tSensitivities of olfactory receptors appear to be log-normally distributed for (A) flies~\\cite{Muench2015} and (B) humans~\\cite{Mainland2015}.\n\t\tThe histograms of the logarithms of $n$ entries of the sensitivity matrix (orange) are compared to a normal distribution (blue) with the same mean and standard deviation~$\\lambda_{\\rm exp}$.\n\t\t\\label{fig:sensitivities}\n\t}\n\\end{figure}%\n\nWe next use the measured log-normal distribution for the sensitivities to compare the concentration resolution~$R$ predicted by \\Eqref{eqn:concentration_resolution} to measured 'just noticeable relative differences'~$R^{-1}$~\\cite{Koulakov2007}.\nFor humans (${N_{\\rm r}} = 300$), the measured values are as low as \\unit[4]{\\%}~\\cite{Cain1977}, which implies $\\eta\\lambda \\approx 4.8$.\nUsing $\\lambda \\approx 1.1$, this suggest that about 4 receptors have to be activated until a change in concentration can be registered.\nAdditionally, our theory predicts that humans can sense concentrations over about $2.6$ orders of magnitude, which follows from \\Eqref{eqn:concentration_range} for $\\lambda = 1.1$, $\\eta = 1$, and ${N_{\\rm r}} = 300$.\nHowever, we are not aware of any measurements of the concentration range for humans.\n\nOur theory also predicts the maximal number of ligands that can be distinguished as a function of the concentration~$c$ of the individual ligands.\nFor $\\lambda \\approx 1.1$, we expect that the maximal number~$s_{\\rm max}$ of ligands in a mixture is around 20 if individual ligands can be detected, see \\figref{fig:mixture_discrimination}A.\nExperimental studies report similar numbers, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, $s_{\\rm max} \\approx 15$~\\cite{Jinks1999} and $s_{\\rm max} < 30$~\\cite{Weiss2012}.\nHowever, \\figref{fig:mixture_discrimination}A shows that $s_{\\rm max}$ strongly depends on the concentration of the individual ligands and thus on experimental details.\nSimilarly, how well mixtures can be discriminated also depends strongly on the ligand concentration.\n\\figref{fig:mixture_discrimination}B shows that the concentration range over which mixtures can be distinguished is less than an order of magnitude for $\\lambda \\approx 1.1$.\n\n\\section{Discussion}\n\nWe studied how arrays of olfactory receptors can be used to measure odor mixtures, focusing on the combinatorial code of olfaction, \\mbox{i.\\hspace{0.125em}e.}\\@\\xspace, how the combined response of multiple receptors can encode the composition (quality) and the concentration (quantity) of odors.\nSuch arrays are optimal if each receptor responds to about half of the encountered odors and the receptors have distinct ligand binding profiles to minimize correlations.\n\nOur simple model of binary receptors can in principle distinguish a huge number of odors, since there are $\\sim10^{90}$ different output combinations for ${N_{\\rm r}}=300$.\nHowever, it is not clear whether all outputs are achievable and how they are used to distinguish odors.\nWe showed that the mean receptor sensitivity must be tailored to the mean concentration to best use the large output space.\nAnother important parameter of receptor arrays is the fraction of receptors that is activated by a single ligand, which is equivalent to the sparsity~$\\xi$ in the simple case of binary sensitivities.\nIf $\\xi$ is small, combining different ligands typically leads to unique output patterns that allow to identify the mixtures, but the concentration of isolated ligands cannot be measured reliably, since only few receptors are involved.\nConversely, if $\\xi$ is large, mixtures of multiple ligands will excite almost all receptors, such that neither the odor quality nor the odor quantity can be measured reliably.\nHowever, here, the concentration of an isolated ligand can be measured precisely.\nWe discussed this property in detail for sensitivities that are log-normally distributed, where the width~$\\lambda$ controls whether mixtures can be distinguished well or concentrations can be measured reliably.\nInterestingly, experiments find that individual ligands at moderate concentration only excite few glomeruli~\\cite{Saito2009}, but natural odors at native concentrations can excite many~\\cite{Vincis2012}.\nThis could imply that the sensitivities are indeed adapted such that each receptor is excited about half the times for natural odors.\n\nOur model implies that having more receptor types can improve all properties of the receptor array.\nIn particular, both the concentration resolution $R$ and the typical distance $h$ between mixtures are proportional to ${N_{\\rm r}}$, a prediction that can be tested experimentally.\nFor instance, mice, with ${N_{\\rm r}} \\approx 1000$ receptor types, are very good at identifying a single odor in a mixture~\\cite{Rokni2014}, but flies, with ${N_{\\rm r}} = 52$~\\cite{Muench2015}, should perform much worse.\nHowever, quantitative comparisons might be difficult since the discrimination performance strongly depends on the normalized concentration~$c\\bar S$ at which odors are presented.\nIn fact, we predict that mixtures can hardly be distinguished if the concentration of the individual ligands is changed by an order of magnitude, see \\figref{fig:mixture_discrimination}B.\n\nOur results also apply to artificial chemical sensor arrays known as 'artificial noses'~\\cite{Albert2000, Stitzel2011}.\nHaving more sensors improves the general performance of the array, but it is also important to tune the sensitivity of individual sensors.\nHere, sensors should be as diverse as possible while still responding to about half the incoming mixtures.\nUnfortunately, building such chemical sensors is difficult and their binding properties are hard to control~\\cite{Stitzel2011}.\nIf the sensitivity matrix of the sensor array is known, our theory can be used to estimate the information~$I_n$ that receptor $n$ contributes as\n$\nI_n \\approx \n\t\tH_{\\rm b}(\\mean{a_n})\n\t\t- \\frac{4}{\\ln 2} \\sum_{m \\neq n} \\! \\cov(a_n, a_m)^2\n$\nwhere $H_{\\rm b}(p) = - p\\log_2 p - (1 - p)\\log_2(1 - p)$, such that $I = \\sum_n I_n$, see \\Eqref{eqn:MI_est}.\nThis can then be used for identifying poor receptors that contribute only little information to the overall results.\n\nOur focus on the combinatorial code of the olfactory system certainly neglects intricate details of the system.\nFor instance, we consider sensitivity matrices with independent entries, but biophysical constraints will cause chemically similar ligands to excite similar receptors~\\cite{Malnic1999, Hallem2006}.\nThis is important because it makes it difficult to distinguish similar ligands~\\cite{Perez2015} and it might thus be worthwhile to dedicate more receptors to such a part of chemical space.\nAdditionally, receptors or glomeruli might interact with each other, \\mbox{e.\\hspace{0.125em}g.}\\@\\xspace, causing inhibition reducing the signal upon binding a ligand~\\cite{Ukhanov2010}.\nWe can in principle discuss inhibition in our model by allowing for negative sensitivities, but more complicated features cannot be captures by the linear relationship in \\Eqref{eqn:excitation}.\nOne important non-linearity is the dose-response curve of individual receptor neurons~\\cite{Reisert2009}, which we approximate by a step function, see \\Eqref{eqn:receptor_activity}.\nThis simplification reduces the information capacity of a single glomerulus to $\\unit[1]{bit}$, while it is likely higher in reality. \nHowever, we expect that allowing for multiple output levels would only increase the concentration resolution and not change the discriminability of mixtures very much~\\cite{Koulakov2007}.\nIt would be interesting to see how such an extended model can measure heterogenous mixtures with ligands at different concentrations.\n\n\\begin{acknowledgments}\nWe thank Michael Tikhonov and Carl Goodrich for helpful discussions and a critical reading of the manuscript.\nThis research was funded by\nthe National Science Foundation through DMR-1435964, \nthe Harvard Materials Research Science and Engineering Center DMR-1420570,\nthe Division of Mathematical Sciences DMS-1411694, and\nthe German Science Foundation through ZW 222\/1-1.\nMPB is an investigator of the Simons Foundation. \n\n\\end{acknowledgments}\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nFullerenes, hollow clusters made up of carbon atoms bonded by sp$^2$ orbitals, have interesting conducting properties. The origin of such a behavior \nis their delocalized $\\pi$ frontier molecular orbitals, \nwhat gives rise to a high conductance when an electric field is applied on the molecule, e.g. by an external potential bias. It has been discussed that free-electron and tight-binding (TB) models can capture the main features of the electronic transport through nearly spherical fullerenes \\cite{Manousakis91, Mizorogi03}. The high conductivity of C$_{60}$ has lead to speculate on the possibility of considering it as a conducting spherical shell \\cite{Amusia06}.\nParticularly, several authors have studied the molecular junctions of the C$_{60}$ fullerene under different types of connections, such as, \na substrate and a STM tip \\cite{Paulsson99, Neel08}, one-dimensional leads \\cite{Saffarzadeh08}, carbon nanotubes \\cite{Shokri11}, gold clusters \\cite{Bilan12} or break junctions \\cite{Lortscher13}. \nFurthermore, the stability and strong hybridization of C$_{60}$ with metallic surfaces make it also a feasible anchoring group with high conductance \\cite{Bilan12}. \nIn the search of similar suitable molecular junctions, other larger icosahedral fullerenes C$_n$ from the same family $n=60 k^2$ ($k$ integer) have also been shown to be stable \\cite{Yu09, Dunlap06}, while the conductance of others, \nsuch as C$_{20}$ or its complexes have also been explored \\cite{Otani04, An09, Ji12}. On the other hand, by doping with boron and nitrogen, fullerene-based molecular junctions were found to have negative resistance \\cite{Yaghobi11}. \n\nA number of methods have been applied to the study of energetics and stability of buckyonions, namely, icosahedral fullerenes encapsulated by larger ones \\cite{Maiti93, Guerin97, Heggie97, Heggie98, Dodziuk00, Glukhova05, Baowan07, Enyashin07, Pudlak09, Xu08, Charkin13}.\nCarbon nano-onions are interesting structures between fullerenes and multi-wall carbon nanotubes, having high thermal stability and chemical reactivity compared to CNTs. They also have the characteristic high contact area and affinity for noble metals, what make them interesting as anchoring groups for molecular electronics. \nFor instance, it has also been shown that onion-like nanoparticles can be used as electrochemical capacitors, also called supercapacitors, with high discharge rates of up to three order of magnitud higher than conventional supercapacitors \\cite{Pech10}.\nThe static polarizability, closely related to the response of the electronic charges to applied static fields, have been studied for onions formed by members of the icosahedral ${60 k^2}$ family, using both phenomenological models and first-principle methods \\cite{Iglesias-Groth03, Zope08, Gueorguiev04}, showing their capability to partially screen static external electric fields. The conductance of onion-like structures functionalized with sulfide-terminated chains has been measured between a gold substrate and a gold STM tip \\cite{Sek13}.\n\nThe present work is aimed to study the electronic transmission of single-wall and multi-wall fullerenes weakly attached to metallic leads. \nIn the next Section we discuss the TB \nmodel and the influence of the curvature and finite size of the layers on it, \nas well as the Green function-based method for the calculation of the transmission.\nIn Section III, we present our results for the dependence of the transmission function on the electron energy $T(E)$. \nWe study how $T(E)$ is affected by the relative angular orientation of the shells, the number of intershell connections included in the TB \nmodel and the number of shells of the onions. Finally in Section IV, we summarize our conclusions on the systems studied.\n\\section{Model and calculation method}\n\\subsection{Single-wall fullerenes: curvature and finite size \\label{one shell model}}\nThe Hamiltonian of a $n$-atom fullerene $C_n$ is described in the TB\napproximation with one $\\pi$ orbital per site\n\\begin{equation}\nH_n = t_n \\sum_{\\langle ij\\rangle} c_i^\\dagger c_j + {\\rm H.c.},\n\\end{equation}\nwhere the summation runs on nearest neighbor atom pairs $\\langle ij\\rangle$, and the operator $c_i^\\dagger$ ($c_j$) creates (annihilates) an electron in the $\\pi$ orbital centered at the atom $i$ ($j$). The constant on-site energy has been taken as zero.\n\nA single parameter $t_n$ is used for the hopping integral between nearest neighbor atoms for a given fullerene.\nAlthough no bond dimerization effect is included, it has been shown that its sole effect is to slightly break some degeneracies in the spectrum with no major qualitative effect \\cite{Manousakis91}. \nIn order to take into account the effect of the curvature of the shell for the various fullerenes, we consider the\nhopping parameter $t_n$ to be a function of the mean radius $R_n$ of the (nearly) spherical shell of $n$ C atoms and the mean inter-atomic distance $d_n$ \\cite{Dresselhaus02}\n\\begin{equation}\nt_n=t \\left[1-\\frac{1}{2}\\left(\\frac{d_n}{R_n}\\right)^2\\right].\n\\end{equation}\nThe value $t =-2.73$ eV is a suitable hopping for graphene and is chosen to correctly reproduce the HOMO-LUMO gap for C$_{60}$, $E_g=-1.90$ eV, as obtained from DFT calculations \\cite{Saito91}. Although the TB \nHamiltonian only depends on the topology of the molecule (i.e., on the atoms bonded), the molecular geometry affects the hopping integral. \n\\begin{comment}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline \\hline \nSystem & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) \\\\ \\hline\nC$_{60}$ & 1.43 & 3.54 & 2.51 \\\\\nC$_{240}$ & 1.42 & 7.05 & 2.68 \\\\\nC$_{540}$ & 1.42 & 10.58 & 2.71 \\\\ \n \\hline\nC$_{20}$ & 1.42 & 1.99 & 2.04 \\\\\nC$_{180}$ & 1.42 & 6.10 & 2.66 \\\\\nC$_{500}$ & 1.42 & 10.18 & 2.70 \\\\ \n\\hline \\hline\n\\end{tabular}\n\\caption{\\label{d R t} Mean interatomic C-C distances, mean radii and hopping integral for the single-wall icosahedral fullerenes of the families $60k^2$ and $20k^2$ ($k$ integer).}\n\\end{center}\n\\end{table}\n\\end{comment}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccc|ccccc}\n\\hline \\hline \nC$_n$ & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) &&& C$_n$ & $d_n$ (\\AA) & $R_n$ (\\AA) & $t_n$ (eV) \\\\ \\hline\nC$_{60}$ & 1.43 & 3.54 & 2.51 &&& C$_{20}$ & 1.42 & 1.99 & 2.04 \\\\\nC$_{240}$ & 1.42 & 7.05 & 2.68 &&& C$_{180}$ & 1.42 & 6.10 & 2.66 \\\\\nC$_{540}$ & 1.42 & 10.58 & 2.71 &&& C$_{500}$ & 1.42 & 10.18 & 2.70 \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{\\label{d R t} Mean interatomic C-C distances, mean radii and hopping integral for the single-wall icosahedral fullerenes of the families $60k^2$ and $20k^2$ ($k$ integer).}\n\\end{center}\n\\end{table}\nTable \\ref{d R t} shows that the mean radius of the shell is the main geometrical variable, with the mean inter-atomic distance being approximately constant for the various fullerenes C$_n$ of the families $n=60k^2$ and $n=20k^2$ ($k$ integer). The inter-shell separations are approximately constant for successive fullerenes of each family ($\\sim3.5$\\AA, \\ close to the inter-layer separation in bulk graphite, for the former, and $\\sim 2$\\AA \\ for the latter).\n\\subsection{Intershell interactions in two-wall fullerenes \\label{two-shells}}\nWe shall study single-wall fullerenes and bilayer and trilayer onions composed by two and three concentric shells of icosahedral symmetry ($I_h$), C$_{n_1}$@C$_{n_2}\\ldots$. \nIn onions, it has been shown that the resulting structures for each shell are much the same those ones of the isolated fullerenes. Therefore, we assume the geometry for each shell in the onion to be the same as that of isolated C$_n$, taken from Yoshida data base. The most stable mutual orientation is the one that preserves the $I_h$ symmetry of the composite system \\cite{Heggie97, Heggie98}. \nFurthermore, in the onion-like family C$_{60k^2}$, it is reasonable to assume the strength of their intershell interactions as similar to those between the layers of graphite, due to the similarity in their interlayer separation.\nFigure \\ref{bilayer}a shows a scheme of a part of two adjacent parallel layers in graphite separated a distance of 3.35 \\AA. Each layer is constituted by two triangular lattices whose sites are denoted as A and B. The subscript 1 or 2 indicates which layer each site belongs. In graphite, the distances and number of nearest neighbors are exactly determined by the relative position between the plane infinite layers of identical geometry. \nIn two-wall buckyonions, nevertheless, the inner and outer fullerenes have finite sizes with different number of atoms, different geometrical structures due of their different number of hexagons (e.g., in the family 60$n^2$, every member have 12 pentagons) and eventually different orientations from each other (even though both have $I_h$ symmetry), as shown in Figures \\ref{bilayer}b and \\ref{bilayer}c. \nTherefore, although it is usually thought of intershell interactions in buckyonions as locally similar to interlayer interactions in bilayer graphene, a more thorough consideration of the intershell hopping is needed due to the differences mentioned above.\n\\begin{figure*}\n\\includegraphics[angle=-90,scale=0.5]{figure1.eps}\n\\caption{\\label{bilayer} (a) Scheme of a part of a two adjacent layers of graphite showing the interlayer connections between the lattices A and B from the upper and lower layers, (b)-(c) Schemes of two portions of the three layers of the onions considered in this work and their connections: (b) with the centers of pentagons aligned, and (c) with the centers of hexagons aligned. These and others configurations occurs in all the multiwall fullerenes considered here, making variable the number of nearest neighbors of the atoms of the shells. (d) Average number of connections $f$ in two-wall buckyonions per atom belonging to the inner sheel as a function of a maximum range for including neighbors $R_{\\rm cutoff}$, Eq. (\\ref{def f_links}), for C$_{60}$@C$_{240}$ with $I_h$ symmetry, C$_{240}$@C$_{540}$ with $I_h$ symmetry, C$_{60}$@C$_{240}$ without symmetry (symmetry $C_1$) and for two layers of graphite, for comparison purpose. In graphite only $f=0.5$ and $f=5$ are possible for the range of $R_{\\rm cutoff}$ shown, and the step-like variation of $f$ is due to the constrain imposed by the geometrical alignment between the two infinite layers. The three shadowed regions highlight ranges of $R_{\\rm cutoff}$ where $f$ has also a step-like behavior for the two-wall onions. (d) Scheme of the single-wall C$_{60}$, C$_{240}$ and C$_{540}$ fullerenes.}\n\\end{figure*}\nIn graphite the hopping parameter $\\gamma_0=t$ describes the covalent bonding arising from $sp^2$ hybridization in a layer.\nInter-layer Van der Waals interactions are described by the parameters $\\gamma_1 = t_\\perp\\approx 0.4$ eV (hopping energy between atoms A1 and A2), $\\gamma_3 \\approx 0.3$ eV (between B1 and B2 atoms) and $\\gamma_4 \\approx 0.04$ eV (hopping energy for A1-B2 and A2-B1 pairs) \\cite{CastroNeto09, Castro10}. \nIn bilayer onions, no such two lattices exists due to the finite size and the curvature of the layers. Nevertheless, similar Van de Waals interactions between nearest neighbor (NN) and next-nearest neighbors (NNN) atoms exist. Taking into account that for graphite $\\gamma_1\\approx \\gamma_3$ and $\\gamma_4\\approx 0$, we take a single value $t_\\perp=0.35$ eV for the hopping between pairs of the NN and NNN between layers \\cite{Pudlak09}. \nThe intershell interaction is included through the hopping integral $t_\\perp$ between pairs of NN and NNN atoms as follows: \nwe define a cutoff radius $R_{\\rm cut off}$ such that every pair of atoms, at ${\\bf r}_i$ and ${\\bf r}_j$, belonging to adjacent shells and separated a distance shorter than $R_{\\rm cut off}$ is assigned a hopping $t_\\perp$, i.e.,\n\\begin{equation}\nt_{ij} = \\left\\{ \n\\begin{array}{cc}\nt_\\perp, & |{\\bf r}_i-{\\bf r}_j|\\le R_{\\rm cut off} \\\\\n0 , & {\\rm otherwise}.\n\\end{array}\n\\right.\n\\label{def t_ij}\n\\end{equation}\nIt should be noted that, due to the faceting of the icosahedral symmetry, the number of intershell connections is not isotropic. \nAs an illustration, Figures \\ref{bilayer}b and \\ref{bilayer}c show two portions of a trilayer onion within a solid angle around the directions joining the centers of the pentagons and hexagons, respectively. The lines joining atoms in adjacent shells represent the intershell connections $t_{ij}=t_\\perp$ for a given cutoff radius. Due to the non sphericity of the shells, the outermost pentagon is not connected but the outermost hexagon it is.\nWe characterize the number of pairs of atoms connected at a given $R_{\\rm cutoff}$, by the mean number of neighbors (in the outer shell) `felt' by atoms in the inner shell, \n\\begin{equation}\nf(R_{\\rm cutoff})=N_{\\rm connect}\/N_{\\rm inner},\n\\label{def f_links}\n\\end{equation}\nwhere $N_{\\rm connect}$ is the number of intershell connections for a given $R_{\\rm cutoff}$, and $N_{\\rm inner}$ is the number of atoms in the inner shell.\nFigure \\ref{bilayer}d shows the variation of $f$ with $R_{\\rm cut off}$ for C$_{60}$@C$_{240}$ with $I_h$ symmetry, C$_{240}$@C$_{540}$ with $I_h$ symmetry, C$_{60}$@C$_{240}$ without symmetry (symmetry $C_1$) and for two adjacent layers of graphite. In the latter, there are wide ranges of $R_{\\rm cut off}$ where the number of connections keeps constant. Thus, due to the parallel orientation of the infinite layers, there are only some discrete values at which the number of connections increases due to the inclusion of neighbors farther to a given atom. For the onions with $I_h$ symmetry, \nthere are common ranges of $R_{\\rm cutoff}$ that show step-like behavior for both onions (see regions shadowed in Figure \\ref{bilayer}d). In the next section we present results of calculations with a number of intershell connections (characterized by $f$) chosen in those regions. Such a step-like dependence is in contrast to the approximately linear dependence for C$_{60}$@C$_{240}$ with symmetry $C_1$, when the symmetry axes of the $I_h$ fullerenes are misaligned. \nThe faceting of the fullerenes induced by the $I_h$ symmetry is particularly noticeable for the largest layers, as can be seen en Figure \\ref{bilayer}e for C$_{60}$, C$_{240}$ and C$_{540}$. Therefore, the fraction of carbon atoms having a NN or NNN within the range $|{\\bf r}_i-{\\bf r}_j|\\le R_{\\rm cut off}$ is smaller for the larger fullerenes. \nHence, the TB Hamiltonian for the onions becomes that of the isolated layers with an inter-layer interaction term\n\\begin{equation}\nH_{\\rm onion} = \\sum_n^{N} H_n + t_\\perp \\sum_{\\langle ij\\rangle} (c_i^\\dagger c_j + c_j^\\dagger c_i ),\n\\label{H_onion}\n\\end{equation}\nwhere site $i$ belongs to a inner shell, and site $j$ is its NN or NNN on the adjacent outer shell.\n\\subsection{Electronic transport}\nWhen a molecule is attached between two metallic leads and subject to a potential bias, the charge current flowing through it can be calculated with the Landauer equation \\cite{Landauer86}\n\\begin{equation}\nI=\\frac{2e}{h}\\int dE \\ T(E) \\left[f_L(E)-f_R(E) \\right],\n\\end{equation}\nwhere $f_L$ and $f_R$ are the Fermi distributions at the left (L) and right (R) leads. At low temperatures, the transmission function represents the dimensionless conductance (in units of the quantum $e^2\/2h$) and is calculated as\n\\begin{equation}\nT(E) = 4{\\rm Tr}({\\bf \\Gamma}^L {\\bf G}^r(E) {\\bf \\Gamma}^R {\\bf G}^a(E)),\n\\end{equation}\nwhere ${\\bf G}^a$ and ${\\bf G}^r$ are the matrix representation of the advanced and retarded Green functions ${\\bf G}^{r,a}=(E {\\bf 1}-{\\bf H}\\pm i0)^{-1}$, and ${\\bf \\Gamma}^L$ and ${\\bf \\Gamma}^R$ are the spectral densities of the leads \\cite{Cuevas10}.\n\nIn the wide band approximation, the Green function of the connected system can be obtained by using Dyson equation, as\n\\begin{equation}\nG_{1n}^r = \\frac{g_{1n}}{1-\\Gamma^2(g_{11}g_{nn}-|g_{1n}|^2)-i\\Gamma(g_{11}+g_{nn})}.\n\\label{connected G1n}\n\\end{equation}\nwhere $g_{ij}$ is the retarded Green function of the isolated system, and $\\Sigma_L=\\Sigma_R=i\\Gamma$ are the self-energies of the leads, considered to be energy independent. Throughout this work, we connect the fullerenes to the leads, using $\\Gamma=0.05$ eV, through two carbon atoms located diametrally opposite to each other, one atom in the vertex of a pentagon and its corresponding one obtained by applying the operation of inversion with respect to the center of symmetry.\n\\section{Results and discussion}\n\\subsection{Effect of the relative orientation between adjacent shells}\nThe approximately spherical form of the fullerenes, particularly the smaller ones, allows to treat onions as a family of concentric spherical shells for the calculation of some properties, such as the determination of radii of equilibrium, intershell distances, static polarizability or photoionization cross section \\cite{Xu96, Ruiz04, Kidun06, Dolmatov08}. For other applications, however, a more accurate description of their geometry is relevant. \nIn particular, the trend in larger fullerenes to approach faceted icosahedral forms makes very relevant the relative orientation of adjacent layers, even for concentric shells having individually $I_h$ symmetry, as already shown in Figure \\ref{bilayer}d for the average number of connections per atom $f$. \nThe influence of the relative orientation on the quantum transmission is shown in Figure \\ref{orientation} for two bilayer onions: C$_{60}$@C$_{240}$, and C$_{240}$@C$_{540}$. Two orientations were considered: one where the onion has overall $I_h$ symmetry, that is, with the symmetry axes of the individual shell aligned; and one where both shells are concentric but the individual symmetry axes are rotated an arbitrary relative angle from each other ($C_1$ symmetry).\n\\begin{figure*}\n\\includegraphics[scale=0.9]{figure2.eps} \n\\vspace{-0.0cm}\n\\caption{\\label{orientation} Transmission function for (a) C$_{60}$@C$_{240}$, and (b) C$_{240}$@C$_{540}$ buckyonions for the concentric shells rotated an arbitrary angle with respect to each other ($C_1$ symmetry), and with their symmetry axes aligned ($I_h$ symmetry).}\n\\end{figure*}\nThe differences in $T(E)$ between the icosahedral ($I_h$) and the non symmetrical ($C_1$) onion are visible in Figure \\ref{orientation}. In the onion with misalignment between the shells, some degeneracies in the energy spectrum are broken, as reflected in the occurrence of multiple resonant states with many peaks and antiresonances close to each other leading to a trend to the formation of narrow bands for the larger onion C$_{240}$@C$_{540}$. The transmission function of the non symmetrical onions have rapid variations from perfect to vanishing transmission with slight variations of the Fermi energy $E$, while $T(E)$ for the $I_h$ onions is a well behaved smooth function with a few peaks of perfect transmission in the range around the HOMO-LUMO gap. The gap itself increases when the shells become disoriented from each other what also corresponds to a less stable configuration, as previously reported \\cite{Heggie97}.\n\\subsection{Dependence on the number of intershell hopping connections \\label{connections}}\nAs mentioned in Section \\ref{two-shells}, the relative orientation of hexagons belonging to adjacent shells, the choice of the cutoff radius for defining NN and NNN sites of a given atom and the faceting of the larger fullerenes preclude a unique definition of the number of intershell connections to be included in the Hamiltonian (\\ref{H_onion}). Both the interlayer distance in graphite and intershell distance in the $60 k^2$ family are close to 3.5 \\AA. Thus, we took two cutoff radii close to this value and a larger one (see shadowed regions of Figure \\ref{bilayer}d), to include a bigger number of NN and NNN pairs in the intershell interaction Hamiltonian.\nIn the following we show the results of calculated $T(E)$ for icosahedral onions ($I_h$@$I_h$) and the aforementioned three choices of $R_{{\\rm cutoff}}$. In those three regions, $f=1$, 1 and 4 for C$_{60}$@C$_{240}$, and $f=0.5$, 1 and 2 for C$_{240}$@C$_{540}$. \nIn Figure \\ref{R_cutoff}, the transmission $T(E)$ is depicted for the three values of $f$, as indicated in the legends. It should be noticed that for the smaller onion C$_{60}$@C$_{240}$, the three $f$ values give almost the same curve (Fig. \\ref{R_cutoff}a). \nTherefore, this increase of connectivity does not affect the transmission throughout the composite system. Therefore, the main paths of transmission between shells are along the pairs formed by the atoms of the inner shell with its closest neighbor in the outer one.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure3.eps} \n\\vspace{-0.1cm}\n\\caption{\\label{R_cutoff} Transmission function $T(E)$ of the bilayer and trilayer onions (a) C$_{60}$@C$_{240}$, (b) C$_{240}$@C$_{540}$ and (c) C$_{60}$@C$_{240}$@C$_{540}$ for three different cutoff radii chosen in the shadowed regions of Figure \\ref{bilayer}. The resulting mean number of intershell connections $f$, Eq. (\\ref{def f_links}), are indicated in each curve.}\n\\end{figure*}\nFigure \\ref{R_cutoff}b shows that the dependence of the transmission on the number of intershell connections is more important for the larger onion C$_{240}$@$C_{540}$. \nFor larger fullerenes the deviation from the spherical shape becomes more noticeable as the number of atoms increases. Thus the C$_{240}$ shell is less spherical than C$_{60}$, and C$_{540}$ is clearly faceted. This departure from sphericity favours the hopping from the inner to the outer shell along certain directions, namely, those in which the icosahedral faces of both shells are closer to each other. Therefore, the larger the number of intershell links the better the quantum transmission. Interestingly, the increasing in the number of connections mainly affect the states above the highest occupied-lowest unoccupied (HOMO-LUMO) gap, noticeably the LUMO and LUMO+1 ones.\n\nFinally, Figure \\ref{R_cutoff}c shows $T(E)$ for the trilayered onion C$_{60}$@C$_{240}$@$C_{540}$ which is notoriously similar to the one of the bilayer onion of Fig \\ref{R_cutoff}b, thus showing that the two external shells are the most relevant for the transmission, with only small corrections coming from the innermost C$_{60}$ shell. This effect is discussed in greater detail in Section \\ref{shells}. \nThe bilayer onion have four peaks above the gap in range shown, namely, those corresponding to LUMO, LUMO+1, LUMO+2 and LUMO+3. The two central peaks (LUMO+1, LUMO+2) are almost degenerates in C$_{240}$@$C_{540}$ but become better resolved in the trilayered onion.\n\\subsection{Influence of the number of onion shells \\label{shells}}\nWe shall show here that the outermost shell greatly determines the most relevant features of the transmission spectra of bilayered and trilayered onions. The importance of the influence of the inner shells on $T(E)$ decreases inwards.\nIn Figure \\ref{R_cutoff}c we observed that the most noticeable effects when adding C$_{60}$ to the two external shells are the occurrence of an antiresonance between the LUMO and LUMO+1 peaks, and the widening of the narrow Fano-like profile after the LUMO+2 peak (at $E \\sim0.2$ eV.). Such a type of effects are analogous to the modifications to the transmission function through a chain introduced by adding a lateral site or chain to the system.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure4.eps} \n\\vspace{-0.0cm}\n\\caption{\\label{shell number} Transmission function for (a)C$_{60}$@C$_{240}$ and (b) C$_{240}$@C$_{540}$ compared to those for the single-wall fullerenes.}\n\\end{figure*}\nIn the following we fix the number of connections in trilayer onions by chosing $f=4$ ($f=2$) between the two outermost (innermost) shells.\nFigure \\ref{shell number} shows the transmission for the bilayer onions C$_{60}$@C$_{240}$ and C$_{240}$@C$_{540}$ as compared to those for the corresponding single-wall fullerenes. It can be seen that for energies below the gap, $T(E)$ for the onions is very similar to the one of the outer fullerene, i.e., the one for C$_{240}$ in Figure \\ref{shell number}(a), and for C$_{540}$ in Figure \\ref{shell number}(b). \nThe peaks above the gap preserves similar features both in C$_{60}$@C$_{240}$ and C$_{240}$, although with a relative energy shift of the peaks.\n\\begin{figure*}\n\\includegraphics[scale=0.90]{figure5.eps}\n\\vspace{-0.0cm}\n\\caption{\\label{family 20} transmission function for (a)C$_{20}$@C$_{180}$ and C$_{180}$@C$_{500}$, and (b) C$_{20}$@C$_{180}$@C$_{500}$ compared to those for the single-wall fullerenes.}\n\\end{figure*}\nIn Figure \\ref{shell number}b similar considerations can be made about the comparison the transmission through C$_{240}$@C$_{540}$ and C$_{540}$; the latter provides most of the features observed in the former, particularly for energies below the gap. The intershell connections in C$_{240}$@C$_{540}$ eventually contributes to the occurrence of antiresonances, such as that at $\\sim-1.4$ eV, not present in $T(E)$ for C$_{540}$.\nIn other cases, it softens the vanishing of transmission, such as in the antiresonance of C$_{540}$ at $\\sim-1.1$ eV which becomes in a finite transmission for C$_{240}$@C$_{540}$. \nRoughly speaking, introducing C$_{60}$ as a third innermost shell does not greatly modify the transmission of the bilayered onion.\nThe influence of the external shell on the transmission can be interestingly shown in the onions formed from fullerenes belonging to the family $20 k^2$, which also have icosahedral symmetry. Thus, C$_n$ fullerenes with $n=20$, 80, 180, 320, $500\\ldots$ are predicted to be stable, with a radii increasing by approximately 2 \\AA \\ from each member from the family to the next one. The radii of equilibrium for the family $60 k^2$ were shown to be accurately determined by a continuous spherically symmetric Lennard-Jones model \\cite{Baowan07}. Application of the same model for the multiwall onions of the $20 k^2$ family (with $k$ odd integer), results an intershell distance of about 4 \\AA, not far from the intershell distance for the $60 k^2$ family or the interlayer distance in graphite. Therefore, we show calculations for the single-wall, two-wall and three-wall fullerenes obtained from C$_{20}$, C$_{180}$ and C$_{500}$.\nOur TB\ncalculations show that C$_{20}$ and C$_{500}$ are gapless, while C$_{180}$ presents a gap of $\\approx 1.5$ eV. In Figures \\ref{family 20}a and \\ref {family 20}b, C$_{180}$ is the outer and inner shell respectively. As a consequence, in Figure \\ref{family 20}a, C$_{20}$@C$_{180}$ shows a region of vanishing transmission within the gap, while it keeps finite for C$_{180}$@C$_{500}$, except for the already discussed narrow antiresonances arising from the intershell connections. Hence, the transmission function is strongly sensitive to the electronic structure of the external shell. As a final example of this property, Figure \\ref{family 20}b, shows the transmission for the trilayered C$_{20}$@C$_{180}$@ C$_{500}$ as compared to those of the single-wall fullerenes. It is seen that the most of the features of the onion are reproduced by the transmission through C$_{500}$, with some Fano-like antiresonances originated in the transmission spectrum of the inner C$_{180}$ and, to less extent, in the innermost C$_{20}$.\nIt can bee seen that the three-wall onion (Figure Figure \\ref{family 20}b) is well described by the two-wall one (Figure \\ref{family 20}a). Interestingly, the Fano-like resonance of C$_{20}$@C$_{180}$@ C$_{500}$ at $E\\approx 0.1$ eV is not present in C$_{180}$@ C$_{500}$ but it is in C$_{20}$@C$_{180}$; therefore, such a peak of conductance is an effect from the coupling of the two inner sheells.\n\\section{Conclusions}\nIn this work we have studied theoretically the quantum transmission through single-wall fullerenes and bilayered and trilayered onions of icosahedral symmetry, when attached to metallic leads, by using a TB\nHamiltonian and Green functions methods. Although the Van der Waals interactions between onion shells are supposed to be similar to those between graphite layers, the finite size of the fullerenes, their curvature and relative orientations need some analysis for including the intershell hopping parameter. We include in the model the effect of finite size and curvature through a parametrization of the hopping integral as a function of the number of atoms of the shell. The number of connections from a given atom to others belonging to adjacent shells was studied by introducing a cutoff radius for the interaction. \nWe found that misalignment of the symmetry axes produces breaking of the level degeneracies of the individual shells, giving rise some narrow quasi-continuum bands instead of the localized discrete peaks of the individual fullerenes. Most of the features of the transmission through the onions are already visible in the transmission function of the single-wall fullerene forming the outer shell. The main modifications between them are antiresonances arising from the coupling between the outer layer with the next innermost one. For three-wall onions, the transmission becomes barely sensitive to the most internal shell. Interestingly, when the fullerene of the external shell is gapless, the transmission of the onion does not vanish along finite ranges of energy. This property could be useful for designing multilayered fullerenes with tailored conductance by properly growing the outermost layers.\n\\section*{Acknowledgments}\nWe acknowledge financial support for this project from CONICET (PIP 11220090100654\/2010) and SGCyT(UNNE) through grant PI F007\/11.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nTwisted 2D van der Waals materials have emerged as an elegant platform to engineer and study correlated quantum phases with unprecedented experimental control \\cite{Andrei2020, Novoselov2016, Cao2018, Balents2020, Wu2018, Regan2020}. At certain \\textit{magic} angles, the electronic structure is deformed into exceedingly narrow bands in a moir\\'e Brillouin zone. The small bandwidth yields pronounced correlation effects and generally makes multiple neighboring quantum phases experimentally accessible in a single fabricated device through application of electrostatic gating \\cite{Cao2018}. \n\nRecently, a related paradigm of twisted bilayer structures was introduced that does not rely on engineering of correlated flat bands, but can produce interesting new phases by combining known non-trivial properties of constituent monolayers \\cite{Can2021,volkov2021}. It was shown that two cuprate monolayers, stacked at a twist angle $\\theta$, can give rise to a spontaneously time-reversal symmetry broken state by virtue of simple electron tunneling between the layers. Most notably, in a finite range of angles around the critical twist $\\theta_c=45^\\circ$, the ground state of an otherwise nodal $d$-wave superconductor becomes fully gapped and acquires a finite Chern number. At exactly $45^\\circ$ the gapped phase persists up to the native critical temperature of the cuprate monolayer, thus furnishing the first known example of a high-$T_c$ topological superconductor. \n\nPioneering experimental work on very thin twisted Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ (BSCCO) flakes succeeded in fabricating bilayers at various twist angles \\cite{Zhao2021}. Measurements of the interlayer Josephson current, Fraunhofer interference patterns and half-integer Shapiro steps in samples close to the $45^\\circ$ are suggestive of a $\\mathcal{T}$-broken phase \\cite{Tummuru2022}. Strong twist angle dependence of the critical current has been reported elsewhere \\cite{Lee2021}. \n\nIt was later noted by Song, Zhang and Vishwanath \\cite{Song2021} that twist angle and momentum dependence of the interlayer tunneling matrix element $g_{\\bm{k}}$, arising from the symmetry of the copper active orbitals, can play an important role in the emergence of the $\\mathcal{T}$-broken phase. As originally argued by Xiang and Wheatley \\cite{Xiang_1996}, the matrix element has the form\n\\begin{equation}\\label{gk}\n g_{\\bm{k}}=g_0 \\cos{2\\theta}+g_1\\mu_{\\bm{k}}(\\theta\/2)\\mu_{\\bm{k}}(-\\theta\/2),\n\\end{equation}\nwhich we generalized here to a twisted bilayer geometry following \\cite{Song2021}. The $g_0$ term represents the direct tunneling between copper $d_{x^2-y^2}$ orbitals while the $g_1$ term describes the `oxygen-assisted' tunneling process with form factor $\\mu_{\\bm{k}}(\\theta)=[\\cos{(R_\\theta k_x)}-\\cos{(R_\\theta k_y)}]\/2$ and $R_\\theta$ the rotation matrix. Crucially, the part of $g_{\\bm{k}}$ that remains non-vanishing at $\\theta=45^\\circ$ contains the form factor $\\mu_{\\bm{k}}$ that suppresses tunneling at the nodes of the $d$-wave order parameter. While spontaneous $\\mathcal{T}$-breaking is still found to occur in this case, the ground state remains gapless and hence does not support the gapped topological phase with non-zero Chern number predicted in Ref.\\,\\cite{Can2021}.\n\nIn the present work we consider the twisted bilayer problem within a family of incoherent tunneling models \\cite{Graf1993,Radtke1995,Radtke1996,Radtke_1997,Turlakov2001} in which the transfer of electrons between two adjacent CuO$_2$ layers is mediated by impurities that are inherently present in the otherwise inert `spacer' layers. Such incoherent tunneling models have been shown to yield better agreement with experimentally measured $c$-axis transport properties of nominally clean single-crystal cuprates than models where momentum is strictly conserved \\cite{Ginsberg1994, Sheehy_2004}. Because random impurities break all spatial symmetries of the system, the form of the interlayer coupling is required to respect the crystal symmetry constraints only on average. One may thus expect that incoherent tunneling models will evade the difficulties noted above and produce a fully gapped topological phase near $\\theta=45^\\circ$. \n\nBased on a perturbative diagrammatic treatment within a simplified continuum model we show that the incoherent tunneling model indeed delivers the same phenomenology as the coherent model of Ref.\\,\\cite{Can2021} while respecting with the point group symmetries of the physical system. Importantly, we show that for sufficiently slowly varying disorder the ground state near $\\theta=45^\\circ$ is gapped in the $\\mathcal{T}$-breaking region and topologically non-trivial. These results are then confirmed in a more realistic setting through a full numerical diagonalization of a lattice model with parameters chosen to reproduce the actual cuprate band structure in the vicinity of the Fermi level. The effect of interface inhomogeneity on Josephson effects in twisted bilayers was recently considered in Ref.\\,\\cite{volkov2021} where it was found that sufficiently strong disorder can leave the system in a topologically trivial state around 45$^\\circ$. This is consistent with our deductions.\n\nThe paper is structured as follows. After summarizing the origin of the $\\mathcal{T}$-broken phase in the language of group theory (Sec.~\\ref{sec:group theory}), in Sec.~\\ref{sec:model} we introduce a model of incoherent interlayer tunneling within a continuum framework and show that a fully-gapped $\\mathcal{T}$-broken phase emerges in the vicinity of $\\theta=45^\\circ$. The phase is topological as evidenced by chiral edge modes traversing the gap that appear in the disorder-averaged boundary spectral function. In Sec.~\\ref{sec:latticemodel}, we supplement our continuum results with a lattice model that simultaneously captures the characteristic cuprate Fermi surface geometry, the moir\\'e effects and disorder in interlayer coupling. Concluding remarks appear in Sec.~\\ref{sec:conclusions}.\n\n\n\n\\section{Group theoretical discussion of $\\mathcal{T}$-breaking in twisted cuprates}\n\\label{sec:group theory}\n\nThe phenomenology of $\\mathcal{T}$-breaking in twisted cuprates can be captured by a two-component Landau-Ginzburg theory with complex order parameters $\\Psi_1$ and $\\Psi_2$ with a relative phase $\\varphi$. The $\\varphi$-dependent part of free energy is of the general form \n\\begin{align}\n \\mathcal{F}(\\varphi) = -B \\, |\\Psi_1\\Psi_2| \\cos \\varphi + C \\, |\\Psi_1 \\Psi_2|^2 \\cos 2\\varphi.\n \\label{eq:LG}\n\\end{align}\nTime reversal symmetry will be spontaneously broken whenever the free energy develops two minima that are related by $\\mathcal{T}: \\varphi \\rightarrow -\\varphi$. The Josephson coupling term, proportional to $\\cos \\varphi$, has only a single minimum at $\\varphi=0$ or $\\pi$, depending on the sign of $B$.\nPresence the fourth order term proportional to $C\\cos 2\\varphi$ is therefore necessary to break $\\mathcal{T}$. Additionally, one must have $C>0$, since otherwise the two minima of $\\cos 2\\varphi$ occur at $\\varphi=0,\\pi$ which map to themselves under $\\mathcal{T}$. In Ref.\\,\\cite{Can2021} it was argued that $C$ is indeed positive based on microscopic mean-field calculations. We confirm that $C$ remains positive in the case of incoherent interlayer coupling in Sec.~\\ref{sec:freeenergy}.\n\nGiven $C>0$, the fourth order term is minimized at $\\varphi=\\pm\\pi\/2$. Then, $\\mathcal{T}$-breaking occurs as a consequence of the competition among the two terms in Eq.~\\eqref{eq:LG}. Specifically, $\\mathcal{T}$ will be broken when \n\\begin{align}\n 4 C \\left| \\Psi_1 \\Psi_2 \\right| > \\left| B\\right| \\,.\n \\label{eq:trsbreaking}\n\\end{align}\nA special situation clearly arises if symmetry requires $B$ to vanish; then Eq.~\\eqref{eq:trsbreaking} is guaranteed to be satisfied for any $C>0$. \n\nNext we describe a set of symmetry requirements under which the coefficient $B$ vanishes and the system is forced into the $\\mathcal{T}$-broken phase. The order parameters $\\Psi_1,\\Psi_2$ transform according to irreducible representations (irreps) of the point group of the crystal. Two cases must be distinguished: (a) $\\Psi_1$ and $\\Psi_2$ transform under two \\textit{different} 1D irreps or (b) $(\\Psi_1,\\Psi_2)$ transform under a 2D irrep \\cite{Annett1990,Poniatowski2022}. The latter case is considered a generic pathway to $\\mathcal{T}$-breaking that occurs immediately upon entering the SC phase. The former case generically yields two successive phase transitions with distinct critical temperatures, $T_{c}$ and $T_c'$, with $\\mathcal{T}$-breaking setting at the lower one $T_c'$. Note that $T_c'$ can be zero or negative, in which case the $\\mathcal{T}$-broken phase is physically not accessible \\cite{Annett1990, Kaba.2019}.\n\nThe point symmetries of an untwisted cuprate bilayer form the point group $D_{4h}$. Here, the $d_{x^2-y^2}$ and $d_{xy}$ order parameters transform according to the 1D irreps $B_{1g}$ and $B_{2g}$, respectively. At arbitrary twist angle, inversion and mirror symmetries are broken and the point group reduces to $D_4$ with $d$-wave irreps $B_1$ and $B_2$. Thus, given the $d$-wave nature of the order parameter in cuprates, only pathway (b) to $\\mathcal{T}$-breaking is possible and no definite symmetry-based arguments can be made.\n\nPrecisely at ${45^{\\circ}}$, however, the symmetry group is enlarged to the non-crystallographic point group $D_{4d}$ which contains an additional $8$-fold improper rotation $S_8$ of the quasicrystalline lattice. Most notably, among the irreps of $D_{4d}$ \\textit{only} the 2D $E_2$-irrep supports $d$-wave order. Thus, the Josephson coupling term $ -B \\, |\\Psi_1\\Psi_2| \\cos \\varphi$ must necessarily be absent at $45^\\circ$. This is so because it descends from the $-B(\\Psi_1\\Psi_2^\\ast +{\\rm c.c.})$ term in the free energy which is not invariant under $S_8: (\\Psi_1,\\Psi_2)\\rightarrow (\\Psi_2,-\\Psi_1)$. Thus, $\\mathcal{T}$-breaking can be viewed as a fundamental consequence of the point group at $\\theta= 45^\\circ$. We summarize our key arguments as follows: Two-component order parameters that transform under a 2D irrep naturally break $\\mathcal{T}$. At $45^\\circ$ twist angle, because the point group of the bilayer is $D_{4d}$, any $d$-wave order parameter must necessarily transform under a 2D irrep. Therefore, the superconducting state breaks $\\mathcal{T}$ right below $T_c$. \n\nThe phase diagram of twisted bilayer cuprates derived in Ref.~\\cite{Can2021} can then be understood from continuity arguments. It is expected that the $\\mathcal{T}$-breaking phase will not be limited to the exact $45^\\circ$-twist, but will extend to a range of twist angles in its vicinity. Since at twists slightly away from $45^\\circ$ the order parameters transform under two 1D-irreps, two distinct critical temperatures are permitted and $\\mathcal{T}$-breaking will no longer coincide with the critical temperature $T_c$ of the spontaneous $U(1)$-symmetry breaking. This naturally leads to the wedge-shaped $\\mathcal{T}$-broken domain in the phase diagram explicitly computed in Ref.~\\cite{Tummuru2022}. Our symmetry arguments will be manifest in a microscopic description of the bilayer system. \n\n\n\n\n\\section{Incoherent tunneling}\n\\label{sec:model}\n\n\\subsection{Background and model definition}\n\nExperimental measurements of the $c$-axis transport in bulk crystals of BSCCO and other cuprates, summarized for example in Ref.\\,\\cite{Ginsberg1994}, have been interpreted as evidence of interlayer tunneling dominated by disorder-mediated, incoherent processes. The $c$-axis superfluid stiffness, accessible through the measurement of the $c$-axis London penetration depth \\cite{Hosseini1998, Panagopoulos2003}, provides a particularly clear evidence. Experimentally, the temperature dependence of the $c$-axis superfluid stiffness in clean single crystals was observed to follow an approximate power-law behavior $\\rho_c = a-b T^\\alpha$ with $\\alpha\\simeq2$ at low temperatures, whereas the in-plane stiffness showed a $T$-linear dependence \\cite{Hardy1993}. The latter is the canonical behavior expected of a clean $d$-wave superconductor, reflecting the presence of low-energy excitations with a Dirac spectrum \\cite{Hirschfeld1993}. Models with coherent tunneling between CuO$_2$ predict the same linear $T$-dependence for the $c$-axis stiffness \\cite{Klemm1995}, in clear disagreement with experimental data. If the interlayer tunneling were dominated by the oxygen-assisted processes (the $g_1$ term in Eq.\\ \\eqref{gk}) then theory predicts $\\rho_c = a-b T^5$ \\cite{Xiang_1996}, again at variance with experiment. \n \nAs demonstrated in Refs.\\ \\cite{Radtke1995,Radtke1996,Radtke_1997,Sheehy_2004} a description that captures the correct $\\sim T^2$ scaling along the $c$-axis (while preserving the $T$-linear behavior in the $ab$ plane) can be given using the incoherent $c$-axis tunneling approach. In the following we shall review the relevant model and then apply it to the problem of a twisted bilayer.\n\nA minimal model of the uncoupled bilayer system consists of the second-quantized Hamiltonian\n\\begin{align}\n\t\\mathcal{H}_0 = \\sum_{\\mathbf{k}l}\\Psi_{\\mathbf{k}l}^\\dagger\n\tH_{\\mathbf{k}l}\n\t\\Psi_{\\mathbf{k}l}\n\\end{align}\nwhere $l=1,2$ denotes the layer index, Nambu-Gorkov spinors $\\Psi_{\\mathbf{k}l}=(c_{\\mathbf{k}l\\uparrow },\\, c_{-\\mathbf{k}l\\downarrow })^T$. In the BCS approximation, we have\n\\begin{align}\n\tH_{\\mathbf{k}l}\n\t=\n\t\\xi_{\\mathbf{k}} \\sigma_z \n\t+ \\Delta_{\\mathbf{k}l}' \\sigma_x\n\t- \\Delta_{\\mathbf{k}l}'' \\sigma_y,\n\\end{align}\nwith Pauli matrices $\\sigma_j$ acting in the Nambu space and $\\Delta_{\\mathbf{k}l}'$, $\\Delta_{\\mathbf{k}l}''$ denoting real and imaginary parts of the superconducting gap function. We adopt units such that $\\hbar=e=k_B=m_e=a_0=1$, where mass is measured in units of electron mass $m_e$ and length scales in units of lattice constant $a_0$. To make the model tractable we assume a simple parabolic band dispersion given by $\\xi_{\\mathbf{k}}=\\mathbf{k}^2\/2m-\\mu$ in each layer. (In Sec.\\ \\ref{sec:latticemodel} we consider a more realistic band structure and show that it leads to similar results.) The two superconducting $d$-wave order parameters are\n\\begin{align}\n\t\\Delta_{\\mathbf{k}1} &= \\Delta e^{i\\varphi\/2} \\cos(2\\alpha_\\mathbf{k} -\n\t\\theta) \n\t\\nonumber\n\t\\\\\n\t\\Delta_{\\mathbf{k}2} &= \\Delta e^{-i\\varphi\/2} \\cos(2\\alpha_\\mathbf{k} +\n\t\\theta)\n\t\\label{eq:op}\n\t\\,,\n\\end{align}\nwhere $\\varphi$ is the phase difference and $\\alpha_\\mathbf{k}$ denotes the polar angle of $\\mathbf{k}$. \n\nThe layers are coupled by the term\n\\begin{align}\\label{tun}\n\t\\mathcal{H}' = \\sum_{\\mathbf{kq}} \\gamma_{\\mathbf{q}} c_{\\mathbf{k},1}^\\dagger\n\tc_{\\mathbf{k-q},2} + \\textrm{h.c.}\n\\end{align}\nand $\\mathcal{H}=\\mathcal{H}_0 + \\mathcal{H}'$ constitutes the full model. The lack of momentum conservation in Eq.\\ \\eqref{tun} is the defining feature of the incoherent tunneling models and originates, physically, from the disorder present in the spacer layers separating the copper-oxygen planes. The disorder is captured via a set of Gaussian-distributed random variables $\\gamma_\\mathbf{q}$ of average $\\overline{\\gamma_\\mathbf{q}}=0$ and variance given by\n\\begin{align}\n\t\\overline{\\gamma_\\mathbf{q}^* \\gamma_\\mathbf{q+p}} &=\n\t\\frac{1}{N}\\frac{4\\pi g^2}{3\\Lambda^2}\n\t\\delta_{\\mathbf{p},0}e^{-\\mathbf{q}^2\/\\Lambda^2} \\,.\n \\label{eq:incotunnterm}\n\\end{align}\nThe scale $\\Lambda$ defines the characteristic momentum change that an electron undergoes when tunneling between the two layers. The factor $1\/3$ is chosen to reproduce the phase diagram of the coherent model of Ref.~\\cite{Can2021} in the limit $\\Lambda\\rightarrow 0$ for the same value of $g$. For simplicity we have neglected any $\\theta$-dependence of the interlayer coupling although we expect the randomness to be stronger in twisted samples due to the increase in interface roughness, added strain, and moir\\'e lattice modulations. \n\nThe above form of incoherent interlayer tunneling is consistent with all lattice symmetries because $\\gamma_\\mathbf{q}$ vanishes on average. This constitutes the key difference to a coherent coupling of the form $\\sum_{\\mathbf{k}}(g\\, c_{\\mathbf{k},1}^\\dagger c_{\\mathbf{k},2} + \\textrm{h.c.})$. As was pointed out in Ref.~\\cite{Song2021}, at $45^\\circ$ twist the two participating Cu $d$-orbitals transform under a 2D representation of $D_{4d}$ and a coherent tunneling term is therefore not invariant under $S_8: (c_{\\mathbf{k},1},c_{\\mathbf{k},2})\\rightarrow (c_{\\mathbf{k},2}, -c_{\\mathbf{k},1})$. It thus vanishes by virtue of the same argument as the Josephson coupling in Eq.~\\eqref{eq:trsbreaking}. This is indeed owed to the coincidence of the atomic Cu orbitals transforming under the same representation as the superconducting order parameters.\n\nIt is instructive to consider the limit $\\Lambda \\rightarrow 0$ of the incoherent tunneling in Eq.~\\ref{eq:incotunnterm}. Here, one has $\\overline{\\gamma_\\mathbf{q}^* \\gamma_\\mathbf{q+p}} = g^2\\delta_{\\mathbf{q},0}\\delta_{\\mathbf{p},0}\/3$ and momentum is conserved in the interlayer tunneling process. Yet, $\\Lambda \\rightarrow 0$ is not the clean limit in the sense that, in real space, it corresponds to the case where macroscopic regions are correlated with the same random value of interlayer tunneling $g\/\\sqrt{3}$. From the viewpoint of disorder-induced incoherence, the random values of $g$ should only be correlated in the vicinity of an impurity which sets the appropriate scale for $1\/\\Lambda$. In the discussion below, the $\\Lambda\\rightarrow 0$ thus serves as an abstract but convenient reference point that connects the present model to the calculations in the original work \\cite{Can2021}. We will refer to it as the \\textit{coherent} limit.\n\n\n\n\\subsection{Free energy and phase diagram}\n\\label{sec:freeenergy}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{diagram-figure}\n\t\\caption{Diagrammatic expansion of the interlayer current (a) at order $g^2$ and (b-c) at order $g^4$. Full lines correspond to electronic propagators $G$ and dashed lines correspond to impurity vertices paired by disorder average. The open circle denotes the current vertex $j_\\mathbf{q}$ defined in the main text and impurity vertices $\\gamma_\\mathbf{q}$ are given by black dots.}\n\t\\label{fig:diagrams}\n\\end{figure}\n\nThe physics of $\\mathcal{T}$-breaking is captured by the $\\varphi$-dependence of the free energy. To determine the free energy we begin by implementing the global gauge transformation \n$\n\\left( c_{\\mathbf{k}1},\\,c_{\\mathbf{k}2} \\right)\n\\rightarrow\n\\left( c_{\\mathbf{k}1}e^{i\\varphi\/4},\\,c_{\\mathbf{k}2}e^{-i\\varphi\/4} \\right)\n$\nwhich moves the superconducting phase difference from the order parameters in Eq.~\\eqref{eq:op} to $\\mathcal{H}'$ according to\n\\begin{align}\n\t\\mathcal{H}' \\rightarrow \n\t\\mathcal{H}' =\n\t\\sum_{\\mathbf{kq}} \\gamma_{\\mathbf{q}} e^{i\\varphi\/2}c_{\\mathbf{k},1}^\\dagger\n\tc_{\\mathbf{k-q},2} + \\textrm{h.c.} \\,.\n\\end{align}\nIn this gauge, the disorder-averaged interlayer current is given by\n\\begin{align}\n\tJ &= \\sum_{\\mathbf{kq}} i e^{i\\varphi\/2}\\overline{\\gamma_\\mathbf{q}\n\t\\langle c_{\\mathbf{k},1}^\\dagger c_{\\mathbf{k-q},2} \\rangle} + \\textrm{h.c.} \n\t\\nonumber\n\t\\\\\n\t&= \\text{Tr}\n\t\\left[ \\overline{\n\t\tj_\\mathbf{q} G(\\mathbf{k},\\mathbf{k-q},\\omega_n)}\n\t\\right]\\,,\n\t\\label{eq:current}\n\\end{align}\nwhere $G(\\mathbf{k},\\mathbf{k'},\\tau)=\\langle T_\\tau c_{\\mathbf{k}}(\\tau) c_{\\mathbf{k'}}^\\dagger(0) \\rangle$ is the full imaginary time ordered Green's function of the disordered system and the current vertex is\n\\begin{align}\n\tj_\\mathbf{q} =i\\gamma_\\mathbf{q} \n\t\\begin{pmatrix}\n\t\t0 & e^{i\\sigma_z\\varphi \/2}\\\\\n\t\t-e^{-i\\sigma_z\\varphi \/2} & 0\n\t\\end{pmatrix} \\,.\n\\end{align}\nNote that the trace is to be performed over all momenta $\\mathbf{k},\\mathbf{q}$ and Matsubara frequencies $\\omega_n=(2n+1)\\pi\/\\beta$, in addition to interlayer and Nambu indices.\n\nFrom the Josephson relation $J(\\varphi)=2\\partial \\mathcal{F}(\\varphi)\/\\partial \\varphi$ one then obtains the functional dependence of the free energy on the interlayer phase difference $\\varphi$ by simple integration. We expand Eq.~\\eqref{eq:current} up to fourth order in $g$ while treating $\\mathcal{H}'$ as a perturbation. Three different terms arise, which are diagrammatically represented in Fig.~\\ref{fig:diagrams}. Panel (a) corresponds to the term \n\\begin{align}\n J_c^{(2)} = \\text{Tr}\\left[\\overline{j_{\\mathbf{q}} G_0(\\mathbf{k-q},\\omega_n)\n \tu_{-\\mathbf{q}} G_0(\\mathbf{k},\\omega_n)}\n \\right] \\,\n \\label{eq:j1}\n\\end{align}\nwhich is quadratic in $g$. Here, $G_0(\\mathbf{k},\\omega_n)=\\left( i\\omega_n - H_{\\mathbf{k}} \\right)^{-1}$ is the unperturbed, translationally invariant Green's function with $H_{\\mathbf{k}}={\\rm diag}(H_{\\mathbf{k}1},H_{\\mathbf{k}2})$ and\n\\begin{align}\n\tu_\\mathbf{q} =\\gamma_\\mathbf{q} \n\t\\begin{pmatrix}\n\t\t0 & \\sigma_ze^{i\\sigma_z\\varphi \/2}\\\\\n\t\t\\sigma_z e^{-i\\sigma_z\\varphi \/2} & 0\n\t\\end{pmatrix} \\,.\n\\end{align}\nis the impurity vertex. The disorder average acts on $\\gamma_\\mathbf{q}$ factors and is performed according to Eq.\\ \\eqref{eq:incotunnterm}.\nThe diagrams in Fig.~\\ref{fig:diagrams}(b-c) represent terms of order $g^{4}$:\n\\begin{equation}\n\\begin{aligned}\n\tJ_c^{(4)} &= \n\t2\\, \\text{Tr} \\left[ j_{\\mathbf{q}} G_0(\\mathbf{k},\\omega_n)\n\t\tu_{\\mathbf{q}'} G_0(\\mathbf{k-q'},\\omega_n)\n\t\tu_{-\\mathbf{q}'}\n\t\t\\right.\n\t\t\\nonumber\n\t\t\\\\\n\t\t& \\quad\\quad\n\t\t\\times\n\t\t\\left.\n\t\tG_0(\\mathbf{k},\\omega_n)\n\t\tu_{-\\mathbf{q}}G_0(\\mathbf{k+q},\\omega_n)\n\t\\right] \\\\\n\t&\\quad +\n\t\\text{Tr}\\left[ \n\t j_{\\mathbf{q}} \n\t G_0(\\mathbf{k},\\omega_n)\n\t\tu_{\\mathbf{q}'} \n\t\tG_0(\\mathbf{k-q'},\\omega_n)\n\t\tu_{-\\mathbf{q}}\n\t\t\\right.\n\t\t\\nonumber\n\t\t\\\\\n\t\t& \\quad\\quad\n\t\t\\times\n\t\t\\left.\n\t\tG_0(\\mathbf{k-q'+q},\\omega_n)\n\t\tu_{-\\mathbf{q}'}G_0(\\mathbf{k+q},\\omega_n)\n\t\\right]\n\\end{aligned}\n\\end{equation}\nwhere impurity averaging is assumed but not explicitly shown for clarity of notation. Evaluating the traces, we obtain the current of the form \n\\begin{align}\n\tJ = J_c^{(2)} + J_c^{(4)} = J_{c1}(\\theta) \\sin \\varphi - J_{c2}(\\theta) \\sin 2\\varphi \\,,\n\t\\label{eq:jc}\n\\end{align}\nwith coefficients, to lowest order of $g$,\n\\begin{align}\n J_{c1}&=\n\t4\n\t\\sum_{n\\mathbf{k}} j_{n\\mathbf{k}}\n\n\n \n \n \n \n \n \n \n \\label{eq:jc1}\n \\\\\n\tJ_{c2}&=\n\t8\n\t\\sum_{n\\mathbf{k}}\n\tj_{n\\mathbf{k}}^2\n\t+\n\t4\\sum_{n\\mathbf{kqq'}} \n\t\\left|\\gamma_\\mathbf{q} \\right|^2\n \\left|\\gamma_\\mathbf{q'} \\right|^2\n \\label{eq:jc2}\n \\\\\n &\n \\times\n f_{n\\mathbf{k},1}f_{n\\mathbf{k+q+q'},1}\n f_{n\\mathbf{k+q},2}f_{n\\mathbf{k+q'},2}\n\t\\nonumber\n\\end{align}\nHere, we have defined\n\\begin{align}\n\\label{eq:convolution}\n j_{n\\mathbf{k}} &= \\left(f_{n\\mathbf{k},1} * \\left|\\gamma_{n\\mathbf{k}} \\right|^2 \\right)\n f_{n\\mathbf{k},2}\n \\\\\n f_{n\\mathbf{k},l} &= \\frac{\\Delta_{\\mathbf{k},i}}{\\omega_n^2+E_{\\mathbf{k},l}^2}\n \\label{eqn:f-function}\n\\end{align}\nand $(*)$ denotes a convolution integral, $a_{\\mathbf{k}}*b_{\\mathbf{k}} = \\sum_{\\mathbf{q}}a_{\\mathbf{q}}b_{\\mathbf{k-q}}$. The quasiparticle dispersion of the unperturbed bands is given by $E_{kl} = \\sqrt{\\xi_{\\mathbf{k}}^2+\\Delta_{\\mathbf{k}l}^2}$. \n\n\nFrom Eq.~\\eqref{eq:jc} one obtains the free energy\n\\begin{align}\n 2\\mathcal{F}=-J_{c1}(\\theta) \\cos \\varphi + \\frac{J_{c2}(\\theta)}{2} \\cos 2\\varphi +\\textrm{const}\\,.\n\\end{align}\nThe $\\mathcal{T}$-breaking phase transition occurs as a consequence of competition between $\\cos \\varphi$ and $\\cos 2 \\varphi$ terms. Clearly, $J_{c2}>0$ and the ground state acquires a finite phase difference for \n\\begin{align}\n 2J_{c2}>|J_{c1}|,\n \\label{eq:crit}\n\\end{align}\nwhere it spontaneously breaks $\\mathcal{T}$. From our discussion in Sec.~\\ref{sec:group theory} it follows that that $J_{c1}$ must vanish at twist of $\\theta=\\pi\/4$. Explicitly, one can see this result as follows. The functions $E_{\\mathbf{k},l}, \\, \\left|\\gamma_\\mathbf{k} \\right|^2$ transform under the $A_{1g}$ irrep of $D_{4h}$ whereas the $f_{\\mathbf{k},1},f_{\\mathbf{k},2}$ transform under $B_{1g}$ and $B_{2g}$, respectively. We note that convolution with the $A_{1g}$-symmetric impurity distribution $\\left|\\gamma_\\mathbf{k} \\right|^2$ does not change the symmetry of the convolution integral. Hence, $j_{\\mathbf{k}}$ transforms under $B_{1g}\\otimes B_{2g}=A_{2g}$ and all terms in Eq.~\\eqref{eq:jc1} average to zero at ${45^{\\circ}}$-twist. However, $j_{\\mathbf{k}}^2$ is $A_{1g}$-symmetric and $J_{c2}$ will consequently be finite and positive. Thus it is clear that condition \\eqref{eq:crit} is generally satisfied at $\\theta=45^\\circ$ and $\\mathcal{T}$ will always be broken as soon as the system enters the SC state below $T_c$.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{phase-diagram.pdf}\n\t\\caption{Phase diagram of incoherently coupled twisted bilayer cuprates. For a given $\\Lambda$, the inside of the cone-shaped region breaks $\\mathcal{T}$. Black-dashed lines mark the phase boundary in the clean limit, previously introduced in \\cite{Can2021}. For increasing degree of momentum non-conservation $\\Lambda$, the $\\mathcal{T}$-breaking phase boundaries shrink towards $45^\\circ$.}\n\t\\label{fig:phasediagram}\n\\end{figure}\n\nWe conclude that impurity-mediated tunneling must not qualitatively change the $\\mathcal{T}$-breaking phase diagram relative to the model of Ref.\\ \\cite{Can2021}. The incoherent tunneling, however, shifts the phase boundaries. As shown in Appendix \\ref{apdx:lambda-scaling}, $J_{c1}\\sim 1\/\\Lambda$ and $J_{c2}\\sim 1\/\\Lambda^2$. Since $J_{c1}$ is only weakly dependent on $\\theta$, and $J_{c1}$ vanishes linearly around $45^\\circ$ twist, it follows from Eq.~\\eqref{eq:crit} that the width of the $\\mathcal{T}$-breaking phase space is proportional to $J_{c2}(\\theta=0)\/J_{c1}(\\theta=0) \\sim 1\/\\Lambda$. In the perfectly incoherent limit, $\\Lambda\\rightarrow\\infty$, the free energy becomes independent of $\\varphi$ and the $\\mathcal{T}$-breaking phase disappears.\n\nTo quantitatively ascertain the effect of incoherent tunneling on the phase diagram, we numerically evaluate the coefficients $J_{ci}$. In principle, all Matsubara sums can be evaluated analytically, at the cost of removing the simple convolution structure in Eq.~(\\ref{eq:convolution}). This leaves three remaining momentum integrals to be numerically evaluated at complexity $\\mathcal{O}(N^3)$ where $N$ is the number of $\\mathbf{k}$-points of the 2D mesh used to perform the integrals. A more efficient approach is to exploit the convolution structure of Eq.~(\\ref{eq:convolution}) using the fast Fourier transform (fft) algorithm and numerically evaluate $M$ Matsubara frequencies, affording evaluation of diagrams Fig.~\\ref{fig:diagrams}(a-b) at order $\\mathcal{O}(MN\\log N)$. The crossed diagram Fig.\\ \\ref{fig:diagrams}(c) does not possess a convolution structure. As we show in Appendix \\ref{apdx:complexity}, it can be evaluated at a cost of $\\mathcal{O}(MN^2)$.\n\nThe resulting phase diagram is shown in Fig.~\\ref{fig:phasediagram} for coupling strength $g=\\SI{10.5}{\\milli\\electronvolt}$ and several values of $\\Lambda$. We see that the $\\mathcal{T}$-breaking phase space is largest in the `clean' limit $\\Lambda\\rightarrow 0$ where it extends between $(45\\pm 6)^\\circ$ at $T=0$. Increasing $\\Lambda$ gradually reduces the extent of the $\\mathcal{T}$-broken phase which eventually vanishes in the perfectly incoherent limit when $\\Lambda \\sim k_F$, i.e. when impurity correlations are on the scale of the lattice constant. Physically, this occurs because at this level of incoherence the Cooper pair essentially looses all memory of its momentum structure in the process of tunneling between layers.\n\n\n\n\\subsection{Spectral gap and topological superconductivity}\n\nHaving discerned the fate of the $\\mathcal{T}$-breaking phase in the presence of impurity-mediated tunneling we proceed to examine the topological properties of the resulting ground state. In the clean limit, $\\mathcal{T}$-breaking establishes a topological phase with Chern number $\\mathcal{C}=4$ \\cite{Can2021}. Since the disordered model is connected to the clean case by taking the limit $\\Lambda \\rightarrow 0$, it is reasonable to expect the same $\\mathcal{C}=4$ phase as long as the quasiparticle gap does not close.\n\nHere, we show that these expectations are indeed met. To this end, we evaluate the Green's function\n\\begin{align}\n G(\\mathbf{k},\\omega_n) = [G_0-\\Sigma(\\mathbf{k},\\omega_n)]^{-1}\n \\label{eq:gfd}\n\\end{align}\nin the Born approximation where\n\\begin{align}\n \\Sigma_{\\tau\\tau'} &=\n \\sum_{\\mathbf{q}} u_{\\mathbf{q}} \\, G_0(\\mathbf{k-q},\\omega_n)\\, u_{\\mathbf{-q}}\n \\\\\n &=-\\delta_{\\tau,\\tau'}\\,f_{\\mathbf{k},\\bar{\\tau}} (i\\omega_n + \\xi_{\\mathbf{k}} \\sigma_z + e^{i\\tau\\sigma_z \\varphi}\\Delta_{\\mathbf{k},\\bar{\\tau}} \\sigma_x ) \n * \\left|\\gamma_{\\mathbf{k}}\\right|^2\n \\nonumber \\,.\n \\end{align}\n with layer-indices $\\tau=\\pm 1$. Here, we regularized the continuum model on a square lattice using \n \\begin{align}\n \\xi_\\mathbf{k}&=-2t (\\cos k_x +\\cos k_y) - \\mu\n \\nonumber\n \\\\\n \\Delta_{\\mathbf{k},\\tau}&= \\Delta [(\\cos k_x - \\cos k_y) \\cos \\theta \n \\label{eq:alignedgap}\n \\\\\n & \\qquad\\quad\n + \\,\\tau \\sin k_x \\sin k_y \\, e^{-i\\varphi}\\sin \\theta ]\n \\nonumber\n \\end{align}\n with parameters chosen to match the continuum model.\n \nFollowing the method introduced in Ref.~\\cite{Pinon.2020}, we compute a spatially resolved Green function\n\\begin{align}\n G_B(x, k_y,\\omega_n) &= G(x, k_y) - G(x, k_y) \n T(k_y)\n G(-x, k_y) \n \\nonumber\n \\\\\n T(k_y) &= \\left[ \n \\frac{1}{\\sqrt{N}} \\sum_{k_x} G(k_x, k_y)\n \\right]^{-1} \n \\label{eq:transfm}\n\\end{align}\nin the presence of a strong repulsive potential at $x=0$ which simulates an edge and thus allows us to inspect the edge modes of the disordered system. We outline the method and give a derivation of Eq.~\\eqref{eq:transfm} in Appendix~\\ref{apdx:surfacegf}. \n\nIn Fig.~\\ref{fig:edgemode} we plot the analytically continued boundary spectral function\n\\begin{align}\n A_B(x,k_y,\\omega) = -\\frac{1}{\\pi} \\text{Im} \\left[G_B(x,k_y,-i\\omega+\\eta)\\right]\n\\end{align}\nat the edge $(x=1)$ as well as the bulk spectral function. We clearly observe two chiral edge modes traversing the bulk gap thus confirming the non-trivial topology of the system. The edge modes display a degeneracy in the layer degree of freedom, suggesting that the model is in a topological phase with Chern number $\\mathcal{C}=4$. The bulk gap is reduced but remains finite as $\\Lambda$ increases.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{spectrum.pdf}\n\t\\caption{Bulk (a-b) and boundary (c-d) spectrum for incoherently coupled cuprate bilayers with $\\Lambda=0$ (left) and $\\Lambda\/k_F=0.08$ (right) at $45^\\circ$ twist angle. The spectrum shows chiral edge modes traversing the bulk gap which is reduced but finite for increased $\\Lambda$. Edge modes, which are in fact degenerate, indicate a Chern number $\\mathcal{C}=4$. }\n\t\\label{fig:edgemode}\n\\end{figure}\n\n\n\n\\section{Lattice model}\n\\label{sec:latticemodel}\n\n\\begin{figure*}[t]\n \\includegraphics[width=16cm]{latt_summ.pdf}\n \\caption{(a) Illustration of the geometry of the bilayer lattice model at an incommensurate angle of $\\sim 43^\\circ$. (b) Disorder averaged free energy of the bilayer at zero temperature as a function of the phase difference. The minima $\\varphi_{\\rm min}$ are situated away zero at small disorder strengths. (c) Dependence of the order parameter amplitude and phase as a function of temperature for $\\tilde{\\Lambda}=0.2$. One could view it as a vertical cut at a specific twist in the phase diagram of Fig.~\\ref{fig:phasediagram}, with the onset of a non-zero phase marking the phase boundary.}\n \\label{fig:latt_F}\n\\end{figure*}\n\nSo far we have looked at the role of disorder in a continuum formulation of a twisted bilayer. The two-site unit cell of the regularized model allowed for analytical expressions for the layer Green's function to which we have systematically added incoherent interlayer tunneling and calculated the free energy up to fourth order in $g$. Another approach to tackle the problem and corroborate the results in a more general setting is to perform a BCS mean field theory on two twisted square lattices that represent the two CuO planes. In this case one can incorporate more realistic band structures but it is also harder to obtain the Green's functions analytically using Feynman diagrams. The reason is twofold: For one, at an arbitrary twist the lattice model is not commensurate. Secondly, at commensurate twist angles close to $45^\\circ$, the moir\\'{e} unit cell contains many sites. To get around this, we perform a brute force disorder average wherein several disorder realizations with the same microscopic parameters are taken into account. While it limits us to real space, such a treatment is exact because all orders in perturbation theory are implicitly accounted for.\n\nFor each layer, we consider a square lattice Hubbard model with nearest neighbor density-density interactions such that a mean-field decoupling produces a $d$-wave order parameter. Including the interlayer tunneling processes with amplitudes $g_{ij}$, the bilayer is described by\n\\begin{eqnarray}\n &{\\cal H} = & -t \\sum_{\\langle ij \\rangle \\sigma l} c^\\dag_{i \\sigma l} c_{j \\sigma l} - t'\\sum_{\\langle \\langle ij \\rangle \\rangle \\sigma l} c^\\dag_{i \\sigma l} c_{j \\sigma l} \n - \\mu \\sum_{i \\sigma l} n_{i \\sigma l} \\nonumber \\\\\n &+&\\sum_{\\langle ij \\rangle l}\\left(\\Delta_{ij, l}c^\\dag_{i\\uparrow\n l}c^\\dag_{j\\downarrow l}+{\\rm h.c.}\\right) \n - \\sum_{i j \\sigma} g_{ij} c^\\dag_{i \\sigma 1} c_{j \\sigma 2},\n \\label{eq:hm_latt}\n\\end{eqnarray}\nwhere $l$ is a layer index, $t$ ($t'$) is the (next-)nearest-neighbor hopping amplitude, $\\mu$ is the chemical potential that controls on-site particle density $n_{i \\sigma l}$ and $\\Delta_{ij,l}$ denotes the complex order parameter on the bond connecting sites $i$ and $j$ on layer $l$. Considering a fully coherent interlayer tunneling, Ref.~\\cite{Can2021} employs a circularly symmetric, exponentially decaying form $g_{ij} = e^{-(r_{ij}-c)\/\\rho}$ which connects sites $i$ and $j$ separated by $\\bm{r}_{ij}$. Therein $c$ in the interlayer separation and $\\rho$ is defined by the radial extent of the participating orbitals. The twist angle $\\theta$ between the layers determines connectivity and the strength of the interlayer tunnelings. The free energy of this model shows a double-well structure for twist angles around $45^\\circ$ \\cite{Can2021}. \n\nTo incorporate incoherent processes, we introduce a random tunneling factor that vanishes on average, but encodes the correlation between different processes depending on spatial separation. That is, \n\\begin{equation}\n g_{ij} = g_{\\bm{R}} ~ e^{-(r_{ij}-c)\/\\rho} \n \\label{eq:g_inco}\n\\end{equation}\nwhere $\\bm{R} = (\\bm{r}_{i} + \\bm{r}_{j})\/2$ denotes the center of mass location of the hopping and \n\\begin{align}\n \\overline{g_{\\bm{R}}} &= 0, \\nonumber \\\\\n \\overline{g_{\\bm{R}}g_{\\bm{R}'}} &= g^2 \\exp\\left[-\\frac{\\tilde{\\Lambda}^2}{4} (\\bm{R}-\\bm{R}')^2 \\right].\n\\end{align}\nAnalogous to the parameter $\\Lambda$ in the continuum model, $\\tilde{\\Lambda}$ sets the length scale for the correlation between different tunneling amplitudes and is indicative of disorder strength. We distinguish the two simply because of the slightly differing definitions. To simulate the Fermi surface of optimally doped BSCCO with hole pocket around $(\\pi,\\pi)$, we set $t=153$meV, $t' = -0.45t$ and $\\mu = -1.35t$ \\cite{Bille2001}. Further, we choose $c=2.2$ and $\\rho=0.4$ (in units of the lattice constant) to set interlayer distances. The $d$-wave order parameters in cuprates originates in the CuO planes and the interlayer coupling is a minor perturbation that does not influence the order parameter magnitude. In other words, temperature dependence of the gap in each layer is independent of twist and coupling strength strength $g$, which we peg at 20meV. Therefore, we use a $\\Delta$ calculated self-consistently in a monolayer, which has a maximum of $\\sim 40$meV at 0K in accordance with experimental findings in cuprates \\cite{Dama2003, Fischer2007}.\n\nTo look for ${\\cal T}$-breaking we examine the free energy of the system, which can be calculated from the eigenvalues $E_i$ of the BdG Hamiltonian \\eqref{eq:hm_latt}:\n\\begin{equation}\n {\\cal F}_{\\rm BdG}= \\sum_i E_i - 2 k_B T \\sum_{i}\\ln\\left[2\\cosh{(E_{i}\/2 k_B T)}\\right].\n\\end{equation}\nIn particular, for a given twist $\\theta$ and disorder parameter $\\tilde\\Lambda$, we draw from the distribution \\eqref{eq:g_inco} and average the free energy over 50 independent realizations. We choose a square bilayer sample as shown in Fig.~\\ref{fig:latt_F}(a), but the results are independent of the shape. Further, the exact number of sites in the system depends on the cut and the twist angle, but the free energy does not show an appreciable change beyond $\\sim 900$ sites per layer. In agreement with the continuum model, Fig.~\\ref{fig:latt_F}(b) shows that the presence of ${\\cal T}$-breaking free energy minima is controlled by $\\tilde\\Lambda$. Namely, small values of $\\tilde\\Lambda$ support the ${\\cal T}$-broken ground state while larger values do not. \n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nTwisted bilayers of high-$T_c$ cuprates hold the potential for realizing topological superconductivity, wherein a topological gap is spontaneously induced. As per the symmetry informed momentum space form factors, which determine the electron hopping between interlayer Cu atoms, the tunneling amplitude vanishes along the nodal directions and a spectral gap may not appear. In this work we highlight that an important aspect to consider in such an analysis is the disorder mediated tunneling. Not only does disorder appear naturally due to oxygen doping and interfacial defects, but incorporating momentum non-conservation has been shown to better represent experimental data in clean single crystals.\n\nUsing perturbative diagrammatic calculations and disorder averaging on the lattice, we find that an experimentally motivated incoherent tunneling model that respects all point group symmetries of the physical system gives rise to a qualitatively similar phase diagram as obtained in Ref.\\,\\cite{Can2021}. Specifically we find a substantial range of twist angles around $45^\\circ$ and temperatures where spontaneous ${\\cal T}$-breaking occurs and produces a fully gapped topological phase with non-zero Chern number. The angular extent of the ${\\cal T}$-broken phase depends on the disorder length scale $\\Lambda^{-1}$ where the coherent limit $\\Lambda\\to 0$ recovers the phase diagram of Ref.\\ \\cite{Can2021} and increasing $\\Lambda$ corresponds to a shrinking extent of the topological phase. Only when the incoherence length scale is comparable to the Fermi momentum, the twist angle for spontaneous ${\\cal T}$ breaking is reduced to exactly ${45^{\\circ}}$. \n\nFrom an experimental point of view, the inhomogeneity due to oxygen doping the BiO planes of untwisted BSCCO was found to be correlated over $\\approx 14 \\si{\\angstrom}$ \\cite{Pan2001}. Since the CuO plane lattice constant is $\\approx 5 \\si{\\angstrom}$, that amounts to a correlations over 3 unit cells, i.e., $\\tilde{\\Lambda} \\approx 0.3$. In a twisted geometry, one may expect the characteristic length scale to decrease and, hence, the estimate for $\\tilde{\\Lambda}$ could shift up. That said, the role of complex atomic arrangements, moir\\'e length scales and strong correlations are difficult to incorporate into such a heuristic reasoning. One would probably have to await data from complementary experimental probes, such as transport and optical response, to discern the nature of the superconducting state around ${45^{\\circ}}$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{jc.pdf}\n\t\\caption{Critical current $J_c$ of the twisted bilayer as a function as interlayer coherence scale $\\Lambda$. Incoherence significantly reduces $J_c$. The color scale denotes temperature $T$ in panel (a) and twist angle $\\theta$ in (b).}\n\t\\label{fig:jc}\n\\end{figure}\n\nIt was noted in Ref.\\,\\cite{Song2021} that the measured critical current density $J_c$ in both twisted and untwisted Bi2212 is about factor of 500 smaller than the theory prediction based on the slave-boson mean field theory of a $t$-$J$ model used in that study. We checked that a similar discrepancy occurs in the calculation using BCS mean field theory of Ref.\\,\\cite{Can2021}. As indicated in Fig.\\,\\ref{fig:jc} the discrepancy is somewhat reduced in the incoherent tunneling model (by about one order of magnitude at large $\\Lambda$) but nevertheless significant disagreement with experiment persists. As noted in Ref.\\,\\cite{Sheehy_2004} this is a known problem that affects superconductors in the cuprate family and becomes increasingly severe in the underdoped part of their phase diagram. A phenomenological fix can be implemented \\cite{Sheehy_2004} by restricting the momentum sums in the expression for $J_c$ to patches of linear size $\\sim x$ (the hole doping) around the nodal points of the $d$-wave order parameter. This modification leaves the temperature dependence of $\\rho_{ab}(T)$ and $\\rho_{c}(T)$ unchanged, but reduces their $T=0$ magnitude to experimentally observed values. It similarly fixes the problem with $J_c$. As with many aspects of cuprates a truly microscopic understanding of this phenomenon remains a challenge to the theory community. With regards to twisted cuprate bilayers it would be interesting to explore the effect of the phenomenological fix outlined above on the phase diagram but we leave this to future work. \n\n\n\\section*{Acknowledgments}\n\nWe are grateful to S. Egan, O. Can, X. Cui, \\'E. Lantagne-Hurtubise, X.-Y. Song, A. Vishwanath, P. Volkov and S.Y.F. Zhao for helpful discussions and correspondence. This research was supported in part by NSERC and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. We thank the Max Planck-UBC-UTokyo Center for Quantum Materials for fruitful collaborations and financial support. R.H. acknowledges the Joint-PhD program of the University of British Columbia and the University of Stuttgart.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\nAt the LHC, top quark pairs ($t\\bar{t}$) will be produced mainly via gluon fusion \\mbox{($\\sim$ 90\\%)}. A recent prediction for the cross section at $\\sqrt{s}=14$ TeV is a next-to-leading order (NLO) calculation with soft-gluon next-to-leading-log (NLL) resummation \\cite{xsecNLONLL}: $\\sigma_{t\\bar{t}}^{NLO+NLL}=908^{+82}_{-85}$ (factorisation and renormalisation scales) $^{+30}_{-29}$ (parton distribution function (PDF)) pb. As $V_{tb} \\approx 1$, top quarks nearly always decay into a $W$-boson and a $b$-quark. A $W$-boson decays roughly 2\/3$^{rd}$ of the cases into two quarks and 1\/3$^{rd}$ into a lepton and a neutrino. Therefore, $t\\bar{t}$ decay can be divided into three channels: 4\/9$^{th}$ fully hadronic, 4\/9$^{th}$ semileptonic and 1\/9$^{th}$ dileptonic. The fully hadronic channel suffers from large QCD multi-jet background. The presence of missing transverse energy (from undetectable neutrino's) and at least one lepton in the two other channels creates the possibility to select $t\\bar{t}$ events while reducing background considerably. These two channels are used in ATLAS for $t\\bar{t}$ cross section measurements with the first 100pb$^{-1}$ integrated luminosity of data \\citep{ATLAS-CSC} and will be discussed in the following sections.\n\n\n\\section{SINGLE LEPTON CHANNEL}\nTwo complementary measurements in the single lepton channel are investigated: a counting method and a likelihood fit. Events are selected that passed the electron (muon) trigger {\\texttt{e22i}} ({\\texttt{mu20}}) and contain missing transverse energy $\\not\\negthickspace{E}_{T} > 20$ GeV, an isolated electron (muon) with transverse momentum $p_{T} > 20$ GeV\/c and pseudo rapidity $\\lvert \\eta \\rvert < 2.5$, three jets with $p_{T} > 40$ GeV\/c and $\\lvert \\eta \\rvert < 2.5$ and an additional fourth jet with $p_{T} > 20$ GeV\/c and $\\lvert \\eta \\rvert < 2.5$. The hadronic top is reconstructed by taking the invariant mass $M_{jjj}$ of the three jet combination with the highest $p_{T}$. To reduce background from jet combinatorics and other processes, at least one di-jet combination is required to be compatible with the $W$-boson mass $\\lvert M_{jj}-M_{W} \\rvert < 10$ GeV\/c$^{2}$. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{TopMassPlot_muon_Figure2_100pb.eps}\n\\hspace{.1\\textwidth}\n\\includegraphics[width=.3\\textwidth]{CSC_MassPlot_Mwcut_muon_Fit_100pb.eps}\n\\caption{(a) Expected invariant three jet mass distribution of correctly reconstructed $t\\bar{t}$ events (white), incorrectly reconstructed events due to jet combinatorics (dark shaded) and background processes (light shaded). The histogram is normalised to 100pb$^{-1}$ using the total available statistics (1 fb$^{-1}$). (b) Fit of a Gaussian and Chebychev polynomial to a Monte Carlo generated pseudo-experiment corresponding to 100pb$^{-1}$ of data.} \n\\label{fig:top_mass_plot}\n\\end{figure*}\n\n\\begin{samepage}\nIn the counting method the cross section is determined by subtracting the number of expected background events from the number of observed events in the $M_{jjj}$ distribution divided by the selection efficiency and integrated luminosity. The main background comes from $W$+ jets and single top. When tagging of $b$-quark jets is possible, the purity can be increased by a factor four, while the efficiency is only reduced by a factor two. The largest systematic uncertainty ($\\sim$10\\%) in this analysis is the background normalisation.\n\\end{samepage}\n\nIn the likelihood fit, only the correctly reconstructed $t\\bar{t}$ events that end up in the peak of the distribution are used for the cross section measurement. The peak is fitted with a Gaussian, while the combined background shape from jet combinatorics and other processes such as $W$+ jets is described by a Chebychev polynomial. The cross section is calculated from the number of $t\\bar{t}$ events in the peak divided by the overall efficiency for a $t\\bar{t}$ event to end up in the peak and the integrated luminosity. The fit is sensitive to the shapes of the distribution and therefore this is the main systematic uncertainty ($\\sim$10\\%). The expected uncertainties on the cross section for 100 pb$^{-1}$ of data are:\n\\begin{center}\n\\begin{tabular}{l r r r r}\n Likelihood method & 7 (stat) & $\\pm 15$ (syst) & $\\pm 3$ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Counting method & 3 (stat) & $\\pm 16$ (syst) & $\\pm 3$ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n\\end{tabular}\n\\end{center}\n\n\n\\section{DIFFERENTIAL CROSS SECTION}\nThe measurement of the differential cross section as function of the invariant mass $M_{t\\bar{t}}$ of the $t\\bar{t}$ system provides an important check of the Standard Model. Deviations from the $t\\bar{t}$ continuum indicate the presence of new physics, for example new heavy resonances decaying into a $t\\bar{t}$ pair. Semileptonic events are selected using the same criteria as mentioned in the previous section, i.e. without $b$-tagging. The $t\\bar{t}$ pair is reconstructed by adding up the vectors of the four highest $p_{T}$ jets, the lepton and $\\not\\negthickspace{E}_{T}$. By using a $W$- and top mass constraint in a least square fit, assigning jets to the (anti-)top, a better result can be obtained than with the default event reconstruction only combining the four jets with the lepton and $\\not\\negthickspace{E}_{T}$ using a $W$-mass constraint on the leptonic side. The expected mass resolution ranges from 5\\% to 9\\% between 200 and 850 GeV\/c$^{2}$. \n\nAlso, the double differential cross section for $t\\bar{t}$ is sensitive to possible new physics. In this measurement $b$-tagging is used in addition to the default single lepton $t\\bar{t}$ selection criteria to improve purity up to 45\\%. The hadronic top is reconstructed by finding the highest $p_{T}$ combination of a $b$-tagged jet with a di-jet combination $\\lvert M_{jj} - M_{W}\\rvert < 20$ GeV\/c$^{2}$ close to it. The region which is covered by this method is 50 GeV\/c $< p_{T} <$ 280 GeV\/c and rapidity $\\lvert y \\rvert < 2$. The statistical error on the distribution will be around 30\\% for 100 pb$^{-1}$ (10\\% for 1 fb$^{-1}$). The main systematic uncertainty comes from the jet energy scale and initial\/final state radiation ($\\sim$15\\%).\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{ditopmasscomp_linscale_simple.eps} \n\\hspace{.1\\textwidth}\n\\includegraphics[width=.3\\textwidth]{AllHadTopPtYReco_1fb.eps}\n\\caption{(a) Normalised di-top mass distribution for the default (dotted line) and the improved (dashed line) reconstruction and (b) reconstructed $p_{T}$ and $y$ distribution of hadronically decaying top quarks, normalised to 1 fb$^{-1}$.} \n\\label{fig:differential_xsec}\n\\end{figure*}\n\n\n\\section{DILEPTON CHANNEL}\nFor the cross section determination in the dilepton channel three measurements are considered. The three measurements start with the preselection of events with two high-$p_{T}$ opposite signed leptons ($e^{+}e^{-}$, $\\mu^{+}\\mu^{-}$ and $e^{\\pm}\\mu^{\\mp}$) and then have different additional requirements. These events are efficiently selected by a combination of single and dilepton triggers. The uncertainty is expected to be small due to the trigger {\\texttt{OR}} condition between the channels and the high statistics available for efficiency measurements such as the 'tag-and-probe' method using events with $Z$-bosons.\n\nIn the cut and count method the cross section is determined by comparing the number of observed events with the number of expected background events (from Monte Carlo). The optimum selection criteria are determined to be: two isolated opposite signed leptons with $p_{T}>20$ GeV\/c, a veto on events with $M_{\\ell^{+}\\ell^{-}}$ around $Z$ peak (85-95 GeV\/c$^{2}$), two jets of at least 20 GeV\/c and $\\not\\negthickspace{E}_{T}>30$ GeV. With a S\/B of 4.3 the main background processes remaining are $Z\\rightarrow \\tau^{+} \\tau^{-}$ and semi-leptonic $t\\bar{t}$ where one of the jets fakes a lepton. The jet energy scale introduces the largest systematic uncertainty.\n\nThe inclusive template method is based on the observation that the three dominant sources of isolated leptons which can be selected in the $e\\mu$ channel are $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$. These three processes can be separated by looking at the 2-D plane spanned by $\\not\\negthickspace{E}_{T}$ and $N_{jet}$. After scanning different configurations of background templates, the template with the highest probability is selected. The fit has free parameters including the $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$ cross section. Contamination from processes with a fake lepton is reduced by using tight isolation criteria for the electron and a veto on events with $\\not\\negthickspace{E}_{T}$ aligned along the reconstructed muon. The acceptance of the two leptons and the shapes of the 2-D templates determine the systematic uncertainties.\n\nFor the third method a log-likelihood function is used to extract the parameters $N_{sig}$ and $N_{bkg}$ given the fixed total number of events $N_{tot}$. The signal and background input functions are determined by fitting Chebychev polynomials to Monte Carlo generated distributions. The sum of the semi-leptonic $t\\bar{t}$, $W^{+}W^{-}$ and $Z\\rightarrow\\tau^{+}\\tau^{-}$ is considered as a single background distribution. The two variables for the distributions are $\\Delta\\varphi$ between: (i) the highest $p_{T}$ lepton and $\\not\\negthickspace{E}_{T}$ and (ii) the highest $p_{T}$ jet and $\\not\\negthickspace{E}_{T}$. Like for the cut and count method, the jet energy scale is the dominant systematic uncertainty. Expected uncertainties on the cross section for 100 pb$^{-1}$ of data are:\n\\begin{center}\n\\begin{tabular}{l r r r r}\n Cut and Count method & 4 (stat) & $^{+5}_{-2}$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Template method & 4 (stat) & $\\pm 4$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminosity) $\\%$ \\\\\n Likelihood method & 5 (stat) & $^{+8}_{-5}$ (syst) & $\\pm 2 $ (PDF) $\\pm 5$ (luminostiy) $\\%$ \\\\\n\\end{tabular} \n\\end{center}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.3\\textwidth]{zpeak_resized.eps} \n\\includegraphics[width=.3\\textwidth]{tt_nj_met.eps}\n\\includegraphics[width=.3\\textwidth]{L0_MET_edited_resized.eps}\n\\caption{(a) Invariant di-lepton mass distribution in the cut \\& count method, (b) $t\\bar{t}$ template of the $N_{jet}$ vs $\\not\\negthickspace{E}_{T}$ distribution for the inclusive template fit and (c) a likelihood fit to the signal (dotted blue line) and background (dotted red line).} \n\\label{fig:dilepton_channel}\n\\end{figure*}\n\n\n\\begin{acknowledgments}\nThe author has been supported by the Netherlands Organisation of Scientific Research (NWO) under VIDI research grant 680.47.218.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\nKirchberg proved in 1994 that tensorial absorption of $\\mathcal{O}_\\infty$ and pure infiniteness were equivalent for simple separable nuclear C$^*$-algebras. This theorem can be reinterpreted as the equivalence of $\\mathcal{Z}$-stability and strict comparison for simple separable nuclear traceless C$^*$-algebras (see \\cite{Ro}), an equivalence that can be considered in stably finite algebras, too. This and the results of \\cite{TW} led to the following conjecture:\n\n\\begin{conjs}[T-Winter, 2008]\\label{conjecture}\nLet $A$ be a simple unital nuclear separable C$^*$-algebra. The following are equivalent:\n\\begin{enumerate}\n\\item[(i)] $A$ has finite nuclear dimension;\n\\item[(ii)] $A$ is $\\mathcal{Z}$-stable;\n\\item[(iii)] $A$ has strict comparison of positive elements.\n\\end{enumerate}\n\\end{conjs}\n\n\\noindent\nIt is expected that these conjecturally equivalent conditions will characterize those algebras which are determined up to isomorphism by their Elliott invariants. In the absence of even the weakest condition, (iii), one cannot classify AH algebras using only the Elliott invariant (\\cite{T3}). While it is possible that this can be corrected with an enlarged invariant, the jury is still out. Combining the main result of \\cite{W} with that of \\cite{R} yields (i) $\\Rightarrow$ (ii), while R\\o rdam proves (ii) $\\Rightarrow$ (iii) in \\cite{Ro}. This note yields the following result.\n\\begin{thms}\\label{main}\nConjecture \\ref{conjecture} holds for AH algebras.\n\\end{thms}\n\\noindent\nWe proceed by proving (ii) implies (iii) for AH algebras (Corollary \\ref{maincor}) and appealing to \\cite[Corollary 6.7]{W}. It should be noted that our contribution is quite modest: we simply verify the hypotheses of the main result of \\cite{W}. \n\n\n\\section{Strict comparison and almost divisibility}\n\nLet $A$ be a unital C$^*$-algebra, and $\\mathrm{T}(A)$ its simplex of tracial states. Let $\\mathrm{Aff}(\\mathrm{T}(A))$ denote the continuous $\\mathbb{R}$-valued affine functions on $\\mathrm{T}(A)$, and let $\\mathrm{lsc}(\\mathrm{T}(A))$ denote the set of bounded lower semicontinuous strictly positive affine functions on $\\mathrm{T}(A)$. Set $\\mathrm{M}_{\\infty}(A) = \\cup_n \\mathrm{M}_n(A)$. For positive \n$a,b \\in \\mathrm{M}_{\\infty}(A)$, we write $a \\precsim b$ if there is a sequence $(v_n)$ in $\\mathrm{M}_{\\infty}(A)$ such that $v_nbv_n^* \\to a$ in norm (for the norm, view $\\mathrm{M}_\\infty(A)$ sitting naturally inside $A \\otimes \\mathcal{K}$). We write $a \\sim b$ if $a \\precsim b$ and $b \\precsim a$. Set $W(A) = \\{a \\in \\mathrm{M}_\\infty(A) \\ | \\ a \\geq 0 \\}\/\\sim$, and let $\\langle a \\rangle$ denote the equivalence class of $a$. $W(A)$ can be made into an ordered Abelian monoid by setting\n\\[\n\\langle a \\rangle + \\langle b \\rangle = \\langle a \\oplus b \\rangle \\ \\ \\mathrm{and} \\ \\ \\langle a \\rangle \\leq \\langle b \\rangle \\Leftrightarrow a \\precsim b.\n\\]\n$W(A)$ is the original {\\it Cuntz semigroup} of $A$. \n\nWe say that $W(A)$ is {\\it almost divisible} if for any $x \\in W(A)$ and $n \\in \\mathbb{N}$, there exists $y \\in W(A)$ such that \n\\[\nny \\leq x \\leq (n+1)y.\n\\]\nIf $\\tau \\in \\mathrm{T}(A)$, we define $d_\\tau: W(A) \\to \\mathbb{R}^+$ by\n\\[\nd_\\tau(\\langle a \\rangle) = \\lim_{n \\to \\infty} \\tau(a^{1\/n}).\n\\]\nThis map is known to be well-defined, additive, and order preserving, and for a fixed positive $a \\in \\mathrm{M}_{\\infty}(A)$, $A$ simple, the map $\\tau \\mapsto d_\\tau(a)$ belongs to $\\mathrm{lsc}(\\mathrm{T}(A))$. If $a \\precsim b$ whenever $d_\\tau(a) < d_\\tau(b)$ for every $\\tau \\in \\mathrm{T}(A)$, then we say that $A$ has {\\it strict comparison}. \n\nIt is implicit in Conjecture \\ref{conjecture} that a unital simple separable nuclear C$^*$-algebra with strict comparison of positive elements should have almost divisible Cuntz semigroup, but no general method has yet been found to establish this fact. Positive results have been limited to particular classes of C$^*$-algebras. Here we handle the case of AH algebras.\n\n\n\n\\begin{props}\\label{dense}\nLet $A$ be a unital, simple, stably finite C$^*$-algebra with strict comparison of positive elements. Suppose that for any $f \\in \\mathrm{Aff}(\\mathrm{T}(A))$ and $\\epsilon>0$ there is positive $a \\in \\mathrm{M}_\\infty(A)$ such that\n\\[\n|f(\\tau) - d_\\tau(a)| < \\epsilon, \\ \\forall \\tau \\in \\mathrm{T}(A).\n\\]\nIt follows that $W(A)$ is almost divisible.\n\\end{props}\n\n\\begin{proof}\nLet $g \\in \\mathrm{lsc}(\\mathrm{T}(A))$ be given. Then there is a strictly increasing sequence $(f_i)$ of strictly positive functions in $\\mathrm{Aff}(\\mathrm{T}(A))$ with the property that $\\sup_i f_i(\\tau) = g(\\tau)$. The function $f_i - f_{i-1}$ is strictly positive and continuous, and so achieves a minimum value $\\epsilon_i >0$ on the compact set $\\mathrm{T}(A)$. Passing to a subsequence, we may assume that $\\epsilon_i < \\epsilon_{i-1}$. By hypothesis, we can find, for each $i$, a positive $a_i \\in \\mathrm{M}_\\infty(A)$ such that\n\\[\n|f_i(\\tau)-d_\\tau(a_i)|<\\epsilon_{i+1}\/3.\n\\]\nIt follows that $(\\tau \\mapsto d_\\tau(a_i))_{i \\in \\mathbb{N}}$ is a strictly increasing sequence in $\\mathrm{lsc}(\\mathrm{T}(A))$ with supremum\n$g$. By strict comparison, we have $a_i \\precsim a_{i+1}$, i.e., $(\\langle a_i \\rangle)_{i \\in \\mathbb{N}}$ is an increasing sequence in $W(A)$. \\cite[Theorem 1]{cei} then guarantees the existence of a supremum $y$ for this sequence in $W(A \\otimes \\mathcal{K}) \\supseteq W(A)$. The map $d_\\tau$ is supremum preserving for each $\\tau$, and we conclude that $d_\\tau(y) = g(\\tau), \\ \\forall \\tau \\in \\mathrm{T}(A)$. \n\nNow let $x \\in W(A)$ and $n \\in \\mathbb{N}$ be given, and set $h(\\tau) = d_\\tau(x)$ for each $\\tau \\in \\mathrm{T}(A)$. It is straightforward to find $g \\in \\mathrm{lsc}(\\mathrm{T}(A))$ with the property that \n\\[\nng < h < (n+1)g.\n\\]\nWe may moreover find $x \\in W(A \\otimes \\mathcal{K})$ such that $d_\\tau(x) = g(\\tau)$, as in the first part of the proof. By strict comparison, any representative $a$ for $x$ (that is, $\\langle a \\rangle =x$ in $W(A \\otimes \\mathcal{K})$) will satisfy\n\\[\nn\\langle a \\rangle \\precsim x \\precsim (n+1)\\langle a \\rangle,\n\\]\nso it remains only to prove that $a$ can be chosen to lie in $\\mathrm{M}_\\infty(A)$, rather than $A \\otimes \\mathcal{K}$. Let $\\mathbf{1_k}$ denote the unit of $\\mathrm{M}_k(A)$. Since $g$ is bounded, we have $g(\\tau) < k = d_\\tau(\\mathbf{1_k})$ for some $k \\in \\mathbb{N}$ and all $\\tau$. By strict comparison, then, $\\langle a \\rangle$ is dominated by a Cuntz class in $W(A)$. It then follows from \\cite[Theorem 4.4.1]{5guys} that there is positive $b \\in \\mathrm{M}_\\infty(A)$ such that $\\langle b \\rangle = \\langle a \\rangle$, completing the proof.\n\\end{proof}\n\n\\begin{cors}\\label{maincor}\nLet $A$ be a unital simple AH algebra with strict comparison. It follows that $A$ is $\\mathcal{Z}$-stable.\n\\end{cors}\n\n\\begin{proof}\n$A$ is stably finite, and satisifes the hypothesis of Proposition \\ref{dense} concerning the existence of suitable $a$ for each $f$ and $\\epsilon$ by \\cite[Theorem 5.3]{bpt}. It therefore has strict comparison and almost divisible Cuntz semigroup. These hypotheses, together with the fact that $A$ is simple, nuclear, separable, and has locally finite nuclear dimension allow us to appeal to \\cite[Theorem 6.1]{W} and conclude that $A$ is $\\mathcal{Z}$-stable.\n\\end{proof}\n\nWe must concede that Proposition \\ref{dense} closes a gap in the proof of \\cite[Theorem 1.2]{T2}. There, we proved that the hypotheses of Propostion \\ref{dense} were satisfied for a simple unital ASH algebra with slow dimension growth but neglected to explain how this guarantees almost divisibility for $W(A)$ as opposed to $W(A \\otimes \\mathcal{K})$. While this could have been done in several ways, our appeal here to the recent article \\cite{5guys} was the most efficient one. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}