diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeogt" "b/data_all_eng_slimpj/shuffled/split2/finalzzeogt" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeogt" @@ -0,0 +1,5 @@ +{"text":"\\section*{Appendix}\n\\section{Training and evaluation details}\nWe use the multi-layered neural network model provided by the authors of \\cite{drozdov2019diora} to initialize our model's weights. We have a learning rate of $10^{-4}$ and a batch size of $32$. Using these settings we train our models for $500000$ steps. We use weights $\\lambda_{ranking} = 10^{-1}$ and $\\lambda_{ce} = 1.0$ for the rule based losses. All other model parameters are same as the ones set in \\cite{drozdov2019diora}. We run all our experiments on Nvidia RTX 2080Ti GPUs 12 GB RAM over Intel Xeon Gold 5120 CPU having 56 cores and 256 GB RAM. It takes about 2 days to train the model on NLI data. \\\\\nFor evaluation, we have reported the tree F-1 score for MNLI dev and WSJ test set. The metric computes the F-1 score for each tree based on the constituent spans induced in the predicted tree against the constituent spans in the ground truth. We further binarize the WSJ test set using the Stanford CoreNLP Parser and report scores on unlabelled binary trees. \\\\\nWe find that training with AR helps us achieve better results on both MNLI as well as on WSJ. This could be because extracted rules from SNLI have wider coverage on the training set than HR resulting in a stronger training signal and better performance. Further, our ranking loss performs better for HR extracted rules, indicating its efficacy with non-extensive rule sets, i.e. in the cases where the training signal is not rich. In such cases when some cells may not have any triggering rules, the ranking loss ensures that the model's decision is guided by the reconstruction loss. \\\\\nWe also find that the generic background knowledge of English grammar (HR) helps the model to better chunk constituents that are rarer (e.g. SBAR), while dataset-specific rules (AR) might benefit its overall tree structures more, leading to higher unlabelled F1 scores.\n\n\n\\section{Rule sets}\nIn this work we utilise two distinct rule sets - (i) The first rule set (HR) consists of 2500 human created CNF production rules ii) the other set (AR) consisted of 2500 most frequently occurring CNF production rules extracted from the trees of automatically parsed SNLI corpus. All rules in both these sets consist only of non-terminals. The rules in (HR) come from observing human annotated parse trees from the PTB train set and consists of 2500 rules in the Chomsky Normal Form. The rules in (AR) are programatically extracted from the parse trees generated by running the Stanford Parser on the SNLI train set. We only retain the 2500 most frequently occurring productions from the set to match the size of the HR set. We note however that these rules have a higher coverage on the train data. We also provides \\texttt{rules-AR.txt} and \\texttt{rules-HR.txt} in the github repository. \\\\ \nIn our training procedure, we aim to learn a weight $r_p$ for each production rule $p$ in our train set. Table \\ref{tab:top_rules} shows the top 10 most important (i.e. the ones with the highest $r_p$) rules from the grammar as determined by our models.\n\\begin{table*}[]\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\n\\textbf{Model} & \\textbf{Top Rules} & & & \\\\ \\midrule\nAR ranking loss & \\begin{tabular}[c]{@{}l@{}}NP ---\\textgreater PRP NP$|$CD-JJ-VBN-NNS\\textgreater \\\\ NP ---\\textgreater PRP NP$|$\\textless{}NNP-NNP-NN\\textgreater \\\\ S ---\\textgreater S S$|$\\textless{}CC-SINV-.\\textgreater \\\\ ADVP ---\\textgreater IN ADVP$|$\\textless{}CC-JJ\\textgreater \\\\ VP ---\\textgreater VBN VP$|$\\textless{}``-NP-''-PP\\textgreater \\\\ NP ---\\textgreater NP NP$|$\\textless{}NN-NN-''\\textgreater \\\\ PP ---\\textgreater IN , \\\\ NP ---\\textgreater NP NP$|$\\textless{}ADJP-PP-SBAR\\textgreater \\\\ NP ---\\textgreater NP NP$|$\\textless{}NNS-S\\textgreater \\\\ PP ---\\textgreater `` PP$|$\\textless{}IN-NP\\textgreater{}\\end{tabular} & & & \\\\\n\\hline\nHR cross entropy & \\begin{tabular}[c]{@{}l@{}}NP ---\\textgreater CD NNS \\\\ S-CLR ---\\textgreater VP \\\\ PP-PRP ---\\textgreater IN NP \\\\ QP$|$\\textless{}CD-TO-CD-CD\\textgreater ---\\textgreater CD QP$|$\\textless{}TO-CD-CD\\textgreater \\\\ S-2 ---\\textgreater NP S-2$|$\\textless{}VP-.\\textgreater \\\\ VP ---\\textgreater VB VP$|$\\textless{}NP-ADVP-S\\textgreater \\\\ NP$|$\\textless{},-''-SBAR\\textgreater ---\\textgreater , NP$|$\\textless{}''-SBAR\\textgreater \\\\ S ---\\textgreater NP S$|$\\textless{}NP-VP-.\\textgreater \\\\ NP$|$\\textless{}JJS-NNS\\textgreater ---\\textgreater JJS NNS \\\\ S-2$|$\\textless{}VP-.\\textgreater ---\\textgreater VP .\\end{tabular} & & & \\\\\n\\hline\nHR ranking loss & \\begin{tabular}[c]{@{}l@{}}NP ---\\textgreater PRP NP$|$\\textless{}NNP-CD-NN\\textgreater \\\\ NP ---\\textgreater DT NP$|$\\textless{}JJ-NN-NN\\textgreater \\\\ S$|$\\textless{}NP-VP-.---\\textgreater ---\\textgreater NP S$|$\\textless{}VP-.---\\textgreater \\\\ VP$|$\\textless{}NP-S\\textgreater ---\\textgreater NP S \\\\ NP$|$\\textless{}NNP-NNP-NNP-NN-NN\\textgreater ---\\textgreater NNP NP$|$\\textless{}NNP-NNP-NN-NN\\textgreater \\\\ NP ---\\textgreater JJ NP$|$\\textless{}NN-POS\\textgreater \\\\ VP$|$\\textless{}CC-,-VP\\textgreater ---\\textgreater CC VP$|$\\textless{},-VP\\textgreater \\\\ NP ---\\textgreater NNP NP$|$\\textless{}CC-NNS\\textgreater \\\\ NP$|$\\textless{}:-SBAR\\textgreater ---\\textgreater : SBAR \\\\ ADJP ---\\textgreater \\$ CD\\end{tabular} & & & \\\\\n\\hline\nAR cross entropy & \\begin{tabular}[c]{@{}l@{}}NP ---\\textgreater DT NP$|$\\textless{}JJ-NNP-NNP-POS\\textgreater \\\\ VP ---\\textgreater VB VP$|$\\textless{}NP-PRT\\textgreater \\\\ S$|$\\textless{}S-:\\textgreater ---\\textgreater S : \\\\ VP ---\\textgreater VBZ VP \\\\ PRN ---\\textgreater , PRN$|$\\textless{}CC-PP\\textgreater \\\\ VP$|$\\textless{}CC-VBG-NP-PP\\textgreater ---\\textgreater CC VP$|$\\textless{}VBG-NP-PP\\textgreater \\\\ NP$|$\\textless{},-NP-,-VP-.\\textgreater ---\\textgreater , NP$|$\\textless{}NP-,-VP-.\\textgreater \\\\ S$|$\\textless{}PP-,-VP-.\\textgreater ---\\textgreater PP S$|$\\textless{},-VP-.\\textgreater \\\\ PP ---\\textgreater RB PP$|$\\textless{}CC-RB-NP\\textgreater \\\\ NP ---\\textgreater JJ NP$|$\\textless{}NNP-NNP\\textgreater{}\\end{tabular} & & & \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Rules with the highest weights as learnt by our models } \n\\label{tab:top_rules}\n\n\\end{table*}\n\n\\begin{figure*}[!t]\n\\begin{tabular}{| c |@{}c@{} | @{}c@{}|}\n\\hline\n\nOur Results &\n{\\includegraphics[height=50mm,width=50mm,valign=b]{Figures\/oursTree_cropped.jpg}} \n&\n{\\includegraphics[height=60mm, width=65mm,valign=b]{Figures\/ours2_cropped.jpg}}\n\\\\\n\\hline\nDIORA's Results &\n{\\includegraphics[width=50mm]{Figures\/Diora_cropped.jpg}}\n&\n{\\includegraphics[width=60mm]{Figures\/Diora2_cropped.jpg}}\n\\\\\n\\hline\nGround Truth &\n{\\includegraphics[width=50mm]{Figures\/ground_cropped.jpg}}\n &\n{\\includegraphics[width=60mm]{Figures\/ground2_cropped.jpg}}\n \\\\\n\\hline\n\\end{tabular}\n\n\\caption{Comparison of induced trees by our model and DIORA with the ground truth trees}\n\\label{fig:trees}\n\\end{figure*}\n\n\n\n\\section{Learnt Trees}\nIn this section we present examples of trees induced by DIORA and our model. The first row of fig. 1 shows an example where our tree matches the ground truth exactly while DIORA does not, and the second row of fig. 1 shows an example where both models do not provide exact matches, but our model is able to capture the syntax better.\n\n\n\\section{Introduction}\nSyntactic parse trees have demonstrated their importance in several downstream NLP applications such as machine translation~\\cite{eriguchi2017learning, zaremoodi2017incorporating}, natural language inference (NLI) \\cite{choi2018learning}, relation extraction \\cite{gamallo2012dependency} and text classification~\\cite{tai2015improved}. Based on linguistic theories that have promoted the usefulness of tree-based representation of natural language text, tree-based models such as Tree-LSTM have been proposed to learn sentence representations~\\cite{socher2011semi}. Inspired by the Tree-LSTM based models, many approaches were proposed do not require parse tree supervision~\\cite{yogatama2016learning,choi2018learning,Maillard_2019,drozdov2019diora}. However, \\citep{williams2018latent, sahay} have shown that these methods cannot learn meaningful semantics (not even simple grammar), though they perform well on NLI tasks. \nRecently, there has been surge in approaches using weak supervision in the form of rules for various tasks such as sequence classification \\cite{safranchik2020weakly}, text classification \\cite{chatterjee2020robust, maheshwari}, {\\em etc.} These approaches have demonstrated the importance of external knowledge in both unsupervised and supervised setup. To the best of our knowledge, previous works on syntactic parse tree has not utilized such external information. \nIn this paper, we propose an approach that leverages linguistic (and potentially domain agnostic) knowledge in the form of explicit syntactic grammar rules while building upon a state of the art, deep and unsupervised inside-outside recursive autoencoder (DIORA; \\cite{drozdov2019diora}). DIORA is an unsupervised model that uses inside-outside dynamic programming to compose latent representations from all possible binary trees. We extend DIORA and propose a framework that harness grammar rules to learn constituent parse trees. We use context free grammar (CFG) productions for English language (like NP \\textrightarrow VP NP, PP \\textrightarrow IN NP, \\textit{etc}) as rules. Note that the construction of such a rule set is a one time effort and our method is independent of any underlying dataset. The rule sets used are available in our github repository.\\\\\nSummarily, our main contributions are : (a) a framework ({\\em cf.}, Section~\\ref{sec:approach}) that uses (potentially domain agnostic), off-the-shelf CFG to learn to produce constituent parse trees (b) two rule-aware loss functions ({\\em cf.}, Section~\\ref{sec:loss}) that maximize some form of agreement between the unsupervised model and the rule-based model (c) experimental analysis ({\\em cf.}, Section~\\ref{sec:expt}), demonstrating improvements on unsupervised constituency parsing over previous state-of-the art by over \\textbf{3}\\% on two benchmark datasets.\n\n\\section{Background and Related Work}\n\\label{sec:back}\nA brief survey of latent tree learning models is covered in~\\cite{williams2018latent}. Several prior works have explored the unsupervised learning of constituency trees \\cite{brill1990deducing, ando2000mostly} using dependency parsers~\\cite{klein2004corpus} and inside-outside parsing algorithm \\cite{drozdov2019diora}. \nRecently, \\cite{drozdov2019diora} proposed an unsupervised latent chart tree parsing algorithm, {\\em viz.}, DIORA, that uses the inside-outside algorithm for parsing and has an autoencoder-based neural network trained to reconstruct the input sentence. DIORA is trained end to end using masked language model via word prediction. As of date, DIORA is the state-of-the-art approach to unsupervised constituency parsing. \n\n\nExploiting additional semantic and syntactic information that acts as a source of additional guidance rather than the primary objective function has been discussed since 1990s~\\cite{sun2017neural}. Recently, \\citet{kim2020compound} proposed to learn CFG rules and their probabilities by the parameterizing terminal or non-terminal symbols with neural networks. However, our approach leverages pre-defined language CFG rules and provisions for augmenting an existing (state-of-the-art) inside-outside algorithm with such external knowledge.\n\n More specifically, we augment DIORA~\\cite{drozdov2019diora} with CFG rules to reconstruct the input by exploiting syntactic information of the language. \nWe next provide some technical details of the inside-outside algorithm of DIORA.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/CYK.pdf}\n \\caption{For the input `The cat sat', DIORA computes $e(i_1, j_1)$ compatibility score for each pair of neighboring constituents. $l(i_1,j_1)$ is computed using triggered rules for each span and it interacts with the compatibility score in our loss function as explained in Section~\\ref{sec:loss}.}\n \\label{fig:arch}\n\\end{figure}\n\n\\subsection{DIORA}\n\\label{sec:diora}\nDIORA learns constituency trees from the raw input text using an unsupervised training procedure that operates like a masked language model or denoising autoencoder. It encodes the entire input sequence into a single vector analogous to the encoding step in an autoencoder. Thereafter, the decoder is trained to reconstruct and reproduce each input word. We next describe the inside and the outside pass of DIORA, respectively. \n\\textit{Inside Pass:} Given an input sentence with $T$ tokens $x_0, x_1, x_2 \\ldots x_{T-1}$, DIORA computes a \\textit{compatibility} score $e$ and a \\textit{composition} vector $\\overline{a}$ for each pair of neighboring constituents $i$ and $j$. It composes a vector $\\overline{a}$ weighing over all possible pairs of constituents of $i$ and $j$:\n$\n\\overline{a}(k) = \\sum_{{i,j} \\epsilon \\{k\\} } e(i,j) a(i,j)\n$ \\& $\n\\overline{e}(k) = \\sum_{{i,j} \\epsilon \\{k\\} } e(i,j) \\hat{e}(i,j)\n$\nThe composition vector, $\\overline{a}(k)$ is a weighted sum of all possible constituent pairs, ${k}$. Here, $\\hat{e}$ is a bilinear function of the vectors from neighboring spans, $\\overline{a}(i)$ and $\\overline{a}(j)$. Composition vector $\\overline{a}(k)$ is learnt using a TreeLSTM or multi-layer neural network (MLP).\n\n\\textit{Outside Pass:} The outside pass of DIORA computes an outside vector $\\overline{b}(k)$ representing the constituents not in $x_{i:j}$. It computes the values for a target space $(i,j)$ recursively from its sibling $(j+1,k)$ and outside spans $(0,i-1)$ and $(k+1, T-1)$.\n\n\\textit{Training and Inference:} DIORA is trained end to end using masked language model via word prediction. The missing token $x_i$ is predicted from the outside vector $\\overline{b}(k)$. The training objective uses reconstruction based max-margin loss to predict the original input $x_i$: \\\\ \\\\ \n$L_{rec} = \\sum\\limits_{i=0}^{T-1} \\sum\\limits_{i^*=0}^{N-1} \\max (0, 1 -\\overline{b}(i). \\overline{a}(i) + \\overline{b}(i). \\overline{a}(i^*))\n\\vspace{0.5cm} $\n\nThe chart filling procedure of DIORA is used to extract binary unlabeled parse trees. It uses the CYK algorithm to find the maximal scoring tree in a greedy manner. For each cell of the parse table, the algorithm computes the span $(i,j)$ with the maximal net compatibility score, computed recursively by summing the maximum compatibility score $e(a,b)$ for each constituent of the span.\n\n\n\\section{Our Approach to Rule Augmentation} \n\\label{sec:approach}\nOur goal is to learn to produce constituency parse trees using input sentences alone and in the absence of ground truth parse trees. We introduce a rule-augmented unsupervised model that leverages generic (potentially domain agnostic) production rules of the language grammar to infer constituency trees. Since most grammar rules for constituency parsing are generic, designing them can be a onetime effort, while being able to leverage their benefits across domains as background knowledge (as we will see in our experiments in Section~\\ref{sec:expt}). As described in Section~\\ref{sec:diora}, the induction of latent trees in DIORA is based on a CYK-like parsing algorithm that uses the compatibility scores $e(i,j)$ at each cell to merge two constituents in the final tree. We impart supervision through the production rules of English language grammar. \n\nFor each sentence, we associate CFG production rules with constituents $i$ and $j$ in a CYK parse table format. We curate a set of domain-agnostic rules of the form $X \\rightarrow Y Z$ and a dictionary of the form $X \\rightarrow x$, where $X,Y,Z$ are non-terminals while $x$ is a terminal. Concretely, $X$ represents constituent tags such as S, NP, VP, etc., while $x$ represents words in the vocabulary. \nUsing the CYK parsing algorithm on our rule set and each sentence, we first determine which rules are triggered at each cell for a particular sentence. \nWhenever a rule $r$ is triggered for a span $(i,j)$, we weakly associate label $\\delta_{(i,j)}(r) = 1$ otherwise 0. We use these weak labels to guide the rule scores $l(i,j)$ for the constituents. Compatibility score observed for the span $(i,j)$ is defined as :\n\\begin{equation*}\nl(i,j) = \\frac{\\exp\\Big(\\sum\\limits_{p=0}^{P} r_p \\delta_{(i,j)}(p) \\Big)}{\\mathlarger{\\sum}\\limits_{(a,b) \\in \\{k\\}}\\exp\\Big(\\sum\\limits_{p=0}^{P} r_p \\delta_{(a,b)}(p) \\Big)}\n\\end{equation*}\n\nwhere $r_p$ are the \\textit{learned} weights associated with each of the production rules and $P$ is the total number of rules. The score sums to $1$ over all spans belonging to a particular cell in the CYK parse table. Intuitively, we aim to align $e(i,j)$ and $l(i,j)$ score to maximize the agreement between model and rules. We note that we use rules only to augment the training objective, and our inference procedure is identical to that of DIORA.\n\n\n\\subsection{Training Objective}\n\\label{sec:loss}\nWe learn a model that minimizes the overall loss $L$ that is a composition of the reconstruction loss $L_{rec}$ and the rule based agreement loss or $L_{rule}$:\n$\nL = L_{rec} + \\lambda L_{rule}.\n$\nWe propose two alternatives for the loss function $L_{rule}$. \\\\\n\\textbf{Cross entropy (CE)} - For each cell $k$ in the CYK parse table, this loss (CE) tries {\\em to match the distribution (score)} of $e(i,j)$ induced by DIORA with the distribution $l(i,j)$ induced by the background knowledge: \n\\begin{align*}\nL_{ce} = \\sum_{k} \\sum_{(i,j) \\in \\{k\\}} - l(i,j) \\log(e(i,j))\n\\end{align*}\n\\textbf{Ranking Loss (RL)} -\nWe recall from Section~\\ref{sec:diora} that the CYK algorithm finds the maximal scoring tree in a greedy manner based on the highest compatibility score $e(i,j)$ among all spans. Since the final parse tree output by DIORA relies only on the \\textit{relative order} of the $e(i,j)$ to decide which span to merge, we propose an alternative rule-based loss that aims {\\em to match the relative order} induced by compatibility scores $e(i,j)$ of DIORA at each cell with the order induced by the scores of the rules $l(i,j)$ at that cell. We achieve this through a pairwise ranking loss defined as \n\\begin{equation*}\nL_{rank} = \\sum_{\\text{k}} \\sum\\limits_{\\small \\substack{(i,j) \\\\ (i',j')}\\in \\{\\text{k}_{trig}\\}} \\bigg( \\mathlarger{\\Delta}_{\\substack{(i,j),\\\\(i',j')}}^l - \\mathlarger{\\Delta}_{\\substack{(i,j),\\\\(i',j')}}^e \\bigg)^2\n\\end{equation*}\n\\normalsize\nwhere $\\{\\text{k}_{trig}\\} = \\{(i,j) \\in \\{k\\} | \\sum_{p=1}^P \\delta_{i,j}(p) \\neq 0\\}$, $\\Delta_{(i,j),(i',j')}^f = \\tanh(f(i,j) - f(i',j'))$ and $p$ is index into the rule set. The set $\\{k_{trig}\\}$ consists of all spans which have at least one rule triggered in its cell. In cases where our rule set is not extensive enough, we would like our model's compatibility score to rely more on the reconstruction loss, and $\\{\\text{k}_{trig}\\}$ ensures that a sparse rule set does not lead to bad performance. \n\n\\begin{table}[]\n\\begin{adjustbox}{width=\\linewidth}\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{Model} & \\multicolumn{1}{c}{\\textbf{F1}} & \\multicolumn{1}{c}{$\\delta$} \\\\\n\\midrule\n300D Gumbel Tree-LSTM & 25.2 & 4.2\\\\\n~~w\/o Leaf GRU & 29.0& 4.7\\\\\n300D RL-SPINN & 19.0& 8.6\\\\\n~~w\/o Leaf GRU & 18.2&8.6\\\\\nStructural Attentive(Gumble Tree LSTM) & 31.3 & 4.7\\\\\n~~w\/o Leaf GRU & 31.0 & 5.3\\\\\n\\midrule\n300D SPINN & \\textbf{74.5$^\\dagger$} & 6.2\\\\\n~~w\/o Leaf GRU & 65.7$^\\dagger$ & 6.4\\\\\n\nDIORA with PP & 58.3 & 5.6\\\\ \n\\midrule\nOurs: Rule augmented (HR) + RL + PP & 60.5 & 5.7\\\\ \nOurs: Rule augmented (HR) + CE + PP & 59.0 & 5.6 \\\\\nOurs: Rule augmented (AR) + RL + PP & 60.3 & 5.7 \\\\\nOurs: Rule augmented (AR) + CE + PP & \\textbf{61.7} & 5.7 \\\\\n\\midrule \n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\caption{F1-scores of trees wrt ground truth on the MultiNLI development set. The depth ($\\delta$) is the average tree height. All reported numbers are maximum F1-score. PP refers to post-processing heuristic. HR and AR refer to the training rule sets as per Sec \\ref{sec:rule_set}. RL and CE refer to the losses from section \\ref{sec:loss}. $^\\dagger$ indicates scores reported by \\cite{williams2018latent} for a fully supervised model}\n\\label{tab:quantMultiNLI}\n\\end{table}\n\n\n\n\\begin{table}[h]\n\n\\centering\n\\begin{adjustbox}{width=\\linewidth}\n\\begin{tabular}{lcccc}\n\\hline\n\\multicolumn{1}{c}{\\textbf{Model}} & \\multicolumn{1}{c}{\\textbf{F1}} & \\multicolumn{1}{c}{\\textbf{$\\delta$}} \\\\ \\hline\nLB& 13.1 & 12.4 \\\\\nRB& 16.5 & 12.4 \\\\\nRandom & 21.4 & 5.3 \\\\\nBalanced & 21.3 & 4.6 \\\\\nRL-SPINN \\cite{choi2018learning} & 13.2 & - \\\\\nST-Gumbel - GRU \\cite{yogatama2016learning}& 22.8 \u00b11.6 & - \\\\ \\hline\nPRPN-UP \\cite{shen2018neural} & 38.3 & 5.9 \\\\\nPRPN-LM& 35.0 & 6.2 \\\\\nON-LSTM \\cite{shen2018ordered}& 47.7 & 5.6 \\\\\nDIORA& 48.9 & 8.0 \\\\ \\hline\nPRPN-UP+PP & 45.2 & 6.7 \\\\\nPRPN-LM+PP & 42.4 & 6.3 \\\\\nDIORA+PP & 55.7 & 8.5 \\\\ \\hline\nNeural PCFG \\cite{kim2020pretrained} $^*$ & 50.8 & - \\\\\nCompound PCFG \\cite{kim2020compound}$^*$\n& 55.2 & -\\\\ \n300D SPINN & \\textbf{59.6$^\\dagger$} & -\\\\ \\hline\n\nOurs: Rule augmented (HR)\n+ RL + PP & 56.5 & 7.1\\\\ \nOurs: Rule augmented (HR)\n+ CE + PP & 55.3 & 7.2 \\\\\nOurs: Rule augmented (AR) \n+ RL + PP & 55.9 & 7.1 \\\\\nOurs: Rule augmented (AR)\n+ CE + PP & \\textbf{58.3} & 7.3 \\\\ \\hline \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{ Performance on WSJ test set for binary constituency parsing including punctuation characters. HR and AR refers to handcrafted and automated rules respectively. RL and CE are rule loss and cross entropy loss respectively. ($\\delta$) is the average tree height. PP refers to post-processing heuristic. $^\\dagger$ indicates scores reported by \\cite{williams2018latent} for a fully supervised model. $^*$ are reported by \\cite{kim2020pretrained}.}\n\\label{tab:wsj}\n\\end{table}\n \n\\section{Experiments}\n\\label{sec:expt}\nWe evaluate our rule augmented model and compare it against baselines on the tasks of unsupervised parsing, unsupervised segment recall, and phrase similarity.\n\n\\subsection{Data}\nWe evaluate our model on two data sets: The Wall Street Journal (WSJ) and MultiNLI. WSJ is an extraction of PennTree Bank~\\cite{article} containing human-annotated constituency parse trees. MultiNLI consists of Stanford generated parse trees ~\\cite{manning2014stanford} as the ground truth. MultiNLI is originally designed for evaluating NLI tasks, but is often also utilized to evaluate constituency parse trees. We train on the complete NLI dataset, which is a composition of the MultiNLI and SNLI train sets. We evaluate model performance on the MultiNLI dev set and WSJ test set (split 23) following the experimental setting and evaluation metrics in~\\cite{drozdov2019diora}. Further details are provided in the appendix.\nWe initialize our model with the trained weights of DIORA and evaluate on unsupervised constituency parsing and segment recall. We also perform the post-processing (PP) of generated trees by attaching the trailing punctuation to the root node, exactly as carried out by~\\cite{drozdov2019diora}. \n\n\\subsection{Rule Set}\n\\label{sec:rule_set}\nWe consider two rule-sets: (i) {\\bf Set of Handcrafted Rules (HR)} consists of 2500 human created CNF production rules ii) To assess robustness of the rule-augmented method to the preciseness of the rule set, we present comparison by instead using a {\\bf set of Automated Rules' (AR)} which consists of the 2500 most frequently occurring CNF production rules extracted from the trees of automatically (using the Stanford CoreNLP parser) parsed SNLI corpus. Further details about these rule sets can be found in the appendix.\nWe also use a train-set specific dictionary containing the POS (part-of-speech) tags of words in the training vocabulary for the terminal CFG productions for CYK parsing.\n\n\\begin{table*}[!h]\n\\begin{tabular}{l|ccclll}\n\\hline\n\\multicolumn{1}{c|}{\\textbf{Model}} & \\textbf{SBAR} & \\textbf{NP} & \\textbf{VP} & \\textbf{PP} & \\textbf{ADJP} & \\textbf{ADVP} \\\\ \\hline\nLB $\\dagger$ & 5\\% & 11\\% & 0\\% & 5\\% & 2\\% & 8\\% \\\\\nRB $\\dagger$ & 68\\% & 24\\% & 71\\% & 42\\% & 27\\% & 38\\% \\\\\nRandom $\\dagger$ & 8\\% & 23\\% & 12\\% & 18\\% & 23\\% & 28\\% \\\\\nBalanced $\\dagger$ & 7\\% & 27\\% & 8\\% & 18\\% & 27\\% & 25\\% \\\\ \\hline\nPRPN-UP \\cite{shen2018neural} & 55.4\\% & 59.8\\% & 31.6\\% & 60.2\\% & 36.0\\% & 50\\% \\\\\nPRPN-LM & 40.3\\% & 68.7\\% & 39.3\\% & 49.7\\% & 34.2\\% & 39.2\\% \\\\\nDIORA & 61.3\\% & 76.7\\% & 62.8\\% & 59.5\\% & 60.4\\% & 69.3\\% \\\\ \\hline\nPRPN (tuned)$\\dagger$ & 50\\% & 59\\% & 46\\% & 57\\% & 44\\% & 32\\% \\\\\nON (tuned) \\cite{shen2018ordered} & 51\\% & 64\\% & 41\\% & 54\\% & 38\\% & 31\\% \\\\\nNeural PCFG \\cite{kim2020pretrained} & 52\\% & 71\\% & 33\\% & 58\\% & 32\\% & 45\\% \\\\\nCompound PCFG \\cite{kim2020compound} & \\multicolumn{1}{l}{56\\%} & \\multicolumn{1}{l}{74\\%} & \\multicolumn{1}{l}{41\\%} & 68\\% & 40\\% & 52\\% \\\\ \\hline\nOurs: Rule augmented (HR)+ RL & \\textbf{71.1\\%} & 77.2\\% & 65.8\\% & 59.4\\% & \\textbf{62.9\\%} & 69.5\\% \\\\\nOurs: Rule augmented (HR)+ CE & \\multicolumn{1}{l}{68.3\\%} & \\multicolumn{1}{l}{75.4\\%} & \\multicolumn{1}{l}{66.5\\%} & \\textbf{60.5\\%} & 61\\% & \\textbf{70.8\\%} \\\\\nOurs: Rule augmented (AR)+ RL & \\multicolumn{1}{l}{71\\%} & \\multicolumn{1}{l}{76.4\\%} & \\multicolumn{1}{l}{\\textbf{69.1\\%}} & 58.6\\% & 61\\% & 64.8\\% \\\\\nOurs: Rule augmented (AR)+ CE & \\multicolumn{1}{l}{70\\%} & \\multicolumn{1}{l}{\\textbf{77.5\\%}} & \\multicolumn{1}{l}{67\\%} & 58.6\\% & \\textbf{62.2\\%} & 70\\% \\\\ \\hline \\hline\n\\end{tabular}\n\\caption{Segment recall from WSJ by phrase type; $\\dagger$ are reported by~\\citet{kim2020pretrained}. } \n\\label{tab:segment}\n\n\\end{table*}\n \n\n\n\\begin{table*}\n\\centering\n\\begin{adjustbox}{width=0.7\\linewidth}\n\n\n\\begin{tabular}{l|ccc|cc}\n\\hline\n\\multicolumn{1}{c|}{\\textbf{Model}} & \\multicolumn{3}{c|}{\\textbf{CONLL 2000}} & \\multicolumn{2}{c}{\\textbf{CONLL 2012}} \\\\\n& \\multicolumn{1}{l}{\\textbf{P@1}} & \\multicolumn{1}{l}{\\textbf{P@10}} & \\multicolumn{1}{l|}{\\textbf{P@100}} & \\multicolumn{1}{l}{\\textbf{P@1}} & \\multicolumn{1}{l}{\\textbf{P@10}} \\\\ \\hline\nDIORA & 0.974 & 0.969 & 0.943 & 0.815 & 0.759 \\\\\nOurs: Rule augmented(AR)+ RL & 0.976 & 0.968 & 0.941 & 0.813 & 0.755 \\\\\nOurs: Rule augmented(HR)+ CE & \\textbf{0.978} & \\textbf{0.97} & 0.941 & 0.786 & 0.717 \\\\\nOurs: Rule augmented(AR)+ CE & 0.970 & 0.965 & 0.937 & 0.809 & 0.745 \\\\\nOurs: Rule augmented(HR)+ RL & 0.976 & 0.970 & \\textbf{0.944} & \\textbf{0.824} & \\textbf{0.760} \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{Phrase similarity scores on CoNLL2000 and CoNLL 2012 tasks.} \n\\label{tab:similarity}\n\\end{table*}\n\n\\subsection{Unsupervised Parsing}\n\nIn Tables~\\ref{tab:quantMultiNLI} and \\ref{tab:wsj}, we present comparison between different approaches on the MultiNLI dev set and WSJ test set. We observe that our rule augmented approach outperforms the state of the art with respect to the max-F1 score. registering a maximum increase of 3.4 and 3.1 F1 points over DIORA respectively. \nThe HR trained models outperform DIORA on both datasets, demonstrating that rule creation is indeed a one-time process and independent of domain. \nWe also report parsing scores of a fully supervised model SPINN from \\cite{williams2018latent} as an upper bound, and RL-SPINN \\cite{choi2018learning}, a distantly supervised model.\n\n\\subsection{Constituency Segment Recall}\nIn Table \\ref{tab:segment}, we present the breakdown of constituent recall across the 6 most common types. Our approach achieves the highest recall across all the types and is the only model to perform effectively on SBAR and NP. Unlike other approaches, our approach consistently close to or the best recall score. \\\\\nWe observe that rule augmentation using HR is more beneficial than AR with respect to precise evaluation measures such as Constituency, Segment Recall and Phrase Recall but yields smaller improvements than AR with respect to looser evaluation measures such as max F1 of Unsupervised Parsing. This can be possibly attributed to our observation that the extracted (most frequent) rules from SNLI, have (around 25\\%) higher coverage on the training set than HR, but appear to be semantically less precise. \n\n\\subsection{Phrase Similarity}\nWe also employed the phrase similarity strategy followed by ~\\cite{drozdov2019diora}. Phrase Similarity scores measures the models capability to learn meaningful representation for spans of the text. Generally, most models focus more on generating the tokens representation and then use some ad-hoc arithmetic operations to generate representation for the larger spans of text thus losing the essence of the context that ties the words of the span. \n\nTo evaluate on the phrase similarity task we consider two data sets of labeled phrases: 1) CoNLL 2000 ~\\cite{tjong2000introduction}, which is a shallow parsed dataset and contains spans of verb phrases, noun phrases, preposition phrases \\emph{etc.}, and 2) CoNLL 2012 ~\\cite{pradhan2012conll} which is a named entity dataset containing 19 different entity types. For the evaluation routine, we first generated the phrase representation of labeled spans whose length is greater than one. Cosine similarity is then used to obtain the similarity score of it with respect to all other labeled spans. We then calculate if the label for that query span matches the labels for each of\nthe K most similar other spans in the dataset.In Table \\ref{tab:similarity} we report precision@K for both datasets and various values of K. The baseline numbers are reported using the weights of DIORA provided by the authors.\n\\section{Conclusion}\nIn this work, we leverage linguistically grounded and domain agnostic CFG rules for language to induce parse trees and representations of constituent spans. We show that our approach augmented with generic, linguistically grounded grammatical rules, is easily able to outperform previous methods on constituency parsing and obtain higher segment recall. \n\n\\section*{Acknowledgements}\nWe thank anonymous reviewers for providing constructive feedback. Ayush Maheshwari is supported by a Fellowship from Ekal Foundation (www.ekal.org). We are also grateful to IBM Research, India (specifically the IBM AI Horizon Networks - IIT Bombay initiative) for their support and sponsorship.\n\\bibliographystyle{acl_natbib}\n\n\\section{Introduction}\nSyntactic parse trees have demonstrated their importance in several downstream NLP applications such as machine translation~\\cite{eriguchi2017learning, zaremoodi2017incorporating}, natural language inference (NLI) \\cite{choi2018learning}, relation extraction \\cite{gamallo2012dependency} and text classification~\\cite{tai2015improved}. Based on linguistic theories that have promoted the usefulness of tree-based representation of natural language text, tree-based models such as Tree-LSTM have been proposed to learn sentence representations~\\cite{socher2011semi}. Inspired by the Tree-LSTM based models, many approaches were proposed do not require parse tree supervision~\\cite{yogatama2016learning,choi2017learning,Maillard_2019,drozdov2019diora}. However, \\citet{williams2018latent} have shown that these methods cannot learn meaningful semantics (not even simple grammar), though they perform well on NLI tasks. \nRecently, there has been surge in approaches using weak supervision in the form of rules for various tasks such as sequence classification \\cite{safranchik2020weakly}, text classification \\cite{ratner2017snorkel, chatterjee2020robust}, {\\em etc.} These approaches have demonstrated the importance of external knowledge in both unsupervised and supervised setup. To the best of our knowledge, previous works on syntactic parse tree has not leveraged such external information. \nIn this paper, we propose an approach that leverages linguistic (and potentially domain agnostic) knowledge in the form of explicit syntactic grammar rules while building upon a state of the art, deep and unsupervised inside-outside recursive autoencoder (DIORA; \\cite{drozdov2019diora}). DIORA is an unsupervised model that uses inside-outside dynamic programming to compose latent representations from all possible binary trees. We extend DIORA and propose a framework that harness grammar rules to learn constituent parse trees. We use context free grammar (CFG) productions for English language (like NP \\textrightarrow VP NP, PP \\textrightarrow IN NP, \\textit{etc}) as rules. Note that the construction of such a rule set is a one time effort and our method is independent of any underlying dataset. The rule sets used are available \\href{https:\/\/anonymous.4open.science\/r\/ffd19508-6be9-4e41-b7b3-ee2c4e4f5979\/}{here}.\\\\\nSummarily, our main contributions are : (a) a framework ({\\em c.f.}, Section~\\ref{sec:approach}) that uses (potentially domain agnostic), off-the-shelf CFG to learn to produce constituent parse trees (b) two rule-aware loss functions ({\\em c.f.}, Section~\\ref{sec:loss}) that maximize some form of agreement between the unsupervised model and the rule-based model (c) experimental analysis ({\\em c.f.}, Section~\\ref{sec:expt}), demonstrating improvements on unsupervised constituency parsing over previous state-of-the art by over \\textbf{3}\\% on two benchmark datasets.\n\n\\section{Background and Related Work}\n\\label{sec:back}\nA brief survey of latent tree learning models is covered in~\\citep{williams2018latent}. Several prior works have explored the unsupervised learning of constituency trees \\cite{brill1990deducing, ando2000mostly} using dependency parsers~\\cite{klein2004corpus} and inside-outside parsing algorithm \\cite{sdiora}. \nRecently, \\cite{drozdov2019diora} proposed an unsupervised latent chart tree parsing algorithm, {\\em viz.}, DIORA, that uses the inside-outside algorithm for parsing and has an autoencoder-based neural network trained to reconstruct the input sentence. DIORA is trained end to end using masked language model via word prediction. As of date, DIORA is the state-of-the-art approach to unsupervised constituency parsing. \n\n\nExploiting additional semantic and syntactic information that acts as a source of additional guidance rather than the primary objective function has been discussed since 1990s~\\cite{sun2017neural}. Recently, \\citet{kim2020compound} proposed to learn CFG rules and their probabilities by the parameterizing terminal or non-terminal symbols with neural networks. However, our approach leverages pre-defined language CFG rules and provisions for augmenting an existing (state-of-the-art) inside-outside algorithm with such external knowledge.\n\n More specifically, we augment DIORA~\\cite{drozdov2019diora} with CFG rules to reconstruct the input by exploiting syntactic information of the language. \nWe next provide some technical details of the inside-outside algorithm of DIORA.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\linewidth, height=3cm]{Figures\/CYK.pdf}\n \\caption{For the input `The cat sat', DIORA computes $e(i_1, j_1)$ compatibility score for each pair of neighboring constituents. $l(i_1,j_1)$ is computed using triggered rules for each span and it interacts with the compatibility score in our loss function as explained in Section~\\ref{sec:loss}.}\n \\label{fig:arch}\n\\end{figure}\n\n\\subsection{DIORA}\n\\label{sec:diora}\nDIORA learns constituency trees from the raw input text using an unsupervised training procedure that operates like a masked language model or denoising autoencoder. It encodes the entire input sequence into a single vector analogous to the encoding step in an autoencoder. Thereafter, the decoder is trained to reconstruct and reproduce each input word. We next describe the inside and the outside pass of DIORA, respectively. \n\\textit{Inside Pass:} Given an input sentence with $T$ tokens $x_0, x_1, x_2 \\ldots x_{T-1}$, DIORA computes a \\textit{compatibility} score $e$ and a \\textit{composition} vector $\\overline{a}$ for each pair of neighboring constituents $i$ and $j$. It composes a vector $\\overline{a}$ weighing over all possible pairs of constituents of $i$ and $j$:\n$\n\\overline{a}(k) = \\sum_{{i,j} \\epsilon \\{k\\} } e(i,j) a(i,j)\n$ \\& $\n\\overline{e}(k) = \\sum_{{i,j} \\epsilon \\{k\\} } e(i,j) \\hat{e}(i,j)\n$\nThe composition vector, $\\overline{a}(k)$ is a weighted sum of all possible constituent pairs, ${k}$. Here, $\\hat{e}$ is a bilinear function of the vectors from neighboring spans, $\\overline{a}(i)$ and $\\overline{a}(j)$. Composition vector $\\overline{a}(k)$ is learnt using a TreeLSTM or multi-layer neural network (MLP).\n\n\\textit{Outside Pass:} The outside pass of DIORA computes an outside vector $\\overline{b}(k)$ representing the constituents not in $x_{i:j}$. It computes the values for a target space $(i,j)$ recursively from its sibling $(j+1,k)$ and outside spans $(0,i-1)$ and $(k+1, T-1)$.\n\n\\textit{Training and Inference:} DIORA is trained end to end using masked language model via word prediction. The missing token $x_i$ is predicted from the outside vector $\\overline{b}(k)$. The training objective uses reconstruction based max-margin loss to predict the original input $x_i$:\\\\\n$L_{rec} = \\sum\\limits_{i=0}^{T-1} \\sum\\limits_{i^*=0}^{N-1} \\max (0, 1 -\\overline{b}(i). \\overline{a}(i) + \\overline{b}(i). \\overline{a}(i^*))\n$\nThe chart filling procedure of DIORA is used to extract binary unlabeled parse trees. It uses the CYK algorithm to find the maximal scoring tree in a greedy manner. For each cell of the parse table, the algorithm computes the span $(i,j)$ with the maximal net compatibility score, computed recursively by summing the maximum compatibility score $e(a,b)$ for each constituent of the span.\n\n\n\\section{Our Approach to Rule Augmentation} \n\\label{sec:approach}\nOur goal is to learn to produce constituency parse trees using input sentences alone and in the absence of ground truth parse trees. We introduce a rule-augmented unsupervised model that leverages generic (potentially domain agnostic) production rules of the language grammar to infer constituency trees. Since most grammar rules for constituency parsing are generic, designing them can be a onetime effort, while being able to leverage their benefits across domains as background knowledge (as we will see in our experiments in Section~\\ref{sec:expt}). As described in Section~\\ref{sec:diora}, the induction of latent trees in DIORA is based on a CYK-like parsing algorithm that uses the compatibility scores $e(i,j)$ at each cell to merge two constituents in the final tree. We impart supervision by through the production rules of English language grammar. \n\nFor each sentence, we associate CFG production rules with constituents $i$ and $j$ in a CYK parse table format. We curate a set of domain-agnostic rules of the form $X \\rightarrow Y Z$ and a dictionary of the form $X \\rightarrow x$, where $X,Y,Z$ are non-terminals while $x$ is a terminal. Concretely, $X$ represents constituent tags such as S, NP, VP, etc., while $x$ represents words in the vocabulary. \nUsing the CYK parsing algorithm on our rule set and each sentence, we first determine which rules are triggered at each cell for a particular sentence. \nWhenever a rule $r$ is triggered for a span $(i,j)$, we weakly associate label $\\delta_{(i,j)}(r) = 1$ otherwise 0. We use these weak labels to guide the rule scores $l(i,j)$ for the constituents. Compatibility score observed for the span $(i,j)$ is defined as :\\\\\n$\nl(i,j) = \\frac{\\exp(\\sum\\limits_{p=0}^{P} r_p \\delta_{(i,j)}(p) )}{\\sum\\limits_{(a,b) \\in \\{k\\}}\\exp(\\sum\\limits_{p=0}^{P} r_p \\delta_{(a,b)}(p) )}\n$\\\\\nwhere $r_p$ are the \\textit{learned} weights associated with each of the production rules and $P$ is the total number of rules. The score sums to $1$ over all spans belonging to a particular cell in the CYK parse table. Intuitively, we aim to align $e(i,j)$ and $l(i,j)$ score to maximize the agreement between model and rules. We note that we use rules only to augment the training objective, and our inference procedure is identical to that of DIORA.\n\n\n\\subsection{Training Objective}\n\\label{sec:loss}\nWe learn a model that minimizes the overall loss $L$ that is a composition of the reconstruction loss $L_{rec}$ and the rule based agreement loss or $L_{rule}$:\n$\nL = L_{rec} + \\lambda L_{rule}.\n$\nWe propose two alternatives for the loss function $L_{rule}$. \\\\\n\\textbf{Cross entropy (CE)} - For each cell $k$ in the CYK parse table, this loss (CE) tries {\\em to match the distribution (score)} of $e(i,j)$ induced by DIORA with the distribution $l(i,j)$ induced by the background knowledge: \n\\begin{align}\nL_{ce} = \\sum_{k} \\sum_{(i,j) \\in \\{k\\}} - l(i,j) \\log(e(i,j)) \\nonumber\n\\label{eq:ce}\n\\end{align}\n\\textbf{Ranking Loss (RL)} -\nWe recall from Section~\\ref{sec:diora} that the CYK algorithm finds the maximal scoring tree in a greedy manner based on the highest compatibility score $e(i,j)$ among all spans. Since the the final parse tree output by DIORA relies only on the \\textit{relative order} of the $e(i,j)$ to decide which span to merge, we propose an alternative rule-based loss that aims {\\em to match the relative order} induced by compatibility scores $e(i,j)$ of DIORA at each cell with the order induced by the scores of the rules $l(i,j)$ at that cell. We achieve this through a pairwise ranking loss defined as \n\\begin{align}\nL_{rank} = \\sum_{k} \\sum\\limits_{\\small \\substack{(i,j) \\\\ (i',j')}\\in \\{k_{trig}\\}} ( \\Delta_{(i,j),(i',j')}^l - \\Delta_{(i,j),(i',j')}^e )^2\n\\label{eq:rl} \\nonumber\n\\end{align}\n\\normalsize\nwhere $\\{k_{trig}\\} = \\{(i,j) \\in \\{k\\} | \\sum_{p=1}^P \\delta_{i,j}(p) \\neq 0\\}$, $\\Delta_{(i,j),(i',j')}^f = tanh(f(i,j) - f(i',j'))$ and $p$ is index into the rule set. The set $\\{k_{trig}\\}$ consists of all spans which have atleast one rule triggered in its cell. In cases where our rule set is not extensive enough, we would like our model's compatibility score to rely more on the reconstruction loss, and $\\{k_{trig}\\}$ ensures that a sparse rule set does not lead to bad performance. \n\n\\begin{table}[]\n\\begin{adjustbox}{width=\\linewidth}\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{Model} & \\multicolumn{1}{c}{\\textbf{F1}} & \\multicolumn{1}{c}{$\\delta$} \\\\\n\\midrule\n300D Gumbel Tree-LSTM & 25.2 & 4.2\\\\\n~~w\/o Leaf GRU & 29.0& 4.7\\\\\n300D RL-SPINN & 19.0& 8.6\\\\\n~~w\/o Leaf GRU & 18.2&8.6\\\\\nStructural Attentive(Gumble Tree LSTM) & 31.3 & 4.7\\\\\n~~w\/o Leaf GRU & 31.0 & 5.3\\\\\n\nDIORA with PP & 58.3 & 5.6\\\\ \n\\midrule\nOurs: Rule augmented (HR) + RL + PP & 60.5 & 5.7\\\\ \nOurs: Rule augmented (HR) + CE + PP & 59.0 & 5.6 \\\\\nOurs: Rule augmented (AR) + RL + PP & 60.3 & 5.7 \\\\\nOurs: Rule augmented (AR) + CE + PP & \\textbf{61.7} & 5.7 \\\\\n\\midrule \n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\caption{F1-scores of trees wrt ground truth on the MultiNLI development set. The depth ($\\delta$) is the average tree height. All reported numbers are maximum F1-score. PP refers to post-processing heuristic. HR and AR refer to the training rule sets as per Sec \\ref{sec:rule_set}. RL and CE refer to the losses from section \\ref{sec:loss}.}\n\\label{tab:quantMultiNLI}\n\\end{table}\n\n\n\n\\begin{table}[h]\n\n\\centering\n\\begin{adjustbox}{width=\\linewidth}\n\\begin{tabular}{lcccc}\n\\hline\n\\multicolumn{1}{c}{\\textbf{Model}} & \\multicolumn{1}{c}{\\textbf{F1}} & \\multicolumn{1}{c}{\\textbf{$\\delta$}} \\\\ \\hline\nLB& 13.1 & 12.4 \\\\\nRB& 16.5 & 12.4 \\\\\nRandom & 21.4 & 5.3 \\\\\nBalanced & 21.3 & 4.6 \\\\\nRL-SPINN \\cite{choi2018learning} & 13.2 & - \\\\\nST-Gumbel - GRU \\cite{yogatama2016learning}& 22.8 \u00b11.6 & - \\\\ \\hline\nPRPN-UP \\cite{shen2018neural} & 38.3 & 5.9 \\\\\nPRPN-LM& 35.0 & 6.2 \\\\\nON-LSTM \\cite{shen2018ordered}& 47.7 & 5.6 \\\\\nDIORA& 48.9 & 8.0 \\\\ \\hline\nPRPN-UP+PP & 45.2 & 6.7 \\\\\nPRPN-LM+PP & 42.4 & 6.3 \\\\\nDIORA+PP & 55.7 & 8.5 \\\\ \\hline\nNeural PCFG \\cite{kim2020pretrained} $^*$ & 50.8 & - \\\\\nCompound PCFG \\cite{kim2020compound}$^*$\n& 55.2 & -\\\\ \\hline\n\nOurs: Rule augmented (HR)\n+ RL + PP & 56.5 & 7.1\\\\ \nOurs: Rule augmented (HR)\n+ CE + PP & 55.3 & 7.2 \\\\\nOurs: Rule augmented (AR) \n+ RL + PP & 55.9 & 7.1 \\\\\nOurs: Rule augmented (AR)\n+ CE + PP & \\textbf{58.3} & 7.3 \\\\ \\hline \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{ Performance on WSJ test set for binary constituency parsing including punctuation characters. $^*$ are reported by \\cite{kim2020pretrained}.\n ($\\delta$) is the average tree height. PP refers to post-processing heuristic.}\n\\label{tab:wsj}\n\\end{table}\n \n\\section{Experiments}\n\\label{sec:expt}\nWe evaluate our rule augmented model and compare it against baselines on the tasks of unsupervised parsing, unsupervised segment recall, and phrase similarity. Results on phrase similarity are reported in the appendix.\n\n\\subsection{Data}\nWe evaluate our model on two data sets: The Wall Street Journal (WSJ) and MultiNLI. WSJ is an extraction of PennTree Bank~\\cite{article} containing human-annotated constituency parse trees. MultiNLI consists of Stanford generated parse trees ~\\cite{manning2014stanford} as the ground truth. MultiNLI is originally designed for evaluating NLI tasks, but is often also utilized to evaluate constituency parse trees. We train on the complete NLI dataset, which is a composition of the MultiNLI and SNLI train sets. We evaluate model performance on the MultiNLI dev set and WSJ test set (split 23) following the experimental setting and evaluation metrics in~\\cite{drozdov2019diora}. Further details are in the appendix.\nWe initialize our model with the trained weights of DIORA and evaluate on unsupervised constituency parsing and segment recall. We also perform the post-processing (PP) of generated trees by attaching the trailing punctuation to the root node, exactly as carried out by~\\cite{drozdov2019diora}. \n\n\\subsection{Rule Set}\n\\label{sec:rule_set}\nWe consider two rule-sets: (i) {\\bf Set of Handcrafted Rules (HR)} consists of 2500 human created CNF production rules ii) To assess robustness of the rule-augmented method to the preciseness of the rule set, we present comparison by instead using a {\\bf set of Automated Rules' (AR)} which consists of the 2500 most frequently occurring CNF production rules extracted from the trees of automatically (using the Stanford CoreNLP parser) parsed SNLI corpus. Further details about these rule sets can be found in the appendix.\nWe also use a train-set specific dictionary containing the POS (part-of-speech) tags of words in the training vocabulary for the terminal CFG productions for CYK parsing.\n\\subsection{Unsupervised Parsing}\n\nIn Table~\\ref{tab:quantMultiNLI}, we present comparison between different approaches on the MultiNLI dev set. We observe that our rule augmented approach outperforms DIORA and Tree-LSTM based methods with respect to the max-F1 score for both the ranking and cross-entropy losses. \nOn max-F1, our approach registers a maximum increase of 3.4 points. In Table \\ref{tab:wsj}, we observe that our approach outperforms the existing state-of-the-art approaches on the WSJ Full test set. Our approach registers a maximum increase of over 3.1 F1 points over DIORA. \n\n\\subsection{Constituency Segment Recall}\nIn Table \\ref{tab:segment}, we present the breakdown of constituent recall across the 6 most common types. Our approach achieves the highest recall across all the types and is the only model to perform effectively on SBAR and NP. Unlike other approaches, our approach consistently has the best recall or close to the best recall score. \\\\\nWe observe that rule augmentation using HR is more beneficial than AR with respect to precise evaluation measures such as Constituency, Segment Recall and Phrase Recall but yields smaller improvements than AR with respect to looser evaluation measures such as max F1 of Unsupervised Parsing. This can be possibly attributed to our observation that the extracted (most frequent) rules from SNLI, have (around 25\\%) higher coverage on the training set than HR, but appear to be semantically less precise. More description can be found in the appendix.\n\\begin{table}[!t]\n\\begin{adjustbox}{width=\\linewidth}\n\\begin{tabular}{l|ccclll}\n\\hline\n\\multicolumn{1}{c|}{\\textbf{Model}} & \\textbf{SBAR} & \\textbf{NP} & \\textbf{VP} & \\textbf{PP} & \\textbf{ADJP} & \\textbf{ADVP} \\\\ \\hline\nLB $\\dagger$ & 5\\% & 11\\% & 0\\% & 5\\% & 2\\% & 8\\% \\\\\nRB $\\dagger$ & 68\\% & 24\\% & 71\\% & 42\\% & 27\\% & 38\\% \\\\\nRandom $\\dagger$ & 8\\% & 23\\% & 12\\% & 18\\% & 23\\% & 28\\% \\\\\nBalanced $\\dagger$ & 7\\% & 27\\% & 8\\% & 18\\% & 27\\% & 25\\% \\\\ \\hline\nPRPN-UP \\cite{shen2018neural} & 55.4\\% & 59.8\\% & 31.6\\% & 60.2\\% & 36.0\\% & 50\\% \\\\\nPRPN-LM & 40.3\\% & 68.7\\% & 39.3\\% & 49.7\\% & 34.2\\% & 39.2\\% \\\\\nDIORA & 61.3\\% & 76.7\\% & 62.8\\% & 59.5\\% & 60.4\\% & 69.3\\% \\\\ \\hline\nPRPN (tuned)$\\dagger$ & 50\\% & 59\\% & 46\\% & 57\\% & 44\\% & 32\\% \\\\\nON (tuned) \\cite{shen2018ordered} & 51\\% & 64\\% & 41\\% & 54\\% & 38\\% & 31\\% \\\\\nNeural PCFG \\cite{kim2020pretrained} & 52\\% & 71\\% & 33\\% & 58\\% & 32\\% & 45\\% \\\\\nCompound PCFG \\cite{kim2020compound} & \\multicolumn{1}{l}{56\\%} & \\multicolumn{1}{l}{74\\%} & \\multicolumn{1}{l}{41\\%} & 68\\% & 40\\% & 52\\% \\\\ \\hline\nOurs: Rule augmented (HR)+ RL & \\textbf{71.1\\%} & 77.2\\% & 65.8\\% & 59.4\\% & \\textbf{62.9\\%} & 69.5\\% \\\\\nOurs: Rule augmented (HR)+ CE & \\multicolumn{1}{l}{68.3\\%} & \\multicolumn{1}{l}{75.4\\%} & \\multicolumn{1}{l}{66.5\\%} & \\textbf{60.5\\%} & 61\\% & \\textbf{70.8\\%} \\\\\nOurs: Rule augmented (AR)+ RL & \\multicolumn{1}{l}{71\\%} & \\multicolumn{1}{l}{76.4\\%} & \\multicolumn{1}{l}{\\textbf{69.1\\%}} & 58.6\\% & 61\\% & 64.8\\% \\\\\nOurs: Rule augmented (AR)+ CE & \\multicolumn{1}{l}{70\\%} & \\multicolumn{1}{l}{\\textbf{77.5\\%}} & \\multicolumn{1}{l}{67\\%} & 58.6\\% & \\textbf{62.2\\%} & 70\\% \\\\ \\hline \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{Segment recall from WSJ by phrase type; $\\dagger$ are reported by~\\citet{kim2020pretrained}. } \n\\label{tab:segment}\n\n\\end{table}\n \n\n\n\\section{Conclusion}\nIn this work, we leverage linguistically grounded and domain agnostic CFG rules for language to induce parse trees and representations of constituent spans. We show that our approach augmented with generic, linguistically grounded grammatical rules, is easily able to outperform previous methods on constituency parsing and obtain higher segment recall. \n\n\\bibliographystyle{acl_natbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\n\nThe Liouville equation in two dimensions, which has the form\n\\begin{equation}\\label{eq:prescribing curvature equation}\n -\\Delta u= \\widetilde{K} e^{2u}-K,\n\\end{equation}\nfor some given functions~$K,\\widetilde{K}$ on a surface~$M$, has been extensively studied and has wide applications in geometry and physics. \nA typical example is the prescription of curvature.\nLet~$g$ be a Riemannian metric on a surface~$M$ with Gaussian curvature~$K=K_g$ and let~$\\widetilde{K}$ be a given function on~$M$. \nThe question is whether there exists a functions~$u \\in C^\\infty(M)$ such that the conformal metric~$\\widetilde{g}=e^{2u}g$ has Gaussian curvature~$\\widetilde{K}$, \nsee e.g. \\cite{changyang1987prescribing, kazdan1975scalar}. \n\nSince the Gaussian curvature for~$\\widetilde{g}$ is given by~$e^{-2u}(K_g-\\Delta_g u)$, the problem is equivalent to the solvability of \nequation~\\eqref{eq:prescribing curvature equation}. \nObserve that the conformal factor~$u$ within the conformal class of~$[g]$ can be found as a critical point of the following functional:\n\\begin{equation}\nI(u):=\\int_M \\left( |\\nabla u|^2+2K_gu-\\widetilde{K}e^{2u} \\right) \\dv_g.\n\\end{equation}\n\nWhen~$M$ is a closed Riemann surface, which is the case we are interested in for this paper, \nthe function~$\\widetilde{K}$ has to satisfy the Gauss--Bonnet formula with respect to the new metric~$\\widetilde{g}$.\nWhen~$\\widetilde{K}$ is constant with the sign compatible with the Gauss--Bonnet formula, the equation is always solvable, according to the~\\emph{uniformization theorem}. \nFor non-constant~$\\widetilde{K}$, though not being totally solved, we have a good understanding of the problem in most cases, see e.g.~\\cite[Chapter 5]{schoenyau1994lectureOnDG-I} and ~\\cite[Chapter 6]{aubin1998somenonlinear}. \n\n\\medskip\n\nMore recently, equation~\\eqref{eq:prescribing curvature equation} has been studied in the context of hyperelliptic curves and of the Painlev\\'e equations, see \\cite{chai-lin-wang} and \\cite{chen-kuo-lin}, respectively. \n\n\\\n\nEquation~\\eqref{eq:prescribing curvature equation} plays also an important role in mathematical physics. On one hand, it arises in Electroweak and Chern-Simons self-dual vortices, see \\cite{spruck-yang,tarantello,yang}. On the other hand, it appears in the Liouville field theory with applications to string theory, see \\cite{ops1988extremals,polya1,polya2}. See also \\cite{hawking} for a recent connection between \\eqref{eq:prescribing curvature equation} and the Hawking mass.\n \n\\\n\nMotivated by the supersymmetric extension of the Liouville theory, the authors in~\\cite{jost2007superLiouville} \nintroduced the following so-called \\emph{super-Liouville functional}:\n\\begin{equation}\n \\widetilde I(u,\\psi):=\\int_M \\biggr( |\\nabla u|^2+ 2K_g u-e^{2u} +2\\left<(\\D+e^u)\\psi,\\psi\\right> \\biggr) \\dv_g, \n\\end{equation}\nwhere $\\D$ is the Dirac operator acting on spinors $\\psi$, see Subsection~\\ref{subsec:spinor} for precise definitions. In a series of works they performed blow-up analysis and studied the compactness of the solution spaces under weak assumptions and in various setting; see e.g.~\\cite{jost2007superLiouville, jost2009energy,jost2014qualitative,jost2015LocalEstimate} and the references therein. For the role of the super-Liouville equations in physics we refer to \\cite{super1,super2,super3}. One should note that the sign conventions adopted above are adapted to the sphere case. \n\nIn this paper we consider the problem posed on a closed Riemann surface of genus~$\\genus> 1$. \nIn this case the coefficients in the action functional need to be adapted to the Gauss--Bonnet formula. \nLet~$g$ be a Riemannian metric compatible with the given complex structure.\nWe are going to consider the following functional: \n\\begin{equation}\\label{eq:super-Liouville functional}\n J_\\rho(u,\\psi)\\coloneqq \\int_M \\biggr( |\\nabla u|^2+2K_gu+e^{2u}\n +2\\left<(\\D-\\rho e^u)\\psi,\\psi\\right> \\biggr) \\dv_g,\n\\end{equation}\nwhere the parameter~$\\rho$ is a positive constant. \nWe are adopting a different notation from that in~\\cite{jost2007superLiouville}, making our choice compatible with equation~\\eqref{eq:prescribing curvature equation}.\nThe Euler--Lagrange equation for~$J_\\rho$ is \n\\begin{equation}\\label{eq:EL for super-Liouville}\n\\tag{EL}\n \\begin{cases}\n \\Delta_g u{}=&{}e^{2u}+K_g-\\rho e^u|\\psi|^2, \\vspace{0.2cm}\\\\\n \\D_g\\psi{}=&{}\\rho e^u\\psi,\n \\end{cases}\n\\end{equation}\nwhich takes the name of \\emph{super-Liouville equations}. The system~\\eqref{eq:EL for super-Liouville} clearly admits the {\\em trivial solution} $(u_*,0)$, where~$u_*$ satisfies \n\\begin{equation}\n -\\Delta u_*= -e^{2u_*}-K_g\n\\end{equation}\nand whose existence is given by the uniformization theorem. \nThis is also a~\\emph{ground state solution} in the sense that it has minimal critical level: this follows from the fact that the spinorial part does not affect the critical levels, while the scalar component of the functional is coercive and convex. The latter properties \nalso yield uniqueness of such a trivial solution. \nThe aim of the present paper is to find a solution with non-zero spinor part, a so-called \\emph{non-trivial solution}. \n\n\\\n\n\\textbf{Conformal symmetry and reduction to uniformized case.}\nSystem~\\eqref{eq:EL for super-Liouville} admits a conformal symmetry in the following sense. \nSuppose that~$(u,\\psi)$ is a solution of~\\eqref{eq:EL for super-Liouville}, \nlet~$v\\in C^\\infty(M)$ and consider the metric~$\\widetilde{g}\\coloneqq e^{2v}g$. \nThere exists an isometric isomorphism~$\\beta\\colon S_g\\to \\widetilde{S}_{\\widetilde{g}}$ of the spinor bundles corresponding to different metrics such that\n\\begin{equation}\\label{eq:D-conf}\n \\widetilde{\\D}_{\\widetilde{g}}\\left(e^{-\\frac{v}{2}}\\beta(\\psi)\\right)\n =e^{-\\frac{3}{2}v}\\beta(\\D_g\\psi),\n\\end{equation}\nsee e.g.~\\cite{ginoux2009dirac, hitchin1974harmonicspinors}, where we are using the notation from~\\cite{jost2018symmetries}.\nThus the pair\n$$\n \\begin{cases}\n \\widetilde{u}=u-v, \\vspace{0.2cm}\\\\\n \\widetilde{\\psi}= e^{-\\frac{u}{2}}\\beta(\\psi),\n \\end{cases}\n$$\nsolves the system \n\\begin{align}\n \\Delta_{\\widetilde{g}}\\widetilde{u} \n ={}&e^{-2v}\\Delta_g(u-v)= e^{-2v}(e^{2u}+K_g-\\rho e^{u}|\\psi|^2-\\Delta_g v) \\\\\n ={}&e^{2(u-v)}+e^{-2v}(K_g-\\Delta_g v)-\\rho e^{u-v}|e^{-\\frac{v}{2}} \\beta(\\psi)|^2 \\\\\n ={}&e^{2\\widetilde{u}}+{K}_{\\widetilde{g}}\n -\\rho e^{\\widetilde{u}}|\\widetilde{\\psi}|^2, \\\\ \n\\widetilde{\\D}_{\\widetilde{g}} \\widetilde{\\psi}\n ={}&\\rho e^{-\\frac{3}{2}v}\\beta(e^u\\psi)\n =\\rho e^{u-v}\\left(e^{-\\frac{1}{2}v}\\beta(\\psi)\\right)\n =\\rho e^{\\widetilde{u}}\\widetilde{\\psi}, \n\\end{align}\nanalogous to \\eqref{eq:EL for super-Liouville}. \nTherefore, we can work with a convenient background metric inside the given conformal class.\nW.l.o.g., recalling that the genus is larger than one, we assume that the background metric~$g_0$ is uniformized, meaning that~$K_{g_0}\\equiv -1$: notice that \nsuch a metric is unique. \nIn this case the trivial solution is given by~$(0,0)$: \nthe main result of the paper is the existence of a non-trivial min-max solution obtained via a variational approach. \n\n\\begin{thm} \\label{thm}\n Let~$M$ be a closed Riemann surface of genus~$\\genus>1$ with Riemannian metric~$g$.\n Let~$g_0\\in [g]$ be a conformal uniformized metric, i.e.~$K_{g_0}\\equiv -1$, and suppose that the spin structure is chosen so that~$0\\notin \\Spec(\\D_{g_0})$.\n Then for any~$\\rho\\notin\\Spec(\\D_{g_0})$, there exists a non-trivial solution to~\\eqref{eq:EL for super-Liouville}. \n\\end{thm}\n\nWe stress that this is the first non-trivial existence result for this class of problems. Moreover, observe that by \\eqref{eq:D-conf} ~$\\dim\\ker(\\D_g)$ is a conformal invariant, and the condition~$0\\notin \\Spec(\\D_{[g]})$ is valid for many spin structures and conformal structures, as it will be explained later. \n\n\\begin{rmk}\n Note that the spinor bundle~$S\\to M$ admits global automorphisms, e.g. the quaternionic structures, which form a group.\n These are parallel with respect to~$\\nabla^s$ and commute with the Clifford multiplications by tangent vectors, see~\\cite[Sect. 2]{jost2016regularity}.\n The functional~$J_\\rho$ is thus invariant under the actions of such isometries. \nIt follows that there exist more than one non-trivial solution (at least eight, which is the cardinality of the quaternion group). \nGiven a solution ~$(U,\\Psi)$, an intuitive example is the antipodal solution~$(U,-\\Psi)$, which is in the orbit of the quaternionic structure group actions. \n\\end{rmk}\n\nConcerning the case of genus one, i.e. when the base surface is a torus, the problem might not be well-defined. \nIndeed, if we take~$\\widetilde{K}$ and~$K$ to be zero, then the system~\\eqref{eq:EL for super-Liouville} has only trivial solutions of the form~$(a,0)$ where~$a\\in\\R$. \nMeanwhile in the sphere case, where both~$\\widetilde{K}$ and~$K$ should be~$1$, the functional turns out to be even more strongly indefinite, and admits \nneither the classical mountain pass nor the linking geometry. \nIn the genus-one case it might be interesting to consider the case of changing-sign $\\tilde{K}$, as it was done in \n\\cite{kazdan1975scalar} for the prescribed Gaussian curvature problem. \n\n\\medskip\n\nThe main difficulty in studying \\eqref{eq:EL for super-Liouville} is that the Dirac operator is strongly indefinite: the spectrum of~$\\D$ is real and symmetric with respect to the origin. \nThe classical theory for variational problems involving Laplacians or Schr\\\"odinger operators, where the positive parts usually dominates the behavior of the functional, fails to work for Dirac type actions.\nThere were methods developed for general strongly indefinite variational problems, see e.g.~\\cite{benci1982oncritical, benci1979critical, hulshof1993differential}, but they are not directly applicable to Dirac operators.\nDirac operators usually relates more closely to the geometry and topology of the spin manifolds.\nRecently several attempts have been made to attack such problems.\nWith suitable nonlinearities as perturbation adding to the geometric equations, T. Isobe made remarkable progress in adapting the classical theory of calculus of variations to the Dirac setting~\\cite{isobe2010existence,isobe2011nonlinear, isobe2019onthemultiple}. \nCombined with the methods of Robinowitz-Floer homology, A. Maalaoui and V. Martino also obtained existence results of some nonlinear Dirac type equations, see~\\cite{maalaoui2013rabinowitz,maalaoui2015therabinowitz,maalaoui2017characterization} and the references therein.\nIn the case of super-Liouville equations we have to deal with an exponential nonlinearity, which does not fit in the above settings. \nMoreover, we are directly facing a geometric problem without auxiliary nonlinear perturbations, which is usually harder to deal with. \n\n\\\n\nThe article is organized in the following way.\nIn the second section we introduce some preliminaries in spin geometry and discuss existence of harmonic \nspinors depending on the genus and on the conformal class. We also introduce suitable Sobolev spaces to work \nwith and the Moser-Trudinger inequality. \nIn the third section we tackle the strong-indefiniteness of the functional by building a natural constraint which defines a generalized Nehari manifold~$N$. \nWe then verify the Palais--Smale condition for~$J_\\rho|_N$ by showing first some a-priori bounds and then \nproving strong subsequential convergence. \nFor suitable $\\rho$ we finally show either mountain pass or linking geometry on the Nehari manifold which yield\nthe existence of a min-max critical point for~$J_\\rho$: the \ndetails of this construction are given in the last section. \n\n\\\n\n\\noindent {\\bf Acknowledgments.}\nA.M. has been partially supported by the projects {\\em Geometric Variational Problems} and {\\em Finanziamento a supporto della ricerca di base} from Scuola Normale Superiore. \nA.J. and A.M. has been partially supported by MIUR Bando PRIN 2015 2015KB9WPT$_{001}$. They are also members of GNAMPA as part of INdAM.\nA.J. and R.W. are supported by the Centro di Ricerca Matematica 'Ennio de Giorgi'. \n\n\n\\\n\n\\section{Preliminaries}\n\nWe will assume some background in spin geometry and Sobolev spaces.\nFor detailed material one can refer to~\\cite{ammann2003habilitation, ginoux2009dirac, gilbarg2001elliptic, lawson1989spin}. \n\n\\\n\n\\subsection{Spinor bundles and Dirac operator} \\label{subsec:spinor}\nHere we introduce our setting and fix the notation. \nLet~$M$ be a closed Riemann surface with a fixed conformal structure and of genus~$\\genus$. \nLet~$g$ be a Riemannian metric in the given conformal class and denote the Gaussian curvature by~$K_g$. \nThe orthonormal frame bundle~$P_{\\SO}(M,g)\\to M$ is then a principal~$\\SO(2)$ bundle.\nLet~$\\Spin(2)= U(1)\\to \\SO(2)$ be the two-fold covering of the circle. \nA \\emph{spin structure} is given by a principal~$\\Spin(2)$ bundle~$P_{\\Spin}(M,g)\\to M$ together with an equivariant two-fold covering\n\\begin{equation}\n P_{\\Spin}(M,g)\\to P_{\\SO}(M,g).\n\\end{equation}\nIn dimension two such double coverings always exist; moreover they are in one-to-one correspondence with the elements in~$H^1(M;\\mathbb{Z}_2)$, see e.g. \\cite[Chapter 2]{lawson1989spin}.\nThis cohomology group has cardinality~$2^{2\\genus}$. \n\nLet~$S\\equiv S_g\\to M$ be the associated spinor bundle with a real Riemannian structure~$g^s$ and induced spin connection~$\\nabla^s$:\nsections of~$S$ are called~\\emph{spinors}.\nRecall that the \\emph{Dirac operator}~$\\D$ acting on spinors is defined as the composition of the following chain\n\\begin{equation}\n \\Gamma(S)\\xrightarrow{\\nabla^s} \\Gamma(T^*M\\otimes S)\\xrightarrow{\\cong} \\Gamma(TM\\otimes S)\\xrightarrow{\\cliff}\\Gamma(S),\n\\end{equation}\nwhere the second isometric isomorphism is given by the identification via the metric~$g$, the third arrow~$\\cliff$ denotes the Clifford multiplication, and the~$\\End(S)$-valued map~$\\cliff\\colon TM\\to \\End(S)$ satisfies the following Clifford relation:\n\\begin{equation}\n \\cliff(X)\\cliff(Y)+\\cliff(Y)\\cliff(X)=-2g(X,Y), \\qquad \\forall X,Y\\in\\Gamma(TM).\n\\end{equation}\nLater, for simplicity, we will write~$X\\cdot\\psi$ for~$\\cliff(X)\\psi$, where~$X\\in\\Gamma(TM)$ and~$\\psi\\in\\Gamma(S)$.\nIn terms of a local orthonormal frame~$(e_i)_{i=1,2}$ we then have the {\\em Dirac operator}\n\\begin{equation}\n \\D\\psi=\\sum_{i}\\cliff(e_i)\\nabla^s_{e_i}\\psi,\\quad \\forall \\psi\\in \\Gamma(S).\n\\end{equation}\nThis is a self-adjoint elliptic operator of first order, and it has a finite-dimensional kernel consisting of \\emph{harmonic spinors}.\nThe dimension of the space of harmonic spinors is a conformal invariant, but it depends on the choice of spin structures and the conformal structures in general.\nThe Bochner-Lichnerowicz formula\n\\begin{equation}\n \\D^2=(\\nabla^s)^* \\nabla^s+\\frac{Scal}{4}\n\\end{equation}\nimplies that there is no non-trivial harmonic spinor if~$Scal\\ge0$ and~$Scal\\not\\equiv 0$. \nIn particular, there is no harmonic spinor on the 2-sphere with arbitrary metric (since there is only one conformal structure on the 2-sphere).\nHowever, when the genus~$\\genus$ is greater than or equal to $1$, there might exist non-trivial harmonic spinors for some choice of spin structures. \nThe dimensions of the spaces of harmonic spinors have been computed in literature e.g.~\\cite{hitchin1974harmonicspinors, bar1992harmonic, bores1994harmonic}. \nWe summarize some facts here to have a picture of the different cases.\n\n\\\n\n\\subsection{Examples of Riemann surfaces with no non-trivial harmonic spinors}\nHere we give some examples of Riemann surfaces having negative Euler characteristic~$2\\pi\\chi(M)=4\\pi(1-\\genus)<0$ but admitting no non-trivial harmonic spinors. \n\nAny element~$\\alpha\\in H^1(M,\\mathbb{Z}_2)$ determines a spin structure~$\\xi(\\alpha)$, as well as a holomorphic line bundle~$\\mathscr{L}_\\alpha$ such that~$\\mathscr{L}_\\alpha\\otimes_\\C \\mathscr{L}_\\alpha= \\mathscr{K}_M$, where~$\\mathscr{K}_M$ denotes the canonical line bundle of~$M$, see e.g. \\cite{hitchin1974harmonicspinors, lawson1989spin}.\nDenote by~$\\mathscr{O}(\\mathscr{L_\\alpha})$ the sheaf of germs of holomorphic sections of the holomorphic line bundle~$\\mathscr{L}_\\alpha$, and set~$h_{\\alpha,g}^0 =\\dim H^0(M, \\mathscr{O}(\\mathscr{L}_\\alpha))$. \nIf the associated spinor bundle~$S\\equiv S(\\alpha,g)$ admits a space of harmonic spinors of dimension~$h_{\\xi(\\alpha),g}$, then\n\\begin{equation}\n h_{\\xi(\\alpha),g}=2h^0_{\\alpha,g}. \n\\end{equation}\nIt is known that, for a Riemann surface~$M$ of genus~$\\genus$, there are precisely~$2^{\\genus-1}(2^\\genus+1)$ spin structures~$\\alpha$ on~$M$ for which ~$h^0_{\\alpha,g}$ is an even number (such spin structures are called \\emph{even spin structures} on~$M$), and for the other~$2^{\\genus-1}(2^{\\genus}-1)$ spin structures the number~$h^0_{\\alpha,g}$ is odd (\\emph{odd spin structures}). \n \nFor~$\\genus=1$ ~$M$ is topologically a torus, and for any conformal structure~$[g]$ we have four spin structures: three even spin structures with no non-trivial harmonic spinors and one odd spin structure (the trivial one~$\\alpha= 0$) with one-dimensional space of positive harmonic spinors (hence~$h_{\\xi(0);g}=2$). \n\nFor~$\\genus=2$ the description is similar, namely for any conformal structure~$[g]$ there are ten even spin structures with no non-trivial harmonic spinors and six odd spin structures with one-dimensional space of positive harmonic spinors (hence~$h_{\\xi(0)}=2$).\n\nThese are the known cases where the dimension of~$\\ker(D)$ is independent of the choice of metric~$g$ (i.e. the choice of the Riemann surface structure on~$M$). \nWhen the genera become larger, the dimension of the kernels generally depends on the conformal class.\nEven in this case we still have many examples where there are no non-trivial harmonic spinors.\n\nRecall that a hyperelliptic Riemann surface is a complex projective curve admitting a rational surjective map onto~$\\C P^1$ which is 2-to-1 up to a finite set of branching points.\nAll Riemann surfaces of genera~$\\genus\\le 2$ are hyperelliptic, while there exist non-hyperelliptic surfaces of all genera~$\\genus\\ge 3$. \n\nFor the hyperelliptic case, C. B\\\"ar~\\cite{bar1992harmonic} showed that the spin structures correspond one-to-one to the pairwise inequivalent square roots of the canonical divisor, and in terms of a suitably defined \\emph{weight} of the divisors, he also clarified the dimensions~$h^0$ of the kernels: \n\\begin{enumerate}\n \\item if~$M$ is hyperelliptic with~$\\genus=2k+1$, \n \\begin{itemize}\n \\item there is exactly one spin structure of weight~$\\genus-1$ and in this case~$h^0=\\frac{\\genus+1}{2}=k+1$;\n \\item for~$w=1,3,5,\\cdots,\\genus-2$, there are exactly~$\\binom{2\\genus+2}{\\genus-w}$ spin structures of weight~$w$ and in this case~$h^0=\\frac{w+1}{2}$;\n \\item there are exactly~$\\binom{2\\genus+1}{\\genus}$ spin structures of weight~$-1$ and in this case~$h^0=0$;\n \\end{itemize}\n \\item if~$M$ is hyperelliptic with~$\\genus=2k$,\n \\begin{itemize}\n \\item there is exactly~$2\\genus+2$ spin structure of weight~$\\genus-1$ and in this case~$h^0=[\\frac{\\genus+1}{2}]=k$;\n \\item for~$w=1,3,5,\\cdots,\\genus-1$, there are exactly~$\\binom{2\\genus+2}{\\genus-w}$ spin structures of weight~$w$ and in this case~$h^0=\\frac{w+1}{2}$;\n \\item there are exactly~$\\binom{2\\genus+1}{\\genus}$ spin structures of weight~$-1$ and in this case~$h^0=0$;\n \\end{itemize}\n\n\\end{enumerate}\nFor non-hyperelliptic surfaces, there are also known examples where the dimensions of kernels are computed. \n\nFor a genus~$\\genus=3$ non-hyperelliptic surface, among the~$2^{2\\genus}=64$ spin structures there are~$28$ odd ones with~$h^0=1$ and~$36$ even ones with~$h^0=0$. \n\nThe case for~$\\genus=4$ non-hyperelliptic surfaces is different: there are in total~$2^{2\\genus}= 256$ spin structures, 120 of them are odd with~$h^0=1$, and for the other~$136$ even spin structures, one of the followings may happen:\n\\begin{enumerate}\n \\item[(I)] there exists a unique even spin structure with~$h^0=2$, while the other~$135$ even spin structures have~$h^0=0$;\n \\item[(II)] all the 136 spin structures have~$h^0=0$.\n\\end{enumerate}\nA non-hyperelliptic Riemann surface is called of type~(I) or~(II) if it satisfies the corresponding above conditions. \nBoth classes are non-empty. \n\n\\\n\n\\subsection{Sobolev spaces for spinors}\n\nThe spinor bundle~$S=S_g$ has a Riemannian structure~$g^s$ and a spin connection~$\\nabla^s$ induced from the Levi-Civita connection.\nThen we can define the usual Sobolev spaces with integer differentiability, namely ~$W^{k,p}(S)$ consists of the spinors whose~$k$-th covariant derivatives are in~$L^p$ for~$k\\in \\mathbb{N}$ and~$p\\in [1,+\\infty]$ and~$W^{-k,q}(S)\\coloneqq (W^{k,p}(S))^*$ where~$q$ is the H\\\"older conjugate of~$p$. \nHere we will consider also fractional Sobolev exponents, see the discussion in the sequel. \n\nRecall that~$\\D=\\D_g$ is a first order elliptic operator which is essentially self-adjoint. \nThe spectrum~$\\Spec(\\D)$ is discrete and consists of real eigenvalues,~$\\Spec_0(\\D)\\cup\\{\\lambda_k\\}_{k\\in\\mathbb{Z}\\backslash\\{0\\}}$, where~$\\Spec_0(\\D)$ stands for the zero element in the spectrum (or the empty set) while the lambda's are the non-zero eigenvalues, indexed by~$\\mathbb{Z}_*\\equiv \\mathbb{Z}\\backslash\\{0\\}$ in an increasing order (in absolute value) and counted with multiplicities:\n\\begin{equation}\n -\\infty \\leftarrow\\cdots\\le \\lambda_{-l-1}\\le \\lambda_{-l}\\le\\cdots\\le \\lambda_{-1}\\le 0 \n \\le \\lambda_1\\le \\cdots \\le \\lambda_k \\le \\lambda_{k+1}\\le \\cdots \\to +\\infty.\n\\end{equation}\nMoreover, the spectrum is symmetric with respect the the origin when~$\\dim M=2$. \nLet~$\\varphi_k$ be the eigenspinor corresponding to~$\\lambda_k$, ~$k\\in\\mathbb{Z}_*$ with~$\\|\\varphi_k\\|_{L^2(M)}=1$, and let ~$\\varphi_{_{0,j}}$, ~$1\\le j\\le h^0$, be an orthonormal basis of~$\\ker(\\D)$. \nThese together form a complete orthonormal basis of~$L^2(S)$: \n any spinor~$\\psi\\in\\Gamma(S)$ can be expressed in terms of this basis as \n\\begin{equation}\\label{eq:spinor in basis}\n \\psi=\\sum_{k\\in \\mathbb{Z}_*}a_k\\varphi_k\n +\\sum_{1\\le j\\le h^0} a_{0,j}\\varphi_{_{0,j}}, \n\\end{equation}\nand the Dirac operator acts as\n\\begin{equation}\n \\D\\psi= \\sum_{k\\in\\mathbb{Z}_*} \\lambda_k a_k \\varphi_k. \n\\end{equation}\nFor any~$s>0$, the operator~$|\\D|^s\\colon \\Gamma(S)\\to \\Gamma(S)$ is defined as \n\\begin{equation}\n |\\D|^s\\psi =\\sum_{k\\in\\mathbb{Z}_*} |\\lambda_k|^s a_k\\varphi_k,\n\\end{equation}\nprovided that the right-hand side belongs to~$L^2(S)$. The domain of~$|\\D|^s$ is \n\\begin{equation}\n H^s(S)\\coloneqq \\left\\{\\psi\\in L^2(S)\\mid \\int_M\\left<|\\D|^s\\psi,|\\D|^s\\psi\\right>\\dv_g <\\infty \\right\\}, \n\\end{equation}\nwhich is a Hilbert space with inner product\n\\begin{equation}\n \\left<\\psi,\\phi\\right>_{H^s} \n = \\left<\\psi,\\phi\\right>_{L^2}\n +\\left<|\\D|^s\\psi, |\\D|^s\\phi\\right>_{L^2}.\n\\end{equation}\nFor~$s=k\\in\\mathbb{N}$,~$H^k(S)=W^{k,2}(S)$ and the above norm is equivalent to the Sobolev~$W^{k,2}$ norm.\nFor~$s<0$,~$H^{s}(S)$ is by definition the dual space of~$H^{-s}(S)$. \n\nSince~$S$ has finite rank, the general theory for Sobolev's embedding on closed manifold continues to hold here.\nIn particular, for~$0\\dv_g\n\\end{equation}\nis well-defined. \nNote that for~$\\psi\\in H^{\\frac{1}{2}}(S)$ we have~$\\D\\psi\\in H^{-\\frac{1}{2}}(S)$, which is defined in the distributional sense. \nThus we can define the duality pairing\n\\begin{equation}\n \\left<\\D\\psi,\\psi\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}} \\in\\R.\n\\end{equation}\nOn the other hand, by the expression~\\eqref{eq:spinor in basis} we see that the function\n\\begin{equation}\n g^s_x \\left(\\D\\psi(x),\\psi(x)\\right)\n\\end{equation}\nis in~$L^1(M)$, whose integral is exactly given by~$\\sum_{k\\in\\mathbb{Z}_*} \\lambda_k a_k^2 <\\infty$.\nBy this we validate the Dirac action in the equivalent form\n\\begin{equation}\n \\left<\\D\\psi,\\psi\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}}\n =\\int_M \\left<\\D\\psi(x),\\psi(x)\\right>_{g^s(x)}\\dv_g(x). \n\\end{equation}\n\n\nSuppose~$h^0=0$, i.e. there are no non-trivial harmonic spinors.\nThen the Dirac operator~$\\D$ is invertible. \nSplitting into the positive and negative parts of the spectrum~$\\Spec(\\D)$, we have the decomposition \n\\begin{equation}\\label{eq:split}\n H^{\\frac{1}{2}}(S)= H^{\\frac{1}{2},+}(S)\\oplus H^{\\frac{1}{2},-}(S).\n\\end{equation}\nLet~$\\psi=\\psi^+ +\\psi^-$ be decomposed accordingly: then, \n\\begin{align}\n \\int_M\\left<\\D\\psi^+,\\psi^+\\right>\\dv_g \n =\\int_M\\left<|\\D|^{\\frac{1}{2}}\\psi^+,|\\D|^{\\frac{1}{2}}\\psi^+\\right>\\dv_g \\ge \\lambda_1(\\D_g)\\|\\psi^+\\|_{L^2(M)}^2, \n\\end{align}\nwhere~$\\lambda_1$ is the first positive eigenvalue of~$\\D=\\D_g$.\nHence\n\\begin{align}\n \\|\\psi^+\\|^2_{H^{\\frac{1}{2}}}\n =&\\|\\psi^+\\|^2_{L^2}+\\||\\D|^{\\frac{1}{2}}\\psi^+\\|_{L^2}^2 \\\\ \n \\le&(\\lambda_1(\\D_g)+1)\\||\\D|^{\\frac{1}{2}}\\psi^+\\|^2_{L^2}\n \\le(\\lambda_1(\\D_g)+1)\\|\\psi^+\\|^2_{H^{\\frac{1}{2}}}. \n\\end{align}\nThat is, for a given~$g$, the integral~$\\int_M \\left<\\D\\psi^+,\\psi^+\\right>\\dv_g$ defines a norm on~$H^{\\frac{1}{2},+}(S)$ equivalent to the Hilbert's. \nSimilarly, on~$H^{\\frac{1}{2},-}(S)$ there is an equivalent norm given by\n\\begin{equation}\n -\\int_M \\left<\\D\\psi^-,\\psi^-\\right>\\dv_g\n =\\||\\D|^{\\frac{1}{2}}\\psi^-\\|^2_{L^2}.\n\\end{equation}\nConsequently,\n\\begin{equation}\n \\int_M \\left[ \\left<\\D\\psi^+,\\psi^+\\right>-\\left<\\D\\psi^-,\\psi^-\\right> \\right] \\dv_g\n =\\||\\D|^{\\frac{1}{2}}\\psi^+\\|^2_{L^2}\n +\\||\\D|^{\\frac{1}{2}}\\psi^-\\|^2_{L^2}\n\\end{equation}\ndefines a norm equivalent to the~$H^{\\frac{1}{2}}$-norm. \n\n\\\n\n\\subsection{Moser--Trudinger embedding}\nAnother space we use frequently is~$H^1(M)=W^{1,2}(M,\\R)$. \nConsider the subspace in $H^1(M)$ of the functions with zero average \n\\begin{equation}\n H^1_0(M)\\coloneqq \\left\\{u\\in H^1(M)\\mid \\int_M u\\dv_g=0\\right\\}.\n\\end{equation}\nThen~$H^1(M)= \\R\\oplus H^1_0(M)$, and any $u \\in H^1(M)$ can be written as~$u=\\bar{u}+\\widehat{u}$ where~$\\bar{u}=\\fint_M u\\dv_g $ denotes the average of~$u$. By Poincar\\'e's inequality,~$\\|\\nabla \\widehat{u}\\|_{L^2}$ defines a norm equivalent to~$\\|\\widehat{u}\\|_{H^1}$ on~$H^1_0(M)$, and \n\\begin{equation}\n |\\bar{u}|+\\|\\nabla\\widehat{u}\\|_{L^2}\n\\end{equation}\n a norm equivalent to~$\\|u\\|_{H^1}$. \nThe Sobolev embedding theorems imply that for any~$p<\\infty$, ~$H^1(M)$ embeds into~$L^p(M)$ continuously and compactly. \nFurthermore, the Moser--Trudinger inequality states that there exists~$C>0$ such that \n\\begin{equation}\n \\int_M \\exp\\left(\\frac{4\\pi|\\widehat{u}|^2}{\\|\\nabla\\widehat{u}\\|^2_{L^2(M)}}\\right)\\dv_g\\le C. \n\\end{equation}\nAs a consequence \n\\begin{equation}\n 8\\pi\\log\\int_M e^{\\widehat{u}}\\dv_g\n \\le\\frac{1}{2}\\int_M |\\nabla\\widehat{u}|^2\\dv_g+ C.\n\\end{equation}\nThis implies that~$e^{u}$ is~$L^p$ integrable for any~$p> 0$. \nMoreover, the map\n\\begin{equation}\n H^1(M)\\ni u\\mapsto e^u\\in L^1(M)\n\\end{equation}\nis compact (see e.g.~\\cite[Theorem 2.46]{aubin1998somenonlinear}). \nIt follows that the maps~$H^1(M)\\ni u\\mapsto e^u\\in L^p(M)$ are compact for all~$p > 0$. \n\n\\section{A natural constraint and the Palais-Smale condition}\n\nIt is standard to prove that the functional\n\\begin{equation}\n J_\\rho\\colon H^1(M)\\times H^{\\frac{1}{2}}(S)\\to \\R\n\\end{equation}\ndefined in formula \\eqref{eq:super-Liouville functional} is of class~$C^1$. \nThe critical points of~$J_\\rho$, which are weak solutions of~\\eqref{eq:EL for super-Liouville}, are actually smooth.\nTo see this we can use the argument from~\\cite{jost2007superLiouville}.\nNote that, although the authors there are using different Banach spaces, the proof goes quite similarly and is omitted here.\nAlternatively, note that~$u\\in H^1(M)$ implies~$e^{u}\\in L^p(M)$ for any~$p<\\infty$,i.e. the equation is actually subcritical and we can appeal to a bootstrap argument to obtain the full regularity. \n \nTo obtain a non-trivial solution to the system~\\eqref{eq:EL for super-Liouville} we employ a min-max approach.\nAs observed, thanks to the conformal covariance of the system, it is sufficient to consider the uniformized metric. \nFrom now on we assume that~$g$ has constant Gaussian curvature~$K\\equiv-1$. \nFor this choice we then look for non-trivial critical points of the functional\n\\begin{equation}\\label{eq:functional-uniformized metric}\n J_\\rho(u,\\psi)\n =\\int_M \\biggr( |\\nabla u|^2-2u+e^{2u}+2\\left(\\left<\\D\\psi,\\psi\\right>-\\rho e^u|\\psi|^2\\right) \\biggr)\\dv_g,\n\\end{equation}\nwhich are non-trivial solutions of the system\n\\begin{equation}\\label{eq:EL-uniformized metric}\\tag{$EL_0$} \n \\begin{cases}\n \\Delta_g u=e^{2u}-1-\\rho e^u|\\psi|^2, \\vspace{0.2cm}\\\\\n \\D_g\\psi= \\rho e^u\\psi. \n \\end{cases} \n\\end{equation}\nThe argument in the sequel is simplified by this assumption, but it can be modified and adapted to a general metric. \nNote that in the uniformized case the Gauss-Bonnet formula yields\n\\begin{equation}\n vol(M,g)=-2\\pi\\chi(M)=4\\pi(\\genus-1). \n\\end{equation}\n\nObserve that in the functional~$J_\\rho$ the first part is coercive and convex. The main difficulty is due to the spinorial part which is strongly indefinite. \nTo overcome this issue we are inspired by an idea from~\\cite{maalaoui2017characterization} and we consider a natural constraint: in the next section we will find critical points of the restricted functional. \n\n\\\n\n\\subsection{A Nehari type manifold}\nRoughly speaking, the space~$H^{\\frac{1}{2},-}(S)$ defined in \\eqref{eq:split} contains infinitely-many directions decreasing the functional $J_\\rho$ to negative infinity and the usual variational approaches can not be applied. \nHence we introduce a natural constraint in order to exclude most of these directions, obtaining a submanifold in~$H^1(M)\\times H^{\\frac{1}{2}}(S)$, which we still call it a \\emph{Nehari manifold}, though it may not fit into the classical definition as in~\\cite{ambrosetti2007nonlinear}.\nThis may be considered to be a Nehari manifold in the generalized sense, as in~\\cite{pankov2005periodic, szulkin2009ground, szulkin2010themethod}. \n\nLet~$P^\\pm\\colon H^{\\frac{1}{2}}(S)\\to H^{\\frac{1}{2},\\pm}(S)$ be the orthonormal projection according to the splitting in \\eqref{eq:split}.\nConsider the map\n\\begin{align}\n G\\colon H^1(M)\\times H^{\\frac{1}{2}}(S)&\\to H^{\\frac{1}{2},-}(S), \\\\\n (u,\\psi)&\\mapsto P^-\\left[(1+|\\D|)^{-1}(\\D\\psi-\\rho e^{u}\\psi)\\right]. \n\\end{align}\nSome explanations are in order. \nRecall that~$H^{\\frac{1}{2},-}$ is a Hilbert space, with inner product\n\\begin{align}\n \\left<\\psi,\\varphi\\right>_{_{H^{1\/2}}}\n =&\\left<\\psi,\\varphi\\right>_{L^2}\n +\\left<|\\D|^{\\frac{1}{2}}\\psi,|\\D|^{\\frac{1}{2}}\\varphi\\right>_{L^2}\\\\\n =&\\left<(1+|\\D|)\\psi,\\varphi\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}}.\n\\end{align}\nNow, let ~$G(u,\\psi)$ be the element in~$H^{\\frac{1}{2}}(S)$ such that, for any~$\\varphi\\in H^{\\frac{1}{2}}(S)$, \n\\begin{equation}\n \\left_{_{H^{1\/2}}}\n =\\left<\\D\\psi-\\rho e^u\\psi,P^- (\\varphi)\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}}.\n\\end{equation}\nIt follows that~$G(u,\\psi) \\in H^{\\frac{1}{2},-}(S)$, and it is given by the Riesz representation theorem as \n\\begin{equation}\n G(u,\\psi)=P^-\\left[(1+|\\D|)^{-1}(\\D\\psi-\\rho e^u\\psi)\\right]. \n\\end{equation}\n\nDefine the {\\em Nehari manifold} ~$N=G^{-1}(0)$, which is non-empty since~$(u,0)\\in N$ for any~$u\\in H^1(M)$. \nNote that, for each~$u$ fixed, the subset \n\\begin{equation}\n N_u\\coloneqq\\left\\{\\psi\\in H^{\\frac{1}{2}}(S)\\mid (u,\\psi)\\in N\\right\\}=\\ker \\left[ P^-\\circ(1+|\\D|)^{-1}\\circ (\\D- \\rho e^u) \\right]\n\\end{equation}\nis a linear subspace (of infinite dimension).\nHence we have a fibration~$N\\to H^1(M)$ with fiber~$N_u$ over~$u\\in H^1(M)$.\nThe total space~$N$ is contractible. \n\\begin{lemma}\n The Nehari manifold $N$ is a natural constraint for~$J_\\rho$, namely every critical point of $J_\\rho|_N$ is \n an unconstrained critical point of $J_\\rho$. \n\\end{lemma}\n\\begin{proof}\n To see that $N$ is a manifold, we show for any~$(u,\\psi)$ the surjectivity of the differential~$\\dd G(u,\\psi)$, which is given by\n \\begin{equation}\n \\dd G(u,\\psi)[v,\\phi]=P^-\\left[(1+|\\D|)^{-1}(\\D\\phi-\\rho e^{u}\\phi-\\rho e^u v\\psi)\\right]. \n \\end{equation}\n Restricting to those vectors with~$v=0$ and~$\\phi\\in H^{\\frac{1}{2},-}(S)$, we have \n \\begin{align}\n \\left<\\dd G(u,\\psi)[0,\\phi],\\phi\\right>_{H^{1\/2}}\n =&\\left<(1+|\\D|)^{-1}(\\D\\phi-\\rho e^u\\phi),\\phi\\right>_{H^{1\/2}}\\\\\n =&\\int_M \\left<\\D\\phi-\\rho e^u\\phi,\\phi\\right>\\dv_g \\\\\n =&-\\||\\D|^{\\frac{1}{2}}\\phi\\|_{L^2}^2-\\rho\\int_M e^u|\\phi|^2\\dv_g. \n \\end{align} \n Thus~$\\left<\\dd G(u,\\psi)[0,\\phi],\\phi\\right>_{H^{1\/2}}$ yields a negative-definite quadratic form on~$H^{\\frac{1}{2},-}(S)$.\n In particular,~$\\dd G(u,\\psi)$ is surjective onto~$H^{\\frac{1}{2},-}(S)$, for any~$(u,\\psi)$. \n It follows from the regular value theorem (for an infinite dimensional version, see e.g. \\cite{glockner2016fundamentals}) that~$N=G^{-1}(0)$ is a submanifold of~$H^1(M)\\times H^{\\frac{1}{2}}(S)$. \n \n Next, we need to show that {if~$(u_0,\\psi_0)$ is a critical point of~$J_\\rho|_N$, then it is also a critical points of~$J_\\rho$ on the full space~$H^1(M)\\times H^{\\frac{1}{2}}(S)$. }\n\n Recall that the orthonormal basis~$(\\varphi_k)$ for~$H^{\\frac{1}{2}}(S)$ consists of eigenspinors. \n Note that\n \\begin{align}\n (u,\\psi)\\in N\n &\\Leftrightarrow G(u,\\psi)=0 \n \\Leftrightarrow \\int_M \\left<\\D\\psi-\\rho e^u \\psi, h\\right>\\dv_g=0, \\quad \\forall h \\in H^{\\frac{1}{2},-}\\\\\n &\\Leftrightarrow G_j(u,\\psi)\\coloneqq \\int_M \\left<\\D\\psi-\\rho e^u\\psi,\\varphi_j\\right>\\dv_g=0, \\quad \\forall j<0, \n \\end{align}\n that is\n \\begin{equation}\n N= G^{-1}(0)=\\bigcap_{j<0} G_j^{-1}(0). \n \\end{equation}\n Now let~$(u_0,\\psi_0)$ be a critical point of~$J_\\rho|_N$: ~$\\nabla^N J(u_0,\\psi_0)=0$. \n Then there exist~$\\mu_j\\in\\R$ such that\\footnote{To see that such an infinite dimensional version of the Lagrange multiplier theory works, we note that\n \\begin{equation}\n \\nabla^N J(u_0,\\psi_0)= \\nabla J(u_0,\\psi_0)-\\left(\\nabla J(u_0,\\psi_0)\\right)^{\\bot} \n \\end{equation}\n where~$\\nabla J(u_0,\\psi_0)$ denotes the unconstrained gradient and~$(\\nabla J(u_0,\\psi_0))^\\bot$ denotes its normal component.\n Since the gradients~$\\{\\nabla G_j(u_0,\\psi_0): j<0\\}$ span the normal space, we can express~$(\\nabla J(u_0,\\psi_0))^\\bot$ in terms of them:\n \\begin{equation}\n (\\nabla J(u_0,\\psi_0))^\\bot=\\sum_{j<0} \\mu_j\\nabla G_j(u_0,\\psi_0) \\in H^{\\frac{1}{2}}(S)\n \\end{equation}\n for some~$\\mu_j\\in\\R$, ~$j<0$. \n }\n \\begin{equation}\\label{eq:Lagrange multiplier}\n \\dd J_\\rho(u_0,\\psi_0)=\\sum_{j<0} \\mu_j \\dd G_j(u_0,\\psi_0).\n \\end{equation}\n Testing both sides with tangent vectors of the form~$(0,h)$, we have\n \\begin{align}\n \\int_M\\left<\\D\\psi_0-\\rho e^{u_0}\\psi_0,h\\right>\\dv_g\n =\\sum_{j<0}\\mu_j\\int_M\\left<\\D h-\\rho e^{u_0}h, \\varphi_j\\right>\\dv_g.\n \\end{align}\n In particular, take~$h = \\varphi=\\sum_{j<0}\\mu_j\\varphi_j\\in H^{\\frac{1}{2},-}$ to obtain\n \\begin{align}\n 0=\\int_M\\left<\\D\\psi_0-\\rho e^{u_0}\\psi_0,\\varphi\\right>\n =\\int_M\\left<\\D\\varphi-\\rho e^{u_0}\\varphi,\\varphi\\right>\\dv_g\n \\le -C\\|\\varphi\\|^2-\\int_M \\rho e^{u_0}|\\varphi|^2\\dv_g.\n \\end{align}\n Thus~$\\varphi=0$, i.e. ~$\\mu_j=0$ for all~$j<0$. \n Hence in~\\eqref{eq:Lagrange multiplier} we have~$\\dd J_\\rho(u_0,\\psi_0)=0$. \n\\end{proof}\n\n\\\n\n\n\\subsection{Verification of the Palais-Smale condition}\nThis subsection is devoted to verifying the~$(PS)$ condition for the constrained functional~$J_\\rho|_{N}$. \nNote that\n\\begin{align}\n \\dd J_\\rho(u,\\psi)[v,\\phi]\n =\\int_M 2(-\\Delta u-1+e^{2u}-\\rho e^u|\\psi|^2)v + 4\\left<\\D\\psi-\\rho e^{u}\\psi,\\phi\\right>\\dv_g\n\\end{align}\nand for each~$j<0$, with~$G_j$ defined as in the above proof: \n\\begin{align}\n \\dd G_j(u,\\psi)[v,\\phi]\n =\\int_M \\left<\\D\\phi-\\rho e^u\\phi,\\varphi_j\\right>\\dv_g -\\int_M \\rho e^u v\\left<\\psi,\\varphi_j\\right>\\dv_g.\n\\end{align}\nFor each~$(u,\\psi)\\in N$, there exist constants~$\\mu_j(u,\\psi)$ such that \n\\begin{equation}\n \\dd^N J_\\rho(u,\\psi)\n =\\dd J_\\rho(u,\\psi)-\\sum_{j<0} \\mu_j(u,\\psi)\\dd G_j(u,\\psi), \n\\end{equation}\nthat is such that for any~$(v,\\phi)\\in H^1(M)\\times H^{\\frac{1}{2}}(S)$\n\\begin{align}\n \\dd^N J_\\rho(u,\\psi)[v,\\phi]\n =\\dd J(u,\\psi)[v,\\phi]-\\sum_{j<0} \\mu_j(u,\\psi)\\dd G_j(u,\\psi)[v,\\phi].\n\\end{align}\nFormally writing~$\\varphi(u,\\psi)\\coloneqq \\sum_{j<0} \\mu_j\\varphi_j$, then \n\\begin{align}\n\\dd^N J(u,\\psi)[v,\\phi]\n=&\\int_M 2(-\\Delta u-1+e^{2u}-\\rho e^u|\\psi|^2+\\rho e^u\\left<\\psi, \\varphi(u,\\psi)\\right>)v\\dv_g \\\\\n &+ \\int_M 4 \\left( \\left<\\D\\psi-\\rho e^{u}\\psi,\\phi\\right>-\\left<\\D\\varphi-\\rho e^u\\varphi,\\phi\\right> \\right) \\dv_g.\n\\end{align}\nNote that this holds for arbitrary~$(v,\\phi)$, not only those tangent vectors to~$N$.\n\nNow let~$(u_n,\\psi_n)\\in N$ be a~$(PS)_c$ sequence for~$J_\\rho|_{N}$: this will satisfy \n\\begin{equation}\\label{eq:PS:at level c}\nJ_\\rho(u_n,\\psi_n)=\\int_M \\left[ |\\nabla u_n|^2-2u_n+e^{2u_n}+2\\left(\\left<\\D\\psi_n,\\psi_n\\right>-\\rho e^{u_n}|\\psi_n|^2\\right) \\right] \\dv_g\\to c, \n\\end{equation}\n\\begin{equation}\\label{eq:PS:natural constraint}\nP^-\\circ(1+|\\D|)^{-1}\\circ (\\D\\psi_n-\\rho e^{u_n}\\psi_n)=0; \n\\end{equation}\nmoreover, since the differential of $J_\\rho$ is tending to zero only when applied \nto vectors tangent to $N$, there exists \n some~$\\varphi_n\\in H^{\\frac{1}{2},-}(S)$ \nsuch that \n\\begin{equation}\\label{eq:PS:function part}\n2(-\\Delta u_n-1+e^{2u_n}-\\rho e^{u_n}|\\psi_n|^2)-\\rho e^{u_n}\\left<\\psi_n,\\varphi_n\\right>=\\alpha_n\\to 0 \\textnormal{ in \\;} H^{-1}(M),\n\\end{equation}\n\\begin{equation}\\label{eq:PS:spinor part}\n4(\\D\\psi_n-\\rho e^{u_n}\\psi_n)-(\\D\\varphi_n-\\rho e^{u_n}\\varphi_n)=\\beta_n \\to 0 \\textnormal{ in } H^{-\\frac{1}{2}}(S).\n\\end{equation}\n\n\n\n\\medspace \n\n\\begin{lemma}\nWith the same notation as above, we have \n\\begin{enumerate}\n \\item The auxiliary spinors~$\\varphi_n$ satisfy~$\\|\\varphi_n\\|_{H^{\\frac{1}{2}}}\\to 0$ as~$n\\to\\infty$. \n \\item The sequence~$(u_n,\\psi_n)$ is uniformly bounded (with bounds depending on the level~$c$) in~$H^1(M)\\times H^{\\frac{1}{2}}(S)$.\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof}\n (1) Testing~\\eqref{eq:PS:natural constraint} against~$\\varphi_n$ we find \n \\begin{equation}\n \\int_M \\left<\\D\\psi_n-\\rho e^{u_n}\\psi_n,\\varphi_n\\right>\\dv_g=0, \n \\end{equation}\n while testing ~\\eqref{eq:PS:spinor part} against~$\\varphi_n$ we get \n\t\\begin{equation}\n\t -\\int_M \\left<\\D\\varphi_n,\\varphi_n\\right>\\dv_g +\\rho\\int_M e^{u_n}|\\varphi_n|^2\\dv_g=\\left<\\beta_n,\\varphi_n\\right>.\n\t\\end{equation}\t \n Since~$\\varphi_n$ lies in the span of the negative eigenspinors, we see that \n \\begin{equation}\n C\\|\\varphi_n\\|_{H^{\\frac{1}{2}}}^2+\\rho\\int_M e^{u_n}|\\varphi_n|^2\\dv_g=o(\\|\\varphi_n\\|_{H^{\\frac{1}{2}}}). \n \\end{equation}\n It follows that as~$n\\to\\infty$, \n \\begin{align}\n \\|\\varphi_n\\|_{H^{\\frac{1}{2}}}\\to 0, \\qquad \\int_M \\rho e^{u_n}|\\varphi_n|^2\\dv_g \\to 0. \n \\end{align} \n \n\\\n\n\n(2) Testing~\\eqref{eq:PS:function part} against~$v\\equiv 1\\in H^1(M)$, we obtain \n\t\\begin{align}\n\t2\\int_M e^{2u_n}\\dv_g -2\\int_M \\dv_g-2\\rho\\int_M \\left( e^{u_n}|\\psi_n|^2-e^{u_n}\\left<\\psi_n,\\varphi_n\\right> \\right)\\dv_g\n\t=\\left<\\alpha_n,1\\right>_{H^{-1}\\times H^1},\n\t\\end{align}\n\twhich can be read as \n\t\\begin{equation}\\label{eq:estimate of e-2u}\n\t\\int_M e^{2u_n}\\dv_g\n\t=4\\pi(\\genus-1)+\\rho\\int_M e^{u_n}|\\psi_n|^2\\dv_g+\\frac{1}{2}\\rho\\int_M e^{u_n}\\left<\\psi_n,\\varphi_n\\right>\\dv_g +o(1).\n\t\\end{equation}\n\tNow we can control the second integral on the right-hand side by\n\t\\begin{equation}\\label{eq:estimate of mixed term:spinor-multiplier-fucntion}\n\t\\left|\\frac{\\rho}{2}\\int_M e^{u_n}\\left<\\psi_n,\\varphi_n\\right>\\dv_g\\right|\n\t\\le \\varepsilon\\int_M \\rho e^{u_n}|\\psi_n|^2\\dv_g+\\varepsilon\\int_M e^{2u_n}\\dv_g+ C(\\varepsilon,\\rho)\\|\\varphi_n\\|^4, \n\t\\end{equation}\n where~$\\varepsilon>0$ is some small number. \n Substituting this into~\\eqref{eq:estimate of e-2u} and noting that~$\\|\\varphi_n\\|=o(1)$, we get\n \\begin{equation}\n \\int_M e^{2u_n}\\ge \\frac{4\\pi(\\genus-1)}{1+\\varepsilon}+\\frac{1-\\varepsilon}{1+\\varepsilon}\\int_M \\rho e^{u_n}|\\psi_n|^2\\dv_g+o(1).\n \\end{equation}\n \n\tTesting ~\\eqref{eq:PS:spinor part} against~$\\psi_n$ we deduce \n\t\\begin{equation}\n\t4\\int_M \\left( \\left<\\D\\psi_n,\\psi_n\\right>-\\rho e^{u_n}|\\psi_n|^2 \\right)\\dv_g -\\int_M \\left<\\D\\varphi_n-\\rho e^{u_n}\\varphi_n,\\psi_n\\right>\\dv_g\n\t=\\left<\\beta_n,\\psi_n\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}}.\n\t\\end{equation}\n\tSince the second integral vanishes because of~\\eqref{eq:PS:natural constraint}, we thus get\n\t\\begin{equation}\n\t\\int_M \\left( \\left<\\D\\psi_n,\\psi_n\\right>-\\rho e^{u_n}|\\psi_n|^2 \\right) \\dv_g =o(\\|\\psi_n\\|).\n\t\\end{equation}\n\tCombining these estimates with~\\eqref{eq:PS:at level c} we see that\n\t\\begin{align}\n\tc+o(1)\n\t=&\\int_M|\\nabla\\widehat{u}_n|^2\\dv_g-8\\pi(\\genus-1)\\bar{u}_n+\\int_M e^{2u_n}\\dv_g\n\t+2\\int_M \\left(\\left<\\D\\psi_n,\\psi_n\\right>-\\rho e^{u_n}|\\psi_n|^2 \\right)\\dv_g \\\\\n\t\\ge&\\int_M|\\nabla\\widehat{u}_n|^2\\dv_g-4\\pi(\\genus-1)(2\\bar{u}_n-\\frac{1}{1+\\varepsilon})\n\t+\\frac{1-\\varepsilon}{1+\\varepsilon}\\rho\\int_M e^{u_n}|\\psi_n|^2\\dv_g\n\t +C(\\varepsilon,\\rho)o(1)+o(\\|\\psi_n\\|),\n\t\\end{align}\n\twhich is to say,\n\t\\begin{align}\\label{eq:estimate of Dirichlet and mix-spinor-function}\n\t\\int_M |\\nabla \\widehat{u}_n|^2+\\frac{1-\\varepsilon}{1+\\varepsilon}\\rho e^{u_n}|\\psi_n|^2\\dv_g\n\t\\le c+4\\pi(\\genus-1)(2\\bar{u}_n-\\frac{1}{1+\\varepsilon})+C(\\varepsilon,\\rho)o(1)+o(\\|\\psi_n\\|).\n\t\\end{align}\n\t\n\t\\medskip\n\t\n\tNow we estimate the averages~$\\bar{u}_n$. \n\tNote that by~\\eqref{eq:estimate of e-2u} and~\\eqref{eq:estimate of mixed term:spinor-multiplier-fucntion} we also obtain\n\t\\begin{align}\n\t \\int_M e^{2u_n}\\dv_g \n\t \\le \\frac{4\\pi(\\genus-1)}{1-\\varepsilon}\n\t\t\t+\\frac{1+\\varepsilon}{1-\\varepsilon}\\int_M \\rho e^{u_n}|\\psi_n|^2\\dv_g +C(\\varepsilon,\\rho)o(1).\t \n\t\\end{align}\n\tThen by Jensen's inequality,\n\t\\begin{align}\n\te^{2\\bar{u}_n}\n\t\\le & e^{2\\bar{u}_n}\\fint_M e^{2\\widehat{u}_n}\\dv_g =\\frac{1}{4\\pi(\\genus-1)}\\int_M e^{2u_n}\\dv_g \\\\\n\t\\le& \\frac{1}{1-\\varepsilon}+\\frac{1}{4\\pi(\\genus-1)}\\frac{1+\\varepsilon}{1-\\varepsilon}\\int_M \\rho e^{u_n}|\\psi_n|^2\\dv_g+ C(\\varepsilon,\\rho,\\genus) o(1) \\\\\n\t\\le&\\frac{1}{1-\\varepsilon}+\\left(\\frac{1+\\varepsilon}{1-\\varepsilon}\\right)^2 \\left(\\frac{c}{4\\pi(\\genus-1)}+2\\bar{u}_n-\\frac{1}{1+\\varepsilon}\\right)+ C(\\varepsilon,\\rho,\\genus) o(1)+ C(\\varepsilon,\\genus) o(\\|\\psi_n\\|). \n\t\\end{align}\n\tThus there exists ~$C=C(\\varepsilon,\\rho,\\genus)>0$ such that \n\t\\begin{equation}\n\t|\\bar{u}_n|\\le C\\bigr(1+c+o(\\|\\psi_n\\|)\\bigr).\n\t\\end{equation}\n\t\n\t\\medskip\n\t\n\tThe spinors can be controlled by the above growth estimates. \n\tTesting~\\eqref{eq:PS:spinor part} against~$\\psi_n^+$, we find \n\t\\begin{align}\n\t4\\int_M \\left( \\left<\\D\\psi_n,\\psi_n^+\\right>-\\rho e^{u_n}\\left<\\psi_n,\\psi_n^+\\right> \\right) \\dv_g\n\t-\\int_M \\left<\\D\\varphi_n-\\rho e^{u_n}\\varphi_n,\\psi_n^+\\right>\\dv_g\n\t=\\left<\\beta_n,\\psi_n^+\\right>_{H^{-\\frac{1}{2}}\\times H^{\\frac{1}{2}}}.\n\t\\end{align}\n\tIt follows that\n\t\\begin{align}\n\tC\\|\\psi_n^+\\|^2\n\t\\le& \\int_M\\left<\\D\\psi_n,\\psi^+\\right>\\dv_g\n\t =\\int_M\\rho e^{u_n}\\left<\\psi_n,\\psi_n^+\\right>\\dv_g\n\t +\\frac{1}{4}\\int_M\\left<\\D\\varphi_n-\\rho e^{u_n}\\varphi_n,\\psi_n^+\\right>\\dv_g+o(\\|\\psi_n^+\\|)\\\\\n\t\\le& \\left(\\int_M e^{u_n}|\\psi_n|^2\\dv_g\\right)^{\\frac{1}{2}}\n\t\t\t\\left(\\int_M e^{2u_n}\\dv_g\\right)^{\\frac{1}{4}}\\left(\\int_M|\\psi_n^+|^4\\dv_g\\right)\\\\\n\t\t&\t+\\|\\varphi_n\\|\\|\\psi_n^+\\|\n\t\t\t+\\rho\\left(\\int_M e^{2u_n}\\dv_g\\right)^{\\frac{1}{2}}\\|\\varphi_n\\| \\|\\psi_n^+\\| +o(\\|\\psi_n^+\\|) \\\\\n \\le& C\\left(1+c+o(\\|\\psi_n\\|)\\right)^{\\frac{3}{4}}\\|\\psi_n^+\\| + o(\\|\\psi_n^+\\|).\n\t\\end{align}\n \tFor what concerns the other component~$\\psi_n^-$, we use~\\eqref{eq:PS:natural constraint} to get\n \t\\begin{align}\n \tC\\|\\psi_n^-\\|^2\n \t\\le &-\\int_M \\left<\\D\\psi_n^-,\\psi_n\\right>\\dv_g\n \t\t=-\\rho\\int_M e^{u_n}\\left<\\psi_n,\\psi_n^-\\right>\\dv_g \\\\\n \t\\le& \\rho\\left(\\int_M e^{2u_n}\\dv_g\\right)^{\\frac{1}{4}} \\left(\\int_M e^{u_n}|\\psi_n|^2\\dv_g\\right)^{\\frac{1}{2}}\\|\\psi_n^-\\|\\\\\n \t\\le& C(1+c+o(\\|\\psi_n\\|))^{\\frac{3}{4}} \\|\\psi_n^-\\|.\n \t\\end{align}\n \tConsequently,\n \t\\begin{equation}\n \t\\|\\psi_n\\|^2=\\|\\psi_n^+\\|^2+\\|\\psi_n^-\\|^2 \n \t\\le C\\bigr(1+c+o(\\|\\psi_n\\|)\\bigr)^{\\frac{3}{4}}\\|\\psi_n\\| + o(\\|\\psi_n\\|).\n \t\\end{equation}\n Thus there exists some constant~$C=C(c,\\genus,\\rho)>0$ such that\n \t\\begin{equation}\n \t\\|\\psi_n\\|\\le C(c,\\genus,\\rho)<+\\infty.\n \t\\end{equation}\n \tThis uniform bound (depending on the level~$c$) in turn gives bounds on ~$\\bar{u}_n$ and thus\n \t\\begin{equation}\n \t\\int_M \\left( |\\nabla \\widehat{u}_n|^2+\\rho e^{u_n}|\\psi_n|^2 \\right) \\dv_g \n \t\\le C'(c,\\genus,\\rho)<\\infty. \n \t\\end{equation}\n\\end{proof}\n\n\nNow, passing to a subsequence if necessary, we may assume that there exist~$u_\\infty\\in H^1(M)$ and~$\\psi_\\infty\\in H^{\\frac{1}{2}}(S)$ such that\n\\begin{align}\n &u_n\\rightharpoonup u_\\infty \\textnormal{ weakly in } H^1(M), \\\\\n &\\psi_n\\rightharpoonup \\psi_\\infty \\textnormal{ weakly in } H^{\\frac{1}{2}}(S). \n\\end{align}\n\n\\medspace \n\n\\begin{lemma}\n The pair~$(u_\\infty,\\psi_\\infty)$ is a smooth solution of~\\eqref{eq:EL-uniformized metric}. \n\\end{lemma}\n\\begin{proof}\n According to the compactness of the Moser--Trudinger embedding (\\cite[theorem 2.46]{aubin1998somenonlinear}), we see that\n \\begin{equation}\n e^{u_n} \\to e^{u_\\infty} \\textnormal{ strongly in } L^p(M),\\quad (p<\\infty).\n \\end{equation}\n Meanwhile, thanks to Rellich-Kondrachov compact embedding Theorem (see e.g.~\\cite{gilbarg2001elliptic})\n \\begin{equation}\n \\psi_n\\to \\psi_\\infty \\textnormal{ strongly in } L^q(S), \\quad(q<4).\n \\end{equation}\n Hence~$e^{u_n}|\\psi_n|^2$ converges weakly in~$L^p$ to~$e^{u_\\infty}|\\psi_\\infty|^2$, for any~$p<2$.\n\n It follows that $(u_\\infty, \\psi_\\infty)$ is a weak solution to~\\eqref{eq:EL-uniformized metric}.\n As remarked before, any weak solution is a classical, hence smooth, solution. \n\\end{proof}\n\nIn particular, this implies that the weak limit~$(u_\\infty,\\psi_\\infty)$ is in the Nehari manifold~$N$. \nConsider the differences\n\\begin{align}\n &v_n\\coloneqq u_n-u_\\infty, \\\\\n &\\phi_n\\coloneqq \\psi_n-\\psi_\\infty.\n\\end{align}\nThen~$(v_n,\\phi_n)\\rightharpoonup (0,0)$ weakly in~$H^1(M)\\times H^{\\frac{1}{2}}(S)$. \nThe functions~$v_n$ satisfy\n\\begin{align}\n \\Delta_g v_n\n =(e^{2u_n}-e^{2u_\\infty})-\\rho\\left(e^{u_n}|\\psi_n|^2-e^{u_\\infty}|\\psi_\\infty|^2\\right), \n\\end{align}\nwhere the right-hand sides converge to~$0$ in~$L^q(M)$ for any~$q<2$. \nThis implies that the~$v_n$'s are uniformly bounded in~$H^1(M)$ and converge strongly to a limit function~$v_\\infty\\in H^1(M)$ satisfying\n\\begin{equation}\n \\Delta_g v_\\infty=0.\n\\end{equation}\nThus~$v_\\infty=const.$, which has to be zero since~$v_n\\rightharpoonup 0$. \nThis implies the strong convergence of~$u_n$ to~$u_\\infty$ in~$H^1(M)$. \nLet us look next at the equations for~$\\phi_n$'s:\n\\begin{equation}\n \\D_g\\phi_n=\\rho e^{u_n}\\psi_n- \\rho e^{u_\\infty}\\psi_\\infty, \n\\end{equation}\nwhere the right-hand sides converges to~$0$ in~$L^q(S)$ for any~$q<4$.\nThus there exists ~$\\phi_\\infty\\in H^{\\frac{1}{2}}(S)$ such that~$\\phi_n$ converges strongly in~$H^{\\frac{1}{2}}(S)$ to~$\\phi_\\infty$, which satisfies\n\\begin{equation}\n \\D_g\\phi_\\infty=0.\n\\end{equation}\nBy the assumption of trivial kernel on $\\D_g$, we have that~$\\phi_\\infty=0$, that is ~$\\psi_n$ converges strongly to~$\\psi_\\infty$ in~$H^{\\frac{1}{2}}(S)$. \nSince~$N$ is a submanifold of~$H^1(M)\\times H^{\\frac{1}{2}}(S)$, the sequence~$(u_n,\\psi_n)$ also converges inside~$N$ to~$(u_\\infty,\\psi_\\infty)$; in other words,~$(u_\\infty,\\psi_\\infty)$ lies in the closure of~$\\{(u_n,\\psi_n)\\}$ relative to~$N$. \nThus we verified the following \n\n\\\n\n\\begin{prop}\n The functional~$J_\\rho|_N$ satisfies the Palais-Smale condition. \n\\end{prop}\n\n\\\n\n\\section{Mountain pass and linking geometry}\nIn this section we will show that the functional~$J_\\rho|_N$, for suitable~$\\rho$'s, possesses either a mountain pass or linking geometry around the trivial solution~$(0,0)$, which will yield existence of a non-trivial min-max critical point. \n\nFor later convenience let us introduce the notation\n\\begin{align}\n F(u)\\coloneqq\\int_M \\left( |\\nabla u|^2-2u+e^{2u} \\right) \\dv_g, & & \n Q(u,\\psi)\\coloneqq2\\int_M \\left( \\left<\\D\\psi,\\psi\\right>-\\rho e^{u}|\\psi|^2 \\right) \\dv_g.\n\\end{align}\nThen we have\n\\begin{enumerate}\n \\item[(i)] $J_\\rho(u,\\psi)=F(u)+Q(u,\\psi)$.\n \\item[(ii)] $F(u) \\geq 4\\pi(\\genus-1)$, and this lower bound is achieved by the unique minimizer~$u_{min}\\equiv0$. \n \\item[(iii)] $Q(u,\\psi)$ is quadratic in~$\\psi$ and strongly indefinite. \n\\end{enumerate}\n\n\\\n\n\\subsection{Local behavior near (0,0)}\n\nLet~$(u,\\psi)\\in N$ be \\emph{close} to~$(0,0)$. \nThe constraint that defines $N$, i.e.~$P^-(1+|\\D|)^{-1}(\\D\\psi-\\rho e^u\\psi)=0$, implies\n\\begin{equation}\n \\int_M\\left<\\D\\psi-\\rho e^u\\psi,P^-\\psi\\right>\\dv_g=0.\n\\end{equation}\nHence we get\n\\begin{align}\n -\\int_M\\left<\\D\\psi,\\psi^-\\right>\\dv_g\n =&-\\rho\\int_M e^{u}\\left<\\psi^++\\psi^-,\\psi^-\\right> \\\\\n =&-\\rho\\int_M e^{u}|\\psi^-|^2\\dv_g \n -\\rho\\int_M e^{u}\\left<\\psi^+,\\psi^-\\right>\\dv_g.\n\\end{align}\nSince~$\\|e^u\\|_{L^p}\\le C(1+\\|u\\|_{H^1})\\le C$ for~$\\|u\\|$ uniformly bounded, \nwe have\n\\begin{equation}\\label{eq:positive spinor dominates negative part}\n \\|\\psi^-\\|\\le C\\rho\\|\\psi^+\\|.\n\\end{equation}\n\nNow consider the functional\n\\begin{align}\\label{eq:local behavior of J rho}\n J_\\rho(u,\\psi)\n & = F(u)+Q(u,\\psi)=F(u)+2\\int_M \\left( \\left<\\D\\psi,\\psi\\right>-\\rho e^u|\\psi|^2 \\right) \\dv_g \\\\\n & = F(u)+2\\int_M\\left<(\\D-\\rho)\\psi, \\psi^+\\right>\\dv_g\n +2\\int_M \\rho (1-e^u)\\left<\\psi,\\psi^+\\right>\\dv_g.\n\\end{align}\nThe last integral is now of cubic order in $(u,\\psi)$, i.e.:\n\\begin{align}\n 2\\int_M \\rho (1-e^u)\\left<\\psi,\\psi^+\\right>\\dv_g\n \\le C\\|u\\|_{H^1} \\|\\psi\\|\\|\\psi^+\\|.\n\\end{align}\nFor the first term, if we take the equivalent norm~$|\\bar{u}|^2+\\|\\nabla \\widehat{u}\\|_{L^2}^2\\sim \\|u\\|_{H^1}^2$, then for~$t^2=|\\bar{u}|^2+\\|\\nabla \\widehat{u}\\|_{L^2}^2>0$\n\\begin{itemize}\n \\item if~$|\\bar{u}|^2\\ge\\frac{t^2}{2}\\ge \\|\\nabla\\widehat{u}\\|_{L^2}^2$, then \n \\begin{equation}\n F(u)\\ge \\int_M \\left( e^{2\\bar{u}}-2\\bar{u} \\right) \\dv_g \\ge 4\\pi(\\genus-1)+Ct^2,\n \\end{equation}\n \\item if~$\\|\\nabla\\widehat{u}\\|_{L^2}^2\\ge\\frac{t^2}{2}\\ge|\\bar{u}|^2$, then \n \\begin{equation}\n F(u)\\ge \\int_M \\left( |\\nabla\\widehat{u}|^2+1 \\right) \\dv_g\\ge4\\pi(\\genus-1)+\\frac{1}{2}t^2, \n \\end{equation}\n\\end{itemize}\nthus in either case we have\n\\begin{equation}\n F(u)\\ge 4\\pi(\\genus-1)+C^{-1}\\|u\\|_{H^1}^2. \n\\end{equation}\nIt remains to analyze the middle integral term in the r.h.s. of~\\eqref{eq:local behavior of J rho}. \nAs before, we write~$\\psi=\\sum_{j\\in \\mathbb{Z}_*} a_j\\varphi_j$: \nthen\n\\begin{align}\n 2\\int_M\\left<(\\D-\\rho)\\psi, \\psi^+\\right>\\dv_g\n =\\sum_{j>0} 2(\\lambda_j-\\rho)a_j^2.\n\\end{align}\n\nFrom now on we assume that~$\\rho\\notin\\Spec(\\D)$. Thus the above summation can be split into two parts\n\\begin{equation}\n 2\\int_M\\left<(\\D-\\rho)\\psi, \\psi^+\\right>\\dv_g\n =-\\sum_{0<\\lambda_j<\\rho} 2(\\rho-\\lambda_j)a_j^2\n +\\sum_{\\lambda_j>\\rho} 2(\\lambda_j-\\rho)a_j^2. \n\\end{equation}\n\n\\subsubsection{Mountain pass geometry}\nFirst we consider the easier case~$0<\\rho<\\lambda_1$, so the first part of the above summation vanishes. \nThen, locally near~$(0,0)$ in ~$N$, we have\n\\begin{align}\n J_\\rho(u,\\psi) \n \\ge& 4\\pi(\\genus-1)+C^{-1}\\|u\\|_{H^1}^2+ C^{-1}\\left(1-\\frac{\\rho}{\\lambda_1}\\right)\\|\\psi^+\\|^2-C\\|u\\|_{H^1}\\|\\psi\\|\\|\\psi^+\\| \\\\\n \\ge& 4\\pi(\\genus-1)+C^{-1}\\|u\\|_{H^1}^2+ C^{-1}\\left(1-\\frac{\\rho}{\\lambda_1}-C^2\\|\\psi\\|^2\\right)\\|\\psi^+\\|^2, \n\\end{align}\nwhere we have used Cauchy-Schwarz inequality for the last term, of cubic order. \nIt follows that when\n$\\|u\\|_{H^1}^2+\\|\\psi\\|_{H^{\\frac{1}{2}}}^2=r^2>0$ is small, there exists a continuous function~$\\theta(r)>0$ such that\n\\begin{equation}\\label{eq:l-bd}\n J_\\rho(u,\\psi)\\ge J(0,0)+\\theta(r). \n\\end{equation}\nOn the other hand, we can choose a large constant~$\\bar{u}_1\\in H^1(M)$ such that~$\\rho e^{\\bar{u}_1}>\\lambda_1+1$ and then take~$s>0$ large such that\n\\begin{align}\n J(\\bar{u}_1, s\\varphi_1) \n =&vol(M,g)(e^{2\\bar{u}_1}-2\\bar{u}_1) \n +2(\\lambda_1-\\rho e^{\\bar{u}_1})s^2 \\\\\n =&4\\pi(\\genus-1)(e^{2\\bar{u}_1}-2\\bar{u}_1)\n -2(\\rho e^{\\bar{u}_1}-\\lambda_1)s^2\n\\end{align}\nis negative. \nThus we have the mountain pass geometry locally near ~$(0,0)$ in the Nehari manifold~$N$. \nLet~$\\Gamma$ be the space of paths connecting~$(0,0)$ and~$(\\bar{u}_1, s\\varphi_1)$ inside~$N$ (notice that \n$\\Gamma \\neq \\emptyset$ since $N$ is contractible, and hence connected), parametrized by~$t\\in [0,1]$, and define\n\\begin{equation}\n c_1\\coloneqq \\inf_{\\alpha\\in\\Gamma} \\sup_{t\\in [0,1]} J_\\rho(\\alpha(t)).\n\\end{equation}\nFrom the above arguments we have that~$c_1>4\\pi(\\genus-1)$.\nIt follows that~$c_1$ is a critical value for~$J_\\rho$ with a critical point at this level, which is different from the trivial one. \nThis concludes the proof of Theorem~\\ref{thm} in this case. \n\n\\medskip\n\n\\subsubsection{Linking geometry}\nNext we consider the case~$\\rho\\in(\\lambda_k,\\lambda_{k+1})$ for some~$k\\ge 1$. \nNow there are more directions in which~$J_\\rho$ becomes negative, but these are at most finitely-many, and we will apply a linking method to \nexploit the geometry of the functional. \n\n Decomposing first the space~$H^{\\frac{1}{2}}(S)$ into two parts: \n\\begin{align}\n H^{\\frac{1}{2},k+}\\coloneqq &\n \\left\\{\\phi_1\\in H^{\\frac{1}{2}}(S)\\mid \\phi_1=\\sum_{j>k}a_j\\varphi_j\\right\\}, \\\\\n H^{\\frac{1}{2},k-}\\coloneqq &\n \\left\\{\\phi_2\\in H^{\\frac{1}{2}}(S)\\mid \\phi_2=\\sum_{j\\le k}a_j\\varphi_j\\right\\},\n\\end{align}\nwe have then the orthogonal decomposition \n\\begin{equation}\n H^{\\frac{1}{2}}(S)=H^{\\frac{1}{2},k+}\\oplus \n \\left(H^{\\frac{1}{2},k-}\\cap H^{\\frac{1}{2},+}(S)\\right)\n \\oplus H^{\\frac{1}{2},-}(S). \n\\end{equation}\n\nNow consider the set\n\\begin{equation}\n \\mathscr{N}_k\\coloneqq\n \\{0\\}\\times \\left(H^{\\frac{1}{2},k-}\\cap H^{\\frac{1}{2},+}(S)\\right)\\subset H^1(M)\\times H^{\\frac{1}{2}}(S).\n\\end{equation}\nIt is easy to see that~$\\mathscr{N}_k$ is a linear subspace inside~$N$, and along this subspace the functional~$J_\\rho$ is not larger than the minimal critical value:\n\\begin{equation}\n J_\\rho(0,\\phi_1)\n =4\\pi(\\genus-1)-2\\sum_{0 0$ let us consider the following cone around~$\\mathscr{N}_k$:\n\\begin{multline}\n \\mathcal{C}_\\tau(\\mathscr{N}_k)\\coloneqq \n \\Big\\{(u,\\psi)\\in N\n \\mid u\\in H^1(M), \\psi=\\phi_1+\\phi_2+\\psi^-\\in H^{\\frac{1}{2},k+}\\oplus (H^{\\frac{1}{2},k-}\\cap H^{\\frac{1}{2},+}(S))\\oplus H^{\\frac{1}{2},-}(S), \\\\\n \\|u\\|_{H^1}+\\|\\phi_1\\|^2+\\|\\psi^-\\|^2< \\tau \\|\\phi_2\\|^2 \\Big\\}. \n\\end{multline} \nWe claim that for~$\\tau$ suitably chosen this cone contains all the decreasing directions, in the sense that outside the cone we can find a region on which the functional is strictly above the ground state level.\n\nLetting~$(u,\\psi)\\in N\\setminus \\mathcal{C}_\\tau(\\mathscr{N}_k)$, i.e., with~$\\psi=\\phi_1+\\phi_2$ decomposed as above, we have\n\\begin{equation}\n \\|u\\|^2_{H^1}+\\|\\phi_1\\|^2+\\|\\psi^-\\|^2\\ge \\tau \\|\\phi_2\\|^2.\n\\end{equation}\nBy~\\eqref{eq:positive spinor dominates negative part}, which can be now interpreted as\n\\begin{equation}\n \\|\\psi^-\\|^2\\le C\\rho^2(\\|\\phi_1\\|^2+\\|\\phi_2\\|^2), \n\\end{equation}\nwe see that \n\\begin{equation}\\label{eq:lower part of spinor is dominated by higher part}\n \\|\\phi_2\\|^2\\le \\frac{1}{\\tau-C\\rho^2}\\left(\\|u\\|^2_{H^1}+(1+C\\rho^2)\\|\\phi_1\\|^2\\right).\n\\end{equation}\nMoreover, this also implies that \n\\begin{equation}\n \\|u\\|_{H^1}^2+ \\|\\phi_1\\|^2 \\ge C(\\|u\\|^2 +\\|\\psi\\|^2),\n\\end{equation}\nfor some~$C=C(\\rho,\\tau)>0$. \n\n \n\nThen in~\\eqref{eq:local behavior of J rho} for the scalar component we have as before the control\n\\begin{equation}\n F(u)\\ge 4\\pi(\\genus-1)+C\\|u\\|_{H^1}^2.\n\\end{equation}\nFor the spinorial part, since~$\\psi=\\phi_1+\\phi_2+\\psi^-$ is an orthogonal decomposition, we have\n\\begin{align}\n Q(u,\\psi)\n =&2\\int_M\\left<(\\D-\\rho)\\psi,\\psi^+\\right>\\dv_g\n +2\\rho\\int_M (1-e^u)\\left<\\psi,\\psi^+\\right>\\dv_g \\\\\n =&2\\int_M\\left<(\\D-\\rho)\\phi_1,\\phi_1\\right>\\dv_g\n +2\\int_M\\left<(\\D-\\rho)\\phi_2,\\phi_2\\right>\\dv_g \\\\\n &+2\\rho\\int_M (1-e^u)\\left<\\psi,\\phi_1+\\phi_2\\right>\\dv_g \\\\\n \\ge&C\\left(1-\\frac{\\rho}{\\lambda_{k+1}}\\right)\\|\\phi_1\\|^2\n -C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)\\|\\phi_2\\|^2 \n -C\\rho\\|u\\|_{H^1}\\|\\psi\\|(\\|\\phi_1\\|+\\|\\phi_2\\|).\n\\end{align}\nAssuming~$\\|u\\|^2_{H^1}+\\|\\psi\\|^2=r^2$ is small and noting~\\eqref{eq:lower part of spinor is dominated by higher part}, we get\n\\begin{align}\n J_\\rho(u,\\phi_1+\\phi_2)\n \\ge& 4\\pi(\\genus-1)+C\\|u\\|_{H^1}^2\n + C\\left(1-\\frac{\\rho}{\\lambda_{k+1}}-Cr^2\\right)\\|\\phi_1\\|^2\\\\\n & -C\\left(\\frac{\\rho}{\\lambda_k}-1-Cr^2\\right)\\|\\phi_2\\|^2 \\\\\n \\ge&4\\pi(\\genus-1)+C\\|u\\|_{H^1}^2\n + C\\left(1-\\frac{\\rho}{\\lambda_{k+1}}-Cr^2\\right)\\|\\phi_1\\|^2\\\\\n & -C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)\\frac{\\|u\\|_{H^1}^2+(1+C\\rho^2)\\|\\phi_1\\|^2}{\\tau-C\\rho^2}.\n\\end{align}\nWe can first choose~$r$ small enough and then choose~$\\tau$ large enough such that\n\\begin{align}\n J_\\rho(u,\\phi_1+\\phi_2)\n \\ge& 4\\pi(\\genus-1)+ C(\\|u\\|^2+\\|\\phi_1\\|^2) \\\\\n \\ge& 4\\pi(\\genus-1)+C r^2\n\\end{align}\noutside the cone~$\\mathcal{C}(\\mathscr{N}_k)$. \nThus the claim is confirmed. \n\n\\medskip\n\nFor~$r$ as above, consider the set\n\\begin{equation}\n L_1\\coloneqq\\bigr( \\p B_r(0,0)\\backslash \\mathcal{C}(\\mathscr{N}_k)\\bigr) \\cap N, \n\\end{equation}\nwhich is non-empty since~$(0,r\\varphi_{k+1})\\in L_1$.\nRecall that~$N$ is locally modeled by a Hilbert space, e.g. $T_{(0,0)} N$. \nWe can assume that in a local chart,~$\\mathscr{N}_k$ is some {\\em coordinate subspace}, while~$L_1$ is homeomorphic to a collar neighborhood of the sphere (of infinite dimension) which lies in a subspace complementary to~$\\mathscr{N}_k$ and intersects $\\mathcal{C}_\\tau(\\mathscr{N}_k)$ only at $\\{(0,0)\\}$. \n\nNext we introduce a set~$L_2$ on which the functional attains low values and such that it links with~$L_1$, see Figure~\\ref{fig:linking}. \nThe construction of such a set is performed in several steps. First we take the ball\n\\begin{equation}\n B_R^{0,k}(0)\\coloneqq\\Bigr\\{(0,\\phi_2)\\in\\mathscr{N}_k \\mid \\|\\phi_2\\|\\le R\\Bigr\\}\\subset \\mathscr{N}_k\n\\end{equation}\nwith~$R>0$ a large constant to be fixed later. \nNote that for any~$(0,\\phi_2)$,~$J_\\rho(0,\\phi_2)\\le 4\\pi(\\genus-1)$ and for~$(0,\\phi_2)\\in \\p B_R^{0,k}(0)$, \n\\begin{equation}\n J_\\rho(0,\\phi_2)\n \\le 4\\pi(\\genus-1)-C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)\\|\\phi_2\\|^2\n \\le 4\\pi(\\genus-1)-C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)R^2.\n\\end{equation}\nFor any~$(0,\\phi_2)\\in\\p B_R^{0,k}(0)$, we consider the following curves. \nFirst let\n\\begin{equation}\n \\sigma_{1}\\colon [0,T]\\to N, \\quad\n \\sigma_1(t)\\coloneqq(t, \\phi_2+At\\varphi_{k+1}),\n\\end{equation}\nwhere~$A>0$ is again a constant to be fixed later.\nOne easily sees that this is a curve in~$N$ and \n\\begin{align}\n J_\\rho(t,\\phi_2+&At\\varphi_{k+1})\n =vol(M,g)(e^{2t}-2t)+2\\int_M\\left<(\\D-\\rho e^t)\\phi_2,\\phi_2\\right>\\dv_g\\\\\n &\\qquad\\qquad+2A^2t^2\\int_M\\left<(\\D-\\rho e^t)\\varphi_{k+1},\\varphi_{k+1}\\right>\\dv_g\\\\\n \\le&4\\pi(\\genus-1)(e^{2t}-2t)\n +2\\int_M\\left<(\\D-\\rho)\\phi_2,\\phi_2\\right>\\dv_g \n +2A^2t^2 (\\lambda_{k+1}-\\rho e^t) \\\\\n \\le&4\\pi(\\genus-1)(e^{2t}-2t)\n -C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)R^2\n +2A^2t^2 (\\lambda_{k+1}-\\rho e^t). \n\\end{align}\nNow we fix some constants:\n\\begin{itemize}\n \\item we choose~$T>0$ such that~$\\rho e^T-\\lambda_{k+1}\\ge 1$;\n \\item then we choose~$A>0$ such that\n \\begin{equation}\n 4\\pi(\\genus-1)(e^{2T}-2T)-2A^2T^2(\\rho e^T-\\lambda_{k+1})<4\\pi(\\genus-1); \n \\end{equation}\n \\item finally, choose~$R>0$ such that for any~$t\\in [0,T]$\n \\begin{equation}\n 4\\pi(\\genus-1)(e^{2t}-2t)\n -C\\left(\\frac{\\rho}{\\lambda_k}-1\\right)R^2\n +2A^2t^2 (\\lambda_{k+1}-\\rho e^t)<4\\pi(\\genus-1).\n \\end{equation}\n\\end{itemize}\nThen we consider the curve\n\\begin{align}\n \\sigma_2\\colon [-1,1]\\to N, \\quad\n \\sigma_2(r)\\coloneqq \\bigr(T,(-r)\\phi_2+AT\\varphi_{k+1}\\bigr),\n\\end{align} \nwhich joins~$(T,\\phi_2+AT\\varphi_{k+1})$ to~$(T,-\\phi_2+AT\\varphi_{k+1})$ inside~$N$. \nThereafter we can come back to~$\\mathscr{N}_k$ via the curve\n\\begin{equation}\n \\sigma_3\\colon [0,T]\\to N, \\quad \n \\sigma_3(t)\\coloneqq \\bigr((T-t), \\phi_2+A(T-t)\\varphi_{k+1}\\bigr). \n\\end{equation}\nFinally, consider the subset\n\\begin{equation}\n \\mathcal{D}\\coloneqq\n \\left\\{(t,At\\varphi_{k+1}+\\phi_2) \\mid t\\in [0,T], (0, \\phi_2)\\in B_R^{0,k}(0) \\right\\}, \n\\end{equation} \nwhich is compact and homeomorphic to a finite-dimensional cylindrical segment \n\\begin{equation}\n [0,T]\\times B_R^{0,k}(0).\n\\end{equation}\nNote that~$\\mathcal{D}\\subset N$ and\nlet~$L_2=\\p \\mathcal{D}$, see Figure~\\ref{fig:linking}. \nThe curves $\\sigma_1, \\sigma_2, \\sigma_3$ constructed above pass through every point of~$L_2\\backslash B_R^{0,k}(0)$. It follows that on~$L_2$ the functional attains low values. \nOne can shrink~$L_2$ (in an homotopically equivalent way) into the coordinate chart to see that~$L_1$ and~$L_2$ actually link, see e.g. \\cite{ambrosetti2007nonlinear, struwe2008variational} for a rigorous definition of this concept. \n\n\\begin{figure}[h] \n\\centering\n\\includegraphics[width=0.5\\linewidth]{.\/linking}\n\\caption{}\n\\label{fig:linking}\n\\end{figure}\n\n\n\\medskip\n\nNow we define the linking level.\nLet~$\\Gamma$ be the space of continuous maps\n\\begin{equation}\n \\alpha\\colon \\mathcal{D}\\to N, \n\\end{equation}\nsuch that~$\\alpha(v,h)=(v,h)$ for any~$(v,h)\\in L_2=\\p \\mathcal{D}$. \nThis set~$\\Gamma$ is clearly non-empty since~$\\Id_\\mathcal{D}\\in\\Gamma$. \nThen we define the linking level\n\\begin{equation}\n c_1\\coloneqq \\inf_{\\alpha\\in\\Gamma}\\max_{(v,h)\\in \\mathcal{D}}J_{\\rho}(\\alpha(v,h)) .\n\\end{equation}\nAs~$L_1$ and~$L_2$ link, we have from the above arguments that\n\\begin{equation}\n c_1 \\ge 4\\pi(\\genus-1)+\\theta(r). \n\\end{equation}\nIt follows that~$c_1$ is a critical value for~$J_\\rho$, and again we obtain a critical point for~$J_\\rho $ which is different from the trivial one. \nThis concludes the proof of Theorem~\\ref{thm} in this case as well.\n \n\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe massless Dirac fermions in the quasi-two-dimensional organic conductor \n$\\alpha$-(BEDT-TTF)$_2$I$_3$,\\cite{Katayama2006ZGS} \nwhich obey the tilted Weyl equation\\cite{Kobayashi2007,Goerbig2008}, have attracted much interest \nbecause of the mysteries of the experimental findings such as the weak temperature ($T$) dependence of resistivity \n(close to $h\/e^2$), the strong $T$-dependence of the Hall coefficient,\n\\cite{KajitaFirst,Tajima2006JPSJ,Tajima2009PRL,Kobayashi2008} \nand the two-step increase of resistivity with decreasing $T$ in the presence of the magnetic field.\n\\cite{Tajima2006JPSJ} \n\nIn the absence of magnetic field and under pressure $P=18$kbar, \nthe in-plane resistivity decreases weakly with decreasing $T$ from the room temperature \nand turns to increase below the onset temperature, $10$K, and then it saturates in the limit of $T\\rightarrow 0$.\n\\cite{Tajima2006JPSJ}\nThe origin of such weak but obvious $T$-dependence has not been elucidated yet.\nIn the presence of magnetic field, $H$, perpendicular to the conducting plane, \nthe two-step increase of resistivity is observed.\\cite{Tajima2006JPSJ}\nFor example at $H=10$T, with decreasing $T$, the resistivity decreases weakly from room temperature and turns to increase below $T_0 \\cong 20$K with the plateau in the lower temperature region.\nAnother relative sharp increase is observed around $T_l \\cong 5$K leading to apparent saturation \nas $T\\rightarrow 0$.\nBoth $T_0$ and $T_l$ increase with increasing magnetic field.\n\nIn the absence of tiling it has been established that the energy spectrum of the massless Dirac fermions \nbecomes discrete by the Landau quantization owing to the orbital motion in the magnetic field, \n$E_N={\\it sgn} (N) \\sqrt{2 \\hbar v^2 eH |N|\/c}$, where $v$ is the velocity, and $N$ is an integer. \nEach Landau level has large degeneracy proportional to $H$, and is split into two states \nwith up and down spins by the Zeeman energy, as shown in Fig. 1. \nThe effects of tilting have been investigated recently and it is found that the Landau-level structure is \nqualitatively the same and that the effects of tilting are incorporated as modifications of effective velocity. \nBy use of values of relevant parameters appropriate for $\\alpha$-(BEDT-TTF)$_2$I$_3$,\n\\cite{Goerbig2008,Morinari2008} \n$E_1 \\cong 5$meV and $2E_Z \\equiv g\\mu_{\\rm B}H \\cong 1$meV with the g-factor $g=2$ for H=10T.\nHence we see that the energy scale seen in resistivity measurement, $T_0$ ($\\cong 2$meV) and $T_l$, \nare smaller than $E_1$.\nThus, the observed two-step increase of resistivity in $\\alpha$-(BEDT-TTF)$_2$I$_3$ may be attributed \nto the $N=0$ Landau levels, whose causes will be studied theoretically in this paper.\n\nThe long range Coulomb interaction plays an important role for massless Dirac fermions. \nThe effective Coulomb interaction under magnetic field, $I$, is estimated as \n$I \\cong e^2\/\\epsilon l_{\\rm H} \\cong 50 \\sqrt{H{\\rm [T]}}\/\\epsilon$ meV, \nwhere $l_{\\rm H}$ is the magnetic length, $l_{\\rm H} =\\sqrt{\\hbar c\/eH}$.\nAlthough the polarizability $\\epsilon$ of $\\alpha$-(BEDT-TTF)$_2$I$_3$ under high pressure \nhas not been identified so far and then there is some ambiguity, \nit is demonstrated later that $N=0$ Landau states have instability toward the pseudo-spin ferromagnetism \nsince the effective Coulomb interaction $I$ can exceed $2E_z$.\nIn the presence of tilting, it is shown that the electron correlation can give rise to \nthe quantum Hall ferromagnet of the pseudo-spin \n(the degree of freedom on the valleys) with the help of the large degeneracy of the Landau levels.\nThe easy plane anisotropy of the pseudo-spin ferromagnet results from the back scattering processes \nwhich is the inter-valley scattering terms exchanging large momentum.\nMoreover, it is shown that the effects of fluctuations of phase variables of the order parameters \ncan be described by XY Heisenberg model leading to Kosterlitz-Thouless (KT) transition \n\\cite{Kosterlitz-Thouless1972} at lower temperature.\n\n\\begin{figure}[htb]\n\\begin{center}\n \\vspace{2mm}\n \\leavevmode\n\\epsfysize=7cm\\epsfbox{fig1.eps}\n \\vspace{-3mm}\n\\caption[]{Schematic figure of the Landau levels as a function of magnetic field.\nThe energy scale of the onset temperature of the first increase of resistivity, $T_0$, is located \nbetween the Zeeman gap $2E_Z$ and the $E_1 -E_0$ gap, where the Boltzmann factor $k_{\\rm B}$ is taken as unit.\n{\\color{black} The energy scale of the second increase of resistivity at low temperatures, $T_l$, is smaller than the Zeeman gap.}\n}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Formulation}\n\n\n\n\\subsection{Hamiltonian {\\color{black} for} massless Dirac fermions with tilting}\n\\label{Hamiltonian}\n\n\n\nIn the absence of magnetic field, the Hamiltonian of the massless Dirac fermions is given by,\n{\\color{black} \n\\begin{eqnarray}\n{\\cal H}&=&{\\cal H}_0 +{\\cal H}^\\prime , \\nonumber \\\\ \n{\\cal H}_{\\rm 0} &=&\n\\sum_{{\\bf k} \\gamma \\gamma^\\prime \\sigma \\tau} \n[ {\\cal H}_0^{\\sigma \\tau} ]_{\\gamma \\gamma^\\prime}\nc^\\dagger_{{\\bf k} \\gamma \\sigma \\tau} c_{{\\bf k} \\gamma^\\prime \\sigma \\tau} \\nonumber \\\\ \n{\\cal H}^\\prime &=& \\frac{1}{2} \\int \\int {\\rm d}{\\bf r} {\\rm d}{\\bf r}^\\prime \nV_0 ({\\bf r}-{\\bf r}^\\prime ) n({\\bf r}) n({\\bf r}^\\prime )\n\\end{eqnarray}\n} with the long-range Coulomb interaction $V_0 ({\\bf r})= e^2 \/\\epsilon r$ and the density operator $n({\\bf r})$.\n{\\color{black} \nThe degree of freedom on the spins are represented as $\\sigma =\\pm$ \ncorresponding to $\\uparrow$, $\\downarrow$.\nThe degree of freedom of the pseudo-spins, $\\tau =\\pm$, corresponds to the valleys $R$, $L$. \nThe valleys are located at the crossing points of the conduction and valence bands, \n$\\pm {\\bf k}_0$, where ${\\bf k}_0$ is an incommensurate momentum in the first Brillouin zone.\\cite{Katayama2006ZGS}\nThe creation and annihilation operators, \n$c^\\dagger_{{\\bf k} \\gamma \\sigma \\tau}$ and $c_{{\\bf k} \\gamma \\sigma \\tau}$, respectively, \nare based on the Luttinger-Kohn representation\\cite{LuttingerKohn} using the Bloch's functions \nat the crossing points as the basis of wave functions,\\cite{Kobayashi2007} \nand then $\\gamma =1,2$ denotes the basis of the Luttinger-Kohn representation.\nThe relation between the Luttinger-Kohn representation and the site representation based on the molecular orbitals \nare described in Appendix.}\nThe low-energy properties around the two crossing points, labeled by $\\tau=\\pm$, are described in \nterms of the two tilted Weyl Hamiltonians\\cite{Kobayashi2007,Goerbig2008} \n\\begin{eqnarray}\\label{eq02}\n&&H_0^{\\sigma , \\tau =+}=\\hbar ( v {\\bf k}\\cdot\\mbox{\\boldmath $\\sigma $} + {\\bf w}_0\\cdot {\\bf k} \\sigma_0) \\nonumber \\\\ \n&&H_0^{\\sigma , \\tau =-}=-\\hbar ( v {\\bf k}\\cdot\\mbox{\\boldmath $\\sigma $}^* + {\\bf w}_0\\cdot {\\bf k} \\sigma_0) \n\\end{eqnarray}\nwith respect to the time-reversal symmetry, where \n$v$ represents the velocity of the cone and ${\\bf w}_0$ represents the tilting velocity.\nHere, we have neglected the anisotropy of the velocity of the cone.\nIn the case of the cone with anisotropic the velocity, \none needs to replace $v {\\bf k}\\rightarrow (v_x k_x, v_y k_y)$.\n\nOnce a magnetic field perpendicular to the conducting plane is taken into account, \nthe momentum ${\\bf k}$ is replaced by ${\\bf k}+(e\/c){\\bf A}$ with the vector potential \nin the Landau gauge ${\\bf A}=(0, Hx, 0)$, and then we obtain \n\\begin{eqnarray}\\label{eq02}\n&&H_0^{\\sigma , \\tau =+}=\\left[\n\\begin{array}{cc}\n\\hbar \\{ w_{0x} \\frac{1}{\\rm i} \\frac{\\partial}{\\partial x} + w_{0y} (\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} \n+\\frac{eHx}{c} ) \\} -\\sigma E_Z \n& \\hbar v \\{ \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial x} \n-{\\rm i} ( \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\} \\\\\n\\hbar v \\{ \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial x} \n+{\\rm i} ( \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\}\n& \\hbar \\{ w_{0x} \\frac{1}{\\rm i} \\frac{\\partial}{\\partial x} \n+ w_{0y} (\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\} -\\sigma E_Z \\\\ \n\\end{array}\n\\right] \\nonumber \\\\\n&&H_0^{\\sigma , \\tau =-}=\\left[\n\\begin{array}{cc}\n\\hbar \\{ w_{0x} \\frac{1}{\\rm i} \\frac{\\partial}{\\partial x} \n+ w_{0y} (\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\} -\\sigma E_Z \n& \\hbar v \\{ -\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial x} \n-{\\rm i} ( \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\} \\\\\n\\hbar v \\{ -\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial x} \n+{\\rm i} ( \\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\}\n& \\hbar \\{ w_{0x} \\frac{1}{\\rm i} \\frac{\\partial}{\\partial x} \n+ w_{0y} (\\frac{1}{{\\rm i}} \\frac{\\partial}{\\partial y} +\\frac{eHx}{c} ) \\} -\\sigma E_Z \\\\ \n\\end{array}\n\\right]\n\\end{eqnarray}\nin terms of the Zeeman energy $E_Z$.\n\n\n\n\\subsection{Zero-energy Landau level for the case with tilting}\n\\label{appC}\n\n\\YS{\nThe eigen equations for the zero-energy Landau level are given by \n\\begin{equation}\\label{WFT00}\nH_0^{\\sigma , \\tau} \\phi_X^{\\tau}({\\bf r}) =0\n\\end{equation}\nwhich gives the eigen functions of the tilted Weyl Hamiltonians in the presence of magnetic field.\nThe wave functions are given by\\cite{Goerbig2008} }\n\\begin{equation}\\label{WFT}\n\\phi_X^{\\tau}({\\bf r})=\\frac{1}{\\sqrt{L}}e^{-{\\rm i}X y\/l_H^2}\\varphi^{\\tau}(x-X) e^{-{\\rm i}\\tau {\\bf k}_0\\cdot {\\bf r}}\n\\end{equation}\nwith \n\\begin{equation}\\label{WF+}\n\\varphi^\\tau (x)=\\frac{1}{\\sqrt{\\sqrt{\\pi\/\\gamma}l_H}}\n\\chi_\\tau\ne^{-\\gamma x^2\/2l_H^2} ,\n\\end{equation}\nwhere $X$ is the {\\color{black} guiding-center} coordinate and $L$ is the length of the system.\nThe spinor parts $\\chi_\\tau$ are given by \n\\begin{eqnarray}\n\\label{spin+}\n\\chi_{\\tau=+} &=& \\frac{1}{\\sqrt{\\tilde{w}_0^2+(1+\\gamma)^2}}\\left(\n\\begin{array}{c} \n-\\tilde{w}_0 e^{-i\\varphi} \\\\ 1+\\gamma \n\\end{array}\\right)\\ , \\\\\n\\label{spin-}\n\\chi_{\\tau=-} &=& \\frac{1}{\\sqrt{\\tilde{w}_0^2+(1+\\gamma)^2}}\\left(\n\\begin{array}{c} \n 1+\\gamma \\\\ -\\tilde{w}_0 e^{+i\\varphi} \n\\end{array}\\right)\\ .\n\\end{eqnarray}\nHere, we have defined\n\\begin{equation}\n\\tilde{w}_0 e^{i\\varphi}\\equiv \\frac{w_{0x}+i w_{0y}}{v} ,\n\\end{equation}\nin terms of the effective tilting parameter \n\\begin{equation}\\label{tiltparam}\n\\tilde{w}_0\\equiv\\sqrt{\\left(\\frac{w_{0x}}{v}\\right)^2+\n\\left(\\frac{w_{0y}}{v}\\right)^2}\n\\end{equation}\nwith ${\\bf w}_0 =(w_{0x},w_{0y})$, and $\\gamma=\\sqrt{1-\\tilde{w}_0^2}$.\nWe note that one recovers the usual result for the $n=0$ wave function in graphene \nwhen we use in the limit $\\tilde{w}_0\\rightarrow 0$\n($\\gamma\\rightarrow 1$), i.e. in the case without tilting.\n\n\n\n\\subsection{Effective Hamiltonian on fictitious magnetic lattice}\n\n\n\nWe consider the present model by using the bases of the Wannier functions \nfor the magnetic rectangular lattice which is introduced fictitiously.\nThe wave functions in the Landau gauge \nare not localized in the $y$-direction, {\\color{black} but in the $x$-direction around the position $X$.}\nBy applying the periodic boundary condition in the $y$-direction with the system length $L$, \nthe $X$ is discretized as $X=-2 \\pi l_{\\rm H}^2 j \/L$ with an integer $j$.\nThe Wannier functions, which satisfy orthonormality and are localized \naround ${\\bf R} =(ma,nb)$ with integers $m$ and $n$ \nas shown in Fig. 2, are constructed by the linear combination of $\\phi_{X} ({\\bf r})$,\\cite{Fukuyama1977} \n\\begin{equation}\\label{wannier}\n\\Phi_{{\\bf R}}^{\\tau}({\\bf r})=\\frac{\\sqrt{L}}{a\\sqrt{b}}\\int_{-a\/2}^{a\/2} dX\\, e^{i X nb\/l_H^2}\\phi_{X+ma}^{\\tau}({\\bf r})\\ ,\n\\end{equation}\nwhere $a$ is arbitrary but $a \\gg 2 \\pi l_{\\rm H}^2 \/L$ and $b=2 \\pi l_{\\rm H}^2 \/a$. \nWe note that $\\vert \\Phi_{{\\bf R}}^{\\tau}({\\bf r}) \\vert$ exhibits an exponential-like decrease in the $x$-direction, but decreases \n{\\color{black} algebraically} as $\\vert y \\vert^{-1}$ in the $y$-direction.\n\n\\begin{figure}[htb]\n\\begin{center}\n \\vspace{2mm}\n \\leavevmode\n\\epsfysize=7cm\\epsfbox{fig2.eps}\n \\vspace{-3mm}\n\\caption[]{\nThe ${\\bf r}$-dependence of $\\vert \\Phi_{{\\bf R}}^{\\tau}({\\bf r}) \\vert^2$ with ${\\bf R}={\\bf 0}$, \nwhere $\\Phi_{{\\bf R}}^{\\tau}({\\bf r})$ is the Wannier functions for the magnetic rectangular lattice.\n}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nIn the basis of these Wannier functions, the effective interaction Hamiltonian is given by\n\\begin{eqnarray}\n&& {\\cal H}_{N=0}^\\prime = \\frac{1}{2} \\sum_{{\\bf R}_{1,2,3,4}} \n\\sum_{\\sigma \\sigma^\\prime \\tau_{1,2,3,4}} \n\\int \\int {\\rm d}{\\bf r} {\\rm d}{\\bf r}^\\prime V_0 ({\\bf r}-{\\bf r}^\\prime ) \\nonumber \\\\\n&& \\times {\\color{black} \\left[ \\Phi_{{\\bf R}_1}^{\\tau_1} ({\\bf r})^\\dagger \\cdot \\Phi_{{\\bf R}_2}^{\\tau_2} ({\\bf r}) \\right] \n\\left[ \\Phi_{{\\bf R}_3}^{\\tau_3} ({\\bf r}^\\prime)^\\dagger \\cdot \\Phi_{{\\bf R}_4}^{\\tau_4} ({\\bf r}^\\prime) \\right]} \\nonumber \\\\\n&& \\times \nc_{{\\bf R}_1 \\sigma \\tau_1}^\\dagger c_{{\\bf R}_2 \\sigma \\tau_2} \nc_{{\\bf R}_3 \\sigma^\\prime \\tau_3}^\\dagger c_{{\\bf R}_4 \\sigma^\\prime \\tau_4} \n\\end{eqnarray}\nunder the assumption that the density operator is effectively determined by the field operator \nfor $N=0$ Landau states and the contributions to the Hamiltonian from $N \\ne 0$ states are negligible, \n{\\it i. e.} we use the density operator\n\\begin{eqnarray}\nn ({\\bf r})= \\sum_{{\\bf R} {\\bf R}^\\prime \\sigma \\tau \\tau^\\prime} \n {\\color{black} \\left[ \\Phi_{\\bf R}^\\tau ({\\bf r})^\\dagger \\cdot \\Phi_{{\\bf R}^\\prime}^{\\tau^\\prime} ({\\bf r}) \\right]}\n c_{{\\bf R} \\sigma \\tau}^\\dagger c_{{\\bf R}^\\prime \\sigma \\tau^\\prime}.\n\\end{eqnarray}\n\nThus the effective Hamiltonian for the $N=0$ Landau states in the magnetic rectangular lattice is given by \n\\begin{eqnarray}\n{\\cal H}_{\\rm eff} &=& \n\\sum_{i \\sigma \\tau} (-\\sigma E_Z) c^\\dagger_{i \\sigma \\tau} c_{i \\sigma \\tau} \\nonumber \\\\\n&+& \\sum_{i j k l \\sigma \\sigma^\\prime \\tau \\tau^\\prime} V_{i j k l} \nc^\\dagger_{i \\sigma \\tau} c_{j \\sigma \\tau} \nc^\\dagger_{k \\sigma^\\prime \\tau^\\prime} c_{l \\sigma^\\prime \\tau^\\prime} \\nonumber \\\\\n&+& \\sum_{i j k l \\sigma \\sigma^\\prime \\tau} W_{i j k l} \nc^\\dagger_{i \\sigma \\bar{\\tau}} c_{j \\sigma \\tau} \nc^\\dagger_{k \\sigma^\\prime \\tau} c_{l \\sigma^\\prime \\bar{\\tau}} ,\n\\end{eqnarray}\nwhere $i$, $j$, $k$, and $l$ denote the unit cells of the magnetic rectangular lattice \nat ${\\bf R}_i$, ${\\bf R}_j$, ${\\bf R}_k$, and ${\\bf R}_l$, respectively, and $\\bar{\\tau} =-\\tau$.\n\nThe forward-scattering term, $V_{i j k l}$, is given by\n\\begin{equation}\nV_{i j k l} = \\frac{1}{2} \\int \\int {\\rm d}{\\bf r} {\\rm d}{\\bf r}^\\prime \\,\nV_0 ({\\bf r}-{\\bf r}^\\prime ) \\,\n{\\color{black} \\left[ \\Phi_i^\\tau ({\\bf r})^\\dagger \\cdot \\Phi_j^\\tau ({\\bf r}) \\right] \n\\left[ \\Phi_k^{\\tau^\\prime} ({\\bf r}^\\prime)^\\dagger \\cdot \\Phi_l^{\\tau^\\prime} ({\\bf r}^\\prime) \\right]}\n\\end{equation}\nfrom the long wave length part of ${\\cal H}_{N=0}^\\prime$.\nThis term does not depend on the spin and pseudo-spin, and then \nit does not break the SU(4) symmetry, {\\color{black} neither in the spin subspace nor in that of the pseudo-spin}.\nWe find that the forward-scattering term is not affected by the tilting, \nbecause $( \\chi_{\\tau}^{\\dagger} \\cdot \\chi_{\\tau} )( \\chi_{\\tau^\\prime}^{\\dagger} \\cdot \\chi_{\\tau^\\prime} )=1$.\n\n\n\nOn the other hand, the backscattering term, $W_{i j k l}$, \nwhich is the inter-valley scattering term exchanging large momentum $2k_0$ and \nbreaks the SU(2) symmetry in the subspace of the pseudo-spin, \nis given by\n\\begin{equation}\nW_{i j k l} = \\frac{1}{2} \\int \\int {\\rm d}{\\bf r} {\\rm d}{\\bf r}^\\prime \\,\nV_0 ({\\bf r}-{\\bf r}^\\prime ) \\,\n\\color{black}{\\left[ \\Phi_i^{\\bar{\\tau}} ({\\bf r})^\\dagger \\cdot \\Phi_j^\\tau ({\\bf r}) \\right] \n\\left[ \\Phi_k^\\tau ({\\bf r}^\\prime)^\\dagger \\cdot \\Phi_l^{\\bar{\\tau}} ({\\bf r}^\\prime) \\right]}\n\\end{equation}\nfrom the short wave length part of ${\\cal H}_{N=0}^\\prime$.\nIn the absence of tilting, {\\color{black} as e.g. in graphene,} the backscattering term vanishes because \n$( \\chi_{\\bar{\\tau}}^{\\dagger} \\cdot \\chi_{\\tau} )=0$.\\cite{Goerbig2006}\nWe find that the tilting is essential to have a non-zero backscattering term.\nThe tilting dependence of the backscattering term is given by the spinor part, \n\\begin{equation}\\label{BW}\n( \\chi_{\\bar{\\tau}}^{\\dagger} \\cdot \\chi_{\\tau} )( \\chi_{\\tau}^{\\dagger} \\cdot \\chi_{\\bar{\\tau}} )\n=\\frac{4\\tilde{w}_0^2(1+\\gamma)^2 }\n{[\\tilde{w}_0^2+(1+\\gamma)^2]^2}\\simeq 4\\tilde{w}_0^2 + O (\\tilde{w}_0^4) ,\n\\end{equation}\nwhere the last step has been obtained from the limit $\\tilde{w}_0\\ll 1$.\n{\\color{black} \nThe ratio between the forward and the backscattering terms, $W_{ijkl}\/V_{ijkl}$, is given by \n\\begin{equation}\\label{BW}\nW_{ijkl}\/V_{ijkl} \n\\simeq \\frac{\\tilde{w}_0^2 a_{\\rm L}}{l_{\\rm H}} ,\n\\end{equation}\n}where $a_{\\rm L}$ is the lattice constant in the conducting plane.\nThe backscattering term is proportional to $a_{\\rm L}$, \nsince the large momentum $\\vert 2 k_0 \\vert \\cong \\pi \/a_{\\rm L}$ is exchanged.\\cite{Goerbig2006}\nWe note that the lattice constant of $\\alpha$-(BEDT-TTF)$_2$I$_3$, $a_{\\rm L} \\cong 10 {\\rm \\AA}$, \nis much larger than that of graphene.\nThus it is expected that the backscattering term {\\color{black} plays an important role for electron-correlation effects \nin $\\alpha$-(BEDT-TTF)$_2$I$_3$}.\nThe typical value of the ratio $W_{ijkl}\/V_{ijkl}$ is approximately $0.07$ for $\\alpha$-(BEDT-TTF)$_2$I$_3$ \nat $H=10$T using the tilting parameter $\\tilde{w}_0 \\cong 0.8$.\nThe Umklapp scattering term ($\\tau \\bar{\\tau} \\tau \\bar{\\tau}$-term) can be neglected, \nbecause it is exponentially smaller than the other terms as a function of $a_{\\rm L}\/l_{\\rm H}$, \nwhich is estimated as $0.1$ at $H=10T$ in $\\alpha$-(BEDT-TTF)$_2$I$_3$.\n\n\n\n\\section{Pseudo-spin ferromagnet and Kosterlitz-Thouless transition}~\n\n\n\nPossible spin and pseudo-spin ferromagnetic states in the zero-energy Landau level in graphene have\nbeen extensively studied in recent years.\\cite{Goerbig2006,NomuraMacDonald2006,Alicea2006,Yang2006,Gusynin2006,Herbut2007,doretto,Ezawa2007}\nGenerically, the ferromagnetic ordering may be understood within an interaction model with no explicit\nspin or pseudo-spin symmetry breaking; in order to minimize their exchange energy, the global $N$-particle \nwave function should be fully antisymmetric in its orbital part, the (pseudo-)spin part needs to be fully\nsymmetric in order to fulfil fermionic statistics. Whereas in a normal metal this ordering is only partial, due\nto the increase in the kinetic energy, a single Landau level may be viewed as an infinitely flat energy band,\nand the ferromagnetic ordering may therefore be complete. In the absence of an explicit symmetry breaking, such as\nthe Zeeman effect that naturally tends to polarize the physical spin or the above-mentioned backscattering term\nthat affects the pseudo-spin, no particular spin or pseudo-spin channel is selected, and one may even find \nan entangled spin-pseudo-spin ferromagnetic state.\\cite{Doucot2008} The symmetry-breaking terms may, thus, be\nviewed as ones that choose a particular channel (spin or pseudo-spin) and direction of a pre-existing ferromagnetic\nstate by explicitly breaking the original SU(4) symmetry.\n\n\n\n\\subsection{Mean-field solution}\n\n\n\nThe mean-field Hamiltonian for the pseudo-spin ferromagnetic state is given by\n\\begin{eqnarray}\n&& {\\cal H}_{\\rm MF} = \\sum_{j \\sigma \\tau} \\left[-\\sigma E_Z -2\\sum_i \n(V_{ijji} n_{i\\sigma \\tau} +W_{ijji} n_{i\\sigma \\bar{\\tau}}) \\right]\nc_{j\\sigma \\tau}^\\dagger c_{j\\sigma \\tau} \\nonumber \\\\\n&& -2\\sum_{ij\\sigma \\tau} (V_{ijji} +W_{iijj}) (\\langle c_{i\\sigma \\tau}^\\dagger c_{i\\sigma \\bar{\\tau}} \\rangle \nc_{j\\sigma \\bar{\\tau}}^\\dagger c_{j\\sigma \\tau} +{\\rm h.c})\n\\end{eqnarray}\nand the order parameter of the pseudo-spin ferromagnetic state, $\\Delta$, \nwhich is independent of $i$ and $\\sigma$, is defined by \n\\begin{equation}\n\\Delta = 2I \\langle c_{i\\sigma -}^\\dagger c_{i\\sigma +} \\rangle \n\\end{equation}\nwith the effective interaction $I=\\sum_i (V_{i00i} +W_{ii00})$.\nUsing the spin polarization, $m = \\sum_{\\sigma \\tau} \\sigma n_{i\\sigma \\tau}$, which is also independent of $i$, \nand the renormalized Zeeman energy, $\\tilde{E}_Z = E_Z +mI$, the mean-field Hamiltonian is given by \n\\begin{eqnarray}\n&& {\\cal H}_{\\rm MF} = \\sum_{j} {\\bf c}_j^\\dagger \\hat{{\\cal H}} {\\bf c}_j \\nonumber \\\\ \n&& \\hat{{\\cal H}} = \\left[\n\\begin{array}{cccc}\n-\\tilde{E}_Z & -\\Delta^\\ast & 0 & 0 \\nonumber \\\\ \n-\\Delta & -\\tilde{E}_Z & 0 & 0 \\nonumber \\\\ \n0 & 0 & \\tilde{E}_Z & -\\Delta^\\ast \\nonumber \\\\ \n0 & 0 & -\\Delta & \\tilde{E}_Z \\nonumber \\\\ \n\\end{array}\n\\right]\n\\end{eqnarray}\nwith ${\\bf c}_j =(c_{j \\uparrow +}, c_{j \\uparrow -}, c_{j \\downarrow +}, c_{j \\downarrow -} )$.\nThe mean-field solution is calculated from \n\\begin{equation}\n\\vert \\Delta \\vert =\\frac{2I}{\\pi} \\int {\\rm d}x f(x) {\\rm Im} \n\\frac{\\vert \\Delta \\vert}{(x+{\\rm i}\\delta -\\sigma \\tilde{E}_Z)^2 -\\vert \\Delta \\vert^2} \n\\end{equation}\nand \n\\begin{equation}\nm =\\frac{1}{2} \\sum_{\\sigma \\tau} \\sigma f(-\\sigma \\tilde{E}_Z +\\tau \\vert \\Delta \\vert )\n\\end{equation}\nwith the Fermi distribution function, $f(x)$. \n\nThe ground state in the case with $I < E_Z$ is a spin polarized state \nwithout pseudo-spin polarization ($m=1$ and $\\Delta =0$ at $T=0$), \nwhere electrons reside in the spin-up branches of the $N=0$ Landau levels \n(see the left hand side of fig. 3(a)).\nThe ground state in the case with $I > E_Z$ is, on the other hand, \na pseudo-spin ferromagnetic state ($m=0$ and $\\vert \\Delta \\vert =I$ at $T=0$), \nwhere the easy-plane pseudo-spin polarization lifts \nthe pseudo-spin degeneracy and then the spin polarization is suppressed \n(see the right hand side of fig. 3(a)).\n\nThe mean-field phase diagram in the $I$-$T$ plane scaled by $E_Z$ is shown in fig. 3(b).\nThe transition temperature for the easy-plane pseudo-spin ferromagnetic state, $T_{\\rm c}$, \nis finite in the case with $I > E_Z$, and increases with increasing $I$.\nBelow $T_{\\rm c}$, the spin polarization vanishes at $T \\rightarrow 0$, \nalthough it is still finite at finite temperatures.\n\nOne notices that this competition between a spin polarized state and an easy-plane pseudo-spin ferromagnetism \nis original to the filling factor $\\nu=0$, \nwhere necessarily two (of four) subbranches of the zero-energy Landau levels are occupied. In contrast\nto this particular filling factor, this competition is absent at $\\nu=\\pm 1$, \nwhere only one subbranch is occupied and where,\ntherefore, a spin polarization does not exclude a simultaneous pseudo-spin ordering \nin a coherent superposition of both pseudo-spin states. \n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n \\vspace{2mm}\n \\leavevmode\n\\epsfysize=7cm\\epsfbox{fig3.eps}\n \\vspace{-3mm}\n\\caption[]{\n(a) Schematic figure of the energy levels in the spin-polarized state (left hand side) \nand the pseudo-spin ferromagnetic state (right hand side).\n(b) Phase diagram in the plane of the interaction $I$ and temperature $T$ scaled by the Zeeman energy $E_Z$.\n}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\subsection{Phase fluctuations and Kosterlitz-Thouless transition}\n\nIn the presence of {\\color{black} an order parameter with a finite amplitude} below $T_c$,\nphase fluctuation exists with the characteristic length of spatial variation \nmuch longer than the fictitious lattice spacing.\nThe effect of these phase fluctuation, {\\color{black} which has so far been ignored} in the mean-field approximation, is treated \non the basis of the Wannier functions and the resulting model is similar to the {\\color{black} XY model}\nleading to the KT transition.\nUsing the pseudo-spin operator, $\\tilde{S}_{i-}^\\sigma \\equiv c_{i\\sigma L}^\\dagger c_{i\\sigma R}$, \nand the real spin operator, $S_{jz} =\\frac{1}{2} \\sum_{\\sigma \\tau} \\sigma c_{j\\sigma \\tau}^\\dagger c_{j\\sigma \\tau}$, \nthe mean-field Hamiltonian is given by \n\\begin{eqnarray}\n{\\cal H}_{\\rm MF} &=& -2(E_Z + mI) \\sum_j S_{jz} \\nonumber \\\\\n&-& 2\\sum_{ij\\sigma} I_{ij} \\left(\\langle \\tilde{S}_{i-}^\\sigma \\rangle \n\\tilde{S}_{j+}^\\sigma + \\langle \\tilde{S}_{i+}^\\sigma \\rangle \n\\tilde{S}_{j-}^\\sigma \\right) .\n\\end{eqnarray}\nThe real spin polarization, $m$, remains finite at finite temperatures, \nalthough it vanishes at $T \\rightarrow 0$ in the pseudo-spin ferromagnetic state.\nHowever, it can be shown that the spin polarization is independent of $i$, \nsince the pseudo-spin can fluctuate only in the easy plane and then \nthe occupation numbers of electrons are independent of $i$.\nThe interactions between the pseudo-spins on the magnetic rectangular lattice, \n$I_{ij} \\equiv V_{ijji}+W_{iijj}$.\nThe interaction $I_{ij}$ rapidly decreases with increasing $\\vert {\\bf R}_j -{\\bf R}_i \\vert$.\nIt is numerically found {\\color{black} as seen in Fig. 4} that the nearest-neighbor $I_{ij}$ \nis approximately isotropic and the ratio \n$I_{i,i+1} \/I_{i,i}=0.10$ with the arbitrary choice of $b=\\sqrt{2}a$ \n(but leading to the almost isotropic localization of the Wannier function on the fictitious lattice), \n\\YS{ where\n\\begin{equation}\\label{AD00}\n\\frac{I_{i,j}}{I_{i,i}} = \\frac{V_{ijji}+W_{iijj}}{V_{iiii}+W_{iiii}} \\cong \\frac{V_{ijji}}{V_{iiii}}\n\\end{equation} }\nThe phase of the order parameter corresponds to the angles of the pseudo-spin.\nThe $x$- and $y$-components of the pseudo-spins are given by \n$\\langle \\tilde{S}_{ix}^\\sigma \\rangle ={\\rm Re} \\langle \\tilde{S}_{i-}^\\sigma \\rangle$ and \n$\\langle \\tilde{S}_{iy}^\\sigma \\rangle =-{\\rm Im} \\langle \\tilde{S}_{i-}^\\sigma \\rangle$, respectively, \nwith $\\langle \\tilde{S}_{i+}^\\sigma \\rangle =\\langle \\tilde{S}_{i-}^\\sigma \\rangle^\\ast$. \nThus $\\langle \\tilde{S}_{i-}^\\sigma \\rangle$ can be represented using the angle of the pseudo-spin from the $x$-direction \nin the $x$-$y$ plane, $\\phi_i^\\sigma$, as \n$\\langle \\tilde{S}_{i-}^\\sigma \\rangle = \\vert \\langle \\tilde{S}_{i-}^\\sigma \\rangle \\vert \n\\exp (-{\\rm i}\\phi_i^\\sigma ) $.\nWhen the characteristic length of spatial variation of the phases is much longer than the lattice spacing, \nwe can expand the free energy by the fluctuations of the phases \n$f_{ij}^\\sigma =1-\\cos (\\phi_j^\\sigma -\\phi_i^\\sigma ) \\ll 1$ \nunder the assumption that the amplitude, $\\vert \\langle \\tilde{S}_{i-}^\\sigma \\rangle \\vert$, \ndoes not change within the characteristic length of the phase fluctuation, and then we obtain \n\\begin{eqnarray}\nF_f -F_n & \\cong & (F_f -F_n )\\vert_{f_{ij}^\\sigma=0} +\\sum_{\\langle i \\ne j \\rangle \\sigma} \n\\frac{\\partial (F_f -F_n )}{\\partial f_{ij}^\\sigma} f_{ij}^\\sigma \\nonumber \\\\\n&=& F_0 -\\sum_{\\langle i \\ne j \\rangle \\sigma} J_{ij}^\\sigma \\cos (\\phi_j^\\sigma -\\phi_i^\\sigma )\n\\end{eqnarray}\nwith $J_{ij}^\\sigma =4 \\vert \\langle \\tilde{S}_{-}^\\sigma \\rangle \\vert^2 I_{ij}$, \nwhere $F_f$ and $F_n$ denote the free energies of the pseudo-spin ferromagnetic and normal states, respectively, \nand $F_0$ is independent of the phases.\nThe effects of the phase fluctuations in $J_{ij}^\\sigma$ are neglected since those effects are the higher-order terms, \nand $J_{ij}^\\sigma \\rightarrow I_{ij}$ at temperatures much lower than $T_{\\rm c}$.\nThen the physics of the phase fluctuation is equivalent to that of the two-dimensional $XY$ Heisenberg model \nwith the nearest-neighbor exchange interaction, $J \\cong I_{i,i+1} \\cong 0.087 I$. \n{\\color{black} In the two-dimensional XY model, the KT transition occurs \ndue to the onset of bound pairs of the vortices.\\cite{Kosterlitz-Thouless1972}}\nIt is known that $T_{\\rm KT} \\cong 1.54 J$ by the renormalization group analysis.\\cite{Kosterlitz1974}\nHere the effects of spin polarization are negligible, because \n$J$ is much larger than the Zeeman energy and the phase fluctuation on each electron with up or down spin \nis described by the same interaction, $J$.\n\nLastly we discuss the role of the long-range part of $I_{ij}$ farther than the nearest neighbor one.\nFigure 4 shows the distance dependences of $I_{0,j}\/I_{0,0}$ with ${\\bf R}_j =(na,0)$ (the closed circles) and \n${\\bf R}_i =(0,nb)$ (the open circles) defined on an integer $n$, where we take $b=\\sqrt{2}a$.\nThe interaction $I_{0j}$ decays very rapidly along the $x$-axis but slowly along $y$-axis.\nThen the role of the long range part of $I_{ij}$ along $y$-axis should be considered for improving \nthe effective Hamiltonian. \nThe KT transition temperature $T_{\\rm KT}$, however, is determined by the competition \nbetween the excitation energy of a vortex and the entropy effect coming from the degree of freedom \nfor the position of the vortex core.\nSince the length scale of the vortex is much longer than $a$ and \nthe interaction is ferromagnetic, \nthe long-range part of $I_{ij}$ does not disturb the KT transition essentially. \n\n\\begin{figure}[htb]\n\\begin{center}\n \\vspace{2mm}\n \\leavevmode\n\\epsfysize=7cm\\epsfbox{fig4.eps}\n \\vspace{-3mm}\n\\caption[]{\nThe $n$-dependences of $I_{0,j}\/I_{0,0}$ with ${\\bf R}_j =(na,0)$ (the closed circles) and \n${\\bf R}_i =(0,nb)$ (the open circles), where we take $b=\\sqrt{2}a$.\nThe dashed and dotted lines are {\\color{black} a guide to the eye}.\n}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Relation between Experimental Findings and Theoretical Results}\n\nIn the presence of magnetic field perpendicular to the conducting plane, \nthe two-step increase of resistivity, for example at $T_0 \\cong 20$K and $T_l \\cong 5$K at $H=10$T, \nis observed in $\\alpha$-(BEDT-TTF)$_2$I$_3$, \nwhere both $T_0$ and $T_l$ increase with increasing magnetic field.\\cite{Tajima2006JPSJ}\nWe may be able to associate two stepwise changes of resistivity \nas due to the easy-plane pseudo-spin ferromagnetic transition at $T_{\\rm c}$ and the KT transition at $T_{\\rm KT}$, \nsince our estimate indicates that $T_{\\rm c} \\cong 4 T_{\\rm KT}$, \nwhere $T_{\\rm c} \\cong 0.5 I$ as seen in Fig. 3(b) in the region of $I\/E_Z >>1$ of interest and \n$T_{\\rm KT} \\cong 1.54J \\cong 0.13 I$.\nWe emphasize that the tilting of the Dirac cone is essential to the appearance of the easy-plane \npseudo-spin ferromagnet, and thus, \nto the appearance of the KT transition, due to the long range Coulomb interaction.\n\n\n\n\\section{Conclusion and Discussion}\n\n\n\nIn the present paper, motivated by the experimental observation of the particular temperature dependences \nof resistivity in $\\alpha$-(BEDT-TTF)$_2$I$_3$ under magnetic field, the\npossibility of the pseudo-spin quantum Hall ferromagnet at $\\nu =0$ has been investigated \nin the massless Dirac fermion system. \nThe pseudo-spin ferromagnetic transition occurs when the electron correlation exceeds the Zeeman energy.\nThe tilting of the Dirac cone induces the backscattering terms resulting in the easy-plane pseudo-spin ferromagnet.\nThere will be intrinsic fluctuations and $T_{\\rm c}$ should be considered \nonly as a crossover temperature for the growing amplitude of order parameters with remaining large phase fluctuations \nin the two-dimensional system.\nTo treat such phase fluctuations, {\\color{black} a} spatially localized basis set similar to ``Wannier function'' \nare introduced, which indicates that the model is similar to {\\color{black} the XY model} which is known to lead to \nthe KT transition at lower temperature, $T_{\\rm KT}$.\nIn comparison with experiments, the two-step increase of resistivity with decreasing temperature are observed \n at around $20$K and $5$K at $H= 10$T. \nPresent theory has revealed $T_{\\rm KT} \\cong 5$K on the choice of the parameters giving $T_{\\rm c} \\cong 20$K, \nand then there are reasonable correspondences to identify two stepwise changes of resistivity \nas due to the amplitude growing and the phase coherence of the order parameters.\n\nObviously there are remaining problems to be clarified. \nThe experimental data indicates the saturation of resistivity in the low temperature. \nThe saturation indicates the existence of dilute carriers which may originate from \nweak three-dimensionality or disorder. \n\nIn graphene,\\cite{graphene} the the quantum Hall ferromagnet at $\\nu =0$ has been investigated,\n\\cite{Goerbig2006,Alicea2006,NomuraMacDonald2006,Gusynin2006,Ezawa2007} \nand very recently it is suggested that the electron-phonon interaction breaking the pseudo-spin SU(2) symmetry, \nwhich may be characteristic of graphene, induces the easy-plane pseudo-spin ferromagnet \nresulting in the KT transition \\cite{Nomura2009} in order to explain the possible KT transition \nobserved in graphene.\\cite{Ong2008} \nWe emphasize that the backscattering term, which is the key factor for the easy-plane pseudo-spin ferromagnet \nin our paper, is characteristic of $\\alpha$-(BEDT-TTF)$_2$I$_3$, but can be realized in graphene \nby distorting the honeycomb lattice.\nIn addition, we note that a lattice model describing the fluctuations of the pseudo-spins \nshould be based on Wannier functions which satisfy orthonormality under magnetic field, \nsince the bases on the original crystal lattice are no longer the eigenstates of the $N=0$ Landau levels.\n\nThe effects of the short range parts of the Coulomb interaction in graphene also have been investigated.\nIf once the pseudo-spin ferromagnetism occurs, the Hubbard-$U$-type on-site interaction favors \nthe easy-plane ferromagnetism (uniform charge density), \nwhile the nearest-neighbor interaction $V$ favors the easy-axis ferromagnetism \n(the sublatice CDW), in the case of graphene.\\cite{Alicea2006}\nIn $\\alpha$-(BEDT-TTF)$_2$I$_3$, however, the easy-axis ferromagnetism does not \ncorrespond to CDW directly, because the bases of the Weyl Hamiltonian is not the sublattice \nbut the Bloch states at ${\\bf k}=\\pm {\\bf k}_0$. \nThe Bloch states at ${\\bf k}=\\pm {\\bf k}_0$ consist of the linear combination \nof the contributions from four BEDT-TTF molecules. \nThus, although the easy-axis ferromagnetism may modify \nthe intrinsic charge disproportionation in $\\alpha$-(BEDT-TTF)$_2$I$_3$, \nthe effect of $V$ on such state may be weaker than than of graphene.\nIt is an interesting difference between graphene and $\\alpha$-(BEDT-TTF)$_2$I$_3$, \nand it will be investigated intensively in future.\n\nLastly, we discuss the renormalization of fluctuation in the pseudo-spin ferromagnet, \nwhich are very complicated and not captured on the mean-field level. \nIn the absence of the Zeeman effect, \nthe ferromagnetic moment may fluctuate in the SU(4) space at temperatures \nbetween $T_{\\rm c}$ and a symmetry breaking temperature, $T_{\\rm sb}$, which is essentially given by \nthe symmetry-breaking interaction energy, $W_{ijkl}$.\nThe ferromagnetic moment may be forced in the easy-plane below $T_{\\rm sb}$.\nIn the presence of the Zeeman effect, the situation is more complicated \nowing to the competition between $W_{ijkl}$ and the Zeeman energy, $E_Z$.\nThe results in the present paper, thus, identify a new research target, {\\it i. e.} \na two-dimensional SU(4) model with the symmetry-breaking terms.\n\n\\section*{ Acknowledgments }\n\nThe authors are thankful to N. Tajima for fruitful discussions.\nThis work has been financially supported by Grant-in-Aid for \nSpecial Coordination Funds for Promoting Science and Technology (SCF), \nScientific Research on Innovative Areas 20110002, and \nScientific Research 19740205 \nfrom the Ministry of Education, Culture, Sports, Science and Technology in Japan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\noindent Excited-state transitions in atomic systems are finding an increasing range of applications including quantum information~\\cite{HarocheBook}, optical filters~\\cite{ESFADOF}, electric field sensing~\\cite{moha08,sedl12,holl14} and quantum optics~\\cite{prit13,dudi12}. They are also used for state lifetime measurements~\\cite{shen08}, frequency up-conversion \\cite{meij06}, the search for new stable frequency references~\\cite{bret93,abel09} and multi-photon laser cooling~\\cite{wu09}. However excited-state transitions are inherently more difficult to probe than ground-state transitions, especially if the lower state is short-lived. It is possible to probe an excited-state transition directly if the dipole moment is large enough~\\cite{tana12}, but more commonly excited-state character is observed by mapping onto ground-state transitions using electromagnetically induced transparency (EIT) in a ladder configuration. Using EIT it is possible to probe even relatively weak excited-state transitions, such as those to highly excited Rydberg states~\\cite{moha07,weat08,carr12a}. Nanosecond timescales have also been probed, effectively `freezing out' the motion of thermal atoms~\\cite{hube11}.\n\nEIT involving Rydberg states has paved the way to recent advances in non-linear and quantum optics \\cite{prit13} as the strong interactions among the Rydberg atoms lead to large optical non-linearities, even at the single-photon level \\cite{dudi12,peyr12,maxw13}. In room temperature Rydberg gases, the atomic interactions can lead to a non-equilibrium phase transition \\cite{carr13} and evidence for strong van der Waals interactions has been observed~\\cite{balu13}. Despite the considerable successes of ladder EIT, there is a particular class of energy level schemes for which ladder EIT cannot be observed in a Doppler-broadened medium. Specifically, when the upper transition wavelength is longer than the lower (`inverted-wavelength' system)~\\cite{Boon99,urvo13}, the transparency window is absent as it is smeared out by velocity averaging. \n\n\\begin{figure}\n\\includegraphics[width=3.4in]{Make_Fig1.pdf}\n\\caption[]{\\label{fig:toy_model}(Color Online)\nToy model of our experiment. Left: Atoms prepared in a superposition of closely spaced excited states $|e\\rangle$ and $|e'\\rangle$ demonstrate quantum beats at a frequency corresponding to the difference in their energies. Right: Driving an excited-state transition splits state $|e\\rangle$ into two dressed states $|e^+\\rangle$ and $|e^-\\rangle$ and the dynamics of the excited-state transition are written into the quantum beats. We show the level scheme (top), the Fourier spectrum, $|\\mathcal{F}(\\omega)|$, of the fluorescence (middle) and the time-dependent fluorescence into an appropriately chosen polarization mode (bottom). The Fourier spectra are calculated by taking the magnitude of the Fourier transform of the fluorescence signals.\n}\n\\end{figure}\n\nIn this paper we make novel use of hyperfine quantum beats~\\cite{haro73,hack91} to probe the excited-state transition dynamics of an `inverted-wavelength' ladder system in a thermal vapor. We find strong evidence for both Rabi oscillations and sub-Doppler Autler-Townes splitting. \n\nThe paper is organized as follows:\nIn Section~\\ref{sec:toy_model} we construct a toy model of our experiment, giving an overview of the physics involved. Section~\\ref{sec:experiment} details our experimental procedure and in Section~\\ref{sec:results} we present results in both the time-domain and the frequency domain. Section~\\ref{sec:theory} outlines a computer model that we developed to understand the signals, which we compare to the data in Section~\\ref{sec:analysis}. The model yields good qualitative agreement which allows us to interpret features that we observe in the frequency domain.\n\n\n\n\n\\section{Principle of Perturbed Quantum Beats in a Ladder System}\n\\label{sec:toy_model}\n\n\n\\noindent In this section we outline a toy model of our ladder system which includes the minimum possible complexity to illustrate the physical principle (Figure~\\ref{fig:toy_model}). The toy model considers a zero-velocity atom with ground state $|g\\rangle$, an intermediate excited state $|e\\rangle$ and an upper excited state $|u\\rangle$. There is also a reference state $|e'\\rangle$ which is close in energy to $|e\\rangle$. The transition from $|g\\rangle~\\to~|e\\rangle$ is driven by a short pulse whilst a continuous wave (CW) laser drives the excited-state transition from $|e\\rangle~\\to~|u\\rangle$. For a sufficiently short excitation pulse the bandwidth exceeds the energy interval between $|e\\rangle$ and $|e'\\rangle$, and a coherent superposition of the two states is prepared by the pulse. The dynamics of the excited-state transition are read out by measuring the fluorescence from states $|e\\rangle$ and $|e'\\rangle$.\n\nWe begin our explanation by considering the simple case when the excited-state transition driving field is switched off (left column of Figure~\\ref{fig:toy_model}). Once the coherent superposition of states $|e\\rangle$ and $|e'\\rangle$ has been prepared, the total fluorescence decays exponentially according to the state lifetime. However, the fluorescence into an appropriately chosen mode, characterized by polarization and propagation direction, is modulated by beating~\\cite{haro76}. These `quantum beats' represent interference between the two different quantum pathways associated with $|e\\rangle$ and $|e'\\rangle$. The interference is erased if information regarding which pathway was taken is recovered (e.g. spectroscopically resolving the fluorescence from each state). In our toy model the time-dependent fluorescence into a particular mode has the form of an exponentially decaying envelope modulated by beating. The modulus of the Fourier transform of this time-dependent fluorescence, $|\\mathcal{F}( \\omega)|$, allows us to read off the beat frequency (see middle row of Figure~\\ref{fig:toy_model}). States $|e\\rangle$ and $|e'\\rangle$ have energy $\\hbar \\omega_{e}$ and $\\hbar \\omega_{e'}$ respectively, and the beat frequency $\\omega_{{\\rm b}} = \\omega_{e} - \\omega_{e'}$ corresponds to the difference in energy. The visibility of the beats from a zero-velocity atom is set by a number of factors including the distribution of the population between the two states and the relative strengths with which the states couple to the selected fluorescence mode. In our experiment the visibility is limited by velocity averaging as well.\n\n\nTo understand the effects of driving the excited-state transition it is easiest to consider the dressed state picture (right hand column of Figure~\\ref{fig:toy_model}). CW driving of the excited-state transition splits $|e\\rangle$ into two dressed states, $|e^+\\rangle$ and $|e^-\\rangle$, separated according to the Rabi frequency of the driving field, $\\Omega$. The original beat at frequency $\\omega_{{\\rm b}}$, is split into two distinct beats with frequencies, $\\omega^+~=~\\omega_{{\\rm b}}~+~\\Omega\/ 2$ and $\\omega^-~=~\\omega_{{\\rm b}}~-~\\Omega \/ 2$. Furthermore, a new beat frequency is introduced with frequency $\\Omega$. This beat frequency relates to Rabi oscillations with atoms cycling on the excited-state transition. We note that unlike the initial quantum beat, this cycling leads to a modulation of the total fluorescence, not just a particular polarization mode. The Fourier spectrum, $|\\mathcal{F}( \\omega)|$, includes all information regarding Autler-Townes splitting of the state $|e\\rangle$ and Rabi oscillations on the excited-state transition, $|e\\rangle~\\to~|u\\rangle$. The more complicated form of the time-dependent fluorescence is shown in the lower right panel of Figure~\\ref{fig:toy_model}.\n\n\\begin{figure}\n\\includegraphics[width=3.4in]{Make_Fig2.pdf}\n\\caption[]{\\label{fig:Exp}(Color Online)\n(a) Level scheme of our experiment: A short pulse of light excites several states in the 6P$_{3\/2}$ manifold and a CW laser drives an excited-state transition 6P$_{3\/2}$\\,$F$\\,=\\,5~$\\to$~7S$_{1\/2}$\\,$F$\\,=\\,4 (b) Schematic of experiment: Vertically polarized beams counter-propagate through a cesium vapor cell and fluorescence from the D$_2$ transition is detected with a single photon counter.\n}\n\\end{figure}\n\n\n\n\n\\section{Experiment}\n\\label{sec:experiment}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{Make_Fig3.pdf}\n\\caption[]{\\label{fig:TwoCol}(Color Online)\nTop row (a-d): Measurements of fluorescence showing unperturbed hyperfine quantum beats. Bottom row (e-h): Measurements of fluorescence showing quantum beats that are modified by CW driving field with intensity $I_{\\rm d}~=~4~$W\\,cm$^{-2}$. We present measurements of time-dependent vertically polarized fluorescence~(a,e) and horizontally polarized fluorescence~(b,f). In panels (c,g) we present the Fourier spectra, calculated by taking the magnitude of the Fourier transform of the time-dependent fluorescence signals. The solid line (blue) shows the vertically polarized fluorescence and the dashed line (red) shows the horizontally polarized fluorescence. In panels (d,h) we show the spectrum of the difference between the two polarization signals. The spectra are normalized such that the peak in the difference signal relating to the unperturbed $F'\\,=\\,5\\,\\to\\,4$ beat (d) has a height of~1 (See the Appendix for full details). The dashed vertical lines correspond to the 6P$_{3\/2}$ hyperfine splitting~\\cite{das05} and the shaded bands correspond to regions presented as color plots in Figure~\\ref{fig:RawData}.\n}\n\\end{figure*}\n\n\\noindent The simplified level scheme and experimental setup are shown in Figure~\\ref{fig:Exp}a and Figure~\\ref{fig:Exp}b respectively. We use cesium atoms in a vapor cell (length 2~mm) at room temperature (19\\,$^{\\circ}$C). The ladder scheme comprises the 6S$_{1\/2}$\\,$F$\\,=\\,4 state as the ground state, 6P$_{3\/2}$\\,$F'$~=\\,5 as the intermediate state and 7S$_{1\/2}$\\,$F''$~=\\,4 as the upper excited state. The other 6P$_{3\/2}$ hyperfine states play the role of the reference state described in Section~\\ref{sec:toy_model}.\n\nWe excite the first transition using a short pulse (FWHM of 1~ns) of 852~nm light generated by a CW diode laser stabilized to the $F\\,=\\,4\\,\\to\\,F'\\,=\\,5$ hyperfine transition and modulated by a Pockels cell between two crossed, high extinction polarizers. The short pulse duration means that the pulse bandwidth spans the hyperfine energy splitting of the 6P$_{3\/2}$ manifold and therefore prepares a coherent superposition of several hyperfine states. It is this coherent superposition of states that leads to quantum beats in our system. The excited-state transition is driven by a counter-propagating, CW laser beam locked to the 6P$_{3\/2}\\ F'\\,=\\,5\\,\\to 7$S$_{1\/2}\\,F''\\,=\\,4$ (1469 nm) transition using excited-state polarization spectroscopy~\\cite{carr12}. \n\nTo best control the effects of driving the excited-state transition, it is desirable to minimize the spread of intensity of the excited-state transition driving field that the atoms experience. To achieve this, we only sample the center of the CW driving laser beam ($1\/{\\rm e}^2$ radius 0.3~mm) where the intensity is most uniform, by virtue of tighter focusing of the preparation pulse ($1\/{\\rm e}^2$ radius 0.06~mm). Both the laser beams are vertically polarized, and we detect fluorescence propagating in the horizontal plane. A narrow-band filter is used to select only fluorescence from the D$_2$ transition and a polarizer selects a particular mode of this fluorescence. \n\nThe fluorescence is measured using a single photon detector module which generates a TTL level pulse for each photon. The pulses are timed and counted by a high-bandwidth oscilloscope and in this way we achieve nanosecond timing resolution. To avoid saturating the counting module, we ensure that the expected delay between photons is much longer than the dead time of the counting module ($\\approx\\, $35~ns).\n\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\noindent We begin by considering the case of unperturbed hyperfine quantum beats. Figure~\\ref{fig:TwoCol}(a) and Figure~\\ref{fig:TwoCol}(b) show measurements of vertically and horizontally polarized fluorescence respectively, with the center of the excitation pulse incident at time $t=0$\\,ns. As we noted in Section~\\ref{sec:toy_model}, the total fluorescence is not modulated and so we see that the beating of the vertically and horizontally polarized fluorescence is out of phase. In Figure~\\ref{fig:TwoCol}(c) we present the magnitudes of the normalized Fourier transforms of these fluorescence measurements and in Figure~\\ref{fig:TwoCol}(d) we remove frequency components relating to the exponential decay envelope by subtracting the two signals (See the Appendix for details of normalization and subtraction). Because the beating of the two polarization signals is out of phase we retain the quantum beat frequency components and so we observe peaks at 201,~251~and~452~MHz, corresponding to the 6P$_{3\/2}$ hyperfine splitting~\\cite{das05} (highlighted with vertical dashed lines). The peak relating to the $F'=3\\to F'=4$ quantum beat (201~MHz) is very weak as the population in these two states is limited. This restricted population is a result of both weaker coupling to the ground state and also detuning from the middle of the excitation pulse bandwidth which is centered on the $F=4\\to F'=5$ transition.\n\nWhen we drive the excited-state transition the quantum beats are modified. For the driving field intensity at the center of the laser beam $I_{\\rm d}\\,=\\,4\\,$~W\\,cm$^{-2}$, we present the vertically and horizontally polarized fluorescence measurements in Figure~\\ref{fig:TwoCol}(e) and Figure~\\ref{fig:TwoCol}(f) respectively, along with their Fourier spectra in Figure~\\ref{fig:TwoCol}(g) and the spectrum of the difference signal in Figure~\\ref{fig:TwoCol}(h). We can see the changes to the Fourier spectra that we expected from considering the toy model in Section~\\ref{sec:toy_model}. Firstly the peak relating to the $F'=5\\to F'=4$ beat (251~MHz) is split in two (Figure~\\ref{fig:TwoCol}(g,h)). The origin of this effect is Autler-Townes splitting of the 6P$_{3\/2}~F'=5$ atomic state, caused by driving the excited-state transition. Secondly a new oscillation is present, leading to a peak at 100~MHz in this example (Figure~\\ref{fig:TwoCol}(g)). This represents atoms performing Rabi oscillations on the excited-state transition. The absence of this peak from the difference signal in Figure~\\ref{fig:TwoCol}(h) is because the Rabi oscillations modulate the entire 852~nm fluorescence. Therefore the oscillation is in phase between the vertically and horizontally polarized fluorescence and is removed in the difference signal.\n\nIt is interesting to note that whilst the simple model outlined in Section~\\ref{sec:toy_model} predicts that the splitting of the beat frequency would be equal to the frequency of the Rabi oscillation, it is clear from Figure~\\ref{fig:TwoCol}(g) that this is not the case. The cause of this discrepancy stems from Doppler effects which we explore and explain in Section~\\ref{sec:analysis} using a comprehensive computer simulation outlined in Section~\\ref{sec:theory}. \n\nIn Figure~\\ref{fig:RawData} we present colorplots covering a range of excited-state transition laser driving intensities $I_{\\rm d}\\,=\\,0\\to\\,7\\,$~W\\,cm$^{-2}$, constructed from nine individual sets of intensity measurements. Figure~\\ref{fig:RawData}(a) shows the modulus of the Fourier transform of the vertically polarized fluorescence measurements. The diagonal feature corresponds to the Rabi oscillation, which increases in frequency with increasing laser power. Figure~\\ref{fig:RawData}(b) shows the modulus of the Fourier transform of the difference signal and the splitting of the $F'=5\\to F'=4$ beat into two separate branches is clear.\n\n\\begin{figure}\n\\includegraphics[width=3.4in]{Make_Fig4.pdf}\n\\caption[]{\\label{fig:RawData}(Color Online) \nColorplots of the magnitude of the Fourier transforms of the fluorescence measurements. (a)~Vertically polarized fluorescence shows a diagonal feature that corresponds to Rabi oscillations. (b)~The difference signal demonstrates a branched feature relating to Autler-Townes splitting. Parts (a) and (b) relate to the highlighted regions of Figure~\\ref{fig:TwoCol}(c,g) and Figure~\\ref{fig:TwoCol}(d,h) respectively.\n}\n\\end{figure}\n\nWe also point out some further, more subtle effects. First, the fluorescence decays more slowly as the longer lived 7S$_{1\/2}~F''\\,=\\,4$ state is mixed into the 6P$_{3\/2}$ states. Second, the total amount of measured 852~nm fluorescence decreases. This is partly because the atoms can now decay from the 7S$_{1\/2}~F''\\,=\\,4$ state via the 6P$_{1\/2}$ manifold as well as the 6P$_{3\/2}$ manifold, but could also be due to hyperfine optical pumping caused by light leaking through the Pockels cell between pulses. Further consequences of this effect are discussed in Section~\\ref{sec:analysis}. Finally we note that the higher frequency branch of the split $F'=5\\to F'=4$ quantum beats is stronger than the low frequency branch (Figure~\\ref{fig:RawData}(b)). This effect is even more exaggerated in the $F'=5\\to F'=3$ (452 MHz) beat where we do not observe the low frequency branch at all (Figure~\\ref{fig:TwoCol}(d,g)). The absence of the lower branch originates from Doppler effects that we also discuss in Section~\\ref{sec:analysis}. \n\n\n\n\n\\section{Computer Model}\n\\label{sec:theory}\n\n\\noindent Here we develop a theoretical simulation to predict the behavior of our system. Conceptually, it involves two steps. First the optical Bloch equations for the system are solved numerically; second, the time-dependent expectation value of a `detection operator', $\\mathcal{B}$, is calculated, giving the expected fluorescence \\cite{haro76}. The operator has the form,\n\n\\begin{equation}\n\\label{eq:det}\n\\mathcal{B} = C \\sum_{f} \\textbf{e. \\^{D}} |f\\rangle \\langle f|\\textbf{e*. \\^{D}},\n\\end{equation}\n\n\\noindent where $C$ is a coefficient relating to detection efficiency, \\textbf{e} is a unit vector describing the polarization of the detected fluorescence and \\textbf{\\^{D}} is the electric dipole operator for the D$_2$ transition of the atom. The final states $f$ include all of the magnetic sub-levels ($m_{{\\rm F}}$) of the two 6S$_{1\/2}$ hyperfine ground states. We note that the expectation value of the operator $\\mathcal{B}$ is proportional to the square of the atomic dipole projected onto the detected polarization angle and measures the coupling of the atomic state to the field modes. The calculation process is repeated for a sample of velocity classes which are then summed and weighted according to a Boltzmann distribution.\n\nThe computation basis is fixed such that the linearly polarized excitation lasers drive only $\\pi$ transitions, allowing our calculation to be performed in a set of mutually uncoupled $m_{{\\rm F}}$ subspaces. Note that this basis might not be the energy eigenbasis due to uncompensated laboratory magnetic fields. However, any coherence developed between the $m_{{\\rm F}}$ subspaces as a consequence of this can be neglected since the duration of our experiment is much shorter than the relevant Larmor precession timescale. The time evolution of the density matrix $\\hat{\\rho}_{m_{{\\rm F}}}$ in each of the nine $m_{{\\rm F}}$ subspaces, is calculated using a set of optical Bloch equations,\n\n\\begin{equation}\n\\label{eq:Lou}\n\\dot{\\hat{\\rho}}_{m_{\\rm F}} = \\frac{i}{\\hbar}\\begin{bmatrix}\\hat{\\rho}_{m_{\\rm F}},\\hat{H}_{m_{\\rm F}}\\end{bmatrix} - \\hat{\\Gamma},\n\\end{equation}\n\n\\noindent where $\\hat{H}_{m_{\\rm F}}$ is the Hamiltonian for each subspace, and $\\hat{\\Gamma}$ is a decay operator. The Rabi frequencies are calculated individually for each $m_{\\rm F}$ subspace and each subspace includes one state from each of the hyperfine levels: 6S$_{1\/2}~F =~3,4$; 6P$_{3\/2}~F'~=~3,4,5$ and 7S$_{1\/2}~F''~=~4$. This convenient sub-division offers a computational speed up that permits the simulation to be run on a desktop computer.\n\nAlthough the subspaces are not coupled by the driving laser fields, we note that they are not truly separate, since atoms can undergo spontaneous $\\sigma^{\\pm}$ transitions resulting in a change of $m_{\\rm F}$ quantum number. Instead of modeling this full behavior, we attribute the total rate of spontaneous decay of each state to $\\pi$ transitions, thus conserving the total population in each subspace. In this way, we are able to capture the lifetimes of the states and retain the computational efficiency. Furthermore, because the timescale of our experiment is set by a single atomic state lifetime, we are confident that the effect of these angular momentum changing processes is negligible, as there is insufficient time to redistribute atomic population amongst the $m_{\\rm F}$ subspace.\n\nIn the final step of our model, we collate the populations and coherences from the nine subspaces into a single density matrix, giving the complete state of the atom as it changes in time. Using the `detection operator', we project the atomic dipole at each time step and hence infer both the linear and circularly polarized 852~nm fluorescence. \n\n\\begin{figure}\n\\includegraphics[width=3.4in]{Make_Fig5.pdf}\n\\caption[]{\\label{fig:Comp}(Color Online)\nVertically polarized fluorescence: We compare the model (black line) and experimental data points (blue) for measured excited-state transition laser intensity $I_{{\\rm d}}$\\,=\\,(0,\\,3,\\,6,\\,7)~${\\rm W\\,cm}^{-2}$ (top to bottom). We note that the visibility of the peaks is always smaller than the model predicts. The incident light pulse occurs at time $t$~=~0~ns and the error bars are calculated from Poissonian photon counting statistics~\\cite{hughes}.\n}\n\\end{figure}\n\n\n\n\n\\section{Analysis}\n\\label{sec:analysis}\n\n\\noindent In this section we make a direct comparison between the computer simulation and the measured data. In Figure~\\ref{fig:Comp} we present the results of the vertically polarized fluorescence for both the experiment and simulation. The unperturbed hyperfine quantum beat signal fits well, and we see at least qualitative agreement for the perturbed beats. Although the features are often more pronounced in the simulation than the data, there is a qualitative match between the data and theory. On the strength of this we can draw additional physical insight about the system.\n\nIn Section~\\ref{sec:results} we noted that the splitting of the $F'\\,=\\,5\\to\\,F'\\,=\\,4$ hyperfine quantum beat was unexpectedly smaller than the measured frequency of the Rabi oscillation. We suggest this is similar to narrowed EIT windows in thermal vapors \\cite{Natarajan08,baso09} where off-resonant velocity classes partially fill the transparency window left by resonant atoms. Calculated contributions to the splitting of the $F'~=~4~\\to~F'~=~5$ quantum beat from different velocity classes are shown in Figure~\\ref{fig:FillIn}. The zero-velocity class (bold, green) shows a splitting that is consistent with the simulated excited-state transition Rabi frequency, yet this is much larger than the splitting which appears in the total signal. Figure~\\ref{fig:FillIn} shows how contributions from off-resonant velocity classes fill in the gap. The inset compares this best fit calculated spectrum with our data and we see qualitative agreement, although we acknowledge a significant discrepancy in the simulated ($I_{\\rm d}^{{\\rm Sim}}\\,=\\,0.9$~W\\,cm$^{-2}$) and measured ($I_{\\rm d}\\,=\\,3$\\,W\\,cm$^{-2}$) excited-state transition laser intensities.\n\n\\begin{figure}\n\\includegraphics[width=3.4in]{Make_Fig6.pdf}\n\\caption[]{\\label{fig:FillIn}(Color Online)\nCalculated magnitude of the normalized Fourier transform of the vertically polarized fluorescence: We show a break down of velocity class contributions, with dashed (red) lines showing individual velocity classes traveling away from the 852~nm laser, and solid (blue) lines showing velocity classes traveling towards the 852~nm laser. The velocity classes are spaced at 33~m\\,s$^{-1}$ intervals and the bold (green) line shows the contribution from zero-velocity atoms. The gray shaded area shows the scaled sum of the signals and the calculation relates to probing a region with a uniform excited-state transition driving field intensity $I_{\\rm d}^{{\\rm Sim}}\\,=\\,0.9\\,$\\,W\\,cm$^{-2}$. The inset compares the data taken for measured CW driving field intensity $I_{\\rm d}\\,=\\,3\\,$W\\,cm$^{-2}$ shown in bold (blue) and the summed model (gray shaded area and black line).}\n\\end{figure}\n\nWe also noted in Section~\\ref{sec:results} that the high-frequency branch of the split $F'~=~4~\\to~F'~=~5$ quantum beat makes a stronger contribution to the Fourier spectrum than the low-frequency branch. This unexpected asymmetry can be explained by constructing a two-step argument: First, we note that the strongest beats arise from atoms experiencing a red Doppler shift of the 852~nm laser. This moves the center of the frequency profile of the pulse between the two beating transitions, promoting the excitation of both levels as required for quantum beats. Second, the atoms which experience a red-shift for the 852~nm laser see a blue-shift of the 1469~nm laser because the laser beams are counter-propagating. This blue-shift means that the dressed states represented in the higher frequency branch of the split quantum beat have a greater admixture of the 6P$_{3\/2}~F'$\\,=\\,5 state, and as such a stronger coupling to the ground state. Thus the imbalance between branches of the quantum beat comes from a bias towards a particular velocity class, and a bias within this velocity class to a particular branch. The asymmetry between branches is even stronger for the $F'~=~3~\\to~F'~=~5$ quantum beat as the $F'\\,=\\,3$ hyperfine state is further in energy from the $F'\\,=\\,5$ hyperfine state. Consequently, only the high frequency branch of the splitting was observed and the lower-frequency branch is absent. (Section~\\ref{sec:results}).\n\nThere are some remaining discrepancies between the simulation and the data. We found that when we used the CW driving field intensity as a fit parameter in the model, the best fit did not match or even scale linearly with the intensity we measured in the experiment. We believe that this might originate from optical pumping between excitation pulses. The high extinction polarizers each side of the Pockels cell (Figure~\\ref{fig:Exp}b) still allowed a few hundred nanowatts of 852~nm light to leak into the vapor cell for the 1~ms duration between pulses. For resonant velocity classes, this could have lead to an initial state other than the uniform distribution over the ground states that the computer simulation assumes.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\n\\noindent We have demonstrated a novel method using hyperfine quantum beat spectroscopy for observing sub-Doppler Autler-Townes type splitting in an `inverted wavelength' ladder scheme which would not be observable in a continuously excited room-temperature vapor. A comprehensive model of the fluorescence gives qualitative agreement with our data, and we use it to gain physical insight into the process. By exploiting our method to its full potential it would be possible to combine information from both the Autler-Townes splitting and the Rabi oscillations to achieve a complete read out of excited-state transition dynamics. Our work on ladder excitation schemes contributes to a general effort towards the exploitation of Rydberg atoms in a room temperature atomic vapor using multi-photon, step-wise excitation. In a wider context, our method offers a new means for investigating excited-state transitions in a room-temperature vapor.\n\t\n\n\n\n\n\\begin{acknowledgements}\nThe authors would like to thank Ifan Hughes for stimulating discussions and acknowledge financial support from EPSRC [grant EP\/K502832\/1] and Durham University. The data in this paper are provided in the Supplemental Material\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCopulas, which will be defined in Section \\ref{intro_sec}, describe the\ndependence of a multivariate distribution that is invariant under\nmonotone (increasing) transformations of each coordinate. In this\npaper, we investigate the dependence that arises in a one-dimensional\nMarkov process. Darsow \\textit{et al.} \\cite{dno} began the study of\ncopulas related to Markov processes; see also \\cite{nelsen}, Chapter\n6.3. More precisely, they showed what the Kolmogorov--Chapman equations\nfor transition kernels translate to in the language of copulas and\nintroduced some families of copulas $(C_{st})_{s\\leq t}$ that are\nconsistent in the sense that $C_{st}$ is the copula of $(X_s,X_t)$ for\na Markov process $(X_t)_{t\\geq0}$.\n\nIn Section \\ref{intro_sec}, we will introduce a Markov product of\ncopulas $C\\ast D$ such that if $C$ gives the dependence of $(X_0,X_1)$\nand $D$ the dependence of $(X_1,X_2)$, then $C\\ast D$ gives the\ndependence of $(X_0,X_2)$ for a Markov chain $X_0,X_1,X_2$. An analogy\nis that of a product of transition matrices of finite-state Markov\nchains, in particular, doubly stochastic matrices (whose column sums\nare all 1) since they have uniform stationary distribution.\n\nThis approach might, at first, seem like a sensible way of introducing\nthe machinery of copulas into the field of stochastic processes:\nMikosch \\cite{mikosch}, for example, has criticized the widespread use\nof copulas in many areas and, among other things, pointed out a lack of\nunderstanding of the temporal dependence, in terms of copulas, of most\nbasic stochastic processes.\n\nThis paper builds on that of Darsow \\textit{et al.}~\\cite{dno}, but\nwith a heavier emphasis on probabilistic, rather than analytic or\nalgebraic, understanding. Our main results are negative, in that we\nshow how:\n\\begin{enumerate}[(3)]\n\\item[(1)] a proposed characterization of the copulas of time-homogeneous\nMarkov processes fails (Section \\ref{families_sec});\n\\item[(2)] Fr{\\'{e}}chet copulas imply quite strange Markov processes\n(Section \\ref{frechet_sec});\n\\item[(3)] Archimedean copulas are incompatible with the dependence of\nMarkov chains (Section~\\ref{arch_sec});\n\\item[(4)] a conjectured characterization of idempotent copulas,\nrelated to\nexchangeable Markov chains, fails (Section \\ref{idem_sec}).\n\\end{enumerate}\n\n\\section{Copulas and the Markov product}\\label{intro_sec}\n\n\\begin{definition}\nA \\textit{copula} is a distribution function of a multivariate random\nvariable whose univariate marginal distributions are all uniform on $[0,1]$.\n\\end{definition}\n\nWe will mostly concern ourselves with two-dimensional copulas. In the\nfollowing, all random variables denoted by $U$ have a uniform\ndistribution on $[0,1]$ (or, sometimes, $(0,1)$).\n\n\\begin{definition}\n$\\Pi(x,y)=xy$ is the copula of independence: $(U_1,U_2)$ has the\ndistribution $\\Pi$ if and only if $U_1$ and $U_2$ are independent.\n\\end{definition}\n\n\\begin{definition}\n$M(x,y)=\\min(x,y)$ is the copula of complete positive dependence:\n$(U_1,U_2)$ has the distribution $M$ if and only if $U_1=U_2$ almost\nsurely (a.s.).\n\\end{definition}\n\n\\begin{definition}\n$W(x,y)=\\max(x+y-1,0)$ is the copula of complete negative dependence:\n$(U_1,U_2)$ has the distribution $W$ if and only if $U_1=1-U_2$ a.s.\n\\end{definition}\n\nNote that a mixture $\\sum_ip_i C_i$ of copulas $C_1,C_2,\\ldots$ is also\na copula if $p_1,p_2,\\ldots$ is a probability distribution since one can\ninterpret the mixture as a randomization: first, choose a copula\naccording to the distribution $p_1,p_2,\\dots$ and then draw from the\nchosen distribution.\n\nIt is well known that if $X$ is a (one-dimensional) continuous random\nvariable with distribution function $F$, then $F(X)$ is uniform on\n$[0,1]$. Thus, if $(X_1,\\dots,X_n)$ is an $n$-dimensional continuous\nrandom variable with joint distribution function $F$ and marginal\ndistributions $(F_1,\\dots,F_n)$, then the random variable\n$(F_1(X_1),\\dots,F_n(X_n))$ has uniform marginal distributions, that\nis,\\ its joint distribution function is a copula, say $C$.\n\n\\textit{Sklar's theorem} (see \\cite{nelsen}, Theorem 2.10.9) states that\nany $n$-dimensional distribution function~$F$ with marginals\n$(F_1,\\dots\n,F_n)$ can be ``factored'' into $F(x_1,\\dots,x_n)=C(F_1(x_1),\\dots\n, F_n(x_n))$ for a copula $C$, which is, furthermore, unique if the\ndistribution $F$ is continuous. We say that the $n$-dimensional\ndistribution $F$, or the random variable $(X_1,\\dots,X_n)$, has the\ncopula $C$.\n\n\\begin{remark}\nWhen $(X_1,\\dots,X_n)$ does not have a unique copula, all copulas of\nthis random variable agree at points $(u_1,\\dots,u_n)$ where $u_i$ is\nin the range $R_i$ of the function $x_i\\mapsto F_i(x_i)$. One can\nobtain a unique copula by an interpolation between these points which\nis linear in each coordinate, and we will, as Darsow \\textit{et al.}\\\n\\cite{dno}, speak of this as the copula of such random variables.\n\\end{remark}\n\nCopulas allow for a study of the dependence in a multivariate\ndistribution separately from the marginal distributions. It gives\nreasonable information about dependence in the sense that the copula is\nunchanged if $(X_1,\\dots,X_n)$ is transformed into $(g_1(X_1),\\dots\n,g_n(X_n))$, where $g_1,\\dots,g_n$ are strictly increasing.\n\n\\begin{example}\nThe notion of copulas makes it possible to take a copula from, say, a\nmultivariate $t$-distribution and marginal distributions from, say, a\nnormal distribution and combine them into a multivariate distribution\nwhere the marginals are normal, but the joint distribution is not\nmultivariate normal. This is sometimes desirable in order to have\nmodels with, in a sense, ``stronger'' dependence than what is possible\nfor a multivariate normal distribution.\n\\end{example}\n\n\\begin{example}\n$(X_1,X_2)$ has the copula $\\Pi$ if and only if $X_1$ and $X_2$ are\n\\textit{independent}. $(X_1,X_2)$ has the copula $M$ if and only if\n$X_2=g(X_1)$ for a strictly \\textit{increasing} function $g$. $(X_1,X_2)$\nhas the copula $W$ if and only if $X_2=h(X_1)$ for a strictly \\emph\n{decreasing} function $h$. (When $X_1$ and $X_2$ furthermore have the\nsame marginal distributions, they are usually called \\textit{antithetic}\nrandom variables.)\n\\end{example}\n\nIn this paper, we are in particular interested in the dependence that\narises in a Markov process in $\\R$, for example,\\ the copula of\n$(X_0,X_1)$ for a stationary Markov chain $X_0,X_1,\\ldots.$ By \\cite\n{kallenberg}, Proposition 8.6, the sequence $X_0,X_1,\\ldots$ constitutes\na Markov chain in a Borel space $S$ if and only if there exist\nmeasurable functions $f_1,f_2,\\ldots\\dvtx S\\times[0,1]\\to S$ and i.i.d.\nrandom variables $V_1,V_2,\\dots$ uniform on $[0,1]$ and all independent\nof $X_0$ such that $X_n=f_n(X_{n-1},V_n)$ a.s. for $n=1,2,\\ldots.$ One\nmay let $f_1=f_2=\\cdots= f$ if and only if the process is time-homogeneous.\n\nWe can, without loss of generality, let $S=[0,1]$ since we can\ntransform the coordinates $X_0,X_1,\\dots$ monotonically without\nchanging their copula. The copula is clearly related to the function\n$f$ above. We have $f_{\\Pi}(x,u)=u$, $f_M(x,u)=x$ and $f_W(x,u)=1-x$\nwith obvious notation.\n\nDarsow \\textit{et al.}\\ \\cite{dno} introduced an operation on copulas\ndenoted $\\ast$ which we will call the \\textit{Markov product}.\n\n\\begin{definition}\nLet $X_0,X_1,X_2$ be a Markov chain and let $C$ be the copula of\n$(X_0,X_1)$, $D$ the copula of $(X_1,X_2)$ and $E$ the copula of\n$(X_0,X_2)$ (note that $X_0,X_2$ is also a Markov chain). We then write\n$C\\ast D=E$.\n\\end{definition}\n\nIt is also possible to define this operation as an integral of a\nproduct of partial derivatives of the copulas $C$ and $D$;\nsee \\cite{dno}, formula (2.10), or \\cite{nelsen}, formula (6.3.2),\nbut, in this paper, the\nprobabilistic definition will suffice.\n\nFrom the definition, it should be clear that the operation $\\ast$ is\nassociative, but not necessarily commutative and for all $C$:\n\\begin{eqnarray*}\n\\Pi\\ast C &=& C\\ast\\Pi= \\Pi,\\\\\nM\\ast C &=& C\\ast M = C\n\\end{eqnarray*}\nso that $\\Pi$ acts as a null element and $M$ as an identity. We write\n$C^{\\ast n}$ for the $n$-fold Markov product of $C$ with itself and\ndefine $C^{\\ast0}=M$. We have $W^{\\ast2}=M$, so $W^{\\ast n}=M$ if $n$\nis even and $W^{\\ast n}=W$ is $n$ is odd. In Section \\ref{idem_sec}, we\nwill investigate \\textit{idempotent} copulas $C$, meaning $C^{\\ast2}=C$.\n\n\\begin{example}\nIf $X_0,X_1,\\dots$ is a time-homogeneous Markov chain where $(X_0,X_1)$\nhas copula $C$, then $C^{\\ast n}$ is the copula of $(X_0,X_n)$ for all\n$n=0,1,\\dots.$\n\\end{example}\n\n\\begin{definition}\nFor any copula $C(x,y)$ of the random variable $(X,Y)$, we define its\n\\textit{transpose} $C^T(x,y)=C(y,x)$, the copula of $(Y,X)$.\n\\end{definition}\n\nWe can say that $W$ is its own \\textit{inverse} since $W\\ast W=M$.\n\\begin{definition}\nIn general, we say that a copula $R$ is \\textit{left-invertible} or a\n\\textit{right-inverse} if there exists a copula $L$ such that $L\\ast R=M$\nand we say that $L$ is \\textit{right-invertible} or a \\textit{left-inverse}.\n\\end{definition}\n\nThe equation $L\\ast R=M$ implies that any randomness in the transition\ndescribed by $L$ is eliminated by $R$ and thus $f_R(x,u)$ must be a\nfunction of $x$ alone. A rigorous proof of the last proposition may be\nfound in \\cite{dno}, Theorem 11.1. Furthermore, if $L$ is a\nright-invertible copula of $(X,Y)$, then its right-inverse $R$ can be\ntaken as the transpose of $L$, $R=L^{\\mathrm{T}}$, since $M$ is the copula of\n$(X,X)$ and thus $R$ should be the copula of $(Y,X)$ so that $L,R$\ncorrespond to the Markov chain $X,Y,X$. A proof of this can also found\nin \\cite{dno}, Theorem 7.1.\n\n\\begin{example}\nLet $L_{\\theta}$ be the copula of the random variable $(X,Y)$ whose\ndistribution is as follows: $(X,Y)$ is uniform on the line segment\n$y=\\theta x$, $0\\leq x\\leq1$, with probability $0\\leq\\theta\\leq1$\nand $(X,Y)$ is uniform on the line segment $y=1-(1-\\theta)x$, $0\\leq\nx\\leq1$, with probability $1-\\theta$. The function\n\\[\nf_{L_{\\theta}}(x,u)=\\theta x\\mathbf{1}(u\\leq\\theta)+\\bigl\n(1-(1-\\theta\n)x\\bigr)\\mathbf{1}(u>\\theta)\n\\]\ncan be used to describe the transition from $X$ to $Y$. Let $R_{\\theta\n}=L_{\\theta}^{\\mathrm{T}}$. One can take\n\\[\nf_{R_{\\theta}}(y,v)=\\frac{y}{\\theta}\\mathbf{1}(y\\leq\\theta)+\\frac\n{1-y}{1-\\theta}\\mathbf{1}(y>\\theta)\n\\]\nto describe the transition from $Y$ to $X$. Note that $f_{R_{\\theta\n}}(y,v)$ is a function of $y$ only. We also get $f_{R_{\\theta\n}}(f_{L_{\\theta}}(x,u),v)=f_M(x,w)=x$ so that, indeed, $L_{\\theta\n}\\ast\nR_{\\theta} = M$.\n\\end{example}\n\nThe Markov product is linear:\n\\begin{equation}\\label{linear}\n\\sum_ip_iC_i \\ast\\sum_jq_j D_j = \\sum_{ij}p_iq_j C_i\\ast D_j\n\\end{equation}\nsince the left-hand side can be interpreted as first choosing a $C_i$\nwith probability $p_i$ as transition mechanism from $X_0$ to $X_1$ and\nthen independently choosing a $D_j$ with probability $q_j$ as\ntransition mechanism from $X_1$ to $X_2$, whereas the right-hand side\ncan be interpreted as choosing a combined transition mechanism $C_i\\ast\nD_j$ from $X_0$ to $X_2$ with probability $p_iq_j$.\n\nFor a given Markov process $(X_t)_{t\\geq0}$ in continuous time, we\nwill denote the copula of $(X_s,X_t)$ by $C_{st}$ for $s\\leq t$. For\ntime-homogeneous processes, we only write $C_t$ for the copula of\n$(X_s,X_{s+t})$ for $t\\geq0$. Note that, for all $t$,\n\\[\nC_{tt} = C_{00} = C_0 = M.\n\\]\nCopulas for continuous-time Markov processes must obey a\nKolmogorov--Chapman-type relationship:\n\\begin{equation}\\label{kol-chap}\nC_{rt}=C_{rs}\\ast C_{st},\\qquad r\\leq s\\leq t.\n\\end{equation}\n\n\\section{Some families of copulas}\\label{families_sec}\n\nLet $(X_t)_{t\\geq0}$ be a Markov process with transition kernel\n$P_{st}(x,\\cdot)$ and marginal distributions $(F_t)_{t\\geq0}$. Now,\n\\begin{equation}\\label{krangligt}\nC_{st}(F_s(x),F_t(y))=\\Prob(X_s\\leq x,X_t\\leq y)=\\int_{-\\infty\n}^xP_{st}(u,(-\\infty,y])\\,\\mathrm{d}F_s(u)\n\\end{equation}\nand, from this, $C_{st}$ may be derived in principle.\n\nThe expression \\eqref{krangligt} becomes more manageable if the\nmarginal distributions are uniform and if the transition kernel\nfurthermore has a density $f_{st}(x,y)$, then we get that the density\n$c_{st}(x,y)=\\frac{\\partial^2}{\\partial x\\,\\partial y}C_{st}(x,y)$ of the\nMarkov copula equals the transition density: $c_{st}=f_{st}$.\n\n\n\\begin{example}\\label{ex_tva}\nLet $(U_t)_{t\\geq0}$ be a Brownian motion reflected at 0 and 1, with\n$\\sigma=1$ and with $U_0$ uniform on $(0,1)$. This process is\nstationary and time-homogeneous, with\n\\[\nc_t(x,y)=\\frac{1}{\\sqrt{2\\curpi t}}\\sum_{n\\in\\Z}\\bigl(\\mathrm{e}^{-\n(2n+y-x)^2\/(2t)}+\\mathrm{e}^{-(2n-y-x)^2\/(2t)}\\bigr).\n\\]\nIt is clear that $C_t\\to M$ as $t\\to0$ and $C_t\\to\\Pi$ as $t\\to\n\\infty$.\n\\end{example}\n\nIt is usually hard to compute transition densities for interesting\nprocesses, so another way of obtaining families of Markov copulas is to\nconstruct them directly from copulas so that \\eqref{kol-chap} holds. A\nproblem with this approach is that a probabilistic understanding of the\nprocess may be lost.\n\n\n\n\\begin{example}\nDarsow \\textit{et al.} \\cite{dno} pose the question of\nwhether all time-homo\\-geneous Markov copulas may be expressed as\n\\begin{equation}\\label{inte_all_hom}\nC_t = \\mathrm{e}^{-at}\\Biggl(E+\\sum_{n=1}^{\\infty}\\frac{a^nt^n}{n!}C^{\\ast\nn}\\Biggr),\n\\end{equation}\nwhere $a$ is a positive constant and $E$ and $C$ are two copulas\nsatisfying $C\\ast E=E\\ast C=C$ and $E$ is idempotent ($E\\ast E=E$). We\nimmediately observe that $C_0=E$ according to equation \\eqref\n{inte_all_hom} and thus $E$ cannot be taken to be arbitrary, but must\nequal $M$. However, since $M$ commutes with all copulas, $C$ may be\narbitrary. As $M=C^{\\ast0}$, we can rewrite\n\\begin{equation}\\label{inte_all_hom_po}\nC_t = \\sum_{n=0}^{\\infty}\\frac{(at)^n}{n!}\\mathrm{e}^{-at}C^{\\ast n} = \\Ex\n\\bigl[C^{\\ast N(t)}\\bigr],\n\\end{equation}\nwhere $N$ is a Poisson process with intensity $a$. We can thus give the\nfollowing probabilistic interpretation: a Markov process has the Markov\ncopula of equation \\eqref{inte_all_hom_po} if it jumps according to the\nPoisson process $N$ with intensity $a$ and, at each jump, it jumps\naccording to the copula $C$. Between jumps, it remains constant. This\nclearly does not cover all possible time-homogeneous Markov processes\nor Markov copulas; see\\ the previous Example \\ref{ex_tva}.\n\\end{example}\n\n\\section{Fr{\\'{e}}chet copulas}\\label{frechet_sec}\n\nIn this section, we only consider Markov processes in continuous time.\n\nA copula $C$ is said to be in the Fr{\\'{e}}chet family if $C=\\alpha W +\n(1-\\alpha-\\beta)\\Pi+ \\beta M$ for some non-negative constants\n$\\alpha$\nand $\\beta$ satisfying $\\alpha+\\beta\\leq1$; see \\cite{nelsen}, page 12.\nDarsow \\textit{et al.} \\cite{dno} found conditions on the functions\n$\\alpha(s,t)$ and $\\beta(s,t)$ in\n\\[\nC_{st}=\\alpha(s,t) W + \\bigl(1-\\alpha(s,t)-\\beta(s,t)\\bigr)\\Pi+ \\beta(s,t) M\n\\]\nsuch that $C_{st}$ satisfies equation \\eqref{kol-chap}. By equation\n\\eqref{linear}, we find\n\\begin{eqnarray}\n\\label{alpha_eq}\n\\beta(r,s)\\alpha(s,t)+\\alpha(r,s)\\beta(s,t)&=&\\alpha(r,t),\\\\\n\\label{beta_eq}\n\\alpha(r,s)\\alpha(s,t)+\\beta(r,s)\\beta(s,t)&=&\\beta(r,t).\n\\end{eqnarray}\nDarsow \\textit{et al.} \\cite{dno} solved these equations by putting\n$r=0$ and defining $f(t)=\\alpha(0,t)$ and $g(t)=\\beta(0,t),$ which yields\n\\begin{eqnarray*}\n\\alpha(s,t)&=&\\frac{f(t)g(s)-f(s)g(t)}{g(s)^2-f(s)^2},\\\\\n\\beta(s,t) &=&\\frac{g(t)g(s)-f(s)f(t)}{g(s)^2-f(s)^2}.\n\\end{eqnarray*}\nThis solution in terms of the functions $f$ and $g$ does not have an\nimmediate probabilistic interpretation and it is therefore hard to give\nnecessary conditions on the functions $f$ and $g$ for \\eqref{alpha_eq}\nand \\eqref{beta_eq} to hold.\n\nWe will first investigate the time-homogeneous case, where $\\alpha\n(s,t)=a(t-s)$ and $\\beta(s,t)=b(t-s)$ for some functions $a$ and $b$.\nThe equations \\eqref{alpha_eq} and \\eqref{beta_eq} are then\n\\begin{eqnarray}\n\\label{alpha_hom}\nb(s)a(t)+a(s)b(t)&=&a(s+t),\\\\\n\\label{beta_hom}\na(s)a(t)+b(s)b(t)&=&b(s+t).\n\\end{eqnarray}\nLetting $\\rho(t)=a(t)+b(t)$, we find, by summing the two equations\n\\eqref{alpha_hom} and \\eqref{beta_hom}, that\n\\begin{equation}\\label{rho_hom}\n\\rho(s)\\rho(t)=\\rho(s+t).\n\\end{equation}\nSince $\\rho$ is bounded and $\\rho(0)=1$ (since $C_0=M$), we necessarily\nhave $\\rho(t)=\\mathrm{e}^{-\\lambda t}$, where $\\lambda\\geq0$ or $\\rho\n(t)=\\mathbf\n{1}(t=0)$. Note that $\\rho(t)$ equals the probability that a Poisson\nprocess $N_{\\Pi}$ with intensity~$\\lambda$ has no points in the\ninterval $(0,t]$.\n\nFor the moment, we disregard the possibility $\\rho(t)=\\mathbf{1}(t=0)$.\nSince $\\rho$ is positive, we can define $\\sigma(t)=a(t)\/\\rho(t)$. By\ndividing both sides of \\eqref{beta_hom} by $\\rho(s+t)$ and using\n\\eqref{rho_hom}, we get\n\\begin{equation}\\label{sigma_hom}\n\\sigma(s)\\sigma(t)+\\bigl(1-\\sigma(s)\\bigr)\\bigl(1-\\sigma(t)\\bigr)=\\sigma(s+t).\n\\end{equation}\nIf we now let $\\tau(t)=1-2\\sigma(t)$, equation \\eqref{sigma_hom} yields\n\\begin{equation}\n\\tau(s)\\tau(t)=\\tau(s+t)\n\\end{equation}\nand, by the same reasoning as for $\\rho$, we get $\\tau(t)=\\mathrm{e}^{-2\\mu t}$\nfor some $\\mu\\geq0$ or $\\tau(t)=\\mathbf{1}(t=0)$. We disregard the\nlatter possibility for the moment. Thus, $\\sigma(t)=\\frac12-\\frac\n12\\mathrm{e}^{-2\\mu t}=\\mathrm{e}^{-\\mu t}\\sinh\\mu t$ for some constant $\\mu\\geq0$.\nNote that\n\\begin{equation}\\label{odd}\n\\sigma(t)=\\mathrm{e}^{-\\mu t}\\sinh\\mu t = \\sum_{k=0}\\mathrm{e}^{-\\mu t}\\frac{(\\mu\nt)^{2k+1}}{(2k+1)!},\n\\end{equation}\nthat is,\\ $\\sigma(t)$ equals the probability that a Poisson process\n$N_{W}$ with intensity $\\mu$ has an odd number of points in $(0,t]$.\n\nThus, we have\n\\begin{eqnarray}\\label{hom}\nC_t &=& \\sigma(t)\\rho(t)W+\\bigl(1-\\rho(t)\\bigr)\\Pi+\\bigl(1-\\sigma(t)\\bigr)\\rho(t)M\n\\nonumber\\\\\n&=& \\mathrm{e}^{-(\\lambda+\\mu)t}\\sinh(\\mu t) W+(1-\\mathrm{e}^{-\\lambda t})\\Pi+\\mathrm{e}^{-(\\lambda\n+\\mu)t}\\cosh(\\mu t) M\\nonumber\\\\[-8pt]\\\\[-8pt]\n&=&\\Prob\\bigl(N_W(t)\\mbox{ is odd}, N_{\\Pi}(t)=0\\bigr)W+\\Prob\\bigl(N_{\\Pi}(t)\\geq1\\bigr)\\Pi\\nonumber\\\\\n&&{}+\\Prob\\bigl(N_W(t)\\mbox{ is even}, N_{\\Pi}(t)=0\\bigr)M,\\nonumber\n\\end{eqnarray}\nwhere the aforementioned Poisson processes $N_{\\Pi}$ and $N_W$ are independent.\n\n\\subsection*{Probabilistic interpretation}\n\nThe time-homogeneous Markov\nprocess with $C_t$ as copula is therefore rather special. We may,\nwithout loss of generality, assume that all marginal distributions are\nuniform on $[0,1]$. It ``restarts'' -- becoming independent of its\nhistory -- according to a Poisson process $N_{\\Pi}$. Independently of\nthis process, it ``switches'' by transforming a present value $U_{t-}$\nto $U_t=1-U_{t-}$, and this happens according to a Poisson process\n$N_W$. Note that the intensity of either process may be zero.\n\nIf $\\tau(t)=\\mathbf{1}(t=0)$, then $\\sigma(t)=\\frac12\\mathbf{1}(t>0)$\nso that $C_t = \\rho(t)(\\frac12W+\\frac12M)+(1-\\rho(t))\\Pi$ for \\mbox{$t>0$}.\nThe process can be described as follows. Between points $t_i0$ so\nthat the process is, at each moment, independent of the value at any\nother moment, that is,\\ $(U_t)_{t\\geq0}$ is a collection of\nindependent random variables.\n\nWith the probabilistic interpretation, it is easy to rewrite equation\n\\eqref{hom} in the form \\eqref{inte_all_hom_po} when $\\rho,\\sigma>0$.\nThe process makes a jump of either ``restart'' or ``switch'' type with\nintensity $\\lambda+\\mu$ and each jump is of ``restart'' type with\nprobability $\\lambda\/(\\lambda+\\mu)$ and of ``switch'' type with\nprobability $\\mu\/(\\lambda+\\mu)$. Thus,\n\\[\nC_t = \\sum_{n=0}^{\\infty}\\frac{((\\lambda+\\mu)t)^n}{n!}\\mathrm{e}^{-(\\lambda\n+\\mu\n)t}\\biggl(\\frac{\\lambda}{\\lambda+\\mu}\\Pi+\\frac{\\mu}{\\lambda+\\mu\n}W\\biggr)^{\\ast n} = \\Ex\\bigl[C^{\\ast N(t)}\\bigr],\n\\]\nwhere $C=\\frac{\\lambda}{\\lambda+\\mu}\\Pi+\\frac{\\mu}{\\lambda+\\mu\n}W$ and\n$N$ is a Poisson process with intensity $\\lambda+\\mu$.\n\nIt is clear that the time-homogeneous process can be generalized to a\ntime-inhomogeneous Markov process by taking $N_{\\Pi}$ and $N_W$ to be\nindependent inhomogeneous Poisson processes. With\n\\begin{eqnarray*}\n\\rho(s,t)&=&\\Prob\\bigl(N_{\\Pi}(t)-N_{\\Pi}(s)=0\\bigr),\\\\\n\\sigma(s,t)&=&\\Prob\\bigl(N_W(t)-N_W(s)\\mbox{ is odd}\\bigr),\n\\end{eqnarray*}\nwe get a more general version of the Fr{\\'{e}}chet copula:\n\\[\nC_{st}=\\sigma(s,t)\\rho(s,t)W+\\bigl(1-\\rho(s,t)\\bigr)\\Pi+\\bigl(1-\\sigma(s,t)\\bigr)\\rho(s,t)M,\n\\]\nwith essentially the same probabilistic interpretation as the\ntime-homogeneous case.\n\nIn the time-inhomogeneous case, it is also possible to let either or\nboth of the two processes consist of only one point, say $\\tau_{\\Pi}$\nand\/or $\\tau_{W}$ that may have arbitrary distributions on $(0,\\infty\n)$. In addition to this, both in the Poisson case and the single point\ncase, it is possible to add deterministic points to the processes\n$N_{\\Pi}$ and $N_{W}$ and still retain the (time-inhomogeneous) Markov\nproperty.\n\n\n\\section{Archimedean copulas}\\label{arch_sec}\n\nIf a copula of an $n$-dimensional random variable $(X_1,\\dots,X_n)$ is\nof the form\n\\begin{equation}\\label{arch}\n\\phi^{-1}\\bigl(\\phi(u_1)+\\cdots+\\phi(u_n)\\bigr),\n\\end{equation}\nit is called \\textit{Archimedean} with generator $\\phi$.\nNecessary and sufficient conditions on the generator to produce a\ncopula are given in \\cite{mn}. We will only use the following necessary\nproperties, which we express with the inverse $\\psi=\\phi^{-1}$. The\nfunction $\\psi$ is non-increasing, continuous, defined on $[0,\\infty)$\nwith $\\psi(0)=1$ and $\\lim_{x\\to\\infty}\\psi(x)=0$, and decreasing when\n$\\psi>0$; see \\cite{mn}, Definition~2.2.\n\n\\begin{example}\\label{arch_ex}\nLet $(U_1,\\dots,U_n)$ be distributed according to \\eqref{arch} and let\n$X_i = -\\phi(U_i)$ for $i=1,\\dots,n$. Since $-\\phi$ is an increasing\nfunction, $(X_1,\\dots,X_n)$ has the same copula as $(U_1,\\dots,U_n)$.\nAll components of $(X_1,\\dots,X_n)$ are non-positive and\n\\begin{eqnarray*}\n\\Prob(X_i\\leq-x_i, i=1,\\dots,n)&=&\\Prob\\bigl(-\\phi(U_i)\\leq\n-x_i,i=1,\\dots\n,n\\bigr)\\\\\n&=&\\Prob\\bigl(U_i\\leq\\psi(x_i), i=1,\\dots,n\\bigr) \\\\\n&=& \\psi(x_1+\\cdots+x_n)\n\\end{eqnarray*}\nfor $x_1,\\dots,x_n\\geq0$.\n\\end{example}\n\nLet us now consider Markov processes with Archimedean copulas.\n\\begin{proposition*}\nIf $X_1,\\dots,X_n$ is a Markov chain, where $(X_1,\\dots,X_n)$ has an\nArchimedean copula with generator $\\phi$ and $n\\geq3$, then all\n$X_1,\\dots,X_n$ are independent.\n\\end{proposition*}\n\n\\begin{pf}\nWithout loss of generality, we may, and will, assume that the marginal\ndistribution of each single $X_i$ is that of Example \\ref{arch_ex}\nabove, that is,\\ $P(X_i\\leq x)=\\psi(-x)$ for $x\\leq0$, since we can\nalways transform each coordinate monotonically so that it has the\nproposed distribution after transformation. Thus, with $x_1,x_2,x_3\\geq0$,\n\\begin{eqnarray*}\n\\Prob(X_3\\leq-x_3|X_2\\leq-x_2,X_1\\leq-x_1)&=&\\frac{\\Prob(X_3\\leq\n-x_3,X_2\\leq-x_2,X_1\\leq-x_1)}{\\Prob(X_2\\leq-x_2,X_1\\leq-x_1)}\\\\\n&=&\\frac{\\psi(x_1+x_2+x_3)}{\\psi(x_1+x_2)},\\\\\n\\Prob(X_3\\leq-x_3|X_2\\leq-x_2)&=&\\frac{\\Prob(X_3\\leq-x_3,X_2\\leq\n-x_2)}{\\Prob(X_2\\leq-x_2)}\\\\\n&=&\\frac{\\psi(x_2+x_3)}{\\psi(x_2)},\\\\\n\\end{eqnarray*}\nso, by the Markov property,\n\\[\n\\frac{\\psi(x_2+x_3)}{\\psi(x_2)}=\\frac{\\psi(x_1+x_2+x_3)}{\\psi(x_1+x_2)}.\n\\]\nLet $f(x)=\\psi(x_2+x)\/\\psi(x_2)$ so that the above equation is\nequivalent to\n\\[\nf(x_3) = \\frac{f(x_1+x_3)}{f(x_1)}\n\\]\nand thus $f(x_1+x_3)=f(x_1)f(x_3)$, which implies that $f(x) = \\mathrm{e}^{-cx}$\nfor some constant $c$. Putting $x=-x_2$ yields\n\\[\n\\mathrm{e}^{cx_2}=f(-x_2)=\\frac{\\psi(0)}{\\psi(x_2)} = \\frac{1}{\\psi(x_2)}\n\\]\nand thus $\\psi(t) = \\mathrm{e}^{-ct}$. Hence, $\\phi(s)=-\\frac{1}{c}\\log s$ so\nthat the copula\n\\[\n\\psi\\bigl(\\phi(u_1)+\\cdots+\\phi(u_n)\\bigr) = u_1\\cdots u_n,\n\\]\nthat is,\\ all $X_1,\\dots,X_n$ are independent.\n\\end{pf}\n\n\\section{Idempotent copulas}\\label{idem_sec}\n\nA copula is said to be \\textit{idempotent} if $C\\ast C=C$. In this\nsection, we will investigate Markov chains with idempotent copulas\nwhose probabilistic structure will turn out to be quite peculiar.\n\n\\begin{example}\\label{ord_sum}\nLet $I_i=[a_i,b_i]$, $i=1,2,\\dots,$ be a set of disjoint intervals in\n$[0,1]$. Let $I_0=[0,1]\\setminus\\bigcup_{i\\geq1}I_i$ and let\n$p_i=\\lambda\n(I_i)$ be the Lebesgue measure of each set $I_i$, $i=0,1,\\ldots.$\nConsider the random variable $(U,V)$ that has the following\ndistribution: $(U,V)$ is uniform on $I_i\\times I_i$ with probability\n$p_i$ for $i=1,2,\\ldots$ and $U=V$ with $U$ uniform on $I_0$ with\nprobability $p_0$. Let $D$ be the copula of $(U,V)$. ($D$ is a\nso-called ``ordinal sum'' of copies of $\\Pi$ and $M$; see\n\\cite{nelsen}, Chapter~3.2.2.) We have\n\\[\nf_D(x,u) = x\\mathbf{1}(x\\in I_0)+\\sum_{i\\geq\n1}\\bigl((b_i-a_i)u+a_i\\bigr)\\mathbf{1}(x\\in I_i).\n\\]\nIt is easy to check that $f_D(f_D(x,u),v)=f_D(x,v)$ so that $D\\ast\nD=D$, that is,\\ $D$ is idempotent. If $U_0,U_1,\\dots$ is a Markov chain\ngoverned by the copula $D$, then all $U_0,U_1,\\dots$ lie in the same\nset $I_{\\iota}$, where the random index $\\iota$ differs from\nrealization to realization.\n\\end{example}\n\nIf $C$ is idempotent and $L$ and $R=L^{\\mathrm{T}}$ are two copulas satisfying\n$L\\ast R=M$, then $R\\ast C\\ast L$ is also idempotent. Darsow \\textit{et\nal.}\\ \\cite{dno} conjectured that all idempotent copulas could be\nfactored in this form with $C$ as in Example \\ref{ord_sum}. We will\nshow that the class of idempotent copulas, even though they correspond\nto quite a restricted kind of dependence, is richer than what can be\ncovered by that characterization. If a Markov chain $X_0,\\dots,$ which,\nwithout loss of generality, we assume is in $[0,1]$ is governed by the\ncopula $R\\ast D\\ast L$, then all $f_R(X_0,u_0),f_R(X_1,u_1),\\dots$ are\nin the same set $I_{\\iota}$ for some random $\\iota$ and there are only\ncountably many such possible sets. Note that $f_R(x,u)$ is a function\nof $x$ only.\n\nWe start with some background on spreadable and exchangeable sequences,\nwith notation from \\cite{kallenberg}, which will be useful.\n\n\\begin{definition}\nAn infinite sequence $\\xi_1,\\xi_2,\\dots$ is said to be \\emph\n{exchangeable} if\n\\[\n(\\xi_1,\\xi_2,\\dots)\\stackrel{d}{=}(\\xi_{k_1},\\xi_{k_2},\\dots)\n\\]\nfor all permutations $(1,2,\\dots)\\mapsto(k_1,k_2,\\dots)$ which affect\na finite set of numbers. The sequence is said to be \\textit{spreadable}\nif the equality in distribution is required only for strictly\nincreasing sequences $k_1