diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzyrq" "b/data_all_eng_slimpj/shuffled/split2/finalzyrq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzyrq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIt is not an uncommon perception among the practitioners of machine learning and of theoretical many-body physics that some ideas of physics, most notably those of equilibrium and nonequilibrium statistical physics, might have significance in the fundamental understanding of the machine learning dynamics. Such sentiment and progress along the direction has continued for some time and still in active pursuit, mostly by the researchers in the machine learning community~\\cite{welling11,fox15,mandt16,chaudhari18,baldassi19,ganguli13,ganguli19,ganguli20,yaida18}. The belief in the statistical-physics foundation of the machine learning will be strengthened obviously by more examples of ideas originating from statistical physics and then manifesting themselves in the machine learning. Here we establish one such connection, relating a fundamental theorem in near-equilibrium statistical physics~\\cite{ao04,KAT05,ao06,kwon11a,kwon11} to the theory of learning dynamics~\\cite{welling11,fox15,mandt16,yaida18,ganguli13,ganguli19}, in particular where the learning process is {\\it linear} and described by a stochastic equation similar to what governs the Ornstein-Uhlenbeck processes~\\cite{risken96}. The theorem in question is the fluctuation-dissipation theorem (FDT). \n\nThe FDT in a strict sense refers to specific relations that hold between correlation functions and response functions of physical systems under equilibrium~\\cite{risken96}. Here we use the term in a more relaxed sense, referring to mathematical identities among the observable quantities under the stationary state condition. The difference between the equilibrium and the stationary state is revealed by the existence of an anti-symmetric matrix $\\v Q$~\\cite{ao04,KAT05,ao06,kwon11a,kwon11}, which will be defined shortly. The FDT is illustrated most simply in the Langevin dynamics of a single particle subject simultaneously to dissipative and stochastic forces \n\\begin{eqnarray} \\dot{x} = -\\gamma x + f(t) \\end{eqnarray}\nwhere, in the context of Newtonian motion, $x$ represents the velocity of a particle in one dimension, $-\\gamma x$ is the resistive force, and $f(t)$ is the random force coming from the environment. On integrating the first-order differential eqution we obtain the formally exact solution $x(t) = e^{-\\gamma t} [ x(0) + \\int_0^t e^{\\gamma t'} f(t') ]$ which, in the long-time limit ($t\\rightarrow \\infty$) yields the average \n\\begin{eqnarray} \n\\langle x^2 \\rangle = 2D e^{-2\\gamma t} \\int_0^t dt' e^{2\\gamma t'} = D\/\\gamma \\end{eqnarray}\nassuming the white-noise correlation\n$\\langle f(t) f(t') \\rangle = 2D \\delta (t - t' )$. The competing tendencies of the dissipation ($\\gamma$) and fluctuation ($D$) finds balance through the identity. \n\nMulti-dimensional generalization of the Langevin dynamics finds expression in \n\\begin{eqnarray} \\dot{\\v x} = -{\\bf \\Gamma} {\\v x} + {\\v f} ( t) \\label{eq:1.1} \\end{eqnarray}\nwith $n$-dimensional variables $\\v x = (x_1 , \\cdots x_n )$, the $n\\times n$ dissipation matrix $\\bf \\Gamma$, and the $n$-dimensional stochastic force vector $\\v f$ obeying the zero mean $\\langle \\v f \\rangle =0$ and the variance $\\langle \\v f ( t) \\v f^T (t' ) \\rangle = 2 \\v D \\delta (t-t')$, in terms of the $n\\times n$ diffusion matrix $\\v D$. From the exact solution $\\v x(t) = e^{-\\v \\Gamma t} [ \\v x (0) + \\int_{0}^{t} e^{\\v \\Gamma t'} \\v f (t') dt' ]$ we derive the long-time correlation average \n\\begin{eqnarray}\n\\bm \\Sigma(t) & = & \\langle \\v x(t) \\v x^T(t) \\rangle \\nonumber \\\\\n& = & 2 \\int_{0}^{t} dt' e^{\\v \\Gamma (t'-t)} \\v D e^{\\v \\Gamma^T(t'-t)}\n\\end{eqnarray}\nand the following identity for $\\bm \\Sigma = \\bm \\Sigma (t \\rightarrow \\infty)$: \n\\begin{eqnarray}\n\\v \\Gamma \\bm \\Sigma + \\bm \\Sigma \\v \\Gamma^T = 2\\v D . \\label{eq:2.13} \n\\end{eqnarray}\nThis identity relates the diffusion matrix $\\v D$ with the dissipation matrix $\\bf \\Gamma$ through the correlation matrix $\\bm \\Sigma$ in the stationary-state, for the Ornstein-Uhlenbeck processes with constant $\\bf \\Gamma$ and $\\v D$~\\cite{ao04,KAT05}. Extensions and applications of the theorem both in physical systems and machine learning have since appeared~\\cite{kwon11,fox15,mandt16}. Thanks to the identity, one can write the matrix $\\bf \\Gamma \\bm \\Sigma$ as the sum of the symmetric ($\\v D$) and anti-symmetric ($\\v Q$) matrix:\n\\begin{eqnarray} \\bm \\Gamma \\bm \\Sigma = \\v D + \\v Q . \\label{eq:decomposition} \\end{eqnarray}\nIt was pointed out in Ref. \\onlinecite{kwon11a} that $\\v Q =0$ implies the detailed balance, otherwise one should allow the possibility $\\v Q \\neq 0$ in the decomposition, Eq. (\\ref{eq:decomposition}). \n\n\nIn Sec. \\ref{sec:FDT-for-W}, we derive an analogous mathematical identity for the stochastic linear learning dynamics. This is then verified, in Sec. \\ref{sec:experiments}, through numerical experiments on several well-known machine learning datasets. Implications of our work are discussed in Sec. \\ref{sec:discussion}. \n\n\n\\section{FDT in Learning Dynamics} \n\\label{sec:FDT-for-W}\n\nIn the learning dynamics one is confronted with a collection of input vectors $\\v x_\\alpha$ (e.g. pixels in a jpg file re-formatted as a one-dimensional vector) and output vectors $\\v y_\\alpha$ (e.g. classification of the picture as an image of a cat or a dog), where $1 \\le \\alpha \\le N$ runs over the entire dataset called the {\\it batch}. In the linear learning dynamics one is interested in finding the matrix $\\v W$ that minimizes the error\n\\begin{eqnarray} E & = & \\frac{1}{2N} \\sum_{\\alpha=1}^N ({\\bf y}_\\alpha - \\v W \\v x_\\alpha )^T ({\\bf y}_\\alpha - \\v W \\v x_\\alpha ) \\nonumber \\\\\n& \\equiv & \\frac{1}{2} {\\rm Tr} [ {\\bf \\Sigma}_{xx} {\\bf W}^T {\\bf W} - {\\bf W}^T {\\bf \\Sigma}_{yx} - {\\bf \\Sigma}^T_{yx} {\\bf W} ] . \\label{eq:error-function}\n\\end{eqnarray}\nThe two correlation functions appearing in the second line are \n\\begin{eqnarray} {\\bf \\Sigma}_{xx} = \\frac{1}{N} \\sum_{\\alpha=1}^N {\\bf x}_\\alpha {\\bf x}_\\alpha^T , ~~ {\\bf \\Sigma}_{yx} = \\frac{1}{N} \\sum_{\\alpha=1}^N {\\bf y}_\\alpha {\\bf x}_\\alpha^T. \\label{eq:Sigma-xx-definition}\\end{eqnarray}\nThe gradient descent (GD) method of finding the optimal $\\v W$ results in the first-order differential equation for $\\v W$~\\cite{ganguli13,ganguli19}:\n\\begin{eqnarray} \\frac{d \\bf W }{dt} = - \\frac{\\delta E}{\\delta \\v W} = - {\\bf W} {\\bf \\Sigma}_{xx} + { \\bf \\Sigma}_{yx} . \\label{eq:1.10} \\end{eqnarray}\nThe full solution is given by ${\\bf W}(t) = {\\bf W}(0) e^{-{\\bf \\Sigma}_{xx} t} + \\v W_0 (1 - e^{-{\\bf \\Sigma}_{xx} t} )$ where $\\v W_0 = {\\bf \\Sigma}_{yx} {\\bf \\Sigma}_{xx}^{-1}$ offers the equilibrium solution. \n\nAn interesting connection to the Langevin dynamics and FDT arises when we treat $\\bm \\Sigma_{xx}$ and $\\bm \\Sigma_{yx}$ in the dynamics of Eq. (\\ref{eq:1.10}) as a mini-batch (not a full-batch) average. At each stage of $\\v W$-evolution one picks a different, randomly chosen mini-batch to compute the average $\\bm \\Sigma_{xx} (t) = N_m^{-1} \\sum_{\\alpha \\in B(t)} {\\bf x}_\\alpha {\\bf x}_\\alpha^T$ and ${\\bf \\Sigma}_{yx} (t) = N_m^{-1} \\sum_{\\alpha \\in B(t) } {\\bf y}_\\alpha {\\bf x}_\\alpha^T$, where $N_m$ is the mini-batch size and $B(t)$ is the particular mini-batch chosen at the time $t$. The $\\v W$-dynamics according to the stochastic gradient descent (SGD) scheme becomes \n\\begin{eqnarray} \\frac{d \\v W}{dt} = - \\v W \\bm \\Sigma_{xx}(t) + \\bm \\Sigma_{yx} (t) . \\label{eq:stochastic-W-dynamics} \\end{eqnarray}\nPhrased in the language of Langevin dynamics, both the dissipative ($\\bm \\Sigma_{xx} (t)$) and the stochastic ($\\bm \\Sigma_{yx} (t)$) forces are time-dependent. We can re-write the variables in the equation explicitly as the sum of the stationary (time-independent) and the fluctuating (time-dependent) parts,\n\\begin{eqnarray} \\v W (t) & \\rightarrow & \\v W_0 + \\v W (t), \\nonumber \\\\\n\\bm \\Sigma_{xx} (t) & \\rightarrow & \\bm \\Sigma_{xx} + \\bm \\Sigma_{xx} (t), \\nonumber \\\\\n\\bm \\Sigma_{yx} (t) & \\rightarrow &\\bm \\Sigma_{yx} + \\bm \\Sigma_{yx} (t), \\label{eq:re-definition} \\end{eqnarray}\nand work with the equation\n\\begin{eqnarray}\n\\frac{d\\v W }{dt}\\! =\\! -\\v W ( \\bm \\Sigma_{xx} \\!+\\! \\bm \\Sigma_{xx} (t) ) \\!+\\! \\bm \\Sigma_{yx} (t) \\!-\\! \\v W_0 \\bm \\Sigma_{xx} (t) . \\nonumber \\\\ \\label{eq:modified-stochastic-DE} \\end{eqnarray}\nAlthough the exact solution to this equation can be found in the form of Wiener integral (see Appendix \\ref{appendix:A}), we will here assume a simplified situation where $\\bm \\Sigma_{xx} (t) =0$ on the right-hand side of the equation. Relaxing the assumption will not change the overall conclusion as long as $\\bm \\Sigma_{xx} (t)$ is small - see Appendix \\ref{appendix:A}. The stochastic learning dynamics is now reduced to an Ornstein-Uhlenbeck process~\\cite{risken96} and allows a simple solution\n\\begin{eqnarray} \\v W(t) = \\left[ \\v W (0) + \\int_0^t \\bm \\Sigma_{yx} (t') e^{\\bm \\Sigma_{xx} t' } \\right] e^{-\\bm \\Sigma_{xx} t } . \\label{eq:approximate-W} \\end{eqnarray}\n\n\nWe can write down the long-time correlation matrix\n\\begin{eqnarray}\n\\bm \\Sigma_{WW} (t) & = & \\langle \\v W^T(t) \\v W(t) \\rangle \\nonumber \\\\\n& = & \\int_{0}^{t} dt' \\int_{0}^{t} dt'' e^{\\bm \\Sigma_{xx} (t'-t)} \\langle \\bm \\Sigma_{yx}^T(t')\\bm \\Sigma_{yx} (t'') \\rangle e^{\\bm \\Sigma_{xx} (t''-t)} \\nonumber \\\\\n& = & \\int_{0}^{t} dt' e^{\\bm \\Sigma_{xx} (t'-t)} 2\\v D e^{\\bm \\Sigma_{xx} (t'-t)}\n\\end{eqnarray}\nassuming $ \\langle \\bm \\Sigma_{yx}^T(t')\\bm \\Sigma_{yx} (t'') \\rangle = 2\\v D\\delta(t'-t'')$. From this follows the identity \n\\begin{eqnarray}\n\\bm \\Sigma_{xx} \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx} = 2\\v D \\label{eq:FDT-for-W}\n\\end{eqnarray}\nfor $\\bm \\Sigma_{WW} \\equiv \\bm \\Sigma_{WW} (t \\rightarrow \\infty )$. This is the FDT type identity in the stochastic linear learning dynamics and our central result (a more refined form of FDT exists - see Appendix \\ref{appendix:B}). In the expression (\\ref{eq:FDT-for-W}), $\\bm \\Sigma_{xx}$ is the full-batch correlation matrix given in Eq. (\\ref{eq:Sigma-xx-definition}). Restoring the original definition, we have \n\\begin{eqnarray} \n\\langle ( \\bm \\Sigma_{yx} (t) \\!-\\! \\bm \\Sigma_{yx} )^T ( \\bm \\Sigma_{yx} (t') \\!-\\! \\bm \\Sigma_{yx} ) \\rangle & = & 2\\v D \\delta(t -t') \\nonumber \\\\\n\\langle [ \\v W (t) - \\v W_0 ]^T [\\v W (t) - \\v W_0 ] \\rangle & = & \\bm \\Sigma_{WW} . \n\\label{eq:definitions} \n\\end{eqnarray}\n\nThe full-batch input-input correlation matrix $\\bm \\Sigma_{xx}$ provides a sort of dissipative force while (fluctuating part of) the input-output correlation function plays the stochastic force in the learning dynamics, according to Eq. (\\ref{eq:modified-stochastic-DE}). The correlator of the learning matrix, i.e. $\\bm \\Sigma_{WW}$, is obtained as the balance between the two tendencies. \n\n\n\\section{Numerical Experiments}\n\\label{sec:experiments} \n\nFor sufficiently small time $t=h$ we can solve the stochastic equation (\\ref{eq:stochastic-W-dynamics}) approximately\n\\begin{eqnarray}\n\\v W (h) &\\approx & \\left[ \\v W (0) + \\int_{0}^{h} \\bm \\Sigma_{yx} (t')e^{\\bm \\Sigma_{xx} (0) t'} dt' \\right]e^{-\\bm \\Sigma_{xx} (0) h} \\nonumber \\\\\n& \\approx & \\v W (0) [1 - \\bm \\Sigma_{xx} (0) h ] + \\int_{0}^{h} \\bm \\Sigma_{yx} (t') dt' . \\label{eq:W-h} \n\\end{eqnarray}\nWe can further divide up the interval $t \\in [0, h]$ into $M$ equal segments, each of width $\\varepsilon \\equiv h\/M$, and use the discrete formula\n$\\Sigma_{xx} (0) \\rightarrow M^{-1} \\sum_{i = 1}^M \\Sigma_{xx} (i \\cdot \\Delta)$ and \n$ \\int_{0}^{h} \\bm \\Sigma_{yx} (t') dt' \\rightarrow \\varepsilon \\sum_{i=1}^M \\Sigma_{yx} ( i \\cdot \\Delta )$. In the end, Eq. (\\ref{eq:W-h}) turns into a recursive formula\n\\begin{eqnarray} \\v W^{(n+1)} = \\v W^{(n)} [ 1- \\varepsilon \\bm \\Sigma_{xx}^{(n)} ] + \\varepsilon \\bm \\Sigma_{yx}^{(n)}\n\\label{eq:W-n} \\end{eqnarray}\nwhere $\\bm \\Sigma_{xx}^{(n)}$ and $\\bm \\Sigma_{yx}^{(n)}$ are averages over the mini-batch of size $M N_m$. At sufficiently large $n$, $\\v W^{(n)}$ executes a steady-state fluctuation around the minimum $\\v W_0$. \n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{fig1.png}\n\\caption{Fluctuation analysis for (a) MNIST (b) CIFAR-10 and (c) EMNIST datasets. (top) Plots of $\\v D$ obtained from each dataset. (middle) Plots of $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx}$. (bottom) Normalized Fourier components for $\\v D$ (red) and $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx}$ (blue) plotted along $\\v k = (k_x , 0)$ with $k_0 = 2\\pi\/a$.} \n\\label{fig:MNIST-CIFAR}\n\\end{figure*}\n\nTo test out the validity of the FDT in stochastic linear learning derived in Eq. (\\ref{eq:FDT-for-W}), we employ three representative datasets: MNIST, CIFAR-10 and EMNIST Letters (abbreviated as EMNIST from here on)~\\cite{cohen17}. MNIST and CIFAR-10 consist of ten different objectives or output vectors $\\v y^\\alpha$, represented by one-hot vectors $(1,0, \\cdots, 0)$ through $(0, \\cdots, 0, 1)$. Twenty-six alphabets are represented by as many output vectors in the case of EMNIST. The pixel sizes are $28\\times28$ for both MNIST and EMNIST, and $32\\times32$ for CIFAR-10. Updating $\\v W (t)$ according to the SGD algorithm outlined in Eqs. \n(\\ref{eq:W-h}) and (\\ref{eq:W-n}), we found good convergence to the error-minimizing value $\\v W_0 = \\bm \\Sigma_{xy} \\bm \\Sigma_{xx}^{-1}$ by measuring the inner product of the $\\v W(t)$ and $\\v W_0$ divided by their norms approaching unity: $\\cos \\theta(t) = \\v W (t) \\cdot \\v W_0 \/ \\| \\v W (t) \\| \\| \\v W_0 \\|$. The inner product of two matrices is defined by taking a product of the matrix elements sharing the same $(ij)$ index and making a sum over all $(ij)$'s.\n\nOnce the steady state is reached, e.g. $\\cos \\theta \\gtrsim 0.999$, we begin analyzing the small fluctuations by calculating the two correlators in Eq. (\\ref{eq:definitions}) by taking averages $\\langle \\cdots \\rangle$ over several tens of thousands of $\\v W^{(n)}$'s and $\\bm \\Sigma_{yx}^{(n)}$'s. To deduce the diffusion matrix $\\v D$ in Eq. (\\ref{eq:definitions}) we take the equal-time correlator $t=t'$ and compute the average of $\\bm \\Sigma_{yx}^{(n)}$. This gives the $\\v D$ matrix up to an overall constant. In the end, a good proportionality between $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx}$ and $\\v D$ is found as shown in Fig. \\ref{fig:MNIST-CIFAR} for all the datasets tested. It turns out the correlators exhibit a highly periodic structure with period $a$ coming from the $a\\times a$ pixel size of each dataset. (The original $a=28$ dimension of the MNIST and EMNIST was chopped at the boundary to $a=24$. Otherwise it was difficult to get the full-batch inverse $\\bm \\Sigma_{xx}^{-1}$.) \n\nDue to the highly periodic structure of the real-space images of $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx}$ and $\\v D$, only a handful of Fourier peaks at $\\v k= (k_x , k_y)$ given by multiples of $2\\pi\/a$ were significant. Figure \\ref{fig:MNIST-CIFAR} shows the Fourier components along $\\v k = (k_x , 0)$ normalized by the value at $\\v k = (0,0)$. \nThe near-perfect match in the Fourier analysis of both $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} + \\bm \\Sigma_{WW} \\bm \\Sigma_{xx}$ and $\\v D$ is not {\\it a priori} obvious, and must be attributed to the FDT theorem at work in the stochastic linear learning dynamics. \n\n\\section{Discussion}\n\\label{sec:discussion}\n\nOur work addresses a FDT type relation in the stochastic linear learning dynamics. The relation derived in Eq. (\\ref{eq:FDT-for-W}) is found to hold quite well for a number of machine learning datasets. The analogy to the Langevin dynamics naturally gives rise to an interpretation of the input covariance matrix $\\bm \\Sigma_{xx}$ as the effective friction, and the input-output variance $\\bm \\Sigma_{yx}$ as the effective stochastic force in the learning dynamics. \n\nWe have made several attempts to go beyond the simple stochastic linear learning scheme. For one, we tried placing a CNN layer before the neural network layer $\\v W$. As shown in Appendix \\ref{appendix:C}, this formulation naturally leads to FDT in terms of the CNN-filtered input data sets $\\v X^\\alpha = \\v C \\otimes \\v x^\\alpha$, where $\\otimes$ represents the CNN operation. The FDT holds with respect to the renormalized datasets $\\v X^\\alpha$. In another attempt, we tried introducing non-linearity explicitly by using an alternative error function $E = (2N)^{-1} \\sum_{\\alpha =1}^N \\sum_{i=1}^n (y^\\alpha_i - z^\\alpha_i )^2$ with the sigmoid function $z^\\alpha_i = [ e^{-\\sum_{j=1}^n W_{ij} x_j^\\alpha} + 1 ]^{-1}$ parameterized by the learning matrix $\\v W$. Such formulation leads to the dynamics $d \\v W \/dt$ that is, unfortunately, highly non-linear and defies further analytical treatment. \n\nThe FDT type relation in the stochastic learning was noticed some years earlier by Yaida~\\cite{yaida18}. His derivation of the so-called FDT relation avoids any use of an explicit error function and relies solely on the stationary property of observables after the learning process has saturated. It is a powerful formulation in the sense that the relations apply to an arbitrary learning architecture with non-linearities. On the other hand, by avoiding the stochastic differential equation formulation, the connection that his relations have with the FDT in statistical physics becomes somewhat vague. More seriously, when our error function is used to work out his formulas, the outcome does not match our FDT formula derived in Eq. (\\ref{eq:FDT-for-W}). This leads us to suspect that there may be multiple FDT type theorems governing the stationary states of learning, with both our formula and his addressing different facets. \n\nWe have investigated whether, writing $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW}$ in Eq. (\\ref{eq:FDT-for-W}) as the sum $\\bm \\Sigma_{xx} \\bm \\Sigma_{WW} = \\v D + \\v Q$, there will be a significant contribution of the anti-symmetric matrix $\\v Q$. A crude measure of the significance of $\\v Q$ relative to $\\v D$ is the maximum value of the matrix elements in $\\v Q$ divided by that of $\\v D$. The results are 0.12, 0.096, 0.045 for MNIST, CIFAR-10, and EMNIST, respectively, suggesting that the anti-symmetric components are probably very small and insignificant. \n\n\\acknowledgments \nThe Python code used in the numerical experiment can be found at https:\/\/github.com\/lemonseed117\/FDT-Stochastic.git J. H. H. acknowledges fruitful discussion with and input on the manuscript from Ping Ao, J. H. Jo, S. B. Lim, J. D. Noh, Vinit Singh, and Hayong Yun.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:Intro}\n\nIn the study of the geometry of Hitchin's fibration a recurring problem has been to determine how much of this geometry is determined by the smooth part of the fibration. Ng\\^ o's support theorem provides a tool to formulate and sometimes to prove a precise version of this question for general fibrations equipped with an action of a family of polarized abelian group schemes (see \\cite{ngoaf}). In particular, for variants of the fibration parameterizing Higgs bundles with poles, Chaudouard and Laumon in \\cite{ch-la} proved that the only perverse cohomology sheaves appearing in the decomposition of the direct image of the constant sheaf are the intermediate extensions of the local systems on the smooth locus, that is the perverse cohomology sheaves are supported over the whole base. In particular, all of the cohomology is determined, in principle, by the monodromy of the cohomology of the smooth fibers. As is explained in the last section of \\cite{ch-la},\nunfortunately, this method does not apply to the original symplectic (no poles) version of the Hitchin fibration.\nMotivated by the $P=W$ conjecture \\cite{dCHM}, one would like to understand the perverse filtration of the fibration better and for this it is important to determine whether this result extends to this case as well. Surprisingly, we do find new supports for every rank $n\\geq 2$, and new cohomological contributions for any rank $n>2$.\n\n\nBefore we explain the general strategy of our approach, let us concisely state our main result. In order to do this, let us briefly introduce the standard notation that we use, which is recalled in more detail in \\cref{sec:NotationAndBackground}. We fix a smooth projective curve $C$ and denote by $h_{n}\\colon \\mathcal{M}_{n}^d \\to \\ensuremath{\\mathcal{A}_n}$ Hitchin's fibration for $\\textrm{GL}_n$ and $d$ an integer coprime to $n$, i.e., $\\mathcal{M}_{n}^d$ is the moduli space of semistable Higgs bundles of rank $n$ and degree $d$ on $C$. The base $\\mathcal{A}_n$ is an affine space parameterizing spectral curves $C_a\\in T^*C$ that are of degree $n$ over $C$.\nFor any partition $\\un{n}=(n_i)_{i=1,\\dots,k}$ of $n$ there is a closed subvariety $S_{\\un{n}}\\subset \\mathcal{A}_n$, which is the closure of the subset $S_{\\un{n}}^\\times\\subset S_{\\ve{n}}$ of reducible nodal curves having smooth irreducible components of degree $n_i$ over $C$ (see \\cref{rem:bertini}). Also we denote by $\\ensuremath{\\mathcal{A}_n}^{\\red}\\subset \\ensuremath{\\mathcal{A}_n}$ the open subset parameterizing reduced spectral curves. Using these notions our main results can be summarized as follows (note that according to our convention in \\S\\ref{sec:NotationAndBackground}, the local systems given by the $r$-th cohomology of \nthe smooth fibers of $h_{n}$ contribute to $^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R}{h_{n}}_{*}\\mathbbm{Q})$):\n\n\\begin{theorem*}[\\Cref{prop:SuppOnlySn} and \\Cref{thm:main1}]\nLet $h_{n}:\\mathcal{M}_{n}^d \\to \\ensuremath{\\mathcal{A}_n}$ be the Hitchin map.\n\tIf $S\\subset \\ensuremath{\\mathcal{A}_n}$ is a support of $^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R}{h_{n}}_{*}\\mathbbm{Q})$ for any $r$ with $S\\cap \\ensuremath{\\mathcal{A}_n}^{\\red}\\neq \\varnothing$ then $S=S_{\\ve{n}}$ for some partition $\\un{n}$.\n\nMoreover, for every partition $\\ve{n}$ of $n$, the stratum $S_{\\ve{n}}$ is a support for all of the sheaves\n\t\\begin{equation*}\\label{eq:range}\n\t^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R} {h_{n}}_{*}\\mathbbm{Q}) \n\t\\hbox{ with } \\delta^{\\aff}(\\ve{n}) \\leq r \\leq {2\\dim \\ensuremath{\\mathcal{A}_n} -\\delta^{\\aff}(\\ve{n})} \n\t\\end{equation*}\n\twhere $\\delta^{\\aff}(\\ve{n})=\\sum_{i l &\\hbox{ for } l> \\codim Y,\n\\end{align*}\ni.e., this is the usual t-structure, but shifted by $\\dim X$. This will be useful for us, as we will study restrictions of perverse sheaves to closed subvarieties and we can then avoid to shift the constant sheaf. \n\nA semisimple perverse sheaf on a complex variety $X$ is a complex of the form $P=\\bigoplus_\\alpha \\IC(Y_\\alpha, L_\\alpha)$,\nwhere $Y_\\alpha \\subseteq X$ are irreducible closed subvarieties and $L_\\alpha$ are semisimple local systems defined \non dense open subsets of the $Y_\\alpha$'s. The generic points of the $Y_\\alpha$'s\nare called the {\\em supports } of $P$. \n\nIf \n$h\\colon M \\to X$ is a proper map between smooth varieties, the decomposition theorem of \\cite{bbd} says that \n$${\\mathbbm R} h_*\\mathbbm{Q} \\simeq \\bigoplus_{k\\geq 0}\t \\,\\,\n^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q})[-k]$$\nwhere all $^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q})$ are semisimple perverse sheaves. The union of supports of the perverse sheaves $^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q})$\nis the set of {\\em supports} of the map $h$ (see \\cite[\\S 7]{NgoLEmme}).\n\nIf $Y$ is a support of ${\\mathbbm R} h_*\\mathbbm{Q} \\simeq \\bigoplus_k \\,\\,\n^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q})[-k]$\nwe denote by \n\\begin{align*}\nn^+_Y({\\mathbbm R} h_*\\mathbbm{Q})&:= \\max\\{ k | Y \\text{ is a support of } ^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q}) \\}\\\\\nn^-_Y({\\mathbbm R} h_*\\mathbbm{Q})&:= \\min\\{ k | Y \\text{ is a support of } ^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q}) \\}\n\\end{align*}\n\nWe say that a semisimple complex $K=\\bigoplus_k \\,\\,\n^p\\!\\!{\\mathscr H}^k({\\mathbbm R} h_*\\mathbbm{Q})[-k]$ has {\\em no proper supports} if $X$ is the only support of $K$.\n\\subsection{The Hitchin fibration}\\label{subsec:HitchinFibration}\nWe fix a nonsingular, connected, projective curve $C$ of genus $g\\geq 2$, an integer $n \\in \\zed_{\\geq 1}$, and \nan integer $d \\in \\zed$ such that ${\\rm gcd}(n,d)=1$. We denote by $K_C$ the canonical bundle of $C$. \n\nWe denote by $\\Higgs_n^d$ the moduli stack of Higgs bundle of rank $n$ and degree $d$ on $C$, i.e., it parametrizes pairs $(E,\\phi) $ where $E$ is a vector bundle of rank $n$ and degree $d$ on $C$ and $\\phi\\in H^0(C,\\End(E)\\otimes K_C)$. \n\nWe denote by $\\mathcal{M}_{n}^{d}$ the coarse moduli space of stable Higgs bundles of rank $n$ and degree $d$, where as usual, stability\nis defined by imposing the inequality $\\deg (F)\/{\\rm rank}(F) < \\deg (E)\/{\\rm rank} (E)$ for every $\\phi$-invariant\nproper sub-bundle $F \\subseteq E$. \n\nBecause of our assumption that $n$ and $d$ are coprime $\\mathcal{M}_{n}^{d}$ is an irreducible, nonsingular, quasi-projective variety of dimension \n\\begin{equation}\\label{eq:dimMn}\n\\dim (\\mathcal{M}_{n}^{d})= n^2(2g-2)+2=: 2d_{n}.\n\\end{equation}\nThe cotangent space $T^*\\mathcal{N}_{n}^{d}$ of the moduli space $\\mathcal{N}_{n}^{d}$ of stable rank $n$ and degree $d$ vector bundles on $C$ is a dense open subvariety of $\\mathcal{M}_{n}^{d}$.\nThe Hitchin base is defined to be the vector space\n\\begin{equation}\\label{anz}\n\\ensuremath{\\mathcal{A}_n}:= \\prod_{i=1}^n H^0(C, K_C^{\\otimes i}),\n\\end{equation}\nwhich has dimension\n\\begin{equation}\\label{dan}\n \\dim (\\ensuremath{\\mathcal{A}_n})= \\frac{1}{2} \\dim (\\mathcal{M}_n^d) = n^2(g-1)+1= d_{n}.\n\\end{equation}\nThe Hitchin morphisms\n\\begin{equation}\\label{himo}\n\\un{h}_n^d\\colon \\Higgs_n^d \\to \\ensuremath{\\mathcal{A}_n} \\text{ and }\nh_{n}^d: \\mathcal{M}_n^d \\to \\ensuremath{\\mathcal{A}_n}\n\\end{equation}\nassigns to any Higgs bundle $(E,\\phi)$, the coefficients of the characteristic polynomial of $\\phi$.\nThe morphism $h_{n}^d$ is proper, flat of relative dimension $d_{n}=n^2(g-1)+1$, it has connected fibers, and it is often called the Hitchin fibration.\n\nSince the degree $d$ doesn't play any role in what follows, as long as it is coprime to the rank $n$, we will not indicate it from now on, and simply write $\\mathcal{M}_{n}$ for $\\mathcal{M}_n^d$\nand $h_{n}$ for $h_{n}^d$.\n\n\\subsection{Spectral curves and the BNR-correspondence}\\label{spcv}\n\nAs the key to the geometry of the fibers of the Hitchin fibration $h_n$ is their description as compactified jacobians of spectral curves through the Beauville--Narasimhan--Ramanan--correspondence we also recall this briefly.\n\nAny $a\\in \\ensuremath{\\mathcal{A}_n}$ defines a curve $C_a$, called spectral curve, in the total space of the cotangent bundle $T^*C={\\rm Tot}_C(K_C)$ by viewing $a$ as a monic polynomial of degree $n$ with coefficient of the degree $n-i$ term in $H^0(C,K_C^{\\otimes i})$. This defines a flat family $C_{\\mathcal{A}}\\to \\mathcal{A}$ of projective curves. \n\n\nThe natural projection $\\pi: C_a \\to C$, exhibits the spectral curve as a degree $n$ cover of $C$, but $C_a$ can be singular, non-reduced and reducible.\n\nWe denote by $\\ensuremath{\\mathcal{A}_n}^{\\mathrm{red}} \\subset \\ensuremath{\\mathcal{A}_n}$ the subset corresponding to reduced spectral curves, by $\\ensuremath{\\mathcal{A}_n}^{\\mathrm{int}} \\subset\\ensuremath{\\mathcal{A}_n}^{\\mathrm{red}} $ the subset corresponding to integral spectral curves and by $\\ensuremath{\\mathcal{A}_n}^{{\\times}}\\subset \\ensuremath{\\mathcal{A}_n}$ the open subset corresponding to nodal spectral curves. For us reducible spectral curves will be of particular interest.\n\nWhen viewed as an effective divisor on the surface $T^*C$, any spectral curve $C_a$\ncan be written uniquely as \n\\begin{equation}\\label{eq:spectralcurvedecomp}\nC_a = \\sum_{k=1}^s m_k{C}_{a_k}, \n\\end{equation}\nwhere the $a_k$ are the distinct irreducible factors of the characteristic polynomial and $m_k$ their multiplicities.\nIn particular, the $C_{a_k}$ are integral and pairwise distinct curves which are\nspectral curves of some degree $n_k$. We then have \n\\begin{equation}\\label{eq:DecompositionOfN}\nn=\\sum_{k=1}^r m_k n_k.\n\\end{equation}\nFor $\\un{n}=(n_k)_k \\in {\\mathbbm Z}_{>0}^r$ we write\n$$\\mathcal{A}_{\\un{n}} := \\prod_{k=1}^r \\mathcal{A}_{n_k}$$\nThen, for $\\un{n},\\un{m} \\in {\\mathbbm Z}_{>0}^r$ satisfying (\\ref{eq:DecompositionOfN}), multiplication of polynomials $(p_k)_k \\mapsto \\prod p_k^{m_k}$ defines a finite morphism \n$$ \\mult_{\\un{m},\\un{n}}\\colon \\mathcal{A}_{\\un{n}} \\to \\mathcal{A}_n$$\nand we denote by $S_{\\un{m},\\un{n}}$ its image. For $\\un{m}=\\un{1}=(1,\\dots,1)$, the generic point of the image consists of reduced spectral curves and we abbreviate $$S_{\\un{n}}:= S_{\\un{1},\\un{n}} \\text{ and } \\mult_{\\un{n}}:=\\mult_{\\un{1},\\un{n}}.$$\nThe generic spectral curves defined by points in these subsets are rather simple.\n\n\\begin{lemma}\\label{rem:bertini}\n\\begin{enumerate}\n\t\\item[]\n\t\\item For every $a\\in \\mathcal{A}$ the spectral curve $C_a$ is connected.\n\t\\item For every $\\un{n},\\un{m}$ satisfying $\\sum m_kn_k=n$ there is a dense open subset $$S_{\\un{m},\\un{n}}^{\\times} \\subset S_{\\un{m},\\un{n}}$$ such that for $a\\in S^{\\times}_{\\un{m},\\un{n}}$ the reduced curve $C_a^{\\red} \\subset C_a$ is nodal and with nonsingular irreducible components. \n\t\n\tIn particular, since every irreducible component of $C_a$ has genus $g\\geq 2$, the curves $C_a^{\\red}$ are stable curves in the sense of Deligne-Mumford \\cite{DM} for all $a\\in S^{\\times}_{\\un{m},\\un{n}}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tThis is a consequence of Bertini's theorems, because the spectral curves $C_{a_k} \\subset T^*C \\subset {\\mathbbm P}:={\\mathbbm P}_C(\\mathcal{O}_C \\oplus K_C)$ are defined by general sections of the relative $\\mathcal{O}_{{\\mathbbm P}}(n_k)$ which is \n\tbig and generated by global sections in our case. This implies that the subset of $\\mathcal{A}_{n_k}$ defining smooth curves is dense and open and $S^{\\times}_{\\un{m},\\un{n}}$ is the image of the dense open subset of $\\mathcal{A}_{\\un{n}}$ where the curves intersect transversally.\n\\end{proof}\n\nGiven a Higgs bundle $(E,\\phi)$ with $h(E,\\phi)=a$ we can consider $E$ as a coherent sheaf on $C_a$, because sheaves on $T^*C$ can be viewed as $\\mathcal{O}_C$-modules equipped with an action of the $\\mathcal{O}_C$-algebra $\\oplus_{i\\geq 0} K_C^{-\\otimes i}$. The Cayley--Hamilton theorem then says that the module $\\mathcal{F}_{\\mathcal{E},\\phi}$ defined by $\\phi$ is supported on $C_a$ and it is a torsion free sheaf of rank $1$ on $C_a$ (a notion that in the case of non-reduced curves was introduced by Schaub \\cite{Schaub}). Conversely given $\\mathcal{F}$ a torsion free sheaf of rank $1$ on $C_a$ the sheaf $\\mathcal{E}=p_{a,*}\\mathcal{F}$ is a vector bundle of rank $n$ on $C$ that comes equipped with a Higgs field $\\phi$.\n\nThat this procedure in fact induces an equivalence was proved in increasing generality by Hitchin, Beauville--Narasimhan--Ramanan, and Schaub. To state this result let us denote by $\\Coh_{1,C_\\mathcal{A}}^{tf} \\to \\mathcal{A}$ the stack of torsion free sheaves of rank $1$ on spectral curves.\n\\begin{theorem}[\\cite{Hitchin}\\cite{BNR}\\cite{Schaub}]\n\tThe functor $(E,\\phi) \\mapsto \\mathcal{F}_{\\mathcal{E},\\phi}$ induces an equivalence $\\Higgs_n\\cong \\Coh_{1,C_\\mathcal{A}}^{tf}$. Under this equivalence the stack $\\Higgs_n^d$ is identified with the substack of torsion free sheaves of rank $1$ and Euler-characteristic $\\chi=d+n(1-g)$.\n\\end{theorem}\nIn \\cite{Schaub} (see \\cite[Remarque 4.2]{ch-la}) it was explained how stability of Higgs bundles translates into a stability condition for sheaves on spectral curves.\n\nLet $a\\in S_{\\un{n}}$ be a point that defines a reducible, reduced spectral curve $C_a$ with irreducible components $C_{a_1},\\dots,C_{a_k}$. \nIn this case a torsion-free rank $1$ sheaf $\\mathcal{F}$ on $C_a$ defines a stable Higgs bundle of degree $d$ if and only if for all proper subcurves $C_I=\\cup_{i\\in I} C_{a_i} \\subsetneq C$ we have\n$$ \\chi(\\mathcal{F}_{C_I}) \\geq \\sum_{i \\in I} (n_i\\cdot (\\frac{d}{n}+1-g)),$$\nwhere $\\chi$ is the Euler characteristic and $\\mathcal{F}_{C_I}$ is the maximal torsion-free quotient of $\\mathcal{F}|_{C_I}$. \n\n\\begin{remark}\\label{rem:BNRstable}\n\tThis notion of stability coincides with a stability notion for compactified jacobians (see e.g., \\cite{MRV1}) with respect to the polarization $ \\un{q}:=(n_i\\cdot (\\frac{d}{n}+1-g))_i$, which is a general polarization as $\\gcd(n,d)=1$.\n\t\n\tIn particular the restriction of the Hitchin fibration $h_{n} \\colon \\mathcal{M}_{n}\\to \\ensuremath{\\mathcal{A}_n}$ to $\\mathcal{A}^{\\red}$ is a fine relative compactified jacobian for the family $C_{\\mathcal{A}}|_{\\mathcal{A}^{\\red}}$ in the sense of Esteves \\cite[Theorem A]{EstevesTAMS}.\n\\end{remark}\nFinally we recall the $\\delta^{\\aff}$-invariant of our spectral curves. For any spectral curve $C_a$ we denote by $J_a:=\\Pic^{\\un{0}}_{C_a}$ the generalized jacobian of $C_a$, which is the group scheme parameterizing line bundles on $C_a$ that have degree $0$ on all irreducible components of $C_a$. The $J_a$ are the fibers of a group scheme $J_{\\mathcal{A}} \\to \\mathcal{A}_n$ over $\\mathcal{A}$ which acts on $\\mathcal{M}_{n}$.\n\nFor every $a$ the connected group scheme $J_a$ has a canonical filtration\n$$ 0 \\to J_a^{\\aff} \\to J_a \\to J_a^{\\mathrm{proj}} \\to 0$$\nwhere $J_a^{\\aff}$ is affine, $J_a^{\\mathrm{proj}}$ is projective and both are connected. One defines\n$$ \\delta^{\\aff}(C_a):= \\dim( J_a^{\\aff}).$$\n\n\\begin{remark}\\label{rem:deltanode}\n\tIf $C_a$ is a reduced, connected curve and $\\nu\\colon \\widetilde{C}_a \\to C_a$ is the normalization, then $\\nu^*$ defines an isomorphism $J_a^{\\mathrm{proj}}\\cong \\Pic_{\\widetilde{C}_a}^0$. In this case \n\\begin{equation}\\label{eq:deltaff}\n\\delta^{\\aff}(C_a) =\\dim H^0(C,\\nu_*\\mathcal{O}_{\\widetilde{C}_a}\/\\mathcal{O}_C)+1-\\#(\\pi_0(\\widetilde{C}_a)).\n\\end{equation}\nIf furthermore, the only singularities of $C_a$ are nodes, we have\n\\begin{equation}\\label{deltaff_nodes}\n\\delta^{\\aff}(C_a) =\\# (\\text{nodes})+1-\\#(\\pi_0(\\widetilde{C}_a))=1-\\chi(\\Gamma)=\\dim H^1(\\Gamma),\n\\end{equation}\t\nwhere $\\Gamma$ is the dual graph of the curve $C_a$.\n\\end{remark}\n\nThe function $a \\mapsto \\delta^{\\aff}(C_a)$ is upper semi-continuous by \\cite[X, Remark. 8.7]{sga3.2.2}, i.e. there are closed subsets\n$$\\mathcal{A}_n^{\\geq \\delta}:=\\{ a \\in \\mathcal{A}_n | \\delta^{\\aff}(C_a)\\geq \\delta\\} \\subseteq \\mathcal{A}_n.$$\n\n\\begin{remark}\nFor a flat family $C_Y \\to Y$ of projective curves over an irreducible scheme $Y$ with generic point $\\eta_Y$, we will denote by $\\delta^{\\aff}(Y):= \\delta^{\\aff}(C_{\\eta_Y})$ and call it generic $\\delta^{\\aff}$-invariant on $Y$.\t\n\\end{remark}\t\n\n\\begin{lemma}\\label{lem:deltasn}\n\tLet $\\un{n}$ be a partition of $n$ and $a\\in S_{\\un{n}}$ the generic point then we have \n\t$$\\codim S_{\\un{n}}= \\delta^{\\aff}(C_a)=:\\delta^{\\aff}(\\un{n}).$$\n\\end{lemma}\n\\begin{proof}\n\tWe know that $\\dim S_{\\un{n}} = \\dim \\mathcal{A}_{\\un{n}} = \\sum_{i=1}^k (n_i^2 (g-1)+1).$\n\t\n\tBy \\cref{rem:bertini} for a general $a=(a_k)\\in \\mathcal{A}_{\\un{n}}$ the spectral curve $C_a$ has $k$ smooth components intersecting transversally. As each component is defined by a polynomial in $\\mathcal{A}_{n_i}=\\oplus_{r=1}^{n_i} H^0(C, K_C^{\\otimes r})$ we have \n$$ \\# (C_{a_i}\\cap C_{a_j}) = n_in_j (2g-2).$$\nBy Remark \\ref{rem:deltanode} for any nodal curve $D$ the $\\delta$-invariant is equal to the number of nodes minus the number of irreducible components plus the number of connected components. Therefore \n \\begin{align*}\n \\dim S_{\\un{n}} + \\delta^{\\aff}(C_a) &= \\left(\\sum_{i=1}^k n_i^2 (g-1)\\right) + k + \\sum_{i0$ would contribute a summand in \n\t$\\, ^p\\!\\!{\\mathscr H}^{k_0 +j}(i^*K)$ which violates the RHL symmetry.\n\\end{proof}\n\\begin{lemma}\n\tLet $\\ensuremath{\\mathcal{L}}$ be a semisimple local system on an open dense subset $U\\subset X$ and let $P=\\IC(\\ensuremath{\\mathcal{L}})$ be its intersection cohomology complex. \n\tLet $i:Z \\hookrightarrow X$ be a closed subvariety such that: \n\t\\begin{enumerate}\n\t\t\\item \n\t\t$U \\bigcap Z$ is Zariski dense in $Z$.\n\t\t\\item\n\t\tThe complex $i^*P$ is perverse semisimple.\n\t\\end{enumerate}\n\tThen \n\t$$ i^*P = \\oplus_k \\IC(Z^{k},\\ensuremath{\\mathcal{L}}^{k})$$\n\twhere $Z^k$ is the union of the irreducible components of $\\mathrm{Supp}\\, \\mathcal{H}^{k}(P)\\bigcap Z$ of codimension $k$ in $Z$ and $\\ensuremath{\\mathcal{L}}^k = \\mathcal{H}^k(i^*P)$ on the smooth part of the dense open subset of $Z^k$ where this sheaf is a local system.\t\n\\end{lemma}\n\\begin{proof}\nRecall that, if $Q$ be a perverse semisimple sheaf on $Z$, \nthen we have a canonical decomposition \n\\begin{equation}\\label{candec}\nQ=\\bigoplus_{k=0}^n \\IC(Z^k, \\ensuremath{\\mathcal{L}}^k)\n\\end{equation}\nwhere, for every $k$,\n$Z^k$ is a closed subvariety of $Z$ of codimension $k$, and $\\ensuremath{\\mathcal{L}}^k$ is a semisimple local systems on an open set $Z^{k,\\circ}$ of $Z_k$.\nNote that $Z^k$ is allowed to be reducible and $\\ensuremath{\\mathcal{L}}^k$ may have different rank on the different components of $Z^{k,\\circ}$.\n\nThe subsets $Z^k$ and local systems $\\ensuremath{\\mathcal{L}}^k$ afford an easy characterization, which follows immediately from the strong support condition\nfor the intersection cohomology complex (\\Cref{subsec:ConventionIC}), i.e.: \nFor every $k$, the closed subset $Z^k$ is the union of the $k$-codimensional components of $\\mathrm{Supp}\\, \\mathcal{H}^{k}(Q)$ and if $x \\in Z^{k,\\circ}$, then there is a canonical isomorphism $\\ensuremath{\\mathcal{L}}^k_{x}= \\mathcal{H}^{k}(K)_{x}.$\n\\end{proof}\n\n\n\t\\begin{remark}\\label{rem:transverseorsplit}\n\t\tNotice that, by the support condition (see \\cref{subsec:ConventionIC}) for the intersection cohomology complex, we have that\n\t\t$\\codim \\mathrm{Supp}\\, \\mathcal{H}^{k}(\\IC(\\ensuremath{\\mathcal{L}})) \\geq k+1$. Thus, \n\t\t if $Z$ intersects $\\mathrm{Supp}\\, \\mathcal{H}^{k}(\\IC(\\ensuremath{\\mathcal{L}}))$ properly, we have\n\t\t\t$\\mathrm{Supp}\\, \\mathcal{H}^{k}(\\IC(\\ensuremath{\\mathcal{L}}))\\bigcap Z $ has codimension at least $k +1$ in $Z$ and therefore it cannot contribute a perverse summand. On the other hand,\n\t\t\tif $\\codim \\mathrm{Supp}\\, \\mathcal{H}^{k}(\\IC(\\ensuremath{\\mathcal{L}}))\\bigcap Z < k$, then $i_*i^*P$ is not perverse.\n\t\tSo if $i^*P$ is perverse on $Z$ then the splitting off of the restriction of an intersection cohomology complex is governed by a precise failing of transversality \tof $Z$ to the supports of the cohomology sheaves.\n\t\\end{remark}\n\\begin{remark}\\label{purity} In the situation of Proposition \\ref{prop:splitperverse}\n\tassume $K$ is pure of weight $0$ so that by our assumptions $i^*K$ is pure of weight $0$ too. \n\tThen the local systems $\\ensuremath{\\mathcal{L}}_{i}^{k} $ are pure of weight $i+k$.\n\tNotice that since $\\mathcal{R}_i$ is of weight $i$, by purity we have \n\t$$\n\t\\mathrm{weight} (\\mathcal{H}^k(\\IC(\\mathcal{R}_i)))_x \\leq i+k,\n\t$$\n\ttherefore the local systems $\\ensuremath{\\mathcal{L}}_{i}^{k}$ are the maximal weight quotients of the cohomology sheaves.\n\\end{remark}\n\n\\begin{remark}\\label{forex}\n\tThe assumptions of the \\cref{prop:splitperverse} are met when $K={\\mathbbm R} f_*\\mathbbm{Q}$ for $f$ a projective map satisfying the support theorem,\n\tand $Z\\subset X$ is a local complete intersection such that $f^{-1}(Z)$ is nonsingular. \n\\end{remark}\n\n\\subsection{The Kodaira--Spencer map for spectral curves}\n\nWe want to apply Remark \\ref{forex} to compare the cohomology of the Hitchin fibration to the cohomology of relative compactified jacobians for versal families of spectral curves. To verify the assumptions that $Z$ is a local complete intersection we need to describe the Kodaira--Spencer map for the family of spectral curves over $\\mathcal{A}_n$. \n\nFor any point $a\\in \\mathcal{A}_n$ we denote by $\\mathcal{I}_{C_a} \\subset \\mathcal{O}_{T^*C}$ the ideal sheaf defining $C_a \\subset T^*C$. \nRecall that embedded deformations of $C_a \\subset T^*C$ are described by the cotangent complex \n$$ {\\mathbbm L}_{C_a\/T^*C} =[\\mathcal{I}_a\/\\mathcal{I}_a^2 \\to 0]$$ \nwhich concentrated in degree $[-1,0]$. Considering the composition $C_a \\hookrightarrow T^*C \\to \\Spec k$ we see that the cotangent complex of $C_a$ is \n$$ {\\mathbbm L}_{C_a} = \\left[\\mathcal{I}_a\/\\mathcal{I}_a^2 \\to \\left(\\Omega_{T^*C}|_{C_a}\\right)\\right].$$\nNow the universal spectral curve over $\\mathcal{A}_n$ defines a Kodaira--Spencer map\n\\[ KS_a\\colon T_a\\mathcal{A}_n \\to H^1(C_a,{\\mathbbm L}_{C_a}^\\vee) = \\Ext^1({\\mathbbm L}_{C_a},\\mathcal{O}_{C_a}). \\]\nWe know that the ${\\mathbbm G}_m$-action on $\\mathcal{A}$ and the translation action $H^0(C,K_C)\\times \\mathcal{A}_n \\to \\mathcal{A}_n$ lift to the universal spectral curve $C_{\\mathcal{A}_n}\\to \\mathcal{A}_n$ and therefore induce trivial deformations of $C_a$. \nLet us denote by \n\\[ \\dmult\\colon k= \\Lie({\\mathbbm G}_m) \\to T_a\\mathcal{A}\\cong \\oplus_{i=1}^n H^0(C,K_C^{\\otimes i}) \\]\nthe derivative of the ${\\mathbbm G}_m$-action and by\n\\[ \\dshift \\colon H^0(C,K_C) \\to T_a\\mathcal{A}\\cong \\oplus_{i=1}^n H^0(C,K_C^{\\otimes i}) \\]\nthe derivative of the translation. We will show in \\Cref{Lem:KodairaSpencerComputation} below that the span of the image of these maps is the kernel of the Kodaira--Specner map.\n\nLet us also recall that $S_{n,1}\\subset \\mathcal{A}_n$ is the locus of spectral curves that are given by the $n-$th infinitesimal neighborhood of a section in $T^*C$.\n\\begin{lemma}[Kodaira-Spencer map for $C_a$]\\label{Lem:KodairaSpencerComputation}\n\t\\begin{enumerate}\n\t\t\\item[]\n\t\t\\item For any point $a\\in \\mathcal{A}_n-S_{n,1}$ the kernel of the Kodaira--Spencer $K_a$ is the direct sum of the images of $\\dmult$ and $\\dshift$, i.e., the map $KS_a$ factors as \n\t\t$$ T_a \\mathcal{A} \\twoheadrightarrow (T_a \\mathcal{A})\/(H^0(C,\\mathcal{O}_C\\oplus K_C)) \\hookrightarrow H^1(C_a,{\\mathbbm L}_{C_a}^\\vee).$$\n\t\t\\item \tFor $a \\in S_{n,1} \\subset \\mathcal{A}_n$ the kernel of the Kodaira--Spencer $K_a$ is equal the image of $\\dshift$, i.e., the map $KS_a$ factors as \n\t\t$$ T_a \\mathcal{A} \\twoheadrightarrow (T_a \\mathcal{A})\/(H^0(C,K_C)) \\hookrightarrow H^1(C_a,{\\mathbbm L}_{C_a}^\\vee).$$\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tLet us first describe the sheaves occurring in ${\\mathbbm L}_{C_a}$ more explicitly.\n\tThe cotangent bundle $T^*C$ is the relative spectrum of the $\\mathcal{O}_C$ algebra \n\t$$\\Sym^\\bullet K_C^{\\otimes -1} = \\oplus_{r=0}^\\infty K_C^{\\otimes -r}$$ and the spectral curve $\tC_a\\subset T^*C$ is defined by the ideal generated by $K_C^{\\otimes -n} \\to \\oplus_{r=0}^\\infty K_C^{\\otimes -r}$ defined by $\\alpha \\mapsto \\alpha + a_1\\alpha + \\dots + a_n\\alpha$. Therefore, denoting by $\\pi_a \\colon C_a \\to C$ the projection we see that\n\t\\[\\pi_{a,*} \\mathcal{O}_{C_a} \\cong \\oplus_{r=0}^{n-1}K_C^{\\otimes -r}\\]\n\tand\n\t\\[ \\mathcal{I}_a|_{C_a}= \\mathcal{I}_{C_a}\/\\mathcal{I}_{C_a}^2 \\cong \\pi_a^* K_C^{\\otimes -n}.\\] \n\t \n\n\tThe dual of the canonical map \n\t$$ {\\mathbbm L}_{C_a}= \\left[\\mathcal{I}_a\/\\mathcal{I}_a^2 \\to \\left(\\Omega_{T^*C}|_{C_a}\\right)\\right] \\to [\\mathcal{I}_a\/\\mathcal{I}_a^2 \\to 0]= {\\mathbbm L}_{C_a\/T^*C} $$\n\tis given by\n\t$$ [0 \\to \t(\\mathcal{I}_a\/\\mathcal{I}_a^2)^\\vee ] \\to [ T_{T^*C}|_{C_a} \\to (\\mathcal{I}_a\/\\mathcal{I}_a^2)^\\vee ].$$\n\tNote that\n\t$$ H^0(C_a,(\\mathcal{I}_a\/\\mathcal{I}_a^2)^\\vee) \\cong H^0(C, \\oplus_{r=1}^n K_C^{\\otimes{r}})=T_a\\ensuremath{\\mathcal{A}_n}$$\n\tis the space of embedded deformations of $C_a\\subset T^*C$.\n\t\n\tTaking cohomology of the exact triangle of complexes:\n\t$$ \\to [0 \\to \t(\\mathcal{I}_a\/\\mathcal{I}_a^2)^\\vee ] \\to {\\mathbbm L}_{C_a}^\\vee \\map{p} [ T_{T^*C}|_{C_a} \\to 0] \\to $$\n\twe obtain a long exact sequence:\n\t\\begin{equation}\\label{seq:kslong} 0 \\to H^0(C_a,{\\mathbbm L}_{C_a}^\\vee) \\map{H^0(p)} H^0(C_a, T_{T^*C}|_{C_a} ) \\map{\\delta} H^0(C_a, (\\mathcal{I}_a\/\\mathcal{I}_a^2)^\\vee) \\map{KS_a} H^1(C_a,{\\mathbbm L}_{C_a}^\\vee) \\to \\dots\\end{equation}\n\tTo conclude we will compute the dimension of $ H^0(C_a, T_{T^*C}|_{C_a})$ and then compare it to the dimension of the image of $\\dmult$ and $\\dshift$.\n\t\n\tRestricting the relative tangent sequence $0\\to \\pi^* K_C \\to T_{T^*C} \\to \\pi^* TC \\to 0$ on $T^*C$ to $C_a$ we get \n\t$$ 0\\to \\pi_a^* K_C \\to \\Omega_{T^*C}|_{C_a} \\to \\pi_a^* K_C^{\\otimes -1} \\to 0.$$\n\tApplying $\\pi_{a,*}$ and the projection we find: \n\t$$ 0 \\to \\oplus_{r=0}^{n-1} K_C^{\\otimes (1-r)} \\to \\pi_* T_{T^*C}|_{C_a} \\to \\oplus_{r=0}^{n-1} K_C^{\\otimes (-1-r)} \\to 0.$$\n\tIn particular we see that \\[H^0(C_a, T_{T^*C}|_{C_a} ) \\cong H^0(C_a,\\pi_a^*K_C)=H^0(C,K_C) \\oplus H^0(C,\\mathcal{O}_C).\\]\n\tThus we have an exact sequence:\n\t\\begin{equation}\\label{eq:KSmap}\n H^0(C_a,\\pi_a^*K_C)=H^0(C,K_C) \\oplus H^0(C,\\mathcal{O}_C) \\map{\\delta} T_a\\ensuremath{\\mathcal{A}_n} \\map{KS_a} H^1(C_a,{\\mathbbm L}_{C_a}^\\vee),\n\t\\end{equation}\n\twhere the map $\\delta$ is determined by the differential in the cotangent complex ${\\mathbbm L}_{C_a}^\\vee$.\n\t \n\tNow let us determine the dimension of the image of $\\dmult$ and $\\dshift$. The ${\\mathbbm G}_m$ action is given by the action of weight $i$ on $H^0(C,K_C^{\\otimes i})$, so at $a$ the element $c \\in {\\mathbbm C}=Lie({\\mathbbm G}_m)$ defines the tangent vector $(a_i+ i a_i\\cdot\\epsilon ) \\in T_a\\ensuremath{\\mathcal{A}_n} \\subset \\ensuremath{\\mathcal{A}_n}({\\mathbbm C}[\\epsilon]\/(\\epsilon^2))$. \n\t\n\tSimilarly as $a$ is given by the coefficients of a characteristic polynomial the translation by an element $\\omega \\in H^0(C,K_C)$ sends a polynomial $p(t)$ to $p(t-\\omega)$. Thus the derivative at $a=(a_i)$ is $(a_i - (n-i+1) \\omega a_{i-1} \\cdot\\epsilon)$ where we put $a_0:=1$.\n\t\n\tIn particular these vector fields are linearly independent unless $a_{i}= (-1)^i {n \\choose i} \\omega^i$, i.e., $a\\in S_{n,1}$. This shows that the the kernel of $KS_a$ has dimension $\\geq g+1$ for $a\\not\\in S_{n,1}$. By equation (\\ref{eq:KSmap}) we know that the dimension is $\\leq g+1$, so this shows the first claim.\n\t\n\tIf $a$ is the $n-$fold multiple of a section, then the spectral curve $C_a$ admits a continuous family of automorphisms, given by multiplication of the nilpotent generator therefore $ H^0(C_a,{\\mathbbm L}_{C_a}^\\vee)$ which is the tangent space to the automorphism group of $C_a$ is at least $1$ dimensional. Thus the second claim follows from (\\ref{seq:kslong}).\n\\end{proof}\n\nLet us now apply this result to the restriction of the Hitchin fibration to the subset of nodal curves. Let $g_n=d_n= \\dim \\ensuremath{\\mathcal{A}_n}$ be the arithmetic genus of the spectral curves $C_a$. We denote by $\\overline{\\mathcal{M}}_{g_n}$ the stack of stable curves of genus $g_n$. Then by \\cref{rem:bertini} the universal family of spectral curves $C_{\\ensuremath{\\mathcal{A}_n}}$ induces a morphism \n\\[f_{\\nod}\\colon \\ensuremath{\\mathcal{A}_n}^{\\nod} \\to \\overline{\\mathcal{M}}_{g_n}.\\]\nRecall from Remark \\ref{rem:BNRstable} that for any $a\\in \\mathcal{A}^{\\nod}$ the stability condition for Higgs bundles defines a general polarization $\\un{q}$ for rank $1$ torsion free sheaves on $C_a$ in the sense of \\cite{MRV1}.\nAs this polarization only depends on the fixed values $n,d$ and the genus of the irreducible components of $C_a$ the induced polarization on the fibers of a semiuniversal deformation again has this property and therefore is independent of the choice of $a$. Thus we obtain a polarization $\\un{q}$ on an open neighborhood $\\mathcal{U}\\subset \\overline{\\mathcal{M}}_{g_n}$ of $f_{\\nod}(\\ensuremath{\\mathcal{A}_n}^{\\nod})$.\n\n\n\nThe following is a consequence of the work of Esteves \\cite[Theorem A]{EstevesTAMS} and Melo--Rapagnetta--Viviani \\cite[Theorem C]{MRV1}.\n\\begin{proposition}\n\tThere exists an open substack $\\mathcal{U} \\subset \\overline{\\mathcal{M}}_{g_n}$ containing $f_{\\nod}(\\ensuremath{\\mathcal{A}_n}^{\\nod})$ such that {the compactified jacobian parametrizing $\\un{q}$-stable torsion free sheaves defined on any cover of $U$ that admits a universal family descends to} a regular and irreducible Deligne--Mumford stack $u\\colon \\overline{J}_{\\mathcal{U}}(\\un{q}) \\to \\mathcal{U}$. The map $u$ is representable and locally projective.\n\\end{proposition}\n\\begin{proof}\n\tBy \\cite[Theorem C]{MRV1} for any general polarization $\\un{q}$ the algebraic stack of $\\un{q}$-stable rank $1$ torsion free sheaves admits a projective geometric coarse moduli space for any versal deformation of a reduced curve with planar singularities and these spaces are regular and irreducible. We therefore obtain relative compactified jacobians $\\overline{J}_{\\Spec R}(\\un{q})$ on \\'etale neighborhoods of any point $a\\in \\mathcal{U}$. As these spaces are geometric coarse moduli spaces they are canonically isomorphic on the intersections of these neighborhoods and therefore define an \\'etale covering of {a} stack $\\overline{J}_{\\mathcal{U}}(\\un{q})$.\n\\end{proof}\t\n\nCombining this result with our computation of the Kodaira--Spencer map for the family $C_{\\mathcal{A}}$ (\\cref{Lem:KodairaSpencerComputation}) we deduce:\n\n\\begin{corollary}\\label{cor:map_to_moduli}\n\tFor every partition $\\ve{n} $ of $n$, let $a \\in S_{\\ve{n}}^{\\times}$ (see \\cref{rem:bertini}).\n\tGiven a subvariety $ \\Sigma_a $ passing through $a$ and intersecting $S_{\\ve{n}}$ transversally, \n\tthe classifying map $f_{\\Sigma_a}\\colon\\Sigma_a \\to \\mathcal{U} \\subset \\overline{\\mathcal{M}}_{g_n}$ is unramified on an open neighborhood of $a$ in $\\Sigma_a$.\n\tFurthermore we have a cartesian diagram \n\t\\begin{equation}\\label{equ:cartesian}\n\t\\xymatrix{\n\t\t{h_{n}}^{-1}(\\Sigma_a) \\ar[rd]_{{h_{n}}_|}\\ar[r]^{\\simeq} & \\ov{J}_{\\mathcal{C}}\\times_U \\Sigma_a \\ar[r]\\ar[d] & \\ov{J}_{\\mathcal{U}} \\ar[d]^{u}\\\\\n\t\t&\\Sigma_a \\ar[r]^-{f_{\\Sigma_a}} & \\mathcal{U}.\n\t}\n\t\\end{equation}\n\\end{corollary}\n\n\\begin{remark}\\label{rem:compwithversal}\n\tAs the map $f_{\\Sigma_a}$ is unramified (i.e. it is \\'etale over a closed embedding) it is a local complete intersection morphism. By Remark \\ref{forex} the restriction result for semisimple complexes \\ref{prop:splitperverse} we can therefore compute the supports of ${\\mathbbm R} h_{n,*}\\mathbbm{Q}$ on $\\Sigma_a$ as the restrictions of ${\\mathbbm R} u_* \\mathbbm{Q}$.\n\\end{remark}\n\n\n\n\\section{The partition strata are supports}\\label{sec:sn_supports}\n\n\\subsection{Description of the $\\IC$ complexes for families with full support}\nHere we recall a few facts on the Cattani--Kaplan--Schmid (CKS) complex, introduced in \\cite{cks, KK}, see also (\\cite{MSV} \\S 3). {This will finally allow us to reduce our problem to a homology calculation of an explicit combinatorially defined complex attached to the dual graph of a nodal curve.} \n\nAssume $B$ is a complex manifold of dimension $n$, $D \\subset {B}$ is a normal crossing divisor $D$, and\n$\\ensuremath{\\mathcal{L}}$ is a local system on $B \\setminus D$ with unipotent monodromies $\\{T_i\\}$ around the components of $D$. We work locally, near a point $p \\in B$. After picking a holomorphic chart $U \\subset B$ in a neighborhood of $p$, we may assume $p$ to be the origin in a polydisc $\\Delta^n$ and the divisor $D$ to have equation $\\prod_{i=1}^l z_i=0$. Thus $U \\bigcap (B \\setminus D)\\simeq (\\Delta^*)^l \\times \\Delta^{n-l} $, where $\\Delta^*$ is the punctured unit-disc.\nUp to taking a slice transverse to the stratum of $D$ to which $p$ belongs, we may assume $l=n$, and denote $i_p: \\{*\\} \\to B$ the closed embedding. The local system on $(\\Delta^*)^n$ is described by the stalk at a base point, a vector space $L$, and $n$ commuting nilpotent endomorphisms $N_i =\\log T_i: L \\to L$.\nGiven a subset $\\{i_1, \\cdots i_k\\}=I \\subset \\{1, \\cdots , n\\}$, with $1 \\leq i_1 \\delta^{\\aff}= \\dim H^1(\\Gamma)$ for every $i$.\n\\end{lemma}\nLemma \\ref{lem:bond_matroid} justifies the following definition:\n\\begin{definition}\\label{def:GraphComplex}\nGiven a connected graph $\\Gamma$, with set of edges $\\mathrm{E}$,\n\twe write $\\mathscr{C}(\\Gamma)$ for the collection of subsets of $\\mathrm{E}$ whose removal does not disconnect $\\Gamma$.\n\tIn other words, a subset $I\\subseteq \\mathrm{E}$ belongs to $\\mathscr{C}(\\Gamma)$ if and only if $\\Gamma\\setminus I$ is connected. \t\n\\end{definition} \n\\begin{remark}\\label{rem:matr}\nIn the literature $\\mathscr{C}(\\Gamma)$ is the family of independent subsets in what is known as the {\\em bond, or cographic, matroid } of the graph $\\Gamma$. \nThe set $\\mathscr{C}(\\Gamma)$ is partially ordered with respect to inclusions and we denote by $|\\mathscr{C}(\\Gamma)|$ the associated simplicial complex, i.e.,\nthe complex whose $k$-dimensional faces are the $(k+1)$-ples of edges belonging to $\\mathscr{C}(\\Gamma)$. \n\\end{remark}\nFor any matroid $M$ the simplicial complex $|\\In(M)|$ of its independent subsets has special properties:\n\\begin{theorem}\\cite[Theorem 7.3.3, Theorem 7.8.1]{bjorner}\\label{shellable_mat}\nThe simplicial complex $|\\In(M)|$ of independent subsets associated with a rank $r$ matroid \nis shellable and has the homotopy type of a bouquet of \n$r-1$-dimensional spheres.\n\\end{theorem}\n\nThe rank of $\\mathscr{C}(\\Gamma)$ is the cardinality of the complement of a spanning tree, namely\n$|\\mathrm{E}|-|\\mathrm{V}|+1$, which, in the case of the dual graph of a nodal curve $C_\\times$ equals \n$ \\delta^{\\aff}(C_\\times)$, hence:\n\n\\begin{corollary}\\label{shellable}\nLet $\\Gamma$ be the dual graph of a nodal curve $C_\\times$. \nThen $|\\mathscr{C}(\\Gamma)|$ is homotopic to a bouquet of $(\\delta^{\\aff}-1)$-dimensional spheres:\n\\[ |\\mathscr{C}(\\Gamma)| \\simeq \\bigvee S^{\\delta^{\\aff}-1},\\].\n\\end{corollary} \n\n\\begin{corollary}\\label{cor:highest_weight_cohomology}\nLet $C_\\times$ be a nodal stable curve, and ${\\mathbf C}^{\\bullet}( \\{ N_j \\}, \\bigwedge^i H^1(\\mathcal{C}_\\eta))$ be the associated CKS complex.\nThen, for every $i=0, \\cdots 2g$ we have\n\\begin{enumerate}\n\\item $H^{r}({\\mathbf C}^{\\bullet}( \\{ N_j \\}, \\bigwedge^i H^1(\\mathcal{C}_\\eta)))=0 \\text{ for } r > \\delta^{\\aff}.$\n\\item\\label{item:highestweight}\nFor the highest weight quotient of ${\\mathbf C}^{\\bullet}( \\{ N_j \\}, \\bigwedge^i H^1(\\mathcal{C}_\\eta))$ \nwe have \n$$Gr_{i+\\delta^{\\aff}}^WH^{r}({\\mathbf C}^{\\bullet}( \\{ N_j \\}, \\bigwedge^i H^1(\\mathcal{C}_\\eta))) = 0 \\hbox{ if }r < \\delta^{\\aff},$$ \n$$Gr_{i+\\delta^{\\aff}}^WH^{\\delta^{\\aff}}({\\mathbf C}^{\\bullet}( \\{ N_j \\}, \\bigwedge^i H^1(\\mathcal{C}_\\eta)))= \n\\bigwedge^{i-\\delta^{\\aff}}H^1(\\widetilde{C}_\\times)\\otimes H^{\\delta^{\\aff}-1}(|\\mathscr{C}(\\Gamma)|)(-\\delta^{\\aff}).$$\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nThe first statement follows from \\cref{lem:bond_matroid}.\nTo prove (\\ref{item:highestweight}) notice that $\\bigwedge^{\\delta^{\\aff}} H_1(\\Gamma)$ is one-dimensional, since $\\delta^{\\aff}=\\dim H_1(\\Gamma)$. Similarly the image of $N_I$ is isomorphic to $\\bigwedge^{\\delta^{\\aff}-|I|} (H_1(\\Gamma-I))$ which is also one dimensional. So for every $k$, the degree $k$ part of the complex has a basis consisting of the non-disconnecting cardinality $k$ subsets of the edges set, namely precisely the $(k-1)$-cells of $|\\mathscr{C}(\\Gamma)| $. It is easy to check that the boundary maps coincide with the maps of the complex computing the reduced homology of $|\\mathscr{C}(\\Gamma)| $. It then follows from \\cref{eq:weight_exterior} that, for $i \\geq \\delta^{\\aff}$, the cohomology of the CKS complex in highest weight is computed by a complex which is the tensor product with the vector space\n$\\bigwedge^{i-\\delta^{\\aff}}H^1(\\widetilde{C}_{\\times}) $ of the complex computing the reduced homology of $|\\mathscr{C}(\\Gamma)|$ twisted by $(-\\delta^{\\aff})$ and shifted by one.\n\\end{proof}\n\\begin{remark}\\label{rem:purity}\nThe vanishing of the cohomology groups in the range above can be also derived from the purity result \\cref{e:purity}.\n\\end{remark}\n\n\\subsection{Description of the summands supported on the partition strata}\n\nWe need the following well known, elementary estimate:\n\\begin{lemma}\\label{lem:estimate}\nLet $\\pi: \\mathcal{C} \\to B$ be a flat projective family of locally planar reduced curves of arithmetic genus $g$ such that \ncompactified jacobian family $\\pi^J: \\ov{J}_{\\mathcal{C}} \\to B$, \nrelative to a choice of a fine polarization exists and has nonsingular total space. Let $B^\\circ \\subset B$ be a dense open set such that the restriction\n$\\pi\\colon \\mathcal{C}_{B^\\circ} \\to B^\\circ$ is smooth, and denote by $R^1$ the local system on $B^\\circ$:\n$$\nR^1:={R^1\\pi_* \\mathbbm{Q}}_{|B^\\circ}.\n$$\nThen\n\\begin{equation}\\label{equ: estimate}\n\\mathcal{H}^r(\\IC(\\bigwedge^i R^1 ))=0 \\text{ for } r>i.\n\\end{equation}\n\n\\end{lemma}\n\\begin{proof}\nIt follows from the decomposition theorem for $\\pi^J: \\ov{J}_{\\mathcal{C}} \\to B$ that $\\IC(\\bigwedge^i R^1 ))[-i]$ is a direct summand in ${\\mathbbm R} \\pi^J\\mathbbm{Q}$:\n\\begin{equation}\\label{eq:dtpi}\n\\IC(\\bigwedge^i R^1 ))[-i] \\subset {\\mathbbm R} \\pi^J\\mathbbm{Q}\n\\end{equation}\nAssume $\\mathcal{H}^r(\\IC(\\bigwedge^i R^1 ))_b\\neq 0$ for $b\\in B$ and some $r >i$. It then follows from the relative Hard Lefschetz theorem that we may assume $i\\geq g$.\nTaking stalks of the cohomology sheaf $\\mathcal{H}^{r+i}$ at $b$ in the above \\cref{eq:dtpi} we have\n\\begin{equation*}\n0 \\neq \\mathcal{H}^{r}(\\IC(\\bigwedge^i R^1 ))\\subset H^{r+i}(\\ov{J}_{\\mathcal{C}}(b)).\n\\end{equation*} \nwhich is a contradiction since $r+i>2i \\geq 2g=2\\dim \\ov{J}_{\\mathcal{C}}(b)$.\n\\end{proof}\nWe will rely on the following result (\\cite[Theorem 5.11]{MSV}):\n\\begin{theorem}\\label{thm:hd_cj}\n\tLet $\\pi:\\mathcal{C} \\to B$ be a projective versal family of curves with locally planar singularities and arithmetic genus $g$, and let $\\pi^J:\\ov{J}_{\\mathcal{C}}\\to B$ \n\tbe a relative fine compactified Jacobian. \n\tThen no summand of ${\\mathbbm R} \\pi^J_* \\mathbbm{Q}$ has positive codimensional support,\n\tnamely, \n\t\\begin{equation*}\n\t{\\mathbbm R} \\pi^J_* \\mathbbm{Q}= \\bigoplus_{i=0}^{2g} \\IC(\\bigwedge^i R^1)[-i]\n\t\\end{equation*}\n\twhere $R^1$ is defined as in \\cref{lem:estimate}.\n\\end{theorem}\n\nThe following is one of the main results of this paper: \n\\begin{theorem}\\label{thm:main1}\nLet $h_{n}:\\mathcal{M}_{n} \\to \\ensuremath{\\mathcal{A}_n}$ be the Hitchin map.\n\\begin{enumerate}\n\\item \nFor every partition $\\ve{n}$ of $n$, the stratum $S_{\\ve{n}}$ is a support for all the sheaves\n\\begin{equation*}\\label{eq:range_supp}\n^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R} {h_{n}}_{*}\\mathbbm{Q}) \n\\hbox{ with } \\delta^{\\aff}(\\ve{n}) \\leq r \\leq {2\\dim \\ensuremath{\\mathcal{A}_n} -\\delta^{\\aff}(\\ve{n})}\n\\end{equation*}\n\\item\\label{item:loc_sys}\nMore precisely, for every $r$ in the range of (\\ref{eq:range_supp}), there is a direct summand in \n$^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R} {h_{n}}_{*}\\mathbbm{Q}) $ which is the intermediate extension of the \nlocal system $\\ensuremath{\\mathcal{L}}_{r, \\ve{n}}$ on the open set $ {S_{\\ve{n}}^{\\times}} \\subset S_{\\ve{n}}$, \nwhose stalk at a point $a\\in S_{\\ve{n}}^{\\times}$ is \n\\begin{equation*}\\label{eq:loc_sys}\n({\\ensuremath{\\mathcal{L}}_{r, \\ve{n}}})_a = H^{\\delta^{\\aff}(\\ve{n})-1}\\left( |\\mathscr{C}(\\Gamma_{\\ve{n}})| \\right)(-\\delta^{\\aff})\\otimes \\bigwedge^{r-\\delta^{\\aff}}H^1(\\widetilde{C}_a)\n\\end{equation*}\nand underlying a variation of pure Hodge structures of weight $r+\\delta^{\\aff}(\\ve{n})$.\n\\end{enumerate}\n\\end{theorem}\n\n{\n\\begin{remark}\nTheorem \\ref{thm:main1} holds in the context of M. Saito's mixed Hodge modules (see \\cite[Appendix]{drs}).\nIn particular, the resulting direct summands of the pure Hodge structures given by the cohomology groups $H^k(\\mathcal{M}_{n},\\mathbbm{Q})$ (these are pure since they coincide with the cohomology of the nilpotent cone (fiber of the Hitchin map over the origin)) are pure Hodge substructures. \n\\end{remark}\n}\n\n\\begin{proof}\nSince, by \\cref{thm:hidiscrHitc}, the stratum $S_{\\ve{n}}$ is a $\\delta^{\\aff}(\\ve{n})$-codimensional component of $\\Delta^{\\delta^{\\aff}(\\ve{n})}(h_{n})$, either \n$^p\\!\\!{\\mathscr H}^{r}({\\mathbbm R} {h_{n}}_{*}\\mathbbm{Q})$ has a summand which is {\\em fully} supported at $S_{\\ve{n}}$ or none of its summands intersect $S_{\\ve{n}}$, therefore\nit is enough to consider a general point $a\\in S_{\\ve{n}}$ corresponding to a nodal spectral curve $C_a$ with $l(\\ve{n})$ smooth components, by \\cref{rem:bertini}.\nLet $\\Sigma_a$ as in \\cref{cor:map_to_moduli} be a transversal slice to $S_{\\ve{n}}$ at $a$. Since $S_{\\ve{n}}$ has codimension $\\delta^{\\aff}(\\ve{n})$, we have $\\dim \\Sigma_a= \\delta^{\\aff}(\\ve{n})$. Furthermore,\nby transversality, $h_{n}^{-1}(\\Sigma_a)$ is nonsingular, and we have the diagram (\\ref{equ:cartesian}). Let $U^o\\subset U$ be an open set where the universal curve \n$\\pi: \\mathcal{C}_{U^o} \\to U^o$ is smooth, and denote by $R^1$ the local system \n$$\nR^1:={R^1\\pi_* \\mathbbm{Q}}_{U^o}.\n$$\nSince the family $\\pi: \\mathcal{C}_{U} \\to U$ is versal, we have, by \\cite[Theorem 5.11]{MSV}, which we recalled in \\cref{thm:hd_cj}, that \n\\begin{equation}\\label{equ:dec_thm_vsl}\n{\\mathbbm R} \\pi^J_* \\mathbbm{Q} \\simeq \\bigoplus_{i=0}^{2d_{n}} \\IC(\\bigwedge^iR^1)[-i]. \n\\end{equation}\nBy proper base change and the isomorphism in Diagram (\\ref{equ:cartesian}) we have that\n\\begin{equation}\n{{\\mathbbm R} {h_{n}}_* \\mathbbm{Q}}_{|\\Sigma_a} \\simeq i^*({\\mathbbm R} \\pi^J_* \\mathbbm{Q} )\n\\end{equation}\nis split semisimple. Since furthermore, $\\Sigma_a $ intersect the open set $U^o$, the hypotheses of \\cref{prop:splitperverse} are met.\nIn particular\n\\begin{equation}\\label{equ:perv_coh}\n^p\\!\\!{\\mathscr H}^i({{\\mathbbm R} {h_{n}}_* \\mathbbm{Q}}_{|\\Sigma_a}) = ^p\\!\\!{\\mathscr H}^i(i^*{\\mathbbm R} \\pi^J_* \\mathbbm{Q}) = i^* \\IC(\\bigwedge^iR^1).\n\\end{equation}\nBy stratification theory it is clear that\n$S_{\\ve{n}}$ is a support for $^p\\!\\!{\\mathscr H}^i({{\\mathbbm R} {h_{n}}_* \\mathbbm{Q}})$ if and only if $a$ is a support for $^p\\!\\!{\\mathscr H}^i({{\\mathbbm R} {h_{n}}_* \\mathbbm{Q}}_{|\\Sigma_a})$.\nSince $\\dim \\Sigma_a= \\delta^{\\aff}(\\ve{n})$, by \\cref{cor:point} and \\cref{equ:perv_coh}, this happens if and only if\n\\begin{equation}\\label{equ:new_sprt}\n\\mathcal{H}^{\\delta^{\\aff}(\\ve{n})}(\\IC(\\bigwedge^iR^1))_a \\neq 0.\n\\end{equation}\nBy \\cref{lem:estimate} this is possible only if $i>\\delta^{\\aff}(\\ve{n})$. On the other hand, \nwhen this is the case, \\cref{cor:highest_weight_cohomology} tells us that \nthat \\cref{equ:new_sprt} holds.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:fin_mon}\nFor every $\\ve{n}$, and for $r=\\delta^{\\aff}(\\ve{n})$ and $r={2\\dim(\\ensuremath{\\mathcal{A}_n})- \\delta^{\\aff}(\\ve{n})}$, the pull back of the local systems \n${\\ensuremath{\\mathcal{L}}_{r, \\ve{n}}}$ to $\\mathcal{A}^{\\nod}_{\\un{n}}$ has trivial monodromy . \n\\end{corollary}\t\n\\begin{proof}\nThe irreducible components of the jacobian of the spectral curve $C_a$ are indexed by the degrees of the restriction of the line bundles to the components of $C_a$. \nTherefore, the sheaf of irreducible components of $C_a$ is constant on $\\mathcal{A}^{\\nod}_{\\un{n}}$. The local system in maximal perversity\n\\[\n({\\ensuremath{\\mathcal{L}}_{{2\\dim(\\ensuremath{\\mathcal{A}_n})- \\delta^{\\aff}(\\ve{n})}, \\ve{n}}})_a = H^{\\delta^{\\aff}(\\ve{n})-1}\\left( |\\mathscr{C}(\\Gamma_{\\ve{n}})| \\right)(-\\delta^{\\aff})\n\\otimes \\bigwedge^{h^1(\\ve{n})}H^1(\\widetilde{C_a})\n\\]\nis, by the decomposition theorem, a subsheaf of the $\\mathbbm{Q}$-linearizaton of the sheaf of irreducible components, see the argument in \\cref{prop:SuppOnlySn}, therefore its pullback to $\\mathcal{A}^{\\nod}_{\\un{n}}$\nis constant. This is also true for ${\\ensuremath{\\mathcal{L}}_{\\delta^{\\aff}(\\ve{n}), \\ve{n}}}$ which is isomorphic to it (up to a Tate twist). \n\\end{proof}\t\n\n\\begin{remark}\\label{rem:fin_mon}\n\tThe generic Galois group of the finite map \n$\\mult_{\\un{n}}: \\mathcal{A}^{\\nod}_{\\un{n}} \\longrightarrow S^{\\nod}_{\\un{n}}$\nis the subgroup of the symmetric group $S_r$ stabilizing the partition $\\ve{n}$ of $n$. Writing $\\ve{n}=1^{\\alpha_1}\\cdots n^{\\alpha_n}$, i.e. letting $\\alpha_i$ be the number of elements in $\\ve{n}$ equal to $i$,\nthis subgroup is \n\\begin{equation}\\label{subgroup}\n\\prod_i S_{\\alpha_i} \\subseteq S_r.\n\\end{equation}\nIn particular the sheaves $\\ensuremath{\\mathcal{L}}_{\\delta^{\\aff}( \\ve{n}), \\ve{n}}$ and \n\t$\\ensuremath{\\mathcal{L}}_{{2\\dim\\ensuremath{\\mathcal{A}_n} - \\delta^{\\aff}(\\ve{n})}, \\ve{n}}$ are constant if $n_i\\neq n_j$ for all $i\\neq j$.\n\\end{remark}\n\n\n\n\\section{The monodromy of the new local systems}\\label{sec:mon_loc_sys}\nIn this section we determine the monodromy of the local systems ${\\ensuremath{\\mathcal{L}}_{{2\\dim\\ensuremath{\\mathcal{A}_n} - \\delta^{\\aff}(\\ve{n})}, \\ve{n}}}$ and \n${\\ensuremath{\\mathcal{L}}_{\\delta^{\\aff}(\\ve{n}), \\ve{n}}}$. Notice that, by the relative Hard Lefschetz theorem, these local system differ only by a Tate twist, as it can also be seen directly from \\cref{thm:main1} (\\ref{item:loc_sys}).\n\nIt follows from \\cref{cor:fin_mon}\nand \\cref{rem:fin_mon} that it is enough to compute the associated representation of \nthe subgroup of \\cref{subgroup}. {In the main theorem \\cref{thm:main1} we already reduced this to the computation of the action on a combinatorially defined complex constructed from the dual graph of a nodal curve, which was a special case of the simplicial complex defined by a matroid (for terminology about matroids see \\cite{white}).}\n\nLet us denote by $\\Gamma_r$ the complete graph on $r$ vertices for $r\\geq 2$. As before we will denote by $|\\mathscr{C}(\\Gamma_r)|$ the simplicial complex defined by the cographic matroid of $\\Gamma_r$, i.e., its $k$-simplices are the subsets of $k+1$ edges of $\\Gamma_r$ that do not disconnect the graph. \n\nThe following result is a combination of well known results on matroids and a result of Stanley.\n\\begin{proposition}\\label{repsym}\n\tFor any $r \\geq 2$ the cohomology group \n$H^{\\mathrm{ top}}(|\\mathscr{C}(\\Gamma_{r})|)$, has rank $(r-1)!$, and, with its natural structure of $S_r$-module, is isomorphic to the representation induced by a primitive character of a maximal cyclic subgroup\n\\end{proposition}\nTo deduce this result let us introduce the simplical complex $\\Nspan(\\Gamma_r)$ defined by the subsets of edges that do not span $\\Gamma_r$. Let us denote by $\\Flat(\\Gamma_r)$ the lattice of partitions of $\\{1,\\dots,r\\}$ which in the language of matroids correspond to the poset of flats of the cographic matroid, because a flat in this case is a partition into complete subgraphs. To this lattice one attaches the simplicial complex $\\Delta(\\Flat(\\Gamma_r))$ whose $k-$simplices are chains of partitions $p_{disc} < p_1 < \\dots < p_k < p_{triv}$ where $p_{disc}$ is the discrete partition and $p_{triv}$ is the trivial partition. \n\nWe need the following result which is a general fact on matroids: \n\\begin{lemma}\\label{lem:cographicflat}\nLet $N={r \\choose 2}$ denote the number of edges of $\\Gamma_r$ We have natural isomorphisms\n\\begin{equation}\nH^i(|\\mathscr{C}(\\Gamma_{r})|) \\simeq H_{N-3-i}(|\\Nspan(\\Gamma_r)|) \\simeq H_{N-3-i}(|\\Delta(\\Flat(\\Gamma_r))|)\n\\end{equation}\nThe second isomorphism is $S_r$-equivariant, while in the first isomorphism the $S_r$-rep\\-re\\-sen\\-ta\\-tions differ by the sign character. \n\\end{lemma}\n\\begin{proof}\nConsider the boundary of the complex of all subsets of the edges of $\\Gamma_r$. Its geometric realization is the boundary of an $N-1$-simplex, i.e.\\ an $N-2$-dimensional sphere.\n\nThe first isomorphism is the content of Exercise 7.43 of \\cite{bjorner} and amounts to combinatorial Alexander duality, once one notices that $\\mathscr{C}(\\Gamma_{r})$ and $\\Nspan(\\Gamma_r)$ are Alexander dual complexes in $\\partial \\Delta^{N-1}$ (see \\cite{bjotan} for a quick proof if Alexander duality which is adapted to this context). We see that the isomorphism is twisted by the sign representation considering the action of $S_r$ on the top cohomology of the ambient sphere (\\cite[Theorem 2.4]{Stanley}).\n\nThe second isomorphism, due to Folkman, is (\\cite[Theorem 3.1]{folk}), using that the set of edges of $\\Gamma_r$ forms a crosscut of the partition lattice. To see that this isomorphism is $S_r$-equivariant we briefly recall Folkman's argument. \n\nNote that for any edge $e$ the subcomplex $L_e$ of $\\Delta(\\Flat(\\Gamma_r))$ formed by the simplices that are contained in a simplex that satisfies $p_1=e$ is contractible. Moreover for any non-spanning subset $I$ of edges the intersection $\\cap_{e\\in I} L_e$ is contractible to the $0$-simplex given by the partition defined by the subgraph $I$ (see \\cite[Section 3]{folk}). \n\nThus the cohomology of $\\Delta(\\Flat(\\Gamma_r))$ can be computed from the nerve of the covering given by the subcomplexes $L_e$ and this agrees with the cohomology of $|\\Nspan(\\Gamma_r)|$. \n\\end{proof}\n\\begin{proof}(of Proposition \\ref{repsym})\nApplying the previous lemma, the computation reduces to the computation of the homology of the lattice of partitions which was determined in \\cite[Theorem 7.3]{Stanley} to be the representation induced by a primitive character of a maximal cyclic subgroup tensored with the sign representation.\n\\end{proof}\n\nThe dual graph $\\Gamma$ of a spectral curve in $S_{\\ve{n}}^\\times$ contains a complete graph on the vertices, but it will have multiple edges between the vertices.\n\nLet us therefore fix some notation. Given a graph $\\Gamma$ and $I$ a subset of edges let us denote by $\\widehat{\\Gamma}_{I}$ the graph obtained by doubling the edges in $I$, i.e. for every edge $e\\in I$ we add an edge $\\widehat{e}$ connecting the same vertices as $e$. \n \n\\begin{proposition}\nLet $\\Gamma $ be a graph, let $I$ be a non-empty subset of edges. Let $ |\\mathscr{C}(\\Gamma)|$ and $|\\mathscr{C}(\\widehat{\\Gamma}_{I})|$ be the simplicial complexes associated to $\\Gamma$ and $\\widehat{\\Gamma}_{I}$. \nThen, for every $\\ell$, there is a canonical isomorphism\n\\begin{equation}\nH_{\\ell}\\left( |\\mathscr{C}(\\Gamma)| \\right)\\simeq H_{\\ell+|I|} ( |\\mathscr{C}(\\widehat{\\Gamma}_{I})| ).\n\\end{equation}\nIf a finite group $G$ acts on $\\Gamma$ preserving $I$, then these isomorphisms are $G$-equivariant.\n\\end{proposition}\n\n\\begin{proof}\nIt is a direct application of the deletion-contraction sequence: Let us first assume that $I=\\{e\\}$ consists of a single edge.\nThen the set of faces in $\\mathscr{C}(\\widehat{\\Gamma}_{I})$ is the disjoint union of the set of those which contain a doubled edge $\\widehat{e}$ and those who don't. The subcomplex of those faces not containing $\\widehat{e}$ is the simplicial complex of the graph $\\widehat{\\Gamma}_{I}\/{\\widehat{e}}$ obtained by removing $\\widehat{e}$ and collapsing the vertices joined by $\\widehat{e}$. We therefore get an exact sequence of chain complexes \n\\begin{equation}\n0 \\longrightarrow C_\\bullet(\\widehat{\\Gamma}_{I}\/\\widehat{e}) \\longrightarrow C_\\bullet(\\widehat{\\Gamma}_{I}) \\longrightarrow C_{\\bullet-1}(\\Gamma) \\longrightarrow 0.\n\\end{equation}\nNote that the edge $e$ becomes a loop in the graph $\\widehat{\\Gamma}_{I}\/{\\widehat{e}}$, hence $|\\mathscr{C}(\\widehat{\\Gamma}_{I}\/\\widehat{e})|$ is a cone and has vanishing homology.\n\nBy induction this shows that the morphism $C_\\bullet(\\widehat{\\Gamma}_{I}) \\to C_{\\bullet-|I|}(\\Gamma)$ induced by mapping those faces in $\\widehat{\\Gamma}_{I}$ that contain all of the doubled edges to its intersection with $\\Gamma$ induces an isomorphism in homology. \n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:rank_loc_syst}\nLet $\\ve{n}= n_1 \\geq n_2 \\geq \\cdots \\geq n_r=1^{\\alpha_1}\\cdots n^{\\alpha_n}$ be a partition of $n$. The rank of the local system $\\ensuremath{\\mathcal{L}}_{{\\delta^{\\aff}(\\ve{n})+i}, \\ve{n}}$ is \n\\begin{equation}\n\\mathrm{rank}\\,\\ensuremath{\\mathcal{L}}_{{\\delta^{\\aff}(\\ve{n})+i}, \\ve{n}}=\n(r-1)!\\binom{{2(\\dim\\ensuremath{\\mathcal{A}_n}-\\delta^{\\aff})}}{i}.\n\\end{equation}\nThe monodromy of the (isomorphic) local systems ${\\ensuremath{\\mathcal{L}}_{\\delta^{\\aff}(\\ve{n}), \\ve{n}}}$ and \n${\\ensuremath{\\mathcal{L}}_{{2\\dim\\ensuremath{\\mathcal{A}_n} - \\delta^{\\aff}(\\ve{n})}, \\ve{n}}}$ is given by the restriction to the subgroup \n$\\prod_i S_{\\alpha_i} \\subseteq S_r$ \nof the representation of $S_r$ induced by a primitive character of a maximal cyclic subgroup. \n\\end{corollary}\n\n\\begin{remark}\\label{n=2case}\nIf $n=2$ and $\\Gamma_2$ is the graph with two vertices joined by $2g-2$ edges, it is immediately seen that $ |\\mathscr{C}({\\Gamma }_2)|$ is a sphere of dimension $2g-4$. The corresponding representation is, for $g\\neq 2$, the sign representation. For $g=2$ we have a zero-dimensional sphere, namely two points, and the relevant representation is that on {\\em reduced} homology.\n\\end{remark}\n\\section{Appendix: The derivative of the Hitchin morphism is dual to the derivative of the action}\\label{appe}\n\nThe duality statement from the title of the section is certainly known, but we could not find a reference for it. Although we only apply the result for the group $\\textrm{GL}_n$ it turns out that the proof is most easily explained in the more general setting of Higgs bundles for reductive groups. This is because in the case of $\\textrm{GL}_n$ it is easy to loose track of implicit identifications between the Lie algebra and its dual.\n\n\\subsection{Reminder on $G$-Higgs bundles}\n\nWe keep working over ${\\mathbbm C}$ and use our fixed smooth projective curve $C$. In addition let $G$ be a connected reductive group with Lie algebra $\\mathfrak{g}=\\Lie(G)$. We will denote the dual of $\\mathfrak{g}$ by $\\mathfrak{g}^*$.\n\nGiven a $G$-torsor $\\mathcal{P}\\to C$ and a representation $\\rho\\colon G \\to V$ we will denote by $\\mathcal{P}(V):=\\mathcal{P}\\times^G V$ the associated vector bundle.\n\nA $G$-Higgs bundle on $C$ is a pair $(\\mathcal{P},\\phi)$ where $\\mathcal{P}\\to C$ is $G$-torsor and $\\phi \\in H^0(C,\\mathcal{P}(\\mathfrak{g}^*)\\otimes K_C)$ is a global section of the coadjoint bundle twisted by $K_C$. We denote by \n$$ \\Higgs_G := \\left\\langle (\\mathcal{P},\\phi) | \\mathcal{E} \\in \\Bun_G, \\phi \\in H^0(C,\\mathcal{P}(\\mathfrak{g}^*)\\otimes K_C) \\right\\rangle$$\nthe stack of $G$-Higgs bundles over $C$, which is the cotangent stack to the stack of $G$-bundles on $C$.\n\n\\begin{remark}\n\tThe above definition follows the convention of \\cite{BeilinsonDrinfeld}. In the literature on $G$-Higgs bundles it is also common to choose a $G$-invariant inner product $(,)$ on $\\mathfrak{g}$ and use it to identify $\\mathfrak{g} \\cong \\mathfrak{g}^*$. To state the results in an invariant form it seems to be most convenient to avoid this choice. As a consequence we will formulate some notions for the dual $\\mathfrak{g}^*$ that are commonly used for $\\mathfrak{g}$ for Higgs bundles, i.e., to use coadjoint orbits instead of adjoint orbits. \n\\end{remark}\n\nLet us recall from \\cite{NgoHitchin} how to view $G$-Higgs bundles as sections of a morphism of stacks.\n\n\\begin{lemma}\\label{Lem:NgoHiggsDescription} The category of Higgs bundles $(\\mathcal{P},\\phi)$ on $C$ is equivalent to the category of 2-commutative diagram diagrams\n$$\\xymatrix{\n& [\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m]\\ar[d]\\\\\nC \\ar[r]_-{K_C}\\ar[ur]^-{(\\mathcal{P},\\phi)} & B{\\mathbbm G}_m,\n}$$\nwhere $B{\\mathbbm G}_m$ is the classifying stack of line bundles, $K_C$ is the map defined by the canonical bundle on $C$ and $[\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m]$ is the quotient stack defined by the product of the coadjoint action of $G$ on $\\mathfrak{g}^*$ and the standard scaling action of ${\\mathbbm G}_m$ on the vector space $\\mathfrak{g}^*$.\n\\end{lemma}\n\\begin{proof}\nThis is not hard to unravel: By definition a $G$-torsor on $C$ is the same as a map $C \\to BG =[\\Spec{{\\mathbbm C}}\/G]$, so the pair $\\mathcal{P},K_C$ defines a map $C\\to [B (G\\times {\\mathbbm G}_m)].$ Now for any representation $\\rho \\colon G\\times {\\mathbbm G}_m \\to \\textrm{GL}(V)$ the associated bundle is the pull back of the morphism $[V\/G\\times{\\mathbbm G}_m]$ and applying this to the representation on $\\mathfrak{g}^*$ we see that $\\mathcal{P}(\\mathfrak{g}^*)\\otimes K_C = C\\times_{B(G\\times {\\mathbbm G}_m)} [\\mathfrak{g}^*\/(G\\times {\\mathbbm G}_m)]$. Therefore the datum of a section of this bundle is equivalent to a section of \n$$\\xymatrix{\n\t& [\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m]\\ar[d]\\\\\n\tC \\ar[r]_-{(\\mathcal{P},K_C)}\\ar[ur]^-{(\\mathcal{P},\\phi)} & B(G\\times {\\mathbbm G}_m).\n}$$\n\\end{proof}\n\n\\subsection{Deformations of $G$-Higgs bundles}\nAs the main aim of the section is to compare derivatives of morphisms from an to $\\Higgs_G$ we need to recall the basic results on deformations of Higgs bundles.\n\nTo a Higgs bundle $(\\mathcal{P},\\phi)$ we attach the complex of vector bundles on $C$\n\t$$\\mathcal{C}(\\mathcal{P},\\phi):=[ \\mathcal{P}(\\mathfrak{g}) \\map{\\ad^*()(\\phi)} \\mathcal{P}(\\mathfrak{g}^*) \\otimes K_C],$$\nwhere $\\ad^*\\colon \\mathfrak{g} \\times \\mathfrak{g}^* \\to \\mathfrak{g}^*$ denotes the coadjoint action of $\\mathfrak{g}$ on $\\mathfrak{g}^*$.\n\n\\begin{lemma}[\\cite{Nitsure}]\nThe tangent space of the deformation functor of $G$-Higgs bundles at $(\\mathcal{P},\\phi)\\in \\Higgs_G$ is given by $H^1(C,\\mathcal{C}(\\mathcal{P},\\phi))$ and automorphisms of deformations that extend the identity of $(\\mathcal{P},\\phi)$ are given by $H^0(C,\\mathcal{C}(\\mathcal{P},\\phi))$.\n\\end{lemma}\n\\begin{proof}\n\tThe deformation theory argument for the computation of the tangent space to $\\Higgs_G$ can be found in \\cite{Nitsure}. In the language of \\cref{Lem:NgoHiggsDescription} we have a cartesian diagram:\t\n\t$$\\xymatrix{\n\t\t[\\mathfrak{g}^* \\otimes K_C\/G]\\ar[d]^{p_{K_C}}\\ar[r] & [\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m] \\ar[d]^p\\\\\n\t\tC \\ar[r]^-{K_C} & B{\\mathbbm G}_m =[\\Spec k\/{\\mathbbm G}_m]\n\t}$$ \n\tand Higgs bundles are sections of the map $p_{K_C}$. \n\t\n\tNow the tangent stack to any quotient stack $[X\/G]$ can be described as the quotient of the complex of $G$-vector bundles $\\Lie(G) \\times X \\to TX$ on $X$, which we think of a complex in degree $[-1,0]$. \n\t\n\tTherefore the tangent complex to the stack $[\\mathfrak{g}^*\/G]$ (which lives in degree $[-1,0]$) is given by the $G$-equivariant complex \n\t$$[\\mathfrak{g} \\map{\\ad^*} \\mathfrak{g}^*]$$\n\ton $\\mathfrak{g}^*$ and thus the tangent complex to $p_{K_C}$ over $\\mathfrak{g}^*\\otimes K_C$ is given by $$[\\mathfrak{g}\\otimes \\mathcal{O}_C \\to \\mathfrak{g}^* \\otimes K_C].$$ \n\t\n\tDeformations of $(\\mathcal{P},\\phi)$ are deformations of the corresponding section $(\\mathcal{P},\\phi) \\colon C\\to [\\mathfrak{g}^* \\otimes K_C\/G]$ and the pull back of the tangent complex at this section is $$[\\mathcal{P}(\\mathfrak{g}) \\map{\\ad^*()(\\phi)} \\mathcal{P}(\\mathfrak{g}^*) \\otimes K_C].$$ \n\\end{proof}\t\t\n\n\\begin{remark}\n\tFor any Higgs bundle $(\\mathcal{P},\\phi)$ the complex $\\mathcal{C}(\\mathcal{P},\\phi)$ is self-dual with respect to the duality defined by $\\cHom(\\,\\cdot\\, , K_C[1])$.\n\tTherefore Serre-duality induces pairings \n\t$$H^i(C,\\mathcal{C}(\\mathcal{P},\\phi)) \\times H^{2-i}(C,\\mathcal{C}(\\mathcal{P},\\phi)) \\to {\\mathbbm C}$$\n\tthat for $i=1$ define the standard 2-form $\\omega_{\\Higgs}$ on $\\Higgs_G=T^*\\Bun$.\n\\end{remark}\n\n\n\\subsection{The Hitchin morphism}\n\nThe Hitchin morphism for $G$-Higgs bundles is defined as follows. Denote by $\\chi$ the quotient map $$\\chi\\colon \\mathfrak{g}^* \\to \\mathfrak{g}^*\/\\!\/G =\\car^*,$$ where $\\car^*=\\Spec (\\Sym^\\bullet \\mathfrak{g})^G$. \n\n\\begin{remark}\n\tAs usual, a choice of homogeneous invariant polynomials would give an isomorphism $\\car^* \\cong \\Spec k[f_1,\\dots,f_r] \\cong {\\mathbbm A}^r$ identifying $\\car^*$ with an affine space. The map $\\chi$ is equivariant with respect to the ${\\mathbbm G}_m$ action on $\\mathfrak{g}^*$ and the induced action on $\\Spec (\\Sym^\\bullet \\mathfrak{g})^G$, whose weights are given by the degrees of the invariant polynomials $f_i$.\n\\end{remark}\n\nWe denote by $\\car^*_{K_C}=(\\mathfrak{g}^*\\times K_C\/\\!\/G) \\to C$ the corresponding affine bundle and by \n$$\\mathcal{A}_G := H^0 (C, \\car^*_{K_C})$$\nthe base of the Hitchin morphism. Again, any choice of invariant polynomials for $G$ defines an isomorphism $\\mathcal{A}_G \\cong \\oplus_i H^0(C,K_C^i)$, but it will be more convenient to avoid such a choice.\n\nThe map $\\overline{\\chi}\\colon[\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m] \\to [\\car^*\/{\\mathbbm G}_m]$ then induces a map\n$$h_G\\colon \\Higgs_G \\to \\mathcal{A}= H^0(C,\\car^*_{K_C}),$$\nwhich is often denoted as $h_G(\\mathcal{P},\\phi)=:\\chi(\\phi)$.\n\n\\subsection{The regular centralizer}\n\nTo define the Hitchin morphism and the analog of the action of the Jacobian of the spectral curve we now recall the construction of the regular centralizer groups from \\cite{NgoHitchin}. \n\nLet us fix the standard notations. The group $G$ acts on $\\mathfrak{g}$ via the adjoint action, which we will denote by $\\Ad\\colon G \\to \\textrm{GL}(\\mathfrak{g})$ the derivative of this action is denoted $\\ad\\colon \\mathfrak{g} \\to \\End(\\mathfrak{g})$. Similarly $\\Ad^*\\colon G \\to \\textrm{GL}(\\mathfrak{g}^*)$ denotes the dual action given by $\\Ad^*(g)(\\phi)(\\underline{\\quad}):=\\phi(\\Ad(g)^{-1}.\\underline{\\quad})$, so that its derivative is $ad^*(X)=-\\ad(X)^t$. \n\nFor an element $\\varphi\\in \\mathfrak{g}^*$ we denote its centralizer in $G$ by \n$C(\\varphi):=\\{ g\\in G | \\Ad^*(G)(\\varphi)=\\varphi\\}$ and by $\\mathfrak{g}^{\\varphi}:=\\{A\\in \\mathfrak{g} | \\ad^*(A)(\\varphi)=0\\}$ its Lie algebra. The groups $C_G(\\varphi)$ define a group scheme \n$$C_{\\mathfrak{g}^*} :=\\{(g,\\varphi)\\in G\\times \\mathfrak{g}^*| \\Ad^*(g)(\\varphi)=\\varphi\\} \\to \\mathfrak{g}^*$$ over $\\mathfrak{g}^*$. The set of regular elements $\\mathfrak{g}^{*,\\reg}\\subset \\mathfrak{g}^*$ is defined to be the subset of those elements for which $\\dim C_G(\\varphi)=\\rank(G)$ is minimal.\n\nThe restriction $C_{\\mathfrak{g}{*,reg}}$ of $C_{\\mathfrak{g}^*}$ to the space of regular elements descends to a group scheme $J_{\\car^*}$ on $\\car^*=\\mathfrak{g}^*\/\\!\/ G$, called the regular centralizer. The group scheme $J_{\\car^*}$ comes equipped with a natural map \n$$m\\colon \\chi^*J_{\\car^*} \\to C_{\\mathfrak{g}^*} \\subset G\\times \\mathfrak{g}^*$$ \nwhich is defined to be the unique regular map extending the natural isomorphism $\\chi^*J_{\\car^*}|_{\\mathfrak{g}^{*,\\reg}} \\cong C_{\\mathfrak{g}^{*,reg}}$. We denote by $dm$ the induced map on Lie algebras $$dm \\colon \\chi^*Lie(J_{\\car^*}) \\to \\Lie(C_{\\mathfrak{g}^*} )\\to \\mathfrak{g} \\times \\mathfrak{g}^*.$$\n\n\\begin{notation}\n\tAs in \\cite{NgoHitchin} we will need to keep track of the action of the multiplicative group ${\\mathbbm G}_m$ on our objects. We will denote by ${\\mathbbm C}(n)$ the one dimensional vector space with the ${\\mathbbm G}_m$ action given by the $n$-th power of the standard action. For any vector bundle $E$ with a ${\\mathbbm G}_m$-action we will denote by $E(n):=E\\otimes {\\mathbbm C}(n)$.\n\\end{notation}\n\n\\begin{remark}\\label{rem:dm}\n\tOn $\\mathfrak{g}^*$ the group ${\\mathbbm G}_m$ acts by scalar multiplication which induces an action on $\\car^*=\\mathfrak{g}^*\/\\!\/G$. The action on $\\mathfrak{g}^*$ also preserves centralizers and thus induces an action on $C_{G,\\mathcal{G}}$, given by $$t.(g,\\varphi):=(g,t\\varphi).$$ In particular this action preserves $\\mathfrak{g}^{*,\\reg}$ and thus $C|_{\\mathfrak{g}^{*,\\reg}}$ even descends to a group $\\overline{J}$ over $[\\car^*\/{\\mathbbm G}_m]$.\n\t\n\tNote that the formula for the ${\\mathbbm G}_m$ action shows that the derivative\n\t\\begin{equation*}\ndm\\colon \\chi^*Lie(J_{\\car^*}) \\to \\mathfrak{g} \\times \\mathfrak{g}^*\n\t\\end{equation*} is equivariant for the ${\\mathbbm G}_m$--action that on $\\mathfrak{g} \\times \\mathfrak{g}^*$ is given by the trivial action on the first factor $\\mathfrak{g}$ and the standard action on the second factor $\\mathfrak{g}^*$. Therefore, identifying $\\mathfrak{g}(-1) \\times \\mathfrak{g}^* \\cong T^*\\mathfrak{g}^*$ we can interpret $dm$ as a morphism\n\t\\begin{equation}\\label{eq:dm}\n\tdm\\colon \\chi^*Lie(J_{\\car^*})(-1) \\to T^*\\mathfrak{g}^* \n\t\\end{equation}\n\tThe restriction of this map to $\\mathfrak{g}^{*,\\reg}$ is injective, as $m$ was injective over $\\mathfrak{g}^{*,\\reg}$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:dchi}\t\n\tThe map $\\chi\\colon \\mathfrak{g}^{*} \\to \\car^*$ is by definition $G$-invariant and equivariant with respect to the ${\\mathbbm G}_m$ action, therefore its derivative\n\t\\begin{equation}\\label{eq:dchi}\n\td\\chi \\colon \\mathfrak{g}^* \\times \\mathfrak{g}^* = T\\mathfrak{g}^* \\to \\chi^* T\\car^*\n\t\\end{equation}\n\tis also equivariant with respect to the induced ${\\mathbbm G}_m$ action and the restriction \n\t$$d\\chi|_{\\mathfrak{g}^{*,\\reg}} \\colon T \\mathfrak{g}^{*,\\reg} = \\mathfrak{g}^* \\times \\mathfrak{g}^{*,\\reg} \\to \\chi^* T\\car^*|_{\\mathfrak{g}^{*,\\reg}}$$ is surjective, because the map $\\chi\\colon \\mathfrak{g}^{*} \\to \\car^*$ admits a section $\\kappa\\colon \\car^* \\to \\mathfrak{g}^{*,\\reg}\\subset \\mathfrak{g}^*$ called the Kostant section.\t\t\n\\end{remark}\n\nThe following observation is the group theoretic origin of the duality result for the Hitchin fibration.\n\\begin{lemma}\\label{lem:LocalPairing}\n\tThe canonical pairing $$\\langle \\,, \\, \\rangle \\colon T\\mathfrak{g}^* \\times_{\\mathfrak{g}^*} T^*\\mathfrak{g}^* \\to {\\mathbbm C}$$\n\tinduces a $G\\times{\\mathbbm G}_m$-equivariant perfect pairing \n $$\\chi^*Lie(J_{\\car^*})|_{\\mathfrak{g}^{*,\\reg}}(-1) \\times_{\\mathfrak{g}^*} \\chi^* T\\car^*|_{\\mathfrak{g}^{*,\\reg}} \\to {\\mathbbm C}(0)$$\n and thereby an isomorphism\n $$\\Lie(J)^*(1) \\cong T\\car^*.$$ \n\\end{lemma}\n\\begin{proof}\n\tFrom s \\cref{rem:dm},\\cref{rem:dm} we know that $\\chi^*Lie(J_{\\car^*})|_{\\mathfrak{g}^{*,\\reg}}(-1)$ is a subbundle of $T^*\\mathfrak{g}^{*,\\reg}$ and $\\chi^* T\\car^*|_{\\mathfrak{g}^{*,\\reg}}$ is a quotient of $T\\mathfrak{g}^{*,\\reg}$ and both have the same dimension.\n\t\n\tAs the map $\\chi$ is constant on $G$-orbits, the tangent space to a $G$ orbit is in the kernel of $d\\chi$, i.e., for every $\\varphi\\in \\mathfrak{g}^*$\n\t$$ V_\\varphi:=\\mathrm{Im}\\,( \\mathfrak{g} \\map{\\ad^*({\\quad})(\\varphi)} T_\\varphi\\mathfrak{g}^* =\\mathfrak{g}^* )\\subset \\ker(d\\chi).$$\n\tIf $\\varphi\\in\\mathfrak{g}^{*,\\reg}$ is regular we have $\\dim V_\\varphi = \\dim \\mathfrak{g}\/\\mathfrak{g}^\\varphi = \\dim \\mathfrak{g} - \\dim \\car^*$. As $d\\chi$ is surjective in this case we find $V_\\varphi=\\ker(d\\chi)$ for $\\varphi\\in \\mathfrak{g}^{*,\\reg}$. \n\t\n\tNow $G$-invariance of the pairing $\\langle \\,, \\, \\rangle$ i.e., $\\langle g.\\varphi,g.A\\rangle=\\langle\\varphi,A\\rangle$ for all $g\\in G,\\varphi\\in \\mathfrak{g}^*,A\\in\\mathfrak{g}$ implies that for all $X\\in\\mathfrak{g}$ we have\n\t$$\\langle \\ad^*(X)(\\varphi),A\\rangle=\\langle\\varphi,-\\ad(X)(A)\\rangle=-\\langle\\ad^*(A)(\\varphi),X\\rangle.$$ \n\tThis implies that $V_\\varphi^\\perp=\\mathfrak{g}^\\varphi$ and this implies our claim. \n\\end{proof}\t\n\n\n\n\\begin{remark}\\label{LocalComputationForGLn}\n\tFor $G=\\textrm{GL}_n$ the above can be rephrased in terms of coordinates. In this case $\\car^*\\cong {\\mathbbm A}^n$ is the space of characteristic polynomials of matrices. In order to compute the differential $d\\chi$ of the map $\\chi \\colon \\mathfrak{gl}_n \\to \\car^*$ it is convenient to choose the coordinates $\\chi(\\varphi):=(\\frac{1}{i}\\Trace(\\varphi^i))_{i=1\\dots n}$. Then $d\\chi_\\varphi\\colon \\mathfrak{gl}_n \\to k^n$ is given by $X \\mapsto (\\Trace(\\varphi^{i-1}X))_{i=1 \\dots n}$.\n\t\n\tThe regular centralizer group scheme can also be described explicitly: For any monic polynomial $p(t)\\in k[t]$ we define $J_p:=(k[t]\/p(t))^*$ as the unit group of the algebra $k[t]\/p(t)$, which defines an $n$-dimensional commutative group scheme $J$ over ${\\mathbbm A}^n$. As a matrix $\\varphi$ is regular if and only if its characteristic polynomial $p_\\varphi$ is its minimal polynomial, we see that the assignment $J_{p_\\varphi} \\to \\textrm{GL}_n$ given by $f(t) \\mapsto f(\\varphi)$ is injective for regular matrices $\\varphi$ and therefore identifies $J_{p_\\varphi}$ with the centralizer of $\\varphi$. By definition of the regular centralizer the map $\\chi^*(J)\\to I \\subset \\textrm{GL}_n \\times \\mathfrak{gl}_n$ is given by the unique extension of the canonical map on $\\mathfrak{gl}^{\\reg}$. As the formula $f(t)\\mapsto f(\\varphi)$ is well defined for all $\\varphi$ it gives this extension.\n\t\n\tWe also observe that $s \\in {\\mathbbm G}_m$ acts on $\\car^*$ by $p\\mapsto s.p$, where $s.p$ is the polynomial given by multiplying the coefficient of $t^{n-i}$ by $s^i$. This lifts to an action $J_p \\to J_{s.p}$, given by $t\\mapsto st$ and this is compatible with the above map $f(t) \\mapsto f(\\varphi)$. \n\t\n\tNote that $\\Lie(J_p)\\cong k[t]\/(p(t))$ (as $1+\\epsilon f(t)$ is an invertible element of $k[\\epsilon,t]\/(\\epsilon^2,p(t))$ for all $f$). Finally the standard basis $1,t,\\dots t^{n-1}$ of $k[t]\/p(t)$ defines an isomorphism $\\Lie(J) \\cong k^n \\times \\car$.\n\t\n\tThus for any $\\varphi$ the map $dm \\colon k^n \\cong \\Lie(J_p) \\to \\mathfrak{gl}_n$ is given by $(a_i)\\mapsto \\sum_{i=0}^{n-1} a_i \\varphi^i$. \n\t\n\tFinally, we use the pairing $(A,B):=\\Trace(AB)$ on $\\mathfrak{gl}_n$. With respect to this form the dual of the map $k^n \\cong \\Lie(J_p) \\to \\mathfrak{gl}_n$ is therefore given by $$X \\mapsto (\\Trace(\\varphi^iX))_{i=0\\dots n-1}$$ which is $d\\chi_\\varphi$.\t\n\\end{remark}\n\n\\begin{remark}\\label{rem:LocalVersionOfDuality}\n\tWe can reformulate the above Lemma as a duality statement on $[\\mathfrak{g}^*\/G]$: As for any quotient stack, the tangent stack to this quotient is defined by the complex \n\t$$ [\\mathfrak{g} \\times \\mathfrak{g}^* \\map{(A,\\phi) \\mapsto (\\ad^*(A)(\\phi),\\phi)} T\\mathfrak{g}^* = \\mathfrak{g}^* \\times \\mathfrak{g}^*],$$\n\ti.e., the quotient stack of these bundles is the pull-back of the tangent stack to $\\mathfrak{g}^*$. This complex is self-dual up to a shift by $1$.\n\t\n\tConsidering $\\chi$ as a morphism $\\overline{\\chi}\\colon [\\mathfrak{g}^*\/G] \\to \\car^*$ the differential becomes the morphism \n\t$$ [\\mathfrak{g} \\times \\mathfrak{g}^* \\map{(A,\\phi) \\mapsto (\\ad^*(A)(\\phi),\\phi)} \\mathfrak{g}^* \\times \\mathfrak{g}^*] \\map{(0,d\\chi)} [ 0 \\to T\\car^*].$$\n\t\n\tSimilarly as the morphism $\\chi^*J \\to I$ is $G$-equivariant (because $J$ was defined by descending $I|_{\\mathfrak{g}^{*,\\reg}}$), the morphism $dm$ defines a $G$-equivariant morphism $$[\\chi^*\\Lie(J) \\to 0] \\map{(dm,0)} [\\mathfrak{g} \\times \\mathfrak{g}^* \\map{(A,\\phi) \\mapsto (\\ad^*(A)(\\phi),\\phi)} \\mathfrak{g}^* \\times \\mathfrak{g}^*].$$\n\t\n\t \\cref{lem:LocalPairing} says, that these morphisms are ${\\mathbbm G}_m$-equivariantly dual to each up to a shift by $1$ of the complex and twisting the action by $(1)$.\n\\end{remark}\n\\subsection{The regular centralizer}\n\nLet us recall the global version of the regular centralizer as explained in \\cite[Section 4]{NgoHitchin}: We saw that the regular centralizer defines a group scheme $\\overline{J}$ on $[\\car^*\/{\\mathbbm G}_m]$, that we can pull back to a group scheme $J_{\\car^*_{K_C}}$ on $\\car^*_{K_C}=C \\times_{B{\\mathbbm G}_m} [\\car^*\/{\\mathbbm G}_m]$ which we pull back via the tautological map $\\mathcal{A}_G\\times C\\to \\car^*_{K_C}$ to define a group scheme $J_{\\mathcal{A}_G}$ on $\\mathcal{A}_G\\times C$. \n\nSimilarly, the pull back of the sheaf of centralizers $\\overline{C}_{\\mathfrak{g}^*}$ on $[\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m]$ under the classifying map $\\Higgs_G \\times C \\to [\\mathfrak{g}^*\/G\\times {\\mathbbm G}_m]$ is denoted $C_{\\Higgs\\times C}$. By construction $C_{\\Higgs\\times C}=\\Aut(\\mathcal{E}_{\\textrm{\\tiny univ}},\\phi_{\\textrm{\\tiny univ}})$ is identified with the group of $G$-automorphisms of the universal Higgs bundle $(\\mathcal{E}_{\\textrm{\\tiny univ}},\\phi_{\\textrm{\\tiny univ}})$ that preserve $\\phi_{\\textrm{\\tiny univ}}$.\n\nThe map $\\chi^*J \\to C_{\\mathfrak{g}^*}$ therefore induces a natural morphism $$\\iota\\colon(h\\times id_C)^*J_{\\mathcal{A}} \\to \\Aut(\\mathcal{E}_{\\textrm{\\tiny univ}},\\phi_{\\textrm{\\tiny univ}}).$$ \n\nOver $\\mathcal{A}_G$ one defines the group scheme $P_{\\mathcal{A}_G}$ of $J_{\\mathcal{A}_G}$-torsors on $C$, i.e., at a point $a\\colon C \\to \\car^*_{\\Omega}$ is given by the torsors of the group scheme of $a^*J_{\\mathcal{A}_G}$ on $C$. Then $\\iota$ induces an action $\\act\\colon P_{\\mathcal{A}_G}\\times_{\\mathcal{A}_G} \\Higgs_G \\to \\Higgs_G$.\n\n\n\\subsection{The duality statement}\nWe can now formulate the main result of this section:\n\\begin{proposition}\\label{prop:dactdhdual}\n\tThere exists a canonical isomorphism $\\Lie(P_{\\mathcal{A}_G}\/\\mathcal{A}_G) \\cong T^*\\mathcal{A}_G$ such that the morphisms \n\t$$d\\act\\colon h^*\\Lie (P_{\\mathcal{A}_G}\/\\mathcal{A}_G) \\to T\\Higgs_G$$\n\tand $$dh\\colon T\\Higgs_G \\to h^*(T\\mathcal{A}_G)$$\n\tbecome dual to each other with respect to the symplectic form $\\omega_{\\Higgs}$.\n\\end{proposition}\n\n\\begin{proof}\n\tThe result follows from the local statement \\cref{lem:LocalPairing} as follows:\n\tThe regular centralizers $P_a$ are defined to be $a^*J_{\\mathcal{A}}$-torsors, so $\\Lie(P_a)=H^1(C,\\Lie(a^*J_{\\mathcal{A}_G}))$. \n\t\n\tFor any Higgs bundle $(\\mathcal{E},\\phi)$ the action of $\\act_{(\\mathcal{E},\\phi)}\\colon \\Lie(P_a) \\to h^{-1}(a)$ is induced from $\\iota\\colon(h\\times id_C)^*J_{\\mathcal{A}_G} \\to C_{(\\mathcal{E},\\phi)}\\subset \\Aut(\\mathcal{E}\/C)$. \n\t\n\tTherefore applying $\\Lie$ we find that the differential of the action is induced from the morphism of complexes \n\t$$ [\\Lie(a^*J_{\\mathcal{A}}) \\to 0 ] \\map{(d\\act,0)} [ \\ad(\\mathcal{E}) \\map{\\ad^*()(\\phi)} \\ad(\\mathcal{E})^* \\otimes \\Omega]$$\n\tafter passing to $H^1$.\n\t\n\n\n\n\n\t\n\tBy \\cref{lem:LocalPairing} we know that this map of complexes is up to tensoring with $K_C[-1]$ this map is dual to the map\n\t$$[ \\ad(\\mathcal{E}) \\map{\\ad^*()(\\phi)} \\ad(\\mathcal{E})^* \\otimes K_C] \\map{(0,d\\chi)} [0 \\to T\\car^*_{K_C}]$$\n\tthat induces $dh$ by \\cref{rem:LocalVersionOfDuality}. Therefore applying Serre-duality to $H^1$ of the above complexes we obtain the proposition. \n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nChange in the topological structure of the ground state, driven by disorders, has been intensively investigated recently and it is thought to be responsible for potential novel criticality which involves interplay of topological structures, disorders, and interactions \\cite{Ruy12}. Such a change of topological structure is reflected in magneto-electrical transport phenomena such as anomalous Hall and spin Hall effects \\cite{Nagaosa10}, negative longitudinal magnetoresistance \\cite{Nielsen83,HJKim13}, chiral magnetic effect \\cite{Fuku08}, and so on. Even if topology of the ground state is trivial, its anomalous geometric (local) structure described by the Berry curvature or determined by spin chirality can be also affected by disorders \\cite{Xiao10}, showing an interesting variation of magneto-electrical transport, for example such as a crossover from weak anti-localization to weak localization driven by randomness. BiTeI may be an appropriate platform to investigate the interplay between Berry phase and disorder, originating from the unique electronic structure with broken inversion symmetry.\n\nInversion symmetry breaking in BiTeI splits a single degenerate band near the hexagonal face center of the Brillouin zone, referred to as the A point, into an inner band with a left-handed or ``positive'' spin-chiral configuration and an outer one with a right-handed or ``negative'', whose spin structures are intimately locked with momentum\\cite{Ish11,Cre12,Lan12}. As a result, low energy physics of this inversion-symmetry-broken material is governed by two distinct Fermi surfaces when the Fermi energy $E_F$ lies near the band-touching point generated by the Rashba spin-orbit interaction. See Fig. 1(a). Dynamics of electrons on the inner Fermi surface (IFS) is described by the Weyl equation, exhibiting the change of the Fermi surface character from electron-like to hole-like across the Weyl point. Indeed, a nontrivial Berry phase of $\\pi$ has been detected for both the IFS and outer Fermi surface (OFS) in the Shubnikov-de Haas measurements \\cite{Mura13,Park}. Thus, this system is expected to show physics of a Weyl\/Dirac metal \\cite{Balents11} with interesting response to disorder \\cite{Hosur12,Bjorn14,Roy14}. \n\nUp to now, however, this important point in BiTeI has been overlooked. Most electrical transport studies have focused on measurements at high magnetic fields to detect Shubnikov-de Haas or quantum oscillations \\cite{Mura13,Park,Bell13,Martin13}. Probably, this is because Shubnikov-de Haas or quantum oscillations are considered to be few experimental techniques to provide essential information about nontrivial Berry phase in this system \\cite{Mura13,Park}. Another direction of research in BiTeI, in connection with its nontrivial topology is to induce a topological quantum phase transition and a topological insulator under pressure, first proposed by Nagaosa and his colleagues \\cite{Bah12}. Indeed, closing of the energy gap and some indirect signatures of the topological quantum phase transition were observed experimentally \\cite{Xi13,Tran14}, but the nature of this topological critical point and of a topological insulator under pressure are still elusive, particularly in the experimental point of view. \n\nIn this paper, we investigate the interplay between the Berry phase and randomness in magneto-electrical transport properties of BiTeI. By analyzing Hall and magneto resistivity of Fermi-energy-tuned BiTeI single crystals, particularly at low magnetic fields, we reveal extreme disparity of the mobility values between IFS and OFS near the Weyl point, where the IFS mobility becomes colossally enhanced, intimately related with anti-localization in electrical transport. Based on the self-consistent Born approximation, we explain this disparity and ``divergent'' IFS mobility near the Weyl point quantitatively. We identify this fixed-point solution for BiTeI as a diffusive helical Fermi liquid, characterized by a pair of concentric spin-chiral Fermi surfaces with negligible inter-valley scattering. Our theoretical analysis indicates the existence of a crossover in the ``topological'' structure or geometric phase toward a conventional diffusive Fermi liquid when the stronger-disorder-enhanced inter-valley scattering destroys the spin-chiral property. However, we realize that this mean-field theory for disorders fails to describe the universal scaling in Hall resistivity, which is another main experiment result. We speculate that this failure in the self-consistent Born analysis implies the existence of mass renormalization of the IFS near the Weyl point, possibly resulting from enhanced interactions between electrons near the Weyl point.\n\nMain experimental observations made on six Fermi-energy-tuned BiTeI single crystals are (1) an anomalous weak-field feature in Hall resistivity $\\rho_H(B)$, (2) unconventional magnetic field $B$ dependence of magnetoresistance (MR), which is in stark contrast with the usual $B$-quadratic MR in a metal, and (3) a universal scaling of Hall resistivity. The first experimental result is analyzed and understood based on a picture that two types of charge carriers exist in BiTeI: one with small mobility and the other with very large mobility. \nIndeed, we find that the overall negative slop in $\\rho_H$($B$) is determined by electrons on the OFS\nHowever, we also observe the deviation of $\\rho_H$ from the linear dependence at the low $B$ region.\nWe assign it a contribution from the Weyl fermions in the IFS. \n\nThe second result about MR is also consistent with the existence of two kinds of charge carriers in that the total electrical conductivity \n$\\sigma_{total}$ in a magnetic field is decomposed by two channels of conduction given by $\\sigma_{total} =\\sigma_{OFS}+\\sigma_{WF}$,\n where $\\sigma_{OFS}$ and $\\sigma_{WF}$ are the conductivity contributions of OFS and IFS, respectively. One can rewrite $\\sigma_{total}$ into $\\sigma_{total} = \\sigma_c +\\Delta\\sigma^N_{out} +\\Delta\\sigma^N_{in} +\\Delta\\sigma_{WAL}$ with $\\Delta\\sigma_{WAL} \\propto \\sqrt{B}$, where $\\sigma_c$, $\\Delta\\sigma^N_{out(in)}$, and $\\Delta\\sigma_{WAL}$ are the field-independent conductivity, the conductivity contribution of OFS (IFS), and the weak anti-localization correction in three dimensions (3D), respectively. The explicit form of each component will be given later. One important outcome in this MR analysis is the confirmation of the 3D weak anti-localization contribution in $\\sigma_{total}(B)$.\n The analysis of $\\rho_H(B)$ and $\\sigma_{total}$ enables us to extract separately the mobility values of the charge carriers in the OFS and IFS for all six samples. We plot the mobility values as a function of $E_F$ in Fig. 1(b). This data shows extreme disparity of the mobility values between IFS and OFS and ``divergent'' IFS mobility near the Weyl point. The detailed procedure how the IFS and OFS mobility values \n are obtained will be presented in subsequent sections. \n Here, we emphasize that mobility disparity and ``divergent'' IFS mobility near the Weyl point are determined not by the scattering time but by the transport time. As the transport time is a scattering time weighed more by backward scattering processes, chiral nature is more reflected in the transport time.\n \n\nThe rest of the paper is organized as follows. In Sec. II, we discuss the sample synthesis, magneto-electrical transport experiments, and analysis of the data in detail. In this section, we introduce two-carrier models for $\\rho_H(B)$ and $\\sigma_{total}(B)$, which is necessary to \nquantitatively explain the low-field features observed in $\\rho_H(B)$ and $\\sigma_{total}(B)$. As an outcome of the analysis, we determine the OFS and IFS mobility values of six BiTeI single crystals with different $E_F$ [Fig. 1(b)]. In Sec. III, we calculate the IFS and OFS mobility values based on the Rashba model within the self-consistent Born approximation. Here we consider two different cases: one considers only the intra-valley forward scattering and the other includes both intra- and inter-valley scattering. It is revealed that the experimental IFS and OFS mobility values are quite well reproduced within this model in the absence of the inter-valley scattering or in the weak inter-valley scattering. Besides, we predict how the ground state of BiTeI changes as the disorder increases by using renormalization group (RG) arguments. According to these, \nthe inter-valley scattering smears out the spin-chirality with increasing disorder, leading to a topological crossover or a weak version of topological phase transition driven by disorder. This also accompanies the change of quantum correction in electrical transport from weak antilocalization to weak localization. \nWithin this picture, the BiTeI single crystals which we investigate are in a weakly disordered region with negligible inter-valley scattering called diffusive helical Fermi liquid. In Sec. IV, we discuss implication of the experimental results based on the theory introduced in Sec. III. In fact, we find the existence of universal scaling in Hall resistivity from the experimental results. \nThis scaling, however, is not reproduced within the self-consistent Born approximation. This necessitates mass renormalization in the IFS beyond the independent electron picture, especially near the Weyl point. We conclude in Sec. V with a brief discussion of our main results. \n\n\n\n\n\\section{Experiment}\n\\subsection{Sample synthesis}\nSingle crystals of BiTeI were grown by a modified Bridgman method. We prepared more than 20 samples \nand tried to vary the carrier density $n$ by adding a small amount of extra Bi. \nAs the amount of additionally inserted Bi is quite small, X-ray diffraction measurements do not detect any change of structure\n in the doped samples.\nWe selected six single crystals ($\\#$1 $-$ $\\#$6). \nAll the as-grown single crystals were degenerate semiconductors, exhibiting a metallic behavior. \nCarrier densities $n$ were determined by the linear part of the Hall resistivity. \nTheir signs are all negative, implying that dominant charge carriers are electrons, which presumably determined by the OFS. \nCarrier densities from the linear part were determined to be 0.10, 0.30, 0.35, 0.80, 3.9, and \n6.4$\\times10^{20}$ cm$^{-3}$ for $\\#$1 $-$ $\\#$6, respectively. \nEstimated from the linear part of $\\rho_H$, the Fermi energies from the bottom of the conduction band are 40, 90, 100, 170, 550, and 760 meV for all six samples. As the Weyl point is located at 113 meV from the bottom of the conduction band \\cite{Bahramy11}, the former three (the latter three) should have positive (negative) charge carriers in the IFS. Later we will show that this, in fact is consistent with the sign change in the deviation of $\\rho_H$ from the linear dependence.\nTemperature dependence of the resistivity $\\rho(T)$ are presented in Fig. 2, \nshowing the overall decrease of the $\\rho(T)$ curves with the increase of $n$. \nThis behavior confirms that our samples are in the region of a typical degenerate semiconductor.\nSpecifying the distribution of the excess Bi and volatile I in the BiTeI samples can provide an important clue about the nature of disorder in this system,\n especially in connection with the results obtained in the present \ntransport experiments. Even though we verified that our single crystals are homogeneous and uniform on macroscopic scale, probably because of small amount of excess Bi, it is considered that nanoscale inhomogeneity can still exist. What type of local disorders or defects can promote intra- or inter-valley scattering in BiTeI is a very important question which should be addressed in future studies. The six samples investigated in the present study are considered to be in weakly disordered region based on our final results, which suggests that effects of disorder are nearly equal at least in those samples for electrical transport. \n\n\n\\subsection{Analysis of magnetoresistivity and Hall resistivity}\nThe transverse MR = $(\\rho(B) - \\rho(0))\/\\rho(0)$ with $\\rho(B)$ and $\\rho(0)$, resistivity at $B$ and $B = 0$, respectively, \nand the Hall resistivity $\\rho_H(B)$ for $\\#1 - \\#6$ are measured at 4.2 K and up to $B$ = 4 T. \nWhile the magnitude of MR is only few percent even at $B$ = 4 T for all samples, $\\#1 - \\#5$ show weak field anomalies, which deviate from the conventional $B$ quadratic behavior significantly, except for $\\#6$ with the largest $n$, as presented in Fig. 3(a). In particular, the MR for $\\#1$ \npossesses a pronounced dip in the weak field region. Even beyond the region of the dip, MR does not recover the $B$ quadratic behavior. \nThe sample $\\#1 - \\#5$ exhibit essentially same features. \n\n\nHall resistivity curves are almost linear with negative slops, suggesting the existence of ``normal'' negative charge carriers. \nHowever, a more careful inspection for the low-field region reveals tiny weak-field anomalies, displayed in Fig. 3(b), \nwhich plots the deviation $\\Delta \\rho_H$, where the overall linear dependence is subtracted from Hall resistvity $\\rho_H(B)$. \nThis data indicates that Hall resistivity curves deviate from the linearity significantly in the field region where a corresponding dip in MR is observed. While the deviations in $\\#2 - \\#5$ are confined \nfor $-1$ T $<$ $B < 1$ T, they extend to $-4$ T $< B < 4$ T for $\\#1$ and $\\#6$. The shape of $\\Delta \\rho_H$ in Fig. 3(b)\n is reminiscent of the general case for the Hall resistivity with two types of charge carriers \\cite{Wang14}. \n In the limit that one mobility is much larger than the other, the formula at the low field region is simplified into\n\\begin{equation}\n\\rho_H \\approx \\frac{1}{n_1ec}\\frac{B}{1+(\\mu_1B)^2}+\\frac{B}{n_2ec},\n\\end{equation}\nwhere $n_1$ and $n_2$ are carrier densities with larger and smaller mobility, \nand $\\mu_1$ is the larger mobility. If this simple expression explains the origin of the weak-field anomaly well, \nit suggests existence of a charge carrier with extremely high mobility, whose value corresponds to \nthe maximum or minimum of $\\Delta \\rho_H$. Indeed, the first term of Eq. (1) fits the $\\Delta \\rho_H(B)$ data quite well, giving the mobility\nvalue of the high mobility carrier as shown in Fig. 4.\nWe also observe that the sign of $\\Delta \\rho_H$ is positive for $\\#1 - \\#3$ \nand negative for $\\#4 - \\#6$, respectively, which implies that the charge carrier with extremely high mobility \nis hole-type for $\\#1 - \\#3$ and electron-type for $\\#4 - \\#6$.\n\nConsidering the band structure of BiTeI near $E_F$, \nwe assign the charge carrier with extremely high and the other one to be the Weyl fermion in the IFS and the OFS electron, respectively. \nWhile the OFS mobility is estimated from the linear part of the Hall resistivity and the residual resistivity, the mobility of the Weyl fermions can be obtained from the fitting of $\\Delta\\rho_H$ to the first term of Eq. (1). Our analysis based on Eq. (1) turns out to explain $\\Delta\\rho_H$ in a quantitative level. The mobility values of the Weyl fermion and the OFS carrier are plotted for the six samples in Fig. 1(b) as a function of $E_F$.\n\nIt is appealing that the simple formula of Eq. (1) for the Hall resistivity explains the low-field region quite well. However, one might speculate that there must be anomalous Hall effect either intrinsic (Berry curvature) or extrinsic (side jump or skew scattering) \\cite{Nagaosa10} because there are Weyl fermions in BiTeI. Although we cannot rule out the appearance of the extrinsic anomalous Hall effect, we strongly believe that the anomalous Hall effect induced by Berry curvature does not exist. The intrinsic anomalous Hall conductivity can be classified into two contributions, one of which results from the contribution of all states below the Fermi energy, given by the distance of momentum space between a pair of Weyl points \\cite{Chen14,Goswami13}, and the other of which originates from the contribution of Fermi surfaces with Berry phase. The second is non-universal \\cite{Haldane04,Xiao10,Nagaosa10}. Since a pair of Weyl points exists at the same momentum point, the first contribution vanishes. On the other hand, the second contribution from IFS and OFS may still exist, giving rise to an offset near the zero-field region. However, both contributions from the OFS and IFS will be cancelled because the sum of their Berry phases vanishes.\n\n\nAs in Hall resistivity, we also assume the existence of two conductivity channels. \nThen, the total contribution of electrical conductivity in BiTeI is given by $\\sigma_{total}=\\sigma_{OFS}+\\sigma_{WF}$, \nwhere $\\sigma_{OFS}=\\frac{\\sigma_{out}+\\Delta\\sigma^{out}_{WAL}}{\\sigma_{out}^{-2}(\\sigma_{out}+\\Delta\\sigma^{out}_{WAL})^2+\\omega^2_{out}\\tau^2_{out}}$\nis the conductivity from the OFS and $\\sigma_{WF}=\\frac{\\sigma_{in}+\\Delta\\sigma^{in}_{WAL}}{\\sigma_{in}^{-2}(\\sigma_{in}+\\Delta\\sigma^{in}_{WAL})^2+\\omega^2_{in}\\tau^2_{in}}$ is that from Weyl fermions of the IFS. These expressions can be derived from the Boltzmann-equation approach, where the role of the Berry phase is introduced into the Boltzmann equation via the weak anti-localization correction phenomenologically \\cite{Ki14}.\n$\\sigma_{in(out)}$ and $\\Delta\\sigma_{WAL}^{in(out)}$ are the residual conductivity at zero magnetic field and the weak anti-localization correction, respectively. $\\omega_{in(out)}$ is the cyclotron frequency, and $\\tau_{in(out)}$ is the transport time. \nEmploying $\\Delta\\sigma_{WAL}^{in(out)}=a_{in(out)}\\sqrt{B}$ in three dimensions, we are allowed to assume $\\sigma_{in(out)} >> \\Delta\\sigma_{WAL}^{in(out)}$ in the weak-field region. Then, these equations become simplified as follows, \n$\\sigma_{OFS} \\approx \\frac{\\sigma_{out}+\\Delta\\sigma^{out}_{WAL}}{1+\\omega^2_{out}\\tau^2_{out}} \\approx \\sigma_{out}^N+\\Delta\\sigma^{out}_{WAL}$ and\n$\\sigma_{WF} \\approx \\frac{\\sigma_{in}+\\Delta\\sigma^{in}_{WAL}}{1+\\omega^2_{in}\\tau^2_{in}} \\approx \\sigma_{in}^N+\\Delta\\sigma^{in}_{WAL}$, respectively, where $\\sigma_{out}^N = (\\rho_{out}+A_{out}B^2)^{-1} \\approx \\rho_{out}^{-1}+\\rho_{out}^{-2}A_{out}B^2$ \nwith $\\rho_{out} >> A_{out}B^2$ and $\\sigma_{in}^N = (\\rho_{in}+A_{in}B^2)^{-1}$ . \nThe total magneto-electrical conductivity is finally written as $\\sigma_{total} = \\sigma_c + \\Delta\\sigma_{out}^N + \\Delta\\sigma_{in}^N + \\Delta\\sigma_{WAL}$ with $\\Delta\\sigma_{WAL} = \\Delta\\sigma_{WAL}^{out}+\\Delta\\sigma_{WAL}^{in}$,\n where all field-independent constants are expressed as $\\sigma_c$.\n \nThe Fig. 5(a) show the decomposition of the magneto-electrical conductivity, $\\Delta\\sigma = \\sigma_{total} - \\sigma_c$ for the sample $\\#1$. \nThe sample $\\#6$ with the highest $n$ is described only with $\\Delta\\sigma_{out}^{N}$ presumably because $E_F$ is far away from the Weyl point. On the other hand, for other samples, all the other terms are necessary to describe the magneto-electrical conductivity properly. Performing successful decompositions, we isolate the weak anti-localization correction in Fig. 5(b), where all samples except for $\\#6$ exhibit the scaling behavior with $\\sqrt{b}$ dependence, where $b$ is a dimensionless reduced magnetic field given by $b = \\hbar \\omega\/E_F$, where \n$\\omega$ is the cyclotron frequency. The existence of $\\Delta\\sigma_{WAL}$ for $\\#1 - \\#5$ \njustifies the validity of our data analysis. \n\nOur analysis on the magneto-electrical conductivity demonstrates existence of two types of charge carriers, one of which has an extremely high mobility, identified with Weyl fermions on the IFS, given by $\\mu^2_{WF} = A_{in}\/\\rho_{in}$. Fig. 1(b) \ndisplays the mobility as a function of the Fermi energy $E_F$, \nwhere the value of $\\mu_{out} = \\sqrt{A_{out}\/\\rho_{out}}$ [(black) open circles] is in the order of $0.01 \\sim 0.03$ m$^2$\/Vs \nwhile that of $\\mu_{WF}$ [(red) open squares] is two or three orders of magnitude larger than $\\mu_{out}$. In particular, $\\mu_{WF}$ touches its maximum when $E_F$ is closest to the Weyl point. The enhancement of $\\mu_{WF}$ , compared to $\\mu_{out}$, is partially a consequence of a reduced phase space available for the scattering in the IFS, which is an intrinsic property of the Weyl metal as derived in the following theoretical sections. It is noted that the mobility values deduced from $\\Delta\\rho_H$ are very similar to those from the MR analysis. \n\n\n\n\\section{Calculation of mobility values within the Rashba model}\n\\subsection{Effective model Hamiltonian} \n\nWe start from the Rashba model with potential randomness:\n\\begin{eqnarray*}\nS[\\bar{\\Psi}_{i\\alpha}(\\tau,\\bm{x}),\\Psi_{i\\alpha}(\\tau,\\bm{x})]&=&\\frac{1}{2}\\int^{\\beta}_{0}d\\tau\\int d^{d}\\bm{x}\\biggr[\\bar{\\Psi}_{i\\alpha}(\\tau,\\bm{x})\\left(\\partial_{\\tau}-\\frac{\\hbar^{2}\\nabla^{2}}{2m}-E_{F}\\right)\\Psi_{i\\alpha}(\\tau,\\bm{x})\\\\\n&&+\\bar{\\Psi}_{i\\alpha}(\\tau,\\bm{x})\\lambda_{R}\\bm{\\sigma}^{\\textrm{spin}}_{\\alpha\\beta}\\cdot\\left(\\bm{E}\\times(-\\imath\\nabla)\\right)\\Psi_{i\\beta}(\\tau,\\bm{x})+\\bar{\\Psi}_{i\\alpha}(\\tau,\\bm{x})V(\\bm{x})\\Psi_{i\\alpha}(\\tau,\\bm{x})\\biggl] ,\n\\end{eqnarray*}\nwhere $\\lambda_{R}$ is the Rashba coupling constant and $V(\\bm{x})$ is a random potential. The indices of $``i\"$ and $``\\alpha\"$ stand for time-reversal and spin component, repectively. ``Time-reversal symmetrized\" basis is given by\n\\begin{eqnarray*}\n\\Psi(\\tau)=\\begin{pmatrix}\\psi(\\tau)\\\\\\imath\\sigma^{\\textrm{spin}}_{y}\\psi^{*}(-\\tau)\\end{pmatrix}=\\begin{pmatrix}\\psi_{\\uparrow}(\\tau)\\\\\\psi_{\\downarrow}(\\tau)\\\\\\psi^{*}_{\\downarrow}(-\\tau)\\\\-\\psi^{*}_{\\uparrow}(-\\tau)\\end{pmatrix} ~ \\& ~~ \\bar{\\Psi}(\\tau)=\\Psi^{\\dagger}(\\tau)I^{\\textrm{spin}}\\otimes\\sigma^{\\textrm{tr}}_{z}=\\begin{pmatrix}\\psi^{*}_{\\uparrow}(\\tau)\\\\\\psi^{*}_{\\downarrow}(\\tau)\\\\-\\psi_{\\downarrow}(-\\tau)\\\\\\psi_{\\uparrow}(-\\tau)\\end{pmatrix}^{T}.\n\\end{eqnarray*}\n\nTaking into account the BiTeI band structure given by $\\bm{E}=E\\hat{z} ~ \\left(\\alpha_{R}=\\lambda_{R}E\\right)$ and moving on the momentum and frequency space, we obtain\n\\fontsize{10}{10}\n\\begin{eqnarray*}\n&&S[\\bar{\\Psi}_{i\\alpha A(n)}(\\bm{k}),\\Psi_{i\\alpha A(n)}(\\bm{k})]\\\\\n&=&\\frac{1}{2}\\sum_{n}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\biggr[\\bar{\\Psi}_{i\\alpha A(n)}(\\bm{k})\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)\\Psi_{i\\alpha A(n)}(\\bm{k})+\\bar{\\Psi}_{i\\alpha n}(\\bm{k})\\alpha_{R}\\left(k_{x}(\\sigma_{y})_{\\alpha\\beta}-k_{y}(\\sigma_{x})_{\\alpha\\beta}\\right)\\Psi_{j\\beta A(n)}(\\bm{k})\\\\\n&&+\\int\\frac{d^{d}\\bm{q}}{(2\\pi)^{d}}\\bar{\\Psi}_{i\\alpha A(n)}(\\bm{k}+\\bm{q})V(\\bm{q})\\Psi_{i\\alpha A(n)}(\\bm{k})\\biggl] ,\n\\end{eqnarray*}\n\\normalsize\nwhere $A$ stands for ``retarded\" ($\\mathcal{R}$) or ``advanced\" ($\\mathcal{A}$). For example, $\\mathcal{A}(n)$ corresponds to a negative frequency whose magnitude is $\\left |\\omega_{n}\\right |$. Diagonalizing this effective Rashba Hamiltonian based on the following momentum-dependent unitary matrix \n\\fontsize{11}{11}\n\\begin{eqnarray*}\nU^{\\dagger}(\\bm{k})I^{\\textrm{spin}}U(\\bm{k})=I^{\\textrm{spin}} ~ \\& ~ U(\\bm{k})\\left(k_{x}\\sigma^{\\textrm{spin}}_{y}-k_{y}\\sigma^{\\textrm{spin}}_{x}\\right)U^{\\dagger}(\\bm{k})=\\sigma^{\\textrm{spin}}_{z}~\\Rightarrow~U(\\bm{k})=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} e^{\\imath\\frac{\\phi(\\bm{k})}{2}}&-\\imath e^{-\\imath\\frac{\\phi(\\bm{k})}{2}}\\\\ e^{\\imath\\frac{\\phi(\\bm{k})}{2}}&\\imath e^{-\\imath\\frac{\\phi(\\bm{k})}{2}}\\end{pmatrix} ,\n\\end{eqnarray*}\n\\normalsize\nwe obtain \n\\begin{eqnarray*}\n&&S[\\bar{\\Phi}_{i\\alpha n}(\\bm{k}),\\Phi_{i\\alpha n}(\\bm{k})]\\\\\n&=&\\frac{1}{2}\\sum_{n}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\biggr[\\bar{\\Phi}_{i\\alpha n}(\\bm{k})\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)\\Phi_{i\\alpha n}(\\bm{k})+\\bar{\\Phi}_{i\\alpha n}(\\bm{k})\\alpha_{R}(\\sigma^{\\textrm{spin}}_{z})_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\Phi_{j\\beta n}(\\bm{k})\\\\\n&&~~~+\\int\\frac{d^{d}\\bm{q}}{(2\\pi)^{d}}\\bar{\\Phi}_{i\\alpha n}(\\bm{k}+\\bm{q})U_{\\alpha\\beta}(\\bm{k}+\\bm{q})V(\\bm{q})U^{\\dagger}_{\\beta\\gamma}(\\bm{k})\\Phi_{i\\gamma n}(\\bm{k})\\biggl]\n\\end{eqnarray*}\nwhere $\\Phi_{i\\alpha n}(\\bm{k})=U_{\\alpha\\beta}(\\bm{k})\\Psi_{i\\beta n}(\\bm{k})$ is an eigenfunction field and the index of $\\alpha$ represents spin chirality, identified with ``$+$\" or ``$-$\". \n\nPerforming the disorder average within the replica trick, the effective replicated Rashba action becomes\n\\fontsize{9.4}{9.4}\n\\begin{eqnarray*}\n&&S[\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k}),\\Phi_{i\\alpha n}^{a}(\\bm{k})]\\\\\n&=&\\frac{1}{2}\\sum_{n}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\biggr[\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k})\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)\\Phi^{a}_{i\\alpha n}(\\bm{k})+\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k})\\alpha_{R}\\left(\\sigma_{z}^{\\textrm{spin}}\\right)_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\Phi^{a}_{i\\beta n}(\\bm{k})\\biggl]\\\\\n&&-\\sum_{nm}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\int\\frac{d^{d}\\bm{k'}}{(2\\pi)^{d}}\\int\\frac{d^{d}\\bm{q}}{(2\\pi)^{d}}\\frac{\\Gamma}{8}\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k}+\\bm{q})M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k})\\Phi^{a}_{i\\alpha' n}(\\bm{k})\\bar{\\Phi}^{b}_{j\\beta m}(\\bm{k}'-\\bm{q})M_{\\beta\\beta'}(\\bm{k}'-\\bm{q},\\bm{k}')\\Phi^{b}_{j\\beta'm}(\\bm{k}') ,\n\\end{eqnarray*}\n\\normalsize\nwhere the free energy is given by $\\mathcal{F}=-T\\lim_{R \\to 0}\\frac{1}{R}\\left(\\int\\mathcal{D}(\\bar{\\Phi}^{a}_{i\\alpha},\\Phi^{a}_{i\\alpha})e^{-S[\\bar{\\Phi}^{a}_{i\\alpha},\\Phi^{a}_{i\\alpha}]}-1\\right)$. The product of unitary matrices is \n\\fontsize{10}{10}\n\\begin{eqnarray*}\nM(\\bm{k}+\\bm{q},\\bm{k})\\equiv U(\\bm{k}+\\bm{q})U^{\\dagger}(\\bm{k})= \\begin{pmatrix}\\cos{\\left(\\frac{\\phi(\\bm{k}+\\bm{q})-\\phi(\\bm{k})}{2}\\right)}&\\imath\\sin{\\left(\\frac{\\phi(\\bm{k}+\\bm{q})-\\phi(\\bm{k})}{2}\\right)}\\\\\\imath\\sin{\\left(\\frac{\\phi(\\bm{k}+\\bm{q})-\\phi(\\bm{k})}{2}\\right)}&\\cos{\\left(\\frac{\\phi(\\bm{k}+\\bm{q})-\\phi(\\bm{k})}{2}\\right)}\\end{pmatrix}.\n\\end{eqnarray*}\n\\normalsize\n\n\\subsection{A self-consistent Born approximation}\n\nWe perform the Hubbard-Stratonovich transformation in the particle-hole singlet channel of $\\Phi^{a}_{i\\alpha n}\\bar{\\Phi}^{b}_{j\\beta m}$, and obtain\n\\fontsize{10.5}{10.5}\n\\begin{eqnarray*}\n&&S[\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k}),\\Phi^{a}_{i\\alpha n}(\\bm{k});Q^{ab}_{ij;\\alpha\\beta;nm}(\\bm{q})]\\\\\n&=&\\frac{1}{2}\\sum_{n}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\left[\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k})\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)\\Phi^{a}_{i\\alpha n}(\\bm{k})+\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k})\\alpha_{R}\\left(\\sigma^{\\textrm{spin}}_{z}\\right)_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\Phi^{a}_{i\\beta n}(\\bm{k})\\right]\\\\\n&&+\\sum_{nm}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\int\\frac{d^{d}\\bm{q}}{(2\\pi)^{d}}\\biggr[-\\frac{\\imath}{2}\\bar{\\Phi}^{a}_{i\\alpha n}(\\bm{k}+\\bm{q})M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k})Q^{ab}_{ij;\\alpha'\\beta';nm}(\\bm{q})M_{\\beta'\\beta}(\\bm{k}-\\bm{q},\\bm{k})\\Phi^{b}_{j\\beta m}(\\bm{k})\\\\\n&&+\\frac{1}{2\\Gamma}M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k})Q^{ab}_{ij;\\alpha'\\beta;nm}(\\bm{q})M_{\\beta\\beta'}(\\bm{k}-\\bm{q},\\bm{k})Q^{ba}_{ji;\\beta'\\alpha;mn}(-\\bm{q})\\biggl].\n\\end{eqnarray*}\n\\normalsize\nIntegrating over fermionic degrees of freedom, we obtain \n\\fontsize{10.5}{10.5}\n\\begin{eqnarray*}\n&&S[Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(\\bm{q})]\\\\\n&=&-\\frac{1}{2}\\textrm{tr}\\textrm{ln}\\biggr[\\delta^{ab}\\delta_{AB}\\left\\{\\delta_{ij}\\delta_{\\alpha\\beta}\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)+\\alpha_{R}(\\sigma^{\\textrm{spin}}_{z})_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\right\\}\\\\\n&&-\\imath M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k})Q^{ab}_{ij;\\alpha'\\beta';A(n)B(m)}(\\bm{q})M_{\\beta'\\beta}(\\bm{k}-\\bm{q},\\bm{k})\\biggl]\\\\\n&&+\\sum_{nm}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\int\\frac{d^{d}\\bm{q}}{(2\\pi)^{d}}\\biggr[\\frac{1}{2\\Gamma}M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k}) Q^{ab}_{ij;\\alpha'\\beta;A(n)B(m)}(\\bm{q})M_{\\beta\\beta'}(\\bm{k}-\\bm{q},\\bm{k})Q^{ba}_{ji;\\beta'\\alpha;B(m)A(n)}(-\\bm{q})\\biggl].\n\\end{eqnarray*}\n\\normalsize\nMinimizing this effective free energy with respect to the matrix field $Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(\\bm{q})$, we obtain the saddle-point equation given by\n\\fontsize{10.5}{10.5}\n\\begin{eqnarray*}\n\\frac{2\\imath}{\\Gamma}Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(\\bm{q})=\\textrm{tr}\\left[{G^{ab}_{ij;\\alpha\\beta;A(n)B(m)}}^{-1}(\\bm{k})-\\imath M_{\\alpha\\alpha'}(\\bm{k}+\\bm{q},\\bm{k})Q^{ab}_{ij;\\alpha'\\beta';A(n)B(m)}(\\bm{q})M_{\\beta'\\beta}(\\bm{k}-\\bm{q},\\bm{k})\\right]^{-1} ,\n\\end{eqnarray*}\n\\normalsize\nwhere\n\\begin{eqnarray*}\n\\left[G^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(\\bm{k})\\right]^{-1}=\\delta^{ab}\\delta_{ij}\\delta_{AB} \\left\\{\\delta_{\\alpha\\beta}\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)+\\alpha_{R}(\\sigma^{\\textrm{spin}}_{z})_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\right\\} \n\\end{eqnarray*}\nis the inverse of the fermion Green's function.\n\n\\subsubsection{Saddle-point analysis I}\n\nFocusing on the forward scattering described by $Q_{MF}=Q(\\bm{0})$,\nwe obtain mean-field equations of\n\\begin{eqnarray*}\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{++}}&=&\\frac{G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}}{\\left(G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\right)\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)+Q_{\\scriptscriptstyle{+-}}Q_{\\scriptscriptstyle{-+}}}\\\\\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{+-}}&=&\\frac{\\imath Q_{\\scriptscriptstyle{+-}}}{\\left(G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\right)\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)+Q_{\\scriptscriptstyle{+-}}Q_{\\scriptscriptstyle{-+}}}\\\\\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{-+}}&=&\\frac{\\imath Q_{\\scriptscriptstyle{-+}}}{\\left(G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\right)\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)+Q_{\\scriptscriptstyle{+-}}Q_{\\scriptscriptstyle{-+}}}\\\\\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{--}}&=&\\frac{G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}}{\\left(G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\right)\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)+Q_{\\scriptscriptstyle{+-}}Q_{\\scriptscriptstyle{-+}}},\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\nG=\\begin{pmatrix} G_{\\scriptscriptstyle{++}}&0\\\\0&G_{\\scriptscriptstyle{--}}\\end{pmatrix} ~ \\& ~ Q=\\begin{pmatrix} Q_{\\scriptscriptstyle{++}}&Q_{\\scriptscriptstyle{+-}}\\\\Q_{\\scriptscriptstyle{-+}}&Q_{\\scriptscriptstyle{--}}\\end{pmatrix} \n\\end{eqnarray*}\nand the forward scattering doesn't change spin orientations, resulting in $M(\\bm{k},\\bm{k})=I$. It is straightforward to see $Q_{\\scriptscriptstyle{+-}}=0$ due to spin chirality. \nThen, we reach the following expression\n\\fontsize{10.5}{10.5}\n\\begin{eqnarray*}\n\\frac{2\\imath}{\\Gamma}Q^{ab}_{ij;\\pm\\pm;A(n)A(n)}(\\bm{0})=\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\frac{1}{\\delta^{ab}\\delta_{AB}\\left\\{\\delta_{ij}\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)\\pm\\alpha_{R}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\right\\}-\\imath Q^{ab}_{ij;\\pm\\pm;A(n)A(n)}(\\bm{0})}.\n\\end{eqnarray*}\n\\normalsize\n\nIn order to solve the above equation, we introduce the mean-field ansatz of\n\\begin{eqnarray*}\n(Q_{\\textrm{MF}})^{ab}_{ij;\\alpha\\beta;AB}=\\frac{\\pi}{2}N_{F}\\Gamma\\delta^{ab}\\delta_{ij}\\delta_{\\alpha\\beta}F_{\\alpha}(r)\\Lambda_{AB} ,\n\\end{eqnarray*}\nwhere $N_{F}=mk_{F}\/2\\pi^{2}\\hbar^{2}$ is the density of the states (without the factor $2$ of spin-degeneracy) at the Fermi level with $\\alpha_{R}=0$ and $\\Lambda_{AB}=\\textrm{diag}(1,-1)$ is the diagonal matrix for the retarded and advanced sectors. $F_{\\alpha}(r)$ is a function of $r=\\frac{2\\hbar^{2}E_{F}}{m\\alpha_{R}^{2}}$, regarded as an order parameter to be determined from this self-consistent equation. Considering $\\alpha=\\beta=+$ and $A=B=\\mathcal{R}$, we obtain\n\\begin{eqnarray*}\n\\imath \\pi N_{F}F_{+} = \\int\\frac{d^{3}\\bm{k}}{(2\\pi)^{3}}\\frac{1}{-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}+\\alpha_{R}\\sqrt{k_{x}^{2}+k_{y}^{2}}-\\frac{\\imath\\pi N_{F}\\Gamma F_{+}}{2}}.\n\\end{eqnarray*}\nSince this integrand is not rotationally invariant along the $z-$direction, we need to be cautious for the $k_{z}$ integration. We will not show this procedure. Performing the momentum integration, we find\n\n\\begin{eqnarray*}\n&& F_{+}(r)\n= \\frac{\\pi}{2}\\frac{1}{\\sqrt{r}\\sqrt{1+r}}\\biggr[\\frac{8}{3}+2r-\\frac{4}{3}\\sqrt{r+1}\\biggr\\{\\left(r+2\\right)\\mathcal{E}\\left(\\frac{1}{r+1}\\right)-r\\mathcal{K}\\left(\\frac{1}{r+1}\\right)\\biggl\\}\\biggl],\n\\end{eqnarray*}\nwhere $\\mathcal{K}(x)$ and $\\mathcal{E}(x)$ are complete elliptic integrals of the first kind and the second kind \\cite{Mathematica}. In the same way, we find \n\\begin{eqnarray*}\n&& F_{-}(r)= \\frac{\\pi}{2}\\frac{1}{\\sqrt{r}\\sqrt{1+r}}\\biggr[\\frac{8}{3}+2r+\\frac{4}{3}\\sqrt{r+1}\\biggr\\{\\left(r+2\\right)\\mathcal{E}\\left(\\frac{1}{r+1}\\right)-r\\mathcal{K}\\left(\\frac{1}{r+1}\\right)\\biggl\\}\\biggl],\n\\end{eqnarray*}\nwhere the sign in front of the elliptic integrals has been changed. As a result, two kinds of scattering times are given by\n\\begin{eqnarray*}\n\\tau_{+}=\\frac{1}{2Q_{+}(r)}=\\frac{1}{\\pi N_{F}\\Gamma F_{+}(r)} ~ \\& ~ \\tau_{-}=\\frac{1}{2Q_{-}(r)}=\\frac{1}{\\pi N_{F}\\Gamma F_{-}(r)}\n\\end{eqnarray*}\nfor inner and outer Fermi surfaces, respectively. Considering that the scattering time is expressed as $\\tau=\\frac{1}{\\pi N_{F}\\Gamma}$ for the normal diffusive Fermi-liquid state, one may regard that the appearance of the additional factor of $F_{\\pm}(r)$ results from the presence of the Rashba spin-orbit coupling, modifying the density of states for inner and our Fermi surfaces, respectively. Finally, we obtain diffusion constants, given by\n\\begin{eqnarray*}\nD_{\\pm}=\\hbar v_{F}^{2}\\tau_{\\pm} = \\frac{2\\pi\\alpha_{R}\\hbar^{3}}{m^{2}\\Gamma}\\frac{(1+r)}{\\sqrt{r}F_{\\pm}(r)}.\n\\end{eqnarray*}\n\nAlthough we did not show the integration procedure in a detail, these diffusion coefficients are justified only when $r \\geq 0$. Since there are no density of states for the inner Fermi surface (FIG. \\ref{FS_3d}), we divide this case when the Fermi energy is below the Weyl point from the other. As a result, we find the general formula valid for both cases of $r \\geq 0$ and $r < 0$, given by\n\\fontsize{8.5}{8.5}\n\\begin{eqnarray*}\nD_{\\pm}(r)=\\frac{4\\alpha_{R}\\hbar^{3}}{m^{2}\\Gamma}\\frac{(1+r)^{\\frac{3}{2}}}{\\textrm{Re}\\left[\\frac{8}{3}+2r-\\frac{4}{3}\\sqrt{r+1}\\left\\{(r+2)\\mathcal{E}\\left(\\frac{1}{r+1}\\right)-r\\mathcal{K}\\left(\\frac{1}{r+1}\\right)\\right\\}\\right]-\\Theta(-r)\\left(\\frac{8}{3}-\\frac{8}{3}\\sqrt{1-\\left | r\\right |}+\\frac{2}{3}\\left | r\\right |\\sqrt{1-\\left | r\\right |}-2\\left | r\\right |\\right)} .\n\\end{eqnarray*}\n\\normalsize\nIn order to compare our analytic expressions with the experimental data, we need to obtain the mobility. Resorting to the Einstein's relation $D=\\mu k_{B}T\/e$, we have\n\\fontsize{4.7}{2}\n\\begin{eqnarray*}\n\\mu_{\\pm}(E_{F})=A\\frac{(1+bE_{F})^{\\frac{3}{2}}}{\\textrm{Re}\\left[\\frac{8}{3}+2bE_{F}-\\frac{4}{3}\\sqrt{bE_{F}+1}\\left\\{(bE_{F}+2)\\mathcal{E}\\left(\\frac{1}{bE_{F}+1}\\right)-bE_{F}\\mathcal{K}\\left(\\frac{1}{bE_{F}+1}\\right)\\right\\}\\right]-\\Theta(-bE_{F})\\left(\\frac{8}{3}-\\frac{8}{3}\\sqrt{1-\\left | bE_{F}\\right |}+\\frac{2}{3}\\left | bE_{F}\\right |\\sqrt{1-\\left | bE_{F}\\right |}-2\\left | bE_{F}\\right |\\right)} ,\n\\end{eqnarray*}\n\\normalsize\nwhere $A=\\frac{2\\alpha_{R}\\hbar^{3}}{m^{2}\\Gamma ek_{B}T}$ and $b=\\frac{2\\hbar^{2}}{m\\alpha_{R}^{2}}$. In the experiment, the Weyl point was observed at 113 meV from the bottom of the conduction band mimnimum. In our model, we set $E_{W}=0$ and the conduction band minimum is $-\\frac{m\\alpha_{R}^{2}}{2\\hbar^{2}}$, so $b=\\frac{2\\hbar^{2}}{m\\alpha_{R}^{2}}=\\frac{1}{0.113eV}\\simeq$ 8.85 $(eV)^{-1}$. Based on the formula with this value, we fit the experimental data and obtain the result of FIG. \\ref{diffusion_constant_3d}, where $A=$ 0.984 $[m^{2}\/Vs]$. \n\n\n\\subsubsection{Saddle-point analysis II}\n\nPreviously, we did not take into account effects of inter-valley scattering. Taking $\\bm{q}=-2\\bm{k}-\\bm{a}$ where $\\bm{a}=\\frac{2m\\alpha_{R}}{\\hbar^{2}}\\frac{k_{x}\\hat{x}+k_{y}\\hat{y}}{\\sqrt{k_{x}^{2}+k_{y}^{2}}}$, the effective Rashba action becomes\n\\fontsize{10}{10}\n\\begin{eqnarray*}\n&&S[Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(-2\\bm{k}-\\bm{a})]\\\\\n&=&-\\frac{1}{2}\\textrm{tr}\\textrm{ln}\\biggr[\\delta^{ab}\\delta_{AB}\\left\\{\\delta_{ij}\\delta_{\\alpha\\beta}\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)+\\alpha_{R}(\\sigma^{\\textrm{spin}}_{z})_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\right\\}\\\\\n&&-\\imath M_{\\alpha\\alpha'}(-\\bm{k}-\\bm{a},-\\bm{k})Q^{ab}_{ij;\\alpha'\\beta';A(n)B(m)}(-2\\bm{k}-\\bm{a})M_{\\beta'\\beta}(\\bm{k}+\\bm{a},\\bm{k})\\biggl]\\\\\n&&+\\sum_{nm}\\int\\frac{d^{d}\\bm{k}}{(2\\pi)^{d}}\\frac{1}{2\\Gamma}\\biggr[M_{\\alpha\\alpha'}(-\\bm{k}-\\bm{a},-\\bm{k})Q^{ab}_{ij;\\alpha'\\beta;A(n)B(m)}(-2\\bm{k}-\\bm{a})M_{\\beta\\beta'}(\\bm{k}+\\bm{a},\\bm{k}) Q^{ba}_{ji;\\beta'\\alpha;B(m)A(n)}(2\\bm{k}+\\bm{a})\\biggl].\n\\end{eqnarray*}\n\\normalsize\nSince $\\bm{k}+\\bm{a}$ and $\\bm{k}$ are in the same direction on the $xy-$plane, we still have $M(-\\bm{k}-\\bm{a},-\\bm{k})=M(\\bm{k}+\\bm{a},\\bm{k})=I$. Then, the above expression is simplified as follows\n\\fontsize{10.5}{10.5}\n\\begin{eqnarray*}\n&&S[Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(-2\\bm{k}-\\bm{a})]\\\\\n&=&-\\frac{1}{2}\\textrm{tr}\\textrm{ln}\\biggr[\\delta^{ab}\\delta_{AB}\\left\\{\\delta_{ij}\\delta_{\\alpha\\beta}\\left(-\\imath\\omega_{n}+\\frac{\\hbar^{2}\\bm{k}^{2}}{2m}-E_{F}\\right)+\\alpha_{R}(\\sigma^{\\textrm{spin}}_{z})_{\\alpha\\beta}\\sqrt{k_{x}^{2}+k_{y}^{2}}\\right\\}-\\imath Q^{ab}_{ij;\\alpha\\beta;nm}(-2\\bm{k}-\\bm{a})\\biggl]\\\\\n&&+\\sum_{nm}\\frac{1}{2\\Gamma}\\biggr[Q^{ab}_{ij;\\alpha\\beta;A(n)B(m)}(-2\\bm{k}-\\bm{a})Q^{ba}_{ji;\\beta\\alpha;B(m)A(n)}(2\\bm{k}+\\bm{a})\\biggl].\n\\end{eqnarray*}\n\\normalsize\nUnfortunately, this effective action is not diagonal in the presence of such a $Q(2\\bm{k}+\\bm{a})$ matrix. We can resolve this difficulty, choosing a better basis as\n\\fontsize{9.5}{9.5}\n\\begin{eqnarray*}\n&&\\bar{\\phi}(\\bm{k})\\left[G^{-1}(\\bm{k})\\right]\\phi(\\bm{k})+\\bar{\\phi}(-\\bm{k}-\\bm{a})\\left[G^{-1}(-\\bm{k}-\\bm{a})\\right]\\phi(-\\bm{k}-\\bm{a})+\\bar{\\phi}(-\\bm{k}-\\bm{a})\\left[-\\imath Q(-2\\bm{k}-\\bm{a})\\right]\\phi(\\bm{k})\\\\\n&&+\\bar{\\phi}(\\bm{k})\\left[-\\imath Q(2\\bm{k}+\\bm{a})\\right]\\phi(-\\bm{k}-\\bm{a})\\\\\n&=&\\begin{pmatrix}\\bar{\\phi}(\\bm{k}),&\\bar{\\phi}(-\\bm{k}-\\bm{a})\\end{pmatrix}\\begin{pmatrix} G^{-1}(\\bm{k})&-\\imath Q(\\bm{2\\bm{k}+\\bm{a}})\\\\-\\imath Q(-2\\bm{k}-\\bm{a})&G^{-1}(-\\bm{k}-\\bm{a})\\end{pmatrix}\\begin{pmatrix}\\phi(\\bm{k})\\\\\\phi(-\\bm{k}-\\bm{a})\\end{pmatrix}\\\\\n&=&\\begin{pmatrix}\\bar{\\phi}_{+}(\\bm{k})\\\\\\bar{\\phi}_{-}(\\bm{k})\\\\\\bar{\\phi}_{+}(-\\bm{k}-\\bm{a})\\\\\\bar{\\phi}_{-}(-\\bm{k}-\\bm{a})\\end{pmatrix}^{T}\\begin{pmatrix} G^{-1}_{\\scriptscriptstyle{++}}(\\bm{k})&0&0&-\\imath Q_{\\scriptscriptstyle{+-}}(\\bm{2\\bm{k}+\\bm{a}})\\\\0&G^{-1}_{\\scriptscriptstyle{--}}(\\bm{k})&-\\imath Q_{\\scriptscriptstyle{-+}}(2\\bm{k}+\\bm{a})&0\\\\0&-\\imath Q_{\\scriptscriptstyle{+-}}(-2\\bm{k}-\\bm{a})&G^{-1}_{\\scriptscriptstyle{++}}(-\\bm{k}-\\bm{a})&0\\\\-\\imath Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})&0&0&G^{-1}_{\\scriptscriptstyle{--}}(-\\bm{k}-\\bm{a})\\end{pmatrix}\\begin{pmatrix}\\phi_{+}(\\bm{k})\\\\\\phi_{-}(\\bm{k})\\\\\\phi_{+}(-\\bm{k}-\\bm{a})\\\\\\phi_{-}(-\\bm{k}-\\bm{a})\\end{pmatrix}.\n\\end{eqnarray*}\n\\normalsize\n\nThis expanded matrix can be made to be a block-diagonal form, so we are allowed to consider two components only:\n\\begin{eqnarray*}\n\\begin{pmatrix}\\bar{\\phi}_{+}(\\bm{k}),&\\bar{\\phi}_{-}(-\\bm{k}-\\bm{a})\\end{pmatrix}\\begin{pmatrix} G^{-1}_{\\scriptscriptstyle{++}}(\\bm{k})&-\\imath Q_{\\scriptscriptstyle{+-}}(\\bm{2\\bm{k}+\\bm{a}})\\\\-\\imath Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})&G^{-1}_{\\scriptscriptstyle{--}}(-\\bm{k}-\\bm{a})\\end{pmatrix}\\begin{pmatrix}\\phi_{+}(\\bm{k})\\\\\\phi_{-}(-\\bm{k}-\\bm{a})\\end{pmatrix}.\n\\end{eqnarray*}\nAs a result, we find self-consistent equations for inter-valley scattering\n\\fontsize{11}{11}\n\\begin{eqnarray*}\n\\frac{2\\imath}{\\Gamma}\\cdot0&=&\\int\\frac{d^{3}\\bm{k}}{(2\\pi)^{3}}\\frac{G_{\\scriptscriptstyle{--}}^{-1}(-\\bm{k}-\\bm{a})}{G^{-1}_{\\scriptscriptstyle{++}}(\\bm{k})G^{-1}_{\\scriptscriptstyle{--}}(-\\bm{k}-\\bm{a})+Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})}\\\\\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})&=&\\int\\frac{d^{3}\\bm{k}}{(2\\pi)^{3}}\\frac{\\imath Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})}{G_{\\scriptscriptstyle{++}}^{-1}(\\bm{k})G_{\\scriptscriptstyle{--}}^{-1}(-\\bm{k}-\\bm{a})+Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})}\\\\\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})&=&\\int\\frac{d^{3}\\bm{k}}{(2\\pi)^{3}}\\frac{\\imath Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})}{G_{\\scriptscriptstyle{++}}^{-1}(\\bm{k})G_{\\scriptscriptstyle{--}}^{-1}(-\\bm{k}-\\bm{a})+Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})}\\\\\n\\frac{2\\imath}{\\Gamma}\\cdot0&=&\\int\\frac{d^{3}\\bm{k}}{(2\\pi)^{3}}\\frac{G_{\\scriptscriptstyle{++}}^{-1}(\\bm{k})}{G_{\\scriptscriptstyle{++}}^{-1}(\\bm{k}) G_{\\scriptscriptstyle{--}}^{-1}(-\\bm{k}-\\bm{a})+Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})Q_{\\scriptscriptstyle{-+}}(-2\\bm{k}-\\bm{a})},\n\\end{eqnarray*}\n\\normalsize\nwhere $Q_{\\scriptscriptstyle{++}(\\scriptscriptstyle{--})}(\\pm(2\\bm{k}+\\bm{a}))$ turn out to vanish due to spin chirality. Note that $G_{\\scriptscriptstyle{++}}(\\bm{k})$ being on the Fermi surface means $G_{\\scriptscriptstyle{--}}(-\\bm{k}-\\bm{a})=G_{\\scriptscriptstyle{--}}(\\bm{k}+\\bm{a})$ should also be on the Fermi surface.\nThus, $Q_{\\scriptscriptstyle{+-}}$ doesn't have to vanish in this case. \nLinearizing the energy spectrum around the inner Fermi surface, we obtain\n\\begin{eqnarray*}\n\\frac{2\\imath}{\\Gamma}Q_{\\scriptscriptstyle{+-}}&=&\\int^{2\\pi}_{0}\\frac{d\\phi}{2\\pi}\\int^{\\pi}_{0}\\frac{d\\theta J_{+}(\\sin{\\theta})}{2\\pi}\\int^{\\infty}_{-\\infty}\\frac{d\\varepsilon}{2\\pi}\\frac{\\imath Q_{\\scriptscriptstyle{+-}}}{J(\\sin{\\theta})\\left(\\hbar v_{F}\\varepsilon\\right)^{2}+\\left | Q_{\\scriptscriptstyle{+-}}\\right |^{2}}\n\\end{eqnarray*}\nwhere $J_{+}(\\sin{\\theta})$ is a Jacobian factor from expanding $\\bm{k}$ around the inner Fermi surface and $J(\\sin{\\theta})$ is a Jacobian factor from connecting the integral on the outer Fermi surface to the integral on the inner fermi surface. Other two equations are satisfied automatically, identically zero. Straightforward calculations give rise to the final expression\n\\begin{eqnarray*}\n\\left | Q_{\\scriptscriptstyle{+-}}\\right |=\\frac{\\Gamma}{2}\\frac{1}{2\\hbar v_{F}}\\left(\\frac{m\\alpha_{R}}{2\\pi\\hbar^{4}}\\right)^{2}\\left(ar+br^{2}+cr^{3}\\right)=\\frac{\\pi N_{F}\\Gamma}{4}\\sqrt{\\frac{r}{1+r}}\\left(a+br+cr^{2}\\right)\n\\end{eqnarray*}\nwhere $a=9.42\\times10^{-3},~b=2.36\\times10^{-1}$ and $c=-3.62\\times10^{-2}$.\n\n\n\nIn the presence of the off-diagonal term of $Q_{\\scriptscriptstyle{+-}}$, the fermion propagator is altered as follows\n\\fontsize{9}{9}\n\\begin{eqnarray*}\n\\begin{pmatrix} G^{-1}_{\\scriptscriptstyle{++}}-\\imath Q_{\\scriptscriptstyle{++}}&-\\imath Q_{\\scriptscriptstyle{+-}}\\\\-\\imath Q_{\\scriptscriptstyle{-+}}&J\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)\\end{pmatrix}^{-1}=\\frac{1}{\\left(G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\right)J\\left(G^{-1}_{\\scriptscriptstyle{--}}-\\imath Q_{\\scriptscriptstyle{--}}\\right)+\\left | Q_{\\scriptscriptstyle{+-}}\\right |^{2}}\\begin{pmatrix} J\\left(G_{\\scriptscriptstyle{--}}^{-1}-\\imath Q_{\\scriptscriptstyle{--}}\\right)&\\imath Q_{\\scriptscriptstyle{+-}}\\\\\\imath Q_{\\scriptscriptstyle{-+}}&G_{\\scriptscriptstyle{++}}^{-1}-\\imath Q_{\\scriptscriptstyle{++}}\\end{pmatrix}.\n\\end{eqnarray*}\n\\normalsize\nThen, the effective inner-fermion propagator is given by\n\\begin{eqnarray*}\nG_{\\scriptscriptstyle{++}\\textrm{eff}}^{-1}(\\bm{k})&=&\\frac{1}{G^{-1}_{\\scriptscriptstyle{++}}(\\bm{k})-\\imath Q_{\\scriptscriptstyle{++}}(\\bm{0})+\\frac{\\left | Q_{\\scriptscriptstyle{+-}}(2\\bm{k}+\\bm{a})\\right |^{2}}{J(\\Omega)\\left(G_{\\scriptscriptstyle{--}}^{-1}(-\\bm{k}-\\bm{a})-\\imath Q_{\\scriptscriptstyle{--}}(\\bm{0})\\right)}}.\n\\end{eqnarray*}\nTaking $\\bm{k}=\\bm{k}^{+}_{F}$, where $\\bm{k}_{F}^{+}$ is the Fermi momentum of the inner Fermi surface,\nwe find \n\\begin{eqnarray*}\nQ_{\\scriptscriptstyle{++}\\textrm{eff}}(\\bm{0})=Q_{\\scriptscriptstyle{++}}(\\bm{0})\\left[1-\\frac{\\left | Q_{\\scriptscriptstyle{+-}}(2\\bm{k}_{F}^{+}+\\bm{a})\\right |^{2}}{Q_{\\scriptscriptstyle{++}}(\\bm{0})Q_{\\scriptscriptstyle{--}}(\\bm{0})}\\right]=\\frac{\\pi}{2}N_{F}\\Gamma F_{+}\\left[1-\\frac{r}{4(1+r)}\\frac{\\left(a+br+cr^2\\right)^{2}}{F_{+}F_{-}}\\right].\n\\end{eqnarray*}\nAccordingly, scattering times are modified as\n\\begin{eqnarray*}\n\\tau_{\\pm\\textrm{eff}}=\\frac{1}{2Q_{\\scriptscriptstyle{\\pm\\pm}\\textrm{eff}}}=\\frac{1}{2\\frac{\\pi}{2}N_{F}\\Gamma F_{\\pm}\\left[1-\\frac{r}{4(1+r)}\\frac{\\left(a+br+cr^2\\right)^{2}}{F_{+}F_{-}}\\right]}.\n\\end{eqnarray*}\nAs a result, diffusion constants are given by\n\\begin{eqnarray*}\nD_{\\pm\\textrm{eff}}(r)=\\hbar v_{F}^{2}\\tau_{\\pm\\textrm{eff}} = \\frac{2\\pi\\alpha_{R}\\hbar^{3}}{m^{2}\\Gamma}\\frac{1+r}{\\sqrt{r}}\\frac{1}{F_{\\pm}\\left(1-\\frac{r}{4(1+r)}\\frac{\\left(a+br+cr^2\\right)^{2}}{F_{+}F_{-}}\\right)}\n\\end{eqnarray*}\nwhere $F_{\\pm}(r)$ are \n\\begin{eqnarray*}\nF_{\\pm}(r)&=&\\frac{\\pi}{2}\\frac{1}{\\sqrt{r}\\sqrt{1+r}}\\textrm{Re}\\left[\\frac{8}{3}+2r-\\frac{4}{3}\\sqrt{r+1}\\left\\{(r+2)\\mathcal{E}\\left(\\frac{1}{r+1}\\right)-r\\mathcal{K}\\left(\\frac{1}{r+1}\\right)\\right\\}\\right]\\\\\n&&-\\Theta(-r)\\left(\\frac{8}{3}-\\frac{8}{3}\\sqrt{1-\\left | r\\right |}+\\frac{2}{3}\\left | r\\right |\\sqrt{1-\\left | r\\right |}-2\\left | r\\right |\\right) ,\n\\end{eqnarray*}\nthe same as before. These results are summarized and compared to experiments in FIG. \\ref{diffusion_constant_3d_with_offQ}, where this fixed-point solution (the presence of the off-diagonal component) turns out to reduce the mobilities of both fermions at inner and outer Fermi surfaces.\n\n\\subsection{Two different cases within the self-consistent Born approximation}\n\nPreviously, we considered two types of solutions for the fermion Green's function within the Born approximation: One contains effects of only the intra-valley forward scattering and the other introduces both effects of intra- and inter-valley scattering into the fermion Green's function. It is natural to expect that the former solution would be justified when effects of disorder scattering are not strong. On the other hand, the second solution is expected to be realized when disorder scattering becomes more relevant than the first case. A crucially different point between these two solutions lies in spin chirality. Intra-valley scattering preserves the spin chirality while inter-valley scattering destroys it. As a result, we predict that the weak antilocalization turns into the weak localization in the presence of inter-valley scattering, increasing disorder strength. This crossover behavior may be regarded as a weak version of a topological phase transition driven by disorder although BiTeI is a topologically trivial metallic state.\n\n\\subsection{Self-consistent Born approximation as a fixed-point solution}\n\nThe solution based on the self-consistent Born approximation can be regarded as an effective theory for the corresponding diffusive Fermi-liquid fixed point. In order to understand this statement, we consider the following renormalization group equation for disorder strength up to one-loop order\n\\begin{eqnarray*}\n\\frac{d\\Gamma_{ss'}}{dt}=\\Gamma_{ss'}-\\Gamma_{ss_{1}}\\mathcal{C}_{s_{1}s_{2}}\\Gamma_{s_{2}s'} .\n\\end{eqnarray*}\n$\\Gamma_{\\scriptscriptstyle{++}(\\scriptscriptstyle{--})}$ is the scattering rate or variance within the inner (outer) Fermi surface and $\\Gamma_{\\scriptscriptstyle{+-}(\\scriptscriptstyle{-+})}$ is that between the inner and outer Fermi surfaces. The first term ensures the relevance of disorder scattering in the tree level when there is a Fermi surface. Such relevant disorder scattering becomes weak through quantum fluctuations, where the disorder potential is screened by particle-hole excitations. $\\mathcal{C}_{s_{1}s_{2}}$ are positive constants, computed in quantum corrections of the one-loop level. $t$ is the renormalization-group transformation scale.\n\nThese renormalization group equations can be rewritten as follows\n\\begin{eqnarray*}\n\\frac{d\\Gamma_{\\scriptscriptstyle{++}}}{dt}&=&\\Gamma_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{++}}\\Gamma_{\\scriptscriptstyle{++}} -\\Gamma_{\\scriptscriptstyle{+-}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{-+}}\\\\\n\\frac{d\\Gamma_{\\scriptscriptstyle{+-}}}{dt}&=&\\Gamma_{\\scriptscriptstyle{+-}}-\\Gamma_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{++}}\\Gamma_{\\scriptscriptstyle{+-}} -\\Gamma_{\\scriptscriptstyle{+-}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{--}}\\\\\n\\frac{d\\Gamma_{\\scriptscriptstyle{-+}}}{dt}&=&\\Gamma_{\\scriptscriptstyle{-+}}-\\Gamma_{\\scriptscriptstyle{-+}}C_{\\scriptscriptstyle{++}}\\Gamma_{\\scriptscriptstyle{++}} -\\Gamma_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{-+}}\\\\\n\\frac{d\\Gamma_{\\scriptscriptstyle{--}}}{dt}&=&\\Gamma_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{--}}\\Gamma_{\\scriptscriptstyle{--}} -\\Gamma_{\\scriptscriptstyle{-+}}C_{\\scriptscriptstyle{++}} \\Gamma_{\\scriptscriptstyle{+-}} .\n\\end{eqnarray*} \nFixed points are determined by $d \\Gamma_{ss'} \/ d t = 0$, resulting in\n\\begin{eqnarray*}\n&&\\Gamma_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{++}}^{2}C_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{+-}}\\Gamma_{\\scriptscriptstyle{-+}}C_{\\scriptscriptstyle{--}} =0\\\\\n&&\\Gamma_{\\scriptscriptstyle{+-}}\\left(1-\\Gamma_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{--}}\\right)=0\\\\\n&&\\Gamma_{\\scriptscriptstyle{-+}}\\left(1-\\Gamma_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{--}}\\right)=0\\\\\n&&\\Gamma_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{--}}^{2}C_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{-+}}\\Gamma_{\\scriptscriptstyle{+-}}C_{\\scriptscriptstyle{++}} =0.\n\\end{eqnarray*}\n\nFirst, we consider the case with the absence of inter-valley scattering, given by $\\Gamma_{\\scriptscriptstyle{+-}}=\\Gamma_{\\scriptscriptstyle{-+}}=0$. Then, we obtain\n\\begin{eqnarray*}\n\\Gamma_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{++}}^{2}C_{\\scriptscriptstyle{++}}=0 ~ \\& ~ \\Gamma_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{--}}^{2} C_{\\scriptscriptstyle{++}}=0.\n\\end{eqnarray*}\n$(\\Gamma_{\\scriptscriptstyle{++}},\\Gamma_{\\scriptscriptstyle{--}})=\\left\\{(0,0),~(0,1\/C_{\\scriptscriptstyle{--}}),~(1\/C_{\\scriptscriptstyle{++}},0)\\right\\}$ are unstable fixed points, and $(\\Gamma_{\\scriptscriptstyle{++}},\\Gamma_{\\scriptscriptstyle{--}})=(1\/C_{\\scriptscriptstyle{++}},1\/C_{\\scriptscriptstyle{--}})$ is the only stable fixed point. This stable fixed point is described by the first self-consistent Born approximation without inter-valley scattering, where spin chirality is well defined.\n\nNext, we consider the presence of inter-valley scattering, given by $\\Gamma_{\\scriptscriptstyle{+-}}=\\Gamma_{\\scriptscriptstyle{-+}}$ and $\\Gamma_{\\scriptscriptstyle{+-}}\\neq0$. Then, we obtain \n\\begin{eqnarray*}\n&&\\Gamma_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{++}}+\\Gamma_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{--}}=1\\\\\n&&\\Gamma_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{++}}^{2}C_{\\scriptscriptstyle{++}}-\\Gamma_{\\scriptscriptstyle{+-}}^{2}C_{\\scriptscriptstyle{--}}=0\\\\\n&&\\Gamma_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{--}}^{2}C_{\\scriptscriptstyle{--}}-\\Gamma_{\\scriptscriptstyle{+-}}^{2}C_{\\scriptscriptstyle{++}}=0.\n\\end{eqnarray*}\nSolving these equations, we find two fixed points:\n\\begin{eqnarray*}\n(\\Gamma_{\\scriptscriptstyle{++}},\\Gamma_{\\scriptscriptstyle{--}})&=&\\left(\\frac{1+\\sqrt{1-4C_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{+-}}^{2}}}{2C_{\\scriptscriptstyle{++}}},\\frac{1-\\sqrt{1-4C_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{++}}\\Gamma_{\\scriptscriptstyle{+-}}^{2}}}{2C_{\\scriptscriptstyle{--}}}\\right)\\\\\n(\\Gamma_{\\scriptscriptstyle{++}},\\Gamma_{\\scriptscriptstyle{--}})&=&\\left(\\frac{1-\\sqrt{1-4C_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{+-}}^{2}}}{2C_{\\scriptscriptstyle{++}}},\\frac{1+\\sqrt{1-4C_{\\scriptscriptstyle{--}}C_{\\scriptscriptstyle{++}}\\Gamma_{\\scriptscriptstyle{+-}}^{2}}}{2C_{\\scriptscriptstyle{--}}}\\right) .\n\\end{eqnarray*} \nWhen $\\Gamma_{\\scriptscriptstyle{+-}}$ satisfies $1-4C_{\\scriptscriptstyle{++}}C_{\\scriptscriptstyle{--}} \\Gamma_{\\scriptscriptstyle{+-}}^{2} \\approx 0$, we find that these two fixed points emerge into $(\\Gamma_{\\scriptscriptstyle{++}},\\Gamma_{\\scriptscriptstyle{--}}) \\approx (1\/C_{\\scriptscriptstyle{++}},1\/C_{\\scriptscriptstyle{--}})$. This fixed point is described by the second solution of the Born approximation in the presence of inter-valley scattering, in which the spin chirality is smeared out. This may be regarded as an intermediate solution with spin chirality before the ``topological phase transition\" toward normal diffusive Fermi liquids without spin chirality appears. Table. \\ref{table1} summarizes our results, where ``WAL\" and ``WL\" represent weak antilocalization and weak localization, respectively. \n\n\n\\begin{table}[ht]\n\\centering\n\\caption{Two ground states of self-consistent Born approximation and two fixed points of the renormalization group analysis.}\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{tabular}{>{\\centering}m{2.1in} >{\\centering}m{2.1in} >{\\centering\\arraybackslash}m{1.6in}}\n\\hline\n& Diffusive Helical Fermi Liquid & Diffusive Fermi Liquid \\\\\n\\hline\nFixed point & \\shortstack[c]{~\\\\$\\Gamma_{\\scriptscriptstyle{++}}\\neq0$,~$\\Gamma_{\\scriptscriptstyle{--}}\\neq0$,\\\\and $\\Gamma_{\\scriptscriptstyle{+-}}=0$} & \\shortstack[c]{$\\Gamma_{\\scriptscriptstyle{++}}\\neq0$,~$\\Gamma_{\\scriptscriptstyle{--}}\\neq0$,\\\\and $\\Gamma_{\\scriptscriptstyle{+-}}\\neq0$} \\\\\n\\shortstack[c]{~\\\\Ground State\\\\~~(Self-consistent Born analysis)~~} & \\shortstack[c]{~\\\\$Q_{\\scriptscriptstyle{++}}\\neq0$,~$Q_{\\scriptscriptstyle{--}}\\neq0$,\\\\and $Q_{\\scriptscriptstyle{+-}}=0$} & \\shortstack[c]{~\\\\$Q_{\\scriptscriptstyle{++}}\\neq0$,~$Q_{\\scriptscriptstyle{--}}\\neq0$,\\\\and $Q_{\\scriptscriptstyle{+-}}\\neq0$}\\\\\nTransport property & WAL & \\shortstack[c]{~\\\\WAL$\\rightarrow$WL\\\\(Crossover)}\\\\\n\\hline\n\\end{tabular}\n\\label{table1}\n\\end{table}\n\n\n\\section{Discussion}\nConsidering that the only relevant energy scales are the cyclotron energy $\\hbar \\omega$ and the Fermi energy $E_F$ in the IFS, it is natural to introduce a single parameter $b = \\hbar \\omega\/E_F = (\\hbar e\/m_{WF}E_F)B$ for the Hall resistivity contribution from the Weyl fermions, anticipating the scaling behavior for $\\Delta \\rho_H$ \\cite{HJKim11}, where $m_{WF}$ is an effective mass of the Weyl fermion and $\\Delta \\rho_H$ is the Hall resistivity component deviating from the linearity. Indeed, we found a scaling property in $\\Delta \\rho_H$, presented in Fig. 9(a), where the y-axis should be also scaled as the magnitude of $\\Delta \\rho_H$ is inversely proportional to the carrier density. This scaling analysis enables us to estimate $m_{WF}$, whose values for all six samples are plotted in Fig. 9(b) as a function of the corresponding $E_F$. $m_{WF}$ is in the order of \n$10^{-6} - 10^{-4} m_0$, where $m_0$ is the mass of an electron and they exhibit a singular behavior with a minimum at the Weyl point.\n\nIt is straightforward to find $m_{WF} = m - \\frac{\\alpha_R}{\\sqrt{\\alpha_R^2+\\frac{2 \\hbar^2}{m}E_F}}m$ from the Rashba Hamiltonian with degenerate parabolic bands, where $m$ is the bare band mass given by the curvature of the parabolic band and the Rashba coupling constant $\\alpha_R$ determines the energy of the Weyl point from the bottom of the conduction band, given by $E_{WF} =\\frac{1}{2}\\frac{m}{\\hbar^2}\\alpha_R^2$. Considering an overall shift for the Fermi energy and taking the limit of $\\alpha_R^2 >> \\frac{2 \\hbar^2}{m}E_F$, we obtain $m_{WF} \\approx \\frac{\\hbar^2}{\\alpha_R^2} \\mid E_{F}-E_{WF} \\mid$. This equation describes zero mass at the Weyl point quite well. However, compared to the experimental result, the mass increases rather steeply as $E_F$ deviates from the Weyl point. In the opposite limit of the strictly linear dispersion, the mass is zero even away from the Weyl point. According to the density functional theory (DFT) \\cite{Bahramy11}, the dispersion near the Weyl point is neither quadratic nor linear. Therefore, to parameterize the degree of deviation from the linear dispersion, we introduce a phenomenological equation,\n$\\frac{m_{WF}}{m} \\approx \\frac{1}{2} | 1-(\\frac{v_{linear}}{v_{real}} )^2 |\\frac{\\mid E_F-E_{WF} \\mid}{E_{WF}}$. \nAssuming $\\frac{v_{linear}}{v_{real}} \\approx 1+\\varepsilon$, we obtain $\\varepsilon \\sim 2.2 \\times 10^{-2}$. This result implies that all higher order terms of the curvature in the dispersion is only few \\% and thus, the real dispersion in BiTeI near the Weyl point is considerably linear. Though the dispersion is mainly determined by periodic ionic potentials, we do not exclude any contribution to the linear dispersion resulting from electron interaction.\n\nThe universal scaling of the Hall resistivity discussed above is quite consistent with the extreme disparity of the mobility and divergent IFS mobility. Representing the Hall resistivity $\\Delta \\rho_H(B) =\\frac{1}{nec} \\frac{B}{1+\\mu^2B^2}$ as\n $\\Delta \\rho_H(b)=\\frac{1}{nec}\\frac{\\frac{m_{WF}E_F}{\\hbar e}b}{1+\\mu^2(\\frac{m_{WF}E_F}{\\hbar e})^2b^2}$ with the dimensionless magnetic field $b$ discussed before, we obtain the scaling expression of $\\frac{\\Delta \\rho_H(b)}{\\Delta \\rho_H(b=1)}=\\frac{(1+\\mu_{sc}^2)b}{1+\\mu_{sc}^2b^2}$, \n where $\\mu_{sc}=\\frac{m_{WF}E_F}{\\hbar e}\\mu$ is a scaled mobility. This scaled mobility, being a universal constant does not depend on the Fermi energy. Introducing the empirical formula introduced above for Weyl-fermion mass into the mobility, we find the following expressions of IFS and OFS mobility, given by $\\mu_{IFS}(E_F)=2\\frac{\\hbar e \\mu_{sc}}{m|1-(\\frac{v_{linear}}{v_{real}})^2|} \\frac{E_F}{|E_F||E_F-E_{WF}|}$ \n and $\\mu_{OFS}(E_F)=\\frac{\\hbar e\\mu_{sc}}{2m|E_F|}$, respectively. As $E_F$ is inversely proportional to $m$ for the charge carrier on OFS,\n $\\mu_{OFS}$ is constant and $\\mu_{IFS}$ follows $\\mu_{IFS}(E_F) \\propto \\frac{1}{|E_F-E_{WF}|}$ \n with the ratio of $\\frac{\\mu_{IFS}(E_F)}{\\mu_{OFS}(E_F)}=\\frac{m_{OFS}}{m_{WF}}=\\frac{4}{|1-(\\frac{v_{linear}}{v_{real}})^2|}\\frac{E_{WF}}{|E_F-E_{WF}|} \\approx 10^3 \\sim 10^4$. Indeed, this scaling argument is consistent with the ``divergent'' $\\mu_{IFS}$ at the Weyl point shown in the experimental result [Fig. 1(b)]. Note that while the mass ratio between IFS and OFS in this argument is mostly determined by the empirical factor\n $\\varepsilon \\approx1-\\frac{v_{linear}}{v_{real}}$ introduced above, the ``divergent'' behavior of $\\mu_{IFS}$ at the Weyl point is given by \n $\\frac{1}{|E_F-E_{WF}|}$ and in fact, this term is inherent in the Rashba model.\n\nIn order to understand the origin of the divergent IFS mobility and the extreme disparity between IFS and OFS mobility near the Weyl point, \nwe have performed the self-consistent Born analysis for the Rashba Hamiltonian, which is a mean-field theory in the presence of disorder with no consideration of electron correlation. Here we summarize main results of the perturbative renormalization group analysis to understand how a fixed-point phase is determined. In general, disorder strength increases at the short-distance scale in three dimensions because huge number of electrons on the Fermi surfaces are affected by disorder potentials. On the other hands, it decreases at the long-distance scale because disorder potentials are effectively screened. As a result, balancing is achieved and it gives rise to a finite-disorder fixed point, which is known as a diffusive Fermi liquid. In the present problem, we found two types of fixed points: One contains the effects of the intra-valley forward scattering only and the other considers both intra- and inter-valley scattering. We have performed the self-consistent Born approximation and found a fixed-point solution for the electron Green's function in both cases. Then, we have calculated transport coefficients, evaluating current-current correlation functions with this mean-field-theory propagator. \n\nFig. 8 shows that the self-consistent Born analysis describes our experimental data quantitatively, where lines and discrete points represent theoretical curves and experimental results, respectively. It is natural to expect that the presence of inter-valley scattering reduces the mobility. However, effects of the inter-valley scattering are not relevant in describing the experimental data in the present case\n because it suppresses spin chirality and the weak anti-localization. This fixed point is distinguished from a conventional diffusive Fermi liquid because of definite chirality and we name it a diffusive helical Fermi liquid. Thus, our BiTeI single crystals are weakly disordered with negligible inter-valley scattering whose ground state is considered to be a diffusive helical Fermi liquid. We would like to emphasize that only one fitting parameter, related with the variance of disorder potential at the fixed point is used in this comparison, whereas all other parameters are determined by the experiment.\n\nIt is straightforward to understand the divergent IFS mobility within the framework of the self-consistent Born approximation. As the IFS density of states vanishes, approaching the Weyl point, the scattering rate also becomes zero at the Weyl point. However, it is difficult to explain the experimentally confirmed scaling of Hall resistivity within the same framework. In fact, we find that scaled mobility \n$\\mu_{sc} = \\frac{m_{WF}E_F}{\\hbar e} \\mu$ is not independent of the Fermi energy $E_F$ when the mobility $\\mu$ evaluated from the self-consistent mean-field analysis is used. The independence of $\\mu_{sc}$ on $E_F$ is achieved only when the divergence of the IFS mobility is exactly cancelled by the mass reduction near the Weyl point. In the scaling argument, we obtained $m_{WF} \\propto |E_F - E_{WF}|\/E_F$. On the other hand, Born mean-field theory gives $\\mu_{IFS}(E_F) \\propto \\frac{1}{|E_F-E_{WF}|^{\\kappa}}$ with $\\kappa$ larger than 1. Thus, the $\\mu_{sc}$ \n is not independent of $E_F$ in the self-consistent mean-field analysis.\n \n One way to reconcile this inconsistency is to take into account the role of effective interactions between electrons near the Weyl point. As the IFS density of states vanishes at the Weyl point, effective interactions can be enhanced due to weaker screening effect. In fact, this is what happens in graphene. Correlation effects indeed reshape the linear band dispersion of graphene \\cite{Elias11,Siegel11,Kotov12,Chae12}. Possible interplay among inversion symmetry breaking (spin chirality), disorders, and effective interactions may lead to a novel interacting diffusive fixed point, which allows the universal scaling in the Hall resistivity.\n \nWhen disorders become stronger, it is possible that a topological structure (geometric phase) in the ground-state wave-function changes. Previously, we considered two types of fixed points, corresponding to the absence and presence of inter-valley scattering, respectively. The former solution would be justified when effects of disorder scattering are not strong, called a diffusive helical Fermi-liquid state. On the other hand, the second solution is expected to be realized when disorder scattering becomes more relevant than the first, identified with a diffusive Fermi-liquid state. A crucial difference is spin chirality. Intra-valley scattering preserves the spin chirality while inter-valley scattering destroys it. As a result, we predict that the weak anti-localization turns into the weak localization in the presence of strong inter-valley scattering, increasing disorder strength. This crossover behavior from the diffusive helical Fermi liquid to the conventional diffusive Fermi liquid may be regarded as a weak version of a topological phase transition driven by disorder although BiTeI is topologically trivial.\n\n\\section{Conclusion}\nIn conclusion, we uncovered that the interplay between disorder and inversion symmetry breaking is responsible for (1) divergent mobility in the inner chiral Fermi surface (FS), (2) extreme disparity of the mobility values between the inner and outer chiral FS, and (3) universal scaling in the Hall resistivity. Based on the self-consistent Born approximation, we could consistently explain the observation (1) and (2), quantitatively reproducing mobility values of the inner and outer FS as a function of the Fermi energy. However, the universal scaling of the Hall resistivity cannot be accounted for within this mean-field theory, which indicates the existence of mass renormalization of the inner Fermi-surface near the Weyl point, possibly originated from electron correlation due to weaker screening near the Weyl point. \\\\\n\n\\acknowledgements\nThis study was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (No. 2014R1A1A1002263). KS was also supported by the Ministry of Education, Science, and Technology (No. 2012R1A1B3000550 and No. 2011-0030785) of the National Research Foundation of Korea (NRF) and by TJ Park Science Fellowship of the POSCO TJ Park Foundation. MS was also supported by YO-COE Foundation from Yamagata University. MS wishes to express his thanks to Prof. T. Iwata for his support. \n\n$^{\\ast}$ Both authors equally contribute to the present paper \\\\\n$^{\\dagger}$ Corresponding author; hjkim76@daegu.ac.kr \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOn October 2019, we were invited to join a computing challenge on 3-XORSAT problems launched by some colleagues at University of Southern California \\cite{USCchallenge}.\nThe idea behind the challenge was to compare actual performance of the best available computing platforms, including quantum computers, in solving a particularly hard optimization problem.\nQuantum computing is becoming practical these days, and many different computing devices based on quantum technologies are becoming available (D-Wave, Google and IBM just to cite the most known).\nSo it is a natural question to ask, whether any of these quantum devices available today can do better than classical (i.e.\\ non-quantum) computing machines.\n\nWe decided to join this 3-XORSAT challenge with a proposal combining new algorithmic ideas and a highly optimized GPU implementation.\nWe are not going to discuss in detail the results of the 3-XORSAT challenge, that will appear elsewhere \\cite{USCchallenge}. We just remark that the performances of our algorithm running on commercial Nvidia GPUs are at least 2 orders of magnitude better than those of the other devices that entered the 3-XORSAT challenge: D-Wave quantum annealing processor \\cite{DWave}, Memcomputing machine \\cite{Memcomputing}, Fujitsu digital annealer \\cite{fujitsu} and Toshiba's simulated bifurcation machine \\cite{toshiba}.\n\nThis is clearly not the end of the story, as quantum technologies are evolving very fast and presumably will become competitive soon (eventually getting what is called a quantum advantage). Nonetheless, we believe it is very important to clarify what is today the state of art in the ``classical vs.\\ quantum computation challenge''.\n\nIn the present manuscript we report the ideas and the technical details that make our solving algorithm ranking first as a solver of the hard optimization problems presented at the 3-XORSAT challenge.\n\nThe manuscript is organized as follows. First we recap the known physical properties of these hard optimization problems (especially their energy landscape). Then we describe the algorithm we decided to use with a particular emphasis on the use of large number of clones and how this can be implemented efficiently for classical computers. We also provide a description of the technical choices that made our GPU implementation extremely efficient, even if the problem, being defined on a random graph topology, would in principle not be ideal for a platform like GPU. Finally, we discuss the numerical results about the time-to-solution (TTS), proposing an improved way of measuring the largest percentiles of TTS. We finish with a few concluding remarks.\n\n\\section{The model and its energy landscape}\n\nThe optimization problem that has been presented to the contenders at the 3-XORSAT challenge is the search for the ground state of a model based on Ising variables. The model is well known in statistical physics under the name of \\textit{diluted 3-spin ferromagnetic model} \\cite{franz2001ferromagnet}. In the computer science literature, it corresponds to a constraint satisfaction problem known under the name of \\textit{3-xorsat} \\cite{dubois20023}.\nIn this paper, we use the statistical physics formulation of the model, but switching to its computer science formulation requires just a change of variables.\n\nThe model is defined by the Hamiltonian\n\\begin{equation}\\label{eq:Hamiltonian}\n H[\\sigma]\\equiv -\\sum_{(i,j,k)\\in E} s_i s_j s_k\\;,\n\\end{equation}\nwhere $s_i=\\pm 1$ are $N$ Ising spins. The sum over the set $E$ of triplets $(i,j,k)$ is what defines the interaction topology. The instances provided in the 3-XORSAT challenge were generated on a random regular graph of fixed degree 3. In other words, the set $E$ is made of $N$ triplets randomly chosen under the constraints that in each triplet the 3 indices are different and each index appears exactly in 3 triplets.\nFrom the definition of the Hamiltonian $H[s]$ in Eq.~(\\ref{eq:Hamiltonian}), it is clear that the ground state is the configuration $s^*$ with all $s^*_i=1$. However our algorithm, publicly available at \\cite{algo}, searches for the ground states without computing the magnetization and having access only to the energy, so there is no need to ``hide'' the solution by a gauge transformation. The organizers of the competition are expected to check that the same is true for all the others contenders.\n\nOne may argue that such a model should be easy to optimize, because all interactions are ferromagnetic. However it is well known that such a model shows the same glassy physics of a disordered model \\cite{franz2001ferromagnet,ricci2010being} because the 3-spin interaction can be satisfied in many ways and this generates frustration in the system during the optimization.\\footnote{The careful reader may have noticed that the problem of satisfying all interactions in $H[s]$, i.e.\\ $s_j s_j s_k = 1$, is equivalent to the problem of solving linear equations modulo 2, $(x_i+x_j+x_k)(\\text{mod } 2)=0$ where $s_i=(-1)^{x_i}$. This problem can be solved in polynomial time, e.g.\\ by Gaussian elimination. However, as discussed in previous publications \\cite{barthel2002hiding}, the problem can be slightly modified preserving the same physical behavior, and making the polynomial algorithm no longer useful. The competition was restricted to algorithms which are robust with respect to such a change: using Gaussian elimination, or algorithms derived from it, was forbidden.} \n\nActually, in the 3-XORSAT challenge, an equivalent formulation has been used, where variables are twice in number (one $\\eta$ variable is added per each constraint) and variables interact only pairwise \\cite{hen2019equation}. This has been done to allow devices implementing only pairwise interactions to enter the competition. The resulting Hamiltonian $H_2[s,\\eta]$ is such that $\\sum_\\eta H_2[s,\\eta]=H[s]$. We have performed such a marginalization on the instances provided, so our algorithm minimizes the cost function $H[s]$ given in Eq.~(\\ref{eq:Hamiltonian}).\n\nThe 3-spin on 3-regular random graph (3S3R) model offers a paradigmatic example of a \\emph{golf course} energy landscape.\nThe thermodynamics of the model has been exactly solved \\cite{mezard2003two,montanari2003nature,krzakala2010following} and its dynamics has been accurately studied numerically \\cite{montanari2004cooling,krzakala2010following}.\nThe picture that comes out from these studies is exactly the one that goes under the name of ``random first order transition'' in the physics of glasses \\cite{kirkpatrick1987connections,kirkpatrick1989scaling,castellani2005spin,biroli2012random}.\nThere exists an exponential (in $N$) number of metastable states that dominate the Gibbs measure below the dynamical critical temperature $T_d\\simeq 0.51$, such that for $T0$ it is in a paramagnetic phase. This means that in thermal equilibrium, the dynamics will unlikely reach the ferromagnetic ground state in $s^*$.\nEven if, by chance, $s^*$ is reached, the dynamics will soon leave that configuration to thermalize again in the paramagnetic state.\n\nIn Ref.~\\cite{bellitti2021entropic}, a new class of heuristic algorithms has been introduced with the aim of proving that entropic barriers are the main source of computational complexity in the optimization of the 3S3R model. These \\emph{quasi-greedy} (QG) algorithms perform with high probability, when this is possible, a step decreasing the energy, but when they reach a local minimum they keep flipping a spin that enters in at least one violated interaction.\nThese algorithms have several advantages: (i) they converge fast to the interesting low-energy part of the configurational space; (ii) they keep the system evolving even in presence of many local minima, but without increasing too much the energy (that would make the search ineffective since it would be run in an uninteresting region); (iii) once the ground state $s^*$ is found, the algorithm stops and thus does not escape from the solution of the problem (without the need of checking it after every single spin flip).\n\nCalling $w_k$ the probability of flipping a spin entering $k$ unsatisfied interactions, in Ref.~\\cite{bellitti2021entropic} the QG algorithm with $w_0=0$ and $w_2=w_3=1$ was studied numerically. The probability of finding the solution $s^*$ was found to reach a maximum close to $w_1=0.05$, with a median TTS growing approximately as $\\exp(0.0835 N)$.\nStarting on these very promising results, we have built here a very optimized version of the QG algorithm.\n\nThe QG algorithm can also be viewed as an imperfect Metropolis algorithms not satisfying detailed balance, since by setting $w_0=w_1^3$ we would have an algorithm satisfying detailed balance for the 3S3R model. Setting $w_0=0$ breaks detailed balance, but brings two advantages: large energy jumps are forbidden (they are not strictly required in a search limited for entropic reasons), and once the ground state $s^*$ is found, the algorithm stops.\n\nThe latter property is extremely useful because any efficient implementation of the QG algorithm must perform a large number of steps before checking for the energy (that takes a time comparable to the one needed for a sweep of the QG algorithm).\nThe condition $w_0=0$ ensures that a ground state found during the dynamical evolution will not be lost\nbetween two successive measurements.\n\n\\section{The search by rare events requires many clones}\n\nAs discussed in \\cite{bellitti2021entropic}, the search for the ground state is slowed down by entropic barriers, i.e.\\ the search for the right well bringing from the $\\eth$ manifold to the ground state configuration $s^*$ is like ``finding a needle in a haystack''.\nFor this reason, instead of having a single copy of the system evolving for a very long time, it turns out to be more appropriate to follow a large number of copies of the system (evolving independently) for a shorter time, starting each one from a different random initial condition: we call these \\emph{clones}.\n\nThe rationale beyond this choice is that the evolution on the marginal manifold at $\\eth$ is not fast enough to allow a single clone to visit the entire manifold in a reasonable time.\nSo if the clone starts from an unfavourable initial condition, his search is bound to fail even if it keeps evolving for a very long time.\n\nWe are facing a typical phenomenon ruled by rare events: in the large $N$ limit, for a typical initial condition, the QG algorithm gets stuck at $\\eth$ and fails to find $s^*$, but there are rare initial conditions that allow the QG algorithm to find the solution $s^*$ in a short time.\nThe probability of such rare initial conditions (that roughly coincide with the basin of attraction of $s^*$) is exponentially small in $N$, as in any large deviation process.\n\nOne has to make a choice between the following two extreme strategies: running a single clone for a time scaling exponentially in $N$ or running a number of clones scaling exponentially with $N$ for a finite time. In principle, one should optimize over all choices in between these extremes, at a fixed total amount of computing time.\n\nOur choice has been to run the largest possible number of clones. This turns out to be the best choice for several reasons. It reduces fluctuations and, if the number of clones is large enough, we can derive analytic predictions for setting the running time to an optimal value (see below). Moreover, it is very unlikely for a single clone (or few clones) to find the solution, while when running a huge number of clones some of them can find the solution, thus allowing us to estimate the mean TTS.\nFinally, running a large number of clones is highly beneficial from the coding point of view, since the clones can evolve in parallel, leading to a drastic reduction in running times.\n\n\\section{Basic information about the GPU implementation}\n\nWe implemented the algorithm described above in a CUDA code that looks for the ground state of the given problem instance using concurrently thousands of clones.\nSince spins can only have two values, we use a multi-spin coding technique by packing values from different clones into 32-bit words \\cite{jacobs1981multi}. This allows to update 32 distinct clones in parallel by using Boolean operations on the spin words (we use the same random number for the 32 clones in the same word). Moreover we use the natural thread parallelism, evolving different clones in each GPU core. Finally, multiple GPUs can be used simultaneously by executing the same code with different random seeds.\nSo, in the end, we have three levels of parallelism: $i)$ multi-spin coding; $ii)$ thread level; $iii)$ multiple independent executions on distinct GPUs. Although the code is able to fit the number of threads to the actual number of cores available on the GPU in use, most of the runs have been executed on Volta 100 GPUs featuring 5120 cores. On those GPUs, the total number of clones was $N_\\text{cl}=327680$.\n\nOne more crucial aspect of our GPU implementation is the partitioning of variables in independent sets. In this way, the spin update procedure can be performed in parallel inside each independent set.\n\nFurther details on the GPU implementation, on the optimizations and fine tuning of the code are reported in the Supplementary Material.\n\n\\section{Numerical results}\n\nWe have been provided with 100 instances for different problem sizes. After some preliminary runs, we decided to focus our attention on sizes 256, 512 and 640, that, once transformed back to the form of a 3S3R model, correspond to $N=128,\\,256,\\,320$.\n\n\\begin{figure}\n \\onefigure[width=0.8\\columnwidth]{w1.pdf}\n \\caption{Preliminary runs on instance \\#38 of size $N=256$ allowed us to estimate of the optimal value $w_1=0.054(1)$ for the only parameter of the QG algorithm.}\n \\label{fig:w1}\n\\end{figure}\n\nThe QG algorithm depends on a single parameter, $w_1$, which is the probability of flipping a variable entering in one unsatisfied interaction and two satisfied interactions. The other parameters are fixed: $w_0=0$ (the ground state is a fixed point of the QG algorithm) and $w_2=w_3=1$ (the QG algorithm decreases the energy whenever possible).\nOur preliminary runs also served to optimize over $w_1$. In Fig.~\\ref{fig:w1} we show the mean TTS in a given instance of size $N=256$. The quadratic interpolation to the data estimates an optimal value $w_1=0.054(1)$, with a negligible dependence on the problem size (at least for $N\\ge 128$). So hereafter we fix $w_1=0.055$.\nAlthough the QG algorithm does not satisfy the detailed balance, we can associate a pseudo-temperature to the value of $w_1$ by the relation $w_1=\\exp(-2\/T_\\text{run})$, where the latter is the probability of flipping the spin in the Metropolis algorithm running at temperature $T_\\text{run}$. It is worth noticing that for $w_1=0.055$ we have $T_\\text{run}\\simeq 0.69$, which is slightly above the dynamical transition temperature $T_d\\simeq 0.51$ \\cite{krzakala2010following}.\n\nBeing the QG algorithm stochastic, the TTS is a random variable whose probability distribution is often measured via its percentiles $\\text{TTS}_p$, defined by $\\mathbb{P}[\\text{TTS}<\\text{TTS}_p] = p\/100$.\nThe organizers of the 3-XORSAT competition asked the participants to estimate the 99-th percentile $\\text{TTS}_{99}$ for each of the 100 instances of a given size and to report the median value (over the instances). This is the time required to solve with 99\\% probability an instance of median hardness. As we will discuss in detail we are not confident that this is the best metric for evaluating the performance of the proposed algorithms.\n\nMost of our simulations have been executed on Nvidia V100 GPUs running in parallel $N_\\text{cl}=327680$ clones. \nThe TTS for a single run of our QG algorithm is given by the shortest among the $N_\\text{cl}$ times each clone requires to reach the solution. Under the assumption, that we have checked numerically with great accuracy, that the cumulative distribution of the single clone time to reach a solution starts linearly in the origin, we have that the TTS (the best of the $N_\\text{cl}$ clones) is exponentially distributed as (see SM)\n\\begin{equation}\\label{eq:expTTS}\n \\mathbb{P}[\\text{TTS}>t] = \\exp(-t\/\\tau)\\,.\n\\end{equation}\nA check of the above equation is reported in Fig.~\\ref{fig:N256_fig1} where we plot in a semilogarithmic scale the probability that the TTS is larger than a given time $t$ (in seconds) for 20 instances of size $N=256$. Data have been obtained running the QG algorithm 1008 times and sorting the corresponding 1008 values of the TTS. We observe that the exponential distribution (which is linear in a semilogarithmic scale) describes very well the data down to a small probability.\nA consequence of this observation is that the TTS of our QG algorithm with a very large number of clones can be perfectly described in terms of the single timescale $\\tau$, \\emph{the mean TTS}, that depends solely on the particular instance under study.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{N256_fig1.pdf}\n \\caption{Cumulative probability distribution of TTS for 20 instances of size $N=256$.}\n \\label{fig:N256_fig1}\n\\end{figure}\n\nIn Fig.~\\ref{fig:N256_fig1} the crossing of the data with the horizontal dotted line determines the value of $\\text{TTS}_{99}$. We believe that this is not the best estimate for the time such that the QG algorithm finds the solution with probability 99\\%. As a matter of fact, a better estimate, that is affected by much smaller fluctuations, is given by $-\\log(0.01)\\;\\tau$. The latter estimate is much more robust than $\\text{TTS}_{99}$ since it is obtained from all the measured TTS values. Moreover, $\\text{TTS}_{99}$ requires the execution of the algorithm at least 100 times, whereas $\\tau$ can be safely estimated from a much smaller number of measures.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{N256_fig2.pdf}\n \\caption{Different estimates of $\\tau$, the mean TTS, for the 100 instances of size $N=256$.}\n \\label{fig:N256_fig2}\n\\end{figure}\n\nIn general, the value of $\\text{TTS}_p$ can be better computed via $-\\log(1-p\/100)\\tau$, after the values of $\\tau$ have been estimated. In Fig.~\\ref{fig:N256_fig2}, we plot for each of the 100 instances of size $N=256$ the mean TTS $\\tau$ and three equivalent estimates obtained from $\\text{TTS}_{50}$ (the median), $\\text{TTS}_{90}$ and $\\text{TTS}_{99}$.\nFor each instance, the four estimates are very close, whereas they vary a lot when changing the instance. A more careful inspection of the data in Fig.~\\ref{fig:N256_fig2} highlights that the mean TTS $\\tau$ is always in the middle of the group of the four estimates, while the estimate based on $\\text{TTS}_{99}$ is sometimes far from the other.\nThe above observations suggest that using $\\tau$ instead of $\\text{TTS}_{99}$ would provide more reliable and stable results in the analysis of algorithms performance.\n\n\\section{Running with a short timeout}\n\nWhat is the best possible way to estimate $\\tau$? Obviously, having an unbounded computing power, one could simply execute the search $N_\\text{run}$ times and just take the average of all the TTS values.\nBut for a process that requires a computing time growing exponentially with the problem size $N$, this naive approach becomes soon unfeasible.\n\nNonetheless, we deduce from the data shown in Fig.~\\ref{fig:N256_fig1} and from the cumulative distribution in Eq.~(\\ref{eq:expTTS}) that runs with a very short TTS always exists for any $\\tau$, although they become very rare for large values of $\\tau$.\n\nSo we can adopt a different search strategy. Instead of letting every run to finish reaching the solution $s^*$ (that sooner or later is found, since the dynamical process we simulate is ergodic for finite $N$ values), we can set a timeout $t_\\text{max}$ such that the QG algorithm reports a failure if the solution is not found in a time shorter than $t_\\text{max}$.\n\nThis \\emph{early stop} strategy has several advantages.\nThe use of a timeout prevents very long runs: this is very useful, not only because it stops in advance those unfortunate runs that would take an atypically long time, but also because makes all runs of a similar time duration (and this is very useful when planning a large group of parallel runs).\nMore importantly, the algorithm with a timeout can also be run for very large sizes, when the algorithm without any timeout would take too long to finish. The data for problems of size $N=320$ have been obtained with this strategy, and a sensible estimate would have been otherwise impossible to get.\n\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{timeout.pdf}\n \\caption{Estimates of mean TTS $\\tau$ in all the 100 instances of size $N=256$ obtained from runs with a short timeout.}\n \\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{99perc.pdf}\n \\caption{99-th percentiles $\\text{TTS}_{99}$ for the median instances of several problem sizes.}\n \\label{fig:99perc}\n\\end{figure}\n\nIf the timeout $t_\\text{max}$ is much shorter than the mean TTS, $t_\\text{max} \\ll \\tau$, only a small fraction of runs will find the solution. By running $N_\\text{run}$ runs with a timeout $t_\\text{max}$, we can estimate $\\tau$ from the number $n$ of successful runs as follows.\nThe posterior distribution on $\\tau$ given that we observe $n$ successful runs among $N_\\text{run}$ is proportional to\n\\begin{equation*}\nP(\\tau|n) \\propto \\frac{1}{\\tau} \\binom{N_\\text{run}}{n} \\left(1-e^{-t_\\text{max}\/\\tau}\\right)^n \\left(e^{-t_\\text{max}\/\\tau}\\right)^{N_\\text{run}-n}\\;,\n\\end{equation*}\nwhere the factor $1\/\\tau$ before the binomial coefficient is the prior on $\\tau$ and it is such that, before taking any measurement, the probability measure is uniform on the variable $\\ln\\tau$.\nSince $t_\\text{max}\\ll\\tau$ we can simplify the posterior to the following normalized distribution\n\\begin{equation*}\nP(\\tau|n)=\\frac{\\mathcal{T}_\\text{tot}^n}{(n-1)!}\\frac{e^{-\\mathcal{T}_\\text{tot}\/\\tau}}{\\tau^{n+1}}\\;,\n\\end{equation*}\nwith $\\mathcal{T}_\\text{tot}=N_\\text{run}t_\\text{max}$ being the total running time. Getting an estimate of $\\tau$ from this posterior distribution is straightforward and the results are shown in Fig.~\\ref{fig3} for timeouts of 5, 10 and 20 seconds.\n\nWe see from the data in Fig.~\\ref{fig3} that the estimates of $\\tau$ from runs with a rather short timeout are very accurate for most of the samples: only for samples whose mean TTS is almost 2 orders of magnitude larger than the timeout did the estimate turn out to be larger, but still compatible within error bars.\nIn particular we notice that for the median instance the estimates of $\\tau$ (and thus of the 99-th percentile $\\text{TTS}_{99}$) obtained from runs with a very short timeout are perfectly fine and allow to save a great amount of time.\n\nWe summarize in Table \\ref{table1} our best estimates for $\\text{TTS}_{99}$ in the median instance, and in Fig.~\\ref{fig:99perc}, we plot the values reported in the table together with the best fitting exponential growth, $\\text{TTS}_{99} \\propto \\exp(a N)$ with $a=0.0786(4)$.\n\n\\begin{table}[!ht]\n \\centering\n \\begin{tabular}{|llll|} \n \\hline\n & from all runs & from runs & timeout \\\\\n $N$ & (no timeout) & with timeout & value \\\\ [0.5ex]\n \\hline\n 128 & 0.0275 $\\pm$ 0.0005 & --- & --- \\\\\n 256 & 640 $\\pm$ 20 & 700 $\\pm$ 100 & 4.5 \\\\\n 320 & --- & 130k $\\pm$ 30k & 450 \\\\ [1ex] \n \\hline\n \\end{tabular}\n \\caption{Values of the 99-th percentile $\\text{TTS}_{99}$ for the median instances of several sizes. All times are expressed in seconds.}\n \\label{table1}\n\\end{table}\n\n\\section{Analytic prediction for exponent $a$ in $\\ln\\tau\\sim a N$}\n\nAlthough the QG algorithm is heuristic and it does not satisfy the detailed balance condition, we can still obtain an approximate analytical estimate of the exponent $a$ ruling the growth of $\\tau$ with $N$, assuming the dynamics takes place in contact with a thermal bath at an effective temperature $T_\\text{run}=-2\/\\log(w_1)$.\nIn thermal equilibrium, we expect the time to visit the ground state $s^*$ to be related to the free-energy barrier between the paramagnetic state and the ordered state around $s^*$.\nWe need to compute the free-energy as a function of the magnetization $m=\\sum_i s_i \/ N$.\n\nWe consider a $K$-spin model on a $K$-regular random graph (the model we simulated has $K=3$, but is worth presenting analytic computations for a generic $K$ value). In order to set the magnetization to an arbitrary value, we add an external field $b$ to the Hamiltonian: $H[s]-b \\sum_i s_i$. Using the cavity method for sparse models \\cite{mezard2001bethe,mezard2003cavity,mezard2009information} we can write the free-energy at temperature $T=1\/\\beta$ in the following variational form\n\\begin{eqnarray}\n -\\beta f &=& \\log(Z_i) + \\log(Z_a) + K \\log(Z_{ai}) - b\\,m\\;,\\label{eq:f}\\\\\n Z_i &=& \\frac{2 \\cosh(\\beta(Ku+b))}{(2 \\cosh(\\beta u))^K}\\;,\\nonumber\\\\\n Z_a &=& \\cosh(\\beta) \\left(1+\\tanh(\\beta) \\tanh(\\beta h)^K\\right)\\;,\\nonumber\\\\\n Z_{ai} &=& \\frac{\\cosh(\\beta(u+h))}{2\\cosh(\\beta u)\\cosh(\\beta h)}\\;,\\nonumber\n\\end{eqnarray}\nthat needs to be extremized with respect to the external field $b$ and the cavity fields $u$ and $h$. The saddle point equations read\n\\begin{eqnarray}\n m &=& \\tanh(\\beta(Ku+b))\\;,\\nonumber\\\\\n h &=& (K-1)u+b\\;,\\label{eq:spe}\\\\\n \\tanh(\\beta u) &=& \\tanh(\\beta) \\tanh(\\beta h)^{K-1}\\;.\\nonumber\n\\end{eqnarray}\nThe paramagnetic solution to Eq.~(\\ref{eq:spe}) has $u=h=m=0$ and $f=f_\\text{para}\\equiv-\\log(2\\cosh(\\beta))\/\\beta$.\nThe ferromagnetic solution has $u,h,m>0$ and it exists only for temperatures $T 2$:\n\\begin{align*}\nr = \\sum_{M_2\\le a \\le M_1} c_a x_2^ax_1^{n-a},\n\\end{align*}\nwhere $M_1 = \\min \\{n, N-1\\}$, $M_2 = \\max \\{0, n + 1 - N\\}$. Then\n\\begin{align*}\n\\Delta(r)&= \\sum_{M_2\\le a \\le M_1} c_a \\Big(\\sum_{i=0}^a \\binom{a}{i}_{k^2} x_2^i\\otimes x_2^{a-i}\\Big)\n\\Big(\\sum_{j=0}^{n-a} \\binom{n-a}{j}_{k^2} x_1^j\\otimes x_1^{n-a-j}\\Big) \\\\\n&= \\sum_{M_2\\le a \\le M_1}\\sum_{i=0}^a\\sum_{j=0}^{n-a} c_a \\binom{a}{i}_{k^2}\\binom{n-a}{j}_{k^2} (kp)^{(a-i)j} x_2^ix_1^j\\otimes x_2^{a-i}x_1^{n-a-j}.\n\\end{align*}\nThus\n\\begin{align*}\n\\partial_1(r) &= \\sum_{M_2\\le a \\le \\min \\{M_1, n-1\\}} c_a (n-a)_{k^2} (kp)^{a}\\, x_2^{a}x_1^{n-a-1},\n\\\\ \\partial_2(r) &= \\sum_{\\max\\{M_2, 1\\} \\le a \\le M_1} c_a (a)_{k^2} x_2^{a-1}x_1^{n-a}.\n\\end{align*}\nBy minimality of $n$, $\\partial_1(r) = \\partial_2(r) =0$, hence\n\\begin{align*}\nc_a (n-a)_{k^2} &= 0,& M_2 &\\le a \\le \\min \\{M_1, n-1\\}, \\\\ c_a (a)_{k^2} &= 0,& \\max\\{M_2, 1\\} &\\le a \\le M_1.\n\\end{align*}\nIf any $ c_a\\neq 0 $, then either $(a)_{k^2}$ or $(n-a)_{k^2} = 0$, but this contradicts \nthe definition of $N$. Therefore the coefficients $ c_a $ are trivial and $\\pi$ is bijective.\n\\end{proof}\n\nWe close this Subsection by a discussion of the Nichols algebras arising from the equivalence in \\cite{H}. \nFirst, \\ref{item:hiet-b} and \\ref{item:hiet-c} give rise to the same braiding, that is\n\\begin{align}\\label{eq:braiding21bc}\n(c'(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nk^2 x_1\\otimes x_1 & kp x_2\\otimes x_1 \\\\\nkq x_1\\otimes x_2 + (k^2-pq) x_2\\otimes x_1 & k^2 x_2\\otimes x_2\n\\end{pmatrix}.\n\\end{align}\nSince \\ref{item:hiet-b} is a change of basis, the Nichols algebras are isomorphic.\nSecond, \\ref{item:hiet-a} gives rise to the braiding\n\\begin{align}\\label{eq:braiding21a}\n(c''(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nk^2 x_1\\otimes x_1 & kq x_2\\otimes x_1 \\\\\nkp x_1\\otimes x_2 + (k^2-pq) x_2\\otimes x_1 & k^2 x_2\\otimes x_2\n\\end{pmatrix}.\n\\end{align}\nBut \\eqref{eq:braiding21a} is \\eqref{eq:braiding21bc} up to $p \\leftrightarrow q$, so no new Nichols algebra arises.\n\nThird, \\ref{item:hiet-a} composed with \\ref{item:hiet-c} gives the initial $ \\mathfrak{R}_{2, 1}$ \nup to $p \\leftrightarrow q$, so no new Nichols algebra arises.\n\n\\subsection{Case $ \\mathfrak{R}_{2, 2}$}\\label{subsec:hit22} \nWe assume that $ k, p, q\\neq 0 $ and $ k^2\\neq pq $. The associated braiding is\n\\begin{align*}\n(c(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nk^2 x_1\\otimes x_1 & kq x_2\\otimes x_1 + (k^2-pq) x_1\\otimes x_2 \\\\\nkp x_1\\otimes x_2 & -pq x_2\\otimes x_2 \n\\end{pmatrix}.\n\\end{align*}\nLet $N_1 = \\begin{cases} \\ord k^2, &\\text{if } 1 \\neq \\ord k^2; \\\\ \\infty, &\\text{otherwise.}\n\\end{cases}$ and $N_2 = \\begin{cases} \\ord (-pq), &\\text{if } 1 \\neq \\ord (-pq); \\\\ \\infty, &\\text{otherwise.}\n\\end{cases}$\n\n\\begin{pro} If $ k^2 \\neq -1 $ and $ pq \\neq 1$, then there are no quadratic relations.\n\tOtherwise, the Nichols algebras are\n\\begin{align}\\label{eqn:Nichols2,2}\n{\\mathcal B}(V) &= T(V) \/ \\langle x_1x_2 - kqx_2x_1, r_1, r_2\\rangle \n\\end{align}\nwhere $ r_i= x_i^{N_i}, \\, i\\in \\I_2$, only if $ N_i<\\infty $. Also $ \\{ x_2^{a_2}x_1^{a_1}: a_i \\in \\I_{0, N_i-1}\\}$ \nis a PBW-basis and $ \\GK {\\mathcal B}(V) = |\\{i\\in\\I_2: N_i =\\infty\\}|$; if 0, then $\\dim {\\mathcal B}(V) = N_1N_2$.\n\\end{pro}\n\n\\begin{proof} Similar to the proof of Proposition \\ref{prop:case2,1}.\n\\end{proof}\n\nWe next discuss the Nichols algebras arising from the equivalence in \\cite{H}. \n First, \\ref{item:hiet-a} gives rise to the braiding\n\\begin{align}\\label{eq:braiding22a}\n(c'(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nk^2 x_1\\otimes x_1 & kq x_2\\otimes x_1 \\\\\nkp x_1\\otimes x_2 + (k^2-pq) x_2\\otimes x_1 & -pq x_2\\otimes x_2 \n\\end{pmatrix}.\n\\end{align}\n\nLet $N_1$ and $N_2$ be as above.\n\n\\begin{pro} If $ k^2 \\neq -1 $ and $ pq \\neq 1$, then there are no quadratic relations.\n\tOtherwise, the Nichols algebras are\n\t\\begin{align}\\label{eqn:Nichols2,2a}\n\t{\\mathcal B}(V) &=T(V)\/ \\langle x_2x_1 - kpx_1x_2, r_1, r_2\\rangle \n\t\\end{align}\nwhere $ r_i= x_i^{N_i}, \\, i\\in \\I_2$, only if $ N_i<\\infty $. Also $ \\{ x_2^{a_2}x_1^{a_1}: a_i \\in \\I_{0, N_i-1}\\}$ \nis a PBW-basis and $ \\GK {\\mathcal B}(V) = |\\{i\\in\\I_2: N_i =\\infty\\}|$; if 0, then $\\dim {\\mathcal B}(V) = N_1N_2$.\n\\end{pro}\n\n\\begin{proof} Similar to the proof of Proposition \\ref{prop:case2,1}.\n\\end{proof}\n\nSecond, \\ref{item:hiet-c} gives rise to the braiding\n\\begin{align}\\label{eq:braiding22c}\n(c''(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nk^2 x_1\\otimes x_1 & kp x_2\\otimes x_1 \\\\\nkq x_1\\otimes x_2 + (k^2-pq) x_2\\otimes x_1 & -pq x_2\\otimes x_2 \n\\end{pmatrix}.\n\\end{align}\nBut \\eqref{eq:braiding22c} is \\eqref{eq:braiding22a} up to $p \\leftrightarrow q$, so no new Nichols algebra arises here.\n\nThird, \\ref{item:hiet-a} composed with \\ref{item:hiet-c} gives the initial $ \\mathfrak{R}_{2, 1}$ \nup to $p \\leftrightarrow q$, so no new Nichols algebra arises.\n\n\n\n\\subsection{Case $ \\mathfrak{R}_{2, 3}$}\\label{subsec:hit23} We assume that $ k\\neq 0$, and either $p\\neq 0$, or $q\\neq 0$, or $s\\neq 0$. \nThe associated braiding is $(c(x_i \\otimes x_j))_{i, j \\in \\I_2} =$\n\\begin{align*}\n= \\begin{pmatrix}\nk x_1\\otimes x_1 & k x_2\\otimes x_1 + q x_1\\otimes x_1 \\\\\nk x_1\\otimes x_2 + p x_1\\otimes x_1 & k x_2\\otimes x_2 + s x_1\\otimes x_1 + p x_2\\otimes x_1 + q x_1\\otimes x_2\n\\end{pmatrix}.\n\\end{align*}\n\n\n\\begin{rem}\nThe braided vector spaces $\\mathcal{V}(\\epsilon, 2)$ considered in \\cite[\\S 1.2]{AAH} \nfit in this case taking $k=\\epsilon$, $q=1$ and $p = s = 0$. \nIn particular, the Jordan plane $\\mathcal{V}(1, 2)$ and the super Jordan plane \n$\\mathcal{V}(-1, 2)$ belong to this case. \n\\end{rem}\n\nTo state the next result, we need the notation \n\\begin{align*}\nx_{21} = [x_2, x_1]_c = x_2x_1 - \\mu (c(x_2\\otimes x_1)).\n\\end{align*}\n\n\\begin{table}[ht]\n\t\\caption{Nichols algebras of type $ \\mathfrak{R}_{2, 3} $}\\label{tab:h23}\n\t\\begin{center}\n\t\t\\begin{tabular}{| c | p{1cm} | c | p{4,9cm} | c | c |}\\hline\n\t\t\t$k$ & $p$ & $ s $ & $ {\\mathcal J}(V)$ & Basis &$\\GK$\n\t\t\t\\\\\\hline\n\t\t\t\\begin{small}\n\t\t\t\t$-1$ \\end{small} & $ -q $ & $ q^2 $ &$ \\langle x_1^2, x_2^2 - qx_1x_2, x_1x_2 + x_2x_1 \\rangle$ & $ (*_{2, 3})_1 $ \n\t\t\t& \\begin{small} $0, \\dim = 4$ \\end{small}\n\t\t\t\\\\\\cline{3-6}\n\t\t\t & & $ \\neq q^2 $ &$\\langle x_1^2, x_1x_2 + x_2x_1 \\rangle$ & $ (*_{2, 3})_2 $ & $1$\n\t\t\t\\\\\\cline{2-6}\n\t\t\t & \\begin{small} $ \\neq -q $\\end{small} & & \\begin{small} \n\t\t\t \t$\\langle x_1^2, x_2x_{21} + (p-q)x_1x_{21} - x_{21}x_2 \\rangle$\\end{small} & $ (*_{2, 3})_3 $& $2$\n\t\t\t\\\\\\hline\n\t\t\t$1$ & & &$\\langle \\dfrac{q-p}{2}x_1^2 - x_1x_2 +x_2x_1 \\rangle$ & $ (*_{2, 3})_4 $ & $2$\n\t\t\t\\\\\\hline \n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{pro} \\label{prop:Nichols23}\nIf $k \\neq \\pm 1$, then there are no quadratic relations. \nOtherwise, the Nichols algebras are as in Table \\ref{tab:h23}, where\n\\begin{align*}\n (*_{2, 3})_1 &= \\{x_1^{a_1}x_2^{a_2}: 0\\leq a_i \\leq 1 \\} ;\\\\ \n(*_{2, 3})_2 &= \\{x_1^{a_1}x_2^{a_2}: 0\\leq a_1 \\leq 1, 0\\leq a_2 < \\infty \\};\\\\\n(*_{2, 3})_3 &= \\{x_1^{a}x_{21}^{b}x_2^{c}: 0\\leq a \\leq 1, 0\\leq b, c < \\infty \\}; \\\\\n(*_{2, 3})_4 &= \\{x_1^{a_1}x_2^{a_2}: 0\\leq a_i < \\infty \\}.\n\\end{align*}\n\\end{pro}\n\n\\begin{proof} Set $u = \\lambda_1x_1^2+\\lambda_2x_1x_2+\\lambda_3x_2x_1+\\lambda_4x_2^2$. Then\n\\begin{align*}\n\\Delta(u) &= u\\otimes 1 + 1\\otimes u + (\\lambda_1(1+k)+\\lambda_2q+\\lambda_3p+\\lambda_4s)x_1\\otimes x_1 \\\\\n&+ \\lambda_4(1+k)x_2\\otimes x_2 + (\\lambda_2k+\\lambda_3+\\lambda_4p)x_2\\otimes x_1 \\\\\n&+ (\\lambda_2+\\lambda_3k+\\lambda_4q)x_1\\otimes x_2.\n\\end{align*}\nThen all the assertions on quadratic relations hold. The claim in the \nfirst row of Table \\ref{tab:h23} follows then easily. \nTo simplify the discussion of the rest, we consider three cases:\n\\begin{enumerate}[leftmargin=*,label=\\rm{(\\roman*)}] \n\\item \\label{case1:23} $k=-1$, $ p=-q $ and $ s\\neq q^2 $;\n\\item \\label{case2:23} $k=-1$, $ p\\neq -q $;\n\\item \\label{case3:23} $k=1$.\n\\end{enumerate}\nCase \\ref{case1:23}: We first prove by induction that for $n\\geq 2$\n\\begin{align*}\n\\partial_1(x_2^n) &= \\begin{cases}\n\\dfrac{n}{2}qx_2^{n-1}+(\\dfrac{n(n-2)}{4}q^2 +\\dfrac{n}{2}s)x_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is even};\\\\\n-\\dfrac{n-1}{2}qx_2^{n-1}+ \\Big(\\dfrac{(n-1)(n-3)}{4}q^2 & \\hspace{-20pt}+\\dfrac{n-1}{2}s\\Big)x_1x_2^{n-2}, \n\\\\& \\textrm{if } n \\textrm{ is odd}.\n\\end{cases} \\\\\n\\partial_2(x_2^n)&= \\begin{cases}\n-\\dfrac{n}{2}q x_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is even};\\\\\nx_2^{n-1}-\\dfrac{n-1}{2}qx_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is odd}.\n\\end{cases}\n\\end{align*}\nFrom this, we prove again by induction that, for $n\\geq 2$\n\\begin{align*}\n\\partial_1(x_1x_2^{n-1})&= \\begin{cases}\nx_2^{n-1}+\\dfrac{n}{2}qx_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is even};\\\\\nx_2^{n-1}-\\dfrac{n-1}{2}qx_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is odd}.\n\\end{cases}\\\\\n\\partial_2(x_1x_2^{n-1})&= \\begin{cases}\n-x_1x_2^{n-2} , & \\textrm{if } n \\textrm{ is even};\\\\\n0, & \\textrm{if } n \\textrm{ is odd}.\n\\end{cases}\n\\end{align*}\nLet $\\mathfrak{B} = T(V)\/ \\langle x_1^2, x_1x_2 + x_2x_1\\rangle$ and let $\\pi: \\mathfrak{B}\\to {\\mathcal B}(V)$\nbe the natural projection. A standard argument shows that $(*_{2, 3})_2$ generates linearly $\\mathfrak{B}$. \nAssume that the image $B = \\pi(B)$ is not linearly independent\nand pick a linear homogeneous \nrelation of minimal degree $n > 2$\n\\begin{align*}\nr = c_1x_2^n + c_2x_1x_2^{n-1}.\n\\end{align*}\n\nAssume that $n$ is odd. Then\n\\begin{align*}\n0 &= \\partial_2(r) = c_1 \\left(x_2^{n-1}-\\dfrac{n-1}{2}qx_1x_2^{n-2}\\right)\n\\implies c_1=0 \\implies \\\\\n0 &= \\partial_1(r) = c_2\\left(x_2^{n-1}-\\dfrac{n-1}{2}qx_1x_2^{n-2}\\right) \\implies r = 0.\n\\end{align*}\nAssume that $n$ is even. Then\n\\begin{align*}\n0 = \\partial_1(r) &= c_1\\left(\\dfrac{n}{2}qx_2^{n-1}+\\left(\\dfrac{n(n-2)}{4}q^2 +\\dfrac{n}{2}s\\right) x_1x_2^{n-2}\\right)\\\\\n&+c_2\\left(x_2^{n-1}+\\dfrac{n}{2}qx_1x_2^{n-2}\\right)\\\\\n&= \\left(\\dfrac{n}{2}qc_1+c_2\\right)x_2^{n-1} + \\left(\\left(\\dfrac{n(n-2)}{4}q^2 +\\dfrac{n}{2}s\\right)c_1+\\dfrac{n}{2}qc_2\\right)x_1x_2^{n-2};\\\\\n0 = \\partial_2(r) &= -c_1\\dfrac{n}{2}q x_1x_2^{n-2}-c_2 x_1x_2^{n-2} =(-\\dfrac{n}{2}qc_1 -c_2)x_1x_2^{n-2}.\n\\end{align*}\nHence\n\\begin{align*}\n\\dfrac{nq}{2}c_1+c_2 &= 0, &\n\\left(\\dfrac{n(n-2)q^2}{4} +\\dfrac{ns}{2}\\right) c_1 + \\dfrac{nq}{2}c_2 &= 0.\n\\end{align*}\n\nThe system above has non trivial solution iff $q^2 = s$. The claim in row 2 of Table \\ref{tab:h23} is established. \n\n\\smallbreak\n\nCase \\ref{case2:23}: By analogy with the super Jordan plane $\\mathcal{V}(1, 2)$, \nwe look for cubic relations and obtain the following one by \\eqref{eq:deriv-criteria}:\n\\begin{align*}\n0=x_2^2x_1 + (p-q)x_1x_2x_1 - x_1x_2^2 = x_2x_{21} + (p-q)x_1x_{21} - x_{21}x_2.\n\\end{align*}\nLet $\\mathfrak{B} = T(V)\/ \\langle x_1^2, x_2x_{21} + (p-q)x_1x_{21} - x_{21}x_2\\rangle$. \nObserve that $x_1x_{21}= x_{21}x_1$ in $\\mathfrak{B}$. \nArguing as in \\cite{AAH}, that is using the commutation relations, \nwe see that $(*_{2, 3})_3$ is a system of linear generators of $\\mathfrak{B}$. \nWe need the formulae of the derivations on the elements of this basis.\nFirst,\n\\begin{align*}\n\\partial_1(x_{21}^b)&=b(p+q)x_1x_{21}^{b-1}, &\\partial_2(x_{21}^b)&=0,& b&\\geq 1.\n\\end{align*}\nFor $i\\in \\I_2$, $c\\geq 0$, set\n\\begin{align*}\n\\partial_i(x_{2}^c) &=\\partial_{i,c}=\\partial_{i,c,0}+x_1\\partial_{i,c,1},&\n\\text{where } \\partial_{i,c,j} \\in \\Bbbk \\{x_{21}^bx_2^d: b,d \\ge 0\\}.\n\\end{align*}\n\nStraightforward calculations show that, for $ b\\geq 1$, $c\\geq 0$,\n\\begin{align}\n\\partial_1(x_{21}^bx_2^c)&=x_{21}^b(\\partial_{1,c}-2bq\\partial_{2,c})+b(p+q)x_1x_{21}^{b-1}x_2^c, \\\\\n\\partial_2(x_{21}^bx_2^c)&=x_{21}^{b}\\partial_{2,c},\\\\\n\\partial_1(x_1x_{21}^bx_2^c)&=x_{21}^bx_2^c-x_1x_{21}^b\\partial_{1,c,0}+(2b+1)qx_1x_{21}^b\\partial_{2,c,0}, \\\\\n\\partial_2(x_1x_{21}^bx_2^c)&=-x_1x_{21}^{b}\\partial_{2,c,0},\n\\\\\n\\label{eqn:superjordan}\n\\partial_{2,c,0} &= \\begin{cases}\n0, & \\textrm{if } n \\textrm{ is even},\\\\\nx_2^{c-1}, & \\textrm{if } n \\textrm{ is odd}.\n\\end{cases}\n\\end{align}\n\nAssume that the image of $B$ under the projection $\\pi: \\mathfrak{B}\\to {\\mathcal B}(V)$ is not linearly independent.\nPick $r$ a non-trivial linear combination homogeneous of minimal degree $N\\geq 4$. \n\nSuppose first that $ N $ is odd. Then there are scalars $\\lambda_{b}$, $\\mu_{t}$ such that\n\\begin{align*}\nr = \\sum_{0 \\le b \\le \\frac{N-1}{2}} \\lambda_{b}\\,x_{21}^bx_{2}^{N-2b} +\\sum_{0 \\le t \\le \\frac{N-1}{2}} \\mu_{t} \\, x_1x_{21}^{t}x_{2}^{N-1-2t}.\n\\end{align*}\nApplying $ \\partial_2 $ to $r$, we obtain\n\\begin{align*}\n0 &= \\sum_{0 \\le b \\le \\frac{N-1}{2}} \\lambda_{b}x_{21}^b\\partial_{2,N-2b} -\\sum_{0 \\le t \\le \\frac{N-1}{2}} \\mu_{t}x_1x_{21}^{t}\\partial_{2,N-1-2t,0} \\\\\n&= \\sum_{0 \\le b \\le \\frac{N-1}{2}} \\lambda_{b}x_{21}^b(\\partial_{2,N-2b,0} + x_1\\partial_{2,N-2b,1}) -\\sum_{0 \\le t \\le \\frac{N-1}{2}} \\mu_{t}x_1x_{21}^{t}\\partial_{2,N-1-2t,0}\\\\\n&\\stackrel{\\eqref{eqn:superjordan}}{=} \\sum_{0 \\le b \\le \\frac{N-1}{2}} \\lambda_{b}x_{21}^bx_2^{N-1-2b} + \\lambda_{b}x_1x_{21}^b\\partial_{2,N-2b,1}.\n\\end{align*}\nFrom this, we see that $\\lambda_{b}= 0$, $b= 0, 1, \\cdots, \\frac{N-1}{2}$. Therefore\n\\begin{align*}\n0 &= \\partial_1(r) \\stackrel{\\eqref{eqn:superjordan}}{=} \\sum_{0 \\le t \\le \\frac{N-1}{2}} \\mu_{t}(x_{21}^{t}x_2^{N-1-2t}-x_1x_{21}^{t}\\partial_{1,N-1-2t,0});\n\\end{align*}\nhence $r=0$. \n\nAssume next that $N$ is even. Then there are scalars $\\lambda_{b}$, $\\mu_{t}$ such that\n\\begin{align*}\nr = \\sum_{0 \\le b \\le \\frac{N}{2}} \\lambda_{b}x_{21}^bx_{2}^{N-2b} +\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_1x_{21}^{t}x_{2}^{N-1-2t}.\n\\end{align*}\nThus\n\\begin{align}\\label{eqn:1superjordan}\n\\begin{split}\n0 &= \\partial_2(r) \\stackrel{\\eqref{eqn:superjordan}}{=} \\sum_{0 \\le b \\le \\frac{N}{2}} \\lambda_{b}x_1x_{21}^b\\partial_{2,N-2b,1} -\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_1x_{21}^{t}x_2^{N-2-2t}\\\\\n&= \\sum_{0 \\le b \\le \\frac{N -2}{2}} \\lambda_{b}x_1x_{21}^b\\partial_{2,N-2b,1} -\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_1x_{21}^{t}x_2^{N-2-2t}.\n\\end{split}\n\\end{align}\nApplying $\\partial_1$ to $r$, we obtain for some $z \\in {\\mathcal B}^{N-2}(V)$\n\\begin{align*}\n0&= \\sum_{0 \\le b \\le \\frac{N}{2}} \\lambda_{b}x_{21}^b(\\partial_{1,N-2b,0}-2bq\\partial_{2,N-2b,0}) +\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_{21}^{t}x_{2}^{N-1-2t}+x_1z\\\\\n&\\stackrel{\\eqref{eqn:superjordan}}{=} \\sum_{0 \\le b \\le \\frac{N -2}{2}} \\lambda_{b}x_{21}^b\\partial_{1,N-2b,0} +\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_{21}^{t}x_{2}^{N-1-2t}+x_1 z.\n\\end{align*}\nIn particular,\n\\begin{align}\\label{eqn:2superjordan}\n0&= \\sum_{0 \\le b \\le \\frac{N -2}{2}} \\lambda_{b}x_{21}^b\\partial_{1,N-2b,0} +\\sum_{0 \\le t \\le \\frac{N -2}{2}} \\mu_{t}x_{21}^{t}x_{2}^{N-1-2t}.\n\\end{align}\nAlso observe that if $n\\geq 2$ is even, then we obtain that for some $w_i\n\\in \\Bbbk \\{x_{21}^bx_2^d: b,d \\ge 0, \\, b+d =n+i-5\\}$, $ i\\in \\I_2 $, \n\\begin{align*}\n\\partial_{2,n,1}&= \\frac{n}{2}px_{2}^{n-2} +x_{21} w_1, &\\partial_{1,n,0}&= \\frac{n}{2}qx_{2}^{n-1} +x_{21} w_2.\n\\end{align*}\nLooking at the terms $ x_1x_2^{N-2} $ in \\eqref{eqn:1superjordan} and $ x_2^{N-1} $\nin \\eqref{eqn:2superjordan}, we get \n\\begin{align*}\n\\begin{cases}\n\\frac{N}{2}p\\lambda_{0} - \\mu_{0} =0\\\\\n\\frac{N}{2}q\\lambda_{0} + \\mu_{0} =0\n\\end{cases}\n\\end{align*}\nwhose determinant is $ \\frac{N}{2}(p+q)\\neq 0 $. Thus $ \\lambda_{0} = \\mu_{0} =0 $. Similarly, we prove that \n$\\lambda_{i} = \\mu_{i} =0$, $i=0, 1, \\cdots, \\frac{N-2}{2} $. It remains $\\lambda_{\\frac{N}{2}}$, but \n\\begin{align*}\n\\partial_1(x_{21}^{\\frac{N}{2}})= \\frac{N(p+q)}{2}x_1x_{21}^{\\frac{N-2}{2}}\n\\end{align*} \nhence $r=0$. \n\n\\smallbreak\nCase \\ref{case3:23}: \nWe start by the following claim, whose proof is straightforward:\n\t\\begin{align}\\label{eqn:1of2,3}\nc(x_1^n\\otimes x_2)&= x_2\\otimes x_1^n +nq x_1\\otimes x_1^n,\\quad n\\geq 1.\n\t\\end{align}\nLet $\\mathfrak{B} = T(V)\/ \\langle \\frac{q-p}{2}x_1^2 - x_1x_2 +x_2x_1\\rangle$. \nBy a standard argument, $(*_{2, 3})_4$ generates linearly $\\mathfrak{B}$. \nWe need the formulae of the derivations on $(*_{2, 3})_4$.\nFirst, we set \n\\begin{align*}\n\\partial_2(x_2^n) &= \\sum_{0\\le j\\le n-1} d_{j}^{(n-1)} x_1^jx_2^{n-1-j}, \\quad n\\geq 1,\n\\end{align*}\nand claim that \n\\begin{align}\\label{eq:3of2.3}\nd_{0}^{(n-1)}&=n, \\quad n\\geq 1.\n\\end{align}\nIndeed, the case $ n=1 $ is clear. Assume that \\eqref{eq:3of2.3} holds for $n$.\nProjecting $\\Delta(x_2^{n+1})$ to $V \\otimes {\\mathcal B}^{n}(V)$, we get\n\\begin{align*}\n\\sum_{i \\in \\I_2} x_i\\otimes \\partial_i(x_2^{n+1})&= (x_2\\otimes 1)(1\\otimes x_2^n) + (1\\otimes x_2)(x_1\\otimes \\partial_1(x_2^n) + x_2\\otimes \\partial_2(x_2^n))\\\\\n&= x_1\\otimes (x_2\\partial_1(x_2^n)+px_1\\partial_1(x_2^n)+sx_1\\partial_2(x_2^n)+qx_2\\partial_2(x_2^n)) \\\\\n&+x_2\\otimes (x_2^n+x_2\\partial_2(x_2^n)+px_1\\partial_2(x_2^n)).\n\\end{align*}\nHence, by the inductive hypothesis,\n\\begin{align*}\n\\partial_2(x_2^{n+1})&= x_2^n+x_2\\partial_2(x_2^n)+px_1\\partial_2(x_2^n) \\\\\n&= x_2^n+x_2(nx_2^{n-1}+\\sum_{1\\le j\\le n-1} d_{j}^{(n-1)} x_1^jx_2^{n-1-j})+px_1\\partial_2(x_2^n) \\\\\n&= (n+1)x_2^n +x_1(x_2+\\dfrac{p-q}{2}x_1)\\sum_{1\\le j\\le n-1} d_{j}^{(n-1)} x_1^{j-1}x_2^{n-1-j} \\\\\n&+px_1\\partial_2(x_2^n),\n\\end{align*}\nand the claim is proved.\n\nObserve that, by \\eqref{eqn:1of2,3}, for $ n\\geq 3 $\n\\begin{align}\\label{eq:1of2.3}\n\\partial_2(x_1^ix_2^{n-i})&=\\begin{cases}\n0, & \\textrm{if } i=n;\\\\\nx_1^{i}\\partial_2(x_2^{n-i}), & \\textrm{if } 0\\leq i2$. By \\eqref{eq:1of2.3}\n\\begin{align}\\label{eq:2of2.3}\n0 = \\partial_2(r) = \\sum_{0\\le i\\le n-1} c_i x_1^{i}\\partial_2(x_2^{n-i}).\n\\end{align}\nObserve that the term $ x_2^{n-1} $ appears only one time in \\eqref{eq:2of2.3}. Furthermore, by \\eqref{eq:3of2.3}, we can rewrite \\eqref{eq:2of2.3} as\n\\begin{align}\\label{eq:4of2.3}\n0= \\partial_2(r) = c_0 nx_2^{n-1} + \\sum_{1\\le j\\le n-1}m_jx_1^jx_2^{n-1-j} \n\\end{align}\nfor some $ m_j\\in \\Bbbk $. By minimality of $n$, we obtain $ c_0 = 0 $. Similarly, we can replace \\eqref{eq:4of2.3} by\n\\begin{align}\n0= \\partial_2(r) = c_1 (n-1)x_1x_2^{n-2} + \\sum_{2\\le j\\le n-1}m_j'x_1^jx_2^{n-1-j} \n\\end{align}\nwhat gives us $ c_1=0 $. Inductively, we get $ c_i = 0, \\, 0\\leq i 2$. By \\eqref{eq:1of1.3},\n\\begin{align}\\label{eq:2of1.3}\n0 = \\partial_2(r) = \\sum_{0\\le i\\le n-1} c_i x_1^{i}\\partial_2(x_2^{n-i}).\n\\end{align}\nLooking at the monomials $ x_2^{n-i} $in \\eqref{eq:2of1.3} as in case \\ref{case3:23} of the proof of Proposition \\ref{prop:Nichols23}, we get $ c_i = 0, \\, 0\\leq i 0.\n\\end{align*}\n\nThe claim is proved. Then we check, by an inductive argument, that \n\\begin{align*}\n\\partial_1(x_1^a) &= \\begin{cases}\n(\\frac{a-2}{2})_k \\, x_1^{a-1}, \\qquad \\,\\, & \\textrm{if } a \\textrm{ is even;}\\\\\n(\\frac{a-1}{2})_k \\, x_1^{a-1}, & \\textrm{if } a \\textrm{ is odd.}\n\\end{cases} \\\\\n\\partial_2(x_1^a) &= \\begin{cases}\nq(\\frac{a-2}{2})_k \\, x_1^{a-2}x_2, & \\textrm{if } a \\textrm{ is even;}\\\\\nq(\\frac{a-3}{2})_k \\, x_1^{a-3}x_2x_1, & \\textrm{if } a \\textrm{ is odd, } a\\geq 3;\\\\\n0, & \\textrm{if } a =1,\n\\end{cases}\n\\end{align*}\n$ a\\in \\mathbb{N}$.\nAlso, we have that, for $ b\\geq 1$\n\\begin{align*}\n\\partial_1((x_2x_1)^b) &= 0, && \\partial_2((x_2x_1)^b) = (2b-1)_k \\, x_1(x_2x_1)^{b-1}.\n\\end{align*}\nTherefore, for $ a,b\\geq 1 $ \n\\begin{align*}\n\\partial_1(x_1^a(x_2x_1)^b) &= \\begin{cases}\n(\\frac{a-2}{2})_k \\, x_1^{a-1}(x_2x_1)^b, & \\textrm{if } a \\textrm{ is even;}\\\\\n(\\frac{a-1}{2}+2b)_k \\, x_1^{a-1}(x_2x_1)^b, & \\textrm{if } a \\textrm{ is odd.}\n\\end{cases} \\\\\n\\partial_2(x_1^a(x_2x_1)^b) &= \\begin{cases}\n(\\frac{a-2}{2}+2b)_k \\, x_1^{a+1}(x_2x_1)^{b-1}, & \\textrm{if } a \\textrm{ is even;}\\\\\nq(\\frac{a-3}{2})_k \\, x_1^{a-3}(x_2x_1)^{b+1}, & \\textrm{if } a \\textrm{ is odd, } a\\geq 3;\\\\\n0, & \\textrm{if } a =1.\n\\end{cases}\n\\end{align*}\nand then\n\\begin{align*}\n\\partial_1(x_1^a(x_2x_1)^bx_2) &= \\begin{cases}\n\\partial_1(x_1^a(x_2x_1)^b)x_2, & \\textrm{if } a \\textrm{ is even;}\\\\\n(\\frac{a+1}{2}+2b)_k \\, x_1^{a-1}(x_2x_1)^bx_2, & \\textrm{if } a \\textrm{ is odd.}\n\\end{cases} \\\\\n\\partial_2(x_1^a(x_2x_1)^bx_2) &= \\begin{cases}\n(\\frac{a}{2}+2b)_k \\, x_1^{a+1}(x_2x_1)^{b-1}x_2, & \\textrm{if } a \\textrm{ is even;}\\\\\n\\partial_2(x_1^a(x_2x_1)^b)x_2, & \\textrm{if } a \\textrm{ is odd, } a\\geq 3;\\\\\n0, & \\textrm{if } a =1.\n\\end{cases}\n\\end{align*}\nWe then proceed as in the previous case. Namely, let $r \\in \\ker (\\widetilde{{\\mathcal B}} \\to {\\mathcal B}(V))$\nbe an homogeneous relation of degree $n$ with $n\\geq 3$ minimal.\nWe consider separately the cases $n$ odd and $n$ even.\nUsing the derivations, we see that the relations $r_{1,N}$ and $r_{2,N}$ hold.\nAlso, any other relation arises in higher degree.\nIn this way, rows $3$ and $4$ of Table \\ref{tab:h14} are established.\n\\end{proof}\n\n\nWe finally discuss the Nichols algebras arising from the equivalence in \\cite{H}. \nFirst, \\ref{item:hiet-a} and \\ref{item:hiet-a} composed with \\ref{item:hiet-c} give rise to the same braiding\n\\begin{align*}\n(c'(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nq x_2\\otimes x_2 & k x_1\\otimes x_2 \\\\\nk x_2\\otimes x_1 & p x_1\\otimes x_1\n\\end{pmatrix}\n\\end{align*}\nwhich is $ \\mathfrak{R}_{1, 4}$ up to $p \\leftrightarrow q$, so no new Nichols algebra arises.\n\nSecond, \\ref{item:hiet-c} gives the initial $ \\mathfrak{R}_{1, 4}$, so no new Nichols algebra arises.\n\n\n\n\n\\subsection{Case $ \\mathfrak{R}_{0, 1}$}\\label{subsec:hit01} The associated braiding is\n\\begin{align*}\n(c(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nkx_1\\otimes x_1 & -kx_2\\otimes x_1 \\\\\n-kx_1\\otimes x_2 & kx_2\\otimes x_2 + kx_1\\otimes x_1\n\\end{pmatrix}.\n\\end{align*}\n\n\\begin{pro} \nIf $k^2 \\neq 1$, then there are no quadratic relations. \n\n\\smallbreak\nIf $k=1$, then \n\\begin{align}\\label{eqn:Nichols10,k1}\n{\\mathcal B}(V) =T(V)\/ \\langle x_1x_2 + x_2x_1 \\rangle,\n\\end{align}\n$B_1 = \\{x_1^{a_1}x_2^{a_2}: a_i \\in \\mathbb{N}_0 \\}$ is a PBW-basis of ${\\mathcal B}(V)$ and \n$\\GK {\\mathcal B}(V) = 2$.\n\n\\smallbreak\nIf $k=-1$, then \n\\begin{align}\\label{eqn:Nichols10,k-1}\n{\\mathcal B}(V) =T(V)\/ \\langle x_1^2, x_1x_2 - x_2x_1 \\rangle,\n\\end{align}\n$B_2 = \\{x_1^{a_1}x_2^{a_2}: a_1 \\in \\I_{0,2}, \\, a_2 \\in \\mathbb{N}_0 \\}$ is a PBW-basis of $ {\\mathcal B}(V) $. Hence \\newline\n$\\GK {\\mathcal B}(V) = 1$.\n\\end{pro}\n\n\\begin{proof} Set $u = \\lambda_1x_1^2+\\lambda_2x_1x_2+\\lambda_3x_2x_1+\\lambda_4x_2^2$. Then\n\\begin{align*}\n\\Delta(u) &= u\\otimes 1 + 1\\otimes u + ((1+k)\\lambda_1+k\\lambda_4)x_1\\otimes x_1 +(1+k)\\lambda_4 x_2\\otimes x_2 \\\\\n&+ (\\lambda_2-k\\lambda_3)x_1\\otimes x_2 + (\\lambda_3-k\\lambda_2)x_2\\otimes x_1.\n\\end{align*}\nFrom here, all assertions about quadratic relations hold. \n\nCase $k =1$: Let $\\mathfrak{B} = T(V)\/ \\langle x_1x_2 + x_2x_1 \\rangle$. \nBy a standard argument, $B_1$ is a basis of $\\mathfrak{B}$. Note that, for $n\\geq 2$\n\\begin{align*}\n\\partial_1(x_2^n)&= \\binom{n}{2} x_1x_2^{n-2}, & \\partial_2(x_2^n)&=n x_2^{n-1}.\n\\end{align*}\nLet $ n\\geq 3 $ and $i \\in \\I_{2, n-2}$. Then\n\\begin{align}\\label{eqn:prop:0.1}\n\\begin{split}\n\\partial_1(x_1^ix_2^{n-i})&= \\binom{n-i}{2} x_1^{i+1}x_2^{n-i-2}+ix_1^{i-1}x_2^{n-i}, \\\\\n\\partial_2(x_1^ix_2^{n-i})&=(-1)^i (n-i) x_1^ix_2^{n-i-1}.\n\\end{split}\n\\end{align}\nObserve that the formulae \\eqref{eqn:prop:0.1} also hold for $i\\in \\{0, 1, n-1, n\\} $ taking by $0$ \nthe terms that are not well defined. \n\nAssume that the image of $B_1$ under the projection $\\pi: \\mathfrak{B}\\to {\\mathcal B}(V)$ is not linearly independent. \nPick $0 \\neq r = \\sum_{i=0}^N c_i x_1^ix_2^{N-i}$ a homogeneous relation of minimal degree $ N>2$. Thus\n\\begin{align*}\n0 &= \\partial_2(r) = \\sum_{i=0}^{N-1} c_i (-1)^i(N-i) x_1^ix_2^{N-i-1}.\n\\end{align*}\nThen, $ c_i = 0 $ for $i \\in \\I_{0, N-1}$. But $ \\partial_1(x_1^N) = N x_1^{N-1} $, hence $r = 0$.\n\n\\smallbreak\nCase $k = -1$: Let $\\mathfrak{B} = T(V)\/ \\langle x_1^2, x_1x_2 - x_2x_1 \\rangle$. \nBy a standard argument, $B_2$ is a basis of $\\mathfrak{B}$. Note that, for $n\\geq 2$\n\\begin{align*}\n\\partial_1(x_2^n) &= \\begin{cases}\n-\\frac{n}{2} x_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is even;}\\\\\n-\\frac{n-1}{2} x_1x_2^{n-2}, & \\textrm{if } n \\textrm{ is odd.}\n\\end{cases}, & \\partial_2(x_2^n)&=\\begin{cases}\n0, & \\textrm{if } n \\textrm{ is even;}\\\\\nx_2^{n-1}, & \\textrm{if } n \\textrm{ is odd.}\n\\end{cases}.\n\\end{align*}\nThen, for $n\\geq 2$,\n\\begin{align*}\n\\partial_1(x_1x_2^n) &= x_2^{n}, & \\partial_2(x_1x_2^n)&=\\begin{cases}\n0, & \\textrm{if } n \\textrm{ is even;}\\\\\nx_1x_2^{n-1}, & \\textrm{if } n \\textrm{ is odd.}\n\\end{cases}.\n\\end{align*}\n\n\nSuppose that the image of $B$ under the projection $\\pi: \\mathfrak{B}\\to {\\mathcal B}(V)$ is not linearly independent. \nPick $r = c_1 x_2^{N} + c_2 x_1x_2^{N-1}$ a homogeneous non-trivial relation of minimal degree $ N>2$. Applying $ \\partial_1 $ to $ r $, we obtain $r = 0$.\n\\end{proof}\n\nWe next discuss the Nichols algebras arising from the equivalence in \\cite{H}. \nFirst, \\ref{item:hiet-a}, \\ref{item:hiet-b} and \\ref{item:hiet-a} composed with \\ref{item:hiet-c} give rise to the same braiding\n\\begin{align*}\n(c'(x_i \\otimes x_j))_{i, j \\in \\I_2} = \\begin{pmatrix}\nkx_1\\otimes x_1 + kx_2\\otimes x_2 & -kx_2\\otimes x_1 \\\\\n-kx_1\\otimes x_2 & kx_2\\otimes x_2 \n\\end{pmatrix}.\n\\end{align*}\nSince \\ref{item:hiet-b} is a change of basis, the Nichols algebras are isomorphic.\n\nFinally, \\ref{item:hiet-c} gives the initial braiding\n$ \\mathfrak{R}_{0, 1}$, so no new Nichols algebra arises.\n\n\n\\section{Appendix}\n\nHere we collect all isomorphism classes of algebras arising as Nichols algebras in Theorem \\ref{th:main}, \nsee the information in Table \\ref{tab:general}. In many cases a change of variables is needed, and we leave to the reader its explicit calculation. All the algebras are of the form $T(W)\/ {\\mathcal J}$, where $W$ has a basis $y_1, y_2$. \n\n\\newcommand{\\zeta}{\\zeta}\n\\newcommand{\\eta}{\\eta}\n\\begin{table}[ht]\n\t\\caption{Algebras arising as Nichols algebras of rank 2 }\\label{tab:appendix}\n\t\\begin{center}\n\t\t\\begin{tabular}{|p{3cm} | p{5cm} | p{3cm} | } \n\t\\hline\tAlgebra & ${\\mathcal J}$ \t& Parameters\t\\\\\t\\hline\n\\begin{small}quantum plane \\end{small} &\t$\\langle y_1y_2 - \\zeta y_2 y_1 \\rangle$\t\t& $\\zeta \\in \\Bbbk^{\\times}$ \n\t\t\t\\\\ \\hline\n\\begin{small}quantum plane \\end{small} &\t$\\langle y_1^M, y_1y_2 - \\zeta y_2 y_1 \\rangle$\t\t& $\\zeta \\in \\Bbbk^{\\times}$,\\newline $M \\in \\mathbb{N}_{\\ge2}$ \n\\\\ \\hline\n\\begin{small}quantum plane \\end{small} &\t$\\langle y_1^M, y_1y_2 - \\zeta y_2 y_1, y_2^N \\rangle$\t\t& $\\zeta \\in \\Bbbk^{\\times}$, \\newline $M, N \\in \\mathbb{N}_{\\ge2}$ \n\\\\ \\hline\n\\begin{small}deformation of\\newline a quantum plane \\end{small} &\t$\\langle y_2^2 - \\zeta y_1^2, y_1y_2 -\\eta y_2 y_1, y_1^N \\rangle$\t\t& $\\eta, \\zeta \\in \\Bbbk^{\\times}$, \\newline\n$N \\in \\mathbb{N}_{\\ge 3}$ \n\\\\ \\hline\n\\begin{small}deformation of\\newline a quantum plane \\end{small} &\t$\\langle y_2^2 - \\zeta y_1^2, y_1y_2 -\\eta y_2 y_1 \\rangle$\t\t& $\\eta, \\zeta \\in \\Bbbk^{\\times}$ \n\\\\ \\hline\n\\begin{small}deformation of\\newline an exterior algebra \\end{small}&\t$\\langle y_1^2, y_2^2 - y_1 y_2, y_1 y_2 + y_2 y_1 \\rangle$\t\t& \n\\\\ \\hline\n&\t$\\langle y_1^2 - \\zeta y_2^2, y_1y_2 + \\epsilon y_2 y_1, y_1y_2^N \\rangle$\t\t& $\\zeta \\in \\Bbbk^{\\times}$, \n $\\epsilon \\in \\mathbb{G}_2$, \\newline $N \\in \\mathbb{N}_{\\ge2}$\n\\\\ \\hline\n\\begin{small}Jordan plane \\end{small} &\t$\\langle y_1^2 -y_1y_2 + y_2 y_1 \\rangle$\t\t& \n\\\\ \\hline\n\\begin{small} super Jordan plane \\end{small} &\t$\\langle y_1^2, y_2^2y_1 - y_1 y_2 y_1 - y_1 y_2^2 \\rangle$ & \n\\\\ \\hline\n&\t$\\langle y_1 y_2, y_2 y_1 \\rangle$\t\t&\n\\\\ \\hline\n &\t$\\langle y_1 y_2, y_2 y_1, y_1^{2N}-\\zeta y_2^{2N} \\rangle$\t\t& $\\zeta \\in \\Bbbk^{\\times}$, $N \\in \\mathbb{N}$ \n\\\\ \\hline\n &\t$\\langle y_1^2 - \\zeta y_2^2 \\rangle$\t& $\\zeta \\in \\Bbbk^{\\times}$ \n\\\\ \\hline\n &\t$\\langle y_1^2 - \\zeta y_2^2, r_{1,N}, r_{2,N} \\rangle$, \\newline\n cf. \\eqref{eqn:relation1of1.4}, \\eqref{eqn:relation2of1.4}\t\t& $\\zeta \\in \\Bbbk^{\\times}$, $N \\in \\mathbb{N}_{\\ge 3}$ \n\\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}