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arXiv:1402.6767 · Journal of the Physical Society of Japan

Snapshot Spectrum and Critical Phenomenon for Two-Dimensional Classical Spin Systems

Yukinari Imura, Tsuyoshi Okubo, Satoshi Morita, Kouichi Okunishi

Published Oct 15, 2014• 12 citations
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Abstract

We investigate the eigenvalue distribution of the snapshot density matrix (SDM) generated by Monte Carlo simulation for two-dimensional classical spin systems. We find that the distribution in the high-temperature limit is well explained by the random-matrix theory, while that in the low-temperature limit can be characterized by the zero-eigenvalue condensation. At the critical point, we obtain the power-law distribution with a nontrivial exponent $α\equiv(2-η)/(1-η)$ and the asymptotic form of the snapshot entropy, on the basis of the relationship of the SDM with the correlation function matrix. The aspect-ratio dependence of the SDM spectrum is also mentioned.

DOI
10.7566/jpsj.83.114002

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We investigate the eigenvalue distribution of the snapshot density matrix (SDM) generated by Monte Carlo simulation for two-dimensional classical spin systems. We find that the distribution in the high-temperature limit is well explained by the random-matrix theory, while that in the low-temperature limit can be characterized by the zero-eigenvalue condensation. At the critical point, we obtain the power-law distribution with a nontrivial exponent α≡(2−η)/(1−η)α\equiv(2-η)/(1-η)α≡(2−η)/(1−η) and the asymptotic form of the snapshot entropy, on the basis of the relationship of the SDM with the correlation function matrix. The aspect-ratio dependence of the SDM spectrum is also mentioned.
We introduce a snapshot density matrix and snapshot spectrum for world-line (WL) quantum Monte Carlo simulations, by integrating out the continuous-imaginary-time index of WL snapshots. For the transverse-field Ising chain, we reveal fundamental properties of the snapshot spectrum as follows. (i) In the ordered phase, the isolated peak corresponding to the maximum eigenvalue appears, indicating the classical nature of WL configurations. (ii) In the disordered phase, the spectrum converges to the dome-like distribution characterized by an effective aspect ratio of the snapshots. (iii) At the critical point, the power-law distribution is observed with an appropriate scaling analysis of the fixed aspect ratio. For the XXZ chain, we also obtain numerical results consistent with the above properties. In addition, the representation-basis dependence of the snapshot spectrum is also discussed.
In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. They indicates that the matrix dimension of the MPS represents a length scale of spin correlation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones. Motivated by the scaling and the mapping, we introduce new entanglement entropy for 2D classical spin configuration as well as entanglement spectrum, and examine their basic properties in Ising and 3-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations concerned with length scales in the snapshot at TcT_{c}Tc​. There, the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scale. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, we find multiple gaps in the entanglement spectrum, and in contrast to standard topological phases, the low-lying entanglement levels below the gap represent spontaneous symmetry breaking. Based on these observations, we discuss about the amount of information contained in one snapshot in a viewpoint of the CFT scaling.
In this letter, the random matrix theory representation of a bond-percolation model on square lattice is presented. The behavior of random matrix model can be determined only by its two largest eigenvalues. The second largest eigenvalue sits exactly on the edge of semicircle part of eigenvalue’s distribution and its position is a function of p, on the other hand the first largest eigenvalue is disjointed from other eigenvalues and its distribution is Gaussian. Also the first largest eigenvalue is responsible for scaling properties near criticality. Numerical simulations show power-law divergences emerged from coalescence of two largest eigenvalues near critical point at the thermodynamic limit. In this letter a scaling formalism is presented which describes complete scaling behavior of largest eigenvalue’s fluctuations with a set of scaling exponents in finite-size systems.
We study spectral densities for systems on lattices, which, at a phase transition display, power-law spatial correlations. Constructing the spatial correlation matrix we prove that its eigenvalue density shows a power law that can be derived from the spatial correlations. In practice time series are short in the sense that they are either not stationary over long time intervals or not available over long time intervals. Also we usually do not have time series for all variables available. We shall make numerical simulations on a two-dimensional Ising model with the usual Metropolis algorithm as time evolution. Using all spins on a grid with periodic boundary conditions we find a power law, that is, for large grids, compatible with the analytic result. We still find a power law even if we choose a fairly small subset of grid points at random. The exponents of the power laws will be smaller under such circumstances. For very short time series leading to singular correlation matrices we use a recently developed technique to lift the degeneracy at zero in the spectrum and find a significant signature of critical behavior even in this case as compared to high temperature results which tend to those of random matrix models.
We reexamine the snapshot entropy of the Ising and three-states Potts models on the $L\ifmmode\times\else\texttimes\fi{}L$ square lattice. Focusing on how to encode the spin snapshot, we find that the entropy at Tc{T}_{c}Tc​ scales asymptotically as $S\ensuremath{\sim}(1/3)lnL$ that strongly reminds us of the entanglement entropy in one-dimensional quantum critical systems. This finding seems to support that the snapshot entropy after the proper encoding is related to the holographic entanglement entropy. On the other hand, the anomalous scaling ${S}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\eta}}ln\ensuremath{\chi}$ for the coarse-grained snapshot entropy holds even for the proper encoding. These features originate in the fact that the largest singular value of the snapshot matrix is regulated by the proper encoding.
The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{\underline{}}{\mathrm{Tr}{\mathrm{\ensuremath{\rho}}}_{\mathrm{A}}^{\mathrm{\ensuremath{\alpha}}}}$ of the reduced density matrix ${\ensuremath{\rho}}_{A}$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random XX\mathit{XX}XX model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the XX\mathit{XX}XX case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.