Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in the configurations of classical critical systems, whose computation do not require full knowledge of the wave function. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an Ansatz ensemble of random fractal images. By virtue of the central limit theorem, our Ansatz accurately reproduces the entanglement spectra of actual Potts snapshots without any fine tuning of parameters or artificial restrictions on ensemble choice. It provides a microscopic interpretation of the results of previous studies, which established a relation between the scaling behavior of snapshot entropy and the critical exponent. More importantly, it elucidates the role of ensemble disorder in restoring conformal invariance, an aspect previously ignored. Away from criticality, the breakdown of scale invariance leads to a renormalization of the parameter Σ in the random fractal Ansatz, whose variation can be used as an alternative determination of the critical exponent. We conclude by providing a recipe for the explicit construction of fractal unit cells consistent with a given scaling exponent.
Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in the configurations of classical critical systems, whose computation do not require full knowledge of the wave function. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an Ansatz ensemble of random fractal images. By virtue of the central limit theorem, our Ansatz accurately reproduces the entanglement spectra of actual Potts snapshots without any fine tuning of parameters or artificial restrictions on ensemble choice. It provides a microscopic interpretation of the results of previous studies, which established a relation between the scaling behavior of snapshot entropy and the critical exponent. More importantly, it elucidates the role of ensemble disorder in restoring conformal invariance, an aspect previously ignored. Away from criticality, the breakdown of scale invariance leads to a renormalization of the parameter Σ in the random fractal Ansatz, whose variation can be used as an alternative determination of the critical exponent. We conclude by providing a recipe for the explicit construction of fractal unit cells consistent with a given scaling exponent.
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 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.
We examine the snapshot entropy of general fractal images defined by their singular values. Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension. These fermions allow for a holographic interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the Calabrese-Cardy formula. However, the coarse-grained entropy exhibits a linear scaling due to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in one-dimensional quantum near-critical systems.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
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. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones by the Suzuki-Trotter decomposition. Motivated by the scaling and the mapping, we introduce information entropy for 2D classical spin configurations as well as a spectrum, and examine their basic properties in the Ising and the three-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 of the entropy compatible with the CFT and the MPS results. Thus, we propose that the entropy is a kind of "holographic" entanglement entropy. At T(c), 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 scales, respectively. 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, the spectrum, which is a set of logarithms of the singular values, also seems to be a holographic entanglement spectrum. We find multiple gaps in the spectrum, and in contrast to the topological phases, the low-lying levels below the gap represent spontaneous symmetry breaking. These contrasts are strong evidence of the dual nature of the holography. Based on these observations, we discuss the amount of information contained in one snapshot.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for n-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.
We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L<16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.