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 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 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.
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.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability
p, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
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.
We study the scaling of the entanglement entropy in different classes of one-dimensional fermionic quasiperiodic systems with and without pairing, focusing on multifractal critical points/phases. We find that the entanglement entropy scales logarithmically with the subsystem size
NA with a proportionality coefficient
C, as in homogeneous critical points, apart from possible additional small oscillations. In the absence of pairing, we find that the entanglement entropy coefficient
C is nonuniversal and depends significantly and nontrivially both on the model parameters and electron filling, in multifractal critical points. In some of these points,
C can take values close to the homogeneous (or ballistic) system, although it typically takes smaller values. We find a close relation between the behavior of the entanglement entropy and the small-
q (long-wavelength) dependence of the momentum structure factor
S(q).
S(q) increases linearly with
q as in the homogeneous case, with a slope that grows with
C. In the presence of pairing, we find that even the addition of small anomalous terms affects very significantly the scaling of the entanglement entropy compared to the unpaired case. In particular, we focused on topological phase transitions for which the gap closes with either extended or critical multifractal states. In the former case, the scaling of the entanglement entropy mirrors the behavior observed at the critical points of the homogeneous Kitaev chain, while in the latter, it shows only slight deviations arising at small length scales. In contrast with the unpaired case, we always observe $\mathcal{C}\ensuremath{\approx}1/6$ for different critical points, the known value for the homogeneous Kitaev chain with periodic boundary conditions.