Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

Abstract

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.

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

“…Since this relation successfully explains the numerical result in a wide range of », we think that it is a correct asymptotic form of S rather than the naive logarithmic dependence proposed in Ref. 5. The snapshot spectrum may be a different concept from the entanglement spectrum in the quantum system, although our motivation originally came from the entanglement for the quantum system.…”

“…(20) shows the correct asymptotic behavior of the snapshot entropy, rather than the naive logarithmic behavior proposed in Ref. 5. As » approaches N, the theoretical curve deviates from the numerical result.…”

“…(17) could be S $ ð1=6Þ log þ const for a small-» region and S $ log þ const for a large-» region. 5) However, the theoretical justification for these asymptotic behaviors is missing in the context of statistical mechanics, so that a precise verification is desired. As discussed in Sect.…”

Tensor network and a black hole

“…þ 1Þ is the finiteentanglement scaling exponent, and is the matrix dimension [35][36][37]. It has been shown that the correlation length is given by ¼ .…”

Snapshot Spectra in the World-Line Quantum Monte Carlo for One-Dimensional Quantum Spin Systems

“…For two-dimensional (2D) classical spin systems, very recently, a similar but different idea of the quantum entanglement was proposed for classical MC; a snapshot spectrum and snapshot entropy were defined for the singular value spectrum of a snapshot generated by a MC simulation. [13][14][15] It was shown that the snapshot spectrum gave hierarchical decomposition of snapshot images in Ref. [13], and a characteristic behavior of a snapshot entropy was also observed.…”