Entanglement Entropy and Entanglement Spectrum for TwoDimensional Classical Spin Configuration
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. They indicates that the matrix dimension of the MPS represents a length scale of spin correlation. On the other hand, the quantum spinchain models can be mapped onto twodimensional (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 3state 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 $T_{c}$. 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, longrange 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 lowlying 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.0104
 Bibcode:
 2011arXiv1109.0104M
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 13 pages, 14 figures