Identification of Degeneracy and Criticality of Two-Dimensional Statistical and Quantum Systems by the Boundary States of Tensor Networks

TL;DR

This paper introduces a method to analyze 2D statistical and quantum systems via their boundary states in tensor networks, revealing degeneracy, criticality, and universality classes through effective 1D theories.

Contribution

It establishes a systematic scheme connecting 2D systems to effective 1D theories using tensor networks, enabling characterization of degeneracy and criticality via boundary states.

Findings

Degeneracy determined by boundary thermal state purity

Criticality reflected in entanglement entropy scaling law

Universal critical behavior characterized by central charge and correlation length scaling

Abstract

We propose a systematic scheme to reach the properties of two-dimensional (2D) statistical and quantum systems by studying the effective (1+1)-dimensional theory that is constructed from the tensor network representation. On on hand, we discover that the degeneracy of the 2D system can be determined by the purity of the boundary thermal state, which is the density operator of the effective theory at zero (effective) temperature. On the other hand, we find that the gapped (or critical) 2D system leads to a gapped (or critical) effective (1+1)-dimensional theory whose criticality can be accessed by the entanglement entropy $S$ of its ground state dubbed as boundary pure state. We also uncover that for the critical systems, $S$ obeys the same logarithmic law as that found in the critical 1D quantum chains, which reads $S = (\kappa c/6)\log_2 D + const.$, with $c$ the central charge and $\kappa$ a constant related to the scaling property of the correlation length $\xi$ as $\xi \sim D^{\kappa}$. Such a scaling law presents an efficient way to characterize the critical universality class of the original 2D systems. An important implication of our work is that many well-established theories for 1D quantum chains become available for studying 2D systems with the help of the proposed lower dimensional correspondence.