Developments in the Tensor Network — from Statistical Mechanics to Quantum Entanglement

Abstract

Tensor networks (TNs) have become one of the most essential building blocks\nfor various fields of theoretical physics such as condensed matter theory,\nstatistical mechanics, quantum information, and quantum gravity. This review\nprovides a unified description of a series of developments in the TN from the\nstatistical mechanics side. In particular, we begin with the variational\nprinciple for the transfer matrix of the 2D Ising model, which naturally leads\nus to the matrix product state (MPS) and the corner transfer matrix (CTM). We\nthen explain how the CTM can be evolved to such MPS-based approaches as density\nmatrix renormalization group (DMRG) and infinite time-evolved block decimation.\nWe also elucidate that the finite-size DMRG played an intrinsic role for\nincorporating various quantum information concepts in subsequent developments\nin the TN. After surveying higher-dimensional generalizations like tensor\nproduct states or projected entangled pair states, we describe tensor\nrenormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson\ntype real-space renormalization groups, focusing on their fixed point\nstructures. We then discuss how the difficulty in TRGs for critical systems can\nbe overcome in the tensor network renormalization and the multi-scale\nentanglement renormalization ansatz.\n