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

Abstract

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

Heaps of Fish: arrays, generalized associativity and heapoids

“…Our main insight comes from the development of a diagrammatic calculus based on hypergraphs that generalizes the diagrams of category theory [Lei14] and tensor networks [ONU21]. The basic idea is to represent multi-index objects, such as higher order matrices and higher arity relations, as hyperedges of a hypergraph whose vertices represent indices.…”