Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

TL;DR

This paper provides a comprehensive overview of methods for diagonalizing transfer matrices and matrix product operators, covering both classic and new results in tensor network approaches for many-body physics.

Contribution

It introduces novel computational techniques and unifies existing methods for analyzing transfer matrices and matrix product operators in statistical physics.

Findings

Exact solutions for transfer matrices in equilibrium and non-equilibrium systems

Development of tensor network algorithms for eigenvector computation

Insights into matrix product operator algebras and their applications

Abstract

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.