Transfer Matrices and Excitations with Matrix Product States

TL;DR

This paper explores how the transfer matrix of matrix product states encodes information about low energy excitations and correlation functions in quantum many-body systems, linking static properties to dynamic spectra.

Contribution

It establishes a connection between the MPS transfer matrix spectrum and the dispersion relations of low energy excitations, supported by numerical data across various models.

Findings

MPS transfer matrix minima relate to low energy dispersion minima

The eigenspectrum of the MPS transfer matrix reflects the low energy spectrum structure

Finite bond dimension MPS can approximate the quantum transfer matrix via renormalization

Abstract

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilson's Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.