Continuum limits of Matrix Product States

TL;DR

This paper investigates which translationally invariant matrix product states can be viewed as discretizations of continuum states, analyzing their limits through renormalization procedures and characterizing the set of such states.

Contribution

It introduces a detailed analysis of continuum limits of matrix product states, expanding understanding of their relation to continuum quantum states and renormalization flows.

Findings

The set of states with a continuum limit strictly contains continuous matrix product states.

Certain states attain a continuum limit after finite coarse-graining steps.

Examples illustrate states with and without continuum limits.

Abstract

We determine which translationally invariant matrix product states have a continuum limit, that is, which can be considered as discretized versions of states defined in the continuum. To do this, we analyse a fine-graining renormalization procedure in real space, characterise the set of limiting states of its flow, and find that it strictly contains the set of continuous matrix product states. We also analyse which states have a continuum limit after a finite number of a coarse-graining renormalization steps. We give several examples of states with and without the different kinds of continuum limits.