Charting the space of ground states with tensor networks

TL;DR

This paper uses tensor network methods to classify and understand the topological properties of ground state spaces in gapped many-body quantum systems, revealing obstructions to continuous representations and extending to higher dimensions.

Contribution

It introduces a gerbe-based framework to analyze topological features of ground state families using tensor networks, generalizing previous line bundle approaches.

Findings

Nontrivial gerbes indicate obstructions to continuous MPS representations.

Examples over $S^3$ and $ ext{RP}^2 \times S^1$ demonstrate the framework.

Extension to higher dimensions includes a nontrivial 2-gerbe over $S^4$.

Abstract

We employ matrix product states (MPS) and tensor networks to study topological properties of the space of ground states of gapped many-body systems. We focus on families of states in one spatial dimension, where each state can be represented as an injective MPS of finite bond dimension. Such states are short-range entangled ground states of gapped local Hamiltonians. To such parametrized families over $X$ we associate a gerbe, which generalizes the line bundle of ground states in zero-dimensional families (\emph{i.e.} in few-body quantum mechanics). The nontriviality of the gerbe is measured by a class in $H^3(X, \mathbb{Z})$, which is believed to classify one-dimensional parametrized systems. We show that when the gerbe is nontrivial, there is an obstruction to representing the family of ground states with an MPS tensor that is continuous everywhere on $X$. We illustrate our construction with two examples of nontrivial parametrized systems over $X=S^3$ and $X = \mathbb{R} P^2 \times S^1$. Finally, we sketch using tensor network methods how the construction extends to higher dimensional parametrized systems, with an example of a two-dimensional parametrized system that gives rise to a nontrivial 2-gerbe over $X = S^4$.