Pure gapped ground states of spin chains are short-range entangled

TL;DR

This paper proves that unique gapped ground states in infinite one-dimensional spin chains are short-range entangled and can be transformed into product states via finite, quasi-local Hamiltonian evolutions, confirming their topological triviality.

Contribution

It rigorously establishes that all unique gapped ground states in such systems are short-range entangled, using advanced techniques to connect them to product states.

Findings

Unique gapped ground states are short-range entangled.

Such states can be transformed into product states via finite quasi-local evolutions.

Supports the belief that 1D gapped systems are topologically trivial.

Abstract

We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.