A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization II

TL;DR

This paper characterizes a class of gapped Hamiltonians on quantum spin chains, showing that Hamiltonians satisfying certain properties are approximated by MPS Hamiltonians, aiding in their classification.

Contribution

It proves the converse of previous results, demonstrating that Hamiltonians with specific properties are approximated by MPS Hamiltonians, facilitating classification.

Findings

Ground state spaces coincide on infinite intervals.

Spectral projections are exponentially well approximated.

Application to Hamiltonian classification.

Abstract

We give a characterization of the class of gapped Hamiltonians introduced in PartI [O]. The Hamiltonians in this class are given as MPS (Matrix product state) Hamiltonians. In [O], we list up properties of ground state structures of Hamiltonians in this class. In this Part II, we show the converse. Namely, if a (not necessarily MPS) Hamiltonian $H$ satisfies five of the listed properties, there is a Hamiltonian $H'$ from the class in [O], satisfying the followings: The ground state spaces of the two Hamiltonians on the infinite intervals coincide. The spectral projections onto the ground state space of $H$ on each finite intervals are approximated by that of $H'$ exponentially well, with respect to the interval size. The latter property has an application to the classification problem.