Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

TL;DR

This paper establishes conditions under which gapped ground states of quantum spin systems are automorphically equivalent, demonstrating that such equivalence can be implemented via a rapidly decaying flow using quasi-adiabatic evolution, preserving phase structure.

Contribution

It provides a rigorous framework for automorphic equivalence of gapped ground states using spectral flow and extends Hastings' quasi-adiabatic evolution to infinite-dimensional systems.

Findings

Automorphic equivalence can be implemented by a quasi-local flow.

Spectral flow satisfies a Lieb-Robinson bound.

Ground state phase structure is preserved in the thermodynamic limit.

Abstract

Gapped ground states of quantum spin systems have been referred to in the physics literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on $s \in [0,1]$, such that for each $s$, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an $s$-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the {\em spectral flow}, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models $H(s), 0\leq s\leq 1$.