# Automorphic equivalence within gapped phases in the bulk

## TL;DR

This paper introduces a new adiabatic theorem for unique gapped ground states that relies on a bulk gap and smoothness of expectation values, relaxing the need for a local Hamiltonian gap.

## Contribution

It establishes a weaker condition for adiabatic evolution in gapped phases, focusing on bulk properties rather than local Hamiltonian gaps.

## Key findings

- The new theorem applies without requiring a local Hamiltonian gap.
- It demonstrates the sufficiency of bulk gap and smoothness for adiabatic evolution.
- The approach broadens the understanding of phase equivalence in quantum many-body systems.

## Abstract

We develop a new adiabatic theorem for unique gapped ground states which does not require the gap for local Hamiltonians. We instead require a gap in the bulk and a smoothness of expectation values of sub-exponentially localized observables in the unique gapped ground state $\varphi_s(A)$. This requirement is weaker than the requirement of the gap of the local Hamiltonians, since a uniform spectral gap for finite dimensional ground states implies a gap in the bulk for unique gapped ground states, as well as the smoothness.