Adiabatic Theorem for Quantum Spin Systems

TL;DR

This paper proves a version of the quantum adiabatic theorem applicable to many-body quantum spin systems, addressing limitations of previous proofs and supporting applications like linear response theory.

Contribution

It provides the first proof of the adiabatic theorem for gapped quantum spin systems in the thermodynamic limit, extending its applicability.

Findings

Proves adiabatic theorem for gapped quantum spin systems in the thermodynamic limit

Validates Kubo linear response formula for broad class of gapped systems

Addresses previous limitations in many-body quantum adiabatic proofs

Abstract

The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g. in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation $\varepsilon$ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this letter, we prove a version of the adiabatic theorem for gapped ground states of quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo linear response formula for a broad class of gapped interacting systems.