# The adiabatic theorem and linear response theory for extended quantum systems

## TL;DR

This paper proves an adiabatic theorem with error bounds independent of system size for quantum spin systems with spectral gaps, validating linear response theory in extended interacting quantum systems like those relevant to the quantum Hall effect.

## Contribution

It establishes an adiabatic theorem with size-independent error bounds and applies it to confirm linear response theory in gapped quantum spin systems.

## Key findings

- Error bound independent of degrees of freedom
- Validation of linear response theory for extended systems
- Application to quantum Hall effect models

## Abstract

The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $\epsilon$. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g.~for the integer quantum Hall effect.