Adiabatic currents for interacting electrons on a lattice

TL;DR

This paper establishes a rigorous adiabatic theorem for interacting fermions on a lattice, enabling precise analysis of current responses and quantum Hall effects without periodicity constraints.

Contribution

It introduces a general adiabatic theorem for interacting fermions with uniform error estimates, applicable to degenerate eigenstates and non-periodic systems.

Findings

Provides the first rigorous derivation of the linear response formula for interacting fermions.

Offers uniform error estimates in system size for adiabatic evolution.

Discusses applications to quantum Hall systems.

Abstract

We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from a recent work of Bachmann et al. who proved an adiabatic theorem for interacting spin systems. As one important application of this adiabatic theorem, we provide the first rigorous derivation of the so-called linear response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.