Adiabatic currents for interacting fermions on a lattice

• Published in: Reviews in mathematical physics Vol. 31; no. 3; p. 1950009
• Main Authors: Monaco, Domenico, Teufel, Stefan
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.04.2019; World Scientific Publishing Co. Pte., Ltd
• Subjects: Adiabatic flow; Eigenvalues; Fermions; Periodic variations; Theorems; adiabatic response; quantum Hall conductivity; quantum Hall conductance; interacting fermions; adiabatic current; Adiabatic theorem
• ISSN: 0129-055X; 1793-6659; 1793-6659
• DOI: 10.1142/S0129055X19500090

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 the initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [10], where Bachmann et al. proved an adiabatic theorem for interacting spin systems. As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic 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.