The generalized adiabatic theorem for extended lattice systems

TL;DR

This paper establishes a generalized adiabatic theorem for extended lattice fermion systems with gapped ground states, allowing for perturbations that may close the gap, and constructs super-adiabatic states with controlled errors.

Contribution

It introduces a local construction of super-adiabatic states for extended lattice systems under less restrictive decay conditions and allows for gap-closing perturbations.

Findings

Constructs super-adiabatic states with errors smaller than any power of parameters

Provides a rigorous basis for linear response and Ohm's law in gapped systems

Works under super-polynomial decay of interactions, not requiring exponential decay

Abstract

We prove an adiabatic theorem for infinitely extended lattice fermion systems with gapped ground states, allowing perturbations that may close the gap. The Heisenberg dynamics on the CAR-algebra is generated by a time dependent two-parameter family of Hamiltonians $H^{\varepsilon,\eta}_t=\eta^{-1}(H_t+\varepsilon(H^1_t+V_t))$, where $H_t$ is assumed to have a gapped ground state $\omega_t$, $\eta \in (0,1]$ is the adiabatic parameter and $ \varepsilon \in [0,1]$ controls the strength of the perturbation. We construct a quasi-local dressing transformation $\beta^{\varepsilon,\eta}_t=\exp(i \mathcal{L}_{S^{\varepsilon,\eta}_t})$ that yields super-adiabatic states $\omega^{\varepsilon,\eta}_t =\omega_t \circ \beta^{\varepsilon,\eta}_t$ which, when tested against local observables, solve the corresponding time-dependent Schr\"odinger equation up to errors asymptotically smaller than any power of $\eta$ and $\varepsilon$. The construction is local in space and time, does not assume uniqueness of the ground state, and works under super-polynomial decay of the interactions $H_t$ and $H_t^1$ rather than exponential decay. If the Hamiltonian is time-independent on an interval, the dressed state is $\eta$-independent and forms a non-equilibrium almost-stationary state with lifetime of order $\varepsilon^{-\infty}$. The result provides a rigorous basis for linear response to macroscopic changes in gapped systems, including a proof of Ohm's law for macroscopic Hall currents.