The generalized adiabatic theorem for extended lattice systems
Lennart Becker, Stefan Teufel, Marius Wesle
TL;DR
The paper develops a rigorous framework for adiabatic evolution in infinitely extended lattice fermion systems with gapped ground states, allowing perturbations that may close the spectral gap. It introduces a quasi-local dressing β^{ε,η}_t generated by a time-dependent S^{ε,η}_t that produces super-adiabatic states ω^{ε,η}_t = ω_t ∘ β^{ε,η}_t which approximate the true dynamics to arbitrarily high order in the small parameters η and ε. The main contribution is a constructive proof of a generalized adiabatic theorem under super-polynomial decay (B_inf) without requiring ground-state uniqueness, along with a detailed localization analysis and a resummation to an ε- and η-uniform, n-independent generator. This yields a solid mathematical basis for linear response in extended gapped systems and underpins Ohm’s law for macroscopic Hall currents in infinite-volume interacting lattice fermions. The results broaden previous adiabatic theorems to include long-range decays and non-unique ground states, using modern automorphic equivalence results and LR bounds for fermions, thereby impacting the theoretical foundations of transport in quantum many-body systems.
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 $H^{\varepsilon,η}_t=η^{-1}(H_t+\varepsilon(H^1_t+V_t))$, where $H_t$ is assumed to have a gapped ground state $ω_t$, $η\in (0,1]$ is the adiabatic parameter and $ \varepsilon \in [0,1]$ controls the strength of the perturbation. We construct a quasi-local dressing transformation $β^{\varepsilon,η}_t=\exp(i \mathcal{L}_{S^{\varepsilon,η}_t})$ that yields super-adiabatic states $ω^{\varepsilon,η}_t =ω_t \circ β^{\varepsilon,η}_t$ which, when tested against local observables, solve the corresponding time-dependent Schrödinger equation up to errors asymptotically smaller than any power of $η$ 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 $η$-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.