The generalized adiabatic theorem for extended lattice systems

• URL: http://arxiv.org/abs/2510.20914v1
• Date: Thu, 23 Oct 2025 18:10:40 GMT
• Title: The generalized adiabatic theorem for extended lattice systems
• Authors: Lennart Becker, Stefan Teufel, Marius Wesle,
• Abstract summary: We prove an adiabatic theorem for infinitely extended lattice fermion systems with gapped ground states, allowing perturbations that may close the gap.<n>The result provides a rigorous basis for linear response to macroscopic changes in gapped systems, including a proof of Ohm's law for Hall currents.
• Score: 0.9558392439655014
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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