Adiabatic currents for interacting electrons on a lattice

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.

arXiv comment46 pages; v1->v2: typos corrected, references added, Remark 4 after Thm 2 slightly reworded, v2->v3: major revision of the presentation of the result, 3 figures added

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Cited by in corpus (19)

• Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
• The adiabatic theorem and linear response theory for extended quantum systems
• A many-body index for quantum charge transport
• Persistence of exponential decay and spectral gaps for interacting fermions
• Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results
• A new approach to transport coefficients in the quantum spin Hall effect
• Quantization of the interacting Hall conductivity in the critical regime
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Canonical Drude weight for non-integrable quantum spin chains
• Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap
• Exactness of linear response in the quantum Hall effect
• Local stability of ground states in locally gapped and weakly interacting quantum spin systems
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• From charge to spin: analogies and differences in quantum transport coefficients
• Trotter product formulae for $\ast$-automorphisms of quantum lattice systems
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Purely linear response of the quantum Hall current to space-adiabatic perturbations
• The index of a pair of pure states and the interacting integer quantum Hall effect
• On adiabatic theory for extended fermionic lattice systems