Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk

Henheik, Joscha; Teufel, Stefan

doi:10.48550/arxiv.2012.15239

Cited by 5 publications

(8 citation statements)

References 22 publications

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“…As mentioned in the introduction, we expect a similar strong LPPL-principle to hold also for fermionic lattice systems with weak finite range interactions. As discussed in [8,9], this would have important consequences for linear response and adiabatic theorems for systems with a gap only in the bulk.…”

Section: Theorem 3 the Strong Lppl-principlementioning

confidence: 99%

“…It would imply that a gapped ground state for such a system with periodic boundary conditions remains unchanged in the bulk when introducing open boundary conditions that may close the global gap due to the emergence of edge states. And as a consequence, it would also explain why the adiabatic response to external fields in the bulk of such systems is not affected by edge states that close the gap, see [1,10,15,8,9] for related results. However, it is known that the strong LPPL-principle cannot hold in general, but requires further conditions on the unperturbed ground state sector such as local topological quantum order (LTQO) [11,14].…”

Section: Introductionmentioning

confidence: 96%

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Local stability of ground states in locally gapped and weakly interacting quantum spin systems

Henheik,

Teufel,

Wessel

2021

Preprint

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Based on a result by Yarotsky (J. Stat. Phys. 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space.While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.

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Section: Theorem 3 the Strong Lppl-principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Local stability of ground states in locally gapped and weakly interacting quantum spin systems

Henheik,

Teufel,

Wessel

2021

Preprint

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“…One then expects that the adiabatic theory will remain valid for times comparable with the lifetime of the resonance, [2,20]. For state of the art assertions on this topic, we refer the reader to [29] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Locobatic theorem for disordered media and validity of linear response

Roeck¹,

Elgart²,

Fraas³

2022

Preprint

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Spectral localization is intrinsically unstable under perturbation. As a result, the adiabatic theorem of quantum mechanics cannot generally hold for localized eigenstates. However, it turns out that a remnant of the adiabatic theorem, which we name the "locobatic theorem", survives: The physical evolution of a typical eigenstate ψ for a random system remains close, with high probability, to the spectral flow for ψ associated with a restriction of the full Hamiltonian to a region where ψ is supported. We make the above statement precise for a class of Hamiltonians describing a particle in a disordered background. Our argument relies on finding a local structure that remains stable under the small perturbation of a random system.An application of this work is the justification of the linear response formula for the Hall conductivity of a two-dimensional system with the Fermi energy lying in a mobility gap. Additional results are concerned with eigenvector hybridization in a one-dimensional Anderson model and the construction of a Wannier basis for underlying spectral projections.

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“…We therefore propose to avoid the time modulation of the perturbing field altogether, and follow ideas inspired by what is known in physics and chemistry as Rayleigh-Schrödinger perturbation theory [24], and which have been made rigorous in the mathematical physics community in the context of space-adiabatic perturbation theory (see [23,30] and references therein). This alternative method has been advanced by S. Teufel and collaborators in a series of works [20,31,9,10] and applied in the present context of quantum transport also in collaboration with the authors [18,15,19]. The approach allows to identify a so-called non-equilibrium almost stationary state (NEASS), which in the adiabatic regime well approximates the physical state once the dynamical switching drives the system out of equilibrium, but which can be defined without resorting to a time-dependent modulation of the perturbing field (see Theorem 2.1 for a precise stament).…”

Section: Introductionmentioning

confidence: 99%

From charge to spin: analogies and differences in quantum transport coefficients

Marcelli,

Monaco

2022

Preprint

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We review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be in transport coefficients like conductivities and conductances. As for the former, those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which arises from the perturbation of an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), we obtain formulas for appropriately-defined spin conductivities which are still explicit but more involved. Certain "Kubo-like" terms in these formulas are also shown to agree with corresponding contributions to the spin conductance.In addition to that, we employ similar techniques to show a new result, namely that even in systems with non-conserved spin there is no generation of spin torque, namely the spin torque operator has an expectation in the NEASS which vanishes faster than any power of the intensity of the perturbing field. Contents

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Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk

Cited by 5 publications

References 22 publications

Local stability of ground states in locally gapped and weakly interacting quantum spin systems

Local stability of ground states in locally gapped and weakly interacting quantum spin systems

Locobatic theorem for disordered media and validity of linear response

From charge to spin: analogies and differences in quantum transport coefficients

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