Exactness of Linear Response in the Quantum Hall Effect

Bachmann, Sven; Roeck, Wojciech De; Fraas, Martin; Lange, Markus

doi:10.1007/s00023-020-00989-z

Cited by 6 publications

(9 citation statements)

References 30 publications

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“…The existing literature on this property was initiated by the heuristic magnetic flux insertion argument proposed by Laughlin in a cylindrical geometry [18], which was later elaborated in a rigorous way for many-body electron gases in the continuum [16] or discrete [4] setting. These proofs focus on a related quantity, namely the Hall conductance, defined as the (linear) response of the current intensity to the voltage drop: in two dimensions, this quantity agrees with the Hall conductivity σ Hall defined above, see [1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Klein and Seiler [16] then make use of the geometric interpretation of σ Hall to conclude the validity of (the Hall-conductance analogue of) (1.1), at least up to averaging over time and over the inserted magnetic flux. Instead, Bachmann et al [4] obtain an analogous statement (in the context of lattice spin systems with local interactions and observables) avoiding magnetic-flux averaging and the geometric argument, at the expense of exploiting the integrality of a certain Fredholm index related to the Hall conductance. Both approaches rely on the assumption that this magnetic flux insertion does not close the gap of the unperturbed Hamiltonian.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…By definition, the NEASS is unitarily conjugated to the equilibrium Fermi projection (see property (SA 1 ) below): this structure is reminiscent of the "magnetic gauge transformed projection" of [16], as well as of the "dressed ground state" of [4], with the main difference that the unitary conjugation is defined here by employing space-adiabatic rather than time-adiabatic perturbation theory. The NEASS was constructed in [23] up to first order in ε in the same context that we will employ; using arguments from [29], we extend this construction to arbitrarily high orders in ε (Theorem 3.1), a result which is interesting in its own right.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We say that an operator A acting in H is trace class on compact sets if and only if χ K Aχ K is trace class for all compact sets K ⊂ X (4) .…”

Section: Appendix a Trace Per Unit Volumementioning

confidence: 99%

“…Tr(χ 1 Aχ 1 ). (4) Notice that in the discrete case this condition is automatically satisfied for any operator A because the range of χ K is finite-dimensional.…”

Section: Appendix a Trace Per Unit Volumementioning

confidence: 99%

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Purely linear response of the quantum Hall current to space-adiabatic perturbations

Marcelli,

Monaco

2021

Preprint

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Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We say that an operator A acting in H is trace class on compact sets if and only if χ K Aχ K is trace class for all compact sets K ⊂ X (4) .…”

Section: Appendix a Trace Per Unit Volumementioning

confidence: 99%

“…Tr(χ 1 Aχ 1 ). (4) Notice that in the discrete case this condition is automatically satisfied for any operator A because the range of χ K is finite-dimensional.…”

Section: Appendix a Trace Per Unit Volumementioning

confidence: 99%

See 3 more Smart Citations

Purely linear response of the quantum Hall current to space-adiabatic perturbations

Marcelli,

Monaco

2021

Preprint

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Adiabatic Evolution of Low-Temperature Many-Body Systems

Greenblatt,

Lange,

Marcelli

et al. 2024

Commun. Math. Phys.

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We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

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Purely linear response of the quantum Hall current to space-adiabatic perturbations

Marcelli

Monaco

2022

Lett Math Phys

View full text Add to dashboard Cite

Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost-stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect.

show abstract

Exactness of Linear Response in the Quantum Hall Effect

Cited by 6 publications

References 30 publications

Purely linear response of the quantum Hall current to space-adiabatic perturbations

Purely linear response of the quantum Hall current to space-adiabatic perturbations

Adiabatic Evolution of Low-Temperature Many-Body Systems

Purely linear response of the quantum Hall current to space-adiabatic perturbations

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