The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems

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

doi:10.1007/s00220-018-3117-9

Cited by 57 publications

(124 citation statements)

References 61 publications

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“…We take the Kubo formula (1) as our starting point, and we do not discuss its validity or derivation. Recently, remarkable progress has been made on the foundations of Kubo formula, in the case of gapped Hamiltonian [11,12,48], which validate its use in the present context.…”

Section: )supporting

confidence: 59%

Order, Disorder and Phase Transitions in Quantum Many Body Systems

Giuliani

2020

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In this paper, I give an overview of some selected results in quantum many body theory, lying at the interface between mathematical quantum statistical mechanics and condensed matter theory. In particular, I discuss some recent results on the universality of transport coefficients in lattice models of interacting electrons, with specific focus on the independence of the quantum Hall conductivity from the electron-electron interaction. In this context, the exchange of ideas between mathematical and theoretical physics proved particularly fruitful, and helped in clarifying the role played by quantum conservation laws (Ward Identities) together with the decay properties of the Euclidean current-current correlation functions, on the interaction-independence of the conductivity coefficients. arXiv:1711.06991v1 [cond-mat.stat-mech] 19 Nov 2017In this paper I will review some of the latest developments in the universality theory of quantum transport coefficients, which allowed to clarify certain debated issues connected with the optical conductivity in graphene [31]. The key new technical tool that emerged from the combination of theoretical and mathematical physics ideas, is the implementation of Ward Identities within the constructive scheme ('multiscale fermionic cluster expansion') that is currently able to control the analyticity and decay of correlation functions for the ground state of several two-dimensional interacting electron systems. Note that a formal use of Ward Identities in the effective field theory description of quantum phenomena, can easily lead to inconsistent results, particularly as far as the computation of transport coefficients is concerned, cf., e.g., with [39].In order to make the ideas behind these recent applications as transparent as possible, I will restrict my attention to the study of the universality properties of the Kubo conductivity, and, in particular, of its transverse component (Hall conducitivity) in weakly interacting lattice fermions characterized by a gapped ('massive') reference non-interacting Hamiltonian. For these systems, the construction of the ground state correlation functions and the proof of their analyticity properties is particularly simple and, strictly speaking, does not require a multiscale expansion at all. The extension of the proof to the gapless case, in particular in the case of graphene-like systems, requires the use of a multiscale analysis (constructive fermionic renormalization group), which goes beyond the purpose of this review.The argument presented here is based on [32], which I will refer to for some technical aspects of the proof. However, compared to [32], the proof presented here has some important simplifications in the proof of analyticity and exponential decay of correlations.The plan is to first introduce the context, the model and the main results. Next, I will present the proof, first giving an overview of the structure of the proof, and then explaining in some details the different steps. The quantum Hall effectBefore presenting the mai...

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Section: )supporting

confidence: 59%

Order, Disorder and Phase Transitions in Quantum Many Body Systems

Giuliani

2020

SCIENZE

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“…Their proof exploits locality of interactions in the form of Lieb-Robinson propagation bounds [35] and the local inverse of the Liouvillian introduced by Hastings and Wen [28] (see also [9]). However, Bachmann et al [8] require the spectral gap not only for H 0 , but also for the perturbed Hamiltonians H ε (t). In order to apply their result to slowly varying potential perturbations that close the spectral gap (small fields over large regions), one needs to use the alternative gauge with a time-dependent vector potential and consider the adiabatic response instead, i.e.…”

Section: Justifying Kubo's Formula: Mathematical Resultsmentioning

confidence: 99%

“…the first order deviation from ideal adiabatic behavior. This could be done using the results of [39], which are a slight generalization of [8] in several directions: a super-adiabatic version of the theorem is formulated and proved which covers also the trace per unitvolume of extensive observables. This version is then used to derive Kubo's formula for conductance and conductivity not from adiabatic switching of a small potential, but as the adiabatic response for a Hamiltonian with time-dependent fluxes.…”

Section: Justifying Kubo's Formula: Mathematical Resultsmentioning

confidence: 99%

Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results

Henheik

Teufel

2020

Rev. Math. Phys.

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We first review the problem of a rigorous justification of Kubo's formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo's formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.

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“…Newer methods for higher-dimensional spin systems are also in development [25]. There has also been several results concerning stability of topological invariants such as the Hall conductance in interacting fermion systems [7,8,9,33,37,47].…”

Section: Introductionmentioning

confidence: 99%

On ℤ2-indices for ground states of fermionic chains

Bourne

Schulz-Baldes

2020

Rev. Math. Phys.

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For parity-conserving fermionic chains, we review how to associate Z 2 -indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z 2 -valued spectral flow provides a topological obstruction for two systems to have the same Z 2 -index. A rudimentary definition of a Z 2 -phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

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The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems

Cited by 57 publications

References 61 publications

Order, Disorder and Phase Transitions in Quantum Many Body Systems

Order, Disorder and Phase Transitions in Quantum Many Body Systems

Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results

On ℤ2-indices for ground states of fermionic chains

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