General bulk-edge correspondence at positive temperature

Cornean, Horia D.; Moscolari, Massimo; Teufel, Stefan

doi:10.48550/arxiv.2107.13456

Cited by 5 publications

(14 citation statements)

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“…However these model would violate the Assumption 2 by having zero energy bulk bands. This can be seen by contradiction using the relation (7), as the edge and bulk indices should be bounded whereas n A,θ − n B,θ would increase linearly with the distance of the cut-off from the edge. Remark 4.…”

Section: Bulk and Edge Indicesmentioning

confidence: 99%

“…Therefore, as in condensed matter experiments, the temperature is small but never zero, this trade-off may naturally be satisfied as the thermal energy is often much smaller than the size of the gap, whereas the size of the sample is often much larger than the typical thermal correlation length. See also some recent mathematical work about extending two-dimensional bulk and edge quantities to (physical) finite temperature [7].…”

Section: Bulk and Edge Indicesmentioning

confidence: 99%

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Estimating bulk and edge topological indices in finite open chiral chains

Jezequel,

Tauber,

Delplace

2022

Preprint

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We develop a formalism to extend, simultaneously, the usual definition of bulk and edge indices from topological insulators to the case of a finite sample with open boundary conditions, and provide a physical interpretation of these quantities. We then show that they converge exponentially fast to an integer value when we increase the system size, and also that bulk and edge quantities coincide at finite size. The theorem applies to any non-homogeneous system such as disordered or defect configurations. We focus on one-dimensional chains with chiral symmetry, such as the Su-Schrieffer-Heeger model, but the proof actually only requires the Hamiltonian to be short-range and with a spectral gap in the bulk. The definition of bulk and edge indices relies on a finite-size version of the switch-function formalism where the Fermi projector is smoothed in energy using a carefully chosen regularization parameter.

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Section: Bulk and Edge Indicesmentioning

confidence: 99%

Section: Bulk and Edge Indicesmentioning

confidence: 99%

Estimating bulk and edge topological indices in finite open chiral chains

Jezequel,

Tauber,

Delplace

2022

Preprint

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“…The second operator is the Dirichlet realization of H b in the half-space E = {x ∈ R 2 | x 2 > 0} and is denoted by H E b . This setting is motivated by the study of the bulk-edge correspondence in the field of topological insulators, and in particular the recent paper [15] giving a very general bulk-edge correspondence. For this reason H b is referred to as the bulk Hamiltonian and H E b is referred to as the edge Hamiltonian.…”

Section: Paper Bmentioning

confidence: 99%

“…For this reason H b is referred to as the bulk Hamiltonian and H E b is referred to as the edge Hamiltonian. The aim of Paper B is to extend certain regularity properties of the edge Hamiltonian H E b established in [15]. In particular, it is shown in [15] that if F : σ(H E b ) → R is a function which can be extended to a function in S (R), then for any…”

Section: Paper Bmentioning

confidence: 99%

“…In Paper B we establish that these kernels are not merely continuous but in fact smooth. For the bulk Hamiltonian H b this can be done using the Beals criterion for the magnetic Weyl quantization [15], but for the edge Hamiltonian this type of argument no longer works. From this regularity result, it is then established that the bulk and edge particle current densities J 1,b and J E 1,b defined as…”

Section: Paper Bmentioning

confidence: 99%

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Spectral, scattering, and regularity properties related to various functional and differential equations

Støttrup¹

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Waves occur in many real world phenomena and much of the scientific literature concerns the study of waves. Different types of waves are described by different wave equations, for example Schrödinger's equation describing the time evolution of the wave function in a quantum mechanical system, or the Euler-Bernoulli equation describing bending waves in beams. This dissertation mainly considers these two types of waves and related topics. However, the dissertation also has a secondary focus unrelated to the topic of wave equations, which is the study of random variables with stationary digits. The dissertation consists of two parts. The first gives an introduction to the subjects listed above and an overview of the papers included in the second part. The second part consists of 6 papers, labeled A-F, concerning the aforementioned subjects.Paper A and B deal with topics related Schrödinger's equation. Specifically, in Paper A spectral results for the magnetic Weyl quantization of S 0 0,0 v male profil h bestemmes ved at anvende metoder fra variationsregningen, og det vises, at det optimale højdeprofil h er en generalisering af højdeprofilen h(x) = εx 2 , der har fået meget opmaerksomhed i litteraturen.Artikel E og F omhandler emner relateret til stokastiske variabler X ∈ [0, 1], hvis cifre {X n } n≥1 udgør en stationaer stokastisk proces. Bemaerk at dette emne ikke har nogen direkte sammenhaeng med de foregående emner relateret til bølgefaenomener. I Artikel E bevises en funktionalligning for den kumulative fordelingsfunktion F tilhørende X. Fra funktionalligningen bevises yderligere karakteriseringer af stationaritet af cifrene {X n } n≥1 ud fra egenskaberne ved funktionen F. I Artikel F betragtes bestemte stationaere stokastiske modeller for cifrene. Konkret betragtes stationaere Markovkaeder og stationaere fornyelsesprocesser. Det vises, at for sådanne processer vil F enten vaere givet ved F(x) = x for x ∈ [0, 1], eller vaere singulaer, dvs. F (x) = 0 for naesten alle x ∈ [0, 1]. Miksturer af stationaere Markovkaeder eller stationaere fornyelsesprocesser behandles ligeledes i Artikel F, og helt fundamentalt for denne analyse er egenskaberne ved normale tal.

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Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

Mastropietro

Porta

2022

Commun. Math. Phys.

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We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory.

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General bulk-edge correspondence at positive temperature

Cited by 5 publications

References 0 publications

Estimating bulk and edge topological indices in finite open chiral chains

Estimating bulk and edge topological indices in finite open chiral chains

Spectral, scattering, and regularity properties related to various functional and differential equations

Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

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