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From orbital magnetism to bulk-edge correspondence

Horia D. Cornean, Massimo Moscolari, Stefan Teufel

Abstract

By extending the gauge covariant magnetic perturbation theory to operators defined on half-planes, we prove that for $2d$ random ergodic magnetic Schrödinger operators, the zero-temperature bulk-edge correspondence can be obtained from a general bulk-edge duality at positive temperature involving the bulk magnetization and the total edge current. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a ``zero-temperature" limit.

From orbital magnetism to bulk-edge correspondence

Abstract

By extending the gauge covariant magnetic perturbation theory to operators defined on half-planes, we prove that for random ergodic magnetic Schrödinger operators, the zero-temperature bulk-edge correspondence can be obtained from a general bulk-edge duality at positive temperature involving the bulk magnetization and the total edge current. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a ``zero-temperature" limit.

Paper Structure

This paper contains 25 sections, 14 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The setting
  3. General bulk-edge correspondence at positive temperature
  4. Physical implications of Theorem \ref{['thm-positive']}
  5. The bulk magnetization
  6. The total edge current
  7. Bulk-edge correspondence of transport coefficients
  8. The zero-temperature limit
  9. Open questions
  10. Two key technical results
  11. The content of the rest of the paper
  12. Connections with the existing literature
  13. Proof of Proposition \ref{['AR1']}
  14. Proof of Lemma \ref{['AR2']}
  15. Proof of Theorem \ref{['thm-positive']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\Omega=[0,1]\times [0,1]\subset {\mathbb R}^2$ and denote by $\chi_{\Omega}$ its indicator function. Let also $\chi_{L}$ be the indicator function of the strip $\mathcal{S}_{L}:=[0,1]\times [0,L]$, $L\geq 1$. Let $F$ be any real valued function whose restriction to $[\inf \Sigma(0),\infty)$ can

Figures (1)

  • Figure 1: The averaged density of the edge particle-current $j_1^E$ (blue curve) as a function of the distance $x_2$ to the edge approaches the oscillatory persistent averaged density of the bulk particle-current $j_1^B$ (dashed red curve).

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 21 more