[Paper Review] General bulk-edge correspondence at positive temperature.

Horia D. Cornean, Massimo Moscolari+1 more|arXiv (Cornell University)|Jul 28, 2021

Spectral Theory in Mathematical Physics61 references4 citations

TL;DR

This paper establishes a general bulk-edge correspondence for two-dimensional random ergodic magnetic Schrödinger operators at positive temperature by extending gauge-covariant magnetic perturbation theory to half-plane operators. It shows that the derivative of bulk partition functions with respect to an external magnetic field equals the expectation of an edge distribution function for the velocity component parallel to the edge, without requiring spectral or mobility gaps, and recovers the conventional bulk-boundary correspondence as a zero-temperature limit under a gap condition.

ABSTRACT

By extending the gauge covariant magnetic perturbation theory to operators defined on half planes, we prove that for general $2d$ random ergodic magnetic Schr\odinger operators the celebrated bulk-edge correspondence is just a particular case of a much more general paradigm, which also includes the theory of diamagnetic currents and of Landau diamagnetism. 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-boundary correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a zero temperature limit.

Motivation & Objective

• To extend gauge-covariant magnetic perturbation theory to operators on half-planes for two-dimensional systems.
• To establish a general bulk-edge correspondence valid at positive temperature, beyond topological or gap conditions.
• To unify the theory of diamagnetic currents and Landau diamagnetism within a single analytical framework.
• To derive the conventional bulk-boundary correspondence as a limiting case under a gap condition and zero-temperature limit.

Proposed method

• Application of gauge-covariant magnetic perturbation theory to operators defined on half-planes.
• Derivation of a general formula linking the derivative of bulk partition functions to edge velocity distribution functions.
• Use of random ergodic Schrödinger operators to model disordered two-dimensional systems.
• Analysis of the magnetic field dependence of partition functions via functional calculus and trace-class perturbations.
• Introduction of an edge distribution function for the velocity component parallel to the boundary.
• Taking the zero-temperature limit to recover the conventional bulk-boundary correspondence under a spectral gap condition.

Experimental results

Research questions

• RQ1 How can bulk-edge correspondence be generalized beyond topological invariants and spectral gaps in two-dimensional quantum systems at positive temperature?
• RQ2 What is the precise analytical relationship between the derivative of the bulk partition function and edge observables in the presence of a magnetic field?
• RQ3 How does the theory of diamagnetic currents and Landau diamagnetism emerge as special cases of a broader bulk-edge principle?
• RQ4 In what sense does the conventional bulk-boundary correspondence arise as a limiting case of this general framework?
• RQ5 What role does the velocity component parallel to the edge play in connecting bulk and edge responses?

Key findings

• The derivative of a large class of bulk partition functions with respect to the external magnetic field equals the expectation of an edge distribution function for the velocity component parallel to the edge.
• The general bulk-edge correspondence holds without requiring spectral gaps, mobility gaps, or topological invariants.
• The theory unifies the description of diamagnetic currents and Landau diamagnetism within a single analytical framework.
• The conventional bulk-boundary correspondence is recovered as a corollary by imposing a spectral gap and taking the zero-temperature limit.
• The result is derived purely through analytical methods, without relying on K-theory or index theorems.