Field theoretical computations of static conductivity for a white noise magnetic system

TL;DR

This paper develops a field-theoretical approach to compute static conductivity in a system with a random magnetic field, revealing quantized Hall conductance steps related to spin projections.

Contribution

It introduces a non-hermitian Hamiltonian and a functional supermatrix formalism to analyze static transport in disordered magnetic systems, including effects of external fields.

Findings

Quantized Hall conductance steps between spin projections.

Unstable limit of vanishing external magnetic field due to E=0 states.

Method incorporates ladder diagrams for conductivity calculation.

Abstract

We consider an electron coupled to a random magnetic field with local correlations and eventually non-zero average value. Starting from the source-term formalism of Levine, Libby and Pruisken, we define a generating function for static transport coefficients. We use a non-hermitian representation of the Hamiltonian, and a functionnal representation of the generating function, where the integration variable consists of non-local supermatrices, due to the non-locality of the Aharanov-Bohm interaction.The conductivity coefficients are computed using a specific saddle-point procedure, which incorporates ladder diagrams contributions. The situation with an external uniform magnetic field is considered with the two different spin projection possibilities. The limit of vanishingly small external magnetic field is unstable with respect to spin flip, because of E=0 states. In particular a quantized step is obtained when comparing the Hall conductance between the two possible spin projections.