Self-isospectrality, special supersymmetry, and their effect on the band structure

TL;DR

This paper explores a novel supersymmetry in a 2D electron system with periodic fields, revealing its impact on the band structure and introducing a new nonlinear supersymmetry related to finite-gap Lame equations.

Contribution

It introduces a new type of special nonlinear supersymmetry in a periodic electron system modeled by finite-gap Lame equations, linking supersymmetry to band structure features.

Findings

Identification of a self-isospectral pair of finite-gap Lame equations

Discovery of a new nonlinear supersymmetry generated by three integrals of motion

Analysis of supersymmetry breaking in the infinite period limit

Abstract

We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lame equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parity-odd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.