Self-Isospectrality, Special Supersymmetry, and their Effect on the Band Structure

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 Lamé 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 parityodd 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.

Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

“…As both families of quantum systems are characterized by nontrivial, higher derivative integrals of motion, one could expect that supersymmetric extensions of them should possess some peculiar properties. This is indeed the case [16,17,18,19,20,21], and exotic supersymmetric structures of reflectionless and finite-gap systems found recently some interesting physical applications [22,23,24,25,26].…”

“…Such a kind of supersymmetry of the pairs of reflectionless systems was not investigated yet in the literature, but, instead, supersymmetry of the pairs (H + = H j , H − = H j+l ), l ≥ 1, belonging to the same Darboux chain (2.12) is usually considered. In particular, the pairs of reflectionless Pöschl-Teller systems, see below, appear in the context of shape-invariance [43,44,7], they also emerge in the infinite-period limit of finite-gap periodic crystal structures [22,24]. Supersymmetry of reflectionless Pöschl-Teller pairs (H j , H j+l ) was studied recently from the perspective of AdS/CFT holography and Aharonov-Bohm effect [45].…”

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

“…For example, the exact solutions of the Schrödinger equation for a hydro-gen atom and for a harmonic oscillator in three dimensions are an important milestone at the beginning stage of quantum mechanics, which provided a strong evidence for supporting the correctness of the quantum theory [1][2][3]. The investigation of exactly solvable systems has been drawing people's attention all the time, for example the recent discovery of the hidden bosonized suppersymmetry in reflectionless hyperbolic Pöschl-Teller system [4][5][6]. However, the exact solutions are rare such that many quantum systems have to be treated by approximate methods.…”

Twist deformations of the supersymmetric quantum mechanics

“…As it will become clear, this nonlinear superalgebra is in correspondence with the = 2 case of the nonlinear supersymmetry algebra {Q Q } = δ P (H) introduced in [46], which has been shown to have important physical applications, as, for instance, in the context of periodic finite-gap systems [47]. The multiplication on the module [1] acts as follows…”