Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry

TL;DR

This paper explores a class of $ ext{PT}$-symmetric quantum models that are perfectly invisible zero-gap systems with unique bound states, revealing their connection to conformal field theories, integrable systems, and exotic nonlinear supersymmetry.

Contribution

It introduces a new class of $ ext{PT}$-symmetric zero-gap systems, links them to conformal field theories and integrable models, and uncovers their exotic nonlinear supersymmetry structures.

Findings

Identification of perfectly invisible zero-gap $ ext{PT}$-symmetric systems.

Connection of these systems to conformal field theories and KdV hierarchy.

Discovery of extended $ ext{N}=4$ nonlinear supersymmetry involving Lax-Novikov integrals.

Abstract

We investigate a special class of the $\mathcal{PT}$-symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the $\mathcal{PT}$-regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the $\mathcal{PT}$-regularized kinks arising as traveling waves in the field-theoretical Liouville and $SU(3)$ conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional $\mathcal{N}=2$ supersymmetry is extended here to the $\mathcal{N}=4$ nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.