Non-Hermitian multi-particle systems from complex root spaces

TL;DR

This paper introduces a general method for constructing complex root spaces invariant under antilinear symmetries, enabling the development of non-Hermitian yet physically consistent models like Calogero and affine Toda theories with novel duality properties.

Contribution

It presents a universal construction technique for complex root spaces applicable to any Weyl group, leading to new non-Hermitian models and q-deformed roots with duality features.

Findings

New non-Hermitian models based on complex roots are physically viable.

Construction of q-deformed roots for affine Toda theories.

Classical strong-weak duality observed in deformed models.

Abstract

We provide a general construction procedure for antilinearly invariant complex root spaces. The proposed method is generic and may be applied to any Weyl group allowing to take any element of the group as a starting point for the construction. Worked out examples for several specific Weyl groups are presented, focusing especially on those cases for which no solutions were found previously. When applied in the defining relations of models based on root systems this usually leads to non-Hermitian models, which are nonetheless physically viable in a self-consistent sense as they are antilinearly invariant by construction. We discuss new types of Calogero models based on these complex roots. In addition we propose an alternative construction leading to q-deformed roots. We employ the latter type of roots to formulate a new version of affine Toda field theories based on non-simply laced roots systems. These models exhibit on the classical level a strong-weak duality in the coupling constant equivalent to a Lie algebraic duality, which is known for the quantum version of the undeformed case.