The Integrable (Hyper)eclectic Spin Chain

TL;DR

This paper introduces a class of integrable, non-Hermitian spin chains with complex spectra characterized by Jordan blocks, revealing limitations of the quantum inverse scattering method and uncovering universal spectral patterns.

Contribution

It refines the notion of eclectic spin chains by incorporating maximal deformation parameters and analyzes their spectral properties, including non-diagonalizability and universality.

Findings

Spectra consist of intricate Jordan block structures.

Quantum inverse scattering method fails to fully capture the spectrum.

Universal spectral patterns observed across deformation parameters.

Abstract

We refine the recently introduced notion of eclectic spin chains by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n>2), where their "spectrum" consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not "random", but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above "generic" spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of N=4 Super Yang-Mills Theory.