The integrable (hyper)eclectic spin chain

We refine the notion of eclectic spin chains introduced in [1] 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.

Automated summary

We refine the concept of eclectic spin chains introduced in [1] by incorporating a maximal number of deformation parameters. These models, integrable nearest-neighbor n-state spin chains, have simple non-hermitian Hamiltonians and show non-diagonalizability in the multiparticle sector (n > 2). Despite the failure of the quantum inverse scattering method to reproduce the spectrum details, we provide evidence for the non-random, subtle, and regular patterns exhibited by the spectrum for n=3. Our models, inspired by the one-loop dilatation operator of a strongly twisted, double-scaled deformation of N = 4 Super Yang-Mills Theory, also include a new model, the hypereclectic spin chain.