Spectrum of the Hypereclectic Spin Chain and P\'olya Counting

TL;DR

This paper improves the explicitness and elegance of generating functions that encode the spectrum of the Hypereclectic spin chain, connecting integrable models with combinatorial enumeration techniques.

Contribution

It provides more explicit and elegant expressions for the generating functions of the spin chain spectrum, including cases with and without cyclicity constraints, using Pólya counting and q-binomial coefficients.

Findings

Enhanced generating functions for the spin chain spectrum.

Explicit treatment of cyclic and non-cyclic states.

Application of Pólya enumeration in spectral analysis.

Abstract

In earlier work we proposed a generating function that encodes the Jordan block spectrum of the integrable Hypereclectic spin chain, related to the one-loop dilatation operator of the dynamical fishnet quantum field theory. We significantly improve the expressions for these generating functions, rendering them much more explicit and elegant. In particular, we treat the case of the full spin chain without imposing any cyclicity constraints on the states, as well as the case of cyclic states. The latter involves the P\'olya enumeration theorem in conjunction with q-binomial coefficients.