Fast analytic solver of rational Bethe equations

TL;DR

This paper introduces a rapid analytic method for solving rational Bethe equations in integrable spin chains, efficiently determining all eigenstates via Baxter Q-functions without missing exceptional solutions.

Contribution

It presents a novel approach that leverages the Young diagram structure to solve Bethe equations analytically for various GL(N|M) spin chains, regardless of the rank.

Findings

All eigenstates can be computed quickly using the proposed method.

Exceptional solutions are automatically included in the analysis.

The approach is flexible across different ranks and representations.

Abstract

In this note we propose an approach for a fast analytic determination of all possible eigenstates of rational GL(N|M) integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted. The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the GL symmetry. Hence we can choose arbitrary N and M that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac-Dynkin path.