Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-$\frac{1}{2}$ chain

TL;DR

This paper investigates the scaling limit of Bethe roots in an inhomogeneous XXZ spin-$rac{1}{2}$ chain, revealing their connection to differential equations generalizing the Schrödinger equation, extending the ODE/IM correspondence.

Contribution

It introduces a multiparametric generalization of the XXZ chain and demonstrates that the scaled Bethe roots satisfy multi-parametric differential equations in the critical regime.

Findings

Scaled Bethe roots relate to multi-parametric Schrödinger-like equations

Extension of the ODE/IM correspondence to inhomogeneous models

Ground state roots follow specific differential equations in the scaling limit

Abstract

It is known that for the Heisenberg XXZ spin-$\frac{1}{2}$ chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schr\"{o}dinger equation for the anharmonic oscillator.