Double Scaling and Finite Size Corrections in sl(2) Spin Chain

TL;DR

This paper derives explicit finite size corrections for the distribution of Bethe roots, energy, and conserved charges in the sl(2) spin chain, using a double scaling approach related to integrability in super-Yang-Mills theory.

Contribution

It introduces a double scaling technique applied to the Baxter equation to compute finite size corrections in the sl(2) spin chain, generalizable to higher orders and other models.

Findings

Explicit expressions for first finite size corrections to Bethe root distribution.

Bethe root positions near the edge described by zeros of Airy function.

Method applicable to other quantum integrable models.

Abstract

We find explicit expressions for two first finite size corrections to the distribution of Bethe roots, the asymptotics of energy and high conserved charges in the sl(2) quantum Heisenberg spin chain of length J in the thermodynamical limit J->\infty for low energies E\sim 1/J. This limit was recently studied in the context of integrability in perturbative N=4 super-Yang-Mills theory. We applied the double scaling technique to Baxter equation, similarly to the one used for large random matrices near the edge of the eigenvalue distribution. The positions of Bethe roots are described near the edge by zeros of Airy function. Our method can be generalized to any order in 1/J. It should also work for other quantum integrable models.