Fluctuations and Energy Shifts in the Bethe Ansatz

TL;DR

This paper investigates fluctuations and finite size effects in the Bethe ansatz for the Heisenberg XXX1/2 spin chain, relevant to AdS/CFT correspondence, deriving formulas for energy shifts and analyzing specific solutions.

Contribution

It derives a quantum mechanical formula for energy shifts as a sum over fluctuation energies in the Bethe ansatz framework for the Heisenberg spin chain.

Findings

Derived the standard quantum mechanical formula for energy shifts.

Applied the formula to a one-cut solution to find its fluctuation spectrum.

Provided insights into finite size corrections in the Bethe ansatz context.

Abstract

We study fluctuations and finite size corrections for the ferromagnetic thermodynamic limit in the Bethe ansatz for the Heisenberg XXX1/2 spin chain, which is the AdS/CFT dual of semiclassical spinning strings. For this system we derive the standard quantum mechanical formula which expresses the energy shift as a sum over fluctuation energies. As an example we apply our results to the simplest, one-cut solution of this system and derive its spectrum of fluctuations.