Fluctuations and Energy Shifts in the Bethe Ansatz

TL;DR

The paper analyzes fluctuations and finite-size corrections in the Bethe Ansatz for the ferromagnetic thermodynamic limit of the Heisenberg XXX$_{1/2}$ spin chain, the gauge/string dual to semiclassical spinning strings. It derives a quantum-mechanical formula expressing the first-order energy shift as a sum over fluctuation energies, with a refined regularization to handle large-mode contributions. The authors apply the method to a simple one-cut solution, obtaining the spectrum of fluctuations and showing consistency with known gauge-theory and string-theory results. This provides two complementary routes to compute $1/L$ corrections and clarifies the role of fluctuations in the AdS/CFT correspondence, while outlining limitations near instabilities and avenues for extension to higher loops and other sectors.

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.

Figures (1)

• Figure 1: The one-cut solution $\mathcal{C},\rho$ with $n=6$ for some fixed value of $\alpha=K/L$ and the distribution of fluctuation roots $\mu_k^\pm$. The mode number of $\mu^+_k$ is $6+k$ for $k> k_{\mathrm{c}}=4$ for all other $\mu^+_k,\mu^-_k$ it is $6-k$. Some of these mode numbers are indicated in the figure. As compared to the vacuum with $\alpha=0$, the roots between $\mu_4^-$ and $\mu_4^+$, as well as their mode numbers, are distorted strongly by the cut.