Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings

TL;DR

This paper analyzes finite-size (1/L) corrections to the Bethe ansatz for ferromagnetic SU(2) and SU(1,1) spin chains, connecting them to quantum corrections on the string theory side of AdS/CFT. By deriving corrected Bethe equations and computing the corresponding energy corrections for circular-string-like states, the authors extend the well-established thermodynamic-limit matches to include leading quantum effects. Notably, the SU(1,1) sector yields results that align with one-loop string energies, reinforcing the AdS/CFT correspondence, while highlighting subtleties in the SU(2) case. The work thus lays groundwork for verifying the complete tower of conserved charges at the quantum level and deepens the link between finite-size spin chains and quantum string dynamics.

Abstract

N=4 supersymmetric Yang-Mills operators carrying large charges are dual to semiclassical strings in AdS_5xS^5. The spectrum of anomalous dimensions of very large operators has been calculated solving the Bethe ansatz equations in the thermodynamic limit, and matched to energies of string solitons. We have considered finite size corrections to the Bethe equations, that should correspond to quantum effects on the string side.

Figures (3)

• Figure 1: Check of \ref{['fsize']} for a density $\rho\! =\! 3x \sqrt{1\!-x^2}$ for which $x_j\!\! =\!\! \sqrt{1-(1\!-\!j/L)^{2/3}}$ with $j=1,..,L\!-\!1$ and $L=15$. $(a)$ The dotted line is the LHS of \ref{['fsize']}, while the $+$ are the RHS; the $\times$ denote the first term in the RHS. $(b)$ The $\ast$ represent the difference between the LHS and the first term in the RHS of \ref{['fsize']}, and the $+$ are the $1/L$ correction.
• Figure 2: Contour $\Gamma$ for a) $SU(1,1)$ and b) $SU(2)$ cases. The dots, $\bullet$, are the points $\rho=0, \rho_{M}$. The crosses, $\times$, in figure b) represent the poles of $\pi \rho \coth \pi \rho$.
• Figure :