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

TL;DR

The paper develops a double-scaling framework to compute finite-size corrections in the sl(2) spin chain at large length $J$ for low-energy states. By expanding the Baxter equation and exploiting an Airy-edge universality, it derives explicit $1/J$ and $1/J^2$ corrections to the Bethe-root distribution, energy, and both local and non-local charges, expressing results in terms of the underlying algebraic curve and edge data. The approach reveals a universal edge behavior governed by Airy functions, enabling accurate near-edge descriptions and providing insights transferable to random-matrix theory and AdS/CFT integrability. The work offers a systematic path to higher-order corrections and suggests applicability to other quantum integrable models and sectors of ${\ m N}=4$ SYM.

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.

Figures (5)

• Figure 1: Hyperelliptic Riemann surface
• Figure 2: Density of roots. The dots correspond to numerical 3-cut solution with total number of Bethe roots $S=300$ and equal fractions $\alpha_i=1/6$, and $n_i=\{-1,3,1\}$. They are fixed from the numerical values of the roots by the eq.(\ref{['defvro']}). Solid line is the density at $J=\infty$ computed analytically from the corresponding hyper-elliptic curve. $x$ coordinates of the dots are $\frac{u_{j}+u_{j+1}}{2J}$ so that the solitary points in the middle of empty cuts are artifacts of this definition.
• Figure 3: Quasi-momentum near branch point as a function of the scaling variable $v$ for $S=200$. The poles corresponds to the positions of Bethe roots $u_i$. Red dashed line - "exact" numerical value, light grey - zero order approximation given by Airy function ${\rm Ai}(a^{1/3}x)$, grey - first order and black - second order approximation.
• Figure 4: Resolvent far from branch point as a function of $x$. Red dashed line - "exact" numerical value for one cut solution with $S=10,\;n=2,\;m=1$, light grey - zero order approximation, grey - first order given by eq.(\ref{['p1res']}) and black - second order approximation given by eq.(\ref{['p2Gx']}). Note that near branch point ($x_0=0.02$) the approximation does not work and instead of it we should use the Airy function of eq.(\ref{['AiryF']}), like in the usual WKB near a turning point.
• Figure 5: Relative deviation $\delta E(S)/E(S)$ of analytical computations of the energy $E(S)$ from its "exact" value $E_{exact}(S)$ for the one cut distribution found numerically by Mathematica (solid line corresponds to $\delta E(S)=0$), for a finite number of roots $S$ and a finite length $J$ for zero order (light gray), first order (gray) and second order (black) approximation. Details are summarized in the table