Hubbard's Adventures in ${\cal N}=4$ SYM-land? Some non-perturbative considerations on finite length operators

TL;DR

This work develops two coupled non-linear integral equations (NLIEs) to exactly describe the highest-energy state of the half-filled Hubbard model, connecting finite-length operator dimensions in ${\cal N}=4$ SYM to a non-perturbative lattice framework. The energy is obtained from NLIE solutions and compared to the all-loop BDS Ansatz, with the key finding that at large length $L$ the two energies share the same asymptotic expansion up to exponentially small corrections, while wrapping effects appear as non-perturbative damped terms. Detailed analyses in strong and weak coupling, along with order-of-limit studies, demonstrate that both models agree at leading order but differ in subleading finite-size corrections, reinforcing the utility of NLIEs for non-perturbative tests of AdS/CFT. Numerically, the Konishi case ($L=4$) validates the approach against exact results, and intermediate-to-large lengths show that BDS is an excellent finite-$L$ approximation, with exponential damping controlling deviations as $L$ grows. The results lay groundwork for applying NLIEs to the BES dressing and larger sectors, enabling precise non-perturbative tests of gauge/string duality.

Abstract

As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU(2) sector of the planar $ {\cal N}=4$ SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, $L$, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, $λ$. For instance, we will show that the large $L$ (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on $L$ and $λ$ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easier-looking theories.

Figures (7)

• Figure 1: Comparison of the highest Hubbard and BDS energies (anomalous dimensions) for a system with $L=4$ sites, corresponding to the anomalous dimension of the Konishi operator. The curves indicated by "NLIE" are obtained by the non-linear integral equation -- (\ref{['Eexp']}) for Hubbard and eq. 3.24 of FFGR for BDS -- and those indicated by "power" are obtained with the power expansions (\ref{['powexpH']} and \ref{['powexpBDS']}) up to the thirtieth order. The value of $g$ where they get far away gives an idea of the convergence radius. The small image is a zoom of the surrounded area of the largest one.
• Figure 2: Comparison of numerical NLIE data (the same as in Fig. \ref{['konishi1']}) and the exact albeit implicit solution of MIN, from weak to strong coupling. The two curves perfectly overlap in the common interval $g\in[0,1.5]$.
• Figure 3: Comparison of Hubbard and BDS energies (anomalous dimensions) for $L=12$. The two curves are almost indistinguishable in this range of $g$ and start to separate at the right border of the plot.
• Figure 4: The behaviour of the energies $E(g)$ and $E_{\text{BDS}}$ from small to strong coupling is plotted here for a lattice of 12 sites. The left branches of the curves are the same as in Fig. \ref{['antif12']} while the right branches are given by the strong coupling expansions (\ref{['strong']}). In the small picture there is a zoom of the region where the branches overlap.
• Figure 5: Comparison of Hubbard and BDS energies (anomalous dimensions) for $L=40$. The two curves are visibly hard to distinguish in this range of $g$. The largest reached absolute difference is 0.0038 namely a relative difference of $0.019\%$.