Even-length Bethe strings and the antiferromagnetic vacuum of the all-loop su(2) Bethe equations in planar N=4 supersymmetric Yang-Mills theory

For the all-loop Bethe equations of Beisert, Dippel and Staudacher (BDS) describing the su(2) sector of planar N=4 supersymmetric Yang-Mills theory, Bytsko and Shenderovich proved that in the thermodynamic limit the Bethe roots group into strings, but discovered that even-length strings acquire, through the discontinuity of the phase of $x(u)$ across the cut $[-\chi,\chi]$, an additional contribution to their density and energy whose sign is controlled by a parameter $\nu_\epsilon$ fixed by the exponentially small deviations $\epsilon_m=O(e^{-\omega_m L})$ of the finite-length roots from perfect strings. A uniformly filled state of 2-strings in the $\nu_\epsilon=+1$ regime would have a larger maximal scaling dimension per site than the accepted 1-string (real-root) antiferromagnetic vacuum, while in the $\nu_\epsilon=-1$ regime the single-string energy is negative on parts of the real axis and the vacuum could only be filled on subintervals. Which regime is realized in the thermodynamic limit was left open, its resolution requiring a subtle beyond-all-orders analysis of the exponential corrections. The open problem is to determine the signs and magnitudes of the $\epsilon_m$ as $L\to\infty$, the realized value of $\nu_\epsilon$, and the resulting structure (filling pattern) and energy $\Delta_{\max}/L$ of the true antiferromagnetic vacuum of the BDS equations.

Background

In planar $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory, single-trace operators built from two complex scalar fields, $O=\mathrm{tr}(Z^{J_1}W^{J_2}+\text{permutations})$, close under renormalization and mix as an integrable spin chain of length $L=J_1+J_2$, with each $Z$ a spin up and each $W$ a spin down (Minahan and Zarembo, 2003). The eigenvalues of the planar dilatation operator, i.e. the scaling dimensions $\Delta$ of the operators, are found from Bethe ansatz equations. At one loop these are the Bethe equations of the Heisenberg XXX spin chain; Beisert, Dippel and Staudacher (BDS, 2004) proposed the all-loop deformation $\left(\frac{x(u_j+i/2)}{x(u_j-i/2)}\right)^L=\prod_{k\neq j}\frac{u_j-u_k+i}{u_j-u_k-i},\qquad x(u)=\frac{1}{2}\left(u+\sqrt{u^2-\chi^2}\right),\qquad \chi^2=\frac{\lambda}{4\pi^2},$ where $\lambda$ is the 't Hooft coupling, $u_j$ ($j=1,\dots,M=J_2$) are Bethe rapidities, and the equations are asymptotic, i.e. valid up to the so-called wrapping order, the loop order at which finite-size corrections first appear, in the energy. The structural novelty compared to the XXX chain is that $x(u)$ has a square-root branch cut on the segment $[-\chi,\chi]$ of the real rapidity axis: $|x(u)|$ is continuous across it, but the phase of $x(u)$ jumps. Because the number of su(2) operators at fixed $L$ is finite, the scaling dimensions are bounded, $L\le\Delta\le\Delta_{\max}$. The upper bound is attained by the antiferromagnetic vacuum, the state of maximal anomalous dimension, in which up and down spins occur in equal numbers ($M=L/2$). By the string hypothesis (Takahashi; Faddeev and Takhtajan), the roots of the exact finite-length equations of an XXX-type chain group, at large $L$, into strings: nearly collinear arrays $u_{j,m}=u_j+i m$ with real center $u_j$ and integer or half-integer $m$, exact only up to exponentially small deviations. For the XXX chain the antiferromagnetic vacuum is the state whose roots are all real (strings of length 1) filling every momentum level, with energy per site $2\log 2$ (Hulthén's result). Zarembo (2006) computed the analogous $\Delta_{\max}$ for the BDS equations in the thermodynamic limit $L\to\infty$ assuming the same 1-string (real-root) vacuum, and Rej, Serban and Staudacher (2006) showed that the resulting energy per site coincides with minus the ground-state energy of the half-filled one-dimensional Hubbard model found by Lieb and Wu (1968), identifying the BDS chain with the strong-coupling Hubbard model; this match was itself derived under the 1-string assumption. Meanwhile, the example of the higher-spin XXX chains (Takhtajan, 1982) shows that when the left-hand side of the Bethe equations is modified, the antiferromagnetic vacuum can be built from strings of length greater than 1. Bytsko and Shenderovich (2007) proved for the BDS equations that strings of any length do exist in the thermodynamic limit (via convexity of $|x(u)|$ in $\mathrm{Im}\,u$), that Yang's action (the functional whose stationarity yields the thermodynamic equations for the string centers) is convex, which implies uniqueness of the solution for the centers, and that among odd-length string fillings the energy per site decreases with the string length, so that length 1 is maximal among odd strings. The subtlety is specific to even lengths. At any finite $L$ the roots form strings only up to deviations $u_{j,m}=u_j+i(m+\epsilon_m)$ with $\epsilon_m=O(e^{-\omega_m L})$, $\omega_m>0$. For an even-length string whose center lies on the real axis, the two central roots produce energy contributions whose would-be cancellation depends on the jump of the phase of $x(u)$ across the cut: for $u\in[-\chi,\chi]$ one has $\lim_{\epsilon\to 0}\ln\frac{x(u-i\epsilon)}{x(u+i\epsilon)}=2i\,\nu_\epsilon\arctan\frac{\sqrt{\chi^2-u^2}}{u},$ where the sign $\nu_\epsilon=\pm 1$ is fixed by the side from which the deviation $\epsilon$ of the central pair tends to zero as $L\to\infty$ ($\nu_\epsilon=-1$ if $\epsilon$ tends to zero from the right and $\nu_\epsilon=+1$ if it tends from the left). In the regime $\nu_\epsilon=+1$ the extra contribution is positive, and Bytsko and Shenderovich showed that a uniformly filled state of strings of length 2 then yields a strictly larger $\Delta_{\max}/L$ than the 1-string state. In the regime $\nu_\epsilon=-1$ the single-string energy $\mathcal{E}_j$ is negative on parts of the real axis, so that the vacuum, if built of even strings, could only be filled on the intervals where $\mathcal{E}_j>0$. Which regime is actually realized in the thermodynamic limit was left open by Bytsko and Shenderovich, who note that resolving it requires a subtle analysis of the exponential corrections $\epsilon$ for $L\to\infty$.

Problem Statement

For the all-loop (BDS) Bethe equations of the su(2) sector of planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory, displayed in the Background, in the antiferromagnetic regime ($M=L/2$ roots) and the thermodynamic limit $L\to\infty$ at fixed $\chi>0$: (i) determine the asymptotic behavior of the deviations $\epsilon_m=O(e^{-\omega_m L})$ of the finite-$L$ roots from perfect even-length strings, $u_{j,m}=u_j+i(m+\epsilon_m)$ - their magnitudes, their signs, and their dependence on the string center $u_j$ and on $\chi$; (ii) determine thereby which regime, $\nu_\epsilon=+1$ or $\nu_\epsilon=-1$, is realized for the central pair of an even-length string in the thermodynamic limit, i.e. from which side the deviation of the central pair approaches zero in the discontinuity formula above; (iii) establish the structure of the true antiferromagnetic vacuum of these equations - the state of maximal $\Delta/L$: the uniformly filled 1-string (real-root) state assumed by Zarembo and by Rej, Serban and Staudacher, a uniformly filled 2-string state (which in the $\nu_\epsilon=+1$ regime yields a strictly larger $\Delta_{\max}/L$), or, in the $\nu_\epsilon=-1$ regime where the single-string energy $\mathcal{E}_j$ is positive only on parts of the real axis, a state with even strings filled only on the intervals where $\mathcal{E}_j>0$ - and compute the corresponding value of $\Delta_{\max}/L$. These three items form one coherent objective: the exponentially small corrections of (i) fix the sign $\nu_\epsilon$ of (ii), which in turn decides the filling pattern and the maximal energy of (iii). An answer must settle the realized regime and the vacuum structure for the BDS equations as stated; deriving the consequences within one assumed regime (as Bytsko and Shenderovich did for $\nu_\epsilon=+1$) is not by itself a resolution.

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