Analytic-in-anisotropy condensed Bethe solutions for the symmetric six-vertex model up to exact half filling

For the symmetric six-vertex model (weights a, b, c; anisotropy $\Delta=(a^2+b^2-c^2)/(2ab)<1$) on the torus with periodic boundary conditions and n up arrows per row, eigenvalues of the transfer matrix are parameterized by solutions of a discrete system of Bethe equations. Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia (Comm. Math. Phys. 395:1383-1430, 2022) proved that condensed Bethe solutions exist for every $n\le N/2$ and $\Delta\ne -1$, and that they can be organized into families analytic in $\Delta$: on $(-\infty,\Delta_0)$ for $\Delta_0<-1$ with $N\ge N_0(\Delta_0)$ and $n\le N/2$, on $(\Delta_0,1)\subset(-1,1)$ under a half-filling margin $n\le N/2-C_0(\Delta_0)$ and a size threshold (both constants vanishing for one exceptional $\Delta_0\in(-1,0)$ near 0), and on the whole $(-1,1)$ only under $n\le N/2-C_1\log^2 N$. Whether an analytic-in-$\Delta$ family of condensed solutions exists for arbitrary $n\le N/2$ - including exact half filling $n=N/2$ - over the whole intervals $(-\infty,-1)$ and $(-1,1)$ remains open; the authors link the question to the expected infinite-order (Kosterlitz-Thouless-type) transition at $\Delta=-1$. A positive answer would identify the Bethe-ansatz eigenvalue as the Perron-Frobenius eigenvalue for all $\Delta<1$ and complete the free-energy asymptotics at half filling.

Background

The six-vertex model is a two-dimensional lattice model of arrow configurations on the edges of the square lattice: at each vertex exactly two of the four incident edges point in and two point out (the ice rule), which leaves six admissible vertex types carrying weights a, b, c. In the symmetric case (one weight a for both vertex types with two horizontal incoming arrows, similarly b and c), the physics is governed by the anisotropy parameter $\Delta=(a^2+b^2-c^2)/(2ab)$: $\Delta>1$ is ferroelectric, $\Delta<-1$ antiferroelectric (massive, exponentially decaying correlations), and $-1\le\Delta\le 1$ disordered (massless, algebraically decaying correlations); the point $\Delta=-1$ is expected to host a phase transition of infinite order, of Kosterlitz-Thouless type (the smooth, essential-singularity transition mechanism known from the two-dimensional XY model). On the $N\times M$ torus with periodic boundary conditions, the partition function restricted to configurations with n up arrows per row ($n\le N/2$; the density is $m=n/N$) is governed by the Perron-Frobenius eigenvalue - the maximal positive eigenvalue - of the transfer matrix $V_N^{(n)}(a,b,c)$ (the finite matrix summing the Boltzmann weights over the arrow configurations of a single row, whose iterated trace generates the partition function) in that sector, from which the free energy is extracted in the thermodynamic limit.

The Bethe ansatz diagonalizes this transfer matrix: eigenvalues are built from n real rapidities $\lambda_1<\dots<\lambda_n$ solving the discrete Bethe equations

$\frac{N}{2\pi}\,\mathfrak{p}(\lambda_i)-\frac{1}{2\pi}\sum_{j=1}^n\vartheta(\lambda_i-\lambda_j)=I_i,\qquad 1\le i\le n,$

where $\mathfrak{p}$ and $\vartheta$ are explicit $\Delta$-dependent functions (momentum and two-particle scattering phase, given in closed form in the appendix of the cited work) and the $I_i$ are the prescribed half-integers $I_i=i-\tfrac{n+1}{2}$ (the canonical choice used throughout that work). One restricts to solutions that are symmetric ($\lambda_{n+1-i}=-\lambda_i$) and strictly ordered. In the thermodynamic limit the empirical distribution of a condensed solution converges to a limiting root density $\rho(\cdot|\mathfrak{q})$ supported on $[-\mathfrak{q},\mathfrak{q}]$, where the Fermi boundary $\mathfrak{q}=Q(m)$ (the endpoint of the roots' limiting support) is fixed by $\int_{-\mathfrak{q}}^{\mathfrak{q}}\rho(\lambda|\mathfrak{q})d\lambda=m$; condensation is quantitative: $\big|\tfrac1N\sum_j f(\lambda_j)-\int_{-\mathfrak{q}}^{\mathfrak{q}}f\,d\rho\big|\le (C/\zeta N)\|f'\|_{L^1}$ for every test function f with integrable derivative, with $\Delta=-\cos\zeta$ for $-1<\Delta<1$ and $\Delta=-\cosh\zeta$ for $\Delta<-1$. A key structural notion is $(k,\mathfrak{q})$-interlacement: each root $\lambda_i$ lies between the $(i-k/2)$-th and $(i+k/2)$-th quantiles of $\rho(\cdot|\mathfrak{q})$; for $k=1$ the roots perfectly interlace with these quantiles.

After Lieb's exact solution of square ice and Sutherland's finite-density extension, Yang and Yang developed the Bethe-ansatz thermodynamics of the equivalent XXZ spin-$1/2$ chain (the one-dimensional quantum spin chain with anisotropic nearest-neighbour exchange, whose coordinate Bethe ansatz leads to the same equations). Kozlowski (2018), through a rigorous theory of the non-linear integral equations (NLIE) satisfied by the Bethe counting function (the integrated form of the Bethe equations recording the integers $I_i$), established condensation and analyticity in $\Delta$ for $\Delta<-1$ at all densities $m\in[0,1/2]$ and for $-1\le\Delta<1$ at $m\in[0,1/2)$. Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia (2022) then gave new proofs of existence and condensation valid for all $n\le N/2$ and $\Delta\ne -1$, and proved: (i) an analytic-in-$\Delta$ family of condensed solutions, unique in its class, on $(\Delta_0,1)$ for $N\ge N_0(\Delta_0)$ and $n\le N/2-C_0(\Delta_0)$ when $\Delta_0>-1$ - the two constants vanishing for one exceptional $\Delta_0\in(-1,0)$ near 0, so that there the family exists for every $N$ and every $n\le N/2$, exact half filling included - on $(-\infty,\Delta_0)$ for $N\ge N_0(\Delta_0)$ and $n\le N/2$ when $\Delta_0<-1$, and (Theorem 6 of that paper) on the whole interval $(-1,1)$ only under the half-filling deficit $n\le N/2-C_1\log^2 N$ and a size threshold; (ii) that the transfer-matrix eigenvalue constructed from this analytic family is the Perron-Frobenius eigenvalue of $V_N^{(n)}$; and (iii) sharp asymptotics of the free energy as the density approaches $1/2$, under the condition $n\le N/2-C\min\{\zeta^{-2},\log^2N\}$. The remaining obstruction is concentrated at half filling in the anisotropy range $-1<\Delta<0$: for $|\Delta|\le 1$ the Fermi boundary at half filling is infinite ($Q(1/2)=\infty$, the roots spread over the whole real line), and there the Jacobian of the root map is proven invertible only under the deficit $n\le N/2-Ck^2$ with $k(\Delta)\sim C/\zeta$ blowing up as $\Delta\to -1^+$ - a deficit which excludes $n=N/2$ for every fixed $\Delta\in(-1,0)$ outside the exceptional neighborhood of $\Delta=0$ (at $\Delta=0$ itself the Jacobian is diagonal). The authors expect $(1,\mathfrak{q})$-interlaced condensed solutions to exist for every $\Delta<1$ - numerical plots at $n=N/2$ suggest $(1,+\infty)$-interlacement for $\Delta\ge -1$ - but cannot prove this for n close to N/2, and they explicitly state they are unable to prove the existence of an analytic solution for arbitrary $n\le N/2$ over the whole intervals $(-\infty,-1)$ and $(-1,1)$, a fact they expect to be closely related to the infinite-order transition at $\Delta=-1$.

Problem Statement

Prove or refute the following statement, left open by Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia (Comm. Math. Phys. 395:1383-1430, 2022): for the symmetric six-vertex model with periodic boundary conditions and anisotropy $\Delta<1$, for every even $N$ and every $n\le N/2$ there exists a family $\Delta\mapsto\boldsymbol{\lambda}(\Delta)=(\lambda_1,\dots,\lambda_n)(\Delta)$ of symmetric, strictly ordered solutions of the discrete Bethe equations stated in the background, which is analytic in $\Delta$ on the whole intervals $(-\infty,-1)$ and $(-1,1)$ and whose members are condensed (satisfy the quantitative root-distribution bound with respect to the limiting density $\rho(\cdot|\mathfrak q)$, $\mathfrak q=Q(n/N)$; at exact half filling with $|\Delta|\le 1$ this is read with $\mathfrak q=Q(1/2)=\infty$, i.e. as the corresponding whole-line statement, exactly as in the source) for every $\Delta$ in those intervals. Equivalently: remove from Theorem 4 and Theorem 6 of that paper the restrictions that fail at half filling - the density deficit $n\le N/2-C_0(\Delta_0)$ resp. $n\le N/2-C_1\log^2 N$, and the $\Delta$-dependent size threshold $N\ge N_0(\Delta_0)$ diverging as $\Delta_0\downarrow -1$ (these constants already vanish for the one exceptional $\Delta_0\in(-1,0)$ near 0 singled out in Theorem 4) - so that the statement covers exact half filling $n=N/2$ on all of $(-\infty,-1)\cup(-1,1)$, including the approach to $\Delta=-1$ from both sides.

The objective is uniform analytic control of the half-filled Bethe equations across the anisotropy: a positive resolution is expected, via the mechanism of Theorem 5 of the cited work, to imply that the eigenvalue built from $\boldsymbol{\lambda}(\Delta)$ is the Perron-Frobenius eigenvalue of the transfer matrix $V_N^{(n)}(a,b,c)$ for every $\Delta<1$, $\Delta\ne -1$, and every $n\le N/2$, and to allow removal of the half-filling condition $n\le N/2-C\min\{\zeta^{-2},\log^2 N\}$ from the free-energy asymptotics as $m\uparrow 1/2$ at $-1\le\Delta<1$, closing the gap where the non-linear integral equation method reaches no estimate beyond $o(1)$. A refutation - e.g. a certified collision of condensed solution branches with degenerate Jacobian at some $\Delta^*$ inside $(-1,1)$ precluding any analytic selection - would instead show the expectation is false and would clarify how the infinite-order transition at $\Delta=-1$ is mirrored in the finite-N Bethe equations.

References

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