Origin of the full nonuniversality of the electric critical exponents of the symmetric 16-vertex model on the square lattice

The symmetric two-state 16-vertex model on the square lattice, whose vertex weights are invariant under any permutation of the four adjacent edge states, possesses a self-dual critical line under the O(2) gauge transformation of vertex weights, parametrized by an energy parameter $\varepsilon$. The line connects two exactly solvable points: a square-lattice Ising point at $\varepsilon\approx 0.6275$ and, as $\varepsilon\to\infty$, a Baxter eight-vertex point with $a=1$, $b=c=d$, with the polarization as order parameter throughout. Numerical work by Pospíšilová, Krčmár, Gendiar and Šamaj (2020), using the Corner Transfer Matrix Renormalization Group with independent tensor-network benchmarks, indicates that the electric critical exponents $\beta_e,\delta_e,\gamma_e,\nu_e,\eta_e$ vary continuously (and for $\beta_e$ nonmonotonically) with $\varepsilon$, violating weak universality. The finding is, however, purely numerical: $\nu_e$ and $\eta_e$ were obtained only through scaling relations, the extrapolations in the truncation dimension and in the field/temperature steps were not carried to full numerical proof, and no analytic or field-theoretic derivation of the exponent functions — nor of the microscopic mechanism producing the full nonuniversality, including its relation to the gauge/self-dual structure — exists. No resolution is known: the symmetric 16-vertex model has no known exact solution beyond its two solvable endpoints, and the solvable subfamilies found so far lie on staggered, free-fermion-type submanifolds that do not touch the symmetric self-dual line. The problem of deriving the exponent functions and the mechanism of their continuous variation along the self-dual critical line, or of demonstrating that the reported variation is a numerical artifact, therefore remains open.

Background

In a two-state vertex model on the square lattice, every edge carries a two-valued state, drawn as an arrow pointing right/up ($+$) or left/down ($-$), and each of the $2^4=16$ configurations of the four edges meeting at a vertex carries a Boltzmann weight. The classical six- and eight-vertex models impose the ice rule (an even number of arrows points into each vertex), which expresses arrow conservation; the 16-vertex model lifts this constraint completely and is the most general two-state vertex model on the square lattice. It maps onto a "magnetic" Ising model on the dual (also square) lattice, the vertex weight becoming an interaction among the four Ising spins around a plaquette: the even, ice-rule-respecting sector keeps only nearest-neighbor pair and four-spin plaquette terms, while the general model adds odd field-like and three-spin terms and subsumes the two-dimensional Ising model in a magnetic field, so no exact solution of the general model is known. The 'symmetric' 16-vertex model requires the weight $w_i$ ($i=0,\ldots,4$) to depend only on the number $i$ of adjacent edges in the $-$ state, i.e. invariance under any permutation of the four edge states; with the additional state-flip symmetry $w_i=w_{4-i}$ one may fix $w_0=w_4=1$, $w_1=w_3=\exp(-\varepsilon/T)$, $w_2=\exp(-1/T)$ with a real energy parameter $\varepsilon\ge 0$ and dimensionless temperature $T$. The order parameter is the mean polarization $P$ (the average edge state); a uniform external electric field $E$ conjugate to it defines the remaining exponents.

The partition function of a vertex model is invariant under a gauge (weak-graph) transformation of its weights; for two-state models this is the O(2) gauge transformation, an orthogonal two-by-two mixing of the two edge states controlled by a single real parameter. For the symmetric 16-vertex model this transformation was analyzed by Wu and Wu: the self-dual manifold — points mapped onto themselves by a gauge transformation with nontrivial parameter — is a codimension-one algebraic subset, and all second-order critical points lie on it. The parametrization above satisfies the self-duality condition identically, so for each $\varepsilon\ge 0$ there is a single critical temperature $T_c(\varepsilon)$, known numerically to about $10^{-4}$.

Two points of this critical line are exactly solvable. At $\varepsilon^{(I)}=1-\ln(1+\sqrt{2})/\ln(5+4\sqrt{2})\approx 0.6275$ the model maps onto the square-lattice Ising model, and the electric exponents equal the Ising ones: $(\beta_e,\delta_e,\gamma_e,\nu_e,\eta_e)=(1/8,15,7/4,1,1/4)$. As $\varepsilon\to\infty$ the vertices with an odd number of $-$ edges freeze out and the model reduces to Baxter's zero-field eight-vertex model with $a=1$, $b=c=d=\exp(-1/T)$, whose second-order ferroelectric transition at $T_c=1/\ln 3$ has, combining Baxter's exact solution with the two-dimensional scaling relations, $(\beta_e,\delta_e,\gamma_e,\nu_e,\eta_e)=(1/8,11,5/4,3/4,1/3)$; in terms of the eight-vertex parameter $\mu=2\pi/3$ at this point, $\beta_e=(\pi-\mu)/(4\mu)$, $\delta_e=(3\pi+\mu)/(\pi-\mu)$, $\gamma_e=(\pi+\mu)/(2\mu)$, $\nu_e=\pi/(2\mu)$, $\eta_e=1-\mu/\pi$. Here $\beta_e$ governs the spontaneous polarization $P\propto(T_c-T)^{\beta_e}$, $\delta_e$ the field dependence $P\propto E^{1/\delta_e}$ at $T_c$, $\gamma_e$ the divergence of the polarization susceptibility, $\nu_e$ the divergence of the correlation length $\xi\propto|T_c-T|^{-\nu_e}$, and $\eta_e$ the power-law decay of the arrow–arrow correlation function at criticality.

The universality backdrop is the following. Baxter's exact solution of the zero-field eight-vertex model gave the first violation of simple universality: its magnetic exponents vary continuously with the weights yet obey weak universality in Suzuki's sense — the renormalized ratios $\alpha/\nu$, $\beta/\nu$, $\gamma/\nu$ and the exponents $\delta$, $\eta$ defined at $T_c$ remain constant. For the electric formulation of the same model, Krčmár and Šamaj later showed the opposite: the electric exponents depend continuously on the weights, so the electric formulation is fully nonuniversal — but there the variation is analytically derived from the exact solution. For the symmetric 16-vertex model no such analytic handle exists.

Pospíšilová, Krčmár, Gendiar and Šamaj studied the model along its self-dual line with the Corner Transfer Matrix Renormalization Group (CTMRG) method — a density-matrix-renormalization-group-type algorithm that iteratively truncates the corner-transfer-matrix degrees of freedom of a two-dimensional classical model to the dominant eigenspace of a reduced density matrix, controlled by a truncation dimension $m$ — with independent benchmarks from the Higher-Order Tensor Renormalization Group (HOTRG). They found: a line of critical points with polarization order parameter connecting the two solvable endpoints; $T_c(\varepsilon)\to 0$ linearly as $\varepsilon\to 0$ and $T_c\to 1/\ln 3$ at large $\varepsilon$; $\beta_e(\varepsilon)$ nonmonotonic in the transition region $2\lesssim\varepsilon\lesssim 14$ with variations greatly exceeding the method's error bars while equal to $1/8$ at both ends; $\delta_e$ interpolating between 15 and 11; $\gamma_e$ between 7/4 and 5/4; and $\nu_e,\eta_e$ — obtained only by inserting the fitted $\beta_e,\delta_e$ into the scaling relations $\gamma_e=\beta_e(\delta_e-1)$, $\nu_e=\beta_e(\delta_e+1)/2$, $\eta_e=4/(\delta_e+1)$ — between 1 and 3/4 and between 1/4 and 1/3. Since $\beta_e/\nu_e$ and $\gamma_e/\nu_e$ take different values at the two endpoints, weak universality is violated: the electric criticality is fully nonuniversal. What is missing is any analytic understanding. The ice-rule conservation that underpins the Coulomb-gas and Gaussian field-theory treatments of the six- and eight-vertex families is absent here, so no off-the-shelf continuum description applies, and the mechanism generating the continuous variation of the exponents — and its relation to the O(2) gauge/self-dual structure that organizes the critical line — is not understood.

Problem Statement

Consider the symmetric 16-vertex model on the square lattice with permutation-invariant, state-flip-symmetric vertex weights $w_0=w_4=1$, $w_1=w_3=\exp(-\varepsilon/T)$, $w_2=\exp(-1/T)$ ($\varepsilon\ge 0$), whose self-dual critical line $T_c(\varepsilon)$ carries the polarization $P$ as order parameter and connects the exactly solvable Ising point $\varepsilon^{(I)}=1-\ln(1+\sqrt{2})/\ln(5+4\sqrt{2})\approx 0.6275$ to the exactly solvable Baxter eight-vertex point $\varepsilon\to\infty$. Along this line, CTMRG/HOTRG numerics indicate that the electric critical exponents $\beta_e,\delta_e,\gamma_e,\nu_e,\eta_e$ vary continuously with $\varepsilon$ (nonmonotonically for $\beta_e$ in $2\lesssim\varepsilon\lesssim 14$), in violation of weak universality.

Provide an analytic account of this electric critical behavior. Specifically: derive, from the vertex weights or from an explicitly constructed continuum/field-theoretic description of the model on its self-dual line, the exponent functions $\beta_e(\varepsilon),\delta_e(\varepsilon),\gamma_e(\varepsilon),\nu_e(\varepsilon),\eta_e(\varepsilon)$; for an analytic theory in which the standard two-dimensional scaling relations hold identically, an analytic derivation of two independent exponent functions together with those relations constitutes a derivation of all five — in contrast to the source numerics, where $\nu_e$ and $\eta_e$ were obtained only by inserting numerically fitted $\beta_e,\delta_e$ into the scaling relations; reproduce the exact endpoint values $(1/8,15,7/4,1,1/4)$ at $\varepsilon^{(I)}$ and $(1/8,11,5/4,3/4,1/3)$ at $\varepsilon\to\infty$; and identify the microscopic mechanism that generates the continuous full-nonuniversal variation of the electric exponents, in particular its relation to the O(2) gauge/self-dual structure on which the critical line lies. An exact solution of the model on the critical line (free energy, spontaneous polarization, correlation length, and $T_c(\varepsilon)$) from which these exponents and the mechanism follow is an equally valid resolution.

Alternatively — and equally decisive for the question as posed, since the source claims only numerical indications — a complete and controlled demonstration that the reported continuous variation is an artifact of finite truncation dimension $m$ and finite field/temperature steps, i.e. that in the controlled limits $m\to\infty$, $E\to 0$, $\Delta T\to 0$ (and thermodynamic $L\to\infty$) the electric exponents do not vary continuously along the line as reported — for instance, because they are constant in the regions of small and large $\varepsilon$ with a discontinuous change at intermediate values of $\varepsilon$, the two-universality-classes scenario explicitly contemplated in the source — would resolve the question; such a demonstration must cover the transition region $2\lesssim\varepsilon\lesssim 14$ where the reported variation is concentrated and must include direct correlation-function determinations of $\nu_e$ and $\eta_e$.

References

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