Electric critical exponents of the symmetric 16-vertex model on its self-dual critical line: analytic determination and mechanism of full nonuniversality

The symmetric two-state 16-vertex model on the square lattice — vertex weights invariant under any permutation of adjacent edge states and, in the subfamily studied, under the global flip of all edge states — possesses a line of second-order phase transitions on the self-dual manifold of the gauge transformation, with the mean polarization as order parameter. The line connects two exactly solvable anchors: an Ising point at $\varepsilon^{\rm(I)}\approx 0.6275$ with electric critical exponents $(\beta_{\rm e},\delta_{\rm e},\gamma_{\rm e},\nu_{\rm e},\eta_{\rm e})=(1/8,15,7/4,1,1/4)$, and the $\varepsilon\to\infty$ limit, a Baxter eight-vertex point with $(1/8,11,5/4,3/4,1/3)$. Corner-transfer-matrix renormalization group numerics indicate that between these anchors the electric exponents vary continuously with $\varepsilon$ in a way that violates the weak-universality hypothesis; however, no analytic derivation, exact solution, or continuum field-theoretic description of this variation exists, the numerical determination has not been extrapolated in a controlled way to the $m\to\infty$, zero-field, and zero-temperature-step limits, and $\nu_{\rm e}$ and $\eta_{\rm e}$ were obtained only indirectly from scaling relations — the $\nu_{\rm e}$ data alone even remain compatible with two universality classes joined by a discontinuous change. The problem is to determine the five exponent functions $\beta_{\rm e}(\varepsilon)$, $\delta_{\rm e}(\varepsilon)$, $\gamma_{\rm e}(\varepsilon)$, $\nu_{\rm e}(\varepsilon)$, $\eta_{\rm e}(\varepsilon)$ along the self-dual line — analytically if possible — together with an independent determination of $\nu_{\rm e}$ and $\eta_{\rm e}$ that decides whether the variation is genuinely continuous for all five exponents or whether part of it corresponds to distinct universality classes joined by a discontinuous change; should an analytic treatment be found, it would also elucidate how the gauge/self-duality structure of the weights produces the full nonuniversality on this line.

Background

A two-state vertex model on the square lattice places an arrow variable on each edge and assigns a Boltzmann weight to each lattice site depending only on the configuration of its four adjacent edge states; there are $2^4=16$ possible vertex configurations, and the general model of this type is the 16-vertex model. Its restricted subclasses, the six-vertex model and the zero-field eight-vertex model, are exactly solvable, but no exact solution is known for the full 16-vertex model. The 'symmetric' 16-vertex model postulates that the 16 weights are invariant under any permutation of the adjacent edge states, which groups them into five classes $w_0,\dots,w_4$ labelled by the number of arrows flowing into the vertex. The source work further restricts to weights invariant under the flip of all edge states ('state-flip symmetry', $w_i=w_{4-i}$), giving the two-parameter family $w_0=w_4=1,\qquad w_1=w_3=e^{-\varepsilon/T},\qquad w_2=e^{-1/T},\qquad \varepsilon\ge 0,$ where $T$ is the temperature in energy units.

A gauge transformation of vertex weights reverses the arrows on all edges touching a chosen set of vertices; points of the parameter space that are mapped onto themselves by a nontrivial gauge transformation form the self-dual manifold. For the general vertex model this manifold was derived by Wu and Wu (1989), and it contains all second-order critical points, because the free energy is invariant along gauge orbits. The state-flip parametrization above lies on it automatically, so for every $\varepsilon\ge 0$ the model has a critical temperature $T_{\rm c}(\varepsilon)$ at which a second-order transition occurs. The order parameter is the mean polarization $P=\frac{1}{4}\sum_{i=0}^4(4-2i)c_i$, where $c_i$ is the mean concentration of vertex type $i$; an applied isotropic electric field $E$ couples to the arrow dipoles. The transition is 'electric': it breaks the state-flip symmetry and orders the polarization, by contrast with the 'magnetic' Ising-type transitions of the dual spin representations.

Near $T_{\rm c}$ the electric critical exponents are defined by the standard singularities: $P\propto(T_{\rm c}-T)^{\beta_{\rm e}}$, $P(E)\propto E^{1/\delta_{\rm e}}$ at $T=T_{\rm c}$, susceptibility $\chi\propto|T-T_{\rm c}|^{-\gamma_{\rm e}}$, correlation length $\xi\propto|T-T_{\rm c}|^{-\nu_{\rm e}}$, and large-distance decay of the arrow-arrow correlation function $G_{\rm e}(r)\propto r^{-\eta_{\rm e}}$ at criticality. Weak universality (Suzuki 1974) is the observation that on many lines of critical points the exponents defined at $T_{\rm c}$ ($\delta$, $\eta$) and the ratios $\beta/\nu$, $\gamma/\nu$ stay constant while $\beta$, $\gamma$, $\nu$ vary; it holds for the magnetic exponents of Baxter's exactly solved zero-field eight-vertex model. Baxter's solution, however, fixes only one electric exponent ($\beta_{\rm e}$). Krčmár and Šamaj (2018) showed that the electric correlation-length exponent equals the magnetic one, $\nu_{\rm e}=\nu$, and thereby derived exact formulas for all electric exponents of the symmetric eight-vertex model — with $\mu$ the angle parameter that labels points on the critical line of Baxter's zero-field eight-vertex solution, $\beta_{\rm e}=(\pi-\mu)/4\mu$, $\delta_{\rm e}=(3\pi+\mu)/(\pi-\mu)$, $\gamma_{\rm e}=(\pi+\mu)/2\mu$, $\nu_{\rm e}=\pi/2\mu$, $\eta_{\rm e}=1-\mu/\pi$ — establishing that the electric exponents of the eight-vertex model vary with $\mu$ in violation of weak universality ('full nonuniversality' of the electric formulation).

The 16-vertex state-flip line inherits two exactly solvable endpoints. At $\varepsilon^{\rm(I)}=1-\ln(1+\sqrt{2})/\ln(5+4\sqrt{2})=0.627516\dots$ the model maps onto the square-lattice Ising model on the same lattice (Šamaj and Kolesík 1991, 1992), with the polarization linear in the Ising magnetization, so the electric exponents equal the Ising ones, $(1/8,15,7/4,1,1/4)$, and $T_{\rm c}^{\rm(I)}=1/\ln(5+4\sqrt{2})$. As $\varepsilon\to\infty$ the weights $w_1,w_3$ vanish and the model reduces to the zero-field eight-vertex model with $a=1$, $b=c=d=e^{-1/T}$; there $T_{\rm c}=1/\ln 3$ and $\mu=2\pi/3$, giving $(1/8,11,5/4,3/4,1/3)$. Since the weak-universality combinations differ between the two endpoints ($\beta_{\rm e}/\nu_{\rm e}=1/8$ versus $1/6$, $\gamma_{\rm e}/\nu_{\rm e}=7/4$ versus $5/3$, $\delta_{\rm e}=15$ versus $11$, $\eta_{\rm e}=1/4$ versus $1/3$), weak universality must fail somewhere on the line.

Pospíšilová, Krčmár, Gendiar and Šamaj (2020) studied the interior of the line numerically with the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi (1996) — a deterministic tensor-network algorithm that iteratively grows the lattice while truncating the corner-transfer-matrix density-matrix spectrum to its $m$ dominant eigenvalues. Earlier, Kolesík and Šamaj (1993) had reached the same qualitative conclusion with low-temperature series expansions analyzed by the coherent-anomaly method (a technique in which the divergence of a prefactor shared by a sequence of mean-field-type approximants is fitted to extract the true, generally nonuniversal critical exponents). The 2020 work found $T_{\rm c}(\varepsilon)$ vanishing linearly as $\varepsilon\to 0$ and tending to $1/\ln 3$ as $\varepsilon\to\infty$, and exponent data ($\Delta T=10^{-4}$, $m=200$ for $\beta_{\rm e}$; $E=2.5\times 10^{-5}$, $m=100$ for $\delta_{\rm e}$) showing $\beta_{\rm e}\approx 1/8$ for $\varepsilon\lesssim 2$, a non-monotone variation for $2\lesssim\varepsilon\lesssim 14$ whose amplitude exceeds the error bars by orders of magnitude, and a return to $1/8$ for $\varepsilon\gtrsim 14$, with $\delta_{\rm e}$ interpolating between 15 and 11 and $\gamma_{\rm e}$ between 7/4 and 5/4. The exponents $\nu_{\rm e}$ and $\eta_{\rm e}$ were evaluated only by inserting $\beta_{\rm e}$ and $\delta_{\rm e}$ into the two-dimensional scaling relations $\gamma_{\rm e}=\beta_{\rm e}(\delta_{\rm e}-1)$, $\nu_{\rm e}=\frac{1}{2}\beta_{\rm e}(\delta_{\rm e}+1)$, $\eta_{\rm e}=4/(\delta_{\rm e}+1)$; no correlation functions were measured directly. The authors note that the variation of $\nu_{\rm e}$ alone would still permit a scenario of two universality classes joined by a discontinuous change. No analytic derivation, exact solution, or continuum description of the exponent functions exists, and the known eight-vertex formulas cannot simply be transcribed to this line: at the Ising endpoint the exact tuple would require simultaneously $\mu=3\pi/4$ (from $\eta_{\rm e}=1/4$) and $\mu=\pi/2$ (from $\nu_{\rm e}=1$), which is impossible, so the interior demands a genuinely new analytic structure if one exists.

Problem Statement

For the symmetric two-state 16-vertex model on the square lattice with state-flip-invariant, permutation-symmetric vertex weights $w_0=w_4=1$, $w_1=w_3=e^{-\varepsilon/T}$, $w_2=e^{-1/T}$ ($\varepsilon\ge 0$), which lies on the self-dual manifold of the gauge transformation and has a line of second-order critical points $T_{\rm c}(\varepsilon)$ with polarization order parameter, determine the electric critical exponents $\beta_{\rm e}(\varepsilon)$, $\delta_{\rm e}(\varepsilon)$, $\gamma_{\rm e}(\varepsilon)$, $\nu_{\rm e}(\varepsilon)$, $\eta_{\rm e}(\varepsilon)$ as functions of $\varepsilon$ along the self-dual line, covering the full $\varepsilon\ge 0$ range including the small-$\varepsilon$ regime where $T_{\rm c}\to 0$ and the $\varepsilon\to\infty$ eight-vertex limit. Concretely, an answer should determine the five exponent functions and resolve the question the source work itself leaves open: whether all five exponents vary continuously along the line, or whether part of the variation — in particular that of $\nu_{\rm e}$ — is compatible with distinct universality classes joined by a discontinuous change. Since $\nu_{\rm e}$ and $\eta_{\rm e}$ were obtained in the source work only by inserting $\beta_{\rm e}$ and $\delta_{\rm e}$ into the two-dimensional scaling relations, resolving this question requires determining $\nu_{\rm e}$ and $\eta_{\rm e}$ independently of those relations, e.g. from the large-distance behavior of correlation functions. An analytic determination of the exponent functions — an exact solution of the line, a mapping onto a solvable model, or a renormalization-group/continuum field-theoretic construction — remains open; if achieved, it would also elucidate how the gauge/self-duality structure of the weights produces the observed full nonuniversality (violation of the weak-universality hypothesis, under which the ratios $\beta_{\rm e}/\nu_{\rm e}$, $\gamma_{\rm e}/\nu_{\rm e}$ and the exponents $\delta_{\rm e}$, $\eta_{\rm e}$ would be constant along the line). The published CTMRG data were obtained at fixed truncation dimension, temperature step, and field values without systematic extrapolation to the $m\to\infty$, zero-field, and zero-temperature-step limits, so the interior exponent functions are not yet numerically settled. Any answer must reproduce the exactly known endpoint values: at $\varepsilon^{\rm(I)}=1-\ln(1+\sqrt{2})/\ln(5+4\sqrt{2})\approx 0.6275$ the Ising tuple $(\beta_{\rm e},\delta_{\rm e},\gamma_{\rm e},\nu_{\rm e},\eta_{\rm e})=(1/8,15,7/4,1,1/4)$, and in the $\varepsilon\to\infty$ eight-vertex limit the tuple $(1/8,11,5/4,3/4,1/3)$.

References

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