Rigorous derivation of the electric critical exponents $\eta_e=1-\mu/\pi$ and $\delta_e=(3\pi+\mu)/(\pi-\mu)$ of the symmetric eight-vertex model

The symmetric (zero-field) eight-vertex model on the square lattice can be formulated either in the original electric (arrow/vertex) representation or in an equivalent magnetic (Ising) representation with plaquette interactions. The two representations share the same partition function, yet Baxter's exact solution shows that the magnetic critical exponents vary continuously with the parameter $\mu$ while $\eta=1/4$ and $\delta=15$ remain constant (weak universality). Krčmár and Šamaj (2018) combined Baxter's exact spontaneous-polarization exponent $\beta_e=(\pi-\mu)/(4\mu)$ with the electric scaling relations and the identification $\nu_e=\nu$ to suggest closed-form electric exponents $\eta_e=1-\mu/\pi$ and $\delta_e=(3\pi+\mu)/(\pi-\mu)$, verified them numerically with corner-transfer-matrix renormalization-group calculations, and concluded that the electric formulation of the model is fully non-universal; their conclusion explicitly calls for a rigorous analytic derivation of these formulas, for instance by the quantum inverse-scattering method. The problem is to provide such a lattice-level derivation (or a refutation): to establish the critical-point decay $G_e(r)\propto r^{-(1-\mu/\pi)}$ of the polarization correlation function and the critical isotherm $P(E)\propto E^{(\pi-\mu)/(3\pi+\mu)}$ of the polarization, including a justification or bypass of the scaling relations and of $\nu_e=\nu$ on which the suggested formulas rest.

Background

The eight-vertex model on the square lattice places arrows on lattice edges subject to the ice rule that an even number of arrows point into each vertex; each of the eight allowed vertex configurations carries a Boltzmann weight. In the symmetric (zero-electric-field) version the weights are invariant under global reversal of all arrows, leaving four independent weights $a,b,c,d$; the order parameter is the polarization $P$, the mean arrow asymmetry on the edges. Baxter solved this model exactly in 1971 using commuting transfer matrices whose consistency condition is the Yang–Baxter equation (a functional equation guaranteeing that the row-to-row transfer matrices at different spectral parameters commute); this solution became the prototype of the quantum inverse-scattering method (QISM), the standard machinery for deriving exact correlation functions of integrable lattice models. The partition function can equivalently be rewritten, by a change of variables to Ising spins on the dual lattice, as an Ising model with two-spin couplings $J,J'$ and a four-spin (plaquette) coupling $J''$ — the magnetic formulation, in which the model is equivalent to the Ashkin–Teller model of two coupled Ising lattices.

Baxter's exact magnetic critical exponents depend continuously on a single parameter $\mu\in(0,\pi)$, an angle fixed by the critical vertex weights (equivalently by the elliptic modulus of the exact solution): $\beta=\pi/(16\mu)$ and $\nu=\pi/(2\mu)$, while $\eta=1/4$ and $\delta=15$ stay constant. Here $\eta$ governs the decay $G(r)\propto r^{-\eta}$ of the spin correlation function exactly at criticality, $\delta$ the critical isotherm $M\propto H^{1/\delta}$, $\beta$ the vanishing of the order parameter below $T_c$, and $\nu$ the divergence of the correlation length. Suzuki's weak universality explains this pattern: exponents defined away from the critical point (such as $\beta,\nu$) may vary because the distance from criticality can be rescaled by the correlation length, but $\eta$ and $\delta$, which are defined at the critical point itself, must be constant. The value $\mu=\pi/2$ corresponds to two decoupled Ising models ($J''=0$) and $\mu=3\pi/4$ to the four-state Potts point, where the magnetic exponents $\beta=1/12$ and $\nu=2/3$ take the exactly known four-state Potts values.

In the original electric formulation the natural local observables are different: the pair correlation function $G_e(r)$ of the arrow (edge) variables, whose critical-point decay defines $\eta_e$, and the response of the polarization to a small electric field $E$ conjugate to the arrows, $P(E)\propto E^{1/\delta_e}$ at $T=T_c$. Baxter's exact solution yields only one electric exponent, the spontaneous-polarization exponent $\beta_e=(\pi-\mu)/(4\mu)$. For the six-vertex limit of the model (a subset of the weights set to zero, equivalent to the spin-1/2 XXZ chain, the anisotropic Heisenberg quantum spin chain), Luther and Peschel derived the electric exponents by bosonization of the continuum chain — representing its low-energy excitations by a free bosonic field — with numerical confirmation by Takada and Kubo; the corresponding question for the eight-vertex model has no lattice-level derivation. On the field-theory side, Kadanoff and Brown evaluated the electric correlation functions on the critical line in terms of a Gaussian free field (the Coulomb-gas picture); this universality argument supports the expected exponent values but is not a derivation from the exact lattice solution. Krčmár and Šamaj (2018) assembled the suggested closed-form electric exponents $\eta_e=1-\mu/\pi$ and $\delta_e=(3\pi+\mu)/(\pi-\mu)$ from Baxter's $\beta_e$, the electric scaling relations $\eta_e=2\beta_e/\nu_e$ and $\delta_e=4/\eta_e-1$, and the identification $\nu_e=\nu$ (argued from the exact equality of the magnetic and electric internal energies for equivalent boundary conditions), verified them by the corner-transfer-matrix renormalization group (CTMRG, a tensor-network method that contracts the planar partition function by iteratively renormalizing Baxter's corner transfer matrices and retaining the $D$ dominant eigenvectors of the associated density matrix), and concluded that the electric formulation is fully non-universal: equivalent partition functions can carry different critical behavior. Their conclusion explicitly requests a rigorous derivation of the suggested formulas, maybe by using the QISM machinery.

Problem Statement

Provide a rigorous analytic derivation, at the level of the exact lattice solution of the symmetric zero-field eight-vertex model on the square lattice — for instance by the quantum inverse-scattering method (QISM) or an equivalent integrable or free-field technique — of the closed-form electric critical exponents

$\eta_e=1-\frac{\mu}{\pi},\qquad \delta_e=\frac{3\pi+\mu}{\pi-\mu},\qquad \mu\in(0,\pi),$

as explicitly requested in the conclusion of Krčmár and Šamaj (2018). Concretely, establish the critical-point decay of the pair arrow–arrow (polarization) correlation function,

$G_e(r)\propto r^{-\eta_e}\quad (T=T_c),$

and the critical isotherm of the polarization in a small electric field $E$,

$P(E)\propto E^{1/\delta_e}\quad (T=T_c,\ E\to 0),$

for the symmetric model on its critical surface, where $\mu$ is Baxter's parameter. The derivation must either justify the two electric scaling relations $\eta_e=2\beta_e/\nu_e$ and $\delta_e=4/\eta_e-1$ together with the identification $\nu_e=\nu$ of the electric and dominant correlation-length exponents that were used to assemble the suggested formulas, or bypass them by direct computation of the electric correlations. A rigorous demonstration that the true exponents differ from these formulas — for example an exact evaluation at an accessible point of the critical line yielding different values, with the convention for $\mu$ fixed unambiguously — refutes the suggestion and equally resolves the problem. As a consistency check, the suggested formulas reproduce the independently known endpoint values at $\mu=\pi/2$ (two decoupled Ising models, where the arrow variable is the product of one spin from each Ising layer, each correlating as $r^{-1/4}$, so $\eta_e=1/2$) and at $\mu=3\pi/4$ (the four-state Potts point, where the electric exponents coincide with the exactly known four-state Potts magnetic exponents $\eta=1/4$ and $\delta=15$, consistent with Baxter's exact $\beta_e=1/12$ there).

References

1. R. Krčmár and L. Šamaj, "Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully nonuniversal", Phys. Rev. E 97, 012108 (2018). DOI: 10.1103/PhysRevE.97.012108; arXiv:1708.06115; https://arxiv.org/abs/1708.06115
2. R. J. Baxter, "Eight-Vertex Model in Lattice Statistics", Phys. Rev. Lett. 26, 832 (1971). DOI: 10.1103/PhysRevLett.26.832; https://doi.org/10.1103/PhysRevLett.26.832
3. M. Suzuki, "New Universality of Critical Exponents", Prog. Theor. Phys. 51, 1992 (1974). DOI: 10.1143/PTP.51.1992; https://doi.org/10.1143/PTP.51.1992
4. L. P. Kadanoff and A. C. Brown, "Correlation functions on the critical lines of the Baxter and Ashkin-Teller models", Ann. Phys. (NY) 121, 318 (1979). DOI: 10.1016/0003-4916(79)90100-3; https://doi.org/10.1016/0003-4916(79)90100-3
5. A. Luther and I. Peschel, "Calculation of critical exponents in two dimensions from quantum field theory in one dimension", Phys. Rev. B 12, 3908 (1975). DOI: 10.1103/PhysRevB.12.3908; https://doi.org/10.1103/PhysRevB.12.3908
6. S. Takada and K. Kubo, "Critical Properties of S=1/2 XXZ Chain", J. Phys. Soc. Jpn. 55, 1671 (1986). DOI: 10.1143/JPSJ.55.1671; https://doi.org/10.1143/JPSJ.55.1671
7. M. Jimbo, T. Miwa, and A. Nakayashiki, "Difference equations for the correlation functions of the eight vertex model", J. Phys. A: Math. Gen. 26, 2199 (1993). DOI: 10.1088/0305-4470/26/9/015; arXiv:hep-th/9211066; https://arxiv.org/abs/hep-th/9211066
8. M. Lashkevich and Y. Pugai, "Free field construction for correlation functions of the eight vertex model", Nucl. Phys. B 516, 623 (1998). DOI: 10.1016/S0550-3213(98)00086-8; arXiv:hep-th/9710099; https://arxiv.org/abs/hep-th/9710099
9. L.-F. Ko and B. M. McCoy, "Some Exact Results for the Electric Correlation Functions of the Eight-Vertex Model", Phys. Rev. Lett. 56, 2645 (1986). DOI: 10.1103/PhysRevLett.56.2645; https://doi.org/10.1103/PhysRevLett.56.2645
10. L.-F. Ko and B. M. McCoy, "Electric correlation function for the eight-vertex model for T→Tc− to first order in the four-spin coupling", Phys. Lett. A 118, 287 (1986). DOI: 10.1016/0375-9601(86)90361-0; https://doi.org/10.1016/0375-9601(86)90361-0
11. E. Pospíšilová, R. Krčmár, A. Gendiar, and L. Šamaj, "Full nonuniversality of the symmetric 16-vertex model on the square lattice", Phys. Rev. E 102, 012125 (2020). DOI: 10.1103/PhysRevE.102.012125; arXiv:2002.02757; https://arxiv.org/abs/2002.02757
12. I. Mukherjee and P. K. Mohanty, "Hidden superuniversality in systems with continuous variation of critical exponents", Phys. Rev. B 108, 174417 (2023). DOI: 10.1103/PhysRevB.108.174417; arXiv:2307.15022; https://arxiv.org/abs/2307.15022
13. A. Banerjee, P. Jana, and P. K. Mohanty, "Geometric percolation of spins and spin dipoles in the Ashkin-Teller model", Phys. Rev. B 111, 014403 (2025). DOI: 10.1103/PhysRevB.111.014403; arXiv:2411.11644; https://arxiv.org/abs/2411.11644
14. D. Rossini and E. Vicari, "Stacked quantum Ising systems and quantum Ashkin-Teller model", Phys. Rev. B 113, 155143 (2026). DOI: 10.1103/y9n7-d6vr; arXiv:2601.18922; https://arxiv.org/abs/2601.18922
15. C. Bonati, A. Pelissetto, and E. Vicari, "Deconfinement transitions in three-dimensional compact lattice Abelian Higgs models with multiple-charge scalar fields", Phys. Rev. E 109, 044146 (2024). DOI: 10.1103/PhysRevE.109.044146; arXiv:2402.06374; https://arxiv.org/abs/2402.06374