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

The symmetric (zero-field) eight-vertex model on the square lattice has two equivalent descriptions: the original electric (arrow/vertex-weight) formulation and a magnetic formulation as an Ising model with two- and four-spin couplings. Baxter's exact solution gives all magnetic exponents, which satisfy Suzuki's weak universality (the critical-point exponents $\eta=1/4$ and $\delta=15$ are constant), but only one electric exponent, $\beta_e=(\pi-\mu)/(4\mu)$. Krčmár and Šamaj (Phys. Rev. E 97, 012108 (2018)) combined Baxter's $\beta_e$, the identification $\nu_e=\nu$, and scaling relations to suggest $\eta_e=1-\mu/\pi$ and $\delta_e=(3\pi+\mu)/(\pi-\mu)$, verified them numerically by CTMRG, and concluded that the electric formulation is fully non-universal; they posed a rigorous derivation of these formulas from the lattice model (e.g., via the quantum inverse-scattering method) as an open problem. Prior Coulomb-gas-level derivations (Kadanoff–Brown 1979, Knops 1980) fix the expected answer, and no subsequent rigorous derivation has been found in the literature, so the problem remains open.

Background

The eight-vertex model places arrows on the edges of a square lattice with the ice rule (an even number of arrows points into each vertex), which leaves eight allowed vertex configurations. In the symmetric, zero-electric-field version, the eight configurations come in four pairs related by flipping all arrows, with Boltzmann weights $a$, $b$, $c$, $d$. The order parameter is the electric polarization $P$, the mean fraction of arrows pointing along a chosen direction on horizontal edges. Baxter solved this model exactly (Phys. Rev. Lett. 26, 832 (1971)) using commuting transfer matrices whose consistency condition is the Yang–Baxter equation, the master algebraic consistency relation of exactly solved lattice models. The same partition function maps onto a magnetic formulation: an Ising model on the dual square lattice with nearest-neighbor couplings $J$, $J'$ and a four-spin plaquette coupling $J''$ (Wu, Phys. Rev. B 4, 2312 (1971); Kadanoff and Wegner). On the second-order critical surface separating the ferroelectric phase (uniformly aligned arrows, nonzero spontaneous polarization) from the disordered phase one introduces the parameter $\mu = 2\arctan\sqrt{a_c b_c/(c_c d_c)} = 2\arctan(e^{2J''/T_c})$, with $\mu=\pi/2$ corresponding to $J''=0$, i.e., two decoupled Ising models.

The magnetic critical exponents are exactly known and vary continuously along the surface: $\beta = \pi/(16\mu)$ and $\nu = \pi/(2\mu)$. Suzuki's weak universality (Prog. Theor. Phys. 51, 1992 (1974)) explains this variation as an artifact of measuring the deviation from criticality by temperature rather than by the inverse correlation length; exponents defined at the critical point itself, namely the correlation decay exponent $\eta$ (from $\langle s_0 s_r\rangle \propto r^{-\eta}$ at $T=T_c$) and the critical-isotherm exponent $\delta$ (from $M \propto H^{1/\delta}$ at $T=T_c$), must then be constant, and indeed $\eta = 1/4$ and $\delta = 15$, the 2D Ising values.

On the electric side the situation is poorer. The only electric exponent known from the exact solution is $\beta_e = (\pi-\mu)/(4\mu)$, governing the spontaneous polarization below $T_c$. For the six-vertex subcase (weight $d=0$), the arrow correlation length was computed by Johnson, Krinsky and McCoy, and Luther and Peschel derived electric exponents from the corresponding continuum limit, a Luttinger model (the effective low-energy field theory of the one-dimensional fermions associated with the six-vertex model). For the full eight-vertex model, Kadanoff and Brown (Ann. Phys. (NY) 121, 318 (1979)) evaluated the electric correlation functions using a universality argument mapping the critical line onto the two-dimensional Gaussian model (the Coulomb-gas picture), and Knops (Ann. Phys. (NY) 128, 448 (1980)) gave a renormalization-group version of the same connection; these physics-level arguments fix the expected $\mu$-dependence but are not derivations from the lattice model.

Krčmár and Šamaj (Phys. Rev. E 97, 012108 (2018)) studied both formulations numerically with the corner transfer matrix renormalization group (CTMRG), a tensor-network method that builds on Baxter's corner transfer matrices and truncates the density-matrix spectrum to $D$ dominant states. Their magnetic results reproduce Baxter's exponents and weak universality. For the electric formulation they argued that the magnetic and electric correlation-length exponents coincide ($\nu_e = \nu$, from the equality of the Ising and vertex internal energies at equivalent boundary conditions) and combined this with Baxter's exact $\beta_e$ and the scaling relations $\eta_e = 2\beta_e/\nu_e$, $\delta_e = 4/\eta_e - 1$ to obtain the closed forms $\eta_e = 1-\mu/\pi$ and $\delta_e = (3\pi+\mu)/(\pi-\mu)$; here $\eta_e$ governs the critical decay of the arrow-pair correlation $G_e(r) = \langle\phi_0\phi_r\rangle \propto r^{-\eta_e}$ and $\delta_e$ the response $P \propto E^{1/\delta_e}$ of the polarization to a small isotropic electric field $E$ (equal fields on the horizontal and vertical edges, $E_x=E_y=E$) at $T_c$. Their CTMRG data (for both $c=d$ and $c\neq d$ weights) agree with these formulas at the $0.1$–$1\%$ level, and the von Neumann entropy of the electric density matrix yields a central charge $c_e \approx 1$ for $c=d$. Since $\eta_e$ and $\delta_e$ are defined at the critical point and are therefore immune to any weak-universality rescaling, their dependence on $\mu$ means the electric-vertex formulation is fully non-universal, even though the equivalent magnetic formulation is weakly universal.

What is missing is rigor: the closed-form exponents rest on scaling relations and on $\nu_e=\nu$, neither of which has been proven for the electric variables, and the supporting evidence is numerical. The quantum inverse-scattering method (QISM) — the algebraic Bethe-ansatz framework generalizing Baxter's commuting-transfer-matrix technique, for which correlation functions of the six-vertex model but not of the eight-vertex model are computable — is the natural candidate machinery for a first-principles derivation, which the authors explicitly pose as the open direction.

Problem Statement

For the symmetric (zero-field) eight-vertex model on the square lattice with vertex weights $a$, $b$, $c$, $d$ on its ferroelectric critical surface, provide a rigorous analytic derivation, from the lattice definition of the model, of the critical behavior of the electric (polarization) variables — specifically of the arrow-pair correlation function at criticality, $G_e(r)=\langle\phi_0\phi_r\rangle \propto r^{-\eta_e}$, and of the critical isotherm of the polarization, $P(E) \propto E^{1/\delta_e}$ under a weak isotropic electric field $E$ (equal fields on the horizontal and vertical lattice directions, $E_x=E_y=E$) at $T=T_c$ — establishing (or refuting) the closed forms $\eta_e = 1-\mu/\pi$ and $\delta_e = (3\pi+\mu)/(\pi-\mu)$, where $\mu = 2\arctan\sqrt{a_c b_c/(c_c d_c)} \in (0,\pi)$ parametrizes the critical surface. The derivation, for which the quantum inverse-scattering method (algebraic Bethe ansatz on Baxter's commuting transfer matrices) is the suggested machinery, must not assume the electric scaling relations $\eta_e = 2\beta_e/\nu_e$ and $\delta_e = 4/\eta_e - 1$ or the identification $\nu_e=\nu$ unless these are themselves proven en route. The two requested results (critical correlation decay and critical-isotherm response) jointly support one objective: placing the full non-universality of the electric-vertex formulation — critical-point exponents that vary continuously with $\mu$, in contrast to the weakly universal magnetic exponents $\eta=1/4$, $\delta=15$ of the identical partition function — on an exact analytic footing. A derivation yielding different exponent values would equally answer the question, provided it reconciles them with Baxter's exact $\beta_e=(\pi-\mu)/(4\mu)$ and with the existing numerical and Coulomb-gas evidence.

References

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