Universality classes and electric critical exponents of the eight-vertex model in external fields beyond the $c=d$ subspace

The eight-vertex model on the square lattice with external electric fields is non-integrable, and its critical behaviour is known numerically only inside the $c=d$ subspace of vertex weights, where corner-transfer-matrix renormalization-group (CTMRG) studies by Krčmár and Šamaj found 2D Ising universality for generic fields but persisting weak universality along the two partially symmetric families $E=\pm E'$ — contradicting the Kadanoff–Wegner–Baxter conjecture — while the 'electric' critical exponents of the polarization and of arrow correlation functions were left undetermined. The open problem, stated in EPL 115, 56001 (2016) and untouched by subsequent literature (the 2018 follow-up settled the electric exponents only for the zero-field model), is to extend the analysis to the full vertex-weight parameter space: decide whether the partially symmetric weak universality survives breaking the $c=d$ symmetry, map the universality classes across the full in-field critical surface, and determine the electric critical exponents of the in-field model.

Background

The two-dimensional eight-vertex model places a two-state arrow (dipole) variable on each edge of the square lattice, with the local rule that an even number of arrows point into every vertex; the eight admissible vertex configurations carry Boltzmann weights $a_1,a_2,b_1,b_2,c,d$ . In this original 'electric' formulation, unequal weights within the pairs describe uniform vertical and horizontal external fields $E$ and $E'$ acting on the arrows: $a_1=C\exp[-(\epsilon_a-E-E')/T]$ , $a_2=C\exp[-(\epsilon_a+E+E')/T]$ , $b_1=C\exp[-(\epsilon_b+E-E')/T]$ , $b_2=C\exp[-(\epsilon_b-E+E')/T]$ , $c=C\exp(-\epsilon_c/T)$ , $d=C\exp(-\epsilon_d/T)$ , so that $E=-E'$ gives $a_1=a_2$ , $E=E'$ gives $b_1=b_2$ , and generic fields break both equalities. The model maps to an Ising ('magnetic') representation on the dual lattice with nearest-neighbour couplings $J_h,J_v$ , diagonal couplings $J,J'$ and a four-spin plaquette coupling $J''$ ; the zero-field model corresponds to $J_h=J_v=0$ , and the constraint $c=d$ to $J=J'$ .

Baxter solved the zero-field model exactly; along the second-order transition between the ferroelectric and disordered phases (critical surface $a_c=b_c+c_c+d_c$ ) the critical exponents vary continuously with the parameter $\mu=2\arctan\sqrt{a_c b_c/(c_c d_c)}$ : $\alpha=2-\pi/\mu$ , $\beta=\pi/(16\mu)$ , $\gamma=7\pi/(8\mu)$ , $\nu=\pi/(2\mu)$ , while $\eta=1/4$ and $\delta=15$ . This was the first known violation of the universality hypothesis. Suzuki's weak universality repairs it partially: if singularities are expressed through the inverse correlation length rather than the temperature difference, the rescaled exponents $(2-\alpha)/\nu=2$ , $\beta/\nu=1/8$ , $\gamma/\nu=7/4$ are fixed (Ising-like), and the exponents $\eta$ and $\delta$ , defined exactly at criticality, are constant. Kadanoff and Wegner argued that the varying exponents owe to hidden symmetries of the zero-field model and conjectured that external fields destroy them, restoring 2D Ising universality for the magnetic exponents; the same expectation is recorded in Baxter's monograph, so the statement is referred to as the Kadanoff–Wegner–Baxter conjecture. Since the model with fields is non-integrable, this conjecture must be tested numerically.

Krčmár and Šamaj performed such a test with the Corner Transfer Matrix Renormalization Group (CTMRG) method, a tensor-network renormalization built on Baxter's corner transfer matrices in which each quadrant of the lattice is represented by a corner matrix $C$ and degrees of freedom are truncated to the dominant eigenvectors of the density matrix $\rho=C^4$ ; the standard truncation requires $\rho$ to be symmetric, which is guaranteed by the $c=d$ constraint. Working inside $c=d$ they found: generic fields $E\neq\pm E'$ give the 2D Ising universality class (central charge $c=1/2$ , $\nu=1$ , $\beta=1/8$ , $\eta=1/4$ ; the central charge, denoted $c$ following the source papers, is that of the conformal field theory describing the critical point and is distinct from the vertex weight $c$ ), confirming the conjecture there; but the two partially symmetric families $E=\pm E'$ still exhibit weak universality, with central charge $c=1$ and parameter-dependent $\nu$ and $\beta$ satisfying $\beta/\nu=1/8$ — a violation of the conjecture inside the $c=d$ subspace.

Besides these magnetic (Ising-representation) exponents, the model has electric observables native to the vertex formulation: the spontaneous polarization (the mean arrow, the order parameter of the ferroelectric phase) and the arrow–arrow correlation function, with exponents $\beta_e,\gamma_e,\delta_e,\eta_e,\nu_e$ ; note that the polarization of a single arrow is not the magnetization of the Ising representation (which is a product of two arrows). For the zero-field model the electric exponents are known: Baxter's exact $\beta_e=(\pi-\mu)/(4\mu)$ , and Krčmár and Šamaj subsequently obtained $\nu_e=\nu$ , $\eta_e=1-\mu/\pi$ and $\delta_e=(3\pi+\mu)/(\pi-\mu)$ semi-analytically, by combining $\nu_e=\nu$ with the two-dimensional scaling relations $\eta=2\beta/\nu$ and $\delta=4/\eta-1$ and verifying the result with high-precision CTMRG (a rigorous derivation of the last two formulas is still lacking). This shows that the electric sector is fully non-universal — its exponents vary with the model's parameters without even a weak-universal rescaling. With fields, however, nothing beyond the $c=d$ subspace has ever been explored, and the electric exponents in field have not been determined.

Problem Statement

Determine the critical behaviour of the eight-vertex model in external fields $E,E'$ (the uniform vertical and horizontal fields acting on the arrow variables) on the full vertex-weight parameter space $\{a_1,a_2,b_1,b_2,c,d\}$ , removing the $c=d$ restriction under which all existing results were obtained. Two linked questions, stated by Krčmár and Šamaj as the open issues of their study, define the objective.

(1) Universality classes with fields beyond $c=d$ . Determine the critical exponents ( $\nu,\beta,\eta$ and the central charge) across the critical hypersurface of the in-field model with generic weights $c\neq d$ — which requires diagonalizing an a-priori non-symmetric density matrix within CTMRG or using an alternative high-precision scheme — and in particular decide the source paper's crucial question: whether the partially symmetric in-field cases ( $a_1=a_2$ or $b_1=b_2$ , i.e. $E=\pm E'$ , so far studied only with $c=d$ ) remain weakly universal when the $c=d$ symmetry is broken, or whether generic points of the full critical surface belong to the 2D Ising universality class, as the Kadanoff–Wegner–Baxter conjecture predicts.

(2) Electric critical exponents in fields. Determine the values and universality properties of the electric critical exponents — the spontaneous-polarization exponent $\beta_e$ , the arrow-correlation exponent $\eta_e$ , the critical-field exponent $\delta_e$ , together with $\gamma_e$ and $\nu_e$ — for the eight-vertex model in external fields (both the partially symmetric and the fully non-symmetric cases), in analogy with the determination available for the zero-field model.

Both deliverables jointly support one common objective: a complete universality-class and critical-exponent characterization of the archetypal eight-vertex model in fields, in its natural generality.

References

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