The square-lattice eight-vertex model in external electric fields was studied numerically by Krčmár and Šamaj (EPL 115, 56001 (2016)) with the corner transfer matrix renormalization group (CTMRG), but only inside the c=dc=d subspace of vertex weights, where the truncated density matrix is symmetric. There, generic field combinations fall into the Ising universality class — confirming a conjecture attributed to Kadanoff and Baxter — except for two partially symmetric field combinations that still exhibit weak universality (fixed rescaled exponents), contradicting the conjecture. The authors left two questions open: whether the weak-universal exceptions survive when the c=dc=d symmetry is broken, which requires diagonalizing non-symmetric density matrices over the full vertex-weight space; and the values of the 'electric' critical exponents of the polarization and arrow correlation functions. A 2018 follow-up by the same authors settled the electric exponents only for the symmetric zero-field model, where they vary continuously with parameters. The surviving open problem is the full critical characterization of the eight-vertex model in fields beyond c=dc=d: the robustness of weak universality and the electric exponents in the presence of fields.

Background

In the eight-vertex model on the square lattice, a two-state arrow variable sits on every lattice edge, and only vertices with an even number of arrows pointing in are allowed. The eight admissible vertex configurations carry Boltzmann weights a1,a2,b1,b2,c,da_1,a_2,b_1,b_2,c,d; vertical and horizontal 'electric' fields EE and E′E' enter by splitting the weights pairwise, e.g. a1∝exp⁡[−(ϵa−E−E′)/T]a_1 \propto \exp[-(\epsilon_a-E-E')/T] and a2∝exp⁡[−(ϵa+E+E′)/T]a_2 \propto \exp[-(\epsilon_a+E+E')/T], so the field-free 'symmetric' model has a1=a2a_1=a_2 and b1=b2b_1=b_2. Baxter (1972) solved the symmetric model exactly: along its critical surface the magnetic critical exponents, such as β=π/(16μ)\beta=\pi/(16\mu) and ν=π/(2μ)\nu=\pi/(2\mu) for an angle parameter μ\mu set by the critical weights, vary continuously, violating naive universality (the Ising limit is μ=π/2\mu=\pi/2). Suzuki (1974) noted that when temperature-like deviations are replaced by the correlation length, the exponent ratios such as β/ν=1/8\beta/\nu=1/8, together with the critical-point exponents η=1/4\eta=1/4 and δ=15\delta=15, retain their two-dimensional Ising values — a property called weak universality. A long-standing conjecture, attributed by Krčmár and Šamaj to Kadanoff and Baxter, holds that this weakly universal behavior is exceptional to the zero-field model: in external fields the eight-vertex model should fall into the ordinary Ising universality class. The vertex model is equivalent (the Wu–Kadanoff–Wegner mapping) to an Ising model on the dual lattice with two-spin couplings J,J′J,J', a four-spin plaquette coupling J′′J'', and, when fields are on, additional nearest-neighbor couplings; the constraint c=dc=d is equivalent to J=J′J=J'. Krčmár and Šamaj (2016) tested the conjecture numerically with CTMRG, a tensor-network method that represents each lattice quadrant by a corner transfer matrix CC (so the partition function is Z=Tr C4Z=\mathrm{Tr}\,C^4) and truncates the growing number of degrees of freedom to the dominant eigenvectors of the density matrix ρ=C4\rho=C^4, keeping a number DD of them. This truncation presupposes a symmetric ρ\rho, which is guaranteed precisely when c=dc=d; all their results are therefore restricted to that subspace, and extending the analysis beyond c=dc=d requires handling an a priori non-symmetric density matrix. Within c=dc=d they found Ising universality for generic field combinations (E≠±E′E\neq\pm E'), but for two partially symmetric choices (fields E=−E′E=-E' with a1=a2a_1=a_2, b1≠b2b_1\neq b_2) the exponents ν\nu and β\beta vary with the vertex weights while the ratio β/ν≃1/8\beta/\nu\simeq 1/8 and the central charge (the parameter of the conformal field theory describing the critical point, which equals 1/21/2 for Ising and 11 for the symmetric eight-vertex model) equals 1: weak universality survives in a field, contradicting the conjecture. The model is intimately related to the Ashkin–Teller model (two coupled Ising systems), which shares this coexistence of weak-universal magnetic exponents and continuously varying polarization exponents.

Problem Statement

As stated at the end of Krčmár and Šamaj (2016), the problem is to determine the critical behavior of the square-lattice eight-vertex model in external electric fields over the whole space of vertex weights {a1,a2,b1,b2,c,d}\{a_1,a_2,b_1,b_2,c,d\} with arbitrary fields E,E′E,E' and c≠dc\neq d. Within this full space, two linked questions must be answered. First, the crucial question: do the partially symmetric cases that exhibit weak universality within c=dc=d (field-dependent exponents ν,β\nu,\beta with β/ν≃1/8\beta/\nu\simeq1/8 and central charge 1) remain weakly universal when the c=dc=d symmetry is broken — i.e., is the failure of the Kadanoff–Baxter Ising-universality conjecture robust rather than an artifact of the c=dc=d subspace, or does generic c≠dc\neq d restore full Ising universality? Second, what are the 'electric' critical exponents — those of the polarization and of the arrow correlation function, the natural observables of the vertex formulation — in the presence of fields, both their numerical values along the critical surface and whether they vary continuously with the vertex weights as they do in the zero-field model (where they are known: Baxter's exact βe\beta_e and the 2018 results for νe,ηe,δe,γe\nu_e,\eta_e,\delta_e,\gamma_e)?

References

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