The eight-vertex model on the square lattice with general vertex weights in external electric fields is non-integrable, and its ferroelectric-to-disordered criticality has been characterized numerically only inside the c=d subspace of weights, where the corner-transfer-matrix renormalization group (CTMRG) density matrix is symmetric. There, generic fields drive the model to the 2D Ising universality class, confirming the Kadanoff-Wegner-Baxter conjecture, except on the partially symmetric lines E=±E' where weak universality with continuously varying exponents survives. The open problem is to complete this characterization over the full six-weight parameter space: (i) determine whether the weak-universal behavior of the partially symmetric cases survives when the c=d symmetry is broken, which requires handling non-symmetric density matrices in CTMRG or an equivalent high-precision method; and (ii) determine the 'electric' critical exponents governing the polarization and the arrow correlation functions of the model in a field, whose values and universality properties are unknown.

Background

In the eight-vertex model on the square lattice, two-state arrow variables live on the edges, and only vertex configurations with an even number of arrows pointing into each vertex are allowed (the 'ice rule'), leaving eight admissible vertices with Boltzmann weights a1,a2,b1,b2,c,da_1,a_2,b_1,b_2,c,d. Vertical and horizontal electric fields E,E′E,E' and vertex energies ϵa,ϵb,ϵc,ϵd\epsilon_a,\epsilon_b,\epsilon_c,\epsilon_d parametrize the six weights, e.g. a1=Cexp⁡[−(ϵa−E−E′)/T]a_1=C\exp[-(\epsilon_a-E-E')/T]. The model can be mapped onto Ising spins on the dual lattice with two-spin couplings J,J′,Jh,JvJ,J',J_h,J_v and a four-spin plaquette coupling J′′J'' (the 'magnetic' representation, due to Wu and to Kadanoff-Wegner), so that 'magnetic' critical exponents refer to these Ising-spin observables.

The symmetric (zero-field, E=E′=0E=E'=0) model has a1=a2≡aa_1=a_2\equiv a, b1=b2≡bb_1=b_2\equiv b and was solved exactly by Baxter. Its ferroelectric phases (all arrows aligned, realized when one vertex weight dominates the others, e.g. a>b+c+da>b+c+d) and the disordered phase meet at a second-order transition on the critical surface ac=bc+cc+dca_c=b_c+c_c+d_c, where the exponents vary continuously with the parameter μ=2arctan⁡(acbc/(ccdc))\mu=2\arctan(\sqrt{a_cb_c/(c_cd_c)}): α=2−π/μ\alpha=2-\pi/\mu, β=π/(16μ)\beta=\pi/(16\mu), γ=7π/(8μ)\gamma=7\pi/(8\mu), ν=π/(2μ)\nu=\pi/(2\mu). Suzuki observed that although α,β,γ,ν\alpha,\beta,\gamma,\nu vary, the combinations (2−α)/ν(2-\alpha)/\nu, β/ν\beta/\nu, γ/ν\gamma/\nu and the exponents η\eta and δ\delta defined at criticality stay at their 2D Ising values ('weak universality'). Kadanoff and Wegner conjectured that this behavior is a peculiarity of the zero-field symmetries: in the presence of external fields the model should fall into the ordinary 2D Ising universality class (a conjecture repeated by Baxter and supported by early renormalization-group work).

The model in a field is non-integrable, so the conjecture must be tested numerically. Krčmár and Šamaj applied the corner transfer matrix renormalization group (CTMRG) — a variant of the density matrix renormalization group (DMRG) that builds the environment of a site from four corner transfer matrices (the quadrants of the row-to-row transfer matrix introduced by Baxter) and truncates via the eigenvalues of a density matrix — restricted to the c=dc=d subspace, because only there is the density matrix symmetric and safely diagonalizable. Within c=dc=d they found: for non-symmetric fields, E≠±E′E\neq\pm E', Ising exponents and central charge cconf=1/2c_{\mathrm{conf}}=1/2 (conjecture confirmed); but on the partially symmetric lines E=±E′E=\pm E', exponents varying with parameters, weak universality, and central charge cconf=1c_{\mathrm{conf}}=1 (conjecture violated). Here cconfc_{\mathrm{conf}} denotes the conformal-field-theory invariant labeling the critical universality class, extracted from the entanglement entropy of the CTMRG density matrix, and is distinct from the vertex weight cc.

Separately from these 'magnetic' results, the natural observables of the original vertex model are electric: the polarization (the mean arrow, the order parameter of the ferroelectric phases) and the arrow-arrow correlation function. Their 'electric' critical exponents behave differently from the magnetic ones: for the zero-field model the same authors later showed, using νe=ν\nu_e=\nu, Baxter's exact polarization exponent βe\beta_e, and scaling relations, that the electric exponents ηe\eta_e and δe\delta_e vary continuously with parameters, i.e. the electric formulation is fully non-universal even though the magnetic one is weakly universal. For the model in a field, no electric exponent has been computed.

Problem Statement

For the eight-vertex model on the square lattice in external electric fields (six independent vertex weights a1,a2,b1,b2,c,da_1,a_2,b_1,b_2,c,d), the ferroelectric-to-disordered criticality has been characterized only within the c=dc=d subspace. The problem is to complete this characterization over the whole space of vertex weights and for the model's own (electric) observables:

  1. Determine the critical behavior of the ferroelectric-to-disordered transition without the c=dc=d restriction — critical temperatures, magnetic exponents (ν\nu, β\beta, η\eta), and central charge as functions of the field and weight parameters, using a non-symmetric CTMRG (or an equivalent high-precision method) that handles the a priori non-symmetric density matrix. The crucial question is whether the partially symmetric cases (E=±E′E=\pm E'), which exhibit weak universality within c=dc=d, remain weakly universal when the c=dc=d symmetry is broken, i.e. whether the observed weak-universal exceptions to Ising universality are robust beyond the c=dc=d subspace or whether breaking c=dc=d generically restores the Kadanoff-Wegner-Baxter expectation.

  2. Determine the 'electric' critical exponents associated directly with the polarization and with the arrow correlation functions of the eight-vertex model in a field — their numerical values and their universality or non-universality properties — in direct analogy to what is now known exactly/numerically for the symmetric zero-field model.

Both parts serve one objective: a complete critical characterization of the non-integrable field-coupled eight-vertex model over its full parameter space and in both its magnetic and electric observables.

The completion standard for both parts is a decisive, bounded characterization rather than an exhaustive tabulation over the six-weight manifold: what must be delivered is (a) a verdict on whether the partially symmetric lines E=±E′E=\pm E' retain weak universality when c≠dc\neq d, established over a representative range of the remaining parameters, and (b) the critical temperatures, magnetic exponents (ν\nu, β\beta, η\eta) and central charge cconfc_{\mathrm{conf}}, together with the electric exponents and their universality or non-universality status, at a comparable representative set of parameter points covering generic non-symmetric fields and the partially symmetric lines outside c=dc=d. 'Complete characterization' therefore means this universality classification over the full parameter space plus such representative exponent data, not a point-by-point map of the parameter manifold.

References

  1. R. Krčmár and L. Šamaj, 'Critical properties of the eight-vertex model in a field', EPL (Europhysics Letters) 115, 56001 (2016). DOI: 10.1209/0295-5075/115/56001, https://doi.org/10.1209/0295-5075/115/56001 (arXiv:1610.08657)
  2. R. Krčmár and L. Šamaj, 'Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully nonuniversal', Physical Review E 97, 012108 (2018). DOI: 10.1103/PhysRevE.97.012108, https://doi.org/10.1103/PhysRevE.97.012108 (arXiv:1708.06115)
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