Analytic renormalization-group characterization of the self-dual generalized Baxter-Wu models: marginal-field dependence on coupling asymmetry and the q-dependent electric-charge fugacity

Deng, Guo, Heringa, Blöte and Nienhuis (Nucl. Phys. B 827 (2010) 406) showed numerically that the self-dual generalizations of the Baxter-Wu model -- with unequal couplings $K_1\neq K_2$ on the up and down triangles of the triangular lattice, or with $q>2$ spin states coupled by three-spin interactions -- undergo first-order transitions on their exactly known self-dual lines, and placed them in the Potts renormalization flow diagram by inference. The placement rests on two undischarged steps: the marginal scaling field $v$ is only 'expected' to scale as $(K_1-K_2)^2$ by symmetry, and the nonzero fugacity of the $e=4$ Coulomb-gas electric charges for $q>2$ is only inferred ('apparently') from the observed discontinuous transitions. The open problem is the corresponding controlled analytic characterization: a microscopic derivation of $v(K_1-K_2)$ near the four-state Potts fixed point, a quantitative computation of the $q$-dependent electric-charge fugacity, and the consequent derived flow placement and phase-boundary structure near the self-dual line, from which the growth of the first-order strength (latent heat) with $q$ and asymmetry follows as a consequence. A 2017 continuation by two of the authors determined $|v|\approx 0.38\,\Delta K^2$ numerically but not microscopically; the $q>2$ sector has remained untouched since 2010.

Background

The Baxter-Wu model (Baxter and Wu, 1973) is an exactly solved two-dimensional lattice model: Ising spins on a triangular lattice coupled by three-spin interactions on every elementary triangle, with reduced Hamiltonian $\beta H=-K\sum_{\triangle}s_i s_j s_k$. Its critical behavior lies in the universality class of the two-dimensional four-state Potts model, yet its leading singularities carry no logarithmic correction factors. The accepted explanation uses the renormalization-group solution of the Potts model by Nienhuis, Berker, Riedel and Schick (1979): in the plane parametrized by the number of Potts states $q_{\mathrm{Potts}}$ and a vacancy fugacity $v$ (a temperature-like variable of the equivalent dilute random-cluster model), a line of fixed points has an attractive branch describing critical models and a repulsive branch describing tricritical ones; the branches meet at $q_{\mathrm{Potts}}=4$, where the scaling field $v$ becomes marginal -- marginally irrelevant on one side, producing the Potts logarithmic corrections, and marginally relevant on the other, driving the flow to a discontinuity fixed point describing a first-order transition. The Baxter-Wu model sits exactly at the marginal point $v=0$ -- the Baxter-Wu point ($q=2$ spin states) is the four-state-Potts marginal point $q_{\mathrm{Potts}}=4$ of this flow plane -- which is why it shows pure power-law criticality. In the Coulomb-gas representation of two-dimensional criticality, in which critical lattice models are described as a Gaussian free-boson theory decorated by mutually interacting electric and magnetic charges, the marginal temperature field translates into the fugacity (activity) of electric charges of charge $e=4$; at the Baxter-Wu point these charges are absent. In the normalization used by Blöte, Guo and Nightingale (2017) for the numerical benchmark of the marginal field, $v$ is set to zero at the four-state Potts fixed point and the scaling field coupled to $v$ is normalized so that its two-point correlation decays as $r^{-4}$ at that fixed point; in this convention $v<0$ is marginally irrelevant and $v>0$ is marginally relevant at $q_{\mathrm{Potts}}=4$.

Deng, Guo, Heringa, Blöte and Nienhuis (2010) studied two self-dual generalizations of the Baxter-Wu model. (Self-duality is a transformation, analogous to the Kramers-Wannier duality of the two-dimensional Ising model, that exchanges the ordered and disordered descriptions of the same model; on the self-dual manifold, which the generalizations map onto themselves, a phase transition is expected to lie.) The first generalization assigns different couplings $K_1$ and $K_2$ to the up-pointing and down-pointing triangles; for $q=2$ its self-dual line is $\sinh(2K_1)\sinh(2K_2)=1$. The second replaces Ising spins by $q$-state spins $\sigma_i\in\mathbb{Z}_q$ with three-spin interactions $\delta_q(\sigma_i+\sigma_j+\sigma_k)$ (with $\delta_q(x)=1$ for $x\equiv 0 \bmod q$ and $0$ otherwise); for $q>2$ the full permutation symmetry of the Potts model is broken to $\mathbb{Z}_q\otimes\mathbb{Z}_q\otimes\mathbb{Z}_2\otimes S_3$, and for $q=3$ the model coincides with the three-spin clock model earlier studied by Alcaraz, Cardy and Ostlund (1983). Using transfer matrices (a method that computes the partition function of a cylinder of circumference $L$ exactly, here up to $L\le 18$ for $q=3$ and $L\le 12$ for $q=4$) and Monte Carlo simulations, they established that transitions occur on the self-dual line and are first order for $K_1\neq K_2$ (extremely weakly for small asymmetry -- even at $K_1/K_2=5$ the hysteresis loop is very narrow) and for $q=3$ and $q=4$.

Crucially, the placement of these models in the $(q_{\mathrm{Potts}},v)$ flow diagram was inferred rather than derived. The dependence of the marginal field $v$ on the asymmetry $K_1-K_2$ was argued only by symmetry -- since $v$ cannot depend linearly on $K_1-K_2$ under the $K_1\leftrightarrow K_2$ interchange, one 'expects' a contribution proportional to $(K_1-K_2)^2$ -- and the nonzero fugacity of the $e=4$ electric charges for $q>2$ was inferred ('apparently') from the observed discontinuous transitions, since the marginal charges at $q=2$ are expected to become relevant as the Coulomb-gas coupling grows with $q$. The strength of the first-order character (e.g. the latent heat, the energy discontinuity at the transition) is expected to grow with $q$ and with asymmetry, but no quantitative scaling was established.

Problem Statement

Replace the numerically inferred renormalization-group placement of the self-dual generalized Baxter-Wu models by a controlled analytic characterization. Concretely: (1) derive microscopically the marginal scaling field $v$ of the Nienhuis-Berker-Riedel-Schick flow diagram as a function of the coupling asymmetry $K_1-K_2$ near the four-state Potts fixed point -- establishing the lowest-order dependence $v\propto(K_1-K_2)^2$ (presently only argued by the $K_1\leftrightarrow K_2$ symmetry) and its amplitude (presently only fitted numerically, $|v|\approx 0.38\,\Delta K^2$ with $\Delta K=(K_1-K_2)/2$, in the normalization of Blöte, Guo and Nightingale (2017) introduced in the Background); (2) compute quantitatively the fugacity of the $e=4$ Coulomb-gas electric charges of the $q$-state three-spin models as a function of the spin-state count $q$ (here $q$ is an integer: the lattice $\mathbb{Z}_q$ models are defined for integer $q$, and the existing numerical evidence covers $q=3$ and $q=4$), so that their relevance for $q>2$ and the resulting first-order character of the self-dual transitions follow from first principles rather than from the observed discontinuities; (3) determine, from these derivations, the renormalization flow and the phase-boundary structure near the self-dual line -- how first-order behavior sets in as a function of $q$ and of asymmetry -- so that the growth of the first-order strength (e.g. the latent heat, the energy discontinuity at the transition) with $q$ and with $K_1-K_2$ follows as a consequence of the derived flow rather than remaining a qualitative conjecture. These three items jointly serve the one objective of a derived, rather than inferred, renormalization-group characterization. Items (1) and (2) are its mandatory core: item (1) must give the microscopic $v(K_1-K_2)$ -- both its lowest-order quadratic form and its amplitude -- and item (2) must determine the charge fugacity quantitatively at integer $q$, i.e. its value or its explicit analytic dependence on $q$, not merely its sign or qualitative relevance, with values at least at $q=3$ and $q=4$ where numerical evidence exists; item (3) is required not as a further independent input but as the consequence that must be derived from (1) and (2), the flow and the resulting phase-boundary structure and first-order-strength behaviour having to follow from the derived $v$ and fugacity rather than being inferred from the numerics. Any answer must remain consistent with the exactly known self-dual transition locations and with the existing finite-size numerical data (the transfer-matrix and Monte Carlo results of Deng et al. (2010) and Blöte, Guo and Nightingale (2017)).

References

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