Universal amplitude ratios of the two-dimensional 4-state Potts universality class: reconciling two-kink field-theory predictions with lattice determinations

For the two-dimensional 4-state Potts model, which sits at the marginal end of the critical q-state Potts family, universal combinations of critical amplitudes in the susceptibility sector are still not known at the few-percent level. A 2009 Monte Carlo and series re-analysis of the square-lattice model gave $\Gamma_+/\Gamma_L\approx6.2$--$6.9$, $\Gamma_T/\Gamma_L\approx0.15$ and $R_C^+\approx0.034$, in serious disagreement with the analytic two-kink form-factor predictions $4.013$, $0.129$ and $0.0204$, while a 2010 Monte Carlo study of the Baxter-Wu model, in the same universality class, gave $\Gamma_+/\Gamma_L=3.9(1)$ and $R_C^+=0.0201(5)$, supporting the analytic values. Under universality these numbers cannot all be correct. The open problem is to determine $\Gamma_+/\Gamma_L$, $\Gamma_T/\Gamma_L$ and $R_C^+$ for this universality class at the few-percent level and to identify the origin of the conflict: either uncontrolled systematics (additive backgrounds, logarithmic-correction modelling) in the lattice analyses of the 4-state Potts model itself, or the uncontrolled two-kink truncation of the form-factor expansion near the mass-degenerate $q=4$ spectrum.

Background

Near a continuous phase transition the singular part of a thermodynamic quantity follows a power law in the reduced temperature $t=(T-T_c)/T_c$: the specific heat behaves as $C\sim A_{\pm}|t|^{-\alpha}$, the susceptibility as $\chi\sim\Gamma_{\pm}|t|^{-\gamma}$, and the spontaneous magnetization as $M\sim B|t|^{\beta}$, with different amplitudes on the high-temperature ($+$) and low-temperature ($-$) sides of the transition. Below $T_c$ one distinguishes the longitudinal susceptibility $\chi_L$ (response to a magnetic field parallel to the spontaneous magnetization) from the transverse susceptibility $\chi_T$. The amplitudes themselves are nonuniversal, but ratios and combinations such as $A_+/A_-$, $\Gamma_+/\Gamma_L$, $\Gamma_T/\Gamma_L$ and $R_C^{\pm}=\alpha A_{\pm}\Gamma_{\pm}/B^{2}$ are universal numbers: they are identical for all microscopic models in the same universality class and are among the sharpest quantitative tests of renormalization-group and field-theory methods.

The ferromagnetic $q$-state Potts model (spins taking $q$ values, with nearest-neighbour couplings favoring equal values) on the square lattice is critical for $q\le4$; $q=4$ is the marginal end of the critical family (the correction-to-scaling exponent vanishes there), with exponents $\alpha=2/3$, $\beta=1/12$, $\gamma=7/6$ and an exact self-duality (a duality transformation that maps the model onto itself while exchanging the high- and low-temperature phases). Marginality produces multiplicative logarithmic corrections with universal exponents, for example $\chi\sim\Gamma_{+}|t|^{-7/6}[\ln(1/|t|)]^{3/4}$, whose structure was quantified by Salas and Sokal. These corrections, together with additive nonuniversal background terms, make few-percent amplitude extraction at $q=4$ far harder than at $q=2$ or $q=3$.

On the analytic side, the scaling limit of the two-dimensional Potts model is an integrable quantum field theory (a field theory with infinitely many conserved charges, which makes its dynamics exactly solvable) whose elementary excitations are kinks (domain walls between the $q$ degenerate ordered vacua), with the exact kink scattering matrix found by Chim and Zamolodchikov. Delfino and Cardy computed the one- and two-kink form factors (matrix elements, taken on kink states, of the energy operator, of the order operator — the magnetization density — and of the disorder operator, its dual under the duality transformation) and evaluated the spectral expansion of correlation functions (the representation of a correlation function as a sum over intermediate multiparticle states) keeping only this lowest term, the two-kink approximation, an uncontrolled truncation. For $q=4$ this yields $\Gamma_+/\Gamma_L=4.013$, $\Gamma_T/\Gamma_L=0.129$ (Delfino, Barkema and Cardy), $R_C^+=0.0204$, $R_C^-=0.00508$ and $A_+/A_-=1$. Delfino, Barkema and Cardy themselves stressed that the derivation of $\Gamma_+/\Gamma_L$ rests on a relative normalization of order- and disorder-operator form factors that may be unjustified for $q=3$ and $q=4$, and their own Monte Carlo check for $q=4$ was inconclusive because of large corrections to scaling. For $q=3$, a later joint Monte Carlo and series analysis settled $\Gamma_+/\Gamma_L=13.83(8)$, in agreement with the two-kink value $13.85$ — a precedent for the reliability of the method, and the reason the two-kink numbers are widely trusted.

The $q=4$ numerics are contentious. Caselle, Tateo and Vinti's Monte Carlo study gave $\Gamma_+/\Gamma_L=3.14(70)$ and $R_C^+=0.021(5)$, and Enting and Guttmann's series analysis gave $\Gamma_+/\Gamma_L=3.5(4)$ and $\Gamma_T/\Gamma_L=0.11(4)$, both compatible with the two-kink values within generous errors. Shchur, Berche and Butera (2009), a joint Monte Carlo and series re-analysis imposing the exact logarithmic-correction structure and controlling backgrounds through effective-amplitude ratios (ratios of measured amplitudes extrapolated to $t=0$, in which the logarithms cancel), found systematically larger values: $\Gamma_+/\Gamma_L\approx6.2$--$6.9$, $\Gamma_T/\Gamma_L\approx0.148$--$0.167$ and $R_C^+\approx0.033$--$0.035$, while confirming $A_+/A_-=1.000(5)$ and $R_C^-\approx0.005$. This is a $40$--$70$ percent discrepancy with the analytic predictions, far outside the quoted errors.

The same ratios must also hold for the other known realizations of the 4-state Potts universality class: the Baxter-Wu model (Ising spins on a triangular lattice coupled through three-spin interactions, with four degenerate ground states) and the Ashkin-Teller model (two coupled Ising models) at its four-state Potts point, for which Delfino and Grinza obtained $\Gamma_+/\Gamma_L=4.02$ from the same two-kink approximation. In 2010 Shchur and Janke measured the Baxter-Wu amplitudes by Monte Carlo and obtained $\Gamma_+/\Gamma_L=3.9(1)$, $R_C^+=0.0201(5)$, $R_C^-=0.00517(7)$ and $A_+/A_-=0.995(5)$, consistent with the analytic values and inconsistent with the 2009 four-state Potts analysis. The published determinations are therefore mutually contradictory under universality, and the contradiction — which side is right, and why the other side went wrong — has never been resolved in print.

Problem Statement

Determine the universal amplitude combinations of the two-dimensional 4-state Potts universality class in the susceptibility sector at the few-percent level, and resolve the origin of the conflict between the analytic and lattice determinations. Any well-controlled argument that settles the question is acceptable; the two lines of attack already present in the literature are (i) to compute $\Gamma_+/\Gamma_L$, $\Gamma_T/\Gamma_L$ and $R_C^+$ for the scaling 4-state Potts field theory beyond the two-kink truncation, or otherwise to supply a controlled error bound on that truncation — addressing the mass degeneracy of the kink spectrum at $q=4$ and the relative normalization of order- and disorder-operator form factors flagged by Delfino, Barkema and Cardy; and (ii) to produce a lattice determination (Monte Carlo or series expansion) for the 4-state Potts model itself, with quantified control of additive backgrounds and of the multiplicative logarithmic corrections carrying their exact $q=4$ exponents, that is consistent within combined errors with an independent determination for at least one other realization of the same universality class (Baxter-Wu or Ashkin-Teller); neither line is exhaustive. The resolution must establish which of the published determinations is in error and demonstrate which systematic error — analytic truncation, form-factor normalization, or background and logarithmic-correction modelling on the lattice — produced the rejected values. Confirming the already-converged ratios $A_+/A_-$ and $R_C^-$ is part of the same determination, not a separate objective.

References

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