Logarithmic finite-size corrections at the marginal transition of the triangular lattice gas with first- and second-neighbor exclusions

The hard-core lattice gas on the triangular lattice with exclusion of both first and second neighbors orders by occupying one of four sublattices, and its continuous transition is believed to lie in the marginal 4-state Potts universality class, where the leading correction-to-scaling field is marginally irrelevant and observables acquire logarithmic finite-size corrections. Zhang and Deng (2008) located the critical point precisely, but their finite-size-scaling fits could not include logarithmic terms: magnetic observables yielded the 4-state Potts values, while energy-like observables (particle density, specific-heat-like quantities) gave an effective thermal exponent of $1.440(5)$ to $1.470(5)$ instead of $3/2$, implying uncontrolled multiplicative logarithmic corrections. The open problem is to determine quantitatively the logarithmic correction structure of energy-like observables in this realization (multiplicative powers of $\ln L$, additive terms, or both, with their powers and amplitudes), to build a controlled log-corrected finite-size-scaling analysis that settles whether the thermal exponent converges to $3/2$, and to deliver definitive critical parameters and exponents with controlled systematic errors. The most recent dedicated study of this model (2022) still fits without logarithmic corrections and reports exponent deviations it explicitly attributes to them, so the problem remains open.

Background

In a hard-core lattice gas, particles occupy sites of a lattice subject to the rule that all sites within a specified exclusion range of an occupied site must remain empty; the chemical potential $\mu$ controls the average particle density $\rho$, and entropy alone can drive an ordering transition. On the triangular lattice with only nearest-neighbor exclusion the model is Baxter's exactly solved hard-hexagon model, whose transition lies in the 3-state Potts universality class. Extending the exclusion to second neighbors (the '2-NN' or first- and second-neighbor exclusion model) leaves a lattice that decomposes into four interpenetrating sublattices, so the dense phase selects one of them, and the density-driven melting transition is expected - but not proven - to lie in the 4-state Potts universality class. Early simulations by Orban and Bellemans (1968) even suggested a first-order transition; transfer-matrix phenomenological renormalization (exact diagonalization of the model on strips of finite width and infinite length, combined with finite-size scaling of the leading eigenvalues) by Bartelt and Einstein (1984) supported a continuous 4-state-Potts-like transition with effective exponents $y_t = 1.400$ and $y_h = 1.885$ that converge only slowly toward the exact 4-state Potts values $y_t = 3/2$ (the thermal exponent, equal to $1/\nu$) and $y_h = 15/8$ (the magnetic exponent). Here $y_t$ and $y_h$ are the renormalization-group eigenvalues governing, respectively, the scaling of the distance from criticality and of a symmetry-breaking field with system size $L$, and $\nu$ is the correlation-length exponent.

The 4-state Potts point is marginal: in two dimensions $q = 4$ is the largest number of Potts spin states admitting a continuous ferromagnetic transition, and there the leading correction-to-scaling operator is marginally irrelevant. The exponents keep their values, but observables acquire logarithmic corrections. Perturbative renormalization-group analyses (Cardy, Nauenberg and Scalapino 1980; Black and Emery 1981) and transfer-matrix studies (Hamer, Batchelor and Barber 1988) established these corrections, and Salas and Sokal (1997) worked out the complete structure for the pure 4-state Potts model: multiplicative powers of $\ln L$ (for example, at criticality the specific heat scales as $C_H \sim L(\ln L)^{-3/2}$ and the susceptibility as $\chi \sim L^{7/4}(\ln L)^{-1/8}$), together with additive corrections of order $\ln\ln L/\ln L$ and $1/\ln L$, some of them universal. Two features make these corrections treacherous. First, over accessible system sizes a multiplicative logarithm is nearly indistinguishable from a small power-law or additive correction, so naive power-law fits are strongly biased (Salas and Sokal measured an effective $\alpha/\nu$ of $0.770(8)$ where the exact value is $1$). Second, the marginal field need not be present in every realization of the same exponents: the pure Baxter-Wu model (a triangular-lattice Ising model with three-spin interactions on the elementary triangles, exactly solved and also lying in the 4-state Potts class) shows 4-state Potts exponents with no logarithmic corrections at all, and the logarithms switch on only when the model is deformed away from that special subspace (Blöte, Guo and Nightingale 2017). Whether a given realization carries the marginal field, and with which sign and strength, is therefore a genuine model-specific question.

Monte Carlo simulation of the 2-NN triangular gas is dynamically slow, and Zhang and Deng (2008) combined Metropolis grand-canonical updates (local single-site insertion and removal moves) with the geometric cluster algorithm, a cluster method based on lattice reflection and inversion symmetries that transports whole regions of a configuration while conserving particle number (they formulated a Swendsen-Wang-like version of it, in which whole symmetry-related clusters are updated collectively, as in the classic Swendsen-Wang algorithm for spin models). They determined $\mu_c = 1.75682(2)$ and $\rho_c = 0.180(4)$, and from magnetic quantities - the Binder ratio (a dimensionless fluctuation ratio of the order parameter) and the susceptibility - obtained $y_t = 1.51(1)$ and $y_h = 1.8748(8)$, consistent with 4-state Potts scaling. Energy-like quantities, however, whose fits probe the thermal exponent, gave $y_t$ between $1.440(5)$ and $1.470(5)$, significantly below $3/2$. Because their system sizes (up to $400\times 400$) did not support the extra fit parameters, they analyzed the data without explicit logarithmic terms and concluded that additive logarithmic corrections are probably small but that multiplicative logarithmic corrections cannot be neglected in energy-like quantities.

Problem Statement

For the hard-core lattice gas on the triangular lattice with exclusion of first and second neighbors at its continuous ordering transition (believed to lie in the marginal 4-state Potts universality class), determine quantitatively the logarithmic finite-size corrections and thereby settle the finite-size scaling of energy-like observables. Concretely: (1) establish a controlled finite-size-scaling ansatz for the particle density and specific-heat-like observables that incorporates the logarithmic structure of the marginal fixed point - multiplicative powers of $\ln L$ and/or additive $\ln L$ terms, including the subleading $\ln\ln L/\ln L$ and $1/\ln L$ corrections - and determine, from new finite-size simulations of this model, the leading logarithmic powers, their signs and amplitudes in this realization, i.e. whether the corrections to energy-like quantities are multiplicative, additive, or both; (2) determine, from the resulting log-corrected fits, whether the thermal exponent extracted from energy-like observables converges to $y_t = 3/2$ (equivalently $\nu = 2/3$), and produce definitive critical parameters ($\mu_c$, $\rho_c$) and exponents ($y_t$, $y_h$) with controlled systematic errors. This requirement is satisfied by either of two independently checkable outcomes: (2a) the log-corrected fits give $y_t = 3/2$ (equivalently $\nu = 2/3$) within the quoted systematic errors, together with the reported critical parameters and exponents; or (2b) they do not, in which case the resolution is the quantitative demonstration - reporting which standard 4-state Potts (Salas-Sokal) logarithmic powers, signs and amplitudes the energy-like data exclude and at what statistical significance - that no member of the standard correction class reproduces the data, i.e. how the marginal correction structure of this realization departs from the standard 4-state Potts predictions.

References

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