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arXiv:1301.5219 · Journal of Physics A Mathematical and Theoretical

The two-dimensional 4-state Potts model in a magnetic field

Bertrand Berche, P. Butera, Щур Лев Николаевич

Published Feb 12, 2013• 11 citations
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Abstract

We present a solution of the nonlinear renormalization group equations leading to the dominant and subdominant singular behaviors of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non-vanishing magnetic field. The solutions (i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and (ii) prove recently proposed relations among logarithmic exponents.

DOI
10.1088/1751-8113/46/9/095001

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We present a solution of the nonlinear renormalization group equations leading to the dominant and subdominant singular behaviors of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non-vanishing magnetic field. The solutions (i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and (ii) prove recently proposed relations among logarithmic exponents.
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections and of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.
Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios A + / A - , Γ + /Γ - , Γ T /Γ - and R C ± are given, which arouse new questions with respect to previous works.
A theory of the scaling behavior near the qc=4{q}_{c}=4qc​=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for qqq near qc{q}_{c}qc​. For the pure Potts model with q=qcq={q}_{c}q=qc​ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At Tc{T}_{c}Tc​ we find ln(r)\mathrm{ln}(r)ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with q=qcq={q}_{c}q=qc​ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as TTT approaches the multicritical point from the first-order side. Results for q>qcq>{q}_{c}q>qc​ and q<qcq<{q}_{c}q<qc​ are also given.
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical