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Physical review. B, Condensed matter

Extended scaling relations for the magnetic critical exponents of the Potts model

Marcel den Nijs

Published Feb 1, 1983• 152 citations
Publisher

Abstract

The extended scaling relations for the leading and next-to-leading magnetic critical exponents of the two-dimensional Potts model are derived from the spin-spin correlation function in the Coulomb-gas representation.
DOI
10.1103/physrevb.27.1674

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The fractal dimensions of the largest clusters of like spins connected by nearest-neighbor bonds in the critical two-dimensional Q-state Potts model have been determined via Monte Carlo simulations to be df{\mathit{d}}_{\mathit{f}}df​=1.946(3), 1.922(5), and 1.909(8) for Q=2, 3, and 4, respectively. These dimensions are close to the magnetic tricritical exponents of the (Q-1)-state dilute Potts model (d-x=187/96=1.947..., 77/40=1.925, and 40/21=1.904... for Q-1=1, 2, and 3, respectively). The critical Q-state model and the tricritical (Q-1)-state model are related to a two-component Potts lattice gas; this model generates percolative susceptibilities of critical Potts clusters, and suggests a renormalization-group argument for df{\mathit{d}}_{\mathit{f}}df​=d-x.
We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent η = 1/4 at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at β 2 = 8π, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure. KEY WORDS: soliton correlation functions, sine-Gordon theory, XY-model, 2D Coulomb gas, Kosterlitz-Thouless transition
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality 2βν+γν=d\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = dν2β​+νγ​=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.