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Physical Review Letters

Renormalization Group Results for Lattice-Gas Phase Boundaries in Two Dimensions

K. R. Subbaswamy, Gerald Dennis Mahan

Published Sep 13, 1976• 44 citations
Publisher

Abstract

Lattice-gas phase boundaries have been calculated for the simple quadratic and honey-comb lattices from the corresponding antiferromagnetic critical curves. Results were obtained using nonlinear renormalization group methods with periodic cell clusters. The result for the simple quadratic lattice is in excellent agreement with the existing temperature-series result.

DOI
10.1103/physrevlett.37.642

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Lattice-gas phase boundaries have been calculated for the simple quadratic and honey-comb lattices from the corresponding antiferromagnetic critical curves. Results were obtained using nonlinear renormalization group methods with periodic cell clusters. The result for the simple quadratic lattice is in excellent agreement with the existing temperature-series result.
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Ising antiferromagnets in a near-critical magnetic field at low temperatures are equivalent to hard-core lattice gases. Using this connection and the existing series-expansion results for hard-core lattice gases, we determine the slope of the phase boundary at T=0T=0T=0 for the square (sq), plane-triangular (pt), simple cubic (sc), and body-centered cubic (bcc) antiferromagnets. The slope is negative for the sq and pt lattices and nearly zero for the sc case. For the bcc lattice a positive slope is obtained, indicating that the phase boundary bulges above the zero-temperature critical field. We also test M\"uller-Hartmann and Zittarz's postulate for the critical curve of the sq Ising antiferromagnet. A renormalization-group treatment of the hard-square lattice gas yields a critical activity ${z}^{*}=3.7959\ifmmode\pm\else\textpm\fi{}0.0001$, which is in agreement with series-expansion and finite-lattice estimates but at variance with the postulated z∗=4{z}^{*}=4z∗=4. The same calculation gives $\ensuremath{\nu}=0.999\ifmmode\pm\else\textpm\fi{}0.001$ for the correlation-length exponent, thus supporting the conjecture that the transition of the hard-square lattice gas belongs to the Ising universality class.
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We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.