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Physical review. B, Condensed matter

Determination of multicritcal points for lattice-gas models by finite-size scaling of the susceptibility

Per Arne Rikvold

Published Oct 1, 1985• 13 citations
Publisher

Abstract

The finite-size scaling properties of the susceptibility ${\ensuremath{\chi}}_{N}$ with respect to the nonordering field and the ``second correlation length'' \ensuremath{\xi}${^}_{N}$ for a lattice-gas system of strip width N are related by a sum-rule argument, and used to determine multicritical temperatures. For a simple system, the estimates based on ${\ensuremath{\chi}}_{N}$ and \ensuremath{\xi}${^}_{N}$ are consistent. ${\ensuremath{\chi}}_{N}$ depends only on the largest eigenvalue of the transfer matrix and provides improved estimates for a model whose eigenvalue spectrum is too complicated to guarantee accurate determination of \ensuremath{\xi}${^}_{N}$, which depends on the ratio of the largest and third largest eigenvalues.

DOI
10.1103/physrevb.32.4756

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The finite-size scaling properties of the susceptibility \ensuremathχN{\ensuremath{\chi}}_{N}\ensuremathχN​ with respect to the nonordering field and the ``second correlation length'' \ensuremath{\xi}{^}_{N} for a lattice-gas system of strip width N are related by a sum-rule argument, and used to determine multicritical temperatures. For a simple system, the estimates based on \ensuremathχN{\ensuremath{\chi}}_{N}\ensuremathχN​ and \ensuremath{\xi}{^}_{N} are consistent. \ensuremathχN{\ensuremath{\chi}}_{N}\ensuremathχN​ depends only on the largest eigenvalue of the transfer matrix and provides improved estimates for a model whose eigenvalue spectrum is too complicated to guarantee accurate determination of \ensuremath{\xi}{^}_{N}, which depends on the ratio of the largest and third largest eigenvalues.
We present a finite-size-scaling study of a two-dimensional square-lattice-gas model with nearest-neighbor repulsion and next-nearest-neighbor attraction. This system may be used as a simplified model of adsorbates on transition-metal (100) surfaces. An accurate determination of the tricritical temperature is obtained by the utilization of the degeneracy of the three largest eigenvalues of the transfer matrix at a tricritical point. We obtain the phase diagram and critical and tricritical exponents, all quite consistent with previously known results. We also find that a recent conjecture relating the critical exponent $\ensuremath{\eta}$ to the correlation-length amplitude seems to work well even at the tricritical point.
We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.