We present a method of computing coverage discontinuities in two-dimensional lattice-gas phase diagrams using transfer matrices. By applying the method to Baxter's generalized hard-square model, we find good agreement with exact results. This method also can be used to estimate the position of multicritical points, and we again find good agreement with exact results and with previous work on the Ising metamagnet. We discuss the transfer-matrix eigenvalue spectrum around an Ising tricritical point and verify the prediction of conformal invariance that the finite-size scaling behavior of each of the leading eigenvalues is governed by a different critical exponent at the critical point. We show numerically that the finite-size convergence of the free energy at the (Ising-like) tricritical point of Baxter's model is consistent with a conformal anomaly of (7/10). We show that the justification of a commonly used method to locate multicritical points using simultaneous scaling of the correlation length and the ``persistence length'' is misleading. Finally we suggest a method of estimating the position of multicritical points using information from only one strip width.
We present a method of computing coverage discontinuities in two-dimensional lattice-gas phase diagrams using transfer matrices. By applying the method to Baxter's generalized hard-square model, we find good agreement with exact results. This method also can be used to estimate the position of multicritical points, and we again find good agreement with exact results and with previous work on the Ising metamagnet. We discuss the transfer-matrix eigenvalue spectrum around an Ising tricritical point and verify the prediction of conformal invariance that the finite-size scaling behavior of each of the leading eigenvalues is governed by a different critical exponent at the critical point. We show numerically that the finite-size convergence of the free energy at the (Ising-like) tricritical point of Baxter's model is consistent with a conformal anomaly of (7/10). We show that the justification of a commonly used method to locate multicritical points using simultaneous scaling of the correlation length and the ``persistence length'' is misleading. Finally we suggest a method of estimating the position of multicritical points using information from only one strip width.
We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent $\ensuremath{\nu}$ governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude $\ensuremath{\eta}$, (iii) the conformal anomaly c, (iv) the finite-size susceptibility exponent $\ensuremath{\gamma}/\ensuremath{\nu}$, and (v) the set of multifractal exponents ${{\ensuremath{\eta}}_{k}}$ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.
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.
The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-para-magnetic phase boundary is probed, with the following estimates for the associated critical indices: $\ensuremath{\nu}=1.49(2)$, $\ensuremath{\gamma}=2.71(4)$, ${\ensuremath{\eta}}_{1}=0.183(3)$, which are distinctly different from the percolation values.
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 the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=O, which reproduces exactly the critical-point behavior of the full Ising model.
We use numerical transfer-matrix methods to investigate properties of the multicritical point of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and nonlinear zero-field uniform susceptibilities, on strip of widths $4\ensuremath{\leqslant}L\ensuremath{\leqslant}16$ sites, from which we estimate the conformal anomaly c, the decay-of-correlations exponent $\ensuremath{\eta}$, and the linear and nonlinear susceptibility exponents $\ensuremath{\gamma}∕\ensuremath{\nu}$ and ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}$, with the help of finite-size scaling and conformal invariance concepts. Our results are c=0.46(1); $0.187\ensuremath{\lesssim}\ensuremath{\eta}\ensuremath{\lesssim}0.196$; $\ensuremath{\gamma}∕\ensuremath{\nu}=1.797(5)$; ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\ensuremath{\eta}=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=2\ensuremath{\gamma}∕\ensuremath{\nu}+d$ (with space dimensionality d=2) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.