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Physical review. B, Condensed matter

Approximate critical surface of the bond-mixed square-lattice Ising model

Silvio V. F. Levy, Constantino Tsallis, Evaldo M. F. Curado

Published Apr 1, 1980• 35 citations
Publisher

Abstract

The critical surface of the quenched bond-mixed square-lattice spin-$\frac{1}{2}$ first-neighbor-interaction ferromagnetic Ising model (with exchange interactions ${J}_{1}$ and ${J}_{2}$) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely ${p}_{c}=\frac{1}{2}$, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for $p=\frac{1}{2}$. Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the $t$ variables) for any point in the surface.

DOI
10.1103/physrevb.21.2991

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The critical surface of the quenched bond-mixed square-lattice spin-12\frac{1}{2}21​ first-neighbor-interaction ferromagnetic Ising model (with exchange interactions J1{J}_{1}J1​ and J2{J}_{2}J2​) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4 in the variables ti\ensuremath≡tanh(JikBT){t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})ti​\ensuremath≡tanh(kB​TJi​​)] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely pc=12{p}_{c}=\frac{1}{2}pc​=21​, tc=2\ensuremath−1{t}_{c}=\sqrt{2}\ensuremath{-}1tc​=2​\ensuremath−1, both limiting slopes in these points, and t2=(1\ensuremath−t1)(1+t1){t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}t2​=(1+t1​)(1\ensuremath−t1​)​ for p=12p=\frac{1}{2}p=21​. Furthermore an analytic approximation [namely, (1\ensuremath−p)ln(1+t1)+pln(1+t2)=12ln2(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2(1\ensuremath−p)ln(1+t1​)+pln(1+t2​)=21​ln2] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than 10\ensuremath−3{10}^{\ensuremath{-}3}10\ensuremath−3 (in the ttt variables) for any point in the surface.
The ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds J1{J}_{1}J1​ and J2{J}_{2}J2​ occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}J1{J}_{1}J1​/J2{J}_{2}J2​), and the critical bond concentrations pc{p}_{c}pc​ at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.
A real-space rescaling method is applied to the mixed, spin- 1 / 2 , Ising ferromagnet. Both bond and site disorder are considered where, respectively, bonds J A , J B or sites A, B occur with probabilities p, 1-p. For the bond-disordered case, critical curves are obtained for the square lattice for a range of alpha identical to J B /J A . Slopes at p=0,1 are in very good agreement with exact results previously obtained and T c (p- 1 / 2 ) agrees to within a few per cent with an exact result obtained by Fisch. For the more realistic site-disordered problem only the case where bond strengths are related by tanh J AA tanh J BB =tanh 2 J AB is considered. Critical curves are plotted for various alpha for the square and triangular lattices. The results agree well with exact limiting slopes available for the square lattice. Thermal, percolation and crossover exponents are also calculated for both the site- and bond-disordered cases.
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Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.