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Physical review. B, Condensed matter

Extrapolated renormalization-group calculation of the surface tension in square-lattice Ising model

Evaldo M. F. Curado, Constantino Tsallis, Silvio V. F. Levy, Mário J. de Oliveira

Published Feb 1, 1981• 19 citations
Publisher

Abstract

By using self-dual clusters (whose sizes are characterized by the numbers $b=2,3,4,5$), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature ${T}_{c}$ is recovered for any value of $b$; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.

DOI
10.1103/physrevb.23.1419

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Physical review. B, Condensed matter•Apr 1, 1980•35 citations
By using self-dual clusters (whose sizes are characterized by the numbers b=2,3,4,5b=2,3,4,5b=2,3,4,5), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature Tc{T}_{c}Tc​ is recovered for any value of bbb; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 (which, for two dimensions, coincides with the surface-tension critical exponent \ensuremathμ\ensuremath{\mu}\ensuremathμ) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit T\ensuremath→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​. We develop four different numerical procedures for extrapolating to b\ensuremath→\ensuremath∞b\ensuremath{\rightarrow}\ensuremath{\infty}b\ensuremath→\ensuremath∞ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
A simple renormalisation-group approach based on self-dual clusters is proposed for the two-dimensional nearest-neighbour spin- 1 / 2 Ising model on the square lattice; it reproduces the exact critical point. The authors calculate the internal energy and the specific heat for vanishing external magnetic field, spontaneous magnetisation and the thermal (y T ) and magnetic (y H ) critical exponents. The results obtained from the first four smallest cluster sizes strongly suggest the convergence towards the exact values when the cluster size increases. Even for the smallest cluster, where the calculation is very simple, the results are quite accurate, particularly in the neighbourhood of the critical point.
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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{} ${10}^{\ensuremath{-}4}$ 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 pc=12{p}_{c}=\frac{1}{2}pc​=21​, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for p=12p=\frac{1}{2}p=21​. 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 ttt variables) for any point in the surface.