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Physica Scripta

Extrapolated renormalization group calculation of surface tension critical exponent in Ising model

T Kotowski, Maciej Suffczynski

Published Mar 1, 1988• 2 citations
Publisher

Abstract

A real-space renormalization-group calculation of critical exponent of correlation length and surface tension of the square-lattice nearest-neighbor ferromagnetic Ising model is extended to the cluster size b = 6. The extension allows a more reliable extrapolation for large cluster sizes. Values of the approximate critical exponent are obtained nearer to the exact value.

DOI
10.1088/0031-8949/37/3/004

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A real-space renormalization-group calculation of critical exponent of correlation length and surface tension of the square-lattice nearest-neighbor ferromagnetic Ising model is extended to the cluster size b = 6. The extension allows a more reliable extrapolation for large cluster sizes. Values of the approximate critical exponent are obtained nearer to the exact value.
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 $\ensuremath{\nu}=1$ (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.
We obtain an approximation of the phase diagram of an Ising model with both nearest-neighbor and next-nearest-neighbor interactions on the square lattice. We use the Fisher zeros of the partition functions of a sequence of finite-sized systems along with various extrapolation methods to obtain phase transition points. In addition, we obtain an approximate value of the correlation length critical exponent nu. Our results are compared to previous results for this system using a wide variety of other approaches.
Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of [Formula: see text] for small [Formula: see text], where K is the nearest neighbor coupling strength and [Formula: see text] is the nearest neighbor spins degrees of freedom around a central spin. The values of the critical coupling strengths are obtained as 0.4830 for square lattice and 0.2225 for simple cubic (SC) lattice. The corresponding critical exponents values [Formula: see text] and [Formula: see text] are also calculated within very acceptable agreement with those values obtained from numerical works.
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.