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Physical Review Letters

Ising-Model Surface Tension Using Real-Space Renormalization-Group Methods

Mário J. de Oliveira, D. Furman, Robert B. Griffiths

Published Apr 10, 1978• 19 citations
Publisher

Abstract

Approximate real-space (cell-type) renormalization-group methods are used to calculate the surface tension between coexisting phases in two- and three-dimensional Ising models. The results are physically reasonable and show the expected scaling properties near a critical point. However, there are some problems in reconciling the three-dimensional results with the existence of a roughening transition.

DOI
10.1103/physrevlett.40.977

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Comment on "Ising-model surface tension using real-space renormalization-group methods"

Douglas B. Abraham

Attention has been drawn again by de Oliveria et al. to an interesting relationship between the surface tension and correlation length proposed originally by Widom. In this addendum the relation is derived and discussed for two- and three-dimensional Ising ferromagnets using duality.

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Physical Review Letters•May 31, 1976•51 citations
Approximate real-space (cell-type) renormalization-group methods are used to calculate the surface tension between coexisting phases in two- and three-dimensional Ising models. The results are physically reasonable and show the expected scaling properties near a critical point. However, there are some problems in reconciling the three-dimensional results with the existence of a roughening transition.
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.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.