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Physical review. B, Condensed matter

Surface thermodynamic functions of the Ising model from a renormalization group

Nenad M. Svrakic, Rahul Anil Pandit, Michael Wortis

Published Aug 1, 1980• 33 citations
Publisher

Abstract

The surface thermodynamic functions of the semi-infinite square and simple-cubic $s=\frac{1}{2}$ nearest-neighbor Ising models are computed numerically from renormalization groups based on simple cell clusters. It is shown how the exact high- and low-temperature limits may be incorporated into the calculations by (i) building in proper site coordination via an Ursell expansion for the free energy and (ii) treating the ground state exactly. In $d=2$ a 4 \ifmmode\times\else\texttimes\fi{} 4 cell-cluster approximation is in excellent agreement with the exact results of Fisher and Ferdinand. In $d=3$ a 2 \ifmmode\times\else\texttimes\fi{} 8 cell-cluster approximation gives plausible results with enough structure to be interesting.

DOI
10.1103/physrevb.22.1286

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The surface thermodynamic functions of the semi-infinite square and simple-cubic s=12s=\frac{1}{2}s=21​ nearest-neighbor Ising models are computed numerically from renormalization groups based on simple cell clusters. It is shown how the exact high- and low-temperature limits may be incorporated into the calculations by (i) building in proper site coordination via an Ursell expansion for the free energy and (ii) treating the ground state exactly. In d=2d=2d=2 a 4 \ifmmode\times\else\texttimes\fi{} 4 cell-cluster approximation is in excellent agreement with the exact results of Fisher and Ferdinand. In d=3d=3d=3 a 2 \ifmmode\times\else\texttimes\fi{} 8 cell-cluster approximation gives plausible results with enough structure to be interesting.
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.
A semi-infinite Ising model is studied by renormalization-group techniques in the position-space formulation due to Niemeijer and van Leeuwen. We show how to set up the calculation of thermodynamic properties (both critical and noncritical) of a sample with a free surface by iteration of a set of recursion relations for spatially inhomogeneous couplings and magnetic fields. Surface and bulk properties are clearly distinguished. The method is illustrated by a calculation (using a two-cell cluster approximation) of the magnetic-surface critical exponent in two dimensions, where exact results are available for comparison.
We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.