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Physical review. B, Solid state

Renormalization-group calculation for the equation of state of an Ising ferromagnet

Bernard Nienhuis, Michael Nauenberg

Published Jun 1, 1975• 53 citations
Publisher

Abstract

The renormalization-group series expansion for the free energy of the Ising model is applied to obtain the equation of state in an external magnetic field. Numerical calculations for a square lattice in a four-cell-cluster approximation are compared in the scaling limit with results of high- and low-temperature series expansions.
DOI
10.1103/physrevb.11.4152

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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.
The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature Tc{T}_{c}Tc​ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for T<TcT<{T}_{c}T<Tc​. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.
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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 mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
We consider a ferromagnet in an external magnetic field with $T\ensuremath{\le}{T}_{c}$. Using a variational approximation based on the zero-field solution, we obtain an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. Explicit numerical calculations have been carried out for the two-dimensional Ising model, and we find that our results provide a good approximation to the results of series expansions throughout the region $T\ensuremath{\le}{T}_{c}$. We also apply our procedure to the eight-vertex model. Metastability can be discussed within our approximation in a manner analogous to the way it is dealt with in mean-field theory. In our treatment, the critical exponents have nonclassical values, but the pseudocritical exponents all represent square-root singularities, which is the same result as found within mean-field theory.