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Journal of Physics C Solid State Physics

Self-dual cluster renormalisation-group approach for the square lattice Ising model: specific heat and magnetisation

H. O. Mártin, Constantino Tsallis

Published Dec 30, 1981• 21 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/14/36/008

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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.
The free and internal energies and specific heat of the q-state Potts ferromagnet are discussed. A real-space renormalisation-group approach is presented, which reproduces a considerable amount of exact particular results for all dimensionalities (hypercubic lattices). The case of the square lattice is investigated in detail using self-dual clusters (which provide the exact critical point for all q). Comparison with Onsager results (q=2) is satisfactory; the general tendencies for q not=2 (1<or=q<or=4) are exhibited, and the bound percolation limit (q to 1) is worked out in some detail.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
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-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.
The antiferromagnetic Ising model in a uniform field on a square lattice is studied by a position-space renormalisation group approach. A self-dual cluster that preserves sublattice symmetry is employed: at each site of the cluster, the coupling to the field is chosen to be proportional to the coordination number of that site. The author obtains the physically correct result that the magnetic field is irrelevant (in the renormalisation group sense) near the zero-field fixed point. The phase diagram he obtains compares well with experimental results for CoCs 3 Br 5 and with the conjectured analytic form postulated by Muller-Hartmann and Zittartz. He discusses possible applications of this approach to other situations of interest.
We analyse in some detail a recently proposed transfer matrix mean-field approximation which yields the exact critical point for several two-dimensional nearest-neighbour Ising models. For the square lattice model we show explicitly that this approximation yields not only the exact critical point, but also the exact boundary magnetization of a semi-infinite Ising model, independent of the size of the strips used. Then we develop a new mean-field renormalization-group strategy based on this approximation and make connections with finite size scaling. Applying our strategy to the quadratic Ising and three-state Potts models we obtain results for the critical exponents which are in very good agreement with the exact ones. In this way we also clarify some advantages and limitations of the mean field renormalization group approach.