James L. Monroe

Phase diagrams of Ising models on Husimi trees. I. Pure multisite interaction systems

Lattice spin systems with multisite interactions have rich and interesting phase diagrams. We present some results for such systems involving Ising spins (σ=±1) using a generalization of the Bethe lattice approximation. First, we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems. Second, a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made. We exhibit a full bifurcation diagram, chaos, period-3 windows, etc., for the magnetization of the base site of this system.

A new criterion for the location of phase transitions for spin systems on recursive lattices

Abstract Lattice spin systems on Bethe lattices or Husimi trees can be studied recursively. We present a new criterion for the location of phase transitions for such systems. We then apply the criterion to three different systems with complex phase diagrams, in particular with transitions at non-zero magnetic fields resulting from multi-site interactions.

Frustrated Ising systems on Husimi trees

We consider two frustrated Ising model systems. The first is the full frustrated antiferromagnetic Ising model on the triangle lattice. We approximate the system by a Husimi tree. By a “sequential” build up of the tree we get a qualitatively correct phase diagram which quantitatively is close to other approximation methods. Most closed form approximations of this system such as mean field theory give qualitatively incorrect phase diagrams. As a further test of the Husimi tree approach we look at a frustrated Ising model on a checkerboard type lattice. This system has been solved exactly by Azaria et al., Phys. Rev. Lett. 59 (1987) 1629, when h=0. Again the Husimi tree approach gives qualitatively correct results approximating a rather complex phase diagram with e.g. reentrant phases. And in addition this approach allows one to determine the phase diagram for h≠0. Finally, this method should be easily extended to a number of other frustrated lattice spin systems such as the fully frustrated system on the simple cubic lattice.

The coherent anomaly method and long-range one-dimensional ising models

The authors present the results of the coherent anomaly method when applied to Ising models in one dimension with long-range interactions. This class of systems acts as an interesting and challenging test for the method because the critical exponents as well as the critical temperature values of which are given by the method, depend on the rate of fall-off of the interactions. Thus one can see how accurately the method gives results which correctly reflect this dependency. The results obtained from this method are compared with results obtained by a variety of other methods.

Long-range one-dimensional Potts models: a cluster mean-field and extrapolation approach

We obtain approximations for the critical temperature (Tc) of q-state ferromagnetic Potts models on one-dimensional lattices with algebraically decaying ferromagnetic pair interactions, i.e. decaying as 1/r, with 1<2. Initially we use a cluster mean-field method to get approximations to the critical temperature and we look at increasingly large clusters. This gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms. We see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate Tc estimates. Our best estimates we believe to be correct to four-figure accuracy.

The coherent anomaly method applied to a variety of Ising models

Abstract The coherent anomaly method of Suzuki is used to obtain values of the critical exponents γ and β for ferromagnetic nearest-neighbor Ising models on a wide variety of lattices in two, three, four, and higher dimensions. In addition values for γ and β were obtained for higher spin systems. Together these results further support the coherent anomaly method.

Correlation Inequalities for Vector Spin Models

Correlation inequalities forn-vector spin models (n ⩾ 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).

Upper bounds on the critical temperature for various ising models

Upper bounds are obtained for spin ±1 systems. In the case of only nearestneighbor interactions on, for example, the square lattice we obtainΒcJ>0.3592. The methods strength is seen when considering systems with longer-range interactions. For example, we obtainΒcJ>0.360 compared to the previous best bound ofΒc J⩾ 0.345 for the one-dimensional lattice with 1/r2 interactions. The method relies upon an identity between correlation functions and then the use of correlation inequalities to obtain the final bounds.

One-dimensional Ising models with long-range interactions

We consider Ising models with long-range ferromagnetic pair interactions decaying as for . We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For we find which can be compared with recent results of Luijten and Blote who found , and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of decreases significantly. In addition to we obtain approximate values for the critical exponents , and using the coherent anomaly method. For we obtain , , and - all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of .

Bethe lattice approximation of long-range interaction Ising models

Abstract An extension or modification of the usual Bethe lattice is used to approximate one-dimensional Ising models with long-range pair interactions of the form Σ ( J / m Θ )σ i σ i + m . In particular an expression is obtained which allows one to one to obtain the critical temperature, T c , as a function of Θ, in the approximation. Our results are compared to a number of other T c approximations.