[Physics] Antiferromagnetic and Ferromagnetic Ising Model on triangular lattice

Question

Recently I heard a report about the antiferromagnetic Ising model in triangular lattice. It's interesting and I never realized that the result in triangular lattice would be so different from square lattice. I'm curious about whether the following questions have been solved or what's the recent progress about the following question：

1. For homogeneous ferromagnetic Ising model in triangular lattice , I found the critical temperature can be exactly solved by Kramers-Wannier duality and star-triangle relation. $T_c= 3.642…K$, where $K$ is the coupling constant. But I haven't found the result of partion function. Has partition function of ferromagnetic Ising model in triangular lattice with and without external magnetic field been solved?

2. For homogeneous antiferromagnetic Ising model in triangular lattice, what's the ground state and ground state entropy? Does it exist phase transition? What's the partition function with or without external magnetic field?

I want to know whether these have been solved. If it's solved, please recommend some references.

Answer 1

1. The partition function of the Ising model on a triangular model has been computed by Plechko using Grasmann variables to decouple the spins. Some references are:

   • Plechko, V. N. « Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models ». http://arxiv.org/abs/hep-th/9607053.
   • id. « Fermionic Integrals and Analytic Solutions for Two-Dimensional Ising Models ». http://arxiv.org/abs/cond-mat/9812434
   • id. « Fermionic Path Integrals and Analytic Solutions for Two-Dimensional Ising Models ». http://arxiv.org/abs/hep-th/9609044.

   I am not aware of any solution in presence of a magnetic field.

2. In the anti-ferromagnetic Ising model on a triangular lattice, each triangular plaquette is frustrated. There is no spin configuration on a triangle that satisfies simultaneously the three anti-ferromagnetic couplings. In the ground state, exactly one bond per plaquette is frustrated. In the thermodynamic limit, there is an infinite number of ways to choose these frustrated bonds. The problem is actually related to the covering of the dual lattice by dimers (the dimer crosses the frustrated bond). The entropy per site is finite. The model is critical only at zero-temperature, where spin-spin correlation functions decay algebraically with a critical exponent $\eta=1/2$. As far as I remember, the partition function can be computed for another model in the same universality class, the Ising model on a fully-frustrated square lattice (for each square plaquette, there are 3 ferromagnetic bonds and one anti-ferromagnetic), using free fermion techniques. You may check

   • André, G., R. Bidaux, J.-P. Carton, R. Conte, et L. de Seze. « Frustration in periodic systems : exact results for some 2D Ising models ». Journal de Physique p 40, 5 (1979): 10. doi:10.1051/jphys:01979004005047900.

   For a finite magnetic field, the antiferromagnetic Ising model in triangular lattice undergoes a Kosterlitz-Thouless-Berezinski phase transition.