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## Frustration

If the interaction energy $J$ in the Ising model is negative, then neighboring spins lower their energy by being antiparallel. Such a system is called an antiferromagnet. The 2d Ising antiferromagnet on a triangular or hexagonal lattice has some unusual properties. Each site has six nearest neighbors, and it is impossible to create a microstate where every spin has only antiparallel neighboring spins. We say the spin is frustrated. The goal of this simulation is to explore the effect of frustration on the possibility for a phase transition.

Problem: Ising Antiferromagnetic on a hexagonal lattice

Consider the Ising antiferromagnetic on a hexagonal lattice, for which each spin has six nearest neighbors. The ground state in this case is not unique because of frustration Convince yourself that there are multiple ground states. Is the entropy zero or nonzero at $T=0$? ($S(T=0) = 0.3383kN$. See G. H. Wannier, “Antiferromagnetism. The triangular Ising net,” Phys. Rev. 79, 357-364 (1950); erratum, Phys. Rev. B 7, 5017 (1973).) Use Program Frustration to simulate the antiferromagnetic Ising model on a hexagonal lattice at various temperatures and describe its qualitative behavior. This system does not have a phase transition for $T>0$. Are your results consistent with this behavior?

## Resource

Problem 5.41 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).