Ising Antiferromagnet on a Square Lattice
So far we have considered the ferromagnetic Ising model for which J > 0. What is the ground state of the ferromagnetic Ising model on an infinite lattice? What is the ground state energy per spin?
Suppose that we again consider the Ising model, but take J < 0. This model is called the antiferromagnetic Ising model. What is the ground state(s) of the infinite system? What is the ground state energy per spin?
In the following problems we explore the similarities and differences of the ferromagnetic and antiferromagnetic Ising model.
- Confirm the nature of the ground state by doing some simulations at low temperature. How would you describe the ground state of the antiferromagnetic Ising model? (We choose units such that |J|/k = 1.)
- Determine the mean energy for several temperatures and compare your results to the temperature dependence of the mean energy for the ferromagnetic Ising model. Are your results for the mean energy similar or qualitatively different?
- Determine the specific heat as a function of temperature. Is there any evidence of a phase transition?
- Determine the mean magnetization and the susceptibility χ as a function of the temperature. Does the magnetization
an antiferromagnetic phase and a paramagnetic phase? Does χ show any evidence of a phase transition?
- A quantity that does differ between the low temperature antiferromagnetic phase
and the high temperature paramagnetic phase is the staggered magnetization. Imagine your lattice to be two sublattices such as the red and black squares of a checkerboard. The staggered magnetization Ms is given by
Ms = ∑i sgn(i)si,
where sgn(i) = ± 1 depending on whether the site is red (+1) or black (-1). We can obtain the zero field staggered susceptibility χs from the fluctuations in Ms:
χs = (1/kT)[<Ms2> - <Ms>2].
Does χs exhibit any evidence of a phase transition?
- Compare the ground state energy per spin for a system of 3 × 3 spins and a system of 4 × 4 spins. Discuss the importance of finite size effects for small systems.
Updated 4 March 2009.