## Ising Antiferromagnet

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 square lattice behaves almost identically to the ferromagnetic Ising model if we focus on the staggered magnetization, which is defined as $$m_s = (1/N)\sum c_i s_i,$$ where we imagine a checkerboard lattice with red and black squares; $c_i = +1$ on the black squares and $c_i =-1$ on the red squares. The goal of this simulation is to explore the similarities between the ferromagnetic Ising model and the antiferromagnetic version.

Problem: Ising antiferromagnet

We have considered the ferromagnetic Ising model for which the energy of interaction between two nearest-neighbor spins is $J>0$. Hence all spins are parallel in the ground state in the ferromagnetic Ising model. In contrast, if $J < 0$, nearest-neighbor spins must be antiparallel to minimize their energy of interaction.

1. Sketch the ground state of the one-dimensional antiferromagnetic Ising model. Then do the same for the antiferromagnetic Ising model on a square lattice. What is the value of $m$ for the ground state of an Ising antiferromagnet?
2. Use Program IsingAntiferromagnet to simulate the antiferromagnetic Ising model on a square lattice for various temperatures and describe its qualitative behavior. Does the system have a phase transition at $T>0$? Does the value of $m$ show evidence of a phase transition?
3. In addition to the usual thermodynamic quantities the program calculates the staggered magnetization and the staggered susceptibility. The staggered magnetization is calculated by considering the square lattice as a checkerboard with black and red sites so that each black site has four red sites as nearest neighbors and vice versa. The staggered magnetization is calculated from $\sum c_i s_i$ where $c_i = +1$ for a black site and $c_i = -1$ for a red site. Describe the behavior of these quantities and compare them to the behavior of $m$ and $\chi$ for the ferromagnetic Ising model.

## Resource

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

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