## 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.

- 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?
- 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? - 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).