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 antiferromagnetWe 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.
