Goal: The goal of this project is map out the ordered magnetic states of the Heisenberg model on exotic lattices.
The way to find the ordered magnetic pattern at zero temperature is to minimize the energy. However, for competing interactions this is often difficult to do analytically. The standard way is to use the Luttinger-Tisza (LT) method where the minimization problem is simplified by reducing the number of Lagrange multipliers used when minimizing the energy. This results in a set of equations that the true minimal energy solution must satisfy.
Method:
This project starts with learning the Luttinger-Tisza method (LT). To be concrete one should first consider Heisenberg models with up to third nearest neighbor interactions on simple Bravais lattices like the square and cubic lattices. Then add more atoms per unit cell to study various exotic lattices of possible experimental interest, such as any of the two-dimensional Archimedean lattices, or the three-dimensional diamond, or octohedral lattices for instance.
Dependent on time and interests it will thereafter be interesting to characterize the symmetries of the ordered states mathematically, and to complement the LT-study by performing numerical Monte Carlo simulations/relaxations.
The concrete tasks that must be addressed are:
- Get familiar with classical Heisenberg models, different lattices and their reciprocal space.
- Learn the Luttinger-Tisza method.
- Characterize ordered states.
- Make a Monte Carlo program to check if there are other competing states.