The way to find the ordered magnetic pattern at zero temperature is to minimize the energy of the Hamiltonian. However, for competing interactions this is difficult to do analytically.
The standard way is to use the Luttinger-Tisza 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, but they are not sufficient. That is, the LT-method can give solutions which are not the true minimal energy solutions.
One usually picks particularly simple solutions to the LT-equations: planar spin states with a single periodicity, socalled single-q planar spirals. However, in certain cases, these are not the only possible solutions, and one can find additional nonplanar solutions having multiple q-values. By construction these states have the same energy. However, all experiments are carried out at finite temperatures. Thus what matters is which solution has the lowest free energy. Therefore one needs also to calculate the entropy of these states, and the one with highest entropy will be realized at finite temperatures. There is little systematics known about the multi-q states. Such multi-q states encompasses also the skyrmion lattice state, which has been subject to a lot of recent studies.
Goal: The goal of this project is to asses the stability of multi-q states by calculating their entropy and comparing to ordinary single-q spirals.
Method:
This project starts with learning the LT-method. To be concrete we will first do Bravais lattices like the square and triangular lattices. Then add more atoms per unit cell to study honeycomb, diamond and perhaps kagome lattices. Special emphasis will be laid on when or where one can construct multi-q solutions to the LT equations.
Then having obtained candidate multi-q states, a calculation of small fluctuations around these states will be carried out to calculate their entropy. These will be compared to the entropy of the single-q spirals. The project is mainly analytical, but some numerics is needed in order to compare the entropies.
The concrete tasks that must be addressed are:
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Get familiar with classical Heisenberg models, different lattices and their reciprocal space.
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Learn the Luttinger-Tisza method.
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IIdentify possible multi-q states.
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Calculate entropy by considering small fluctuations about ordered spin states.