Reflection Positivity in Heisenberg and Ice-Type Models

Abstract

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.