Path Integral Monte Carlo Study of a Model 2D Quantum Paraelectric

We have begun a study of quantum ferroelectrics and paraelectrics. Simple 2D short-range lattice model hamiltonians are constructed, keeping in mind the phenomenology of real perovskite systems, like $SrTiO_{3}$ and $KTaO_{3}$. Pertinent quantum tunneling phenomena, and the presence of an ice-like constraint are demonstrated. The two simplest models, namely a plain quantum four-state clock model, and a constrained one, are then studied in some detail. We show the equivalence of the former, but not of the latter, to a quantum Ising model. For the latter, we describe a very good analytical wavefunction valid in the special case of zero coupling ($J = 0$). In order to study the full quantum statistical mechanics of both models, a Path Integral Monte Carlo calculation is set up, and implemented with a technique, which even in the constrained case permits a good convergence for increasing time slice number $m$. The method is applied first to the unconstrained model, which serves as a check, and successively to the constrained quantum four-state clock model. It is found that in both cases, a quantum phase transition still takes place at finite coupling J, between a ferroelectric and a quantum paraelectric state, even when the constraint hinders disordering of the ferroelectric state. This model paraelectric state has a finite excitation gap, and no broken symmetry. The possible role of additional ("oxygen hopping") kinetic terms in making closer contact with the known phenomenology of $SrTiO_{3}$ is discussed.