Numerical investigation of the superfluid transition in low dimensions and in anisotropic systems

Abstract

The main theme of this thesis is the superfluid transition in reduced dimensions and in anisotropic and inhomogeneous systems. Using state-of-the-art computational methodology, we carry out large-scale, numerically exact computer simulations to study this topic, making use of classical and quantum lattice models. In particular, we investigate the behavior of the specific heat in two-dimensional (2D) superfluids, using the classical $x$-$y$ Hamiltonian on the square lattice as a minimal, paradigmatic model. The specific heat is found to exhibit a well-defined peak in the thermodynamic limit, at a temperature above the superfluid transition temperature. We then attempt to explore a possible dimensional crossover in a 2D superfluid in the presence of an externally imposed density modulation, in the context of the $|\psi|^4$ classical field theory. As the strength of the modulation increases, the physics of the system becomes more similar to that of the anisotropic $x-y$ model, characterized by a decreased superfluid transition temperature and an anisotropic response, but with no dimensional crossover. Finally, we examine the phase diagram of lattice hard core bosons with anisotropic nearest-neighbor interactions that can vary between repulsion and attraction in different directions. This phase diagram includes a superfluid phase, as well as two crystalline phases at half-filling, either checkerboard or striped, but no ``supersolid'' phase; which is similar to the case of isotropic interactions. Our predictions appear to be in principle testable experimentally, for example by performing measurements on thin films of superfluid helium or cold atom assemblies.