Strongly correlated one-dimensional systems and applications to super-cooled gases

In this thesis I have corroborated earlier findings showing that in a one-dimensional system we have a direct correspondence between some of the physical properties of bosons and fermions. I have shown there is a very good agreement with the length of two-body correlations between a many body Bose system and an exact Fermi system in one dimension. I have also calculated the energy per particle of the Bose system and found that as we approach the Tonk–Girardeau regime of impenetrable bosons that this energy is comparable to the kinetic energy of a non-interacting Fermi gas. The point-like interaction or hard-core condition acts like a pseudo-Pauli exclusion principle in the Bose system. As we decrease the density the interaction potential becomes negligible and we find an almost direct agreement with the kinetic energy of the Bose fluid and the kinetic energy of a system of non-interacting fermions. In high density regimes we find that we have a direct agreement with the mean field approximation and as we tend to low density systems we see that our energy levels cross over to those determined by Fermi statistics. The results gained within the hypernetted-chain scheme show a direct relationship with those gained within the framework of the Quantum Monte Carlo simulations for the same system of impenetrable bosons. We go on to develop further analytical models in one-dimension, including a ring of spinless fermions or Josephson Junctions and to calculate ground state energies and subsequent currents induced. Here we find Aharonov–Bohm oscillations in our one-dimensional investigation dependant on the magnetic flux applied to the systems studied.