Odd-Frequency Superfluidity in the Fermionic Tonks-Girardeau Gas

The fermionic Tonks-Girardeau (FTG) gas is known to exhibit conventional superfluidity, but may also be a vessel for unconventional odd-frequency superfluidity that is not revealed when studying only equal-time pairing correlations. In this thesis, we investigate odd-frequency superfluidity in the FTG gas. Specifically, this system is explored as a possible example of situations where odd-frequency pairing arises alongside conventional, even-frequency superfluidity. Previous works have extensively studied macroscopic quantum phenomena in the FTG gas via reduced density matrices, but this work is the first to investigate the hidden dynamical pairing in this system. We search for even fermion-exchange symmetry in the time-dependent two-body correlation matrix (T2bCM) that was formalised and shown to be synonymous with odd-frequency pair correlations by previous work. The eigenfunctions and eigenvalues of this matrix are studied for the homogeneous and harmonically trapped FTG gas. In the homogeneous case, it is found that the T2bCM eigenfunction is entirely antisymmetric under exchange of fermion coordinates and therefore no symmetric pairing occurs and no odd-frequency superfluidity exists within the system. Furthermore, the conventional macroscopic order present at zero time continues to exist at finite values of the pairing-time difference, but the magnitude of the macroscopic eigenvalue decreases monotonically with this time parameter. In the harmonically trapped case, the T2bCM eigenfunction contains a finite symmetric component that becomes concentrated at the boundary of the system for a large number of particles. It is found that, like the untrapped scenario, the macroscopic eigenvalue is finite and increases with particle number. We then consider a symmetric pairing order parameter and find that it decays with the particle number according to a power law, indicating that the harmonically trapped FTG gas exhibits non-macroscopic interface-localised odd-frequency order.