| Summary: | This thesis is mostly based on the research presented in [1, 2, 3]. We introduce a novel efficient framework to treat infinite periodic vortex lattices in rotating superfluids under a mean-field Gross-Pitaevskii description. In doing so, we introduce a generalisation of the Fourier transform which correctly diagonalises the kinetic energy terms while respecting the required twisted boundary conditions. We call this integral transform a Magnetic Fourier transform. Testing the method, we re-obtain known results in the lowest-Landau-level regime, and further extend to stronger interacting regimes. We provide an extension of the above method to treat multicomponent systems, demonstrating that new degrees of freedom need to be introduced for each new component. We then employ this method to investigate the ground states of binary superfluid systems whose constituents have equal masses, thereby extending previous work carried out in the lowest-Landau-level limit to arbitrary interactions within Gross-Pitaevskii theory. In particular, we find that the interactions depauperate the phase diagram, with only the triangular lattice phase surviving in the limit of strong interactions. Withal we prove this applies regardless of the mass ratio of the constituents. We further investigate binary superfluid systems with non unitary mass ratios, obtaining a range of novel and exotic vortex lattice configurations. Finally we derive a linear relation which accurately describes the phase boundaries in the strong interaction regime.
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