Renormalisation Group for Dense Matter

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Abstract

This thesis presents a theoretical study of homogeneous Bose and Fermi gases in their superfluid phases in one, two and three dimensions. This quantum many-body problem is formulated as a field theory and it is treated nonperturbatively using the functional renormalisation group (FRG) approach. We propose the use of phase fields at low momenta, which describe massless Goldstone fluctuations around the local condensate. This thesis starts with a study of the classical O(2)-model in two and three dimensions, which describes weakly-interacting Bose gases near the superfluid phase transition. We propose a parametrisation for the boson fields which interpolates between a Cartesian representation for high-momentum scales and an amplitude-phase one for low-momentum scales. We refer to this parametrisation as the interpolating representation. The response of the superfluid and boson densities under this parametrisation is studied to test its validity. The results show that the use of the interpolating representation improves the treatment of Goldstone fluctuations at low momenta and stabilises the RG flow in low dimensions. Having demonstrated the usefulness of the interpolating representation, we then apply this approach to the study of the thermodynamics of weakly-interacting Bose gases in one, two and three dimensions. In addition, we propose a novel method to calculate the pressure, which can be difficult to extract from the RG flow. Our results at zero temperature are in good agreement with both analytic results and Monte Carlo simulations. In particular, we obtain stable results in one dimension. This has not been previously achieved with the FRG. At finite temperatures, we are able to give a good description of the three-dimensional gas. In contrast, our calculations in two dimensions are not accurate in the region near the phase transition due to missing vortex physics. Despite this, the flows are stable and recover a physical superfluid phase in contrast to previous FRG studies. Finally, we study Fermi gases in two and three dimensions in the strongly-interacting BCS-BEC crossover regime at zero temperature. Boson fields, which represent pairs of fermions, are introduced to the theory through a Hubbard-Stratonovich transformation. The boson fields are described using the interpolating representation. This improves the description of Fermi gases in two dimensions. We obtain a good qualitative description of Fermi gases at zero temperature, showing the advantage of using the interpolating representation in fermionic systems.

Keyword(s)

Cold atoms; Quantum Gases; Renormalisation Group

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