Effective theories of phase transitions

In this thesis we study systems undergoing a superfluid phase transition at finite temperature and chemical potential. We construct an effective description valid at late times and long wavelengths, using both the holographic duality and the Schwinger-Keldysh formalism for non-equilibrium field theories. In particular, in chapter 2 we employ analytic techniques to find the leading dissipative corrections to the energy-momentum tensor and the electric current of a holographic superfluid, away from criticality. Our method is based on the symplectic current of Crnkovic and Witten [1] and extends on previous results [2, 3]. We assume a general black hole background in the bulk, with finite charge density and scalars fields turned on. We express one-point functions of the boundary field theory solely in terms of thermodynamic quantities and data related to the black hole horizon in the bulk spacetime. Matching our results with the expected constitutive relations of superfluid hydrodynamics, we obtain analytic expressions for the five transport coefficients characterising superfluids with small superfluid velocities. In chapter 3 we examine the hydrodynamics of holographic superfluids arbitrarily close to the critical point. The main difference in this case is that, close to the critical point, the amplitude of the order parameter is an additional hydrodynamic degree of freedom and we have to include it in our effective theory. For simplicity, we choose to work in the probe limit. Utilising the symplectic current once again, we find the equations that govern the critical dynamics of the order parameter and the charge density and show that our holographic results are in complete agreement with Model F of Hohenberg and Halperin [4]. Through this process, we find analytic expressions for all the parameters of Model F, including the dissipative kinetic coefficient, in terms of thermodynamics and horizon data. In addition, we perform various numerical checks of our analytic results. Finally, in chapter 4 we consider critical superfluid dynamics within the Schwinger-Keldysh formalism. As in chapter 3, we focus on the complex order parameter and the conserved current of the spontaneously broken global symmetry, ignoring temperature and normal fluid velocity fluctuations. We construct an effective action up to second order in the a-fields and compare the resulting stochastic system with Model F and our holographic results in chapter 3. A crucial role in this construction is played by a time independent gauge symmetry, called “chemical shift symmetry”. We also integrate out the amplitude mode and obtain the conventional equations of superfluid hydrodynamics, valid for energies well below the gap of the amplitude mode.