Effective theories of phase
transitions

Abstract

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.