Abstract
Extending the quantum effective approach of Son and Nicolis and incorporating dissipation, we develop a MIS formalism for describing a superfluid out of equilibrium by including the Goldstone boson and the condensate together with the hydrodynamic modes as the effective degrees of freedom. We find that the evolution of the superfluid undergoing Bjorken flow is governed by the conventional hydrodynamic attractor with unbroken symmetry and an even number of novel non-dissipative fixed points with broken symmetry. If the initial temperature is super-critical, then the condensate becomes exponentially small very rapidly and the system is trapped by the hydrodynamic attractor for a long intermediate time before it reheats rapidly and switches to one of the symmetry-breaking fixed points eventually. Finally, we show that the fixed points are unstable against inhomogeneous perturbations that should lead to spinodal decomposition. We conclude that these features should be generic beyond the MIS formalism.
Highlights
Phase transitions in out-of-equilibrium systems are of interest for the physics of heavy-ion collisions [1,2], cold atoms [3], and the early Universe [4]
Our work demonstrates that the evolution of superfluid matter out of equilibrium is governed by the conventional hydrodynamic attractor with unbroken symmetry and nontrivial symmetry-breaking fixed points which are determined by the potential of the condensate and the equation of state
We set up the MIS formalism for superfluids, extending the quantum effective action approach, and study the Bjorken flow
Summary
Phase transitions in out-of-equilibrium systems are of interest for the physics of heavy-ion collisions [1,2], cold atoms [3], and the early Universe [4]. Energy-momentum tensor and conserved currents even in the presence of large pressure gradients [5,6,7,8] It is, pertinent to ask if one can construct similar frameworks that incorporate Goldstone boson(s) and order parameter(s) along with hydrodynamic modes for superfluid states far from equilibrium. We include the order parameter and add dissipation by merging the Son-Nicolis framework with the Müller-Israel-Stewart (MIS) formalism [17,18], enabling a causal description We use this to study symmetry breaking in out-of-equilibrium situations in the context of Bjorken flow. Our work demonstrates that the evolution of superfluid matter out of equilibrium is governed by the conventional hydrodynamic attractor with unbroken symmetry (which is rather like a saddle curve in the extended phase space) and nontrivial symmetry-breaking fixed points which are determined by the potential of the condensate and the equation of state.
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