Abstract
We investigate the low-energy dynamics of systems with pseudo-spontaneously broken U(1) symmetry and Goldstone phase relaxation. We construct a hydrodynamic framework which is able to capture these, in principle independent, effects. We consider two generalisations of the standard holographic superfluid model by adding an explicit breaking of the U(1) symmetry by either sourcing the charged bulk scalar or by introducing an explicit mass term for the bulk gauge field. We find agreement between the hydrodynamic dispersion relations and the quasi-normal modes of both holographic models. We verify that phase relaxation arises only due to the breaking of the inherent Goldstone shift symmetry. The interplay of a weak explicit breaking of the U(1) and phase relaxation renders the DC electric conductivity finite but does not result in a Drude-like peak. In this scenario we show the validity of a universal relation, found in the context of translational symmetry breaking, between the phase relaxation rate, the mass of the pseudo-Goldstone and the Goldstone diffusivity.
Highlights
Symmetries may be broken; we will concern ourselves with aspects of spontaneous symmetry breaking and explicit symmetry breaking
Several physical phenomena, such as superfluidity and elasticity, arise as a consequence of spontaneous symmetry breaking [9]; in their hydrodynamic descriptions, the conservation equations are supplemented by the so-called Josephson relation, which governs the dynamics of the Goldstone boson
Physical situations arise where spontaneous symmetry breaking occurs in conjunction with explicit symmetry breaking; if the parameter causing the explicit breaking is small compared to the amount of spontaneous breaking, the total breaking is said to be pseudo-spontaneous [17]
Summary
Hydrodynamics is an effective theory which describes the long-wavelength, late-time dynamics of a system. We will consider relativistic hydrodynamics and its applications to quantum field theory, where a spontaneously broken global U(1) symmetry gives rise to superfluidity. Superfluids support propagating modes even when the dynamics of the normal component is frozen, i.e. when the energy-momentum tensor and fluctuations of the temperature and normal fluid velocity are neglected — we will refer to this regime as the probe limit. For simplicity and for applications to holography, we will only consider probe-limit hydrodynamics — some further results beyond the probe limit may be found in appendix A. We will consider a superfluid in two spatial dimensions. Greek indices span the full spacetime components while Latin indices cover the spatial components
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