Pseudo-spontaneous $U(1)$ Symmetry Breaking in Hydrodynamics and Holography

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.

Approximate symmetries, pseudo-Goldstones, and the second law of thermodynamics

“…Looking at the mode spectrum, we obtain a damped sound mode in the longitudinal and transverse sectors similar to (8). We also find a crystal diffusion mode in the longitudinal sector.…”

“…Pseudo-spontaneous symmetry breaking is common across the phase space of matter, due to inherent defects, inhomogeneities, and impurities in materials. Examples include pinned crystals [1,2], charge density waves [3][4][5][6], pinned superfluids [7,8], electrons in graphene [9], pinned nematics [10,11], and pions in chiral perturbation theory [12][13][14], among many others.…”

“…where the speed of sound v s can be read off using the singularity structure of the dispersion relations (8). This gives λ a concrete physical meaning.…”

“…The discussion can be extended to account for temperature fluctuations and conserved momenta, leading to a theory of explicitly broken superfluids. This theory was recently considered in [8], but a careful analysis of the second law constraints was not carried out. In particular, the authors missed λ and similar coefficients that can appear in the energy flux and stress tensor.…”

Damping of Pseudo-Goldstone Fields

“…Damping of pseudo-Goldstones at finite temperature has been studied for some time [2][3][4][5][6][7][8][9][10][11][12][13], although the relation (1) was not recognized at the time. Instead, to the best of our knowledge, (1) was first demonstrated in a Gauge/Gravity duality model of translation symmetry breaking, [14], see also for subsequent studies [15][16][17][18][19][20][21][22][23][24][25][26][27]. (1) was also noted in the hydrodynamic description of soft pions in [28].…”