Non-vanishing zero-temperature normal density in holographic superfluids
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Springer
Received: September 11, 2020 Accepted: October 6, 2020 Published: November 18, 2020
Blaise Goutéraux and Eric Mefford CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 PALAISEAU, France
E-mail: [email protected], [email protected] Abstract: The low energy and finite temperature excitations of a d + 1-dimensional system exhibiting superfluidity are well described by a hydrodynamic model with two fluid flows: a normal flow and a superfluid flow. In the vicinity of a quantum critical point, thermodynamics and transport in the system are expected to be controlled by the critical exponents and by the spectrum of irrelevant deformations away from the quantum critical point. Here, using gauge-gravity duality, we present the low temperature dependence of thermodynamic and charge transport coefficients at first order in the hydrodynamic derivative expansion in terms of the critical exponents. Special attention will be paid to the behavior of the charge density of the normal flow in systems with emergent infrared conformal and Lifshitz symmetries, parameterized by a Lifshitz dynamical exponent z > 1. When 1 ≤ z < d + 2, we recover (z = 1) and extend (z > 1) previous results obtained by relativistic effective field theory techniques. Instead, when z > d + 2, we show that the normal charge density becomes non-vanishing at zero temperature. An extended appendix generalizes these results to systems that violate hyperscaling as well as systems with generalized photon masses. Our results clarify previous work in the holographic literature and have relevance to recent experimental measurements of the superfluid density on cuprate superconductors. Keywords: Holography and condensed matter physics (AdS/CMT), AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2008.02289
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)091
JHEP11(2020)091
Non-vanishing zero-temperature normal density in holographic superfluids
Contents 1
2 Holographic superfluid model, generalities 2.1 IR geometries with emergent conformal invariance 2.2 Lifshitz symmetric IR geometries 2.3 Nonzero temperature 2.4 Horizon fluxes
4 5 6 8 9
3 Transport 3.1 Incoherent conductivity 3.2 Normal and superfluid densities (0) 3.2.1 Vanishing ρn (0) 3.2.2 Non-vanishing ρn 3.2.3 Competing broken symmetries
10 11 12 15 17 17
4 Numerical examples
18
5 Connection to previous literature
20
6 Discussion
22
A Landau-Tisza hydrodynamics 24 A.1 Conservation equations and constitutive relations for small superfluid velocities 24 A.2 Linear response a la Kadanoff-Martin 25 B Proof of (3.34)
28
C Scale-covariant geometries C.1 IR geometries C.1.1 Marginal deformations, z 6= 1 C.1.2 Irrelevant deformations, z = 1 C.2 Specific example
30 33 34 35 36
D Translation breaking
37
E Semi-local quantum critical geometries E.1 Zero temperature solution
40 42
(0)
F Vanishing ρ(0) n but finite ρin
43
–i–
JHEP11(2020)091
1 Introduction a
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