Gapless and gapped holographic phonons
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Springer
Received: November 5, 2019 Accepted: December 23, 2019 Published: January 10, 2020
Gapless and gapped holographic phonons
a
Dipartimento di Fisica, Universit` a di Genova and I.N.F.N. — Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy b Physique Th´eorique et Math´ematique and International Solvay Institutes, Universit´e Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium c Instituto de F´ısica Te´ orica, UAM/CSIC, Calle Nicol´ as Cabrera 13–15, Cantoblanco, 28049 Madrid, Spain d CPHT, CNRS, Ecole polytechnique, IP Paris, Route de Saclay, F-91128 Palaiseau, France e Universidade de Santiago de Compostela (USC) and Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), R´ ua de Xoaqu´ın D´ıaz de R´ abago, Campus Vida, 15705 Santiago de Compostela, A Coru˜ na, Spain
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We study a holographic model where translations are both spontaneously and explicitly broken, leading to the presence of (pseudo)-phonons in the spectrum. The weak explicit breaking is due to two independent mechanisms: a small source for the condensate itself and additional linearly space-dependent marginal operators. The low energy dynamics of the model is described by Wigner crystal hydrodynamics. In absence of a source for the condensate, the phonons remain gapless, but momentum is relaxed. Turning on a source for the condensate damps and pins the phonons. Finally, we verify that the universal relation between the phonon damping rate, mass and diffusivity reported in [1] continues to hold in this model for weak enough explicit breaking. Keywords: Effective Field Theories, Holography and condensed matter physics (AdS/CMT), Space-Time Symmetries, Spontaneous Symmetry Breaking ArXiv ePrint: 1910.11330
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)058
JHEP01(2020)058
Andrea Amoretti,a,b Daniel Are´ an,c,d Blaise Gout´ erauxd and Daniele Mussoe
Contents 1 Introduction
1
2 The holographic model 2.1 Background geometry and thermodynamics
4 6 7 7 9 11
4 Gapped phonons 4.1 Spectrum at zero wavevector 4.2 Transverse spectrum at nonzero wavevector
13 13 16
5 Outlook
16
1
Introduction
When translations are spontaneously broken in an otherwise translation invariant system, new gapless modes appear in the spectrum. These are the Nambu-Goldstone bosons of broken translations (which we will call phonons, by a slight abuse of terminology). The low energy effective theory describing the dynamics around equilibrium at long wavelengths and late times is that of Wigner crystals, see [2] for a review. The phonons have a dramatic impact on the transverse spectrum: the transverse phonon mixes with transverse momentum to give two shear sound modes s ωshear = ±
G i 2 η q − q ξ⊥ + + O(q 3 ) , χP P 2 χP P
(1.1)
with a velocity proportional to the square root of the elastic shear modulus G and an attenuation controlled by the shear viscosity η and the transverse p
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