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.
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References
R. Penco, An Introduction to Effective Field Theories, arXiv:2006.16285 [INSPIRE].
A. Beekman, L. Rademaker and J. van Wezel, An introduction to spontaneous symmetry breaking, SciPost Phys. Lect. Notes (2019).
C.P. Burgess, Goldstone and pseudo Goldstone bosons in nuclear, particle and condensed matter physics, Phys. Rept. 330 (2000) 193 [hep-th/9808176] [INSPIRE].
P. Hohenberg and A. Krekhov, An introduction to the ginzburg-landau theory of phase transitions and nonequilibrium patterns, Phys. Rept. 572 (2015) 1.
A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff, JHEP 06 (2015) 155 [arXiv:1501.03845] [INSPIRE].
Y. Nambu, Quasiparticles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117 (1960) 648 [INSPIRE].
J. Goldstone, Field Theories with “Superconductor” Solutions, Nuovo Cim. 19 (1961) 154 [INSPIRE].
H. Leutwyler, Phonons as goldstone bosons, Helv. Phys. Acta 70 (1997) 275 [hep-ph/9609466] [INSPIRE].
P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, (1995), [DOI].
H. Watanabe and H. Murayama, Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance, Phys. Rev. Lett. 108 (2012) 251602 [arXiv:1203.0609] [INSPIRE].
Y. Hidaka, Counting rule for Nambu-Goldstone modes in nonrelativistic systems, Phys. Rev. Lett. 110 (2013) 091601 [arXiv:1203.1494] [INSPIRE].
I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].
Y. Minami and Y. Hidaka, Spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems, Phys. Rev. E 97 (2018) 012130 [arXiv:1509.05042] [INSPIRE].
M. Hongo, S. Kim, T. Noumi and A. Ota, Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems, Phys. Rev. D 103 (2021) 056020 [arXiv:1907.08609] [INSPIRE].
Y. Hidaka, Y. Hirono and R. Yokokura, Counting Nambu-Goldstone Modes of Higher-Form Global Symmetries, Phys. Rev. Lett. 126 (2021) 071601 [arXiv:2007.15901] [INSPIRE].
D. Hofman and N. Iqbal, Goldstone modes and photonization for higher form symmetries, SciPost Phys. 6 (2019) 006.
S. Weinberg, Approximate symmetries and pseudoGoldstone bosons, Phys. Rev. Lett. 29 (1972) 1698 [INSPIRE].
M. Gell-Mann, R.J. Oakes and B. Renner, Behavior of current divergences under SU(3) × SU(3), Phys. Rev. 175 (1968) 2195 [INSPIRE].
J.P. Boon and S. Yip, Molecular hydrodynamics, McGraw-Hill (1980).
S. Grozdanov, A. Lucas and N. Poovuttikul, Holography and hydrodynamics with weakly broken symmetries, Phys. Rev. D 99 (2019) 086012 [arXiv:1810.10016] [INSPIRE].
S. Grozdanov, D.M. Hofman and N. Iqbal, Generalized global symmetries and dissipative magnetohydrodynamics, Phys. Rev. D 95 (2017) 096003 [arXiv:1610.07392] [INSPIRE].
S. Grozdanov and N. Poovuttikul, Generalized global symmetries in states with dynamical defects: The case of the transverse sound in field theory and holography, Phys. Rev. D 97 (2018) 106005 [arXiv:1801.03199] [INSPIRE].
L.V. Delacrétaz, D.M. Hofman and G. Mathys, Superfluids as Higher-form Anomalies, SciPost Phys. 8 (2020) 047 [arXiv:1908.06977] [INSPIRE].
M. Baggioli, M. Vasin, V.V. Brazhkin and K. Trachenko, Gapped momentum states, Phys. Rept. 865 (2020) 1 [arXiv:1904.01419] [INSPIRE].
M. Baggioli, M. Landry and A. Zaccone, Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory, Phys. Rev. E 105 (2022) 024602 [arXiv:2101.05015] [INSPIRE].
L.V. Delacrétaz, B. Goutéraux, S.A. Hartnoll and A. Karlsson, Theory of hydrodynamic transport in fluctuating electronic charge density wave states, Phys. Rev. B 96 (2017) 195128 [arXiv:1702.05104] [INSPIRE].
B.I. Halperin and D.R. Nelson, Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978) 121.
J. Bardeen and M.J. Stephen, Theory of the Motion of Vortices in Superconductors, Phys. Rev. 140 (1965) A1197 [INSPIRE].
B.I. Halperin and D.R. Nelson, Resistive transition in superconducting films, J. Low Temp. Phys. 36 (1979) 599.
R.A. Davison, L.V. Delacrétaz, B. Goutéraux and S.A. Hartnoll, Hydrodynamic theory of quantum fluctuating superconductivity, Phys. Rev. B 94 (2016) 054502 [Erratum ibid. 96 (2017) 059902] [arXiv:1602.08171] [INSPIRE].
A. Amoretti, D. Areán, B. Goutéraux and D. Musso, Universal relaxation in a holographic metallic density wave phase, Phys. Rev. Lett. 123 (2019) 211602 [arXiv:1812.08118] [INSPIRE].
M. Baggioli and S. Grieninger, Zoology of solid & fluid holography — Goldstone modes and phase relaxation, JHEP 10 (2019) 235 [arXiv:1905.09488] [INSPIRE].
M. Ammon, M. Baggioli and A. Jiménez-Alba, A Unified Description of Translational Symmetry Breaking in Holography, JHEP 09 (2019) 124 [arXiv:1904.05785] [INSPIRE].
T. Andrade and A. Krikun, Coherent vs incoherent transport in holographic strange insulators, JHEP 05 (2019) 119 [arXiv:1812.08132] [INSPIRE].
A. Donos, D. Martin, C. Pantelidou and V. Ziogas, Hydrodynamics of broken global symmetries in the bulk, JHEP 10 (2019) 218 [arXiv:1905.00398] [INSPIRE].
A. Donos, D. Martin, C. Pantelidou and V. Ziogas, Incoherent hydrodynamics and density waves, Class. Quant. Grav. 37 (2020) 045005 [arXiv:1906.03132] [INSPIRE].
A. Amoretti, D. Areán, D.K. Brattan and N. Magnoli, Hydrodynamic magneto-transport in charge density wave states, JHEP 05 (2021) 027 [arXiv:2101.05343] [INSPIRE].
T. Andrade, M. Baggioli and A. Krikun, Phase relaxation and pattern formation in holographic gapless charge density waves, JHEP 03 (2021) 292 [arXiv:2009.05551] [INSPIRE].
M. Baggioli, Homogeneous holographic viscoelastic models and quasicrystals, Phys. Rev. Res. 2 (2020) 022022 [arXiv:2001.06228] [INSPIRE].
M. Baggioli and M. Landry, Effective Field Theory for Quasicrystals and Phasons Dynamics, SciPost Phys. 9 (2020) 062 [arXiv:2008.05339] [INSPIRE].
E. Grossi, A. Soloviev, D. Teaney and F. Yan, Transport and hydrodynamics in the chiral limit, Phys. Rev. D 102 (2020) 014042 [arXiv:2005.02885] [INSPIRE].
C.P. Herzog, N. Lisker, P. Surowka and A. Yarom, Transport in holographic superfluids, JHEP 08 (2011) 052 [arXiv:1101.3330] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
R. Argurio, A. Marzolla, A. Mezzalira and D. Musso, Analytic pseudo-Goldstone bosons, JHEP 03 (2016) 012 [arXiv:1512.03750] [INSPIRE].
A. Donos, P. Kailidis and C. Pantelidou, Dissipation in holographic superfluids, JHEP 09 (2021) 134 [arXiv:2107.03680] [INSPIRE].
A. Jimenez-Alba, K. Landsteiner, Y. Liu and Y.-W. Sun, Anomalous magnetoconductivity and relaxation times in holography, JHEP 07 (2015) 117 [arXiv:1504.06566] [INSPIRE].
A. Jimenez-Alba, K. Landsteiner and L. Melgar, Anomalous magnetoresponse and the Stückelberg axion in holography, Phys. Rev. D 90 (2014) 126004 [arXiv:1407.8162] [INSPIRE].
D.T. Son and M.A. Stephanov, Real time pion propagation in finite temperature QCD, Phys. Rev. D 66 (2002) 076011 [hep-ph/0204226] [INSPIRE].
E. Grossi, A. Soloviev, D. Teaney and F. Yan, Soft pions and transport near the chiral critical point, Phys. Rev. D 104 (2021) 034025 [arXiv:2101.10847] [INSPIRE].
A. Florio, E. Grossi, A. Soloviev and D. Teaney, Dynamics of the O(4) critical point in QCD, arXiv:2111.03640 [INSPIRE].
G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60 (1988) 1129 [INSPIRE].
L.V. Delacrétaz, B. Goutéraux and V. Ziogas, Damping of Pseudo-Goldstone Fields, arXiv:2111.13459 [INSPIRE].
C.P. Enz, Two-fluid hydrodynamic description of ordered systems, Rev. Mod. Phys. 46 (1974) 705 [INSPIRE].
R.J. Donnelly, The two-fluid theory and second sound in liquid helium, Phys. Today 62 (2009) 34.
D. Areán, M. Baggioli, S. Grieninger and K. Landsteiner, A holographic superfluid symphony, JHEP 11 (2021) 206 [arXiv:2107.08802] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
A. Amoretti, D. Areán, R. Argurio, D. Musso and L.A. Pando Zayas, A holographic perspective on phonons and pseudo-phonons, JHEP 05 (2017) 051 [arXiv:1611.09344] [INSPIRE].
I. Amado, M. Kaminski and K. Landsteiner, Hydrodynamics of Holographic Superconductors, JHEP 05 (2009) 021 [arXiv:0903.2209] [INSPIRE].
P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal. 251 (2007) 573.
G. Gaeta, Bifurcation and symmetry breaking, Phys. Rept. 189 (1990) 1 [INSPIRE].
T. Vojta, Phases and phase transitions in disordered quantum systems, AIP Conf. Proc. 1550 (2013) 188.
D. Areán, L.A. Pando Zayas, I.S. Landea and A. Scardicchio, Holographic disorder driven superconductor-metal transition, Phys. Rev. D 94 (2016) 106003 [arXiv:1507.02280] [INSPIRE].
M. Ammon, M. Baggioli, A. Jiménez-Alba and S. Moeckel, A smeared quantum phase transition in disordered holography, JHEP 04 (2018) 068 [arXiv:1802.08650] [INSPIRE].
L. Alberte, M. Ammon, M. Baggioli, A. Jiménez and O. Pujolàs, Black hole elasticity and gapped transverse phonons in holography, JHEP 01 (2018) 129 [arXiv:1708.08477] [INSPIRE].
A. Donos and C. Pantelidou, Holographic transport and density waves, JHEP 05 (2019) 079 [arXiv:1903.05114] [INSPIRE].
T. Andrade, M. Baggioli, A. Krikun and N. Poovuttikul, Pinning of longitudinal phonons in holographic spontaneous helices, JHEP 02 (2018) 085 [arXiv:1708.08306] [INSPIRE].
P. Phillips and D. Dalidovich, The elusive bose metal, Science 302 (2003) 243.
Y. Korovin, K. Skenderis and M. Taylor, Lifshitz as a deformation of Anti-de Sitter, JHEP 08 (2013) 026 [arXiv:1304.7776] [INSPIRE].
Y. Korovin, K. Skenderis and M. Taylor, Lifshitz from AdS at finite temperature and top down models, JHEP 11 (2013) 127 [arXiv:1306.3344] [INSPIRE].
M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].
L. Alberte, M. Baggioli, A. Khmelnitsky and O. Pujolàs, Solid Holography and Massive Gravity, JHEP 02 (2016) 114 [arXiv:1510.09089] [INSPIRE].
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
I.R. Klebanov, P. Ouyang and E. Witten, A Gravity dual of the chiral anomaly, Phys. Rev. D 65 (2002) 105007 [hep-th/0202056] [INSPIRE].
I. Iatrakis, S. Lin and Y. Yin, The anomalous transport of axial charge: topological vs non-topological fluctuations, JHEP 09 (2015) 030 [arXiv:1506.01384] [INSPIRE].
F. Bigazzi, A.L. Cotrone and F. Porri, Universality of the Chern-Simons diffusion rate, Phys. Rev. D 98 (2018) 106023 [arXiv:1804.09942] [INSPIRE].
I. Iatrakis, S. Lin and Y. Yin, Axial current generation by P-odd domains in QCD matter, Phys. Rev. Lett. 114 (2015) 252301 [arXiv:1411.2863] [INSPIRE].
K. Fukushima, D.E. Kharzeev and H.J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D 78 (2008) 074033 [arXiv:0808.3382] [INSPIRE].
R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
M. Baggioli and K. Trachenko, Low frequency propagating shear waves in holographic liquids, JHEP 03 (2019) 093 [arXiv:1807.10530] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
M. Baggioli and O. Pujolàs, Electron-Phonon Interactions, Metal-Insulator Transitions, and Holographic Massive Gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
M. Baggioli and K. Trachenko, Maxwell interpolation and close similarities between liquids and holographic models, Phys. Rev. D 99 (2019) 106002 [arXiv:1808.05391] [INSPIRE].
M. Baggioli, How small hydrodynamics can go, Phys. Rev. D 103 (2021) 086001 [arXiv:2010.05916] [INSPIRE].
M. Baggioli, U. Gran, A.J. Alba, M. Tornsö and T. Zingg, Holographic Plasmon Relaxation with and without Broken Translations, JHEP 09 (2019) 013 [arXiv:1905.00804] [INSPIRE].
S. Mondkar, A. Mukhopadhyay, A. Rebhan and A. Soloviev, Quasinormal modes of a semi-holographic black brane and thermalization, JHEP 11 (2021) 080 [arXiv:2108.02788] [INSPIRE].
R.A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, JHEP 01 (2015) 039 [arXiv:1411.1062] [INSPIRE].
M. Stephanov, H.-U. Yee and Y. Yin, Collective modes of chiral kinetic theory in a magnetic field, Phys. Rev. D 91 (2015) 125014 [arXiv:1501.00222] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, The complex life of hydrodynamic modes, JHEP 11 (2019) 097 [arXiv:1904.12862] [INSPIRE].
D. Areán, R.A. Davison, B. Goutéraux and K. Suzuki, Hydrodynamic Diffusion and Its Breakdown near AdS2 Quantum Critical Points, Phys. Rev. X 11 (2021) 031024 [arXiv:2011.12301] [INSPIRE].
N. Wu, M. Baggioli and W.-J. Li, On the universality of AdS2 diffusion bounds and the breakdown of linearized hydrodynamics, JHEP 05 (2021) 014 [arXiv:2102.05810] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, The breakdown of magneto-hydrodynamics near AdS2 fixed point and energy diffusion bound, JHEP 02 (2022) 006 [arXiv:2105.03882] [INSPIRE].
Y. Liu and X.-M. Wu, Breakdown of hydrodynamics from holographic pole collision, JHEP 01 (2022) 155 [arXiv:2111.07770] [INSPIRE].
K.-B. Huh, H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Upper bound of the charge diffusion constant in holography, arXiv:2111.07515 [INSPIRE].
T. Hartman, S.A. Hartnoll and R. Mahajan, Upper Bound on Diffusivity, Phys. Rev. Lett. 119 (2017) 141601 [arXiv:1706.00019] [INSPIRE].
M. Baggioli and W.-J. Li, Universal Bounds on Transport in Holographic Systems with Broken Translations, SciPost Phys. 9 (2020) 007 [arXiv:2005.06482] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, Convergence of the Gradient Expansion in Hydrodynamics, Phys. Rev. Lett. 122 (2019) 251601 [arXiv:1904.01018] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
M. Baggioli, K.-Y. Kim, L. Li and W.-J. Li, Holographic Axion Model: a simple gravitational tool for quantum matter, Sci. China Phys. Mech. Astron. 64 (2021) 270001 [arXiv:2101.01892] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
A. Amoretti, D. Areán, B. Goutéraux and D. Musso, Gapless and gapped holographic phonons, JHEP 01 (2020) 058 [arXiv:1910.11330] [INSPIRE].
R. Currat, E. Kats and I. Luk’yanchuk, Sound modes in composite incommensurate crystals, Eur. Phys. J. B 26 (2002) 339.
J. Ollivier, C. Ecolivet, S. Beaufils, F. Guillaume and T. Breczewski, Light scattering by low-frequency excitations in quasi-periodic n-alkane/urea adducts, EPL 43 (1998) 546.
B. Toudic et al., Mixed acoustic phonons and phase modes in an aperiodic composite crystal, Phys. Rev. Lett. 107 (2011) 205502.
J.I. Kapusta, E. Rrapaj and S. Rudaz, Sphaleron transition rates and the chiral magnetic effect, Int. J. Mod. Phys. E 31 (2022) 2250010 [arXiv:2012.13784] [INSPIRE].
I.A. Shovkovy, D.O. Rybalka and E.V. Gorbar, The overdamped chiral magnetic wave, PoS Confinement2018 (2018) 029 [arXiv:1811.10635] [INSPIRE].
S. Grieninger and A. Shukla, Second order equilibrium transport in strongly coupled \( \mathcal{N} \) = 4 supersymmetric SU(Nc) Yang-Mills plasma via holography, JHEP 08 (2021) 108 [arXiv:2105.08673] [INSPIRE].
P. Kovtun and A. Shukla, Kubo formulas for thermodynamic transport coefficients, JHEP 10 (2018) 007 [arXiv:1806.05774] [INSPIRE].
E. Megias and O. Pujolàs, Naturally light dilatons from nearly marginal deformations, JHEP 08 (2014) 081 [arXiv:1401.4998] [INSPIRE].
D. Elander, M. Piai and J. Roughley, Probing the holographic dilaton, JHEP 06 (2020) 177 [Erratum ibid. 12 (2020) 109] [arXiv:2004.05656] [INSPIRE].
D. Elander and M. Piai, The decay constant of the holographic techni-dilaton and the 125 GeV boson, Nucl. Phys. B 867 (2013) 779 [arXiv:1208.0546] [INSPIRE].
B.A. Campbell, J. Ellis and K.A. Olive, Phenomenology and Cosmology of an Electroweak Pseudo-Dilaton and Electroweak Baryons, JHEP 03 (2012) 026 [arXiv:1111.4495] [INSPIRE].
R. Contino, Y. Nomura and A. Pomarol, Higgs as a holographic pseudoGoldstone boson, Nucl. Phys. B 671 (2003) 148 [hep-ph/0306259] [INSPIRE].
J.G. Rau, P.A. McClarty and R. Moessner, Pseudo-Goldstone Gaps and Order-by-Quantum Disorder in Frustrated Magnets, Phys. Rev. Lett. 121 (2018) 237201 [arXiv:1805.00947] [INSPIRE].
I. Amado, D. Areán, A. Jimenez-Alba, K. Landsteiner, L. Melgar and I.S. Landea, Holographic Type II Goldstone bosons, JHEP 07 (2013) 108 [arXiv:1302.5641] [INSPIRE].
M. Baggioli, S. Grieninger and L. Li, Magnetophonons & type-B Goldstones from Hydrodynamics to Holography, JHEP 09 (2020) 037 [arXiv:2005.01725] [INSPIRE].
A. Donos, C. Pantelidou and V. Ziogas, Incoherent hydrodynamics of density waves in magnetic fields, JHEP 05 (2021) 270 [arXiv:2101.06230] [INSPIRE].
A. Amoretti, D. Areán, D.K. Brattan and L. Martinoia, Hydrodynamic magneto-transport in holographic charge density wave states, JHEP 11 (2021) 011 [arXiv:2107.00519] [INSPIRE].
S.L. Grieninger, Non-equilibrium dynamics in Holography, Ph.D. thesis, Jena University, Germany (2020). arXiv:2012.10109 [DOI] [INSPIRE].
J.K. Ghosh, S. Grieninger, K. Landsteiner and S. Morales-Tejera, Is the chiral magnetic effect fast enough?, Phys. Rev. D 104 (2021) 046009 [arXiv:2105.05855] [INSPIRE].
M. Ammon, S. Grieninger, A. Jimenez-Alba, R.P. Macedo and L. Melgar, Holographic quenches and anomalous transport, JHEP 09 (2016) 131 [arXiv:1607.06817] [INSPIRE].
J.P. Boyd, Chebyshev and Fourier Spectral Methods (Second Edition, Revised), Dover Publications, New York, U.S.A. (2001).
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Ammon, M., Areán, D., Baggioli, M. et al. Pseudo-spontaneous U(1) symmetry breaking in hydrodynamics and holography. J. High Energ. Phys. 2022, 15 (2022). https://doi.org/10.1007/JHEP03(2022)015
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DOI: https://doi.org/10.1007/JHEP03(2022)015