Abstract
We study the dynamics of spontaneous translation symmetry breaking in holographic models in presence of weak explicit sources. We show that, unlike conventional gapped quantum charge density wave systems, this dynamics is well characterized by the effective time dependent Ginzburg-Landau equation, both above and below the critical temperature, which leads to a “gapless” algebraic pattern of metal-insulator phase transition. In this framework we elucidate the nature of the damped Goldstone mode (the phason), which has earlier been identified in the effective hydrodynamic theory of pinned charge density wave and observed in holographic homogeneous lattice models. We follow the motion of the quasinormal modes across the dynamical phase transition in models with either periodic inhomogeneous or helical homogeneous spatial structures, showing that the phase relaxation rate is continuous at the critical temperature. Moreover, we find that the qualitative low-energy dynamics of the broken phase is universal, insensitive to the precise pattern of translation symmetry breaking, and therefore applies to homogeneous models as well.
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References
S.A. Kivelson et al., How to detect fluctuating stripes in the high-temperature superconductors, Rev. Mod. Phys. 75 (2003) 1201 [cond-mat/0210683] [INSPIRE].
M. Vojta, Lattice symmetry breaking in cuprate superconductors: stripes, nematics, and superconductivity, Adv. Phys. 58 (2009) 699.
E. Berg, E. Fradkin, S.A. Kivelson and J. Tranquada, Striped superconductors: How the cuprates intertwine spin, charge and superconducting orders, arXiv:0901.4826.
C.-W. Chen, J. Choe and E. Morosan, Charge density waves in strongly correlated electron systems, Rept. Prog. Phys. 79 (2016) 084505.
G. Gruner, The dynamics of charge-density waves, Rev. Mod. Phys. 60 (1988) 1129 [INSPIRE].
L.V. Delacrétaz, B. Goutéraux, S.A. Hartnoll and A. Karlsson, Bad Metals from Fluctuating Density Waves, SciPost Phys. 3 (2017) 025 [arXiv:1612.04381] [INSPIRE].
A. Amoretti, D. Areán, B. Goutéraux and D. Musso, DC resistivity of quantum critical, charge density wave states from gauge-gravity duality, Phys. Rev. Lett. 120 (2018) 171603 [arXiv:1712.07994] [INSPIRE].
T. Andrade, A. Krikun, K. Schalm and J. Zaanen, Doping the holographic Mott insulator, Nature Phys. 14 (2018) 1049 [arXiv:1710.05791] [INSPIRE].
R. Yusupov et al., Coherent dynamics of macroscopic electronic order through a symmetry breaking transition, Nature Phys. 6 (2010) 681.
H. Schaefer, V.V. Kabanov and J. Demsar, Collective modes in quasi-one-dimensional charge-density wave systems probed by femtosecond time-resolved optical studies, Phys. Rev. B 89 (2014) 045106.
M.D. Thomson, K. Rabia, F. Meng, M. Bykov, S. van Smaalen and H.G. Roskos, Phase-channel dynamics reveal the role of impurities and screening in a quasi-one-dimensional charge-density wave system, Sci. Rep. 7 (2017) 2039.
J. Zaanen, Y.-W. Sun, Y. Liu and K. Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press, (2015).
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
M. Baggioli, Applied Holography: A Practical Mini-Course, SpringerBriefs in Physics, Springer, (2019).
A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].
A. Donos, Striped phases from holography, JHEP 05 (2013) 059 [arXiv:1303.7211] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic charge density waves, Phys. Rev. D 87 (2013) 126008 [arXiv:1303.4398] [INSPIRE].
B. Withers, Black branes dual to striped phases, Class. Quant. Grav. 30 (2013) 155025 [arXiv:1304.0129] [INSPIRE].
B. Withers, The moduli space of striped black branes, arXiv:1304.2011 [INSPIRE].
S. Nakamura, H. Ooguri and C.-S. Park, Gravity Dual of Spatially Modulated Phase, Phys. Rev. D 81 (2010) 044018 [arXiv:0911.0679] [INSPIRE].
H. Ooguri and C.-S. Park, Holographic End-Point of Spatially Modulated Phase Transition, Phys. Rev. D 82 (2010) 126001 [arXiv:1007.3737] [INSPIRE].
A. Donos and J.P. Gauntlett, Black holes dual to helical current phases, Phys. Rev. D 86 (2012) 064010 [arXiv:1204.1734] [INSPIRE].
G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].
G.T. Horowitz, J.E. Santos and D. Tong, Further Evidence for Lattice-Induced Scaling, JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].
M. Rangamani, M. Rozali and D. Smyth, Spatial Modulation and Conductivities in Effective Holographic Theories, JHEP 07 (2015) 024 [arXiv:1505.05171] [INSPIRE].
A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].
A. Donos, B. Goutéraux and E. Kiritsis, Holographic Metals and Insulators with Helical Symmetry, JHEP 09 (2014) 038 [arXiv:1406.6351] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [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].
L. Alberte, M. Baggioli, A. Khmelnitsky and O. Pujolàs, Solid Holography and Massive Gravity, JHEP 02 (2016) 114 [arXiv:1510.09089] [INSPIRE].
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].
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].
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. Amoretti, D. Areán, B. Goutéraux and D. Musso, Gapless and gapped holographic phonons, JHEP 01 (2020) 058 [arXiv:1910.11330] [INSPIRE].
T. Andrade and A. Krikun, Coherent vs incoherent transport in holographic strange insulators, JHEP 05 (2019) 119 [arXiv:1812.08132] [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].
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. Ammon, M. Baggioli and A. Jiménez-Alba, A Unified Description of Translational Symmetry Breaking in Holography, JHEP 09 (2019) 124 [arXiv:1904.05785] [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, B. Goutéraux and D. Musso, Diffusion and universal relaxation of holographic phonons, JHEP 10 (2019) 068 [arXiv:1904.11445] [INSPIRE].
M. Ammon, M. Baggioli, S. Gray and S. Grieninger, Longitudinal Sound and Diffusion in Holographic Massive Gravity, JHEP 10 (2019) 064 [arXiv:1905.09164] [INSPIRE].
J. Armas and A. Jain, Viscoelastic hydrodynamics and holography, JHEP 01 (2020) 126 [arXiv:1908.01175] [INSPIRE].
J. Armas and A. Jain, Hydrodynamics for charge density waves and their holographic duals, Phys. Rev. D 101 (2020) 121901 [arXiv:2001.07357] [INSPIRE].
M. Ammon, M. Baggioli, S. Gray, S. Grieninger and A. Jain, On the Hydrodynamic Description of Holographic Viscoelastic Models, Phys. Lett. B 808 (2020) 135691 [arXiv:2001.05737] [INSPIRE].
M. Baggioli, S. Grieninger and L. Li, Magnetophonons & type-B Goldstones from Hydrodynamics to Holography, JHEP 09 (2020) 037 [arXiv:2005.01725] [INSPIRE].
R.A. Davison and B. Goutéraux, Dissecting holographic conductivities, JHEP 09 (2015) 090 [arXiv:1505.05092] [INSPIRE].
R.A. Davison, B. Goutéraux and S.A. Hartnoll, Incoherent transport in clean quantum critical metals, JHEP 10 (2015) 112 [arXiv:1507.07137] [INSPIRE].
S. Grozdanov, A. Lucas, S. Sachdev and K. Schalm, Absence of disorder-driven metal-insulator transitions in simple holographic models, Phys. Rev. Lett. 115 (2015) 221601 [arXiv:1507.00003] [INSPIRE].
M. Baggioli and O. Pujolàs, On holographic disorder-driven metal-insulator transitions, JHEP 01 (2017) 040 [arXiv:1601.07897] [INSPIRE].
Y.-S. An, T. Ji and L. Li, Magnetotransport and Complexity of Holographic Metal-Insulator Transitions, JHEP 10 (2020) 023 [arXiv:2007.13918] [INSPIRE].
A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].
P. Coullet, Commensurate-incommensurate transition in nonequilibrium systems, Phys. Rev. Lett. 56 (1986) 724.
A. Romero-Bermúdez, A. Krikun, K. Schalm and J. Zaanen, Anomalous attenuation of plasmons in strange metals and holography, Phys. Rev. B 99 (2019) 235149 [arXiv:1812.03968] [INSPIRE].
L.P. Gor’kov and G. Eliashberg, Generalization of the ginzburg-landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, in 30 Years Of The Landau Institute — Selected Papers, World Scientific, (1996), pp. 16–22.
M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851 [INSPIRE].
A. Krikun, Holographic discommensurations, JHEP 12 (2018) 030 [arXiv:1710.05801] [INSPIRE].
T. Andrade and A. Krikun, Commensurate lock-in in holographic non-homogeneous lattices, JHEP 03 (2017) 168 [arXiv:1701.04625] [INSPIRE].
S.A. Hartnoll and D.M. Hofman, Locally Critical Resistivities from Umklapp Scattering, Phys. Rev. Lett. 108 (2012) 241601 [arXiv:1201.3917] [INSPIRE].
A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
M. Tinkham, Introduction to superconductivity, Courier Corporation, (2004).
M. Cyrot, Ginzburg-landau theory for superconductors, Rept. Prog. Phys. 36 (1973) 103.
J.-H. She et al., Observing the origin of superconductivity in quantum critical metals, Phys. Rev. B 84 (2011) 144527 [arXiv:1105.5377] [INSPIRE].
N.W.M. Plantz, H.T.C. Stoof and S. Vandoren, Order parameter fluctuations in the holographic superconductor, J. Phys. B 50 (2017) 064001 [arXiv:1511.05112] [INSPIRE].
T. Andrade and A. Krikun, Commensurability effects in holographic homogeneous lattices, JHEP 05 (2016) 039 [arXiv:1512.02465] [INSPIRE].
A. Bagrov, N. Kaplis, A. Krikun, K. Schalm and J. Zaanen, Holographic fermions at strong translational symmetry breaking: a Bianchi-VII case study, JHEP 11 (2016) 057 [arXiv:1608.03738] [INSPIRE].
H. Leutwyler, Phonons as goldstone bosons, Helv. Phys. Acta 70 (1997) 275 [hep-ph/9609466] [INSPIRE].
P. Chaikin and T. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, (2000).
P.C. Martin, O. Parodi and P.S. Pershan, Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids, Phys. Rev. A 6 (1972) 2401.
A. Amoretti, D. Areán, B. Goutéraux and D. Musso, Effective holographic theory of charge density waves, Phys. Rev. D 97 (2018) 086017 [arXiv:1711.06610] [INSPIRE].
M. Baggioli and A. Buchel, Holographic Viscoelastic Hydrodynamics, JHEP 03 (2019) 146 [arXiv:1805.06756] [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].
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].
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].
R. Currat, E. Kats and I. Luk’yanchuk, Sound modes in composite incommensurate crystals, Eur. Phys. J. B 26 (2002) 339.
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. Alberte, M. Ammon, A. Jiménez-Alba, M. Baggioli and O. Pujolàs, Holographic Phonons, Phys. Rev. Lett. 120 (2018) 171602 [arXiv:1711.03100] [INSPIRE].
M. Baggioli and S. Grieninger, Zoology of solid & fluid holography — Goldstone modes and phase relaxation, JHEP 10 (2019) 235 [arXiv:1905.09488] [INSPIRE].
S.S. Gubser and I. Mitra, Instability of charged black holes in Anti-de Sitter space, Clay Math. Proc. 1 (2002) 221 [hep-th/0009126] [INSPIRE].
R. Emparan and M. Martinez, Black Branes in a Box: Hydrodynamics, Stability, and Criticality, JHEP 07 (2012) 120 [arXiv:1205.5646] [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].
M.J. Landry, Second sound and non-equilibrium effective field theory, arXiv:2008.11725 [INSPIRE].
A. Esposito, S. Garcia-Saenz, A. Nicolis and R. Penco, Conformal solids and holography, JHEP 12 (2017) 113 [arXiv:1708.09391] [INSPIRE].
A. Donos and J.P. Gauntlett, Minimally packed phases in holography, JHEP 03 (2016) 148 [arXiv:1512.06861] [INSPIRE].
B. Withers, Holographic Checkerboards, JHEP 09 (2014) 102 [arXiv:1407.1085] [INSPIRE].
A. Donos, J.P. Gauntlett and C. Pantelidou, Holographic Abrikosov Lattices, JHEP 07 (2020) 095 [arXiv:2001.11510] [INSPIRE].
W.R. Inc., Mathematica, Version 12.0, Champaign, IL, U.S.A. (2019).
F. Balm, A. Krikun, A. Romero-Bermúdez, K. Schalm and J. Zaanen, Isolated zeros destroy Fermi surface in holographic models with a lattice, JHEP 01 (2020) 151 [arXiv:1909.09394] [INSPIRE].
A. Krikun, Numerical Solution of the Boundary Value Problems for Partial Differential Equations. Crash course for holographer, (2018) [arXiv:1801.01483] [INSPIRE].
A. Donos and J.P. Gauntlett, Navier-Stokes Equations on Black Hole Horizons and DC Thermoelectric Conductivity, Phys. Rev. D 92 (2015) 121901 [arXiv:1506.01360] [INSPIRE].
A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, DC Conductivity of Magnetised Holographic Matter, JHEP 01 (2016) 113 [arXiv:1511.00713] [INSPIRE].
M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry, Commun. Math. Phys. 67 (1979) 205.
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Andrade, T., Baggioli, M. & Krikun, A. Phase relaxation and pattern formation in holographic gapless charge density waves. J. High Energ. Phys. 2021, 292 (2021). https://doi.org/10.1007/JHEP03(2021)292
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DOI: https://doi.org/10.1007/JHEP03(2021)292