Abstract
We study the linear response of relativistic superfluids with a non-zero superfluid velocity. For sufficiently large superflow, an instability develops via the crossing of a pole of the retarded Green’s functions to the upper half complex frequency plane. We show that this is caused by a local thermodynamic instability, i.e. when an eigenvalue of the static susceptibility matrix (the second derivatives of the free energy) diverges and changes sign. The onset of the instability occurs when ∂ζ(nsζ) = 0, with ζ the norm of the superfluid velocity and ns the superfluid density. The Landau instability for non-relativistic superfluids such as Helium 4 also coincides with the non-relativistic version of this criterion. We then turn to gauge/gravity duality and show that this thermodynamic instability criterion applies equally well to strongly-coupled superfluids. In passing, we compute holographically a number of transport coefficients parametrizing deviations out-of-equilibrium in the hydrodynamic regime and demonstrate that the gapless quasinormal modes of the dual planar black hole match those predicted by superfluid hydrodynamics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Landau and E. Lifshitz, Fluid Mechanics: Course of Theoretical Physics, Pergamon Press (2013).
L. Landau, E. Lifshitz and L. Pitaevskii, Course of Theoretical Physics: Statistical Physics, Part 2, E.M. Lifshitz and L.P. Pitaevskii eds., vol. 9, Pergamon Press (1980).
P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press (1995) [https://doi.org/10.1017/CBO9780511813467].
L.V. Delacrétaz, D.M. Hofman and G. Mathys, Superfluids as Higher-form Anomalies, SciPost Phys. 8 (2020) 047 [arXiv:1908.06977] [INSPIRE].
J. Armas and A. Jain, Approximate higher-form symmetries, topological defects, and dynamical phase transitions, Phys. Rev. D 109 (2024) 045019 [arXiv:2301.09628] [INSPIRE].
L. Meyer and F. Reif, Ion Motion in Superfluid Liquid Helium under Pressure, Phys. Rev. 123 (1961) 727.
P.V.E. McClintock, Ions in superfluid helium, Physica B+C 127 (1984) 300.
A.I. Ahonen et al., Mobility of negative ions in superfluid 3He, Phys. Rev. Lett. 37 (1976) 511.
J. Bardeen, Critical Fields and Currents in Superconductors, Rev. Mod. Phys. 34 (1962) 667.
B. Goutéraux, E. Mefford and F. Sottovia, Critical superflows and thermodynamic instabilities in superfluids, Phys. Rev. D 108 (2023) L081903 [arXiv:2212.10410] [INSPIRE].
J. Armas and E. Have, Ideal fracton superfluids, SciPost Phys. 16 (2024) 039 [arXiv:2304.09596] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser and F.D. Rocha, The gravity dual to a quantum critical point with spontaneous symmetry breaking, Phys. Rev. Lett. 102 (2009) 061601 [arXiv:0807.1737] [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].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
C.P. Herzog, P.K. Kovtun and D.T. Son, Holographic model of superfluidity, Phys. Rev. D 79 (2009) 066002 [arXiv:0809.4870] [INSPIRE].
C.P. Herzog, An Analytic Holographic Superconductor, Phys. Rev. D 81 (2010) 126009 [arXiv:1003.3278] [INSPIRE].
J. Sonner and B. Withers, A gravity derivation of the Tisza-Landau Model in AdS/CFT, Phys. Rev. D 82 (2010) 026001 [arXiv:1004.2707] [INSPIRE].
D. Areán, M. Bertolini, C. Krishnan and T. Prochazka, Type IIB Holographic Superfluid Flows, JHEP 03 (2011) 008 [arXiv:1010.5777] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya and S. Minwalla, Dissipative Superfluid dynamics from gravity, JHEP 04 (2011) 125 [arXiv:1101.3332] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A theory of first order dissipative superfluid dynamics, JHEP 05 (2014) 147 [arXiv:1105.3733] [INSPIRE].
C.P. Herzog, N. Lisker, P. Surowka and A. Yarom, Transport in holographic superfluids, JHEP 08 (2011) 052 [arXiv:1101.3330] [INSPIRE].
I. Amado et al., Holographic Superfluids and the Landau Criterion, JHEP 02 (2014) 063 [arXiv:1307.8100] [INSPIRE].
S. Lan, H. Liu, Y. Tian and H. Zhang, Landau Instability and soliton formations, arXiv:2010.06232 [INSPIRE].
I.M. Khalatnikov and V.V. Lebedev, Relativistic hydrodynamics of a superfluid liquid, Phys. Lett. A 91 (1982) 70.
V. Lebedev and I. Khalatnikov, Relativistic hydrodynamics of a superfluid liquid (in Russian), Zh. Eksp. Teor. Fiz. 83 (1982) 1601.
B. Carter and I.M. Khalatnikov, Equivalence of convective and potential variational derivations of covariant superfluid dynamics, Phys. Rev. D 45 (1992) 4536 [INSPIRE].
B. Carter and I.M. Khalatnikov, Momentum, vorticity, and helicity in covariant superfluid dynamics, Annals Phys. 219 (1992) 243.
W. Israel, Covariant superfluid mechanics, Phys. Lett. A 86 (1981) 79.
W. Israel, Equivalence of two theories of relativistic superfluid mechanics, Phys. Lett. A 92 (1982) 77.
D. Areán, M. Baggioli, S. Grieninger and K. Landsteiner, A holographic superfluid symphony, JHEP 11 (2021) 206 [arXiv:2107.08802] [INSPIRE].
N. Banerjee, S. Dutta and A. Jain, First Order Galilean Superfluid Dynamics, Phys. Rev. D 96 (2017) 065004 [arXiv:1612.01550] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, The eightfold way to dissipation, Phys. Rev. Lett. 114 (2015) 201601 [arXiv:1412.1090] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].
S. Bhattacharyya, Entropy current and equilibrium partition function in fluid dynamics, JHEP 08 (2014) 165 [arXiv:1312.0220] [INSPIRE].
S. Bhattacharyya, Entropy Current from Partition Function: One Example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].
N. Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on Superfluid Hydrodynamics from Equilibrium Partition Functions, JHEP 01 (2013) 040 [arXiv:1206.6106] [INSPIRE].
J. Armas and A. Jain, Viscoelastic hydrodynamics and holography, JHEP 01 (2020) 126 [arXiv:1908.01175] [INSPIRE].
J. Armas, A. Jain and R. Lier, Approximate symmetries, pseudo-Goldstones, and the second law of thermodynamics, Phys. Rev. D 108 (2023) 086011 [arXiv:2112.14373] [INSPIRE].
L. Landau, Theory of the Superfluidity of Helium II, Phys. Rev. 60 (1941) 356 [INSPIRE].
A.J. Clark, On The Hydrodynamics of Superfluid Helium, Ph.D. Thesis, Massachusetts Institute of Technology (1963).
S.J. Putterman, Superfluid Hydrodynamics, vol. 3, North-Holland/American Elsevier (1974).
S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
A. Haber, A. Schmitt and S. Stetina, Instabilities in relativistic two-component (super)fluids, Phys. Rev. D 93 (2016) 025011 [arXiv:1510.01982] [INSPIRE].
N. Andersson and A. Schmitt, Dissipation triggers dynamical two-stream instability, Particles 2 (2019) 457 [arXiv:1908.04275] [INSPIRE].
A. Schmitt, Superfluid two-stream instability in a microscopic model, Phys. Rev. D 89 (2014) 065024 [arXiv:1312.5993] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
C.P. Herzog and A. Yarom, Sound modes in holographic superfluids, Phys. Rev. D 80 (2009) 106002 [arXiv:0906.4810] [INSPIRE].
K.K. Kim, M. Park and K.-Y. Kim, Ward identity and Homes’ law in a holographic superconductor with momentum relaxation, JHEP 10 (2016) 041 [arXiv:1604.06205] [INSPIRE].
L.P. Kadanoff and P.C. Martin, Hydrodynamic equations and correlation functions, Annals Phys. 24 (1963) 419 [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
A. Donos, P. Kailidis and C. Pantelidou, Dissipation in holographic superfluids, JHEP 09 (2021) 134 [arXiv:2107.03680] [INSPIRE].
A. Donos and P. Kailidis, Dissipative effects in finite density holographic superfluids, JHEP 11 (2022) 053 [arXiv:2209.06893] [INSPIRE].
S. Grozdanov, A. Lucas and N. Poovuttikul, Holography and hydrodynamics with weakly broken symmetries, Phys. Rev. D 99 (2019) 086012 [arXiv:1810.10016] [INSPIRE].
R.A. Davison, B. Goutéraux and E. Mefford, Zero sound and higher-form symmetries in compressible holographic phases, JHEP 12 (2023) 040 [arXiv:2210.14802] [INSPIRE].
B. Goutéraux and E. Mefford, Normal charge densities in quantum critical superfluids, Phys. Rev. Lett. 124 (2020) 161604 [arXiv:1912.08849] [INSPIRE].
B. Goutéraux and E. Mefford, Non-vanishing zero-temperature normal density in holographic superfluids, JHEP 11 (2020) 091 [arXiv:2008.02289] [INSPIRE].
D.T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199 [INSPIRE].
G.E. Volovik, On the Kelvin-Helmholtz instability in superfluids, JETP Lett. 75 (2002) 418.
R. Blaauwgeers et al., Vortex lines at a phase boundary between different quantum vacua, Phys. Rev. Lett. 89 (2002) 155301 [cond-mat/0111343] [INSPIRE].
Y.-P. An, L. Li, C.-Y. Xia and H.-B. Zeng, Interface Dynamics of Strongly interacting Binary Superfluids, arXiv:2401.09189 [INSPIRE].
Ó.J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav. 33 (2016) 133001 [arXiv:1510.02804] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
I. Amado, M. Kaminski and K. Landsteiner, Hydrodynamics of Holographic Superconductors, JHEP 05 (2009) 021 [arXiv:0903.2209] [INSPIRE].
Acknowledgments
The work of D.A. is supported through the grants CEX2020-001007-S and PID2021-123017NB-100, PID2021-127726NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”. The work of B.G. and F.S. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No758759). The work of E.M. was supported in part by NSERC and in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No758759). We all gratefully acknowledge Nordita’s hospitality during the Nordita program ‘Recent developments in strongly-correlated quantum matter’ where part of this work was carried out. Part of this work was performed
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2312.08243
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Areán, D., Goutéraux, B., Mefford, E. et al. Hydrodynamics and instabilities of relativistic superfluids at finite superflow. J. High Energ. Phys. 2024, 272 (2024). https://doi.org/10.1007/JHEP05(2024)272
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)272