Superfluid Kubo formulas from partition function

  • Shira ChapmanEmail author
  • Carlos Hoyos
  • Yaron Oz
Open Access


Linear response theory relates hydrodynamic transport coefficients to equilibrium retarded correlation functions of the stress-energy tensor and global symmetry currents in terms of Kubo formulas. Some of these transport coefficients are non-dissipative and affect the fluid dynamics at equilibrium. We present an algebraic framework for deriving Kubo formulas for such thermal transport coefficients by using the equilibrium partition function. We use the framework to derive Kubo formulas for all such transport coefficients of superfluids, as well as to rederive Kubo formulas for various normal fluid systems.


Anomalies in Field and String Theories Spontaneous Symmetry Breaking Phenomenological Models 


Open Access

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  1. [1]
    N. Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    G.D. Moore and K.A. Sohrabi, Kubo Formulae for Second-Order Hydrodynamic Coefficients, Phys. Rev. Lett. 106 (2011) 122302 [arXiv:1007.5333] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    G.D. Moore and K.A. Sohrabi, Thermodynamical second-order hydrodynamic coefficients, JHEP 11 (2012) 148 [arXiv:1210.3340] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A Theory of first order dissipative superfluid dynamics, arXiv:1105.3733 [INSPIRE].
  6. [6]
    Y. Neiman and Y. Oz, Anomalies in Superfluids and a Chiral Electric Effect, JHEP 09 (2011) 011 [arXiv:1106.3576] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    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].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational Anomaly and Transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    I. Amado, K. Landsteiner and F. Pena-Benitez, Anomalous transport coefficients from Kubo formulas in Holography, JHEP 05 (2011) 081 [arXiv:1102.4577] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    K. Landsteiner, E. Megias and F. Pena-Benitez, Anomalous Transport from Kubo Formulae, Lect. Notes Phys. 871 (2013) 433 [arXiv:1207.5808] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Bhattacharya, S. Bhattacharyya and S. Minwalla, Dissipative Superfluid dynamics from gravity, JHEP 04 (2011) 125 [arXiv:1101.3332] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    H. Guo, Ch.-Ch. Chien and Y. He, Gauge-invariant linear response theory of relativistic Bardeen-Cooper-Schrieffer superfluids, Phys. Rev. D 85 (2012) 074025 [arXiv:1202.5234].ADSGoogle Scholar
  14. [14]
    M. Luzum and P. Romatschke, Conformal Relativistic Viscous Hydrodynamics: Applications to RHIC results at \( \sqrt{{{s_{{N\,N}}}}} \) = 200-GeV, Phys. Rev. C 78 (2008) 034915 [Erratum ibid. C 79 (2009)039903] [arXiv:0804.4015] [INSPIRE].ADSGoogle Scholar
  15. [15]
    N. Banerjee, S. Dutta, S. Jain, R. Loganayagam and T. Sharma, Constraints on Anomalous Fluid in Arbitrary Dimensions, JHEP 03 (2013) 048 [arXiv:1206.6499] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A. Meyer and Y. Oz, Constraints on Rindler Hydrodynamics, JHEP 07 (2013) 090 [arXiv:1304.6305] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    Y. Neiman and Y. Oz, Relativistic Hydrodynamics with General Anomalous Charges, JHEP 03 (2011) 023 [arXiv:1011.5107] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez, Holographic Gravitational Anomaly and Chiral Vortical Effect, JHEP 09 (2011) 121 [arXiv:1107.0368] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    S. Chapman, Y. Neiman and Y. Oz, Fluid/Gravity Correspondence, Local Wald Entropy Current and Gravitational Anomaly, JHEP 07 (2012) 128 [arXiv:1202.2469] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    S. Golkar and D.T. Son, Non-Renormalization of the Chiral Vortical Effect Coefficient, arXiv:1207.5806 [INSPIRE].
  22. [22]
    D.-F. Hou, H. Liu and H.-c. Ren, A Possible Higher Order Correction to the Vortical Conductivity in a Gauge Field Plasma, Phys. Rev. D 86 (2012) 121703 [arXiv:1210.0969] [INSPIRE].ADSGoogle Scholar
  23. [23]
    K. Jensen, Triangle Anomalies, Thermodynamics and Hydrodynamics, Phys. Rev. D 85 (2012) 125017 [arXiv:1203.3599] [INSPIRE].ADSGoogle Scholar
  24. [24]
    K. Jensen et al., Parity-Violating Hydrodynamics in 2+1 Dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Physics and AstronomyTel-Aviv UniversityTel-AvivIsrael

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