Comments on Hall transport from effective actions

  • Felix Haehl
  • Mukund Rangamani


We consider parity-odd transport in 2+1 dimensional charged fluids restricting attention to the class of non-dissipative fluids. We show that there is a two parameter family of such non-dissipative fluids which can be derived from an effective action, in contradistinction with a four parameter family that can be derived from an entropy current analysis. The effective action approach allows us to extract the adiabatic transport data, in particular the Hall viscosity and Hall conductivity amongst others, in terms of the thermodynamic functions that enter as ‘coupling constants’. Curiously, we find that Hall viscosity is forced to vanish, whilst the Hall conductivity is generically a non-vanishing function of thermodynamic data determined in terms of the hydrodynamic couplings.


Thermal Field Theory Holography and quark-gluon plasmas Quantum Dissipative Systems Sigma Models 


  1. [1]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, arXiv:1107.5780 [INSPIRE].
  3. [3]
    D.T. Son and P. Surowka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    E.M. Lifshitz and L.D. Landau, Course of theoretical physics. Volume 6: fluid mechanics, Pergamon Press, Oxford U.K. (1959).Google Scholar
  5. [5]
    N. Banerjee et al., Constraints on fluid dynamics from equilibrium partition functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    R. Jackiw, V. Nair, S. Pi and A. Polychronakos, Perfect fluid theory and its extensions, J. Phys. A 37 (2004) R327 [hep-ph/0407101] [INSPIRE].ADSGoogle Scholar
  8. [8]
    S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].ADSGoogle Scholar
  9. [9]
    S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, Null energy condition and superluminal propagation, JHEP 03 (2006) 025 [hep-th/0512260] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: first order effects, arXiv:1211.6461 [INSPIRE].
  11. [11]
    S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, arXiv:1305.3670 [INSPIRE].
  12. [12]
    L. Andrianopoli, R. D’Auria, P. Grassi and M. Trigiante, Entropy current formalism for supersymmetric theories, arXiv:1304.2206 [INSPIRE].
  13. [13]
    L. Andrianopoli, R. D’Auria, P. Grassi and M. Trigiante, A note on the field-theoretical description of superfluids, arXiv:1304.6915 [INSPIRE].
  14. [14]
    S. Dubovsky, L. Hui and A. Nicolis, Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions, arXiv:1107.0732 [INSPIRE].
  15. [15]
    J. Bhattacharya, S. Bhattacharyya and M. Rangamani, Non-dissipative hydrodynamics: effective actions versus entropy current, JHEP 02 (2013) 153 [arXiv:1211.1020] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [INSPIRE].ADSGoogle Scholar
  17. [17]
    A. Nicolis and D.T. Son, Hall viscosity from effective field theory, arXiv:1103.2137 [INSPIRE].
  18. [18]
    K. Jensen et al., Parity-violating hydrodynamics in 2 + 1 dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A theory of first order dissipative superfluid dynamics, arXiv:1105.3733 [INSPIRE].
  20. [20]
    R.G. Leigh, A.C. Petkou and P.M. Petropoulos, Holographic fluids with vorticity and analogue gravity, JHEP 11 (2012) 121 [arXiv:1205.6140] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    S. Bhattacharyya, S. Minwalla and S.R. Wadia, The incompressible non-relativistic NavierStokes equation from gravity, JHEP 08 (2009) 059 [arXiv:0810.1545] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    J. Avron, R. Seiler and P. Zograf, Viscosity of quantum Hall fluids, Phys. Rev. Lett. 75 (1995) 697 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    J.E. Avron, Odd viscosity, J. Stat. Phys. 92 (1998) 543 [physics/9712050].MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    N. Read, Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p(x) + ip(y) paired superfluids, Phys. Rev. B 79 (2009) 045308 [arXiv:0805.2507] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    F. Haldane, ‘Hall viscosityand intrinsic metric of incompressible fractional Hall fluids, arXiv:0906.1854 [INSPIRE].
  27. [27]
    B. Bradlyn, M. Goldstein and N. Read, Kubo formulas for viscosity: Hall viscosity, Ward identities and the relation with conductivity, Phys. Rev. B 86 (2012) 245309 [arXiv:1207.7021] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    O. Saremi and D.T. Son, Hall viscosity from gauge/gravity duality, JHEP 04 (2012) 091 [arXiv:1103.4851] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    T.L. Hughes, R.G. Leigh and E. Fradkin, Torsional response and dissipationless viscosity in topological insulators, Phys. Rev. Lett. 107 (2011) 075502 [arXiv:1101.3541] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    T.L. Hughes, R.G. Leigh and O. Parrikar, Torsional anomalies, Hall viscosity and bulk-boundary correspondence in topological states, Phys. Rev. D 88 (2013) 025040 [arXiv:1211.6442] [INSPIRE].ADSGoogle Scholar
  31. [31]
    C. Hoyos, S. Moroz and D.T. Son, Effective theory of chiral two-dimensional superfluids, arXiv:1305.3925 [INSPIRE].
  32. [32]
    N. Poplawski, Intrinsic spin requires gravity with torsion and curvature, arXiv:1304.0047 [INSPIRE].
  33. [33]
    J.-W. Chen, N.-E. Lee, D. Maity and W.-Y. Wen, A holographic model for Hall viscosity, Phys. Lett. B 713 (2012) 47 [arXiv:1110.0793] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J.-W. Chen, S.-H. Dai, N.-E. Lee and D. Maity, Novel parity violating transport coefficients in 2 + 1 dimensions from holography, JHEP 09 (2012) 096 [arXiv:1206.0850] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    H. Liu, H. Ooguri, B. Stoica and N. Yunes, Spontaneous generation of angular momentum in holographic theories, Phys. Rev. Lett. 110 (2013) 211601 [arXiv:1212.3666] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    F. Haehl, R. Loganayagam and M. Rangamani, work in progress.Google Scholar
  37. [37]
    F. Hehl, P. Von Der Heyde, G.-D. Kerlick and J. Nester, General relativity with spin and torsion: foundations and prospects, Rev. Mod. Phys. 48 (1976) 393 [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    F. Hehl, On the energy tensor of spinning massive matter in classical field theory and general relativity, Rept. Math. Phys. 9 (1976) 55 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Centre for Particle Theory & Department of Mathematical Sciences, Science LaboratoriesDurhamU.K

Personalised recommendations