Abstract
We reconsider general aspects of Galilean-invariant thermal field theory. Using the proposal of our companion paper, we recast non-relativistic hydrodynamics in a manifestly covariant way and couple it to a background spacetime. We examine the concomitant consequences for the thermal partition functions of Galilean theories on a time-independent, but weakly curved background. We work out both the hydrodynamics and partition functions in detail for the example of parity-violating normal fluids in two dimensions to first order in the gradient expansion, finding results that differ from those previously reported in the literature. As for relativistic field theories, the equality-type constraints imposed by the existence of an entropy current appear to be in one-to-one correspondence with those arising from the existence of a hydrostatic partition function. Along the way, we obtain a number of useful results about non-relativistic hydrodynamics, including a manifestly boost-invariant presentation thereof, simplified Ward identities, the systematics of redefinitions of the fluid variables, and the positivity of entropy production.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Fluid mechanics, Pergamon Press, Oxford U.K. (1987).
D.T. Son and P. Surowka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [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].
G.D. Moore and K.A. Sohrabi, Kubo formulae for second-order hydrodynamic coefficients, Phys. Rev. Lett. 106 (2011) 122302 [arXiv:1007.5333] [INSPIRE].
K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
N. Banerjee et al., Constraints on fluid dynamics from equilibrium partition functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
J. Luttinger, Theory of thermal transport coefficients, Phys. Rev. 135 (1964) A1505.
L.P. Kadanoff and P.C. Martin, Hydrodynamic equations and correlation functions, Ann. Phys. 24 (1963) 419.
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [INSPIRE].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
C. Duval and H.P. Kunzle, Minimal gravitational coupling in the newtonian theory and the covariant Schrödinger equation, Gen. Rel. Grav. 16 (1984) 333 [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
O. Andreev, M. Haack and S. Hofmann, On nonrelativistic diffeomorphism invariance, Phys. Rev. D 89 (2014) 064012 [arXiv:1309.7231] [INSPIRE].
R. Banerjee, A. Mitra and P. Mukherjee, Localization of the galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].
T. Brauner, S. Endlich, A. Monin and R. Penco, General coordinate invariance in quantum many-body systems, Phys. Rev. D 90 (2014) 105016 [arXiv:1407.7730] [INSPIRE].
A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [INSPIRE].
B. Bradlyn and N. Read, Low-energy effective theory in the bulk for transport in a topological phase, Phys. Rev. B 91 (2015) 125303 [arXiv:1407.2911] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, arXiv:1409.1519 [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, arXiv:1409.1522 [INSPIRE].
K. Jensen et al., Parity-violating hydrodynamics in 2 + 1 dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
X.G. Wen and A. Zee, Shift and spin vector: new topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69 (1992) 953 [Erratum ibid. 69 (1992) 3000] [INSPIRE].
W. Goldberger and N. Read, unpublished.
J.E. Avron, R. Seiler and P.G. Zograf, Viscosity of quantum Hall fluids, Phys. Rev. Lett. 75 (1995) 697 [INSPIRE].
J.E. Avron, Odd viscosity, physics/9712050.
M. Kaminski and S. Moroz, Nonrelativistic parity-violating hydrodynamics in two spatial dimensions, Phys. Rev. B 89 (2014) 115418 [arXiv:1310.8305] [INSPIRE].
N. Banerjee, S. Dutta, A. Jain and D. Roychowdhury, Entropy current for non-relativistic fluid, JHEP 08 (2014) 037 [arXiv:1405.5687] [INSPIRE].
M. Geracie and D.T. Son, Hydrodynamics on the lowest Landau level, arXiv:1408.6843 [INSPIRE].
S. Bhattacharyya, Entropy current and equilibrium partition function in fluid dynamics, JHEP 08 (2014) 165 [arXiv:1312.0220] [INSPIRE].
R. Loganayagam, Anomaly Induced transport in arbitrary dimensions, arXiv:1106.0277 [INSPIRE].
H.P. Künzle, Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics, Ann. Inst. Henri Poincaré A 17 (1972) 337.
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].
T.S. Evans, N point finite temperature expectation values at real times, Nucl. Phys. B 374 (1992) 340 [INSPIRE].
C. Hoyos, B.S. Kim and Y. Oz, Lifshitz hydrodynamics, JHEP 11 (2013) 145 [arXiv:1304.7481] [INSPIRE].
K. Jensen, Triangle anomalies, thermodynamics and hydrodynamics, Phys. Rev. D 85 (2012) 125017 [arXiv:1203.3599] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
Y. Neiman and Y. Oz, Relativistic hydrodynamics with general anomalous charges, JHEP 03 (2011) 023 [arXiv:1011.5107] [INSPIRE].
K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational anomaly and transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE].
S. Golkar and D.T. Son, (Non)-renormalization of the chiral vortical effect coefficient, JHEP 02 (2015) 169 [arXiv:1207.5806] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [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].
S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [INSPIRE].
R. Loganayagam, Entropy current in conformal hydrodynamics, JHEP 05 (2008) 087 [arXiv:0801.3701] [INSPIRE].
M. Rangamani, S.F. Ross, D.T. Son and E.G. Thompson, Conformal non-relativistic hydrodynamics from gravity, JHEP 01 (2009) 075 [arXiv:0811.2049] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1411.7024
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jensen, K. Aspects of hot Galilean field theory. J. High Energ. Phys. 2015, 123 (2015). https://doi.org/10.1007/JHEP04(2015)123
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2015)123