Characterizing the behaviour of strongly coupled quantum systems out of equilibrium is a cardinal challenge for both theory and experiment. With diverse applications ranging from the dynamics of the quark–gluon plasma to transport in novel states of quantum matter, establishing universal results and organizing principles out of equilibrium is crucial. We present a universal description of energy transport between quantum critical heat baths in arbitrary dimension. The current-carrying non-equilibrium steady state (NESS) is a Lorentz-boosted thermal state. In the context of gauge/gravity duality this reveals an intimate correspondence between far-from-equilibrium transport and black hole uniqueness theorems. We provide analytical expressions for the energy current and the generating function of energy current fluctuations, together with predictions for experiment.
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum Newton’s cradle. Nature 440, 900–903 (2006).
Gring, M. et al. Relaxation and prethermalization in an isolated quantum system. Science 337, 1318–1322 (2012).
Sadler, L. E., Higbie, J. M., Leslie, S. R., Vengalattore, M. & Stamper-Kurn, D. M. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).
Baumann, K., Mottl, R., Brennecke, F. & Esslinger, T. Exploring symmetry breaking at the Dicke quantum phase transition. Phys. Rev. Lett. 107, 140402 (2011).
Smith, R. P., Beattie, S., Moulder, S., Campbell, R. L. D. & Hadzibabic, Z. Condensation dynamics in a quantum-quenched Bose gas. Phys. Rev. Lett. 109, 105301 (2012).
Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484–487 (2012).
Braun, S. et al. Emergence of coherence and the dynamics of quantum phase transitions. Proc. Natl Acad. Sci. USA 112, 3641–3646 (2015).
Brantut, J-P. et al. A thermoelectric heat engine with ultracold atoms. Science 342, 713–715 (2013).
Schmidutz, T. F. et al. Quantum Joule–Thomson effect in a saturated homogeneous Bose gas. Phys. Rev. Lett. 112, 040403 (2014).
Calabrese, P. & Cardy, J. Time-dependence of correlation functions following a quantum quench. Phys. Rev. Lett. 96, 136801 (2006).
Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863–883 (2011).
Rigol, M., Dunjko, V., Yurovsky, V. & Olshanii, M. Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007).
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).
Rigol, M. Breakdown of thermalization in finite one-dimensional systems. Phys. Rev. Lett. 103, 100403 (2009).
Calabrese, P., Essler, F. H. L. & Fagotti, M. Quantum quench in the transverse-field Ising chain. Phys. Rev. Lett. 106, 227203 (2011).
Esposito, M., Harbola, U. & Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665–1702 (2009); erratum 86, 1125 (2014)
Bernard, D. & Doyon, B. Energy flow in non-equilibrium conformal field theory. J. Phys. A 45, 362001 (2012).
Bernard, D. & Doyon, B. Non-equilibrium steady states in conformal field theory. Ann. Henri Poincaré 16, 113–161 (2015).
Bernard, D. & Doyon, B. Time-reversal symmetry and fluctuation relations in non-equilibrium quantum steady states. J. Phys. A 46, 372001 (2013).
Karrasch, C., Ilan, R. & Moore, J. E. Nonequilibrium thermal transport and its relation to linear response. Phys. Rev. B 88, 195129 (2013).
Karrasch, C., Bardarson, J. H. & Moore, J. E. Finite-temperature dynamical density matrix renormalization group and the Drude weight of spin-1/2 chains. Phys. Rev. Lett. 108, 227206 (2012).
Karrasch, C., Bardarson, J. H. & Moore, J. E. Reducing the numerical effort of finite-temperature density matrix renormalization group calculations. New J. Phys. 15, 083031 (2013).
Huang, Y., Karrasch, C. & Moore, J. E. Scaling of electrical and thermal conductivities in an almost integrable chain. Phys. Rev. B 88, 115126 (2013).
Jezouin, S. et al. Quantum limit of heat flow across a single electronic channel. Science 342, 601–604 (2013).
Mazur, P. Non-ergodicity of phase functions in certain systems. Physica 43, 533–545 (1969).
Cardy, J. The ubiquitous “c”: From the Stefan–Boltzmann law to quantum information. J. Stat. Mech. 2010, P10004 (2010).
Hartnoll, S. A. Lectures on holographic methods for condensed matter physics. Class. Quantum Gravity 26, 224002 (2009).
McGreevy, J. Holographic duality with a view toward many-body physics. Adv. High Energy Phys. 2010, 723105 (2010).
Sachdev, S. What can gauge-gravity duality teach us about condensed matter physics? Annu. Rev. Condens. Mat. 3, 9–33 (2012).
Danielsson, U. H., Keski-Vakkuri, E. & Kruczenski, M. Spherically collapsing matter in AdS, holography, and shellons. Nucl. Phys. B 563, 279–292 (1999).
Bhattacharyya, S. & Minwalla, S. Weak field black hole formation in asymptotically AdS spacetimes. JHEP 0909, 034 (2009).
Albash, T. & Johnson, C. V. Evolution of holographic entanglement entropy after thermal and electromagnetic quenches. New J. Phys. 13, 045017 (2011).
Das, S., Nishioka, T. & Takayanagi, T. Probe branes, time-dependent couplings and thermalization in AdS/CFT. JHEP 1007, 071 (2010).
Chesler, P. M. & Yaffe, L. G. Horizon formation and far-from-equilibrium isotropization in a supersymmetric Yang–Mills plasma. Phys. Rev. Lett. 102, 211601 (2009).
Auzzi, R., Elitzur, S., Gudnason, S. B. & Rabinovici, E. On periodically driven AdS/CFT. JHEP 1311, 016 (2013).
Murata, K., Kinoshita, S. & Tanahashi, N. Non-equilibrium condensation process in a holographic superconductor. JHEP 1007, 050 (2010).
Sonner, J. & Green, A. G. Hawking radiation and nonequilibrium quantum critical current noise. Phys. Rev. Lett. 109, 091601 (2012).
Bhaseen, M. J., Gauntlett, J. P., Simons, B. D., Sonner, J. & Wiseman, T. Holographic superfluids and the dynamics of symmetry breaking. Phys. Rev. Lett. 110, 015301 (2013).
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).
Fischetti, S. & Marolf, D. Flowing funnels: Heat sources for field theories and the AdS3 dual of CFT2 Hawking radiation. Class. Quantum Gravity 29, 105004 (2012).
Figueras, P. & Wiseman, T. Stationary holographic plasma quenches and numerical methods for non-Killing horizons. Phys. Rev. Lett. 110, 171602 (2013).
Fischetti, S., Marolf, D. & Santos, J. AdS flowing black funnels: Stationary AdS black holes with non-Killing horizons and heat transport in the dual CFT. Class. Quantum Gravity 30, 075001 (2013).
Fazio, R., Hekking, F. W. J. & Khmelnitskii, D. E. Anomalous thermal transport in quantum wires. Phys. Rev. Lett. 80, 5611–5614 (1998).
Rego, L. G. C. & Kirczenow, G. Quantized thermal conductance of dielectric quantum wires. Phys. Rev. Lett. 81, 232–235 (1998).
Schwab, K., Henriksen, E. A., Worlock, J. M. & Roukes, M. L. Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000).
Kraus, P. Lectures on black holes and the AdS3/CFT2 correspondence. Lecture Notes Phys. 755, 193–247 (2008).
Baier, R., Romatschke, P., Son, D. T., Starinets, A. O. & Stephanov, M. A. Relativistic viscous hydrodynamics, conformal invariance, and holography. JHEP 0804, 100 (2008).
Green, S. R., Carrasco, F. & Lehner, L. Holographic path to the turbulent side of gravity. Phys. Rev. X 4, 011001 (2014).
Adams, A., Chesler, P. M. & Liu, H. Holographic turbulence. Phys. Rev. Lett. 112, 151602 (2014).
We thank B. Benenowski, D. Bernard, P. Chesler, D. Haldane, C. Herzog, D. Marolf, B. Najian, C-A. Pillet, S. Sachdev and A. Starinets for helpful comments; we especially thank A. Green for suggesting the interpretation of TL, R in d = 1 as Doppler-shifted radiation. M.J.B. and K.S. thank the Kavli Royal Society Center Chicheley Hall and the Isaac Newton Institute, Cambridge for hospitality. M.J.B. and B.D. thank The Galileo Galilei Institute for Theoretical Physics. B.D. thanks Université Paris Diderot, where part of this work was done, for financial support through a visiting professorship. A.L. is supported by the Smith Family Science and Engineering Graduate Fellowship and thanks the Perimeter Institute for hospitality. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. This work was supported in part by a VICI grant of the Netherlands Organization for Scientific Research (NWO), by the Netherlands Organization for Scientific Research/Ministry of Science and Education (NWO/OCW) and by the Foundation for Research into Fundamental Matter (FOM).
The authors declare no competing financial interests.
About this article
Cite this article
Bhaseen, M., Doyon, B., Lucas, A. et al. Energy flow in quantum critical systems far from equilibrium. Nature Phys 11, 509–514 (2015). https://doi.org/10.1038/nphys3320
Reports on Progress in Physics (2020)
SciPost Physics (2020)
Generalized hydrodynamic approach to charge and energy currents in the one-dimensional Hubbard model
Physical Review B (2020)
Annales Henri Poincaré (2020)
Physical Review B (2020)