Abstract
Equilibrium thermodynamics is characterized by two fundamental ideas: thermalization—that systems approach a late time thermal state; and phase structure—that thermal states exhibit singular changes as various parameters characterizing the system are changed. We summarize recent progress that has established generalizations of these ideas to periodically driven, or Floquet, closed quantum systems. This has resulted in the discovery of entirely new phases which exist only out of equilibrium, such as the π-spin glass/Floquet time crystal.
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References
Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).
Schollwock, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).
Nandkishore, R. & Huse, D. A. Many body localization and thermalization in quantum statistical mechanics. Ann. Rev. Condensed Matter Phys. 6, 15–38 (2015).
Bordia, P., Lüschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 466–470 (2017).
Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).
Grifoni, M. & Hänggi, P. Driven quantum tunneling. Phys. Rep. 304, 229–354 (1998).
Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).
Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet–Bloch systems. New J. Phys. 17, 125014 (2015).
Roy, R. & Harper, F. Periodic table for floquet topological insulators. Preprint at http://arXiv.org/abs/1603.06944 (2016).
Eckardt, A., Weiss, C. & Holthaus, M. Superfluid–insulator transition in a periodically driven optical lattice. Phys. Rev. Lett. 95, 260404 (2005).
Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406 (2009).
Eckardt, A. Atomic quantum gases in periodically driven optical lattices. Preprint at http://arXiv.org/abs/1606.08041 (2016).
Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).
Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).
Abanin, D., De Roeck, W. & Huveneers, F. Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115, 256803 (2015).
Ponte, P., Chandran, A., Papic, Z. & Abanin, D. A. Periodically driven ergodic and many-body localized quantum systems. Ann. Phys. 353, 196–204 (2015).
D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. B 43, 2046–2049 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).
Chandran, A. & Sondhi, S. L. Interaction stabilized steady states in the driven O(N) model. Phys. Rev. B 93, 174305 (2016).
Prosen, T. Time evolution of a quantum many-body system: transition from integrability to ergodicity in the thermodynamic limit. Phys. Rev. Lett. 80, 1808–1811 (1998).
Citro, R. et al. Dynamical stability of a many-body Kapitza pendulum. Ann. Phys. 360, 694–710 (2015).
Lazarides, A., Das, A. & Moessner, R. Periodic thermodynamics of isolated systems. Phys. Rev. Lett. 112, 150401 (2014).
Gritsev, V. & Polkovnikov, A. Integrable Floquet dynamics. Preprint at http://arXiv.org/abs/1701.05276 (2017).
Abanin, D., De Roeck, W. & Huveneers, F. Theory of many-body localization in periodically driven systems. Ann. Phys. 372, 1–11 (2016).
Lazarides, A., Das, A. & Moessner, R. Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115, 030402 (2015).
Ponte, P., Papic, Z., Huveneers, F. & Abanin, D. A. Many-body localization in periodically driven systems. Phys. Rev. Lett. 114, 140401 (2015).
Bukov, M., Heyl, M., Huse, D. A. & Polkovnikov, A. Heating and many-body resonances in a periodically driven two-band system. Phys. Rev. B 93, 155132 (2016).
Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization protected quantum order. Phys. Rev. B 88, 014206 (2013).
Pekker, D., Refael, G., Altman, E., Demler, E. & Oganesyan, V. The Hilbert-glass transition: new universality of temperature-tuned many-body dynamical quantum criticality. Phys. Rev. X 4, 011052 (2014).
Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. On the phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).
Senthil, T. Symmetry protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Khemani, V., von Keyserlingk, C. W. & Sondhi, S. L. Defining time crystals via representation theory. Preprint at http://arXiv.org/abs/1612.08758 (2016).
Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).
Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).
von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal long-range order in Floquet systems. Phys. Rev. B 94, 085112 (2016).
Russomanno, A., Santoro, G. E. & Fazio, R. Entanglement entropy in a periodically driven Ising chain. J. Stat. Mech. 2016, 073101 (2016).
Sen, A., Nandy, S. & Sengupta, K. Entanglement generation in periodically driven integrable systems: dynamical phase transitions and steady state. Phys. Rev. B 94, 214301 (2016).
von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of 1D interacting Floquet systems I: Abelian SPTs. Phys. Rev. B 93, 245145 (2016).
von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of one-dimensional interacting Floquet systems. II. Symmetry-broken phases. Phys. Rev. B 93, 245146 (2016).
Potter, A. C., Morimoto, T. & Vishwanath, A. Classification of interacting topological Floquet phases in one dimension. Phys. Rev. X 6, 041001 (2016).
Else, D. V. & Nayak, C. Classification of topological phases in periodically driven interacting systems. Phys. Rev. B 93, 201103(R) (2016).
Roy, R. & Harper, F. Abelian Floquet symmetry-protected topological phases in one dimension. Phys. Rev. B 94, 125105 (2016).
Fidkowski, L. & Kitaev, A. Effects of interactions on the topological classification of free fermion systems. Phys. Rev. B 81, 134509 (2010).
Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. The anomalous Floquet–Anderson insulator as a non-adiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).
Bordia, P. et al. Coupling identical 1D many-body localized systems. Phys. Rev. Lett. 116, 140401 (2016).
Choi, J.-y. et al. Exploring the many-body localization transition in two dimensions. Science 352, 1547–1552 (2016).
Else, D. V., Bauer, B. & Nayak, C. Prethermal phases of matter protected by time-translation symmetry. Phys. Rev. X 7, 011026 (2017).
Weidinger, S. A. & Knap, M. Floquet prethermalization and regimes of heating in a periodically driven, interacting quantum system. Preprint at http://arXiv.org/abs/1609.09089 (2016).
Abanin, D. A., De Roeck, W., Ho, W. W. & Huveneers, F. Effective Hamiltonians, prethermalization and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017).
Acknowledgements
We would like to thank A. Das, D. Huse, C. von Keyserlingk, V. Khemani, A. Lazarides and A. Polkovnikov for many useful discussions and for comments on the manuscript. This work was supported by the NSF-DMR via Grant No. 1311781 and the Alexander von Humboldt foundation via a Humboldt award (S.L.S.) as well as the Deutsche Forschungsgemeinschaft via SFB 1143 (R.M.).
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Moessner, R., Sondhi, S. Equilibration and order in quantum Floquet matter. Nature Phys 13, 424–428 (2017). https://doi.org/10.1038/nphys4106
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DOI: https://doi.org/10.1038/nphys4106
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