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  • Open Access

Flow Equation Approach to Periodically Driven Quantum Systems

Michael Vogl*,†, Pontus Laurell*, and Aaron D. Barr

Gregory A. Fiete

  • Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

  • Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA; Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

  • *These authors contributed equally to this work.
  • †Corresponding author. ssss133@googlemail.com
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Phys. Rev. X 9, 021037 – Published 23 May, 2019

DOI: https://doi.org/10.1103/PhysRevX.9.021037

Abstract

We present a theoretical method to generate a highly accurate time-independent Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which renormalization-group–like flow equations are derived to produce the effective Hamiltonian. Our tractable method has a range of validity reaching into frequency—and drive strength—regimes that are usually inaccessible via high-frequency ω expansions in the parameter h/ω, where h is the upper limit for the strength of local interactions. We demonstrate exact properties of our approach on a simple toy model and test an approximate version of it on both interacting and noninteracting many-body Hamiltonians, where it offers an improvement over the more well-known Magnus expansion and other high-frequency expansions. For the interacting models, we compare our approximate results to those found via exact diagonalization. While the approximation generally performs better globally than other high-frequency approximations, the improvement is especially pronounced in the regime of lower frequencies and strong external driving. This regime is of special interest because of its proximity to the resonant regime where the effect of a periodic drive is the most dramatic. Our results open a new route towards identifying novel nonequilibrium regimes and behaviors in driven quantum many-particle systems.

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Popular Summary

One long-standing goal in physics is the ability to control the properties of quantum systems composed of many particles. This could allow researchers, for example, to tune a material to be insulating, superconducting, or something else entirely. A promising approach is to subject the system to periodically time-varying external fields, which can trigger phase transitions in the material. To predict and understand novel phases, researchers need effective mathematical models that accurately describe periodically driven systems. Here, we introduce a new method that extends current theoretical techniques into a new regime.

Much effort to date has focused on finding effective Hamiltonians in the analytically accessible high-frequency regime. However, this accessibility comes at a price: The necessary approximations are only valid in certain restricted situations. Going beyond these approximations is a long-standing problem, which hampers predictions of novel phases.

Our method provides a framework for developing Hamiltonians that work even when driving a system with a strong low-frequency external field. This advance means the low-frequency regime is more theoretically accessible than before. We apply our method to various quantum many-particle systems and compare the evolution of the system with the effective Hamiltonian obtained from our method and with the exact Hamiltonian to demonstrate its power.

We expect our method will help researchers map out the phase diagrams of various periodically driven quantum systems outside of the high-frequency regime, and we believe that it will provide a foundation for numerical and analytical approximation schemes.

Article Text

References (103)

  1. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium Dynamics of Closed Interacting Quantum Systems, Rev. Mod. Phys. 83, 863 (2011).
  2. A. Eckardt, Colloquium: Atomic Quantum Gases in Periodically Driven Optical Lattices, Rev. Mod. Phys. 89, 011004 (2017).
  3. I. Bloch, J. Dalibard, and W. Zwerger, Many-Body Physics with Ultracold Gases, Rev. Mod. Phys. 80, 885 (2008).
  4. J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Colloquium: Artificial Gauge Potentials for Neutral Atoms, Rev. Mod. Phys. 83, 1523 (2011).
  5. D. N. Basov, R. D. Averitt, and D. Hsieh, Towards Properties on Demand in Quantum Materials, Nat. Mat. 16, 1077 (2017).
  6. J. Zhang and R. D. Averitt, Dynamics and Control in Complex Transition Metal Oxides, Annu. Rev. Mater. Res. 44, 19 (2014).
  7. D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, and K. Haule, Electrodynamics of Correlated Electron Materials, Rev. Mod. Phys. 83, 471 (2011).
  8. C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani, and D. Mihailovic, Ultrafast Optical Spectroscopy of Strongly Correlated Materials and High-Temperature Superconductors: A Non-equilibrium Approach, Adv. Phys. 65, 58 (2016).
  9. M. Gandolfi, G. L. Celardo, F. Borgonovi, G. Ferrini, A. Avella, F. Banfi, and C. Giannetti, Emergent Ultrafast Phenomena in Correlated Oxides and Heterostructures, Phys. Scr. 92, 034004 (2017).
  10. J. H. Mentink and M. Eckstein, Ultrafast Quenching of the Exchange Interaction in a Mott Insulator, Phys. Rev. Lett. 113, 057201 (2014).
  11. J. H. Mentink, K. Balzer, and M. Eckstein, Ultrafast and Reversible Control of the Exchange Interaction in Mott Insulators, Nat. Commun. 6, 6708 (2015).
  12. D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Light-Induced Superconductivity in a Stripe-Ordered Cuprate, Science 331, 189 (2011).
  13. A. Cavalleri, Photo-Induced Superconductivity, Contemp. Phys. 59, 31 (2018).
  14. L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, and D. Mihailovic, Ultrafast Switching to a Stable Hidden Quantum State in an Electronic Crystal, Science 344, 177 (2014).
  15. Y. A. Gerasimenko, I. Vaskivskyi, and D. Mihailovic, Long Range Electronic Order in a Metastable State Created by Ultrafast Topological Transformation, arXiv:1704.08149.
  16. J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe, Observation of a Discrete Time Crystal, Nature (London) 543, 217 (2017).
  17. S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, and M. D. Lukin, Observation of Discrete Time-Crystalline Order in a Disordered Dipolar Many-Body System, Nature (London) 543, 221 (2017).
  18. A. C. Potter, T. Morimoto, and A. Vishwanath, Classification of Interacting Topological Floquet Phases in One Dimension, Phys. Rev. X 6, 041001 (2016).
  19. V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phase Structure of Driven Quantum Systems, Phys. Rev. Lett. 116, 250401 (2016).
  20. M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal High-Frequency Behavior of Periodically Driven Systems: From Dynamical Stabilization to Floquet Engineering, Adv. Phys. 64, 139 (2015).
  21. J. H. Shirley, Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time, Phys. Rev. 138, B979 (1965).
  22. R. Moessner and S. L. Sondhi, Equilibration and Order in Quantum Floquet Matter, Nat. Phys. 13, 424 (2017).
  23. A. Lazarides, A. Das, and R. Moessner, Periodic Thermodynamics of Isolated Quantum Systems, Phys. Rev. Lett. 112, 150401 (2014).
  24. W. Berdanier, M. Kolodrubetz, R. Vasseur, and J. E. Moore, Floquet Dynamics of Boundary-Driven Systems at Criticality, Phys. Rev. Lett. 118, 260602 (2017).
  25. X. Wen and J.-Q. Wu, Floquet Conformal Field Theory, arXiv:1805.00031.
  26. P. Ponte, A. Chandran, Z. Papi, and D. A. Abanin, Periodically Driven Ergodic and Many-Body Localized Quantum Systems, Ann. Phys. (Amsterdam) 353, 196 (2015).
  27. P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, Many-Body Localization in Periodically Driven Systems, Phys. Rev. Lett. 114, 140401 (2015).
  28. A. Lazarides, A. Das, and R. Moessner, Equilibrium States of Generic Quantum Systems Subject to Periodic Driving, Phys. Rev. E 90, 012110 (2014).
  29. Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Observation of Floquet-Bloch States on the Surface of a Topological Insulator, Science 342, 453 (2013).
  30. L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Majorana Fermions in Equilibrium and in Driven Cold-Atom Quantum Wires, Phys. Rev. Lett. 106, 220402 (2011).
  31. P. Molignini, E. van Nieuwenburg, and R. Chitra, Sensing Floquet-Majorana Fermions via Heat Transfer, Phys. Rev. B 96, 125144 (2017).
  32. K. Jiménez-García, L. J. LeBlanc, R. A. Williams, M. C. Beeler, C. Qu, M. Gong, C. Zhang, and I. B. Spielman, Tunable Spin-Orbit Coupling via Strong Driving in Ultracold-Atom Systems, Phys. Rev. Lett. 114, 125301 (2015).
  33. J. Klinovaja, P. Stano, and D. Loss, Topological Floquet Phases in Driven Coupled Rashba Nanowires, Phys. Rev. Lett. 116, 176401 (2016).
  34. L. Du, X. Zhou, and G. A. Fiete, Quadratic Band Touching Points and Flat Bands in Two-Dimensional Topological Floquet Systems, Phys. Rev. B 95, 035136 (2017).
  35. Q. Chen, L. Du, and G. A. Fiete, Floquet Band Structure of a Semi-Dirac System, Phys. Rev. B 97, 035422 (2018).
  36. P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H. Lindner, Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump, Phys. Rev. X 6, 021013 (2016).
  37. L. D’Alessio and M. Rigol, Dynamical Preparation of Floquet Chern Insulators, Nat. Commun. 6, 8336 (2015).
  38. I. Martin, G. Refael, and B. Halperin, Topological Frequency Conversion in Strongly Driven Quantum Systems, Phys. Rev. X 7, 041008 (2017).
  39. T. Oka and H. Aoki, Photovoltaic Hall Effect in Graphene, Phys. Rev. B 79, 081406(R) (2009).
  40. N. H. Lindner, G. Refael, and V. Galitski, Floquet Topological Insulator in Semiconductor Quantum Wells, Nat. Phys. 7, 490 (2011).
  41. T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Transport Properties of Nonequilibrium Systems under the Application of Light: Photoinduced Quantum Hall Insulators without Landau Levels, Phys. Rev. B 84, 235108 (2011).
  42. Z. Gu, H. A. Fertig, D. P. Arovas, and A. Auerbach, Floquet Spectrum and Transport through an Irradiated Graphene Ribbon, Phys. Rev. Lett. 107, 216601 (2011).
  43. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems, Phys. Rev. X 3, 031005 (2013).
  44. G. Usaj, P. M. Perez-Piskunow, L. E. F. F. Torres, and C. A. Balseiro, Irradiated Graphene as a Tunable Floquet Topological Insulator, Phys. Rev. B 90, 115423 (2014).
  45. A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E. Arimondo, Coherent Control of Dressed Matter Waves, Phys. Rev. Lett. 102, 100403 (2009).
  46. A. Eckardt, C. Weiss, and M. Holthaus, Superfluid-Insulator Transition in a Periodically Driven Optical Lattice, Phys. Rev. Lett. 95, 260404 (2005).
  47. N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. Vishwanath, Discrete Time Crystals: Rigidity, Criticality, and Realizations, Phys. Rev. Lett. 118, 030401 (2017).
  48. I.-D. Potirniche, A. C. Potter, M. Schleier-Smith, A. Vishwanath, and N. Y. Yao, Floquet Symmetry-Protected Topological Phases in Cold-Atom Systems, Phys. Rev. Lett. 119, 123601 (2017).
  49. A. Lerose, J. Marino, A. Gambassi, and A. Silva, Quantum Many-Body Kapitza Phases of Periodically Driven Spin Systems,” arXiv:1803.04490.
  50. J. Rehn, A. Lazarides, F. Pollmann, and R. Moessner, How Periodic Driving Heats a Disordered Quantum Spin Chain, Phys. Rev. B 94, 020201(R) (2016).
  51. L. D’Alessio and M. Rigol, Long-Time Behavior of Isolated Periodically Driven Interacting Lattice Systems, Phys. Rev. X 4, 041048 (2014).
  52. D. A. Abanin, W. De Roeck, and F. Huveneers, Exponentially Slow Heating in Periodically Driven Many-Body Systems, Phys. Rev. Lett. 115, 256803 (2015).
  53. D. A. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, Effective Hamiltonians, Prethermalization, and Slow Energy Absorption in Periodically Driven Many-Body Systems, Phys. Rev. B 95, 014112 (2017).
  54. T. Mori, T. Kuwahara, and K. Saito, Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems, Phys. Rev. Lett. 116, 120401 (2016).
  55. D. V. Else, B. Bauer, and C. Nayak, Prethermal Phases of Matter Protected by Time-Translation Symmetry, Phys. Rev. X 7, 011026 (2017).
  56. E. Canovi, M. Kollar, and M. Eckstein, Stroboscopic Prethermalization in Weakly Interacting Periodically Driven Systems, Phys. Rev. E 93, 012130 (2016).
  57. F. Machado, G. D. Meyer, D. V. Else, C. Nayak, and N. Y. Yao, Exponentially Slow Heating in Short and Long-Range Interacting Floquet Systems, arXiv:1708.01620.
  58. S. A. Weidinger and M. Knap, Floquet Prethermalization and Regimes of Heating in a Periodically Driven, Interacting Quantum System, Sci. Rep. 7, 45382 (2017).
  59. T. Kuwahara, T. Mori, and K. Saito, Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems, Ann. Phys. (Amsterdam) 367, 96 (2016).
  60. M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model, Phys. Rev. Lett. 110, 135704 (2013).
  61. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System, Phys. Rev. Lett. 119, 080501 (2017).
  62. N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Observation of a Dynamical Topological Phase Transition, Nat. Phys. 14, 265 (2018).
  63. L. Du and G. A. Fiete, Dynamical Recovery of SU(2) Symmetry in the Mass-Quenched Hubbard Model, Phys. Rev. B 97, 085152 (2018).
  64. Yu.-A. Chen, S. Nascimbène, M. Aidelsburger, M. Atala, S. Trotzky, and I. Bloch, Controlling Correlated Tunneling and Superexchange Interactions with ac-Driven Optical Lattices, Phys. Rev. Lett. 107, 210405 (2011).
  65. C. Dutreix and M. I. Katsnelson, Dynamical Control of Electron-Phonon Interactions with High-Frequency Light, Phys. Rev. B 95, 024306 (2017).
  66. M. Bukov, S. Gopalakrishnan, M. Knap, and E. Demler, Prethermal Floquet Steady States and Instabilities in the Periodically Driven, Weakly Interacting Bose-Hubbard Model, Phys. Rev. Lett. 115, 205301 (2015).
  67. M. Bukov, M. Kolodrubetz, and A. Polkovnikov, Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields, Phys. Rev. Lett. 116, 125301 (2016).
  68. F. Peronaci, M. Schiró, and O. Parcollet, Resonant Thermalization of Periodically Driven Strongly Correlated Electrons, Phys. Rev. Lett. 120, 197601 (2018).
  69. N. H. Lindner, E. Berg, and M. S. Rudner, Universal Chiral Quasisteady States in Periodically Driven Many-Body Systems, Phys. Rev. X 7, 011018 (2017).
  70. K. Plekhanov, G. Roux, and K. Le Hur, Floquet Engineering of Haldane Chern Insulators and Chiral Bosonic Phase Transitions, Phys. Rev. B 95, 045102 (2017).
  71. S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The Magnus Expansion and Some of Its Applications, Phys. Rep. 470, 151 (2009).
  72. E. B. Fel’dman, On the Convergence of the Magnus Expansion for Spin Systems in Periodic Magnetic Fields, Phys. Lett. A 104, 479 (1984).
  73. W. Magnus, On the Exponential Solution of Differential Equations for a Linear Operator, Commun. Pure Appl. Math. 7, 649 (1954).
  74. S. Rahav, I. Gilary, and S. Fishman, Effective Hamiltonians for Periodically Driven Systems, Phys. Rev. A 68, 013820 (2003).
  75. N. Goldman and J. Dalibard, Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields, Phys. Rev. X 4, 031027 (2014).
  76. A. P. Itin and M. I. Katsnelson, Effective Hamiltonians for Rapidly Driven Many-Body Lattice Systems: Induced Exchange Interactions and Density-Dependent Hoppings, Phys. Rev. Lett. 115, 075301 (2015).
  77. A. Eckardt and E. Anisimovas, High-Frequency Approximation for Periodically Driven Quantum Systems from a Floquet-Space Perspective, New J. Phys. 17, 093039 (2015).
  78. T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, Brillouin-Wigner Theory for High-Frequency Expansion in Periodically Driven Systems: Application to Floquet Topological Insulators, Phys. Rev. B 93, 144307 (2016).
  79. P. Mohan, R. Saxena, A. Kundu, and S. Rao, Brillouin-Wigner Theory for Floquet Topological Phase Transitions in Spin-Orbit-Coupled Materials, Phys. Rev. B 94, 235419 (2016).
  80. M. M. Maricq, Application of Average Hamiltonian Theory to the NMR of Solids, Phys. Rev. B 25, 6622 (1982).
  81. S. Vajna, K. Klobas, T. Prosen, and A. Polkovnikov, Replica Resummation of the Baker-Campbell-Hausdorff Series, Phys. Rev. Lett. 120, 200607 (2018).
  82. P. Bordia, H. Lüschen, U. Schneider, M. Knap, and I. Bloch, Periodically Driving a Many-Body Localized Quantum System, Nat. Phys. 13, 460 (2017).
  83. N. Goldman, J. Dalibard, M. Aidelsburger, and N. R. Cooper, Periodically Driven Quantum Matter: The Case of Resonant Modulations, Phys. Rev. A 91, 033632 (2015).
  84. A. Haldar, R. Moessner, and A. Das, Onset of Floquet Thermalization, Phys. Rev. B 97, 245122 (2018).
  85. R. Shankar, Renormalization-Group Approach to Interacting Fermions, Rev. Mod. Phys. 66, 129 (1994).
  86. F. Wegner, Flow-Equations for Hamiltonians, Ann. Phys. (Berlin) 506, 77 (1994).
  87. S. Kehrein, The Flow Equation Approach to Many-Particle Systems, Springer Tracts in Modern Physics (Springer, Berlin, Heidelberg, 2007).
  88. S. J. Thomson and M. Schiró, Time Evolution of Many-Body Localized Systems with the Flow Equation Approach, Phys. Rev. B 97, 060201(R) (2018).
  89. A. Verdeny, A. Mielke, and F. Mintert, Accurate Effective Hamiltonians via Unitary Flow in Floquet Space, Phys. Rev. Lett. 111, 175301 (2013).
  90. H. Sambe, Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field, Phys. Rev. A 7, 2203 (1973).
  91. T. Giamarchi, Quantum Physics in One Dimension, International Series of Monographs on Physics (Oxford University, New York, 2004).
  92. K. Nomura and K. Okamoto, Phase Diagram of S=1/2 Antiferromagnetic XXZ Chain with Next-Nearest-Neighbor Interactions, J. Phys. Soc. Jap. 62, 1123 (1993).
  93. F. D. M. Haldane, Spontaneous Dimerization in the S=12 Heisenberg Antiferromagnetic Chain with Competing Interactions, Phys. Rev. B 25, 4925 (1982).
  94. C. Tomaras and S. Kehrein, Scaling Approach for the Time-Dependent Kondo Model, Europhys. Lett. 93, 47011 (2011).
  95. M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal High-Frequency Behavior of Periodically Driven Systems: From Dynamical Stabilization to Floquet Engineering, Adv. Phys. 64, 139 (2015).
  96. G. Birkhoff and G. Rota, Ordinary Differential Equations (Wiley, New York, 1989), ISBN [Amazon][WorldCat].
  97. M. Rodriguez-Vega, M. Lentz, and B. Seradjeh, Floquet Perturbation Theory: Formalism and Application to Low-Frequency Limit, New J. Phys. 20, 093022 (2018).
  98. M. Vogl, P. Laurell, A. D. Barr, and G. A. Fiete, Analogue of Hamilton-Jacobi Theory for the Time-Evolution Operator, arXiv:1902.07237.
  99. P. Weinberg and M. Bukov, QuSpin: A Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems Part I: Spin Chains, SciPost Phys. 2, 003 (2017).
  100. A. W. Sandvik, Computational Studies of Quantum Spin Systems, AIP Conf. Proc. 1297, 135 (2010).
  101. C. Zhang, F. Pollmann, S. L. Sondhi, and R. Moessner, Density-Matrix Renormalization Group Study of Many-Body Localization in Floquet Eigenstates, Ann. Phys. (Berlin) 529, 1600294 (2017).
  102. H. De Raedt and B. De Raedt, Applications of the Generalized Trotter Formula, Phys. Rev. A 28, 3575 (1983).
  103. E. Polizzi, Density-Matrix-Based Algorithm for Solving Eigenvalue Problems, Phys. Rev. B 79, 115112 (2009).

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Phys. Rev. X 9, 021037– Published 23 May, 2019
  • Received 8 August 2018
  • Revised 19 March 2019
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