Topological Phenomena in Periodically Driven Systems: Disorder, Interactions, and Quasi-Steady States Erez Berg
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1 Topological Phenomena in Periodically Driven Systems: Disorder, Interactions, and Quasi-Steady States Erez Berg In collaboration with: Mark Rudner (Copenhagen) Netanel Lindner (Technion) Paraj Titum (Caltech Maryland) Gil Refael (Caltech)
2 New types of non-equilibrium topological phases unique to periodically-driven systems?
3 Outline Review: Floquet, Bloch, Floquet-Bloch Disorder, the Anomalous Floquet- Anderson insulator (AFAI), and non-adiabatic quantized pumping Interactions, thermalization in Floquet bands, and universal current carrying quasi-steady states
4 Floquet states and the quasi-energy Floquet formalism:
5 Floquet band topology induced by periodic driving T. Oka and H. Aoki, Phys. Rev. B 79, (2009). T. Kitagawa, et al., Phys. Rev. B 84, (2011). Z. Gu et al., Phys. Rev. Lett. 107, (2011).
6
7 Quasi-energy winding and adiabatic quantized transport, PRB 27, 6083 (1983)
8 Quasi-energy winding and adiabatic quantized transport, PRB 27, 6083 (1983) D. J. Adiabatic Thouless pump (83 ), PRB 27, 6083 (1983) Recent experiments of Thouless pump in an optical lattice S. Nakajima et. al., Nature Physics (2016) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, I. Bloch, Nature Physics (2016)
9 T. Kitagawa, EB, M. Rudner, and E. Demler PRB (2010) M. Rudner, N. Lindner, EB, and M. Levin, PRX (2013)
10 Outline Review: Floquet, Bloch, Floquet-Bloch Disorder, the Anomalous Floquet- Anderson insulator (AFAI), and non-adiabatic quantized pumping Interactions, thermalization in Floquet bands, and universal current carrying quasi-steady states
11
12 P. Titum, EB, M. Rudner, G. Refael, and N. Lindner, PRX (2015)
13 Non-adiabatic Quantized pumping current in the AFAI
14 Outline Review: Floquet, Bloch, Floquet-Bloch Disorder, the Anomalous Floquet- Anderson insulator (AFAI), and non-adiabatic quantized pumping Interactions, thermalization in Floquet bands, and universal current carrying quasi-steady states
15 See examples (plus others) A. Lazarides, A. Das, R. Moessner (PRL, 2014) L. D`Alessio, M. Rigol (PRX, 2014) P. Ponte, A. Chandran, Z. Papic, D. Abanin (Ann. Phys., 2014)
16 Topological features in a closed, interacting Floquet system? 1D system with non-zero winding numbers (Thouless pump) Initial state: partially filled right moving band (or filled with bosons) Gapless system: adiabatic theorem not applicable!
17 Chiral thermalization? If interband scattering is suppressed, particles can thermalize only within one of the chiral bands Intraband scattering Interband scattering
18 Chiral quasi-steady state Intermediate time quasi-steady state : Thermalization in a single Floquet band t t t intra inter current 1 T density M. Rudner, N. Lindner and EB, arxiv:
19 Outline Review: quantum driven systems: Floquet, Bloch, Floquet-Bloch Interactions in closed systems and thermalization Current-carrying quasi steady states
20 Simulation Single particle quasienergy spectrum: 0.5 Quasienergy k/
21 Simulation Intra-unit cell interaction, U = 3J 0
22 Inter-band relaxation rate
23 Frequency space analysis Matrix element for scattering controlled by the overlap of Fourier components of the scattering Floquet states i t () t e e z iz t z
24 Frequency space analysis Matrix element controlled by the overlap of Fourier components of the scattering Floquet states Intraband scattering Interband scattering Generically significant Suppressed for
25 Intraband scattering Significant overlap
26 Interband scattering Overlap exponentially suppressed
27 Interband scattering: Born approximation Γ 2P U 2 : does not account for observed Γ inter U Δ Δ ω
28 High order scattering T matrix formulation in extended (Fourier harmonic) Hilbert space: T Ω = U + UG 0 U + UG 0 UG 0 U + Born Approx.: Nth order: Optimal order: N Virtual states Fourier harmonic, m Δ Δ/(δmω) δmω Γ inter U Δ
29 Summary Periodically driven systems host a variety of topological phenomena, with no analogues in static systems. Chiral edge states with no Chern numbers Disorder: chiral edge states with fully localized bulk, non-adiabatic quantized charge pumping Interactions: in closed systems, driving generically leads to indefinite heating But unusual long-lived quasi-steady states are possible Thank you.
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