Aditi Mitra New York University

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1 Entanglement dynamics following quantum quenches: pplications to d Floquet chern Insulator and 3d critical system diti Mitra New York University Supported by DOE-BES and NSF- DMR Daniel Yates, PhD student Yonah Lemonik, Postdoc 1

2 Quantum quench Ψ( t) = e ih f t Φ H i What happens when Hi is topological but Hf is not, what if Hi supports one phase (magnetically disordered.) and Hf supports another phase (critical or magnetically ordered). Density matrix: ρ( t) = Ψ( t) Ψ( t) IDEL CLOSED QUNTUM SYSTEM C C = + B B useful concept: Reduced density matrix ρ ( t ) = Tr B [ ρ] Ergodic systems: Region B acts as an effective reservoir for region

3 What does the entanglement entropy tell us? B ρ S = Tr B = Tr [ ψ ψ ] B [ ρ ln ρ ] L Ground state of Hamiltonians: rea law for short-range interaction, area law violated for long-range interactions. S S = = O( L O( L d 1 d ) ) Excited states: Volume law d: spatial dimensions rea-law => efficient numerical simulations using matrix product states (eg: DMRG) Non-ergodic states (many body localized states) show area law entanglement entropy even for excited eigen-states. 3

4 Entanglement Spectrum has even more information Entanglement spectrum: Eigenvalues of the reduced density matrix ρ ( t ) = Tr B [ ] ρ spectroscopic tool when there is no conventional order-parameter, examples being topological order. Reflects the bulk-boundary correspondence in topological systems where an entanglement cut in a spatially extended system now hosts edge-states. MORE ON THIS LTER! 4

5 What about dynamics such as pure-states evolving due to a quantum quench at time t=0? Concrete results only in 1-dimension and that too for short-ranged interactions. Entanglement growth in time reflects Lieb-Robinson bounds (the maximum speed v at which information propagates). S S L = a + vt, t < v L = O( L), t > v Quasi-particle picture: fter a quench, entangled quasi-particles pairs are emitted everywhere. -v B v -v v -v v L B t time such that L=vt, the region has got completely entangled with the region B. 5

6 Outline of talk: Quenches in higher spatial dimensions (d>1). ). Quenches in d= when final Hamiltonian has topological order due to periodic drive (Floquet Chern Insulator). Signatures of it in transport, RPES as well as in the entanglement statistics. Daniel Yates, Yonah Lemonik, M, PRB (016), in print B). Quenches near a critical point (d>). Looking for universal physics both in conventional quantities such as correlation functions, as well as in entanglement entropy and spectrum Yonah Lemonik, M, PRB (016) 6

7 Chern insulator: Quantum Hall effect in zero magnetic field Haldane, 1988 Time reversal symmetry broken by a staggered (but net zero) magnetic flux. Bands with nonzero Chern number and protected chiral edge modes. Haldane model can be realized experimentally using time-periodic drives! 7

8 Graphene irradiated with circularly polarized laser G. Jotzu, T. Esslinger et al.: Experimental realization of the topological Haldane model Nature 515, (014). Oka and oki PRB 009 in graphene Other models: Kitagawa et al PRB 011, Lindner et al Nature 011 Yao, MacDonald, Niu et al PRL 007 H U ( T eff + t, t) = k x σ x e τ z ih eff T + k y σ 0 σ:sublattice index y + σ τ z z Ω τ :K,K points Breaks time-reversal H Maps onto the Haldane model eff

9 STTIC CHERN INSULTOR: Energy bands Emax Emin C=-1 C=1 Quasi-momentum Floquet Brillouin zone. Width= Drive frequency FLOQUET CHERN INSULTOR: Quasi-energy band C=-1 C=1 Quasi-momentum For C=1, there are two possible ways in which topological edge-states connect the two bands. For Floquet Chern Insulators, C= Difference between the number of chiral edge-modes above and below the quasi-energy band (see: Rudner, Lindner, Berg, Levin PRX, 013) 9

10 Examples of some topological phases of the Floquet Chern Insulator C=1 Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries Graphene band width = 6 C=3 10

11 Dehghani and Mitra, 016 Band occupation for closed system for a laser quench Graphene band width = 6 High frequency limit a lot like Quenched Haldane Model: Ciao, Cooper, Bhaseen PRL 015 Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries Size of filled circles: population 11

12 Physical Observables : dc Hall conductance/conductivity Conductance = conductivity* d L Kubo formula for dc conductance (after time-averaging over laser cycle): Similar to nomalous Hall conductance except that Berry curvature is time-averaged over a laser cycle. σ Ω Ω xy k k ( ω = 0) = d kωk ( ρ ρ ) 1 = T T 0 1 π dtω e h k ( t) ( t) = Im[ ϕ( t) ϕ( t) ] x y kd ku ku kd xy ( ω = 0) ρ ρ = 1, σ Ce h Dehghani, Oka, and Mitra, 015 1

13 Hall conductance: chern number vs quench case (closed system) C=3 Dehghani, Oka, and Mitra, 015 Graphene band width = 6 th 13

14 What about topological protection of Floquet edge states? Next: picture in terms of entanglement properties. 14

15 Entanglement spectrum: Diagonalization of reduced density matrix or equivalently the correlation matrix (Wick s theorem) y B r For a spatially invariant system, eigenvalues of C are simply the occupation probabilities. Entanglement cut: Semi-infinite strip, Translationally invariant in y. arxiv: Entanglement properties of Floquet Chern insulators Daniel J. Yates, Yonah Lemonik, diti Mitra part of the entanglement spectrum is simply the occupation of the bulk bands but now projected on the ky axis. 15

16 Entanglement spectrum: off-resonant laser. Off-resonant laser Entanglement -gap Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries arxiv: Entanglement properties of Floquet Chern insulators 16 Daniel J. Yates, Yonah Lemonik, diti Mitra

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18 Resonant laser Entanglement spectrum: not all edge-states visible for the resonant laser. Entanglement -gap Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries Floquet Quench 18

19 Closing of the entanglement gap due to laser induced excitations Resonant laser Entanglement -gap Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries For a resonant laser, edge states co-exist with bulk excitations. Thus these states are no longer protected as they can hybridize with the bulk states. 19

20 Resonant laser Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries 0

21 Entanglement spectrum for slow quench: No adiabatic theorem for the resonant laser. Solid line: Edges at the center of the Floquet zone Dashed line: Edge-states at at Floquet zone-boundaries 1

22 Next topic: Non-topological but interacting system. Dynamics in a critical prethermal phase t * O( N)

23 Model t< 0 H i = d d 1 x Π + 1 ( ϕ) + 1 Ω ϕ 0 t> 0 H f = d d 1 x Π + 1 ( ϕ) quantum quench at t=0 where the mass has been suddenly changed, and interactions (anharmonicities) become important. + 1 rϕ We will be interested in the deep quench limit: + u 4 ϕ 4! N Ω 0 >> r N=Number of! components of ϕ 0 The model is capable of thermalizing, yet the system can get trapped in a metastable state described by an effective H*. Universal features in the timeevolution controlled by H* and memory of the initial state. ging in a quantum system: power-law rather than exponential decay of correlations controlled by universal exponents. 3

24 Results at long-wavelengths (qt, qt <<1). Quench at time=0 Origin of exponent: Scaling dimension of a small magnetic field applied before quench. Can also be thought of as a boundary (in time) critical exponent. Response function: Correlation function: G R ig ( q K ( q = 0, t = 0, t >> t') t >> t') ( t' ) t θ N ( ) ( ) θ t' t ' N θ t N θ 1 N O( ε ); ε = d = ε N N ging: No single time-scale determines the relaxation. Relaxation depends on the waiting time t at which the system has been perturbed. Quantum aging due to ballistic rather than diffusive excitations, leading to differences in the temporal behavior of aging. Chiocchetta, Tavora, Gambassi, M, PRB 015 and 016 4

25 Boson distribution function at the non-thermal fixed point shows universal power-laws Chiocchetta, Tavora, Gambassi, M, PRB 015 and 016 n( k) 1 k Short times kt <1 Increasing time θ N O( ε ); ε = d = ε N N n( k) 1 k θ N Long times kt >1 5

26 Light-cone dynamics in the interacting problem in d>-spatial Dimensions: Universal exponent related to aging. ε N time θ N Chiocchetta, Tavora, Gambassi, M, PRB 015 and 016 t r = t r << t O( ε ); ε = d = N G G G Response function (for quench at time=0) R R R ( r [ r [ r ( r, t; r' 0, t' ) G R = t t' ) = 0 = = t t',( Λt' >> 1)] = t t',( Λt' << 1)] = 1 r ( t' ) r ( d 1) / θ N ( d 1) / r >> t d-spatial dimensions Equal-time correlation function (for quench at time 0) G G G K K K ( t << r) = 0 (t = r) r (t >> r) r 1 ( d )/ 1 d + θ N 6

27 Entanglement dynamics for O(N) bosons in d=3: ρ ( t) = Tr B [ ψ ψ ] S ( t) = Tr B [ ρ ln ρ ] Main idea: For N= infty, Wicks theorem holds so that the reduced density matrix can be constructed from the two point correlation functions in region-. v -v v -v L v -v -v B v -v -v v v 7

28 Entanglement entropy after a quench in d=3: Good agreement with quasi-particle picture and qualitatively different from d=1 due to difference in geometry (cube vs line.) dispersion-1 dispersion- ρ ( t) S( t) = Tr B = Tr [ ψ ( t) ψ ( t) ] [ ρ ln ρ ] 8

29 Time-evolution of the entanglement spectrum (eigenvalues of the reduced density matrix) Remnants of the initial state 9

30 Entanglement spectrum (d=3) : sensitive to aging critical- exponent Free gapless system θ = 0 Interacting critical quench θ = 1 4 λ N ( L, t) L 1 = θ W( t / L) 30

31 Summary: Floquet systems are now being realized in many different ways. Closed quantum systems using cold-atoms, surface states of 3D topological insulators, photonic waveguides. Ideal quantum limit for the Hall conductance achievable provided the laser is highly off-resonant. Entanglement spectrum indicates whether edge-states might be protected or not from scattering off bulk states. For a quench to a critical state, universal exponents are revealed by studying low-lying entanglement eigenvalues. The entanglement entropy itself is dominated by microscopic non-universal features and can be understood in terms of ballistically propagating quasi-particles. 31

32 Hall conductance: chern number vs open case 3