Quantum Quenches in Chern Insulators

Transcription

1 Quantum Quenches in Chern Insulators Nigel Cooper Cavendish Laboratory, University of Cambridge CUA Seminar M.I.T., November 10th, 2015 Marcello Caio & Joe Bhaseen (KCL), Stefan Baur (Cambridge) M.D. Caio, NRC & M.J. Bhaseen, arxiv: S. Baur & NRC, Phys. Rev. A 88, (2013).

2 Topological Invariants Gaussian curvature κ = 1 R 1 R 2 negative, zero and positive κ 1 κ da = (2 2g) Gauss-Bonnet Theorem 2π closed surface genus g = 0, 1, 2,... for sphere, torus, 2-hole torus... Topological invariant: g cannot change under smooth deformations

3 Topological Features of 2D Bands E [Thouless, Kohmoto, Nightingale & den Nijs (1982)] Chern number ν = 1 d 2 k Ω k 2π BZ k Berry curvature Ω k = i k u k u ẑ crystal momentum k, Bloch state u k Topological invariant: ν cannot change under smooth variations of the energy band

4 Physical Consequences E Chern band filled with fermions ( Chern insulator ): k quantized Hall effect, σ xy = ν e2 h [TKNN (1982)]

5 Physical Consequences E Chern band filled with fermions ( Chern insulator ): k quantized Hall effect, σ xy = ν e2 h [TKNN (1982)] ν gapless chiral edge states

6 Dynamical changes in band topology? time, t E E E k k k ν=0 ν=? ν=1

7 Dynamical changes in band topology? E time, t E E k k k ν=0 ν=? ν=1 Bosons / BEC: Adiabatic Adiabatic formation of vortex lattice [S. Baur & NRC, Phys. Rev. A 88, (2013)]

8 Dynamical changes in band topology? time, t E E E k k k ν=0 ν=? ν=1 Bosons / BEC: Adiabatic Adiabatic formation of vortex lattice [S. Baur & NRC, Phys. Rev. A 88, (2013)] Fermionic Band Insulator: Non-adiabatic Effects of quenching between band topologies? [M.D. Caio, NRC & M.J. Bhaseen, arxiv: ]

9 Outline Bosons: Adiabatic Formation of Vortex Lattice Harper-Hofstadter Model Adiabatic Route

10 Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Model Effective magnetic field flux density n φ = α 2π

11 Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Model Effective magnetic field flux density n φ = α 2π Imprint phases on tunneling matrix elements [Jaksch & Zoller 03; Mueller 04; Sørensen, Demler & Lukin 05; Gerbier & Dalibard 2010; Struck et al. (2012)...] e.g. tilted lattice [MIT,LMU] ω 1 ω 2 J eff Ω 1Ω 2 inherits phase difference of the Raman beams

12 Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Spectrum 4 Flux density n φ = α 2π 2 E/J 0-2 n φ = p/q: α/2π q bands with (in general) non-zero Chern numbers weakly interacting BEC in band minimum vortex lattice with vortex density n φ

13 Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Spectrum 4 Flux density n φ = α 2π 2 E/J 0-2 n φ = p/q: α/2π q bands with (in general) non-zero Chern numbers weakly interacting BEC in band minimum vortex lattice with vortex density n φ Can one adiabatically create such high vortex densities?

14 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) 2π/3 2π/3 magnetic unit cell 2π/3

15 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) magnetic unit cell 2π/3 2π/3 2π/3 α 2α α Je iα Je iα J

16 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) magnetic unit cell 2π/3 2π/3 2π/3 α 2α α Je iα Je iα J For fixed unit cell, vary phase α = 0 2π/3

17 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) magnetic unit cell 2π/3 2π/3 2π/3 α 2α α Je iα Je iα J For fixed unit cell, vary phase α = 0 2π/3 e.g. RF + Raman Je iφ 2π i = J RF + J Raman e 3a y

18 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) magnetic unit cell 2π/3 2π/3 2π/3 α 2α α Je iα Je iα J For fixed unit cell, vary phase α = 0 2π/3 e.g. RF + Raman Je iφ 2π i = J RF + J Raman e 3a y adiabatic path from uniform BEC to vortex lattice

19 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, (2013)] e.g. n φ = 1/3 (α = 2π/3) magnetic unit cell 2π/3 2π/3 2π/3 α 2α α Je iα Je iα J For fixed unit cell, vary phase α = 0 2π/3 e.g. RF + Raman Je iφ 2π i = J RF + J Raman e 3a y adiabatic path from uniform BEC to vortex lattice E k E k E k E k α=0 α=2π/3

20 Harper-Hofstadter Model Adiabatic Route Adiabatic Route: more specifically... Repulsive interactions select a particular vortex lattice unit cell Same periodicity as 2π i(x+y) J x,y = J RF + J Raman e 3a [cf. MIT, LMU] BEC is loaded into the stable vortex lattice as α = 0 2π/3

21 Outline Bosons: Adiabatic Formation of Vortex Lattice Harper-Hofstadter Model Adiabatic Route

22 [F. D. M. Haldane, PRL 61, 2015 (1988)] Cold atom realization: [Jotzu et al. [ETH], Nature 515, 237 (2014)] Honeycomb lattice detection A, of B local sublattices Berry curvature. ) Ĥ= t 1 (ĉ i ĉj + h.c. i,j

23 [F. D. M. Haldane, PRL 61, 2015 (1988)] Cold atom realization: [Jotzu et al. [ETH], Nature 515, 237 (2014)] Honeycomb lattice detection A, of B local sublattices Berry curvature. ) Ĥ= t 1 (ĉ i ĉj + h.c. i,j ) t 2 (e iϕ ij ĉ i ĉj + h.c. i,j nearest neighbour next n.n. +M i A ˆn i M i B ˆn i ϕ ij = ±ϕ breaks time-reversal symmetry M breaks inversion symmetry band gap

24 : Phase Diagram Boundaries where gaps close at two Dirac points [After Jotzu et al. [ETH], Nature 515, 237 (2014)] Non-topological (ν = 0) Topological (ν = ±1)

25 Quenches in the [Marcello Caio, NRC & Joe Bhaseen, arxiv: ] (a) M t 2 6 ν = 0 (b) E ν = 1 π (M 0, ϕ 0 ) ν = 1 Change Chern number of lowest band (sign of m α ) (M, ϕ) 2π ϕ k y k x ( ) +m α c [k x + iαk y ] c [k x iαk y ] m α

26 Quenches in the [Marcello Caio, NRC & Joe Bhaseen, arxiv: ] (a) M t 2 6 ν = 0 (b) E ν = 1 π (M 0, ϕ 0 ) ν = 1 Change Chern number of lowest band (sign of m α ) (M, ϕ) 2π ϕ k y k x ( ) +m α c [k x + iαk y ] c [k x iαk y ] m α non-interacting fermions isolated system (unitary evolution)

27 (1) Momentum space: time-evolution of Chern number, ν Occupied single particle states: ψ α (k) = a α (k)e ie l α(k)t l α (k) + b α (k)e ie u α(k)t u α (k) [l/u denote lower/upper bands]

28 (1) Momentum space: time-evolution of Chern number, ν Occupied single particle states: ψ α (k) = a α (k)e ieα(k)t l l α (k) + b α (k)e ieα(k)t u u α (k) [l/u denote lower/upper bands] Chern number of the state ν(t)= ( ) 1 α sign m α 2 b α(0) 2 α=±1 b α ( ) a α ( ) cos[(e u α( ) E l α( ))t + δ] But, b α ( ) = 0 always... so Chern number of state unchanged

29 Band Hamiltonian describes a spin in an effective magnetic field Ĥ k = h k ˆσ band gap h k 0

30 Band Hamiltonian describes a spin in an effective magnetic field Ĥ k = h k ˆσ band gap h k 0 ν is the number of times h k h k wraps the sphere in the BZ e.g. BZ of the honeycomb lattice The spins precess in new h k, but preserve their winding number [D Alessio & Rigol, arxiv: ]

31 Band Hamiltonian describes a spin in an effective magnetic field Ĥ k = h k ˆσ band gap h k 0 ν is the number of times h k h k wraps the sphere in the BZ e.g. BZ of the honeycomb lattice The spins precess in new h k, but preserve their winding number [D Alessio & Rigol, arxiv: ] Out of equilibrium, the Chern number of the state is different from that of the (new) Hamiltonian

32 (2) Real Space: Edge States and Edge Currents At equilibrium, the Chern insulator has ν gapless edge states

33 (2) Real Space: Edge States and Edge Currents At equilibrium, the Chern insulator has ν gapless edge states How do edge currents change after a quench to new Hamiltonian?

34 (2) Real Space: Edge States and Edge Currents At equilibrium, the Chern insulator has ν gapless edge states How do edge currents change after a quench to new Hamiltonian? Finite-width strip (infinite length) N m N rows of lattice sites n + 1 n n 1 y x {mna} {mnb} ϕ 1

35 Edge States t 1 = 1, t 2 = 1/3, M = 1, N = 20 Non-topological (ϕ = π/6) Topological (ϕ = π/3) 5 4 E E k x k x Topological phase has edge states

36 Dynamics of Edge Currents Quench from topological to non-topological phase [t 1 = 1, t 2 = 1/3, ϕ = π/3, M = (inset, 2.2)] J N (t) 2 J N (t) t N = 30 (circles) N = 40 (crosses) Red Line: Ground state current of final Hamiltonian t

37 Dynamics of Edge Currents Quench from topological to non-topological phase [t 1 = 1, t 2 = 1/3, ϕ = π/3, M = (inset, 2.2)] J N (t) 2 J N (t) t N = 30 (circles) N = 40 (crosses) Red Line: Ground state current of final Hamiltonian fast relaxation to (close to) the ground state edge current of the new Hamiltonian t

38 Dynamics of Edge Currents Quench from non-topological to topological phase J edge (t) t 2 Red Line: Ground state current of final Hamiltonian fast relaxation to (close to) the ground state edge current of the new Hamiltonian

39 Experimental Consequences (1) Momentum space: Preservation of the Chern number ν Measure the wave function, e.g. by time-of-flight [Zhao et al., PRA 84, (2011); Alba et al., PRL 107, (2011); Hauke et al., PRL 113, (2014)] [Fläschner et al.[hamburg], arxiv: ]

40 Experimental Consequences (1) Momentum space: Preservation of the Chern number ν Measure the wave function, e.g. by time-of-flight [Zhao et al., PRA 84, (2011); Alba et al., PRL 107, (2011); Hauke et al., PRL 113, (2014)] [Fläschner et al.[hamburg], arxiv: ] (2) Real space: Relaxation dynamics of edge currents Measure currents, e.g. as for two-leg ladders [M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes & I. Bloch, Nat. Phys. 10, 588 (2014)] [Local version (microscope) should show light-cone spreading]

41 Summary Optical lattices allow dynamical changes in the topology of energy bands. For a BEC, the evolution can be adiabatic: allows adiabatic preparation dense vortex lattices. For fermionic band insulators, the dynamics is non-adiabatic: the Chern number of the state is preserved; edge currents quickly relax to (close to) the equilibrium for the new Hamiltonian. Out of equilibrium there can be a sharp distinction between the topology of the state and local observables.