Quantum Quenches in Chern Insulators

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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:1504.01910 S. Baur & NRC, Phys. Rev. A 88, 033603 (2013).

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

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

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

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

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

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, 033603 (2013)]

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, 033603 (2013)] Fermionic Band Insulator: Non-adiabatic Effects of quenching between band topologies? [M.D. Caio, NRC & M.J. Bhaseen, arxiv:1504.01910]

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

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

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

Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Spectrum 4 Flux density n φ = α 2π 2 E/J 0-2 n φ = p/q: -4 0 0.2 0.4 0.6 0.8 1 α/2π q bands with (in general) non-zero Chern numbers weakly interacting BEC in band minimum vortex lattice with vortex density n φ

Harper-Hofstadter Model Adiabatic Route Harper-Hofstadter Spectrum 4 Flux density n φ = α 2π 2 E/J 0-2 n φ = p/q: -4 0 0.2 0.4 0.6 0.8 1 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?

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (2013)] e.g. n φ = 1/3 (α = 2π/3) 2π/3 2π/3 magnetic unit cell 2π/3

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (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

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (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

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (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

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (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

Harper-Hofstadter Model Adiabatic Route Adiabatic Route: Essential Idea [S. Baur & NRC, Phys. Rev. A 88, 033603 (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

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

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

[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

[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

: 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)

Quenches in the [Marcello Caio, NRC & Joe Bhaseen, arxiv:1504.01910] (a) M t 2 6 ν = 0 (b) E 2-2 -6 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

Quenches in the [Marcello Caio, NRC & Joe Bhaseen, arxiv:1504.01910] (a) M t 2 6 ν = 0 (b) E 2-2 -6 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)

(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]

(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

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

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:1409.6319]

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:1409.6319] Out of equilibrium, the Chern number of the state is different from that of the (new) Hamiltonian

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

(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?

(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

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 4 3 2 1 0 1 E 3 2 1 0 1 2 2 3 1.5 1.0 0.5 0.0 0.5 1.0 1.5 k x 3 1.5 1.0 0.5 0.0 0.5 1.0 1.5 k x Topological phase has edge states

Dynamics of Edge Currents Quench from topological to non-topological phase [t 1 = 1, t 2 = 1/3, ϕ = π/3, M = 1.4 1.6 (inset, 2.2)] J N (t) 2 J N (t) 4 2 0 2 1 2 4 6 t N = 30 (circles) N = 40 (crosses) 0 2 4 6 Red Line: Ground state current of final Hamiltonian t

Dynamics of Edge Currents Quench from topological to non-topological phase [t 1 = 1, t 2 = 1/3, ϕ = π/3, M = 1.4 1.6 (inset, 2.2)] J N (t) 2 J N (t) 4 2 0 2 1 2 4 6 t N = 30 (circles) N = 40 (crosses) 0 2 4 6 Red Line: Ground state current of final Hamiltonian fast relaxation to (close to) the ground state edge current of the new Hamiltonian t

Dynamics of Edge Currents Quench from non-topological to topological phase J edge (t) 4 3 2 1 0 1 2 4 6 t 2 Red Line: Ground state current of final Hamiltonian fast relaxation to (close to) the ground state edge current of the new Hamiltonian

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, 063629 (2011); Alba et al., PRL 107, 235301 (2011); Hauke et al., PRL 113, 045303 (2014)] [Fläschner et al.[hamburg], arxiv:1509.05763]

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, 063629 (2011); Alba et al., PRL 107, 235301 (2011); Hauke et al., PRL 113, 045303 (2014)] [Fläschner et al.[hamburg], arxiv:1509.05763] (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]

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.

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