Topological Phases of Matter Out of Equilibrium Nigel Cooper T.C.M. - PowerPoint PPT Presentation

Topological Phases of Matter Out of Equilibrium Nigel Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge Solvay Workshop on Quantum Simulation ULB, Brussels, 18 February 2019 Max McGinley (Cambridge) Marcello Caio (KCL/Leiden), Gunnar Moller (Kent), Joe Bhaseen (KCL)

Topological Invariants 2 π ∫ closed surface 1 = (2 − 2 g ) κ dA 1 Gaussian curvature κ = R 1 R 2 2D Bloch Bands [Thouless, Kohmoto, Nightingale & den Nijs, PRL 1982] ν = 1 2 π ∫ BZ Chern number: d 2 k Ω k Berry curvature: Ω k = − i ∇ k × ⟨ u k | ∇ k u k ⟩ ⋅ ̂ z • cannot change under smooth deformations ν • Insulating bulk with gapless edge states ν Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Topological Insulators • Many generalisations when symmetries are included: [Hasan & Kane, RMP 2010] topological insulators/superconductors in all spatial dimensions ⇒ bulk gap + gapless surface states — Time reversal symmetry (non-magnetic system in vanishing magnetic field) — “Chiral” (sublattice) symmetry e.g. Su-Schrieffer-Heeger model [ARPES: Xia et al. , 2008] 0 0 0 0 A B A B A B A B ✓ ◆ ⇒ Detailed classification of topological matter at equilibrium [Here for free fermions, but also for strongly interacting systems] Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

⇒ Non-Equilibrium Dynamics? [unitary evolution] e.g. dynamical change in band topology E E E k k k ν =1 ν =0 ν =? time • Preparation of topological phases? • Is there a topological classification of non-equilibrium many-body states? Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Outline • Dynamics of Chern Insulators (2D) • Dynamics of Topological Phases in 1D • Topological Classification Out of Equilibrium Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Outline • Dynamics of Chern Insulators (2D) • Dynamics of Topological Phases in 1D • Topological Classification Out of Equilibrium Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

̂ ̂ Dynamics of Chern Insulators (2D) Quench: start in ground state of then time evolve under H i H f E E E k k k ν =1 ν =0 ν =? time Time-evolving Bloch state of fermion at k | u k ( t ) ⟩ = exp( − i ̂ H f k t ) | u k (0) ⟩ Ω k ( t ) = − i ∇ k × ⟨ u k ( t ) | ∇ k u k ( t ) ⟩ ⋅ ̂ z ⇒ Chern number of the many-body state is preserved [D’Alessio & Rigol, Nat. Commun. 2015; Caio, NRC & Bhaseen, PRL 2015] [“topological invariant” under smooth changes of the Bloch states] Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Dynamics of Chern Insulators: Physical Consequences • Obstruction to preparation of a state with differing Chern number [For slow ramps, , deviations can be small] τ ≫ L / 𝗐 • Chern number can be obtained by tomography of Bloch states [Two-band model: ] | u k ⟩ = cos( θ k /2) | A , k ⟩ + sin( θ k /2) e i ϕ k | B , k ⟩ [Fläschner et al. [Hamburg], Science 2016] • Topological of final Hamiltonian can be obtained by tracking the evolution of the Bloch states in time [Wang et al. [Tsinghua], PRL 2017; Tarnowski et al. [Hamburg], arXiv 2017] Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

̂ Dynamics of Chern Insulators in Real Space [Caio, Möller, NRC & Bhaseen, Nat. Phys. 2019] Haldane model [Haldane, PRL 1988] Local Chern Marker [Bianco & Resta, PRB 2011] – M + M Im ∑ c ( r α ) = − 4 π ⟨ r α s | ̂ x ( ̂ y ̂ P ̂ 1 − P ) ̂ P | r α s ⟩ + φ A c t 2 s = A , B y t 1 a x global conservation b ∫ c ( r ) d 2 r = 0 0 t = 0 c t = 2.5 − 5 t = 5 ∂ c ⇒ ∂ t = − ∇ ⋅ J c 0 10 20 30 40 y 0 y c Quench dynamics involves flow of the Chern marker Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Outline • Dynamics of Chern Insulators (2D) • Dynamics of Topological Phases in 1D • Topological Classification Out of Equilibrium Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Dynamics of Topological Phases in 1D (Free Fermions) In 1D all topological invariants can be determined from CS 1 = 1 2 π ∫ BZ dk ⟨ u k | ∂ k u k ⟩ Equivalently: Berry phase around the Brillouin zone (Zak phase) Only quantized in the presence of symmetries In 1D, topology must be protected by symmetry Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

⇒ Example: Su-Schrieffer-Heeger Model 0 0 0 0 A B A B A B A B H k = − ( ✓ ◆ ) = − h ( k ) ⋅ σ J ′ � + Je − ika 0 J ′ � + Je ika 0 “Chiral” (sublattice) symmetry ⇒ ⇒ h x + ih y = | h ( k ) | e i ϕ ( k ) h = ( h x , h y ,0) h y /J 2 CS 1 = N /2 N = 1 2 π ∫ BZ d ϕ winding number h x /J dk dk 2 integer J 0 = 1 . 5 J J 0 = 0 . 5 J − 2 Is this topological invariant preserved out of equilibrium? No… need to consider symmetries! Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Symmetry-Protected Topology Out of Equilibrium [Max McGinley & NRC, PRL 2018] • Start in ground state of then time evolve under ℋ ℋ i f • breaks symmetry ⇒ topological “invariant” can vary ℋ f [“explicit symmetry breaking”] • What if respects the symmetry? ℋ f Symmetry can still be broken! • Anti-unitary symmetries [ ] ⟨𝒫Ψ , 𝒫Φ⟩ = ⟨Φ , Ψ⟩ * 𝒫 e − i ℋ t 𝒫 − 1 = e + i ℋ t Symmetry broken in the non-equilibrium state [“dynamically induced symmetry breaking”] Topological invariant time-varying even if symmetries respected! Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Time-Varying : Physical Consequences CS 1 ( t ) [Max McGinley & NRC, PRL 2018] • Could be observed in Bloch state tomography [cf. Chern number] d dt CS 1 ( t ) = j ( t ) = · • Directly measure via Q ( t ) [cf. Chern number Hall conductance out of equilibrium] ≠ Example: quenches in a generalised SSH model AIII: chiral symmetry only BDI: time-reversal, particle-hole & chiral How can we define topology out of equilibrium? Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Outline • Dynamics of Chern Insulators (2D) • Dynamics of Topological Phases in 1D • Topological Classification Out of Equilibrium Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Topological Classification Out Of Equilibrium [Max McGinley & NRC, PRL 2018] • Equilibrium topological phase ⇒ gapless surface states • Non-equilibrium topological state ⇒ gapless entanglement spectrum | Ψ ( t ) ⟩ = ∑ e − λ i | ψ i L ⟩ ⊗ | ψ i R ⟩ i [Li & Haldane, PRL 2008] Example: quenches in a generalised SSH model AIII: chiral symmetry only BDI: time-reversal, particle-hole & chiral ⇒ meaningful topological classification out of equilibrium Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Generalization to Interacting SPT Phases [Max McGinley & NRC, PRL 2018] Example: Haldane phase of a S=1 spin chain is an SPT phase that can be protected by a variety of symmetries: • Time-reversal symmetry (anti-unitary) • Dihedral symmetry (unitary) Anti-unitary symmetries lost dynamically TRS only TRS and dihedral Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Generalization to other Spatial Dimensions “Ten-fold way” for free fermions [Chiu, Teo, Schnyder & Ryu, RMP 2016] [Time-reversal, particle-hole & chiral symmetries] symmetries spatial dimension δ Class T C S 0 1 2 3 4 5 6 7 A 0 0 0 Z 0 Z 0 Z 0 Z 0 AIII 0 0 1 0 Z 0 Z 0 Z 0 Z þ AI 0 0 Z 0 0 0 2 Z 0 Z 2 Z 2 þ þ BDI 1 Z 2 Z 0 0 0 2 Z 0 Z 2 þ D 0 0 Z 2 Z 2 Z 0 0 0 2 Z 0 þ DIII 1 0 Z 2 Z 2 Z 0 0 0 2 Z − AII 0 0 2 Z 0 Z 2 Z 2 Z 0 0 0 − CII 1 0 2 Z 0 Z 2 Z 2 Z 0 0 − − 2 Z Z 2 Z 2 Z C 0 0 0 0 0 0 − þ CI 1 0 0 0 2 Z 0 Z 2 Z 2 Z − Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium

Topological Classification Out Of Equilibrium “Ten-fold way” for free fermions [Chiu, Teo, Schnyder & Ryu, RMP 2016] [Time-reversal, particle-hole & chiral symmetries] Non-Equilibrium Classification [Max McGinley & NRC, arXiv:1811.00889] Class Symmetries Spatial dimension d 0 1 2 3 4 5 6 7 T C S A 0 0 0 0 0 0 0 Z Z Z Z AIII 0 0 1 0 Z → 0 0 Z → 0 0 Z → 0 0 Z → 0 AI + 0 0 0 0 0 2 Z 0 Z 2 → 0 Z 2 → 0 Z BDI + + 1 0 0 0 2 Z → 0 0 Z 2 → 0 Z 2 Z → Z 2 D 0 + 0 0 0 0 2 Z 0 Z 2 Z 2 Z DIII + 1 0 Z 2 → 0 Z 2 → 0 Z → 0 0 0 0 2 Z → 0 − AII 0 0 2 Z 0 Z 2 → 0 Z 2 → 0 0 0 0 Z − CII 1 0 2 Z → 0 0 Z 2 → 0 0 0 Z 2 Z → Z 2 − − C 0 0 0 0 2 Z 0 0 Z 2 Z 2 Z − CI + 1 0 0 0 2 Z → 0 0 Z 2 → 0 Z 2 → 0 Z → 0 − Nigel Cooper, University of Cambridge Topological Phases of Matter Out of Equilibrium