Reservoir-induced topological order & quantized transport in - PowerPoint PPT Presentation

Reservoir-induced topological order & quantized transport in open systems Michael Fleischhauer & Dominik Linzner Dept. of Physics & research center OPTIMAS Technische Universität Kaiserslautern Bad Honnef 09.05.2016 picture: wikipedia

topological states exo$c quantum states topological protec$on Abelian & non-Abelian anyons protected edge states & edge transport but: in general no protection against losses

topological order in the steady state of an open system ??

open-system dynamics open dynamics drives the system to a steady state steady state: attractor

robustness of the steady state d d t ρ = L ρ L ρ λ = − λ ρ λ gapped open systems Re[ λ ] damping gap λ = 0 parameter space

outline • topological invariants & open systems • Su-Schrieffer-Heeger model & Thouless pump • quantized topological transport in open spin chain with interactions • detection of topological invariant

outline • topological invariants & open systems • Su-Schrieffer-Heeger model & Thouless pump • quantized topological transport in open spin chain with interactions • detection of topological invariant

outline • topological invariants & open systems • Su-Schrieffer-Heeger model & Thouless pump • quantized topological transport in open spin chain with interactions • detection of topological invariant

outline • topological invariants & open systems • Su-Schrieffer-Heeger model & Thouless pump • quantized topological transport in open spin chain with interactions • detection of topological invariant

topological invariants & open systems

topology Möbius strip: locally indistinguishable

topology Möbius strip: differ by global properties !

topological invariants: geometric phases a Zak (Berry) phase Z π /a x 0 dk h u k | i ∂ k | u k i φ Zak = Zak PRL (1989) − π /a Zak = φ Zak + 2 π k i = e ikx 0 | u k i | u 0 φ 0 choice of origin matters a x 0 Chern number C = i Z Z n o d 2 k h ∂ k y u k | ∂ k x u k i � h ∂ k x u k | ∂ k y u k i ∈ Z 2 π BZ no global gauge C 6 = 0

geometric phases for density matrices Uhlmann connection ρ = w w † gauge degree of freedom: U(N) w † → U † w † w → w U O. Viyuela, et al. Phys. Rev. Lett. (2014) Z. Huang, D. P. Arovas, Phys. Rev. Lett. (2014) U(1) Uhlmann phase I e i φ = w ∂ λ w † ⇤ ⇥ d λ Tr

Berry phases for density matrices J. C. Budich, S. Diehl 1501.04135: X d j ( k ) ˆ H ( k ) = σ j finite-T state of a Chern insulator j d 1 = sin( k x ) d 2 = 3 sin( k y ) d 3 = 1 − cos( k x ) − cos( k y ) k y k x f U f U φ ( k y ) φ ( k x ) 3 3 2 2 1 1 3 k y 3 k x - 3 - 2 - 1 1 2 - 3 - 2 - 1 1 2 - 1 - 1 - 2 - 2 - 3 - 3 C = 1 ✓ ∂φ ( k y ) ◆ 6 = C 0 = 1 ✓ ∂φ ( k x ) ◆ Z Z dk y dk x 2 π 2 π ∂ k y ∂ k x Furthermore without constraints: trivial global gauge w = √ ρ

non-interacting fermions C.E. Bardyn, et al. New J. Phys (2013) Gaussian systems c ( † ) X c † H = h ij ˆ c j i ˆ L j ∼ α ˆ j + β ˆ c j ij covariance matrix � c † ρ w i w j } Γ jk ∼ Im Tr c i ± ˆ w i ∼ ˆ i density matrix ⇢ � − i 2 w j Γ jk w k ρ ∼ exp

non-interacting fermions topological classification in terms of Γ ij ✓ ◆ γ 11 γ 12 ∼ (1 + ~ n ( k ) · ~ � ) γ ( k ) = γ 21 γ 22 topological phase transition (I) closing of the damping gap (criticality) (II) closing of the purity gap = gap of effective Hamiltonian ⇢ � − i X 2 w j Γ jk w k ρ ∼ exp H e ff = i Γ jk w j w k jk � beyond Gaussian systems ??

polarization Thouless, Kohmoto, Nightingale, den Nijs (TKNN) PRL (1982) topology �� �� quantized bulk transport Zak phase & Polarization King-Smith, Vanderbilt PRB (1983) w ( x ) Z P = dx w ∗ ( x ) x w ( x ) ∆ P = a 2 π ∆ φ Zak

quantization of Hall conductance x Z 2 π /a C = 1 dk y ∂ k y φ Zak ( k y ) 2 π ~ 0 E y dk y = E y dt σ xy = j x = dP = dP 1 = C E y dt E y dk y

Su-Schrieffer-Heeger model & Thouless pump Su, Schrieffer, Heeger, PRL (1979) D.J. Thouless, PRB (1983) Attala et al. (I. Bloch), Nature Physics (2013)

SSH Model: free fermions on a superlattice with inversion symmetry t 2 t 1 X c † X c † H = − t 1 c i ˆ i +1 − t 2 c i ˆ i +1 + h.a. ˆ ˆ even odd E g ∼ t 1 − t 2 à half filling = band insulator of lower sub-band

SSH � Rice Mele Hamiltonian X ! X c † X c † c † X c † H = − t 1 c i +1 − t 2 c i +1 + ∆ c i − c i h.a. ˆ i ˆ ˆ i ˆ ˆ i ˆ ˆ i ˆ even even odd odd Inversion symmeric SSH symmetry breaking term M.J. Rice & E.J. Mele, PRL(1982) topological t 1 = t 2 phase transition φ Zak = π φ Zak = 0 locally indistinguishable

Rice-Mele Hamiltonian breaking inversion symmetry & Thouless pump X ! X c † X c † c † X c † c i +1 + ∆ H = − t 1 c i +1 − t 2 c i − c i h.a. ˆ i ˆ ˆ i ˆ ˆ i ˆ ˆ i ˆ even even odd odd

Rice-Mele Hamiltonian breaking inversion symmetry & Thouless pump X ! X c † X c † c † X c † c i +1 + ∆ H = − t 1 c i +1 − t 2 c i − c i h.a. ˆ i ˆ ˆ i ˆ ˆ i ˆ ˆ i ˆ even even odd odd

Rice-Mele Hamiltonian breaking inversion symmetry & Thouless pump X ! X c † X c † c † X c † c i +1 + ∆ H = − t 1 c i +1 − t 2 c i − c i h.a. ˆ i ˆ ˆ i ˆ ˆ i ˆ ˆ i ˆ even even odd odd

Rice-Mele Hamiltonian breaking inversion symmetry & Thouless pump X ! X c † X c † c † X c † c i +1 + ∆ H = − t 1 c i +1 − t 2 c i − c i h.a. ˆ i ˆ ˆ i ˆ ˆ i ˆ ˆ i ˆ even even odd odd ∆ P = a 2 π ∆ φ Zak

quantized topological transport in an open spin chain D. Linzner, F. Grusdt, M. Fleischhauer, arxiv:1605.00756

model L A j j − 1 j j + 1 L B j ⇣ ⌘ X j ρ L µ † j − L µ † j ρ − ρ L µ † 2 L µ j L µ j L µ ρ = L ρ = ˙ j j,µ Lindblad generators h ⇣ ⌘ ⇣ ⌘i √ L A σ + σ + j = 1 + ε (1 − λ ) σ L,j + ˆ ˆ + (1 + λ ) ˆ L,j + ˆ σ R,j R,j h ⇣ ⌘ ⇣ ⌘i √ L B σ + σ + j = 1 − ε (1 − λ ) σ L,j +1 + ˆ ˆ + (1 + λ ) ˆ L,j +1 + ˆ σ R,j R,j

model action of Lindblad generators h ⇣ ⌘ ⇣ ⌘i √ L A σ + σ + j = 1 + ε (1 − λ ) σ L,j + ˆ ˆ + (1 + λ ) ˆ L,j + ˆ σ R,j R,j h ⇣ ⌘ ⇣ ⌘i √ L B σ + σ + j = 1 − ε (1 − λ ) σ L,j +1 + ˆ ˆ + (1 + λ ) ˆ L,j +1 + ˆ σ R,j R,j λ = +1 λ = − 1 j j + 1 j j + 1

symmetries L A j j − 1 j j + 1 L B j h ⇣ ⌘ ⇣ ⌘i √ L A σ + σ + j = 1 + ε (1 − λ ) σ L,j + ˆ ˆ + (1 + λ ) ˆ L,j + ˆ σ R,j R,j h ⇣ ⌘ ⇣ ⌘i √ L B σ + σ + j = 1 − ε (1 − λ ) σ L,j +1 + ˆ ˆ + (1 + λ ) ˆ L,j +1 + ˆ σ R,j R,j particle-hole symmetry h σ z R i + h σ z σ z R → − σ z L i = 0 L σ z j → − σ z h σ z j i = 0 λ = 0 j inversion symmetry ε = 0 or λ = 0

steady-state Thouless pump polarization in finite system with PBC R. Resta PRL 80 , 1800 (1998) ⌧ o� P = 1 i 2 π n X 2 π Im ln exp j ˆ n j L j j j + 1

steady-state Thouless pump periodic cycle in parameter space λ ( ii ) 1 steady state is a pure state (dark state) ( iii ) ε ( i ) − 1 ( iv ) 1 − 1 ✓ 1 ◆ λ P = ⌥ 1 2 + 1 + λ 2 2

steady-state Thouless pump periodic cycle in parameter space 1 / 2 − 1 / 2 winding !!

steady-state Thouless pump X L † H = µ L µ parent Hamiltonian µ ⇣ ⌘ ⇣ ⌘ X X σ + σ + σ + σ + H = − t 1 ˆ R,j + t 2 ˆ R,j + h.a. + ∆ σ − L,j ˆ σ − L,j +1 ˆ ˆ L,j ˆ L,j − ˆ R,j ˆ σ − σ − R,j j j t 1 = 2 Γ (1 + ε )(1 − λ 2 ) t 2 = 2 Γ (1 − ε )(1 − λ 2 ) ∆ = 8 Γ λ = Rice-Mele Hamiltonian: winding à quantized bulk transport topological invariant = Zak phase / Chern number ∆ P = a 2 π ∆ φ Zak

steady-state Thouless pump λ inner part of parameter space 1 L µ = L † λ = 0 µ ε ⇣ ⌘ X 2 L µ j ρ L µ † j − L µ † j L µ j ρ − ρ L µ † j L µ ρ = L ρ = ˙ − 1 j j,µ − 1 1 totally mixed state is also steady state ! à lift degeneracy by (generic) nonlinear term ⇣ ⌘ p L A j → L A σ + σ + j + Γ (1 + ε ) ˆ L,j ˆ R,j − ˆ L,j ˆ σ − σ − R,j ⇣ ⌘ p L B → L B σ + σ + j + Γ (1 − ε ) ˆ L,j +1 ˆ R,j − ˆ L,j +1 ˆ σ − σ − j R,j

steady-state Thouless pump TEBD simulations in inner part of parameter space Zak phase undefined winding defines topological invariant

robustness

robustness Hamiltonian disorder homogeneous local losses λ ε robust to disorder and losses

symmetry protected topological order

symmetry-protected topology inversion symmetric axes ε = 0 λ = 0 polarization constant & jumps at sigularity

symmetry-protected topology Inversion symmetry λ = 0 1/4 0 P -1/4 -1 0 1 P λ =0 ( ε ) = P λ =0 ( − ε ) + 1 " 2 1 x + < R,j x ) 2 i h ( < L,j 0.9 h σ z j i = 0 x + < R,j-1 x ) 2 i h ( < L,j 0.8 0.7 0.6 0.5 0.4 0.3 -1 -0.5 0 0.5 1 "

topological singularity

topological singularity

damping spectrum (4 sites) ρ ν ( t ) = ρ ν (0) e − ν t L ρ ν = − νρ ν Re[ ν ] no closing of damping gap