Adiabatic response of open systems

Transcription

1 Adiabatic response of open systems Yosi Avron Martin Fraas Gian Michele Graf December 17, 2015 Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

2 Question Are topological phase protected form contact with the world? Choosing: Observables and states= immunity Ω system bath osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

3 Question Are topological phase protected form contact with the world? Choosing: Observables and states= immunity Ω system bath osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

4 Ambiguity: Which transport coefficients? Open vs. isolated systems F spring V anchor F friction F as response to V. Which F? Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

5 Ambiguity: Which transport coefficients? Open vs. isolated systems F spring V anchor F friction F as response to V. Which F? Open systems F friction = ν V F spring = ν V F Total = Ṗ = 0 V Isolated system F friction = 0 F spring = F Total Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

6 Sources of ambiguities Tensor product & Observables System? Bath osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

7 Sources of ambiguities Tensor product & Observables System? Bath Ambiguous tensor decomposition H system H bath = H system H bath osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

8 Sources of ambiguities Tensor product & Observables System? Bath Ambiguous tensor decomposition H system H bath = H system H bath Ambiguous observables of sub-system H = H s 1 + H interaction + 1 H b osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

9 Ambiguity in fluxes (aka rates aka currents) Q = j<0 a j a j I left right System I system bath Bath Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

10 Ambiguity in fluxes (aka rates aka currents) Q = j<0 a j a j I left right System I system bath Bath Q 1 I left right = i[h s, Q] I Total = i [H s+b, Q 1] Formulate an I Total as a property of the system osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

11 Krauss maps Krauss maps ψ s+b (0) Unitary ψ s+b (t) Tr b Tr b ρ s (0) Krauss ρ s (t) Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

12 Krauss maps Krauss maps ψ s+b (0) Unitary ψ s+b (t) Tr b Tr b ρ s (0) Krauss ρ s (t) Kraus: Positivity and Trace preserving ρ K j ρk j, K j K j = 1 KρK = (K ρ)(k ρ) 0 Tr ρ Tr K j ρk j = Tr ( ) K j K j ρ Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

13 Lindbladians Generators of Krauss maps ρ ρ + δt Lρ }{{} generator ρ K j ρk j }{{} positive osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

14 Lindbladians Generators of Krauss maps ρ ρ + δt Lρ }{{} generator ρ K j ρk j }{{} positive K 0 = 1 + δt Γ 0, K j = δt Γ j, K j K j = 1 trace preserving: Γ 0 + Γ 0 + Γ j Γ j = 0 Solve for Γ 0 : Γ 0 = ih 1 2 Γ j Γ j osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

15 Lindbladians Generators of Krauss maps ρ ρ + δt Lρ }{{} generator ρ K j ρk j }{{} positive K 0 = 1 + δt Γ 0, K j = δt Γ j, K j K j = 1 trace preserving: Γ 0 + Γ 0 + Γ j Γ j = 0 Solve for Γ 0 : Γ 0 = ih 1 2 Lindbladians L ρ = i[h, ρ] + Dρ Γ j Γ j Dρ = 2Γ j ργ j Γ j Γ jρ ργ j Γ j osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

16 Lindbladians Schrodinger and Heisenberg States and Observables Adjoint (Banach space) Tr ( X Aρ ) = Tr ( (A X )ρ ) Observables X < States Trρ < osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

17 Lindbladians Schrodinger and Heisenberg States and Observables Adjoint (Banach space) Tr ( X Aρ ) = Tr ( (A X )ρ ) Observables X < States Trρ < L ρ = i[h, ρ] + 2ΓρΓ Γ Γρ ργ Γ osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

18 Lindbladians Schrodinger and Heisenberg States and Observables Adjoint (Banach space) Tr ( X Aρ ) = Tr ( (A X )ρ ) Observables X < States Trρ < L ρ = i[h, ρ] + 2ΓρΓ Γ Γρ ργ Γ L X = +i[h, X ] + [Γ, X ]Γ + Γ [X, Γ] osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

19 Lindbladians Schrodinger and Heisenberg States and Observables Adjoint (Banach space) Tr ( X Aρ ) = Tr ( (A X )ρ ) Observables X < States Trρ < L ρ = i[h, ρ] + 2ΓρΓ Γ Γρ ργ Γ L X = +i[h, X ] + [Γ, X ]Γ + Γ [X, Γ] Schrodinger and Heisenberg ρ = Lρ, Ẋ = L X osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

20 Lindbladians Fluxes (aka rates) L Q Two notions of currents in open systems Q I system bath Bath I left right System Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

21 Lindbladians Fluxes (aka rates) L Q Two notions of currents in open systems Q I system bath Bath I left right System Left-right current I left right = i[h, Q] = i Ad(H)Q Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

22 Lindbladians Fluxes (aka rates) L Q Two notions of currents in open systems Q I system bath Bath I left right System Left-right current I left right = i[h, Q] = i Ad(H)Q Flux: Total current I Total = Q = L Q Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

23 Lindbladians Loop currents are not rates Fluxes vanish in stationary states: I loop = φ H L Q φ Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

24 Stationary states Stationary states of L Towards adiabatic theory of fluxes L 1 = 0 = 0 Spectrum(L) Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

25 Stationary states Stationary states of L Towards adiabatic theory of fluxes L 1 = 0 = 0 Spectrum(L) σ 2 σ 3 KerL σ 1 σ 1 generic Ker L If dimh < dim Ker L=1 generically Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

26 Fluxes Fluxes vanish in stationary states Fluxes vanish in stationary states: Tr Q σ = 0 Tr Qσ = Tr (L Q)σ = Tr QLσ = 0 Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

27 Fluxes Adiabatic expansion for fluxes Slow manifold Adiabatic evolutions ɛ ρ = L t ρ, ɛ 1 Adiabatic expansion ρ t = σ t +ɛl 1 σ t +... σ t σ 0 Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

28 Fluxes Adiabatic expansion for fluxes Slow manifold Adiabatic evolutions ɛ ρ = L t ρ, ɛ 1 Adiabatic expansion ρ t = σ t +ɛl 1 σ t +... σ t σ 0 ρ t = ρ 0 + ɛρ Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

29 Fluxes Adiabatic expansion for fluxes Slow manifold Adiabatic evolutions ɛ ρ = L t ρ, ɛ 1 Adiabatic expansion ρ t = σ t +ɛl 1 σ t +... σ t σ 0 ρ t = ρ 0 + ɛρ ɛ 0 : L t ρ 0 = 0 = ρ 0 = σ t σ t instantaneous stationary state Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

30 Fluxes Adiabatic expansion for fluxes Slow manifold Adiabatic evolutions ɛ ρ = L t ρ, ɛ 1 Adiabatic expansion ρ t = σ t +ɛl 1 σ t +... σ t σ 0 ρ t = ρ 0 + ɛρ ɛ 0 : L t ρ 0 = 0 = ρ 0 = σ t σ t instantaneous stationary state ɛ 1 : σ t = L t ρ 1 Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

31 Fluxes Adiabatic expansion for fluxes Slow manifold Adiabatic evolutions ɛ ρ = L t ρ, ɛ 1 Adiabatic expansion ρ t = σ t +ɛl 1 σ t +... σ t σ 0 ρ t = ρ 0 + ɛρ ɛ 0 : L t ρ 0 = 0 = ρ 0 = σ t σ t instantaneous stationary state ɛ 1 : σ t = L t ρ 1 Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

32 Fluxes Q in adiabatically evolving state No need to invert L The magic Tr Q ρ t Tr Q σ t Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

33 Fluxes Q in adiabatically evolving state No need to invert L The magic Tr Q ρ t Tr Q σ t ρ = σ t + ɛl 1 σ t +... L t ρ t = ɛll 1 σ + ɛ σ t Tr Qρ t = 1 ɛ Tr L Qρ t = Tr QL t ρ t Tr Q σ t The irrelevant dynamics Adiabatic fluxes oblivious to L. Only care about σ Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

34 Fluxes Sharing stationary states. Sharing instantaneous stationary states KerL t = Ker(Ad H t ) Example: D = γad 2 (H) photon σ osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

35 Fluxes Geometry of transport Transport coefficients φ control plane σ = ( µ σ) φ µ Tr Qρ }{{} t Tr (Q µ σ) φ µ }{{} response driving Transport coefficient F µ = Tr (Q µ σ) Oblivious to L, cares bout σ osi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

36 Fluxes Transport coefficients & Adiabatic curvature U(φ) = e iqµφµ σ(φ) = U(φ)σU (φ) µ σ = i[q µ, σ] Transport coefficients F µν = itr [Q µ, Q ν ]σ If σ projections F muν = i Tr σ[ µ σ, ν σ] Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18

37 Fluxes Conclusions and Overview Views of the QHE φ 2 φ 1 Macroscopic L Q = A H Index=Chern Gap condition? Multiply connected A H Chern Dissipation: Kahler geometry A. Fraas and Graf, JSP (2012) arxiv A. Fraas, Kenneth and Graf, NJP (2011) arxiv Yosi Avron, Martin Fraas, Gian Michele Graf Adiabatic response of open systems December 17, / 18