Coherence by elevated temperature

Transcription

1 Coherence by elevated temperature Volker Meden with Dante Kennes, Alex Kashuba, Mikhail Pletyukhov, Herbert Schoeller Institut für Theorie der Statistischen Physik

2 Goal dynamics of dissipative quantum systems in dissipation -T -plane? here: unbiased, ohmic spin boson model in scaling limit. (Garg 85; Weiss, Grabert 86; Leggett et al. 87)

3 Goal dynamics of dissipative quantum systems in dissipation -T -plane? here: unbiased, ohmic spin boson model in scaling limit. (Garg 85; Weiss, Grabert 86; Leggett et al. 87)

4 The unbiased, ohmic spin-boson model Hamiltonian H = 2 σ x + k (ohmic) spectral density ω k b k b k k λ k 2 σ z ( ) b k + b k, J(ω) = k λ 2 kδ(ω ω k ) = 2αωΘ(ω c ω) scaling limit universal physics emergent (Kondo) scale ω c larger than any other energy scale T K = ( /ω c ) α/(1 α) to be specific: initial condition and observable ρ(t = 0) e βh bath, P (t) = σ z (t) (Leggett et al. 87, Weiss 12)

5 The interacting resonant level model mapping at low energies (by bosonization) H = k ε k a k a k + Γ0 2πν (d a k +da k ) + U 2ν (d d dd ) : a k a k : kk k lead density of states D(ω) = k δ(ε k ω) = ν for ω ω c mapping of parameters advantage U = 1 2α, Γ 0 = 2 /ω c, g = 2U U 2 = 1 2α α = 1/2 (g = 0) corresponds to U = 0 (Toulouse limit) (Leggett et al. 87, Weiss 12)

6 Two complementary RG mehods: FRG and RTRG FRG (Metzner,...,VM,... 12) aims at time-dependent irreducible (Keldysh) vertex functions auxiliary lead cutoff all orders in 2 /ω c = Γ 0 truncation: keep only lowest order in g on rhs of flow equations analytical results for T K t 1 and (T K t) g 1 RTRG (Schoeller 09) aims at reduced density matrix (of spin) setup in Laplace-Liouville space: Laplace variable E as cutoff ( ) ( [ ] ) approximation: all terms O g 2 ω c E but not O g 2 2 ω c E analytical results for all t

7 . T = 0

8 Spin expectation value for T = exp(-t K t) α=1/ P(t) 10-4 exp(-t K t/2)/t 2-2α T K t

9 Spin expectation value for T = exp(-t K t) α=1/ exp(-t K t) P(t) 10-4 exp(-t K t/2)/t 2-2α T K t

10 Spin expectation value for T = α=1/2 α= P(t) T K t

11 Spin expectation value for T = α=1/2 α= α= P(t) T K t

12 Spin expectation value for T = α=1/2 α= α= α=0.4 P(t) T K t

13 Spin expectation value for T = α=1/2 α= α= α=0.4 P(t) T K t

14 Take a closer look at RTRG flow equations Laplace transform P (t) = i 2π +i0 + Π 1 (E) = [E + iγ 1 (E)] 1 +i0 + dee iet Π 1 (E) IRLM: charge relaxation rate Γ 1 flow equations dγ 1/2 (E) de = gγ 1 (E)Π 2/1 (E) Π 2 (E) = [E + iγ 2 (E)/2] 1 IRLM: level broadening Γ 2 /2 initial conditions Γ n (iω c ) = 2 /ω c, n = 1, 2

15 Numerical solution in complex E-plane

16 Numerical solution in complex E-plane

17 Numerical solution in complex E-plane

18 Numerical solution in complex E-plane

19 Numerical solution in complex E-plane α c 0.3

20 Numerical solution in complex E-plane Im E Re Π 1 + Re E Im Π 1

21 Conclusion three distinct regimes 1/2 < α < 1: incoherent α c < α < 1/2: partially coherent 0 < α < α c : asymptotically coherent still controlled? RTRG for SBM (small α!) gives α c 0.36 (Kashuba, Schoeller 13)

22 Analytical results T K t 1 from RTRG and FRG (compare to NIBA: Leggett et al 87) T K t 1 from RTRG 1 P (t) = (T K t) 1+g /Γ(2 + g) + O ( [T K t] 2+2g) P (t) = P bc (t) + P p (t) P bc (t) 1 ( [1 {e π Im γt E 1 2 Γ 2 γ ] )} t γ = e iπg (T K t) g P p (t) 2 1 g 1 + g Γ [ 2 2 T K cos (Ωt) e Γ1 t Θ(g) πg 2 sin(πg) ] 1 1+g, Ω + iγ 1 T K e (iπ+ln 2) g 1+g, frequency Ω as in NIBA and CFT (Lesage, Saleur 98) for (T K t) g 1: P bc (t) = g[1 + 3Θ( g)] e T K t/2 (T K t) 1+ g (Egger et al. 97)

23 Comparison to numerics α=1/2 α= α= P(t) T K t

24 . T > 0

25 Spin expectation value for T 0 FRG for α = (partially coherent) T/T K = P(t) T K t

26 Spin expectation value for T 0 FRG for α = (partially coherent) T/T K =0 T/T K = P(t) T K t

27 Spin expectation value for T 0 FRG for α = (partially coherent) T/T K =0 T/T K =0.05 T/T K =0.25 P(t) T K t

28 Spin expectation value for T 0 FRG for α = (partially coherent) T/T K =0 T/T K =0.05 T/T K =0.25 T/T K =0.5 P(t) T K t

29 Spin expectation value for T 0 FRG for α = (partially coherent) 10 0 P(t) T/T K =0 T/T K =0.05 T/T K =0.25 T/T K =0.5 T/T K = T K t partially coherent asymptotically coherent incoherent?

30 Numerical solution of RTRG flow equations (α = 0.45)

31 Numerical solution of RTRG flow equations (α = 0.45)

32 Numerical solution of RTRG flow equations (α = 0.45) T c1 (α)

33 Numerical solution of RTRG flow equations (α = 0.45)

34 Numerical solution of RTRG flow equations (α = 0.45) T c2 (α)

35 Numerical solution of RTRG flow equations (α = 0.45)

36 Phase diagram T/T K asymptotically coherent ln P(t) 0 T K t 20 incoherent ln P(t) 0 20 T K t 0 partially coherent 0 T K t α ln P(t)

37 consider quench at T = 0.

38 Does memory matter? perform a (quantum) quench at time t q time evolution up to t = t q as before out of spin up-state at t q change α: α i α f will the quantum system have a memory of the α i -dynamics quench from partially coherent to incoherent

39 Quench from partially coherent to incoherent without mem. with mem P(t) FRG RTRG i T K t pronounced memory effects non-markovian dynamics f T K t

40 Summary three distinct dynamical regimes increase T : transition from partially to asymptotically coherent coherence by elevated temperature pronounced memory effects non-markovian dynamics Refs.: T = 0 and quenches: Kennes, Kashuba, Pletyukhov, Schoeller, VM, Phys. Rev. Lett. 110, (2013) mainly quenches: Kashuba, Kennes, Pletyukhov, VM, Schoeller, Phys. Rev. B 88, (2013) T > 0: Kennes, Kashuba, VM, Phys. Rev. B 88, 24110(R) (2013)