(Dynamical) : What is it and what are its physical and computational implications? Jochen Gemmer University of Osnabrück, Kassel, May 13th, 214
Outline Thermal relaxation in closed quantum systems? Typicality in a nutshell Numerical experiment: model, observables and results Typicality in formulas Spin transport in the Heisenberg chain Eigenstate thermalization hypothesis
Thermal relaxation in closed quantum systems? Why it should occur: We see it in system we assume to be closed. Why it should not occur: There are issues with the underlying theory: (Non-eq.) Thermodynamics autonomous dynamics of a few macrovariables attractive fixed point, equilibrium often describable by master equations, Fokker-Planck equations, stochastic processes, etc. Quantum Mechanics autonomous dynamics of the wave function (number of parameters: insane) no attractive fixed point (Schroedinger equation) Schroedinger equation is no rate equation Quantum systems that explicitly exhibit relaxation but are not of the small system + large bath type appear to be rare in the literature. To cut it short: Why and how do two cups of coffee thermalize each other?
Typicality in a nutshell The naive view on relaxation i.e. 2nd law of thermodynamcis: QM: ˆρ = ψ ψ evolves into ˆρ eq = 1 Ĥ Z e kt or ˆρ eq 1 ˆδ(Ĥ E) Z problem: invariance of Von Neuman-entropy traditional cure: open quantum systems this requires: environment special structure of structure of environment: large, broad band (oscillator) bath adequate weak coupling, applicability of projection techniques specific initial states: factorizing, thermal bath... etc. Typicality: ˆρ = ψ ψ does not evolve into ˆρ eq = 1 Ĥ Z e kt, ˆρ eq = 1 ˆδ(Ĥ E) Z but ψ Â(t) ψ evolves into Tr{ˆρ eq Â} for many (all?) Â... Can this be true?
Numerical experiment: model and observables spin-model L R Heisenberg-type Hamiltonian: A ladder with anisotropic, XXZ-type couplings which are strong along the legs and weak along the rungs: Ĥ = ij J ij (ˆσ i xˆσ j x + ˆσ i yˆσ j y +.6 ˆσ i zˆσ j z), J ij = 1 for solid lines, J ij = κ =.2 for dotted lines and J ij = otherwise. Total number of spins: N = 32. The z-component of total magnetization S z = i ˆσi z is conserved We analyze: magnetization difference ˆx ( ˆx = ˆσ z l ) ˆσ z r r R l L eigenvalues of ˆx within the subspace of vanishing total magnetization, S z = : X = 16, 14,...,+16.
Numerical experiment: results (Phys. Rev. E, 89, 42113, (214) ˆx: z-magnetization difference between legs P X (t): probability to find a certain X P x (t) 1.8.6.4.2 15 1 5-5 2-1 -15 x data from solving the Schroedinger equation (N = 32) for two pure, partially random initial states: ψ X () = e αĥ2ˆp xˆp(s z = ) ω, (remark: taking this picture took 6h on 65 CPU s. Thanks to: 4 6 8 t 1 H. de Raedt, K. Michielsen (Juelich)) <x(t)> σ ² (t) 12 8 4 time shifted ˆx(t) <x()>=2 <x()>=4 <x()>=6 <x()>=8 <x()>=1 <x()>=12 5 1 15 t 12 8 4 variances of x <x()>= <x()>=2 <x()>=4 <x()>=6 <x()>=8 <x()>=1 <x()>=12 typical variance 5 1 15 t
Typicality in formulas (Phys. Rev. Lett., 12, 1143 (29), Phys. Rev. Lett., 86, 1927 (21)) static typicality < A >:= Tr{Â}/d: expectation value of the maximally mixed state ω uniform random states sampled according to the unitary invariant measure HA[ ω Â ω ] =< A > HV[ ω Â(t) ω ] = 1 ( < A 2 > < A > 2) d + 1 In a high dimensional Hilbert space almost all possible states feature very similar expectation values for observables with bound spectra. It is no surprise to find these expectation values eventually. dynamical typicality ψ := ˆρd ω : taylored, non-uniform random states, Â ˆB(t) HA[ ψ ˆB(t) ψ ] = Tr{ˆB(t)ˆρ} HV[ ψ ˆB(t) ψ ] Tr{ˆB 2 (t)ˆρ} d eff d eff : inverse of the largest eigenvalue of ˆρ, interpreted as effective dimension Very many different pure states exhibit very similar dynamics of expectation values
Typicality in formulas correlation functions ψ := ˆρd ω : taylored, non-uniform random states, Â Ĉ(t) := 1 2 (ˆB(t)ˆB + ˆBˆB(t)) HA[ ψ Ĉ(t) ψ ] = Tr{Ĉ(t)ˆρ} HV[ ψ Ĉ(t) ψ ] Tr{Ĉ2 (t)ˆρ} d eff What to do practically? Compute ψ(t) := e iĥt/ ˆρd ω, ψ (t) := e iĥt/ ˆB ˆρd ω ψ(t) ˆB ψ(t) Tr{ˆB(t)ˆρ} Re[ ψ(t) ˆB ψ (t) ] Tr{ 1 2 (ˆB(t)ˆB+ˆBˆB(t))ˆρ} If ˆρ is of exponential form, e.g., ˆρ e β(ĥ Ē)2 α(ˆb B) 2 or ˆρ e βĥ αˆb than everything can be done based on pure state propagation
Spin transport in the anisotropic Heisenberg chain (PRL., 112, 1261, (214) ) Linear response conductivity from current autocorrelation function here: infinite temperature, i.e., ˆρ ˆ1, ˆB Ĵ C(t) Tr{Ĵ(t)Ĵ} C(t) / J 2.1.5 [34,36] =.5 =1. =1.5 norm (a.u.) (a) L=33 (β=) [t 1,t 2 ] 1-1 (a) =1. (β=) t J 1 C(t) / J 2 C(t) / J 2 1-2 1-1 1-2 L (b) =1.5 (β=) 1-3 1 2 t J L=18 (ED) L=18 L=2 L=22 L=24 L=26 L=28 L=3 (k=) L=32 (k=) L=33 (k=) tdmrg C(t 1,t 2 ) / J 2 C(t 1,t 2 ) / J 2.5 [2,21] [35] fit =.5 =1. =1.5 even odd (b) Drude weight (β=) 1/3 1/2 1/1 1/L (β=) Bethe Ansatz lower bound.1 ED, fit RK, L=3 RK, L.5 tdmrg.5 1 1.5 anisotropy
Eigenstate thermalization hypothesis (ETH) ETH: Eigenstates of some Hamiltonian Ĥ that are close in energy feature expectation values of some observable ˆB that are close to each other. E n E m n ˆB n m ˆB m Jochen s formulation: Eigenstates belong to the set of typical states If ETH applies: - Expecation values from microncanonical ensembles are close to expecation values of individual eigenstates - Initial state independent equilibration: ˆB(t) = n,m ρ nmb mne i(en Em) t If the oscillating terms behave like white noise for τ large enough ˆB(τ) = n ρ nnb nn ETH B nn But does ETH apply? Have to check by exact diagonalization, or...
ETH check for the Heisenberg ladder (Phys. Rev. Lett., 112, 1343, (214)) Â := ˆx 2 : width of the magnetization distribution ˆρ e α(ĥ U)2 Ā := n ρ nna nn ψ Â ψ Σ 2 + Ā2 := n ρ nna 2 nn Re[ ψ(τ) Â ψ (τ) ] A nn A nn A nn 6 3 6 3 6 3 (a) L=7 (b) L=9 (c) L=1 ED A±Σ -4-2 2 U / J A (d) J J L L d=2 2L Σ 3 2 1 L=7.3 t J 18 L=8.2 (d eff ) -1/2 ETH applies! t 1.1 L=9 L=1 6 γ(t) Σ scales approx like Ĥ was a random matrix.
The End more people involved in this: R. Steinigeweg, A. Khodja, H. Niemeyer, D. Schmidtke, C. Gogolin... Thank you for your attention! The talk itself as well as the mentioned papers may be found on our webpage.