Dynamics and thermalization in isolated quantum systems Marcos Rigol Department of Physics The Pennsylvania State University New Frontiers in Non-equilibrium Physics 2015 Yukawa Institute for Theoretical Physics (YITP) August 4, 2015 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 1 / 34
Outline Introduction 1 Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions Quantum quenches in the thermodynamic limit 2 Diagonal ensemble and NLCEs Quenches in the t - V - t ′ - V ′ chain Many-body localization Conclusions 3 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 2 / 34
Outline Introduction 1 Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions Quantum quenches in the thermodynamic limit 2 Diagonal ensemble and NLCEs Quenches in the t - V - t ′ - V ′ chain Many-body localization Conclusions 3 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 3 / 34
Foundations of quantum statistical mechanics Quantum ergodicity: John von Neumann ‘29 (Proof of the ergodic theorem and the H-theorem in quantum mechanics) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 4 / 34
Foundations of quantum statistical mechanics Quantum ergodicity: John von Neumann ‘29 (Proof of the ergodic theorem and the H-theorem in quantum mechanics) Recent works: Tasaki ‘98 (From Quantum Dynamics to the Canonical Distribution. . . ) Goldstein, Lebowitz, Tumulka, and Zanghi ‘06 (Canonical Typicality) Popescu, Short, and A. Winter ‘06 (Entanglement and the foundation of statistical mechanics) Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi ‘10 (Normal typicality and von Neumann’s quantum ergodic theorem) MR and Srednicki ‘12 (Alternatives to Eigenstate Thermalization) P . Reimann ‘15 (Generalization of von Neumann’s Approach to Thermalization) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 4 / 34
Absence of thermalization in 1D T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440 , 900 (2006). γ = mg 1 D � 2 ρ g 1 D : Interaction strength ρ : One-dimensional density If γ ≫ 1 the system is in the strongly correlated Tonks-Girardeau regime If γ ≪ 1 the system is in the weakly interacting regime Gring et al. , Science 337 , 1318 (2012). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 5 / 34
μ μ μ ̃ ̃ Coherence after quenches in Bose-Fermi mixtures a Delocalized fermions Bose-Einstein condensate a = λ lat /2 b Quench No dynamics No dynamics Dynamics U FB 1 2 1 2 1 2 † † † ĉ 1 ĉ 2 =0 ĉ 1 ĉ 2 ≠0 ĉ 1 ĉ 2 ≠0 c 0.3 T = h / U FB t = 0 t = T/8 0.2 t = T/2 ĉ j ĉ l ( t ) 0.1 † 0 -0.1 -15 -10 -5 0 5 10 15 Distance between lattice sites, j - l (a) S. Will, D. Iyer, and MR Nat. Commun. 6 , 6009 (2015). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 6 / 34
̃ ̃ Coherence after quenches in Bose-Fermi mixtures a Delocalized fermions Bose-Einstein condensate a - k 0 k 0 2 hk lat 2 g t 1 = 120 μ s Integrated density n (a.u.) Integration t 2 = 200 μ s 2 z = a = λ lat /2 F + + 1 2 1' b 1 Quench x 200 μ m No dynamics No dynamics Dynamics 1 2 1' 1 2 1' 1 2 1' 0 b U FB 0.01 n ( t 1 ) - n ( t 2 ) 0 1 2 1 2 1 2 -0.01 † † † ĉ 1 ĉ 2 =0 ĉ 1 ĉ 2 ≠0 ĉ 1 ĉ 2 ≠0 -3 -2 -1 0 1 2 3 Momentum, k x ( k lat ) c c 0.3 T = h / U FB t = 0 t = T/8 0.2 t = T/2 ĉ j ĉ l ( t ) 0.2 Integrated density n 0.1 (normalized) † 0 . 1 0 -0.1 -15 -10 -5 0 5 10 15 0 0 Distance between lattice sites, j - l (a) -2 2 0 S. Will, D. Iyer, and MR Hold time, t (ms) 4 Mom., k x 2 ( k lat ) 6 Nat. Commun. 6 , 6009 (2015). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 6 / 34
Dynamics and thermalization in quantum systems If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a few-body observable O will evolve following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ/ � | ψ 0 � . O | ψ ( τ ) � where Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34
Dynamics and thermalization in quantum systems If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a few-body observable O will evolve following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ/ � | ψ 0 � . O | ψ ( τ ) � where What is it that we call thermalization? O ( τ ) = O ( E 0 ) = O ( T ) = O ( T, µ ) . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34
Dynamics and thermalization in quantum systems If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a few-body observable O will evolve following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ/ � | ψ 0 � . O | ψ ( τ ) � where What is it that we call thermalization? O ( τ ) = O ( E 0 ) = O ( T ) = O ( T, µ ) . One can rewrite � � C ⋆ α ′ C α e i ( E α ′ − E α ) τ/ � O α ′ α O ( τ ) = | ψ 0 � = C α | α � . where α ′ ,α α ρ DE ≡ � α | C α | 2 | α �� α | ) Taking the infinite time average (diagonal ensemble ˆ � τ � 1 dτ ′ � Ψ( τ ′ ) | ˆ | C α | 2 O αα ≡ � ˆ O | Ψ( τ ′ ) � = O ( τ ) = lim O � DE , τ τ →∞ 0 α which depends on the initial conditions through C α = � α | ψ 0 � . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34
Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 8 / 34
Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j � � ψ 0 | � H 2 | ψ 0 � − � ψ 0 | � H | ψ 0 � 2 is The width of the weighted energy density ∆ E = �� � � α | C α | 2 − ( E α | C α | 2 ) 2 = � ψ 0 | � W 2 | ψ 0 � − � ψ 0 | � ∆ E = E 2 W | ψ 0 � 2 , α α or � � √ N →∞ ∆ E = [ � ψ 0 | ˆ w ( j 1 ) ˆ w ( j 2 ) | ψ 0 � − � ψ 0 | ˆ w ( j 1 ) | ψ 0 �� ψ 0 | ˆ w ( j 2 ) | ψ 0 � ] ∝ N, j 1 ,j 2 ∈ σ where N is the total number of lattice sites. MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 8 / 34
Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j � � ψ 0 | � H 2 | ψ 0 � − � ψ 0 | � H | ψ 0 � 2 is The width of the weighted energy density ∆ E = �� � � α | C α | 2 − ( E α | C α | 2 ) 2 = � ψ 0 | � W 2 | ψ 0 � − � ψ 0 | � ∆ E = E 2 W | ψ 0 � 2 , α α or � � √ N →∞ ∆ E = [ � ψ 0 | ˆ w ( j 1 ) ˆ w ( j 2 ) | ψ 0 � − � ψ 0 | ˆ w ( j 1 ) | ψ 0 �� ψ 0 | ˆ w ( j 2 ) | ψ 0 � ] ∝ N, j 1 ,j 2 ∈ σ where N is the total number of lattice sites. Since the width W of the full energy spectrum is ∝ N ∆ ǫ = ∆ E 1 N →∞ ∝ √ , W N so, as in any thermal ensemble, ∆ ǫ vanishes in the thermodynamic limit. MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 8 / 34
Description after relaxation (lattice models) Hard-core boson (spinless fermion) Hamiltonian � � n i +1 − t ′ � � � L ˆ ˆ i ˆ ˆ i ˆ b † b † + V ′ ˆ H = − t b i +1 + H.c. + V ˆ n i ˆ b i +2 + H.c. n i ˆ n i +2 i =1 Dynamics vs statistical ensembles Nonintegrable: t ′ = V ′ � = 0 Integrable: V = t ′ = V ′ = 0 0.6 0.5 time average n(k) thermal 0.5 GGE n(k) 0.25 0.4 initial state 0.3 time average thermal 0 0.2 - π π 0 - π π - π/2 π/2 - π/2 0 π/2 ka ka MR, PRL 103 , 100403 (2009), MR, Dunjko, Yurovsky, and PRA 80 , 053607 (2009), . . . Olshanii, PRL 98 , 050405 (2007), . . . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 9 / 34
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