Dephasing, relaxation and thermalization in one-dimensional quantum systems

Dephasing, relaxation and thermalization in one-dimensional quantum systems Fachbereich Physik, TU Kaiserslautern 26.7.2012

Outline 1 Introduction 2 Dephasing, relaxation and thermalization 3 Particle injection into a chain 4 The lightcone renormalization group 5 Doublon decay in (extended) Hubbard models 6 Conclusions

Collaborations Tilman Enss (TU München) Nick Sedlmayr (TU KL) Jie Ren (TU KL) Florian Gebhard (U Marburg) Benedikt Ziebarth (U Marburg) Kevin zu Münster (U Marburg)

Classical dynamics Newton s equations describe the time evolution of a given initial state of classical particles. Deterministic: Velocities and positions are known at all times

Classical dynamics Newton s equations describe the time evolution of a given initial state of classical particles. Deterministic: Velocities and positions are known at all times Thermalization: Expectation values of generic observables, for long times, should become time independent. Values depend only on few parameters, e.g. energy, not the initial state itself. They can be equally well described by a statistical average. Ergodicity: Time average statistical average

Classical dynamics Ideal gas: Thermalization by a) Minimal energy exchange with reservoir (dynamic walls) b) Minimal interaction between the particles Ideal gas law: pv = Nk B T

Quantum dynamics Schrödinger equation: time evolution of a given initial state i d Ψ = H Ψ dt Deterministic: The state Ψ(t) is known

Quantum dynamics Schrödinger equation: time evolution of a given initial state i d Ψ = H Ψ dt Deterministic: The state Ψ(t) is known (How) Does a quantum system thermalize?

Quantum dynamics Schrödinger equation: time evolution of a given initial state i d Ψ = H Ψ dt Deterministic: The state Ψ(t) is known (How) Does a quantum system thermalize? Interaction with a bath: Weak coupling to environment

Quantum dynamics Schrödinger equation: time evolution of a given initial state i d Ψ = H Ψ dt Deterministic: The state Ψ(t) is known (How) Does a quantum system thermalize? Interaction with a bath: Weak coupling to environment Thermalization of closed quantum system; Ô observable: 1 τ Ō = lim τ τ dt Ψ(t) Ô Ψ(t) [ 0 = lim Ψ(t) Ô Ψ(t) O (time independent) ] t Time statistical average: Ō = Tr{Ôρ}

Obvious questions Which density matrix ρ (ensemble)? Which observables O do we want to consider? Do we require only Ō = Tr{Ôρ} or in addition Ō = O time independent? Thermodynamic limit; otherwise recurrence True relaxation interacting vs. non-int. system

Thermalization of a closed quantum system Only a subsystem can be in a mixed state described e.g. by a canonical ensemble ρ = exp( βh)/z rest might act as an effective bath closed system subsystem

Thermalization of a closed quantum system Only a subsystem can be in a mixed state described e.g. by a canonical ensemble ρ = exp( βh)/z rest might act as an effective bath closed system subsystem However, coupling is not small, bath can have memory,...

Dynamics in quantum systems: Experiments Ultracold gases: Quenches Closed quantum system, tunable simple Hamiltonians H. Ott

Dynamics in quantum systems: Experiments Ultracold gases: Quenches Closed quantum system, tunable simple Hamiltonians H. Ott Measurements: Position and time resolved measurements of observables and correlations are becoming possible

Long-time mean: Diagonal ensemble and fluctuations 1 τ The long-time mean: Ō = lim τ τ dt Ψ(t) Ô Ψ(t) 0

Long-time mean: Diagonal ensemble and fluctuations 1 τ The long-time mean: Ō = lim τ τ dt Ψ(t) Ô Ψ(t) 0 Lehmann representation: Ψ(0) = c n E n n 1 τ Ō = lim dt e i(em En)t c τ τ nc m E m Ô E n = n,m 0 n Diagonal ensemble (up to degeneracies) c n 2 Ô nn

Long-time mean: Diagonal ensemble and fluctuations 1 τ The long-time mean: Ō = lim τ τ dt Ψ(t) Ô Ψ(t) 0 Lehmann representation: Ψ(0) = c n E n n 1 τ Ō = lim dt e i(em En)t c τ τ nc m E m Ô E n = n,m 0 n Diagonal ensemble (up to degeneracies) c n 2 Ô nn Fluctuations around long-time mean can decay due to dephasing: already for non-interacting systems exponential relaxation: only in interacting systems

Ō = Tr{Ôρ}: What is the right ensemble?

Ō = Tr{Ôρ}: What is the right ensemble? Conserved quantities ˆQ n cannot change in time: Ψ(t) ˆQ n Ψ(t) = Ψ(0) ˆQ n Ψ(0) ˆQ n I Generalized Gibbs ensemble: Choose ρ = exp( n λ n ˆQ n )/Z with Lagrange multipliers λ n determined such that ˆQ n I = Tr{ ˆQ n ρ}

Ō = Tr{Ôρ}: What is the right ensemble? Conserved quantities ˆQ n cannot change in time: Ψ(t) ˆQ n Ψ(t) = Ψ(0) ˆQ n Ψ(0) ˆQ n I Generalized Gibbs ensemble: Choose ρ = exp( n λ n ˆQ n )/Z with Lagrange multipliers λ n determined such that ˆQ n I = Tr{ ˆQ n ρ} Projectors onto eigenstates: Q n P n = n n Free fermions, H = ɛ k n k : Q k = n k

Q n P n = n n H n = E n n, [H, n n ] = 0, ρ GGE = exp( λ n P n )/Z

Q n P n = n n H n = E n n, [H, n n ] = 0, ρ GGE = exp( λ n P n )/Z Lagrange multipliers λ n : Ψ 0 P n Ψ 0 = Ψ 0 n 2 = Tr(ρ GGE P n ) = e λn Z

Q n P n = n n H n = E n n, [H, n n ] = 0, ρ GGE = exp( λ n P n )/Z Lagrange multipliers λ n : Ψ 0 P n Ψ 0 = Ψ 0 n 2 = Tr(ρ GGE P n ) = e λn Z Time average: Diagonal ensemble (up to degeneracies) Ō = n Ψ 0 n 2 O nn

Q n P n = n n H n = E n n, [H, n n ] = 0, ρ GGE = exp( λ n P n )/Z Lagrange multipliers λ n : Ψ 0 P n Ψ 0 = Ψ 0 n 2 = Tr(ρ GGE P n ) = e λn Z Time average: Diagonal ensemble (up to degeneracies) Ō = n Ψ 0 n 2 O nn Statistical average: Tr(ρ GGE O) = n e λn O nn Z

Q n P n = n n H n = E n n, [H, n n ] = 0, ρ GGE = exp( λ n P n )/Z Lagrange multipliers λ n : Ψ 0 P n Ψ 0 = Ψ 0 n 2 = Tr(ρ GGE P n ) = e λn Z Time average: Diagonal ensemble (up to degeneracies) Ō = n Ψ 0 n 2 O nn Statistical average: Tr(ρ GGE O) = n e λn O nn Z Diagonal ensemble reproduced by fixing exponentially many Lagrange multipliers M. Rigol et al., Nature 852, 454 (08)

GGE for free fermion models H = k ɛ k n k ρ GGE = 1 Z exp( k λ k n k ) Ψ 0 n k Ψ 0 = Tr(ρ GGE n k ) Rigol et al. PRA 74, 053616 (2006) + many others

GGE for free fermion models H = k ɛ k n k ρ GGE = 1 Z exp( k λ k n k ) Ψ 0 n k Ψ 0 = Tr(ρ GGE n k ) Rigol et al. PRA 74, 053616 (2006) + many others Works for observables where Ō can be expressed as function of the n k. Does not work in general e.g. for P = {n k } {n k }

GGE for free fermion models H = k ɛ k n k ρ GGE = 1 Z exp( k λ k n k ) Ψ 0 n k Ψ 0 = Tr(ρ GGE n k ) Rigol et al. PRA 74, 053616 (2006) + many others Works for observables where Ō can be expressed as function of the n k. Does not work in general e.g. for P = {n k } {n k } In general, no relaxation: For non-conserved observables we have Ψ(t) O Ψ(t) oscillating (undamped) or power law decay towards O (dephasing)

An example: Particle injection into a chain γ 1 γ 2 γ L 1 γ L J L 1 ( ) L 1 ( ) H = J ĉ l+1ĉl + h.c. + γ ĉ l ŝl + h.c. l=1 initial state : Ψ(0) = l=1 L ŝ l vac l=1 [F. Gebhard, JS et al. Ann. Phys. (2012)]

Particle density in the chain in the TD limit for γ/j = 10 1 (d) 0.8 n c 0.6 0.4 0.2 0 0 100 200 300 400 500 Jt Non-interact.: Slow dephasing of two-level systems γ/j Jt

A k,k for γ/j = 1, L = 1000 at Jt = 5000 1 0.8 A k,k (t) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 k/π A k,k = Ψ(t) c k c k Ψ(t) is collection of undamped oscillators. Size of fluctuations depends on initial state; can be as large as the long-time mean Āk,k Determining mean by ensemble average seems fairly useless in this case

Relaxation in the interacting case 0.5 0.4 n c 0.3 0.2 0.1 0 0 5 10 15 Jt t-dmrg for L = 50 and γ/j = 1 with nearest neighbor interaction V = 1, 2, 3, 4 Interaction leads to exponential relaxation

Relaxation in the interacting case 0.5 0.4 n c 0.3 0.2 0.1 0 0 5 10 15 Jt t-dmrg for L = 50 and γ/j = 1 with nearest neighbor interaction V = 1, 2, 3, 4 Interaction leads to exponential relaxation GGE seems unobservable even in weakly interacting systems slow power-law versus fast exponential relaxation no separation of time scales

One possibly interesting question Can O = lim t O(t) in an interacting system in the TD limit with O a local observable be described by a statistical average including only the local conservation laws Q j?

One possibly interesting question Can O = lim t O(t) in an interacting system in the TD limit with O a local observable be described by a statistical average including only the local conservation laws Q j? If answer is yes this is potentially useful because calculating O = lim t Ψ(t) O Ψ(t) is hard (dynamical problem) whereas O ρ = 1 Z Tr{O exp( j λ j Q j )} is usually much easier (static problem) Universality: Independent of initial state

Integrable versus non-integrable 1D models Every quantum system in the thermodyn. limit has infinitely many non-local conserved quantities, e.g.: [H, E n E n ] = 0

Integrable versus non-integrable 1D models Every quantum system in the thermodyn. limit has infinitely many non-local conserved quantities, e.g.: [H, E n E n ] = 0 (Bethe ansatz) integrable models have infinitely many local conserved quantities ˆQ n = ˆq n,j with a density ˆq n,j acting j on a finite number of lattice sites.

Integrable versus non-integrable 1D models Every quantum system in the thermodyn. limit has infinitely many non-local conserved quantities, e.g.: [H, E n E n ] = 0 (Bethe ansatz) integrable models have infinitely many local conserved quantities ˆQ n = ˆq n,j with a density ˆq n,j acting j on a finite number of lattice sites. Generic quantum system with short-range interactions: Only H is locally conserved canonical ensemble: ρ = e βh /Z T = 1/β fixes initial energy

Dynamical density-matrix renormalization group DMRG: Approximate Ψ(t) in a finite dimensional Hilbert space consisting of matrix product states Dynamical DMRG (DDMRG) (Jeckelmann, 2002)

Dynamical density-matrix renormalization group DMRG: Approximate Ψ(t) in a finite dimensional Hilbert space consisting of matrix product states Dynamical DMRG (DDMRG) (Jeckelmann, 2002) Time-dependent DMRG (White,Feiguin; Daley et al., 2004) sweep U sweep Trotter-Suzuki decomposition of time evolution: U = e iδth i,i+1 Very versatile: Equ. dynamics, quenches, transport finite systems; eigenstates are calculated explicitly

Dynamical density-matrix renormalization group DMRG: Approximate Ψ(t) in a finite dimensional Hilbert space consisting of matrix product states Dynamical DMRG (DDMRG) (Jeckelmann, 2002) Time-dependent DMRG (White,Feiguin; Daley et al., 2004) sweep U sweep Trotter-Suzuki decomposition of time evolution: U = e iδth i,i+1 Very versatile: Equ. dynamics, quenches, transport finite systems; eigenstates are calculated explicitly TEBD and itebd (Vidal, 2004, 2007) Direct matrix product state representation Time evolution by Trotter-Suzuki decomposition Thermodynamic limit if system is translationally invariant

Specifications of an efficient algorithm Thermodynamic limit: For a finite system there will always be a recurrence time t rec where Ψ(t rec ) Ψ(0)

Specifications of an efficient algorithm Thermodynamic limit: For a finite system there will always be a recurrence time t rec where Ψ(t rec ) Ψ(0) Flexibility: Impurities, disorder, trapping potentials; thermodynamic limit should not depend on translational invariance

Specifications of an efficient algorithm Thermodynamic limit: For a finite system there will always be a recurrence time t rec where Ψ(t rec ) Ψ(0) Flexibility: Impurities, disorder, trapping potentials; thermodynamic limit should not depend on translational invariance High computational efficiency

Specifications of an efficient algorithm Thermodynamic limit: For a finite system there will always be a recurrence time t rec where Ψ(t rec ) Ψ(0) Flexibility: Impurities, disorder, trapping potentials; thermodynamic limit should not depend on translational invariance High computational efficiency Long simulation times: At least we want to understand what limits the simulation time

Specifications of an efficient algorithm Thermodynamic limit: For a finite system there will always be a recurrence time t rec where Ψ(t rec ) Ψ(0) Flexibility: Impurities, disorder, trapping potentials; thermodynamic limit should not depend on translational invariance High computational efficiency Long simulation times: At least we want to understand what limits the simulation time We have developed the lightcone renormalization group (LCRG) algorithm which makes progress concerning many of the points listed above [T. Enss, JS, New J. Phys. 14, 023008 (2012)]

The Lieb-Robinson bound Consider a (Spin-)Hamiltonian with finite range interactions and two strictly local operators A, B: (Lieb,Robinson, Comm. Math. Phys 28, 251 (72)) ( [A(L, t), B(0, 0)] C exp L v ) LR t ξ

The Lieb-Robinson bound Consider a (Spin-)Hamiltonian with finite range interactions and two strictly local operators A, B: (Lieb,Robinson, Comm. Math. Phys 28, 251 (72)) ( [A(L, t), B(0, 0)] C exp L v ) LR t ξ If spatial separation L v t (outside the light cone) then operators always commute up to exponentially small tails Information, correlations, and entanglement propagate with a finite velocity v LR

The Lieb-Robinson bound Consider a (Spin-)Hamiltonian with finite range interactions and two strictly local operators A, B: (Lieb,Robinson, Comm. Math. Phys 28, 251 (72)) ( [A(L, t), B(0, 0)] C exp L v ) LR t ξ If spatial separation L v t (outside the light cone) then operators always commute up to exponentially small tails Information, correlations, and entanglement propagate with a finite velocity v LR Define S as the set of sites having distance of at least l from the local operator A: (Bravyi et al PRL 97, 050401 (06)) A l (t) Tr S [A(t)] 1 S

The Lieb-Robinson bound (II) It follows: A(t) A l (t) C exp ( l v ) LR t ξ

The Lieb-Robinson bound (II) It follows: A(t) A l (t) C exp ( l v ) LR t ξ The time evolution of an operator can be restricted to an effective light cone with spatial extent l v LR t Thermodynamic limit without requiring translational invariance!

The Lieb-Robinson bound (II) It follows: A(t) A l (t) C exp ( l v ) LR t ξ The time evolution of an operator can be restricted to an effective light cone with spatial extent l v LR t Thermodynamic limit without requiring translational invariance! t A +t

Recent experimental test Bose-Hubbard model, filling n = 1 Quench: U/J = 40 U/J = 9 Parity correlation: C d (t) = e iπ[n j (t) n] e iπ[n j+d (t) n] conn. Lightcone for doublon/holon propagation [Cheneau et al. Nature 481, 484 (2012)]

Light cone renormalization group algorithm We want to calculate: o [j,j+n] I (t) Ψ I e iht o [j,j+n] e iht Ψ I

Light cone renormalization group algorithm We want to calculate: o [j,j+n] I (t) Ψ I e iht o [j,j+n] e iht Ψ I H = h j,j+1 local Hamiltonian Ψ I initial product state (thermal state also possible) o [j,j+n] local observable

Light cone renormalization group algorithm We want to calculate: o [j,j+n] I (t) Ψ I e iht o [j,j+n] e iht Ψ I H = h j,j+1 local Hamiltonian Ψ I initial product state (thermal state also possible) o [j,j+n] local observable Use Trotter-Suzuki decomposition of time evolution operator: e ith = lim N ( = j even ( e iδth) N = lim e iδth j,j+1 }{{} τ j,j+1 (δt) N j odd ( ) N e iδtheven e iδth odd e iδth j,j+1 ) N

Light cone renormalization group algorithm t=0 time t=δt s 1 s 2... s j s j+1... t=2δt τ(-δt) τ(δt) t=δt t=0 s 1 s 2... s j s j+1... τ j,j+1 (δt)τ j,j+1 ( δt) = id Speed in T-S decomposition v LR Thermodyn. limit

Light cone renormalization group algorithm L R C Light cone grows by adding diagonal transfer matrices Optimal Hilbert space chosen by appropriate reduced density matrix (Density-matrix renormalization group)

Decay of double occupancies in fermionic Hubbard models Experiments on ultracold fermions in 3D optical lattices [Strohmaier et al. PRL 104, 080401 (10)]

Decay of double occupancies in fermionic Hubbard models Experiments on ultracold fermions in 3D optical lattices [Strohmaier et al. PRL 104, 080401 (10)] Metastability: If U 6J (bandwidth) then several scattering processes are necessary to dissipate large energy U lifetime of double occupancies exp(u/6j)

Model, initial state and symmetries We consider the (extended) Hubbard model: H U,V = J j,σ=, (c j,σ c j+1,σ + h.c.) + U (n j 1/2)(n j 1/2) j + V (n j 1)(n j+1 1) j

Model, initial state and symmetries We consider the (extended) Hubbard model: H U,V = J j,σ=, (c j,σ c j+1,σ + h.c.) + U (n j 1/2)(n j 1/2) j + V (n j 1)(n j+1 1) j We want to study the following initial state and observable: Doublon lattice: Ψ D = j c 2j c 2j 0 Double occupancy: du,v D (t) = 1 Ψ D (t) n j n j Ψ D (t) L Symmetries: d D U,V (t) = d D U, V (t) j

Free fermions: A test case 0.4 20000 d 0 D (t) 0.2 χ=100 500 2000 5000 exact δd 0 D (t) 10-4 10-5 10-6 10-7 0 0 1 2 Jt 3 4 5 6 0 1 2 3 4 5 Jt S ent 8 6 4 2 χ=100 500 2000 20000 5000 0 0 1 2 3 Jt 4 5 6 7 Error up to the point where simulation breaks down is controlled by the Trotter error The entanglement entropy grows linearly in time

Limitations of DMRG-type algorithms The eigenvalues of a reduced density matrix ρ s = Tr E ρ are used to determine the states which are kept to approximate Ψ(t)

Limitations of DMRG-type algorithms The eigenvalues of a reduced density matrix ρ s = Tr E ρ are used to determine the states which are kept to approximate Ψ(t) Entanglement entropy S ent = Trρ s ln ρ s ln(dim ρ s ) von Neumann entropy of reduced density matrix

Limitations of DMRG-type algorithms The eigenvalues of a reduced density matrix ρ s = Tr E ρ are used to determine the states which are kept to approximate Ψ(t) Entanglement entropy S ent = Trρ s ln ρ s ln(dim ρ s ) von Neumann entropy of reduced density matrix The entanglement entropy which can be faithfully represented is limited for finite matrix dimension

Doublon decay in the Hubbard model d U D (t) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0 1 2 3 4 Jt 5 6 7 8 2x 5 (b) 4 (c) 1 3 a x xx U=2 α x xx xx x 0 0 20 40 60 80 100 U /J U=10 (a) U=5 U=0 (exact) xx 2 xx xx 1 0 2 4 6 8 10 U /J Fit: d d ±U (t) = d D ±U ( ) + e γt [A + B cos(ωt φ)]/t α γ 0: Pure power law decay (?) S ent = ajt: large U longer simulation times

Doublon decay in the extended Hubbard model d U,V D (t) 0.4 0.35 0.3 0.25 U=V=0 (exact) U=2, V=1 γ 2 1.5 1 0.5 V=U/4 x x x V=U/2 xxx x 0 0 2 4 6 8 10 U /J 0.2 U=10, V=5 U=5, V=5/2 0.15 0 1 2 3 4 Jt 5 6 7 8 9 Exponential relaxation in the extended Hubbard model Extrapolation problematic for large U

Thermalization in the extended Hubbard model We use canonical ensemble, ignore other local conservation laws in integrable case, V = 0 Energy is fixed by: H I = Ψ D H Ψ D /L = U/4 V Temperature determined by H I = H th = Tr{He H/T }/LZ

Thermalization in the extended Hubbard model We use canonical ensemble, ignore other local conservation laws in integrable case, V = 0 Energy is fixed by: H I = Ψ D H Ψ D /L = U/4 V Temperature determined by H I = H th = Tr{He H/T }/LZ The spectrum is bounded Negative temperatures are natural,t Ô U,V th = Ô U, V, T th

Thermalization in the extended Hubbard model We use canonical ensemble, ignore other local conservation laws in integrable case, V = 0 Energy is fixed by: H I = Ψ D H Ψ D /L = U/4 V Temperature determined by H I = H th = Tr{He H/T }/LZ The spectrum is bounded Negative temperatures are natural T > 0 for V > U/4 T < 0 for V < U/4,T Ô U,V th = Vice versa in attractive case Ô U, V, T th Repulsive case

Thermalization in the extended Hubbard model T -1 eff 0.5 0-0.5 (a) xx x x V=U/2 V=U/4 V=0 x 0 2 4 6 8 10 U/J x x (b) x x V=0 V=U/4 x V=U/2 xx xx xx 0 0 2 4 6 8 10 U /J x x x 0.5 0.4 0.3 0.2 0.1 d U D ( ), <d>teff Excellent agreement for Hubbard model (V = 0) Deviations larger the larger V /U and U are Additional relaxation for Jt exp(u/j) not covered by fit?

Conclusions Thermalization in closed interacting quantum systems: Time average statistical average Including all projection op. P n reproduces time average Free-model GGE seems not very useful: no damping or only slow power law decay versus exp. decay in interacting case

Conclusions Thermalization in closed interacting quantum systems: Time average statistical average Including all projection op. P n reproduces time average Free-model GGE seems not very useful: no damping or only slow power law decay versus exp. decay in interacting case Can long-time mean of local observables for int. systems in the TD limit be described by including only local conserved quantities?

Conclusions Thermalization in closed interacting quantum systems: Time average statistical average Including all projection op. P n reproduces time average Free-model GGE seems not very useful: no damping or only slow power law decay versus exp. decay in interacting case Can long-time mean of local observables for int. systems in the TD limit be described by including only local conserved quantities? Doublon decay in 1D extended Hubbard models Analysis of data: pure power-law relaxation of double occ. in the integrable case; exp. relaxation in the non-integrable case Thermalization: Eff. temperatures can be positive/negative Additional relaxation at Jt exp(u/j)? Influence of other conservation laws in integrable case?