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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

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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

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Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 2 / 34

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Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 3 / 34

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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

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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

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Absence of thermalization in 1D

  • T. Kinoshita, T. Wenger, and D. S. Weiss,

Nature 440, 900 (2006).

γ = mg1D 2ρ

g1D: 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

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Coherence after quenches in Bose-Fermi mixtures

ĉ1ĉ2 ≠0

† 1 2

ĉ1ĉ2 =0

†

Delocalized fermions Bose-Einstein condensate

a b

a = λlat/2

1 2 1 2

Quench No dynamics No dynamics Dynamics UFB

ĉ1ĉ2 ≠0

†

c

ĉj ĉl (t)

†

0.3 0.2 0.1

  • 0.1

5 10 15

  • 15
  • 10
  • 5

Distance between lattice sites, j - l (a)

t = 0 t = T/8 t = T/2 T = h/UFB

  • S. Will, D. Iyer, and MR
  • Nat. Commun. 6, 6009 (2015).

μ μ

μ

̃ ̃

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 6 / 34

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Coherence after quenches in Bose-Fermi mixtures

ĉ1ĉ2 ≠0

† 1 2

ĉ1ĉ2 =0

†

Delocalized fermions Bose-Einstein condensate

a b

a = λlat/2

1 2 1 2

Quench No dynamics No dynamics Dynamics UFB

ĉ1ĉ2 ≠0

†

c

ĉj ĉl (t)

†

0.3 0.2 0.1

  • 0.1

5 10 15

  • 15
  • 10
  • 5

Distance between lattice sites, j - l (a)

t = 0 t = T/8 t = T/2 T = h/UFB

  • S. Will, D. Iyer, and MR
  • Nat. Commun. 6, 6009 (2015).

Integrated density n (a.u.)

a b

2 1 t1 = 120 μs t2 = 200 μs

  • k0

k0 1 3

  • 3
  • 1

Momentum, kx (klat)

  • 2

2 0.01

  • 0.01

c

200 μm Integration 2hklat

n(t1)- n(t2)

1 1' 2 =

1 2 1' 2

+ + F z x

2 4 6 Hold time, t (ms) Integrated density n (normalized) . 1 0.2 2

  • 2

Mom., kx (klat)

1 1' 2 1 1' 2 g

̃ ̃

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 6 / 34

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Dynamics and thermalization in quantum systems

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a few-body observable O will evolve following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ/|ψ0.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34

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SLIDE 10

Dynamics and thermalization in quantum systems

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a few-body observable O will evolve following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ/|ψ0.

What is it that we call thermalization? O(τ) = O(E0) = O(T) = O(T, µ).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34

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SLIDE 11

Dynamics and thermalization in quantum systems

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a few-body observable O will evolve following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ/|ψ0.

What is it that we call thermalization? O(τ) = O(E0) = O(T) = O(T, µ). One can rewrite O(τ) =

  • α′,α

C⋆

α′Cαei(Eα′−Eα)τ/Oα′α

where |ψ0 =

  • α

Cα|α. Taking the infinite time average (diagonal ensemble ˆ ρDE ≡

α |Cα|2|αα|)

O(τ) = lim

τ→∞

1 τ τ dτ ′Ψ(τ ′)| ˆ O|Ψ(τ ′) =

  • α

|Cα|2Oαα ≡ ˆ ODE, which depends on the initial conditions through Cα = α|ψ0.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 7 / 34

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Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 8 / 34

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Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

The width of the weighted energy density ∆E =

  • ψ0|

H2|ψ0 − ψ0| H|ψ02 is ∆E =

  • α

E2

α|Cα|2 − (

  • α

Eα|Cα|2)2 =

  • ψ0|

W 2|ψ0 − ψ0| W|ψ02,

  • r

∆E =

j1,j2∈σ

[ψ0| ˆ w(j1) ˆ w(j2)|ψ0 − ψ0| ˆ w(j1)|ψ0ψ0| ˆ w(j2)|ψ0]

N→∞

∝ √ N, 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

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Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

The width of the weighted energy density ∆E =

  • ψ0|

H2|ψ0 − ψ0| H|ψ02 is ∆E =

  • α

E2

α|Cα|2 − (

  • α

Eα|Cα|2)2 =

  • ψ0|

W 2|ψ0 − ψ0| W|ψ02,

  • r

∆E =

j1,j2∈σ

[ψ0| ˆ w(j1) ˆ w(j2)|ψ0 − ψ0| ˆ w(j1)|ψ0ψ0| ˆ w(j2)|ψ0]

N→∞

∝ √ N, where N is the total number of lattice sites. Since the width W of the full energy spectrum is ∝ N ∆ǫ = ∆E W

N→∞

∝ 1 √ 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

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Description after relaxation (lattice models)

Hard-core boson (spinless fermion) Hamiltonian

ˆ H =

L

  • i=1

−t

  • ˆ

b†

iˆ

bi+1 + H.c.

  • + V ˆ

niˆ ni+1 − t′ ˆ b†

iˆ

bi+2 + H.c.

  • + V ′ˆ

niˆ ni+2

Dynamics vs statistical ensembles

Nonintegrable: t′ = V ′ = 0

  • π
  • π/2

π/2 π

ka

0.2 0.3 0.4 0.5 0.6

n(k)

initial state time average thermal

MR, PRL 103, 100403 (2009), PRA 80, 053607 (2009), . . .

Integrable: V = t′ = V ′ = 0

  • π
  • π/2

π/2 π

ka

0.25 0.5

n(k)

time average thermal GGE

MR, Dunjko, Yurovsky, and Olshanii, PRL 98, 050405 (2007), . . .

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 9 / 34

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Eigenstate thermalization

Eigenstate thermalization hypothesis

[Deutsch, PRA 43 2046 (1991); Srednicki, PRE 50, 888 (1994).]

The expectation value α| O|α of a few-body observable O in an eigenstate of the Hamiltonian |α, with energy Eα, of a many-body system is equal to the thermal average of O at the mean energy Eα: α| O|α = OME(Eα).

Nonintegrable

1 2 3

n(kx=0)

  • 10
  • 8
  • 6
  • 4
  • 2

E[J]

1 2

ρ(E)[J

  • 1]

ρ(E) exact ρ(E) microcan. ρ(E) canonical

Integrable (ˆ

ρGGE =

1 ZGGE e−

m λm ˆ

Im)

0.5 1 1.5

n(kx=0)

  • 8
  • 6
  • 4
  • 2

E[J]

0.5 1 1.5

ρ(E)[J

  • 1]

ρ(E) exact ρ(E) microcan. ρ(E) canonical

MR, Dunjko, and Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 10 / 34

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Time fluctuations and their scaling with system size

0.1

δNk L=21 L=24

0.1

δNk

0.1

δNk

20 40 60 80 100

τ

0.1

δNk t’=V’=0 t’=V’=0.03 t’=V’=0.12 t’=V’=0.24

Relative differences (struct. factor) δN(τ) =

  • k |N(k, τ) − Ndiag(k)|
  • k Ndiag(k)

Bounds

(G) P . Reimann, PRL 101, 190403 (2008). (G) Linden et al., PRE 79, 061103 (2009). (N) Cramer et al., PRL 100, 030602 (2008). (N) Venuti&Zanardi, PRE 87, 012106 (2013).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 11 / 34

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Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 12 / 34

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Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Time average of ˆ O

ˆ O =

  • α

|Cα|2Oαα ∼

  • α

1 Nstates Oαα ∼ Otypical

αα

One needs: Otypical

α′α

≪ Otypical

αα

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 12 / 34

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SLIDE 20

Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Time average of ˆ O

ˆ O =

  • α

|Cα|2Oαα ∼

  • α

1 Nstates Oαα ∼ Otypical

αα

One needs: Otypical

α′α

≪ Otypical

αα

MR, PRA 80, 053607 (2009)

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 12 / 34

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Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 13 / 34

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Finite temperature properties of lattice models

Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 14 / 34

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Finite temperature properties of lattice models

Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model!

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 14 / 34

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SLIDE 24

Finite temperature properties of lattice models

Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem ⇒ High temperatures Thermodynamic limit ⇒ Extrapolations to low T Can be used for any model! Can fail (at low T) even when correlations are short ranged!

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 14 / 34

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SLIDE 25

Finite temperature properties of lattice models

Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem ⇒ High temperatures Thermodynamic limit ⇒ Extrapolations to low T Can be used for any model! Can fail (at low T) even when correlations are short ranged! DMFT, DCA, DMRG, . . .

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 14 / 34

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Linked-Cluster Expansions

Extensive observables ˆ O per lattice site (O) in the thermodynamic limit O =

  • c

L(c) × WO(c) where L(c) is the number of embeddings of cluster c

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 15 / 34

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SLIDE 27

Linked-Cluster Expansions

Extensive observables ˆ O per lattice site (O) in the thermodynamic limit O =

  • c

L(c) × WO(c) where L(c) is the number of embeddings of cluster c and WO(c) is the weight

  • f observable O in cluster c

WO(c) = O(c) −

  • s⊂c

WO(s). O(c) is the result for O in cluster c O(c) = Tr

  • ˆ

O ˆ ρGC

c

  • ,

ˆ ρGC

c

= 1 ZGC

c

exp−( ˆ

Hc−µ ˆ Nc)/kBT

ZGC

c

= Tr

  • exp−( ˆ

Hc−µ ˆ Nc)/kBT

and the s sum runs over all subclusters of c.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 15 / 34

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SLIDE 28

Linked-Cluster Expansions

Extensive observables ˆ O per lattice site (O) in the thermodynamic limit O =

  • c

L(c) × WO(c) where L(c) is the number of embeddings of cluster c and WO(c) is the weight

  • f observable O in cluster c

WO(c) = O(c) −

  • s⊂c

WO(s). O(c) is the result for O in cluster c O(c) = Tr

  • ˆ

O ˆ ρGC

c

  • ,

ˆ ρGC

c

= 1 ZGC

c

exp−( ˆ

Hc−µ ˆ Nc)/kBT

ZGC

c

= Tr

  • exp−( ˆ

Hc−µ ˆ Nc)/kBT

and the s sum runs over all subclusters of c. In numerical linked cluster expansions (NLCEs) an exact diagonalization of the cluster is used to calculate O(c) at any temperature.

MR, T. Bryant, and R. R. P . Singh, PRL 97, 187202 (2006).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 15 / 34

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Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 16 / 34

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Numerical linked cluster expansions (square lattice)

i) Find all clusters that can be embedded on the lattice Bond clusters c

2

L(c)

2 3 2 4 4 5 4 6 2 7 4 1 1 8 4 9 8

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 17 / 34

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Numerical linked cluster expansions (square lattice)

i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topo- logical cluster)

  • No. of bonds

topological clusters 1 1 1 2 1 3 2 4 4 5 6 6 14 7 28 8 68 9 156 10 399 11 1012 12 2732 13 7385 14 20665

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 17 / 34

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SLIDE 32

Numerical linked cluster expansions (square lattice)

i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topo- logical cluster) iii) Find all subclusters of a given topological cluster

  • No. of bonds

topological clusters 1 1 1 2 1 3 2 4 4 5 6 6 14 7 28 8 68 9 156 10 399 11 1012 12 2732 13 7385 14 20665

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 17 / 34

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SLIDE 33

Numerical linked cluster expansions (square lattice)

i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topo- logical cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the

  • bservables
  • No. of bonds

topological clusters 1 1 1 2 1 3 2 4 4 5 6 6 14 7 28 8 68 9 156 10 399 11 1012 12 2732 13 7385 14 20665

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 17 / 34

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SLIDE 34

Numerical linked cluster expansions (square lattice)

i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topo- logical cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the

  • bservables

v) Compute the weight of each cluster and compute the di- rect sum of the weights Heisenberg Model in 2D

0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

QMC 100×100 12 bonds 13 bonds MR et al., PRE 75, 061118 (2007).

  • B. Tang et al., CPC 184, 557 (2013).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 17 / 34

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SLIDE 35

Numerical linked cluster expansions

Square clusters c

2

L(c)

1 1 3 4 5 1/2 1 2 1

  • No. of squares

topological clusters 1 1 1 2 1 3 2 4 5 5 11

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 18 / 34

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SLIDE 36

Numerical linked cluster expansions

Square clusters c

2

L(c)

1 1 3 4 5 1/2 1 2 1

Heisenberg Model in 2D

0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

QMC 100×100 12 bonds 13 bonds 0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

4 squares 5 squares 0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

14 sites 15 sites MR et al., PRE 75, 061118 (2007).

  • B. Tang et al., CPC 184, 557 (2013).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 18 / 34

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SLIDE 37

Resummation algorithms

We can define partial sums On =

n

  • i=1

Si, with Si =

  • ci

L(ci) × WO(ci) where all clusters ci share a given characteristic (no. of bonds, sites, etc). Goal: Estimate O = limn→∞ On from a sequence {On}, with n = 1, . . . , N.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 19 / 34

slide-38
SLIDE 38

Resummation algorithms

We can define partial sums On =

n

  • i=1

Si, with Si =

  • ci

L(ci) × WO(ci) where all clusters ci share a given characteristic (no. of bonds, sites, etc). Goal: Estimate O = limn→∞ On from a sequence {On}, with n = 1, . . . , N. Wynn’s algorithm: ε(−1)

n

= 0, ε(0)

n

= On, ε(k)

n

= ε(k−2)

n+1

+ 1 ∆ε(k−1)

n

where ∆ε(k−1)

n

= ε(k−1)

n+1

− ε(k−1)

n

. Brezinski’s algorithm [θ(−1)

n

= 0, θ(0)

n

= On]: θ(2k+1)

n

= θ(2k−1)

n

+ 1 ∆θ(2k)

n

, θ(2k+2)

n

= θ(2k)

n+1 + ∆θ(2k) n+1∆θ(2k+1) n+1

∆2θ(2k+1)

n

where ∆2θ(k)

n

= θ(k)

n+2 − 2θ(k) n+1 + θ(k) n .

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 19 / 34

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SLIDE 39

Resummation results (Heisenberg model)

Specific heat in the square lattice 0.1 1 10

T

0.2 0.4 0.6

Cv

QMC Wynn6 Euler ED 16 ED 9

MR, T. Bryant, and R. R. P . Singh, PRE 75, 061118 (2007).

  • B. Tang, E. Khatami, and MR, Comput. Phys. Commun. 184, 557 (2013).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 20 / 34

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SLIDE 40

Finite size effects

In unordered phases, not all ensemble calculations of finite systems approach the thermodynamic limit the same way

There is a preferred ensemble (the grand canonical ensemble) and preferred boundary conditions (periodic boundary conditions, so that the system is translationally invariant) for which finite-size effects are exponentially small in the system size. All others exhibit power-law convergence with system size.

  • D. Iyer, M. Srednicki, and MR, Phys. Rev. E 91, 062142 (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 21 / 34

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SLIDE 41

Finite size effects

In unordered phases, not all ensemble calculations of finite systems approach the thermodynamic limit the same way

There is a preferred ensemble (the grand canonical ensemble) and preferred boundary conditions (periodic boundary conditions, so that the system is translationally invariant) for which finite-size effects are exponentially small in the system size. All others exhibit power-law convergence with system size.

NLCEs convergence is also exponential, but a faster one!

  • D. Iyer, M. Srednicki, and MR, Phys. Rev. E 91, 062142 (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 21 / 34

slide-42
SLIDE 42

Finite size effects

In unordered phases, not all ensemble calculations of finite systems approach the thermodynamic limit the same way

There is a preferred ensemble (the grand canonical ensemble) and preferred boundary conditions (periodic boundary conditions, so that the system is translationally invariant) for which finite-size effects are exponentially small in the system size. All others exhibit power-law convergence with system size.

NLCEs convergence is also exponential, but a faster one! Kinetic energy in the t-V model

2 4 6 8 10 12 14 16 l 10

  • 7

10

  • 5

10

  • 3

10

  • 1

|Kl-KE|/KE CE-O CE-P GE-O GE-P NLCE

T=1.0

  • D. Iyer, M. Srednicki, and MR, Phys. Rev. E 91, 062142 (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 21 / 34

slide-43
SLIDE 43

Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 22 / 34

slide-44
SLIDE 44

Diagonal ensemble and NLCEs

The initial state is in thermal equilibrium in contact with a reservoir ˆ ρI

c =

  • a e−(Ec

a−µIN c a)/TI|acac|

ZI

c

, where ZI

c =

  • a

e−(Ec

a−µIN c a)/TI,

|ac (Ec

a) are the eigenstates (eigenvalues) of the initial Hamiltonian ˆ

HI

c in c.

MR, PRL 112, 170601 (2014); PRE 90, 031301(R) (2014).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 23 / 34

slide-45
SLIDE 45

Diagonal ensemble and NLCEs

The initial state is in thermal equilibrium in contact with a reservoir ˆ ρI

c =

  • a e−(Ec

a−µIN c a)/TI|acac|

ZI

c

, where ZI

c =

  • a

e−(Ec

a−µIN c a)/TI,

|ac (Ec

a) are the eigenstates (eigenvalues) of the initial Hamiltonian ˆ

HI

c in c.

At the time of the quench ˆ HI

c → ˆ

Hc , the system is detached from the

  • reservoir. Writing the eigenstates of ˆ

HI

c in terms of the eigenstates of ˆ

Hc ˆ ρDE

c

≡ limτ ′→∞ 1 τ ′ τ ′ dτ ˆ ρ(τ) =

  • α

W c

α |αcαc|

where W c

α =

  • a e−(Ec

a−µIN c a)/TI|αc|ac|2

ZI

c

, |αc (εc

α) are the eigenstates (eigenvalues) of the final Hamiltonian ˆ

Hc in c.

MR, PRL 112, 170601 (2014); PRE 90, 031301(R) (2014).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 23 / 34

slide-46
SLIDE 46

Diagonal ensemble and NLCEs

The initial state is in thermal equilibrium in contact with a reservoir ˆ ρI

c =

  • a e−(Ec

a−µIN c a)/TI|acac|

ZI

c

, where ZI

c =

  • a

e−(Ec

a−µIN c a)/TI,

|ac (Ec

a) are the eigenstates (eigenvalues) of the initial Hamiltonian ˆ

HI

c in c.

At the time of the quench ˆ HI

c → ˆ

Hc , the system is detached from the

  • reservoir. Writing the eigenstates of ˆ

HI

c in terms of the eigenstates of ˆ

Hc ˆ ρDE

c

≡ limτ ′→∞ 1 τ ′ τ ′ dτ ˆ ρ(τ) =

  • α

W c

α |αcαc|

where W c

α =

  • a e−(Ec

a−µIN c a)/TI|αc|ac|2

ZI

c

, |αc (εc

α) are the eigenstates (eigenvalues) of the final Hamiltonian ˆ

Hc in c. Using ˆ ρDE

c

in the calculation of O(c), NLCEs allow one to compute

  • bservables in the DE in the thermodynamic limit.

MR, PRL 112, 170601 (2014); PRE 90, 031301(R) (2014).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 23 / 34

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SLIDE 47

Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 24 / 34

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SLIDE 48

Models and quenches

Hard-core bosons in 1D lattices at half filling (µI = 0)

ˆ H =

L

  • i=1

−t

  • ˆ

b†

iˆ

bi+1 + H.c.

  • + V ˆ

niˆ ni+1 − t′ ˆ b†

iˆ

bi+2 + H.c.

  • + V ′ˆ

niˆ ni+2

Quench: TI, tI = 0.5, VI = 1.5, t′

I = V ′ I = 0 → t = V = 1.0, t′ = V ′

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 25 / 34

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SLIDE 49

Models and quenches

Hard-core bosons in 1D lattices at half filling (µI = 0)

ˆ H =

L

  • i=1

−t

  • ˆ

b†

iˆ

bi+1 + H.c.

  • + V ˆ

niˆ ni+1 − t′ ˆ b†

iˆ

bi+2 + H.c.

  • + V ′ˆ

niˆ ni+2

Quench: TI, tI = 0.5, VI = 1.5, t′

I = V ′ I = 0 → t = V = 1.0, t′ = V ′

NLCE with maximally connected clusters (l = 18 sites) Energy: EDE = Tr[ ˆ H ˆ ρDE] Convergence: ∆(ODE)l = |ODE

l

− ODE

18 |

|ODE

18 |

Convergence of EDE with l

2 7 12 17 l 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

∆(E

DE)l

2 7 12 17 l 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

TI=1 TI=5

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 25 / 34

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SLIDE 50

Few-body experimental observables in the DE

Momentum distribution ˆ mk = 1 L

  • jj′

eik(j−j′)ˆ ρjj′

π/4 π/2 3π/4 π k 0.4 0.5 0.6 0.7 (mk)18 Initial t’=V’=0, DE t’=V’=0, GE t’=V’=0.5, DE t’=V’=0.5, GE

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 26 / 34

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SLIDE 51

Few-body experimental observables in the DE

Momentum distribution ˆ mk = 1 L

  • jj′

eik(j−j′)ˆ ρjj′

π/4 π/2 3π/4 π k 0.4 0.5 0.6 0.7 (mk)18 Initial t’=V’=0, DE t’=V’=0, GE t’=V’=0.5, DE t’=V’=0.5, GE

Differences between DE and GE δ(m)l =

  • k |(mk)DE

l

− (mk)GE

18 |

  • k(mk)GE

18

10

  • 3

10

  • 2

δ(m)l t’=V’=0 t’=V’=0.025 10 11 12 13 14 15 16 17 18 19

l

10

  • 4

10

  • 3

t’=V’=0.1 t’=V’=0.5

TI=2 TI=10

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 26 / 34

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SLIDE 52

Outline

1

Introduction Quantum dynamics, quenches, and thermalization Many-body quantum systems in thermal equilibrium Numerical linked cluster expansions

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-V -t′-V ′ chain Many-body localization

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 27 / 34

slide-53
SLIDE 53

NLCEs for disordered systems

Hamiltonian with diagonal disorder ˆ H =

  • i
  • −t(ˆ

b†

iˆ

bi+1 + H.c.) + V

  • ˆ

ni − 1 2 ˆ ni+1 − 1 2

  • + hi
  • ˆ

ni − 1 2

  • binary disorder (equal probabilities for hi = ±h).
  • B. Tang, D. Iyer, and MR, PRB 91, 161109(R) (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 28 / 34

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SLIDE 54

NLCEs for disordered systems

Hamiltonian with diagonal disorder ˆ H =

  • i
  • −t(ˆ

b†

iˆ

bi+1 + H.c.) + V

  • ˆ

ni − 1 2 ˆ ni+1 − 1 2

  • + hi
  • ˆ

ni − 1 2

  • binary disorder (equal probabilities for hi = ±h).

Disorder average restores translational invariance (exactly!) O(c) =

  • Tr[ ˆ

Oˆ ρc]

  • dis ,

where ·dis represents the disorder average.

  • B. Tang, D. Iyer, and MR, PRB 91, 161109(R) (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 28 / 34

slide-55
SLIDE 55

NLCEs for disordered systems

Hamiltonian with diagonal disorder ˆ H =

  • i
  • −t(ˆ

b†

iˆ

bi+1 + H.c.) + V

  • ˆ

ni − 1 2 ˆ ni+1 − 1 2

  • + hi
  • ˆ

ni − 1 2

  • binary disorder (equal probabilities for hi = ±h).

Disorder average restores translational invariance (exactly!) O(c) =

  • Tr[ ˆ

Oˆ ρc]

  • dis ,

where ·dis represents the disorder average. Initial state: tI = 0.5, VI = 2.5, hj = 0, and TI (no disorder) Final Hamiltonian: t = 1, V = 2.0, and different values of h = 0

  • B. Tang, D. Iyer, and MR, PRB 91, 161109(R) (2015).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 28 / 34

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SLIDE 56

Disordered systems and many-body localization

Ratio of consecutive energy gaps

0.5 1 2 4 8 h 0.35 0.4 0.45 0.5 0.55 r L=14 L=14 L=15 L=16 t=1.0, V=2.0 r = 0.53 r = 0.39

Ratio of the smaller and the larger of two consecutive energy gaps rn = min[δE

n−1, δE n ]/max[δE n−1, δE n ],

where δE

n ≡ En+1 − En

we compute r = rdis

n ndis.

Continuous disorder: hc ≈ 7 [A. Pal and D. A. Huse, PRB 82, 174411 (2010).]

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 29 / 34

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SLIDE 57

Disordered systems and many-body localization

Ratio of consecutive energy gaps

0.5 1 2 4 8 h 0.35 0.4 0.45 0.5 0.55 r L=14 L=14 L=15 L=16 t=1.0, V=2.0 r = 0.53 r = 0.39

Diagonal vs Thermal

π/4 π/2 3π/2 π k 0.4 0.5 0.6 (mk)14

h=0.6 h=1.0 h=4.0 h=6.0 2 4 6 8

h

  • 0.2
  • 0.15
  • 0.1
  • 0.05

(K)14 DE GE

h=0.6 h=1.0 h=4.0 h=6.0 Initial

TI=2.0 DE GE

Ratio of the smaller and the larger of two consecutive energy gaps rn = min[δE

n−1, δE n ]/max[δE n−1, δE n ],

where δE

n ≡ En+1 − En

we compute r = rdis

n ndis.

Continuous disorder: hc ≈ 7 [A. Pal and D. A. Huse, PRB 82, 174411 (2010).]

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 29 / 34

slide-58
SLIDE 58

Scaling of the differences and errors

0.06 0.07 0.08 δ (m) l h=3.2 h=3.3 h=3.4 h=3.5 6 8 10 12 14 l 10

  • 4

10

  • 3

10

  • 2

10

  • 1

∆ (m

DE) l

h=3.6 h=3.7 h=3.8 TI = 2.0 (a) (b)

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 30 / 34

slide-59
SLIDE 59

Conclusions

NLCEs provide a general framework to study the diagonal en- semble in lattice systems after a quantum quench in the thermo- dynamic limit. The grand canonical ensemble in translationally invariant sys- tems is special (exponentially small finite size effects vs power law for other cases). NLCEs also converge exponentially, but even faster! NLCE results indicate that few-body observables thermalize in nonintegrable systems while they do not thermalize in integrable

  • systems. Time scale for thermalization as one approaches the

integrable point. Quantum quenches within NLCEs can be used to study the tran- sition between ergodicity and many-body localization (arbitrary dimensions). In one dimension, the NLCE results support the existence of many-body localization in the thermodynamics limit.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 31 / 34

slide-60
SLIDE 60

Collaborators

Deepak Iyer (Penn State) Baoming Tang (Penn State) Deepak Iyer (Penn State) Mark Srednicki (UCSB)

PRB 91, 161109(R) (2015). PRE 91, 062142 (2015).

Supported by:

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 32 / 34

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SLIDE 61

Dispersion of the energy in the DE

Dispersion of the energy ∆E2 = 1 L( ˆ H2 − ˆ H2) Deviations from the GE δ(O)l = |ODE

l

− OGE

18 |

|OGE

18 |

1 10 100 T 0.5 0.6 0.7 0.8 0.9 1 ∆E

2

0.6 0.8 1

18

t’=V’=0 t’=V’=0.5 t’=V’=0.5, diff. init. state 2 6 10 14 18 l 0.05 0.1 δ(∆E

2)l

1 10 100 T 0.5 0.6 0.7 0.8 0.9 1 ∆E

2

0.6 0.8 1

18

DE GE

TI=1

The dispersion of the energy (and of the particle number) in the DE depends

  • n the initial state independently of whether the system is integrable or not.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 33 / 34

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SLIDE 62

Few-body experimental observables in the DE

nn kinetic energy K = −t

  • i

ˆ b†

iˆ

bi+1 Differences between DE and GE δ(K)l = |KDE

l

− KGE

18 |

KGE

18

1 10 100 T 0.005 0.01 0.015 0.02 0.025 0.03 δ(K)18 t’=V’=0 t’=V’=0.025 t’=V’=0.1 t’=V’=0.5 10

  • 4

10

  • 3

10

  • 2

t’=V’=0 t’=V’=0.025 10 11 12 13 14 15 16 17 18 19

l

10

  • 3

10

  • 1

δ(K)l t’=V’=0.1 t’=V’=0.5

TI=2 TI=10

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble August 4, 2015 34 / 34