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Quantum quenches in the thermodynamic limit Marcos Rigol Department of Physics The Pennsylvania State University Quantum dynamics in systems with many coupled degrees of freedom: challenges for theory Center for Free-Electron Laser Science,

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Quantum quenches in the thermodynamic limit

Marcos Rigol

Department of Physics The Pennsylvania State University

Quantum dynamics in systems with many coupled degrees of freedom: challenges for theory Center for Free-Electron Laser Science, Hamburg March 26, 2014

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 1 / 30

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SLIDE 2

Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 2 / 30

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SLIDE 3

Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 3 / 30

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SLIDE 4

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 March 26, 2014 4 / 30

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SLIDE 5

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 March 26, 2014 4 / 30

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SLIDE 6

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 March 26, 2014 4 / 30

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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! 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 March 26, 2014 4 / 30

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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 March 26, 2014 5 / 30

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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 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 March 26, 2014 5 / 30

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Linked-Cluster Expansions

In HTEs O(c) is expanded in powers of β and only a finite number

  • f terms are retained.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30

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Linked-Cluster Expansions

In HTEs O(c) is expanded in powers of β and only a finite number

  • f terms are retained.

In 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). MR, T. Bryant, and R. R. P . Singh, PRE 75, 061118 (2007). MR, T. Bryant, and R. R. P . Singh, PRE 75, 061119 (2007).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30

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Linked-Cluster Expansions

In HTEs O(c) is expanded in powers of β and only a finite number

  • f terms are retained.

In 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). MR, T. Bryant, and R. R. P . Singh, PRE 75, 061118 (2007). MR, T. Bryant, and R. R. P . Singh, PRE 75, 061119 (2007).

2D Hubbard-like models (square and honeycomb), spin models (kagome, checkerboard, pyroclore – experiments)

MR and R. R. P . Singh, PRL 98, 207204 (2007). MR and R. R. P . Singh, PRB 76, 184403 (2007).

  • E. Khatami and MR, PRB 83, 134431 (2011).
  • E. Khatami and MR, PRA 84, 053611 (2011).
  • E. Khatami, R. R. P

. Singh, and MR, PRB 84, 224411 (2011).

  • E. Khatami, J. S. Helton, and MR, PRB 85, 064401 (2012).
  • E. Khatami and MR, PRA 86, 023633 (2012).
  • B. Tang, T. Paiva, E. Khatami, and MR, PRL 109, 205301 (2012).
  • B. Tang, T. Paiva, E. Khatami, and MR, PRB 88, 125127 (2013).

. . .

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30

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Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 7 / 30

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Numerical Linked Cluster Expansions

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 March 26, 2014 8 / 30

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Numerical Linked Cluster Expansions

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 March 26, 2014 8 / 30

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Numerical Linked Cluster Expansions

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 March 26, 2014 8 / 30

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Numerical Linked Cluster Expansions

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 March 26, 2014 8 / 30

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Numerical Linked Cluster Expansions

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) Perform the subgraph substraction to compute the weight of each cluster Heisenberg Model

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 March 26, 2014 8 / 30

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SLIDE 19

Numerical Linked-Cluster Expansions

Site clusters

c

2

L(c)

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

  • No. of sites

topological clusters 1 1 2 1 3 1 4 3 5 4 6 10 7 19 8 51 9 112 10 300 11 746 12 2042 13 5450 14 15197 15 42192

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 9 / 30

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SLIDE 20

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 March 26, 2014 10 / 30

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SLIDE 21

Numerical Linked-Cluster Expansions

Square clusters c

2

L(c)

1 1 3 4 5 1/2 1 2 1

Heisenberg Model

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 March 26, 2014 10 / 30

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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 March 26, 2014 11 / 30

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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

.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 11 / 30

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SLIDE 24

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 March 26, 2014 11 / 30

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SLIDE 25

Resummation results (Heisenberg model)

Energy (square lattice)

0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

QMC 100×100 Brez3 Wynn6 ED 16

Cv (square lattice)

0.1 1 10

T

0.2 0.4 0.6

Cv

QMC Wynn6 Euler ED 16 ED 9 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 12 / 30

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Resummation results (Heisenberg model)

Energy (square lattice)

0.1 1 10

T

  • 0.8
  • 0.6
  • 0.4
  • 0.2

E

QMC 100×100 Brez3 Wynn6 ED 16

Cv (square lattice)

0.1 1 10

T

0.2 0.4 0.6

Cv

QMC Wynn6 Euler ED 16 ED 9

Cv Bare (triang. lattice)

0.1 1 10

T

0.1 0.2 0.3

Cv

BM 11 bonds 12 bonds 0.1 1 10

T

0.1 0.2 0.3

Cv

12 sites 13 sites 7 triangles 8 triangles

Cv Ressum. (triang. lattice)

0.1 1 10

T

0.1 0.2 0.3

Cv

BM Euler

12

Euler

13

Wynn5

2

Wynn5

3

(BM) B. Bernu and G. Misguich, PRB 63, 134409 (2001).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 12 / 30

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SLIDE 27

Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 13 / 30

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SLIDE 28

Quantum Newton’s Cradle

  • 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

Also in: M. Gring et al. (Schmiedmayer’s group), Science 337, 1318 (2012).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 14 / 30

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SLIDE 29

Quenches in one-dimensional superlattices

Quantum dynamics in a 1D superlattice

Trotzky et al. (Bloch’s group), Nature Phys. 8, 325 (2012).

Initial state |01010 . . . 1010 Unitary dynamics under the “Bose-Hubbard” Hamiltonian Experimental results (◦) vs exact t-DMRG calculations (lines) without free parameters local observables (top) vs nonlocal observables (bottom)

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 15 / 30

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SLIDE 30

Unitary dynamics after a sudden quench

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 March 26, 2014 16 / 30

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SLIDE 31

Unitary dynamics after a sudden quench

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 March 26, 2014 16 / 30

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SLIDE 32

Unitary dynamics after a sudden quench

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 March 26, 2014 16 / 30

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SLIDE 33

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: 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 March 26, 2014 17 / 30

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SLIDE 34

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 March 26, 2014 18 / 30

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SLIDE 35

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 March 26, 2014 19 / 30

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SLIDE 36

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 March 26, 2014 20 / 30

slide-37
SLIDE 37

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 March 26, 2014 20 / 30

slide-38
SLIDE 38

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 March 26, 2014 20 / 30

slide-39
SLIDE 39

Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 21 / 30

slide-40
SLIDE 40

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, arXiv:1401.2160.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 22 / 30

slide-41
SLIDE 41

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, arXiv:1401.2160.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 22 / 30

slide-42
SLIDE 42

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, arXiv:1401.2160.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 22 / 30

slide-43
SLIDE 43

Outline

1

Introduction Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches

2

Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quantum quenches in one-dimension

3

Conclusions

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 23 / 30

slide-44
SLIDE 44

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 March 26, 2014 24 / 30

slide-45
SLIDE 45

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: ∆(Oens)l = |Oens

l

− Oens

18 |

|Oens

18 |

Convergence of EDE with l

0.1 1 10 100 TI 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

1 ∆(E

DE)17

t’=V’=0 t’=V’=0.1 t’=V’=0.25 t’=V’=0.5 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 TI=1 TI=5

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 24 / 30

slide-46
SLIDE 46

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 ′

Temperature after the quench: (if in thermal equilibrium) EDE

l=18

= EGE

l=18

EGE = Tr[ ˆ He−( ˆ

H−µ ˆ N)/T ]

Tr[e−( ˆ

H−µ ˆ N)/T ]

Relative energy difference between EDE

18 and EGE 18

is smaller than 10−11 Temperature after the quench

0.1 1 10 100 TI 1 10 100 T t’=V’=0 t’=V’=0.1 t’=V’=0.25 t’=V’=0.5 0.1 1 10 100 TI 1 10 100 T t’=V’=0, diff. init. state

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 24 / 30

slide-47
SLIDE 47

Energy and particle number dispersion in the DE

Energy dispersion ∆E2 = 1 L( ˆ H2 − ˆ H2)

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

Particle number dispersion ∆N 2 = 1 L( ˆ N 2 − ˆ N2)

2 6 10 14 18 l 0.2 0.3 0.4 δ(∆N

2)l

1 10 100 T 0.1 0.15 0.2 0.25 ∆N

2

0.2

18

DE GE 1 10 100 T 0.1 0.15 0.2 0.25 ∆N

2

0.2

18

t’=V’=0 t’=V’=0.5 t’=V’=0.5, diff. init. state

TI=1

δ(O)l = |ODE

l

− OGE

18 |

|OGE

18 |

The dispersion of the energy and 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 March 26, 2014 25 / 30

slide-48
SLIDE 48

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 March 26, 2014 26 / 30

slide-49
SLIDE 49

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 10

  • 5

10

  • 3

10

  • 1

∆(m)17 DE GE 0.1 1 10 100 TI 10

  • 13

10

  • 11

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 March 26, 2014 26 / 30

slide-50
SLIDE 50

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 March 26, 2014 27 / 30

slide-51
SLIDE 51

NLCEs vs exact diagonalization

10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

1 10

2

∆(E

GE)l

V’=0 V’=1 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

1 10

2

∆(∆E

2 GE)l

T=1 T=5 1 5 9 13 17 l 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

1 10

2

∆(K

GE)l

1 5 9 13 17 l 10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

10

  • 4

10

  • 2

1 10

2

∆(m

GE)l

18 20 22 L 0.03 0.1 0.3 δ(E

ED)L

18 20 22 L 0.02 0.01 0.005 δ(m

ED)L

18 20 22 L 0.02 0.04 0.07 δ(K

ED)L

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 28 / 30

slide-52
SLIDE 52

Conclusions

NLCEs provide a general framework to study the diagonal en- semble in lattice systems after a quantum quench in the thermo- dynamic limit. NLCE results suggest that few-body observables thermalize in nonintegrable systems while they do not thermalize in integrable systems. As one approaches the integrable point DE-NLCEs behave as NLCEs for equilibrium systems approaching a phase transition. This suggests that a transition to thermalization may occur as soon as one breaks integrability.

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 29 / 30

slide-53
SLIDE 53

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

DQMC of a 2D system with: U = 6t, V = 0.04t, T = 0.31t and 560 fermions

  • 15 -10 -5

5 10 15 Density 0.0 0.3 0.6 0.9 1.2

  • 15 -10 -5

5 10 15 Density fluctuations ×10-1 0.0 0.9 1.8 2.7

  • 15 -10 -5

5 10 15 Spin correlations ×10-1 0.0 0.8 1.6 2.4 3.2

  • 15 -10 -5

5 10 15 Pairing correlations ×10-1 0.0 0.8 1.6 2.4

  • S. Chiesa, C. N. Varney, MR, and R. T. Scalettar, PRL 106, 035301 (2011).

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 30 / 30