Quantum quenches in the thermodynamic limithawk.fisica.uminho.pt/cccqs/PDFS/Rigol_Evora2014.pdf ·...

Quantum quenches in the thermodynamic limit

Marcos Rigol

Department of PhysicsThe Pennsylvania State University

Correlations, criticality, and coherence in quantum systemsEvora, Portugal

October 9, 2014

Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 2014 1 / 34

Outline

1 IntroductionQuantum quenchesMany-body quantum systems in thermal equilibrium

2 Numerical Linked Cluster ExpansionsDirect sumsResummations

3 Quantum quenches in the thermodynamic limitDiagonal ensemble and NLCEsQuenches in the t-V -t′-V ′ chain (thermalization)Quenches in XXZ chain from a Neel state (QA vs GGE)

4 Conclusions


Quenches in one-dimensional superlatticesQuantum dynamics in a

1D superlatticeTrotzky et al. (Bloch’s group),Nature Phys. 8, 325 (2012).

Initial state |01010 . . . 1010〉

Unitary dynamics under the“Bose-Hubbard” Hamiltonian

Experimental results (◦) vsexact t-DMRG calculations

(lines) without free parameters

local observables (top)vs

nonlocal observables (bottom)


Relaxation dynamics after turning off a superlattice

0

0.06

0.12

0.18

0.24

−20 −10 0 10 20

n

x/a

Density profile

τ=0

0

0.5

1

1.5

2

2.5

π/2−π/2 π −π 0n

k

ka

Momentum profile

τ=0

MR, A. Muramatsu, and M. Olshanii, PRA 74, 053616 (2006).MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007).


/home/mrigol/3_Presentation/Talks/VIDEOS/Superlattice.gif

Unitary dynamics after a sudden quenchIf the initial state is not an eigenstate of Ĥ

|ψ0〉 6= |α〉 where Ĥ|α〉 = Eα|α〉 and E0 = 〈ψ0|Ĥ|ψ0〉,

then a few-body observable O will evolve following

O(τ) ≡ 〈ψ(τ)|Ô|ψ(τ)〉 where |ψ(τ)〉 = e−iĤτ/~|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 ρ̂DE ≡∑α |Cα|2|α〉〈α|)

O(τ) = limτ→∞

1

τ

∫ τ0

dτ ′〈Ψ(τ ′)|Ô|Ψ(τ ′)〉 =∑α

|Cα|2Oαα ≡ 〈Ô〉DE,

which depends on the initial conditions through Cα = 〈α|ψ0〉.


Description after relaxation (lattice models)

Hard-core boson (spinless fermion) Hamiltonian

Ĥ =

L∑i=1

−t(b̂†i b̂i+1 + H.c.

)+ V n̂in̂i+1 − t′

(b̂†i b̂i+2 + H.c.

)+ V ′n̂in̂i+2

Dynamics vs statistical ensembles

Nonintegrable: t′ = V ′ 6= 0

-π -π/2 0 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 0 π/2 π

ka

0

0.25

0.5

n(k

) time averagethermalGGE

MR, Dunjko, Yurovsky, andOlshanii, PRL 98, 050405 (2007), . . .


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

The expectation value 〈α|Ô|α〉 of a few-body observable Ô in aneigenstate of the Hamiltonian |α〉, with energy Eα, of a many-bodysystem is equal to the thermal average of Ô at the mean energy Eα:

〈α|Ô|α〉 = 〈Ô〉ME(Eα).

Nonintegrable

0

1

2

3

n(k

x=

0)

-10 -8 -6 -4 -2 0

E[J]

0

1

2

ρ(E

)[J

-1]ρ(E) exact

ρ(E) microcan.

ρ(E) canonical

Integrable (ρ̂GGE = 1ZGGE e−

∑m λm Îm )

0

0.5

1

1.5

n(k

x=

0)

-8 -6 -4 -2 0

E[J]

0

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).


Time fluctuations and their scaling with system size

0.1δN

kL=21

L=24

0

0.1

δN

k

0

0.1

δN

k

0 20 40 60 80 100τ

0

0.1

δN

k

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)|∑

kNdiag(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).


/home/mrigol/3_Presentation/Talks/VIDEOS/Bosons1D_NKLongT3.0.gif

Time fluctuationsAre they small because of dephasing?

〈Ô(t)〉 − 〈Ô(t)〉 =∑α′,αα′ 6=α

C?α′Cαei(Eα′−Eα )tOα′α ∼

∑α′,αα′ 6=α

ei(Eα′−Eα )t

NstatesOα′α

∼√N2statesNstates

Otypicalα′α ∼ Otypicalα′α

Time average of 〈Ô〉

〈Ô〉 =∑α

|Cα|2Oαα

∼∑α

1

NstatesOαα ∼ Otypicalαα

One needs: Otypicalα′α � Otypicalαα

MR, PRA 80, 053607 (2009)


Outline




4 Conclusions


Finite temperature properties of lattice models

Computational techniques for arbitrary dimensionsQuantum Monte Carlo simulationsPolynomial time⇒ Large systems⇒ Finite size scalingSign problem⇒ Limited classes of models

Exact diagonalizationExponential problem⇒ Small systems⇒ Finite size effectsNo systematic extrapolation to larger system sizesCan be used for any model!

High temperature expansionsExponential problem⇒ High temperaturesThermodynamic limit⇒ Extrapolations to low TCan be used for any model!Can fail (at low T ) even when correlations are short ranged!

DMFT, DCA, DMRG, . . .


Linked-Cluster Expansions

Extensive observables Ô 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 weightof 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{Ô ρ̂GCc

},

ρ̂GCc =1

ZGCcexp−(Ĥc−µN̂c)/kBT

ZGCc = Tr{

exp−(Ĥc−µN̂c)/kBT}

and the s sum runs over all subclusters of c.



Extensive observables Ô 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 weightof 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{Ô ρ̂GCc

},

ρ̂GCc =1

ZGCcexp−(Ĥc−µN̂c)/kBT

ZGCc = Tr{

exp−(Ĥc−µN̂c)/kBT}

and the s sum runs over all subclusters of c.



In HTEs O(c) is expanded in powers of β and only a finitenumber of terms is retained.

In numerical linked cluster expansions (NLCEs) an exactdiagonalization of the cluster is used to calculate O(c) at anytemperature.(Spins models in the square, triangular, and kagome lattices)MR, T. Bryant, and R. R. P. Singh, PRL 97, 187202 (2006).MR, T. Bryant, and R. R. P. Singh, PRE 75, 061118 (2007).(t-J model in the square lattice)MR, T. Bryant, and R. R. P. Singh, PRE 75, 061119 (2007).



In HTEs O(c) is expanded in powers of β and only a finitenumber of terms is retained.

In numerical linked cluster expansions (NLCEs) an exactdiagonalization of the cluster is used to calculate O(c) at anytemperature.(Spins models in the square, triangular, and kagome lattices)MR, T. Bryant, and R. R. P. Singh, PRL 97, 187202 (2006).MR, T. Bryant, and R. R. P. Singh, PRE 75, 061118 (2007).(t-J model in the square lattice)MR, T. Bryant, and R. R. P. Singh, PRE 75, 061119 (2007).

PRL 98, 207204 (2007), PRB 76, 184403 (2007), PRB 83, 134431 (2011), PRA 84, 053611 (2011), PRB 84, 224411 (2011),PRB 85, 064401 (2012), PRA 86, 023633 (2012), PRL 109, 205301 (2012), CPC 184, 557 (2013), PRB 88, 125127 (2013), . . .


Outline




4 Conclusions


Numerical Linked Cluster Expansions

i) Find all clusters that can beembedded on the lattice

ii) Group the ones with thesame Hamiltonian (Topo-logical cluster)

iii) Find all subclusters of agiven topological cluster

iv) Diagonalize the topologicalclusters and compute theobservables

v) Compute the weight of eachcluster and compute the di-rect sum of the weights

Bond clusters

c

2

L(c)

2

3 2

4 4

5 4

6 2

7 4

11

8 4

9 8








No. of bonds topological clusters0 11 12 13 24 45 66 147 288 689 156

10 39911 101212 273213 738514 20665








Heisenberg Model

0.1 1 10

T

-0.8

-0.6

-0.4

-0.2

0

E

QMC 100×100

12 bonds13 bonds

MR et al., PRE 75, 061118 (2007).B. Tang et al., CPC 184, 557 (2013).


Numerical Linked-Cluster Expansions

Square clustersc

2

L(c)

11

3

4

5

1/2

1

2

1

No. of squares topological clusters0 11 12 13 24 55 11


Numerical Linked-Cluster Expansions

Square clustersc

2

L(c)

11

3

4

5

1/2

1

2

1

Heisenberg Model

0.1 1 10

T

-0.8

-0.6

-0.4

-0.2

0

E

QMC 100×100

12 bonds13 bonds

0.1 1 10

T

-0.8

-0.6

-0.4

-0.2

0

E4 squares

5 squares

0.1 1 10

T

-0.8

-0.6

-0.4

-0.2

0

E

14 sites15 sites

MR et al., PRE 75, 061118 (2007).B. Tang et al., CPC 184, 557 (2013).


Outline




4 Conclusions


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 .


Resummation results (Heisenberg model)

Energy (square lattice)

0.1 1 10

T

-0.8

-0.6

-0.4

-0.2

0

E

QMC 100×100

Wynn6

EulerED 16

Cv (square lattice)

0.1 1 10

T

0

0.2

0.4

0.6

Cv

QMC

Wynn6

EulerED 16ED 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).


Outline




4 Conclusions


Diagonal ensemble and NLCEsThe initial state is in thermal equilibrium in contact with a reservoir

ρ̂Ic =

∑a e−(Eca−µIN

ca)/TI |ac〉〈ac|

ZIc, where ZIc =

∑a

e−(Eca−µ

INca)/TI ,

|ac〉 (Eca) are the eigenstates (eigenvalues) of the initial Hamiltonian ĤIc in c.

At the time of the quench ĤIc → Ĥc , the system is detached from thereservoir. Writing the eigenstates of ĤIc in terms of the eigenstates of Ĥc

ρ̂DEc ≡ limτ ′→∞1

τ ′

∫ τ ′0

dτ ρ̂(τ) =∑α

W cα |αc〉〈αc|

whereW cα =


ca)/TI |〈αc|ac〉|2

ZIc,

|αc〉 (εcα) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c.

Using ρ̂DEc in the calculation of O(c), NLCEs allow one to computeobservables in the DE in the thermodynamic limit.

MR, PRL 112, 170601 (2014).


Diagonal ensemble and NLCEsThe initial state is in thermal equilibrium in contact with a reservoir

ρ̂Ic =


ca)/TI |ac〉〈ac|

ZIc, where ZIc =

∑a

e−(Eca−µ

INca)/TI ,

|ac〉 (Eca) are the eigenstates (eigenvalues) of the initial Hamiltonian ĤIc in c.

At the time of the quench ĤIc → Ĥc , the system is detached from thereservoir. Writing the eigenstates of ĤIc in terms of the eigenstates of Ĥc

ρ̂DEc ≡ limτ ′→∞1

τ ′

∫ τ ′0

dτ ρ̂(τ) =∑α

W cα |αc〉〈αc|

whereW cα =


ca)/TI |〈αc|ac〉|2

ZIc,

|αc〉 (εcα) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c.

Using ρ̂DEc in the calculation of O(c), NLCEs allow one to computeobservables in the DE in the thermodynamic limit.MR, PRL 112, 170601 (2014).


Outline




4 Conclusions


Models and quenches

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

Ĥ =

L∑i=1

−t(b̂†i b̂i+1 + H.c.

)+ V n̂in̂i+1 − t′

(b̂†i b̂i+2 + H.c.

)+ V ′n̂in̂i+2

Quench: TI , tI = 0.5, VI = 1.5, t′I = V′I = 0→ t = V = 1.0, t′ = V ′


Models and quenches

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

Ĥ =

L∑i=1

−t(b̂†i b̂i+1 + H.c.

)+ V n̂in̂i+1 − t′

(b̂†i b̂i+2 + H.c.

)+ V ′n̂in̂i+2

Quench: TI , tI = 0.5, VI = 1.5, t′I = V′I = 0→ t = V = 1.0, t′ = V ′

NLCE with maximallyconnected clusters

(l = 18 sites)

Energy: EDE = Tr[Ĥρ̂DE]

Convergence:

∆(Oens)l =|Oensl −Oens18 ||Oens18 |

Convergence of EDE with l

0.1 1 10 100T

I

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

∆(E

DE) 1

7

t’=V’=0

t’=V’=0.1

t’=V’=0.25

t’=V’=0.5

2 7 12 17l

10-14

10-12

10-10

10-8

10-6

10-4

10-2

∆(E

DE) l

2 7 12 17l

TI=1 T

I=5


Few-body experimental observables in the DE

Momentum distribution

m̂k =1

L

∑jj′

eik(j−j′)ρ̂jj′

0 π/4 π/2 3π/4 π

k

0.4

0.5

0.6

0.7

(mk) 1

8

Initial

t’=V’=0, DE

t’=V’=0, GE

t’=V’=0.5, DE

t’=V’=0.5, GE



Momentum distribution

m̂k =1

L

∑jj′

eik(j−j′)ρ̂jj′

0 π/4 π/2 3π/4 π

k

0.4

0.5

0.6

0.7

(mk) 1

8

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)DEl − (mk)GE18 |∑

k(mk)GE18

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


Outline




4 Conclusions


Failure of the GGE based on local quantitiesXXZ (integrable) Hamiltonian

Ĥ = J

(∑i

σxi σxi+1 + σ

yi σ

yi+1 + ∆σ

zi σ

zi+1

)

Quench starting from the Neel state to different values of ∆ ≥ 1Quench action solution differs from GGE based on local quantities:B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, MR, and J.-S. Caux,PRL 113, 117202 (2014).B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács,PRL 113, 117203 (2014).

Can we use NLCEs for ground states (pure states)?Using the parity symmetry of the clusters:

|aec〉 =1√2

(| . . . ↑↓↑↓ . . .〉+ | . . . ↓↑↓↑ . . .〉)

|aoc〉 =1√2

(| . . . ↑↓↑↓ . . .〉 − | . . . ↓↑↓↑ . . .〉)

W c,e/oα = |〈αe/oc |ae/oc 〉|2


Results for spin-spin correlations

0.05

0.1

0.15

0.2

<σ

1

zσ

3

z>

0.73

0.77

0.81

<σ

1

zσ

3>

0.9

0.92

0.94

<σ

1

zσ

3

z>

10 11 12 13 14 15 16 17 18

l

0.95

0.96

0.97<

σ1

zσ

3

z>

-0.45

-0.35

-0.25

<σ

1

zσ

2

z>

DEGE

-0.9

-0.88

-0.86

<σ

1

zσ

2

z>

QA

Resum

-0.962

-0.96

-0.958

-0.956

<σ

1

zσ

2

z>

10 11 12 13 14 15 16 17 18

l

-0.981

-0.98

-0.979

<σ

1

zσ

2

z>

∆=7

∆=4

(a)∆=1

(b)

(c)

(d)

(e)

(f)

(g)

(h)∆=10

MR, PRE 90, 031301(R) (2014)


Quench action, GGE, and NLCE

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9

∆

〈σz1σz3〉

(a)

-0.09

00.02

1 2

δ〈σz1σz3〉

0.11

0.12

0.13

1 1.01 1.02 1.03

∆

〈σz1σz3〉

(b) 〈σz1σz3〉sp〈σz1σz3〉GGE〈σz1σz3〉NLCE

δ〈σz1σz3〉 = 〈σz1σz3〉GGE/〈σz1σz3〉QA − 1

B. Wouters et al., PRL 113, 117202 (2014).


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 innonintegrable systems while they do not thermalize in integrablesystems.

As one approaches the integrable point DE-NLCEs behave asNLCEs for equilibrium systems approaching a phase transition.This suggests that a transition to thermalization may occur assoon as one breaks integrability.

The GGE based on known local conserved quantities does notdescribe observables after relaxation, while the QA does, as sug-gested by the NLCE results. New things to be learned about in-tegrable systems!


Collaborators

Michael Brockmann (ITP Amsterdam)

Jean-Sébastien Caux (ITP Amsterdam)

Jacopo De Nardis (ITP Amsterdam)

Davide Fioretto (ITP Amsterdam)

Bram Wouters (ITP Amsterdam)

PRL 113, 117202 (2014).

Supported by:


Finite temperature properties of lattice models

Computational techniques for arbitrary dimensionsQuantum Monte Carlo simulationsPolynomial time⇒ Large systems⇒ Finite size scalingSign 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 0 5 10 15

Density

0.0

0.3

0.6

0.9

1.2

-15 -10 -5 0 5 10 15

Density fluctuations×10

-1

0.0

0.9

1.8

2.7

-15 -10 -5 0 5 10 15

Spin correlations

×10-1

0.0

0.8

1.6

2.4

3.2

-15 -10 -5 0 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).


Dispersion of the energy in the DE

Dispersion of the energy

∆E2 =1

L(〈Ĥ2〉 − 〈Ĥ〉2)

Deviations from the GE

δ(O)l =|ODEl −OGE18 ||OGE18 |

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 18l

0

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

DEGE

TI=1

The dispersion of the energy (and of the particle number) in the DE dependson the initial state independently of whether the system is integrable or not.



nn kinetic energy

K = −t∑i

〈b̂†i b̂i+1〉

Differences between DE and GE

δ(K)l =|KDEl −KGE18 |

KGE18

1 10 100

T

0

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


NLCEs vs exact diagonalization (equilibrium, t′ = 0)

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

102

∆(E

GE) l

V’=0

V’=1

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

102

∆(∆

E 2

GE) l

T=1

T=5

1 5 9 13 17l

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

102

∆(K

GE) l

1 5 9 13 17l

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

102

∆(m

GE) l

18 20 22L

0.03

0.1

0.3

δ(E

ED

) L

18 20 22L

0.02

0.01

0.005

δ(m

ED

) L

18 20 22L

0.02

0.04

0.07

δ(K

ED

) L

ED from L. Santos and MR, PRE 82, 031130 (2010).Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 2014 34 / 34

IntroductionQuantum quenchesMany-body quantum systems in thermal equilibrium

Numerical Linked Cluster ExpansionsDirect sumsResummations

Quantum quenches in the thermodynamic limitDiagonal ensemble and NLCEsQuenches in the t-V-t'-V' chain (thermalization)Quenches in XXZ chain from a Neel state (QA vs GGE)

Conclusions

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