Page 1

APS March Meeting 2016 March 14-18, 2016 Baltimore, Maryland

Short and long-time dynamics 
 of isolated many-body 
 quantum systems:
 how to predict thermalization from
 powerlaw decay exponents Marco Tavora (Yeshiva University) E.J. Torres-Herrera (Universidad Autónoma de Puebla) Lea F. Santos (Yeshiva University)

arXiv:1601.05807

Supported by: NSF DMR 1147430!


0.5

f(t)

peak or Poisson shape [46] to a Wigner-Dyson form [47]. The initial states that are mainly considered here are site0.4 Outline basis vectors, where the spin on each site either points up or 0.3 down in the z-direction. They include the experimentally accessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain 0.2 Study relaxation process isolated systems wall state, |DWi = of | """ . . . ### MB . . .i, quantum both extensively used after a quench in studies of the dynamics of integrable spin models. 0.1 Case 1: Powerlaw decay caused by spectrum bounds.– The scenario of site-basis vectors evolving under Hamiltonian (4) 0.00 2 corresponds to a strong perturbation, where the anisotropy parameter is quenched from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as inFIG. 2: (Color online) LDOS (a deedthe shown in Fig. 7 (a)offor the Néelsystem state under the initial chaotic state shows a At long times probability finding in its state evolving under the chaotic Hamiltonian with = 1. The consequent Gaussian decay of boundaries. In (a): numerical powerlaw decay (this is a general property of any quantum system) F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well envelope (solid line) with 0 = with the analytical expression F (t) = exp( 02 t2 ) [8]. 1) + (L 2) ]/4; L = 16 [8].

• 

• 

F (t) (solid), its time average ( (dotted), and F̄ = IPR0 (horizo L = 22 (light), L = 24 (dark), a

0

10

-2

2 -4

10

1.5

-6

10

ρ0(Ε)

F(t)

10

-8

10

-10

10

1

0.5

0.3

0.5

1

t

2

4

8 0


0.5

f(t)

peak or Poisson shape [46] to a Wigner-Dyson form [47]. The initial states that are mainly considered here are site0.4 Outline basis vectors, where the spin on each site either points up or 0.3 down in the z-direction. They include the experimentally acNéelthe state, |NSi = | "#"#"#"# . . .i, and the domain Show that cessible value of exponent determines if the system thermalizes 0.2 wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. 0.1 Case 1: Powerlaw decay caused by spectrum bounds.– The scenario of site-basis vectors evolving under Hamiltonian (4) 0.00 2 a strong perturbation, where solely the anisotropy paNew corresponds criterion ofto thermalization based on dynamics rameter is quenched from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as inFIG. 2: (Color online) LDOS (a deed shown in Fig. 7 (a) for the Néel state under the chaotic state evolving under the chaotic Hamiltonian with = 1. The consequent Gaussian decay of boundaries. In (a): numerical F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well envelope (solid line) with 0 = with the analytical expression F (t) = exp( 02 t2 ) [8]. 1) + (L 2) ]/4; L = 16 [8].

• 

• 

F (t) (solid), its time average ( (dotted), and F̄ = IPR0 (horizo L = 22 (light), L = 24 (dark), a

0

10

-2

2 -4

10

1.5

-6

10

ρ0(Ε)

F(t)

10

-8

10

-10

10

1

0.5

0.3

0.5

1

t

2

4

8 0


0.5

f(t)

peak or Poisson shape [46] to a Wigner-Dyson form [47]. The initial states that are mainly considered here are site0.4 Outline basis vectors, where the spin on each site either points up or 0.3 down in the z-direction. They include the experimentally acNéelthe state, |NSi = | "#"#"#"# . . .i, and the domain Show that cessible value of exponent determines if the system thermalizes 0.2 wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. 0.1 Case 1: Powerlaw decay caused by spectrum bounds.– The scenario of site-basis vectors evolving under Hamiltonian (4) 0.00 2 a strong perturbation, where solely the anisotropy paNew corresponds criterion ofto thermalization based on dynamics rameter is quenched from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as inFIG. 2: (Color online) LDOS (a deed shown in Fig. 7 (a) for the Néel state under the chaotic state evolving under the chaotic Hamiltonian with = 1. The consequent Gaussian decay of boundaries. In (a): numerical F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well envelope (solid line) with 0 = with the analytical expression F (t) = exp( 02 t2 ) [8]. 1) + (L 2) ]/4; L = 16 [8].

• 

• 

F (t) (solid), its time average ( (dotted), and F̄ = IPR0 (horizo L = 22 (light), L = 24 (dark), a

0

10

-2

2 -4

10

1.5

-6

10

ρ0(Ε)

F(t)

10

-8

10

-10

10

1

0.5

0.3

0.5

1

t

2

4

8 0


f(t

basis vectors, where the spin on each site either points up or 0.3 down in the z-direction. They include the experimentally acOutline cessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain 0.2 wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. 0.1 Show that value of the exponent determines if the system thermalizes Case 1: Powerlaw decay caused by spectrum bounds.– The scenario of site-basis vectors evolving under Hamiltonian (4) 0.00 2 corresponds to a strong perturbation, where the anisotropy parameter is quenched from ! 1 to a finite value. The NewLDOS criterion of thermalization based solely on dynamics is therefore expected to have a Gaussian shape, as inFIG. 2: (Color online) LDO deed shown in Fig. 7 (a) for the Néel state under the chaotic state evolving under the cha Hamiltonian with = 1. The consequent Gaussian decay of boundaries. In (a): numer F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well envelope (solid line) with 2 2 Experiments coldexpression atoms, trapped ions) 0focus with the(e.g. analytical F (t) = exp( t ) [8].on dynamics 1) + (L 2) ]/4; L = 16 [

• 

• 

• 

F (t) (solid), its time avera (dotted), and F̄ = IPR0 (ho L = 22 (light), L = 24 (dar

0

10

-2

2 -4

10

1.5

-6

10

ρ0(Ε)

F(t)

10

-8

10

-10

10

1

0.5

0.3

0.5

1

t

2

4

8 0


Dynamics Quantity of interest survival probability (probability of finding the system in its initial state later in time)

• 

Expressing the initial state in terms of the eigenstates of the final Hamiltonian

components of initial state


Dynamics

• 

energy distribution

We can write the fidelity in terms of the energy distribution of the initial state weighted by the components

weights

• 

All information about survival probability is contained in the LDOS

ρ0(Ε)

0.3 0.2 0.1 0 -10

-5

0

E

5

10


with the analytical expression F (t) = exp( Multiple time scales

2 2 0t )

[8].

0

10

-2

F(t)

10

-4

10

-6

10

-8

10

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


with the analytical expression F (t) = exp( Multiple time scales

[8].

intermediate regime

0

10

decay depends only on the perturbation strength which determines the LDOS shape

-2

10

F(t)

2 2 0t )

-4

10

-6

10

-8

10

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


with the analytical expression F (t) = exp( Multiple time scales

10

PHYSICAL REVIEW E 85, 036209 (2012)

SANTOS, F. BORGONOVI, AND F. M. IZRAILEV

-2

10

µ=0.1

40 20

2 1 0 0.4

-0.2 -0.1

F(t)

0

0

-4

0.1 0.2

10

µ=0.4

-6

-1 -0.5

10 0

1

0.5

µ=1.5

-8

0.2 0

10 -4

-2

0

2

4

60 40 20 0 2

decay depends only on the perturbation λ=0.1 strength which determines the LDOS -0.2 0 0.1 0.2 -0.1 shape λ=0.4

1 0 0.6

-1 -0.5

0

0.5

1

λ=1.0

0.4 0.2 0

-4

-10

10

[8].

intermediate regime

0

60

2 2 0t )

0.3

-2

0

2

0.5

4

IG. 9. (Color online) Strength functions for model 1 (left) and el 2 (right) obtained by averaging over five even unperturbed s in the middle of the spectrum. The average is performed after ing the center of SFs to zero. Circles give the fitting curves. The short-time regime dle panels show the Breit-Wigner function with ε + δ = −0.015, 0.302 (left) and ε + δ = 0.072, # = 0.345 (right). The bottom

agreement between the SF and the energy shell is another way to find the critical values µcr and λcr . We fitted our numerical data with both functions (circles in Fig. 9) and verified that the transition from Breit-Wigner to Gaussian happens for the same critical values µcr ,λcr ≈ 0.5 obtained before from vn /dn in Fig. 4 and from the transition to a Wigner-Dyson distribution in the case of model 2. At large perturbation we then have excellent agreement between the Gaussian fit and the Gaussian function describing the energy shell, which depends only on the off-diagonal elements of the Hamiltonian matrices. As seen in the bottom panels, these two curves become practically indistinguishable. Notice that even at very large perturbation, the width of the energy shell, and thus of the maximal SF, is narrower than the width of the density of states (cf. Figs. 9 and 5), especially for model 2. This contradicts the equality between Pn (E) and ρ(E) found in previous works [64] and may be due to the fact that here the perturbation acts also along the diagonal (such an effect is typically removed by considering a renormalized MF Hamiltonian that takes into account the diagonal contributions of the perturbation).

1

t

2

4

2. Emergence of chaotic eigenstates

8


with the analytical expression F (t) = exp( Multiple time scales

[8].

intermediate regime

0

10

decay depends only on perturbation strength which determines the LDOS shape

-2

10

F(t)

2 2 0t )

-4

10

-6

10

Very strong perturbations LDOS is Gaussian Decay is Gaussian

-8

10

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


with the analytical expression F (t) = exp( Multiple time scales

10 10

F(t)

• 

decay depends only on perturbation strength which determines the LDOS shape

-2 -4

10

[8].

long-time regime

intermediate regime

0

2 2 0t )

Decay always powerlaw

-6

10

Very strong Strong perturbations perturbations LDOS is Gaussian LDOS is Gaussian Decay is Gaussian Decay is Gaussian

-8

10

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


with the analytical expression F (t) = exp( Multiple time scales

10 10

F(t)

• 

decay depends only on perturbation strength which determines the LDOS shape

-2 -4

10

[8].

long-time regime

intermediate regime

0

2 2 0t )

• 

Decay always powerlaw Exponent depends on shape and filling of LDOS

-6

10

Very strong Strong perturbations perturbations LDOS is Gaussian LDOS is Gaussian Decay is Gaussian Decay is Gaussian

-8

10

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


with the analytical expression F (t) = exp( Multiple time scales

10 10

F(t)

• 

decay depends only on perturbation strength which determines the LDOS shape

-2 -4

10

[8].

long-time regime

intermediate regime

0

2 2 0t )

• 

Decay always powerlaw Exponent depends on shape and filling of LDOS

-6

10

Very strong Strong perturbations perturbations LDOS is Gaussian LDOS is Gaussian Decay is Gaussian Decay is Gaussian

-8

10

central focus of our work

-10

10

0.3

short-time regime

0.5

1

t

2

4

8


Long-time decay, LDOS filling and thermalization

• 

At long times the survival probability (fidelity) shows a powerlaw decay with exponent connected to the LDOS shape and filling

• 

We will consider two different scenarios for the LDOS and for the corresponding powerlaw decay exponent


Long-time decay, LDOS filling and thermalization

•  Case 1: LDOS ergodically filled

ρ0(Ε)

0.3

implies thermalization

Ergodicity

0.2 0.1 0 -10

-5

0

E

5

10

At 2 [51], where A depends on L bounds, it is clear that for the largest probability goes to zero as t ! 1, in is absolutely integrable. The expone indicator of the ergodic filling of the e initial state and consequently of the tion.


Long-time decay, LDOS filling and thermalization

•  Case 1: LDOS ergodically filled • 

Ergodic filling

• 

Measure of delocalization

ρ0(Ε)

0.3

implies thermalization

Initial state is highly delocalized, samples most eigenstates and its components are essentially random numbers

Ergodicity

0.2 0.1 0 -10

-5

0

E

5

10

At 2 [51], where A depends on L bounds, it is clear that for the largest probability goes to zero as t ! 1, in is absolutely integrable. The expone indicator of the ergodic filling of the e initial state and consequently of the tion.


Long-time decay, LDOS filling and thermalization where {nl }0 is the set of L/2 elemen

ropy

•  Case 1:E0LDOS ergodically filled

S = ln Z +

T Ergodic filling

Initial state is highly delocalized, samples most eigenstates and its components are essentially random numbers F (t) / e↵t

is the canonical ensemble partition temperature. Since here E0 = 0 we set Measurethat of delocalization ! 1• implying Z = D. The thermal therefore:

↵ /T

we obtain the diagonal entropy: 0.3 Ergodicity

ρ0(Ε)

/2

(83)

S ' L ln 2

2L/2

↵=1

0.2

2

L/2

0.1

ln 2

L/2

L = ln 2 2

(84)

0 not coincide 0implies that the ropies do -10 10 -5 5 E

0.8

where ↵ is proportional to L. The ex be obtained by scaling analysis. Sinc the domain wall initial state in the X ize.

ln IPR0

• 

(82)

binations of the first L odd numbers i long times with L the largest length implies thermalization envelope of the survival probability F

-8

-6

-10

-8

At 2 [51], where A depends on L -10 -12 bounds, it is clear that for the largest probability goes to zero as t ! 1, in -12 -14 is absolutely integrable. The expone -14e -16 indicator of the ergodic filling of the 11 state 12 and 13 14 initial consequently of the ln D tion.


basis vectors, where the spin on each site either points up or down in the z-direction. They include the experimentally acLong-time decay, LDOS filling and thermalization cessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. Case 1:implies Powerlawthermalization decay caused by spectrum bounds.– The Case 1: LDOS ergodically filled scenario of site-basis vectors evolving under Hamiltonian (4) Initial state is highlycorresponds delocalized, samples most eigenstates •  Ergodic to a strong perturbation, where the anisotropy paand its componentsrameter are essentially numbers filling is quenchedrandom from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as indeed shown in Fig. 7 (a) for the Néel state under the chaotic Hamiltonian with = 1. The consequent Gaussian decay of •  Large decay exponents F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely wel with the analytical expression F (t) = exp( 02 t2 ) [8].

• 

0

10

-2

ρ0(Ε)

0.3

F(t)

10 Ergodicity

0.2

10

-6

10

-8

0.1 0 -10

-4

10

-10

-5

0

E

5

10

10

At 2 [51], where A depends on L bounds, it is clear that for the largest probability goes to zero as t ! 1, in is absolutely integrable. The expone indicator of the ergodic filling of the e 0.3initial 1 consequently 2 4 of8the 0.5state and tion. t


ρ0(Ε)

• 

0.2

0

10

-2

10

-4

10

0

0

0

E

5

4

t

6

8

10

10

10 10

is absolutely integrable. The expone indicator of the ergodic filling of the e -10 10 0.3initial 1 consequently 2 4 of8the 0.5state and tion. t

10

-5

2

FIG. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel -2 under the chaotic H (4) with = 1, = 1/2, open state evolving boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid -4 line) with 0 = L 1/2 and E0 = [ (L 2 [8]. In (b): log-log plot for the numerical 1) + (L 2) ]/4; At L = 16 [51], where A depends on L F (t) (solid), its time average (dashed), analytical Gaussian decay -6F̄ = bounds, it is clear forInthe (dotted), and IPR0 (horizontal line); Lthat = 24. (c): largest f (t) for 2 L = 22 (light), L = 24 (dark), andgoes (1/L) (dashed). probability toln(t zero)as t ! 1, in

-8

0.1 0 -10

f(t)

The system becomes chaotic as increases [43–45] level spacing distribution changes from a Shnirelman A convenient quantityform to [47]. Poisson shape [46] to a Wigner-Dyson study the powerlaw behavior initial states that are mainly considered here are sitethe return which is up or ectors, whereisthe spin on eachrate site either points intensive (for large systems) acn the z-direction. They include the experimentally e Néel state, |NSi = | "#"#"#"# . . .i, and the domain te, |DWi = | """ . . . ### . . .i, both extensively used es of the dynamics of integrable spin models. 1: Powerlaw decay caused by spectrum bounds.– The o of site-basis vectors evolving under Hamiltonian (4) onds to a strong perturbation, where the anisotropy pais quenched from ! 1 to a finite value. The is therefore expected to have a Gaussian shape, as inown in Fig. 3 (a) for the Néel state under the chaotic onian with = 1. The consequent Gaussian decay of seen in Fig. 3 (b) up to t ⇠ 2. It agrees extremely well 0.3expression F (t) = exp( 02 t2 ) [8]. e analytical Ergodicity

F(t)

Long-time decay, LDOS filling

basis vectors, where the spin on each site either points up or down in the z-direction. They include the experimentally ac3 and thermalization cessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. Case 1:0.5 Powerlaw decay caused by spectrum bounds.– The scenario of site-basis vectors evolving under Hamiltonian (4) 0.4 corresponds to a strong perturbation, where the anisotropy parameter is0.3 quenched from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as in0.2in Fig. 7 (a) for the Néel state under⎛ Athe⎞ chaotic deed shown f (t → ∞) ∝ ln ⎜ 2 ⎟decay of Hamiltonian with = 1. The consequent Gaussian 0.1 ⎝ t ⎠ wel F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely with the analytical expression F (t) = exp( 02 t2 ) [8]. 0.0

10


G. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS (shadedfilling area) and Gaussian pLDOS Long-time decay, and thermalization velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay F(t) 2:IPRLDOS sparse otted), Case and F̄ = 0 (horizontal line); L = 24. In (c): f (t) for = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). Thermalization

2

ρ0(Ε)

1.5 1

0.5 0-6

-4

-2

0 E

2

4

6

G. 3: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay otted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for


G. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS (shadedfilling area) and Gaussian pLDOS Long-time decay, and thermalization velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay 2: IPR LDOS sparse powerlaw decay ofF(t) F(t) has small exponent otted), Case and F̄ = 0 (horizontal line); L = 24. In (c): f (t) for = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). Thermalization

the powerlaw exponent for |NSi can scaling analysis of the IPR0 . Since the Stirling approximation for ln D γ <1 ln D ) = 1/2.

2

3

1.5

(a)

1

2

0.5

1

-4

-2

0 E

2

4

6

0

0

0

G. 3: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay otted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for

5

t

10 1

f

0.5 0-6

1

f

ρ0(Ε)

1.5

0.5 0

15

0


G. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS (shadedfilling area) and Gaussian pLDOS Long-time decay, and thermalization velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay 2: IPR LDOS sparse powerlaw decay ofF(t) F(t) has small exponent otted), Case and F̄ = 0 (horizontal line); L = 24. In (c): f (t) for = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). Consequence of non-ergodicity Thermalization Thermalization

the powerlaw exponent for |NSi can scaling analysis of the IPR0 . Since the Stirling approximation for ln D γ <1 ln D ) = 1/2.

2

3

1.5

(a)

1

2

0.5

1

-4

-2

0 E

2

4

6

0

0

0

G. 3: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay otted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for

5

t

10 1

f

0.5 0-6

1

f

ρ0(Ε)

1.5

0.5 0

15

0


G. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS (shadedfilling area) and Gaussian pLDOS Long-time decay, and thermalization velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay 2: IPR LDOS sparse powerlaw decay ofF(t) F(t) has small exponent otted), Case and F̄ = 0 (horizontal line); L = 24. In (c): f (t) for = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). Consequence of non-ergodicity Thermalization Thermalization

the powerlaw exponent for |NSi can scaling analysis of the IPR0 . Since the Stirling approximation for ln D γ <1 ln D ) = 1/2.

2

3

1.5

(a)

1

2

0.5

1

-4

-2

0 E

2

4

6

0

0

0

G. 3: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay otted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for

5

t

10

Non-ergodic1filling

f

0.5 0-6

1

f

ρ0(Ε)

1.5

0.5 0

15

0


the powerlaw exponent for |NSi can scaling analysis of the IPR0 . Since the Stirling approximation for ln D γ <1 ln D ) = 1/2.

te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS (shaded area) and Gaussian pLDOS Long-time decay, filling and thermalization velope (solid line) with 0 = L 1/2 and E0 = [ (L + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay 2: IPR LDOS sparse powerlaw decay ofF(t) has small exponent otted), Case and F̄ = 0 (horizontal line); L = 24. In (c): f (t) for F(t) = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). Consequence of non-ergodicity Thermalization Thermalization

2

3

1.5

(a)

1

0.5 0-6

1

2

0.5

1

6 00

0

f

ρ0(Ε)

1.5

-4

-2

0 E

2

4

5

t

10

15

f

G. 3: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel Non-ergodic1filling te evolving under the chaotic H (4) with = 1, = 1/2, open undaries. In (a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L 0.5 exponent analysis coincides with the powerlaw + (L 2) ]/4; L = 16from [8]. Inthe (b):scaling log-log plot for the numerical t) (solid), its time average (dashed), analytical Gaussian decay 0 otted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for

0


models. The system becomes chaotic as increases [43–45] and the level spacing distribution changes from a Shnirelman or Poisson shape [46] decays to a Wigner-Dyson form [47]. Causes of peak these powerlaw The initial states that are mainly considered here are sitevectors, where the spin on each site either points up or Case 1:basis LDOS is ergodically filled, what causes the powerlaw down in the z-direction. They include the experimentally acare thecessible unavoidable presence of bounds Néel state, |NSi = | "#"#"#"# . . .i,in andthe the spectrum domain wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. Case 1: Powerlaw decay caused by spectrum bounds.– The 0.3 scenario of site-basis vectors evolving under Hamiltonian (4) Ergodic corresponds to a strong perturbation, where the anisotropy parameter is0.2 quenched from ! 1 to a finite value. The LDOS is therefore expected to have a Gaussian shape, as in0.1in Fig. 3 (a) for the Néel state under the chaotic deed shown Hamiltonian with = 1. The consequent Gaussian decay of F (t) is seen0in Fig. 3 (b) up to t ⇠ 2. It agrees extremely well -10 expression -5 F (t)0= exp( 52 t2 ) [8]. 10 with the analytical 0

• 

decay

ρ0

F(t)

ρ0(Ε)

At 2 [51], where bounds, it is clear th probability goes to z is absolutely integra indicator of the ergo initial state and con tion. E Algebraic decays 0 FIG. 2: (Color online) LDOS (a), F ( FIG. 4: (Color 10online) LDOS (a), F (t) (b) and f (t) (c) for the Néel ing of the LDOS. Th state evolving under the chaotic H ( state evolving under the chaotic H (4) with = 1, = 1/2, open -2 teractions, many-bo boundaries. In (a): numerical LDO p 10 boundaries. In (a): numerical LDOS (shaded area) and Gaussian envelope (solid line) with = L 0 p the number of part envelope (solid -4line) with 0 = L 4 1/2 and −2 E0 = [ (L1) + (L 2) ]/4; L = 16 [8]. In (b 10 increasing the num (0) −1 F (t) (solid), its time average (dash t 1) + (L 2) ]/4; L= 16 [8]. In (b): log-log plot for the numerical IPR = C ∝ D 0 α (dotted), and = IPR0 (horizontal α theF̄Hamiltonian ma F (t) (solid),10 its-6time average (dashed), analytical Gaussian decay L = 22 (light), L = 24 (dark), and Gaussian to semicir (dotted), and F̄ -8= IPR0 (horizontal line); L = 24. In (c): f (t) for in the shape of the L 10 L = 24 (dark), and (1/L) ln(t 2 ) (dashed). L = 22 (light), 0.3 of a semicircle lead Ergodic -10 0.2 10 0.3filling 1 2 4 8 0.5 J1 is the 0.1 Bessel fun 0.3 t (a) short times0 is faster


n, where the anisotropy pat 1 to a finite value. The e a Gaussian shape, as inFIG. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open éel state under the chaotic boundaries. In (a): numerical LDOS (shaded area) and Gaussian sequent Gaussian decay of p envelope (solid line) with 0 = L 1/2 and E0 = [ (L 2. It agrees extremely well the2)LDOS (sparse), the 1) + (L ]/4; L =is 16 non-ergodic [8]. In (b): log-log plot for the numerical 2 2Case 2: when = exp( 0 t ) [8].

• 

powerlaw decay

F (t) (solid), its time (dashed), analytical Gaussian decay is due to the presence ofaverage eigenstates correlations

(dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

p

2

ρ0(Ε)

Non-ergodic dynamic limit: (Sd Sth )/L = ln 2. 1.5 Similarly to what is done in disordered systems [37, 38], the powerlaw exponent for |NSi can also be obtained from the 1 correlations L/2 scaling analysis of the IPR . Since IPR = 2 and using 0 0 0.5 the Stirling approximation for ln D, we have that ln IPR0 = 0-6 )-4 = ln D -2 1/2. 0 2 4 6

2

4

8

(b) and f (t) (c) for the Néel with = 1, = 1/2, open (shaded area) and Gaussian 1/2 and E0 = [ (L log-log plot for the numerical ), analytical Gaussian decay ne); L = 24. In (c): f (t) for 1/L) ln(t 2 ) (dashed).

y F (t) during the Gausthe infinite time average destructive interferences of an unbounded LDOS state reconstruction due to

E

1.5online) LDOS (a), F (t) (b) and f (t) (c) for the Néel3 FIG. 3: (Color state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS (shaded area) and Gaussian p envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1 2 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for 0.5L = 24 (dark), and (1/L) ln(t 2 ) (dashed). 1 L = 22 (light),

(a)

(b)

f

t

Causes of these powerlaw decays

upper bounds0leads at long times to:

0

F (t

0

1

)'

5

t

1 2 2

2

X

10 e

15 (Ek E0 )2 /

2 0

0 (5)

0

40

t

80

120


f(t)

basis vectors, where the spin on each site either points up or 0.3 down in the z-direction. They include the experimentally acCase 1 – Start with two analytical examples cessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain 0.2 wall state, |DWi = | """ . . . ### . . .i, both extensively used in studies of the dynamics of integrable spin models. 0.1 Case 1: Powerlaw decay caused by spectrum bounds.– The Gaussian LDOS: characteristic of systems with two-body interactions scenario of site-basis vectors evolving under Hamiltonian (4) 0.00 2 4 6 corresponds to a strong perturbation, where the anisotropy pat rameter is 0.3 quenched from ! 1 to a finite value. The At 2 [51], where A depends on L, E LDOS is therefore expected to have a Gaussian shape, as init is LDOS clear (a), thatF for theandlargest (Color online) (t) (b) f(t) (c)syst for 0.2in Fig. 7 (a) for the Néel state under the chaotic FIG. 2:bounds, deed shown state evolving under the chaotic H (4)as with 1, indic =1 probability goes to zero t !=1, Hamiltonian with = 1. The consequent Gaussian decay of In (a): numerical LDOS (shaded area) and G is absolutely integrable. exponent p The 0.1in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well boundaries. F (t) is seen envelope (solid line) with 0 = L 1/2 and E0 = 2 2 indicator of the ergodic filling of the energ with the analytical expression F (t) = exp( 0 t ) [8]. 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the n

ρ0(Ε)

• 

0

-5

10

5

E

0

10

initialitsstate and consequently of the via F (t) (solid), time average (dashed), analytical Gaussia (dotted),tion. and F̄ = IPR0 (horizontal line); L = 24. In (c): 2 L = 22 (light), L = 24decays (dark), and (1/L) (dashe Algebraic faster thanln(tt 2)also si

F(t)

-2 G. 4: (Color 10 online) LDOS (a), F (t) (b) and f(t) (c) for the Néel ate evolving under the chaotic H (4) with = 1, = 1/2, open oundaries.10In-4(a): numerical LDOS p (shaded area) and Gaussian velope (solid line) with 0 = L 1/2 and E0 = [ (L -6 + (L 2) 10]/4; L = 16 [8]. In (b): log-log plot for the numerical (t) (solid), its time average (dashed), analytical Gaussian decay -8 otted), and F̄ 10 = IPR0 (horizontal line); L = 24.2 In (c): f(t) for = 22 (light), L = 24 (dark), and (1/L) ln(t ) (dashed). -10

10

0.3

0.5 0.3

1

t

2

4 (a)

8

ing of the LDOS. They are possible if ins 2 many-body random interactio teractions, the number of particles that interact sim 1.5 the number of uncorrelated n increasing the Hamiltonian matrix, the density of st 1 to semicircle [12]. This transi Gaussian in the shape of the LDOS [8, 55–57]. Th of a0.5 semicircle leads to F (t) = [J1 (2 J1 is the Bessel function of the first kin

ρ0(Ε)

0 -10

0


Case 1 – Start with two analytical examples

• 

0 -10

-5

0

5

10

E

FIG. 4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open When number of particles interacting simultaneously grows, the decay boundaries. In (a): numerical LDOS p (shaded area) and Gaussian exponent increases - LDOS transitionsenvelope from (solid Gaussian semicircular line) with to L 1/2 and E0 = [ (L 0 = 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0.3

ρ0(Ε)

ρ0(Ε)

2 At0.004 [51], where A depends on L, 0.003 it is clear that for the largest sys bounds, 0.2 probability goes to zero as t ! 1, indic 0.002 is absolutely integrable. The exponent 0.1 0.001 indicator of the ergodic filling of the ener 0 initial 0state of the via -200 and -100consequently 0 100 200 -10 0 10 -5 5 E tion. E 2 faster FIG. 5: (ColorAlgebraic online) LDOSdecays (a), F (t) (b) and fthan (t) (c) tfor thealso Néel s state evolving under the chaotic H (4) with are = 1,possible = 1/2, if open G. 4: (Color online) LDOS (a), F (t) (b) and f(t) (c) for the Néel ing of the LDOS. They in boundaries. In (a): numerical LDOS (shaded area) and Gaussian p e evolving under the chaotic H (4) with = 1, = 1/2, open teractions, envelope (solid line) with many-body L 1/2random and E0 = interactio [ (L 0 = ndaries. In (a): numerical LDOS (shaded area) and Gaussian 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical p the number of particles that interact si analytical Typical of fullGaussian decay elope (solid line) with 0 = L 1/2 and E0 = [ (LF (t) (solid), its time average (dashed), increasing the number uncorrelated (dotted), and F̄ = IPR0 (horizontal line); L of = 24. In (c): f (t) for random matrices 2 + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical L = 22 (light), L = 24 (dark), and (1/L) ln(t ) (dashed). the Hamiltonian matrix, the density of s ) (solid), its time average (dashed), analytical Gaussian decay Gaussian to semicircle [12]. This trans ¯


f

F

0.5

4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel -8 10 evolving under the chaotic H (4) with = 1, = 1/2, open 0 ndaries. In (a): numerical LDOS p (shaded area) and Gaussian lope (solid line) with 0 0= L 1/2 and E0 = [ (L 0.1 1 10 100 0 10 t t (L 2) ]/4; L = 3 16 [8]. In (b): log-log plot for the numerical 3 ) (solid), its time (dashed), analyticalmany-body Gaussian decay simultaneous interactions are Fullaverage matrices: 0random FIG. 6: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the N ed), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for included – they do not 2describe real systems (provide an extreme case) state evolving under the chaotic H (4) with = 1, = 1/2, o 22 (light), L = 24 (dark), and (1/L) ln(t ) (dashed). boundaries. In (a): numerical LDOS p (shaded area) and Gauss envelope (solid line) with 0 = L 1/2 and E0 = [ (L 0.004 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numer 1 F (t) (solid), its time average (dashed), analytical Gaussian de 0.003 3 0 (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) 3 0 L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). 0.002

Case 1 – Start with two analytical examples

!

1

sin(4 •  2⇡ t

t)

(63)

.

illations we get:

ρ0(Ε)

)!

1

2⇡

(64)

t

th a =0.001 1/(2⇡

3 0 ), b

0 -200 -100

= 0 and 0

100

= 3. 200

E

0

F(t)

10 5: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel evolving under the chaotic H (4) with = 1, = 1/2, open ndaries. In (a): numerical LDOS p (shaded area) and Gaussian -4 lope (solid line) 10 with 0 = L 1/2 and E0 = [ (L (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical ) (solid), its time average (dashed), analytical Gaussian decay ed), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for 22 (light), L10 =-824 (dark), and (1/L) ln(t 2 ) (dashed). 200

-3

10

-2

10

-1

t

10

orroborating this result, we found the exponent

= 2

is absolutely integrable. The exponent = 2 is therefore indicator of the ergodic filling of the energy distribution of initial state and consequently of the viability of thermali tion. Algebraic decays faster than t 2 also signal the ergodic fi ing of the LDOS. They are possible if instead of two-body teractions, many-body random interactions are included. the number of particles that interact simultaneously gro increasing the number of uncorrelated nonzero elements the Hamiltonian matrix, the density of states transitions fr Gaussian to semicircle [12]. This transition is reflected a in the shape of the LDOS [8, 55–57]. The Fourier transfo of a semicircle leads to F (t) = [J1 (2 0 t)]2 /( 02 t2 ), wh J1 is the Bessel function of the first kind [8]. The decay short times is faster than Gaussian and the asymptotic exp


Two types of systems

• 

Spin-1/2 systems

• 

Banded random matrices

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0


Spin-1/2 systems

• 

The Hamiltonians we consider may have NN and NNN interactions

• 

Chaotic systems: as the parameter λ increases the system becomes increasingly chaotic

•  Δ = 0, λ = 0 : noninteracting XX model (exactly solvable) •  In our work different initial states were considered but here several results will be shown only for the Néel state


obtain:

Case 1 – Results for Néel state with chaotic Hamiltonian Quench

)

)

• 

where we used ln D ! L ln 2 valid for la

Scenario of strong perturbation: LDOS is expected to have Gaussian shape

0.4

ρ0(Ε)

l

1 f (t ! 1) ' ln D ' ln L

0.3

0.6

Gaussian LDOS

0.2

0.4

0.1

0.2

0 -10 0

10

-5

0

E

5

10

0

0

0 10 arXiv:1601.05807


1 (t Néel ! 1) 'withln D ' Hamiltonian ln 2 Case 1 – Resultsffor state chaotic L

(49)

Quench

where we used ln D ! L ln 2 valid for large L.

ρ0(Ε)

0.4

• 

0.6

0.3

0.4 0.2 We studied the decay and obtained γ = 2 : this is the value expected for a 0.2 obtained analytically 0.1 with ergodic Gaussian LDOS as we system 0 -10 0 10 0 -5 5 0 2 4 6

E

0

10

10

-4

F(t)

E

0

10

-3

10 -8

10

-6

10

-12

10

1

(t)

1

t

10

finite-size effects

100

1

t 10

100

1

arXiv:1601.05807


where with {nl } isEq. the (21) set of L/2 elements consisting of all comparing intensive for. . .large binations of the first L odd numbers in {1, 3, , 2L system 1}. At sizes L. F Case 1 – Results for Néel state with chaotic Hamiltonian long times with L the largest obtain: length scale of the system, the

envelope of the survival probability F (t) is now: •  Performing an scaling analysis we find 2 2 0.8 /(2 0 ) (41) F (t) / e↵t

1 f(t ! 1) ' ln D ' 0.8 can also L (89)

where ↵ is proportional to L. The exponent be obtained by scaling analysis. Since < 1 we conclude that This means the LDOSstate is ergodic the• 2. domain initial inwhere the and XXwe model dothermalization not = Thewall survival weexpect used lnthermalD ! toLoccur ln 2 ize.

nd reads:

2 0t

-16 0

11

12

2 0t

-6

◆ (42) 13

0.4

-8 0.3 0.2

ρ0(Ε)

E-10low + i p -12 2 0 ◆-14

ln IPR0

0

-8

valid

0.6

Gaussian LDOS

0.4

-10 -12

14

-14

0 -10 11

0

ln D

10 0

0.2

0.1 0

-5 12

13

ln D

E14

5

10

0

0

10


Case 1 – Results for Néel state with chaotic Hamiltonian

• 

Thermalization

infinite time averages

diagonal entropy

thermodynamic averages

thermodynamic entropy


Case 2 – Strongly disordered systems

• 

Typical example associated with case 2: system with strong disorder described by the Hamiltonian below (hnSn are random magnetic fields)

DYNAMICS AT THE MANY-BODY LOCALIZATION TRANSITION

0 LDOS sparse, becoming increasingly more sparse 10 (a) for increasing disorder h

10

<F(t)>

• 

ρ

10

10

1

-1

0

ρ 0.8

-2

10

(b)

2

(c)

0.4

h

-3

-4

10

0

10

t

2

ρ 10

4

exponent decay reflecting increasing sparsity

0

h

(d)

0.2 0.1 0

-4

-2

0

E

2

4

E. J. Torres-Herrera and L. F. Santos, PRB 92, 014208 (2015)


Case 2 – Strongly disordered systems

• 

Typical example associated with case 2: system with strong disorder described by the Hamiltonian below (hnSn are random magnetic fields)

DYNAMICS AT THE MANY-BODY LOCALIZATION TRANSITION ρ 10

The powerlaw decay exponent γ < 1 and decreases with h

<F(t)>

• 

10 10

(a)

1

-1

0

ρ 0.8

-2

10 10

0

(b)

2

(c)

0.4

h

-3

-4

10

0

10

t

2

ρ 10

4

exponent decay reflecting increasing sparsity

0

h

(d)

0.2 0.1 0

-4

-2

0

E

2

4

E. J. Torres-Herrera and L. F. Santos, PRB 92, 014208 (2015)


Case 2 – Strongly disordered systems

• 

Exponent can also be obtained by scaling analysis

DYNAMICS AT THE MANY-BODY LOCALIZATION T ρ 10

IPR 0 ∝ D

-2

10

-4 -6 -8

10

5

6

7

lnD

8

9

10

10 10

2 1

-1

0

ρ 0.8

-2

-10 4

(a)

−γ

<F(t)>

ln<IPR0>

0

0

0.4

h

-3

ρ

-4

10

0

10

t

2

10

4

exponent decay reflecting increasing sparsity

0 0.2 0.1 0

-4

-2

0

E

FIG. 1. (Color online) Survival probability averaged data points for h = 0.5,1.0,1.5,2.0,2.7,4.0 from bottom t E. J. Torres-Herrera and L. F. Santos, PRB 92, 014208 (2015) and LDOS for a single realization for the bottom panel h =


dynamicXXlimit: (Sd Case 2 – Néel state in noninteracting Hamiltonian

Sth )/L = Similarly to what is done Quench in d the powerlaw exponent for |NSi c scaling analysis of the IPR0 . Sin the Stirling for ln •  The same analysis done in disordered systems isapproximation applicable in this integrable case – from the dynamics we obtained: ln D ) = 1/2. arXiv:1601.05807

1.5

(a) f

1 0.5 0

0

5

t

10

15


Case 2 – Néel state in noninteracting XX Hamiltonian

arXiv:1601.05807 Quench

•  Smallasexponent reflects eigenstate correlations and lack of ergodicity comes chaotic increases [43–45] tribution changes from a Shnirelman 46] to a Wigner-Dyson form [47]. are mainly considered here are siteLDOS clearly sparse spin on each site either points up or They include the experimentally aci = | "#"#"#"# . . .i, and the domain " . . . ### . . .i, both extensively used cs of integrable spin models. cay caused by spectrum bounds.– The ctors evolving under Hamiltonian (4) perturbation, where the anisotropy paom ! 1 to a finite value. The cted to have a Gaussian shape, as in-


(83)

-6

-10

-8

-12

-10

-14

-12

(84)

ln IPR0

Case 2 – Néel state in noninteracting XX Hamiltonian

ropy:

2

-8

arXiv:1601.05807 Quench

•  As in strongly disordered systems, the exponent can also be obtained -14 -16 by scaling analysis 11 12 13 11 12 13 14 ies that the ln D ln D

omain wall

0

0

IPR 0 ∝ D −1/2

ln IPR0

-5

-5

-10 -15 -20 -25

γ = 1/ 2 5

-10

10

ln D

15

-15

5

10

ln D


Case 2 – Néel state in noninteracting XX Hamiltonian

arXiv:1601.05807 Quench

• 

Example of lack of thermalization: verified by comparing the diagonal and thermodynamic entropies which indeed differ


increases from ⇠ 2 to 3. In contrast, as b decreases below 50, the eigenstates become less spread out, the LDOSarXiv:1601.05807 more Banded Random matrices sparse, and decreases below 2. With PBRMs, we obtain a general picture of the behavior of the survival probability, •  We showed that we all exponents found in lattice covering all values ofcan, cover without any restriction to a many-body specific quantum systems using power-law banded random matrices (PBRM) model. 10

-1

F

10

0

10

-2

10

-3

10

-2

10

-1

10

t

0

10

1

10

2


arXiv:1601.05807

Banded Random matrices

â&#x20AC;˘â&#x20AC;Ż

Elements are Gaussian random numbers with bandwith b which determines how fast elements decrease away from diagonal

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0


increases from ⇠ 2 to 3. In contrast, as b decreases below 50, the eigenstates become less spread out, the LDOSarXiv:1601.05807 more Banded Random matrices sparse, and decreases below 2. With PBRMs, we obtain a general picture of the behavior of the survival probability, •  By increasing b all possible exponents are covering all valuestheofbandwith , without any restriction to areached specific model. 10

-1

F

10

0

10

-2 4

ρ0(Ε)

0.004

ρ0(Ε)

2 1.5 1 0.5 0-6 -4 -2 0 2 E

0.003

0.002

4

6

10

0.001

-3

FIG. 4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0 -200 -100

100

200

FIG. 6: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0.3

10

0.3

ρ0

ρ0(Ε)

0

E

0.2

-2

10

(a)

0.2 0.1

0.1 0 -10

0 10

-5

0

5

0

0

-5

(b)

10

(c)

1

f

F

E FIG. 5: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

10

0.5

-8

0 0.1

1

t

10

100

0

10

t

10

t

5

E

-1

20

FIG. 7: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS (shaded area) and Gaussian

0

10

1

10

2


increases from ⇠ 2 to 3. In contrast, as b decreases below 50, the eigenstates become less spread out, the LDOSarXiv:1601.05807 more Banded Random matrices sparse, and decreases below 2. With PBRMs, we obtain a general picture of the behavior of the survival probability, •  BRM:allprovide of dynamics of survival to probability covering valuesgeneral of ,picture without any restriction a specific model.without model restrictions 10

-1

F

10

0

10

-2 4

ρ0(Ε)

0.004

ρ0(Ε)

2 1.5 1 0.5 0-6 -4 -2 0 2 E

0.003

0.002

4

6

10

0.001

-3

FIG. 4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0 -200 -100

100

200

FIG. 6: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0.3

10

0.3

ρ0

ρ0(Ε)

0

E

0.2

-2

10

(a)

0.2 0.1

0.1 0 -10

0 10

-5

0

5

0

0

-5

(b)

10

(c)

1

f

F

E FIG. 5: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

10

0.5

-8

0 0.1

1

t

10

100

0

10

t

10

t

5

E

-1

20

FIG. 7: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS (shaded area) and Gaussian

0

10

1

10

2


Conclusions

• 

arXiv:1601.05807

We studied the powerlaw decay at long times for isolated many-body quantum systems in:

•  • 

Integrable/chaotic, interacting/noninteracting, clean/disordered systems banded random matrices


Conclusions

• 

We studied the powerlaw decay at long times for isolated many-body quantum systems in:

•  •  • 

arXiv:1601.05807

Integrable/chaotic, interacting/noninteracting, clean/disordered systems banded random matrices

We showed there is a relation between the exponent of the powerlaw decay and the ergodicy (or non-ergodicity) of the LDOS, and from that we proposed a criterion for thermalization based on the dynamics only


1 see inEFig. EThe form of ⇢0E E 0 1. 0 is: can (E) p low p up N indeed = erf erf Conclusions 2 2 02 2 02

• 

(40)

1 ln F (t) L arXiv:1601.05807 f (t) =

delity

Z

⌘ to c This quantity is useful if one needs 1 We studied1 e decay at long times for isolated many-body C(t,since t0 ) ⌘from large deviatio ln F (⌘ different sizes, p ⇢ (E) = ⇥(E E )⇥(E E)(39) erf(x) is the error function. Again comparing with Eq. (21) 0 low up ⌘ sizes ⌘0 L. quantum systems2⇡in:2 ⌘0 From intensive for large system we identify ⇠N= 0, P (E) =0 1 and obtain: where ⌘ = ln t. A way to partially exclu 2 2 Integrable/chaotic, interacting/noninteracting, 1bounded (EbyEE )low , Eup ]. The clean/disordered systems where the⌘(E) spectrum is 2 [E 0 ) /(2 0 1 by [42 =p e (41) is to consider the return rate given 2 f (t ! 1) ' ln D ' l banded random matrices 2⇡N0 N normalization constant is: L E0 )2 /(2 02 ) the(Epowerlaw

•  • 

!#

• 

2⇡ 0 N ✓ caused by 2 ◆ 2 2  and ergodic systems e 0t E0 Elow + i 0 t F (t) = erf bounds in4N the spectrum According to Eq. (28) exponent isp2 = 2. The survival 2 the 0 amplitude can be obtained✓analytically and reads: ◆ 2 E0 Eup + i 02 t erf . (43)  p✓ ◆ 2 2 00 2 2 1 E E + i t 0 plow A(t) = e 0 t /2+iE0 t erf 2 0 Taking t 1/ 2N 0 , F (t) becomes: ✓ ◆ E0 Eup + i 02 t F (t 1/ 0 ) p erf (42) h 2 1 (E E )2 / 2 0 (E E )2 / 2 up

0

0

low

0

0

ρ0(Ε)

t

F(t)

• 

!

L

-8

10

10

-6

10

where-12we used ln D ! L ln 2 valid for la 10

1

t

10

100

0.4 1

ρ0f(t) (Ε)

• 

"

1valid for l According to Eq. (28) the exponent is = 2. The survival where we used ln D ! L ln 2 f(t)decay = ln F (t) We showed relation between 1 there E0 is a Elow E0 Eupthe exponent of the powerlaw amplitude can be obtained analytically and reads: L p p N = erf erf (40) and the ergodicy (or2 non-ergodicity) and from 2 0.4 that we proposed 2 2of02the LDOS, ✓ ◆ 0.6 0  2 dynamics 0.3 a criterion1for thermalization based on the only This quantity is useful if one needs to co 2 2 E E + i t 0 0 0.4 plow A(t) = e 0 t /2+iE0 t erf 0.2 sizes, since from large deviation 2Nerror function. Again comparing2 with 0 erf(x) is the Eq. (21) different 0.2 0.1 ✓ ◆ 2 intensive for large system sizes L. From E 0 Eup + i 0 t 0 we identify ⇠ = 0, Perf(E)E=0 1 and -10 0 10 -5 5 0 p (42) obtain:0 Two scenarios for the algebraic decay: E 2 0 0 10 10 2 2 −2 1 (E E0 ) /(2 0 ) 1 p ⌘(E) = e -4 The corresponding fidelity is then: 10 Large decay exponents associated with (41) 2 f(t ! 1) ' ln D-3 ' ln

0.3

0.61

Ergodic

0.2

0.4

0.5

0.5

0.1

0.2

0

0-10 0 0

10

-5 5

0 10

Et

5 15

00 10 0 0 20 0

10


1 see inEFig. EThe form of ⇢0E E 0 1. 0 is: can p low p up N indeed = erf erf (E)

2 Conclusions

• 

2

2 0

2

(40)

2 0

f (t) =

ln F (t) L arXiv:1601.05807

delity

Z

⌘ to c This quantity is useful if one needs 1 1 e We studied the powerlaw decay at long times for isolated C(t,since t0 ) ⌘from large deviatio ln F (⌘ differentmany-body sizes, p ⇢0 (E) = error function. ⇥(E E )⇥(E erf(x) is the Again comparing with Eq.E)(39) (21) low up ⌘ ⌘ 0 L. quantum systems2⇡in:2 ⌘0 From intensive for large system sizes we identify ⇠N= 0, P (E) =0 1 and obtain: where ⌘ = ln t. A way to partially exclu 2 2 1bounded Integrable/chaotic, interacting/noninteracting, (EbyEE )low , Eup ]. The clean/disordered systems where the⌘(E) spectrum is 2 [E 0 ) /(2 0 1 by [42 =p e (41) is to consider the return rate given 2 f (t ! 1) ' ln D ' l banded constant random 2⇡N0matrices N normalization is: L (E E0 )2 /(2

2 0)

•  • 

!#

'

e

up

0

0

+e

low

0

0

ρ0(Ε)

t

L

-8

100.5

10

1

-6

10

where-12we used ln D ! L ln 2 valid 0for la 0 10

0

1 5

t

0.4 2 1

0.3 1.5 0.2 1

0.5

Ergodic

1010 1

f

Large decay exponents associated with 2⇡ 0 N ✓ caused by 2 ◆ 2 2  and ergodic systems e 0t E0 Elow + i 0 t p = 2. The survival F (t) = erf According to Eq. (28) the exponent bounds in4N the 2 spectrum is 2 0 amplitude be obtained and reads: •  Smallcandecay exponents associated ✓analytically ◆ 2 E0 Eup + i 02 t with non-ergodic erf systems . (43)  p ✓ and ◆ 2 2 0 2 2 1 E0 Elow + i 0 t caused /2+iE0 t p A(t) = byethe0 t eigenstates erf correlations 2N 2 0 Taking t 1/ 0 , F (t) becomes: ✓ ◆ E0 Eup + i 02 t F (t 1/ 0 ) p erf (42) h 2 1 (E E )2 / 2 0 (E E )2 / 2

F(t) f

• 

0

• 

!

ρρ0f(t) (Ε) (Ε)

• 

"

According to Eq. (28) the exponent is = 2. The survival where we used ln Df(t) !L ln 21valid for pl = ln F (t) 1 there E0 is a Elow E0 Eupthe exponent of We showed relation between the powerlaw decay dynamic limit: (S S )/L = ln 2. d th L amplitude and reads: p analytically p N =can be erfobtained erf (40) 2 0.4 thattowe and the ergodicy (or2 non-ergodicity) and from proposed Similarly what is done in disorde 2 2of02the LDOS, ✓ ◆ 0.6 0  2 0.3 thequantity powerlaw exponent for |NSi can als This is useful if one needs to co 2 2 a criterion1for thermalization based on the dynamics only E E + i t 0 low 0 t /2+iE0 t 0 0.4 IPR p A(t) = e erf scaling 0.2 analysis of the IPR0 . Since sizes, since from large deviation 2Nerror function. Again comparing2 with 0 erf(x) is the Eq. (21) different 0.2D, w the0.1 Stirling approximation for ln ✓ ◆ 2 intensive for large system sizes L. From E ln 0D ) = 1/2. E + i 0t 0 we identify ⇠ = 0, Perf(E)E=0 1 and -10 0 10 -5 5 0 pup (42) obtain:1.50 E Two scenarios for the algebraic decay: 2 0 03 10 10 (a) −2 1 (E E0 )2 /(2 02 ) 1 p is then: ⌘(E) = e (41) -41 The corresponding fidelity 10 2 f(t ! 1) ' ln D-32 ' ln

0.5

0

15 100

0.61 0.4 0.5

(

0 0.1 0 5 0.2 10 0.5 t 0 000-6 -10 5 4 10 00 2 15 6 00 0 0 -4 -5 20 5 -2 10 FIG. 3: (Color online) Et Analytical f (t) (soli 0 0

(dashed) for three initial states evolving 10 10 und FI


arXiv:1601.05807

Conclusions

• 

3

We studied the powerlaw decay at long times for isolated many-body quantum systems the largest ratio (Elow,up in: E0 )/ 0 , where Eup is the LDOS for n 6= m, and b > 1. The value of b determines how fast

upper bound [53]. the elements decrease as they move away from the diagonal. For b ! 1, we recover a FRM from asystems Gaussian orthogonal The Fourier transform of a Gaussian LDOS with lower Integrable/chaotic, interacting/noninteracting, clean/disordered 1 ensemble [50]. and upper bounds leads at long times to F (t 0 ) ' P 2 2 banded random matrices In Fig. 2 (a), we show the survival probability for PBRMs (2⇡ 02 t2 N 2 ) 1 k=up,low e (Ek E0 ) / 0 , where N is a norwith different values of b. As b grows from ⇠ 50 to D and the malization constant that depends on L, E0 , and the energy transitions 1(i) to Case decay 1(ii), the exponent bounds (see exactthere expression [54]). In Fig.between 1 (b), despitethe LDOS We showed is ain relation exponent offrom theCase powerlaw increases from ⇠ 2 to 3. In contrast, as b decreases below fluctuations caused by finite-size effects, we indeed observe and the ergodicy (or non-ergodicity) of the LDOS, and from we proposed 50, the eigenstates becomethat less spread out, the LDOS more a powerlaw decay / t 2 until equilibration. This behavior R t sparse, and decreases below 2. With PBRMs, we obtain a iscriterion thermalization dynamics only supported by for the time average 1t t0 F (⌧ )d⌧based shown in on the the a general picture of the behavior of the survival probability, figure with the dashed line. covering all values of , without any restriction to a specific We found = 2 also for periodic boundary conditions; model. chaotic models with different values of and , including Two= 0; scenarios forand the other initial states; for algebraic the XXZ modeldecay: with small 0 10 2 random magnetic fields (see figures in [54]). A t decay has also been speculated for the chaotic Ising model with longitu- with Large decay exponents associated -1 dinal and transverse fields [57]. 10 systems andwecaused by1 (c), Toergodic compare different system sizes, show in Fig. the rescaled survival f(t) = (1/L) ln F (t) [55]. bounds inprobability the spectrum -2 10 For LSmall 1, thisdecay quantity exponents is independent ofassociated L [56]. The t 2 behavior is once again evident. with -3 From thenon-ergodic analytical expression systems above, it is clearand that for the 10 caused by the thesurvival eigenstate largest system size, probability correlations goes to zero as -2 -1 0 1 2 10 10 10 10 10 t ! 1, implying that the LDOS is absolutely integrable. Furt ther support for the ergodic filling of the LDOS comes from We alsocomputation showedofthat using BRM one the full range the direct the inverse participation ratio ofcovers the FIG. 2: (Color online) F (t) for basis vectors evolving under PBRMs 1 ofNéel exponents lattice without choosing any particular model state, which of is IPR , as in full random matri0 / D systems with b = 0.1, 0.5, 1, 2, 5, 10, 20, 50, 100, 3000 (solid lines) from ces. The exponent = 2 therefore indicates that the system top to bottom. They correspond respectively to the fitted ⇠

•  • 

• 

• 

F

•  • 

4

ρ0(Ε)

0.004

ρ0(Ε)

2 1.5 1 0.5 0-6 -4 -2 0 2 E

0.002

4

0.001

0 -200 -100

6

FIG. 4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0

100

200

E

FIG. 6: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

0.3

0.3

ρ0

ρ0(Ε)

0.2

(a)

0.2

0.1

0.1

0 -10

0

10

-5

0

5

0

0

-5

5

E

(b)

10

(c)

1

E

FIG. 5: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

Corroborating this result, we found the exponent = 2 also for periodic boundary conditions, chaotic models with different values of and , including = 0, other initial states, and for the XXZ model with small random magnetic fields (see figures in [51]). A t 2 decay has also been speculated for the chaotic Ising model with longitudinal and transverse fields [54]. The exponent

= 2 indicates that the LDOS is ergodically

f

F

• 

0.003

10

0.5

-8

0

0.1

1

t

10

100

0

10

t

20

FIG. 7: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel state evolving under the chaotic H (4) with = 1, = 1/2, open boundaries. In (a): numerical LDOS p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical F (t) (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed).

probability goes to zero as t ! 1, indicating that the LDOS is absolutely integrable. The exponent = 2 is therefore an indicator of the ergodic filling of the energy distribution of the initial state and consequently of the viability of thermalization.


6, 163601 (2006)

PHYSICAL REVIEW LETTERS

Experimental results

week 28 AP

Violation of the Exponential-Decay Law at Long Times C. Rothe, S. I. Hintschich, and A. P. Monkman Department of Physics, University of Durham, Durham, DH1 3LE, United Kingdom (Received 4 July 2005; published 26 April 2006) First-principles quantum mechanical calculations show that the exponential-decay law for any metastable state is only an approximation and predict an asymptotically algebraic contribution to the decay for sufficiently long times. In this Letter, we measure the luminescence decays of many dissolved organic materials after pulsed laser excitation over more than 20 lifetimes and obtain the first experimental proof of the turnover into the nonexponential decay regime. As theoretically expected, the strength of the nonexponential contributions scales with the energetic width of the excited state density distribution whereas the slope indicates the broadening mechanism. P H Y S I C A L R E V I E W L E T T E R S PRL 96, 163601 (2006) 0

10 DOI: 10.1103/PhysRevLett.96.163601

-1

normalized emission intensity

10

-2

10

in the first-principles treatment of 10the decay of a 10 able state Schrödinger’s equation links the probabil10 litude, p!t", to the energy distribution density, !!E" 10

p!t" #

Z

-3 -4 -5 -6 -7

10

-8

10

-9

10 e$!iEt=@" !!E"dE: 10

-10

10

-11

(1) -9

10

-8

10

mple, taking !!E" as a Lorentzian function for all E

Rhodamine 6G polyfluorene exp τ = 0.35 ns exp τ = 3.9 ns slope -2.3 slope -3.5

PACS numbers: 42.50.Xa, 42.50.Fx ganic molecules, ! is not directly acce

test Eq. (2) directly. On the other han can accurately be measured and we m compute which are give Experimental searches forrelative such widths, deviations from of Table I. These values should linearly ponential law have mainly ofal width offocused the excited on statedecays and as such comparison of thein different tive isotopes [5]. Unfortunately, viewmaterials of thi Relatively low intensities of the non conditions on E and !,butions such systems are in fact not (i.e., late turnovers, narrow w at all because the released energy is very largeinclud (E % observed for small molecules Coumarin 450energy and 500, distributio Rhodamine compared to the width of the oligomer with three repeat 60 10$9 eV). For the radioactive isotope units Co,and the main contribution to the broadenin -7 -6 et al. to 30Because exponential li by theout solvent. of its nonr 10 observed10the decay time (s) torsions will additionally add to the val whereas theoretically the deviation from exponenti dye 2,5-bis(4-biphenyl) oxazole $60 (BBO)


Intermediate decay – regime independence

• 

The intermediate-time decay described above does not depend on the regime (integrable or chaotic) of the final Hamiltonian, but on the shape of the LDOS. Exponential and Gaussian decays occur in both integrable and chaotic models


Ergodic Filling and Thermalization

• 

The


Quantum Chaos

• 

The


Basis for BRM

• 

The

Profile for marcotav65

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