47th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics May 24, 2016 Providence , RI

Powerlaw Decays and Thermalization in Isolated Many-Body Quantum Systems    arXiv: 1601.05807 Marco Tavora (Yeshiva University) E.J. Torres-Herrera (Universidad Autónoma de Puebla) Lea F. Santos (Yeshiva University)

Supported by: NSF DMR 1147430!

1.1

Simple example

F (t) =

P

Starting with the sine function y(x) = sin x we rotate the x-y plane using: " # " #" # ↵ ỹ cos ✓ sin ✓ y = (1) x̃ sin ✓ cos ✓ x

Outline

•

R

(0) |C↵ |2e iE↵t

2

Under what conditions do quantum systems thermalize? | Irreversibility from unit

April 1, 2016 Under what conditions do quantum systems thermalize? | Irreversibility from unitary dynamics

2

Irreversibility from quantum many bo Irreversibility from quantum many body dynam

e iEt 0(E) 1 How to evaluate C(t) Spec dE⇢ In the tiled axis the equation reads: P (0) 68 Quantum Systems Out of Equilibrium ⇢ (E) = |C↵1.1 |2Simple (Etheexample E↵)3y(x) =Isolated 0 We study the relaxation process of isolated interacting many-body quantum systems Starting with sine function sin x we rotate the x-y plane using: ỹ = sin (x̃ cos ✓ ỹ sin ✓) sec ✓ x̃ tan ✓ (2) ↵ " # " #" # Equilibrate?Equilibrate? ỹ cos ✓ sin ✓ y after a quench = (1) F (t ! 1) / t F (t) =

F (t ⌧ 0 1) ' 1 2 2 F (t) = e 0 t

sin ✓

2 2 0 t ỹ = sin (x̃ cos ✓

cos ✓

x

Yes No

Yes

In the tiled axis the equation reads:

ĤI ) ĤF = ĤI + g V̂

| (t)i = e iĤF t | (0)i )

F (t) / t ,

ỹ sin ✓) sec ✓

t

t

t

Thermalize?Thermalize? YesNo No Yes

ĤI ) ĤF

F (t) = e 0t | (0)i = | I i

(2)

x̃ tan ✓

| (t)i = e iĤF t | (0)i

T

T

T

T

2

1

1

Fig. 3.1 Dynamics following a quench. a The generic protocol of a quench. After the preparation of an initial state, the Hamiltonian of the system is rapidly changed, creating a non-equilibrium state. This induces a dynamical evolution. The question under study is whether a steady state emerges and if so, what kind of steady state is established. Figure adapted from [9]. b Several scenarios are conceivable for the dynamics. Following the strictly unitary evolution of quantum mechanics,

No

is quenched from ! 1 (0) to a finite value. The 1.1 Simple example Starting with the sine function y(x) = sin x we rotate the x-yrameter plane using: C = h ↵| (0)i ↵ " #y(x)"= sin x we rotate the #x-y " plane# using: LDOS is therefore expected to have a Gaussian shape, as inStarting with the sine function FI 2 cos ✓ # " sin# ✓ y F (t) = under |h (0)|the(t)i| " # ỹ " deed shown in(1)Fig. 7 (a) for the Néel state chaotic = st ỹ cos ✓ sin ✓ y = (1) x̃ sin ✓ cos ✓ x Hamiltonian with = 1. The consequent Gaussian of 2 bo P (0) decay x̃ sin ✓ cos ✓ x ↵t FIt(t) = extremely |C↵ |2e iE F (t) is seen in Fig. 7 (b) up to t ⇠ 2. agrees well en InInthe tiledaxis axis equation the tiled thethe equation reads: reads: ↵ with the analytical expression F (t) = exp( R 02 t2 ) [8]. 2 iEt 1) We study the relaxation process of isolated interacting many-body quantum systems F (t) = dE⇢ (E) e ỹ = sin (x̃ cos ✓ ỹ sin ✓) sec ✓ x̃ tan ✓ (2) 0 Spec ỹ = sin (x̃ cos ✓ ỹ sin ✓) sec ✓ x̃ tan ✓ (2) F after a quench P (0) 0 ⇢0(E) = |C↵ |2 (E E↵) (d L ↵

Outline

•

Equilibrate?

Yes

No

10

ĤI ) ĤFiĤF t

| (0)i

,

F (t) / t

•  •

F (t ! 1) / t

t

-4

At |long times of finding the (t)i = the e probability | (0)i 10 Thermalize? system in the initial state shows a iĤF(fidelity) t -6 ) Yes No 10 powerlaw decay: this is a general property of any -8 F (t)quantum / t system

| (t)i = e )

•

-2

t 10

T

F(t)

•

ĤI ) ĤF = ĤI + g V̂

10

-10

10

2

T 0.3

0.5

1

t

2

4

8

1

FIG. 1: (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 area) and Gaussian p (shaded We will show that the value of the exponent determines if the system thermalizes envelope (solid line) with 0 = L 1/2 and E0 = [ (L 1 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical New criterion of thermalization based solely dynamics F (t)on (solid), its time average (dashed), analytical Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for 2 These questions are not only theoretical and is Lmuch experimental interest in L = there 22 (light), = 24 (dark), and (1/L) ln(t ) (dashed).

the nonequilibrium/coherent dynamics of many-body quantum systems on current experiments (e.g. optical lattices, ion traps, NMR) Solid-State Theory Division Technical University of Braun-

schweig The minimum value reached by F (t) during the Gaussian decay is significantly below the infinite time average

FI st bo en 1) F (d L

up

u Z 1 uto evaluate C(t) 2' 1 How t W 1@ t P (0) iEtiE t P 0 Dynamics (t) == dE |F (t)i C↵e e ⇢↵0 (E) | ↵i 1.1 Simple example

X

e

(Ek E0 )2 /

2 0

/(2⇡

k=up,low

Starting with the sine function y(x) = sin x we rotate the x-y plane using: " # " #" # ỹ cos ✓ sin ✓ y = (1) x̃ sin ✓ cos ✓ x

HF | ↵ i = E ↵ | ↵ i where Winterest: Lambert W -function. if =for example Eup >the Elow {| •  Quantity of survival probability or fidelity probability of finding ↵isi}the ↵=1,...D (0) (t )|2 ⇠ |A (t )|2 . As shown in Fig. ??, the Gaussian decay dom system in G the C↵|A = hinitial (0)i later P ↵ | state R Pin time (overlap between initial and evolved states) 2 ⇤ (0) [Fig. ?? (a)], the contributions from |A (t)| and 2Re [A n R 2 Z G (t)AR (t)] being | |h (0)i(0)| = C↵(t)i| | ↵i 2 F (t) = ĤFis at the vicinity of tiEt the dynamics for t > tP [Fig. ?? Ĥ (c)]. I ) It P , where the X F (t) = dE e ⇢0 (E) F(t) = |h (0)| (t)i| 2 (0) value similar (0) 4 to |AG (t)|2 and |A of absolute (t)| , as seen in Fig. ?? (b) i Ĥ t IPR |C↵ state | at time t | (t)i = e FR | (0)i system state at time t=0 ⌘ system brought to very ↵small values. 2 •  We canUsing decompose02 tthe initial state in the 1 eigenbasis of HF {| P ⇠ F (t P (0)on iE e X ), we main gain some↵ i} insight the o P ↵=1,...D P (0) 0 t |value iE t ↵ ↵ (0) | (t)i = C e i | (t)i = =C↵ e C || ↵↵ii ↵ | (0)i monotonically with↵ the diﬀerence between the cutoﬀ energies and the ↵initi ↵ ↵ (0) H| F (0)i | ↵i = =E HF | ↵i = E↵↵| ↵i C↵↵| |↵i↵ i |Ek E0 | (0) tP ' , C↵ = h ↵| (0)i In the tiled axis the equation reads: ỹ = sin (x̃ cos ✓

ỹ sin ✓) sec ✓

(2)

x̃ tan ✓

2

1

0

References

components of initial state

•  •

where k is either low or up depending on which the smallest en P one (0) gives iE t| i ↵ | (t)i = time C↵ toe initiate. ↵ takes a long ↵

The time evolution of the initial state can |Ebe |/as: k written 0 , the powerlaw decay [1]E0Blue then Fidelity can then be expressed as:

F (t) =

X ↵

2

|C↵(0) |2 e

iE↵ t

2

1

4

2

=

X

A(t)2 =)ZĤ Ĥ I F

Dynamics X

•

|C↵(0) |2 e

|C↵(0) |2 e iE↵ t

=

Z

dE e

iEt

2

le d

thea L

(4)

⇢0 (E)

iE↵ t iEt = dE e ⇢0 (E) (0) We can also writewhere the fidelity energy distribution initial statestate C↵ in=terms h ↵of | the (0)i are the overlapsofofthethe initial F ↵ weighted by the components or LDOS | ↵ i of H and with the eigenstates energy distribution X ⇢0 (E) ⌘ |C↵(0) |2 (E E↵ ) (5) LDOS

H ˆ i Ĥ t | (t)i = e | (0)i I ) | )

whe form

Hˆ (t ) As t is the local density of states (LDOS). The survival)i amplitude F (t) X / t of the LDOS, Zor equivalently, A(t) whic is theF Fourier transform is = F tion6 the characteristic function energy (0)| (t)i| = |C | e of the weighted = dE e distribution. ⇢ (E) ( e regi All information about the , dynamics t is therefore contained in , )/ gi! ⇢ (E). H The energy and variance of the initial state are important elements in the (a) description of the evolution of the survival t (b) 10 probability. They are respectively given by ↵

weights

2

F (t) = |h

x

2

(0) 2 ↵

2

iE↵ t

iEt

0

0

ln IPR

ρ0

nces 0.3 0.2

0

All information about the fidelity -4 isX contained in the LDOS

1

10

E0 = h (0)|H| (0)i =

0.1 0

-3

-5

10

-5

and 0

E

5 2 0

=

X ↵

|C↵(0) |2 E↵ ,

3

10

|C↵(0) |2 (E↵

4

10 2

E0 ) .

refle Gau tial As a

(6)

5

10

ln D

6

10

(7)

5

C↵ = h ↵↵| (0)i LDOS is therefore expected to have a Gaussian shape, as in(0) 2 and C the local density of states (LDOS) h ↵ | in(0)i deed shown Fig. 7 (a) for the Néel state under the chaotic ↵ = F (t) = |h (0)| (t)i| Multiple time scales

F(t)

e the overlaps of the initial state with the Hamiltonian eigenstates | ↵with i of = 1. The consequent Gaussian decay of 2 P (0) . The survival amplitude A(t) caniEbetexpressed terms of 7 (b) up to t ⇠ 2. It agrees extremely well F (t) is in seen in Fig. 2 ↵ F (t) = |C↵ | e e overlaps: with the analytical expression F (t) = exp( 02 t2 ) [8]. •  At short times the time ↵ X dependence must be 2 A(t)R= |C↵(0) |2 e iE↵ t iEt . (5) Fquadratic, (t) = Spec dE⇢ (E) e independently of 0 0 ↵ 10 initial stateP and Hamiltonian (0) e energy and variance ex⇢0(E) = |Cof↵ the |2 initial (E state E↵)can also be -2 essed in terms of the by ↵ overlaps. The former is given10 X F (t ! 1) / t |C (0) |2 E , -4 (6) E0 = ↵ ↵ 10 2 t2 ↵ 1 F (t ⌧ 0 1) ' 0 -6 d the latter by 10 X 2 |C↵(0) |2 (E↵ E0 )2 . -8 (7) 0 = 1

1.1

How to evaluate C(t)

Starting with the sine function y(x) = sin x we rotate " # " #" # ỹ cos ✓ sin ✓ y

10

In the tiled axis the equation reads:

ĤI ) ĤF | (t)i = e iĤF t | (0) )

F (t) / t ,

cos ✓

April 16, 2016

sin ✓

ỹ sin ✓) sec ✓

x

x̃ tan

Time Average of F (t

=

Simple example

ỹ = sin (x̃ cos ✓

1

-10 The decay of F (t) in many-particle systems is characterenergy variance of the 10[8–14]. ed by different behaviorsinitial in state different time regimes 0.3 0.5 1 2 4 8 1 r very short times t ⌧ 02 the decay is quadratic for any t oice of initial state and Hamiltonian. After this universal adratic behavior, the system switches either exponenFIG.to1:an (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel l or Gaussian decay, depending on the strength g of theunder per- the chaotic H (4) with = 1, = 1/2, open short-time regime state evolving 1 bation. This second time regime is validboundaries. for 0 . tIn . (a): tP numerical LDOS (shaded area) and Gaussian p here tP corresponds to the onset of the powerlaw decay. Fienvelope (solid line) with 0 = L 1/2 and E0 = [ (L lly, at long times t & tP , the dynamics slows down2)and]/4; theL = 16 [8]. In (b): log-log plot for the numerical 1) + (L 6 gebraic behavior / t emerges. F (t) (solid), its time average (dashed), analytical Gaussian decay

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

0

10

[8].

intermediate regime

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 7

(0) (t) = exp( with the analytical expressionC↵F = h ↵| (0)i

Multiple time scales

2 2 0t )

[8].

F (t) = |h (0)| (t)i|2 intermediate regime P (0) ↵t | 2 i | (t)i =P C↵(0) e2 iEiE ↵t ↵ F (t) = ↵ |C↵ | e HF | ↵iR↵= E↵ | ↵i 2 REVIEW E 85, 036209 (2012) SANTOS, F. BORGONOVI, AND F. M. IZRAILEV PHYSICAL iEt decay depends F (t) dE⇢ (E) e (0) = C↵ = h Spec | (0)i 0 ↵ only on the P between (0) 2 the SF and the energy shell is another way agreement 60 2 ⇢ (E) = |C | (E E↵ ) 60 ↵ 0 perturbation = |h (0)| (t)i| µ=0.1 λ=0.1 F (t) to find the critical values µcr and λcr . We fitted our numerical

0

10

-2

10

40 20

2 1

-0.2 -0.1

F(t)

0

0

40

-4

0.1 0.2

10

µ=0.4

-6

20 0 2

strength which 2 with both (circles in Fig. 9) and verified that F (tdata ! 1) t(0)functions P/ 2 iE t ↵ determines the the=transition from Breit-Wigner to Gaussian happens for the F (t) 1)|C'↵ 1| e 2t2 F (t ⌧ same critical values µ0cr ,λcr ≈ 0.5 obtained before from vn /dn LDOS 0↵ -0.2 0 0.1 0.2 -0.1 shape

1

2 R4 and 2t2from the transition in Fig. iEt to a Wigner-Dyson distribution F (t) = dE⇢ (E) e 0 0 2. At large perturbation we then have Spec in = thee case of model λ=0.4 F (t) P0t (0) 2 between the Gaussian fit and the Gaussian excellent F ==e agreement ⇢0(t) (E) |C↵ | (E E↵)

function↵describing the energy shell, which depends only on 0 0 F (tthe !off-diagonal 1) / t elements of the Hamiltonian matrices. As -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 seen in the bottom panels, these two curves become practically 0.4 0.6 2 t2 F (tindistinguishable. ⌧ 0 1) ' 21 0 λ=1.0 µ=1.5 0.4 Notice 2that 2 even at very large perturbation, the width of the 0.2 0t e F (t) = energy shell, and thus of the maximal SF, is narrower than the 0.2 width 0t density of states (cf. Figs. 9 and 5), especially F (t) = eof the 0 0 -4 -2 0 2 4 -4 -2 0 2 4 for model 2. This contradicts the equality between Pn (E) and Eα Eα 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 2 IG. 9. (Color online) Strength functions for model 1 (left) and effect is typically removed by considering a renormalized MF el 2 (right) obtained by averaging over five even unperturbed Hamiltonian that takes into account the diagonal contributions s in the middle of the spectrum. The average is performed after of the perturbation).

10

-8

10

-10

10

0.3

0.5

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

1

t

2

4

2. Emergence of chaotic eigenstates

8 8

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

0

F(t)

ure of sysc decay in stems, the he LDOS. on, in the

[8].

intermediate regime

2

10

not ergodd not there presence e Hamilto-

2 2 0t )

decay depends becomes Gaussian:

-2

10

-4

only on " # 2 perturbation 1 (E E0 ) â&#x2021;˘0 (E)strength = p which exp 2 2 2 2â&#x2021;Ą 0 determines0 the LDOS shape

(11)

The fidelity is then approximately given by:

10

-6

F (t) = exp(

2 2 0t )

(12)

Note that within this time window, the fidelity does not depend on the lower energy bound of the spectrum. The latter will Very strong become relevant at long times, which will be examined in the perturbations next section. LDOS is Gaussian

10

-8

10

-10

Decay is Gaussian C. Long times t

tP

10 Though0.3 the dynamics0.5 at intermediate times 1 does not2depend

on the regime of H but only on the magnitude of g and the entimes, the short-time regime velope of the LDOS, the long time dynamics is more involved ependently and is strongly dependent on the filling of the LDOS. Furtheran H. By

t

4

8 9

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

0

F(t)

ure of sysc decay in stems, the he LDOS. on, in the

decay depends becomes Gaussian:

-2

10

-4

only on " # 2 perturbation 1 (E E0 ) ⇢0 (E)strength = p which exp 2 2 2 2⇡ 0 determines0 the LDOS shape

The fidelity is then approximately given by:

10

-6

F (t) = exp(

HF | ↵regime i = E↵ | ↵ i long-time

intermediate regime

10

not ergodd not there presence e Hamilto-

2 2 t P) [8]. (0) 0 (t)i = C↵ e iE↵t | ↵i

•

(0) C2↵ = h ↵| (0)i F (t) = |h (0)| (t)i|2

Decay always powerlaw P (0) 2 iE↵t 2 F (t) = |C↵ | e

↵ (11) R F (t) = Spec dE⇢0(E) e iEt P (0) 2 ⇢0(E) = |C↵ | (E E↵) ↵

2 2 0t )

(12) F (t ! 1) / t

Note that within this time window, the fidelity does not depend on the lower energy bound of the spectrum. The latter will Very strong become relevant at long times, which will be examined in the perturbations next section. LDOS is Gaussian

10

-8

10

-10

Decay is Gaussian C. Long times t

tP

2

10 Though0.3 the dynamics0.5 at intermediate times 1 does not2depend

on the regime of H but only on the magnitude of g and the entimes, the short-time regime velope of the LDOS, the long time dynamics is more involved ependently and is strongly dependent on the filling of the LDOS. Furtheran H. By

t

4

8 10

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

0

F(t)

ure of sysc decay in stems, the he LDOS. on, in the

decay depends becomes Gaussian:

-2

10

-4

only on " # 2 perturbation 1 (E E0 ) ⇢0 (E)strength = p which exp 2 2 2 2⇡ 0 determines0 the LDOS shape

The fidelity is then approximately given by:

10

-6

F (t) = exp(

HF | ↵regime i = E↵ | ↵ i long-time

intermediate regime

10

not ergodd not there presence e Hamilto-

2 2 t P) [8]. (0) 0 (t)i = C↵ e iE↵t | ↵i

2 2 0t )

•

(0) C2↵ = h ↵| (0)i F (t) = |h (0)| (t)i|2

Decay always powerlaw P (0) 2 iE↵t 2 F (t) = depends |C↵ | e •  (11) Exponent ↵ on shapeRand iEt F (t) = filling of LDOS Spec dE⇢0(E) e

⇢0(E) =

P ↵

(0) |C↵ |2 (E

(12) F (t ! 1) / t

Note that within this time window, the fidelity does not depend on the lower energy bound of the spectrum. The latter will Very strong become relevant at long times, which will be examined in the perturbations next section. LDOS is Gaussian

10

-8

10

-10

Decay is Gaussian C. Long times t

tP

2

10 Though0.3 the dynamics0.5 at intermediate times 1 does not2depend

on the regime of H but only on the magnitude of g and the entimes, the short-time regime velope of the LDOS, the long time dynamics is more involved ependently and is strongly dependent on the filling of the LDOS. Furtheran H. By

t

4

8 11

E↵ )

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

0

F(t)

ure of sysc decay in stems, the he LDOS. on, in the

decay depends becomes Gaussian:

-2

10

-4

only on " # 2 perturbation 1 (E E0 ) ⇢0 (E)strength = p which exp 2 2 2 2⇡ 0 determines0 the LDOS shape

The fidelity is then approximately given by:

10

-6

F (t) = exp(

HF | ↵regime i = E↵ | ↵ i long-time

intermediate regime

10

not ergodd not there presence e Hamilto-

2 2 t P) [8]. (0) 0 (t)i = C↵ e iE↵t | ↵i

2 2 0t )

•

(0) C2↵ = h ↵| (0)i F (t) = |h (0)| (t)i|2

Decay always powerlaw P (0) 2 iE↵t 2 F (t) = depends |C↵ | e •  (11) Exponent ↵ on shapeRand iEt F (t) = filling of LDOS Spec dE⇢0(E) e

⇢0(E) =

P ↵

(0) |C↵ |2 (E

(12) F (t ! 1) / t

Note that within this time window, the fidelity does not depend on the lower energy bound of the spectrum. The latter will Very strong become relevant at long times, which will be examined in the perturbations next section. LDOS is Gaussian central focus of Decay is Gaussian our work

10

-8

10

-10

C. Long times t

tP

2

10 Though0.3 the dynamics0.5 at intermediate times 1 does not2depend

on the regime of H but only on the magnitude of g and the entimes, the short-time regime velope of the LDOS, the long time dynamics is more involved ependently and is strongly dependent on the filling of the LDOS. Furtheran H. By

t

4

8 12

E↵ )

We emphasize that fast fidelity decays, such as exponential

(10)state, |NSi = | "#"#"#"# . . .i, and the domain F (t ! 1)0.2 /t Néel are extensively not exclusive to chaotic postquench Hamile, |DWi = | or """Gaussian, . . . ### . . .i, both used 1) ' 1 2 t2 ssian of the dynamics of integrable F (tin⌧ 0.1 tonians. Theyspin aremodels. found also integrable systems. 2 0 The decay 0 Z X : Powerlaw rate decay is caused by spectrum bounds.– The witch 2tLDOS 2 2 not the (0) determined by the shape of(0)| the regime 2 iE↵ t iEt 0.0 F (t) = |h (t)i| = |C | e = dE e ⇢0 of site-basis vectors evolving under Hamiltonian (4)(t) 0 e 0 2 4 8 6 ↵ F = ases, & ortwo chaotic) of H F. nds to a strong perturbation, where the anisotropy paWe(integrable will consider different scenarios for the LDOS and t corresponding ↵ for the 2 Z 2 Fs quenched (t) powerlaw t from ! 1 to a finite value. The X 0in the limit of strong In Fig. 1,exponent we consider quenches e2: (Color decay Fin(t) = 2 a Gaussian (0) 2 as iE iEt LDOS (a), F (t) (b) and f (t) (c) for the Néel ↵t therefore expected to have shape, FIG. online) (t) = |h (0)| (t)i| = |C | e = dE e ⇢0 (E) integrable ]. In ↵ LDOS In under Fig.the 1(a), the quench is under between state evolving the chaotic H (4) with = 1, = 1/2, open wn in Fig. 4perturbation. (a) for the Néel state chaotic ⇢ (E) , ↵ 0 boundaries. In (a): numerical LDOS (shaded area) and Gaussian ian with = 1. The consequent from Gaussian decay of model Hamiltonians, the XX to the XXZ model p with envelope (solid line) with = L 1/2 and E0 = [ (L 0 een in Fig. 4 % (b) up= to t1.5 ⇠ 2.(we It agrees extremely well1, because this is a critical point), ction avoid % = F F 1) + (L 2) ]/4; (b): log-log 2 2 2 L = 16 [8]. In Case 1 plot for the numerical analytical expression F (t) = exp( LDOS 0 t ) [8].

Long-time decay, LDOS filling and thermalization

•

eracthe 0 10 the -2 ergy 10

F (t) the (solid), its time average (dashed), analytical Gaussian decay and in Fig. 1(b), the quench is from integrable XXZ model

(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).

Case 1

(0)

Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel ving under the chaotic10 H0 (4) with = 1, = 1/2, open s. In (a): numerical LDOS (shaded area) and Gaussian p (solid line) with 0 = L 1/2 and E0 = [ (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical -2 d), its time average10 (dashed), analytical Gaussian decay nd F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for (11)

F

(c)

(b)

1.5

2

1

0.5 0

R

4

2

0

iEt dE⇢ (E) e ρ (Ε) 0 Spec

(0) |C↵ 2 |2e iE↵t

2

2

-6

F (t) =

E

8

F (t) =

F (t) =

C↵ = h ↵| (0)i F (t) = |h (0)| (t)i|2

4 01 t 2 -6 -4 -2 0 ↵

0.5

0.1

HF | ↵ i = E ↵ | ↵ i

ini [1] 0.2Blue

P

P (0) iE t C↵ e ↵ | ↵ i

P

R

F (t) =

(a)

Case 2 References 0.3

P

(0) |C↵ |2e iE↵t

10 sian, -6 elity 10 F (t) -8 10 , the -10 085],0.3

0.4

| (t)i =

-4

iEt Spec dE⇢0(E) e

2

2

2

Case 2

-4 -6 -2 -4 0 -2 2 0 4 2 E

E

4

6

FIG. 3: (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 p 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 13 F (t) (solid), its time average (dashed), analytical Gaussian decay

(d)

or Gaussian, are not exclusive to chaotic postquench Hamilmes. The Gaussian tonians. They are found also in integrable systems. The decay Long-time decay, LDOS filling and thermalization ime and then switch rate is determined by the shape of the LDOS not the regime shown several cases, &F . (integrable or chaotic) of H 2 Z lattices, where F (t) X In Fig. 1, we consider quenches in thermalization the limit 2of strong implies •  Case 1:F (t) Show that LDOS ergodically filled 2 (0) 2 iE↵ t iEt = |h (0)| (t)i| = In |C e dEise between ⇢0 (E) integrable aturation [57–60]. In ↵ |1(a), perturbation. Fig. the = quench Hamiltonians,↵from the XX model to the XXZ model with filling: initial state is highly delocalized, samples most eigenstates and its ussian• isErgodic a reflection %F = 1.5 (we avoid %F = 1, because this is a critical point), LDOS components are essentially random numbers h two-body interacand in Fig. 1(b), the quench is from the integrable XXZ model n such systems, the Case 1 DOS is given by the known as the energy

i Ĥ t F | (t)i = e |

ini

ini

0.3

F (t) =

(a)

(b)

0.2 0.1 0 -6

R

(11)

Case 2

F (t) / t ,

0

10

-2

F 10

1

2

(0) |C↵ |2e iE↵t

2

σini t)] , 2 it

P

0.4

P F (t) =

,

(0)

)2

C↵ = h ↵| (0)i F (t) = |h (0)| (t)i|2

E

HF | ↵ i = E ↵ | ↵ i

ther than Gaussian, an Gaussian fidelity ower bound for F (t) erences trix. In this case, the dlue by Wigner [82–85],

iEt Spec dE⇢0(E) e

2

P (0) iE t | (t)i = C↵ e ↵ | ↵i

2

)

-4

-2

0

E

2

(c)

4

-6

-4

-2

0

E

2

(d)

4

14

↵=1,...D (0) C↵

i Ĥ t F | (t)i =e | 2 Z

| (0)i = | ↵ i filling andFthermalization Long-time (t) = dE e iEt ⇢0 (E) Z decay, LDOS 2 Z

•

F (t) = IPRdE (0) e ⌘

X iEt (0) 4 0 (E) |C | F⇢(t) ↵=

iEt

dE e

Case 1: Show that ↵ LDOS ergodically filled

X {| ↵ i}↵=1,...D | (0)i = C↵(0) |

⇢0 (E)

) {|

↵ i}↵=1,...D implies

2

thermalization

↵i | (0)i = C↵ | ↵ i {| i}↵=1,...D •  Ergodic filling:(0) initial state is↵highly delocalized, samples most eigenstates and its ↵

•

(0)

F (t) / t ,

| (0)i =are C↵essentially | ↵i X components random numbers (0) (0) 4 (0) IPR ⌘ |C | ↵ X | (0)i = | i , C = 0 (0) ↵6 = ↵ IPR(0) ⌘ |C↵(0) |4 | (0)i = C↵ | ↵ i Measure of delocalization – how spread out is the initial X state? ↵ (0) many CX IPR(0) ↵ )(0) | (0)i = C↵ | ↵ i ↵ IPR(0) (0) =1 state (0)i = |very i ,localized C↵6= =

⌧ (0) 1

IPR

|

(0)

X ↵

|C↵(0) |4

| (0)i = |

C↵(0) |

↵i

(0)

i , C↵6= = 0

state very delocalized

0

1

c

Time Avera

15

April 17

cos ✓

sin ✓

1 How to evaluate C(t)

1.1 Simple example

Starting with the sine function y(x) = si " # " ỹ

=

ỹ = sin (x̃ cos ✓ ỹ sin

ĤI ) ĤF | (t)i = e iĤF t )

F(t) / t ,

References IPR(0) = 1

In the tiled axis the equation reads:

many C↵(0) ) IPR(0) ⌧ 1

(0)

| (0)i = | (0)i , C (0) = 1, C↵6= = 0 many C↵ ) IPR ⌧ 1

[1] Blue

| (0)i =

!

2 2π σini

exp −

,

2 2σini

sin ✓

cos ✓

x

IV. GAUSSIAN DECA

i Ĥ t | (t)i = e |

the tiled axis the equation reads: % LDOS fillingIn and \$ decay, Long-time thermalization 2 2 We emphasize that fast fidelity F decays

(10)

FG (t) = exp −σini t ,

ỹ = sin (x̃ cos ỹ sin ✓) sec ✓ x̃ tan or Gaussian, are✓ not exclusive to ✓chaotic Eq. (7) at short times. The Gaussian arethermalization found also in integrable implies •  Case 1: Show that LDOS ergodically filled tonians. They d to hold for some time and then switch rate is determined by the shape of the L ger times. We have shown several cases, &F . (integrable or chaotic) of H •  Ergodic filling:lattices, initial state is highly delocalized,In samples eigenstates its periments in optical where F (t) Fig. most 1, we considerandquenches in components are essentially random numbers I F an all the way to saturation [57–60]. In perturbation. In Fig. 1(a), the quench is is analysis. Hamiltonians, from the XX model to th iĤ t •  Measure of delocalization – how spread out is the initial state? E) can become Gaussian is a reflection %F = 1.5 (we avoidF%F = 1, because th tes of systems with two-body interacand in Fig. 1(b), the quench is from the in P (0) 6 4 Gaussian [79–81]. In systems, the IPRsuch IPR(0) / D 1 (0) = ↵ |C↵ | 2 Ris preading of the LDOS 1 given by the P (0) 4 F (t) = E dE e iEt (E Elow )⇠ P (E)⌘(E) IPR = |C | ↵ Eq. (10), which is known (0) ↵ p as the energy1/2 2 R1 0.4 f (t/ L ! 0) / ln t F (t) = E dE e iEt (E Elow )⇠ P (E)⌘(E) (a) -3 (b) p 1 & states of10HF is other than Gaussian, 0.3 1/2 f ini (t/ L ! 0) / ln t leading to -4 faster than Gaussian fidelity P 0.2 10 the lower bound for F (t) ) is unimodal, -5 is a full random matrix. In this case, the 0.1 10 micircular, as derived by Wigner [82–85], 0 3 4 5 6 1 57,58], -6 -4 -2 0 2 4 -6 -4 10 10 10 10 ' E ( )2ln D 0 E 1 16 16 10 = 1− ,

)

Ĥ ) Ĥ F (t)

| (0)i

iEt dE⇢ (E) e 0 Spec

R

low

2

(0) |C↵ |2e iE↵t

F (t) / t ,

P

h ↵| (0)i |h (0)| (t)i|2

2

low

↵ i = E↵ | ↵ i

P (0) iE t ↵ | = C↵ e ↵i

ln IPR

2

| (t)i ,= e )

/t

1

'

e

2⇡N 2 02 t2

2 Long-time decay, LDOS and thermalization t filling (E 2E

•

2e

2

Large decay exponents

0

0.3

E0 )2 0.2 / 02

+e

0.1 0.00 2 2 +Eup )+Elow +2E02

(E

(Elo 4

)/2t

FIG. 2: (Color online) LDOS (a), F (t) (b state evolving under the chaotic H (4) wi boundaries. In (a): numerical LDOS (sh p envelope (solid line) with 0 = L 1 1) + (L 2) ]/4; L = 16 [8]. In (b): log F (t) (solid), its time average (dashed), (dotted), and F̄ = IPR0 (horizontal line); L = 22 (light), L = 24 (dark), and (1/L

At (34) [51], where A The depends on L,curve E0 , and vectors in (35) and evolving it under Hamiltonian correactual ob -2 10 bounds, it is clear that for the largest system size, 2 where E = E E is the width of the up -4 sponds0.2to a strong perturbation, since the10anisotropy param- low influencedindicating by finite probability goest to2 zero as t ! 1, tha 1.5 -6 corresponds to the decay stage in F eter is0.1 being quenched from ! expression 1 to10finite values. The tions observed in Fig is absolutely integrable. The exponent = 2 is ge2 -8 10 the indicator LDOS is therefore expected to haveing a Gaussian shape of asthe weergodic by C(t, t01 ),energy is term, obtain out oscillations from the filling ofcosine the distrib -10 0.5 0 see in Fig. 1. 10 is: 0.3 initial 1state2 and4consequently 8 delity of the viability of 0.5 can indeed ofbecomes: -10 0 The form 10⇢0 (E) -5 5 t 0-6 -4 -2 0 tion. E E 2 Algebraic decays faster than t also signal the X (E E0 )2 /(2 02 ) 1 1 e 1 the LDOS. They t(E ⌘ FIG. 4: (t) (c) for E the Néel 0of) ktw ing of are possible if C(t, instead p F (t) (b) and f⇥(E ⇢0(Color (E) online) = LDOS (a), )⇥(E E) (39) F (t ) ' e low up 0 2 2 2 state evolving underN the chaotic H2⇡ (4) with = 1, = 1/2, open 2⇡ 0 t2 N teractions, many-body random interactions are in 0 2

ρ0(Ε)

F(t)

ρ0(Ε)

10

FIG. 1: (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 p 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).

FIG. 3: (Color online) LDOS (a), F (t) (b state evolving under the chaotic H (4) wi boundaries. In (a): numerical LDOS (sh p envelope (solid line) with 0 = L 1 1) + (L 2) ]/4; L = 16 [8]. In (b): log F (t) (solid), its time average (dashed), (dotted), and F̄ = IPR0 (horizontal line); L = 22 (light), L = 24 (dark), and (1/L

boundaries. In (a): numerical LDOS p (shaded area) and Gaussian the number of particles thatk=up,low interact simultaneo envelope (solid line) with = L 1/2 and E = [ (L 0 0 where ⌘ = lnnonzero t. A w Solid-State Theory Division Technicaldecay University of is BraunWhen the LDOS isspectrum ergodically filled, whatbycauses the powerlaw the increasing the number of uncorrelated where the is bounded E 2 [E , E ]. The schweig low up 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical The minimum value by F (t) duringmatrix, Gausisthe to consider thewith retu unavoidable presence of (dashed), bounds in the spectrum thereached Hamiltonian density of tran This expression has the thegeneral form (13) upper bounds leads atstates long times to: sian decay is significantly below the infinite time average F (t)normalization (solid), its time average Gaussian decay constant N analytical is: F̄ = IPR . This is caused by destructive interferences X Gaussian to semicircle [12]. This transition is re 1 (dotted), and F̄ = IPR 24. In (c):between f (t) the forpure Gaussian !# decay of an unbounded LDOS F (t )' "0 (horizontal line); L =! 2⇡ t N and the probability forin thethe initialshape state reconstruction due to of the LDOS Fourie X The L = 22 (light), L = 24 (dark), and (1/L) ln(t 2 ) (dashed). 0.3 the energy bounds. The point in time where this so-called 1 At[8,255–57]. [51], where f A 0

0

1

2 2 0

2

k=up,lo

ρ0(Ε)

Ecollapse Eofup (E in 2[51]). 0 monotonically semicircle leads to F(see (t)the=exact [Jexpression /( 1 (2 0 t)] = 2, ratio[48, b(E49]=aoccurs 0,E increases a (40) e fluctuations caused by finite-size effe p erfsurvival with the largest )/ ,= where E is the 2bounds, itdecayis/kind clear that 2powerlaw aof t until equilib 2 LDOS upper bound [50]. J is the Bessel function the first [8]. TF R 2⇡ N 1 2 0 0 supported by the time average k=up,low probability goes to ze (a) Since the referee commented that “there are no rigorous dashed line in and the figure. short times is faster than Gaussian the This quantity isasymp usef analytical results”, maybe the inclusion of the following 0.2 equation inside the body of thefor paperlong could betimes useful. reveals To that the numerics giv is absolutely integrabl sion a emphasize powerlaw decay w different since useful to pull sizes, up a few lines the comm erf(x) is the 0.1error0.1 function. Again comparing with Eq. 1(21) 3 3 The Fourier transform andindicator is where the numerical first sho F of(ta Gaussian0 LDOS ) 'with[1lower sin(4 t fitting ).ergodi This of the 0 t)]/(2⇡ 0large intensive for sy we identify ⇠ 0= 0, P = 1 and 0-10(E) incides with Case 1(ii),initial where state for theand semicircle conse 0 10 0 -5 5 -5 5 obtain: 0 2 1 1/2 ⌘(E) = (2⇡ ) (2 E) , and E = 2 10 0 tion. low E 0 E (c)2 2 trate the increase (b) 11 of the value of the powerlaw exp (E E0 ) /(2 0 ) ⌘(E) = p e (41) Algebraic decays fa 17 f (tmat 2 to 3, we consider a powerlaw band random 2N ! 2⇡ FIG. 4: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel ing of the LDOS. 0.5 0 [58–60]. PBRMs have being extensively usedThe in s ρ0

1 E0 Elow p N = erf 2 0.2 2 02 0.3

low,up

0

0

up

2

1 t

f

3

F

•

(E

cessible Néel state, |NSi = | "#"#"#"# . . .i, and the domain up 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 scenario of site-basis vectors evolving under Hamiltonian (4) corresponds to a strong perturbation, where the anisotropy pa2 rameter is quenched from ! 1 to 0 a finite low value. The up 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 F (t) is seen in Fig. 7 (b) up to t ⇠ 2. It agrees extremely well 2 2 with the analytical expression F (t) = exp( 0 t ) [8].

t t0

I

Gauserage ences DOS ue to alled

(0)

(0)

0.1

iĤF t Long-time decay, LDOS filling and thermalization

| (t)i = e )

•

IPR(0) =

1.5 1

References

0.5

10 10 10

[1] Blue

0-6 -4 -2 0 12 IPR(0) / D E

4

1

| (0)i

t

-80

t

ỹ = sin (x̃ cos ✓

state evolving under the chaotic H (4) with = 1, = 1/2, open 2 R1 boundaries. In (a): numerical iEt p e (shaded F (t) = ElowLDOS dE (E area) Elowand )⇠ PGaussian (E)⌘(E) envelope (solid line) with 0 = LReferences 1/2 and E0 = [ (L 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical 1/2 [1] analytical Blue F (t) (solid), its time average (dashed), Gaussian decay (dotted), and F̄ = IPR0 (horizontal line); L = 24. In (c): f (t) for L= 22 (light), L = 24 (dark), of andthe (1/L) ln(t 2 ) analysis (dashed). Exponent scaling coincides

•

)

f (t/ L ! 0) / ln t <1

x̃ tan ✓

-240

0.1

P

ỹ sin ✓) sec ✓

-160

ĤI )1 ĤF

6

1

t

10

(0)i

non-ergodic filling

2

exponent F (t)with/the powerlaw t When the LDOS is non-ergodic (sparse), the, powerlaw decay is due to the presence of

• upper bounds leads at long times to:

X 1 1 eigenstates F (t ) ' correlations 0

2⇡

2 2 2 0 t N k=up,low

e

(Ek E0 )2 /

2 0

(5)

1

iEt

x-

f (t/ L ! 0 Figure 1: ... < 1

(0) 4 i Ĥ t Figure 1: ... 3: (Color online) LDOS (a),= F (t) (b) and f (t) (c) forthermalization the Néel • FIG.Consequence of non-ergodicity: no F IPR |C | ↵ | (t)i = e | (0) ↵

p

p

0 In the 10 tiled axis the equation reads: 1/2

F(t)

F (t) / t ,

2

8

Néel open ussian (L erical decay t) for

F

many C↵ ) IPR ⌧ 1 FIG. 2: (Color online) LDOS (a), F (t) (b) and f (t) (c) for the Néel R1 1.1 Simple example state evolving under the chaotic H (4) with = 1, = 1/2, open F (t) = Elow dE e boundaries. In (a): numerical LDOS IPR(0) = 1 p (shaded area) and Gaussian envelope (solid line) with 0 = L 1/2 and E0 = [ (L Starting with the sine function y(x) = sin x we rotate the 1) + (L 2) ]/4; L = 16 [8]. In (b): log-log plot for the numerical (0) (0) | (0)i = | i , C "= 1, #C↵6= "= 0 #" # F (t) (solid), its time average (dashed), analytical Gaussian decay ỹ cos ✓ sin ✓ y (dotted), and2: F̄ LDOS = IPR0 (horizontal 24. In (c): f (t)decay for Case sparseline); L = powerlaw of fidelity has small = exponent 2 2 F (t) = |A(t)| L = 22 (light), L = 24 (dark), and (1/L) ln(t ) (dashed). x̃ sin ✓ cos ✓ x

ρ0(Ε)

The as inaotic ay of well

2

18

R(0) =Elow) P↵(E)⌘(E)⇥(E |C↵ |

) = (E

Elow )

| (t)i =

C↵ e | ↵i ↵ In the tiled axis HFthe | ↵equation i = E↵reads: | ↵i y y C (0) = h | (0)i n n+1 ↵ ỹ = sin↵(x̃ cos ✓ ỹ sin ✓) se 2

ĤI ) ĤF

P P 1D spin-1/2 systems z z x = J Sn Sn+1 ) HF = J Snx Sn+1 +S S

iĤF t | F (t)= = |he(0)| (t)i| | (t)i H=H + H P (0) 2 iE t 2 (0)iH == X|NSi = |"#"#"# · · ·i FHamiltonian (t) = |C↵ | e J S S +S S + S S ) XXZ(solvable) ↵ R Ĥ ) Ĥ iE X ⇠ I F F (t) = dE⇢ (E) e Δ 0 E) = (E E ) P (E)⌘(E)⇥(E E ) Spec H = low J S S +S S + S low S F (t) ⇢/ t P (0) 2 (E) = |C↵iĤ | (E t E 0 F | |, (t)i (3)= ↵e 2 f (t ! 1) / ln t •  Simpler case (relevant here) is the noninteracting XX = =0 ) H = H + H Hamiltonian which is also exactly solvable t X n•  An example of Hamiltonian with ntwo-body interactions: NN

NNN

x x n n+1

NN

y y n n+1

z z n n+1

n

= anisotropy parameter z z n n+2 Ising interaction

x x y y Nearest neighbors NNN n n+2 n n+2 n or flip-flop term (move excitations non-interacting along the chain)

NN

X n

H = z NN z

NNN

x y y X J Snx S n+1 + Sxn Sxn+1 +

z z S S n y y n+1

F (t) = e

z •  Adding NNN J Sn Sn+1 )n terms HF =the Hamiltonian J Sn Sn+1can + become Sn Sn+1chaotic: + (1/2)Snz Sn+1 F (t) = e

HNNN =

•

X

+ n

nx J Snx Sn+2 + X x x

y Sny Sn+2 + y y

z Snz Sn+2 z z

J Sn Sn+2 + Sn Sn+2 +

2 2 0

F (t) / t t 1 IPR(0) / D , 0

1

Sn Sn+2

n we study may contain both NN and NNN interactions The Hamiltonians (3)

(4) H = HNN + HNNN X y x x ysystem z •  Chaotic systems: asH the parameter increases the becomes λ = J S S + S S Snz Sn+1 NN n n+1 n n+1 + X X y z x z = J increasingly Snz Sn+1 )chaotic HF = J Snx Snn+1 + Sny Sn+1 + (1/2)Snz Sn+1 X n n y x z J y Snx Sn+2 + Sny Sn+2 + Snz Sn+2 X HNNN =

2

1 19

x x y y z z P 2 (0) H = J S S + S S + S Sn+1 NN n n+1 n n n+1 2 2 0 ⇢0(E) = This|Cquantity E ) n ↵ ↵y | (E X isS useful if one x x withy chaotic Hamiltonian z z Case 1 – Results for= Néel state HNNN J S S + S S + S ↵ n n+2 n n+2 n n+2

needs to different sizes, since from large devia 2 comparing with Eq. (21) (tstate !into1) /Hamiltonian ln t (3) •  Quench from initialfNéel chaotic intensive for large system sizes L. Fro = 1, = 0 ) = 1/2, = 1 obtain: n

E0 )

2

X

X

2 t2

y x z ) HF = J Snx Sn+1 + Sny Sn+1 + (1/2)Snz Sn+1 0 n n P (0) IPR(0) = ↵ |C↵ +|4X J S x S x + S y S y + S z S z n n+2 n n+2 n n+2

/(2H02I )=

F (t) =

J

z Snz Sn+1

F (t) (41)= e

R1

F (t) =Z e iEt

Elow dE e

Elow )⇠ P (E)⌘(E) n

(E

2

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

0t

2

X | (0)i(0) = |NSi = |"#"#"# · · ·i iEt 2 0)| (t)i| = |C↵ |2 e iE↵ t = dE e 1⇢0 (E)

(4)

2

is = 2. The survival where we used ln D ! L ln 2 valid fo IPR / D ↵ (0) lly and reads: ! 1 ) finite 0.4 Gaussian ✓ ◆ 0.6 2 LDOS 2 0.3 E0 Elow LDOS + i 0t 0.4 p 0.2 Scenario of strong perturbation 2LDOS 0 is expected 0.2 0.1 ◆ Case 1 2 to have Gaussian shape 0 + i 0 p 0t -10 0 10 -5 5 (42) Case 2 ⇢0 (E) = (E

Elow )⇠ P (E)⌘(E)⇥(E

X n

z J Snz Sn+1 ) HF =

n

y x z J Snx Sn+1 + Sny Sn+1 + (1/2)Snz Sn+1

n

y x J Snx Sn+2 + Sny Sn+2 +

z Snz Sn+2

(4)

=

E

April

cos ✓

ỹ s

10

sin ✓

0

X

(3)

+

X

ρ0(Ε)

z Snz Sn+2

1 How to evaluate C(t

y x J Snx Sn+2 + Sny Sn+2 +

n

1.1 Simple example

X

Starting with the sine function y(x) = " # "

HNNN =

In the tiled axis the equation reads:

n

ỹ = sin (x̃ cos ✓

HI =

0

z Snz Sn+1

ĤI ) ĤF | (t)i = e iĤF )

F (t) / t ,

H = HNN + HNNN X y x HNN = J Snx Sn+1 + Sny Sn+1 +

Elow )

20

0

10

6 Case 1 – Results for Néel state with chaotic Hamiltonian

1, it again •  We studied the0.3decay and obtained(a)

L

0.1 0

10

ln IPR

where the -direction.

ρ0

-3is the value expected : this (b) for a system with 10 0.2 LDOS as we obtained analytically ergodic Gaussian -4

10

-5

10

-5

powerlaw decay begins after 3 first 0revival 5 10

E

4

5

10

10

ln D

0

Powerlaw decay

2) ],

10

-3

6

10

(c)

-2

22

t

exp(-σ0 t )

(34)

F

Gaussian decay

10

-6

10

Saturation

-9

Survival collapse

10

6) ],

IPR

(0)

-12

1

t

10

100

fidelity decays by several

(35)

sed in the-

FIG. 1: (Color online) Local density of states (a), scaling analysis of order of magnitude of the IPR0 (b) and survival probability decay (c) for the Néel state evolving under H (33) with h = 0, = 1/2, = 1, and open boundaries. In (a): the shaded area is the numerical result and the

21

↵ f(t) = ln F (t) E0 Eup 2 L p erf (40) f (t ! 1) / ln t 2 02 for Néel Case 1 – Results state with chaotic Hamiltonian = 1, = 0 ) = 1/2, = 1 This quantity is useful if one needs to comp 2 t2 different sizes, since from large deviation th 0 e (t) =an scaling analysis •  Result corroborated by performing n. Again comparing with Eq.F(21) for large system sizes L. From Eqs 0t = 1 and F (t) = e intensive obtain: 1

low 2 0

2

(E E0 ) /(2

2 0)

IPR(0) / D

(41)

1 2N f(t ! 1) ' ln D ' ln 2 0 •  This confirms the LDOS is ergodic and we expect thermalization to occur L exponent is = 2. The survival 6where we used ln D ! L ln 2 valid for large analytically and reads: 0.4 ◆ 0.6 Gaussian (a)  ✓ 10-3 2 (b) 0.3 LDOS E0 Elow + i 0 t iE0 t 0.4 erf 10-4 p 0.2 2 0 0.2 0.1 -5 ◆ 10 2 0 E05 Eup + i 0 t3 4 5 6 -10 0 10 0 -5 5 0 p (42) 10 10 10 10 E 2 0 0 ln D 0 10 ρ0(Ε)

10

Powerlaw decay -2

(c)

-4

(t)

is then:

2

ln IPR

e

10

-3

10

22

Sd =

1.0

|C↵(0) |2 ln |C↵(0) |2 = SthF (t) = |h (0)| (t)i|2 = coefficients behave as random

Sd =

↵ variables with LDOS ergodically filled 2 Z X 2 (t)i| = |C↵(0) |2 e iE↵ t = X Sd = ↵ |C↵(0) |2 ln |C↵(0) |2

0.5

dE e

iEt

= Sth

2

0

50

⇢0 (E)

|C↵(0) |2 e

iE↵ t

LDOS ↵

! 11 ) finite Case

References

Case 2 [1] Blue LDOS

=

-2

0.2

0.4

0

= Sth

2

|C↵(0) |2 e iE↵ t

! 1 100 ) finite 150

Relaxat dynami Diagon

=

tJ

Z

Microca

Canoni

dE e

iEt

200

LDOS

1.5

n(kx)

2

h (0)| (t)i| =

Z

ln |C↵(0) |2

equilibration in a probabilistic sense (small temporal fluctuations)

c 2.0

1) finite (diagonal! entropy is the entropy of a quenched system) 2

-4

X

X

|C↵(0) |2

t!1

X

P (0) 2 Ō = hÔiME = |C↵ | O↵↵ ↵ Rt Ō = lim 1t 0 d⌧ hÔ(⌧ )i

H=

|C↵(0) |2 O↵↵ ⇠ = hOiT 0.6

n=1

X

0.8

Ō =

L X ⇥

x = 0) = hn(k Ô(t)i

1.5

h

1

(0)

ni

Cn |

X

n

⇠ = hOiT

| i=

|C↵(0) |2 O↵↵

P

Ō =

X

b 2.0

P

h

| (t)i =

Thermalization

Figure 1: LDOS for full

x hn Snz + J Snx Sn+1

↵6=

P |C↵ |2 O↵↵ +

(0) C↵

•

P

Case 1 – Results for Néel state with chaotic Hamiltonian

ρ0(Ε)

0)|

Initial

dE e

2 iEt

⇢0 (E)

1.0

0.5 –2

Initial st Diagon dynami Microca Canoni

Case 1 Case 2

–1

0 kx[2π/(Lxd)]

1

2 23

Figure 1 | Relaxation dynamics. a, Two-dimensional lattice on w

-12

10

1

P |Weakly i= Cndisordered | ni n

10

t systems

(e)

100

1

t 10

firme here In filled to th LDO we a alisti all en

100 (f)

P

P (0)⇤ (0) i(E↵ E )t P (0) 2 (0)foriEthe (Color •  FIG. Another filling and large exponents the hÔ(t)i =3:example |C↵ ofonline) | ergodic O↵↵ Normalized + Cand C decay e probability O=↵isf (t) ↵ survival | (t)i C↵ case e ↵oft | ↵i systems with weakand disorder described by wall the Hamiltonian below (hn↵S random Néel state for the state (b,d), under the chaotic ↵ (a,c) n are ↵6domain = HF | ↵ i = E ↵ | ↵ i magnetic fields) H (33) with h = 0, = 1/2, = 1. Open boundaries in (a,b) and (0) C =Lh = (0)i dark L ↵| 22, ↵ closed boundaries in (c,d). Light solid lines indicate X⇥ ⇤ y x y z2 Snz +dashed J Snx Sn+1 Snz Sn+1 n Sn+1 + linesHL== 24,hnand line+ S(1/L) ln(t ). F. (t) = |h (0)| (t)i|2 n=1

R 1 t

F (t) =

d⌧ hÔ(⌧ )i t!1 t 1000

(a)

-1

h=0.2

F(t)

Ō = hÔiME10

t 2 ge2

-2

10

-3

10

F (t) = -4

10 0.1

P ↵

f(t)

Ō = lim

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

(0) 2 iE↵ t |C | e ↵ 1 10 2 t

P ↵

R

(0) |C↵ |2e iE↵t

2 (b) iEt dE⇢ (E) e 0 Spec h=0.2 2 P (0) t ⇢0(E) = |C↵ |2 (E E↵)

F (t) =

ge2

2F (t ! 1) / t

1 L ln F (t) 2 2 F (t) = e 0 t 4 8 10 6 F (t) = e 0t

f (t) =

t

2 R iEt FIG. 4: (Color online) Survival probability (a) and normalized surF (t) = Spec dE⇢0(E) e vival probability (b) averaged over 10 data of disorder realizations and initial states with energies E close to the middle of the specP indicated in the panels; the other paramtrum. The value of h is (0) 2

fidelity decay again fall with exponent = 2 5

0

2

In Eq. ( basis < cides 24 the s

| i=

P n

Cn |

10

ni

Spin-1/2 systems P

-240

10

P (0)⇤ (0) i(E↵ E )t (0) 2 hÔ(t)i = |C↵ | O↵↵ + C↵ C e O↵ ↵ = 2: system with strong disorder •  Typical example associated with↵6case described by the Hamiltonian below (hnSn are random magnetic fields) H=

L X ⇥

0.1

100.1

t

Figure 1:z ... y z

x hn Snz + J Snx Sn+1 + Sny Sn+1 + Sn Sn+1

n=1

Rt =LDOS lim 1tsparse, d⌧ hÔ(⌧ )i becoming 0 t!1

Ō•

1

The powerlaw decay exponent decreases with h Ō = hÔiME IPR(0) / D 0 10 •  Decrease of exponent (a) reflects increasing sparsity

R

10

2

-1

(0) 2 iE↵ t |C↵ | e h <F(t)>

F (t) =

.

10

ρ

1 0

ρ 0.8

-2

10 10

ρ

2

10

0

10

t

2

10

(c)

h

0.4

-3

-4

(b) (0) / D IPR

2

0

iEt dE⇢ (E) e 0 Spec P (0) 2 ⇢ (E) = |C | (E E )

F (t) =

Figure 1:

DYNAMICS AT THE MANY-BODY LOCALIZATION TRANSITION t 1/2 1/2 increasingly more sparse for increasing disorder h t

•

P

1

4

2

(d)

0.2

2

0.1 0

-4

-2

0

E

2

4 25

↵,

ỹ = sin (x̃ cos ✓

h

Case 2 – Strongly disordered systems h

•

F (t) ! t

DX

|C | ĤI ) Ĥ F

C(!) =

hIPR(0) i / D

(0) 2

↵,

DX

C(!) =

(0)

|C(0) |2 |C↵(0) |2 (E

↵,

hF (t)i =

EE

! !)↵ d! ei!t C(!)

Z(0)1|2 ) long|2 |C times ↵, |C (E small E↵ ↵

Z 11

|C↵(0) |2

(E

E↵

| E(t)i = eZ iĤF t | (0)i !) ) hF (t)i = d! e C(!)

!)

E

h

Correlation function quantifies overlap long times ) small ! between eigenstates

C(!) = DX

•

x̃ tan ✓

Exponent from scaling analysis coincides with the decay exponent. Why?

F (t) ! t hIPR(0) i / D

•

ỹ sin ✓) sec ✓

1

i!t

1

DX

h 1 (0)1 2 (0) 2 i!t h hF (t)i = d! e C(!) Long times in correspond to small frequency in but we C(!) = |C C(! ! 0) / ! | |C↵ | (E 1 ↵, 1 D 1 know that: XC(! ! 0) / D ! 1 hOiT 1 X Ō = |C↵(0) |2 O↵↵ ⇠ = 1 (0) 2 C(! X ! 0) /

F (t) / t , Ō = |C

! ⇠ ↵ | O↵↵ = hOihT C(! !(0)0)2D/ ⇠ ! 1 ŌX = |C↵ | O↵↵D= hOiT ↵ (0) 2 (0) 2 Sd = X|CX X ↵↵ | ln |C↵ | = Sth (0) 2 (0) 2 (0) 2 ⇠ 2 = hOi Ō =Ō↵ =X |C↵ |C| (0) O↵↵ S = |C | ln |C | = F (t) !Stth ⇠ T d ↵ ↵ =2hOiT ↵ 2| O↵↵(0) (0) Sd = ↵ ↵ |C↵ | ln 2|C↵ Z| = Sth ↵ 2 X 2 1 2 ↵ (0) 2 iE↵ t Z X (t) = |h (0)| (t)i| = X |C | e2 = 2 dE e iEt ⇢0 (E) ↵(0) (0) X hIPR i / D (0) 2 (0) 2 (0) Sd =Sd =↵ |C↵ |C| ↵ln| |ClnF↵|C | = S 2 (0) 2 iE↵ t th | = S 2 th Z ↵ 2 ↵ | e (t) = |h (0)| (t)i| = |C = dE e X 2 ↵ (0) 2 iE t iEt ↵ ↵ ↵ F (t) = |h (0)| (t)i| = = dE e ⇢0 (E) ↵ | e ! 1 )|Cfinite 2 2 Z Z ↵ 2 2 long times ) small ! X X finitePRB 92, 01420826 E. J. Torres-Herrera and ) L. F. Santos, (2015) 2 2 (0) 2 iE↵ t iEt ! 1 (0) 2 iE t iEt ↵ F (t) |h (0)| (E) = |h =(0)| (t)i| (t)i| = = |C |C| ↵e | e = = dEdE e e ⇢0 ⇢(E) 0 ↵

↵,

ỹ = sin (x̃ cos ✓

h

Case 2 – Strongly disordered systems h

•

F (t) ! t

DX

|C | ĤI ) Ĥ F

C(!) =

hIPR(0) i / D

(0) 2

↵,

DX

C(!) =

(0)

|C(0) |2 |C↵(0) |2 (E

↵,

hF (t)i =

EE

! !)↵ d! ei!t C(!)

Z(0)1|2 ) long|2 |C times ↵, |C (E small E↵ ↵

Z 11

|C↵(0) |2

(E

E↵

| E(t)i = eZ iĤF t | (0)i !) ) hF (t)i = d! e C(!)

!)

E

h

Correlation function quantifies overlap long times ) small ! between eigenstates

C(!) = DX

•

x̃ tan ✓

Exponent from scaling analysis coincides with the decay exponent. Why?

F (t) ! t hIPR(0) i / D

•

ỹ sin ✓) sec ✓

1

i!t

1

DX

h 1 (0)1 2 (0) 2 i!t h hF (t)i = d! e C(!) Long times in correspond to small frequency in but we C(!) = |C C(! ! 0) / ! | |C↵ | (E 1 ↵, 1 D 1 know that: XC(! ! 0) / D ! 1 hOiT 1 X Ō = |C↵(0) |2 O↵↵ ⇠ = 1 (0) 2 C(! X ! 0) /

F (t) / t , Ō = |C

! ⇠ ↵ | O↵↵ = hOihT C(! !(0)0)2D/ ⇠ ! 1 ŌX = |C↵ | O↵↵D= hOiT ↵ (0) 2 (0) 2 Sd = X|CX X ↵↵ | ln |C↵ | = Sth (0) 2 (0) 2 (0) 2 ⇠ 2 = hOi Ō =Ō↵ =X |C↵ |C| (0) O↵↵ S = |C | ln |C | = F (t) !Stth ⇠ T d ↵ ↵ =2hOiT ↵ 2| O↵↵(0) (0) Sd = ↵ ↵ |C↵ | ln 2|C↵ Z| = Sth ↵ 2 X 2 1 2 ↵ (0) 2 iE↵ t Z X (t) = |h (0)| (t)i| = X |C | e2 = 2 dE e iEt ⇢0 (E) ↵(0) (0) X hIPR i / D (0) 2 (0) 2 (0) Sd =Sd =↵ |C↵ |C| ↵ln| |ClnF↵|C | = S 2 (0) 2 iE↵ t th | = S 2 th Z ↵ 2 ↵ | e (t) = |h (0)| (t)i| = |C = dE e X 2 ↵ (0) 2 iE t iEt ↵ ↵ ↵ F (t) = |h (0)| (t)i| = = dE e ⇢0 (E) ↵ | e ! 1 )|Cfinite 2 2 Z Z ↵ 2 2 long times ) small ! X X finitePRB 92, 01420827 E. J. Torres-Herrera and ) L. F. Santos, (2015) 2 2 (0) 2 iE↵ t iEt ! 1 (0) 2 iE t iEt ↵ F (t) |h (0)| (E) = |h =(0)| (t)i| (t)i| = = |C |C| ↵e | e = = dEdE e e ⇢0 ⇢(E) 0 ↵

↵,

h

Case 2 – Strongly disordered systems h

•

F (t) ! t

"

ỹ x̃

#

=

" #" # ỹ =cos sin✓(x̃ cossin ✓ ✓ ỹ siny✓) sec ✓ sin ✓

cos ✓

x̃ tan ✓

x

In the with tiled axis equation reads: Why? Exponent from scaling analysis coincides the the decay exponent.

DX

ĤI ) ĤF

F (t) ! t hIPR(0) i / D

C(!) =ỹ = sin (x̃ cos|C ✓ ↵,

(0) 2

(0) 2 | |C ↵ ✓ | x̃(E ỹ sin ✓) sec tan ✓

E↵

!)

E

i Ĥ t F | (t)i = e | (0)i DX E DX E C(!) = DX |C | |CC(!) | (E E E |C!) | |C | Z(E = E !) Z | ) ) Ĥ long| |C times ! !) C(!) = |C (E small E Ĥ I ) hFF(t)i = d! e C(!) hF (t)i = d! e C(!) Z h h iĤDFX 1 t h = d! e C(!) •  Long times in hF (t)i correspond to small frequency in but we C(!) = |C C(! ! 0) / ! | |C | (E | (t)i = e | (0)i F (t) / t 1 D know that: XC(! ! 0) / D ! 1 hOi X Ō = | O ⇠ = 1 F (t) ! t C(!|C ! 0) / ! , Ō = |C | O ⇠ = hOih X ! 0)D/ ! ) C(! ŌX = |C | O D⇠ = hOi S = X|CX| ln |C | = S X ⇠ (t) times /S t= ) small long Ō =Ō =X |C |C| O | O= hOi |C !| ln |CF (t) | = !St ⇠ hOi =F S = |C | ln |C Z| = S X Z 1 , Z X| e (t) = |h (0)| (t)i| = X |C = dE e ⇢ (E) X hIPR i / D S = |C | ln |C | = S •

hIPR(0) i / D

Correlation function quantifies overlap long times ) small ! between eigenstates (0) 2

(0) 2

↵, ↵,

(0) 2 ↵ (0) 2

1 ↵

i!t

1 1

d

d

2

d

(0) 2 ↵↵ (0)

2

↵ i!t

(0)1 2

(0) 2 ↵

1

↵↵

(0) 2 ↵

(0) 2 ↵

↵,

↵↵

T

T

th

(0)↵↵ 2 ↵↵(0) 2 T ↵2 (0) 2 ↵ ↵

↵ (0) 2

1

1

T1

(0) 2 ↵

(0) 2 ↵

1

↵↵

(0) 2

i!t

1

(0) 2 ↵

↵ ↵,

h

T th

(0) 2 ↵

d

2

(0) 2 ↵

iEt 1 2 0 (0) 2 i!t (0) 2 (0) 2 iE↵ t (0) th Sd = ↵ ||h = Sth=(t)i|2 =d! e |C (t)i C(!) 2 hF ↵ |C↵ | ln ↵|C Z ↵ 2 F (t) = (0)| | e = ↵ X 2 ↵ 1 (0) 2 iE t iEt ↵ ↵ ↵ ↵(0)

th

iE↵ t

2(0) 2

dE e

F (t) = |h (0)| (t)i| = = dE e ⇢0 (E) ↵ | e ! 1 )|Cfinite ! 2 2 Z Z ↵ 1 2 12 long times ) small X X 28 finitePRB 92, 014208 (2015) J. Torres-Herrera and ) L. F. Santos, C(! !E.0) / !!1 2 2 F (t) |h (0)| e iEt (E) = |h =(0)| (t)i| (t)i| = = |C (0)|C|2↵(0) e |2iEe ↵ tiE↵ t= = dEdE e iEt ⇢0 ⇢(E) 1 D0

C(!) = h

0.1 |2 |C↵(0) |2 (E 1 E↵ |C

↵,

t

Figure ... Case 2 – Strongly1:disordered systems 0.1

1

10

!)

10

h Figure 1: ... t •  Exponent from scaling analysis coincides with the decay exponent. Why? F (t) ! t

t

DYNAMICS AT THE MANY-BODY LOCALIZATION TR

Figure 1: ... 1/2 hIPR(0) i / D t

F (t) ! t

t

1/2

ρ

1/2

-2

-4 -6

-8 -10

hF (t)i =

(0) 1

d! e

1

i!t

10

C(!)

-1

hF (t)i = <F(t)>

ln<IPR0>

0 long times ) small long ! 10times ) IPR small !D (0) / 0 IPR(0) / IPR DZ / D (a)

10

-2

Z

1

d! e

i!t

C(!)

2

Ō =

|C↵(0) |2

ln |C↵(0) |2

Sd =

X ↵

2

F (t) = |h (0)| (t)i| =

X ↵

Sd =

|C↵(0) |2 e

F (t) 2 = |h (0)|

iE↵ t

10

1 0

ρ 0.8

1

1 h C(! ! 0) / ! 1 1 1 D -3! 0) / C(! ! 4 5 6 7 8 9 10 10 D X lnD Ō = |C↵(0) |2 O↵↵ ⇠ 2 = hOiX T -4

02 4 ⇠2 |C↵(0) 10| O↵↵10= hOiT10

t

2

0.4

ρ

0 0.2 0.1

0 = Sth exponent decay reflecting -4 -2 0 X increasing sparsity E |C↵(0) |2 ln |C↵(0) |2 = Sth 2 Z 1. (Color online) Survival 2 ↵FIG. probability averaged ov iEt

= points dEfor e h =⇢0.5,1.0,1.5,2.0,2.7,4.0 0 (E)2 data from bottom to Z X for a single realization for the bottom panel h 2= 2 and LDOS (0) iE↵ t and L. F. Santos, PRB 92, iEt 29 Torres-Herrera (2015) (t)i| =the middle |C↵h E.= |2J.e1.5 ⇢0 (E) (c), and=the topdE h =e 2.7 014208 (b); L = 16.

1 IPR / D Case 2 – Néel state in(0) noninteracting XX Hamiltonian P (0) 1 4 IPR / D IPR = |C | •  Quench from initial(0) Néel state (0)into XX Hamiltonian ↵ ↵ F (t) = {|

Z

2

iEt

dE e

⇢0 (E)

↵ i}↵=1,...D (0)

| (0)i = C↵ | ↵ i X IPR(0) ⌘ |C↵(0) |4

P= 0 )(0)=40, IPR(0) = ↵ |C↵ | = 1,

| (0)i =

X

C↵(0)

|

=0

↵i

P PP (0)z (0)z 4 x x y y H = J S S ) H = J S S + S Sn+1 | I (0)i= = | i , C|C = 0 IPR(0) | F ↵6 = n n+1 n n+1 n ↵ ↵ R1

n

(0)e iEt (E (0) Elow ) dE C ⌧1 ↵ ) IPR

n

2

⇢0 (E) = (E = Elow|"#"#"# )⇠ P (E)⌘(E)⇥(E Elow ) |(0)i (0)i = |NSi · · ·i = |NSi = |"#"#"# · · ·i

F (t) = many Elow

|

P (E)⌘(E)

IPR(0) = 1

=low(E Elow ) ⇢0 (E)⇢= (E E )⇠ P (E)⌘(E)⇥(E 0 (E) (0)

| (0)ianalysis = | i, C same

(0)

PE(E)⌘(E)⇥(E low )

10 10 10 10

H= + |A(t)| HNNN FH (t) NN= X y x HNN = J Snx Sn+1 + Sny Sn+1 + n

0

HNNN =

X n

z HSnzNN = Sn+1

H = HNN + HNNN

x y y J Snx Sn+2 + SX n Sn+2 +

z Snz Sn+2

X

X

HNNN =

J

z Snz Sn+1

n

) HF = +

X

J

0.1

X n

n

J

+

x Snx Sn+2

y Sny Sn+1

y x J Snx Sn+2 + Sny Sn+2 +

HI = HI =

X

x n Snx Sn+1

n

n

-240

X n

(3)

y + Sny Sn+2 +

+

X n

(3)

z Snz Sn+2

(3) (4)

10

z Snz Sn+2

z (1/2)Snz Sn+1

z J Snz Sn+1 ) HF =

t

Figure 1: ...

+

z Snz Sn+1

z Snz Sn+2

z J Snz Sn+1 ) HF =

1

X

y x x x x z y y J H SNNN Sn+1 Snz+ Sn+1 =+ Sny J Sn+ Sn+2 Sn Sn+2 + n Sn+1

HNN = X

y x J Snx Sn+1 + Sny Sn+1 +

n

n

-80

HI = -160

= 1, C =0 done in↵6=disordered

Elow )

The systems is applicable in this H = HNNwe + obtained: HNNN integrable case – from 2the dynamics

F(t)

•

| (0)i = |NSi = |"#"#"# · · ·i

X n

X

y x z J Snx Sn+1 + Sny Sn+1 + (1/2)Snz Sn+1

n

x y y z z X J Snx S n+1 + Sx n Sx n+1 + (1/2)S y y n Sn+1

+

J Sn Sn+2 + Sn Sn+2 +

n

y x J Snx Sn+2 + Sny Sn+2 +

z Snz Sn+2

z Snz Sn+2

(4)

30

ln IP

-10

-12

L Case 2 - Néel state in noninteracting t 1/2 XX Hamiltonian -12 = ln 2 (84) -14 2

-16 systems, the exponent can also-14 •  As in strongly disordered be obtained 11 12 11 12IPR(0) 13 / D14 de implies that the by scaling analysis

0

IPR(0) / D

ln IPR0

e the domain wall

-20 -25

-5

γ = 1/ 2 5

-10

2

10

15

-15

5

ln D

10

ln D

•

Example of lack of thermalization (85)

•

Verified by comparing the diagonal and thermodynamic entropies which indeed differ

(86)

ln D

1/2

-10 -15

14

0

-5

e

kn j

ln D

13

FIG. 4: ln IPR0 vs ln D.

3. Case 1 

<2

In [21], the authors study a quench of Ising model in a trans31

15

Case 2 - Néel state in noninteracting XX Hamiltonian

This relation allows us to read the ing analysis. 3

The number of nonzero coefficients is much smaller than the number of zero coefficients: as system size increases their ratio goes to zero exponentially

LDOS clearly sparse

0.

0.03

|cα|

2

0.04

|cα|

•

Small exponent reflects eigenstates correlations and lack of ergodicity

2

•

IPR0 / D

0.02 0.01 0 -4

0

2

0.

4

f(t)

f(t)

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

-2

0.

10

t

20

32

30

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

delity

Z

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

2 0)

" ! !# 1valid for lar • According We showed there a relation exponent powerlaw the to Eq. (28)isthe exponentbetween is = 2.the The survival of the where we used decay ln Df(t) !and L ln 2 = ln F (t) 1 E E E E ρ0(Ε)

0 low 0 up amplitude obtained reads: ergodicy (orbe non-ergodicity) of erf theand LDOS, from that we proposed a criterion for L p analytically p and N =can erf (40) 2 2 0.4 2 2 0on the ✓ 2 only thermalization based dynamics (useful for current experiments) ◆ 0.6 0 2 0.3 This quantity is useful if one needs to com 2 2 1 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 t 2Nerror function. Again comparing2 with 0 erf(x) is the Eq. (21) different 0.2 0.1 ✓ ◆ intensive Eq •  Two scenarios theE=algebraic 0 for large system sizes L. From Eup + decay: i 02 t 0 0 1 and we identify ⇠ = 0,for Perf(E) -10 0 10 -5 5 0 p (42) obtain:0 E 2 0 0 10 10 2 2 −2 1 (E E0 ) /(2 0 ) 1 p is then: ⌘(E) = e (41) -4 The corresponding fidelity 10 2 f(t ! 1) ' ln D-3 ' ln 2

A(t) =

Taking t F (t

e

0t

/2+iE0 t

1/ 2N 0 , F (t) becomes:

1/ 0 ) erf h 1 (Eup

E0

E0

erf

Eup + i 02 t p 2 )2 / 2 0 (E 0

p 2 ◆

low

0

(42) 2

E0 ) /

2 0

F(t)

✓ ◆ 2 2  2 0t e E E + i t •  Large decay exponents 0associated with plow= 2. 0The survival F (t)to=Eq. (28) erf According the exponent is 2 ergodic systems and caused2by0 4N amplitude can be obtained analytically ✓ ◆ 2 bounds in the spectrum and reads: E0 Eup + i 02 t erf . (43)  p✓ ◆ 2 2 00 2 2 1 E Elow + i 0 t

L

-8

10

10

-6

10

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

1

t

10

100

0.4 1

ρ0f(t) (Ε)

2⇡ 0 N

t

0.3

1

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 33 0

10

can indeed see in Fig. 1. The form of ⇢0 (E) is:

Conclusions

(E E0 )2 /(2

•

delity

arXiv:1601.05807

2 0)

1e C(t, t0 ) ⌘ Z p ⇢0 (E) = ⇥(E E )⇥(E E)(39) low up F (t) = dE e ⇢ (E) We studiedNthe powerlaw 2⇡ 02 decay at long times for isolated many-body quantum⌘ 2

iEt

systems in:

{|

0

1 ⌘0

↵ i}↵=1,...D

Z

ln F (⌘ ⌘0

where ⌘ = | ln(0)it.= CA |way to partially exclu i where the spectrum is bounded by E 2 [Elow , Eup ]. The X IPR the ⌘ return |C | is to consider rate given by [42 • normalization Integrable/chaotic, interacting/noninteracting, clean/disordered systems constant N is: X | (0)i = C | i " ! !# 1 We showed there is a relation between the exponent of the powerlaw and | (0)i =decay | i , C f(t) = 0 =the ln F (t) 1 E0 Elow E0 Eup ergodicy of the LDOS, that we proposed a criterion forL p p and from (40) N =(or non-ergodicity) erf erf many C ) IPR ⌧ 1 2 2 2 thermalization based2on (useful for current experiments) 0 the dynamics2only 0 =1 This quantityIPR is useful if one needs to co | (0)isizes, = | i , Csince = 1, from C = 0 large deviation erf(x) is the error function. Again comparing with Eq. (21) different intensive for Flarge system sizes L. From E (t) = |A(t)| we identify ⇠ = 0, P (E) = 1 and obtain: 0 Two scenarios for the algebraic decay: 10 2 2 1 -80 1 10 ⌘(E) = p 2 e (E E0 ) /(2 0 ) (41) f(t ! 1) ' ln D ' ln 2⇡ 0 N -160 L 10 (0) ↵

(0)

(0) 4 ↵

•

(0) ↵

(0) ↵6=

(0) ↵

(0)

(0)

(0)

(0) ↵6=

-240

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

0.1

0.4 2

1

t

Figure 1: ...

10

0.6

0.3 Ergodic 1.5 0.4 0.2 1 0.2 0.1 0.5 00-6 -10 -4 -5 -2 0 0 2 5 4 106 0 E 34 0 0

0

According Eq. (28) the exponent is = 2. The survival •  Small todecay exponents associated amplitude can be obtainedsystems analytically and reads: with non-ergodic and caused by the eigenstate  correlations ✓ ◆ 2 2 2 1 E0 Elow + i 0 t p A(t) = e 0 t /2+iE0 t erf 2N 2 0 References [1] Blue ✓ ◆ 2 E0 Eup + i 0 t p erf (42) 2 0

ρρ0(Ε) (Ε)

•

F(t)

2

10

10

0

FI

Damop2016v9

Powerlaw Decays and Thermalization in Isolated Many-Body Quantum Systems (2016)

Damop2016v9

Powerlaw Decays and Thermalization in Isolated Many-Body Quantum Systems (2016)