Kim Department of Physics and Photon Science Gwangju Institute of Science and Technology QIT2018 Yangpyung 21 Dec 2018 Outline ManyBody Localization ID: 816030
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Slide1
OTOC in MBL systems
Dong-Hee Kim
Department of Physics and Photon Science Gwangju Institute of Science and Technology
QIT2018
@
Yangpyung,
21
Dec.
2018
Slide2Outline
Many-Body Localization:
an introductionWhat’s so different in MBL?How can we see
it?
(Models,
Challenges, …Review: F. Alet and N. Laflorencie, C. R. Physique 19, 489 (2018).R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).Out-of-Time-Ordered Commutator / Correlator Ergodic vs. AL vs. MBL Typical behavior of OTOC growth in MBL systems (J. Lee, D. Kim, and D.-H. Kim, arXiv:1812.00357)
Department
of Physics & Photon
Science
Slide3To Thermalize, or Not to
Does a
closed quantum system self-thermalize?Eigenstate Thermalization HypothesisThe eigenstates look alike
in
local
observables.It justifies the microcanonical ensemble.Offdiagonals will vanish in the thermodynamic limit.Reduced density matrix is proportional to exp(-H/T)Initial state information is washed out fast.No thermalization: Disorder + Interactions —> MBL (mostly 1D)cf. disorder w.o. interaction -> Anderson Localizationdisorder : onsite, bond, quasiperiodic (Aubre-Andre)
MBL without disorder: effective
disorder (ex. slow + fast particles)
Department
of Physics & Photon
Science
Slide4Thermal
phase
Single-particle localizedMany-body
localized
Memory
of initial conditions hidden in global operators at long timesSome memory of local initial conditions preserved in local observables at long timesSome memory of local initial conditions preserved in local observables at long times
Eigenstate
thermalization hypothesis (ETH) trueETH false
ETH false
May
have nonzero
DC conductivity
Zero DC conductivityZero DC conductivity
Continuous local spectrum
Discrete
local spectrum
Discrete
local
spectrum
Eigenstates
with
volume-law
entanglement
Eigenstates
with
area-law
entanglement
Eigenstates with
area-law
entanglement
Power-law
spreading
of
entanglement
from nonentangled initial conditionNo spreading of entanglementLogarithmic spreading of entanglement from nonentangled initial conditionDephasing and dissipationNo dephasing, no dissipationDephasing but no dissipation
R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
Thermal phase
Anderson Localization
Many-Body Localization
Phenomenological Comparisons
Department
of Physics & Photon
Science
Slide5Comparison of Eigenstates
ETH
MBL
0
1
2
3
4
5
h
10
―
3
10
―
2
z
z
|
〈
n
|
S
i
|
n
〉
—
〈
n
+
1
|
S
i
|
n
+
1
〉
|
10
0
E
=0.5
10
―
1
L
=
12
L
=
14
L
=
16
L
=
18
L
=
20
L
=
22
X
i
2
[1
,L
]
H
=
[
S
i
i
+1
i
z
i
S
—
h
S
]
F.
Alet,
in
his
lecture
(2016)
Heisenberg
chain
S
z
i
Department
of Physics & Photon
Science
Slide6Spectral statistics
n
min(en, en—1)r =
max(
e
n, en—1)en = En En1Oganesyan, Huse, PRB 2007ETH (GOE)Wigner-DysonriGOE ' 0.5307
1. Level spacing statistics
MBLPoissonriPoisson ' 0.386
0
.
54
0
.
52
0
.
50
0
.
48
0
.
46
0
.
44
0
.
42
0
.
40
0
.
38
r
r
GOE
r
Poisson
=0.5
−
80
−
40
0
40
0
.
40
0
.
45
0
.
50
0
.
55
h
=3.72(6)
ν
=0.91(7)
0 1 2 3 4
5
h
12
14
15
16
17
18
19
20
22
Luitz
et.
al.,
PRB
91,
081103(R) (2015).
Department
of Physics & Photon
Science
Slide7Spectral statistics
X
iiKL =
—
p lnpip0i2pi = |hn|ii|KLiGOE = 2 KLPoisson ⇠ ln(dimH)2. Kullback-Leibler
divergenceETH
(GOE)MBL
12
14
15
16
17
18
19
20
22
3
4
5
h
5
10
15
20
25
30
KL
divergence
KL
GOE
0
10
20
30
h
=1
1
.
8
2
.
0
2
.
2
0
30
60
KL
KL
0
.
00
0
.
02
0
.
04
h(KL)
h
=4.8
3.
Inverse
Participation
Ratio
ln(
I
P
R
)
or
X
i
IP
R
=
p
2
i
0
0 1 2
Luitz
et.
al.,
PRB
91,
081103(R)
(2015).
Department
of Physics & Photon
Science
Slide8Entanglement Entropy
Volume Law (extension, ETH
) vs. Area Law (MBL)
Luitz
et.
al.,
PRB
91,
081103(R)
(2015).
S
(
E
)
=
—
Tr(
⇢
ˆ
A
ln
⇢
ˆ
A
)
⇢
ˆ
A
=
Tr
B
|
n
ih
n
L
=
L
A
+
L
B
A
B
Von
Neumann
EE
ETH:
S
(
E
)
=
s
(
E
)
L
A
S
(
E
)
=
O
(1)
MBL:
Department
of Physics & Photon
Science
Slide9Dynamics after quench
the Néel state, for
example:| Ψ(0)⟩ = | ↑ ↓ ↑ ↓ ↑ ↓ …⟩
L
∑i 2m̂s = (−1)iSzi⟨Ψ(t → ∞)| m̂s | Ψ(t → ∞)⟩ = 0⟨Ψ(t → ∞)| m̂s |
Ψ(t → ∞)⟩
→ m* > 0ETH: It findsthe equilibrium value
very fast.
MBL: it never
does.
(t)i
= exp[—iHt]| (0)Ex. staggered magnetizationExperiments:1D (ultracold gas): Schreiber et.
al., Science 2015; Borida et. al.,
PRL 2016 1D (trapped ion):
Smith et. al, Nat. Phys
20162D (ultracold gas):
Choi et. al., Science 2016; Bordia et.
al., PRX 2017
Department
of Physics & Photon
Science
Slide101D trapped ions
(10 171Yb+ ions)
IsingX
i
<
ji,jx xi jH = J a a +B2iaz
i
+X XiD
i
2
a
z
iQuantum Ising chain with power-law interactions + random transverse fields
| Ψ(0)⟩ = | ↑ ↓ ↑ ↓ ↑ ↓
…⟩
No
disorder
2
4
6
J
max
t
8
−
1.
0
0
−
0.5
0.0
0.5
1.0
W
=
0
J
max
z
Z
magnetization,
i
W
=
8
J
max
Strongest
disorder
2
4
6
J
max
t
8
1
0
10
−
1.
0
0
−
0.5
0.0
0.5
1.0
Smith
et.
al.,
Nat.
Phys.
12,
907
(2016)
Department
of Physics & Photon
Science
Slide11U/J
=4.7(1)
,
U/J
=10.3(1)
/J
=8
/J
=3
/J
=0
Imbalance
0
30
20
Time
(
)
0.2
0.8
0.4
0.6
0
10
e
e
e
e
o
o
o
o
A
initial
state
2
0
0
5
10
15
20
U/J
∞
non-ergodic
localized
ergodic
delocalized
AA
localized
/
J
AA
extended
B
3
C
U
2
J
N
e
—
N
o
Imbalance
I
=
N
+
N
e
o
ˆ
H
=
—
J
X
i,
a
†
i,
a
c
ˆ
c
ˆ
i
+1
,
a
+
h.c.
⇣
⌘
+
�
X
i,
a
†
i,
a
cos(2
⇡{3
i
+
c/
)
c
ˆ
c
ˆ
i,
a
+
U
X
i
i,
"
i,
#
n
ˆ
n
ˆ
.
Aubre-Andre-Hubbard
(
40
K,
0.24T
F
)
Schreiber
et.
al.,
Science
349, 842
(2015)
1D
Fermi gas
+
quasi-periodicity
Department
of Physics & Photon
Science
Slide12J.-Y. Choi et. al.,
Science 352, 1547 (2016).
2D
Bose
Hubbard
(
87
Rb)
Imbalance
I
=
N
L
—
N
R
N
L
+
N
R
Department
of Physics & Photon
Science
Slide13H.
Kim and D. A. Huse, PRL 111, 127205 (2013)
Entanglement Entropy: ETHL
X
x
iH = gO +L—1XhOzzzi 1 LL—
1
X+ (h — J)(O + O )+ JO O
z z
i i+1
S
(t) ⇠
t
J
Nearest
neighbor
interaction:
Time scale
for
the
next
site
get
entangled:
Time scale
for
L
sites
get
entangled:
Jt
1
J
t
LEntanglement spreads at constant speed:i=1 i=2 i=1Initial state: a random product statec.f. S(t) ~ t1/z in some disordered systems
Department
of Physics & Photon
Science
Slide14Entanglement
Entropy:
MBL
S
(
t
)
⇠
ln
t
Bardarson,
Pullman,
Moore,
PRL
109, 017202
(2012)
Znidaric
et.
al.,
PRB
2008;
Bardarson
et.
al.,
PRL
2012; Serbyn
et.
al.,
PRL
2013;
Vosk
et.
al.,
PRL
2013;
Andraschko
PRL
2014.
J
e↵
t
1
Time
scale:
L
⇠
⇠
ln(
J
0
t
)
c.f.
AL:
S(t)
~
constant
EE
spreads
logarithmically
in
time.
Origin:
effective
interaction
J
e↵
⇠
J
0
exp(
—
L/
⇠
)
Department
of Physics & Photon
Science
Slide15R. Nandkishore and
D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15
(2015).Thermal phase Anderson Localization Many-Body Localization
Phenomenological
ComparisonsThermal phaseSingle-particle localizedMany-body localizedMemory of initial conditions hidden in global operators at long timesSome memory of local
initial conditions preserved
in local observables at long timesSome memory of local initial conditions preserved
in local observables
at long times
Eigenstate
thermalization hypothesis
(ETH) trueETH falseETH false
May have nonzero
DC conductivity
Zero
DC conductivity
Zero
DC conductivity
Continuous
local
spectrum
Discrete
local
spectrum
Discrete
local
spectrum
Eigenstates
with
volume-law
entanglement
Eigenstates
with area-law entanglementEigenstates with area-law entanglementPower-law spreading of entanglement from nonentangled initial conditionNo spreading of entanglementLogarithmic spreading of entanglement from nonentangled initial condition
Dephasing and dissipation
No dephasing, no dissipation
Dephasing but no
dissipation
EE
growth
in experiments: NONE, so far.
Only EE
distinguishes MBL from
ALc.f. single-site
EE growth:
Lukin et. al., arXiv:1805.09819.
Department
of Physics & Photon
Science
Slide16A new(?)
measure of MBL:
Out-of-Time-Order Commutator / CorrelatorA. I. Larkin and Y. N. Ovchinnikov,
Zh. Eksp.
Teor.
Fiz. 55, 2262 (1968) .Kitaev, a talk in Fundamental Physics Prize Symposium (2014).Swingle and D. Chowdhury, Phys. Rev. B 95, 060201(R) (2017).R. Fan, P. Zhang, H. Shen, and H. Zhai, Sci. Bull. 62, 707 (2017).X. Chen, T. Zhou, D. A. Huse, and E. Fradkin, Ann. Phys. (Berlin) 529, 1600332 (2017). R.-Q. He and Z.-Y. Lu, Phys. Rev. B 95, 054201
(2017).Y. Chen,
arXiv:1608.02765.Y. Huang, Y.-L. Zhang, and X. Chen, Ann. Phys. (Berlin) 529, 1600318 (2017).K. Slagle, Z. Bi, Y.-Z. You, and C.
Xu, Phys. Rev. B 95, 165136
(2017).P.
Bordia, F. Alet, and P. Hosur,
Phys. Rev. A 97, 030103(R)
(2018).and many other OTOC studies for non-MBL systems.
Department
of Physics & Photon
Science
Slide17OTO
“Correlator”
AL vs. MBL: The OTO correlator works like EE.R. Fan,
P.
Zhang,
H. She, H. Zhai, Sci. Bull. 62, 707 (2017).
Department
of Physics & Photon
Science
Slide18Out-of-
Time-Order Commutator
C(t) =12
⌦
ˆ
[W (t), V†ˆ ˆˆ↵] [W (t), V ] = 1 — Re[F (t)]
W
ˆ
V
ˆ
unitary
“local” operators
OTO
“commutator”
A. I.
Larkin
and
Y.
N.
Ovchinnikov,
Zh. Eksp.
Teor.
Fiz.
55,
2262
(1968)
.
A. Kitaev,
a
talk
in Fundamental
Physics
Prize
Symposium
(2014).F(t)
= hWˆ †(t)
Vˆ†Wˆ (t
)Vˆ
OTO “correlator” (measurable!)
Exp.
: NMR,Trapped ions, Ultracold gases
Department
of Physics & Photon Science
Slide19Measures Quantum Chaos
C
(
t
)
=
12
⌦
†
ˆ ˆ ˆ ˆ
[
W (t), V ] [W (t), V ]
Quantum-to-classical
correspondence
in a chaotic
system
“Quantum” Lyapunov exponent?!
and
Does
Other
Things
:
“
Information
scrambling
”
in
non-chaotic
systems
disordered
systems:
Anderson
localization,
many-body localizationDepartment of Physics & Photon Sciencee>Lt*early time
Slide20Chaotic
Heisenberg XXZ chain in a thermal
phase
L
=
7
,
Wˆ =
CJx,
Vˆ = CJx
, Jz
= 1, ⌘ = 12 4Department of Physics & Photon Science
Slide21Anderson Localized
Heisenberg
XXZ
chain
in
the
Anderson-localized
phaseL =
7, Wˆ =
CJx,
Vˆ = CJ
x, J
z = 0, ⌘ = 102 4Department of Physics & Photon Science
Slide22Many-Body Localized
Heisenberg
XXZ
chain
in
the
MBL
phaseL =
7, Wˆ =
CJx,
Vˆ = CJ
x, J
z = 1, ⌘ = 102 4Department of Physics & Photon Science
Slide23Comparisons of ideal cases
G
r
owth
Department
of Physics & Photon
Science
Particle transport
(time-o
r
de
r
ed)
Models
Note
Chaotic
(Thermal)
C
(
t
)
∝
exp[
𝛌
L
t
]
(early
time)
Y
es
Semiclassical, Large-N
limit.
SYK, black hole.𝛌L ≤ 2𝜋T Real systems?Many-Body Localization (MBL)C(t)∝t2(early time)NoPhenomenological l-bit modelCan we see
it in realistic systems?Anderson Localization (AL)
C(t) ≈ 0
No
All
All frozen
Slide24Phenomenological
l-bit
model
i
i
z
i
H
=
h ⌧ˆ
+
X X
ij
z z
i
j
J
⌧
ˆ
⌧
ˆ
+
X
i
j
k
z z z
i
j
k
K
⌧ˆ ⌧ˆ ⌧ˆ + · · ·
Huse,
Albanin
(phenomenology),
Imbre
(exact),
Ros (pert.), Altman & Vosk
(RG)iz zi
⌧ˆ = a
ˆ +
X X
↵
,/3 ↵jc (i, j, k)aˆ aˆ
/3
k
+ · · ·
l-bit
(localized
bit, LIOM)
vs.
p-bit
(physical
bit)
{
i,j
}
{
i,j,k
}
:
Fully
Many-Body
Localized!
j,k
↵,/3
=
x,y,z
Exponentially
decaying
with
distance
Exponentially
decaying
multispin
interaction
Department
of Physics & Photon
Science
Slide25X
i
izi
H
= h ⌧ˆ +Xijz zi jJ ⌧ˆ ⌧ˆ +X{i,j} {i,j,k}ijk
z z z
i j kK ⌧ˆ ⌧ˆ ⌧ˆ +
· ·
·
AL
:
no
interactions!
All
l-bit
are
independent, No
spin
precessions.
MBL
:
exponentially
decaying
multispin
interactions
P
r
ecessions!
⌧
ˆ
x
(
t
)i ⇠ t—aFrom an unentangled initial state,the offdiagonal elements
of RDM decays slowly with a power law of time.(Sebryn
et. al.)Dephasing!
allows
spreading
of quantum
information.Department of Physics & Photon Science
Slide26Phenomenological l-bit model
i
izi
H
= h ⌧ˆ +X X{i,j}ijz zi jJ ⌧ˆ ⌧ˆ +X{i,j,k}
ijk
K ⌧ˆ ⌧ˆ ⌧ˆz z z
i j k
+ · · ·
ˆ
J
e
↵
ab
ab
=
J
+
X
0
k
abk
z
k
K
⌧
ˆ
+
X
0
{
k
,
l
}
abkl
z z
k l
Q ⌧ˆ ⌧ˆ
+ ·· ·Swingle and Chowdhury, PRB 95, 060201(R) (2017)Wˆ =
⌧ˆx
a
Vˆ
= ⌧ˆ
xbF(t) = hWˆ †(t)Vˆ†W
ˆ (t)Vˆ
OTO
correlator
for
i
H
t
F
(
t
)
=
h
e
⌧
ˆ
a
x
—
i
H
t
x
b
e
⌧
ˆ
e
i
H
t
a
⌧
ˆ
e
⌧
ˆ
x
—
i
H
t
x
b
just
a
Ising
spin
flip
Energy
difference:
ˆ
2
⇥
2
J
e
↵
⌧
ˆ
⌧
ˆ
z z
ab
a
b
Effective
interaction:
F
(
t
)
=
ˆ
exp
it
·
4
J
e
↵
⌧
ˆ
⌧
ˆ
z z
ab
a
b
Department
of Physics & Photon
Science
Slide27i
i
ziH = h
⌧
ˆ +X X{i,j}ijJ ⌧ˆ ⌧ˆz zi jX{i,j,k}+ Ki
jk
⌧ˆ ⌧ˆ ⌧ˆz z zi j
k
+ · · ·
⇥⌦ (
ˆ
C(t) = 1 — Re exp it · 4Je
↵
ab
⌧ˆ
⌧ˆ
z z
a b
)↵⇤
(
ˆ
'
1
—
cos
4
t
h
J
e
↵
ab
)
h
2(ˆi exp —8t h[Jabe↵ 2ˆ] i — hJab
ie↵ 2
)ˆ
C(t
) = 8h[J
ab
e↵
2 24] it + O(t )
Swingle and Chowdhury, PRB 95,
060201(R) (2017)
For a
given disorder realization,
At
very
early
times,
Department
of Physics & Photon
Science
Disorder-independent
t
2
behavior
Measured
with
an
eigenstate,
C
(
t
)
'
1
—
cos
ˆ
4
t
h
J
e
↵
ab
i
Phenomenological
l-bit
model
Slide28TEST: disordered XXZ chain
interaction
disorderh
i
2
[—⌘, ⌘]Operator choice :C(t) =12⌦x30
3
x † x x0[CJˆ
(t),
CJˆ ]
[CJˆ (t
), CJˆ
]Wˆ = (J'ˆx, Vˆ = (J'ˆx3 0
State
choice:
1. Maximally
mixed
state
(
𝜷
=0)
Department
of Physics & Photon
Science
2.
Random
pure
state
|
v
i
=
L
Oi=1
✓
cos
✓
i2
icp
i| "i + e sin
✓
i
2
| #i
◆⇢ˆ = 1/d·· · i ⌘ Tr[
⇢ˆ·· ·
]
H = —
L
—1
X
i
=1
J
S
ˆ
x
S
ˆ
x
i i
+1
i i
+1
曰
)
i i
+1
h
i
+
S
ˆ
y
S
ˆ
y
+
J
z
S
ˆ
z
S
ˆ
z
+
L
X
i
=1
h
i
S
ˆ
z
i
Slide29No t
2 growth in disorder averages
1
0
-5
1
0
-4
1
0
-3
10
-2
1
0
-1
10
0
1
0
1
1
0
0
1
0
1
1
0
2
1
0
3
1
0
4
1
0
5
1
0
6
1
0
7
C(t)
t
Disorder-averaged
over
10000
realizations
Department
of Physics & Photon
Science
No power
law!
L
=
12
,
⌘
=
10
,
(3
=
0
t
2
growth
has
not
been
shown
in
quantum
spin
models
Chen
et
al., Ann.
Phys.
(2017)
He
and
Lu,
PRB
(2017)
Huang
et
al., Ann.
Phys.
(2017)
and
more.
The
t
2
behavior
is
derived
in
the
l-bit
model:
Swingle
and
Chowdhury,
PRB
(2017)
Fan
et
al.,
Sci. Bull.
(2017)
MBL
studies
with
OTOC:
Slide30Previous
work: OTO Correlator
No t2 form found.R. Fan, P. Zhang, H.
She,
H.
Zhai, Sci. Bull. 62, 707 (2017).
Department
of Physics & Photon
Science
Slide31X. Chen,
T. Zhou, D.
A. Huse, and E. Fradkin, Ann. Phys. (Berlin) 529, 1600332 (2017).
Previous
work:
OTO
Correlator
Department
of Physics & Photon
Science
Long-time power-law decaying
behavior
Slide32Disorder average?
What
does
the
distribution
look
like?
1.51.0
0.5
0.0
10
0 101
102 103 104107(a)
0
2
.
5
>
5
C
(
t
)
XXZ
(MBL)
Department
of Physics & Photon
Science
Slide33Distribution of C(t) in H
XXZ
1
0
.
8
0
.60.
40.
20
XXZ
chain ergodic (η =
1)0 5 10 15 20 25 3010.8
0.6
0.4
0.2
0
MBL (η =
10)
0
2
4
6
t
=
3
0
1
2
t
=
5
0
1
21t = 150242 012
t = 3004
8
t = 103
0
2
4
012t = 10
3.5 t =
104
0
1
23t = 105(a)
(b)
8
(c)
(d
)
C
(
t
)
C
(
t
)
P
(
C
)
β
=
0
v
=
0
.
0
v
=
0
.
4
v
=
0
.
8
0 1 2 0 1 2 0
P
(
C
)
10
0
10
2
10
4
10
6
10
8
0
1
2
0
1
2
0
1
2
0
1
2
t
C
C
C
C
An
average
is
meaningless
in
the
MBL
phase.
Department
of Physics & Photon
Science
A
double-peak
distribution
appears
in
the
MBL
phase.
DELAY
TIME?
Slide34XXZ model at a given
disorder∼
t2r
l
n
C(t)ln t
Perturbative
regime
No
disorder
contribution!
Interacton
is
not
essential.
Nothing
to
do
with
MBL
Disorder
dependent
Exponent
<
2
(a
power-law
fit)
∼
t
α
MBL-
r
elated
0
20
40
(e)
occurence
[%]
1
.
0 1
.
2 1
.
4 1
.
6 1
.
8
2
.
0 2
.
2
α
Department
of Physics & Photon
Science
Slide35Early-time growth
cr
ˆ
x
r
+1
(
t
) = crˆ
x
r+1
+ it
[H, crˆx r+1] +(it)
2
2!
[
H, [
H, crˆ
x
r
+1
]]
+
·
·
·
The
lowest-order
term
with
O'
ˆx1appears attr4 1[crˆ , crˆ ] ⇠ tx x 3Squared-commutator (OTOC)6
C(t) ⇠ t
This is an
intrinsic property of the XXZ
model!
No influence of disorderInteraction
is not essential.
10−1410−610−810−10
10−12
10
−2
100
∼ t6(b)
C
(
t
)
10
−
1
t
Department
of Physics & Photon
Science
Slide36Intermediate-time behavior
10
−3101010−4
−
5
10−1−2101100100 101 102 103 104 105 106 107∼ t210−4
10
−2100100 102 104
C
(
t)
(c)
C
(
t
)
t
1-cos(
𝝎
t)
activated with
a
delay
time
(from
the
fixed-point
H)
Early-time
non-MBL
part
DISORDER-DEPENDENT
OFFSET!
Some
disorder
realizations
give
a
power-law
behavior
very
close
to
t
2
.
C
(
t
)
=
c
0
+
at
2
Department
of Physics & Photon
Science
Slide3710−6
10
−410010−2
10
0
103104105106t2210012 3 × 10
5
10−610−410−2
100
100
101
10
2103104105
106
10−3
10
−110
−2
100
10
0
10
1
10
2
10
3
t
2
2
1
0
0
50
0
100
0150010−210−1100100101102
10310010−1
101
eig
C (t) −
c102
0
t2(a)(b)
(c)
(
d)
C
eig
(
t
)
C
eig
(
t
)
10
1
10
2
c
0
+
a
1
(
1
−
cos
ω
1
t
)
C
eig
(
t
)
C
eig
(
t
)
t
,
c
0
+
∑
i
=
1
2
a
i
(
1
−
cos
ω
i
t
)
e
i
g
C
(
t
)
=
1
Re
h
X
/3,
,굽
e
it
(
E
↵
E
+
E
3
E
)
s
↵/3
굽
s
↵/3
굽
x
3
x
0
x
3
x
0
=
h
↵
|
C
ˆ
|
/
3
ih
/
3
|
C
ˆ
|
,
ih
,
|
C
ˆ
|
b
ih
b
|
C
ˆ
|
↵
Eigenstate-
OTOC
Measured
with
an
eigenstate
Dominant modes
with
Department
of Physics & Photon
Science
a few
smallest
!
i
t
2
is
transient
at
interm.
times.
e
i
g
0
X
i
C
(
t
)
⇠
c
+
a
i
(1
—
cos
!
i
t
)
Slide38Mixed-Field Ising chain
P(C)
0.00.5
1
.
01001011021060.01.02.03.0t = 103
0.0
1.0t = 103.30.0
1
.0
t = 10
3.5
(a) 1.5(b)(c)(
d)
0
4
>8
C(
t)
10
3
t
mixed-field
Ising
chain
P
(
C
)
b
=
0
v
=
0
.0 v = 0.4v = 0.8 012 012 012CCC
10
—
6
10
—
4
10
—
2
10
0
10
—
1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
t
2
b
=
0
10
—
2
10
0
10
—
1
10
0
10
2
10
4
C
(
t
)
t
0
.
57
10
—
6
10
—
4
10
—
2
10
0
10
—
1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
—
4
10
—
2
10
0
v
=
1
.
0
1
N
Y
v
Â
Y
v
C
[
Y
v
]
10
0
10
2
10
4
10
6
(a
)
(b
)
C
(
t
)
C
(
t
)
t
v
=
1
.
0
v
=
0
.
0
b
=
0
c
0
+
e
t
2
Essentially
the
same
behavior
Department
of Physics & Photon
Science
Slide39Summary
Measures of Many-Body Localization
Direct measure on eigenstates: ETH satisfied or not Spectral statistics: Wiger-Dyson or Poisson Quench dynamics: ETH satisfied
or
not
Entanglement Entropy: Spreading of entanglementOut-of-time-ordered commutator : a new(?) measure Is the t2 behavior in FMBL the characteristic of MBL? We have to be very careful with disorder averages.ChallengesPhase transition, Anomalous transport, Numerical issues, …