Download Entanglement and dynamics in many

Entanglement and dynamics in many-body localized systems
Frank Pollmann!
Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany!
Together with:
Jonas Kjäll!
!
Kjäll, Bárðarson, FP, PRL 113, 107204 (2014)
Singh, Bárðarson, FP (in progress) Jens Bárðarson!
Rajeev Singh!
TENSOR NETWORK STATES: ALGORITHMS AND APPLICATIONS
Beijing, Dec. 2 2014!
Many-body localization
t
t
Anderson (1958)
Basko, Aleiner, Altshuler (2006)!
Oganesyan and Huse (2007) !
Pal and Huse (2010)!
Bauer and Nayak (2013)
…
Many-body localization
t
µ
t
Anderson (1958)
Basko, Aleiner, Altshuler (2006)!
Oganesyan and Huse (2007) !
Pal and Huse (2010)!
Bauer and Nayak (2013)
…
Many-body localization
|
t
µ
ni
=
† † † †
c↵ c c c |0i
t
Anderson (1958)
Basko, Aleiner, Altshuler (2006)!
Oganesyan and Huse (2007) !
Pal and Huse (2010)!
Bauer and Nayak (2013)
…
Many-body localization
Interactions
|
t
µ
?
ni
=
† † † †
c↵ c c c |0i
t
Anderson (1958)
Basko, Aleiner, Altshuler (2006)!
Oganesyan and Huse (2007) !
Pal and Huse (2010)!
Bauer and Nayak (2013)
…
Many-body localization
✏ > ✏0
Extended >0
MBL
=0
Volume law
Area law
ETH
ETH breaks down
⌘ (disorder)
Transition is hard to detect!
Band insulator!
A
⇠ exp(
B
/kT )
Anderson (1958)
Basko, Aleiner, Altshuler (2006)!
Oganesyan and Huse (2007) !
Pal and Huse (2010)!
Bauer and Nayak (2013)
…
Overview
(1) Many-body localization (MBL) in a disordered quantum Ising chain
Detection of the MBL transition!
Localization protected quantum order
-
(2) Quenches in disordered systems
-
Efficient time evolution using MPS!
Entanglement and bipartite fluctuations
Extended
Disordered quantum Ising chain
•
Model system: Z2 symmetric 1D quantum Ising model ( |h| ⌧ J = 1)
X
X
X
z z
x
z z
H=
(1 + Ji ) i i+1 + h
i + J2
i i+2
i
•
•
FM ground states: |
Excitations are domain walls:
h
|
i,!
i
hJ2 Interactions
i
i
Localization-protected quantum order
Huse, Nandkishore, Oganesyan, Pal, Sondhi (2013)
Many-body localization transition
•
Localized and extended phase: AREA vs. VOLUME law
(Exact Diagonalization)
(Random state)
...
...
A
E=0
B
S = ln 2
| """i ± | ###i
Bauer and Nayak (2013)
Page (1993)
Many-body localization transition
•
Localized and extended phase: AREA vs. VOLUME law
➡ Variance of S diverges at the transition point and
goes to zero for small/large disorder strength
✏=1
s
s
J
= L(1
=
[(L
dL
1
)f [( J
2) ln 2
2
1]
Jc )Lb ]
f [( J
Jc )Lb ]
Many-body localization transition
•
Repeating the scaling for various energy densities yields
the phase diagram
Extended
MBL
Spin-glass transition
Extended
Disordered XXZ model
•
•
Model system: Anisotropic Heisenberg S=1/2 chain!
Equivalent to spinless fermions
•
}
}
}
H = J
y y
x x
z z
z +J
(Si Si+1 + Si Si+1 ) +
S
hi Si
z
i Si+1
i
i
i
hopping
random potential interaction
with hi 2 [ , ]
All single particle states localized for ⌘ 6= 0
Anderson (1958)
Dynamics of entanglement entropy
•
Start from an unentangled product state ( S = 0 )!
|
!
•
0⇤
= | ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥⇤
Measure the entanglement after quench and the time
itH
evolution with U (t) = e
using TEBD ![Vidal 2003]
!
B
A
•
Clean system:!
Disordered
system (?):!
localization
SS ?
⇠ Ssaturation
tJ
tJ??
tJtJ
??
x
Lieb and Robinson (1972)
P. Calabrese and J. Cardy (2006)
0.4
Jz /J?
0.0
0.01
0.1
0.2
Dynamics of entanglement entropy
0.1, L = 20
Time evolution of S in the disordered XXZ model
S
0.2
Jz /J
0.0
0.01
0.1
0.3
0.14
/J? = 5
0.2
0.1, L = 20
S
0.4
0.1
0.2
Logarithmic
growth
0.07 0.1
1
Rapid
expansion!
S
0.1
0
product state!
0
0.1
1
10
Jt
0.07
0
0.14
0
S
•
0.3
0.1
1
10 100
Jz t
100
10
J? t
0.1
1
10 100
Jz t
100
1000
S ⇠ ⇠ ln(Jz t)
Serbyn, Papic, Abanin (2013)
R.Vosk & E. Altmann (2012) 1000
Bárðarson, FP, Moore, PRL 109, 017202 (2012)!
M. Znidaric, T. Prosen, and P. Prelovsek (2008)
Dynamics of entanglement entropy
•
Bipartite fluctuations of the XXZ chain
z 2
F = h |(SA
) | i
with
z
SA
=
PL/2
z
S
i=1 i
z
h |SA
| i2
...
...
A
B
⌘=1
⌘ = 10
H. F. Song, S. Rachel, and K. Le Hur (2010)
M. Endres et al (2011)
Dynamics of the bipartite fluctuations
Summary
Extended
Kjäll, Bárðarson, FP, PRL 113, 107204 (2014)
Thank You!
Singh, Bárðarson, FP (in progress)!