Interacting extension of the
Aubry–André model
Francesca Pietracaprina
Statistical Physics PhD Course
14/03/2013
Francesca Pietracaprina
Interacting AA model
Summary
◦ Introduction: localization and Aubry–André model
◦ Interacting extension
-
the problem and the numerical approximation
numerical results: behaviour of a few quantities
detecting the MBL transition
tentative phase diagram
◦ Conclusions
Francesca Pietracaprina
Interacting AA model
Introduction
Let’s consider systems with on site disorder.
Localization:
one particle:
localized wavefunctions
many particles:
breaking of ergodicity
(exponentially decaying envelope)
(local quantities have localized
correlators)
representative model: Anderson model (fermions)
X †
X
ξα cα† cα
−t
(cβ cα + cα† cβ )
H=
|α {z
{α,β}
}
on site random energies
Francesca Pietracaprina
|
{z
hopping
Interacting AA model
}
Introduction
Let’s consider systems with on site disorder.
Localization:
one particle:
localized wavefunctions
many particles:
breaking of ergodicity
(exponentially decaying envelope)
(local quantities have localized
correlators)
representative model: Anderson model (fermions)
X †
X
ξα cα† cα
−t
(cβ cα + cα† cβ )
H=
|α {z
{α,β}
}
on site random energies
Francesca Pietracaprina
|
{z
hopping
Interacting AA model
}
Aubry–André model
disorder is not necessary: 1D Aubry–André (AA) model
H=
L−1
Xh
†
ξi ci† ci − t(ci† ci+1 + ci+1
ci )
i
i=0
ξi = W cos(2πki + δ),
k : quasiperiodic potential
Localized
Delocalized
t/W=1/2
peculiarity: no mobility edge;
Francesca Pietracaprina
transition point at
Interacting AA model
t
W
=
1
2
Many body localization
1 consider many particles
Let’s now
H=
2 turn on interaction.
L−1
Xh
†
ξi ni + t(ci† ci+1 + ci+1
ci ) + ∆ni ni+1
i
i=0
ξi = W cos(2πki + δ)
(as in the AA model)
What happens with regards to localization?
Many-Body Localization in a Quasiperiodic System,
arXiv:1212.4159v1 (2012)
S. Iyer, G. Refael, V. Oganesyan, D. Huse
Notation: g ≡ t/W , u ≡ ∆/W , transition for g =
Francesca Pietracaprina
Interacting AA model
1
2
(at ∆ = 0)
Many body localization
1 consider many particles
Let’s now
H=
2 turn on interaction.
L−1
Xh
†
ξi ni + t(ci† ci+1 + ci+1
ci ) + ∆ni ni+1
i
i=0
ξi = W cos(2πki + δ)
(as in the AA model)
What happens with regards to localization?
Many-Body Localization in a Quasiperiodic System,
arXiv:1212.4159v1 (2012)
S. Iyer, G. Refael, V. Oganesyan, D. Huse
Notation: g ≡ t/W , u ≡ ∆/W , transition for g =
Francesca Pietracaprina
Interacting AA model
1
2
(at ∆ = 0)
The approach:
Take a random element of the configurations space and let it
evolve. Does the system thermalize?
End result:
the interacting model has a MBL transition. Interactions favours
the delocalized phase.
Numerically simpler: sequentially hop on each bond
†
†
Hm = Hon site + Hint + Lt(cm
cm+1 + cm+1
cm )
U(∆t) =
L−1
Y
Um (∆t),
m=0
ı Hm ∆t
with Um (∆t) = exp −
L
It’s a different model: introduces new transitions; ∆t must be small, or else
qualitatively altered dynamics!
Francesca Pietracaprina
Interacting AA model
The approach:
Take a random element of the configurations space and let it
evolve. Does the system thermalize?
End result:
the interacting model has a MBL transition. Interactions favours
the delocalized phase.
Numerically simpler: sequentially hop on each bond
†
†
Hm = Hon site + Hint + Lt(cm
cm+1 + cm+1
cm )
U(∆t) =
L−1
Y
Um (∆t),
m=0
ı Hm ∆t
with Um (∆t) = exp −
L
It’s a different model: introduces new transitions; ∆t must be small, or else
qualitatively altered dynamics!
Francesca Pietracaprina
Interacting AA model
Notes on the numerics:
◦ Free boundary conditions: no hopping over the boundary
◦ Quasiperiodic potential W cos(2πki + δ) with k = φ−1 , δ RV
◦ Initial configuration: random in half filling configuration space
Lattice sizes are (empirically) chosen to minimize finite size effects: L = 8 ÷ 20, even.
Quantities to look at:
- temporal autocorrelator and its average (over sites and samples)
χj (t) = (2hnj i(t) − 1)(2hnj i(0) − 1)
- participation ratio (V is the size of configuration space)
η=
1
[IPR V ] ,
P
IPR = h c |ψc |4 ismp
- Rényi entanglement entropy
2 S2 (t, L) = h− log2 trA %A (t) ismp
Francesca Pietracaprina
Interacting AA model
Notes on the numerics:
◦ Free boundary conditions: no hopping over the boundary
◦ Quasiperiodic potential W cos(2πki + δ) with k = φ−1 , δ RV
◦ Initial configuration: random in half filling configuration space
Lattice sizes are (empirically) chosen to minimize finite size effects: L = 8 ÷ 20, even.
Quantities to look at:
- temporal autocorrelator and its average (over sites and samples)
χj (t) = (2hnj i(t) − 1)(2hnj i(0) − 1)
- participation ratio (V is the size of configuration space)
η=
1
[IPR V ] ,
P
IPR = h c |ψc |4 ismp
- Rényi entanglement entropy
2 S2 (t, L) = h− log2 trA %A (t) ismp
Francesca Pietracaprina
Interacting AA model
Temporal autocorrelator
Different regimes:
u = 0.32, g = 0.1
−0.08
u = 0.32, g = 0.5
u = 0.04, t = 9999
−1
0.4
0.3
2
8
10
12
14
16
18
20
0.2
−0.14
0.1
−0.16
3
4
0 5
0
6
2000
7 4000 8
ln(time)
g
9
6000
−2
ln( )
−0.1
−0.12
S /L
ln( )
0.5
−3
−4
108000
10000
χ(t, L) constant
−5
3
4
5
6
7
ln(time)
8
9
10
χ(t, L) power law, saturation
Crossover between the two regimes:
u = 0.04, tbin = 9980−9999
The transition happens when
χ(tf , L) shows dependence on L.
1
0.8
0.6
This is a lower bound (due to finite
0.4
size of lattice).
0.2
0
0
0.2
0.4
0.6
g
0.8
1
1.2
1.4
Francesca Pietracaprina
Interacting AA model
Participation ratio
IPR = h
X
|ψc (tf )|4 ismp
c
for many particles IPR always decays exponentially with L:
- localized phase → slower than V −1
- delocalized phase → ∼ V −1
Normalized participation ratio
u = 0.04, t
bin
η(tf , L) =
1
∝ e −κL
V IPR(tf , L)
tbin = 9980−9999
= 9980−9999
0
0.6
0.4
0.55
<r >
ln( )
−5
n
−10
0.2
0.5
0
0.45
−0.2
−0.4
0.4
−15
0
0.2
0.4
0.6
g
0.8
1
0.35
1.2
0.1
1.4
0
0
0.04
0.16
0.32
0.64
200
400
0.2
600
0.3
0.4
800
0.5
g
0.6
1000
0.7
0.8
0.9
Note: the non-interacting AA model has an extended phase which isg not fully ergodic: NPR still depends on L.
L: system size;
V : size of the configurations space
Francesca Pietracaprina
Interacting AA model
Entanglement Entropy
Check if a subsystem is a good heath bath. Take Rényi entropy
(simpler to compute numerically)
A: sites 0, 1, . . . , L/2 − 1;
S2 (t, L) = h− log2
B: sites L/2, . . . , L − 1
2 trA %A (t) ismp , %A (t) = trB [%(t)]
Entanglement for
- localized phase: interactions over the boundary AB
independent of L in the non-interacting case
extensive, but subthermal in the interacting case
- extended phase: ∝ L, → thermal value
(finite size: superextensive)
u = 0.04, t = 9999
u = 0, t = 9999
u = 0.04, t = 9999
0.3
0.05
4000
0
g
60000
0.2
8000 0.4
u = 0.04, t = 9999
0.3
0
0
0.1
0.6
0.8
10000
g
0.2
g
1
0.3
0.1
0.4
1.2
0.2
2
1
S /L
0.1
2
0.15
0.4
u = 0, t = 9999
2
S /L
S2/L
0.2
0.5
8
10
12
14
16
18
20
S2
0.25
1.4
Francesca Pietracaprina
0
0
0.2
0.1
0
0
0.2
0.4
0.6
Interacting AA model
g
0.2
0.8
1
g
0.4
1.2
1.4
Entanglement Entropy
t = 9999
0.8
0.6
0.55
0.4
m
<rn>
0.5
0.45
0.2
0.4
0
0.04
0.16
0.32
0.64
m=0
0
0.35
0
m = 1/2
200
−0.2
400
0.1g
600
0.2
800
0.3
0.4
S2 (t∗ , L) = mL − Sd
Francesca Pietracaprina
1000
0.5
g
0.6
0.7
0.8
Sd = 1.15 ÷ 1.3
Interacting AA model
0.9
at high g
Detecting the transition
To get an estimate of the transition point (without finite size scaling)
◦ autocorrelator: provides a lower bound (due to
u = 0.04, t
bin
L finite, L < ξ)
0.8
transition point → g: size dependence begins.
0.6
0.4
◦ region dominated by finite size effects has
unphysical values (near transition)
0.2
0
0
0.2
0.4
0.6
exponent of normalized participation ratio:
transition point → g: minimal value of κ
entropy:
transition point → g: maximum value of m
tbin = 9980−9999
t = 9999
0.8
0.6
0.6
0.4
<rn>
0
0.45
−0.2
0.4
−0.4
0.35
0
0.1
200
400
m = 1/2
0.55
g
0
0.04
0.16
0.32
0.64
0.4
0.5
0
0.04
0.16
0.32
0.64
m
0.2
0.5
<rn>
0.55
= 9980−9999
1
0.45
0.2
0.4
0.2
600
0.3
0.4
0.5
g
0.6
0.7
0.8
0 0.9
η ∝ e−κL
800
1000
Francesca Pietracaprina
m=0
0
0.35
200
−0.2
400
0.1g
600
0.2
800
0.3
0.4
1000
0.5
g
0.6
S2 = mL − Sd
Interacting AA model
0.7
0.8
0.9
g
0.8
1
1.2
1.4
Phase diagram
u
Many%body)ergodic)
Many%body))
localized)
AA)localized)
Francesca Pietracaprina
AA)extended)
Interacting AA model
g
Conclusion
Interacting AA model:
◦ peculiar 1D model
◦ a MBL transition is spotted
◦ behaviour of some quantities in the MBL phase
Thank you for your attention
References:
1 S. Iyer, G. Refael, V. Oganesyan, D. Huse
Many-Body Localization in a Quasiperiodic System
arXiv: 1212.4159v1 (2012)
2 Thouless, Niu
Wavefunction scaling in a quasiperiodic potential
J. Phys. A 16, 1911-1919 (1983)
3 Tang, Kohmoto
Global scaling properties of the spectrum for a quasiperiodic Schroedinger
equation
Physical Review B 34, 2041 (1986)
Francesca Pietracaprina
Interacting AA model