slow dynamics and aging in isolated quantum many body...
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Slow Dynamics and Aging in Isolated Quantum Many Body Systems Far
From EquilibriumMarco Schiro’
CNRS-IPhT Saclay
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Outline
Motivation: Out-of-Equilibrium Dynamics of Isolated Quantum Many Body Systems
Slow Relaxation Dynamics and “Localization” in 1d Interacting Bose Systems
Aging Dynamics in a Quenched Tomonaga-Luttinger Model
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“Quantum” Quenches
O(t) = ��(t)|O|�(t)⇥t⌧
g(t)
H[g(t)] = H0 + g(t)H1
gi
gf
@t| (t)i = H| (t)i
Calabrese&Cardy(2006), Kollath,Altman&Lauchli(2007),
…..many others!
Unitary Dynamics (Energy is conserved, No Thermal Bath)
Approach to Equilibrium at long-times?Expected in generic systems.
Exceptions: Integrable& Many Body Localized Systems
Experimental settings close to this “ideal” limit
Interesting Transient Phenomena?
| (t = 0)i = | 0i
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Ultra Slow Dynamics of Density Inhomogeneities in 1D Bosons
G. Carleo, F. Becca, M.Schiro’, M. Fabrizio, Scientific Report 2, 243 (2012)
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Bose-Hubbard Model
H = �JX
�ij⇥
⇣b†i bj + h.c.
⌘+
U
2
X
i
ni (ni � 1)
Repulsive Bosonic Particles Hopping on a Lattice
Equilibrium Phase Diagram
Superfluid to Mott Transition
Cold-Atoms Experiment
M. Greiner et al, Nature (2002)
Collapse&Revival OscillationsM. Fisher et al, PRB (1989)
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Dynamics of Inhomogeneous Initial States
1 0 1 0 1 0 1 01
S. Trotzky et al, Nat Phys (2012)
Fast relaxation of even/odd sites but...
…for large U the slow degrees of freedom are the empty/doubly-occupied sites
Exp/Theory: Inhomogeneous Initial State
A. Rosch et al, PRL (2008)
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0 2 0 2 0 2 02
Small Quench: fast relaxation
Large Quench: long-lived plateau, trapping in a metastable (inhomogeneous) state
Q:How fast this state is able to relax?
Exact Dynamics, L=8,10,12
Hamiltonian is translational invariant
Initial State: Inhomogeneous+finite density of doubly-occupied sites
Inverse Relaxation Time
Physical Picture: doublons unable to decay and move due
to effective attraction
D. Petrosyan et al, PRA (2007)
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Localization vs Diffusion in Many Body Hilbert Space
Above a threshold energy the (many body) wave function is localized!
Lanczos Mapping:
Single particle dynamics in a 1D tight-binding ‘‘many body’’ lattice
i⇥t |�t� = H̃L|�t�i⇥t |�t� = H|�t�
|�0⇥ � |1⇥
H̃L Tridiagonal in Lanczos Basis!|�0� = | 2 0 2 0 · · · 2 0�
�n = �n|H|n⇥tn,n+1 = �n|H|n+ 1⇥
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Aging Dynamics in a Quenched Tomonaga-Luttinger Model
M. Schiro & A. Mitra, Phys. Rev. Lett. 112, 246401 (2014)
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Dynamical Response to Local Perturbations
Key Quantity: Time-Dep Overlap (“Transient” Loschmidt Echo)
Vloc
time0
| 0i
time0
| 0i
H+ = H + V+
Experimental Signatures: Non-Eq Ramsey Protocol
tw t = tw + ⌧
| (tw)i = eiH tw | 0i
tw t = tw + ⌧
| (t)i = eiH t| 0i
| tw+(t)i = eiH+ (t�tw)| (tw)i
D(t, tw) = h (t)| tw+(t)i = h (tw)| eiH(t�tw) e�iH+(t�tw)| (tw)i
Connection to Orthogonality Catastrophe & X-Ray Edge problems
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Tomonaga-Luttinger Model (1d gapless systems)
Local Static Potential (impurity)
Quenched Tomonaga-Luttinger Model
V
loc
⌘ V
fs
+ V
bs
= g
fs
@
x
�(x)|x=0
+ g
bs
cos 2�(x = 0)
Forward/Backward Scattering contributions factorize
Quench of the bulk Luttinger parameter
0
Vloc
time
time
Cazalilla(’06),Iucci&Cazalilla(’09), Mitra&Giamarchi(’09), Dora et al(’11)
K
K0
H0 =u0
2⇡
Zdx
K0 (@x✓(x))
2 +1
K0(@
x
�(x))2�
D(t, tw) = Dfs(t, tw)Dbs(t, tw)
tw t = tw + ⌧Bonart&Cugliandolo (’12,’13)
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G = 0
No Quench: Equilibrium Dynamical Correlator
Forward Scattering :
Backward Scattering:
Kane&Fisher(’92),Gogolin(‘93),Kane&al(’94), Fabrizio&Gogolin(’95),Furusaki(’97),Komnik&al(’97)
Power-Law Decay with Interaction-Renormalized
exponent
Dbs(⌧) ⇠ ⌧�1/8
KFinite T turns this into an
exponential, for any K
Strong Coupling K<1: is relevant,
perturbation theory breaks down
Weak Coupling K>1: is irrelevant,
perturbation theory well behaved
Dbs(⌧) ⇠ const
Vbs Vbs
Dfs(⌧) ⇠ ⌧�↵⇤
↵⇤ = K g2fs/2u2
Dbs(⌧ ;T ) ⇠ exp(��T ⌧)
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100 101 102 103 104 105
10-4
10-2
100
Dfs
(o;t w
)
tw = 0tw = 10tw = 100tw = 1000
100 101 102 103 104 105
o
10-9
10-6
10-3
100
Dfs
(o;t w
)
tw = 0tw = 10tw = 100tw = 1000
o<b
oc
o<b
oc
o ¾ tw
neq
new
Quench: Waiting-Time Dependence&Aging
�ocneq
=g2fs
4u2
K0
(1 +K2
K2
0
) �octr
=g2fs
4u2
K0
(1� K2
K2
0
)
K0 > K
K0 < K
Dfs(⌧ ; tw) ⇠1
h1 + (⇤⌧)2
i�ocneq
/2
✓[1 + ⇤2(2tw + ⌧)2]2
[1 + (2⇤ tw)2] [1 + 4⇤2(tw + ⌧)2]
◆�octr
/4
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Scaling in the Aging Regime
0.01 0.1 1 10 100 1000 10000t/tw
0.001
0.01
0.1
1
Dfs
(t,t w
)*(t-
t w)-b
ocne
q
tw = 10tw = 100tw = 1000
Dfs(t, tw) ⇠ (t� tw)�↵
✓t
tw
◆✓
F(tw/t)
Non-Universal Exponents (what about Backscattering term?)
↵ = �ocneq
✓ = �octr
Generalized Fluctuation-Dissipation Ratio?
t, tw � 1/⇤
t/tw = const
F(x) = (1 + x)�oc
tr
Glasses (Cugliandolo&Kurchan,..), Critical Systems (Calabrese, Gambassi,..)Quantum Quenches (Foini,Gambassi,Cugliandolo)
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...What about local back-scattering?
Perturbative time-dependent RG analysis:
K
Cbs(t0, t) = logDbs(t0, t) = h 0|T e�i
R t0t dt1 Vbs(t1)| 0ic
K0
“Thermal Regime” Kneq > 1/2 Dbs(⌧) ⇠ exp(��⇤ ⌧)
Cbs(⌧) ⇠ ⌧2(1�Kneq)
Kneq < 1/2
Short-time PT breaks down: power laws?
Non-Perturbative Solution for special values…in progress!
tw ! 1
Non-Equilibrium “Strong Coupling Regime”
Aging Scaling robust to non-linearities?
K
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Conclusions&Open Questions
Slow Dynamics in clean quantum many body systems
Aging Dynamics of Isolated Quantum Systems
Generality? Other dynamical correlators/models/protocols?Quenches in QFTs, Sciolla&Biroli PRB 2013
Quantum Spin Chains (in progress)
Ergodicity Breakdown in absence of disorder?
See Markus Mueller talk on Thursday!
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Acknowledgements
Giuseppe Carleo (Institut d’Optique) Michele Fabrizio, Federico Becca (SISSA)
Aditi Mitra (NYU)
M. Schiro & A. Mitra, Phys. Rev. Lett. 112, 246401 (2014)
G. Carleo, F. Becca, M.Schiro, M. Fabrizio, Scientific Report 2, 243 (2012)
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Thanks!