Post on 11-Jan-2019
!
Slow Dynamics and Aging in Isolated Quantum Many Body Systems Far
From EquilibriumMarco Schiro’
CNRS-IPhT Saclay
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
“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
Ultra Slow Dynamics of Density Inhomogeneities in 1D Bosons
G. Carleo, F. Becca, M.Schiro’, M. Fabrizio, Scientific Report 2, 243 (2012)
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)
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)
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)
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⇥
Aging Dynamics in a Quenched Tomonaga-Luttinger Model
M. Schiro & A. Mitra, Phys. Rev. Lett. 112, 246401 (2014)
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
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)
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 ⌧)
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
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)
...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
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!
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)
Thanks!