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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!