Chaos meets FrustrationButterfly effect in a classical spin system

Thomas Bilitewski (MPI)Subhro Bhattacharjee (ICTS,Bangalore)

Roderich Moessner (MPI)arXiv:1808.02054

?

Why do we care about chaos?

vv

I ThermalisationI relation to chaos and

ergodicityI regularity seems to

oppose thermalisationI many-body (generic)

chaotic

I classical↔ QM?

Overview and Connections

Classical Chaos

Butterfly effectexponential sensitivityergodicityphase-space averagestime averagesQM as T → 0?

QM-Chaos

EV statisticsETHScarsOTOC’sscramblingclassical limit?

Frustration/Spin-liquids

competing interactionsorder suppressed

General Folk-Lore/Expectations

Classical many-body Dynamics

chaotic at T →∞, order as T → 0Butterfly-Effecttrajectories diverge

QM many-body dynamics

Unitary/Distance-conservingbut: OTOC’sQuantum bound on chaos (Maldacena ’15)

Main Questions

Can we answer these questions in a classical system?

Q: Classical ChaosI Chaos for T ∈ (0,∞)?I low T scaling of chaos?I OTOC’s?

Q: QM ChaosI Semi-classical limit?I Quantum-bound?I OTOC’s?

Q: GeneralI Relations between microscopic chaos

and macroscopic dynamics?

Roadmap

Classical Spin Liquid is ideal platform to explore these issues!

I require chaos at all temperaturesI must suppress order down to T → 0I need large groundstate manifold

Frustration & Classical Spin-liquids

I competing interactions suppress orderI un-ordered, correlated phase

?

ferromagnetismanti-ferromagnetism

ferrimagnetismmetamagnetism

geometrical frustration

Frustration

I cooperative paramagnetfor Tord < T < Tcoop ∼ J

I frustration ratiof = Tcoop/Tord ∼ 1− 100

I e.g.I Kagome (2D)I Pyrochlore (3D)

I typesI U(1)/CoulombI “Z2” (Rehn ’17)I “jammed” SL (TB ’17)

Classical Chaos and Butterfly Effect

I Butterfly effectI exponential dependence on initial conditions (temporal

component)I (ballistic) propagation (spatial component)

I Lyapunov exponent: growth ∼ eλt

I butterfly speed: x ∼ vbt

OTOC’s in QM

OTOC (out-of-time-ordered correlator) (Larkin ’69)

I⟨[

W (t), V (0)]2⟩

I shows butterfly effectI Quantum chaos bound λ . T (Maldacena ’15)

I random circuits (Nahum ’18, Keyserlingk ’18)I random Heisenberg chain (David’s talk)I Luttinger liquids (Dora ’17)I kicked rotor (Galitski ’17)I quantum bound (Maldacena ’15)I black holes (Shenker ’14)I SYK/holography (’93,’15)I large-N theoriesI · · ·

Velocity dependent Lyapunov

D(x , t) ∼ eλ(v=x/t)t

(c)

�(v)

�(v)

Large NSYK Chain

v

v

vB

vB

(a)

v

�(v)

vB

(c)

“Strongly”Quantum

�(v)

v

�(v)�(v)

vB

Large NSYK Chain

vvB vB

(a)

v

�(v)

vB

SemiclassicalWeak Scattering

Classical

vB1vB2

(b)

(d)

Classical/Semiclassical

(c)

�(v)

vvB

d < 4 d � 4

“Fully”Quantum

“Fully” Quantum/Integrable

“Fully” Quantum/Integrable

Nahum,arxiv:1803.05902

What we did

Model

a1

a2

I Kagome latticeI O(3) spins interacts with all spins in a hexagon 7

H = J∑

x,x′∈7Sx · Sx′ =

J2

∑

α

(Lα)2 + const

I Classical “Z2” spin liquidI no order as T → 0

Dynamics

I Precessional DynamicsdSx(t)

dt= −Sx(t)×

∑

x′Jxx′Sx′(t)

I Initial states are sampled via Monte-Carlo

Relaxational dynamicsA(t) ∼ e−κt

10−3 10−2 10−1 100

T

10−3

10−2

10−1

100

κ

A(t) ∼ e−κt

L=25L=51

L=101

No orderspin diffusion

S(q, ω) ∼ 1/[(Dq2)2 + ω2]

−20

0

20q y

T = 10 T = 1

−10 0 10

qx

−20

0

20

q y

T = 0.1

−10 0 10

qx

T = 0.01

Exponential Sensitivity and Chaos

I Perturb initial condition δS0(t = 0) = ε(n× S0)

I Evolve perturbed state S(t) = S(t) + δS(t)

I Measure distance D(t) =∣∣∣S(t)− S(t)

∣∣∣2

D(t)

Intuitive Picture/Classical limit of OTOC

2D(x, t) =⟨[δSx(t)]2

⟩≈ ε2

⟨{Sx(t),n · S0(0)}2

⟩

I intuitive: Distance of perturbed trajectory!

I∣∣∣S− S

∣∣∣2= |δS|2

I neat: classical version of OTOC!I [· · · , · · · ]→ {· · · , · · · }

Spreading of OTOC

I Ballistic propagationI isotropic spreadingI Exponential growth near front

Exponential Growth

0 10 20 30 40 50 60

t

10−2

103

108

1013

1018

1023

1028

D(x

=0,t

)/ε2

D(x = 0, t)/ε2 ∼ e2µt

ε = 0.0001, fit : µ = 0.53

ε = 0, fit : µ = 0.56

I De-correlator D(x = 0, t) ∼ e2µt

Ballistic Propagation

I Arrival times t(x) for which D(x , t) > D0 for threshold D0

I for ballistic propagation t(x) = x/vb with butterfly speed vb

0 20 40 60 80 100

x

10

20

30

40

50

60

t D0

D0

5×10−1

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

Scaling Form/Velocity dependent Lyapunov

0.0 0.5 1.0 1.5 2.0 2.5 3.0

x/t

−0.2

0.0

0.2

0.4

ln[D

(x,t

)/ε2

]/(2t)

0.42[1− (v/2.00)2.00

]

0 50

x

0

1

D(x,t

)

I De-correlator shows scaling collapse

D(x , t) ∼ exp[2µ(1− (v/vb)ν)t ]

along rays v = x/tI Recall: velocity dependent Lyapunov λ(v) = µ(1− v/vb)

ν

Temperature Dependence of Chaos

10−3 10−2 10−1 100 101 102

T

10−3

10−2

10−1

100

101

µ,vb,κ

vb

µ

κ

D 0.04

0.06

0.08

0.10

D

at low T

vb ∼ T 0.25

µ∼ T 0.5

κ ∼ TD ∼ const

Butterfly speed vb ballistic propagation of the wavefrontLyapunov exponent µ exponential growth rate of the OTOCSpin-relaxation rate κ of spin auto-correlation A(t) ∼ e−κt

Spin diffusion constant D from dynamical structure factorS(q, ω) ∼ 1/[(Dq2)2 + ω2]

Chaos in classical many-body systems

Main resultsI Spin-liquid ideal platform to study chaos for T ∈ (0,∞) X

I Powerlaw scaling of relevant quantities: µ, vb, κ X

I Connection between macroscopic transport andmicroscopic chaos D ∼ v2

b /µ XI µ ∼ T 0.5 vanishes slower than quantum bound X/?

Directions for future workI Semi-classical/large-N limit and quantum bound µ ∼ T ,

singular corrections?I Behaviour of information spreading/entropy?