Scrambling vs chaos
East Asia String Webinars, 7/24/2020
Xiangyu Cao (UC Berkeley)
Collaborators
Tianrui XuThomas Scaffidi
Jaewon Kim
Ehud Altman

Chaos/butterfly eﬀect: exponential sensitivity to initial perturbation∂qi(t)λ1t∼e∂qj(0)λ1 (largest) Lyapunov exponent

More generally, in a 2N-dimensional phase space,∂qi(t)( ∂qj(0) )SVDijλ1, λ2, …, λ2N

Lyapunov Spectrum

Thermalization(Entropy growth)

Chaos/butterfly eﬀect∂qi(t)( ∂qj(0) )SVDijλ1, λ2, …, λ2N
Liouville theorem ⇒∑i[Latora,Baranger, PRL 99]

λi = 0i∼

phase space
distribution

t=0dS=λi =: Λ+∑dt
λ >0t>0eΛ+teWith fixed resolution,V∼e

Λ+t⇒ S ∼ Λ+t−Λ+tButterfly eﬀect as (numerical) diagnostics of integrability
Non-integrable∂qi(t)λLt∼ e , λL > 0∂qj(0)

Integrable∂qi(t)α∼ t , λL = 0∂qj(0)(Regular motion on invariant tori)

Note: a saddle point is not chaotic!
[exponential sensitivity only for t ≤ 𝒪(1)]“Old-school” quantum chaos
Casati, Berry-Tabor, Bohigas-Giannoni-Schmit
Level statistics

H | n⟩ = En | n⟩related: spectral form factor

Integrable

Non-integrable

Poisson statistics

Powerful diagnostics
But: no λL (diﬀerent time-scale)Random matrix theory

En − En−1

Defining quantum λL (butterfly eﬀect)
Larkin, Ovchinnikov
Maldacena, Shenker, Stanford

Out-of-time-order correlators

C(t) := ⟨[V(t), W] [V(t), W]⟩†

Semiclassical intuition:⟨…⟩1∂q(t)/∂q(0) = {q(t), p}P.B. ≈ [q(t), p]iℏSomeexpectationvalueIn this talk, “scrambling” means
exponential growth of OTOCsC(t) := ⟨[V(t), W] [V(t), W]⟩ ∼ e†λLt

Does scrambling equal chaos?
(and does it have similar diagnostics power?)
I. In the classical limit (w/ Xu, Scaﬃdi)II. In the large-N limit (w/ Kim, Altman)
III. Away from any limits?

Classical limit: Scrambling ⊋ chaos∂q(t)†C(t) = ⟨[q(t), p][q(t), p] ⟩ =ρ(q)∫ ( ∂q )2= ρ(q)e∫λL ≥ 2λ1λL(q)tλL⟩t2λ1t⟨≥e=eℏ=1[Galitski et. al.]Chaos phase-space-averages over the log.
Scrambling averages before taking the log.

Quantitative detail?

Qualitative diﬀerence!
Scrambling can result from mechanisms
other than chaos, e.g., saddle points.
Demo (2d phase space)q−q±(t) = eq+C(t) ≥±μtq±∫q ≤ϵ+(locally near a saddle)

2μt

e dq+dq− ∼ ϵeϵ∼e2μt=eμt−μt

So that the trajectory stay close at time t.

Hence, λL ≥ μ without chaos!
Finite T: [Hashimoto, Huh, Kim, Wanatabe]Example: Lipkin-Meshkov-Glick model
2H = x + 2z , {x, y} = z, …
2d phase space, trivially integrable,
but has a saddle with μ = 3 .

Saddle-dominated scrambling

λL = μOTOCs are computed inquantized model, withℏ = 1/S . The exponential
growth saturates at Ehrenfest
time tE ∼ ln S.

Infinite T OTOC

Microcanonical OTOCs

Example: Kicked rotor

x, p ↦ x + p, p + K sin(x)(x = x + 2π)

λL(scrambling)

2λ1(chaos)

KK ≲ 1: scrambling without classical chaos

λL = μSaddle (x, p = 0,0)-dominated scrambling[Rozenbaum,Ganeshan,Galitski, PRL17]

Remark
Saddle-dominated scrambling can occur
In higher dimension/many-body phase space;
In presence of chaos.
It remains unclear how generally that happens.

Scrambling in large-!N, low-!T
Example: Sachdev-Ye-Kitaev

H=N∑

ijkl=1

JijklγiγjγkγlC(t) = ⟨{γi(t), γj}{γi(t), γj}⟩ ∼ e , t ≲ ln NλLt

TT ≪ J: λL = 2πT, “fast scrambling”
What are some other behaviors? How do λ! L(T) depend on
the IR fixed point?
2Jγiγj + γjγi = δij , Jijkl Ji′j′k′l′ = 3 δijkl,i′j′k′l′
N[Kitaev][Maldacena,Shenker,Stanford], …

Another example: mass-deformed SYK

H=N

∑
ij=1

κij iγiγj +N∑

ijkl=1

Jijklγiγjγkγl

Relevant perturbation,resulting in weakly-coupled IR
fixed point(Δγ = 1/2 ≠ Δγ,SYK = 1/4)

λL(T) = ?
García-García, Loureiro, Romero-Bermúdez, Tezuka (PRL 18)
λL = 0, T < Tc [transition to no scrambling]
Banerjee, Altman (PRB 17, similar model)
2λL ∝ T [ ≪ T, but non-vanishing]

Can we have a more general understanding byinterpolating between IR fixed points?

Low-rank/Yukawa SYKJij,kl =ρ(λ) =∑nFermi Liquid

Non Fermi liquid

γN

∑
n=1

λnuij,nukl,nγ# of mediating bosons perfermion

δ(λ − λn)Fast scrambler

Fast scrambler
ex: graphene flake[Franz, etc., PRL 18]

λL≪ T, ∝ g∝T2πT[Kim, XC, Altman PRB 2020, preprint 2006.02485, Kim, XC, 2004.05313 ] [Phys. Rev. Lett. 120, 241603 (2017)]

Fast scramblers (class III and IV)
IIIIVρ ∝ δ(λ − λmax) + …

λmax < 0qSYK-like with tunable scaling dimension

G(τ) ∼ | τ |−2Δ, conformal invariance
Maximal Lyapunov exponent λL = 2πT (like SYK)Class IV: contains SUSY SYKClass III: applications to superconductivity

R/N (at the delta peak for class III)Class III: Esterlis and Schmalian PRB 2019,
Yuxuan Wang PRL 2019
Class IV: Zhen Bi et. al. PRB 2017

Fermi and non-Fermi liquids (class I & II)“free-fermion” leading scaling of Green function+ sub-leading self-energy Σ (“quasiparticle decay”)

Spectral density

Fermi liquid η > 0G(ω) ∼ sign(ω) , | Σ(ω) | ∼ | ω |

1+η

≪ |ω|

ωNon- Fermi liquid η < 0G(ω) ∼ sign(ω) , | Σ(ω) | ∼ | ω |

1+η

≫ |ω|

ω

λL(OTOC) is determined by ladder diagrams
generated by stacking kernels∫

K(t1,…,4)F(t1, t2) = F(t3, t4) , F(t1, t2) = f(t2 − t1)eλL(t1+t2)/2i.e., λL solved by requiring the largest eigenvalue be 1.
[Kitaev]

The ladder kernel in class I & IIClass III, IV: Conformal solution gives λL = 2πT after direct calculation (like SYK4)Class I, II: Perturbation theory in the coupling γ∼R/N appearing inλL = γ

Conformal (SYK2)

Kinetic (*)

F0: condensate generated SYK2 coupling

Interaction+decay(K̃: dimensionless kernel)Tη+1k(λ/2πT)L
2η+1F0

Most positive eigenvalue of K̃(*): coming from the kinetic term in G−1= ∂t − ΣλL = γFermi liquid (η > 0):

λL ∝ γTTη+1k(λ/2πT)L
2η+1F0η+1k(λL /2πT) ∼ γT

η+1k(0)η+1≪ T we can take k(0) to leading order in TBecause RHS ∼ Tη
λL ≪ T, λ ∝ γ (scrambling is significantly non-maximal, and perturbative in coupling constant)

Non-Fermi liquid (η < 0):
The LHS (kinetic term) is negligible compared to the interaction termk(λL /2πT) = 0 Determines λL

λL = CηT More universal. Independent of the coupling constantηScrambling and quasiparticle decay in lowrank SYK models

Biased opinion: OTOCs are good
diagnostics of the IR fixed point’s nature.

Back to mass-deformed SYK
∝κ

∝J2

2

2Similar to class I, with 1/τ ∼ T decay rate.

⇒ λL ∝ T

2To ensure a positive prefactor requires further
calculation, which will show
2 2

3T J
λL =,T,J≪κ
κ3

What about away from large N or classical limit?

OTOC ∼ e

λLt

Web of chaos diagnostic [Kudler-Flama, Nie, Ryu][w/ Parker, Scaﬃdi, Advoshkin, Altman, PRX 19] [to appear 2020]

C(t) = ⟨O(t)O(0)⟩T=∞

Φ(ω) = C(t)e dt∫iωtHypothesis à la Bohigas-Giannoni-Schmit For non integrable systems
and non-conserved operators O,

Φ(ω) ∼ exp( − | ω | /ω0) , ω → ∞ .

e|ω|−ω0

For integrable systems, Φ(ω) decay faster.
Theorem The above holds for chaotic Ising chain (ZZ+X+Z).

“Theorem” When λL is well-defined, it is bounded by (at T = ∞)λL ≤ ω0π

But, at low temperature, ω0 is too sensitive to UV details…
Technical note: there are sub-leading log corrections in 1d.

Does scrambling equal chaos?
I. In the classical limit: No
II. In the large-N, low-T limit: works as intended
III. Generic quantum: I don’t know

Thank you!

[H, On] = bnOn−1 + bn+1Qn+1{On}: Krylov basis

O(t) =∑i φn(t)On∑2n1G(z) =b12z−b22z−b32z−z − b42…

ω0π
α=2(n)t :=nnn | φn(t) |“Krylov-complexity”

OTOC ≤ C(n)t⇒ λL ≤ ω0π