Semiclassical Limit of a Measurement-Induced Transition in Many-Body Chaos in Integrable and Nonintegrable Oscillator Chains Sibaram Ruidas and Sumilan Banerjee

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Semiclassical Limit of a Measurement-Induced Transition in Many-Body Chaos in

Integrable and Nonintegrable Oscillator Chains

Sibaram Ruidas and Sumilan Banerjee

Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India∗

(˙Dated: January 23, 2024)

We discuss the dynamics of integrable and nonintegrable chains of coupled oscillators under con-

tinuous weak position measurements in the semiclassical limit. We show that, in this limit, the

dynamics is described by a standard stochastic Langevin equation, and a measurement-induced tran-

sition appears as a noise- and dissipation-induced chaotic-to-nonchaotic transition akin to stochastic

synchronization. In the nonintegrable chain of anharmonically coupled oscillators, we show that the

temporal growth and the ballistic light-cone spread of a classical out-of-time correlator characterized

by the Lyapunov exponent and the butterﬂy velocity, are halted above a noise or below an inter-

action strength. The Lyapunov exponent and the butterﬂy velocity both act like order parameter,

vanishing in the nonchaotic phase. In addition, the butterﬂy velocity exhibits a critical ﬁnite-size

scaling. For the integrable model, we consider the classical Toda chain and show that the Lyapunov

exponent changes nonmonotonically with the noise strength, vanishing at the zero noise limit and

above a critical noise, with a maximum at an intermediate noise strength. The butterﬂy velocity in

the Toda chain shows a singular behavior approaching the integrable limit of zero noise strength.

Chaotic-to-nonchaotic transitions play a prominent

role in our understanding of the dynamical phase dia-

gram of both quantum and classical systems, appear-

ing in many diﬀerent contexts such as non-linear dy-

namics, thermalization and quantum information the-

ory. In quantum many-body systems, a certain kind of

chaotic-nonchaotic transitions, dubbed as “measurement-

induced phase transitions" (MIPT)1–12 have led to a new

paradigm for dynamical phase transitions in recent years.

These transitions are characterized by entanglement and

chaotic properties of the many-body states and time evo-

lution. On the other hand, a prominent example of tran-

sition in chaos in classical dynamical systems13–25 are

the so-called synchronization transitions (ST)26. In this

case, classical trajectories starting from diﬀerent initial

conditions synchronize, i.e. the diﬀerence between the

trajectories approaches zero with time, when subjected

to suﬃciently strong common drive, bias, or even random

stochastic noise. Can there be some connection between

the measurement-induced phase transition in quantum

systems and ST in classical systems?

In this Letter, we establish a possible link between

MIPT and ST by considering models of interacting par-

ticles, whose positions are measured continuously, al-

beit weakly. We show that, in the semiclassical limit,

the dynamics of the system is described by a stochas-

tic Langevin equation where the noise and the dissipa-

tion terms are both controlled by the small quantum pa-

rameter ℏand measurement strength. Speciﬁcally, we

study a nonintegrable oscillator chain, and the classi-

cal integrable Toda chain27–30. In both cases, we ﬁnd

a surprising dynamical transition in many-body chaos

in the Langevin evolution. The chaotic-to-nonchaotic

transition occurs as a function of either interaction or

noise (measurement) strength as two classical trajectories

starting with slightly diﬀerent initial conditions synchro-

nize when subjected to identical noise. The transition is

similar to the stochastic STs. The latter has been previ-

ously studied19–25,31,32, however, only for lattices of cou-

pled non-linear maps, as opposed to interacting Hamil-

tonian systems employed in our Letter.

A few recent works have looked into classical analogs

of MIPT in cellular automaton33, kinetically constrain

spin systems34,35, and semiclassical circuit model36. In

contrast to these works, we derive a direct connection

between a quantum measurement dynamics of a Hamil-

tonian system with the Langevin evolution by analyti-

cally taking the semiclassical limit. For interacting quan-

tum systems, MIPTs have been primarily studied in

quantum circuits evolving under discrete-time projective

and weak measurements1–12 as well as continuous weak

measurements37,38. The MIPTs in these models are typ-

ically characterized by scaling of entanglement entropy

with subsystem size, i.e. transition from a volume-law

to area-law scaling, in the longtime steady state. These

MIPTs can often be mapped to phase transitions in some

classical statistical mechanics models1,2,6,8,10,12. How-

ever, it is hard to directly take the classical limit of the

dynamics in these quantum circuit models. Eﬀects of

measurements and MIPTs have also been studied for non-

interacting fermions7,39,40, interacting bosons41–43, and

Luttinger liquid ground state44.

We characterize the chaos transition in the Langevin

dynamics of the oscillator chains by a classical out-of-

time-order correlator (cOTOC)45–51. The latter is de-

ﬁned by comparing two trajectories that diﬀer by a small

amount initially and are subjected to identical noise. In

the chaotic phase, we extract a Lyapunov exponent λL

and a butterﬂy velocity vB, respectively, from the cO-

TOC. We show the following. (i) λL, vB→0above a

critical noise strength or below an interaction strength

for both integrable and nonintegrable chains. (ii) vB

exhibits a critical scaling with system size, whereas λL

shows almost no system-size dependence. The criti-

cal exponents extracted from the ﬁnite-size scaling of

vBdiﬀers from those in the universality classes typi-

arXiv:2210.03760v2 [cond-mat.stat-mech] 20 Jan 2024

cally found in stochastic STs in coupled-map lattices

(CMLs)19,20,23,25,31,32. (iii) For the stochastic dynamics

of the integrable Toda chain, λLchanges nonmonotoni-

cally with the noise strength, vanishing for zero noise, as

well as above a critical noise; vB, on the other hand,

shows a singular behavior approaching the integrable

limit of zero noise strength.

(a)

(c)

(b)

FIG. 1. Measurement model and cOTOC. (a) Schematic of

the measurement model, where the positions of the coupled

oscillators (i= 1,...,L) on a chain are weakly measured at

time tnby meters prepared in Gaussian states just before the

measurements. (b) Schematic of two initially nearby classi-

cal trajectories, Aand B, subjected to identical noise real-

izations. (c) The classical OTOC D(i= 0, t)as a function

of uacross the chaos transition for γ= 0.10 with uvalues

0.80 (darkest), 0.60, 0.50, 0.40, 0.35, 0.32 and 0.30 (lightest).

As shown by dashed magenta lines, the cOTOC grows ex-

ponentially (∼e2λLt) for u>uc≃0.32, whereas it decays

exponentially for u < ucin the synchronized phase.

Quantum measurement model and the semiclassical

limit.—We generalize the well-known model of contin-

uous weak position measurement of a single particle by

Caves, and Milburn52 to the interacting oscillator chains.

The oscillator chain (system) with i= 1, . . . , L oscillators

and the measurement apparatus (meters) [Fig. 1(a)] are

described by the following time-dependent Hamiltonian:

H(t) = Hs+X

i,n

δ(t−tn)ˆxiˆpin (1)

The Hamiltonian of the system is Hs=Pi(ˆp2

i/2m) +

V({ˆxi}), where ˆxi,ˆpiare the operators for displacement

of the i-th oscillator from the equilibrium position and

its momentum. We apply periodic boundary conditions.

The potential is V({xi}) = Piv(ri)with ri=xi+1 −xi.

We take (i) v(r) = [(κ/2)r2+ (u/4)r4]for the noninte-

grable chain with κspring constant and uthe strength

of the anharmonicity, and (ii) v(r) = [(a/b) exp (−br) +

ar −(a/b)] for the integrable Toda chain27–30,53 with pa-

rameters aand b. The displacement xiof the i-th os-

cillator is weakly measured by the in-th meter at time

t=tn=nτ at regular intervals of τ.ˆpin is the momen-

tum operator of the in-th meter, which is in a Gaussian

state ψ(ξin) = (πσ)−1/4exp (−ξ2

in/2σ)at t−

n. At tn, the

position ξin of the meter is projectively measured, and

its state collapses to a position state |ξin⟩. The eﬀect of

this measurement on the system is described by an oper-

ator Ψi(ξin)=(πσ)−1/4exp [−(ξin −ˆxi)2/2σ]acting on

the system, as described in detail in the Supplemental

Material (SM), Sec.S1. In the continuous measurement

limit τ→0, σ → ∞ such that ∆ = στ is kept ﬁxed52.

The mean momentum and position of the particle jump

by an amount ∝ξin after each measurement52, and they

can wander far away from the initial values at long times.

Thus, to incorporate a feedback mechanism present in

any realistic measurement setup52 a displacement opera-

tor, Di(ξin) = exp [(i/ℏ)γτ ξin ˆpi]is applied on the system

after the in-th measurement, where γ=cγp2ℏ/m∆,

with dimensionless coeﬃcient cγ. We do not need to ap-

ply a displacement operator for the position due to the

periodic boundary condition. The feedback mechanism

on the momentum leads to dissipation52, as discussed

below.

The density matrix of the system at t+

nis given by

ρ({ξ}n, t+

n) = M(ξn)ρ({ξ}n−1, t+

n−1)M†(ξn), which de-

pends on the outcomes of all the measurements {ξ}ntill

n. Here M(ξn) = Qi[Di(ξin)Ψi(ξin)] exp (−iHsτ/ℏ).

For the evolution of an initial pure state, the above

time evolution can be written as a quantum state

diﬀusion54,55. Here we write the longtime evolution as

a Schwinger-Keldysh (SK) path integral56 for τ→0, i.e.

Tr[ρ({ξ(t)})] = ´Dxexp (iS[{ξ(t)}, x(t)]/ℏ)with the ac-

tion,

S[{ξ}, x] = ˆ∞

−∞

dt X

s=±

s[{X

2( ˙xs

i)2+mγ ˙xs

iξi

+ (isℏ/2∆)(xs

i−ξi)2} − V({xs

i})],(2)

where s=±denotes two branches of the SK contour56,

˙xi= (dxs

i/dt). To take the semiclassical limit of small ℏ,

we rewrite the above path integral in terms of classical

(xc

i) and quantum components (xq

i), i.e. x±

i=xc

i±xq

To capture nontrivial eﬀects of the quantum (xq

i) ﬂuctu-

ations, which act as noise in the semiclassical limit, we

need to scale ∆∼ℏ2(SM, Sec.S1). Taking the semi-

classical limit in this manner and keeping O(1/√ℏ)and

O(1) terms, we ﬁnd that a Langevin equation describes

the dynamics of the system,

¨xc

i+γ˙xc

i=1

m−∂V ({xc

i})

∂xc

+ηi,(3)

for the classical component xc

i, denoted by xihence-

forth. Here the ηi(t)is Gaussian random noise that

originates from xq

iand is controlled by the measurement

strength ∆−1such that ⟨ηi(t)ηj(t′)⟩= 2mγTeﬀ δij δ(t−t′).

Teﬀ = (ℏ/4cγ)pℏ/2m∆, which we denote as Tin the

rest of the Letter for brevity, is an eﬀective temper-

ature ∼√ℏthat determines the noise strength along

with γ. The latter is the eﬀective dissipation strength

∼1/√ℏ. In the strict classical limit ℏ→0,T→0and

γ→ ∞. As a result, the dissipative term completely

dominates, and the system becomes static. The nontriv-

ial semiclassical dynamics results from keeping ℏsmall

but nonzero. In this limit, the system reaches a long-

time steady state described by classical Boltzmann-Gibbs

distribution ∼exp [−Hs({xi, pi})/T ]determined by the

eﬀective temperature. However, the temperature here

does not arise from any external baths but solely from

the measurement and feedback process.

Classical dynamics and cOTOC.—We study the dy-

namics [Eq.(3)] of the nonintegrable chain as a function

of both γand ufor a ﬁxed T. The Hamiltonian is triv-

ially integrable and nonchaotic for the harmonic chain

(u= 0). Any nonzero umakes the model nonintegrable

and chaotic. On the contrary, the classical Toda chain

is integrable, albeit interacting27–30. We can tune the

model from a harmonic limit to a hard sphere limit by

changing aand b53. We take the parameters in the in-

termediate regime for the convenience of the numerical

simulations.

We numerically simulate Eq.(3) and generate clas-

sical trajectories for the nonintegrable and integrable

chains using the Gunsteren-Berendsen method57; see SM,

Sec.S2 for details. We characterize many-body chaos by

the following cOTOC,

D(i, t) = ⟨[pA

i(t)−pB

i(t)]2⟩.(4)

Here Aand Bare two trajectories of the system gen-

erated from initial thermal equilibrium conﬁgurations

{xA

i(0), pA

i(0)}for T= 1 with pB

i(0) = pA

i(0) + δi,0ε

(ε= 10−4); ⟨···⟩ denotes average over thermal initial

conﬁgurations (see SM, Sec.S3 for details). We use 105

initial conﬁgurations for all our results. We use identi-

cal noise realization for the two copies at each instant

of time, i.e., ηA

i(t) = ηB

i(t), as in the earlier studies of

stochastic STs and CMLs19–25,31,32.

Results.—For γ= 0, the nonintegrable chain is chaotic,

i.e. the cOTOC grows exponentially for any value of u,

except the harmonic limit u= 0 (Fig.S2 of SM). However,

for γ̸= 0, the system is chaotic only above a critical value

ucof the interaction, as shown in Fig. 1(c). The cOTOC

decays exponentially (λL<0) for u<uc, instead of

growing. This is the stochastic ST. Similar transition is

seen as a function of noise strength γfor a ﬁxed u̸= 0

(Fig.S1 of SM). The exponential growth is concomitant

with a ballistic light cone in cOTOC, whereas the light

cone is destroyed in the nonchaotic phase, as shown in

Figs. 2(a) and 2(b).

(a)

(b)

−512 −256 0 256 512

100

200

300

UC FC

−512 −256 0 256 512

200

400

600

800 (c)

(d)

FIG. 2. Ballistic light-cone spreading and Lyapunov exponent

across the chaos transition. (a) Light cone for u= 0.80 > uc

and γ= 0.10. The color represents the value of the cOTOC

D(i, t)as function of lattice site iand time t, as indicated in

the color bar from small D(i, t)or fully correlated (FC) to

large D(i, t)or uncorrelated (UC). (b) The light-cone spread-

ing ceases in the nonchaotic phase for u= 0.30 < ucand

γ= 0.10. (c) λLas a function of ufor diﬀerent γs and sys-

tem sizes L.λLapproaches zero at the critical interactions

uc= 0.14,0.25 and 0.32 (dashed lines) for γ= 0.05,0.08 and

0.10, respectively, for the chaos transition. The shaded region

marks λL<0. (d) Similar transition is observed as a func-

tion of noise strength γat γc≃0.20 for u= 1, as shown for

diﬀerent L’s.

We extract the Lyapunov exponent λLfrom D(0, t)∼

exp (2λLt). The results for λLas a function of ufor

a few γ, and as a function of γfor u= 1, are shown

in Figs. 2(c) and 2(d), for diﬀerent system sizes L=

256,512,800,1024. It is evident that λLapproaches zero

at a critical value ucor γc, becoming negative for u<uc

(γ > γc), and λLhas little Ldependence. Hence the

semiclassical limit [Eq.(3)] of the quantum measurement

dynamics described by the action in Eq.(2) indeed yields

an ST as a function of the interaction and γ. Since

γ∼1/√∆, ST is controlled by the measurement strength

∆−1, which determines how precisely the oscillator posi-

tions are measured. As a result, based on its microscopic

origin from a quantum measurement model [Eq.(1)], the

ST in this case can be termed as an MIPT, albeit in the

semiclassical limit. The transition appears to be contin-

uous, though it is hard to extract λLaccurately close to

the transition.

The ballistic spreading of cOTOC is quantiﬁed by ex-

tracting butterﬂy velocity vB, e.g., from the light cones

of Fig.2(a) (see SM, Sec.S4 for details). vBdecreases

approaching the transition from the chaotic phase, as

shown in Figs. 3(a) and 3(c), as a function of uand

γ, respectively. However, close to the transition, the

light cone becomes progressively ill deﬁned and we could

not extract vBall the way up to the transition. Un-

like λL,vBshows perceptible and systematic Ldepen-

dence [Fig.3(b)(inset)], especially for the transition as

(a)

(b)

(c)

FIG. 3. Butterﬂy velocity and ﬁnite-size scaling in the non-

integrable chain. (a) vBas a function of ufor diﬀerent noise

strength γ. (b) The system size (L) dependence of vB(u)is

shown for γ= 0.08 in the inset, and the ﬁnite-size scaling col-

lapse is shown in the main panel with exponents ∆v= 1.04

and ν= 0.27 for uc= 0.24. (c) vBas a function of γfor

u= 1.0for diﬀerent Ls.

function of u. Thus we perform a ﬁnite-size scaling

analysis of the data for γ= 0.08, where we collapse

the data for diﬀerent Land δu = (u−uc)>0using

vB(u, L) = L−∆vF((δu)L1/ν ). Here F(x)is a scaling

function (SM, Sec.S5). Reasonably, good scaling collapse

is obtained with ∆v≃1.03±0.03 and ν≃0.30±0.05, for

the range uc= 0.21–0.25, which is close to the uc≃0.25

obtained from λLin Fig. 2(c). The scaling form implies

that for L→ ∞,vB∼(δu)βwith β=ν∆v≃0.28,

and a correlation length ξdiverges as (δu)−νin the

chaotic phase. The correlation length exponent ν≃0.3

is diﬀerent from that for the usual universality classes of

STs, such as multiplicative noise or directed percolation,

found in earlier studies in CMLs19,20,23,25,31,32, cellular

automaton33 and kinetically constrained model34,35. We

note that exponents diﬀerent from the known universal-

ity classes have been found for some cases in previous

works on CMLs as well20.

The dynamical transition in the stochastic evolution of

a nonintegrable oscillator chain is not seen in the usual

dynamical properties of a single trajectory. It can only

be detected through many-body chaos by comparing two

trajectories. To conﬁrm this, we compute the average

mean-square displacement (MSD) for the trajectories,

i.e. ⟨∆q2(t)⟩= (1/N)Pi⟨[xi(t)−xi(0)]2⟩(SM, Sec.S6).

For γ= 0, in the harmonic chain (u= 0) with periodic

boundary condition, ⟨∆q2(t)⟩ ∼ texhibits a diﬀusive be-

havior as shown in ref.58. The diﬀusive behavior persists

for u̸= 0 and γ= 0. However, turning on γ̸= 0, dy-

namics becomes subdiﬀusive with ⟨∆q2(t)⟩ ∼ √teven for

u= 0. This is well understood in the context of monomer

subdiﬀusion in polymers59. Again, for u̸= 0 this subd-

iﬀusive behavior remains without any change across the

ST seen via many-body chaos. We expect the quantum

model [Eq.(1)] to exhibit diﬀusion in the absence of mea-

surements. It will be interesting to explore the subdif-

fusive behavior in the presence of measurements in the

quantum limit and the connection between diﬀusion or

subdiﬀusion with entanglement growth60–62.

We now characterize the many-body chaos in the in-

tegrable classical Toda chain. The results for λLand

vBas a function of γfor the Toda chain with a= 0.07

and b= 15.0are shown in Figs. 4(a) and 4(b) (see SM,

Sec. S4 for more details). As expected, the integrable

limit with γ= 0 does not show any exponential growth,

implying λL= 0. However, the cOTOC still exhibits

ballistic spreading in this limit (Fig. S7 of SM), yielding

a nonzero vBas shown in Fig. 4(b).

As soon as γbecomes nonzero, the dynamics be-

comes chaotic with both exponential growth and ballistic

spreading of cOTOC. As shown in Figs. 4(a) and 4(b),

the extracted λLincreases63,64 rapidly as γ0.3and vB

exhibits a jump near the integrable limit with increas-

ing γ. Thus, the integrable limit appears singular with

respect to vBfor γ→0+. Further increasing γ,vB

monotonically decreases and approaches zero at a criti-

cal γ=γc, indicating a transition to a nonchaotic phase.

In contrast, λLshows a nonmonotonic dependence on γ,

with a maximum at an intermediate γ. Nevertheless, λL

eventually vanishes at γc, becoming negative for γ > γc,

as in the nonintegrable model. Thus, the noise or dis-

sipation, though initially making the integrable model

chaotic, eventually destroys chaos due to the stochastic

synchronization. The fact that λL= 0 for γ= 0 and

γ > γc, and λL>0for small γdue to the breaking of

intigribility63,64, dictates that λL(γ)is nonmonotonic.

(a) (b)

FIG. 4. Transitions in many-body chaos in Toda model. (a)

Lyapunov exponent λLand (b) butterﬂy velocity vBas func-

tion of noise strength γfor diﬀerent system sizes. For γ→0,

λL∼γ0.26 as shown by the dashed line in (a). The shaded

region in (a) corresponds to λL<0. The chaos transition

occurs around γc≃3.30 (dashed line) for both λLand vB.

Discussions.–In summary, we show that eﬀective dy-

namics of the position and momentum of the quantum

oscillators under continuous weak position measurements

maps to standard stochastic Langevin evolution in the

semiclassical limit of small but nonzero ℏ. The Langevin

dynamics for interacting chains, remarkably, exhibit ST

in many-body chaos as a function of noise strength. The

latter is controlled by the measurement strength in the

parent quantum measurement model, implying that the

ST is an MIPT in the semiclassical limit.

The use of stochastic Langevin dynamics might sug-

gest the absence of entanglement in the semicalssical

limit. However, this naive inference is not correct. The

semicalssical limit of entanglement needs to be taken

carefully65–67, where one ﬁrst obtains an SK path integral

for entanglement entropy, e.g., second Rényi entropy68,

in the quantum model and then takes the semiclassical

limit. This results in eﬀective dynamical equations for

entanglement69 diﬀerent from Eq.(3). The latter only de-

scribes the eﬀective dynamics of positions and momenta,

as typically done in semiclassical approximations70, and

is useful for capturing OTOC (4) in the semiclassical

limit. However, the the connection between the growth

of OTOC and entanglement has been shown in vari-

ous situations71–73. Hence the ST transition in OTOC

suggests an entanglement transition in the semiclassical

limit.

The study of the MIPT in the fully quantum limit of

our model will be an interesting future direction since

systems of interacting oscillators under continuous mea-

surements mimic many realistic open quantum systems.

MIPTs have already been shown to exist for continu-

ous weak measurements within more tractable quantum

dynamics, like for quantum circuits37,38, and noninter-

acting fermionic systems7. Moreover, there are several

works41–43 on the Bose-Hubbard model that show MIPT

in the presence of measurements. The oscillator model

in our work can be easily mapped to an interacting bo-

son model. Thus, quite generally, MIPT as a function of

measurement strength is expected to occur in our models

even in the fully quantum limit.

We acknowledge useful suggestions and comments by

Sriram Ramaswamy and Sitabhra Sinha, and discus-

sions with Subhro Bhattacharjee, Sthitadhi Roy, and Sri-

ram Ganeshan. S.B. acknowledges support from SERB

(Grant No CRG/2022/001062), DST, India and QuST,

DST, India.

∗sibaramr@iisc.ac.in;sumilan@iisc.ac.in

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SemiclassicalLimitofaMeasurement-InducedTransitioninMany-BodyChaosinIntegrableandNonintegrableOscillatorChainsSibaramRuidasandSumilanBanerjeeCentreforCondensedMatterTheory,DepartmentofPhysics,IndianInstituteofScience,Bangalore560012,India∗(˙Dated:January23,2024)Wediscussthedynamicsofintegrableandnon...

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Semiclassical Limit of a Measurement-Induced Transition in Many-Body Chaos in Integrable and Nonintegrable Oscillator Chains Sibaram Ruidas and Sumilan Banerjee

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