The Multi-faceted Inverted Harmonic Oscillator:Chaos and Complexity

Arpan Bhattacharyya,1, ∗ Wissam Chemissany,2, † S. Shajidul Haque,3, ‡ Jeff Murugan,3, § and Bin Yan4, 5, ¶

1Indian Institute of Technology,Gandhinagar,Gujarat 382355, India2Institute for Quantum Information and Matter, California Institute of Technology,

1200 E California Blvd, Pasadena, CA 91125, USA.3Department of Mathematics and Applied Mathematics,

University of Cape Town, Private Bag, Rondebosch, 7701, South Africa.4Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87544, USA

5Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544, USA

The harmonic oscillator is the paragon of physical models; conceptually and computationallysimple, yet rich enough to teach us about physics on scales that span classical mechanics to quantumfield theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillatorfrequency is analytically continued to pure imaginary values. In this article we probe the invertedharmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for thedisplacement operator of the IHO with and without a non-Gaussian cubic perturbation to exploregenuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunovspectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents.We also use the Heisenberg group to compute the complexity for the time evolved displacementoperator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonicoscillators to study the behaviour of complexity at the different timescales encoded in dissipation,scrambling and asymptotic regimes.

CONTENTS

I. Introduction 1

II. The IHO Model 3

III. Out-of-Time Order Correlator 3A. OTOC for the Displacement Operator 4B. Quantum Lyapunov Spectrum 5

IV. Complexity for Inverted Harmonic Oscillator 5A. Complexity for the Displacement Operator 6B. Complexity for N-Oscillators and Scrambling 7

V. Discussion 9

Note added in proof 10

Acknowledgements 10

References 10

I. INTRODUCTION

One would be hard-pressed to find a physical systemthat we have collectively learnt more from than the har-

∗ abhattacharyya@iitgn.ac.in† wissamch@caltech.edu‡ shajidhaque@gmail.com§ jeff.murugan@uct.ac.za¶ byan@lanl.gov

monic oscillator. Indeed, from the simple pendulum ofclassical mechanics to mode expansions in quantum fieldtheory, there is no more versatile laboratory than theharmonic oscillator (and its many variants). This is duein no small part to two central properties of harmonicoscillator systems; they are mathematically and physicallyrich and simultaneously remarkably simple. It is also theuniversal physical response in perturbation theory.

This utility has again come into sharp relief in twoseemingly disparate contexts; quantum chaos and theemerging science of quantum complexity. While neithersubject is particularly new, both have seen some remark-able recent developments of late. To see why, note thatconservative Hamiltonian systems come in one of twotypes, they are either integrable or non-integrable. Thelatter in turn can be classified as either completely chaoticor mixed (between chaotic, quasiperiodic or periodic), de-pending on whether the defining Hamiltonian is smoothor not [1, 2]. By far, most non-integrable classical sys-tems are of the latter type. The former however includessome iconic Hamiltonian systems such as the Sinai billiardmodel, kicked rotor and, of particular interest to us inthis article, the inverted harmonic oscillator (IHO).

Classical chaotic systems are characterised by a hyper-sensitivity to perturbations in initial conditions under theHamiltonian evolution. This hypersensitivity is usuallydiagnosed by studying individual orbits in phase space.However, as a result of the Heisenberg uncertainty prin-ciple, the volume occupied by a single quantum state inthe classical phase space is ∼ ~N , for a system with Ndegrees of freedom, and we no longer have the luxury offollowing individual orbits. This necessitates the need fornew chaos diagnostics for quantum systems. One such

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diagnostic, discovered by Wigner in the 1950’s already,is encoded in the statistical properties of energy spectra;quantum chaotic Hamiltonians have eigenvalue spacingdistributions that are given by Gaussian random matrixensembles. Unfortunately though, a direct spectral anal-ysis of the Hamiltonian is computationally taxing forall but the simplest, or exceedingly special, systems. Ittherefore makes sense to develop other, complimentarydiagnostics that probe different aspects of quantum chaos,say, at different times or energy scales.

One such tool, originally considered in the contextof superconductivity, but rapidly gaining traction inthe high energy and condensed matter communities,is the out-of-time-order correlator [3–5], OTOC(t) ≡〈B†(0)A†(t)B(0)A(t)〉β , for Heisenberg operators A(t)and B(t), and where 〈O〉β = Tr

(e−βHO

)/Tr e−βH de-

notes a thermal average at temperature T = 1/β. Onereason for this popularity is its relation to the double com-mutator CT (t) = −〈[A(t), B(0)]2〉β , which is the quantumanalog of the classical expectation value,〈(

∂x(t)

∂x(0)

)2〉β

∼∑n

cne2λnt , (1)

for a chaotic system with Lyapunov exponents λn. In-deed, it will often be more convenient to work with thedouble commutator instead of the four-point functionOTOC(t) and since, for unitary operators, the two arerelated through CT (t) = 2 (1− Re (OTOC(t))), their in-formation content, and exponential growth, is the sameand are usually referred to interchangeably. In a chaoticmany-body system, CT (t) exhibits a characteristic ex-ponential growth from which the quantum Lyapunovexponent can be extracted. In this sense, the OTOC cap-tures the early-time scrambling behaviour of the quantumchaotic system.

However, like any new technology, the OTOC is notwithout its subtlties. Among these are;

• the fact that its reliability breaks down at late times,when the chaotic system starts to exhibit randommatrix behaviour,

• a related mismatch to its classical value, where thecommutator in the definition of CT (t) is replacedby the Poisson bracket, and

• no exponential growth for several single-particlequantum chaotic systems, such as the well-knownstadium billiards model, or chaotic lattice systems,such as spin chains.

All three of these points are related in some sense toEhrenfest saturation where quantum corrections are ofthe same order as classical leading terms1 point to theneed for a deeper understanding of the OTOC.

1 We would like to thank the anonymous referee for their insighton this, and an earlier point. and

On the other hand, it is becoming increasingly clearthat while no single diagnostic captures all the featuresof a quantum chaotic system, there is an emerging webof interconnected tools that offer complementary insightinto quantum chaos [6, 7]. There is, for example, the(annealed) spectral form factor (SFF),

g(t;β) ≡ 〈|Z(β, t)|2〉J

〈Z(β, 0)〉J, (2)

where Z(β, t) is the analytic continuation of the thermalpartition function and the average is taken over differentrealizations of the system. The SFF interpolates betweenthe essentially quantum mechanical OTOC and morestandard random matrix theory (RMT) measures makingit a particularly useful probe of systems transitioning be-tween integrable and chaotic behaviour where it displaysa characteristic dip-ramp-plateau shape [8, 9]. However,except in some special cases like bosonic quantum me-chanics where it can be shown that the two-point SFFis obtained by averaging the four-point OTOC over theHeisenberg group [10], computing the SFF is a difficulttask, compounded by various subtleties inherent to thespectral analysis of the chaotic Hamiltonian.

More recently, this diagnostic toolbox has been furtherexpanded with the introduction of a number of moreinformation-theoretic resources with varying degrees ofutility. One of these is the fidelity [11, 12] of a quantumsystem. Let U be a unitary map and |ψ(0)〉, some fiducialstate in the Hilbert space. Now evolve this initial statewith U to |ψ(n)〉 = Un|ψ(0)〉 and again, but with asequence of small perturbations by some non-specific field

perturbation operator, to |ψ̃(n)〉 =(e−iV δU

)n |ψ(0)〉. Itwas shown in [13] that the fidelity

F(n) = |〈ψ(n)|ψ̃(n)|2 , (3)for a classically chaotic quantum system, exhibits a char-acteristic, and efficiently computable, exponential decayunder a sufficiently strong perturbation. This quantityhas recently been shown [14–16] to be intrinsically relatedto the OTOC.

Closer to the focus of this article, another related tooldrawn from theoretical computer science is the notionof computational (or circuit) complexity [17], which, inthe lingo of computer science, measures the minimumnumber of operations required to implement a specifictask in the following sense: fix a reference state |ΨR〉and target state |ΨT〉 and construct a unitary U from aset of elementary gates by sequential operation on thereference state such that |ΨT〉 = U |ΨR〉. Then the com-plexity of |ΨT〉 is defined to be the minimal number ofgates required to implement the unitary transformationfrom reference to target states. Determining the compu-tational complexity is then essentially an optimisationproblem, one that was more or less solved by Nielsen in[18]. Nielsen’s geometrical approach proceeds by defininga cost functional

D[U(t)] ≡∫ 1

0

dt F(U(t), U̇(t)

)(4)

3

on the space of unitaries which is then optimised subjectto the boundary conditions U(0) = 11 and U(t) = U . Fora long time, the idea of circuit complexity was viewed asa curiosity of computer science, living on the peripheryof theoretical physics. This situation changed dramati-cally with the introduction, by Susskind and collaborators[19–21], of complexity as a probe of black hole physics.Further, the idea of complexity has extended to quantumfield theory in recent time [22–48]. Following this line ofreasoning, two disjoint subsets of the current authors con-jectured that not only does the computational complexityfurnish an equivalent chaos diagnostic to the OTOC [49–52], but in addition, a response matrix may be definedto characterize the fine structure of the complexity inresponse to initial perturbations, giving rise to the fullLyapunov spectrum in the classical limit [53]. Table Isummarises the findings of these studies and comparesthe time development of the complexity to that of theOTOC. It is worth emphasizing that the universal be-haviors summarized in Table I are for generic operatorsand complex chaotic systems; one can always cook upspecial scenarios which are not described by these genericforms. For instance, as shown in the following sections,the OTOC of the IHO for the displacement operatorsdoes not decay at all, whereas the OTOC for canonicalvariables, i.e., x and p, the OTOC exhibits intermediateexponential decay. The early scrambling regime of theOTOC will not be discussed in the work, it has beendemonstrated for coupled IHOs in Ref. [14]. We alsostressed that the presence of the early scrambling is lim-ited, i.e., models exhibiting this regime usually show alarge hierarchy between scrambling and local dissipationtime scales [5]. Many chaotic systems, e.g., spin chains[54], only manifest a pure exponential intermediate decay.On the other hand, our study of the complexity revealsboth the early and intermediate regimes, as well as thelocal dissipation before the scrambling time scale (Fig. 2).

Early scrambling Intermediate regime

OTOC 1− �eλt ∼ exp(−�eλt) e−ΓtComplexity �eλt Γt

TABLE I. Universal correspondence between OTOC and com-plexity, complexity ∼ − log(OTOC). This relation holds atboth the early scrambling and intermediate decay regime.

In both cases, the system chosen to exhibit this rela-tionship between the complexity and quantum chaos wasarguably among simplest conceivable; the inverted har-monic oscillator. The inverted harmonic oscillator doesnot resemble typical large N chaotic systems in that thedecay of the OTOC, or the growth in the complexity,here captures an instability of the system, rather thanchaos.Nevertheless, it remains a useful toy model withwhich to study the various chaos diagnostics. The presentarticle builds on these ideas by returning to the invertedoscillator, developing the treatment of the OTOC as wellas the computational complexity of particular states inthe model and then connecting them. In addition to

its pedagogical value the oscillator again provides a richand intuitively clear example within which to understandfurther the OTOC and computational complexity. It istherefore fitting that we begin with a brief overview ofthe inverted harmonic oscillator.

II. THE IHO MODEL

We start with the harmonic oscillator Hamiltonian

H =p2

2m+mω2

2x2, (5)

where p ≡ −i~ ddx is the momentum operator. We willwork in natural units in which ~ = 1 and, without anyloss of generality, assume that the mass of the oscillatorm = 1. By choosing the value of the frequency ω, threedifferent cases can be obtained:

ω =

Ω harmonic oscillator,

0 free particle,

iΩ inverted harmonic oscillator.

Here Ω is a positive real number. In this work, we will bemainly concerned with the Hamiltonian of the invertedharmonic oscillator.

An important point to note is that the regular andinverted harmonic oscillators are genuinely different. Asa result, one cannot take for granted that formulae knownfor the regular oscillator and extrapolate them to theinverted oscillators by simply replacing Ω with iΩ. Forinstance, the regular harmonic oscillator has a discreteenergy spectrum (n+ 1/2)Ω, while the spectrum for aninverted oscillator is a continuum. However, in some othercases, such as the Heisenberg evolution for the position ormomentum operator, the derivation follows in much thesame way for both the regular and inverted oscillators.In such cases, we will explicitly point this out and usethe variable ω to cover both classes of oscillator. It willbe useful in what follows, to define the annihilation andcreation operators [55],

a±ω =1√2

(∓ip+ ωx) . (6)

III. OUT-OF-TIME ORDER CORRELATOR

We will consider the OTOC for the displacement opera-tors in the IHO, which are defined in terms of the creationand annihilation operators as

D (α) = exp(αa† − α∗a

). (7)

This is a well known operator whose phase space OTOChas been the subject of recent study for continuousvariable (CV) systems [56]. There the authors argued

4

that, for a Gaussian-CV system, the OTOC does notdisplay any genuine scrambling2. In this section, wewould like to explicitly check this claim for the IHO.Since the Hamiltonian of the IHO belongs to the categoryof Gaussian-CV systems, we expect the OTOC for thedisplacement operators to display such quasi-scrambling.More explicitly, the OTOC only changes by an overallphase, while the magnitude remains constant. We followthis by adding a cubic-gate perturbation to the oscillatorpotential and explore the OTOC analytically. In contrastto its Gaussian counterpart, this model does indeeddisplay generic scrambling behavior.

In the second part of this section, we will investigate an-other important feature of chaos; the Lyapunov spectrum.Our goal will be to use the IHO to check whether thequantum Lyapunov exponents exhibit a similar pairingstructure to the classical case.

A. OTOC for the Displacement Operator

To warmup, we will evaluate the OTOC for the dis-placement operator for the regular harmonic oscillatorwith real frequency ω = Ω. Our derivation is esentiallyindependent of the choice of the frequency ω. Therefore,by choosing appropriate values of the frequency, we deriveexpressions for both the regular and inverted harmonic os-cillator. The displacement operator in Eq.(7) for a singlemode harmonic oscillator can be written as

D(α) = exp[i√

2 (Im(α)ωx− Re(α)p)]. (8)

To evaluate the OTOC, we need to find the time evolutionof the displacement operator (8), i.e.,

D(α, t) = eiHtD(α, t = 0) e−iHt, (9)

which can be evaluated by implementing the Hadamardlemma. Some straightforward algebra puts this into theform,

D(α, t) = exp[i√

2 (Im(α) cos(ωt) + Re(α) sin(ωt)ω)x

+ i√

2 (Im(α) sin(ωt)/ω − Re(α) cos(ωt)) p].

(10)

The corresponding OTOC function, C2(α, β; t)ρ is definedas

C2(α, β; t)ρ ≡ 〈D†(α, t)D†(β)D(α, t)D(β)〉= Tr[ρD†(α, t)D†(β)D(α, t)D(β)],

(11)

2 An initial operator will be said to be genuinely scrambling (non-Gaussian) when it is localized in phase space and spreads outunder time evolution of the system. A local ensemble of operatorsis said to be quasi scrambling (Gaussian) when it distorts butthe overall volume of the phase space remains fixed.

where ρ is a given state of the harmonic oscillator. Byusing (10), the OTOC (11) simplifies to the followingform

C2(α, β; t)ρ = exp(iθ), (12)

where

θ = 2ωRe(αβ∗) sin(ωt) + 2Im(αβ∗) cos(ωt). (13)

We can immediately see that θ is real-valued for bothω = Ω and ω = iΩ and we conclude that the magnitudeof the OTOC (12) does not decay in time for either theregular or inverted harmonic oscillators. This implies thatthe harmonic oscillator potential is quasi-scrambling, inagreement with the general conclusion for the Gaussiandynamics found in [56]. We can extend the IHO Hamilto-nian to a simple non-Gaussian one by adding a so-calledcubic-gate as follows:

H =p2

2m+mω2

2x2 + γ

x3

3!

To display more clearly the role of the different contri-butions in the Hamiltonian to the OTOC we rewrite theHamiltonain in the following form

H = kp2 + lx2 + Jx3,

where k = 12m , l =mω2

2 andγ3! = J . As in the previous

Gaussian case, we first find the time evolution of thedisplacement operator (8) for this cubic model. It has thefollowing form

D(α, t) = exp[i(A0 +A1p+A2p

2 +A3x+A4x2

+O(x3, p3, px, t3))] (14)

where Ai’s are functions of k, l , J and t. From this ex-pression, the OTOC can be computed exactly as

C2(α, β; t)ρ = exp (iθ(k, l, J)) χ(12iJtRe(α)Re(β), ρ),(15)

where χ(12iJtRe(α)Re(β), ρ) is the characteristic func-tion [57], which typically decays in time. To illustrate this,consider a thermal state ρnth for which the characteristicfunction has the form

χ(ξ, ρnth) = exp

[−(nth +

1

2)|ξ|2

]. (16)

Using this, the OTOC reads

C2(α, β; t)ρnth = exp(iθ) χ(2iγtRe(α)Re(β), ρnth)

= exp(iθ) exp[−2(2nth + 1)

(Re(ξ)2 + Im(ξ)2

)], (17)

where

Re(ξ) = −6kJRe(α)Re(β)t2,Im(ξ) = −6JRe(α)Re(β)t− 6JkIm(αβ∗)t2.

(18)

5

We can immediately see that there will be an exponentialdecay when the cubic term is added to the Hamiltonian.This essential role of the cubic term is clear from Eq. (17)and Eq. (18). More precisely, the overall minus signof the exponent in the amplitude of eq. (17) signals aGaussian decay in time. When J = 0 (and k 6= 0), theentire exponent vanishes, and the amplitude becomesunity. This is true for any value of k. On the otherhand, when J 6= 0 and k = 0 we still get exponentialdecay. This is the case discussed in [56]. Finally whenboth J 6= 0, k 6= 0, we get contributions from both ofthem, namely from the x3 and p2 terms. However, thecontribution coming from the p2 will never cancel thecontribution coming from the cubic term. We have alsochecked that even if we add the higher order Ai’s, ourconclusion, namely the decaying of the OTOC, remainsintact. It is worth emphasising that our analysis, namelythe structure and the behaviour of the derived equations,are independent of the fact whether the oscillator invertedor regular.

B. Quantum Lyapunov Spectrum

A defining property of a classical chaotic system is itshyper-sensitivity to initial conditions in the phase space.This manifests in the exponential divergence of the dis-tance between two initially nearby trajectories. The rateof this divergence is encoded in the so-called Lyapunov ex-ponent. Technically, this is only the largest of a sequenceof such exponents that constitute the Lyapunov spectrum.For a 2n-dimensional phase space, the Lyapunov spec-trum consists of 2n characteristic numbers captured bythe eigenvalues of the Jacobian matrix

Mij(t) ≡∂zi(t)

∂zj, (19)

where the zi are phase space coordinates. The eigenval-ues si(t) of the Jacobian matrix evolve exponentially intime and the Lyapunov spectrum can is extracted in theasymptotic limit,

λi ≡ limt→∞

1

tln si(t). (20)

If the initial perturbation in the phase space is appliedto the “eigen-direction” with respect to one eigenvaluein the Lyapunov spectrum, the trajectories diverge witha corresponding exponential rate. For generic pertur-bations which involve all exponents in the Lyapunovspectrum, in the asymptotic limit, the exponentialwith the largest exponent will eventually dominate. Inthis case only the maximum Lyapunov exponent is visible.

It is worth emphasizing that the Lyapunov spectrum istypically computed from the eigenvalues of the matrix

L(t) ≡M(t)†M(t). (21)

Due to the intrinsic symplectic structure of the classicalphase space, the M -matrix is symplectic and, hence, theLyapunov exponents always come in pairs with oppositesigns.

Now let’s think about quantum systems. In form ofthe commutator square, the OTOC typically grows expo-nentially, with a rate analogous to the classical Lyapunovexponent, i.e., for generic operators,

〈[W (t), V ]2〉 ∼ �eλt, (22)

up-to the time scale known as the scrambling (or Ehren-fest) time [5]. For systems with a classical counterpart,e.g., quantum kicked rotor, the growth rate of the OTOCindeed matches the maximum classical Lyapunov expo-nent. A natural question to ask is if it possible to fine-tunethe operators in the OTOC and extract a full spectrumof Lyapunov exponents, instead of only the leading one?Some recent attempts [58, 59] to tackle this problem gen-eralized the Jacobian matrix to quantum systems usingthe OTOCs:

Mij(t) ≡ i[zi(t), zj ]. (23)

Here zi ranges over canonical variables, and zi(t) is theHeisenberg evolution. In contrast to the classical case,the quantum instability matrix (23) lacks a symplecticstructure. Known examples such as spin chains and thefinite size SY K-model show that the quantum Lyaponovexponents do not come in pairs [58]. However, thesemodels lack any well-behaved exponential growth in thefirst place.

For the IHO, the Heisenberg evolution of the canonicalpair of variables {x, p} can be computed exactly,

x(t) = x(0) cosh Ωt+1

mΩp(0) sinh Ωt

p(t) = p(0) cosh Ωt+mΩx(0) sinh Ωt.(24)

This in turn allows us to compute the quantum Jacobianmatrix,

M(t) =

(sinh Ωt/(mΩ) − cosh Ωt

cosh Ωt −mΩ sinh Ωt

), (25)

the elements of which are OTOCs that all grow exponen-tially with the largest Lyapunov exponent Ω. Once wediagonalize the above matrix, the hyperbolic functionsarrange in such a way that a pair of exponentials emergewith exponents ±Ω. This coincides with the Lyapunovexponent of the classical inverted oscillator.

IV. COMPLEXITY FOR INVERTEDHARMONIC OSCILLATOR

Recently by using the inverted harmonic oscillator,complexity has been proposed as a new diagnostic of

6

quantum chaos [6, 60]. Depending on the setup anddetails of the quantum circuit, the scope and sensitivityof computational complexity as a diagnostic can vary.Since this is a fairly new diagnostic, its full capacity tocapture chaotic behaviour is not yet fully understood.Therefore, to gain further insight into quantum chaoticsystems, we will extend our investigation in two differentdirections. First, we compute the complexity for thedisplacement operator by using the operator methodof Nielsen [18, 22, 61, 62], which explores how one canconstruct a given operator from the identity. Then wewill develop a construction based on the Heisenberggroup, which provides a natural basis choice for thedisplacement operator. Notice that this displacementoperator description is analogous to the doubly evolvedquantum circuit constructed in Ref. [60]. We would liketo explore if this operator formalism is consistent withthe existing results and whether it can provide us withany additional information about chaos.

In the second part of this section, we will use the wavefunction, or correlation matrix method, for a system ofN-oscillators to study the behaviour of complexity atdifferent timescales, namely, dissipation, scrambling andasymptotic regimes. This particular setup introducestwo new parameters into the problem: the number ofoscillators and the lattice spacing, and we will also deter-mine how complexity and the scrambling time dependson them.

A. Complexity for the Displacement Operator

Now let’s compute the circuit complexity correspondingto the time evolved displacement operator. We evolve thedisplacement operator mentioned in (8) by the invertedharmonic oscillator Hamiltonian. We get the following,

D(α, t) = exp[A(t) i x+B(t) i p

], (26)

where

A(t) =√

2 (Im(α) cosh(Ωt)− Re(α) sinh(Ωt)Ω) ,B(t) =

√2 (Im(α) sinh(Ωt)/Ω− Re(α) cosh(Ωt)) .

(27)

Evidently it is an element of the Heisenberg group.Consequently, we can parametrize the unitary as,

U(τ) =←−P exp(i

∫ τ0

dτ H(τ)), (28)

where,

H(τ) =∑a

Y a(τ)Oa. (29)

Oa = {i x, i p,−i ~ I} generators of Heisenberg groupwhose algebra is defined by,

[i x, i p] = −i~ I, [i x,−i ~ I] = 0, [i p,−i ~ I] = 0. (30)

The associated complexity function is defined as,

C(U) =∫ 1

0

dτ√GabY a(τ)Y b(τ). (31)

We choose Gab = δab. To proceed further we choose towork with following representation of Heisenberg gen-erators. For ease of computation, we start with the 3-dimensional representation of the Heisenberg group gen-erators [63].

M1 =

0 1 00 0 00 0 0

,M2 =0 0 00 0 1

0 0 0

,M3 =0 0 10 0 0

0 0 0

(32)

It can be easily checked that these Ma’s satisfy the samealgebra as that of (30). From (28), and using the expres-sions of Ma’s we can easily show that,

Y a = Tr(∂τU(τ).U−1(τ).MTa ). (33)

This in turn helps us to define a metric on this space ofunitaries,

ds2 = δab(Tr(∂τU(τ).U−1(τ).MTa ))×× (Tr(∂τU(τ).U−1(τ).MTb ))

(34)

The last step is to minimize the complexity functional(31). The minimum value that it takes will then give therequired complexity. It can be shown, following [22, 25,64], that (31) can be minimized by evaluating it on thegeodesics of (34) with the boundary conditions,

τ = 0, U(τ = 0) = I, τ = 1, U(τ = 1) = D(α, t), (35)

where in the representation (32) D(α, t) becomes,

D(α, t) =

1 A(t) 12A(t)B(t)0 1 B(t)0 0 1

. (36)For our case U(τ) is an element of Heisenberg group sowe can parametrize U(τ) as,

U(τ) =

1 x1(τ) x3(τ)0 1 x2(τ)0 0 1

, (37)and, given this parametrization, from (34) we find that,

ds2 = (1 + x22)dx21 + dx

22 + dx

23 − 2x2 dx1dx3, (38)

and the complexity functional,

C(U) =∫ 1

0

dτ√gij ẋi(τ)ẋj(τ), (39)

where, xi = {x1, x2, x3}. We have to minimize (39) usingthe boundary conditions (35). For this we need to solvefor the geodesics of this background, which amounts to

7

solving second order differential equations. Alternatively,we can find the killing vectors of this space and the corre-sponding conserved charges to formulate the system as afirst order one. We first list the Killing vectors below,

k1 =∂

∂x1,

k2 =∂

∂x2+ x1

∂

∂x3,

k3 =∂

∂x3.

(40)

The corresponding conserved charges (cI = (kI)igij ẋ

j)are,

c1 = (1 + x2(τ)2)ẋ1(τ)− x2(τ)ẋ3(τ),

c2 = ẋ2(τ) + x1(τ)ẋ3(τ)− x1(τ)x2(τ)ẋ1(τ),c3 = ẋ3(τ)− x2(τ)ẋ1(τ).

(41)

To solve these first order differential equations we first setc3 = 0 in (41) to get,

ẋ1(τ) = c1,

ẋ2(τ) = c2,

ẋ3(τ) = x2(τ)ẋ1(τ).

(42)

The solutions for these equations are,

x1(τ) = χ1 + c1τ,

x2(τ) = χ2 + c2τ,

x3(τ) = χ3 + c1χ2τ +1

2c1c2τ

2

(43)

From τ = 0 boundary condition, χ1 = χ2 = χ3 = 0. Thenwe are left with,

x1(τ) = c1τ,

x2(τ) = c2τ,

x3(τ) =1

2c1c2τ

2

(44)

Then from the final boundary condition at τ = 1 we have,

c1 = A(t), c2 = B(t). (45)

Then finally we have,

x1(τ) = A(t)τ, x2 = B(t)τ, x3(τ) =1

2A(t)B(t)τ2. (46)

We evaluate the complexity with this solution,

C(U) =√A(t)2 +B(t)2. (47)

This is a remarkably simple expression. We suspect thatthis is a consequence of the simple structure of the Heisen-berg group. One can immediately see the behaviour ofcomplexity at large times where it grows as a simpleexponential

C(U) ≈ Im(α)− Re(α)Ω√2Ω

(√

1 + Ω2)eΩt. (48)

FIG. 1. Time evolution of complexity of the displacementoperator (computed from the operator method), for differentchoice of parameters. The red and the blue dotted curvescorrespond to {Im(α) = 0.1,Re(α) = 0.1,Ω = 0.1} and{Im(α) = 5,Re(α) = 0.1,Ω = 0.5} respectively

Fig. 1 displays the time evolution of complexity fortwo different sets of parameters in the logarithmic scale.Note that the overall behaviour is chaotic as expectedfor IHO and it matches with Ref. [60]. The late timebehavior for both cases are exponential as expected. Theearly times behaviour on the other hand is a bit subtle.Looking closer, we notice that for a particular set of valuesof the parameters there is a minimum in the evolutionof complexity during the scrambling-time regime. Thisstrange feature is absent for the complexity computedfrom the correlation matrix method used in Ref. [60]. Thephysical significance of this minimum is unclear to us atthis point. To understand its implications and whether itis a generic feature for this method will require furtherinvestigations of other models. Naively though, it hintsthat the operator method is perhaps more sensitive thanthe wave function method for computing complexity. Wewould like to investigate these issues elsewhere. Also,we note that conclusion drawn here is not sensitive tothe choice of the cost functional (39). One could havecertainly chosen another cost functional. For a detaileddiscussion of the various choices of interested readers arereferred to [64].

B. Complexity for N-Oscillators and Scrambling

To further investigate the scrambling behaviour for theinverted harmonic oscillator, in this sub-section we willtake a different approach. First of all we will compute thestate complexity instead of operator complexity. Secondly,we will consider a large number of inverted harmonicoscillators. To establish our point we will use the modelused by Ref. [49], where the authors extended the invertedharmonic oscillator model and considered the field theorylimit. Below we start with a review of the model studiedin Ref. [49].

First we will consider two free scalar field theories

8

((1+1)-dimensional c = 1 conformal field theories) de-formed by a marginal coupling as in Ref. [49]. The Hamil-tonian for such model is given by

H = H0 +HI =1

2

∫dx[Π21 + (∂xφ1)

2 + Π22 + (∂xφ2)2

+m2(φ21 + φ22)]

+ λ

∫dx(∂xφ1)(∂xφ2).

(49)We will discretize this theory by putting it on a lattice.Using the following re-definitions

x(~n) = δφ(~n), p(~n) = Π(~n)/δ, ω = m,

Ω =1

δ2, λ̂ = λ δ−4 and m̂ =

m

δ,

(50)

we get the following Hamiltonian

H =δ

2

∑n

[p21,n + p

22,n +

(Ω2 (x1,n+1 − x1,n)2

+ Ω2 (x2,n+1 − x2,n)2 +(m̂2(x21,n + x

22,n)

+ λ̂ (x1,n+1 − x1,n)(x2,n+1 − x2,n))].

(51)

Next we perform a series of transformations,

x1,a =1√N

N−1∑k=0

exp(2π i k

Na)x̃1,k,

p1,a =1√N

N−1∑k=0

exp(− 2π i k

Na)p̃1,k,

x2,a =1√N

N−1∑k=0

exp(2π i k

Na)x̃2,k,

p2,a =1√N

N−1∑k=0

exp(− 2π i k

Na)p̃2,k,

p̃1,k =ps,k + pa,k√

2, p̃2,k =

ps,k − pa,k√2

,

x̃1,k =xs,k + xa,k√

2, x̃2,k =

xs,k − pa,k√2

,

(52)

that lead to the Hamiltonian

H =δ

2

N−1∑k=0

[p2s,k + Ω̄

2kx

2s,k + p

2a,k + Ω

2kx

2a,k

], (53)

where

Ω̄2k =

(m̂2 + 4 (Ω2 + λ̂) sin2

(π kN

)),

Ω2k =

(m̂2 + 4 (Ω2 − λ̂) sin2

(π kN

)).

(54)

Note that the underlying model of interest is still the in-verted harmonic oscillators. It becomes immediately clear

by appropriately tuning the value of λ̂–the frequencies

FIG. 2. Universal growth of the complexity in Eq. (56) at differ-ent time scales. Top, middle, and bottom figures show, respec-tively, the power-law dissipation, the exponential scramblingin semi-log scale, and the intermediate linear growth. Bluedotted, black, and red dashed curves correspond to {δ = 0.4,N = 100}, {δ = 0.5, N = 200}, and {δ = 0.5, N = 100},respectively. Other parameters are fixed as m = 1, λ = 10,δλ = 0.01.

Ωk can be made arbitrarily negative resulting in coupledinverted oscillators. The other frequency, Ω̄k, however,will be always positive. Therefore, effectively this model(53) can be seen as the sum of a regular and invertedoscillator for each value of k. Since we are interested inthe inverted oscillators we will ignore the regular oscillatorpart and we will simply use the inverted oscillator partof the Hamiltonian

H̃(m,Ω, λ̂) =

δ

2

N−1∑k=0

[p2k +

(m̂2 + 4 (Ω2 − λ̂) sin2

(π k

N

))x2k

].

(55)

Even with this Hamiltonian, by tuning λ̂ we can get both

9

regular and inverted oscillators.It is worth stressing that the above Hamiltonian origi-

nates from the discretization of the scalar field (49). Thisimposes a UV cut-off inverse proportional to the latticespacing. As long as only the low energy physics is con-cerned, Hamiltonian (55) for the uncoupled oscillatorsshould describe the original field theory very well.

Now we will talk about the structure of the quantumcircuit we will be using to study the complexity. At t = 0

we start with the ground state of H̃(m,Ω, λ̂ = 0) and then

time evolve it with H̃(m,Ω, λ̂ 6= 0) and H̃ ′(m,Ω, λ̂′ 6= 0)with two slightly different couplings, λ̂ and λ̂′ = λ̂+ δλ̂,

where δλ̂ is small. The complexity of the state evolvedby H with respect to the state evolved by H ′ is given by[26, 28, 60]

Ĉ(Ũ) = 12

√√√√N−1∑k=0

(cosh−1

[ω2r,k + |ω̂k(t)|2

2ωr,k Re(ω̂k(t))

])2, (56)

where

ω̂k(t) = i Ω′k cot(Ω

′kt) +

Ω′2ksin2(Ω′kt) (ωk(t) + iΩ

′k cot(Ω

′kt))(57)

and

Ω′2k = m̂2 + 4 (Ω2 − λ̂− δλ̂) sin2

(π kN

). (58)

The frequencies-squared ωk(t)2, ω2r,k are given by

ωk(t) = Ωk

(Ωk − i ωr,k cot(Ωk t)ωr,k − iΩk cot(Ωk t)

), (59)

Interestingly, this simple model exhibits three universalbehaviors for the complexity growth in three differenttime scales.

As shown in Fig. 2, the complexity starts to grow as apower-law, in a transient time known as the dissipationtime [5]. This is a time scale when local perturbationrelaxes. It corresponds to an exponential decay of time-ordered correlators of local observable. At larger timesthe complexity switches to an exponential growth, i.e.,scrambling, which corresponds to the early exponentialdecay of the OTOC, 1− �eλt. Asymptotically, the com-plexity grows linearly in time. This corresponds to theexponential relaxing of the OTOC. Note that the invertedharmonic oscillators are not bounded, the complexitygrows forever without saturation.

We also identified the scaling of the complexity growthin terms of parameters N and δ in the model. In the dis-sipation and intermediate linear growth regime, the com-plexity scales as C ∼

√Nδ−2t. In the scrambling regime

the complexity grows as C ∼√N exp δ−2t. The scram-

bling time, i.e., the time scale for which the complexitybecomes O(1), can then be extracted as td ∼ δ2 log 1/

√N .

As before, the conclusion drawn here is not sensitive tothe choice of the cost functional (56). This generic fea-tures of the complexity for this system still persists evenif we use different cost functional.

V. DISCUSSION

The harmonic oscillator is one of the most versatiletoy models in all of physics. Largely, this is becausethe oscillator Hamiltonian is quadratic, and Gaussianintegrals are a staple of any physicist’s diet. This articledetails our systematic study of the inverted harmonicoscillator as a vehicle to explore quantum chaos andscrambling in a controlled and tractable setting. Inparticular, since the inverted harmonic oscillator isclassically unstable but not chaotic, our expectationfor the quantum system is to find scrambling, butnot true chaotic behavior. We set out to ask if, andhow, this expectation manifests at the level of somefrequently used diagnotics. Concretely, we focused on tworecently developed probes of chaos; the out-of-time-ordercorrelator and the circuit complexity, both of which wecomputed for the displacement operator in eq.(8). TheOTOC in particular appears to be insensitive to whetheror not the oscillator is inverted. On the other hand,the fact the oscillator Hamitonian is quadratic is a keyfeature of this computation. To test this, we extendedthe Hamiltonian by adding a cubic-gate perturbationand found that, with this additional term in the IHOHamiltonian, the OTOC exhibits a crossover fromno-decay to exponential-decay, consistent with the aboveexpectation. We further computed the full quantumLyapunov spectrum for the IHO, finding that it exhibitsa paired structure among the Lyapunov exponents. Thisin turn leads us to conjecture that as long as the OTOCscrambles exponentially, such a structure will manifest inthe Lyapunov spectrum.

Using the operator method we then computed thecomplexity of a target displacement operator obtainedfrom a simple reference displacement operator by thechaotic Gaussian dynamical evolution expected of theIHO. Our construction is primarily based on Nielsen’sgeometric formalism, making use of the Heisenberggroup as a natural avatar for the displacement operator.The takeaway from this analysis is that the choice ofoperator or wave function approaches in the computationof the complexity really depends on the physicalproblem in question. For example, the wave functionapproach is more convenient for the study of a system ofN -oscillators, where we showed that both the complexityand scrambling time depend on two new parameters,namely, the number of oscillators and lattice spacingbetween them.

As elucidating as this study of the chaos and complexityproperties of the inverted harmonic oscillator is, there are

10

several questions remain to be addressed. Among these,we count:

• A clear recipe for the penalization procedure, whichusually accompanies the circuit complexity, is stillmissing for continuous systems such as the invertedoscillator and, more pressingly, in quantum fieldtheory.

• While progress in understanding circuit complex-ity has come in leaps and bounds since it enteredinto the horizon of high energy theory and blackhole physics, much of this progress has been fo-cused on simple linear systems. For the purposedof understanding interacting systems, it would beof obvious benefit to push the operator complexitycomputation beyond the Heisenberg group. Heretoo, the inverted oscillator offers some hints. For ex-ample, one conceptually straightforward extensionthat may shed some light into this matter wouldbe precisely the cubic-gate perturbation that weconsidered above.

• Finally, and more speculatively, there is the noveland largely unexplored class of Hamiltonians withunbroken PT symmetry which describe non-isolatedsystems in which the loss to, and gain from the en-vironment are exactly balanced. In a sense then, PT-symmetric Hamiltonians interpolate between Her-mitian and non-Hermitian Hamiltonians but withspectra that are real, positive and discrete. An ex-ample relevant to our study here is the 1-parameterfamily of Hamiltonians

H(ε) =p2

2+ x2(ix)ε ,

with real parameter ε. Clearly H(0) = p2/2 + x2 isjust the familiar harmonic oscillator. On the otherhand H(1) = p2/2 + ix3 is not only unfamiliar, itis also complex! Continuing along ε, we find the in-verted quartic Hamiltonian H(2) = p2/2− x4 whichlooks decidedly unstable. Nevertheless, it was rigor-ously shown in [65] that the eigenvalues of H(ε) arereal for all ε ≥ 0. Given the numerous manifesta-tions of PT-symmetric Hamiltonians in for exampleoptics, superconductivity and even graphene sys-tems, it would be interesting to explore both itsquantum chaotic as well complexity properties withsome of the tools that we have explored here.

We leave these and related questions for further study.

NOTE ADDED IN PROOF

After our article appeared on the arXiv, another article[66] was posted, discussing and further motivating thestudy of the inverted harmonic oscillator. There, usinga recently developed technique for computing the ther-mal OTOC in single-particle quantum mechanics [67],the authors argue that the inverted harmonic oscillatoremerges quite generically anytime one isolates one de-gree of freedom in a large N system with a gravity dual,and integrates out the remaining degrees of freedom -essentially forming a quadratic hilltop potential. Theyfind an exponential growth of the OTOC with quantumLyapunov exponent of order the classical Lyapunov ex-ponent generated at the hilltop. In fact, they find thatλOTOC ≤ cT at temperature T and for some constantc ∼ O(1), therby generalizing the MSS chaos bound tosingle-particle quantum mechanics. This may seem to bein conflict with our finding for the OTOC, however, aspointed out above, we compute the OTOC for displace-ment operators which, being built out of ladder operatorsare composite. In this sense, our results are reminiscentof the operator thermalization hypothesis introduced in[68]. The reconciliation of these findings is, no doubt, ofgreat interest and we leave this for future work.

ACKNOWLEDGEMENTS

A.B. is supported by Research Initiation Grant(RIG/0300) provided by IIT-Gandhinagar and Start UpResearch Grant (SRG/2020/001380) by Department ofScience & Technology Science and Engineering ResearchBoard (India). S.H. would like to thank the URC ofthe University of Cape Town for a research developmentgrant for emerging researchers. J.M. was supported inpart by the NRF of South Africa under grant CSUR114599. W.A.C acknowledges support provided by theInstitute for Quantum Information and Matter, Cal- tech,and for the stimulating environment from which the au-thor had been significantly benefited. W.A.C gratefullyacknowledges the support of the Natural Sciences andEngineering Research Council of Canada (NSERC). B.Y.was supported by the U.S. Department of Energy, Officeof Science, Basic Energy Sciences, Materials Sciences andEngineering Division, Condensed Matter Theory Program.B.Y. also acknowledges partial support from the Centerfor Nonlinear Studies at LANL.

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The Multi-faceted Inverted Harmonic Oscillator: Chaos and ComplexityAbstract ContentsI IntroductionII The IHO ModelIII Out-of-Time Order CorrelatorA OTOC for the Displacement OperatorB Quantum Lyapunov Spectrum

IV Complexity for Inverted Harmonic OscillatorA Complexity for the Displacement Operator B Complexity for N-Oscillators and Scrambling

V Discussion Note added in proof Acknowledgements References