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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-
∗ [email protected]† [email protected]‡
[email protected]§ [email protected]¶ [email protected]
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)
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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
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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)
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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
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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