Quantum chaos in supersymmetric quantum mechanics: an exact diagonalizationstudy

P. V. Buividovich1, ∗

1Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK(Dated: July 18th, 2022)

We use exact diagonalization to study energy level statistics and out-of-time-order correlators(OTOCs) for the simplest supersymmetric extension HS = HB⊗I+ x1⊗σ1 + x2⊗σ3 of the bosonic

Hamiltonian HB = p21 + p22 + x21 x22. For a long time, this bosonic Hamiltonian was considered one of

the simplest systems which exhibit dynamical chaos both classically and quantum-mechanically. Itsstructure closely resembles that of spatially compactified pure Yang-Mills theory. Correspondingly,the structure of our supersymmetric Hamiltonian is similar to that of spatially compactified super-symmetric Yang-Mills theory, also known as the Banks-Fischler-Shenker-Susskind (BFSS) model.We present numerical evidence that a continuous energy spectrum of the supersymmetric modelleads to monotonous growth of OTOCs down to the lowest temperatures, a property that is alsoexpected for the BFSS model from holographic duality. We find that this growth is saturated bylow-energy eigenstates with effectively one-dimensional wave functions and a completely non-chaoticenergy level distribution. We observe a sharp boundary separating these low-energy states from thebulk of chaotic high-energy states. Our data suggests, although with a limited confidence, thatat low temperatures the OTOC growth might be exponential over a finite range of time, with thecorresponding Lyapunov exponent scaling linearly with temperature. In contrast, the gapped low-energy spectrum of the bosonic Hamiltonian leads to oscillating OTOCs at low temperatures withoutany signatures of exponential growth. We also find that the OTOCs for the bosonic Hamiltonianare never sufficiently close to the classical Lyapunov distance. On the other hand, the OTOCsfor the supersymmetric system agree with the classical limit reasonably well over a finite range oftemperatures and evolution times.

I. INTRODUCTION

Out-of-time-order correlators (OTOCs), first intro-duced in [1], have recently attracted a lot of attentionas probes of quantum chaos in strongly correlated many-body quantum systems. For a system with a Hamiltonianoperator H at temperature T , the OTOC of two opera-tors A and B is defined as

C (t) = −Tr

(ρ[A (t) , B (0)

]2), (1)

where A (t) = eiHtAe−iHt is the time dependent operator

A in the Heisenberg representation, and ρ = Z−1 e−H/Tis the thermal density matrix.

OTOCs measure the sensitivity of a time evolution ofa physical observable A (t) to small perturbations of ini-

tial quantum state by an operator B (0). If the operator

A (t) corresponds to some canonical coordinate x (t) and

B (0) - to its conjugate momentum p (0), the commuta-tor [x (t) , p] in the OTOC (1) corresponds to the classicalPoisson bracket

{x (t) , p (0)} =∂x (t)

∂x (0)

∂p (0)

∂p (0)=∂x (t)

∂x (0)(2)

that measures the sensitivity of time evolution of a dy-namical system to its initial conditions. For a chaotic

∗ pavel.buividovich@liverpool.ac.uk

system, {x (t) , p (0)} is expected to grow exponentiallyas eλ t, where λ is the leading (largest) Lyapunov expo-nent. In what follows we refer to λ as simply the Lya-punov exponent. Generalizing the thermal average of

the squared partial derivative ∂x(t)∂x(0) in (2), OTOCs (1)

provide us with a quantum definition of the Lyapunovexponent λ. Namely, for a quantum chaotic system theOTOCs (1) are expected to grow as e2λ t for some periodof time.

Many studies of OTOCs were to a large extent moti-vated by the derivation of a rigorous bound

λ ≤ 2π T (3)

on the growth of OTOCs in thermal systems at temper-ature T by Maldacena, Shenker and Stanford [2], andthe demonstration that for sufficiently low temperaturesthis bound is saturated by the Sachdev-Ye-Kitaev (SYK)model [3, 4]. More generally, it was found that the MSSbound (3) is saturated for systems with a holographicdual description. Interestingly, the bound might be alsosaturated by simple experimentally accessible systemssuch as trapped ions in an external potential [5].

However, the demonstration of OTOC growth ingeneric chaotic quantum many-body systems and inquantum field theory turned out to be a very challengingendeavour. The progress was mostly limited to confor-mal field theory, quantum circuits and simple spin chainmodels [6–8], often with quenched disorder [9, 10]. Forexample, so far it was not possible to explicitly demon-strate the saturation of the MSS bound (3) in the BFSSmatrix model [11], which has a well-established holo-

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graphic description in terms of black D0-branes in typeIIA superstring theory or M -theory [12, 13]. Even in theSYK model, the equality λ = 2πT can only be demon-strated using a model-specific diagrammatic technique[3, 4, 14], while generally applicable techniques such asexact diagonalization so far could not reach the relevantlow-temperature, large-system regime [15–17]. Needlessto say, the analysis of OTOCs and quantum Lyapunovexponents in higher-dimensional quantum field theoriessuch as QCD is an even more formidable task.

In this situation, a lot of attention was attractedto simple quantum-mechanical systems in which theOTOCs can be calculated exactly using either analyticor numerical techniques, for example, by explicitly solv-ing the Schrodinger equation. At the level of pure states,exponential OTOC growth was demonstrated in a kickedrotor system [18] and in quantum stadia [19]. ThermalOTOCs for simple quantum systems and for quantumbilliards were extensively considered in [20]. In [21],OTOCs were calculated semi-analytically for the quar-tic anharmonic oscillator. In these cases, no exponen-tial growth was clearly observed. Interestingly, an ex-ponential OTOC growth can be observed in a double-well potential for temperatures or energies that are closethe height of the saddle point separating the two wells[22, 23]. In this case, the saddle point resembles an in-verted harmonic oscillator, which also exhibits an expo-nential OTOC growth [24].

Since real-time dynamics of many-body systems re-mains to a large extent inaccessible for simulations onclassical computers, an important methodological ap-plication of simple chaotic quantum-mechanical systemsmight be the development and testing of numerical meth-ods for diagnosing real-time quantum chaos.

One of the most popular and simple quantum-mechanical models for studying various aspects of quan-tum chaos is a system with two bosonic degrees of free-dom x1 and x2 and the Hamiltonian of the form (up tothe choice of pre-factors for the kinetic and the potentialterms)

HB = p21 + p22 + x21 x22. (4)

This Hamiltonian can be obtained by projecting theHamiltonian of SU (2) bosonic matrix model to the sec-tor with zero angular momentum [25] (see also [26, 27]).In turn, SU (2) bosonic matrix model is a dimensional re-duction of SU (2) Yang-Mills theory. It is also a bosonicpart of the Hamiltonian of N = 2 BFSS matrix model.

Classical dynamics of the Hamiltonian (4) is known tobe chaotic at all temperatures or energies [28, 29], withLyapunov exponent scaling as

λ = c T 1/4, (5)

where c ≈ 1.32 for our definitions of the kinetic andpotential terms in (4). The emergence of fractal struc-tures in the classical configuration space of this Hamilto-nian was demonstrated recently in [30]. A generalization

of the Hamiltonian (4) with N bosonic degrees of free-dom was considered recently in [31], and an exponentialgrowth of OTOCs was demonstrated analytically in thenext-to-leading order of expansion in 1/N .

The flat directions x1 = 0 and x2 = 0 of the classicalHamiltonian are lifted due to quantum effects, so thatthe energy levels of the quantum Hamiltonian (4) arediscrete, and all the corresponding wave functions are lo-calized. At sufficiently high energies, the bosonic Hamil-tonian (4) exhibits a random-matrix-type level statis-tics [32] and an exponential OTOC growth [33] at suf-ficiently high temperatures or energies. In this regime,the energy spectrum can be considered as continuous,and manifestations of quantum chaos are closely relatedto classical chaotic dynamics. However, the discrete en-ergy spectrum of the system (4) implies that quantumchaos cannot be observed at low temperatures of theorder of the gap between the two lowest energy levels.The crossover1 between the regular oscillatory dynamicsat low temperatures and chaotic behavior at large tem-peratures/energies is a purely quantum phenomenon andis similar to the confinement-deconfinement transition inpure Yang-Mills theory or bosonic matrix models [34, 35].

In this paper we consider a minimal supersymmetricextension of the simple bosonic Hamiltonian (4), whichexhibits continuous energy spectrum and OTOC growthall the way down to zero temperatures. In this respect, itis qualitatively similar to supersymmetric matrix modelsand the SYK model, and is very different from boundedquantum mechanical systems with discrete energy spec-trum like (4). The Hamiltonian of our model can bewritten as

HS = HB ⊗ I + x1 ⊗ σ1 + x2 ⊗ σ3 =

=

(HB + x2 x1

x1 HB − x2

), (6)

where σ1, σ2 and σ3 are the 2 × 2 Pauli matrices. TheHilbert space of this model is therefore a direct prod-uct of the Hilbert space of the bosonic model (4) and atwo-dimensional “fermionic” Hilbert space on which thePauli matrices act. The Hamiltonian (6) can also be rep-

resented as a square HS = Q2 of the supersymmetrygenerator

Q = x1x2 ⊗ σ2 + p1 ⊗ σ1 − p2 ⊗ σ3. (7)

This representation makes it obvious that the energyspectrum of the supersymmetric Hamiltonian is boundedfrom below, even though the fermionic terms x1 ⊗ σ1 +x2⊗σ3 in (6) are unbounded. The supersymmetric model(6) was first introduced in [36] as a toy model of super-symmetric membrane. In this work it was demonstratedthat quantum corrections that lift the classical flat di-rections x1 = 0 and x2 = 0 are cancelled out due to

1 We are using the term “crossover” because phase transitions can-not exist for systems with a finite number of degrees of freedom.

3

supersymmetry, so that the directions x1 = 0 and x2 = 0remain flat for the supersymmetric Hamiltonian (6). As

a result, the energy spectrum of HS is continuous, andwave functions extend to infinity along the lines x1 = 0and x2 = 0. This demonstration was further extendedto the BFSS model to argue that supersymmetric mem-branes are intrinsically unstable, in contract to bosonicmembranes 2. If we consider the bosonic Hamiltonian(4) as a minimal model that is similar to the bosonicmatrix model of bosonic membranes, the model (6) is aminimal model that is similar to supersymmetric ma-trix/membrane models such as the BFSS [11] matrixmodel.

We will see that much like in the case of the BFSSmodel, additional “fermionic” terms in the model (6)eliminate the transition (more precisely, the crossover inour case) to the non-chaotic “confinement” regime, sothat the model exhibits monotonously growing OTOCsfor all temperatures 3. Somewhat counterintuitively, wefind that the energy level statistics still exhibits a rathersharp change between chaotic and regular behavior atlarge and low energies. Furthermore, we will demonstratethat at high temperatures the supersymmetric model (6)has much better agreement with classical dynamics thanthe purely bosonic one (4).

II. NUMERICAL METHOD

In this work, we perform numerical diagonalization ofthe supersymmetric Hamiltonian (6). To highlight thedifference between the bosonic and the supersymmet-ric model, we also diagonalize the bosonic Hamiltonian(4). Since both Hamiltonians act on infinite-dimensionalHilbert spaces, the first step is to truncate the Hilbertspace to a finite number of states that can be treated nu-merically. To this end we consider the matrices of both

Hamiltonians in the basis of two-dimensional harmonicoscillator states of the form |k1〉⊗ |k2〉, where the states|k1〉 and |k2〉 belong to the Hilbert space of functions ofx1 and x2, respectively. The corresponding wave func-tions are of the form

Ψk1,k2 (x1, x2) = ψk1 (x1) ψk2 (x2) , (8)

where

ψk (x) =1√

2k k!√πL

exp

(− x2

2L2

)Hk (k, x/L) , (9)

are the wave functions that correspond to eigenstatesof a one-dimensional harmonic oscillator Hamiltonian

H0 = L2 p2

2 + x2

2L2 , Hk (k, z) = (−1)kez

2 dk

dzke−z

2

are theHermite polynomials, and L is the length parameter. Wediscuss the tuning of L a bit later.

Matrix elements of the bosonic Hamiltonian in this ba-sis take the form

(HB)k1,k2;l1,l2 = 〈k1| ⊗ 〈k2| HB |l1〉 ⊗ |l2〉 =

= 〈k1| p2 |l1〉δk2 l2 + 〈k2| p2 |l2〉δk1 l1 +

+〈k1| x2 |l1〉 〈k2| x2 |l2〉, (10)

where

〈k| p2 |l〉 =1

2L2(δk,l (2 k + 1)−

−δk+2,l

√(k + 1) (k + 2)− δk−2,l

√k (k − 1)

),

〈k| x2 |l〉 =L2

2(δk,l (2 k + 1) +

−δk+2,l

√(k + 1) (k + 2) + δk−2,l

√k (k − 1)

)(11)

are the matrix elements of the operators x2 and p2 inthe basis of one-dimensional wave functions (9). Corre-spondingly, the matrix of the supersymmetric Hamilto-nian (6) can be represented in the following block form:

〈k1| ⊗ 〈k2| HS |l1〉 ⊗ |l2〉 =

((HB)k1,k2;l1,l2 + δk1,l1 〈k2| x |l2〉 δk2,l2 〈k1| x |l1〉

δk2,l2 〈k1| x |l1〉 (HB)k1,k2;l1,l2 − δk1,l1 〈k2| x |l2〉

), (12)

where 〈k| x |l〉 = L√2

(δk+1,l

√k + 1 + δk−1,l

√k)

are ma-

trix elements of the coordinate operator x in the basis(9).

2 In [36], the supersymmetric Hamiltonian was written as HS =

HB ⊗ I + x1 ⊗ σ1 + x2 ⊗ σ2. In this paper, we choose a unitaryequivalent Hamiltonian (6) that is manifestly real. A unitary

transformation relating both Hamiltonians is U = I ⊗ I−iσ1√2

3 Note that our SUSY Hamiltonian is very different from fullyintegrable supersymmetric Hamiltonians considered in [37, 38].

To make the matrix representations (10) and (12) finiteand thus numerically tractable, we truncate the infinite-dimensional discrete Hilbert space spanned on the basisvectors |k1〉 ⊗ |k2〉 to a finite number of states with

k1 + k2 < 2M, (13)

where M is the truncation parameter. To reduce thecomputational cost of exact diagonalization, we furtheruse the discrete parity symmetry of the Hamiltonians (4)and (6) to represent the matrices (10) and (12) in blockdiagonal form. We then perform numerical diagonaliza-tion of the matrices (10) and (12). For not very large

4

M . 100 we use the QR algorithm as implemented inthe LAPACK routine dsyev, finding all eigenvectors andeigenvalues of the matrices (10) and (12). For larger val-ues of M we use the Arnoldi algorithm as implementedin the ARPACKPP library to find n � M2 lowest eigen-values, with n taking values between 102 and 103. Ourproduction code that performs exact diagonalization andcalculates the OTOCs is publicly available on GitHub[39]. Numerical data for OTOCs and energy levels areincluded as ancillary files in the ArXiv submission.

To analyze the bosonic Hamiltonian (4), thelength parameter L is adjusted to the value L =21/6 ≈ 1.12246 that minimizes the expectation value〈0| ⊗ 〈0| HB |0〉 ⊗ |0〉 = L−2+L4/4 of HB in the “pertur-bative” vacuum state |0〉 ⊗ |0〉. Another strategy wouldbe to minimize the ground state energy upon exact di-agonalization. However, we have found that our resultsdepend very weakly on the choice of L once the numberof basis vectors is sufficiently large.

ΔE

L0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.02

0.05

0.10

0.20

FIG. 1. The gap ∆E = E3 − E1 between the lowest andthe next-to-lowest energy levels E1 and E3 of the matrix (12)of the supersymmetric Hamiltonian (6) as a function of thelength parameter L in the basis wave-functions (9). The valueof the truncation parameter M changes between M = 40 (topcurve plotted in blue) to M = 200 (lowest curve plotted inred) in steps of 10 (the only missing value is M = 100). Greenpoints indicate the position of the minima, estimated usingspline interpolation.

For the supersymmetric Hamiltonian (6), for eachvalue of the truncation parameter M we choose thevalue of the length parameter L that minimizes the gap∆E = E3 − E1 between its lowest and next-to-lowestenergy levels E3 and E1, obtained by exact diagonal-ization. As explained in Appendix A below, the energylevels of the supersymmetric system are all doubly de-generate, therefore E2 = E1 and the first nonzero energygap is ∆E = E3−E1 = E3−E2. The dependence of ∆Eon M and L is illustrated on Fig. 1. We found that thedependence of the optimal value of L on the truncationparameter M can be well described by the formula

L = 0.961624 + 0.0409431(2M − 1)0.377368. (14)

With this choice of L, the dependence of the energy gap

∆E on M is illustrated on Fig. 2. Fitting suggests thatit can be well described by the power law

∆E = 4.36466M−1.16619, (15)

which is shown on Fig. 2 as a solid red line. For compar-ison, on Fig. 2 we also show the dependence of the gap∆E on M for the free particle Hamiltonians in one andtwo dimensions:

H1D = p2, (16)

H2D = p21 + p22, (17)

which are subject to the same truncations of the Hilbertspace as the supersymmetric Hamiltonian. Namely, weconsider the one-dimensional Hamiltonian (16) on theHilbert space spanned by all one-dimensional basis states(9) with k < 2M and with the length parameter thatdepends on M as in (14). The power-law dependence of

the energy gap of H1D on M appears to be quite closeto the result (15) for the supersymmetric Hamiltonian:∆E = 2.54431M−1.17463. This power-law fit is shown onFig. 2 with a solid line.

For the free two-dimensional Hamiltonian (17), theHilbert space is truncated to states |k1〉 |k2〉 with k1 +k2 < 2M , as in (13), and the length parameter is againgiven by (14). In this case, we also obtain a power lawdependence of ∆E on M , although with a somewhat dif-ferent power: ∆E = 1.6393M−0.987156.

In what follows, we will often compare the results ob-tained for the supersymmetric and the bosonic Hamil-tonians (6) and (4) with results for the one- and two-dimensional free Hamiltonians (16) and (17) at the sametemperature and for the same Hilbert space truncation.We will see that at low temperatures the eigenstates ofthe supersymmetric Hamiltonian (6) are with a good pre-cision one-dimensional, and a meaningful comparison canbe made with the one-dimensional free Hamiltonian (16).Similarly, the two-dimensional free-particle Hamiltonian(17) qualitatively reproduces some features of OTOCs atearly times and high temperatures, which are most likelythe artifacts of a finite infrared cutoff.

The largest eigenvalues of the Hamiltonian matrices(10) and (12) also grow with M . For example, for thematrix of the bosonic Hamiltonian (10) for M = 70 andM = 100 the largest eigenvalues are Emax (M = 70) =27073.5 and Emax (M = 100) = 56734.3. For thematrix of the supersymmetric Hamiltonian (12), wehave Emax (M = 70) = 38426.4 and Emax (M = 100) =91055.4, respectively. The truncation parameter M thusprovides both the infrared and the ultraviolet cutoff forthe continuous Hamiltonians (4) and (6). The OTOCsappear to be not very sensitive to the ultraviolet cutoff athigh energies, which allows us to obtain reliable resultsusing only a relatively small number n � M2 of lowestenergy levels.

On the other hand, the infrared cutoff due to the trun-cation to k1 +k2 ≤ 2M is crucial to obtain a numerically

5

SUSY

Free 2D

Free 1D

50 100 200 5000.001

0.005

0.010

0.050

M

E 2-E 1

FIG. 2. The gap ∆E between the lowest and the next-to-lowest energy levels of the supersymmetric Hamiltonian (6)as a function of the truncation parameter M . For each valueof M , the optimal value of L is chosen according to (14).For comparison, we also show the gap of free one- and two-dimensional Hamiltonians (16) and (17) as functions of M ,obtained in exactly the same way and using the same valuesof L for each M . Solid lines are best fits of the form ∆E =AM−B .

tractable approximation for the supersymmetric Hamil-tonian (6) with a continuous energy spectrum. As dis-cussed in the Introduction, the flat directions x1 = 0and x2 = 0 of the classical Hamiltonian remain flat alsofor the supersymmetric Hamiltonian, so that the sys-tem can still escape to infinity, and at least some of thewave functions have infinite spatial extent. Our trun-cation of the Hilbert space to a finite number of basisstates (8) with k1 + k2 ≤ 2M limits the spatial extent ofwave functions, thus effectively introducing soft bound-aries on spatial coordinates x1 and x2. Indeed, usingat most 2M lowest harmonic oscillator eigenstates (9),we can construct wave functions with spatial extent thatdoes not exceed 〈 x2 〉 ' 2M L2. This is obvious from

equalities 〈n| x2/L2 |n〉 = 〈n| p2 L2 |n〉 = 〈n| H0 |n〉 =(n+ 1/2) for the one-dimensional oscillator Hamiltonian

H0 = L2 p2

2 + x2

2L2 with eigenstates (9). Hence the sub-space spanned on eigenstates with n ≤ 2M can onlycontain states with 〈 x2 〉 . 2M L2. Our approach istherefore similar to numerical regularization of any otherunbounded system with continuous spectrum by puttingit in a finite box.

To illustrate how the finite values of M impose a cutoffon spatial coordinates, on Fig. 3 we show thermal expec-tation values 〈 x22 〉 = Tr

(ρx22)

as functions of the tem-perature T at different values of M for supersymmetric,bosonic, and free one- and two-dimensional Hamiltoni-ans (6), (4), (16) and (17). As discussed above, the datafor the free one- and two-dimensional Hamiltonians (16)are used for comparison at low and high temperatures,respectively.

In the high-temperature regime, we use all eigen-states of the truncated supersymmetric, bosonic and two-

dimensional free Hamiltonians (6), (4) and (17) to cal-culate the expectation value. In the low-temperatureregime, we use only n�M2 eigenstates that correspondto n lowest energy levels. We check that our values of nare big enough by comparing the results obtained with nand n/2 lowest energy levels, which are found to be veryclose to each other.

Fig. 3 shows that for the supersymmetric Hamilto-nian (6) as well as for the one- and two-dimensionalfree Hamiltonians, thermal expectation values 〈 x22 〉 =Tr(ρ x22

)strongly depend on M but have weak temper-

ature dependence. This is an expected behavior for afree particle confined within a region of space with size∼√M L2 determined by the infrared cutoff scale. For

the free Hamiltonians (16) and (17) in one and two di-mensions, the expectation values 〈 x22 〉 are indeed reason-ably close to the estimates 〈 x22 〉 ≈ 2M L2. For the su-persymmetric Hamiltonian, 〈 x22 〉 is considerably smallerthan these estimates. This suggests that potential energyterms have a significant effect on the spatial structure ofwave functions at high energies, despite the flatness ofthe x1 = 0 and x2 = 0 directions.

On the other hand, for the bosonic Hamiltonian (4)the expectation value 〈 x22 〉 exhibits strong temperaturedependence and significantly decreases at low tempera-tures. This is an expected behavior for a gapped systemwith localized wave functions. The dependence on thetruncation parameterM is negligible for sufficiently smalltemperatures, and only becomes important at high tem-peratures. Only at T & 20, the expectation values 〈 x22 〉for the bosonic and the supersymmetric Hamiltonians (4)and (6) become reasonably close to each other, and ex-hibit very weak temperature dependence and strong Mdependence. This suggests that at such temperaturesthe lifting of the flat directions due to quantum effectsbecomes negligible for the bosonic Hamiltonian, and theinfrared cutoff scale becomes important. At the sametime, the effect of fermionic terms also becomes smaller,and the supersymmetric system exhibits classical chaoticdynamics that is similar to the dynamics of the bosonicHamiltonian.

Finally, we should note that the truncation of the fullHilbert space of the supersymmetric Hamiltonian (6) tothe subspace spanned by a finite number 2M (M + 1) ofbasis states of the form (8) breaks the exact supersym-

metry of the model. In particular, the equality HS = Q2

is violated at the upper edge of the energy spectrum ifwe truncate the matrices of the Hamiltonian HS and thesupersymmetry generator Q to have finite dimensions.In practice, we find that all the energy levels of HS re-main positive upon the truncation, so the truncation pre-serves the cancellation of the negative unbounded termsin the fermionic operators in HS . Furthermore, the ef-fect of truncation becomes negligible for thermal expec-tation values in the limit of large M , as anti-periodicboundary conditions for fermions on the thermal circlebreak supersymmetry anyway, and the upper edge of thespectrum is suppressed at finite temperatures. Exact su-

6

Free 1D,M=400,n=400Free 1D,M=800,n=800 Free 2D,M=70

Free 2D,M=100Bosonic,M=70Bosonic,M=100

Bosonic,M=800,n=300

SUSY,M=70SUSY,M=100

SUSY,M=400,n=350SUSY,M=800,n=200

0.1 1 10 1001

510

50100

5001000

T

<x 22 >

FIG. 3. Temperature dependence of the thermal expectationvalues 〈 x22 〉 = Tr

(ρ x22

)for the supersymmetric and bosonic

Hamiltonians (6) and (4) at different values of truncation pa-rameters M . For T ≥ 1, we show the results for M = 70 andM = 100 with all eigenvalues taken into account. For T ≤ 1,we show the results for M = 400 and M = 800, with onlyn�M2 lowest energy levels taken into account. For compar-ison, we also show 〈 x22 〉 for the free one- and two-dimensionalHamiltonians (16) and (17) with the same truncations of theHilbert space.

persymmetry also implies that the ground state of thesupersymmetric Hamiltonian should not be doubly de-generate, in contrast to all higher energy levels. On theother hand, with our truncation of the Hilbert space allenergy levels are doubly degenerate, as discussed in detailin Appendix A. Since the energy spectrum of the super-symmetric Hamiltonian is continuous (which we recoverin the limit M → 0), there are infinitely many energylevels that are infinitely close to the ground state, and asingle non-degenerate energy level should likewise have avanishing contribution to thermal expectation values.

III. EIGENSTATES OF THESUPERSYMMETRIC HAMILTONIAN

Preparing to consider out-of-time-order correlators, inthis Section we consider the spectral properties of the su-persymmetric Hamiltonian (6) and compare them withthose of the bosonic Hamiltonian (4) and the free one-and two-dimensional Hamiltonians (16) and (17) (at lowand high temperatures, respectively). We start with theglobal distribution of energy levels. For both the su-persymmetric and the bosonic Hamiltonians, histogram-ming all eigenvalues of the Hamiltonian matrices (10) and

(12) suggests that the level density dn(E)dE falls off as 1/E.

However, this analysis is not very informative, as all ofthe interesting low-lying eigenvalues are counted within asingle near-zero bin. To properly resolve the distributionof eigenvalues at all scales, we analyze the histograms oflog (E). Such histograms approximate the distributionsdn(E)d log(E) = E dn(E)

dE and are shown on Fig. 4 in logarithmic

SUSY,M=70

SUSY,M=100

SUSY,M=800,n=200

Bosonic,M=70

Bosonic,M=100

Bosonic,M=800,n=650

Free 2D,M=100

Free 1D,M=100

Free 1D,M=800

-5 0 5 101

10

100

1000

104

log(E)

dn/d(log(E))

FIG. 4. Histograms approximating the log-scale energy

level density dn(E)d log(E)

= E dn(E)dE

for the supersymmetric and

bosonic Hamiltonians (6) and (4) for different values of thetruncation parameter M . For comparison, we also show sim-ilar histograms for the free one- and two-dimensional Hamil-tonians (16) and (17) with same Hilbert space truncations.For the supersymmetric and bosonic Hamiltonians with trun-cation parameter M = 800, we only show the histograms ofn�M2 lowest eigenvalues.

scale.The log-scale histograms reveal two different scaling

regimes of the level density. At high energies with

log (E) & 5 (or, equivalently, E & 150), dn(E)d log(E) appears

to be almost constant up to the sharp UV cutoff. This

corresponds to the density of energy levels dn(E)dE ∼ E−1

that does not contain any dimensionful parameter.On the other hand, at low energies all the histogram

plots have an almost constant slope in our double logscale, which corresponds to the power-law scaling of

the form dn(E)d log(E) ∼ Eα = eα log(E), or, equivalently,

log(dn(E)d log(E)

)= α log (E). It is instructive to compare

our results with the scaling law

dn (E)

d log (E)= E

dn (E)

dE∼√E (18)

for the one-dimensional free Hamiltonian (16), where

dn ∼ dp ∼ d√E. On the other hand, for the two-

dimensional free Hamiltonian (17) the number of statesis dn ∼ 2πp dp ∼ dE, which leads to the linear scaling

dn (E)

d log (E)= E

dn (E)

dE∼ E. (19)

From Fig. 4 we can see that for small E the slopesof the histogram plots for the supersymmetric Hamilto-nian are quite close to that for the free one-dimensionalHamiltonian (16) with M = 100 and M = 800. As Mis increased, in both cases the histograms are shifted to-wards lower energies. This is our first argument in favourof effectively one-dimensional, gapless structure of low-lying eigenstates of the supersymmetric model. For the

7

FIG. 5. Density plots of the wave functions of some of the eigenstates of the supersymmetric Hamiltonian (6), with thecorresponding values of energy given in the plot labels. We use truncation parameter M = 70 for these plots. For each plot,the color scale is adjusted such that the brightest color corresponds to the maximal absolute value of the wave function. Blackbackground corresponds to zero wave function. Red and blue regions correspond to positive and negative wave function values.The square region for all plots is 20 ≤ x1 ≤ 20, 20 ≤ x2 ≤ 20. The plots in the top and in the bottom rows correspond tothe components φ (x1, x2) and χ (x1, x2) of the two-component wave function Ψ (x1, x2) = {φ (x1, x2) , χ (x1, x2)}. The squarednorms |φ (x1, x2) |2 and |χ (x1, x2) |2 are given in the plot labels.

bosonic Hamiltonian, low-lying energy levels remain dis-crete even for M → ∞. Correspondingly, the histogramis shifted towards larger E and has a steeper slope.

To get further insights into the structure of eigenstatesof the supersymmetric Hamiltonian at different energyscales, on Fig. 5 we show density plots of some of thewave functions obtained with the truncation parameterM = 70. In the plot labels, we give the correspondingvalues of energy as well as the squared norms |φ (x1, x2) |2and |χ (x1, x2) |2 of the two components of the wave func-tion Ψ (x1, x2) = {φ (x1, x2) , χ (x1, x2)}. Both squarednorms add up to one by virtue of normalization. Wesee that the low-energy states are indeed strongly local-ized along the flat directions x1 = ±0, x2 = ±0 of thepotential energy. The number of times the wave func-tions change sign along the direction of their maximalextent coincides with the serial number of the energylevel, which is also an argument in favour of effectivelyone-dimensional structure.

It might seem that the “T”-shape structure of thefunctions φ (x1, x2) violates the parity symmetry of themodel. However, here we only show the results for theeigenstates with positive x1 parity (see Appendix B).Eigenstates with negative x1 parity will have the “T”shape turned upside down, so that the full parity symme-try will be restored in the sum over both parity sectors.It turns out that the OTOCs of operators x2 and p2 takeexactly the same values in both x1 parity sectors, andhenceforth we only work with eigenstates of the super-symmetric Hamiltonian with positive x1 parity.

To get further insights into quantum chaos exhib-

ited by the bosonic and supersymmetric Hamiltonians(4) and (6), it is useful to analyze microscopic correla-tions between energy levels. Quantum chaos is usuallycharacterized by the repulsion between adjacent energylevels. This repulsion forces the energy level spacings∆Ei = Ei+1 − Ei to have one of the few universal sta-tistical distributions, which correspond to ensembles ofGaussian unitary, orthogonal, or symplectic random ma-trices. For Hamiltonians without any quenched disorder,such analysis assumes that energy levels far away fromthe edges of the spectrum can be considered as quasi-random quantities belonging to a statistical ensemble.

A convenient measure of level repulsion associated withquantum chaos is the r-ratio [40, 41]

ri =min (∆Ei−1,∆Ei)

max (∆Ei−1,∆Ei). (20)

For the Gaussian Orthogonal Ensemble (GOE) of ran-dom real symmetric matrices, statistical average of riover many energy levels in the bulk of the spectrum (or,equivalently, over many random matrices) is rGOE =0.53. On the other hand, for non-chaotic integrable sys-tems the energy levels are typically uncorrelated. Cor-respondingly, the number of energy levels within a fixedinterval is usually well described by Poisson distribution[42], and the average of ri over many energy levels is closeto rPoisson = 0.39.

For Hamiltonians that are invariant under a nontrivialsymmetry group, the level spacings ∆Ei = Ei+1−Ei thatenter the r-ratio (20) should be the differences betweenconsecutive energy levels that correspond to eigenstates

8

FIG. 6. Scatter plots of the ratios ri, defined as in (20), asfunctions of the energy Ei. From top to bottom, we com-pare the results for the bosonic, supersymmetric, and freetwo-dimensional Hamiltonians (4), (6) and (17), respectively.For the truncation parameter M = 100, all energy levels areshown. For M ≥ 500, we only show between 200 and 800lowest energy levels calculated using the Arnoldi algorithm.For the supersymmetric case, vertical dashed line shows theenergy above which the irreps of D4d group associated witheach energy level become irregularly ordered, and thin curvedlines of the same color as data points correspond to the for-mula (22) with best-fit parameters c. For M = 100, horizontalmagenta lines show the values of ri averaged over the energywindow given by the line extent.

transforming under the same irreducible representation(irrep) of the symmetry group. In other words, we makean ordered list of eigenstates that transform under somefixed irrep of the symmetry group, label the elements of

this list by consecutive integer indices i, and calculatethe r-ratio for the elements of this list.

As discussed in detail in Appendix A, the bosonicHamiltonian is invariant under a finite non-Abelian groupC4v with four Abelian and one non-Abelian irreduciblerepresentations (irreps) [43]. The supersymmetric Hamil-tonian is invariant under a larger finite non-Abeliangroup D4d with four Abelian and three non-Abelian ir-reps [44]. For our analysis of the r-ratio for the bosonicHamiltonian (4), we select the energy levels that trans-form under the two-dimensional non-Abelian irrep E0 ofC4v. For the supersymmetric Hamiltonian (6), the eigen-states belong to one of the two-dimensional non-Abelianirreps E1 or E2 of D4d, so we pick the states that trans-form under E1. For comparison, we also calculate the r-ratio for the free two-dimensional Hamiltonian (17) withthe same Hilbert space truncation. While the symmetrygroup of this free Hamiltonian is the full O (2) group, theenergy levels can still be classified according to irreps ofC4v which is a subgroup of O (2). To calculate the r-ratioin this case, we use energy levels that transform underirrep E0 of C4v.

Scatter plots of ri versus the energy Ei are shown onFig. 6. For the truncation parameter M = 100, we showall eigenvalues of the Hamiltonian matrices (10) and (12).For M ≥ 500, we use between 200 and 800 smallesteigenvalues of the Hamiltonian matrices (10) and (12)obtained using the Arnoldi algorithm.

To set the stage, we first discuss the results for thepurely bosonic Hamiltonian (4), shown in the top plot onFig. 6. In this case, lowest energy levels that transformunder E0 irrep are of order of 101, and for energies E &102 the r ratio is fluctuating almost randomly between0 and 1, filling the entire plot area almost uniformly.This is the expected behavior for a system that is chaoticin this energy range, both in quantum and in classicalmechanics. Averaging ri over many energy levels in thewindow between E = 102.5 ≈ 316.2 and E = 103.5 ≈3162.3, we obtain the value r = 0.523±0.006 that is veryclose to the universal value rGOE = 0.53 for the GaussianOrthogonal Ensemble. This expectation value as well asthe extent of the energy window for which it was obtainedare shown on Fig. 6. In full agreement with previousstudies [32, 33], we therefore conclude that the energyspectrum of the bosonic Hamiltonian (4) exhibits chaoticbehavior at least for energies of order E & 101 . . . 102.

For energies E & 102, the behavior of the r-ratio for thesupersymmetric Hamiltonian (6) appears to be very sim-ilar to the one for the bosonic Hamiltonian (4). Namely,the r ratio also fluctuates randomly between 0 and 1, av-eraging to r = 0.530± 0.004 in the same window of ener-gies E = 102.5 . . . 103.5 that we considered for the bosonicHamiltonian. The behavior for the low-energy part ofthe spectrum is, however, completely different. There isa family of low-lying energy levels with E . 100.5 ≈ 3.2for which the r-ratio ri behaves in a smooth and regu-lar way as a function of energy Ei, rising from ri ≈ 0.5up to ri ≈ 1.0. This behavior is observed in the energy

9

M = 100M = 500M = 600M = 800

5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

i

E i

FIG. 7. Low-lying energy levels Ei of the supersymmetricHamiltonian (6) as a function of their serial number i for dif-ferent values of the truncation parameterM . Only eigenstatesthat transform under the irrep E1 of the symmetry group D4d

of the Hamiltonian and have Ei ≤ 1 are considered. Solidlines are fits of the form Ei = a+ b (i+ c)2.

range that agrees well with the extent of the low-energy“tail” with dn

d log(E) =√E in histograms of global energy

level density in Fig. 4. Interestingly, the change betweenthe non-chaotic low-energy states and the chaotic high-energy states, where the r-ratio approaches unit value,appears to be quite sharp.

Such a smooth rising behavior of the r-ratio can beexpected for finite-size one-dimensional systems. As weshow on Fig. 7, the dependence of the energy levels Eion their serial number i with a good precision can bedescribed by a quadratic expression

Ei = a+ b (i+ c)2, i = 0, 1, 2, . . . , (21)

which leads to

ri =i+ c+ 1/2

i+ c+ 3/2. (22)

Solid lines on the middle plot on Fig. 6 correspond tothe expression (22) plotted as a function of Ei given by(21). The parameters a, b, c are extracted from the fitsshown on Fig. 7. We find that the best fit parametersb are with a good precision inversely proportional to thetruncation parameter M . The parameter a is very small,and the shift parameter c quite quickly grows with M :c = 0.79 for M = 100, c = 1.40 for M = 500, andc = 4.05 for M = 800. For few lowest eigenstates, smalldeviations of data points that are hardly noticeable onFig. 7 get amplified (compare thin lines with low-energydata points on the middle plot on Fig. 6), and our fitsappear to be not as good as for somewhat larger energieswith Ei < 1. It is instructive to compare these findingswith similar analysis for the free one-dimensional Hamil-tonian (16). In this case, b is also inversely proportionalto M , and the fit parameters a ≈ 0.25 and c ≈ 1 arealmost independent of M . Another instructive case is a

one-dimensional, infinitely deep potential well of width

L, for which Ei = π2 (i+1)2

L2 and ri = 2i+12i+3 are completely

independent of L. Strong dependence of the parameterc on the infrared cutoff set by the parameter M for thesupersymmetric Hamiltonian (6) is quite different fromthese one-dimensional models. For one-dimensional mod-els, this parameter is controlled by boundary conditionsat the end points of the region to which one-dimensionalmotion is effectively confined.

It is interesting to note that for the supersymmet-ric system the change between the regular, quasi-one-dimensional spectrum and the chaotic, random-matrix-like spectrum can be also observed from the ordering ofirreps under which the eigenstates transform. Namely,for the low-energy part of the spectrum, eigenstates witheven and odd serial numbers transform under the non-Abelian irreps E1 and E2 of D4d, respectively. For ener-gies above some threshold level, this ordering is violated,and now and then there appear several eigenstates in arow that transform under one and the same irrep. ForM = 100, we show the position of the first ordering ir-regularity of this kind in the middle plot on Fig. 6 witha vertical dashed thick line.

For comparison, on the plot in the bottom of Fig. 6we show the energy dependence of the r-ratio (20) forthe free two-dimensional Hamiltonian (17), subject tothe same truncation as the supersymmetric and bosonicHamiltonians (6) and (4). Here all the energies aresignificantly smaller than for the interacting systems.While ri ratio does exhibit some quasi-random behav-ior towards the higher end of the spectrum, averaging riover the energy window between E = 100.5 ≈ 3.16 andE = 101.5 ≈ 31.6, we obtain r = 0.40 ± 0.01, in goodagreement with the expected value r = 0.39 for spectraof integrable systems with Poisson distribution of energylevels.

IV. OUT-OF-TIME-ORDER CORRELATORS

In this Section we consider out-of-time-order correla-tors of operators x2 (t) and p2 (0), which correspond tothe classical Poisson brackets (2) defining the classicalLyapunov exponents4. While the expression (1) in theintroductory Section is the most straightforward quan-tum generalization of Poisson brackets of the form (2),in practice it is more convenient to work with regularizedOTOCs that have the same classical limit but exhibit less

4 We use the operators x2 and p2 because they are invariant withrespect to x1 parity transformations (reflections of x1). We usex1 parity to block-diagonalize the matrix (12) of the supersym-metric Hamiltonian (6) and reduce the computational cost ofexact diagonalization (see Appendix B). The OTOCs of the op-erators x1 and p1 are of course equivalent to that of x2 and p2,but are more complicated to calculate with our choice of the basisstates.

10

M=100,n=20200M=500,n=300M=600,n=200M=700,n=200M=800,n=200

0 100 200 300 400 5000

50

100

150

t

F(t),F0(t)

SUSY, T=0.2

M=70

M=100

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

t

F(t),F0(t)

SUSY, T=20

M=900,n=300

0 5 10 15 20

-0.04

-0.02

0.00

0.02

0.04

t

F(t),F0(t)

Bosonic, T=0.2

M=70

M=100

0 5 10 15 20 25

0

200

400

600

800

1000

1200

t

F(t),F0(t)

Bosonic, T=20

M=500,n=500M=600,n=600M=700,n=700M=800,n=800

100 200 300 400 500 600

100

150

200

250

t

F(t),F0(t)

Free 1D, T=0.2M=70

M=100

0 5 10 15 20 25

0

500

1000

1500

2000

2500

t

F(t),F0(t)

Free 2D, T=20

FIG. 8. Time dependence of the functions F (t) (solid lines with colors changing from blue to red) and F0 (t) (dotted lineswith colors changing from black to green), defined in (24), for different Hamiltonians and for different values of the truncationparameter M . Horizontal dashed lines with colors changing from black to green show the corresponding products of regularized

thermal expectation values 2〈 x22 〉r〈 p22 〉r = 2Tr(ρ

12 x2 ρ

12 x2

)Tr

(ρ

12 p2 ρ

12 p2

).

singular quantum behavior [2]:

C (t) = −Tr

([ρ

18 x2 (t) ρ

18 , ρ

18 p2 (0) ρ

18

]2), (23)

where ρ = e−β H/Z is the thermal density matrix, ρ18 =

e−β H/8/Z18 is its fractional power, and Z = Tr e−β H is

the thermal partition function. We will substitute eitherthe supersymmetric, the bosonic, or the free one- or two-

dimensional Hamiltonians for the abstract HamiltonianH in (23).

To discuss the time dependence of OTOCs, it is alsooften convenient to represent C (t) as a difference of two

11

contributions

C (t) = F0 (t)− F (t) ,

F (t) =

= 2Re Tr(ρ

14 x2 (t) ρ

14 p2 (0) ρ

14 x2 (t) ρ

14 p2 (0)

),

F0 (t) =

= 2Re Tr(ρ

14 x2 (t) ρ

14 p2 (0) ρ

14 p2 (0) ρ

14 x2 (t)

). (24)

All operators are time-ordered in F0 (t), and for suf-ficiently ergodic systems this function is expected tohave a conventional behavior of a finite-temperature,time-ordered correlator. In particular, if the finite-temperature two-point correlators Tr (ρ x2 (t) p2 (0)) de-cay sufficiently quickly with time t, F0 (t) is expectedto approach a constant value f0 = 2〈 p22 〉r 〈 x22 〉r, where

〈 O2 〉r = Tr(ρ

12 O ρ

12 O)

is a regularized thermal expec-

tation value [2, 45].The functions F (t) and F0 (t), calculated according to

(24), are plotted on Fig. 8 for different Hamiltonians andfor different values of the truncation parameter M . Freeone- and two-dimensional Hamiltonians are used for com-parisons at low and at high temperatures, respectively.The temperatures are T = 0.2 and T = 20.0 for the plotson the left and on the right, which are well in the low-and high-temperature regimes.

As discussed above, our truncation imposes an infraredcutoff, limiting the system size to 〈 x2 〉 . 2M L2 andhence regularizing the flat directions x1 = 0 and x2 = 0of the classical and the supersymmetric Hamiltonians.For the free one- and two-dimensional Hamiltonians (16),the truncation parameter M effectively puts our free par-ticle in a box of size ∼

√M L. It is therefore not sur-

prising that on Fig. 8 we observe a strong dependenceof the correlators F (t) and F0 (t) on the truncation pa-rameter M for all Hamiltonians and at all temperatures.The only exception is the low-temperature regime of thebosonic Hamiltonian, where only strongly localized wavefunctions contribute, and the effect of infrared cutoff isnegligible.

To some extent, theM dependence of the OTOCs C (t)is due to the strong M dependence of regularized ther-mal expectation values 〈 x22 〉r and 〈 p22 〉r. As discussedabove, the asymptotic values of the time-ordered corre-lator F0 (t) and the OTOCs C (t) are proportional to theproduct of these two thermal expectation values. As il-lustrated on Fig. 3, the expectation value 〈 x22 〉r growslinearly with M as M → +∞, and F (t) and F0 (t) ex-hibit similar growth.

Except for the case of the bosonic Hamiltonian at lowtemperatures, on all plots on Fig. 8 we observe the mainexpected features of F (t) and F0 (t): the two correla-tors are very close to each other at t = 0 and divergeat later times. Exhibiting some initial oscillations, thefunction F0 (t) approaches a constant value, while F (t)keeps decreasing.

Furthermore, we see that as a result of our Hilbert

space truncation, the correlators F (t) and F0 (t) do ex-hibit the same characteristic features of OTOC growtheven for the free-particle Hamiltonians (16) and (17).Clearly, for free-particle Hamiltonians without any trun-cation of the Hilbert space, the OTOC is just constant:

C (t) = −〈 [x (t) , p (0)]2 〉 =

= −〈 [x (0) + 2 p (0) t , p (0)]2 〉 =

= 〈 [x (0) , p (0)]2 〉 = 1. (25)

Therefore, any nontrivial time dependence of OTOCs forthe free-particle Hamiltonians (16) and (17) is a trunca-tion artifact at finite M .

The timescales characterizing the time dependence ofF (t) and F0 (t) appear to be not too different for thefree-particle and the supersymmetric cases. This couldbe expected based on our previous observation that low-energy eigenstates with a finite infrared cutoff are quitesimilar to the ones for a free particle in a finite one-dimensional box. The fact that even a free particle canexhibit OTOC growth in the presence of spatial bound-aries was also noted in [20]. For this reason, our strategywill be to carefully extrapolate the OTOCs growth ratesto the limit M → +∞ in order to distinguish the behav-ior of OTOCs for the supersymmetric Hamiltonian andthe free-particle one-dimensional Hamiltonian (16) (seeFig. 11 for the final results).

In most cases, however, we also significant deviationsfrom the expected behavior of F (t) and F0 (t): ex-cept for the high-temperature regime of the supersym-metric Hamiltonian, the late-time value of F0 (t) is no-ticeably different from the product f0 = 2〈 p22 〉r 〈 x22 〉rof thermal expectation values. This suggests that ourlow-dimensional Hamiltonians are not entirely ergodic,and some kind of regular motion prevents the finite-temperature correlators of the form Tr (ρ x2 (t) p2 (0))from decaying sufficiently quickly. This is expectable fora system with just a few degrees of freedom. For exam-ple, such a finite-temperature correlator would exhibitoscillations instead of decay for a harmonic oscillator ora two-level system. In fact, the functions F (t) and F0 (t)calculated at low temperatures with the bosonic Hamilto-nian (4) provide a nice example of such an oscillatory be-havior (middle left plot on Fig. 8). Since the energy spec-trum of the bosonic Hamiltonian is discrete (and hencegapped), only the two lowest energy levels will contributeto OTOCs for sufficiently low temperatures. This resultsin non-decaying oscillations of OTOCs and other corre-lators, as illustrated in the middle left plot on Fig. 8.

Interestingly, even though the bosonic and the su-persymmetric Hamiltonian are both expected to bechaotic at high temperatures, only for the supersymmet-ric Hamiltonian the late-time asymptotic value of F0 (t)agrees with f0 = 2〈 p22 〉r 〈 x22 〉r (see upper right plot onFig. 8). As one can see on the middle right plot on Fig. 8,for the bosonic Hamiltonian the function F0 (t) at larget has a noticeable deviation from f0 even at T = 20.This difference might be related to the observation that

12

only for the supersymmetric Hamiltonian the OTOCs ex-hibit an expected agreement with classical dynamics athigh temperatures. In contrast, for the bosonic Hamilto-nian, we never observe a good agreement between quan-tum OTOCs and classical Lyapunov distances. This isdemonstrated on Fig. 10 and explained in more detailsbelow.

For large-N gauge theories and matrix models, F (t) isexpected to behave as F (t) ≈ f0 − c e2λt for sufficientlylate times t at which F0 (t) is already sufficiently closeto its expected asymptotic value f0 [2, 45]. Clearly, withF0 (t) = f0 and F (t) = f0− c e2λt we immediately get anexponential OTOC growth C (t) = F0 (t)−F (t) = c e2λt,with λ being the Lyapunov exponent. Such growth, how-ever, cannot continue forever, and at even later timesF (t) approaches zero and stops decreasing. Correspond-ingly, the OTOC C (t) saturates at a value close to f0[45].

For our numerical data we indeed see how F (t) de-creases towards zero in all cases except for the low-temperature limit of the bosonic Hamiltonian. However,we cannot identify any clear exponential decay of F (t) ofthe form f0−c e2λt. In the low-temperature regime of thesupersymmetric Hamiltonian, the late-time decay of F (t)seems to be linear in t (see upper left plot on Fig. 8). Suchbehavior is even more obvious for the one-dimensionalfree-particle Hamiltonian5 (16), see the lower left plot onFig. 8. At high temperatures, the decay of F (t) towardszero is faster than at low temperatures, but it is still dif-ficult to identify a region with a clear exponential decay.

We illustrate the time dependence of the OTOCsC (t) = F0 (t)−F (t) on Fig. 9, comparing the results ob-tained with the supersymmetric and the bosonic Hamil-tonians (6) and (4) and with the free-particle one- andtwo-dimensional Hamiltonians. Fig. 9 shows that in allcases the OTOCs grow and eventually saturate. Thegrowth rate appears to be non-uniform. In particular,for sufficiently high temperatures T & 1 and at earlytimes all OTOCs exhibit some initial decrease. Also forthe bosonic Hamiltonian the OTOCs feature some os-cillations at T . 1, which become stronger and com-pletely dominate the OTOCs at lower temperatures (seethe middle left plot on Fig. 8).

A common expectation is that for sufficiently high tem-peratures the real-time dynamics of the bosonic Hamil-tonian (4) can be described in terms of the correspond-ing classical equations of motion x1 = −4x1 x

22, x2 =

−4x2 x21. This statement is also expected to apply to

the supersymmetric Hamiltonian (6), for which the role

5 In fact, for the one-dimensional free-particle Hamiltonian, thelinearly decreasing function F (t) re-bounces after approachingzero, and exhibits a linear growth until approaching f0. Thisbehavior resembles a classical motion of a particle that bouncesbetween the two walls of a one-dimensional potential well. Inour case, a one-dimensional particle is confined to an intervalof finite width because of the truncation of Hilbert space whichlimits the coordinate values to 〈 x2 〉 < 2ML2

of “fermionic” terms should become negligible at hightemperatures.

To check whether this is indeed the case, we also com-pare our quantum data for T ≥ 1 with the classical ana-logue of OTOCs. It is defined as a square of the rele-

vant Poisson bracket {x2 (t) , p2 (0)} = ∂x2(t)∂x2(0)

, averaged

over the thermal distribution of the initial conditions{x1 (0) , x2 (0) , p1 (0) , p2 (0)} for the classical evolution:

Ccl (t) = 〈(∂x2 (t)

∂x2 (0)

)2

〉. (26)

The calculation of Ccl (t) is discussed in more details inAppendix C. For T = 5 and T = 1, Ccl (t) is plotted onFig. 9 with a solid black line.

Quantum OTOCs only appear to be reasonably closeto Ccl (t) for the case of the supersymmetric Hamilto-nian (6) at sufficiently high temperature. At T = 1,the OTOCs for the supersymmetric Hamiltonian and theclassical OTOCs are only close to each other for a shortinitial period of time t . 1. The agreement is somewhatbetter at T = 5, where the classical and the quantumOTOCs are reasonably close to each other for t . 1.5.In both cases, the OTOCs exhibit some initial decreaseand start growing at t & 0.5. Interestingly, the agree-ment between the classical and the quantum OTOCs islost exactly when the classical OTOCs enters the regimeof clear exponential growth at t & 1.5. For these times,the quantum OTOCs grows considerably slower. On theother hand, our data suggests that OTOCs calculatedwith the bosonic Hamiltonian (4) are never quite close tothe classical OTOCs. This agrees with the observationmade in [20] that the classical limit is never reached forfour-point OTOCs in simple quantum mechanical sys-tems because quantum interference effects become im-portant earlier than the exponential OTOC growth setsin. The fact that for the supersymmetric Hamiltonianwe get a considerably better agreement with the classicaldynamics suggests that supersymmetry might effectivelysuppress or cancel out the wave packet spread that isresponsible for deviations from classical behavior [20].

Overall, our plots for the quantum OTOCs do not showregions of clear exponential growth that would be sim-ilar to the exponential divergence of the classical Lya-punov distance Ccl (t). It is not surprising, as the ex-ponential OTOCs growth is in general very difficult todetect for finite-size systems amenable to exact diago-nalization studies [8], even for the renowned SYK modelwhich is a paradigmatic example of maximal quantumchaos [16]. We do, however, see a clear difference in thelow-temperature behavior of the OTOCs for the bosonicHamiltonian (4) with gapped energy spectrum and forthe supersymmetric and free Hamiltonians which all havecontinuous energy spectra in the limit M → 0. Forthe bosonic Hamiltonian, the spectrum is discrete andOTOCs are oscillatory at low temperatures. For the su-persymmetric and free Hamiltonians, OTOCs grow withtime down to the lowest temperatures, although thisgrowth is much slower than for the classical dynamics.

13

M=100,n=20200M=500,n=300 M=500,n=150M=600,n=200 M=600,n=100M=700,n=200 M=700,n=100M=800,n=200 M=800,n=100

0 100 200 300 400 500 600

1

5

10

t

C(t)

Supersymmetric, T=0.2

M=500,n=500M=600,n=600M=700,n=700M=800,n=800

0 100 200 300 400 500 600

5

10

20

50

t

C(t)

Free 1D, T=0.2

SUSY,M=70SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0 20 40 60 80

0.1

1

10

100

t

C(t)

T = 1.

SUSY,M=70SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0 1 2 3 4 5 6 7

0.51

510

50100

500

t

C(t)

T = 5.

FIG. 9. Time dependence of the OTOC C (t) for the supersymmetric, bosonic, and free one- and two-dimensional Hamiltonians,for different values of the truncation parameter M and at different temperatures. Solid black line on the plots in the high-

temperature regime shows the expectation value 〈(∂x2(t)∂x2(0)

)2

〉 of the squared classical Lyapunov distance calculated at the same

temperature. For the plots at T = 0.2, the legend also shows the number n of lowest eigenstates that were used to calculateC (t).

Furthermore, there is also an observable difference be-tween the behavior of low-temperature OTOCs for thefree one-dimensional Hamiltonian and the supersymmet-ric Hamiltonian. The latter has more nontrivial featuresand a weaker dependence on the truncation parameterM .

To arrive at some estimates for the growth rate ofOTOCs in the absence of clear exponential growth, weconsider the time derivative of the logarithm of theOTOC C (t)

λ (t) =1

2

d

dtlog (C (t)) , (27)

which should be equal the Lyapunov exponent λ ifC (t) = c e2λ t. The time dependence of λ (t) is shown onFig. 10. At low temperatures, we again compare the datafor the supersymmetric Hamiltonian (6) and the free one-dimensional Hamiltonian (16). At high temperatures, wecompare the results for the bosonic, the supersymmetricand the free two-dimensional Hamiltonians, as well aswith the classical result λcl (t) = 1

2ddt log (Ccl (t)).

At all temperatures, the functions λ (t) feature a num-ber of peaks and sometimes plateaus. The peaks arelabelled by “x” or ”o” symbols, where “x” is used forpeaks that we believe to be dominant/most importantfeatures of OTOCs for a particular Hamiltonian, and “o”is used for the peaks that we believe to be sub-dominantor artifact. Plateaus are labelled by vertical lines anda letter “P”. We will use these peak values of λ (t), de-noted as λmax, as upper bound estimates on the valuesof Lyapunov exponent in our system. Since we observequalitatively different behavior at low and at high tem-peratures, let us separately discuss the main features ofλ (t) in these regimes.

A. High temperatures

At sufficiently high temperatures, real-time dynamicsof both the supersymmetric and the bosonic Hamiltoni-ans is expected to reduce to the classical dynamics. Qual-itatively, this regime corresponds to the range of energies

14

XX

OO

XX

OO

XX

OO

XX

OO

M=100,n=20200M=500,n=300M=600,n=200M=700,n=200M=800,n=200

1 5 10 50 100 500-0.005

0.000

0.005

0.010

0.015

0.020

0.025

t

λ(t)

Supersymmetric, T=0.2

P

X

OXX

OO

XX

OO

XX

OO

XX

OO

M=500,n=500M=600,n=600M=700,n=700M=800,n=800

5 10 50 100 500

0.002

0.004

0.006

0.008

t

λ(t)

Free 1D, T=0.2

XXXX

XXXX

XXXX

SUSY,M=70SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0.1 0.5 1 5 10 50 100-0.5

0.0

0.5

1.0

1.5

t

λ(t)

T = 1.

XXXX

XX XX

XX

XX

SUSY,M=70SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0.01 0.10 1 10 100-1

0

1

2

3

t

λ(t)

T = 5.

XX

OO

XX

OO

XXXXXXXX SUSY,M=70

SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0.01 0.10 1 10

0

5

10

15

t

λ(t)

T = 20.OOOO

XX XX

XX

XX

SUSY,M=70SUSY,M=100Bosonic,M=70

Bosonic,M=100Free 2D,M=70

Free 2D,M=100Classic

0.01 0.10 1 10-10

0

10

20

30

40

t

λ(t)

T = 50.

FIG. 10. Time dependence of the derivatives of the logarithm of OTOC λ (t) = 12ddt

log (C (t)) for different Hamiltonians and at

different temperatures and the values of the truncation parameters M and n. Solid black lines show λcl (t) = 12ddt

log (Ccl (t))for the classical Lyapunov distance Ccl (t), defined in (26). Symbols “x” and ”o” denote the peak values of λ (t) that we use asupper bound estimates of the quantum Lyapunov exponents.

E & 10, for which the energy levels behave similarly tothe ones of Gaussian Orthogonal Ensemble (GOE) of ran-dom matrices. In this case, we use truncation parame-ters M = 70 and M = 100, finding all the eigenvaluesand eigenvectors of the corresponding truncated matrices(12) or (10). At very high temperatures, the correspon-dence with the classical dynamics might be violated due

to our truncations of the full Hilbert space. As we havealready discussed above, we were not able to identify aregime where the OTOCs for the bosonic Hamiltonianwould approach the classical OTOCs (see Appendix C)sufficiently closely. As conjectured in [20], this might bea generic feature of simple quantum mechanical systems.The data shown on Fig. 10 for T & 1 supports these

15

observations. We find a limited agreement between theclassical and the quantum data only for the case of thesupersymmetric Hamiltonian at moderately large tem-peratures, 5 . T . 20, see the middle right and thebottom left plots on Fig. 10. The agreement is observedfor 0.1 . t . 1.5 at T = 5 and for 0.5 . t . 1 atT = 20, thus the time range of quantum-classical corre-spondence shrinks towards high temperatures. Both forT = 5 and T = 20, λ (t) has a distinct peak roughlyat the time at which its classical counterpart λcl (t) ap-proaches its plateau value that corresponds to a steadyexponential growth of Ccl (t). We believe that this peakis a physical feature of a supersymmetric model, there-fore we label it with the “x” symbol on the plots withT = 5 and T = 20 on Fig. 10. Its height is compa-rable with the plateau value of λcl (t), and its positionscales approximately as t? ∼ T−1/2. This characteristic“semi-classical” peak of λ (t) exists also at lower tem-peratures, down to T ∼ 1, but the agreement with clas-sical dynamics is gradually lost. A similar peak struc-ture (labelled with “x” on Fig. 10) exists also for thebosonic Hamiltonian at moderately large temperatures1 . T . 10, but the agreement with the classical dy-namics is nowhere sufficiently good. Interestingly, asone can see on the plot with T = 5, the data for thebosonic and the supersymmetric Hamiltonians agree verywell after both peaks, where λ (t) disagree with λcl (t) forboth Hamiltonians. These moderate-temperature peaksof λ (t) are completely absent for the free two-dimensionalHamiltonian, and appear to be a genuine feature of high-temperature, semi-classical chaotic dynamics.

At very high temperatures T & 20, both the bosonicand the supersymmetric Hamiltonians develop very largepeaks of height λmax ∼ T at early times t ∼ T−1, labelledwith “o” for the supersymmetric Hamiltonian. For thebosonic Hamiltonian, this early-time peak seem to con-tinuously transform into the peak near the plateau onsetof λcl (t), and is therefore labelled with “x”. The freetwo-dimensional Hamiltonian also exhibits a very similarpeak, labelled with “x” on Fig. 10. In contrast to the“semiclassical” peak discussed above, these early-timepeaks appear to have strong dependence on the trunca-tion parameter M for all Hamiltonians. The existence ofsimilar peaks in functions λ (t) for free Hamiltonians aswell as the strong M dependence of their heights suggestthat such early-time peaks are the artifacts of the infraredcutoff due to the Hilbert space truncation. At sufficientlyhigh temperatures T & 50, these early-time peaks com-pletely dominate the OTOCs, and any agreement withthe classical dynamics is lost both for the bosonic andthe supersymmetric Hamiltonians.

B. Low temperatures

At low temperatures T . 1, the dynamics of the su-persymmetric Hamiltonian (6) is dominated by nearlyone-dimensional, regular wave functions, as illustrated

on Fig. 5. We therefore compare the OTOCs for thesupersymmetric model with the ones for the free one-dimensional Hamiltonian (16). The OTOCs for thebosonic Hamiltonian at low temperatures only exhibitoscillations similar to the ones in the middle left plot onFig. 8, and we do not consider them. Likewise, compari-son with the classical dynamics makes little sense at lowtemperatures due to the dominance of quantum effects.

The functions λ (t) for the supersymmetric and the freeone-dimensional systems at a low temperature T = 0.2are shown in the two top plots on Fig. 10 for differentvalues of the truncation parameter M . For the super-symmetric case, λ (t) develops a plateau (labelled with“P” in the top left plot on Fig. 10) followed by the twopeaks (labelled with “x” and “o”). Analyzing the datafor the supersymmetric system at different temperatures,we found that the height of the early-time plateau isroughly proportional to the temperature T and its widthscales as ∼ T−1. We also checked that the large-M ex-trapolation of the plateau height λP using the ansatzλP = A + B/M yields a finite result at M → ∞. Thefact that λ (t) reaches a plateau means that for this timerange the growth of OTOCs is with a good precision ex-ponential. However, a comparison with the behavior ofthe functions F (t) and F0 (t), shown on the top left ploton Fig. 8, shows that this growth happens well before thefunction F0 (t) reaches its constant asymptotic value. Infact, even the positions of both subsequent peaks of λ (t)correspond to much earlier times than the saturation ofF0 (t). We conclude that whatever time range we chooseto define the quantum Lyapunov exponent for the super-symmetric Hamiltonian, we cannot find agreement withthe expected behavior of the functions F (t), F0 (t) andC (t) in large-N systems, where C (t) only grows expo-nentially when F0 (t) saturates. This is not surprising, asthere is no natural large parameter like N for our super-symmetric Hamiltonian (6).

The heights of the two peaks of λ (t), labelled with “x”and ”o” in the top left plot on Fig. 10, can be used asupper bounds on the rate of exponential OTOCs growthfor the supersymmetric Hamiltonian. While both peaksare changing considerably as the truncation parameterM is increasing, the dependence on M appears to besignificantly stronger for the second, subdominant, peak(labelled with “o”). The dependence of the peak heighton M is reasonably well described by the formula λmax =A + B/M , and extrapolations to M → +∞ yield theresults for λmax (M → +∞) = A that are close to zero.It is therefore likely that this second peak is a truncationartifact.

On the other hand, the first, dominant peak of λ (t)(labelled with “x” in the top left plot on Fig. 10) hasa weaker dependence on M , and might well present aphysical feature of OTOCs in the supersymmetric case,together with the plateau that precedes it. The posi-tion of this peak scales approximately as t? = T−1/2.We base our final estimates of the upper bound on low-temperature quantum Lyapunov exponents in the super-

16

T = 0.20T = 0.13T = 0.10T = 0.07T = 0.05

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

0.000

0.005

0.010

0.015

1/M

λmax

Supersymmetric T = 0.20T = 0.13T = 0.10T = 0.07T = 0.05

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

0.000

0.005

0.010

0.015

1/M

λmax

Free 1D

M=400, n=350M=500, n=300M=600, n=200M=700, n=200M=800, n=200

Extrap.,λmax=A+B/ MExtrap.,λmax=A+B/MExtrap. slope λmax=A(M)T

0.00 0.05 0.10 0.15 0.20

0.000

0.005

0.010

0.015

T

λmax

Supersymmetric

Extrap. slope λmax=A(M) T

M=400M=500M=600M=700M=800

Extrap.,λmax=A+B/ MExtrap.,λmax=A+B/M

0.00 0.05 0.10 0.15 0.20

0.000

0.005

0.010

0.015

T

λmax

Free 1D

FIG. 11. Extrapolations of the upper bounds λmax on Lyapunov exponents to M → +∞ for the supersymmetric system(on the left) and for the one-dimensional free Hamiltonian with the same Hilbert space truncation (on the right). Solid and

dashed lines show extrapolations to M → +∞ that use fits of the form λmax (M) = A + B/M and λmax (M) = A + B/√M ,

respectively. Shaded areas around these lines show error estimates obtained by excluding the data points with either smallestor largest value of M from the fits.

symmetric case on the height λmax of this peak. It istherefore important to ensure that this peak is a physi-cal feature and not an artifact.

To this end, we consider two different methods to ex-trapolate λmax to M → +∞. The first method, illus-trated on the top left plot on Fig. 11, is to perform ex-trapolations to M → +∞ separately for each value ofthe temperature. To this end we use least squares fits totwo different models:

λmax (M) = λmax (M → +∞) +B

M(28)

λmax (M) = λmax (M → +∞) +B√M

(29)

(30)

We estimate the extrapolation errors in the valuesλmax (M → +∞) by considering their variation upon theremoval of data points with either the smallest or thelargest values of M from the fit. These bounds are shownon the top plots on Fig. 11 as shaded areas around linesthat correspond to the fits with all data points. Let usstress that since our data comes from exact diagonaliza-tion, the numbers are not plagued with any statistical

uncertainties.

To highlight the difference between the behaviorof OTOCs for the supersymmetric and the free one-dimensional Hamiltonians, on the top right plot onFig. 11 we also show similar extrapolations for the max-imal value λmax of the function λ (t) for the free one-dimensional Hamiltonian (16). For this Hamiltonian, thefunction λ (t) has two distinct peaks with strong depen-dence on M . The first peak, denoted by “o” on the topright plot on Fig. 10, corresponds to very early times,much earlier than the plateau onset for the supersym-metric Hamiltonian. It has no direct counterpart in thesupersymmetric case, and we do not consider it in de-tail. On the other hand, the position of the second peak,labelled with “x” on the top right plot on Fig. 10, is be-tween the two peaks of the function λ (t) obtained withthe supersymmetric Hamiltonian. The height λmax ofthis peak is comparable to the maximal values of λ (t)for the supersymmetric Hamiltonian. We therefore useits height λmax as an upper bound for Lyapunov expo-nent for the free one-dimensional Hamiltonian (16) withtruncated Hilbert space, and illustrate theM dependenceof λmax on Fig. 11.

17

From the top left plot on Fig. 11 we see that forthe supersymmetric Hamiltonian the extrapolations us-ing both fits (28) and (29) yield nonzero results. The re-sults of linear extrapolations in 1/M (28) are expectablyhigher than for the square root extrapolations. Fit un-certainties appear to be not very large. The result-ing temperature dependence of the extrapolated valuesλmax (M → +∞) is illustrated on the bottom left ploton Fig. 11 with black points and solid and dashed blacklines. With the linear extrapolation model (28), this de-pendence appears to be approximately linear. Namely, afit of λmax (M → +∞, T ) with the function αT γ yieldsγ = 0.826 (solid black line), close to the linear scalingof the Lyapunov exponent with temperature, λ (T ) ∼ T .With the square root extrapolation model (29), the samepower law fit yields γ = 0.520 (dashed line), thus favoring

the dependence of the form λ (T ) ∼√T . The square root

extrapolation model in fact yields somewhat smaller val-ues of squared deviations χ2/d.o.f., but not significantlysmaller.

For the free one-dimensional Hamiltonian, the extrap-olations using the square root model (29) yield the resultsthat are either compatible with zero or negative, see thetop right plot on Fig. 11. Extrapolations with the lin-ear model (29) yield finite results which are however sig-nificantly smaller than the corresponding extrapolatedvalues for the supersymmetric Hamiltonian. Squared de-viations χ2/d.o.f. appear to be considerably smaller forthe square root model (29). The resulting temperaturedependence of the extrapolated values λmax is shown onthe bottom right plot on Fig. 11 with black points andsolid black (for the linear extrapolations (28)) and dashed(for the square root extrapolations (29)) lines.

Another strategy to estimate the temperature depen-dence of the maximal values of λ (t) in the limit M →+∞ is to fit the temperature dependence of λmax atfixed M with a suitable fitting function, and to extrap-olate the parameters of this fit to M → +∞ after-wards. For each fixed M , the temperature dependenceof λmax for the supersymmetric Hamiltonian is with agood precision linear. We therefore fit λmax at fixed Mto the linear function λmax (M,T ) = A (M) T of thetemperature T . In turn, the dependence of the fit pa-rameter A (M) on M can be well fitted by the formulaA (M) = A (M → +∞) +B/M , which yields the extrap-olated value A (M → +∞) = 0.05. The resulting ex-trapolation of λmax (M → +∞, T ) is shown on the lowerleft plot on Fig. 11 with solid green line, together withthe values of λmax at each finite M and the linear fitsλmax (M,T ) = A (M) T thereof.

Applying the same procedure to the free one-dimensional Hamiltonian, we find that in this case thedata for λmax (M,T ) at fixed M can be sufficiently well

fitted to the function A√T . In this case, the M de-

pendence of the fit parameter A (M) is better described

by the formula A (M) = A (M → +∞) +B/√M , rather

than A (M) = A (M → +∞) + B/M . Using the formerformula to extrapolate the fit parameter to M → +∞,

we find a small negative result. The resulting functionA (M → +∞)

√T is shown on the bottom right plot

on Fig. 11 as a solid green line. Remarkably, it agreeswell with the result of M → +∞ extrapolations at fixedtemperatures, described above. This agreement suggeststhat the non-chaotic behavior of OTOCs for the free one-dimensional Hamiltonian (16) is indeed recovered as thelimit M → +∞ is taken and the Hilbert space trunca-tion is removed. This is in stark contrast with the caseof the supersymmetric Hamiltonian (6), for which all ourextrapolation methods yield a positive and finite value ofthe OTOC growth rate in the limit M → +∞.

Finally, we illustrate the temperature dependence ofall our estimates of λmax as a function of temperatureon Fig. 12. We show the data obtained for all valuesof the truncation parameter M , and use more opaquelines to distinguish larger values of M . To avoid anyextrapolation ambiguities, on this plot we do not showany extrapolations, only the actual numerical data forfinite M values. Different line colors correspond to dif-ferent Hamiltonians. The data points for the peak valuesof λ (t) that we believe to be most important for eachmodel are labelled with “x” symbols. “o” symbols de-note the peak values for either artifact or sub-dominantbut nevertheless prominent maxima. The same “x” or“o” symbols are used to label the corresponding peaks ofλ (t) on Fig. 10.

V. DISCUSSION AND CONCLUSIONS

In this work we considered the simplest supersymmet-ric extension HS = HB ⊗ I + x1 ⊗ σ1 + x2 ⊗ σ3 of thebosonic Hamiltonian HB = p21 + p22 + x21 x

22. The latter is

known to feature both quantum and classical chaos, andis closely related to the Hamiltonian of spatially compact-ified pure SU (2) Yang-Mills theory. We focused on theenergy level statistics and out-of-time order correlators(OTOCs) C (t) = −〈 [x2 (t) , p2 (0)]

2 〉. The OTOCs arethe quantum counterparts of the classical Lyapunov dis-

tance Ccl (t) = 〈 {x2 (t) , p2 (0)}2 〉 = 〈(∂x2(t)∂x2(0)

)2〉, which

characterizes the sensitivity of one of the coordinatesx2 (t) to its initial value x2 (0).

Since the energy spectrum of the bosonic Hamiltonianis gapped, its real-time dynamics is completely regularat low temperatures. In particular, the OTOCs exhibitregular oscillations and show no signature of exponentialgrowth (see the middle left plot on Fig. 8). For bosonicmatrix models and Yang-Mills theory, this regime wouldcorrespond to the low-energy confinement regime.

On the other hand, the energy spectrum of the super-symmetric Hamiltonian (6) is known to be continuous[36]. One of our main results is that the low-energy spec-trum of the supersymmetric model is completely regular,and shows no signatures of random-matrix-type energylevel statistics. The corresponding wave functions areeffectively one-dimensional and are localized along the

18

FIG. 12. A summary plot with our estimates (upper bounds)of the values of quantum Lyapunov exponents for the su-persymmetric Hamiltonian (6) (red lines/symbols), comparedwith similar estimates for the bosonic Hamiltonian (4) (bluelines) and for the free one-dimensional (green lines) and two-dimensional (brown lines) Hamiltonians (16) and (17). Theestimates are based on the values of λ (t) = 1

2ddt

log (C (t)) inmost characteristic local maxima. The symbol “x” denotesthe heights of the maxima that we consider most importantfor each model. The symbol “o” denotes either artifact orsub-dominant but prominent maxima. The use of “x” and“o” symbols for each model is in one-to-one correspondencewith the plots on Fig. 10. Data points for the bosonic Hamil-tonian labelled as “oscill.” correspond to oscillatory behaviorof OTOCs, with λmax corresponding to the maximal value ofthe oscillating function. We collate the data for all M withoutany extrapolations. More opaque lines/symbols correspond tolarger values of M .

flat directions of the supersymmetric Hamiltonian (seeFig. 5).

Nevertheless, these low-energy states produce amonotonous OTOC growth down to the lowest tempera-tures. We presented numerical evidence that this OTOCgrowth is not an artifact of our Hilbert space truncation.All our extrapolations towards the physical limit sug-gest that the time-dependent Lyapunov divergence rateλ (t) = 1

2ddt log (C (t)) remains finite. We observed that

λ (t) reaches a plateau of a finite time extent at intermedi-ate times t ∼ T−1/2 (labelled with “P” on the upper leftplot on Fig. 10), which suggests an exponential growthof the OTOCs C (t) over a finite time range. However, incontrast to large-N gauge theories and matrix model, inour system there is no intrinsic large parameter and noparametric scale separation that would allow to clearlyidentify the regime of exponential OTOC growth.

The plateau is followed by a distinct peak of λ (t) (la-belled with “x” on the upper left plot on Fig. 10). Webase our estimates on the quantum Lyapunov exponentλ, shown on Fig. 12, on the height of this peak. Whilefor a finite Hilbert space truncation even the free parti-cle Hamiltonian exhibits a monotonous OTOC growth,we demonstrated that in the free case the Lyapunov di-vergence rate λ (t) is likely to extrapolate to zero as the

truncation is removed.

We found limited evidence that the estimates of quan-tum Lyapunov exponent in the supersymmetric model,based either on the plateau or peak heights, scale lin-early with temperature, λ = c T with c ≈ 0.05. Sucha linear scaling would ensure that the model does notviolate the MSS bound λ < 2πT . We cannot howevercompletely exclude a scaling λ ≈ c′ T γ with a fractionalpower 1/2 . γ . 1. While this scaling would formallyviolate the MSS bound λ < 2πT at sufficiently low tem-peratures, there is no deep reason why the MSS boundshould not be violated for our simple Hamiltonians. In-deed, the derivation of the MSS bound [2] is based on theassumption that the OTOCs exhibit exponential growthafter the time-ordered correlator F0 (t) (defined in (24))saturates at a constant value f0 = 2〈 x22 〉〈 p22 〉. This as-sumption is justifiable for large-N matrix models andnon-Abelian gauge theories. However, as one can observeby comparing the data on Figs. 8, 9 and 10, in our caseF0 (t) reaches its asymptotic value at much later timesthan the times at which the OTOCs exhibit maximalgrowth. Again, this could be expected in the absence ofany parametric scale separation.

Our estimates of the quantum Lyapunov exponents aresummarized on Fig. 12 for the whole range of temper-atures, and for all Hamiltonians and all values of thetruncation parameters that we consider. The data onFig. 12 suggests that the quantum Lyapunov exponentfor the supersymmetric Hamiltonian (6) continuously in-terpolates between an approximately linear scaling λ ∼ Tat low temperatures and the fractional power scalingλ ∼ T 1/4 that is characteristic for the classical dynamics(see Eq. (26)). This is also an expected behavior for theBFSS model, which, being holographically dual to blackbranes [12, 13], is suspected to saturate the MSS boundat low temperatures, and reduce to the classical Yang-Mills dynamics at high temperatures. The temperaturedependence of λ at low and high temperatures is alsoquite different from the results obtained with the freeone- and two-dimensional Hamiltonians (16) and (17).

At sufficiently large energies, the energy spectra ofboth the bosonic and the supersymmetric Hamiltoniansare chaotic, with r-ratios in perfect agreement with ran-dom matrix theory predictions for the GOE ensemble. Atsufficiently high temperatures, these energy levels domi-nate the real-time dynamics, which in this regime is ex-pected to be similar to the classical chaotic dynamicsfor both systems. The change between the low-energy,effectively one-dimensional states with non-chaotic levelstatistics and the high-energy chaotic states appears tobe quite sharp (see the middle plot on Fig. 6). It wouldbe interesting to understand what might be the counter-part of this change in the BFSS model, and how it can beinterpreted in terms of the holographic dual theory. Anintriguing possibility is the transition to the M -theoryregime that was discussed recently in [12].

An interesting observation is that the correspondencewith the classical dynamics at (moderately) high tem-

19

peratures can only be established for OTOCs in the su-persymmetric model, but not in the bosonic one. Thissuggests that supersymmetry cancels out some of thequantum corrections to OTOCs at high temperatures,a property that is also likely to hold for supersymmet-ric Yang-Mills theory and the BFSS model. For thebosonic model, low-temperature OTOC oscillations withamplitude ∼ T 1/4 (labelled as “Oscill.” on Fig. 12)are continuously transforming into very quick early-timegrowth, without approaching the classical OTOCs suf-ficiently closely anywhere. At very high temperatures(T & 20), this quick early-time growth also makes itimpossible to establish the quantum-classical correspon-dence for the supersymmetric Hamiltonian. The fact thata similar growth occurs also for the free two-dimensionalHamiltonian suggests that it might be a truncation arti-fact. The invalidity of the quantum-classical correspon-dence for the four-point OTOCs in simple quantum me-chanical systems was also discussed recently in [20].

Our results suggest that the supersymmetric Hamilto-nian (6) has more intricate chaotic dynamics than the

simple bosonic Hamiltonian HB = p21 + p22 + x21 x22, both

at low and at high temperatures. The supersymmetricmodel definitely reproduces more features of the BFSSmodel that are expected from the holographic dual de-scription. It would be interesting to work out an analytic

description of the effectively one-dimensional eigenstatesof this Hamiltonian that saturate the OTOC growth atlow temperatures. Such a description might also help tounderstand the behavior of OTOCs in the BFSS model,which provides one of the most elaborate examples ofholographic duality [11, 46].

ACKNOWLEDGMENTS

This work used the DiRAC@Durham facility managedby the Institute for Computational Cosmology on behalfof the STFC DiRAC HPC Facility (www.dirac.ac.uk).The equipment was funded by BEIS capital funding viaSTFC capital grants ST/P002293/1, ST/R002371/1 andST/S002502/1, Durham University and STFC operationsgrant ST/R000832/1. DiRAC is part of the National e-Infrastructure.

This work was performed using the DiRAC Data In-tensive service at Leicester, operated by the Universityof Leicester IT Services, which forms part of the STFCDiRAC HPC Facility (www.dirac.ac.uk). The equipmentwas funded by BEIS capital funding via STFC capi-tal grants ST/K000373/1 and ST/R002363/1 and STFCDiRAC Operations grant ST/R001014/1. DiRAC is partof the National e-Infrastructure.

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th/9510135.

Appendix A: Symmetries of the supersymmetricand the bosonic Hamiltonians

As discussed in the literature [32, 33], the symmetrygroup of the bosonic Hamiltonian (4) is a finite non-Abelian group C4v with 8 elements. This group is gener-ated by the following transformations:

• Rotation R by π/2: x′1 = −x2, x′2 = x1. Rota-

tions act on wave functions as(RΨ)

(x1, x2) =

Ψ (x2,−x1). The group C4v contains R as well R2

and R3.

• Reflections P1 and P2 with respect to the horizontaland vertical lines x2 = 0 and x1 = 0, respectively:(P1Ψ

)(x1, x2) = Ψ (−x1, x2),

(P2Ψ

)(x1, x2) =

Ψ (x1,−x2).

• Reflections P+ and P− with respect to the diago-

nal lines x2 = ±x1:(P+Ψ

)(x1, x2) = Ψ (x2, x1),(

P−Ψ)

(x1, x2) = Ψ (−x2,−x1).

Together with the identity(IΨ)

(x1, x2) = Ψ (x1, x2),

these operators implement the functional representation

of C4v. It is straightforward to check that all these oper-ators commute with the bosonic Hamiltonian (4).

The group C4v has four Abelian irreps: the trivial irrepA1 as well as Z2-valued irreps A2, B1 and B2, which canbe inferred from the character tables in [43]. There isalso one real, two-dimensional, non-Abelian irrep E0. Itcoincides with the non-Abelian irrep E0 of the symmetrygroup of the supersymmetric Hamiltonian, see the Table(A5).

The symmetry group of the supersymmetric Hamil-tonian (6) is D4d, a finite non-Abelian group with 16elements [44]. The transformations that generate thisgroup are closely related to the transformations thatleave the bosonic Hamiltonian invariant, with some addi-tional operations that arise due to the fermionic structureof the Hilbert space of two-component wave functionsΨ (x1, x2) = {ψ (x1, x2) , χ (x1, x2)}:

1. Rotations R by π/2. Rotations act on the two-component wave functions as

(RΨ

)(x1, x2) =

I − ε√2

(RΨ)

(x1, x2) =

=I − ε√

2Ψ (x2,−x1) , (A1)

where I is the 2 × 2 identity matrix, and ε is the2 × 2 anti-symmetric matrix with ε12 = 1. Theunitary matrix I−ε√

2multiplies the two-component

vector Ψ (x1, x2) = {ψ (x1, x2) , χ (x1, x2)}. In con-

trast to the rotation operator R acting on theHilbert space of the bosonic Hamiltonian (4), for

the supersymmetric Hamiltonian the operator R isa fermionic representation of the rotation operator,for which the rotation by 2π results in a change ofsign: R4 = −I, where I is the identity operator.Correspondingly, the group D4d contains all pow-ers of R up to R7, and only the eighth power of Ryields the identity operator: R8 = I.

2. Reflections P1 and P2 with respect to the horizontaland vertical lines x2 = 0 and x1 = 0, respectively:

(P1Ψ

)(x1, x2) = σ3

(P1Ψ

)(x1, x2) = (A2)

σ3 Ψ (−x1, x2) ,(P2Ψ

)(x1, x2) = σ1

(P2Ψ

)(x1, x2) =

= σ1 Ψ (x1,−x2) , (A3)

where σ1 and σ3 are the Pauli matrices thatmultiply the two-component vector Ψ (x1, x2) ={ψ (x1, x2) , χ (x1, x2)}.

3. Reflections P± with respect to the diagonal lines

21

x2 = ±x1:(P+Ψ

)(x1, x2) =

σ3 + σ1√2

(P+Ψ

)(x1, x2) =

=σ3 + σ1√

2Ψ (x2, x1) ,(

P−Ψ)

(x1, x2) =σ3 − σ1√

2

(P−Ψ

)(x1, x2) =

=σ3 − σ1√

2Ψ (−x2,−x1) . (A4)

Note that because of the non-commutativity of Pauli ma-trices that enter P1,2 and P±, these transformations donot commute with each other. In contrast, reflectionsthat act on the Hilbert space of the bosonic Hamiltoniancommute with each other: P1P2 = P2P1, P+P− = P−P+.

The group D4d has four Abelian irreducible represen-tations (irreps): the trivial irrep A1 as well as Z2-valuedirreps A2, B1 and B2, which can be inferred from thecharacter tables in [44]. There are also three real, two-dimensional, non-Abelian irreps E0, E1 and E2. For com-pleteness, in Table A5 we list all the elements of the D4d

group together with the corresponding matrices of irrepsE0, E1 and E2. The format of this Table is the following:

• In the second column, we give the func-tional representation of the corresponding ele-ment as a direct product of the form UB ⊗UF , where the operator UB acts on each ofthe two components of the wave function as{(UBφ

)(x1, x2) ,

(UBχ

)(x1, x2)

}, and the matrix

UF multiplies these two components as a vector,as in equations (A1), (A2) and (A4). The opera-

tors UB generate a functional representation of thegroup C4v.

• In the third column, we give the symbol that cor-responds to this element in the character tables in[44].

• We note that first factors in the direct product ex-pressions in the second column are the elementsof C4v, the symmetry group of the bosonic Hamil-tonian (4). In the fourth column, we give thesymbol that corresponds to these element of C4v inthe character tables in [43].

• In the fifth column, we show how the opera-tors in the second column act on the basis state|k1, k2, ↑〉 = { |k1〉 |k2〉, 0} with the wave function{ψk1 (x1)ψk2 (x2) , 0}, where 〈x1|k1〉 = ψk1 (x1)and 〈x2|k2〉 = ψk2 (x2) are the one-dimensional har-monic oscillator wave functions (9). Note that wecan still apply the Wigner’s theorem to the finitematrices (12) and (10) of the Hamiltonian oper-ators (6) and (4) in the harmonic oscillator basis,because two-dimensional harmonic oscillator eigen-states |k1〉⊗ |k2〉 inherit an O (2) symmetry groupof the two-dimensional harmonic oscillator whichcontains both C4v and D4d as sub-groups.

• In the sixth, seventh and eighth columns we givethe matrices of non-Abelian irreps E0, E1 and E2that correspond to the transformations listed in thefirst and the second columns. All matrices are givenas combinations of 2×2 identity matrix I, the anti-symmetric matrix ε and the Pauli matrices σ1 andσ3. The matrices in the sixth column also form anon-Abelian irrep E0 of the group C4v (generated

by the operators UB , which are the first factors inthe direct products in the second column.)

22

g G (g) UB (g)⊗ UF (g) D4d C4v G (g) |k1, k2, ↑〉 E0 E1 E2

1 I I ⊗ I E E

|k1〉 |k2〉0

I I I

2 R R⊗ I−ε√2

S8 C4(−1)k1√2

|k2〉 |k1〉|k2〉 |k1〉

−ε I−ε√2

−I+ε√2

3 R2 R2 ⊗ (−ε) C4 C2 (−1)k1+k2

0

|k1〉 |k2〉

−I −ε −ε

4 R3 R3 ⊗(−I−ε√

2

)S38 C4

(−1)k2√2

− |k2〉 |k1〉|k2〉 |k1〉

ε −I−ε√2

I+ε√2

5 R4 I ⊗ (−I) C2 E

− |k1〉 |k2〉0

I −I −I

6 R5 R⊗ −I+ε√2

S38 C4

(−1)k1+1

√2

|k2〉 |k1〉|k2〉 |k1〉

−ε −I+ε√2

I−ε√2

7 R6 R2 ⊗ ε C4 C2 (−1)k1+k2+1

0

|k1〉 |k2〉

−I ε ε

8 R7 R3 ⊗(I+ε√

2

)S8 C4

(−1)k2√2

|k2〉 |k1〉

− |k2〉 |k1〉

ε I+ε√2

−I−ε√2

9 P1 P1 ⊗ σ3 C ′2 σv (−1)k1

|k1〉 |k2〉0

−σ3 σ3 σ3

10 P2 P2 ⊗ σ1 C ′2 σv (−1)k2

0

|k1〉 |k2〉

σ3 σ1 σ1

11 P+ P+ ⊗ σ3+σ1√2

σd σd1√2

|k2〉 |k1〉|k2〉 |k1〉

σ1σ3+σ1√

2−σ3−σ1√

2

12 P− P− ⊗ σ3−σ1√2

σd σd,(−1)k1+k2√2

|k2〉 |k1〉

− |k2〉 |k1〉

−σ1 σ3−σ1√2

−σ3+σ1√2

13 −P1 P1 ⊗ (−σ3) C ′2 σv (−1)k1+1

|k1〉 |k2〉0

−σ3 −σ3 −σ3

14 −P2 P2 ⊗ (−σ1) C ′2 σv (−1)k2+1

0

|k1〉 |k2〉

σ3 −σ1 −σ1

15 −P+ P+ ⊗ −σ3−σ1√2

σd σd − 1√2

|k2〉 |k1〉|k2〉 |k1〉

σ1−σ3−σ1√

2σ3+σ1√

2

16 −P− P− ⊗ −σ3+σ1√2

σd σd(−1)k1+k2√2

− |k2〉 |k1〉|k2〉 |k1〉

−σ1 −σ3+σ1√2

σ3−σ1√2

(A5)

By virtue of Wigner’s theorem, eigenstates of theHamiltonians (6) or (4) should form multiplets |Ψα

i 〉 withdegenerate energy levels that transform under one of the

irreps of the groups D4d (for HS) or C4v (for HB):

G (g) |Ψαi 〉 =

dR∑β=1

GRαβ (g) |Ψβi 〉, (A6)

23

where the index i labels distinct energy levels Ei, α =1 . . . dR labels all linearly independent eigenstates withH |Ψα

i 〉 = Ei |Ψαi 〉, GRαβ is the dR × dR matrix of the

group element g in the irrep R. In our case, the op-erators G (g) are listed in the first column of Table A5and defined in the second column of this Table. For ir-reps with dimension dR = 2 the matrices GRαβ (g) aregiven in the sixth, seventh and eighth columns. For one-dimensional irreps (dR = 1), GRαβ (g) ≡ GR (g) can be

easily reconstructed from character tables in [43, 44].To classify all eigenvectors of the Hamiltonian matrices

(6) and (4) according to irreps of the groups C4v or D4d,we can use the identity (A6) and the Schur orthogonalityrelations∑

g

GRαβ (g) GR′

γδ (g) = δRR′ δαγ δβδ|G|dR

, (A7)

where |G| is the total number of elements in the symme-try group of the Hamiltonian. To this end, we multiplythe equality (A6) by GRγδ (g) (that is equal to GRγδ (g) for

our real irreps) and sum over all group elements g:∑g

GR′

γδ (g)G (g) |Ψαi 〉 =

=

dR∑β=1

∑g

GR′

γδ (g)GRαβ (g) |Ψβi 〉 =

=|G|dR

δRR′ δαγ |Ψδi 〉. (A8)

Therefore if we apply all transformations within the sym-metry group to some eigenvector of the Hamiltonian ma-trix, and sum up the results with the weights given bythe elements of the matrices of some irrep R′, we only getnonzero result if R′ is the irrep of the multiplet to whichthis eigenvector belongs. Of course, for one-dimensionalirreps, the vector transforms into itself, and group trans-formations amount to multiplications by a complex phasefactor.

The identity (A8) can also be used to prove thateigenvectors of the supersymmetric Hamiltonian can onlytransform under the two-dimensional non-Abelian irrepsE1 and E2. Indeed, the matrices that multiply the two-component wave function as a vector (UF factors in thesecond column in Table (A5)) form the irrep E1. Thestructure of irreps E1 and E2 is such that for any groupelement g with representation matrix GRαβ (g) there is an-

other element g′ with GRαβ (g′) = −GRαβ (g). Looking at

the Table (A5) as well as on the character tables in [44],we notice that for all other irreps R′ = A1,2, B1,2, E0we have GR

′

αβ (g′) = +GR′

αβ (g), where g and g′ is thesame pair of group elements. Bosonic transformationsUB that transform the coordinate dependence of wavefunctions are also insensitive to the sign factors of E1.Therefore, for any R′ = A1,2, B1,2, E0 the product

GR′

γδ (g) UB (g) ⊗ UF (g) |Ψ〉 in the first line of Eq. (A8)

will be equal to −GR′

γδ (g′) UB (g′) ⊗ UF (g′) |Ψ〉. In the

sum over all group elements, summands that correspondto g and g′ will cancel each other. Therefore, for anyeigenstate of the supersymmetric Hamiltonian the sumin the first line of Eq. (A8) will be equal to zero forR′ = A1,2, B1,2, E0.

We conclude therefore that eigenstates of the super-symmetric Hamiltonian can only transform under the ir-reps E1 or E2 and therefore should all be doubly degen-erate once we impose the IR cutoff (finite M truncation)that makes the continuous spectrum discrete.

Appendix B: Block diagonal representation ofHamiltonian matrices within parity sectors and

basis state enumeration

In this work we perform numerical diagonalization ofthe Hamiltonian matrices (10) and (12). To achieve fasterconvergence of diagonalization algorithms, it is advan-tageous to reduce the size of matrices being diagonal-ized, for example, by representing them in block diagonalform. The largest reduction of the matrix block size couldbe achieved by considering combinations of basis states|k1〉⊗ |k2〉 that transform under irreps of the groups C4v

(for the bosonic Hamiltonian) or D4d (for the supersym-metric Hamiltonian). By virtue of Wigner’s theorem, theHamiltonian matrices (10) and (12) would then reduce toa block diagonal form such that each block correspondsto one of the irreps. Explicit construction of such basisstates is quite involved, and would result in a significantlymore complicated code.

In this work we choose a different but equally efficientstrategy and use an eigenbasis of a single element P1 =P1⊗σ3 of the symmetry groupD4d of the supersymmetricHamiltonian HS . Since P1 commutes with HS as well aswith the corresponding Hamiltonian matrix and P2

1 =

I, the Hamiltonian matrix should have a block diagonalform with blocks corresponding to ±1 eigenvalues of P1.Note that since the parity operators P1,2 and P± do notcommute with each other, only one of these operators canbe diagonalized simultaneously with the Hamiltonian.

An immediate consequence of the identity (A6) is thatany two states |Ψ1

i 〉, |Ψ2i 〉 that belong to a doublet of

states transforming under irreps E1 or E2 of the D4d sym-metry group should have opposite parities. As one caninfer from Table (A5), for both irreps the parity transfor-

mation P1 is represented by the Pauli matrix σ3. Rewrit-ing (A6) in explicit form, we obtain

P1 |Ψ1i 〉 = (σ3)11 |Ψ

1i 〉+ (σ3)12 |Ψ

2i 〉 = + |Ψ1

i 〉,P1 |Ψ2

i 〉 = (σ3)21 |Ψ1i 〉+ (σ3)22 |Ψ

2i 〉 = − |Ψ2

i 〉. (B1)

Together with the fact that eigenstates of HS can onlytransform under irreps E1 or E2, this observation impliesthat both diagonal blocks of the supersymmetric Hamil-tonian matrix (12) have identical energy spectra. Usingthe fact that the operators x2 and p2 are invariant under

24

P1, one can also show that the products of matrix ele-ments of x2 and p2 that enter the OTOCs C (t) in (23)are identical for both parity sectors. Therefore the contri-butions of both diagonal blocks to OTOCs are identical,which allows us to save CPU time by diagonalizing onlythe positive-parity block of the supersymmetric Hamilto-nian matrix. Note that this does not apply to the bosonicHamiltonian, for which both parity sectors need to be di-agonalized in order to get the full OTOCs.

All harmonic oscillator eigenstates (9) have a defi-nite parity under coordinate reflections, therefore the di-rect product states |k1〉 ⊗ |k2〉 are also eigenstates of

the bosonic parity operator P1 with P1 |k1〉 ⊗ |k2〉 =

(−1)k1 |k1〉 ⊗ |k2〉. Using this fact, it is straightforward

to find the two-component eigenstates of the operatorP1 = P1 ⊗ σ3 with eigenvalues ±1:

|k1, k2,+〉 =

(|k1〉 ⊗ |k2〉

0

), k1 = 2m,

|k1, k2,+〉 =

(0

|k1〉 ⊗ |k2〉

), k1 = 2m+ 1,

|k1, k2,−〉 =

(|k1〉 ⊗ |k2〉

0

), k1 = 2m+ 1,

|k1, k2,−〉 =

(0

|k1〉 ⊗ |k2〉

), k1 = 2m,(B2)

where m = 0, 1, 2, . . .. Matrix elements〈k1, k2,+| HS |k1, k2,−〉 are equal to zero because

HS commutes with P1.To work with the basis states (B2), we need a way to

efficiently enumerate the states |k1〉⊗ |k2〉 with k1+k2 ≤M and a definite parity of k1. We use the enumerationshown on Fig. 13. Red and blue points have positiveand negative parity P1 (even and odd k1). The labelsof points on the plot have the format i : k1, k2, where iis the serial number of the state. We enumerate statesseparately in each of the two parity sectors. If we limitthe values of i to lie in the range i = 0 . . .M2 + M −1, the set of basis states is symmetric with respect tothe interchange x1 ↔ x2. Since this is one of the basicsymmetries of our system, we always use ranges of thisform.

The transformation from k1, k2 to the one-dimensionalindex i can be written as

i = 2(div (k1, 2) + div

(r + r2, 2

))+

+mod (k2, 2) , r = div (k1, 2) + div (k2, 2) , (B3)

where div (a, b) and mod (a, b) are the integer division ofa over b and the integer modulo of a over b. The inversetransformation is

k1 = 2(div (i, 2)− div

(r + r2, 2

))+ p1,

k2 = 2(r + div

(r + r2, 2

)− div (i, 2)

)+

+mod (i, 2) , r =

⌊√2div (i, 2) +

1

4− 1

2

⌋, (B4)

0: 0,0

1: 0,1

2: 0,2

3: 0,3

4: 2,0

5: 2,1

6: 0,4

7: 0,5

8: 2,2

9: 2,3

10: 4,0

11: 4,1

0: 1,0

1: 1,1

2: 1,2

3: 1,3

4: 3,0

5: 3,1

6: 1,4

7: 1,5

8: 3,2

9: 3,3

10: 5,0

11: 5,1

0 1 2 3 4 5

0

1

2

3

4

5

k1

k 2

FIG. 13. Enumeration of two-dimensional harmonic oscillatorbasis states |k1〉⊗ |k2〉. States that correspond to red and bluepoints are even and odd under reflections of x1, respectively.The label “i : k1, k2” near each point shows the serial number iof the state and the corresponding values of k1 and k2. Stateswith even and odd x1-parity are enumerated independently.For this plot, the index i changes between 0 and M2 +M − 1with M = 3.

where b. . .c is the floor function and (−1)p1 is the parity

sign: p1 = 0, p1 = 1 for the positive and negative paritysectors, correspondingly.

Note that for the bosonic Hamiltonian (4), we could

have used the two commuting operators P1 and P2 (or

P+ and P−) to define four parity sectors and thus reducethe size of matrix blocks even further. However, herewe only used P1 to reduce the diagonal block size of thebosonic Hamiltonian matrix (10). The reason is thatthe primary focus of this paper is the supersymmetricHamiltonian (6), and for debugging purposes we wantedto maintain state enumeration that would be similar forboth Hamiltonians. Even with this choice, the bosonicHamiltonian is much easier to treat numerically than thesupersymmetric one, and using only P1 parity is enoughto obtain good-quality numerical data for the bosonicmodel.

Appendix C: Calculating the classical analogue ofthermal OTOCs

To check whether in the high temperature limit thequantum dynamics of the bosonic and supersymmetricHamiltonians (4) and (6) agrees with the classical dy-namics of the bosonic Hamiltonian, we calculate the clas-sical analogue of OTOCs. It is given by the thermal

25

expectation value of the square of the Poisson brackets {x2 (t) , p2 (0)} = ∂x2(t)∂x2(0)

:

Ccl (t) =1

Zcl

∫dx1 (0) dx2 (0) dp1 (0) dp2 (0) exp

(−p

21 (0) + p22 (0) + x21 (0) x22 (0)

T

) (∂x2 (t)

∂x2 (0)

)2

, (C1)

where x1 (0), x2 (0), p1 (0), p2 (0) specify the initial con-ditions for the time-dependent coordinates and momentax1 (t), x2 (t), p1 (t), p2 (t). These in turn satisfy the clas-sical equations of motion

d

dtx1 (t) = 2 p1 (t) ,

d

dtx2 (t) = 2 p2 (t) ,

d

dtp1 (t) = −2x1 (t) x22 (t) ,

d

dtp2 (t) = −2x2 (t) x21 (t) . (C2)

The classical partition function Zcl is

Zcl =

∫dp1 dp2 exp

(−p

21 + p22T

)×

×∫dx1 dx2 exp

(−x

21x

22

T

). (C3)

To obtain the classical Poisson bracket {x2 (t) , p2 (0)} =∂x2(t)∂x2(0)

, we can differentiate the classical equations of mo-

tion with respect to x2 (0), thereby obtaining a system ofdifferential equations that govern the time evolution of∂x1(t)∂x2(0)

, ∂x2(t)∂x2(0)

, ∂p1(t)∂x2(0)

and ∂p2(t)∂x2(0)

:

d

dt

∂x1 (t)

∂x2 (0)= 2

∂p1 (t)

∂x2 (0),

d

dt

∂x2 (t)

∂x2 (0)= 2

∂p2 (t)

∂x2 (0),

d

dt

∂p1 (t)

∂x2 (0)= −2

∂x1 (t)

∂x2 (0)x22 (t)− 4x1 (t)x2 (t)

∂x2 (t)

∂x2 (0),

d

dt

∂p2 (t)

∂x2 (0)= −2

∂x2 (t)

∂x2 (0)x21 (t)− 4x1 (t)x2 (t)

∂x1 (t)

∂x2 (0). (C4)

These equations have to be solved simultaneously withthe classical equations of motion (C2) with the initial

conditions ∂x1(0)∂x2(0)

= 0, ∂x2(0)∂x2(0)

= 1, ∂p1(0)∂x2(0)

= 0, ∂p2(0)∂x2(0)

= 0.

To calculate the “classical OTOC” (C1), we carry outa small-scale Monte-Carlo simulation, generating O

(105)

random initial conditions x1 (0), x2 (0), p1 (0), p2 (0)with the probability distribution that is proportional to

exp(−p

21(0)+p

22(0)+x

21(0) x

22(0)

T

). The evolution equations

(C2) and (C4) are then solved numerically, and(∂x2(t)∂x2(0)

)2is averaged over sufficiently many random initial condi-tions. A technical difficulty is that both the classicalpartition function (C3) and the probability distributionof initial coordinate values x1 (0), x2 (0) contain a non-

normalizable weight function exp(−x

21x

22

T

). To regular-

ize the diverging integrals∫dx1dx2 exp

(−x

21x

22

T

)and to

generate x1 (0) and x2 (0) with the required probabilitydistribution, we express x1 and x2 in terms of “hyper-bolic” coordinates −∞ < r < +∞ and −∞ < φ < +∞as x1 = r eφ, x2 = r e−φ. In terms of the new coordi-

nates, the above integrals can be written as∫dx1dx2 exp

(−x

21x

22

T

)= 4

+∞∫−∞

dφ

+∞∫0

dr2 e−r4

T .(C5)

We see that the integral divergence is related to the in-finite integration limits for the “hyperbolic angle” vari-able φ, and the probability distribution of the “radial”coordinate r is perfectly normalizable. To regularize theintegrals over φ and make the probability distribution in(C1), we introduce a cutoff −φmax < φ < φmax on theφ variable. While the integrals in the classical partitionfunction Zcl and in the Lyapunov distance definition (C1)are divergent, this divergence cancels out in the ratio ofthe two integrals. As a result, Ccl (t) is practically inde-pendent of the cutoff φmax once φmax is large enough. Inpractice, we used φmax in the range 2 < φmax < 5 andobserved that the results are independent of φmax withinstatistical errors of Monte-Carlo expectation values.

The expression (5) is obtained by fitting log (Ccl (t)) atsufficiently late times with a linear function. The corre-sponding slope is twice the classical Lyapunov exponentin (5).