Thermo eld dynamics: Quantum Chaos versus Decoherence2020/08/17 · Complex partition functions take a new meaning in the context of thermo eld dynamics [19]. Given an equi-librium

Thermofield dynamics: Quantum Chaos versus Decoherence

Zhenyu Xu,1 Aurelia Chenu,2, 3, 4 Tomaž Prosen,5 and Adolfo del Campo2, 3, 6

1School of Physical Science and Technology, Soochow University, Suzhou 215006, China2Donostia International Physics Center, E-20018 San Sebastián, Spain

3IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain4Massachusetts Institute of Technology, Cambridge, MA 02139, USA

5Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, 1000 Ljubljana, Slovenia6Department of Physics, University of Massachusetts, Boston, MA 02125, USA

(Dated: August 17, 2020)

Quantum chaos imposes universal spectral signatures that govern the thermofield dynamics of amany-body system in isolation. The fidelity between the initial and time-evolving thermofield doublestates exhibits as a function of time a decay, dip, ramp and plateau. Sources of decoherence giverise to a nonunitary evolution and result in information loss. Energy dephasing gradually suppressesquantum noise fluctuations and the dip associated with spectral correlations. Decoherence furtherdelays the appearance of the dip and shortens the span of the linear ramp associated with chaoticbehavior. The interplay between signatures of quantum chaos and information loss is determinedby the competition among the decoherence, dip and plateau characteristic times, as demoonstratedin the stochastic Sachdev-Ye-Kitaev model.

In an isolated many-body quantum system, quantumchaos imposes universal spectral signatures such as theform of the eigenvalue spacing distribution. The latterchanges from a Poissonian to a Wigner-Dyson distribu-tion as the integrability of the system is broken to makeit increasingly chaotic. Such a change in the propertiesof the system can often be induced, e.g. in many-bodyspin systems, by changing a control parameter [1–4].

The Fourier transform of the eigenvalue distributionwas soon recognized as a convenient tool to diagnosequantum chaos [5–8]. The partition function of the sys-tem analytically continued in the complex-temperatureplane has more recently been considered [9–11], and itreduces to the former for a purely imaginary inverse tem-perature β = it. The quantity Z(β + it) is indeed thecomplex Fourier transform of the density of states andits absolute square value is related to the spectral formfactor. It is also related to the Loschmidt echo [12–14]and quantum work statistics [15–18].

Complex partition functions take a new meaning inthe context of thermofield dynamics [19]. Given an equi-librium thermal state of single copy of a quantum sys-tem, it is often convenient to consider its purificationin an enlarged Hilbert space, which is given by a spe-cific entangled state between the original and a secondcopy of the system. The resulting thermofield double(TFD) state was recognized early on to be useful in es-timating thermal averages of observables [19]. The TFDplays also a prominent role in the description of eternalblackholes and wormholes in AdS/CFT. The fidelity be-tween a given TFD and its time evolution under unitarydynamics is precisely described by the complex Fouriertransform of the eigenvalue distribution, specifically, bythe absolute square value of the partition function withcomplex-valued temperature [11, 20].

Unitarity imposes important constraints on the ther-

mofield dynamics. The time-evolved state may exhibithighly non-trivial quantum correlations, but the infor-mation encoded in the initial state, once scrambled, canin principle be recovered by reversing the dynamics inan idealized setting. As a result, the von Neumann en-tropy of the system remains constant during the evo-lution. This feature remains true for the mixed stateresulting from averaging over a Hamiltonian ensemble.The spectral form factor in an isolated chaotic systemdisplays a decay from unit value leading to a dip, alsoknown as correlation hole, a subsequent ramp, and a sat-uration at an asymptotic plateau, in systems character-ized by a finite Hilbert space dimension [9–11, 21], whileits somewhat simpler structure in Floquet many-bodysystems is only recently becoming analytically explained[22–24]. Yet, isolated quantum systems are an idealiza-tion. Any realistic quantum system is embedded in a sur-rounding environment, the rest of the universe. Decoher-ence stems from the interaction between the system andthe surrounding environment, which leads to the build-up of quantum correlations between the two, and theirentanglement. The environment is generally expected tobe complex and its degrees of freedom unavailable. In-formation loss in the system can be traced back to theleakage of information into the inaccessible environment.The dynamics of the system is non-unitary and its vonNeumann entropy is no longer constant [25, 26].

The interplay between spectral signatures of quantumchaos, decoherence and information loss is thus a long-standing open problem [2, 27–30]. We focus on its rolein thermofield dynamics, with applications ranging fromnon-equilibrium many-body physics to machine learningwith quantum neural networks in noisy intermediate-scale quantum computers and simulators [31]. As weshall see, (energy) decoherence washes out short time sig-natures of quantum chaos, such as the dip in the spectral

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form factor (correlation hole), while it preserves its longtime ramp, conditioned by a competition of characteristictime scales that we elucidate. As a test-bed, we considerSachdev-Ye-Kitaev (SYK) model of Majorana fermionsinvolving all-to-all four-body interactions with quencheddisorder [32, 33] which saturates the bound on chaos andadmits a gravitational dual, making it a prominent ex-ample [34, 35] of holography [36].

Setting.— Consider a system described by a Hamil-tonian H, with spectral decomposition in the systemHilbert space H given by H =

∑dn=1En|n〉〈n|, En be-

ing the energy eigenvalues. A canonical thermal stateof the system at inverse temperature β is described bythe operator ρth = e

−βH/Z(β), the partition functionof the system being Z(β) = Tr(e−βH). The thermaldensity matrix can be obtained from a pure, entan-gled state in an enlarged Hilbert space H̃ = H ⊗ H.Namely, a second copy of the system is used to cre-ate the state known as the thermofield double (TFD)state [19] and defined as |TFD〉 =

∑n

√pn|nn〉 where

pn = e−βEn/Z(β) and |nn〉 = |n〉 ⊗ |n〉 in H̃. The

reduced density matrix obtained by tracing over anyone of the two copies,

∑n〈n|TFD〉〈TFD|n〉, corresponds

to the single-copy canonical thermal state ρth. Notethat the TFD is not invariant under the unitary Ut =exp

[− it(H ⊗ 1 + 1⊗H)

], taking ~ = 1.

The fidelity between the initial TFD state and its evo-lution provides a measure of quantum chaos [11, 20]

Ft = |〈TFD|Ut|TFD〉|2 =∣∣∣∣Z(β + i2t)Z(β)

∣∣∣∣2 . (1)In the presence of decoherence, the evolution is not uni-tary and can generally be associated with a quantumchannel Λt that maps the initial density matrix to thetime-evolved state, i.e., ρt = Λt[ρ0]. The fidelity be-tween two mixed states ρ0 and ρt generalizes the no-tion of overlap between pure states. It is defined as(Tr√√

ρ0ρt√ρ0)2

and takes a particularly simple formwhen the initial state is pure. We shall thus be interestedin the fidelity between the initial (pure) TFD state andits evolution, i.e.,

Ft = 〈TFD|Λt[ρ̃0]|TFD〉, (2)

where ρ̃0 = |TFD〉〈TFD| is of dimension d2. Said dif-ferently, Ft equals the probability to find the state ρ̃t attime t in the initial state, i.e., it is the survival probabil-ity of the TFD state. Note that under unitary evolution,Λt[ρ̃0] = Utρ̃0U

†t and Eq. (1) is recovered.

For the sake of illustration, we shall consider the quan-tum channel associated with energy diffusion processesthat occur independently in each of the copies. Thetotal Hamiltonian H̃ = H ⊗ 1 + 1 ⊗ H is perturbedby independent real Gaussian white noise in each copy,H → (1 + √γξt)H, where γ is a positive real constant,and ξt is the noise parameter. Performing the stochastic

average, the evolution of ρ̃t is described by the masterequation [37, 38]

˙̃ρt = −i[H̃, ρ̃t

]− γ

2

∑k

[Vk, [Vk, ρ̃t]] , (3)

with the Lindblad operators V1 = H⊗1 and V2 = 1⊗H.For the initial TFD state, the exact time evolution of

the density matrix is given by

ρ̃t =∑m,n

e−β(Em+En)

2

Z(β)e−2it(Em−En)−γt(Em−En)

2

|mm〉〈nn| ,

(4)and the fidelity (2) of the evolved mixed state reads

Ft =1

Z(β)2

∑m,n

e−β(Em+En)−2it(Em−En)−γt(Em−En)2

.

(5)From this expression it is apparent that, in the absenceof degeneracies in the energy spectrum, the asymptoticvalue of the fidelity is given by Fp = Z(2β)/Z(β)

2, i.e.,the purity of a single-copy thermal state at inverse tem-perature β. This value also corresponds to the long-timeasymptotics under unitary evolution, which can be ob-tained from Eq. (1) by coarse-graining in time [11, 39].In the infinite temperature case, the value Fp = 1/d re-flects the finite Hilbert space dimension. Thus, Fp isinsensitive to the presence of information loss.

For arbitrary time t, an explicit expression of thefidelity can be obtained using the density of states%(E) =

∑δ(E − En) written in the integral form,

%(E) =∫dτeiτETr(e−iτH)/(2π). Use of the Hubbard–

Stratonovich transformation allows us to recast the fi-delity (5) in terms of the analytic continuation of thepartition function [40],

Ft =1

2√πγt

∫ +∞−∞

dτe−(τ−2t2√γt

)2gβ(τ), (6)

as the spectral form factor is given by

gβ(τ) ≡|Z(β + iτ)|2

Z2(β), (7)

and equals the fidelity under unitary dynamics at τ = 2t,see Eq. (1). The latter is an even function of the pa-rameter τ , i.e., gβ(−τ) = gβ(τ). This quantity containsinformation about the correlation of eigenvalues with dif-ferent energies. At late times, it forms a plateau, with avalue Z(2β)/Z(β)2 in absence of degeneracies in the en-ergy spectrum, that characterizes the discreteness of thespectrum [9].

The expression (6) paves the way to a systematic studyof the interplay between quantum chaos and informa-tion loss, provided the energy spectrum of the system isknown. In addition, it shows that noise-induced decoher-ence is equivalent to coarse-graining in time the spectral

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form factor with a specific Gaussian kernel. The quan-tity√γt determines the contribution of the spectral form

factor to the integral at any time t, i.e., to the fidelity. Inthe unitary limit, γ = 0, the Gaussian is sharply peakedand tends to a Dirac delta around 2t, leading to the re-covery of the spectral form factor in Eq. (1). For γ > 0,information is lost. Yet, at long times of evolution, thebehavior with and without decoherence agree. This isconsistent with the fact that the long-time asymptoticplateau is associated with a state diagonal in the energyeigenbasis. The later is the fixed point under the nonuni-tary dephasing dynamics considered but it also emergeseffectively under unitary dynamics with coarse-grainingin time. The effect of dephasing is thus crucial in the timeregion γt ∼ 1. This behavior is universal in that it arisesfrom the open quantum dynamics considered (3) and isindependent of the specific choice of the nondegeneratesystem Hamiltonian.

The Sachdev-Ye-Kitaev model.— In what follows weshall use as a test-bed the SYK model with Hamiltonian[32, 41]

H =∑

1≤k

4

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γ=1γ=��-� ��

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��

○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ��/�� /� ��

-��

γτ �

○ ○ ○ ○ ○ ○

○ ○ ○ ○○□ □ □ □ □ □ □ □ □ □□ �� /� �� /�

○ □

��

��

�

� �(γ)

○ ○ ○ ○○ ○ ○

○ ○ ○○

□ □ □ □□ □ □

□ □ □ □ γ=� ��

��

�

� �(γ)γτ �/�

�

Numerical Numerical

γ=��0100 realizationsSingle realization

Numerical○ γ=0 γ=10 ○ □γ=0 γ=10

��

��

γ=�

○ ○ ○ ○ ○ ○ ○ ○ ○ ○○��

��-��-�� Numerical○

2��/(�� /�)

FIG. 2. Fidelity of the stochastic SYK model. Top: Alog-log plot of fidelity with different decoherence coefficientsγ in the SSYK model of N = 26 Majorana Fermions. Thedata was taken by single and 100 independent samples and

β = 1. Bottom: The dip time t(γ)d , the plateau time t

(γ)p , the

decoherence time γτD, and γτD/td are shown as a function ofN .

is dominated by correlations in the eigenvalue spacingand leads to (ii) a ramp that eventually saturates in (iii)a plateau with value Fp = Z(2β)/Z(β)

2 onset at theHeisenberg or plateau time tp ∼ d. Specifically, for theSYK model we find [40]

tp ' α√

2π

Nd, (11)

with α = 2 − δ4,N mod 8. This late stage is character-ized by fluctuations around the plateau value, sometimesrefer to as quantum noise in this context [39] to be dis-tinguished from the kind of quantum noise that givesrise to decoherence [59]. The characteristic times τD, tdand tp govern the competition between decoherence andquantum chaos.

Figure 2 shows the evolution of the fidelity for a finite-temperature TFD in a single realization and the disor-der average over Jklmn. As the dephasing strength γ isincreased, the features of the fidelity Ft manifested inthe unitary case are gradually washed out. Prominently,for large dephasing strengths, τD � tp, the existence ofthe dip and ramp are completely suppressed and the fi-delity decays monotonically from unit value towards theasymptotic one Fp. Between these extremes, the featuresthat are more sensitive to information loss are those as-sociated with quantum noise, e.g., the dynamical phaseaccumulated by the total Hamiltonian.

Under unitary dynamics, these fluctuations are exhib-ited around the dip and at long times: The decay to-wards the dip is typically characterized by a power-lawgiven that the density of states is bounded from below,i.e., the existence of a ground state [11] (while this effectcan be removed by smooth spectral filtering [60]). Asa precursor of the dip, an oscillatory behavior is oftenpresent that can be understood as the interference of thepower-law contribution and the ramp contribution stem-ming from correlations in the level spacing distribution.Whenever τD ≤ td, information loss leads to the sup-pression of these fluctuations. Regarding the presence ofquantum noise at long times, whenever τD < tp, equation(5) shows that information loss associated with decoher-ence suppresses the fluctuations around the plateau valueFp. Importantly, the suppression of quantum noise fluc-tuations is already manifest at the level of a single real-ization of the SYK Hamiltonian, without averaging overJklmn or ensembles of system Hamiltonians. As shownin [40] the behavior of the SYK models is in qualitativeagreement with that of random-matrix ensembles. Whileunder unitary dynamics this correspondence is only es-tablished at long times, its onset is facilitated by thepresence of information loss. The decoherence time τDscales with 1/γ and the inverse energy variance. It thusdecreases with temperature and the system size, as shownfor the SYK in Fig 2. In the presence of information loss,the dip not only becomes shallower, but it shifts to latertimes; see Fig 2. For moderate values of the dephasingstrength γ the subsequent ramp is essentially unaffectedwith respect to the unitary dynamics, beyond the sup-pression of quantum noise. The time scale tp in which theplateau appears remains essentially constant. Thus, theramp and plateau are shared by isolated and decoheringsystems exhibiting information loss.

Discussion and summary.— An experimental test ofthe interplay between quantum chaos and decoherencecan be envisioned given advances in the quantum simula-tion of open systems by digital methods [61] and tailorednoise. It could be probed via the quantum simulationof the SYK Hamiltonian [42–45] but it is generally ex-pected in an arbitrary quantum chaotic system. Whilethe preparation of the TFD state is being pursued [62–65] this requirement can be relaxed for the study of someobservables, such as the fidelity of a TFD state, as its ex-pectation value can be related to that of a coherent Gibbsstate |ψβ〉 =

∑n e−βEn/2|n〉/

√Z(β) involving a single

copy of the system. Indeed, under unitary time evolu-tion Ft = |〈ψβ | exp(−itH)ψβ〉|2 = |Z(β+ it)/Z(β)|2 thatcan be measured by single-qubit interferometry [66]. Itsgeneral time-evolution can be described by a quantumchannel ρt = Λt[|ψβ〉〈ψβ |] and the fidelity between theinitial state and its evolved form is analogously given byFt = 〈ψβ |ρt|ψβ〉. The measurement of the later can besimplified using quantum algorithms for the estimationof state overlaps [67, 68].

5

In summary, the ubiquity of noise sources gives riseto a competition between the signatures of quantumchaotic dynamics expected for a many-body system inisolation and the presence of information loss resultingfrom decoherence. Such competition can be quantifiedby the fidelity between a thermofield state at a giventime and its subsequent time evolution. For a quantumchaotic system in isolation this quantity equals the spec-tral form factor showing a dip-ramp-plateau structurewhich is suppressed by the loss of information inducedby decoherence. The interplay between information lossand scrambling in open quantum complex systems shouldfind broad applications in quantum computation, simu-lation and machine learning in the presence of noise.

Acknowledgements.— It is a pleasure to acknowledgediscussions with Julian Sonner, Tadashi Takayanagi, andJacobus Verbaarschot. TP acknowledges support byERC Advanced grant 694544 OMNES and the programP1-0402 of Slovenian Research Agency. This work is fur-ther supported by ID2019-109007GA-I00.

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6

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7

I. Fidelity in terms of spectral form factor

We start with the expression of the fidelity in Eq. (5), and use the Hubbard–Stratonovich transformation

e−γt(Em−En)2

=1

2√πγt

∫ +∞−∞

dye−y2

4γt e−iy(Em−En), (S1)

to obtain a universal expression related to the normalized spectral form factor. The fidelity with Eq. (S1) is of theform

Ft =1

2√πγt

∫ +∞−∞

dye−y2

4γt1

Z2(β)

∑m,n

e−(β+2it+iy)Eme−(β−2it−iy)En

=1

2√πγt

∫ +∞−∞

dye−y2

4γt|Z (β + 2it+ iy)|2

Z2(β). (S2)

By changing the variable τ = y + 2t, Eq. (S2) can be further simplified to

Ft =1

2√πγt

∫ +∞−∞


)2gβ(τ), (S3)

with the spectral form factor

gβ(τ) ≡|Z (β + iτ)|2

Z2(β), (S4)

which is the expression given in Eq. (6) of the main text.

II. Logarithmic negativity and the second Rényi entropy

The logarithmic negativity is defined as

EN (ρ̃t) = log2

∥∥∥ρ̃ΓLt ∥∥∥1, (S5)

in terms of the trace norm of the partial transpose of the density matrix, while the second Rényi entropy is

S2(ρ̃t) = − log2 trρ̃2t . (S6)

If the initial state is the thermal field double state, i.e.,

ρ̃0 = |TFD〉〈TFD| =1

Z(β)

∑k,`

e−β2 (Ek+E`)|k〉|k〉〈`|〈`|, (S7)

the above Eqs. (S5) and (S6) can be written as

EN (ρ̃t) = log2

[1

Z(β)

∑k`

e−β(Ek+E`)/2−γt(Ek−E`)2

], (S8)

and

S2(ρ̃t) = − log2 (Pt) with Pt =1

Z(β)2

∑k,`

e−β(Ek+E`)−2γt(Ek−E`)2

. (S9)

Then we have

EN (ρ̃t(2β, 2γ)) + S2(ρ̃t(β, γ)) = log2Z(β)2

Z(2β). (S10)

Thus, the growth of the logarithmic negativity implies the decay of the second Rényi entropy, and viceversa, i.e.,ĖN (ρ̃t(2β, 2γ)) = −Ṡ2(ρ̃t(β, γ)).

8

III. Fidelity in terms of density of states and form factor

In what follows, we make explicit the connection between the fidelity, the density of states and the spectral formfactor. The density of states is defined as

%(E) =∑n

Nnδ(E − En),

where Nn denotes the degeneracy of the energy level En.The thermal state of a single copy can be written as

ρth =1

Z(β)

∫dEσ(E)e−βE |E〉〈E|.

Its purification is given by the thermofield double state

|Ψ〉 = 1√Z(β)

∫dE√%(E)e−βE/2|E,E〉.

The initial density matrix associated with the thermofield double state can then be written using a basis of continuousenergy eigenstates as

ρ̃0(E,E′) =

∫dEdE′

e−β(E+E′)/2

Z(β)

√%(E)%(E′)|E,E〉〈E′, E′|.

In turn, the time-evolved density matrix reads

ρ̃t(E,E′) =

∫dEdE′

e−β(E+E′)/2

Z(β)

√%(E)%(E′)e−2it(E−E

′)e−γt(E−E)′2/2|E,E〉〈E′, E′| (S11)

=

√1

4πγt

∫dEdE′

∫ ∞−∞

dye−y2

4γte−β(E+E

′)/2

Z(β)

√%(E)%(E′)e−2it(E−E

′)e−iy(E−E′)|E,E〉〈E′, E′| (S12)

and the fidelity becomes

Ft =

√1

4πγt

∫ ∞−∞

dye−y2

4γt1

Z(β)2

∣∣∣∣∫ dEσ(E)e−(β−2it+iy)E∣∣∣∣2 .The evaluation of such expression generally requires the use of numerical methods due to the lack of techniques

to evaluate the average of the quotient of partition functions, each of which involving the Hamiltonian over whichthe average is performed. Under the annealed approximation, this average is approximated by the quotient of theaverages. This approximation is generally valid at high temperature and fails at low temperature. Using it, theaverage fidelity reads

〈Ft〉 =√

1

4πγt

∫ ∞−∞

dye−y2

4γt1

〈Z(β)2〉

∫dEdE′e−ν(y)E−ν̄(y)E

′〈%(2)(E,E′)〉,

where

ν(y) = β − 2it+ iy, ν̄(y) = β + 2it− iy.

The two-level correlation function 〈%(2)(E,E′)〉 = 〈%(E)%(E′)〉 can be expressed in terms of the connected two-levelcorrelation function 〈%(2)c (E,E′)〉

〈%(2)c (E,E′) = 〈%(E,E′)〉 − 〈%(E)〉〈%(E′)〉.

Assuming no degeneracies

Ft =G(β)

Z(β)2+

1

Z(β)2

∑k 6=`

e−β(Ek+E`)+i2t(Ek−E`)−γt(Ek−E`)2

. (S13)

where

G(β) =∑n

N2ne−2βEn =

∫dE〈%(2)(E,E)〉e−2βE ,

reduces to Z(2β) in the absence of degeneracies expected for chaotic systems, i.e., when Nn = 1.

9

IV. Ensemble average of the fidelity

The ensemble average over the fidelity of Eq. (6) in the main text can be written as

〈Ft〉 =1

2√πγt

∫ +∞−∞


)2〈gβ(τ)〉 , (S14)

where the averaged spectral form factor in terms of annealing approximation is given by

〈gβ(τ)〉.=

〈|Z (β + iτ)|2

〉〈Z(β)〉2

. (S15)

Note that the annealing approximation is valid at high temperature (also see discussions in the end of Sec. V-A).The density of states is defined as

%(E) =∑n

Nnδ(E − En),

where Nn denotes the degeneracy of the energy level En. Then the denominator and numerator of Eq. (S15) can bewritten as

〈Z(β)〉 =∫dE〈%(E)〉e−βE , (S16)

and 〈|Z (β + iτ)|2

〉=

∫dE〈%(E)2〉e−2βE +

∫dEdE′〈%(E)%(E′)〉e−(β+iτ)Ee−(β−iτ)E

′, (S17)

where the two-point correlation function can be expressed as

〈%(E)%(E′)〉 = 〈%(2)c (E,E′)〉+ 〈%(E)〉〈%(E′)〉, (S18)

in terms of the connected two-point correlation function %c(E,E′).

In the following, we consider two examples, one is the GUE, and the other is the SYK model.

A. GUE-averaged Fidelity

For GUE ensembles, there are no degeneracy of the energy levels, so Eq. (S17) can be further written as〈|Z (β + iτ)|2

〉GUE

= 〈Z(2β)〉GUE + |〈Z(β + iτ)〉GUE|2

+〈gcβ(τ)

〉GUE

, (S19)

where 〈gcβ(τ)

〉GUE

=

∫dEdE′〈%(2)c (E,E′)〉GUEe−(β+iτ)Ee−(β−iτ)E

′. (S20)

The joint probability density of H ∈GUE is proportional to exp(− 12σ2 trH2), where σ is the standard deviation of

the random (off-diagonal) matrix elements of H. Note that in Ref. [51], σ = 1/√

2, and in Ref. [9], σ = 1/√d. To

calculate Eq. (S19), we have to know the spectral density and the two-point correlation function. The eigenvaluedensity averaged over GUE is given by

〈%(E)〉GUE =1√2σKd

(Ẽ, Ẽ

), and Ẽ :=

E√2σ

, (S21)

with the kernel Kd(x, y) defined by

Kd(x, y) =

d−1∑l=0

φl(x)φl(y), and φl(x) :=e−

x2

2 Hl(x)√√π2ll!

, (S22)

10

where Hl(x) are the Hermite polynomials. Furthermore, the two-point correlation function averaged over GUE takesthe form

〈%(E,E′)〉GUE =1

2σ2det

[(Kd(Ẽ, Ẽ) Kd(Ẽ, Ẽ

′)

Kd(Ẽ′, Ẽ) Kd(Ẽ

′, Ẽ′)

)], (S23)

and thus the connected two-level correlation function averaged over GUE reads

〈%(2)c (E,E′)〉GUE = −1

2σ2

(Kd(Ẽ, Ẽ

′))2. (S24)

According to the orthogonality of Hermite polynomials∫dxe−(x+a)

2

Hk(x)Hl(x)=√π2pp!(−2a)|k−l|L(|k−l|)p (−2a2), (S25)

where L(α)n (·) are the associated Laguerre polynomials, and p:=min{k, l}, the first two items of Eq. (S19) and the

denominator of Eq. (S15) is expressed as

〈Z(x)〉GUE = eσ2x2

2 L(1)d−1

(−σ2x2

). (S26)

By Eqs. (S24) and (S25), the third term of Eq. (S19) is

〈gcβ(τ)

〉GUE

= −eσ2(β2−τ2)

d−1∑k,l=0

p!

q!

(σ2(β2 + τ2)

)|k−l| ∣∣∣L(|k−l|)p (−σ2(β + iτ)2)∣∣∣2 , (S27)where q :=max{k, l}.

With Eqs. (S26) and (S27), the spectral form factor is finally obtained

〈gβ(τ)〉GUE.=

eσ2β2L

(1)d−1

(−4σ2β2

)+ e−σ

2τ2[∣∣∣L(1)d−1 (−σ2β2τ)∣∣∣2 −∑d−1k,l=0 p!q! (σ2 |βτ |2)|k−l| ∣∣∣L(|k−l|)p (−σ2β2τ)∣∣∣2]

L(1)d−1 (−σ2β2)

2,

(S28)with βτ := β+ iτ for short. Note again, one should replace τ with 2t when directly analyzing the spectral form factor.

To have a rough estimation of the dip and plateau time, we will consider an approximated connected two-levelcorrelation function of Eq. (S24) when the dimension of GUE is large, i.e.,

〈%(2)c (E,E′)〉GUE ' −1

π2

(sin((E − E′)

√d/σ)

E − E′

)2. (S29)

By defining new variables r = E − E′ and ω = (E + E′)/2, Eq. (S20) is given by

〈gcβ(t)

〉GUE

' − 1π2

∫dωe−2βω

∫ ∞−∞

dr

(sin(r

√d/σ)

r

)2e−2itr, (S30)

where we have replaced τ with 2t. The first integration is divergent. For estimation, the integration is replaced with∫dω →

∫ ω0−ω0

dω. (S31)

Since the spectral density can be approximated by Wigner’s semicircle in large d limit, i.e.,

〈%(E)〉GUE =√d

σπ

√1−

(E

2σ√d

)2, and |E| ≤ 2σ

√d, (S32)

thus 〈%(0)〉GUE =√d/(σπ). According to the normalization of 〈%(E)〉GUE, we have 2ω0 〈%(0)〉GUE ' d, and

ω0 'σπ√d

2, (S33)

11

10-2 10-1 100 101 10210-2

10-1

100

t

Numerical Analytical Eq. (S28) Analytical Eq. (S41)

10-2

10-1

100Sp

ectra

lFor

mFa

ctor

β=0

10-2 10-1 100 101 10210-310-210-1100

t

Spec

tralF

orm

Fact

or

d=30

10-2

10-1

100

β=0.1

10-2 10-1 100 101 10210-310-210-1100

t10-2 10-1 100 101 10210-2

10-1

100

t

10-1

100

β=1

10-1

100

β=2

d=10

FIG. S1. Spectral form factor of GUE. Analytical Eqs. (S28) and (S41) are in comparison with numerical calculations for

β = 0, 0.1, 1, 2, and d = 10, 30. The standard deviation of the random variables of H is selected as σ = 1/√d. The numerical

calculations use 1000 independent realizations.

with which, the first integration reads

∫dωe−2βω '

sinh(√

dπβσ)

β. (S34)

The second integration in Eq. (S30) is the Fourier transform

∫ ∞−∞

dr

(sin(r

√d/σ)

r

)2e−2itr =

{π(√d/σ − t), t ≤

√d/σ,

0, t >√d/σ.

(S35)

Equation (S30) finally takes the form

〈gcβ(t)

〉GUE

'

{− sinh(

√dπβσ)

πβ (√d/σ − t), t ≤

√d/σ,

0, t >√d/σ.

(S36)

According to the above equation, it is easy to observe that the plateau time is

tp =√d/σ. (S37)

With Wigner’s semicircle law of Eq. (S32), the partition function averaged over the GUE ensembles is approximatedby

〈Z(x)〉GUE =√dI1(2σ

√dx)

σx, (S38)

where In(·) is the modified Bessel function of first kind and order n. When β � 1 and t is large, the asymptoticexpansion of the second part of Eq. (S19) reads

|〈Z(β + i2t)〉GUE|2 '√d(1− sin(8σ

√dt))

16πt3σ3. (S39)

By equating Eq. (S39) and Eq. (S36), the dip time can be estimated as

td '1

2

√dβσ3 sinh

(√dπβσ

) 14 ' 1

2π1/4σ− dπ

7/4σ

48β2 +O(β3). (S40)

12

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�=��

�=��

��-��-��-��-��-��-��

��

��

�=��

��-� ��-� ��-� �� -��-��-��-��-��-��

��

��

��

�=��

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�=��

�=��

NumericalAnalytical

FIG. S2. Spectral form factor of SYK model. Analytical Eq. (S50) is commpared with numerical calculations (β = 0.1).

The spectral form factor in the large d limit is obtained

〈gβ(t)〉GUE '

√dI1(4σ

√dβ)

2σβ +√d(1−sin(8σ

√dt))

16πt3σ3 + Eq.(S36)(√dI1(2σ

√dβ)

σβ

)2 . (S41)In Fig. (S1), the spectral form factor with Eqs. (S28), and (S41) are compared with the numerical calculations. Notethat when the dimension d increases, the valid domain of β by annealing approximation becomes larger.

B. Spectral form factor of the SYK model

For SYK model with N mod 8 6= 0, the energy spectrum has a uniformly double degeneracy (Nn = 2). Equation(S17) can be written as〈

|Z (β + i2t)|2〉

SYK= 2 〈Z(2β)〉SYK + |〈Z(β + i2t)〉SYK|

2+〈gcβ(t)

〉SYK

, (S42)

where 〈gcβ(τ)

〉SYK

=

∫dEdE′〈%c(E,E′)〉SYKe−(β+2ti)Ee−(β−2ti)E

′. (S43)

To calculate the first two items of Eq. (S42), we need to know the spectral density of the SYK model, which has beenderived by the method of moments

〈%(E)〉SYK =1

2π

∫dte−iEt

〈Tr(eiHt)

〉SYK

. (S44)

In this part, we are going to roughly estimate the dip and plateau time, therefore, the spectral density can beapproximated in a Gaussian type when N is large [49, 56, 57]

〈%(E)〉SYK '√

2

πNd exp

(−2E

2

N

). (S45)

The partition function is

〈Z(x)〉SYK ' d exp(Nx2

8

). (S46)

With such equation, the decoherence time can be estimated by

γτD ≥1

4 d2

dβ2 ln [〈Z(β)〉SYK]=

1

N. (S47)

13

Since the late time behavior of the SYK model is governed by GOE, GUE, and GSE statistics according to thenumber of Majorana Fermions modulo 8. For simplicity, we first consider the connected part of the two-pointcorrelation function of GUE (i.e., N mod 8 = 2 or 6) as

〈%(2)c (E,E′)〉SYK ' −(

sin(2πr 〈%(ω)〉SYK)πr

)2, (S48)

with r = E − E′ and ω = (E + E′)/2 defined in above subsection. Then〈gcβ(t)

〉SYK'∫dEdE′〈%c(E,E′)〉SYKe−(β+2it)Ee−(β−2it)E

′,

= −∫dωe−2βω

∫dr

(sin(2πr 〈%(ω)〉SYK)

πr

)2e−irt,

=

√N〈Z(2β)〉SYK√

2πdt− 2 〈Z(2β)〉SYK , t . 2

√2πN d,

0, t > 2√

2πN d.

(S49)

With Eqs. (S46) and (S49), the spectral form factor of the SYK reads

〈gβ(t)〉SYK '|〈Z(β + i2t)〉SYK|

2

〈Z(β)〉2SYK+

√N

2√

2πd〈gβ(∞)〉SYK t, t . 2

√2πN d,

〈gβ(∞)〉SYK t > 2√

2πN d,

(S50)

where 〈gβ(∞)〉SYK is the spectral form factor in the long time limit

〈gβ(∞)〉SYK =2 〈Z(2β)〉SYK〈Z(β)〉2SYK

' 2d

exp

(Nβ2

4

). (S51)

Note that when N mod 8 = 0, i.e., the GOE case, there is no degeneracy, and the plateau height would beexp

(Nβ2/4

)/d. From Eq. (S50), the plateau time is given by

tp ' 2√

2π

Nd. (S52)

For GOE (N mod 8 = 0) and GSE (N mod 8 = 4), the calculations would become rather lengthy. Since we only aimto roughly estimate the time scale, we still use the GUE, and modified the plateau time according to the numericalresults. For GSE, the plateau time is around tp '

√2π/Nd. Unlike the GUE and GSE, the ramp and plateau

connect smoothly for the GOE, so it is hard to strictly define the plateau time, for simplicity, we still use Eq. (S52)for estimation.

Before the dip time, the edges of the spectrum cannot be omitted, thus Eq. (S46) is no longer applicable. Thus,we will replace it with 〈Z(x)〉SYK ' x−3/2 [9, 49, 56], and the first part of Eq. (S50) is given by

|〈Z(β + i2t)〉SYK|2

〈Z(β)〉2SYK' β

3

(β2 + cN t2)3/2, (S53)

with cN ' N/400 fitted by numerical calculations. Then, the dip time is roughly estimated as

td ∼

(√π exp

(−Nβ2/4

)c3/2N

√2N

)1/4√d ∝√d. (S54)

Although Eqs. (S52) and (S54) are derived when β is small, they are still valid for low temperature for estimation,just as shown in Fig. (S3).

14

γ=�γ=�γ=��γ=��-� ��-� ��-� ��

��-��-��-��-��-��-��-��

��

��

β=�γ=�γ=�γ=��γ=��

��-� ��-� ��-� ��

β=�γ=�γ=�γ=��γ=��

��-� ��-� ��-� ��

β=�

FIG. S3. Fidelity of the stochastic SYK model for different temperatures. A log-log plot of the fidelity of SSYKmodel (N = 26) under different temperature. The variation of the temperature has a negligible effect on the dip and plateautime.

Thermofield dynamics: Quantum Chaos versus DecoherenceAbstract References I. Fidelity in terms of spectral form factor II. Logarithmic negativity and the second Rényi entropy III. Fidelity in terms of density of states and form factor IV. Ensemble average of the fidelity A. GUE-averaged Fidelity B. Spectral form factor of the SYK model

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