Quenches and (Pre)Thermalisation in a mixed Sachdev-Ye-Kitaev Model
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Quenches and (Pre)Thermalisation in a mixed Sachdev-Ye-Kitaev Model
Ancel Larzul1 and Marco Schiró1, ∗
1
JEIP, USR 3573 CNRS, Collège de France, PSL Research University,
11, place Marcelin Berthelot,75231 Paris Cedex 05, France
(Dated: July 19, 2021)
We study the nonequilibrium quench dynamics of a mixed Sachdev-Ye-Kitaev model, with com-
peting two bodies random interactions leading to maximally chaotic Non-Fermi Liquid dynamics
and a single body term which dominates at low temperatures and leads to Fermi liquid behavior. For
different quench protocols, including sudden switching of two-body interaction and double quench
protocols, we solve the large N real-time Dyson equation on the Keldysh contour and compute the
dynamics of Green’s functions from which we obtain effective temperature and relaxation rates. We
show that the model thermalizes to a finite temperature equilibrium and that depending on the
arXiv:2107.07781v1 [cond-mat.str-el] 16 Jul 2021
value of the quench parameters the effective temperature can be below or above the Fermi-Liquid to
Non-Fermi Liquid crossover scale, which can then be accessed through the nonequilibrium dynamics.
We identify quench protocols for which the heating dynamics slow down significantly, an effect that
we interpret as a signature of prethermalization.
I. INTRODUCTION While a large attention has been devoted to the low-
energy equilibrium physics of these or related models,
Generic interacting quantum many body systems are to their transport [20, 21] or scrambling properties as
expected to thermalize when let evolved unitarily under encoded in the growth of out-of-time ordered correla-
the action of their own Hamiltonian. The understand- tor [22], comparatively less work has focused on the gen-
ing of this quantum thermalization process, its possi- uine nonequilibrium dynamics of SYK models.
ble slowdown or complete breakdown is still a subject of Dynamics in the pure SYK4 model starting from differ-
large interest and effort, in a broad community ranging ent initial states, including completely uncorrelated ones
from high-energy physics to condensed matter, atomic as well as thermal states of the mixed SYK model, has
physics and quantum information. A particularly inter- been studied [23, 24]. In the large N limit it was shown
esting question concerns how fast a quantum many body that the system thermalizes to an equilibrium state at in-
system can thermalize and therefore scramble the quan- finite temperature, unless the initial state is a correlated
tum information initially encoded in a quantum state. thermal state of the mixed SYK4 +SYK2 which lead to a
The Sachdev-Ye-Kitaev model (SYK) [1–3], describing finite effective temperature. More recent works have fo-
N Majorana fermions with random two-body all-to-all cused on the dynamics of mixed SYK models with com-
interactions, has played in this context an important role plex fermions [25] or deformation of the SYK model poss-
as minimal model capturing thermalization, scrambling esing a quantum phase transition [26]. In the high-energy
and chaos [4–7] or analogously the emergence of strange literature the dynamics of pure states in the SYK model
metals in strongly interacting quantum matter [8]. have attracted some interest [27] in particular the dy-
Deformations of the SYK model have been discussed namics of entanglement entropy [28].
actively. An interesting example is the mixed SYK In this work we study the nonequilibrium dynamics of
model, denoted in the following as SYK4 +SYK2 model, the mixed SYK4 +SYK2 in the large N limit. There are
where an additional random one-body all to all coupling several motivations for this study. From one side we will
(equivalent to an hopping term) is introduced. This investigate how the crossover from NFL to FL behavior
model in thermal equilibrium has been shown to possess will affect the thermalization properties of the system. It
a crossover between a Fermi-Liquid (FL) behavior and a was shown that the addition of a relevant perturbation at
Non-Fermi-Liquid (NFL) regime at low temperatures [9– low energy makes the model less chaotic, with a Liapunov
11]. This feature, first identified in related models of exponent vanishing as a quadratic power-law in temper-
disordered quantum spins coupled to electrons [12–15] ature [21, 29–31] and we will discuss how this affects the
motivated by the physics of high-Tc superconductors, has thermalization time. Finally we note that there is cur-
recently attracted renewed experimental interest [16]. At rently large interest in disordered fully connected models
finite N this crossover in the mixed SYK model has been which, even at the classical level have been shown to pos-
claimed to turn into a transition between chaotic and in- sess intriguing properties [32]. Recently, the periodically
tegrable behavior [17, 18], possibly related to many-body driven mixed SYK has been investigated in Ref. [33].
localization [19]. The paper is structured as follows. In Sec. II we intro-
duce the model, the nonequilibrium protocol and present
the large N solution for real-time Green’s functions. In
Sec. III we present our results, first discussing the case
∗ On Leave from: Institut de Physique Théorique, Université Paris of a sudden switching of the two-body random interac-
Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France tion term. We show that the resulting dynamics leads
2
to thermalization, at long times as shown by the spec- A. Large-N Real-Time Dyson Equation
tral and distribution functions of the system, and discuss
the decay in time of the retarded Green’s function. In The real-time dynamics of the mixed SYK model can
Sec. IV we consider the case of a double quench, in which be obtained in the large-N limit through saddle point
both the interaction and the single particle bandwidth methods on the Keldysh action [23, 34]. In particular
are quenched with respect to the initial condition. We the real-time Green’s function of the Majorana fermions,
show that in this case the dynamics slows down signif- defined as
icantly, an effect that we interpret as prethermalization
due to the proximity to an integrable point. Finally, in i X α
Gαβ (t1 , t2 ) = − hχi (t1 )χβi (t2 )i (2)
Sec. V we discuss these results in light of the thermody- N i
namic of the mixed model and show that a double quench
makes the effective temperature decrease and heating to with α, β = ± Keldysh contour index, can be shown to
slow down and identifies signature of the crossover in the satisfy a real-time Dyson Equation of the form
nonequilibrium decay rate evaluated at the effective tem- h i
perature. In Sec. VI we draw our conclusions. Ĝ−1 − Σ̂ ◦ Ĝ = 1, (3)
0
where [Ĝ−1
0 ]
αβ
(t1 , t2 ) = iαδαβ δ(t1 − t2 )∂t1 is the free Ma-
II. THE MIXED SYK MODEL jorana fermions Green’s function, the self-energy reads
Σαβ (t1 , t2 ) = −αβJ4 (t1 )J4 (t2 )Gαβ (t1 , t2 )3
We study a generalisation of the SYK model with both (4)
quartic and quadratic interactions [9–11]whose Hamilto- + J2 (t1 )J2 (t2 )Gαβ (t1 , t2 )
nian reads
and the symbol ◦ denotes real-time convolutions. For
N N Majorana fermions it is convenient to work with the
i X 1 X
H(t) = J2,ij (t) χi χj − J4,ijkl (t)χi χj χk χl greater (lesser) Green function G>( (t1 , t2 ) = −G< (t2 , t1 ) (5)
δij . J2,ij (t) and J4,ijkl (t) are time dependent random in-
dependent Gaussian variables with zero mean and vari- and from which one can obtain all the relevant Green’s
2
2 (t) = J2 (t) and J 2 6J42 (t) functions,
ances J2,ij N 4,ijkl (t) = N 3 .
As we will discuss more in detail in Sec. III we con-
sider as initial condition the ground state of the pure GR (t1 , t2 ) ≡ θ(t1 − t2 ) G> (t1 , t2 ) − G< (t1 , t2 )
(6)
SYK2 model and the quench protocol J2 (t) = θ(−t)J2,i + A >
<
θ(t)J2,f and J4 (t) = θ(t)J4 , leaving the possibily to have G (t1 , t2 ) ≡ −θ(t2 − t1 ) G (t1 , t2 ) − G (t1 , t2 ) (7)
J2,i = J2,f . In this way we can study both the effect GK (t1 , t2 ) ≡ G> (t1 , t2 ) + G< (t1 , t2 ) (8)
of a pure quench of J4 as well as the combined effect of
switching on the interaction and changing the bandwidth The first Schwinger-Dyson equation can be put into
of the Majorana fermions. a more convenient form known as the Kadanoff-Baym
At equilibrium and in the large N limit the SYK model equations:
with zero hopping J2 = 0 describes a non-Fermi liquid Z +∞
where the single particle √ Green’s function in imaginary i∂t1 G >,<
(t1 , t2 ) = dtΣR (t1 , t)G>,< (t, t2 )+
time decays as G(τ ) ∼ 1/ τ . However this phase is not −∞
stable to the introduction of the hopping term J2 6= 0 Z +∞
which constitutes a relevant perturbation and the sys- + dtΣ>,< (t1 , t)GA (t, t2 ) (9)
−∞
tem turns into a Fermi liquid with single particle Green’s Z +∞
function G(τ ) ∼ 1/τ . A cross-over between the Fermi
−i∂t2 G>,< (t1 , t2 ) = dtGR (t1 , t)Σ>,< (t, t2 )+
liquid and non-Fermi liquid is expected to happen when −∞
the hopping term becomes dominant which corresponds G>,< (t1 , t)ΣA (t, t2 ) (10)
to an energy scale (or temperature) T ∗ ∼ J22 /J4 [12, 20].
As we mentioned in the introduction the situation is even Likewise we define the retarded and advanced self-
more interesting at finite N where the crossover turns energies:
into a transition between chaotic and integrable regimes.
Here we will not consider the finite N case and focus on
the thermodynamic limit and the resulting real-time dy- ΣR (t1 , t2 ) ≡ θ(t1 − t2 ) Σ> (t1 , t2 ) − Σ< (t1 , t2 )
(11)
namics that can be also solved exactly through saddle A > <
point techniques as we briefly discuss below. Σ (t1 , t2 ) ≡ −θ(t2 − t1 ) Σ (t1 , t2 ) − Σ (t1 , t2 ) (12)
3
(t1 , t2 ) to an average and relative time coordinates (T , t)
defined as
t1 + t2
T = , t = t1 − t2 (14)
2
The bounds of the (T , t) grid are shown in blue in Fig. 1.
Notice that on this grid the initial conditions corresponds
to T = −Tmax and the maximum T value to T = Tmax
with Tmax = tmax /2.
Taking the Fourier transform with respect to the rel-
ative time t we define the so-called Wigner transform of
the Green’s functions [23, 34]
Z t t
G(T , ω) = dt eiωt G t1 = T + , t2 = T − (15)
Figure 1. Sketch of the t1 , t2 plane as well as the rotated plane 2 2
T , t.
This has the advantage of showing explicitly the effect of
time-translation symmetry breaking due to the quench
We mention here for future use that the greater Green’s and resulting in an explicit dependence on the average
function of the pure SYK2 model (J2 (t) = J2 , J4 (t) = 0) time T . Furthermore it suggests a picture of slow-varying
is known exactly, at least in the long-time stationary (i.e. quasi-equibrium which connects naturally with the long-
time-translational-invariant limit) time limit in which one expects the approach to thermal
equilibrium.
−i Using the Wigner transform we can define quantities
G>
2 (t) = (J1 [2J2 t] − iH1 [2J2 t]) (13) which have a direct physical interpretation such as the
2J2 t
time-dependent spectral function
where J1 (x), H1 (x) are special (Bessel’s and Henkel’s)
functions of the first kind. A(T , ω) = −2ImGR (T , ω) (16)
or distribution function
1. Numerical Integration of Schwinger-Dyson Equations
iGK (T , ω) = F (T , ω)A(T , ω) (17)
In this paper we solve the Kadanoff-Baym equations
numerically on a t1 − t2 grid of size 2001 × 2001 with using a parametrization of the Keldysh Green’s func-
time step dt = 0.05. The grid has a length 2 tmax in tion which evokes explicitly a fluctuation-dissipation the-
each direction as shown in red in Fig. 1. Initially at orem. In fact in thermal equilibrium, corresponding to
times t1 , t2 < 0 the system is prepared in the ground the initial condition or possibly the long-time behavior
state of the pure SYK2 model with coupling J2 = 0.5. if thermalization is established, those two quantities are
We solve the Kadanoff-Baym equations for G> on the time-independent and related by a universal identity, the
grid by using a predictor-corrector scheme [23] [24], [26]. fluctuation-dissipation theorem [35]
Integrals are computed using the trapezoidal rule. We
verify the consistency of our numerical code by check- βω
ing the conservation of energy and the normalization of iGK (ω) = tanh A(ω) (18)
2
the spectral density. Thermal equilibrium solutions for
a given inverse temperature β are obtained by solving where β is the inverse temperature.
the Schwinger-Dyson equation self-consistently following
[23], further details are discussed in Appendix A.
III. RESULTS: QUENCH OF J4
B. Observable of Interests: Spectral and
Distribution Functions In this section we present our results for the dynamics
of the mixed SYK model as obtained from the solution
In order to interpret the results obtained by the nu- of the real-time Dyson equation. Specifically we consider
merical integration of the real-time Dyson equation it is first a sudden switching of the quartic interaction J4 (t) =
convenient to introduce a mixed time-frequency represen- θ(t)J4 , starting from a pure SYK2 model in its ground
tation for the Green’s functions defined in the previous state and with J2 = 0.5. Later in Sec. IV we will consider
section. Specifically we can change the time coordinates the effect of quenching both J2 and J4 .
4
Figure 2. Transient Spectral Function after a sudden switch- Figure 3. Transient Distribution Function after a sudden
on of J4 in the mixed SYK model. Top: Spectral function switch-on of J4 in the mixed SYK model. Top: Distribution
at initial time (T = −Tmax ) and at long times (T = Tmax ), function at initial and final times and its dynamical evolution
compared to the equilibrium spectral function of the mixed showing the onset of heating (finite temperature).Parameters:
SYK2 +SYK4 model at the final temperature (dashed grey J2 = 0.5, J4 = 1.5.
line), confirming thermalisation and evolution of the spectral
function for different times T .
A. Transient Spectral Function and Thermalisation
of the mixed SYK model
We start discussing the evolution of the spectral func-
tion A(T , ω) after a quench to J4 = 1.5. In Fig. 2 we plot
the initial spectral function of the SYK2 model, which
features the well known sharp edge semicircular density
of state, and its long-time limit after the switching on of
the J4 interaction, featuring a much broader resonance
with tails at higher frequencies. We compare the latter
with the equilibirum spectral function of the mixed SYK
model evaluated at the final temperature (see next sec-
tion) and find a perfect match thus confirming that the Figure 4. Dynamics of the effective temperature Teff (T ), ex-
mixed SYK model reaches thermal equilibrium. In the tracted from the distribution function as discussed in the main
text, for different values of the J4 interaction. We see that
bottom panel we plot the evolution of the spectral func-
the heating dynamics of the system strongly depends on the
tion A(T , ω) as a function of increasing time T . We see value of J4 .
that spectral features at high frequencies reshape rather
rapidly after the quench, with tails forming above the
bandwidth set by J2 , while the low frequency features
development of a higher final temperature. From the
follow at later times.
bottom panel we can see that, as for the spectral function,
the high-frequency features of the distribution are those
that re-adjust the faster after the quench, while the low
B. Distribution Function and Dynamics of frequency sector takes longer time to respond.
Effective Temperature We note that throughout the evolution the low fre-
quency behavior of the distribution function is linear in
A further demonstration of the onset of thermalization frequency, which suggests to extract a time-dependent ef-
in the mixed SYK model comes from the dynamics of the fective temperature Teff (T ) by fitting the low frequency
distribution function, F (T , ω), defined in Eq. 17, that we behavior of F (T , ω) with tanh(βeff (T )ω/2). We plot the
plot in Fig. 3. As previously, we show in the top panel dynamics of this effective temperature for different values
the initial condition and the long-time limit, while in the of J4 in Fig. 4. We see that for small quenches the sys-
bottom panel the evolution of the distribution function tem remains close to the ground-state and the effective
for different average times T . As the initial state is taken temperature changes slowly with time, while increasing
at zero temperature, the distribution function is rather J4 leads to a faster heating dynamics. The approach to
sharp around low frequency, while long time after the the long-time limit, that we identify with the final tem-
quench a smoother behavior is approached indicating the perature reached by the system after the quench,
5
(a) (b)
Figure 5. (a) Real-time retarded Green’s function as a function of the relative time t long after the quench (T = 25) for different
J4 . We compare the decay with the power-law behavior expected for a pure SYK2 (dotted blue line). (b) Exponential decay
rate Γ∞ extracted from the large T limit of the retarded Green function for different J4 (red crosses). The grey triangles are
the decay rates extracted from the equilibrium Green’s function at Tf .
Tf = Teff (∞), is exponential Green’s function remains closer to the one of a pure SYK2
which decays as a power law, the system rapidly enters
Teff (T ) − Tf = A exp (−Γ1 T ) into an exponential decay regime as a function of the rel-
ative time, which is compatible with the Lorentzian line-
from which we can extract a thermalization rate Γ1 that shape found in the frequency-resolved spectral function
depends in general on the quench parameters, in partic- (see Fig. 2). We can therefore extract a decay rate for
ular on the final value of J4 , as we show in the inset. the stationary retarded Green’s function using an ansatz
In particular for quenches of intermediate strength we of the form
find Γ1 ∼ J4 , while upon decreasing we expect a sub-
leading behavior, possibly quadratic in J4 . However the GR (Tmax , t) ∼ exp(−Γ∞ t) (19)
time-scales needed to reach equilibrium exceed our Tmax
therefore we cannot conclude on the nature of this scaling As for the thermalization time discussed before, the
(see however next section). decay rate Γ∞ depends in general on the quench param-
C. Real-time Retarded Green’s functions:
Long-time Decay Rate and Waiting Time
dependence
We now discuss the dynamics of the retarded Green’s
function in the time-domain, focusing in particular on its
long time decay. From the analysis of the spectral func-
tion presented in the previous section it appeared difficult
to discuss the effect of the quench on the low frequency
behavior of the spectrum, a feature that will appear more
clearly in the time domain. In the left panel of Fig. 5 we
plot the retarded Green’s function GR (T , t) long time af-
ter the quench, i.e. for T = Tmax , as a function of the
relative time t and for different values of J4 .
We first note that the Green’s function shows pro- Figure 6. Waiting time dependence of the real-time retarded
Green’s function after a quench of J4 = 1.5. At short average
nounced oscillations, with a period which appears to be
time the decay as a function of the relative time t is powerlaw,
independent of J4 and completely set by the J2 scale. reflecting the initial low temperature state of the SYK2 model.
This can be understood by considering the analytic ex- Upon increasing the time after the quench T the decay crosses
pression for the Green’s function of the pure SYK2 model over to an exponential decay, around a time t ∼ 2T . We
given in Eq. 13, which indeed show oscillations with a fre- note that the decay at long time is exponential, yet with a
quency set by J2 . decay rate which is slower than the long-time limit Γ∞ . This
Concerning the long-time decay we see that with the behavior is compatible with an ansatz of the form 20 for the
exception of the very small quench regime, where the two-times Green’s function.
6
Figure 7. Dynamics of the effective temperature Teff (T ) af- Figure 8. Decay of real-time retarded Green’s function
ter a double quench of J2 and J4 . We consider a sudden GR (T , t), long after the quench, as a function of the rela-
switching-on of J4 from zero to J4 = 2 and different quench tive time t after a double quench of J2 and J4 . We consider a
values of J2,f . We see that the heating dynamics of the system sudden switching-on of J4 from zero to J4 = 2.5 and different
slows down upon increasing the strength of the J2 quench, a quench values of J2,f . We see that the decay rate is slower
sign of extended prethermalization dynamics. upon increasing the quench of J2 .
IV. RESULTS: QUENCH OF J4 AND J2
eters, in particular the value of the interaction J4 , as we
plot in the right panel of Fig. 5. is very small for J4 → 0
reflecting the power-law decay of the pure SYK2 model In this section we consider a different quench proto-
and grows upon increasing the strength of the quench col, where in addition to the sudden switching of the J4
showing a linear behavior at intermediate couplings and interaction the system at time t = 0 undergoes also a
a superlinear regime for large J4 . Finally we expect (not quench of the single particle term, from J2,i = 0.5 →
shown) the decay rate to saturate at larger values of J4 . J2,f 6= J2,i . Protocols of this sort have been discussed
before in the literature on quantum quenches, see for ex-
We now move on to discuss the waiting time depen-
ample Refs. [36–38] with the idea of disentangling the
dence of the real-time retarded Green’s function which
role of integrability-breaking and generic nonequilibrium
further allows to clarify how the initial power law decay
perturbations in the heating production and thermaliza-
regime and the long-time exponential decay regime are
tion pathways of quantum systems. In this respect, we
connected as the average time T is increased. In Fig. 6
emphasize that while the equilibrium properties of the
we plot the real-time retarded Green’s function, GR (T , t)
mixed SYK model are completely controlled by the adi-
as a function of the relative time, for different average
mensional ratio J4 /J2 , for the non-equilibrium dynam-
times T and for a quench to J4 = 1.5. We recognize a
ics there are in principle two independent adimensional
characteristic feature of this two-time Green’s function,
quantities, J2,f /J2,i and J4 /J2,f , the former controlling
namely a short time behavior and a long time regimes
the degree of excitation of single particle states while the
with a crossover at times t ∼ 2T . We also notice that
latter the role of interactions. As we are going to show
the short-time behavior is power law for negative average
this double quench protocol allows to explore different
times T < 0, corresponding to the regime of influence of
heating pathways in the nonequilibrium dynamics of the
the initial condition, while it is exponential for positive
mixed SYK model.
average times T > 0. The numerical results suggest an
ansatz of the form
A. Dynamics of Effective Temperature and
GR (T , t) = GR
∞ (t)f (T , t) (20) Real-time Decay of Retarded Green’s function
where f (T , t) is a function chosen to capture the In order to discuss the effect of this double quench we
crossover and the short/long time behavior of the two- focus on the dynamics of the effective temperature, as ob-
times retarded Green’s function [36]. In particular for tained from the low-frequency regime of the distribution
positive times T > 0 we have that f ' 1 for t 2T while function as discussed in Sec. III. In particular we plot in
it grows exponentially for long times, i.e. f ∼ exp(Γ0 t) Fig. 7 the effective temperature for a quench to J4 = 2
for t 2T , with a rate Γ0 which is in general smaller and different values of the J2,f parameter. We see that
than the long-time asymptotic rate Γ∞ , leading to a slow quenching the J2 term leads to a significant slow down of
down of the decay rate at finite positive waiting times, the effective temperature dynamics as compared to the
as shown in Fig. 6. case in which J2 is kept constant (top line in Fig. 7), and7
Figure 9. Quench phase diagram for the mixed SYK2 +SYK4 Figure 10. Out of equilibrium scattering rate, defined as the
model. Effective final temperature versus the quench param- imaginary part of the retarded self-energy long time after the
eter J4 and for different quenches of the J2 term. We see quench, −Im ΣR (T = Tmax , ω = 0), as a function J4 for a
that quite generically the effective temperature production is single quench (blue points) or for a double quench to J2,f 6=
slower upon quenching also J2 . For comparison we plot in the J2,i (orange and green points). Grey triangles: equilibrium
same plane the crossover scale T ∗ ∼ J2,i
2
/J4 and T ∗ ∼ J2,f
2
/J4 result at Tf . Inset: same quantity plotted versus the effective
corresponding to the equilibrium FL-to-NFL crossover. temperature Tf , showing a linear scaling and deviations from
it at lower effective temperatures.
already for J2,f = 1.5 the effective temperature does not
reach a stationary state on the time scales of our simula- possible to explore the crossover from NFL to FL in the
tion. In addition to a slower down of the thermalization quench dynamics.
time we also note that the long-time limit of the effec- To this extent it is useful to plot a dynamical phase dia-
tive temperature is reduced by the quench, namely the gram for the quench problem, as in Fig. 9, where we show
system heats up less. This can be understood by notic- the dependence of the effective temperature Tf from the
ing that, at least for J2,f J4 when the interactions quench parameter J4 for different values of the J2 quench,
can be neglected to a leading order, the final and initial corresponding respectively to a single quench (J2i = J2f )
Hamiltonian are both solvable SYK2 models, a regime and to a double quench (J2i 6= J2f ). The effective tem-
in which the dynamics is known to thermalise instanta- perature is obtained from the dynamics of the distribu-
neously without heating production. tion function, as discussed in Sec. III. We see that Tf
The slow-down of the dynamics in presence of a J2 increases with J4 , namely the stronger is the value of the
quench can be also clearly seen from the decay in time of interaction quench the more the system heats up, how-
the retarded Green’s function long time after the quench, ever this increase is slower for a double quench protocol.
as discussed for a single quench in Sec. III. In Fig. 8 we Specifically, we see that upon quenching the single par-
plot GR (T , t) long time after the quench T = 25 as a ticle coupling term J2 the final temperature at which
function of the relative time t, after a quench to J4 = 2 the system effectively thermalizes is lower as compared
and different values of the J2 quench. We see that the to a pure J4 quench, consistently with the results dis-
decay is always exponential, as in the case of a single cussed in Sec. IV. This slow down of the heating dynam-
quench discussed in Eq. (19) although with a rate Γ∞ ics can be understood in terms of a prethermalization
(not shown), which decreases upon increasing J2,f /J2,i . phenomenon, where due to the proximity to an integrable
point the time scales for reaching full equilibrium depend
non-perturbatively from the scale breaking the integra-
bility.
V. DISCUSSION: QUENCH-INDUCED NFL TO This result is particularly interesting in connection
FL CROSSOVER with the scale T ∗ ∼ J2,f 2
/J4 that controls the equilib-
rium properties of the final mixed SYK model and in
We now summarize our results for the quench dynam- particular its NFL-to-FL crossover. For comparison, we
ics of the mixed SYK2 +SYK4 model. Our analysis so plot these energy scales on the same plane as the effective
far has focused on the dependence of the nonequilibrium final temperature, see Fig. 9. For a single quench of the
dynamics from the quench parameters, respectively the interaction term J4 we see that most of our data lie above
strength of the interaction J4 and of the single parti- the crossover scale and therefore we expect to see a be-
cle quench J2,f /J2,i . An interesting question we would havior compatible with a pure SYK4 . On the other hand,
like to address here is how the long time limit of the for a double quench the effective temperature decreases
quench problem relates to the equilibrium properties of and the crossover line is also pushed to larger values of
the mixed SYK model and whether in particular it is J4 . As a result, quenching the single particle scale we8
expect to see a behavior which is more compatible with limit. Specifically we have solved the real-time Schwinger
a pure SYK2 . Dyson equations numerically for two different quench
To confirm this expectation we consider the out of protocols, corresponding to a sudden switching of the
equilibrium scattering rate for the Majorana fermions, J4 interaction and a simultaneous quench of the single-
namely the imaginary part of the retarded self-energy particle bandwidth J2 .
long time after the quench, −Im ΣR (T = Tmax , ω = 0). We have shown that quite generically the unitary dy-
This can be readily obtained from the nonequilibrium namics of this model thermalizes to a finite temperature
Green’s functions through Eq. 11 and 4 and after Wigner thermal equilibrium state, as confirmed by both the spec-
transform. In thermal equilibrium this quantity is known tral function and the distribution functions of the Majo-
to be sensitive to the crossover scale T ∗ , as we discuss in rana modes, two quantities that however evolve on much
the Appendix A. In Fig. 10 we plot this quantity as a different time scales. In particular the dynamics of the
function of J4 for different values of J2,f and J2,i , includ- effective temperature as obtained from the effective FDT
ing therefore both the single quench protocol discussed on the distribution function appears to become very slow
in Sec. III as well as the double quench. As expected the for weak quenches, or equivalently for large quenches of
scattering rate, much like the effective temperature, in- the single particle term J2 . We have connected this result
creases with J4 but with a rate that depends strongly on to the onset of prethermalization in the quench dynamics
J2,f /J2,i and decreases for large quenches of the J2 cou- of this system.
pling. To further confirm the overall thermalization of As compared to quenches in the pure SYK4 model, the
the nonequilibrium dynamics we compare this quantity mixed case enjoys a much richer dependence from the
with its equilibrium version at the final effective tem- quench parameters, encapsulated in the nonequilibrium
perature (see grey diamonds) finding perfect agreement. phase diagram shown in Fig. 9. We have shown that
Furthermore in the inset of Fig. 10 we plot the same scat- quite generically a quench of the J4 coupling leads to
tering rate as a function of the effective temperature Tf a finite temperature which can be above or below the
for the three values of J2,f /J2,i considered. We see that crossover scale T ∗ and that quenching the single particle
for a single quench, corresponding to J2,f = J2,i , when term J2 significantly reduce the heating in the system
the effective temperature is above the crossover scale (See and allows to access the NFL to FL crossover through
Fig. 9 ) the scattering rate show a NFL scaling expected the quench dynamics, as we have shown by looking at
from SYK4 , i.e. the nonequilibrium scattering rate at long times.
−Im ΣR ∼ Tf (21) Our results offer therefore a complementary picture,
based on the full out of equilibrium dynamics, to the
while for a double quench, when the effective tempera- studies on the scrambling properties of the mixed SYK
ture decreases while the crossover scale is pushed towards model and point out a possible interesting connection be-
higher values, we are able to see more clearly the devia- tween slow scrambling and prethermalization that could
tions from the linear behavior at low effective tempera- be worth discussing further in the future. Further per-
tures. In particular for J2,f = 1.5, which corresponds to spectives opened by this work includes the investiga-
the lower effective temperature we can achieve, we can tion of different nonequilibrium settings involving mixed
see a behavior which is consistent with the FL scaling SYK-like models, such as those recently considered in
connection with traversal wormholes in the high-energy
−Im ΣR ∼ Tf2 . (22) literature.
We emphasize that reaching lower effective temperatures
in our dynamical approach is challenging since the dy-
namics slow down significantly and the long time station- ACKNOWLEDGMENTS
ary limit becomes unaccessible to our finite time simu-
lation. Nevertheless based on the evidence in Fig. 10 We thank A. Georges for helpful discussions and the
we can safely conclude that the scattering rate is a good Collège de France IPH cluster for computational re-
probe of the quench-induced crossover between NFL and sources. This work was supported by the ANR grant
FL. In conclusion we note that, on the other hand, the ”NonEQuMat” (ANR-19-CE47-0001).
relaxation rate Γ∞ defined from the long-time limit of
the retarded Green’s function is not a good probe of the
crossover already in equilibrium, as we show explicitly in
Appendix A. Appendix A: Equilibrium Properties of the Mixed
SYK Model
VI. CONCLUSIONS In this section we briefly recall some of the equilibrium
properties of the mixed SYK2 +SYK4 model in the large
In this work we have discussed the quench dynam- N limit. In this case the Dyson equation for the retarded
ics of the mixed SYK2 +SYK4 model in the large N single particle Green’s function can be written directly in9
where the Keldysh component GK (t) is related to the re-
tarded one by the fluctuation-dissipation theorem, Eq. 18
of the main text. The two equations above can be solved
iteratively, going back and forth from the frequency to
the time domain, until a converged solution is found. A
key feature of the equilibrium SYK2 +SYK4 model is the
crossover from Non-Fermi-Liquid to Fermi-Liquid scal-
ing as temperature of the system is lowered below the
scale T ∗ ∼ J22 /J4 . This crossover can be clearly seen in
the equilibrium scattering rate of the Majorana fermions,
given by the imaginary part of the retarded self-energy
at zero frequency, that we plot in Fig. 11 as a function
of temperature. We see that the low-temperature T 2
behavior crosses over a linear scaling ∼ T when the tem-
Figure 11. Equilibrium scattering rate, defined as the imag- perature is above the dashed line, indicating the crossover
inary part of the retarded self-energy in equilibrium at zero scale T ∗ at that values of J2 , J4 . At higher temperatures
frequency, −Im ΣR (ω = 0), as a function of temperature. We instead the scattering rate saturates. We further note
see the crossover from FL (−Im ΣR T 2 ) to NFL (−Im ΣR T ) that in Fig. 11 the scattering rate is shifted with respect
scaling as the temperature is raised above T ∗ (dashed line). to a constant value proportional to J2 , the single particle
For comparison we plot in the inset the decay rate of the bandwidth, that is also responsible for a finite imaginary
equilibrium retarded Green’s function, equivalent to Γ∞ de- part of the self-energy from Eq. A2, although not related
fined in Sec. III, that show a featureless dependence from the to the many body interactions. For comparison we plot
temperature.
in the inset another measure of the decay rate, obtained
from the long-time decay of the retarded Green’s func-
tion, analogous to Γ∞ defined in Sec.. III. This quantity
frequency as
on the other hand has a very weak dependence on tem-
1 perature and does not show any signature of the crossover
GR (ω) = (A1) scale.
ω − ΣR (ω)
where the retarded self-energy can be still written in the
time-domain as
J42
ΣR (t) = − GR (t)3 + 3GR (t)GK (t)2 + J22 GR (t)
4
(A2)
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