Dynamical quantum phase transitions in SYK Lindbladians Kohei Kawabata1 2Anish Kulkarni1Jiachen Li1Tokiro Numasawa2and Shinsei Ryu1 1Department of Physics Princeton University Princeton New Jersey 08544 USA

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Dynamical quantum phase transitions in SYK Lindbladians

Kohei Kawabata,1, 2, ∗Anish Kulkarni,1, ∗Jiachen Li,1, ∗Tokiro Numasawa,2, ∗and Shinsei Ryu1, ∗

1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

(Dated: August 7, 2023)

We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model described by the

Lindblad master equation, where the SYK model is coupled to Markovian reservoirs with jump

operators that are either linear or quadratic in the Majorana fermion operators. Of particular

interest for us is the time evolution of the dissipative form factor, which quantiﬁes the average

overlap between the initial and time-evolved density matrices as an open quantum generalization

of the Loschmidt echo. We ﬁnd that the dissipative form factor exhibits dynamical quantum phase

transitions. We analytically demonstrate a discontinuous dynamical phase transition in the limit

of large number of fermion ﬂavors, which is formally akin to the thermal phase transition in the

two-coupled SYK model between the black-hole and wormhole phases. We also ﬁnd continuous

dynamical phase transitions that do not have counterparts in the two-coupled SYK model. While

the phase transitions are sharp in the limit of large number of fermion ﬂavors, their qualitative

signatures are present even for the ﬁnite number of fermion ﬂavors, as we show numerically.

I. INTRODUCTION

The physics of open quantum systems has recently at-

tracted growing interest. Since coupling to the external

environment is unavoidable in realistic physical systems,

an understanding of open quantum systems is important

for quantum technology [1]. Notably, dissipation is not

necessarily a nuisance that destroys quantum coherence

and the concomitant quantum phenomena; rather, dis-

sipation can even lead to new physical phenomena that

have no analogs in closed quantum systems. For exam-

ple, engineered dissipation can be utilized to prepare a

desired quantum state [2–4]. Dissipation can also give

rise to unique non-Hermitian topological phenomena [5].

Furthermore, open quantum systems exhibit phase tran-

sitions that cannot occur in closed quantum systems at

thermal equilibrium [6–14]. Prime recent examples in-

clude the entanglement phase transitions induced by the

competition between the unitary dynamics and the quan-

tum measurements [15–19]. Despite these recent ad-

vances, the interplay of strong many-body interactions

and dissipation, as well as the consequent phase transi-

tions, has yet to be fully understood.

In the theory of phase transitions, it is important to

develop a prototypical model that captures the univer-

sal behavior. Recently, open quantum generalizations of

the Sachdev-Ye-Kitaev (SYK) model [20,21] were pro-

posed in Refs. [22,23] as a prototype of open quan-

tum many-body systems. In this model, dissipation is

formulated by the Lindblad master equation [24,25],

which is diﬀerent from the non-Hermitian SYK Hamil-

tonians [26–29]. The original SYK Hamiltonian is a

fermionic model with fully-coupled random interactions

and exhibits quantum chaotic behavior [20,21,30–36].

Similarly, the SYK Lindbladian is a prototype that ex-

hibits the strongly-correlated chaotic behavior of open

∗The authors are listed in alphabetical order.

quantum systems [37–47]. As an advantage, the SYK

Lindbladian is analytically tractable in the limit of the

large number of fermion ﬂavors even in the presence of

dissipation. In Refs. [22,23], the decay rate was ana-

lytically calculated in this limit, by which a transition

between the underdamped and overdamped regimes was

demonstrated. Still, the open quantum dynamics of the

SYK Lindbladians remains mainly unexplored. As a pro-

totype of open quantum many-body systems, the inves-

tigation into the SYK Lindbladians should deepen our

general understanding of open quantum physics.

In this work, we ﬁnd the dynamical quantum phase

transitions in the SYK Lindbladians. We study the open

quantum dynamics of the SYK Lindbladians and espe-

cially focus on the time evolution of the dissipative form

factor. This quantiﬁes the average overlap between the

initial and time-evolved density matrices and serves as a

partition function of the open quantum dynamics, simi-

larly to the Loschmidt echo for the unitary dynamics of

closed quantum systems. We ﬁnd the singularities of the

dissipative form factor as a function of time, which sig-

nal the dynamical quantum phase transitions similarly to

the unitary counterparts [48–57]. Notably, this quantum

phase transition appears only in the dynamics in contrast

with the conventional phase transitions for thermal equi-

librium or ground states. In particular, we investigate

the SYK Hamiltonian coupled to Markovian nonrandom

linear dissipators and random quadratic dissipators. In

the limit of the large number Nof fermions, we analyti-

cally obtain the dissipative form factor and demonstrate

the discontinuous dynamical phase transition, which is

formally akin to the thermal phase transition in the two-

coupled SYK model between the black-hole and worm-

hole phases [58]. We also show the continuous dynamical

transition that has no counterparts in the original two-

coupled SYK model. Furthermore, we numerically show

that signatures of the dynamical quantum phase tran-

sitions remain to appear even for ﬁnite Nalthough the

singularities are not sharp.

arXiv:2210.04093v2 [cond-mat.stat-mech] 3 Aug 2023

The rest of this work is organized as follows. In

Sec. II, we start by introducing the models, and review

the quantity of our interest, the dissipative form factor.

In Sec. III, we study the SYK Lindbladian with nonran-

dom linear jump operators. We discuss both numerics of

the large Nsaddle point equations and the analytical ap-

proach in the large qlimit. In Sec. IV, we study the SYK

Lindbladian with random quadratic jump operators. We

conclude in Sec. V.

II. SYK LINDBLADIANS AND DISSIPATIVE

FORM FACTOR

We consider Markovian dynamics of the density matrix

ρ(t) described by the Lindblad master equation [1]:

dtρ(t) = L(ρ(t))

≡ −i[H, ρ(t)]

aLaρ(t)La†−1

2{La†La, ρ(t)},(1)

where His the Hamiltonian, and {La}is a set of jump

operators that describe the dissipative process with the

external environment. In our models, the Hamiltonian is

given by the q-body SYK Hamiltonian

HSYK =iq/2X

1≤i1<i2<···<iq≤N

Ji1i2···iqψi1ψi2···ψiq.(2)

Here, ψi=1,...,N are Majorana fermion operators satisfy-

ing {ψi, ψj}=δij .Ji1···iqare real independent Gaussian

distributed random variables with zero mean and vari-

ance given by

(Ji1···iq)2=σ2

J=J2(q−1)!

Nq−1J∈R+,(3)

where ··· denotes the disorder average. We consider two

choices of the jump operators: nonrandom linear and

random quadratic. The nonrandom linear jump opera-

tors are

Li=√µψii= 1,··· , N, µ ∈R+.(4)

On the other hand, the random p-body jump operators

are

La=X

1≤i1<···<ip≤N

i1···ipψi1···ψip(a= 1,2, . . . , M),

(5)

where Ka

i1···ipare complex Gaussian random variables

with zero mean and variance given by

|Ka

i1···ip|2=σ2

K=K2(p−1)!

NpK∈R+.(6)

In this work, we focus on p= 2 for clarity.

Since the Lindbladian is a superoperator that acts on

the density matrix, it is useful to introduce the operator-

state map. Here, we vectorize the density matrix ρ(t) and

regard it as a state |ρ(t)⟩in the doubled Hilbert space

H+⊗H−1. Correspondingly, we regard the Lindbladian

as an operator acting on the doubled Hilbert space. For

the SYK-type models relevant to this work, the Lindbla-

dian acting on the doubled Hilbert space is given by

L=−iH++i(−i)qH−

a(−i)pLa

+La†

−−1

2La†

+La

+−1

2La

−La†

−,(7)

where H±and La

±act on H±, respectively. While the

Hamiltonian part acts only on the individual bra or ket

space, the dissipation term couples these two spaces.

A quantity of our central interest is the disorder-

averaged trace of the exponential of the Lindbladian,

F(TL) = TrH+⊗H−(eTLL),(8)

which we call the dissipative form factor [41,46]. The

trace in Eq. (8) is taken over the doubled Hilbert space

H+⊗ H−. As explained shortly, the dissipative form

factor quantiﬁes the average overlap between the ini-

tial and time-evolved density matrices and serves as the

Loschmidt echo of open quantum systems. In the fol-

lowing, we obtain the dissipative form factor of the SYK

Lindbladians, using both the analytical calculations for

large Nand the numerical calculations for ﬁnite N,

and demonstrate its singularities in the open quantum

dynamics—dynamical quantum phase transitions. More

speciﬁcally, we analyze (a dissipative analog of) the rate

function of the dissipative form factor:

iS(TL)≡lim

N→∞

log F(TL)

N.(9)

Although the spectrum of the Lindbladian is complex in

general, the rate function is always real since the spec-

trum is symmetric about the real axis. We also note that

the dissipative form factor does not depend on initial

conditions but is determined solely by the Lindbladian.

At TL= 0, we always have F(TL= 0) = 2Nand hence

iS(TL= 0) = log 2.

Several motivating comments are in order. First, in

the absence of dissipation, the dissipative form factor in

Eq. (8) coincides with the spectral form factor of Hermi-

tian Hamiltonians. In fact, we have

TrH+⊗H−(eTLL)

= TrH+e−iTLH+TrH−e+iTL(−i)qH−

=TrHe−iTLH

2,(10)

1Strictly speaking, this is a sloppy notation when we discuss the

operator-state map for fermionic systems since states in the +

and −sectors may not commute because of the Fermi statis-

tics [59].

where in the last line we take q≡0 (mod 4) for simplic-

ity. Since the spectral form factor captures the quantum

chaos of the SYK-type Hamiltonians [33], we expect that

the dissipative form factor in Eq. (8) also captures the

quantum chaos of the SYK Lindbladians. Accordingly, in

the absence of the dissipation, the deﬁnition of the rate

function in Eq. (9) coincides with its unitary analog. In

the unitary case, non-analytical behavior in the rate func-

tion as a function of time was proposed as a diagnostic of

the dynamical quantum phase transitions [48–57]. Here,

we extend this idea to the non-unitary case. We also note

that there is another related quantity, dissipative spec-

tral form factor, introduced in Ref. [47]. The dissipative

spectral form factor captures the complex-spectral cor-

relations of non-Hermitian operators. By contrast, the

dissipative form factor in Eq. (8) is more directly rele-

vant to the open quantum dynamics since it gives the

Loschmidt echo and the decoherence rate, as explained

below. These two quantities are thus complementary to

understand the quantum chaos of open systems. While

we focus on the dissipative form factor in this work, it

should be worthwhile to study the dissipative spectral

form factor of the SYK Lindbladians as future work.

Second, the dissipative form factor in Eq. (8) is related

to the Loschmidt echo, the overlap between the initial

and time-evolved states,

TrH[ρ(0)ρ(TL)] .(11)

With the operator-state map, the Loschmidt echo is writ-

ten as the overlap between the two pure states in the

doubled Hilbert space,

⟨ρ(0)|ρ(TL)⟩=⟨ρ(0)|eTLL|ρ(0)⟩,(12)

where |ρ(0)⟩ ∈ H+⊗ H−is the state in the doubled

Hilbert space H+⊗H−mapped from the density matrix

ρ(0). To make a contact with the dissipative form factor,

one needs to average over the initial states |ρ(0)⟩. For

example, if we consider a set of states generated from a

reference state ρ0by a unitary rotation,

ρU=Uρ0U†,(13)

and average over the Haar random measure, we obtain

ZdU ⟨ρU|eTLL|ρU⟩

=Tr (ρ2

0)−1/L

L2−1Tr (eTLL) + L−Tr (ρ2

L2−1,(14)

where Lis the dimensions of the Hilbert space (L= 2N/2

for the SYK-type models). Thus, the Loschmidt echo of

open quantum systems is given by the dissipative form

factor in Eq. (8). Here, if we choose ρ0to be the fully

mixed state ρ0= 1/L, the ﬁrst term in the right hand

side of Eq. (14) vanishes. This is consistent with the

fact that the fully mixed state cannot be decohered any

longer. We have to avoid such a special reference state

to connect the dissipative form factor with the average

Loschmidt echo.

Finally, the dissipative form factor in Eq. (8) is also

related to the decoherence rate [38] averaged over initial

states. The decoherence rate Dquantiﬁes the early-time

decay of purity, deﬁned by

D=−2 Tr [ρ(0)(dρ(t)/dt)]

Tr [ρ(0)2]t=0

.(15)

As before we average over initial states ρU, leading to

Dav =−2

Tr [ρ2

dt ZdU Tr [ρU(0)ρU(t)]t=0

.(16)

Thus, the average decoherence rate Dav is given by the

time derivative of the dissipative form factor at t= 0. In

particular, from Eq. (14), Dav is expressed as

Dav =−2

Tr [ρ2

Tr (ρ2

0)−1/L

L2−1Tr (L).(17)

III. NONRANDOM LINEAR JUMP

OPERATORS

We consider the SYK model with the nonrandom lin-

ear jump operators in Eq. (4). We notice that this open

quantum model resembles the two-coupled SYK model

(Maldacena-Qi model) [58], although the SYK Lindbla-

dian is non-Hermitian while the two-coupled SYK model

is Hermitian. As we will show below, there are many

analogies between these models. In Table I, we sum-

marize the similarities and diﬀerences between the two-

coupled SYK model and the SYK Lindbladian with the

linear jump operators. In fact, one obtains the SYK

Lindbladian from the two-coupled SYK model by an an-

alytical continuation of the coupling µfrom a real to

pure imaginary value. In the Hermitian two-coupled

SYK model, the ﬁnite temperature partition function was

studied and the Hawking-Page transition (the thermal

phase transition between the black hole and wormhole

phases) was identiﬁed [58]. On the other hand, we are

here interested in the dissipative form factor in Eq. (8),

where TLplays the role of the real (rather than imagi-

nary) time.

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DynamicalquantumphasetransitionsinSYKLindbladiansKoheiKawabata,1,2,∗AnishKulkarni,1,∗JiachenLi,1,∗TokiroNumasawa,2,∗andShinseiRyu1,∗1DepartmentofPhysics,PrincetonUniversity,Princeton,NewJersey08544,USA2InstituteforSolidStatePhysics,UniversityofTokyo,Kashiwa,Chiba277-8581,Japan(Dated:August7,2023)Wes...

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Dynamical quantum phase transitions in SYK Lindbladians Kohei Kawabata1 2Anish Kulkarni1Jiachen Li1Tokiro Numasawa2and Shinsei Ryu1 1Department of Physics Princeton University Princeton New Jersey 08544 USA.pdf

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Dynamical quantum phase transitions in SYK Lindbladians Kohei Kawabata1 2Anish Kulkarni1Jiachen Li1Tokiro Numasawa2and Shinsei Ryu1 1Department of Physics Princeton University Princeton New Jersey 08544 USA

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