††thanks: J.L. and S.Y. contributed equally to this work.

# Spectral form factor in chaotic, localized, and integrable
open quantum many-body systems

Jiachen Li∗ Department of Physics, Princeton University, Princeton, NJ 08544,
USA  Stephen Yan∗ Department of Physics, University of California, Santa
Barbara, CA 93106, USA  Department of Physics, Princeton University,
Princeton, NJ 08544, USA  Tomaž Prosen  Department of Physics, Faculty of
Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000
Ljubljana, Slovenia  Institute of Mathematics, Physics and Mechanics,
Jadranska 19, SI-1000, Ljubljana, Slovenia  Amos Chan  Physics Department,
Lancaster University, Lancaster LA1 4YB, United Kingdom  Princeton Center for
Theoretical Science, Princeton University, Princeton, NJ 08544, USA

###### Abstract

We numerically study the spectral statistics of open quantum many-body systems
(OQMBS) as signatures of quantum chaos (or the lack thereof), using the
dissipative spectral form factor (DSFF), a generalization of the spectral form
factor to complex spectra. We show that the DSFF of chaotic OQMBS generically
displays the quadratic ramp-plateau behaviour of the Ginibre ensemble from
random matrix theory, in contrast to the linear ramp-plateau behaviour of the
Gaussian ensemble in closed quantum systems. Furthermore, in the presence of
many-body interactions, such RMT behaviour emerges only after a time scale
$\tau_{\mathrm{dev}}$, which generally increases with system size for
sufficiently large system size, and can be identified as the non-Hermitian
analogue of the many-body Thouless time. The universality of the random matrix
theory behavior is demonstrated by surveying twelve models of OQMBS, including
random Kraus circuits (quantum channels) and random Lindbladians
(Liouvillians) in several symmetry classes, as well as Lindbladians of
paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the
transverse field Ising models. We devise an unfolding and filtering procedure
to remove variations of the averaged density of states which would otherwise
hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond
chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS,
specifically the integrable XX model and a system in the many-body localized
(MBL) regime in the presence of dissipation, which exhibit DSFF behaviours
distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we
study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e.
only the jump operators are present, and demonstrate that the results of RMT
universality and scaling of many-body Thouless time survive even without
coherent evolution. As side results, we compute the density of states,
nearest-neighbour spacing distribution, and complex spacing ratio for the
studied models.

Introduction. – The study of open quantum systems is of importance since
realistic systems cannot be perfectly isolated from their environment. Such
open systems, represented by density matrices, have dynamics described by
quantum channels under the Markovian approximation, where a macroscopic number
of observables can be treated as quantum noise Gardiner and Zoller (2004). Two
prominent descriptions are the Kraus operator formulism Choi (1975); Kraus
_et al._ (1983), and the quantum master equation known as the Lindbladian
Gorini  _et al._ (1976); Lindblad (1976). Spectral statistics have
historically served as a robust diagnostic of quantum chaos in closed systems
Bohigas  _et al._ (1984), and have contributed to the recent revival of the
study of many-body quantum chaos Cotler  _et al._ (2017a, b); Kos  _et al._
(2018a); Chan  _et al._ (2018a, b); Bertini  _et al._ (2018); Saad  _et al._
(2019). For open systems, the spectral statistics has been explored much less,
with recent investigations focusing on the scale of mean-level spacing Akemann
_et al._ (2019); Sá  _et al._ (2019); Hamazaki  _et al._ (2019); Sá  _et al._
(2020); Wang  _et al._ (2020); Huang and Shklovskii (2020); Tzortzakakis  _et
al._ (2020); Peron  _et al._ (2020); Álvaro Rubio-García  _et al._ (2021);
García-García  _et al._ (2022); Xiao  _et al._ (2022); Sá  _et al._ (2022,
2023); Kawabata  _et al._ (2023a). With rapid advancements in non-Hermitian
physics Ashida  _et al._ (2020) and the recent discovery that the late-time
relaxation of open systems is not solely governed by the spectral gap Mori and
Shirai (2020); Haga  _et al._ (2021); Bensa and Žnidarič (2022), it is
pertinent to ask what are the universal features of spectral correlations at
all scales in OQMBS.

In this paper, by surveying a wide range of paradigmatic models, we showed
that the DSFF of chaotic OQMBS generically display the universal quadratic
ramp-plateau behaviour from RMT (Fig. 1). Further, We show that in the
presence of many-body interaction, the DSFF of chaotic OQMBS shows an early-
time deviation from the RMT until a time scale, $\tau_{\mathrm{dev}}$, which
diverges with system size, and which we identify as the analogue of the many-
body Thouless time Chan  _et al._ (2018b); Kos  _et al._ (2018b) in OQMBS
(Fig. 2). We devise a procedure using unfolding transformation and filtering
to remove variations of density of states (DOS) which would otherwise hide the
RMT-like signatures in DSFF for chaotic OQMBS. Lastly, we study the spectral
correlation of non-chaotic OQMBS, specifically integrable spin chains and
(prethermal) many-body localized (MBL) systems in the presence of dissipation,
showing a DSFF behaviour distinctive from the ramp-plateau behaviour of GinUE,
and thereby demonstrating that DSFF diagnoses (the lack of) chaos in OQMBS.

Figure 1:  The DSFF for (a) the Kraus operator models RKO ($\log n=6$), RKC
($L=6$), $U(1)$-RKC ($L=8$) and $\mathbb{Z}_{2}$-RKC ($L=7$); and (b) the
random Lindbladian models $0$D-RL ($\log n=7$), SYK-L ($L=12$), $1$D-RL
($L=7$), dXXZ ($L=7$), dXX ($L=7$), SS-$U(1)$-RL ($L=6$), WS-$U(1)$-RL
($L=7$), dIsing-Chaos ($L=7$), dIsing-MBL ($L=7$), 0D-RL-JO ($L=6$), 1D-RL-1JO
($L=6$), and 1D-RL-2JO ($L=6$). The DSFF is computed along $\theta=\pi/4$ with
5000 samples. For RKC models, $0$D-RL, SYK-L and dXXZ, we apply unfolding and
Gaussian filtering procedures. For RKO, we directly compute the DSFF. For all
other models, we only apply the filtering procedure, see details in SM sup
(a). Note that only the models, dIsing-MBL, dXX and 1D-RL-1JO, do not exhibit
RMT GinUE behaviour.

Dissipative spectral form factor. – The spectral form factor (SFF) is one of
the simplest non-trivial and analytically tractable diagnostic of quantum
chaos Berry (1985); Sieber and Richter (2001); Müller  _et al._ (2004, 2005).
The SFF captures correlations between eigenlevels at all scales, including the
level repulsion and spectral rigidity and has recently been shown to capture
novel signatures particularly in quantum many-body physics Kos  _et al._
(2018a); Chan  _et al._ (2018a, b); Bertini  _et al._ (2018); Friedman  _et
al._ (2019); Moudgalya  _et al._ (2020); Roy and Prosen (2020); Flack  _et
al._ (2020); Liao  _et al._ (2020); Chan  _et al._ (2021); Garratt and Chalker
(2020); Chan  _et al._ (2020); Winer and Swingle (2020); Bertini  _et al._
(2021); Šuntajs  _et al._ (2020), and in the studies of black holes and
holography Cotler  _et al._ (2017a, b); del Campo  _et al._ (2017); Altland
and Bagrets (2018); Saad  _et al._ (2019); Gharibyan  _et al._ (2018). For a
non-Hermitian matrix with complex spectra, however, the SFF is exponentially
growing or decaying in time due to the imaginary parts of the complex
eigenvalues. The DSFF has been introduced to circumvent this problem by
considering the complex spectrum as a two-dimensional gas and probing the
correlation therein Li  _et al._ (2021); Fyodorov  _et al._ (1997).
Specifically, given a $N\times N$ matrix with spectrum
$\\{z_{n}=x_{n}+iy_{n}\\}_{n=1}^{N}$, consider a generalized two-parameter
partition function

$Z(t,s)=\sum_{n}e^{\mathrm{i}x_{n}t+\mathrm{i}y_{n}s}\;,$ (1)

and the connected DSFF Li  _et al._ (2021); Fyodorov  _et al._ (1997) defined
by

$K_{\mathrm{c}}(t,s):=\big{\langle}\left|Z(t,s)\right|^{2}\big{\rangle}-\left|\left\langle
Z(t,s)\right\rangle\right|^{2}\;.$ (2)

We define the complex time $\tau\equiv|\tau|\,e^{i\theta}\equiv t+is$, and
will abusively use the radial coordinates $(|\tau|,\theta)$ as the arguments
of $K$. [Li  _et al._ , 2021] derived the exact analytical solution of DSFF
for the GinUE for all $N$, which simplifies in large-$N$ as

$K_{\mathrm{c},\mathrm{Gin}}(|\tau|,\theta)=N\left(1-e^{-|\tau|^{2}/4N}\right)\,.$
(3)

The role of the GinUE solution of DSFF is analogous to that of the Gaussian
unitary ensemble solution of SFF, namely and loosely that the DSFF behaviour
of sufficiently generic or “chaotic” non-Hermitian matrices display universal
behaviour from the GinUE of RMT Li  _et al._ (2021); Ghosh  _et al._ (2022);
Shivam  _et al._ (2023); Garcí a-García  _et al._ (2023); Cipolloni and
Grometto (2023). Alternative approaches in quantifying spectral correlation in
non-Hermitian matrices beyond the scale of mean-level spacing have been
explored in Can (2019); Kawabata  _et al._ (2023b); Matsoukas-Roubeas  _et
al._ (2023a, b); Yoshimura and Sá (2023); Zhou  _et al._ (2023). A few
comments are in order: Firstly, note that
$K_{\mathrm{c},\mathrm{Gin}}(|\tau|,\theta)$ only depends on the absolute
value of $\tau$, i.e. the DSFF of chaotic systems are rotationally symmetric
in complex time, at least after the onset of RMT behaviour, and for this
reason we opt to focus on $K(|\tau|,\theta=\pi/4)$ in this paper. Secondly, as
a function of $|\tau|$, the DSFF shows a (dip)-ramp-plateau behaviour 111At
early time $|\tau|\lesssim\tau_{\mathrm{edge}}\ll\tau_{\mathrm{Hei}}$, DSFF
dips from $K(0,\theta)=N^{2}$ with a form described by the non-universal
disconnected DSFF, $|\left\langle Z(t,s)\right\rangle|^{2}$, discussed in Li
_et al._ (2021), but excluded in (2) for simplicity, analogous to the SFF for
closed quantum systems: At $|\tau|\lesssim\tau_{\mathrm{Hei}}$, DSFF increases
quadratically $K_{\mathrm{c},\mathrm{Gin}}\simeq|\tau|^{2}/4$ in large $N$
until it reaches late time $|\tau|\gtrsim\tau_{\mathrm{Hei}}$ with
$\tau_{\mathrm{Hei}}\sim\sqrt{N}$, where DSFF reaches a plateau at $N$.
Crucially, the DSFF GinUE ramp behaviour is drastically different from the
corresponding SFF GUE behaviour, which is linear in time.

Models. – We will introduce two classes of models. The first class consists of
zero- and one-dimensional Kraus operators $\Phi$, i.e.
$\Phi[\rho]=\sum_{a}M_{a}\rho M^{\dagger}_{a}$ with density matrix $\rho$ and
operators $M_{a}$ satisfying $\sum_{a}M_{a}^{\dagger}M_{a}=\mathbb{1}$ Nielsen
and Chuang (2010). For one-dimensional $L$-site models with on-site Hilbert
space dimension $q$, we define random Kraus circuits (RKC) with brick-wall
geometry in the superoperator representation [Fig. 1 (a) inset] as

$\displaystyle\begin{split}\Phi=&\left(\,\bigotimes_{i\in
2\mathbb{Z}}V_{i}\right)\left(\,\bigotimes_{i\in
2\mathbb{Z}+1}V_{i}\right)\end{split}\,,$ (4)

where $V_{i}$ is a set of Kraus operators acting on supersite $i$ and $i+1$ as

$V_{i}=\sum_{a}v_{i,a}\otimes_{\mathrm{s}}v^{*}_{i,a}\,.$ (5)

Here $v_{i,a}$ are $q^{2}$-by-$q^{2}$ Kraus operators satisfying
$\sum_{a}v^{\dagger}_{i,a}v_{i,a}=\mathbb{1}$ for each $i$.
$\otimes_{\mathrm{s}}$ denotes the tensor product between operators acting to
the left and right of a density matrix. To incorporate symmetries, consider
the conserved quantity $I$, its associated symmetry operator $S=\exp(i\varphi
I)$ with real parameter $\varphi$, and the adjoint representation of $S$ given
by $\mathcal{S}[\rho]=S\rho S^{-1}$. A Kraus operator which respects our
symmetry admits the block-diagonal decomposition:

$v_{i}=\sum_{\mu}P_{\mu}v_{i}^{\mu}P_{\mu}\,,\;\;\;\text{s.t.}\;\;\;[\mathcal{S},\Phi]=[S,v_{i}]=0\,,$
(6)

where $P_{\mu}$ is a projector to the symmetry sector labelled by $\mu$. The
$v^{\mu}_{a}$ is a Kraus operator generated by a protocol using truncated
random unitary matrices Zyczkowski and Sommers (2000); Bruzda  _et al._
(2009): We take a $nd$-by-$nd$ unitary $T$ from the circular unitary ensemble
(CUE) with $d^{2}$ blocks of size $n$-by-$n$ denoted $t_{ab}$,
$a,b=1,\dots,d$, such that $T_{(a\alpha),(b\beta)}=[t_{ab}]_{\alpha\beta}$,
where $\alpha,\beta=1,\dots,n$ and $(a\alpha)$ is a composite index. We take
$v_{a}=t_{1a}$ such that the Kraus condition is satisfied via the unitarity of
$T$. We consider zero- and one-dimensional variations of RKC (4) listed below,
and written more explicitly in the supplementary material (SM) sup (a).

  1. 1.

0D Random Kraus operator (RKO) is our prototypical zero-dimensional model
without symmetries, where we take $\Phi_{\mathrm{0D}}=V$ defined in (5) with
$n$-by-$n$ matrices and $v_{a}$.

  2. 2.

1D RKC without symmetries (RKC) is a one-dimensional model without symmetries,
i.e. the sum (6) for $v_{i}$ is trivial.

  3. 3.

$\mathbb{Z}_{2}$-symmetric RKC ($\mathbb{Z}_{2}$-RKC) with $q=2$ is a RKC that
conserves $I=\prod_{i=1}^{L}\sigma^{z}_{i}$, the parity operator.

  4. 4.

$U(1)$-symmetric RKC ($U(1)$-RKC) with $q=2$ is a RKC that conserves
$I=\sigma^{z}_{\mathrm{tot}}=\sum_{i=1}^{L}\sigma^{z}_{i}$, the total
magnetization.

The second class of models are the Liouvillian of quantum master equations
$\frac{d\rho(t)}{dt}=\mathcal{L}[\rho]$ in the Lindblad form Lindblad (1976);
Gorini  _et al._ (1976). In the superoperator representation, we have

$\displaystyle\mathcal{L}=$
$\displaystyle-\mathrm{i}(H\otimes_{\mathrm{s}}\mathbb{1}-\mathbb{1}\otimes_{\mathrm{s}}H^{*})$
(7)
$\displaystyle+\sum_{a=1}^{d}(2F_{a}\otimes_{\mathrm{s}}F_{a}^{*}-F_{a}^{\dagger}F_{a}\otimes_{\mathrm{s}}\mathbb{1}-\mathbb{1}\otimes_{\mathrm{s}}F_{a}^{\dagger}F_{a})\,,$

where the Hamiltonian $H$ governs the unitary time evolution, the second term
describes the coupling between the system and the environment via jump
operators $F_{a}$. We study the Lindblad dynamics with variations given below.

  * 5.

0D Random Lindbladian (0D-RL) is our prototypical zero-dimensional model,
where $H\in\mathrm{GUE}(n)$ and $F_{a}\in\mathrm{GinUE}(n)$ are respectively
taken from the Gaussian unitary and complex Ginibre ensembles of $n$-by-$n$
matrices.

  * 6.

Sachdev-Ye-Kitaev Lindbladian (SYK-L): The Sachdev-Ye-Kitaev Hamiltonian is
given by
$H_{\mathrm{SYK}}=\mathrm{i}\sum^{L}_{i_{1}<i_{2}<i_{3}<i_{4}}J_{i_{1}i_{2}i_{3}i_{4}}\psi_{i_{1}}\psi_{i_{2}}\psi_{i_{3}}\psi_{i_{4}}$
with Majorana fermion operators $\psi_{i}=\psi_{i}^{\dagger}$ satisfying
$\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}$. We consider jump operator
$F_{a}=\sum_{1\leq i<j\leq L}K^{a}_{ij}\psi_{i}\psi_{j}$. $J_{ijkl}$ and
$K^{a}_{ij}$ are independently drawn from the normal distribution with
variance $J$ and $k$ respectively.

  * 7.

1D Random Lindbladian (1D-RL) is a one-dimensional chain with length $L$ and
on-site dimension $q$. We choose $H=\sum_{i=1}^{L}h_{i,i+1}$ with two-site
gates $h_{i,i+1}\in\mathrm{GUE}(q^{2})$, and two-site jump operators
$F_{i}\equiv\ell_{i,i+1}\in\mathrm{GinUE}(q^{2})$ acting on site $i$ and
$i+1$.

  * 8.

Strongly-$U(1)$-symmetric 1D-RL (SS-$U(1)$-RL) is defined identically to 1DRL,
except that $h_{i,i+1}=\bigoplus_{\mu=-q+1}^{q-1}h^{(\mu)}_{i,i+1}$ and
$\ell_{i,i+1}=\bigoplus_{\mu=-q+1}^{q-1}\ell_{i,i+1}^{(\mu)}$ where
$h^{(\mu)}_{i,i+1}\in\mathrm{GUE}(q-|\mu|)$ and
$\ell^{(\mu)}_{i,i+1}\in\mathrm{GinUE}(q-|\mu|)$ such that the model is
strongly symmetric, conserving the total magnetization
$I=\sigma^{z}_{\mathrm{tot}}$, i.e.

$[\mathcal{S},\mathcal{L}]=[I,H]=[I,\ell_{i,i+1}]=0.$ (8)

  * 9.

Weakly-$U(1)$-symmetric 1D-RL (WS-$U(1)$-RL) is defined similar to
SS-$U(1)$-RL with the same Hamiltonian, but with different random jump
operators $\ell_{i,i+1}$ such that they satisfy the following commutation
relations

$[\mathcal{S},\mathcal{L}]=[I,h_{i,i+1}]=0\,,\;\;[I,\ell_{i,i+1}^{(m)}]=m\ell_{i,i+1}^{(m)},$
(9)

where again $\mathcal{S}[\rho]=S\rho S^{-1}$. The explicit forms of
$\ell_{i,i+1}^{(m)}$ are given in the SM sup (a).

Furthermore, we study two spin chains with dissipation that display chaotic
and non-chaotic behaviours:

  * 10.

Dissipative XXZ and dissipative XX models  are defined through
$H=J\sum_{l=1}^{N-1}(\sigma_{l}^{x}\sigma_{l+1}^{x}+\sigma_{l}^{y}\sigma_{l+1}^{y}+\Delta\sigma_{l}^{z}\sigma_{l+1}^{z})+J^{\prime}\sum_{l=1}^{N-2}(\sigma_{l}^{x}\sigma_{l+2}^{x}+\sigma_{l}^{y}\sigma_{l+2}^{y}+\Delta^{\prime}\sigma_{l}^{z}\sigma_{l+2}^{z})$
and $F_{\mathrm{L}}^{+}=\sqrt{\gamma_{\mathrm{L}}^{+}}\sigma_{1}^{+}$,
$F_{\mathrm{L}}^{-}=\sqrt{\gamma_{\mathrm{L}}^{-}}\sigma_{1}^{-}$,
$F_{\mathrm{R}}^{+}=\sqrt{\gamma_{\mathrm{R}}^{+}}\sigma_{L}^{+}$,
$F_{\mathrm{R}}^{-}=\sqrt{\gamma_{\mathrm{R}}^{-}}\sigma_{L}^{-}$,
$F_{l}=\sqrt{\gamma}\sigma_{l}^{z}$ where $\sigma_{l}^{\alpha},\alpha=x,y,z$
are Pauli matrices and
$\sigma^{\pm}_{l}=\sigma_{l}^{x}\pm\mathrm{i}\sigma_{l}^{y}$. Following
Akemann  _et al._ (2019), we consider two choices of parameters: (a) Chaotic
dissipative XXZ model (dXXZ), with parameters sampled from normal
distributions $J,J^{\prime}\in\mathcal{N}(1,0.09)$,
$\Delta\in\mathcal{N}(0.5,0.0225)$,
$\Delta^{\prime}\in\mathcal{N}(1.5,0.2025)$, $\gamma=0$, $\gamma_{L}^{+}=0.5$,
$\gamma_{L}^{-}=0.3$, $\gamma_{R}^{+}=0.3$, and $\gamma_{R}^{-}=0.9$; (b)
Integrable dissipative XX model (dXX), with $J\in\mathcal{N}(1,0.09)$,
$J^{\prime}=0$, $\Delta=0$, $\Delta^{\prime}=0$, $\gamma=1$,
$\gamma_{L}^{+}=0.5$, $\gamma_{L}^{-}=1.2$, $\gamma_{R}^{+}=1$, and
$\gamma_{R}^{-}=0.8$.

  * 11.

Dissipative transverse field Ising model in the chaotic and MBL phases
Hamazaki  _et al._ (2022) are defined with
$H=J\sum_{l=1}^{L}\sigma^{z}_{l}\sigma^{z}_{l+1}+g\sum_{l=1}^{L}\sigma^{x}_{l}+\sum_{l=1}^{L}h_{l}\sigma^{z}_{l},$
where the on-site disorder $h_{l}\in[-W,W]$ is drawn from a flat distribution
of width $W$. We take
$F_{a}=\sqrt{\frac{\gamma}{2}}(\sigma^{x}_{a}-\mathrm{i}\sigma^{y}_{a})$ with
$a=1,\dots,L$. Without dissipation, in finite system sizes, $H$ displays many-
body localized phenomenology for sufficiently large $W$. In this paper we
focus on $J=1$, $g=-0.9$, $\gamma=0.5$ and two choices of $W$: (a) $W=.6$
(dIsing-Chaos) and (b) $W=5$ (dIsing-MBL).

Lastly, we study Lindbladians with the Hamiltonian term turned off:

  * 12.

Jump-operator-only random Lindbladians are defined with $H=0$ with three
variations: (a) 0D Random Lindbladian with jump operators only (0D-RL-JO)
which coincides with 0D-RL (model 5) with $H=0$. (b) 1D Random Lindbladian
with 1-site jump operators only (1D-RL-1JO) with
$F_{i}=\ell_{i}\in\text{GinUE}(q=2)$ acting as one-site operator on site $i$
with $d=L$. (c) 1D Random Lindbladian with 2-site jump operators only
(1D-RL-2JO) which coincides with 1D-RL (model 7) with $H=0$.

For each model above with symmetries, we analyse the eigenvalues of the
superoperator after projection into the largest symmetry sector. In addition
to the study of the DSFF in Fig. 1 and 2, we provide numerical data for the
single-realization spectra, DOS, nearest-neighbour spacing distribution, and
complex spacing ratio in the SM sup (a).

Unfolding and filtering. – The density of states (DOS) can be written as a sum
of two terms, $\rho_{\mathrm{av}}$, accounting for the smooth, averaged
behavior, and $\rho_{\mathrm{fluc}}$, for the fluctuations about the average,
i.e. $\rho(z)=\rho_{\mathrm{av}}(z)+\rho_{\mathrm{fluc}}(z)$. The expected
quadratic ramp behaviour for chaotic OQMBS means that the variation in
$\rho_{\mathrm{av}}$ has to be removed before comparing the level fluctuations
$\rho_{\mathrm{fluc}}$ in different regions. This process is called
“unfolding.” Without unfolding, regions of the spectrum with varying
$\rho_{\mathrm{av}}(z)$ give quadratic ramps with different values of
$\tau_{\mathrm{Hei}}$, summing to a smeared-out DSFF behaviour (such problem
does not arise for the linear ramp behaviour of the SFF in closed chaotic
quantum systems). Ideally, we would like to find an “unfolding” function
transforming $z\mapsto g(z)$ such that the resulting $\rho_{\mathrm{av}}(z)$
is perfectly uniform and local relationships between eigenvalues are
preserved, as reflected in, e.g., the nearest neighbour spacing distribution
and the complex spacing ratio Sá  _et al._ (2019). However, unfolding for
complex spectra is considerably more challenging than for the real
counterpart, and such function $g$ might not even exist in principle. We
remedy this problem in two steps. Firstly, for each model, we empirically find
a conformal transformation as the (imperfect) unfolding function $g$ such that
the resulting $\rho_{\mathrm{av}}$ is closer to uniform over a subregion of
the complex plane sup (a). Secondly, we employ a filtering procedure which
favours the DSFF contributions from the flatter region of the unfolded
spectrum $\\{g(z_{n})\equiv\tilde{z}_{n}=\tilde{x}_{n}+i\tilde{y}_{n}\\}$,
analogous to the filtering procedure for SFF Gharibyan  _et al._ (2018).
Therefore, we define a filtered partition function,

$\tilde{Z}_{f}(t,s;\\{\alpha\\})=\sum_{n}e^{\mathrm{i}\tilde{x}_{n}t+\mathrm{i}\tilde{y}_{n}s}f\left(\tilde{x}_{n},\tilde{y}_{n};\\{\alpha\\}\right)\,,$
(10)

where $\\{\alpha\\}$ parameterize the filtering function
$f(\tilde{x}_{n},\tilde{y}_{n};\\{\alpha\\})$. To be concrete, we focus on the
Gaussian filter $f(x;\alpha,\mu)=e^{-\alpha(x-\mu)^{2}}$ in either Cartesian
or radial coordinates, with $\mu$ chosen to be the center of the flatter
region in the unfolded spectrum. The determination of filtering strength
$\alpha$ is discussed below. An example of the DSFF for the unfolded and
filtered spectrum for the RKC is provided in Fig. 3. The filtered connected
DSFF, $\tilde{K}_{\mathrm{c}}(t,s;\\{{\alpha}\\})$, is defined as in (2), with
$Z$ replaced by $\tilde{Z}$.

Figure 2: DSFF and the onset of RMT quadratic ramp for structureless models
(green), many-body chaotic models (blue), and MBL and integrable models (red)
of OQMBS for different system sizes. The DSFF data is taken along
$\theta=\pi/4$ with 5000 samples. DSFF of larger (smaller) system sizes are
labelled with darker (lighter) colors. The theoretical curve of GinUE from
RMT, $K_{\mathrm{c},\mathrm{Gin}}$, is in orange. The unfolding and filtering
procedures are as in Fig. 1. In the insets, we plot $\tau_{\mathrm{dev}}$
against $L$. In contrast to the structure-less models (RKO, 0D-RL, 0D-RK-JO),
we observe an increase in $\tau_{\mathrm{dev}}$ in $L$ for all models with
many-body interaction, with the exceptions of $U(1)$-RKC, $U(1)$-RL and dXXZ,
whose dependence of $\tau_{\mathrm{dev}}$ in $L$ has not stabilized.

Onset of RMT quadratic ramp. – Based on the SFF behaviour in closed chaotic
many-body systems, we expect three energy scales in the chaotic OQMBS: (i)
$\Delta_{\mathrm{D}}$, the energy scale over which the DOS varies, defined
more precisely in SM sup (a); (ii) $\Delta_{\mathrm{Th}}$, the scale within
which the RMT behaviour emerges in the spectral correlation of the OQMBS
(discussed below); and (iii) $\Delta_{\mathrm{Hei}}$, the mean level spacing
in the complex plane. Again, analogous to the closed chaotic systems Chan  _et
al._ (2018b); Kos  _et al._ (2018b), we consider an OQMBS chaotic if it
exhibits RMT DSFF behaviour at sufficiently large time scale. For such
systems, and given filtering strength, we define this time scale as

$\displaystyle\tau_{\mathrm{dev}}(\\{\alpha\\})=\frac{\sqrt{N}}{\tau_{\mathrm{Hei}}}\times\min$
$\displaystyle\Bigg{\\{}\tau:\frac{\tilde{K}_{\mathrm{c}}(|\tau|,\theta;\\{\alpha\\})}{\tilde{K}_{\mathrm{c}}(|\tau|\to\infty,\theta;\\{\alpha\\})}$
$\displaystyle-\frac{K_{\mathrm{c},\mathrm{Gin}}(|\tau|,\theta)}{N}\leq\epsilon\Bigg{\\}}\,,$
(11)

where here $\tau_{\mathrm{Hei}}$ is the time scale at which the unfolded and
filtered DSFF reaches the plateau, and consistently, $\tau_{\mathrm{Hei}}$ can
be estimated in terms of the average level-spacing weighted by the filter (see
sup (a)). Given that the mean level-spacing of a spectra with flat
$\rho_{\mathrm{av}}(z)$ scales as $1/\sqrt{N}$, the factor
$\sqrt{N}/\tau_{\mathrm{Hei}}$ above compensates for the artificial effect of
filtering. See e.g. Fig. 3(d) inset. The early-time deviation from RMT
behaviour can be due to non-physical effects like filtering and non-universal
behaviour like finite-size effects and the “dip” in the disconnected DSFF
Gharibyan  _et al._ (2018). Below, we argue that $\tau_{\mathrm{dev}}$ can
indeed be identified as the many-body Thouless time
$\tau_{\mathrm{Th}}\propto\Delta_{\mathrm{Th}}^{-1}$ for appropriate choices
of filtering strength.

Choice of filtering strength. – In order to choose the appropriate filtering
strength $\alpha$, it is useful to define a dimensionless filtering strength
$\tilde{\alpha}=\alpha\Delta_{\mathrm{D}}^{2}$. We choose $\alpha$ to satisfy
$\Delta_{\mathrm{D}}^{-2}\ll\alpha\ll\Delta_{\mathrm{Th}}^{-2}<\Delta_{\mathrm{Hei}}^{-2}$,
such that our filtering function is strong enough to reduce the effect of
inhomogeneity in the DOS, but weak enough to preserve the physics at the scale
$\Delta_{\mathrm{Th}}$.

The typical dependence of $\tau_{\mathrm{dev}}$ on $\tilde{\alpha}$ for
chaotic OQMBS exhibits a trough-like behaviour, see e.g. the RKC model in Fig.
3(c) inset. For weak filter strengths $\tilde{\alpha}$, $\tau_{\mathrm{dev}}$
is large because the DSFF is distorted by the non-uniform DOS and does not
exhibit universal RMT behavior. For strong filter, $\tau_{\mathrm{dev}}$ is
large because the filter removes the signatures of level repulsion in RMT at
scale $\Delta_{\mathrm{Hei}}$. For intermediate filter strengths, we observe a
trough in $\tau_{\mathrm{dev}}(\tilde{\alpha})$ where OQMBS displays prominent
RMT DSFF behaviour. In practice, we pick smaller values of
$\tau_{\mathrm{dev}}$ in the trough to ensure that
$\alpha\ll\Delta_{\mathrm{Th}}^{-2}$. For all studied chaotic OQMBS, the curve
of $\tau_{\mathrm{dev}}$ versus $\tilde{\alpha}$ displays a trough, whose
width increases in system size, see Fig. 3(c) and sup (b). This trough is
relatively flat, and hence we do not expect the value of $\tau_{\mathrm{dev}}$
to be sensitive to the precise choice of filtering strength within the trough.

Finally, we benchmark our filtering procedure by computing the DSFF of the
spectrum of GinUE, after applying three conformal transformations to introduce
inhomogeneities to the originally flat DOS. The form of the DSFF of GinUE
before the transformation is known exactly (3). For suitably chosen filter
strengths, we recover the GinUE RMT behavior of the flat spectrum from the
filtered DSFF of the transformed spectrum for all three transformations we
considered: (i) $\\{z_{n}\\}$, (ii) $\\{\log z_{n}\\}$, and (iii) $\\{\log
z^{p}\\}$.

Chaotic OQMBS and many-body Thouless time. – In Fig. 1 and Fig. 2, we compute
the connected DSFF for OQMBS of models 1 to 9, the dXXZ (model 10) and dIsing-
Chaos (model 11). Additionally, we compute the nearest-neighbour spacing
distribution, and complex spacing ratio in the SM sup (a). The DSFF, spacing
distribution and ratio of all models converge to RMT GinUE behaviour in
increasingly large system size at sufficiently large times (as parameterised
by $\tau_{\mathrm{dev}}$). Remarkably, chaotic OQMBS generically display the
RMT quadratic ramp, i.e. signatures of GinUE eigenvalue correlation well
beyond the scale of mean-level spacing, or equivalently, at times well before
the Heisenberg time.

Figure 3: Unfolding and filtering. (a) A single-realization spectrum $\\{z\\}$
of RKC for $L=6$. (b) The unfolded spectrum $\\{z^{1/\log_{2}N}\\}$ (of the
spectrum in (a)) is subsequently filtered with the filtering function
$f(z)=e^{-\alpha(|z|-\mu)^{2}}$. (c) DSFF of the RKC for varying filtering
strength $\tilde{\alpha}\in[1/100,20]$ from light to dark blue. Inset:
$\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$ shows a trough. The suitable
range of filtering strength is given at the lower range of $\tilde{\alpha}$ at
the trough. Note that the artificial edge of the unfolded spectrum in (b) is
due to the branch cut in the unfolding function. We perform sanity checks by
unfolding the GinUE spectrum, and find that such edges do not qualitatively
distort the DSFF even in the absence of a filtering scheme, see Appendices E
and G.

Now we investigate the early-time deviation from RMT DSFF behaviour, which we
refer to as the “bump.” Note that the bump should not be confused with the
decaying behaviour from $N^{2}$ to the ramp due to the disconnected part of
DSFF Li  _et al._ (2021) or SFF Gharibyan  _et al._ (2018). We demonstrate
that for OQMBS, (i) a sensible definition of the height of the bump in DSFF
increases in system size, see SM sup (a); and (ii) the region of the bump in
$|\tau|$, as parameterised by $\tau_{\mathrm{dev}}$, increases in system size
for sufficiently large system size. In Fig. 2, we present the DSFF for
increasing $L$ for most models. We see a notable difference between models
without structures (RKO and 0D-RL) and those with many-body interactions (e.g.
SYK) or spatial structure (e.g. RKC, 1D-RL, and dIsing-Chaos). For the models
without structure, $\tau_{\mathrm{dev}}$ does not change appreciably with
system size, while for models with many-body interaction, it increases for
sufficiently large $L$. In other words, due to locality, increasingly long
time is required for the many-body systems to be indistinguishable from
structureless random matrices, as far as the spectral statistics is concerned.
Note that several models (e.g. $U(1)$-RKC, $U(1)$-RL, dXXZ), display RMT DSFF
behaviour for the largest system size, the dependence of $\tau_{\mathrm{dev}}$
with system size has not stabilized due to finite-size effects. Note that, for
the collapsed curves in the main panels of Fig. 2,
$\tau_{\mathrm{dev}}/\tau_{\mathrm{Hei}}$ is decreasing for larger $L$, i.e.,
the bump appears to move towards the left. However, this is due to the
exponential growth of the Heisenberg time; as seen in the insets of Fig. 2,
$\tau_{\mathrm{dev}}$ itself increases sub-exponentially for OQMBS, unlike the
case for 0D systems. (i) and (ii) suggest that the deviation from RMT in DSFF
is a genuine many-body effect in OQMBS, not due to finite-size effect.
Therefore, we identify $\tau_{\mathrm{dev}}$ as the many-body Thouless time
for open quantum many-body systems.

Many-body localized and integrable systems. – In Fig. 1 and Fig. 2, we present
the DSFF of dIsing-MBL and dXX after applying the filtering protocol, as
examples of MBL and integrable systems, respectively. For both systems, the
DSFF increases rapidly to (and above) the plateau, fluctuates wildly even
after ensemble averaging, and does not display the ramp-plateau signature of
RMT. In insets of Fig. 2, we compute the time when the DSFF increases to the
plateau, defined as

$\displaystyle\tau_{\mathrm{plat}}=\frac{\sqrt{N}}{\tau_{\mathrm{Hei}}}\min\Bigg{\\{}\tau:\Big{|}\frac{\tilde{K}_{\mathrm{c}}(|\tau|,\theta;\\{\alpha\\})}{\tilde{K}_{\mathrm{c}}(|\tau|\to\infty,\theta;\\{\alpha\\})}-1\Big{|}\leq\epsilon\Bigg{\\}}\,,$

where an additional time average has been applied to $\tilde{K}_{\mathrm{c}}$,
and where $\tau_{\mathrm{Hei}}$ is defined by the mean level spacing weighted
by the filter sup (a). We observe that $\tau_{\mathrm{plat}}$ in the MBL and
integrable systems does not scale with the inverse mean level spacing, i.e.
$\tau_{\mathrm{plat}}$ does not scale exponentially in system size $L$
(although still increasing in $L$ for the accessible system sizes), in
contrast to the chaotic case, and analogous to the behaviour of SFF in MBL and
certain integrable closed systems Šuntajs  _et al._ (2020); Garratt and
Chalker (2021); Prakash  _et al._ (2021). We therefore conclude that the DSFF
clearly distinguishes chaotic and non-chaotic systems, including MBL and
integrable systems.

Jump-operator-only Lindbladians. – We study the DSFF of Lindbladians with the
Hamiltonian term set to zero, i.e. only the jump operators are present. In
Fig. 2 (n-p), we plot the DSFF for Lindbladians with structureless, 1-site and
2-site jump-operators, and see that they display behaviours similar to the
MBL, structureless chaotic and spatially-extended chaotic models respectively.
These results demonstrate that the RMT universality and scaling of many-body
Thouless time can survive even with only driven-dissipative evolution (without
coherent evolution).

Conclusions. – In this paper, by surveying a wide range of paradigmatic
models, we showed that chaotic OQMBS generically display the quadratic ramp-
plateau behaviour of the Ginibre ensemble from RMT after a time scale
$\tau_{\mathrm{dev}}$, which we identify as the non-Hermitian analogue of the
many-body Thouless time. We further show that the long-range spectral
statistics of MBL and integrable OQMBS behave distinctly from the chaotic
OQMBS. Recently, experimental protocols have been proposed to measure the SFF
in closed quantum many-body systems Vasilyev  _et al._ (2020); Joshi  _et al._
(2022); Dağ  _et al._ (2023). However, in practice, decoherence is an inherent
feature of current noisy intermediate-scale era quantum (NISQ) simulators,
i.e. such systems are naturally open. Therefore, it would be beneficial to
devise a protocol to probe quantum many-body dynamics using DSFF. Such
protocol requires one to overcome certain technical obstacles, e.g. treating
the real and imaginary eigenvalues separately as in Eq. (1). For similar
reasons, it is an interesting but non-trivial task to gain a better analytic
handle on the DSFF in OQMBS , whose closed quantum many-body system analogues
have only recently been understood Kos  _et al._ (2018a); Chan  _et al._
(2018a, b); Bertini  _et al._ (2018).

Acknowledgement. We are thankful to Shinsei Ryu and Yan Fyodorov for helpful
discussions. A.C. acknowledges support from the Royal Society grant
RGS$\backslash$R1$\backslash$231444, and the fellowships from EPSRC
EP/X042812/1, the Croucher foundation and the PCTS at the Princeton
University. T.P. acknowledges ERC Advanced grant 101096208 – QUEST, and ARIS
research program P1-0402 and grant N1-0219.

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Supplementary Material
Spectral form factor in chaotic, localized, and integrable
open quantum many-body systems

In this supplementary material we provide additional details about:

  1. A

Models

     * —

Random Kraus circuits

       * 1.

Random Kraus operators (RKO)

       * 2.

Random Kraus circuits (RKC)

       * 3.

$\mathbb{Z}_{2}$-symmetric Random Kraus circuits ($\mathbb{Z}_{2}$-RKC)

       * 4.

$U(1)$-symmetric Random Kraus circuits ($U(1)$-RKC)

     * —

Random Lindbladians

       * 5.

0D Random Lindbladian (0D-RL)

       * 6.

Sachdev-Ye-Kitaev Lindbladian (SYK-L)

       * 7.

1D Random Lindbladian (1D-RL)

       * 8.

Strongly-$U(1)$-symmetric 1D Random Lindbladian (SS-$U(1)$-RL)

       * 9.

Weakly-$U(1)$-symmetric 1D random Lindbladian (WS-$U(1)$-RL)

       * 10.

Dissipative XXZ and XX models

         * (a)

Chaotic dissipative XXZ model (dXXZ)

         * (b)

Integrable dissipative XX model (dXX)

       * 11.

Dissipative transverse field Ising model

         * (a)

Dissipative transverse field Ising model in the chaotic phase (dIsing-Chaos)

         * (b)

Dissipative transverse field Ising model in the many-body localized phase
(dIsing-MBL)

       * 12.

Jump-operator-only Lindbladian

         * (a)

0D Random Lindbladian with jump operator only (0D-RL-JO)

         * (b)

1D Random Lindbladian with 1-site jump operators only (1D-RL-1JO)

         * (c)

1D Random Lindbladian with 2-site jump operators only (1D-RL-2JO)

  2. B

Spectral properties

  3. C

Nearest neighbour spacing distribution

  4. D

Complex spacing ratio

  5. E

Unfolding

  6. F

Filtering

     * —

Definition

     * —

Early time effect of filtering

     * —

Late time effect of filtering

     * —

Definition of Heisenberg time after filtering

     * —

Choice of filtering strength $\alpha$

  7. G

Sanity checks

     * —

Sanity check 1: DSFF of $\\{z\\}$ of GinUE with filtering

     * —

Sanity check 2: DSFF of $\\{\log z\\}$ of GinUE with filtering

     * —

Sanity check 3: DSFF of $\\{z^{p}\\}$ of GinUE with filtering and
$p=\frac{1}{4}$

     * —

Example: DSFF of RKO with filtering

  8. H

Summary table and additional numerics for dissipative spectral form factor
(DSFF)

  9. I

Height of Bumps

##  Appendix A Models

In this section, we write explicitly the content of the gates used to
construct the zero-dimensional RKO, the one-dimensional RKC, and its variants.
We use the convention where the matrix representation of the superoperators is
of size $N$. For many-body systems, we use $L$ to denote the system size in
one-dimensional systems, or the number of particles in zero-dimensional
systems (see the SYK-L model below). We use $I$ for conserved quantities,
$S=\exp(i\varphi I)$ for its associated symmetry operator with real parameter
$\varphi$, and $\mathcal{S}$ for the adjoint representation of $S$ defined by
$\mathcal{S}[\rho]=S\rho S^{-1}$.

###  A.1 Random Kraus circuits

  * 1.

Random Kraus operator (RKO) – In the superoperator representation, the 0D-RKO
acting on a Hilbert space with dimension $n$ is given by

$\Phi_{\mathrm{0D}}=\sum_{a=1}^{d}v_{a}\otimes_{\mathrm{s}}v^{*}_{a}\,,$
(SA.1)

where $v_{a}$ is a $n$-by-$n$ Kraus operator satisfying
$\sum_{a}v_{a}v^{\dagger}_{a}=\mathbb{1}$ and is generated by a protocol using
truncated random unitary matrices Zyczkowski and Sommers (2000); Bruzda  _et
al._ (2009): We take a $nd$-by-$nd$ Haar-random unitary $T\in U(dn)$ with
$d^{2}$ blocks of size $n$-by-$n$ denoted by $t_{ab}$, $a,b=1,\dots,d$, such
that $T_{(a\alpha),(b\beta)}=[t_{ab}]_{\alpha\beta}$ where
$\alpha,\beta=1,\dots,n$ and $(a\alpha)$ is a composite index. We take
$v_{a}=t_{1a}$ for $a=1,\dots,d$ such that the Kraus operator condition is
satisfied via the unitarity of $T$. For simplicity, to represent Kraus
operators sampled in this manner, we will use the notation

$\\{v_{a}\\}\in\mathrm{TCUE}(n,dn)\,.$ (SA.2)

  * 2.

Random Kraus circuit (RKC) – The RKC without any conserved quantities,
$\Phi_{\mathrm{1D}}$, is defined in the main text by (4) and (5), which we
reproduce below,

$\Phi_{\mathrm{1D}}=\left(\,\bigotimes_{i\in
2\mathbb{Z}}V_{i}\right)\left(\,\bigotimes_{i\in
2\mathbb{Z}+1}V_{i}\right)\,,\qquad\qquad
V_{i}=\sum_{a}v_{i,a}\otimes_{\mathrm{s}}v^{*}_{i,a}\,.$ (SA.3)

For RKC without conserved quantities, we take

$\left\\{v_{i,a}\right\\}\in\mathrm{TCUE}\left(q^{2},dq^{2}\right)\,,$ (SA.4)

i.e., $\\{v_{i,a}\\}$ are taken from the truncated CUE.

  * 3.

$\mathbb{Z}_{2}$-symmetric random Kraus circuit ($\mathbb{Z}_{2}$-RKC) –
Consider the parity operator given by $I=\prod_{i=1}^{L}\sigma^{z}_{i}$, where
$\sigma^{z}_{i}$ is the Pauli-$z$ operator acting on site $i$. To construct a
$\mathbb{Z}_{2}$-RKC preserving the parity of the circuit with on-site
dimension $q=2$, we consider $2$-site gates with the block structure in the
basis of $\sigma^{z}$ as follows,

$v_{i,a}=\bigoplus_{\mu=0}^{1}v_{i,a}^{(\mu)}\;,$ (SA.5)

where

$\left\\{v^{(\mu)}_{i,a}\right\\}\in\mathrm{TCUE}\left(2,2d\right)\,.$ (SA.6)

for $\mu=1,2$, such that

$[\mathcal{S},\Phi]=[I,v_{i,a}]=0$ (SA.7)

We focus on the symmetry sector $I=1$ in this paper, although for $I=-1$, we
expect our observables to behave identically.

  * 4.

$U(1)$-symmetric random Kraus circuit ($U(1)$-RKC): Consider the $U(1)$
charge, $I=\sigma^{z}_{\mathrm{tot}}=\sum_{i=1}^{L}\sigma^{z}_{i}$. To
construct a $U_{1}$-RKC with on-site dimension $q=2$, we consider $2$-site
gates with the block structure in the basis of $\sigma^{z}$ as follows,

$v_{i,a}=\bigoplus_{\mu=-1}^{1}v_{i,a}^{(\mu)}$ (SA.8)

where rlrl { v^(μ=±1)_i,a } &∈TCUE(1,d ) { v^(0)_i,a } ∈TCUE(2,2d ) . such
that

$[\mathcal{S},\Phi]=[I,v_{i,a}]=0\;.$ (SA.9)

We focus on the symmetry sector $I=0$ in this paper.

###  A.2 Random Lindbladians

To make this section self-contained, we reproduce the Lindbladian (7) in the
superoperator representation,

$\displaystyle\mathcal{L}=$
$\displaystyle-\mathrm{i}(H\otimes_{\mathrm{s}}\mathbb{1}-\mathbb{1}\otimes_{\mathrm{s}}H^{*})+\sum_{a=1}^{d}(2F_{a}\otimes_{\mathrm{s}}F_{a}^{*}-F_{a}^{\dagger}F_{a}\otimes_{\mathrm{s}}\mathbb{1}-\mathbb{1}\otimes_{\mathrm{s}}F_{a}^{\dagger}F_{a})\,,$
(SA.10)

where $H$ and $F_{a}$ are the Hamiltonian and jump operators specified by the
variation of the models below. Again, we take the dimension of the Hilbert
space to be $n$, and the size of the matrix representation of the
superoperator $\mathcal{L}$ to be $N=n^{2}$. The choice of $d$ will depend on
the specific model considered below.

  * 5\.

0D Random Lindbladian (0D-RL) – On the superspace of a Hilbert space of size
$n$, the Hamiltonian and jump operators are given by

$H\in\mathrm{GUE}(n),\qquad\qquad F_{a}\in\mathrm{GinUE}(n)\,,$ (SA.11)

where $\mathrm{GUE}(n)$ and $\mathrm{GinUE}(n)$ are the Gaussian and complex
Ginibre unitary ensemble of $n$-by-$n$ random matrices respectively. We take
$d=1$ for this model.

  * 6\.

Sachdev-Ye-Kitaev Lindbladian (SYK-L) – The Sachdev-Ye-Kitaev Hamiltonian is
given by

$H_{\mathrm{SYK}}=\mathrm{i}\sum^{L}_{i_{1}<i_{2}<i_{3}<i_{4}}J_{i_{1}i_{2}i_{3}i_{4}}\psi_{i_{1}}\psi_{i_{2}}\psi_{i_{3}}\psi_{i_{4}}\;,$
(SA.12)

with Majorana fermion operators $\psi_{i}=\psi_{i}^{\dagger}$ satisfying
$\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}$, and real Gaussian random variables
$J_{i_{1}i_{2}i_{3}i_{4}}\in\mathcal{N}(0,J)$. We consider jump operators

$\displaystyle F_{a}$ $\displaystyle=\sum_{1\leq i<j\leq
L}K^{a}_{ij}\psi_{i}\psi_{j}\;,\;\;a=1,2,\dots,d\;,$ (SA.13)

with complex Gaussian random variables
$K^{a}_{ij}\in\mathcal{N}(0,k)+i\mathcal{N}(0,k)$ Kulkarni  _et al._ (2022);
Sá  _et al._ (2022). From Kawabata  _et al._ (2023a), SYK-L has two parity
symmetries $(-1)^{\cal F^{+}}$ and $(-1)^{\cal F^{-}}$ where
$\left(-1\right)^{F^{\pm}}=\left[\prod_{i=1}^{\left(L-1\right)/2}\left(2\mathrm{i}\psi_{2i-1}^{\mp}\psi_{2i}^{\mp}\right)\right]\,(\sqrt{2}\psi_{L}^{\mp})$.
In this paper, we take parameters $d=L$, $J=\frac{3!}{L^{3}}$ and $k=0.04$,
and we focus on the symmetry section $(-1)^{\cal F^{\pm}}=1$.

  * 7\.

1D Random Lindbladian (1D-RL) – On the superspace of a one-dimensional system
with on-site Hilbert space dimension $q$, we take the Hamiltonian and jump
operators to be:

$H=\sum_{i=1}^{L-1}h_{i,i+1}\,,\qquad F_{i}=\ell_{i,i+1}\,,\qquad
i=1,2,\dots,L-1.$ (SA.14)

$h_{i,i+1}$ and $\ell_{i,i+1}$ are two-site random operators acting on site
$i$ and $i+1$, and independently drawn from the GUE and GinUE respectively,
i.e.

$h_{i,i+1}\in\mathrm{GUE}(q^{2}),\qquad\qquad\ell_{i,i+1}\in\mathrm{GinUE}(q^{2})\,.$
(SA.15)

The number of jump operators is just $d=L-1$. In this paper we focus on the
case $q=2$, with open boundary conditions.

  * 8\.

Strongly-$U(1)$-symmetric 1D Random Lindbladian (SS-$U(1)$-RL) – Consider the
$U(1)$ charge $I=\sigma^{z}_{\mathrm{tot}}=\sum_{i=1}^{L}\sigma^{z}_{i}$,
where $\sigma^{z}_{i}$ has eigenvalues
$\\{-\frac{q-1}{2},-\frac{q-1}{2}+1,\dots,\frac{q-1}{2}\\}$ measuring the
magnetization along the $z$-axis on site $i$. SS-$U(1)$-RL is defined
similarly as the 1D-RL, except that

$h_{i,i+1}=\bigoplus_{\mu=-q+1}^{q-1}h^{(\mu)}_{i,i+1},\qquad\qquad\ell_{i,i+1}=\bigoplus_{\mu=-q+1}^{q-1}\ell_{i,i+1}^{(\mu)},\qquad
i=1,2,\dots,L-1,$ (SA.16)

where in the local symmetry sector $\mu=\sigma^{z}_{i}+\sigma^{z}_{i+1}$, we
choose

$h^{(\mu)}_{i,i+1}\in\mathrm{GUE}(q-|\mu|),\qquad\qquad\ell^{(\mu)}_{i,i+1}\in\mathrm{GinUE}(q-|\mu|).$
(SA.17)

such that

$[\mathcal{S},\mathcal{L}]=[I,H]=[I,h_{i,i+1}]=[I,F_{i}]=[I,\ell_{i,i+1}]=0$
(SA.18)

i.e. this model is a “strongly symmetric” Lindbladian with respect to $I$
since individual terms in (SA.10) commute with the symmetry operator Buča and
Prosen (2012); Albert and Jiang (2014). We find that for $q=2$, the
SS-$U(1)$-RL is not chaotic, i.e. probes of the spectral correlations, e.g.
the DSFF and NNSD, do not converge to Ginibre RMT behaviour. The lack of
chaotic behaviour is due to the reduction in degrees of freedom for the
Hamiltonian and jump operators after imposing $U(1)$ symmetry. We focus on
SS-$U(1)$-RL with $q=3$, the minimum chaotic case, and the symmetry sector
$I=0$ in this paper.

  * 9\.

Weakly-$U(1)$-symmetric 1D Random Lindbladian (WS-$U(1)$-RL) – Consider again
the $U(1)$ charge, $I=\sigma^{z}_{\mathrm{tot}}=\sum_{i=1}^{L}\sigma^{z}_{i}$.
As mentioned above, we find that the strongly symmetric SS-$U(1)$-RL is only
chaotic when $q$ is taken to be 3. Another way to impose symmetry while
retaining chaotic behaviour with $q=2$ is to relax the strong symmetry of
Lindbladian to a weak symmetry. To this end, we consider a WS-$U(1)$-RL that
acts on the superspace of a one-dimensional system with on-site Hilbert space
dimension $q=2$. It has the same Hamiltonian as the SS-$U(1)$-RL,

$h_{i,i+1}=\bigoplus_{\mu=-1}^{1}h^{(\mu)}_{i,i+1},\qquad h^{(\pm
1)}_{i,i+1},\in\mathrm{GUE}(1),\qquad
h^{(0)}_{i,i+1}\in\mathrm{GUE}(2)\,,\qquad i=1,2,\dots,L-1$ (SA.19)

with the jump operators

$F_{i}=\ell_{i,i+1}^{(m=0,\pm{1},\pm{2})},\qquad i=1,2,\dots,L-1$ (SA.20)

where

$\displaystyle\ell_{i,i+1}^{(\pm{2})}=$
$\displaystyle\qquad\beta_{i}^{\pm}\sigma_{i}^{\pm}\sigma_{i+1}^{\pm},$
(SA.21) $\displaystyle\ell_{i,i+1}^{(1)}=$
$\displaystyle\begin{pmatrix}0&\mu_{i}^{+}&\nu_{i}^{+}&0\\\
0&0&0&\sigma_{i}^{+}\\\ 0&0&0&\rho_{i}^{+}\\\ 0&0&0&0\end{pmatrix},$ (SA.22)
$\displaystyle\ell_{i,i+1}^{(-1)}=$ $\displaystyle\begin{pmatrix}0&0&0&0\\\
\mu_{i}^{-}&0&0&0\\\ \nu_{i}^{-}&0&0&0\\\
0&\sigma_{i}^{-}&\rho_{i}^{-}&0\end{pmatrix}\;.$ (SA.23)

Here $\sigma_{i}^{+}$ and $\sigma_{i}^{-}$ are the ladder operators on site
$i$,
$\beta_{i}^{\pm},\mu_{i}^{\pm},\nu_{i}^{\pm},\sigma_{i}^{\pm},\rho_{i}^{\pm}\in\mathrm{GinUE}(1)$
and $\ell_{i,i+1}^{(m=0)}$ is just the charge conserving jump operators
defined in (SA.16) and (SA.17). The above Hamiltonians and jump operators are
chosen to satisfy

$\qquad[I,H]=[I,h_{i,i+1}]=0\,,\qquad\qquad[I,\ell_{i,i+1}^{(m)}]=m\ell_{i,i+1}^{(m)},$
(SA.24)

such that the Lindbladian superoperater satisfies

$[{\mathcal{L}},\mathcal{S}]=0\;.$ (SA.25)

By (SA.24) and (SA.25), this model is considered a “weakly symmetric”
Lindbladian Buča and Prosen (2012); Albert and Jiang (2014). We focus on the
largest symmetry section $\mathcal{S}=0$ in this paper.

  * 10.

Dissipative XXZ and XX models – Here we consider a one-dimensional dissipative
XXZ model with nearest neighbor and next-to-nearest neighbor interactions. The
Hamiltonian and jump operators are defined as

$\begin{split}&H=J\sum_{l=1}^{N-1}(\sigma_{l}^{x}\sigma_{l+1}^{x}+\sigma_{l}^{y}\sigma_{l+1}^{y}+\Delta\sigma_{l}^{z}\sigma_{l+1}^{z})\\\
&\quad\quad\quad+J^{\prime}\sum_{l=1}^{N-2}(\sigma_{l}^{x}\sigma_{l+2}^{x}+\sigma_{l}^{y}\sigma_{l+2}^{y}+\Delta^{\prime}\sigma_{l}^{z}\sigma_{l+2}^{z})\\\
&F_{\mathrm{L}}^{+}=\sqrt{\gamma_{\mathrm{L}}^{+}}\sigma_{1}^{+}\\\
&F_{\mathrm{L}}^{-}=\sqrt{\gamma_{\mathrm{L}}^{-}}\sigma_{1}^{-}\\\
&F_{\mathrm{R}}^{+}=\sqrt{\gamma_{\mathrm{R}}^{+}}\sigma_{L}^{+}\\\
&F_{\mathrm{R}}^{-}=\sqrt{\gamma_{\mathrm{R}}^{-}}\sigma_{L}^{-}\,\\\
&F_{l}=\sqrt{\gamma}\sigma_{l}^{z}\;,\end{split}$ (SA.26)

where $\sigma_{l}^{\alpha}$ with $\alpha=x,y,z$ are the Pauli matrices and
$\sigma^{\pm}_{l}=\sigma_{l}^{x}\pm i\sigma_{l}^{y}$. Like in WS-$U(1)$-RL,
this model has the $U(1)$ weak symmetry, and we focus on the $\mathcal{S}=0$
sector. Following Akemann  _et al._ (2019), we consider two choices of
parameters:

    * (a)

Chaotic dissipative XXZ model (dXXZ) is defined by (SA.26) with parameters
drawn from normal distributions $J,J^{\prime}\in\mathcal{N}(1,0.09)$,
$\Delta\in\mathcal{N}(0.5,0.0225)$,
$\Delta^{\prime}\in\mathcal{N}(1.5,0.2025)$, $\gamma=0$, $\gamma_{L}^{+}=0.5$,
$\gamma_{L}^{-}=0.3$, $\gamma_{R}^{+}=0.3$, and $\gamma_{R}^{-}=0.9$.

    * (b)

Integrable dissipative XX model (dXX) is defined by (SA.26) with
$J\in\mathcal{N}(1,0.09)$, $J^{\prime}=0$, $\Delta=0$, $\Delta^{\prime}=0$,
$\gamma=1$, $\gamma_{L}^{+}=0.5$, $\gamma_{L}^{-}=1.2$, $\gamma_{R}^{+}=1$,
and $\gamma_{R}^{-}=0.8$. Note that this model is Bethe ansatz integrable with
a mapping to the Fermi-Hubbard chain with imaginary interaction.

  * 11.

Dissipative transverse field Ising model  – We consider the Hamiltonian

$H=J\sum_{l=1}^{L}\sigma^{z}_{l}\sigma^{z}_{l+1}+g\sum_{l=1}^{L}\sigma^{x}_{l}+\sum_{i=1}^{L}h_{l}\sigma^{z}_{l}\;,$
(SA.27)

where the on-site disorder $h_{l}\in[-W,W]$ is drawn from a flat distribution
of width $W$. We take the jump operators to be

$F_{a}=\sqrt{\frac{\gamma}{2}}\sigma^{-}_{a}\;,$ (SA.28)

with $a=1,\dots,L$. Without dissipation, in finite system sizes, $H$ displays
many-body localized phenomenology for sufficiently large $W$. In this paper we
take $J=1$, $g=-0.9$ and $\gamma=0.5$. Following Hamazaki  _et al._ (2022), we
consider the model in two different phases:

    * (a)

Dissipative transverse field Ising model in the chaotic phase (dIsing-Chaos)
defined by (SA.27) and (SA.28) with $W=0.6$.

    * (b)

Dissipative transverse field Ising model in the many-body localized phase
(dIsing-MBL) defined by (SA.27) and (SA.28) with $W=5$.

  * 12.

Jump-operator-only Lindbladian.  We consider the Lindbladians acting on the
superspace of a Hilbert space of size $n$, where the Hamiltonian is set to be
zero, i.e. $H=0$, and the jump operators are drawn from the GinUE with varying
size of operator support. Again the number of jump operator is set to be $d$.

    * (a)

0D Random Lindbladian with jump operator only (0D-RL-JO): This model is a
variation of the model 0D-RL with $H=0$. We take only one jump operator drawn
from the GinUE of $n$-by-$n$ matrices, i.e. $d=1$ and
$F_{1}\in\text{GinUE}(n)$.

    * (b)

1D Random Lindbladian with 1-site jump operators only (1D-RL-1JO): The jump
operators are taken to be $F_{i}=l_{i}$ with $i=1,2,\dots,L$, where
$l_{i}\in\text{GinUE}(q)$ are one-site random operators acting on site $i$.
The number of jump operator is $d=L$, and the on-site dimension is $q=2$.

    * (c)

1D Random Lindbladian with 2-site jump operators only (1D-RL-2JO): This model
is a variation of the model 1D-RL with $H=0$. The jump operators are taken to
be $F_{i}=l_{i,i+1}$ with $i=1,2,\dots,L-1$, where
$l_{i,i+1}\in\text{GinUE}(q^{2})$ are two-site random operators acting on site
$i$ and $i+1$. The number of jump operator is $d=L-1$, and the on-site
dimension is $q=2$.

##  Appendix B Spectral properties

In this section, we provide the plots for the single realization spectra and
heat maps of the DOS for Krauss operators and Lindbladian models.

###  B.1 Random Kraus circuits

For Kraus circuits, we include the representative examples of the RKO (Figure
S1), the RKC and $U(1)$-RKC (Figure S2).

Figure S1: The spectrum of a single realization (left) and the heat map of the
DOS (right) of the RKO with dimension $N=4096$. Note that there is a single
leading eigenvalue at $z=1$, which we have removed for clarity of
presentation. The spectrum is roughly uniform, displays reflection symmetry
across the real axis, exhibits level repulsion, and lies within a circular
region surrounding the origin of the complex plane.

Figure S2: Spectrum of a single realization (first from left) and DOS (second
from left) of the RKC model with $L=6$. Again, we remove the eigenvalue at
$z=1$. Spectrum of a single realization (third from left) and DOS (forth from
left) of the $U(1)$-RKC model with $L=8,S_{z}=4$. Both of the spectra are
sharply peaked at the origin (note the scale) with tails extending outwards.

###  B.2 Random Lindbladians

For Lindbladians, we include the all the models studied, from (Figure S3) to
(Figure S6).

Figure S3: (a) Spectrum of a single realization and (b) the heat map of the
DOS of the 0D-RL. (c) Spectrum of a single realization and (d) the heat map of
the DOS of the 1D-RL model with dimension $N=4096$ and $L=6$ respectively. The
spectra of the 0D-RL has a shape of a lemon (c.f. S  _et al._ (2019)) and gets
denser to the origin. The DOS of the 1D-RL has a smooth peak at around
$z=-7.5$.

Figure S4:  Spectrum of a single realization (first) and the heat map of the
DOS (second) of the WS-$U(1)$-RL model with weak symmetry and physical
dimension $L=7$. The spectra has a smooth peak at around $z=-50$. Notice that
for the weak symmetry case, there is no steady state, i.e. there is not an
eigenvalue with the value $z=0$. Spectrum of a single realization (third) and
the heat map of the DOS (fourth) of the SS-$U(1)$-RL model with weak symmetry
and physical dimension $L=5$. The spectra has a smooth peak at around $z=-5$.

Figure S5:  Spectrum of a single realization (first row) and the heat map of
the DOS (second row) of, from left to right, the SYK-L with $L=12$, dXXZ with
$L=7$, dXX with $L=7$, dIsing-Chaos with $L=6$ and dIsing-MBL with $L=6$.

Figure S6:  Spectrum of a single realization (first row) and the heat map of
the DOS (second row) of, from left to right, the 0D-RL-JO, the 1D-RL-1JO and
the 1D-RL-2JO with $L=6$.

##  Appendix C Nearest neighbour spacing distribution

For a given complex eigenvalue $z_{n}$ and its $m$-th nearest neighbour
$z_{n}^{(m)}$, we define the $m$-th spacing distance of $z_{n}$ to be

$s^{(m)}_{n}=\left|z^{(m)}_{n}-z_{n}\right|\,.$ (SC.29)

The nearest neighbour spacing distribution (NNSD) is the set
$\left\\{s^{(1)}\right\\}$, and is used in non-Hermitian systems Grobe  _et
al._ (1988) to diagnose chaos in OQMBS, as in Hermitian systems Mehta (2004);
Bohigas  _et al._ (1984). Unfolding procedure has to be applied to the
spectrum so that variation of spectral density in the complex plane is
removed. To this end, we unfold the spectrum $\\{z_{n}\\}$ in three steps: (i)
Apply a conformal transformation described in Appendix E to obtain
$\\{z^{\prime}_{n}\\}$, which in particular removes sharp peaks in the DOS;
(ii) Obtain and rescale the NNSD $\left\\{s^{(1)}_{n}\right\\}_{R}$ of
$\\{z^{\prime}_{n}\\}_{R}$ with a rescaling factor dependent on local spectral
density following Hamazaki  _et al._ (2020):

$d^{(m)}_{n}=s^{(1)}_{n}\sqrt{\frac{m}{\pi{s^{(m)}_{n}}^{2}}}\,;$ (SC.30)

(iii) apply a filter $f(z_{n};\alpha,\mu)$ as described in Appendix F to keep
only the transformed eigenvalues in some region $R$ defined by $\alpha,\mu$
which has approximately uniform spectral density. Specifically, we average
over many realizations by computing:

$P(d^{(m)})=\frac{\Big{\langle}\sum_{n}\delta(d^{(m)}-d_{n}^{(m)})\times
f(z_{n};\alpha,\mu)\Big{\rangle}}{\Big{\langle}\sum_{n}f(z_{n};\alpha,\mu)\Big{\rangle}}\,.$
(SC.31)

###  C.1 Random Kraus circuits

Here we provide the NNSD for the RKO, RKC and $U(1)$-RKC. For RKO and RKC in
Fig. S7, NNSD have converged in the accessible system sizes and exhibit
universal eigenvalue correlations of the RMT GinUE NNSD. Consequently, we
expect the DSFF to exhibit universal behavior at late complex time. Crucially,
we find that for the NNSD to match that of RMT, steps (i)-(iii) are necessary.
This in part justifies the need to use both filtering and unfolding. For
$U(1)$-RKC in Fig. S7 right, NNSD has not yet converged in the accessible
system size, but the distribution is clearly approaching the RMT GinUE NNSD.

Figure S7: Left: The NNSD (SC.30) with $m=5$ for the RKO, where we have
implemented step (ii) and (iii), with increasing $N=4^{3},4^{5},4^{6}$ from
light to dark blue. If we only implemented step (ii), we would see deviations
from the RMT spacing at $d^{(5)}>1$ due to a finite-size edge effect in the
DOS. Middle: The NNSD (SC.30) with $m=5$ for the RKC, where we have
implemented step (i)-(iii), with increasing $L\in[3,6]$ from light to dark
blue. For both cases, the sample size is $5000$. Right: The NNSD (SC.30) after
steps (i)-(iii) with $m=5$ for the $U(1)$-RKC with increasing $L\in[5,8]$ in
darker blue. The sample size is $5000$.

###  C.2 Random Lindbladians

Here we provide the NNSD for the 0DRL, 1DRL, WS-$U(1)$-RL, and SS-$U(1)$-RL in
Fig. S8, NNSD have converged in the accessible system sizes and exhibit
universal eigenvalue correlations of the RMT GinUE NNSD.

Figure S8:  The NNSD (SC.30) with $m=5$ for (a) the 0D-RL with $L\in[3,7]$,
(b) the 1D-RL with $L\in[3,7]$, (c) the WS-$U(1)$-RL with the symmetry sector
$M=L-2$ and $L\in[5,8]$, (d) for the $q=3$ ($q=2$) SS-$U(1)$-RL with
$L\in[4,6]$ ($L\in[6,9]$) in blue (red) and the symmetry section $S_{z}=0$,
(e) SYK-L with $L/2\in[4,6]$ and the symmetry section $p=1$ (f) dXXZ with
$L/2\in[4,6]$ and the symmetry section (g) dXX (h) dIsing-Chaos and (l)
dIsing-MBL. For all cases, the system size increases from light to dark blue,
and the sample size is $2000$.

Figure S9:  The NNSD (SC.30) with $m=5$ for (a) the 0D-RL-JO, (b) the
1D-RL-1JO and (c) the 1D-RL-2JO where we have implemented step (i-ii), with
increasing $L\in[4,6]$ from light to dark blue.

##  Appendix D Complex spacing ratio

We consider the complex spacing ratios (CSR) defined as Sá  _et al._ (2019)

$r_{m}=\frac{z_{m}^{(1)}-z_{m}}{z_{m}^{(2)}-z_{m}}\,,$ (SD.32)

where $z_{m}^{(1)}$ and $z_{m}^{(2)}$ represent the nearest- and next-nearest-
neighboring eigenvalues to $z_{m}$, respectively. The CSR has the advantage
that no unfolding procedure is required. However, we find that the convergence
of CSR (and related quantities) is slower.

The density of the CSR $P(r)$ is plotted for nine example models in Figure S10
and Figure S11. For chaotic open systems, $P(r)$ has lower value around the
origin due to spectral rigidity, and lower value around $r=1$ since
$z_{m}^{(1)}$ and $z_{m}^{(2)}$ tend to lie on the opposite side of $z_{m}$
due to level repulsion. For dIsing-MBL, we see that $P(r)$ is approximately
flat, as consistent with the Poissonian distributed complex spectrum.

Figure S10: The density function $P(r)$ for the CSR (SD.32) plotted for two
example models of Kraus operators, the RKO (first left) and the RKC (second
left), and two example models of Lindbladians, the 0D-RL (third left) and the
1D-RL (fourth left). For all cases, the sample size is $5000$.

Figure S11: The density function $P(r)$ for the CSR (SD.32) plotted for SYK-L,
dXXZ, dXX, dIsing-Chaos ($W=0.6$) and dIsing-MBL ($W=5$) from the left to the
right. For dXX and dIsing-MBL, the signature of level repulsion around $z=0$
and $z=1$ is absent. For all cases, the sample size is $5000$.

Furthermore, there are a few other salient features of the CSR. First, the CSR
exhibits reflection symmetry across the real axis in the complex plane. This
is due to the fact that the eigenvalues of the superoperator occur in
conjugate pairs. Indeed, within the bulk, the nearest- and next-nearest
neighbors for $\overline{z}_{n}$ are given by $\overline{z}_{n}^{(1)}$ and
$\overline{z}_{n}^{(2)}$ such that the CSR is given by $\overline{r}_{n}$,
i.e. for each $r_{n}$ we also have $\overline{r}_{n}$. As a sanity check, we
find that if we compute the CSR for only eigenvalues in the upper half plane
with $\mathrm{Im}(z)>0$, the CSR is no longer reflection symmetric.

Second, the CSR exhibits a concentration along the real axis. This is due to
the fact that the spectrum of has a concentration along the real axis, and
that the ratio of real numbers is real. As a sanity check, if we remove
eigenvalues lying on the real axis before computing the CSR, we eliminate the
concentration along the real axis.

Finally, the density of the spacing ratio exhibits an “arc”-like feature, as
well as a vertical line located at $\mathrm{Re}(r)=\frac{1}{2}$. These are
caused by the case where an eigenvalue’s two nearest neighbors are its complex
conjugate and another eigenvalue lying on the real axis. Let the distance
between the eigenvalue and its complex conjugate be $2y$, and the distance to
the real eigenvalue be $a$. The angle between the lines connecting the two
pairs of eigenvalues, $\theta$, is also the argument of the complex spacing
$r$, and is given by $\cos\theta=y/a$ and $\sin\theta=\sqrt{1-y^{2}/a^{2}}$.
Now there are two cases.

In the first case, $2y>a$, and so we have that:

$\mathrm{Re}(r)=|r|\cos\theta=\frac{a}{2y}\frac{y}{a}=\frac{1}{2}\,,$ (SD.33)

which explains the concentration along the vertical line
$\mathrm{Re}(r)=\frac{1}{2}$.

In the second case, $2y<a$, and so we have that
$\mathrm{Re}(r)=(2y/a)\cos\theta=2y^{2}/a^{2}$. Since $a>2y$, this implies
that $\mathrm{Re}(r)<1/2$. On the other hand, we have that:

$\mathrm{Im}(r)=\frac{2d}{a}\sin\theta=\frac{2d}{a}\sqrt{1-\frac{y^{2}}{a^{2}}}=\pm\sqrt{\mathrm{Re}(r)\left(2-\mathrm{Re}(r)\right)}\,.$
(SD.34)

This explains the concentration along the “arc,” as one finds that it is well-
fit by the curve defined by the above relation for
$0<\mathrm{Re}(r)<\frac{1}{2}$.

##  Appendix E Unfolding

To compute long-range spectral correlation including the DSFF, we need to
“unfold” the complex spectrum, i.e. to remove the variation of spectral
density in the complex plane. The unfolding procedure typically involves a
transformation of the spectra which makes the associated DOS flatter. In fact,
unfolding for complex spectra is considerably more challenging than the
analogous procedure for real spectra. One particular obstacle is that the
complex spectral transformation must be a conformal transformation, which
restrict the possible choices of transformations. As a result, we often cannot
perfectly unfold the spectrum, and a further “filtering” procedure needs to be
applied to favour DSFF contributions from flatter regions in the spectra (see
Appendix F).

Here we comment on the necessity of unfolding (and filtering) in the context
of DSFF. Consider an ensemble of a chaotic open system with a highly non-
uniform DOS. Each sufficiently small section of the DOS will yield a
contribution to DSFF that is well-described by the GinUE ensemble, but with an
effective $\tau_{\mathrm{Hei}}$ that depends locally on the DOS. Because the
“ramp” of the DSFF for the GinUE ensemble grows quadratically Li  _et al._
(2021) as opposed to linearly in the Gaussian unitary ensemble for Hermitian
systems, DSFF for ensembles with non-unform DOS will deviate significantly
from the GinUE DSFF behavior, and hence unfolding is required.

###  E.1 Random Kraus circuits

First we consider the zero-dimensional RKO model. As pictured in Figure S1,
for sufficiently large $d$, the number of jump operators, the DOS of a RKO is
characterized by a roughly flat DOS which has support over a circular region
in the complex plane. The radius of the circle does not depend on the Hilbert
space dimension, and is qualitatively similar to the DOS of a RMT ensemble.
Therefore, unfolding is not needed.

This is to be contrasted with the situation for the one-dimensional model, the
RKC. In this case, the DOS is sharply peaked about the origin in the complex
plane, with the strength of the peak growing as the physical dimension is
increased, i.e. the DOS becomes less uniform. To unfold the spectra, we choose
a similar unfolding function to that used in Akemann and Burda (2012), which
for our case is given by:

$z\mapsto Az^{1/\log{N}}\,,$ (SE.35)

where $A$ is some arbitrary scaling constant, which we chose to be $A=e$. The
branch cuts here and in all Kraus models are chosen to be $(-\infty,0]$. The
results of this unfolding scheme for the RKC and $U(1)$-RKC are presented in
Figure S12 and Figure S13. By comparing with the unfolded spectra in Figure
S2, we see that unfolding simultaneously increases the support and uniformity
of the spectrum. Furthermore, suppose we have a density of state
$\rho(r,\theta)$. If we assume that the DOS before unfolding is rotationally
symmetric, we may define a local radial density function $\rho(r)$ via

$\iint dr\,d\theta\,r\rho(r,\theta)=\int dr\,2\pi r\,\rho(r)\,,$ (SE.36)

such that unfolding is successful if our unfolding function (SE.35) transforms
$\rho(r)$ into a flat function of $r$ (Note that we have a factor of $r$ in
(SE.36)). The radial density function $\rho(r)$ before and after unfolding for
the RKC is provided in Figure S12 right. We see that unfolding creates a local
maxima in the radial density $\rho(r)$. At this maxima, the radial density
does not change to the first order in $r$, and so the DOS can be taken to be
approximately uniform in a small radius window.

Generally, we consider unfolding functions which are multi-valued, and could
potentially introduce artificial edges due to the branch cut. Ideally, we
would like to isolate the bulk of the spectrum by avoiding these edges.
However, in Appendix G we consider the sanity check of unfolding the GinUE
spectrum, and find that such edges do not qualitatively distort the DSFF even
in the absence of a filtering scheme which removes the edge. We expect this to
be true for other models as long as $\rho_{\mathrm{av}}$ is sufficiently
uniform.

Figure S12: Left and middle: Spectrum of a single realization (left) and the
density of state (right) of the RKC model with physical dimension $L=6$ after
unfolding. The spectra is spread out over a sector of an annulus centered at
the origin. Right: Radial density function $\rho(r)$ for the RKC before
(dashed lines) and after (solid lines) unfolding, given for different system
sizes $L\in[3,6]$ from light to dark blue.

Figure S13: Spectrum of a single realization (left) and the DOS (right) of the
$U(1)$-RKC model with physical dimension $L=6$ after unfolding (left). The
spectra is spread out over a sector of an annulus centered at the origin.

###  E.2 Random Lindbladians

As shown in Appendix B.2, the spectrum of 1D-RL and dIsing-MBL are relatively
flat and gives nice DSFF behavior after proper filtering. We find that an
transformation similar to (SE.35) is sufficient to unfold the spectrum of
0D-RL, SYK-L, dXXZ, 0D-RL-JO, 1D-RL-1JO, and 1D-RL-2JO:

$z\mapsto A(z-z_{0})^{\beta}\,.$ (SE.37)

where $\beta=\frac{1}{2}$ for all cases except for SYK-L and 0D-RL-JO, where
$\beta=\frac{1}{3}$. $A=-i$ and $z_{0}$ is chosen at the maximum DOS, which is
model-dependent. The branch cut for each model is chosen to be
$[z_{0},\infty)$. The DOS after unfolding is shown in S14.

Figure S14: The density of state of the 0D- RL model with dimension $N=4096$,
SYK-L with $L=6$ and dXXZ with $L=6$ after unfolding.

##  Appendix F Filtering

###  F.1 Definitions

In Appendix E, we described an unfolding procedure that creates a region in
the complex plane with approximately uniform density. In the context of DSFF,
to further isolate the spectral correlation contributed from regions with
uniform DOS, we apply an additional “filtering” procedure, analogous to the
filtering procedure used in studying SFF for Hermitian systems Gharibyan  _et
al._ (2018),

$\tilde{K}_{\mathrm{c}}(t,s)=\left\langle\left|\sum_{n}e^{ix_{n}t+iy_{n}s}f\left(x_{n},y_{n};\\{\vec{\alpha}\\}\right)\right|^{2}\right\rangle-\left|\left\langle\sum_{n}e^{ix_{n}t+iy_{n}s}f\left(x_{n},y_{n};\\{\vec{\alpha}\\}\right)\right\rangle\right|^{2}\,,$
(SF.38)

where $f$ is the filter function whose strength is controlled by a set of
parameters $\vec{\alpha}$. Note that we use $\tilde{K}$ (as supposed to $K$)
to denote the DSFF for spectra after filtering. Two possible choices of
filtering function we consider are

  1. 1.

Sharp Cutoff: We choose the filter as an indicator function over some region
$R$, i.e.

$f\left(x_{n},y_{n};R\right)=\begin{cases}0,&\text{ if }z_{n}\not\in R\\\
1,&\text{ if }z_{n}\in R\end{cases}\,.$ (SF.39)

where $R$ is the region whose DSFF contribution we want to include.

  2. 2.

Gaussian Filter: The filter function is taken to be Gaussian function(s) with
certain center labelled by $\vec{\mu}$ and strength parametrised by
$\vec{\alpha}$. For a general coordinate ${x_{m}}$, we define

$f\left(\vec{x};\vec{\alpha},\vec{\mu}\right)=e^{-\sum_{m}{\alpha_{m}}({x}_{m}-{\mu}_{m})^{2}}\,,$
(SF.40)

More concretely, for Lindbladians, the coordinate is taken to be the Cartesian
coordinate in the complex plane, we define

$f\left(x_{n},y_{n};\alpha_{x},\alpha_{y},\mu_{x},\mu_{y}\right)=e^{-\alpha_{x}(x_{n}-\mu_{x})^{2}-\alpha_{y}(y_{n}-\mu_{y})^{2}}\,,$
(SF.41)

where $(\mu_{x},\mu_{y})$ is the center of the Gaussian filter, and
$(\alpha_{x},\alpha_{y})$ parametrize the strength in the real and imaginary
directions respectively. For the Kraus circuits, we use Gaussian filters along
the radial axis,

$f\left(z_{n};\alpha,\mu_{r}\right)=e^{-\alpha(|z_{n}|-\mu)^{2}}\,,$ (SF.42)

The center of the Gaussian filter $\vec{\mu}$ is chosen to be the peak of the
averaged DOS $\rho(z)$, and $\vec{\alpha}$ is chosen so that we focused on
data around the peak where $\rho(z)$ is relatively flat. We introduce a single
tunable parameter, the dimensionless filtering strength $\tilde{\alpha}$,
which is chosen to be the same for all direction. In turn, $\alpha_{m}$ for
the $m$-th coordinate is fixed by the relation
$\tilde{\alpha}=\alpha_{m}\Delta_{\mathrm{D},m}^{2}$. $\Delta_{\mathrm{D},m}$
is the half width at half maximum of the peak of $\rho(z)$ along coordinate
$m$, i.e. $\Delta_{\mathrm{D},m}=\frac{x_{m+}-x_{m-}}{2}$ where $\rho(x_{m+})$
and $\rho(x_{m-})$ reach half of its peak value. For Kraus, we have
$\alpha_{\theta}=0$, $\alpha_{r}=\tilde{\alpha}/\Delta_{\mathrm{D},r}^{2}$ and
$\Delta_{D,r}=\frac{r_{+}-r_{-}}{2}$. For Lindbladians, we have
$\alpha_{m}=\tilde{\alpha}/\Delta_{\mathrm{D},m}^{2}(L)$ with $m=x,y$, where
the horizontal and vertical ‘peak width’ are given by
$\Delta_{\mathrm{D},x}=\frac{x_{+}-x_{-}}{2}$ and
$\Delta_{\mathrm{D},y}=\frac{y_{+}-y_{-}}{2}$ respectively.

###  F.2 Early time effect of filtering

The introduction of filtering leads to artifacts in the early-time DSFF due to
edge-effects in the filtering procedure (Figure S15). For the sharp cutoff
filter, the early-time DSFF is roughly given by

$\tilde{K}_{c}(t,s)\big{|}_{t,s\ll 1}\approx\left\langle\left|\sum_{z_{n}\in
R}1\,\,\right|^{2}\right\rangle-\left|\left\langle\sum_{z_{n}\in
R}1\,\right\rangle\right|^{2}\equiv\left\langle\delta\mathcal{N}^{2}(R)\right\rangle\,.$
(SF.43)

That is, the early-time value of the connected DSFF is non-zero, and measures
the variance of the number of eigenvalues
$\left\langle\delta\mathcal{N}^{2}(R)\right\rangle$ within region $R$. Since
our unfolding procedure is constructed such that the uniform part of the DOS
has roughly fixed region size as we increase $L$, the number of eigenvalues in
$R$ grows with the Hilbert space dimension. Thus, the early-time artifacts of
the filtering procedure are a finite-size effect, that disappears with
increasing Hilbert space dimension.

Figure S15: Left(Right) plot is respectively DSFF for RKC (1DRL) with $L=6$ at
$\theta=\pi/4$ with 5000 samples. From light blue to dark blue, the Gaussian
filter strength is varied from $\tilde{\alpha}\in[1/20,1/5,1,20,40,100]$
($[1/5,2/3,2,20,60,200]$). In orange, theoretical DSFF curves for the GinUE
random matrix ensemble are fit to the late-time DSFF for selected values of
$\alpha$.

For the Gaussian filter, the early-time value of DSFF is determined by the
variance of $\sum_{n}e^{-\alpha(|z_{n}|-\mu)^{2}}$ given by

$\tilde{K}_{c}(t,s)\big{|}_{t,s\ll
1}\approx\left\langle\left|\sum_{n}e^{-\alpha(|z_{n}|-\mu)^{2}}\,\,\right|^{2}\right\rangle-\left|\left\langle\sum_{n}e^{-\alpha(|z_{n}|-\mu)^{2}}\,\right\rangle\right|^{2}\,.$
(SF.44)

We observe empirically that the early-time artifacts caused by the Gaussian
filter are smaller in magnitude for the system sizes of interest (see examples
in Appendices G and H). The difference in relative strength of the early-time
artifacts can be understood as a difference in the sharpness of the two
filters. The Gaussian filter can be thought of as a cutoff with smoothened
edges which is less sensitive to fluctuations in the number of eigenvalues.
Since we are interested in the early-time features of the DSFF, namely the
“bump” and the Thouless time, we opt to use the Gaussian filter to minimize
such filtering artifacts.

###  F.3 Late time effect of filtering

The unfolding and the (Gaussian) filtering procedure alter both the late-time
plateau value as well as the scaling of $\tau_{\mathrm{Hei}}$. The late-time
plateau value can be determined by taking $|\tau|\to\infty$ in $\tilde{K}_{c}$
given by (SF.38), which averages the phase differences in the first term to a
delta function $\delta_{nm}$ and in the second term to zero. Thus, the late-
time plateau value is

$\lim_{|\tau|\to\infty}\tilde{K}_{c}(t,s)=\left\langle\sum_{n}e^{-2{\alpha(|z_{n}|-\mu)^{2}}}\right\rangle\,.$
(SF.45)

The Heisenberg time $\tau_{\mathrm{Hei}}$ is determined empirically by
computing the “weighted” nearest-neighbor spacing. For a given filter strength
$\alpha$, we estimate the “weighted” nearest-neighbor spacing as

$\tilde{s}_{n}^{(1)}=|z_{n}^{(1)}-z_{n}|\times
e^{-\alpha(|z_{n}|-\mu)^{2}-\alpha(|z_{n}^{(1)}|-\mu)^{2}}\,.$ (SF.46)

Averaging over all eigenvalues $z_{n}$ across many realizations, we obtain the
averaged weighted nearest-neighbor spacing
$\tilde{s}^{(1)}(\mu(L),\alpha(L))$.

$\tilde{s}^{(1)}(\mu(L),\alpha(L))=\langle\mathbb{E}_{n}[\tilde{s}_{n}^{(1)}]\rangle$
(SF.47)

where $\mathbb{E}$ averages over all eigenvalues within the spectrum of a
single realization.

###  F.4 Definition of Heisenberg time after filtering

For the GinUE, the Heisenberg time in the DSFF is proportional to the mean
level spacing by a constant we call $\chi_{\mathrm{GinUE}}$,

$\tau_{\mathrm{Hei,GinUE}}=\frac{\chi_{\mathrm{GinUE}}}{s^{(1)}_{\mathrm{GinUE}}}\;.$
(SF.48)

We suppose the corresponding “weighted” Heisenberg time in the filtered DSFF
is related to the weighted mean level spacing by the same
$\chi_{\mathrm{GinUE}}$, i.e.

${\tau}_{\mathrm{Hei}}:=\frac{\chi_{\mathrm{GinUE}}}{\tilde{s}^{(1)}}\;.$
(SF.49)

Indeed, this is good definition of the weighted Heisenberg time since it
allows us to collapse DSFF of many-body open quantum systems with different
system sizes into a single curve, see Figure 1. Note that
$\tau_{\mathrm{Hei}}$ refers to a time scale of the filtered DSFF, but we have
suppressed the tilde notation on $\tau_{\mathrm{Hei}}$.

###  F.5 Choice of filtering strength $\alpha$

The introduction of the Gaussian filter means that $\tau_{\mathrm{Hei}}$ and
the general shape of the DSFF depend on the filter strength $\alpha$ in a non-
trivial way. As we reasoned in Appendix E, due to the non-uniformity of the
DOS, our DSFF does not exhibit universal RMT behavior even at late-times, and
thus filtering is required. However, when filtering is too strong, we might
hide physically relevant features. Specifically, there are three energy scales
$\Delta_{\mathrm{D}}\geq\Delta_{\mathrm{Th}}\geq\Delta_{\mathrm{Hei}}$ in our
problem, and we seek for a value of $\alpha$ such that

$\frac{1}{\Delta_{\mathrm{D}}^{2}}\ll\alpha\ll\frac{1}{\Delta_{\mathrm{Th}}^{2}}\leq\frac{1}{\Delta_{\mathrm{Hei}}^{2}}\;.$
(SF.50)

Here $\Delta_{\mathrm{D}}$ represents the energy scale over which the DOS
varies. We describe below an approach to identify $\alpha$ that lies in this
regime described by Equation SF.50.

When the filter is too weak, i.e.
$\tilde{\alpha}\equiv\alpha\Delta_{\mathrm{D}}^{2}\ll 1$, we observe
$\tau_{\mathrm{dev}}\gtrsim\tau_{\mathrm{Hei}}$. This is because the DSFF is
distorted by a non-uniform DOS, which is expected to be dependent on the
microscopic details of the open system (light blue in Figure S15). Thus, the
weakly filtered DSFF largely deviates from that of the Ginibre ensemble, even
at large times. When the filter is too strong, i.e.
$\alpha\Delta_{\mathrm{Hei}}^{2}\gg 1$, we also observe
$\tau_{\mathrm{dev}}\gtrsim\tau_{\mathrm{Hei}}$. This is because the filter
will wash out the signatures of level repulsion, and therefore the filtered
DSFF will not display the ramp-plateau behavior of GinUE (dark blue in Figure
S15).

As $L$ increases, we expect there to be a wider range of $\alpha$ which
satisfies
$\frac{1}{\Delta_{\mathrm{D}}^{2}}\ll\alpha\ll\frac{1}{\Delta_{\mathrm{Hei}}^{2}}\;$,
since the number of eigenvalues grow exponentially in $L$. Further, if
$\Delta_{\mathrm{Th}}$ behaves like its Thouless energy analogue in closed
generic quantum many-body systems, which scales polynomially in $L$ Chan  _et
al._ (2018b); Gharibyan  _et al._ (2018), the window,
$\frac{1}{\Delta_{\mathrm{D}}^{2}}\ll\alpha\ll\frac{1}{\Delta_{\mathrm{Th}}^{2}}\;$,
also increases as $L$ increases. This is consistent with Figure S16, for
example. The above arguments and empirical observation mean that
$\tau_{\mathrm{dev}}$ against $\alpha$ displays a trough shape, and the
appropriate $\alpha$ should be located at the bottom of the trough. For
certain models, we observe empirically that the bottom of the trough is
becoming flatter and increasing in size (as argued above), which suggest that
the ratio $\Delta_{\mathrm{D}}^{2}/\Delta_{\mathrm{Th}}^{2}$ is increasing in
system size. The flatness of the trough in $\tau_{\mathrm{dev}}$ versus
$\alpha$ implies that $\tau_{\mathrm{dev}}$ becomes less and less dependent on
the choice of $\alpha$ (within the trough) as $L$ increases. For example,
consider the $\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$ plot for RKC in
Figure S16, and for SYK-L in Figure S17. The valley shape is increasingly
apparent as $L$ increases, which allows us to identify the regime
$\frac{1}{\Delta_{\mathrm{D}}^{2}}<\alpha<\frac{1}{\Delta_{\mathrm{Hei}}^{2}}$.
Furthermore, we see that in this regime, for most models, neither the value of
$\tau_{\mathrm{dev}}$ at fixed $L$, nor the behavior of $\tau_{\mathrm{dev}}$
as a function of $L$ vary greatly with $\tilde{\alpha}$. Note that however,
for 1D-RL and SYK-L, the dependence of $\tau_{\mathrm{dev}}$ on
$\tilde{\alpha}$ is not stable except for the largest two system sizes. This
sensitivity is present especially since $\tau_{\mathrm{dev}}$ are small, i.e.
these models become RMT-like quickly for small $L$. Therefore, for
sufficiently large system size, we may identify $\tau_{\mathrm{dev}}$ with the
Thouless time $\tau_{\mathrm{Th}}$, independent from details of the filtering
protocol like the precise value of $\tilde{\alpha}$ as long as
$\tilde{\alpha}$ is within the trough. While all models display a trough
structure in $\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$, some do not
display the trough-widening behaviour in larger $L$, e.g. 1D-RL Figure S17,
and the suitable choice of $\tilde{\alpha}$ may depend on $L$ in general. In
practice, we choose $\tilde{\alpha}=1$, which lies roughly in the trough of
the $\tau_{\mathrm{dev}}$ versus $\tilde{\alpha}$ curve. Similar to Fig. 3 in
the main text, in Figure S18 and Figure S19, we provide plots for the single-
realization spectrum, unfolded spectrum, DSFF for varying filtering strength,
and $\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$, which indeed display the
trough structure.

Figure S16: The effect of changing filter strength $\tilde{\alpha}$ and system
size $L$ on $\tau_{\mathrm{dev}}$ for the RKC model. Left: Trend of
$\tau_{\mathrm{dev}}$ for different $\tilde{\alpha}$. Plotted for different
system sizes $L\in[4,6]$ in light to dark blue. Right: Trend of
$\tau_{\mathrm{dev}}$ versus system size for multiple
$\tilde{\alpha}\in[1/2,2]$ in light to dark blue.

Figure S17: The effect of changing filter strength $\tilde{\alpha}$ and system
size $L$ on $\tau_{\mathrm{dev}}$ for the 1D-RL and SYK-L models. First from
the left: $\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$ for 1D-RL for system
size $L\in[5,7]$ from light to dark blue. Second: $\tau_{\mathrm{dev}}$
against $L$ for 1D-RL for multiple $\tilde{\alpha}=0.5,1,2,5$ from light to
dark blue. Third: $\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$ for SYK-L for
$L\in[4,6]$ from light to dark blue. Fourth: $\tau_{\mathrm{dev}}$ against $L$
for SYK-L for $\tilde{\alpha}=1,2$ from light to dark blue.

Figure S18: Left: A single-realization spectrum $\\{z\\}$ of SYK-L model for
$L=14$. Middle: The unfolded spectrum $\\{z^{1/3}\\}$ (of the single-
realization spectrum) is subsequently filtered with the filtering function
$f(z)=e^{-(\mathrm{Re}(z)-3)^{2}/0.09-\mathrm{Im}(z)^{2}/0.02}$. Right: DSFF
of the SYK-L model for varying filtering strength
$\tilde{\alpha}\in[1/100,100]$ from light to dark blue. Inset:
$\tau_{\mathrm{dev}}$ against $\tilde{\alpha}$ shows a trough in which the
suitable filtering strength lies.

Figure S19: Left: A single-realization spectrum $\\{z\\}$ of dXXZ model for
$L=7$. Middle: The unfolded spectrum $\\{(z+8)^{1/2}\\}$ (of the single-
realization spectrum) is subsequently filtered with the filtering function
$f(z)=e^{-(\mathrm{Re}(z)-1.9)^{2}/0.16-(\mathrm{Im}(z)+1.9)^{2}/0.14}$.
Right: DSFF of the dXXZ model for varying filtering strength
$\tilde{\alpha}\in[1/100,75]$ from light to dark blue. $\tau_{\mathrm{dev}}$
against $\tilde{\alpha}$ shows a trough in which the suitable filtering
strength lies.

##  Appendix G Sanity checks

As sanity checks for our filtering procedure, we compute the DSFF of GinUE
after deforming the DOS with a variety of conformal transformations. We also
compare with the DSFF of the RKO, which has a flat spectrum and is expected to
be described by RMT.

Our sanity checks are:

  1. 1.

DSFF of $\\{z\\}$ of GinUE with filtering, and

  2. 2.

DSFF of $\\{\log z\\}$ of GinUE with filtering, and

  3. 3.

DSFF of $\\{z^{p}\\}$ of GinUE with filtering and $p=\frac{1}{4}$.

These checks are particularly instructive because the form of DSFF for GinUE
without filtering is exactly known (3). As we show in Figure S20, for fixed
$\tilde{\alpha}=1$, the DSFF approaches the GlnUE solution (3) as $N$
increases. In Figure S21, for fixed $N$, the DSFF fits the GinUE solution for
sufficiently large ${\tilde{\alpha}}$, until an early-time plateau forms for
very large ${\tilde{\alpha}}$.

Figure S20: DSFF for $\\{z\\}$ of GinUE (left), $\\{\log z\\}$ of GinUE
(middle), and $\\{z^{p}\\}$ of GinUE (right) at $\theta=\pi/4$ with 5000
samples. System size is varied from $N=4^{3},4^{4},4^{5},4^{6}$ from light
blue to dark blue. All presented curves are computed with filter strength
$\tilde{\alpha}=1$. The DSFF analytical solution for GinUE are plotted in
orange. The DSFF behaviour for small $|\tau|/\tau_{\mathrm{Hei}}$ are plotted
in the insets.

Figure S21:  DSFF for $\\{z\\}$ of GinUE (left), $\\{\log z\\}$ of GinUE
(middle), and $\\{z^{p}\\}$ of GinUE (right) at $\theta=\pi/4$ with 5000
samples. The dimension is chosen with $N=4^{6}$. The filter strength is
increased from $\tilde{\alpha}\in[1/20,1/10,1/5,1,10,20,40]$ from light blue
to dark. For the weakest filters (lightest blue),
$\tilde{\alpha}=1/20,1/10,1/5,1$, the DSFF collapses well onto the theoretical
curve (orange).

### Example: DSFF of RKO with filtering

Figure S22: Left: DSFF for RKO at $\theta=\pi/4$ with 5000 samples. System
size is varied from $N=4^{3},4^{4},4^{5},4^{6}$ from light blue to dark blue.
All presented curves are computed with filter strength $\tilde{\alpha}=1$. The
DSFF behaviour for small $|\tau|/\tau_{\mathrm{Hei}}$ are plotted in the
insets. Right: DSFF for RKO at $\theta=\pi/4$ with 5000 samples. The dimension
is chosen with $N=4^{6}$. The filter strength is increased from light blue to
dark, with $\tilde{\alpha}\in[1/20,1/10,1/5,1,10,20,40]$. The DSFF analytical
solution for GinUE are plotted in orange.

##  Appendix H Summary table and additional numerics for dissipative spectral
form factor (DSFF)

Figure S23: DSFF for RKO without filtering at $\theta=\pi/4$ and with 5000
samples. The dimension is increased from $N=4^{3},4^{4},4^{5},4^{6}$, from
light blue to dark blue. The DSFF behaviour for small
$|\tau|/\tau_{\mathrm{Hei}}$ are plotted in the insets. The DSFF analytical
solution for GinUE are plotted in orange.

In this section, we provide additional numerics of the DSFF. Il Figure S23, we
provide the DSFF of RKO with different system sizes $L$ without filtering. In
Figure S24 and Figure S25, we provide the DSFF of different Kraus operators
and Lindbladians with fixed $L$ and varying filtering strength. Lastly, we
provide a table of DSFF of all models with the corresponding unfolding
function, filtering function, DSFF against system sizes $L$, and deviation
time $\tau_{\mathrm{dev}}$ against system sizes $L$.

Figure S24:  The DSFF for the $U(1)$-RKC model after unfolding and filtering
along $\theta=\pi/4$ and 5000 samples. On the left, we fix $\tilde{\alpha}=1$
and vary $L=5,6,7,8$ from light to dark blue respectively. On the right, we
vary, from light to dark blue,
$\tilde{\alpha}\in[1/20,1/10,1/5,1,10,20,40,100]$ for fixed system size $L=8$.

Figure S25:  The DSFF for the 0D-RL, 1D-RL, WS-$U(1)$-RL, SS-$U(1)$-RL after unfolding and filtering along $\theta=\pi/4$ and with 5000 samples. We vary, from light to dark blue, $\tilde{\alpha}\in[0,\frac{1}{5},\frac{2}{3},2,20,60,200]$ for the fixed system size $L=6$ or $L_{\mathrm{eff}}=6$.  Table S1: DSFF with fits from the GinUE, Unfolding and filtering methods, and $\tau_{\mathrm{dev}}$ and $\tau_{\mathrm{plat}}$ for all models. Note that for certain models (e.g. $U(1)$-RKC, dXXZ), DSFF has not converged to GinUE RMT behaviour for the accessible system sizes yet, and consequently the trend of $\tau_{\mathrm{dev}}$ in system size is not representative. | Unfolding | Filtering | DSFF & GinUE fits | $\tau_{\mathrm{dev}}$
---|---|---|---|---
RKO | N/A | N/A |  |
RKC | $z\to z^{1/\log_{2}N}$ | Radial Gaussian |  |
$U(1)$-RKC | $z\to z^{1/\log_{2}N}$ | Radial Gaussian |  |
$\mathbb{Z}_{2}$-RKC | $z\to z^{1/\log_{2}N}$ | Radial Gaussian |  |
0D-RL | $z\to z^{1/2}$ | 2D Gaussian |  |
1D-RL | N/A | 2D Gaussian |  |
WS-$U(1)$-RL | N/A | 2D Gaussian |  |
SS-$U(1)$-RL | N/A | 2D Gaussian |  |

| Unfolding | Filtering | Fits GinUE |  $\tau_{\mathrm{dev}}$ or $\tau_{\mathrm{plat}}$
---|---|---|---|---
SYK-L | $z\to z^{1/3}$ | 2D Gaussian |  |
dXXZ | $z\to(z-z_{0})^{1/2}$ | 2D Gaussian |  |
dXX | N/A | 2D Gaussian |  |
dIsing-Chaos | N/A | 2D Gaussian |  |
dIsing-MBL | N/A | 2D Gaussian |  |
0D-RL-JO | $z\to z^{1/3}$ | 2D Gaussian |  |
1D-RL-1JO | $z\to z^{1/2}$ | 2D Gaussian |  |
1D-RL-2JO | $z\to z^{1/2}$ | 2D Gaussian |  |

##  Appendix I Height of bumps

In this section, for chaotic OQMBS displaying GinUE DSFF behaviours, we
determine the system size dependence of the height of the bump. We observe
that for non-many-body open quantum systems, e.g. RKO and 0D-RL, the height
does not increase with system size. For many-body open quantum systems, e.g.
RKC, $\mathbb{Z}_{2}$-RKC, $U(1)$-RKC, 1D-RL and L=SYK, the height increases.

Note that the “bump” should not to be confused with the “dip” in the
literature, which refers to the decaying behaviour from $N^{2}$ to the ramp
due to the disconnected part of DSFF Li  _et al._ (2021) or SFF Gharibyan  _et
al._ (2018). Here we deal with the deviation from RMT behavior in the
connected part of DSFF. Specifically, we track the dependence of the bump
height of the DSFF, normalized by its late-time plateau value, defined as
follows,

$\mathcal{K}_{\mathrm{peak}}:=\frac{\tilde{K}_{\mathrm{c}}(|\tau_{\mathrm{peak}}|,\theta)}{\tilde{K}_{\mathrm{c}}(|\tau|\to\infty,\theta)}\;,\quad\quad\tau_{\mathrm{peak}}:=\mathrm{arg\,max}_{|\tau|}\left[\frac{\tilde{K}_{\mathrm{c}}(|\tau|,\theta)}{\tilde{K}_{\mathrm{c}}(|\tau|\to\infty,\theta)}-\frac{K_{\mathrm{c},\mathrm{Gin}}(|\tau|,\theta)}{N}\right]\,.$
(SI.51)

We see a distinct difference between the models with no spatial structure
(e.g. RKO) and those with spatial structure (e.g. RKC, $\mathbb{Z}_{2}$-RKC,
$U(1)$-RKC), and we illustrate this difference by comparing, in particular,
the RKO and RKC models Figure S26 and the 0D-RL, 1D-RL and SYK-L Figure S27.
Note that in the SYK model, even though the interaction is all-to-all, i.e.
there is no spatial structure, the peak of the bump is increasing with system
size. This data suggests that the deviation from RMT in DSFF is a generic
many-body effect in open quantum many-body chaotic systems.

Figure S26:  The peak value of the DSFF $\mathcal{K}_{\mathrm{peak}}$ against
the effective system size $L_{\mathrm{eff}}=\frac{1}{2}\log N$ for RKO (left)
and system size $L$ for RKC (right).

Figure S27:  The peak value of the DSFF $\mathcal{K}_{\mathrm{peak}}$ against
the effective system size $L_{\mathrm{eff}}=\frac{1}{2}\log N$ for 0D-RL
(left) and SYK-L (right) and system size $L$ for 1D-RL (middle).