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# Universal Entanglement Transitions of Free Fermions with Long-range Non-
unitary Dynamics

Pengfei Zhang  Institute for Quantum Information and Matter and Walter Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Chunxiao Liu<EMAIL_ADDRESS>Department of
Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
Shao-Kai Jian<EMAIL_ADDRESS>Condensed Matter Theory Center and Joint
Quantum Institute, Department of Physics, University of Maryland, College
Park, MD 20742, USA  Xiao Chen<EMAIL_ADDRESS>Department of Physics, Boston
College, Chestnut Hill, MA 02467, USA

###### Abstract

Non-unitary evolution can give rise to novel steady states classified by their
entanglement properties. In this work, we aim to understand the effect of
long-range hopping that decays with $\bm{r^{-\alpha}}$ in non-Hermitian free-
fermion systems. We first study two solvable Brownian models with long-range
non-unitary dynamics: a large-$N$ SYK2 chain and a single-flavor fermion
chain, and we show that they share the same phase diagram. When
$\bm{\alpha>0.5}$, we observe two critical phases with subvolume entanglement
scaling: (i) $\bm{\alpha>1.5}$, a logarithmic phase with dynamical exponent
$\bm{z=1}$ and logarithmic subsystem entanglement, and (ii)
$\bm{0.5<\alpha<1.5}$, a fractal phase with $\bm{z=\frac{2\alpha-1}{2}}$ and
subsystem entanglement $\bm{S_{A}\propto L_{A}^{1-z}}$, where $\bm{L_{A}}$ is
the length of the subsystem $\bm{A}$. These two phases cannot be distinguished
by the purification dynamics, in which the entropy always decays as
$\bm{L/T}$. We then confirm that the results are also valid for the static
SYK2 chain, indicating the phase diagram is universal for general free-fermion
systems. We also discuss phase diagrams in higher dimensions and the
implication in measurement-induced phase transitions.

##  1 Introduction

Unveiling new phases and novel dynamics of quantum many-body systems is one of
the most important subjects in condensed matter physics. Recent developments
find that a new paradigm exists when we consider non-unitary evolutions. It is
found that for a “hybrid” quantum dynamics composed of both unitary evolution
and projective measurement, if we follow the quantum trajectories, the steady
state exhibits a transition between a volume-law entangled phase and an area-
law entangled phase by varying the measurement strength [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Later studies show the entanglement
transitions also appear in more generalized non-unitary random evolutions [19,
20, 21, 22, 23, 24]. In particular, in free-fermion systems, under non-unitary
random evolution, there is a stable critical phase, in which the steady state
has power-law correlation functions and logarithmic entanglement entropy, with
possible entanglement transitions into area-law phases [18, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35]. This criticality is attributed to the existence
of Goldstone modes from the spontaneous breaking of the continuous symmetry in
the enlarged replicated Hilbert space [18, 28, 29].

Most of these studies focus on quantum systems with local interactions.
However, most of the state-of-the-art experimental platforms to simulate the
quantum many-body dynamics contains intrinsic long-range interactions. For
example, the ultracold atoms in the optical lattices interact with a Van der
Waals potential $\sim 1/r^{6}$ and the dipole-dipole interaction in the NMR
system decays even slower $\sim 1/r^{3}$. These long-range interactions can
significantly change the ground state property [36, 37] and the quantum
dynamics [38, 39, 40, 41, 42, 43, 44] under unitary evolution. For example,
the Lieb-Robinson bound in local spin chains can receive non-trivial
corrections, giving rise to rich lightcone structures depending on the
strength of the interaction and the local Hilbert space dimension [38, 39, 40,
41, 42, 43, 44]. Consequently, it is not only of great interest but also
urgently necessary to correctly incorporate the effect of long-range couplings
in non-unitary evolution. Although several papers of numerical simulations of
long-range interacting [45, 46] or non-interacting [46] systems under non-
unitary evolutions appeared recently, a definite answer of the entanglement
properties of non-unitary long-range-coupled random free fermion dynamics is
still lacking.

Figure 1: Schematics of (a). The long-range non-hermitian SYK2 chain. The
Hamiltonian is either Brownian or static. (b). Single-flavor chain with
imaginary random on-site potential. The randomness is time dependent. In both
cases, we use $J$ or $V$ for terms in the $H_{R}$ or $H_{I}$ respectively. In
particular, $J_{r}$ represents the long-range hopping. (c). The phase diagram
is valid in both models, regardless of the local Hilbert space dimension. Here
$z$ is the dynamical exponent and $S_{A}$ is the entanglement entropy of a
subsystem with length $L_{A}$.

In this work, we present a detailed study on non-unitary free-fermion systems
with long-range hopping $J_{r}\propto 1/r^{\alpha}$, where $r$ labels the
distance between two sites. We investigate this problem by constructing two
analytical solvable models in $1$-D: the quadratic Brownian Sachdev-Ye-Kitaev
(SYK) chain [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] with
$N\rightarrow\infty$ fermions per site, and a single-flavor $N=1$ Brownian
chain. In both models, the non-unitary evolution can be treated as the free
fermion dynamics subject to continuous weak measurements. We will show that
such non-unitary dynamics with non-local hopping can stabilize phases with
non-trivial correlation and entanglement structure.

The SYKq model describes $N$ Majorana modes with random $q$-body couplings
[47, 48, 49], which is solvable under the large-$N$ expansion. The SYK model
has been simulated using nuclear spin chains [59], and a solid-state
realization has been proposed using a graphene flake [60]. Later, different
generalizations of the SYK model have been studied, including non-Hermitian
Brownian SYK chains in which measurement induced phase transitions can be
analyzed using the effective action approach [27, 28, 29]. Motivated by these
developments, here we use Brownian SYK chains to understand entanglement
properties of the steady state under long-range non-unitary evolutions. To
test whether the result is sensitive to the large-$N$ limit, we further study
the single-flavor Brownian chain using nonlinear master equations [61]. We
find that all results match in these models, indicating that the physics is
independent of the local Hilbert space dimension. For $\alpha>0.5$, they show
critical behavior and can be further separated into two phases (see FIG. 1).
For $\alpha\\!>\\!1.5$, the long-range hopping decays rapidly enough and the
result is the same as the short-range hopping case with dynamical exponent
$z=1$ and logarithmic entanglement entropy. For $0.5\\!<\\!\alpha\\!<\\!1.5$,
the system is in a fractal phase [21] with subsystem entanglement
$S_{A}\propto L_{A}^{1-z}$, where $L_{A}$ is the length of the subsystem $A$
and $z=(2\alpha-1)/2$. We further confirm that the same result holds for
static SYK2 chain and we propose the phase diagram to be universal for generic
non-unitary random free-fermion systems. We also analyze the purification
dynamics, where we find the entropy of the system decays as $L/T$ for any
arbitrary $\alpha$.

##  2 Large-$N$ Model

We first consider the long-range non-Hermitian SYK2 chain with $N$ Majorana
fermions $\chi^{i}$ on each site (see FIG. 1). The Hamiltonian
$H=H_{R}-iH_{I}$ reads HR(t)=∑x,iji~Jijx(t)
χixχjx/2+∑r≥1∑x,ijiJijx,x+r(t)rαχixχjx+r,HI(t)=∑x,ij[iVijx,x+1(t)
χixχjx+1+i~Vijx(t) χixχjx/2]. Here $i,j=1,2...N$ labels the Majorana modes
$\chi$ on each site and $r$ labels the hopping distance. Here $H_{I}$ contains
intra-site and nearest neighbor hopping, and $H_{R}$ contains intra-site
coupling and long-range hopping. $\tilde{J}_{ij}^{x}$, $J_{ij}^{x,x+r}$,
$V_{ij}^{x,x+1}$ and $\tilde{V}_{ij}^{x}$ are independent random Gaussian
variables with zero expectation value. To be concrete, we focus on the
Brownian case with variance ¯~Jijx(t)~Jijx(0)=J0δ(t)N,
¯Vijx,x+1(t)Vijx,x+1(0)=V1δ(t)2N,¯~Vijx(t)~Vijx(0)=V0δ(t)N,
¯Jijx,x+r(t)Jijx,x+r(0)=Jlδ(t)2N. For simplicity, we set $J_{l}=J_{0}$.
Physically, quantum trajectories undergoing non-unitary evolution can be
realized by continuous forced measurements as discussed in Appendix E. There
are also other related studies of measurements and feedback in quantum many-
body systems [62, 63, 64, 65, 66, 67, 68]. In particular, in [67], authors
find feedback, which gives rise to effective long-memory couplings, can change
the critical behaviors.

We are interested in analyzing the steady state under non-unitary evolutions.
The system is prepared in some initial state $|\psi_{0}\rangle$. At time $t$,
the wave function evolves as
$|\psi(t)\rangle=e^{-iHt}|\psi_{0}\rangle/\sqrt{\langle\psi_{0}|e^{iH^{\dagger}t}e^{-iHt}|\psi_{0}\rangle}$.
Here we keep any time-ordering operator implicit. Recent studies show phases
can be classified according to their entanglement properties [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
We consider the second Rényi entanglement entropy, the definition of which
includes two replicas of the original system. We choose a subsystem $A$ with
length $L_{A}$. The purity of the subsystem $A$ reads
P_A=tr_A(ρ_A)^2=trA(trBe-iHt|ψ0⟩⟨ψ0|eiH†t)2(⟨ψ0|eiH†te-iHt|ψ0⟩)2, where
$\text{tr}_{A/B}$ denotes the partial trace of subsystem $A$ or $B$, and the
second Rényi entropy can be obtained as $S^{(2)}_{A}(t)=-\overline{\log
P_{A}(t)}$. Previous studies [28, 29] show Keldysh squared correlators, which
probe the correlation between two replicas, can also distinguish different
phases. It is defined as
$F=\sum_{ij}\overline{\left<\chi_{x}^{i}\chi_{0}^{j}\right>^{2}}/N$, where
⟨χ_x^iχ_0^j⟩≡t→∞lim⟨ψ0|eiH†te-iHt/2χxiχ0je-iHt/2|ψ0⟩⟨ψ0|eiH†te-iHt|ψ0⟩.

Using the fact that the saddle-point solution in the large$-N$ limit is the
disorder replica diagonal in the SYK-like models [48, 69, 70], both $F$ [28]
and $S^{(2)}_{A}$ [71, 72, 73, 74, 75, 76, 77, 27, 78, 29] can be expressed as
a path-integral with two replicas. We begin with the evaluation of
$\overline{(\left<\psi_{0}\right|e^{iH^{\dagger}t}e^{-iHt}\left|\psi_{0}\right>)^{2}}$.
The contour contains four branches (labeled by $1$-$4$), with two forward
evolutions ($1$, $3$) and two backward evolutions ($2$, $4$). Here $1$ and $2$
($3$ and $4$) belong to the same replica. The standard derivation [48] leads
to the $G$-$\Sigma$ action:

$\displaystyle\frac{I}{N}=$
$\displaystyle-\sum_{x}\frac{1}{2}\text{tr}\log\Big{(}(-1)^{a+1}\delta^{ab}\partial_{t}-\Sigma_{x}^{ab}\Big{)}$
(2)
$\displaystyle+\int_{t_{1},t_{2}}\Big{[}\frac{1}{2}\Sigma_{x}^{ab}G_{x}^{ab}+\frac{(-1)^{a+b}\delta(t_{12})}{4}$
$\displaystyle\times[J_{0}(G_{x}^{ab})^{2}+\sum_{r}\frac{J_{0}}{r^{2\alpha}}G_{x}^{ab}G_{x+r}^{ab}]$
$\displaystyle-\frac{\delta(t_{12})}{4}[V_{0}(G_{x}^{ab})^{2}+V_{1}G_{x}^{ab}G_{x+1}^{ab}]\Big{]}.$

It exhibits $O(2)\times O(2)$ symmetry $G\rightarrow OGO^{T}$ and
$\Sigma\rightarrow O\Sigma O^{T}$ with
$O=\exp(-\gamma_{13}\theta_{13}-\gamma_{24}\theta_{24})$ and
$(\gamma_{cd})^{ab}=\delta_{ac}\delta_{bd}-\delta_{bc}\delta_{ad}$. This owes
to the enlarge of the permutation symmetry (between $1,3$ or $2,4$) for
quadratic actions [28]. In the large-$N$ limit, the saddle point solution for
Green’s function $G^{ab}_{s}$ is

$\displaystyle
G_{s}^{11}=-G_{s}^{22}=\frac{i\omega}{\omega^{2}+\Gamma^{2}/4},$ (3)
$\displaystyle
G_{s}^{21}=-G_{s}^{12}=\frac{\Gamma/2}{\omega^{2}+\Gamma^{2}/4}.$

We also have $G_{s}^{a,b}=G_{s}^{a-2,b-2}$ for $a,b=3,4$, while other
components of $G_{s}$ are zero. Here $\Gamma\equiv V+J$, with $V=V_{0}+V_{1}$
and $J=J_{0}(1+\sum_{r=1}^{\infty}1/r^{2\alpha})$. When $\alpha<1/2$, the
summation diverges and we need to scale $J_{0}$ with system size $L$ to
obtained a meaningful thermodynamic limit. This indicates the system becomes
all-to-all connected, similar to the single SYK2 model with $NL$ Majorana
modes, and the steady state has volume-law entanglement regardless of the
strength of the $V_{0}$ [71, 74]. For $\alpha>0.5$, we are able to perform the
summation as $J=J_{0}(1+\zeta(2\alpha))$, where $\zeta(x)$ is the Riemann zeta
function. We will focus on this case in the following discussions.

##  3 Effective Action

The saddle-point solution $G_{s}$ breaks the $O(2)\times O(2)$ symmetry down
to $O(2)$. Consequently, the fluctuation around the saddle-point contains a
gapless Goldstone mode [28, 29]. We consider the fluctuation around saddle-
point

$\displaystyle\Sigma(t_{1},t_{2})$
$\displaystyle=\Sigma_{s}(t_{1},t_{2})+\delta\Sigma(t_{1})\delta(t_{12}),$ (4)
$\displaystyle G(t_{1},t_{2})$ $\displaystyle=G_{s}(t_{1},t_{2})+\delta
G(t_{1},t_{2}),$

and keep to the quadratic order of $\delta G$ and $\delta\Sigma$. We will
focus on components between two replicas as in [28]. Motivated by the symmetry
analysis, we define $\theta(t)=\delta G^{14}(t,t)/2$ in the coset space. After
integrating out gapped modes, the effective action derived in Appendix A reads

$\frac{I_{\text{eff}}}{N}=\frac{1}{2}\int\frac{d\omega
dk}{(2\pi)^{2}}\left(\frac{\epsilon(k)}{2}+\frac{\omega^{2}}{4V}\right)|\theta(\omega,k)|^{2},$
(5)

where the dispersion $\epsilon(k)$ is defined as

$\epsilon(k)=V_{1}(1-\cos(k))+\sum_{r=1}^{\infty}\frac{J_{0}}{r^{2\alpha}}(1-\cos(rk)).$
(6)

Here we have set lattice constant $a_{\text{l}}=1$. To determine the low-
energy effective theory, we expand $\epsilon(k)$ at small momentum $k\ll 1$.
Two possibilities exist depending on $\alpha$:

$\epsilon(k)=\begin{cases}C_{1}k^{2}/2&\text{for $\alpha>1.5$,}\\\
C_{2}k^{2\alpha-1}&\text{for $\alpha\leq 1.5$.}\end{cases}$ (7)

Here we have $C_{1}={V_{1}+J_{0}\zeta(2\alpha-2)}$ and
$C_{2}=-J_{0}\cos\left(\frac{1}{2}\pi(1-2\alpha)\right)\Gamma(1-2\alpha)$.
This can be understood by expanding $(1-\cos(rk))\approx k^{2}r^{2}/2$ and
realizing the summation is convergent only for $\alpha>1.5$. The presence of
$k^{2\alpha-1}$ for $\alpha<1.5$ leads to an anoumalous action [79], which
breaks the conformal symmetry in the replicated Hilbert space. The asymptotic
behavior of $\epsilon(k)$ directly determines the dynamical exponent $z$. We
have $z=1$ for $\alpha>1.5$ and $z=\frac{2\alpha-1}{2}$ for $\alpha\leq 1.5$.

Figure 2: A sketch for the spin configuration ($\theta$ is the angle respect
to the x-axis) that contributes to the calculation of (a). the second Rényi
entanglement entropy on the steady state, (b). the purification dynamics. The
cross represents the insertion of the twist operators.

Now we use the effective action (5) to compute the Keldysh squared correlator
$F$ and entanglement entropy $S^{(2)}_{A}$. On the replicated contour, $F$
corresponds to a four-point correlator of fermions, and thus a two-point
function of collective fields $\theta$. More precisely, we have

$\displaystyle F$ $\displaystyle\sim
N\left<\partial_{t}\theta(x)\partial_{t}\theta(0)\right>_{11}$ (8)
$\displaystyle\sim\int_{\omega
k}e^{ikx}\frac{\omega^{2}}{\omega^{2}+k^{2z}}\sim x^{-z-1}.$

Here we have dropped non-universal factors.

For the entanglement entropy $S^{(2)}_{A}(T)$, the path-integral contour (2)
is defined with additional twist operators at $t=T$, as shown in FIG. 2 (a).
In terms of $\theta$, this corresponds to fixing the boundary condition
$\theta(T,x\in A)=\pi/2$ and $\theta(T,x\in B)=0$ [28, 29], which creates a
pair of half-vortices, separated by a distance $L_{A}$. $S^{(2)}_{A}$ is equal
to the excitation energy of this half-vortex pair. This can be computed by the
two-point correlator
$P_{A}\sim\left<e^{i\pi\varphi(L_{A})/2}e^{-i\pi\varphi(0)/2}\right>$. Here
$\varphi(x)$ is defined by $\nabla\varphi=N\partial_{t}\theta/4V$, which
shifts the value of $\theta(y)$ for $y<x$. The effective action of $\varphi$
can be determined by introducing a Lagrangian multipler $r$ to impose the
relation $\nabla\varphi=\partial_{t}\theta$ and then integrate out $\theta$
and $r$. Working out the details as in Appendix A gives

$\displaystyle S_{A}^{(2)}$
$\displaystyle\sim\left<\varphi(L_{A})\varphi(0)\right>$ (9)
$\displaystyle\sim\int_{\omega k}\leavevmode\nobreak\
\frac{N}{k^{2}+k^{2+2z}/\omega^{2}}e^{ikL_{A}}\sim NL_{A}^{1-z}.$

For $z=1$, this should be understood as $\log L_{A}$. We find for
$0.5<\alpha<1.5$, the system is subvolume-law entangled. This is called the
fractal phase in [21]. It is straightforward to extend the above discussion to
compute the mutual information between two small subregions separated by
distance $d$. The result is
$I^{(2)}(d)\sim\left<\nabla\varphi(d)\nabla\varphi(0)\right>\sim Nd^{-z-1}$,
which matches the scaling of the squared correlator $F$.

It is also interesting to ask about the purification dynamics [8] in the free
fermion system and check how long it will take to purify the system [80]. We
prepare the system in the maximally mixed state and evolve it under the non-
unitary dynamics for time $T$. The second Rényi entropy $S^{(2)}(T)$ of the
full system then describes how much the initial quantum information stored in
the system is lost [8]. In the effective theory, $S^{(2)}(T)$ is given by
computing the energy of the $\theta$ field with a boundary condition
$\theta(t=0)=0$ and $\theta(t=T)=\pi/2$. The dominant contribution is
determined by the saddle-point equation $\partial_{t}^{2}\theta=0$, which
gives $\theta(t)=\pi t/2T$. This configuration is shown in FIG. 2 (b), where
the spins rotate smoothly from $0$ to $\pi/2$. This immediately suggests the
purification process is insensitive to the range of the interaction.
Explicitly, we have $S^{(2)}(T)/N\sim TL\times 1/T^{2}\sim L/T$ and the
purification time $\sim O(L)$ for any arbitrary $\alpha\geq 0$.

##  4 Single-flavor Model

Now we ask whether our results derived in the large$-N$ limit hold for systems
with small local Hilbert space dimension. To answer this question, we
introduce a second model with single-flavor of fermion per site [81]: H= ∑_r≥1
(Jx,r+x(t)rα c^†_x+rc_x+H.C.)-iκ_x(t)c^†_xc_x. The first term describes the
long-range hopping and the second term is an on-site imaginary potential.
Similar to the SYK case, $J^{x,r+x}$ and $\kappa_{x}$ are Brownian variables
with variance

$\displaystyle\overline{J^{x,x+r}(t)J^{x,x+r}(0)^{*}}=J_{l}\delta(t),$ (10)
$\displaystyle\overline{\kappa_{x}(t)\kappa_{x}(0)}=\kappa\delta(t).$

We further choose $\kappa=1$ as the energy unit. At time $t$,

$\displaystyle|\psi(t)\rangle=\frac{e^{-iHt}}{\sqrt{\langle\psi_{0}|e^{iH^{\dagger}t}e^{-iHt}|\psi_{0}\rangle}}|\psi_{0}\rangle.$
(11)

where the initial state $|\psi_{0}\rangle$ is chosen as a product state in the
real space. It is known that under the non-unitary quadratic evolution,
$|\psi(t)\rangle$ remains a fermionic Gaussian state [82] and all information
is encoded in the correlation function
$C_{xy}(t)\equiv\left<\psi(t)\right|c^{\dagger}_{x}c_{y}|\psi(t)\rangle$. For
the Brownian model, it is useful to introduce the distribution function as

$\displaystyle f_{n}\equiv\begin{cases}\sum_{x}\frac{|C_{x,x}|^{2}}{L},&n=0\\\
\sum_{x}\frac{|C_{x,x+n}|^{2}+|C_{x,x-n}|^{2}}{L},&n>0\end{cases}.$ (12)

As in [26], $f_{n>1}$ approximately satisfies a set of non-linear master
equation, which takes the form

$\displaystyle\frac{df_{n}}{dt}$
$\displaystyle=\mu_{n}+\sum_{0<r<n}\frac{J_{l}}{r^{2\alpha}}(f_{n+r}+f_{n-r}-2f_{n})$
(13) $\displaystyle+\sum_{r\geq
n}\frac{J_{l}}{r^{2\alpha}}(f_{n+r}-2f_{n})-2f_{n}\sum_{m=1}^{\infty}f_{m}$
$\displaystyle+\sum_{m=1}^{\infty}f_{m}f_{m+n}+\frac{1}{2}\sum_{m=1}^{n-1}f_{m}f_{n-m}.$

Here the first term $\mu_{n}=\mu_{l}/n^{2\alpha}$ is a source term which
captures contributions from the diagonal term $C_{x,x}$. The second and the
third term are contributions from the Hermitian Brownian hopping and describe
a Lévy flight process. Other terms are contributions from the random imaginary
fields (see Appendix B) and is non-linear. For an initial product state, we
have $f_{n>1}(0)=0$.

We aim to understand the dynamics and the long-time behavior of (13). In the
short-range hopping limit where $J^{x,x+r}=0$ for any $r\geq 2$, it is known
that the steady state has $f_{n}\sim 1/n^{2}$ and the dynamical exponent $z=1$
[26]. Now we turn on the long-range hopping term. Since the Lévy flight is
equivalent to a diffusive random walker when $\alpha>1.5$, we expect that the
entanglement/correlation dynamics is the same as the system with only local
hopping term. In this regime, the diffusive term is much slower than the non-
linear terms in the master equation and can be safely removed when we analyze
the dynamics of the master equation [26].

Figure 3: Numerical results obtained by solving the nonlinear master equation
(13) with $\mu_{l}=J_{l}=1/\zeta(2\alpha)$. We take $t\in[60,800]$ with a
cutoff at $n=1000$. The slopes for the curves in (a) and (b) are -1.1 and -1.7
respectively. The results show the validity of the scaling form (15).

To determine the physics with $\alpha<1.5$, we first consider the steady state
where $df_{n}/dt=0$. The R.H.S. of (13) contains three terms. The contribution
from $\mu_{n}$ is proportional to $1/n^{2\alpha}$. Assuming $f_{n}\sim
1/n^{\delta}$, the contribution from the imaginary potential gives

$\int dm\leavevmode\nobreak\ f_{m}(2\cos(m\partial_{n})-2)f_{n}\sim
1/n^{2\delta-1}.$ (14)

The steady state is achieved when the contribution from the imaginary
potential balances the contribution from $\mu_{n}$. This gives
$\delta=\frac{2\alpha+1}{2}$. For any $\alpha>0.5$, this guarantees that the
Lévy flight terms have larger power-law exponents than the other two terms and
can be safely neglected. Assuming the entanglement is contributed by EPR pairs
with length distribution described by $f_{n}$ [30], we can perform a double
integral over it to obtain the entanglement entropy $S^{(2)}_{A}\sim
L_{A}^{{(3-2\alpha)/2}}$. All these results are consistent with the SYK
calculation in the large-$N$ limit.

Now we turn to the determination of the dynamical exponent $z$ in the regime
$0.5<\alpha<1.5$. At long but finite time $t$, we expect $f_{n}$ takes the
scaling form:

$f_{n}(t)=t^{-\eta}F(n/t^{1/z}).$ (15)

Taking the infinite time limit, we should have

$f_{n}(t\to\infty)\sim t^{-\eta}(nt^{-1/z})^{-\eta z}\sim n^{-\eta z}.$ (16)

Consequently, we have the constraint $\eta z=\delta=\frac{2\alpha+1}{2}$. To
determine $z$, we plug the scaling form (15) into equation (13). It is
straightforward to confirm that $df_{n}/dt$ balances the non-linear
convolution term from the imaginary potential 111The last two terms in the
master equation (13) combine into a convolution term in the continuum limit..
We have $df_{n}/dt\sim 1/t^{\eta+1}$ and

$\displaystyle\int dm\leavevmode\nobreak\
f_{m}(2\cos(m\partial_{n})-2)f_{n}\sim 1/t^{2\eta-1/z}.$ (17)

We conclude $2\eta-1/z=\eta+1$ and thus $z=\frac{2\alpha-1}{2}$. This again
matches the large$-N$ result. To verify the scaling form (15), we numerically
solve the differential equation (13). The results are shown in Fig. 3, from
which it is clear that we have $z=1$ for $\alpha>1.5$ and
$z=\frac{2\alpha-1}{2}$ for $0.5<\alpha<1.5$, consistent with our analysis
222We also numerically check the purification dynamics and find that $S(T)\sim
L/T$, the same as the large-$N$ system.. In particular, we find that the Lévy
flight terms can be disregarded (set $J_{l}=0$) and does not change the
dynamics and the steady state, the same as the master equation with local
diffusion term [26].

##  5 Static Hopping Case

Now we examine whether our phase diagram works for time independent
Hamiltonians. The large$-N$ model (2) can be made static by assuming that the
random variables are time-independent:

$\displaystyle\overline{(\tilde{J}_{ij}^{x})^{2}}=\frac{J_{0}^{2}}{N},\ \ \ \
\ \ \ \overline{(V_{ij}^{x,x+1})^{2}}=\frac{V_{1}^{2}}{2N},$ (18)
$\displaystyle\overline{(\tilde{V}_{ij}^{x})^{2}}=\frac{V_{0}^{2}}{N},\ \ \ \
\ \ \ \overline{(J_{ij}^{x,x+r})^{2}}=\frac{J_{0}^{2}}{2N}.$

This model can also be analyzed analogously to the Brownian case (see Appendix
C). The main difference is that now the effective action contains copies of
Goldstone modes because of the enlarged symmetry due to the time translation
symmetry [28]. For each copy, its effective action takes the same form as (5).
Consequently, the scaling of the $F$ in (8), the scaling of $S^{(2)}_{A}$ in
(9), and thus phase diagram in FIG. 1 are still valid. We also numerically
study the static version of $N=1$ single-flavor model in (4). Here we take
hermitian long-range hopping terms to be time independent while keeping the
imaginary potential to be random in the time direction. We manage to
numerically reproduce the same result for $\alpha>1.5$ in Appendix D. Putting
all results together, we conclude our phase diagram FIG. 1 is universal for
both Brownian and static models, regardless of the local Hilbert space
dimension. This is consistent with recent studies [46, 83].

##  6 Discussions

In this work, we consider the long-range non-unitary random dynamics in free-
fermion systems. We analytically show that both the large-$N$ Brownian/static
SYK2 chain and the single-flavor Brownian model exhibit a logarithmic phase
for $\alpha>1.5$ where entanglement is logarithmic in subsystem size, and a
fractal phase with $0.5<\alpha<1.5$ where the entanglement is subvolume-law.
We also show that these two phases cannot be distinguished by the purification
dynamics. We expect that the phase diagram is universal for general non-
interacting random fermionic systems under long-range non-unitary dynamics.
The non-unitary evolution can be interpreted as the dynamics under continuous
forced measurements, as explained in Appendix E. An explicit experimental
scheme with continuous measurement of particle density on each site has been
proposed in [84] using interactions between atoms and photons. Consequently,
both entanglement entropy [85] and squared correlator can, in principle, be
measured in an experiment with certain post-selection.

We finally make a few comments. Firstly, it is interesting to extend the
discussions to general spatial dimension $D$. For the SYK2 model, this
corresponds to changing the summation over $r$ in (6) to a $D$-dimensional
integral. Consequently, for $\alpha>\frac{2+D}{2}$ we have a phase with $z=1$,
$F(x)=1/x^{D+1}$ and $S^{(2)}=L_{A}^{D-1}\log L_{A}$, while for
$\frac{2+D}{2}>\alpha>\frac{D}{2}$, we have $z=\frac{2\alpha-D}{2}$,
$F(x)=1/x^{D+z}$ and $S^{(2)}=L_{A}^{D-z}$. Secondly, in [29], authors study
the measurement effect on SYK chains by introducing models with two copy of
chains with interchain coupling $\mu$. We can construct a similar model with
long-range hopping terms. When $\alpha>0.5$, a transition between the
fractal/logarithmic phase and the area law phase occurs at
$\mu=J_{0}(1+\zeta(2\alpha))$, similar to the observed phase diagram in [46].

Acknowledgment. PZ acknowledges support from the Walter Burke Institute for
Theoretical Physics at Caltech. SKJ is supported by the Simons Foundation via
the It From Qubit Collaboration. CL is supported by the NSF CMMT program under
Grants No. DMR-1818533.

Note added. After this work had been completed, we became aware of an
independent investigation of entanglement transition in long-range hopping
fermion chains [83].

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##  Appendix A Effective action and Entanglement entropy for the Brownian SYK
chain

As we discussed in the main text, we consider the replicated system
$\overline{(\left<\psi_{0}\right|e^{iH^{\dagger}t}e^{-iHt}\left|\psi_{0}\right>)^{2}}$.
After introducing bilocal fields and integrating out Majorana fermions, the
$G$-$\Sigma$ action reads

$\displaystyle-\frac{I}{N}=$
$\displaystyle\sum_{x}\frac{1}{2}\text{tr}\log\Big{(}(-1)^{a+1}\delta^{ab}\partial_{t}-\Sigma_{x}^{ab}\Big{)}-\int_{tt^{\prime}}\Big{[}\frac{1}{2}\Sigma_{x}^{ab}G_{x}^{ab}+\frac{(-1)^{a+b}\delta(t-t^{\prime})}{4}[J_{0}(G_{x}^{ab})^{2}$
(19)
$\displaystyle+\sum_{r}\frac{J_{0}}{r^{2\alpha}}G_{x}^{ab}G_{x+r}^{ab}]-\frac{\delta(t-t^{\prime})}{4}[V_{0}(G_{x}^{ab})^{2}+V_{1}G_{x}^{ab}G_{x+1}^{ab}]\Big{]}.$

We first consider the saddle-point equation for $G_{x}$ and $\Sigma_{x}$,
which reads

$\displaystyle\left[(-1)^{a+1}\delta^{ac}\partial_{t}-\Sigma_{x}^{ac}\right]\circ
G_{x}^{cb}=I^{ab}.$ (20)
$\displaystyle\Sigma_{x}^{ab}(t,t^{\prime})=V_{1}\delta(t-t^{\prime})\frac{G_{x+1}^{ab}(t,t^{\prime})+G_{x-1}^{ab}(t,t^{\prime})}{2}-\sum_{r\geq
1}(-1)^{a+b}\frac{J_{0}}{r^{2\alpha}}\frac{G_{x+r}^{ab}(t,t^{\prime})+G_{x-r}^{ab}(t,t^{\prime})}{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
+V_{0}G_{x}^{ab}(t,t^{\prime})-J_{0}(-1)^{a+b}G_{x}^{ab}(t,t^{\prime}).$

The solution of the equation is translational invariant along space or time
$\Sigma_{x}^{ab}(t,t^{\prime})=G_{s}^{ab}(t-t^{\prime})$. In fact, it takes
exactly the same form as the model without long-range hopping [28]:

$G_{s}^{11}(\omega)=\frac{i\omega}{\omega^{2}+\Gamma^{2}/4},\ \ \ \
G_{s}^{12}(\omega)=-\frac{\Gamma/2}{\omega^{2}+\Gamma^{2}/4},$ (21)

We also have $G_{s}^{a,b}=G_{s}^{a-2,b-2}$ for $a,b=3,4$, while other
components of $G_{s}$ are zero. The effective action can be derived by
consider fluctuations around the saddle-point,

$\Sigma(t_{1},t_{2})=\Sigma_{s}(t_{1},t_{2})+\delta\Sigma(t_{1})\delta(t_{12}),\quad
G(t_{1},t_{2})=G_{s}(t_{1},t_{2})+\delta G(t_{1},t_{2}),$ (22)

and keep everything to the quadratic order. We will focus on components
between two replicas. There are contributions from the $\text{tr}\log$ term,
the $\Sigma G$ term, and the $G^{2}$ terms. Comparing to the short-range
hopping model [28], the only difference is from the contribution from $G^{2}$
terms. Fortunately, these contributions are the simplest ones. As in [28], we
introduce

$\displaystyle\phi_{1}(\omega)$ $\displaystyle=\sqrt{2}\int dt\delta
G^{13}(t,t)e^{i\omega t},$ (23) $\displaystyle\phi_{2}(\omega)$
$\displaystyle=-\sqrt{2}\int dt\delta G^{14}(t,t)e^{i\omega t}.$

The $G^{2}$ term gives a contribution

$-\frac{\delta
I_{\text{eff}}}{N}=\frac{1}{2}\int_{t}\left(V_{k}(\phi_{1,k}\phi_{1,-k}+\phi_{2,k}\phi_{2,-k})-J_{k}(\phi_{1,k}\phi_{1,-k}-\phi_{2,k}\phi_{2,-k})\right).$
(24)

Here $V_{k}=V_{0}+V_{1}\cos(k)$,
$J_{k}=J_{0}(1+\sum_{r}\cos(rk)/r^{2\alpha})$. Adding back contributions from
other terms, we have

$-\frac{I_{eff}}{N}=\frac{1}{2}\sum_{k}\int_{\omega}\phi_{k}(\omega)\left(\begin{array}[]{cccc}\Gamma-
J_{k}+V_{k}&-i\omega\\\
i\omega&-\Gamma+J_{k}+V_{k}\end{array}\right)\phi_{-k}(-\omega).$ (25)

Denoting $\theta=\sqrt{2}\phi_{2}$ and integrating out $\phi_{1}$, we arrive
the effective action used in the main text:

$\frac{I_{\text{eff}}}{N}=\frac{1}{2}\int\frac{d\omega
dk}{(2\pi)^{2}}\left(\frac{\epsilon(k)}{2}+\frac{1}{4V}\omega^{2}\right)|\theta(\omega,k)|^{2}.$
(26)

The squared correlator $F$ corresponds to the two point function of $\phi_{1}$
[28]. In the low energy limit, the equation of motion of $\phi_{1}$ gives
$2V\phi_{1}(\omega)\approx-i\omega\phi_{2}(\omega)$. This justifies the
calculation of $F$ in the main text.

For computing the entanglement entropy, as explained in the main text, we
introduce Lagrangian multiplier $r(\omega,k)$ to impose the relation between
$\theta$ and its dual field $\varphi$:

$I_{\text{eff}}=\int_{\omega
k}\frac{N}{2}\left(k^{2z}+\omega^{2}\right)|\theta(\omega,k)|^{2}+r(\omega,k)(k\theta(\omega,k)+N\omega\varphi(\omega,k)).$
(27)

Here we have dropped non-universal parameters. Integrating out $r$ and
$\theta$ leads to the effective action of $\varphi$ used the in main text:

$\frac{I_{\text{eff}}}{N}=\frac{1}{2N}\int_{\omega
k}\left(k^{2}+k^{2+2z}/\omega^{2}\right)|\varphi(\omega,k)|^{2}.$ (28)

##  Appendix B Non-linear master equation for the single-flavor model

Now we present the derivation of the non-linear master equation for the
single-flavor model. The derivation also follows the short-range hopping case
studied in [61].

We first write out the equation satisfied by the correlation matrix $C_{xy}$.
Since the model is Brownian, we can obtain the evolution of $|C_{xy}|^{2}$ by
using the Itô calculus. The derivation is tedious but straightforward [61].
Here we only cite the result. The contribution from the Hermitian long-range
hopping part reads

$\displaystyle\left(\frac{d|C_{x,y}|^{2}}{dt}\right)^{(1)}=$
$\displaystyle\sum_{r}\frac{J_{l}}{r^{2\alpha}}\Big{\\{}|C_{y+r,x}|^{2}+|C_{y-r,x}|^{2}+|C_{y,x+r}|^{2}+|C_{y,x-r}|^{2}$
(29)
$\displaystyle-2\delta_{x,y+r}C_{y+r,y+r}C_{y,y}-2\delta_{x,y-r}C_{y-r,y-r}C_{y,y}$
$\displaystyle-4|C_{x,y}|^{2}+2(C_{x+r,x+r}C_{x,x}+C_{x-r,x-r}C_{x,x})\delta_{x,y}\Big{\\}}.$

The contribution from the on-site non-Hermiticity reads

$\displaystyle\left(\frac{d|C_{x,y}|^{2}}{dt}\right)^{(2)}=$ $\displaystyle
4|C_{x,y}|^{2}+4|C_{x,y}|^{2}\delta_{a,b}-8|C_{x,y}|^{2}(C_{y,y}+C_{x,x})+4\sum_{m}|C_{y,m}|^{2}|C_{m,x}|^{2}$
(30)
$\displaystyle+4\sum_{m}\left[C_{y,m}C_{m,x}C_{m,m}C_{x,y}+C_{x,m}C_{m,y}C_{m,m}C_{y,x}\right]$
$\displaystyle-2\sum_{m}\left[C_{y,m}C_{m,x}C_{x,y}+C_{x,m}C_{m,y}C_{y,x}\right].$

To study the dynamics of $f_{n}$, we rewrite Eq.(29) and Eq.(30) in terms of
$f_{n}$. Note that this is only possible under certain approximations. The
Hermitian part Eq.(29) contributes a term $\sum
J_{l}(f_{n-r}+f_{n+r}-2f_{n})/r^{2\alpha}$. In particular, for $n=r$ this
leads to a production of long-range correlation directly from the on-site
correlation. For Eq.(30), we follow the discussion in [61] and throw away
terms $C_{y,m}C_{m,x}C_{x,y}$ and $C_{y,m}C_{m,x}C_{m,m}C_{x,y}$ when $m\neq
x$ or $y$. The contribution from the on-site imaginary potential then mainly
contributes a convolution term
$-2f_{n}\sum_{m=1}^{\infty}f_{m}+\sum_{m=1}^{\infty}f_{m}f_{m+n}+\frac{1}{2}\sum_{m=1}^{n-1}f_{m}f_{n-m}$.
Summing up these contributions, we arrive at the master equation used in the
main text

$\displaystyle\frac{df_{n}}{dt}=$
$\displaystyle\mu_{n}+\sum_{0<r<n}\frac{J_{l}}{r^{2\alpha}}(f_{n+r}+f_{n-r}-2f_{n})+\sum_{r\geq
n}\frac{J_{l}}{r^{2\alpha}}(f_{n+r}-2f_{n})$ (31)
$\displaystyle-2f_{n}\sum_{m=1}^{\infty}f_{m}+\sum_{m=1}^{\infty}f_{m}f_{m+n}+\frac{1}{2}\sum_{m=1}^{n-1}f_{m}f_{n-m},$

##  Appendix C The static model

We first consider the static SYK chain with long-range non-Hermitian
Hamiltonian. The self-consistent equation for the Green’s function reads

$\displaystyle\left[(-1)^{a+1}\delta^{ac}\partial_{t}-\Sigma_{x}^{ac}\right]\circ
G_{x}^{cb}=I^{ab}.$ (32)
$\displaystyle\Sigma_{x}^{ab}=V_{1}^{2}\frac{G_{x+1}^{ab}+G_{x-1}^{ab}}{2}+V_{0}^{2}G_{x}^{ab}-(-1)^{a+b}J_{0}^{2}G_{x}^{ab}-\sum_{l\geq
1}(-1)^{a+b}\frac{J_{0}^{2}}{l^{2\alpha}}\frac{G_{x+l}^{ab}+G_{x-l}^{ab}}{2}.$

For the saddle-point solution with translational symmetry
$G^{ab}_{x}=G_{s}^{ab}$. Focusing on the low-energy limit
$|\omega|<\frac{2J^{2}}{\sqrt{J^{2}+V^{2}}}$, $G_{s}(\omega)$ reads

$G_{s}^{11}(\omega)=\frac{i\omega}{2J^{2}},\ \ \ \
G_{s}^{12}(\omega)=-\frac{1}{2J^{2}}\sqrt{\frac{4J^{4}}{J^{2}+V^{2}}-\omega^{2}},$
(33)

together with $G_{s}^{22}(t)=-G_{s}^{11}(t)$ and
$G_{s}^{21}(t)=-G_{s}^{12}(t)$. Here we have defined $V^{2}\equiv
V_{0}^{2}+V_{1}^{2}$ and

$\displaystyle J^{2}=J_{0}^{2}+\sum_{l\geq
1}\frac{J_{0}^{2}}{l^{2\alpha}}=J_{0}^{2}(1+\zeta(2\alpha)).$ (34)

Now we turn to the effective action. The $G-\Sigma$ action reads

$\displaystyle-\frac{I}{N}=$
$\displaystyle\sum_{x}\frac{1}{2}\text{Tr}\log\Big{(}(-1)^{a+1}\delta^{ab}\partial_{t}-\Sigma_{x}^{ab}\Big{)}+\int\Big{[}-\frac{1}{2}\Sigma_{x}^{ab}G_{x}^{ab}$
(35)
$\displaystyle+\frac{1}{4}[V_{0}^{2}(G_{x}^{ab})^{2}+V_{1}^{2}G_{x}^{ab}G_{x+1}^{ab}]-\frac{(-1)^{a+b}}{4}[J_{0}^{2}(G_{x}^{ab})^{2}+\sum_{l}\frac{J_{0}^{2}}{l^{2\alpha}}G_{x}^{ab}G_{x+l}^{ab}]\Big{]},$

Expanding $G^{ab}_{x}(t,t^{\prime})=G^{ab}_{s}(t,t^{\prime})+\delta
G^{ab}_{x}(t,t^{\prime})$. We again focus on fluctuations involving two
replicas $\\{\delta G^{13}$, $\delta G^{14}$, $\delta G^{23}$, $\delta
G^{24}\\}$ and define
$\phi_{\pm}=(\phi_{\pm}^{1},\phi_{\pm}^{2})=\frac{1}{\sqrt{2}}(\delta
G^{13}\pm\delta G^{24},\delta G^{14}\pm\delta G^{23})$ as in [28], where it is
found that only $\phi_{+}$ contributes in the low-energy limit. Keeping to the
quadratic order as in the Brownian case, we find the effective action

$\displaystyle-\frac{I_{\text{eff}}}{N}=\frac{1}{2}\int_{\Omega\omega
k}\phi_{+}(\Omega,\omega,k)\begin{pmatrix}2V^{2}&\frac{i(J^{2}+V^{2})^{3/2}}{2J^{2}}\Omega\\\
-\frac{i(J^{2}+V^{2})^{3/2}}{2J^{2}}\Omega&-\epsilon(k)+\frac{(J^{2}+V^{2})^{2}\Omega^{2}}{8J^{4}}\end{pmatrix}\phi_{+}(\Omega,\omega,k).$
(36)

Here $\Omega$ is the center-of-mass frequency and $\omega$ is the relative
frequency of $t$ and $t^{\prime}$. We have

$\epsilon(k)=\Big{(}1-\cos(k)\Big{)}V_{1}^{2}+\sum_{l\geq
1}\Big{(}1-\cos(lk)\Big{)}J_{l}^{2}\approx k^{2}+k^{2\alpha-1},$ (37)

which takes the same form as the Brownian case. Integrating out $\phi_{+}^{1}$
and using the coset variables [28] leads to the anisotropic XY model:

$\frac{I_{\text{eff}}}{N}=\frac{1}{2}\int_{\omega\Omega
k}\left(\frac{2\epsilon(k)}{J^{2}+V^{2}}+\frac{J^{2}+V^{2}}{4J^{2}V^{2}}\Omega^{2}\right)|\theta^{\omega}(\Omega,k)|^{2}.$
(38)

This is the same for as the Brownian case, with an additional label $\omega$
for different Goldstone modes. Here we should treat $\Omega$ as the frequency
in the Brownian case. Consequently, the scaling of the squared correlators and
entanglement entropy should be the same as the Brownian model.

##  Appendix D Numerical simulation for $N=1$

For numerical convenience, we construct a discrete non-unitary circuit model
for single-flavor free fermion with the evolution operator (unnormalized)

$\displaystyle U=\prod_{t=1}^{T}U_{\beta}(t)U_{\tau}(t).$ (39)

Here $U_{\tau}=\exp(-iH_{R}\tau)$ represents the evolution governed by a long-
range hopping Hermitian Hamiltonian $H_{R}$ and and
$U_{\beta}=\exp(-H_{I}\beta)$ represents the imaginary evolution with
Hamiltonian $H_{I}$. At time $t$, we have the wave function

$\displaystyle|\psi(t)\rangle=\frac{U}{\sqrt{Z}}|\psi(0)\rangle,\quad\text{with}\
\ Z=\langle\psi(0)|U^{\dagger}U|\psi(0)\rangle,$ (40)

where the initial state $|\psi(0)\rangle$ is chosen as a product state in real
space. If $H_{R}$ and $H_{I}$ are both quadratic Hamiltonians,
$|\psi(t)\rangle$ remains a Gaussian state under time evolution. All the
information is encoded in the two point correlation function matrix
$C_{xy}=\langle\psi(t)|c^{\dagger}_{x}c_{y}|\psi(t)\rangle$.

Figure 4: The numerical results at $\alpha=1.8$ with periodic boundary
condition. All the results presented here are averaged over large number of
samples. (a) The steady state second Rényi entropy for various non-unitary
dynamics. For the blue curve, $J^{x,r+x}$ is a time dependent random variable
with the distribution function described by (43). For the red curve,
$J^{x,r+x}$ is a time independent random variable with the distribution
function described by (44) with $p=0.5$. For the yellow curve, $J^{x,r+x}=1$
and $H_{R}$ has no randomness in it. (b) The steady state squared correlator
for various non-unitary dynamics. The color of the curves are the same as that
in panel (a). The purple curve is proportional to $1/\sin^{2}(\pi r/L)$. The
blue, red and yellow curves are all parallel to the purple curve, suggesting
that the squared correlator decays as $1/r^{2}$ in all these three models. (c)
The data collapse of the squared correlator on the log-log scale. The curves
at different times collapse into a single curve, suggesting that $z=1$. (d)
The blue curve is the half system $S_{2}$ vs $\log T-3$ and the red curve is
the steady state $S_{2}$ vs $\log L_{A}$. These two curves have the same
slope, suggesting that $z=1$.

To incorporate the long-range hopping and periodic boundary condition, we take
the following Hamiltonian

$\displaystyle H_{R}$ $\displaystyle=\sum_{r\geq
0}\frac{J^{x,r+x}}{1+\frac{\sin^{\alpha}(\pi
r/L)}{(\pi/L)^{\alpha}}}\left(c^{\dagger}_{x+r}c_{x}+\text{H.C.}\right)$ (41)
$\displaystyle H_{I}$ $\displaystyle=\kappa_{x}(t)c^{\dagger}_{x}c_{x}.$

Here $\kappa_{x}(t)$ is stochastic random variable with a simple two-component
distribution,

$\displaystyle
P(\kappa_{x}(t))=\frac{1}{2}\delta(\kappa_{x}(t)-1)+\frac{1}{2}\delta(\kappa_{x}(t)+1).$
(42)

For the variable $J^{x,r+x}$, we consider two versions: the stochastic and
static.

In the stochastic case, we take $J^{x,r+x}$ to be a time dependent random
variable

$\displaystyle
P(J^{x,r+x}(t))=\frac{1}{2}\delta(J^{x,r+x}(t)-1)+\frac{1}{2}\delta(J^{x,r+x}(t)+1).$
(43)

The numerical results for $\alpha=1.8$ is shown in Fig. 4. We show that this
model has dynamical exponent $z=1$ and the final steady state has
$S_{2}\sim\log L_{A}$ and $C(r)\sim 1/r^{2}$. We also take $H_{R}$ to be a
time independent Hamiltonian with $J^{x,r+x}$ satisfying

$\displaystyle P(J^{x,r+x})=p\delta(J^{x,r+x}-1)+(1-p)\delta(J^{x,r+x}+1).$
(44)

We present the steady state results for $p=0$ and $p=0.5$ in Fig. 4 (a) (b).
Notice that when $p=0$, $H_{R}$ is a clean system without disorder. They all
have the same scaling behavior as the stochastic version. We also try other
$\alpha>1.5$ and we obtain the same results. These numerical simulation
results suggest that the discrete dynamics has the same result as the Brownian
dynamics we discussed in the main text in the regime $\alpha>1.5$. However, we
are unable to obtain consistent numerical results for $\alpha<1.5$ due to
strong finite size effect with $L=1000$.

###  D.1 Purification dynamics

We also take the initial system $A$ as a density matrix and study how fast it
can purify under non-unitary random dynamics. We take the unitary dynamics
$U_{\tau}$ to be random in the spatial and time direction and $H_{I}$ to be a
stochastic random potential term. We consider three different cases for
unitary dynamics : a local Hamiltonian for $H_{R}$, a single-particle Gaussian
orthogonal ensemble (GOE) random matrix for $H_{R}$ and single-particle Haar
random matrix for $U_{\tau}$. As shown in Fig. 5, in all these cases, the
system $A$ has the second Rényi entropy $S_{A}\sim 1/T$, regardless of the
locality structure of $U_{\tau}$.

Figure 5: The purification dynamics of non-unitary random dynamics. We take
three different unitary evolutions $U_{\tau}$ and we show that the entropies
of the system all decay as $1/T$.

##  Appendix E The realization of non-unitary dynamics

In this section, we discuss the experimental relevance of our work by relating
the single quantum trajectory under the non-unitary evolution to evolution
with continuously forced measurements. We focus on the single-flavor model
where

$H_{I}=\sum_{x}\kappa_{x}(t)c^{\dagger}_{x}c_{x}.$ (45)

After having generated a set of random numbers for $\kappa_{x}(t)$, we can
make a constant shift of the potential such that $\kappa_{x}(t)>0$. This is
due to the particle number conservation of the system.

Intuitively, such imaginary potential terms should corresponds to measuring
the on-site particle numbers. We hope to make such relation explicit. To begin
with, we consider continuously monitoring our system: at each time step $dt$,
we apply a measurement at site $x$. The measurement is described by a set of
operators

$\\{M_{x}^{0},M_{x}^{1}\\}=\left\\{1-s_{x}(t)c^{\dagger}_{x}c_{x},\sqrt{2s_{x}(t)-s_{x}(t)^{2}}c^{\dagger}_{x}c_{x}\right\\}.$
(46)

Here we assume $0<s_{x}(t)\ll 1$, which means the measurement strength is
weak. It is straightforward to check the completeness relation
$\sum_{i}M_{x}^{i}(M_{x}^{i})^{\dagger}=I$. For a system in some density
matrix $\rho(t)$ before the measurement, after a single measurement, depending
on the outcome, the unnormalized density matrix becomes

$\rho^{0}(t+dt)=M_{x}^{0}\rho(t)(M_{x}^{0})^{\dagger},\ \ \ \ \ \ \ \
\rho^{1}(t+dt)=M_{x}^{1}\rho(t)(M_{x}^{1})^{\dagger}.$ (47)

And the probability for obtaining $\rho^{i}(t+dt)$ is given by
$p^{i}=\text{tr}[\rho^{i}(t+dt)]$. For forced measurement, we only keep the
result when the outcome is $0$. Then the evolution of the unnormalized density
matrix is given by

$\displaystyle\rho(t+dt)\leavevmode\nobreak\ =\leavevmode\nobreak\
M_{x}^{0}\rho(t)M_{x}^{0}\approx\rho(t)-s_{x}(t)c^{\dagger}_{x}c_{x}\rho(t)-\rho(t)s_{x}(t)c^{\dagger}_{x}c_{x}+o(s_{x})$
(48)

Here we have kept only terms up to $O(s_{x})$. Now we choose
$s_{x}(t)=\kappa_{x}(t)dt$, the evolution can now be written as

$\displaystyle\rho(t+dt)\leavevmode\nobreak\ =\leavevmode\nobreak\
e^{-\kappa_{x}(t)c^{\dagger}_{x}c_{x}dt}\rho(t)e^{-\kappa_{x}(t)c^{\dagger}_{x}c_{x}dt}+O(dt^{2}).$
(49)

After adding up measurements over all sites, this matches the result of non-
Hermitian Hamiltonian dynamics.

The traditional $n$-th subsystem Rényi entropy $S_{A}^{(n)}$ is defined as

$S_{A}^{(n)}=-\frac{1}{(n-1)}\text{tr}\leavevmode\nobreak\
\rho\log\left(\frac{\text{tr}_{A}(\text{tr}_{B}\leavevmode\nobreak\
\rho)^{n}}{(\text{tr}\leavevmode\nobreak\
\rho)^{n}}\right)=-\frac{1}{(n-1)}\log\left(\frac{\text{tr}\leavevmode\nobreak\
[\rho\otimes\rho...\otimes\rho\mathcal{X}_{A}]}{\text{tr}\leavevmode\nobreak\
[\rho\otimes\rho...\otimes\rho]}\right).$ (50)

In particular, the second Rényi entropy $S^{(2)}_{A}$ has been measured in
experiments [85] for systems under unitary evolution. We have written the
Rényi entropy as the expectation of the replicated system. Here
$\mathcal{X}_{A}$ is the cyclic permutation operator for subsystem $A$.

We further show that $S_{A}^{(n)}$ can in principle be measured directly in
experiments by generalization protocols in [85]. For simplicity, we again
focus on the $n=2$ case and analyze the effect of a single measurement. We
prepare two copies of systems in some initial state $\rho(t)\otimes\rho(t)$.
We now measure the system using operators $M_{x}^{i}\otimes M_{x}^{j}$:

$\\{M_{x}^{0}\otimes M_{x}^{0},M_{x}^{1}\otimes M_{x}^{0},M_{x}^{0}\otimes
M_{x}^{1},M_{x}^{1}\otimes M_{x}^{1}\\}.$ (51)

The probability of getting $ij$ is given by $\text{tr}\leavevmode\nobreak\
[\rho^{i}\otimes\rho^{j}]$. We now post select results with $i=j=0$ only. The
probability of getting result $ii$ and the normalized state after getting the
result are then given by

$\rho_{d}=\frac{\rho\otimes\rho}{\text{tr}\leavevmode\nobreak\
[\rho\otimes\rho]}.$ (52)

The swap operator $\mathcal{X}_{A}$ can be measured by applying additional
Rabi oscillations between replicas [85]. This leads to

$\left<\mathcal{X}_{A}\right>=\text{tr}\leavevmode\nobreak\
[\mathcal{X}_{A}\rho_{d}]=\frac{\text{tr}\leavevmode\nobreak\
[\rho\otimes\rho\mathcal{X}_{A}]}{\text{tr}\leavevmode\nobreak\
[\rho\otimes\rho]},$ (53)

which directly gives $e^{-\tilde{S}_{A}^{(2)}}$. Note that this experimental
realization faces the same challenging in the mixed unitary-measurement
dynamics described in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18].