# Entanglement Phases in large-$N$ hybrid Brownian circuits with long-range
couplings

Subhayan Sahu<EMAIL_ADDRESS>Condensed Matter Theory Center and
Joint Quantum Institute, Department of Physics, University of Maryland,
College Park, MD 20742, USA  Shao-Kai Jian  Martin A. Fisher School of
Physics, Brandeis University, Waltham MA, USA  Condensed Matter Theory Center
and Joint Quantum Institute, Department of Physics, University of Maryland,
College Park, MD 20742, USA  Gregory Bentsen  Martin A. Fisher School of
Physics, Brandeis University, Waltham MA, USA  Brian Swingle  Martin A. Fisher
School of Physics, Brandeis University, Waltham MA, USA  Condensed Matter
Theory Center and Joint Quantum Institute, Department of Physics, University
of Maryland, College Park, MD 20742, USA

###### Abstract

We develop solvable models of large-$N$ hybrid quantum circuits on qubits and
fermions with long-range power-law interactions and continuous local
monitoring, which provide analytical access to the entanglement phase diagram
and error-correcting properties of many-body entangled non-equilibrium states
generated by such dynamics. In one dimension, the long-range coupling is
irrelevant for $\alpha>3/2$, where $\alpha$ is the power-law exponent, and the
models exhibit a conventional measurement-induced phase transition between
volume- and area-law entangled phases. For $1/2<\alpha<3/2$ the long-range
coupling becomes relevant, leading to a nontrivial dynamical exponent at the
measurement-induced phase transition. More interestingly, for $\alpha<1$ the
entanglement pattern receives a sub-volume correction for both area-law and
volume-law phases, indicating that the phase realizes a quantum error
correcting code whose code distance scales as $L^{2-2\alpha}$. While the
entanglement phase diagram is the same for both the interacting qubit and
fermionic hybrid Brownian circuits, we find that long-range free-fermionic
circuits exhibit a distinct phase diagram with two different fractal entangled
phases.

Introduction.— Modern quantum technologies facilitate increasingly detailed
access to quantum phases of matter with complex patterns of many-body
entanglement [1, 2]. In particular, long-range interactions decaying with
distance as $r^{-\alpha}$, available in state-of-the-art experiments featuring
Rydberg atoms, trapped ions, and neutral atoms in optical cavities, are
capable of dramatically altering the dynamics of quantum information [3, 4, 5,
6, 7, 8, 9, 10] and rapidly generating complex many-body entanglement [11, 12,
13, 14, 15, 16, 17]. Our understanding of entanglement dynamics in generic
strongly-interacting many-body systems is still under rapid development. A
prime example is the recent discovery of robust phases of many-body
entanglement which survive for exponentially long times in hybrid quantum
circuits consisting of scrambling unitary evolution interspersed with repeated
local measurements [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 2, 46]. At low measurement
rates initially local information is dynamically encoded in many-body
entangled states which are robust to subsequent measurements, resulting in a
volume-law-entangled phase stabilized by a dynamically-generated quantum
error-correcting code (QECC) [24, 27, 31, 4, 48]. At higher measurement rates,
the many-body entanglement is destroyed, leading to an area-law-entangled
phase.

Figure 1: (a) Monitored large-$N$ models with long-range interactions. (b)
Entangled phases for Brownian spin and SYK4 models as a function of
measurement rate $\gamma/J$ and long-range exponent $\alpha$. $\hat{S}_{A}$ is
the quasi-Rényi entropy of a contiguous subsystem of volume $A$. (c) Dynamical
critical exponent $z$ and domain wall tension critical exponent $\nu$ vs
$\alpha$. (d) Error-correcting properties of the measurement-induced phases at
large $L$.

It is highly desirable to develop theoretical tools to easily estimate
entanglement properties of many-body states generated by strongly-interacting
quantum dynamics. In this work we develop exactly solvable models composed of
large-$N$ clusters of qubits or fermions in a 1D chain [Fig. 1(a)], for which
entanglement properties can be computed using path-integral techniques. In
particular, these methods provide simple pen-and-paper calculations for the
dynamics of many-body entanglement that can be immediately applied to problems
of experimental interest. We understand the entanglement phases in these
models in terms of a replica-symmetry-breaking transition of a corresponding
statistical mechanical system, and derive analytical expressions for
entanglement entropies, code properties of the QECC phase, and critical
properties of the phase transition. We summarize our primary findings in Figs.
1(b-d), including a phase diagram, critical properties at the phase
transition, and the error-correcting properties as a function of the
measurement rate and the long-range exponent $\alpha$.

Our analysis of these large-$N$ models leads to several new results and
insights. First, our models go beyond the standard set of tools – Clifford
circuits, matrix product states, and exact diagonalization – typically used to
study entanglement dynamics, and are able to provide an analytical mean-field
understanding of entanglement phases and measurement-induced phase
transitions. Second, the analytical control afforded by our models allows us
to derive new results, namely that long-range interactions generate novel sub-
region entanglement structure, leading to a non-trivial QECC phase with a
tunable sub-extensive code distance $L^{2-2\alpha}$ for $\alpha<1$, where $L$
is the number of clusters in the chain. The entanglement phase diagram we find
is thus even richer than previously known [49, 50, 51], and suggests a recipe
for constructing QECCs with enhanced code distance. Third, the models we study
are experimentally relevant as large-$N$ clusters naturally occur in cold atom
experiments, including ensembles of atomic spins coupled uniformly to optical
cavities [52, 53, 54, 55] and ensembles of Rydberg atoms clustered within a
single blockade radius [56, 57]. Fourth, our results highlight the crucial
role played by interactions in stabilizing the volume-law phase. We
demonstrate this point explicitly by studying a non-interacting circuit on
fermions with long-range hopping and find two distinct fractal entangled
phases [51], but no QECC phases. Lastly, although we focus here on the more
easily computable Rényi entropies, we also expect similar calculations to
allow for analytical calculations of von-Neumann entropies using an
appropriate replica limit [58].

Setup.— Here we consider a system $Q$ of particles (qubits or fermions)
arranged into a 1D chain of $L$ clusters, each containing a large number $N$
of particles as shown in Fig. 2. The particles are subjected to two competing
dynamics: long-range Brownian unitary interactions $U(t)$ that rapidly
generate entanglement leading to a volume-law phase, and continuous weak
monitoring $M(t)$ that tends to destroy entanglement leading to an area-law
phase. To probe the transition between these two phases, we maximally entangle
the system $Q$ with a reference $S$ [Fig. 2(a)], and compute entanglement
Renyi entropies $\hat{S}_{A}^{(n)}$ of system subregions $A\subset Q$. We
shall show that these entropies can be readily computed using path-integral
techniques, leading to straightforward pen-and-paper estimates for salient
physical quantities including code properties and critical exponents.

For an analytical treatment, we focus on the quasi Rényi entropy [1] of the
reduced density matrix $\rho_{A}$ of a contiguous subsystem $A$ of $Q$,
$\hat{S}^{(n)}_{A}=-\log\frac{\mathbb{E}\mathrm{Tr}\left[\rho_{A}^{n}(t)\right]}{\mathbb{E}\mathrm{Tr}\left[\rho_{A}(t)\right]^{n}}$,
where $\mathbb{E}$ is a disorder average over measurement outcomes and circuit
realizations 111The relation between Rényi entropy, quasi Rényi entropy and
the von Neumann entropy is explained in Sec I of the Supplementary Materials
[62].. The denominator is necessary for normalizing the state generated by the
non-unitary evolution, and the ratio of averages is considered for ease of
calculation over the averaged Rényi entropy which involves averaging the ratio
of two multi-replica quantities.

Figure 2: (a) System $Q$ is maximally entangled with reference $R$ and evolves
under monitored dynamics $V(t)$. (b) Quasi Rényi-2 entropy represented as a
quantum circuit. (c) Brownian qubit chain with $L$ clusters, each composed of
$N$ qubits. (d) SYK4 model with two independent chains of length $L$ coupled
by measurement.

$\hat{S}^{(2)}_{A}$ involves tracing over two copies of density matrix
$\mathrm{Tr}\left[\rho^{2}(t)\right]$ with $\rho(t)=V(t)\rho V^{\dagger}(t)$,
which we can interpret in terms of time evolution
$\mathbf{V}(t)=V^{(1)}\otimes V^{(2)*}\otimes V^{(3)}\otimes V^{(4)*}$ on four
replicas $r=1,2,3,4$ [2, 6]. Here, $1,2$ ($3,4$) denote the first (second)
Rényi replica, and $1,3$ ($2,4$) denote forward (backward) time-evolution. In
the replicated Hilbert space, the quasi Rènyi entropy can be expressed as a
ratio of propagators,
$\exp\left(-\hat{S}^{(2)}_{A}\right)=\frac{\langle\left\langle\mathcal{S}_{A}\right|\mathbb{E}\mathbf{V}(t)\left|\mathcal{I}\right\rangle\rangle}{\langle\left\langle\mathcal{I}\right|\mathbb{E}\mathbf{V}(t)\left|\mathcal{I}\right\rangle\rangle}$,
for appropriately defined initial and final states
$|\mathcal{I}\rangle\rangle$, $|\mathcal{S}_{A}\rangle\rangle$ in the
replicated Hilbert space [62]. The numerator has twisted boundary conditions
$|\mathcal{S}_{A}\rangle\rangle$ which swap replicas $1,3$ at the final time
$t=T$ [Fig. 2(b)], reflecting the SWAP test in the trace
$\mathrm{Tr}\left[\rho_{A}^{2}\right]$ [63].

We will construct analytically-tractable models for qubits and fermions for
which the propagator $\mathbb{E}\mathbf{V}(t)$ can be expressed as a large-$N$
path integral with classical action $NI$ that is amenable to saddle point
analysis. Consequently, the quasi Rényi entropy is simply proportional to the
difference between the large-$N$ saddle-point actions with and without the
SWAP boundary condition [2, 6, 8], $\hat{S}^{(2)}_{A}\propto
N\left(I_{\text{SWAP}}-I\right).$

Hybrid Brownian circuit on qubits.—We first consider a system of qubits
$\mathcal{S}_{r,i,\alpha}$ residing in clusters $r$ and labeled by an intra-
cluster index $i$ and spin component $\alpha=x,y,z$. During each time step
$\Delta t$, the qubits evolve under a two-body scrambling unitary matrix
$U(t)=\exp{[-iH(t)\Delta t/2]}$ for half the duration, and under one-body weak
measurement for the rest of the time. The unitary is generated by

$\displaystyle
H(t)=\sum_{\begin{subarray}{c}r,i<j,\alpha\beta\end{subarray}}J_{ij\alpha\beta}(t)\mathcal{S}_{ri\alpha}\mathcal{S}_{rj\beta}$
$\displaystyle+\sum_{\begin{subarray}{c}r^{\prime}\neq
r,ij\alpha\beta\end{subarray}}\tilde{J}^{rr^{\prime}}_{ij\alpha\beta}(t)\mathcal{S}_{ri\alpha}\mathcal{S}_{r^{\prime}j\beta},$
(1)

with intra-cluster coupling $J_{ijuv}$, and inter-cluster coupling
$\tilde{J}^{rr^{\prime}}_{ij\alpha\beta}$ between spins at site $r$ with spins
at $r^{\prime}$ [Fig. 2(c)]. For the weak measurement, a random one-body
operator at the given time,
$\mathcal{O}(t)=\sum_{i,\alpha}n_{i}^{\alpha}(t)\mathcal{S}_{i\alpha}$ is
coupled to an auxiliary qubit, which is then projectively measured followed by
post-selection [65]. This results in a deterministic evolution of the state
with a non-unitary operator $M(t)\sim\left(1-\frac{1}{2}\mathcal{O}\Delta
t-\frac{1}{8}\mathcal{O}^{2}\Delta t^{2}+\cdots\right)$. The system evolves
under a circuit constructed by stacking alternating layers of $U(t)$ and
$M(t)$ gates, $V(t)\equiv\prod_{t=0}^{T}M(t)U(t)$.

We consider Brownian Gaussian couplings and measurements with zero mean and
variance,
$\mathbb{E}\left[n(t)n(t^{\prime})\right]\sim\delta_{tt^{\prime}}/\Delta t$,
$\mathbb{E}\left[J(t)J(t^{\prime})\right]\sim\delta_{tt^{\prime}}/(N\Delta t)$
and $\mathbb{E}\left[\tilde{J}(t)\tilde{J}(t^{\prime})\right]\sim
gJ|r_{1}-r_{2}|^{-2\alpha}\delta_{tt^{\prime}}/(N\Delta t)$ [65]. Here we have
suppressed the indices of $n,J,\tilde{J}$ which should be considered to be
independent and random, and the factor of $N$ is introduced for a meaningful
large-$N$ limit. We also take the continuum time limit $\Delta t\rightarrow
0$. This model allows us to make analytical progress in accessing the quasi-
Rényi entropies.

We now introduce multiple replicas $u,v\in\\{1,2,3,4\\}$ of the system, and
use SWAP tests between the replicas to measure Rényi entropies [66, 67].
Averaging over the random couplings $J,\tilde{J},n$ introduces inter-replica
interaction, which in the large-N limit are mean-field couplings
$G^{uv}_{r}\sim 1/N\sum_{i}\mathcal{S}^{u}_{ri}\cdot\mathcal{S}^{v}_{ri}$
between replicas $u,v$. We decouple the $G$ fields by introducing Hubbard-
Stratonovich replica-fields $iF_{r}^{uv}$. The averaged circuit
$\mathbb{E}\mathbf{V}(t)$ can now be expressed as a path integral in these
fields [65],
$\mathbb{E}\mathbf{V}(t)=\int\mathcal{D}\left[iF_{r}^{uv}\right]\exp\left(-I\left[iF_{r}^{uv}\right]\right)$
with the action,

$\displaystyle\frac{I\left[iF_{r}^{uv}\right]}{N}=\sum_{r}\Big{[}\sum_{u<v}\int_{t}\big{(}(-1)^{u+v+1}\sum_{r^{\prime}}\mathcal{J}_{rr^{\prime}}\left(iF_{r}^{uv}\right)\left(iF_{r^{\prime}}^{uv}\right)$
$\displaystyle-\Gamma\left(iF_{r}^{uv}\right)\big{)}-\log\mathcal{K}_{r}\Big{]}.$
(2)

$\mathcal{K}_{r}\\!=\\!\mathrm{Tr}\left[\exp\left(\int_{t}\sum_{u<v}\frac{-iF_{r}^{uv}}{(\mathcal{S}+1)^{2}}(\mathcal{S}^{u}_{r}\cdot\mathcal{S}^{v}_{r})\right)\right]$
is the spin propagator which determines the bulk theory. The renormalized
interaction between different sites is long-range,
$\mathcal{J}_{rr^{\prime}}\sim|r-r^{\prime}|^{-2\alpha}$ [65], For specific
matrix elements like
$\langle\left\langle\mathcal{S}_{A}\right|\mathbb{E}\mathbf{V}(t)\left|\mathcal{I}\right\rangle\rangle$,
the spin propagator $\mathcal{K}_{r}$ has to be evaluated with fixed boundary
condition instead of the trace.

Replica permutation symmetry breaking.—The action (Entanglement Phases in
large-$N$ hybrid Brownian circuits with long-range couplings) is invariant
under the replica symmetry group $G=(S_{2}\times S_{2})\rtimes\mathbb{Z}_{2}$,
where the two inner $S_{2}\cong\mathbb{Z}_{2}$ denote permutations of the
forward and backward replicas $1\leftrightarrow 3$ and $2\leftrightarrow 4$
[2, 46]. The outer $\mathbb{Z}_{2}$ in the semidirect product is generated by
time-reversal $\mathcal{T}$ on four replicas followed by exchange of even and
odd replicas $1\leftrightarrow 2$, $3\leftrightarrow 4$. The boundary states
for the entropy $\hat{S}_{A}^{(2)}$ explicitly break $1\leftrightarrow 3$ (or
equivalently $2\leftrightarrow 4$) symmetry.

Saddle point analysis [2] of the bulk action in the mean field limit reveals a
phase transition at $\gamma_{c}=\frac{J}{9}\left(1+2g\zeta(2\alpha)\right)$
with the Riemann Zeta function
$\zeta(\alpha)\equiv\sum_{r=1}^{\infty}\frac{1}{r^{\alpha}}$, and an order
parameter field
$\phi\sim\frac{2}{3\sqrt{3}}\left(iF_{12}+iF_{34}-iF_{14}-iF_{23}\right)$. For
$\gamma>\gamma_{c}$ the saddle point is $\phi=0$, while for
$\gamma<\gamma_{c}$, $\phi$ is non-zero and comes in a pair,
$\phi\propto\pm\sqrt{\gamma_{c}-\gamma}$. The replica permutation
$1\leftrightarrow 3$ is equivalent to $\phi\leftrightarrow-\phi$ symmetry,
which is spontaneously broken for $\gamma<\gamma_{c}$. The Landau-Ginzburg
field theory is given by [68]

$\displaystyle\frac{I_{\text{eff}}}{N}=\int_{t,r}[-\phi\partial_{t}^{2}\phi-b\int^{\prime}_{s}\frac{\phi_{r}\phi_{s}}{|r-s|^{2\alpha}}-\phi\partial_{r}^{2}\phi-\frac{\delta}{2}\phi^{2}+\frac{\
\phi^{4}}{4}],$ (3)

which for $2\alpha>1$ has a phase transition effected by the mass term
$\delta\propto\gamma_{c}-\gamma$, with $\delta>(<)0$ being the symmetry-broken
(symmetric) phase, and the $\mathbb{Z}_{2}$ phase transition occurring at
$\delta\to 0$. The long-range term appears with a regulated integral:
$\int^{\prime}_{s}$ to be read as
$\int_{\mathbb{R}\backslash(r-\varepsilon,r+\varepsilon)}ds$, with an
ultraviolet cut-off $\varepsilon$. For $b\neq 0$, this is the long wavelength
theory of an anisotropic long range Ising model [69, 70, 71] in 2D, where the
interaction is long-range along space and short-range along time. For
$2\alpha<1$ the power law contribution diverges, so $J$ must be scaled with
$L$ to take the thermodynamic limit: the system behaves as a single all-to-all
cluster with $NL$ qubits without any volume to area-law transition. There is
however a phase transition in the purification times for the system and its
parts [2].

For $2\alpha\\!>\\!3$ the long-range term is irrelevant and the transition is
governed by the short-ranged fixed point. For $2\alpha\\!<\\!3$, the
underlying Ising model is anisotropic with a non-trivial dynamical critical
exponent,

$\displaystyle z=\begin{cases}1&2\alpha>3\\\
\frac{2\alpha-1}{2}&1<2\alpha<3.\end{cases}$ (4)

In this regime we have two distinct correlation lengths
$\xi_{t}\sim\delta^{-\frac{1}{2}}$ and $\xi_{r}\sim\delta^{-\frac{1}{2z}}$
corresponding to the time-like and space-like directions, respectively.

Figure 3: Domains and domain walls in the anisotropic Ising model
corresponding to the quasi-entropy $\hat{S}_{A}^{(2)}$ of a small subregion
$A$ in the ferromagnetic phase (a, black $\phi>0$ and white $\phi<0$ are
symmetry-broken domains separated by a domain wall), and the paramagnetic
phase $\phi=0$ (b, light gray). The entropy of the complement subregion
$A^{c}$ corresponds to one of two possible competing domain-wall
configurations (c,d).

Entanglement phases.— Using the effective action (3) we now calculate the
quasi entropy of $\rho_{A}(t)$ for a contiguous region $A\subset Q$. The
twisted boundary condition in $I_{\text{SWAP}}$ corresponds to pinning
$\phi>0$ within the subregion $A$ at the future time boundary $t=T$, and
pinning $\phi<0$ on all other boundaries [Fig. 3(a,b)]. In the symmetry-broken
phase $\delta>0$ the bulk organizes into domains separated by a domain wall
[Fig. 3(a)] such that the bulk field is positive $\phi>0$ (black) within a
temporal correlation length $\xi_{t}$ of the final time-boundary of the
subregion $A$ and negative $\phi<0$ (white) throughout the remainder of the
bulk. In the symmetric phase $\delta<0$ only the trivial bulk saddle point
$\phi=0$ [Fig. 3(b), light gray] contributes, and therefore this pinning
effect is only relevant very close to the final time boundary . For $A\ll L$
the excess energy $I_{\text{SWAP}}-I$ of this configuration compared to the
configuration without the twisted boundary condition is simply given by the
energy cost of the power-law interaction acting along a spatial slice of
height $\xi_{t}$:

$\displaystyle J\xi_{t}\int^{\prime}_{r\in A,s\in
A^{c}}drds\frac{1}{|r-s|^{2\alpha}}\sim J\xi_{t}A^{2-2\alpha}+\text{const.}$
(5)

In the symmetry broken phase $\delta>0$ [72], the twisted boundary condition
at $t=T$ supports a bulk domain wall with spatial extent $A$ and time-like
height $\xi_{t}$. The space-like part of the domain wall with spatial extent
$A$ has a domain wall tension
$\sigma_{r}\sim\xi_{t}\left(\phi/\xi_{t}\right)^{2}\propto\delta^{\frac{3}{2}}$.
The time-like part of the domain wall also contributes an energetic term (sub-
extensive in $A$) arising from the power-law just like (5). Combining these
results together we have 222In the absence of power-law terms, we obtain $1/N$
logarithmic corrections to the entropy in the symmetry-broken phase for the
local model, which arise from the fluctuations of the domain wall [4,
SM_secIV].,

$\displaystyle\frac{\hat{S}^{(2)}_{A}}{N}\sim\begin{cases}cJ\xi_{t}A^{2-2\alpha}&\text{
for }\gamma>\gamma_{c}\\\
J\left(\sigma_{r}A+c\xi_{t}A^{2-2\alpha}\right)&\text{ for
}\gamma<\gamma_{c}.\end{cases}$ (6)

We emphasize that the power-law correction to the entropy in the volume-law
(symmetry-broken) phase is a novel result, which leads to enhanced error-
correcting properties for long-range hybrid circuits.

Error correcting properties.— $\hat{S}^{(2)}_{R}$ in the volume-law phase
$\gamma<\gamma_{c}$ can be understood to be the ‘rate’ of the QECC, which
refers to the amount of logical information of $R$ that is encoded in $Q$ and
protected from ‘errors’ due to measurements with a Code Rate
$\sim\hat{S}^{(2)}_{R}\sim\sigma NL$. The mutual information
$\hat{I}^{(2)}(A:R)=\hat{S}^{(2)}_{A}+\hat{S}^{(2)}_{R}-\hat{S}^{(2)}_{A^{c}}$
between a subregion A and the reference R is related to the contiguous ‘code
distance’, which refers to the size of the largest contiguous subsystem of $Q$
whose deletion would not spoil the encoded information of $R$ [4]333This
statement comes with two important caveats. First, the theorem proved in [4]
rigorously applies only to Clifford circuits (stabilizer states), whereas here
our Brownian circuit elements clearly take the quantum state outside the
Clifford group. Second, to make meaningful information-theoretic arguments one
must ultimately work with the disorder-averaged von Neumann entropy
$-\mathbb{E}[\mathrm{Tr}\left[\rho_{A}\log\rho_{A}\right]]$. From (6) we have
$\hat{S}^{(2)}_{A}\sim N\left(\sigma
A+c\xi_{t}A^{\upsilon}\right)+\mathcal{O}\left(1\right)$ and
$\hat{S}^{(2)}_{R}\sim N\sigma L+\mathcal{O}\left(1\right)$, where
$\upsilon\equiv 2-2\alpha$. The quasi entropy $\hat{S}^{(2)}(A^{c})$ is the
minimum of two configurations in Fig. 3(c,d) [4, 48],

$\displaystyle\hat{S}^{(2)}_{A^{c}}\sim\min\left\\{\hat{S}^{(2)}_{A}+\hat{S}^{(2)}_{R},N\left(\sigma(L-A)+c\xi_{t}(L-A)^{\upsilon}\right)\right\\}.$

The cross-over between the two occurs for a critical subregion size
$A^{*}\sim\frac{1}{2\sigma}L^{\upsilon}+\mathcal{O}(N^{-1})$. Thus, for
$A<A^{*}$ we have $\hat{I}^{(2)}(A:R)\approx 0$, and $NA^{*}$ can be
identified as a power-law code distance,

$\displaystyle\text{`Code Distance'}\sim NL^{\upsilon}\text{ for
}\gamma<\gamma_{c}.$ (7)

The distance can be tuned with the long-range exponent $\alpha$, and is sub-
linear but scales favorably with $L$ for $2\alpha<2$. For $2\alpha>2$, the
code distance is $1/N$ suppressed and scales as $\log L$ in our model [72].

Monitored SYK chain.— We now turn to the study of the effects of long-range
couplings on the fermionic monitored Brownian Sachdev-Ye-Kitaev (SYK) chain
circuit introduced in [6]. This allows us to separately consider the effects
of the long-range coupling and the interactions, which highlight the role of
interactions on the entanglement properties of the states generated by hybrid
circuits. The setup contains a left ($\mathcal{L}$) and a right
($\mathcal{R}$) chain with $L$ clusters of $N$ Majorana fermions each [75, 76,
6] [Fig. 2(d)], with chain undergoing intermittent unitary evolution and
monitoring. The unitary evolution is generated by inter-cluster long-range
two-fermion of strength $J_{ij}^{rr^{\prime}}\sim|r-r^{\prime}|^{-\alpha}$ and
on-site four-fermion interaction of strength $U_{ijkl}^{r}$, which are both
independent Brownian variables for each chain. We study the free-fermion limit
$U\to 0$ which allows us to consider the effects of long-range hopping and an
on-site interaction separately. The $\mathcal{L}$ and $\mathcal{R}$ chains are
coupled by a inter-chain parity measurement for each site, described by Kraus
operators,
$\\{M_{1}^{r,i},M^{r,i}_{2}\\}\\!=\\!\left\\{\pi^{-}_{r,i}+\sqrt{1-s^{2}}\pi^{+}_{r,i},s\pi^{+}_{r,i}\right\\}$,
where $\pi^{\pm}_{x,i}=\frac{1}{2}(1\mp i2\psi_{r,L,i}\psi_{r,R,i})$. $s$
denotes the measurement strength, chosen as [77] $s=\sqrt{\gamma\Delta t}$,
with $\Delta t\rightarrow 1$ keeping $\gamma$ fixed. It is convenient to
define $\hat{J}=J\zeta(2\alpha)$ and $\tilde{\gamma}=\gamma/\hat{J}$ and
$\tilde{U}=U/\hat{J}$.

Figure 4: Phases of long-range monitored SYK2 model.

Landau-Ginzburg theory.—The theory for quasi-Rényi entropy for the free-
fermionic case ($U=0$) is invariant under the replica symmetry group
$(O(2)\times O(2))\rtimes\mathbb{Z}_{2}$ [78], where the two $O(2)$
transformations rotate the $1,3$ and $2,4$ contours, respectively. This
continuous symmetry (in contrast to the discrete symmetry for the qubits) is
spontaneously broken for $\gamma\\!<\\!\hat{J}$, resulting in a Goldstone mode
$\theta_{r}$ [79] corresponding to the relative $O(2)$ rotation angle at site
$r$. The effective theory of the Goldstone mode can be derived [78],

$\displaystyle\frac{I_{\text{eff}}}{N}=\frac{\rho}{2}\sum_{k}\int_{\Omega}\left(\frac{\Omega^{2}}{\gamma^{2}}+(1-\epsilon_{k})\right)|\theta_{k}(\Omega)|^{2},$
(8)

where $\theta_{k}=\frac{1}{\sqrt{L}}\sum_{r}\theta_{r}e^{-ikr}$ and
$\epsilon_{k}\equiv\frac{1}{\zeta(2\alpha)}\sum_{r=1}^{\infty}\frac{\cos
kr}{r^{2\alpha}}$ results from the power-law hopping, similar to [8], and the
stiffness $\rho=\hat{J}(1-\tilde{\gamma}^{2})$ vanishes at $\tilde{\gamma}=1$,
indicating a phase transition at $\gamma=\hat{J}$. The long-range couplings
lead to a nontrivial dynamical exponent as before (4).

In the symmetry-broken phase $(\gamma/\hat{J}<1)$, the boundary condition pins
the angle $\theta=\pi/2$ in subsystem $A$, and $\theta=0$ in $A^{c}$ [8],
equivalent to creating half-vortices at the left and right boundaries of $A$.
$\hat{S}^{(2)}_{A}/N$ can be mapped to the correlation function of a vortex
creation operator and scales as $\log A$ for $\alpha>3/2$ (long-range coupling
is irrelevant and we get logarithmic free energy for vertices), and as
$A^{3/2-\alpha}$ for $1/2<\alpha<3/2$ [78]. In the symmetric phase same
analysis as (5) leads to a distinct fractal phase when $\alpha<1$ [Fig. 4].
The free energy of vortices is proportional to the stiffness and its critical
exponent is simply $\nu=1$ for all $\alpha>1/2$. When we turn on interactions
$U>0$, the replica symmetry is reduced to a discrete group
$(C_{4}\\!\times\\!C_{4})\\!\rtimes\\!\mathbb{Z}_{2}$. There is consequently a
$Z_{4}$ symmetry-breaking transition which reproduces the same entanglement
and error-correcting phase diagram as the spin model as shown in Fig. 1(b).

Concluding remarks.—The phase diagram in Fig. 1 can be readily generalized to
Brownian chains in higher dimensions, and demonstrates an entanglement
transition for all $2\alpha>d$, with non-trivial dynamical exponent
$z=(2\alpha-d)/2$ for $d<2\alpha<d+2$. The subextensive correction arising
from (5) is $A^{2d-2\alpha}$. It will be interesting for future studies to
study the Rényi entropy for general $n$ and the entanglement entropy at
$n\rightarrow 1$, and quantify the $1/N$ effects on the phase diagram.

Acknowledgement.— SS is supported in part by AFOSR under Award FA9550-17-1-
0180. SKJ and BGS are supported by the Simons Foundation via the It From Qubit
Collaboration. GB is supported by the DOE GeoFlow program (DE-SC0019380). BGS
is supported in part by the AFOSR under grant number FA9550-19-1-0360. SS and
SKJ have contributed equally to the paper.

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Supplementary Materials for “Entanglement Phases in large-$N$ hybrid Brownian
circuits with long-range couplings”

##  I Setup of the Rényi entropy calculation

In the main part of the paper we focus on a 1-dimensional lattice of size L
with periodic boundary conditions, however the results can be readily extended
to higher dimensions. For the spin model, each spin operator is labeled as
$\mathcal{S}_{ri\alpha}$, with $i$ and $\alpha$ denoting the intra-cluster
label and spin direction respectively, while $r$ refers to the position label
of the cluster in the chain, and a similar labeling applies to the fermions.

The dofs in Q (vectors in the Hilbert space $\mathcal{H}$) undergo a non-
unitary random evolution $V(t)$. The randomness comes from the Brownian nature
of the unitary evolution that we consider in our setup. Under this evolution,
a density matrix (when viewed as vector in the doubled Hilbert space
$\left|\rho\right\rangle\rangle$) evolves as,
$\left|\rho\right\rangle\rangle\to V(t)\otimes
V^{*}(t)\left|\rho\right\rangle\rangle$, which generates an unnormalized state
because of the non-unitary evolution. We want to study the entanglement
properties of the subsystem $A$ for this state, a particular diagnostic of
which is the Rényi-n entropy of the normalized reduced density matrix
$\rho_{A}$,

$\displaystyle
S^{(n)}_{A}=\frac{1}{1-n}\mathbb{E}\log\frac{\mathrm{Tr}\left[\rho_{A}^{n}(t)\right]}{\mathrm{Tr}\left[\rho_{A}(t)\right]^{n}},$
(9)

where $\mathbb{E}$ refers to the averaging over the realizations of the random
circuit $V(t)$.

Estimating the actual Rényi-2 entropy averaged over the randomness requires
taking a non-trivial replica limit since this is an average of a ratio of two
multi-replica quantities. Instead, we will calculate the quasi Rényi-2 entropy
[1], which is the ratio of the averages,

$\displaystyle\hat{S}^{(n)}_{A}=\frac{1}{1-n}\log\frac{\mathbb{E}\mathrm{Tr}\left[\rho_{A}^{n}(t)\right]}{\mathbb{E}\mathrm{Tr}\left[\rho_{A}(t)\right]^{n}}$
(10)

This quantity, while distinct from the disorder averaged Rényi-n entropy, is
easier to calculate. In this work we focus on estimating $\hat{S}^{(2)}_{A}$,
for which it is convenient to consider the dynamics in 4 replicas of the
Hilbert space $\mathcal{H}^{(4)}$. We use a $1,2,3,4$ notation: $1,2$ ($3,4$)
denote the first (second) replica, and $1,3$ ($2,4$) denote the forward
(backward) evolution. In $\mathcal{H}^{(4)}$ the time evolution is given by
$\mathbf{V}(t)=V^{(1)}(t)\otimes V^{(2)*}(t)\otimes V^{(3)}(t)\otimes
V^{(4)*}(t)$, and the normalization factor is given by
$\mathcal{N}(\rho)=\sqrt{\langle\left\langle\mathcal{I}\right|\mathbf{V}(t)\left|\rho\right\rangle\rangle\left|\rho\right\rangle\rangle}$.
To define $\left|\mathcal{I}\right\rangle\rangle\in\mathcal{H}^{(4)}$, we
first define two normalized states,

$\displaystyle\left|+\right\rangle\rangle$
$\displaystyle\sim\sum_{ab=0,1}\left|aabb\right\rangle\rangle$ (11)
$\displaystyle\left|-\right\rangle\rangle$
$\displaystyle\sim\sum_{ab=0,1}\left|abba\right\rangle\rangle,$ (12)

which leads to the definition,
$\left|\mathcal{I}\right\rangle\rangle=\bigotimes_{i,r}\left|\mathcal{+}\right\rangle\rangle$.
This definition ensures that the normalization factor is the usual norm of a
state. In our setup, the system dofs in $Q$ are initially maximally entangled
with $NL$ qubits in a reference $R$. The corresponding initial state for Q in
$\mathcal{H}^{(4)}$ is
$\left|\rho\right\rangle\rangle\otimes\left|\rho\right\rangle\rangle=\left|\mathcal{I}\right\rangle\rangle$.

We further define a SWAP state in $\mathcal{H}^{(4)}$ as,

$\displaystyle\left|\mathcal{S}_{A}\right\rangle\rangle=\bigotimes_{i,r\in
A}\left|-\right\rangle\rangle\bigotimes_{i,r\in
A^{c}=Q-A}\left|+\right\rangle\rangle,$ (13)

and the quasi Rényi-2 entropy is given by,

$\displaystyle\exp\left(-\hat{S}^{(2)}_{A}\right)=\frac{\langle\left\langle\mathcal{S}_{A}\right|\mathbb{E}\mathbf{V}(t)\left|\mathcal{I}\right\rangle\rangle}{\langle\left\langle\mathcal{I}\right|\mathbb{E}\mathbf{V}(t)\left|\mathcal{I}\right\rangle\rangle}.$
(14)

The quasi-entropy can be simulated in a quntum experiment with extra classical
post-processing and no extra quantum resources over estimating the usual Rényi
entropy [2].

###  I.1 Relation between the quasi Rényi, Rényi, and von Neumann entropies

One can show that the Rényi entropy and the quasi Rényi entropy can be treated
as a part of the same family of generalized entropic quantities. Consider the
generalized function,

$\displaystyle\chi^{(nm)}_{A}=\frac{1}{m(1-n)}\log\frac{\mathbb{E}\left(\mathrm{Tr}\left[\rho_{A}^{n}\right]\right)^{m}}{\mathbb{E}\left(\mathrm{Tr}\left[\rho_{A}\right]\right)^{nm}}$
(15)

One can show,

$\displaystyle\chi^{(nm)}_{A}\big{|}_{m\to 0}$ $\displaystyle=S^{(n)}_{A}$
(16) $\displaystyle\chi^{(nm)}_{A}\big{|}_{m\to 1}$
$\displaystyle=\hat{S}^{(n)}_{A}.$ (17)

Intriguingly, one can also extract the von Neumann entropy directly from the
quasi Rényi entropy. First let us define the normalized density matrix,
$\tilde{\rho}_{A}=\rho_{A}/\mathrm{Tr}\left[\rho_{A}\right]$. The probability
of each trajectory is given by $\mathrm{Tr}\left[\rho_{A}\right]$. The
trajectory averaged von-Neumann entropy is given by,

$\displaystyle\overline{S_{A}}=-\mathbb{E}\big{[}\mathrm{Tr}\left[\rho_{A}\right]\mathrm{Tr}\left[\tilde{\rho}_{A}\log{\tilde{\rho}_{A}}\right]\big{]}.$
(18)

We find that the trajectory averaged von-Neumann entropy is the $n\to 1$ limit
of the quasi-Renyi entropy,

$\displaystyle\hat{S}^{(n\to 1)}_{A}=\overline{S_{A}}.$ (19)

Let us sketch this proof.

$\displaystyle\overline{S_{A}}$
$\displaystyle=-\mathbb{E}\big{[}\mathrm{Tr}\left[\rho_{A}\log{\rho_{A}}\right]-\mathrm{Tr}\left[\rho_{A}\right]\log{\mathrm{Tr}\left[\rho_{A}\right]}\big{]}$
$\displaystyle=-\partial_{n}\mathbb{E}\big{[}\mathrm{Tr}\left[\rho_{A}^{n}\right]-\left(\mathrm{Tr}\left[\rho_{A}\right]\right)^{n}\big{]}\bigg{|}_{n\to
1}$
$\displaystyle=-\partial_{n}\left[\frac{\mathbb{E}\mathrm{Tr}\left[\rho^{n}\right]}{\mathbb{E}\left(\mathrm{Tr}\left[\rho_{A}\right]\right)^{n}}\right]\bigg{|}_{n\to
1}\text{ (using $\mathbb{E}\mathrm{Tr}\left[\rho_{A}\right]=1$)}$
$\displaystyle=\frac{1}{1-n}\log\frac{\mathbb{E}\mathrm{Tr}\left[\rho^{n}\right]}{\mathbb{E}\left(\mathrm{Tr}\left[\rho_{A}\right]\right)^{n}}\bigg{|}_{n\to
1}$ $\displaystyle=\hat{S}^{(n\to 1)}_{A}.$

Hence we find that although the quasi Rényi and the averaged Rényi entropies
are different replica limits of a generalized entropy function, a particular
replica limit of the quasi Rényi entropy can access the physical von Neumann
entropy of the state generated by the hybrid circuit.

##  II Path integral representation of the replicated dynamics

In this section of the Appendix we show how to derive the spin path integral
for the replicated dynamics and derive Eqs. 4,5 and 6 in the main text.

###  II.1 Measurement model in the Brownian spin model

A random 1-body operator at a given time,
$\mathcal{O}(t)=\sum_{i,\alpha}n_{i}^{\alpha}(t)\mathcal{S}_{i\alpha}$ is
coupled to an auxiliary qubit via the interaction,

$\exp{\left[-i\frac{\Delta
t}{2}\mathcal{O}(t)\sigma^{x}_{\mathrm{aux}}\right]}\left|\Psi\right\rangle\left|0\right\rangle_{\mathrm{aux}},$
(20)

with $\left|\Psi\right\rangle$ being the state of the system. After this
coupling, the auxiliary qubit is measured in the $\sigma^{y}_{\mathrm{aux}}$
basis, and only $+1$ results are post-selected. Under this dynamics, the
system deterministically evolves with a non-unitary evolution operator $M(t)$,

$\displaystyle\left|\Psi\right\rangle\to
M(t)\left|\Psi\right\rangle=\left(1-\frac{1}{2}\mathcal{O}\Delta
t-\frac{1}{8}\mathcal{O}^{2}\Delta
t^{2}+\cdots\right)\left|\Psi\right\rangle.$ (21)

###  II.2 Integrating out disorder and spin path integral

The Brownian disorders are explicitly given by,

$\displaystyle\mathbb{E}\left[n_{i}^{\alpha}(t)n_{j}^{\beta}(t^{\prime})\right]=\frac{\gamma}{(\mathcal{S}+1)^{2}}\frac{\delta_{tt^{\prime}}}{\Delta
t/2}\delta_{ij}\delta^{\alpha\beta},$
$\displaystyle\mathbb{E}\left[J_{ij\alpha\beta}(t)J_{kl\mu\rho}(t^{\prime})\right]=\frac{J}{N(\mathcal{S}+1)^{4}}\frac{\delta_{tt^{\prime}}}{\Delta
t/2}\delta_{ik}\delta_{jl}\delta_{\alpha\mu}\delta_{\beta\rho},$ (22)
$\displaystyle\mathbb{E}\left[\tilde{J}^{r_{1}r_{2}}_{ij\alpha\beta}(t)\tilde{J}^{r^{\prime}_{1}r^{\prime}_{2}}_{kl\mu\rho}(t^{\prime})\right]=\frac{gJ_{r_{1}r_{2}}}{N(\mathcal{S}+1)^{4}}\frac{\delta_{tt^{\prime}}}{\Delta
t/2}\delta_{r_{1}r_{1}^{\prime}}\delta_{r_{2}r_{2}^{\prime}}\delta_{ik}\delta_{jl}\delta_{\alpha\mu}\delta_{\beta\rho}.$

We want to write down a path integral expression for the averaged circuit
evolution in $n$ replicated copies of the physical system, which includes $2n$
copies of the circuit (counting the time-reversed copies). This quantity,

$\displaystyle\mathbf{V}(t)=\mathbb{E}\left[V^{(1)}(t)\otimes
V^{(2)*}(t)\otimes\ldots V^{(2n-1)}(t)\otimes V^{(2n)*}(t)\right],$ (23)

can give us access to the $n$-th Rényi entropies, and in particular, the $n=2$
result corresponds to the Rényi-2 entropy that we consider in this paper. Let
$u,v$ denote the replica index. The time evolution operator in any given copy
is generated by an interaction Hamiltonian and a non-unitary evolution
generated by a weak measurement followed by post-selection, as described in
the main text. Schematically this can be expressed as,

$\displaystyle V^{(u)}(\Delta t)\sim\exp\left[-\frac{i\Delta
t}{2}\left(H^{(u)}(t)-i\mathcal{O}^{(u)}(t)\right)\right],$ (24)

where the $H^{(u)}$ and $\mathcal{O}^{(u)}$ are the Hamiltonian and the
measured operator (in replica $u$) respectively. These operators are taken to
be Brownian in our model, which implies that the time evolution is
uncorrelated in the time direction. However the randomness is same for each
replica, which implies that in $\mathbf{V}$ different replicas become
correlated when the disorder is integrated away, using the Gaussian nature of
the Brownian variables defined in Eq. 3 of the main text. One way to derive
this is by expanding the evolution in each time step in the product in (24) to
second order in $\Delta t$, and collecting terms like
$\mathbb{E}\left[H^{(u)}H^{(v)}\right]$,
$\mathbb{E}\left[H^{(u)}H^{(u)}\right]$ or
$\mathbb{E}\left[\mathcal{O}^{(u)}\mathcal{O}^{(v)}\right]$ together. This
procedure is discussed in detail in the Appendix B of [2]. Exponentiating
back, one obtains that $\mathbf{V}(\Delta t)\sim\exp\left(-NI_{n}(t)\Delta
t\right)$. The variances of the random terms have been scaled in a way that
the exponential comes with a prefactor of $N$. Dropping some constant terms
(which arise from the intra-replica terms) $I_{n}(t)$ is given by,

$\displaystyle I_{n}(t)$ $\displaystyle=\Bigg{[}\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}(-1)^{u+v}\frac{J}{4(\mathcal{S}+1)^{4}}\left(\frac{1}{N}\sum_{i}\mathcal{S}_{ri}^{u}\cdot\mathcal{S}_{ri}^{v}\right)^{2}$
$\displaystyle+\sum_{\begin{subarray}{c}r\neq r^{\prime}\\\
u<v\end{subarray}}(-1)^{u+v}\frac{gJ_{rr^{\prime}}}{2(\mathcal{S}+1)^{4}}\left(\frac{1}{N}\sum_{i}\mathcal{S}_{ri}^{u}\cdot\mathcal{S}_{ri}^{v}\right)\left(\frac{1}{N}\sum_{i}\mathcal{S}_{r^{\prime}i}^{u}\cdot\mathcal{S}_{r^{\prime}i}^{v}\right)$
$\displaystyle-\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}(-1)^{u+v}\frac{\gamma}{2(\mathcal{S}+1)^{2}}\left(\frac{1}{N}\sum_{i}\mathcal{S}_{ri}^{u}\cdot\mathcal{S}_{ri}^{v}\right)\Bigg{]}.$
(25)

In this expression the spin operator $\mathcal{S}$ refers to the spin operator
at time $t$.

We can now stack $\mathbf{V}_{u}(\Delta t)$ at different times by repeating
sequence of the Brownian interactions and measurements, insert resolutions of
the identity between each layer, and take the limit $\Delta t\rightarrow 0$
with $T$ fixed to express $\mathbb{V}(t)$ as a path integral over $2nN$ unit-
norm $\mathrm{SO}(3)$ spins $\mathbf{\mathcal{S}}_{ri}^{u}$, using spin
coherent states as the basis. The completeness relation for the coherent
states for a single spin is given by,

$\mathbb{I}=\int\frac{2\mathcal{S}+1}{4\pi}d\mathbf{\Omega}_{i}\left|\mathbf{\Omega}_{i}\right\rangle\left\langle\mathbf{\Omega}_{i}\right|.$
(26)

To turn the spins into coherent states, we use the upper symbols for single
spin-$S$ Pauli operators [3],

$\displaystyle\mathcal{S}_{i\alpha}$
$\displaystyle=\int\frac{2\mathcal{S}+1}{4\pi}d\mathbf{\Omega}_{i}\left|\mathbf{\Omega}_{i}\right\rangle\left\langle\mathbf{\Omega}_{i}\right|(\mathcal{S}+1)\Omega_{i\alpha}$
(27) $\displaystyle\left(\mathcal{S}_{i\alpha}\right)^{2}$
$\displaystyle=\int\frac{2\mathcal{S}+1}{4\pi}d\mathbf{\Omega}_{i}\left|\mathbf{\Omega}_{i}\right\rangle\left\langle\mathbf{\Omega}_{i}\right|\left[(\mathcal{S}+1)\left(\mathcal{S}+\frac{3}{2}\right)\left(\Omega_{i\alpha}\right)^{2}-\frac{\mathcal{S}+1}{2}\right],$
(28)

and introduce a measure for the coherent spin states in the path integral,

$\mathcal{D}\Omega_{ri}^{u}=\prod_{t_{n}}\frac{2\mathcal{S}+1}{4\pi}d\mathbf{\Omega}^{u}_{r,i,t_{n}}\langle\mathbf{\Omega}^{u}_{r,i,t_{n+1}}|\mathbf{\Omega}^{u}_{r,i,t_{n}}\rangle,$
(29)

with implicit time dependent terms.

In terms of the coherent states $\mathbb{V}(t)$ is given by a path integral,

$\displaystyle\mathbf{V}(t)=\mathbb{E}\left[V\otimes
V_{\mathcal{T}}\otimes\cdots\otimes V\otimes
V_{\mathcal{T}}\right]=e^{-NI[\mathbf{\Omega}]}$ $\displaystyle
I[\mathbf{\Omega}]=\int_{0}^{T}dt$
$\displaystyle\Bigg{[}\frac{J}{4}\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}(-1)^{u+v}\left(\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}\right)^{2}\
$ $\displaystyle+\sum_{\begin{subarray}{c}r\neq r^{\prime}\\\
u<v\end{subarray}}(-1)^{u+v}\frac{gJ_{rr^{\prime}}}{2}\left(\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}\right)\left(\frac{1}{N}\sum_{i}\mathbf{\Omega}_{r^{\prime}i}^{u}\cdot\mathbf{\Omega}_{r^{\prime}i}^{v}\right)$
$\displaystyle-\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}(-1)^{u+v}\frac{\gamma}{2(S+1)^{2}}\left(\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}\right)\Bigg{]}$
(30)

To decouple the non-linear interactions in
$\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}$, we introduce Hubbard
Stratonovich type fields which couple different replicas $F_{uv}^{(r)}$ and
$G_{uv}^{(r)}$, satisfying the operator identity,

$1=\left(\prod_{r,u<v}\int\mathcal{D}F^{uv}_{r}\mathcal{D}G^{uv}_{r}\right)\exp\left[iN\int_{0}^{T}dtF^{uv}_{r}\left(G^{uv}_{r}-\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}\right)\right].$
(31)

The action can thus be re-written as,

$\displaystyle I[F,G,\mathbf{\Omega}]$ $\displaystyle=\int
dt\Bigg{[}\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}\frac{(-1)^{u+v}J}{4}\left(G^{uv}_{r}\right)^{2}+\sum_{\begin{subarray}{c}r\neq
r^{\prime}\\\
u<v\end{subarray}}\frac{(-1)^{u+v}gJ_{rr^{\prime}}}{2}\left(G^{uv}_{r}\right)\left(G^{uv}_{r^{\prime}}\right)-\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}\frac{(-1)^{u+v}\gamma}{2}\left(G^{uv}_{r}\right)$
$\displaystyle-\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}\left(iF^{uv}_{r}G^{uv}_{r}\right)+\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}iF^{uv}_{r}\left(\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}^{v}_{ri}\right)\Bigg{]}$
(32)

Note that (II.2) is applicable in any general dimensions, where the position
label $r\in\mathbb{Z}^{d}$ can be identified as a $d$ dimensional vector, with
$L$ being the linear size along each dimension.

###  II.3 Field theory with periodic boundary condition

We can further simplify (II.2) by assuming periodic boundary condition and
going to momentum space. We also assume that the interaction $J_{rr^{\prime}}$
is translationally invariant and even, i.e. $J_{rr^{\prime}}\sim
J_{|r-r^{\prime}|}$.

We consider discrete d-dimensional cubic lattice in the space $\mathbb{Z}^{d}$
with $L$ being the linear extent of the cube. In the limit $L\gg 1$, the
momentum space domain is $k\in\\{0,2\pi\\}^{d}$. We use the following
schematic definitions,

$\displaystyle G_{k}=\sum_{r\in\mathbb{Z}^{d}}e^{-ik\cdot r}G_{r}$
$\displaystyle
G_{r}=\frac{1}{(2\pi)^{d}}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}dk_{1}\cdots
dk_{d}e^{ik\cdot r}G_{k}\equiv\int d\bar{}\hskip 1.00006pt^{d}ke^{ik\cdot
r}G_{k}$ $\displaystyle\frac{1}{(2\pi)^{d}}\sum_{r\in\mathbb{Z}^{d}}e^{ik\cdot
r}=\delta(k)\ ,\ \int d\bar{}\hskip 1.00006pt^{d}ke^{ikr}=\delta_{r,0}.$ (33)

We also introduce the notation,

$\displaystyle\Omega_{r}^{uv}=\frac{1}{N}\sum_{i}\mathbf{\Omega}_{ri}^{u}\cdot\mathbf{\Omega}_{ri}^{v}.$
(34)

In momentum space, we can rewrite Eq. II.2 can be rewritten as (dropping the
constant terms),

$\displaystyle I[F,G,\mathbf{\Omega}]$ $\displaystyle=\int dt\int
d\bar{}\hskip
1.00006pt^{d}k\sum_{\begin{subarray}{c}u<v\end{subarray}}\Bigg{[}\frac{(-1)^{u+v}}{4}\left(\hat{J}_{k}G_{k}^{uv}G_{-k}^{uv}-4\pi\gamma
G_{k}^{uv}\delta(k)\right)-iF_{k}^{uv}G_{-k}^{uv}+iF_{k}^{uv}\Omega^{uv}_{-k}\Bigg{]},$
$\displaystyle\ \text{where,
}\hat{J}_{k}=J+\sum_{r\in\mathbb{Z}^{d}}e^{-ik\cdot r}J_{|r|}$ (35)

Note that all the $G^{uv}_{k}$ fields can be integrated out, by satisfying the
equations of motion,

$\displaystyle G^{uv}_{k}=(-1)^{u+v}\frac{2iF^{uv}_{k}}{\hat{J}_{k}}$
$\displaystyle\ \text{for }k\neq 0,$ $\displaystyle
G^{uv}_{0}=\frac{2\pi\gamma}{\hat{J}_{0}}+(-1)^{u+v}\frac{2iF^{uv}_{0}}{\hat{J}_{0}}$
$\displaystyle\ \text{for }k=0.$ (36)

Integrating out the G fields, we get (again dropping constant terms),

$\displaystyle I[F,\mathbf{\Omega}]$ $\displaystyle=\int dt\int d\bar{}\hskip
1.00006pt^{d}k\sum_{\begin{subarray}{c}u<v\end{subarray}}\Bigg{[}-\frac{(-1)^{u+v}}{\hat{J}_{k}}\left(iF^{uv}_{k}\right)\left(iF^{uv}_{-k}\right)-\frac{2\pi\gamma}{\hat{J}_{0}}\left(iF^{uv}_{k}\right)\delta(k)+iF_{k}^{uv}\Omega^{uv}_{-k}\Bigg{]}.$
(37)

Going back to real space, we get,

$\displaystyle I[F,\mathbf{\Omega}]$ $\displaystyle=\int
dt\sum_{\begin{subarray}{c}r\\\
u<v\end{subarray}}\Bigg{[}-(-1)^{u+v}\sum_{r^{\prime}}\mathcal{J}_{rr^{\prime}}\left(iF^{uv}_{r}\right)\left(iF^{uv}_{r^{\prime}}\right)-\frac{\gamma}{\hat{J}_{0}}\left(iF^{uv}_{r}\right)+\left(iF_{r}^{uv}\right)\Omega^{uv}_{r}\Bigg{]},$
(38)

with an effective real space interaction $\mathcal{J}_{rr^{\prime}}$ given by,

$\displaystyle\mathcal{J}_{rr^{\prime}}$ $\displaystyle=\int d\bar{}\hskip
1.00006pt^{d}k\frac{e^{ik\cdot(r-r^{\prime})}}{\hat{J}_{k}}.$ (39)

By identifying the last term in (38) as a partition function for the spins
with external fields $iF$, one gets the Eq. 4 in the main text.

###  II.4 Effective real space interaction

(38) holds for any general number of dimensions. The microscopic interaction
between the spins i.e. $J_{rr^{\prime}}$ can be Fourier transformed to give
the momentum space interaction $\hat{J}_{k}$ between the $G$ fields. This can
then be transformed to an interaction $\mathcal{J}_{rr^{\prime}}$ between the
$iF$ fields via (39).

Here we consider two forms of the interaction, nearest neighbor (NN) or power-
law interacting (PL). In the main text results are used for the PL case.
Including the on-site term, the real space interaction between spins in a
d-dimensional lattice is defined as follows,

$\displaystyle J_{rr^{\prime}}$
$\displaystyle=\begin{cases}J\left(\delta_{r,r^{\prime}}+g\sum_{i}\left(\delta_{r,r^{\prime}+e_{i}}+\delta_{r,r^{\prime}-e_{i}}\right)\right)&\text{
(NN)}\\\
J\left(\delta_{r,r^{\prime}}+g\left(1-\delta_{r,r^{\prime}}\right)\frac{1}{|r-r^{\prime}|^{2\alpha}}\right)&\text{
(PL)}.\end{cases}$ (40)

Here $e_{i}$ is a d-dimensional vector $\\{0,0,..,i,..,0\\}$ with 1 in the
i-th position.

The momentum space interaction is given by its discrete-time Fourier
transform,

$\displaystyle\hat{J}_{k}=\begin{cases}J\left(1+2g\sum_{i}\cos{k_{i}}\right)&\text{
(NN)}\\\ J\left(1+g\sum_{s\neq 0}\frac{e^{-ik\cdot
s}}{|s|^{2\alpha}}\right)&\text{ (PL)}.\end{cases}$ (41)

To get the effective real space interaction $\mathcal{J}$ between the replica
fields $iF$, one has to take the Fourier transform of $\hat{J}_{k}^{-1}$. This
can be done exactly for $d=1$.

####  II.4.1 d=1

In $d=1$ we have,

$\displaystyle\hat{J}_{k}=\begin{cases}J\left(1+2g\cos{k}\right)&\text{
(NN)}\\\
J\left(1+g\left(\text{Li}_{2\alpha}\left(e^{-ik}\right)+\text{Li}_{2\alpha}\left(e^{ik}\right)\right)\right)&\text{
(PL)},\end{cases}$ (42)

where $\text{Li}_{n}\left(z\right)$ is the Polylogarithm function. For large
$\alpha$, $\text{Li}_{2\alpha}(z)\to z$, and $\hat{J}_{k}$ reduces to the
Nearest Neighbor case.

In the NN case, the real space interaction between the replica fields is given
by,

$\displaystyle\mathcal{J}^{\text{NN}}_{rr^{\prime}}=\int_{0}^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(r-r^{\prime})}}{1+2g\cos{k}}.$
(43)

Changing the variables to $z=e^{ik}$, we get contour integral defined along
the unit circle in the complex $z$-plane, with two isolated poles along the
negative real axis. Performing the contour integral picks up the pole within
the unit circle and one obtains,

$\displaystyle\mathcal{J}^{\text{NN}}_{rr^{\prime}}=\frac{(-1)^{r-r^{\prime}}}{\sqrt{1-4g^{2}}}e^{-\text{acosh}\frac{1}{2g}|r-r^{\prime}|}.$
(44)

In the PL case, with the same change of variables, the real space interaction
is given by the following contour integral defined along the unit circle,

$\displaystyle\mathcal{J}^{\text{PL}}_{rr^{\prime}}=\int_{\mathcal{C}}\frac{dz}{2\pi
i}\frac{z^{r-r^{\prime}-1}}{1+g\left(\text{Li}_{2\alpha}(z)+\text{Li}_{2\alpha}(1/z)\right)}.$
(45)

For large enough $g$ (including $g=1$) there is an isolated pole along the
negative real axis within the unit circle, and a branch cut due to the
$\text{Li}_{2\alpha}(1/z)$ term along the positive real axis $z\in(0,\infty)$.
The pole gives an exponential decay like the NN case, while we will show that
the branch cut contribution leads to a power law interaction.

Deforming the contour to hug the branch cut, the integral is proportional to
the discontinuity along the branch cut
($\int_{0+i\varepsilon}^{1+i\varepsilon}-\int_{0-i\varepsilon}^{1-i\varepsilon}$),
which for the function $\text{Li}_{2\alpha}(1/z)$ is proportional to
$\log^{2\alpha-1}(1/z)$. Thus we get,

$\displaystyle\mathcal{J}^{\text{PL}}_{rr^{\prime}}=\text{Pole
contribution}-\int_{0}^{1}dzz^{r-r^{\prime}-1}\log^{2\alpha-1}(z)g(z),$ (46)

where $g(z)$ is a smooth function. For large $|r-r^{\prime}|$ the integrand is
heavily suppressed away from $z=1$, so we can change variables $z\sim e^{-w}$,
and after dropping the regular terms, we have an integral,

$\displaystyle\int_{0}^{\infty}dwe^{-w|r-r^{\prime}|}w^{2\alpha-1}\sim|r-r^{\prime}|^{-2\alpha},\text{
for }|r-r^{\prime}|\gg 1.$

Along with the pole contribution we thus get the effective real space
interaction,

$\displaystyle\mathcal{J}^{\text{PL}}_{rr^{\prime}}\sim(-1)^{r-r^{\prime}}e^{-\mu|r-r^{\prime}|}-(1-\delta_{rr^{\prime}})\frac{1}{|r-r^{\prime}|^{2\alpha}}.$
(47)

At large $|r-r^{\prime}|$,
$\mathcal{J}_{rr^{\prime}}\sim|r-r^{\prime}|^{-2\alpha}$, as was noted in the
main text.

####  II.4.2 General d

In this section we will derive the effective real space interaction in any
general dimension, $d$.

For the NN case, we have,

$\displaystyle\mathcal{J}^{\text{NN}}_{rr^{\prime}}=\int d\bar{}\hskip
1.00006pt^{d}k\frac{e^{ik\cdot(r-r^{\prime})}}{1+2g\sum_{i}\cos{k_{i}}},$ (48)

which can be expanded around $k_{i}=\pi$. Furthermore, we can consider
$r-r^{\prime}$ to be along a particular dimension, say 1. With these
manipulations we get,

$\displaystyle\mathcal{J}^{\text{NN}}_{rr^{\prime}}$
$\displaystyle\sim\int_{-\pi}^{+\pi}d\bar{}\hskip
1.00006pt^{d}k\frac{e^{ik_{1}(r-r^{\prime})}}{1+\tilde{g}\sum_{i}k_{i}^{2}}$
$\displaystyle\sim\int_{0}^{\infty}dkk^{d-2}\int d\bar{}\hskip
1.00006ptk_{1}\frac{e^{ik_{1}(r-r^{\prime})}}{1+k_{1}^{2}+k^{2}}$
$\displaystyle\sim\int_{0}^{\infty}\frac{k^{d-2}}{\sqrt{1+k^{2}}}e^{-\sqrt{1+k^{2}}|r-r^{\prime}|}\sim
e^{-\mu|r-r^{\prime}|}.$ (49)

For the PL case, one needs to evaluate,

$\displaystyle\mathcal{J}^{\text{PL}}_{rr^{\prime}}\sim\int d\bar{}\hskip
1.00006pt^{d}k\frac{e^{ik\cdot s}}{1+g\sum_{u\neq 0}\frac{e^{-ik\cdot
u}}{|u|^{2\alpha}}}.$ (50)

Firstly we have,

$\displaystyle\sum_{u\neq 0}\frac{e^{-ik\cdot
u}}{|u|^{2\alpha}}\sim\int^{\prime}d\bar{}\hskip
1.00006pt^{d}u\frac{e^{-ik\cdot u}}{|u|^{2\alpha}}\sim k^{2\alpha-d}.$ (51)

With this, we can expand the function to be Fourier Transformed at small $k$,

$\displaystyle\mathcal{J}^{\text{PL}}_{rr^{\prime}}\sim\int d\bar{}\hskip
1.00006pt^{d}ke^{ik\cdot
s}\left(1-gk^{2\alpha-d}\right)\sim\delta(s)-g|s|^{-2\alpha}.$ (52)

These results generalize 47 for any general dimension.

##  III Saddle point analysis of the mean field

###  III.1 Effective Bulk Action and saddle points

The mean-field action in Eq. (38) (equivalent to Eq. 4 in the main text),
describes the dynamics of $4L$ spins $\mathbf{S}_{r}^{u}$ interacting via
Heisenberg couplings $\mathbf{S}_{r}^{u}\cdot\mathbf{S}_{r}^{v}$. Because
these coupling terms are manifestly symmetric under global $\mathrm{SU}(2)$
rotations, the action $I=I[iF_{r}^{uv}]$ is also $\mathrm{SU}(2)$ invariant.
In fact, because the interaction terms are separable
$\sum_{r}\ln\mathcal{K}_{r}$ in the space coordinate $r$, the action $I$ is
invariant under all _local_ $\mathrm{SU}(2)$ rotations generated by the total
spin operators $\mathbf{S}_{r}^{\mathrm{Tot}}=\sum_{u}\mathbf{S}_{r}^{u}$
within each cluster $r$. At the same time, the boundary conditions generated
by the EPR pairs and SWAP operator force the system to form $\mathrm{SU}(2)$
spin singlets at times $t=0,T$. Together, these facts constrain the dynamics
within each cluster $r$ to live entirely in the spin-singlet subspace
$\mathbf{S}_{r}^{\mathrm{Tot}}=0$ for each $r=1,\ldots,L$. Thus, each cluster
$r$ supports a single replica qubit or r-bit spanned by the states
$\left|\uparrow\right\rangle_{r},\left|\downarrow\right\rangle_{r}$.

The underlying $\mathrm{SU}(2)$ symmetry significantly simplifies the problem.
For $n=2$ the propagator $\mathcal{K}_{r}$ in each two-dimensional r-bit
subspace simplifies to

$\displaystyle\mathcal{K}_{r}=\left\langle\psi_{T}\right|\exp{\left[\frac{1}{2}\int_{0}^{T}dt\
\big{(}\phi_{r}(t)\sigma^{x}_{r}+\Theta_{r}(t)\sigma^{z}_{r}\big{)}\right]}\left|\psi_{0}\right\rangle
e^{BT/2}$ (53)

where $\sigma^{x,z}_{r}$ are the $2\times 2$ Pauli matrices acting on the
r-bit $\left|\uparrow\right\rangle_{r},\left|\downarrow\right\rangle_{r}$
subspace. The fields $\phi_{r}(t),\Theta_{r}(t)$ are linear combinations of
the mean fields

$\displaystyle\phi_{r}$
$\displaystyle=\frac{2}{3\sqrt{3}}\left(iF^{12}_{r}+iF^{34}_{r}-iF^{14}_{r}-iF^{23}_{r}\right)$
$\displaystyle\Theta_{r}$
$\displaystyle=\frac{2}{9}\sum_{u<v}iF^{uv}_{r}-\frac{2}{3}\left(iF^{13}_{r}+iF^{24}_{r}\right)$
(54)

and where terms proportional to the identity within the r-bit subspace have
been collected into the term

$B=\frac{2}{9L}\sum_{r,u<v}iF^{uv}_{r}.$ (55)

The time- and space-dependent bulk fields $\phi_{r}(t),\Theta_{r}(t)$ encode
the relevant mean-field dynamics of the r-bits in each cluster $r$. In general
these fields must execute nontrivial motions in the bulk in order to satisfy
the non-equal boundary conditions
$\left|\psi_{0}\right\rangle,\left|\psi_{T}\right\rangle$. By contrast, the
remaining fields in the action appear simply as quadratic Gaussian fields and
may therefore be trivially integrated out of the path integral, leading to the
effective action

$\displaystyle\mathbb{E}\mathrm{Tr}\left[\rho_{A}^{2}(t)\right]\ \mathrm{or}\
\mathbb{E}\mathrm{Tr}\left[\rho_{A}(t)\right]^{2}$
$\displaystyle=\int\mathcal{D}\phi\mathcal{D}\Theta\exp{\left[-NI[\phi,\Theta]\right]}$
$\displaystyle I[\phi,\Theta]$
$\displaystyle=\sum_{r}\int_{0}^{T}dt\left[\sum_{r^{\prime}}\frac{27\mathcal{J}_{rr^{\prime}}}{16}\left(\phi_{r}\phi_{r^{\prime}}-3\Theta_{r}\Theta_{r^{\prime}}\right)+\Theta_{r}\left(1+\frac{9\gamma}{\hat{J}_{0}}\right)\right]-\sum_{r}\ln\mathcal{K}_{r}.$
(56)

with the propagator $\mathcal{K}_{r}$ given in Eq. (53), and where we have
dropped additive constant terms.

To understand the saddle points of the action $I[\phi,\Theta]$ we first
consider the bulk mean-field limit in which the fields are independent of
space and time: $\phi_{r}(t)=\phi,\Theta_{r}(t)=\Theta$. In this case the
propagator $\mathcal{K}_{r}$ is dominated at long times by the ground state of
the r-bit effective Hamiltonian $\phi\sigma_{r}^{x}+\Theta\sigma_{r}^{z}$,
which yields

$\sum_{r}\ln\mathcal{K}_{r}=\frac{LT}{2}\sqrt{\phi^{2}+\Theta^{2}}+\frac{BLT}{2}.$
(57)

By substituting this into Eq. (III.1) (and again dropping additive constant
terms) we obtain the time- and space-independent effective bulk action,

$\displaystyle
I_{\text{MF}}[\phi,\Theta]=NTL\left[\mathcal{J}\left(\phi^{2}-3\Theta^{2}\right)-9\left(\Gamma+\frac{1}{9}\right)\Theta-\frac{1}{2}\sqrt{\phi^{2}+\Theta^{2}}\right],$
(58)

where we have defined $\Gamma\equiv\frac{\gamma}{\hat{J}_{0}}$ and
$\mathcal{J}\equiv 27/16\sum_{s\in\mathbb{Z}}\mathcal{J}_{r(r+s)}$.

Saddle point analysis of this action [2] reveals a phase transition at
$\Gamma_{c}=1/9$. For $\Gamma>\Gamma_{c}$ the action is dominated by the
symmetric saddle point with $\phi^{*}=0$ and
$\Theta^{*}=-3(\Gamma_{c}+2\Gamma)/4\mathcal{J}$. By contrast, for
$\Gamma<\Gamma_{c}$, the action is dominated by the symmetry-broken saddle-
point where the field $\phi^{*}$ is non-zero and comes in a pair,
$\phi^{*}\propto\pm\sqrt{\Gamma_{c}-\Gamma}$, while
$\Theta^{*}=-9(\Gamma_{c}+\Gamma)/8\mathcal{J}$. The replica permutation
$1\leftrightarrow 3$ is equivalent to $\phi\leftrightarrow-\phi$ symmetry,
which is spontaneously broken for $\Gamma<\Gamma_{c}$.

Figure 5: Action cost for symmetric and symmetry-broken phases. Below the
critical point $\Gamma<\Gamma_{c}$, the action cost $\mathrm{Re}[I_{MF}]$ is
minimized by the symmetry-broken saddle points (solid orange) relative to the
symmetric saddle point (dashed purple). Above the critical point
$\Gamma>\Gamma_{c}$ the path integral is dominated by the symmetric saddle
point (solid purple) because the symmetry-broken saddles (dashed orange) are
imaginary and therefore do not contribute to the integral (see Fig. 6). Figure
6: Saddle-point integration contours. (a) Below the critical point, both the
symmetry-broken saddle points (i., orange dots) and the symmetric saddle point
(ii., purple dot) lie along the real axis of integration (solid black). In
this case all three saddle points contribute to the integral, but the
symmetry-broken saddle points dominate because they minimize the action
$\mathrm{Re}[I_{\mathrm{MF}}]$. (b) Above the critical point, the symmetry-
broken saddle points (i., orange dots) lie on the imaginary axis and therefore
do not contribute to the integral. Therefore the integral is dominated by the
symmetric saddle point (ii., purple dot). Dotted blue lines show contours of
steepest ascent / descent.

Note, $\gamma_{c}$ is given by
$\Gamma_{c}\hat{J_{0}}\sim\Gamma_{c}\left(1+g\sum_{s\neq
0}|s|^{-2\alpha}\right)\approx\Gamma_{c}\left(1+g\int^{\prime}d^{d}s|s|^{-2\alpha}\right)$.
Thus $\gamma_{c}$ diverges when $2\alpha<d$, when the system effectively
becomes all-to-all.

###  III.2 Effective Field Theory Near Criticality

In the previous analysis we understood the bulk mean-field physics of the
model by ignoring the space- and time- dependence of the fields
$\phi_{r}(t),\Theta_{r}(t)$. Here we restore the space- and time-dependence
and study the model near the critical point. In this limit we can transform
the effective r-bit action Eq. (III.1) into a $\phi^{4}$ field theory with
long-range interactions. In particular, near the critical point the dynamics
is governed entirely by the small fluctuations of the symmetry-breaking field
$\phi_{r}(t)$. In the following we expand the action $I[\phi,\Theta]$ in
fluctuations of this small parameter to obtain an effective long-range
$\phi^{4}$ field theory for the system.

First we evaluate the leading term of the effective action $I[\phi,\Theta]$ in
Eq. (III.1), corresponding to the long-range power-law interactions.
Substituting
$\mathcal{J}_{rr^{\prime}}=\mathcal{J}_{rr^{\prime}}^{\mathrm{PL}}$ from Eq.
(47), we obtain

$\displaystyle\sum_{r^{\prime}}(-1)^{r-r^{\prime}}e^{-\mu|r-r^{\prime}|}\phi_{r}\phi_{r^{\prime}}$
$\displaystyle\approx\sum_{q\in\mathbb{Z}}(-1)^{q}e^{-\mu|q|}\phi_{r}^{2}+\sum_{q\in\mathbb{Z}}(-1)^{q}\frac{q^{2}}{2}e^{-\mu|q|}\phi_{r}\partial_{r}^{2}\phi_{r}+O(\phi_{r}\partial_{r}^{4}\phi_{r})$
$\displaystyle\approx\frac{(1-e^{-\mu})}{(1+e^{-\mu})}\phi_{r}^{2}-\frac{e^{-2\mu}\left(1-e^{-\mu}\right)}{\left(1+e^{-\mu}\right)^{3}}\phi_{r}\partial_{r}^{2}\phi_{r}$
(59)

and

$\displaystyle\sum_{r^{\prime}}(1-\delta_{rr^{\prime}})\frac{1}{|r-r^{\prime}|^{2\alpha}}\phi_{r}\phi_{r^{\prime}}\rightarrow\int^{\prime}_{s}\frac{1}{|r-s|^{2\alpha}}\phi_{r}\phi_{s}$
(60)

in the continuum limit, where the delta-function term $1-\delta_{rr^{\prime}}$
is responsible for the UV cut-off in the continuum integral
$\int^{\prime}_{s}=\int_{\mathbb{R}\backslash(r-\varepsilon,r+\varepsilon)}ds$.
Here we have assumed that the parameter $\mu$ is large such that the
exponential $e^{-\mu\left|q\right|}$ decays rapidly and suppresses higher
derivative terms e.g. $\phi_{r}\partial_{r}^{4}\phi_{r}$. Next, we fix
$\Theta_{r}(t)=\Theta=-3(\Gamma_{c}+2\Gamma)/4\mathcal{J}$ to its saddle-point
value and expand the propagators $\ln\mathcal{K}_{r}$ in the small parameter
$\phi_{r}(t)$ [2]:

$\displaystyle\ln\mathcal{K}_{r}$ $\displaystyle=\ln 2\cosh{\frac{\Theta
T}{2}}+\frac{1}{8}\int
dt_{1}dt_{2}\phi_{r}(t_{1})\phi_{r}(t_{2})e^{-\left|\Theta\right|\left|t_{1}-t_{2}\right|}-\int
dt\frac{\phi_{r}^{4}}{16\left|\Theta\right|^{3}}+O(\phi_{r}^{6})$
$\displaystyle\approx\int
dt\left[\frac{\phi_{r}\partial_{t}^{2}\phi_{r}}{4\left|\Theta\right|^{3}}+\frac{\phi_{r}^{2}}{4\left|\Theta\right|}-\frac{\phi_{r}^{4}}{16\left|\Theta\right|^{3}}\right]+\mathrm{const}.$
(61)

Finally, summing the results, taking the continuum limit, and ignoring
additive constants, we find the near-critical Landau-Ginzburg effective field
theory

$\displaystyle\frac{I_{\text{eff}}[\phi]}{N}=\frac{1}{4\left|\Theta\right|^{3}}\int_{t,r}dtdr\Bigg{[}-\phi_{r}\left(\partial_{t}^{2}+\beta\partial_{r}^{2}\right)\phi_{r}-b\int^{\prime}_{s}\frac{\phi_{r}\phi_{s}}{|r-s|^{2\alpha}}-\frac{\delta}{2}\phi_{r}^{2}+\frac{\
\phi_{r}^{4}}{4}\Bigg{]},$ (62)

for small fluctuations $\phi_{r}(t)$ around the critical point. Here we have
introduced the numerical coefficients

$\displaystyle\beta$
$\displaystyle=4\left|\Theta\right|^{3}\frac{27J}{16}\frac{e^{-2\mu}\left(1-e^{-\mu}\right)}{\left(1+e^{-\mu}\right)^{3}}$
$\displaystyle b$ $\displaystyle=4\left|\Theta\right|^{3}\frac{27J}{16}$
$\displaystyle\delta$
$\displaystyle=8\left|\Theta\right|^{3}\left(\frac{1}{4\left|\Theta\right|}-\frac{27J}{16}\frac{(1-e^{-\mu})}{(1+e^{-\mu})}\right).$
(63)

By rescaling space and $b$ we can get rid of the $\beta$ parameter, and end up
with the Landau-Ginzburg theory in Eq. 6 in the main text.

##  IV Scaling of entanglement entropy in the symmetry broken phase

In this section we derive results for $d=1$.

###  IV.1 Nearest Neighbor model

For the NN case, the power law term in the action is absent and we will be
working with the short-range action,

$\displaystyle I[\phi]$ $\displaystyle=\int dt\int
dr\left[-\phi\left(\partial_{t}^{2}+\beta\partial_{r}^{2}\right)\phi-\frac{\delta}{2}\phi^{2}+\frac{\phi^{4}}{4}\right],$
(64)

####  IV.1.1 Entropy of a maximally mixed initial state

The SWAP action acts on the whole spatial slice, and we need to only solve for
the equations of motion with the time derivative,

$\partial_{t}^{2}\phi=-\delta\phi+\phi^{3}.$ (65)

For $\delta<0$, we have an instanton-like solution,

$\displaystyle\phi^{*}(r,t)=\sqrt{\delta}\tanh\left[\sqrt{\frac{\delta}{2}}\left(t-t_{0}\right)\right].$
(66)

Hence for $\delta>0$, we have the following scaling of the quasi-entropy of
the full system,

$\displaystyle\hat{S}^{(2)}_{Q}\sim
N\left(I[\phi^{*}]-I[\sqrt{\delta}]\right)\sim NL\delta^{3/2}+...$ (67)

The correction to this term will be given by the fluctuations of the domain
wall within periodic boundary condition, which was estimated using Capillary
Wave Theory in [4]. For $T\gg\sqrt{L}$, this will be given by,

$\displaystyle-\log\frac{T}{\sqrt{L}}-\log\sqrt{\frac{\beta
N\delta^{3/2}}{12\pi}}\sim-\log\frac{T}{\sqrt{L}}+\text{const.}$ (68)

####  IV.1.2 Sub-system entropy

For a subsystem between the region $r_{1}$ and $r_{2}$, the domain wall must
be pinned to the future time boundary at $r_{1}$ and $r_{2}$. We assume the
solution of the fields is of the form,

$\displaystyle\phi^{*}(r,t)=\sqrt{\delta}\tanh\left[\sqrt{\frac{\delta}{2}}\left(t-y(r)\right)\right],$
(69)

where $y(r)$ is the ‘height’ of the domain wall (or equivalently the position
of the instanton). Due to the boundary pinning effect, $y(r)$ must be
$\epsilon\to 0$ at $r=r_{1}$ and $r=r_{2}$. The action for the quasi-entropy
is given by a functional of $y(r)$,

$\displaystyle I[y(r)]$ $\displaystyle=I[\phi^{*}]-I[\sqrt{\delta}]$
$\displaystyle=\int dt\int dr\delta^{2}\Bigg{[}\left(1+\beta
y^{\prime}(r)^{2}\right)\text{sech}^{2}\sqrt{\frac{\delta}{2}}(t-y)-\frac{1}{4}\left(3+4\beta
y^{\prime}(r)^{2}\right)\text{sech}^{4}\sqrt{\frac{\delta}{2}}(t-y)$
$\displaystyle+\frac{\beta}{\sqrt{2\delta}}y^{\prime\prime}(r)\text{sech}^{2}\sqrt{\frac{\delta}{2}}(t-y)\text{tanh}\sqrt{\frac{\delta}{2}}(t-y)\Bigg{]}$

Rescale the time variable,

$\displaystyle z=\sqrt{\frac{\delta}{2}}(t-y(r)).$

The limits for z integration are approximately $0\to\infty$. The action for
quasi-entropy is thus given by,

$\displaystyle I[y(r)]$ $\displaystyle\sim\delta^{3/2}\int dr\int
dz\left[\left(1+\beta
y^{\prime}(r)^{2}\right)\text{sech}^{2}z-\frac{1}{4}\left(3+4\beta
y^{\prime}(r)^{2}\right)\text{sech}^{4}z+\mathcal{O}(y^{\prime\prime}(r))\right]$
$\displaystyle\sim\delta^{3/2}\int
dr\left(1+\frac{\beta}{6}y^{\prime}(r)^{2}+\mathcal{O}(y^{\prime\prime}(r)\right).$
(70)

We can ignore the $\mathcal{O}(y^{\prime\prime}(r))$ term as it is irrelevant
under RG flow, since $[y]=1$ and $[y^{\prime\prime}]=-1$. Thus, the quasi-
entropy is thus given by,

$\displaystyle\hat{S}^{(2)}(A)$
$\displaystyle=-\ln\int\mathcal{D}(y(r))\exp\left[-N\delta^{3/2}\int_{A}dr\left(1+\frac{\beta}{6}y^{\prime}(r)^{2}\right)\right]$
(71)
$\displaystyle=N\delta^{3/2}|A|-\ln\int\mathcal{D}(y(r))\exp\left[-\frac{\beta
N\delta^{3/2}}{6}\int_{A}dr\left(y^{\prime}(r)^{2}\right)\right]$ (72)

This is exactly the action conjectured by Li and Fisher and solved using
Capillary Wave Theory in [4]. Using their result, the quasi-entropy is given
by,

$\displaystyle\hat{S}_{A}^{(2)}$ $\displaystyle\approx
N\delta^{3/2}|A|+\frac{3}{2}\ln|A|-\ln\left[\sqrt{\frac{2}{\pi}}\epsilon^{2}\left(\frac{\beta
N\delta^{3/2}}{6}\right)^{3/2}\right]$ (73)
$\displaystyle=N\delta^{3/2}|A|+\frac{3}{2}\ln|A|+\text{const...}$ (74)

The approximations a la Li and Fisher [4] behind this are $|A|\ll L$,
$\sqrt{A}\ll T$, and for a spatial lattice cut-off, $\epsilon$, such that,
$\epsilon\ll\sqrt{|A|/(\beta N\delta^{3/2})}$. Note, we further have our large
N approximation, $T\lesssim\text{poly}(N)$. These can all be satisfied in the
regime, $\sqrt{A}\ll T\lesssim\text{poly}(N)$. The logarithmic correction term
is $1/N$ suppressed in our model.

###  IV.2 Power Law model

In the symmetry broken phase, the field theory is given by,

$\displaystyle I[\phi]$ $\displaystyle=\int dtdr\Bigg{[}-\phi\left(a\
\partial_{t}^{2}+\beta\
\partial_{r}^{2}\right)\phi-b\int^{\prime}ds\frac{\phi_{r}\phi_{s}}{|r-s|^{2\alpha}}-\frac{\delta\
\phi^{2}}{2}+\frac{u\ \phi^{4}}{4}\Bigg{]},$ (75)

for $\delta>0$.

By appropriate rescaling of the fields, we can rewrite the field theory as,

$\displaystyle I[\phi]$ $\displaystyle=\int
dtdr\Bigg{[}-\phi\left(\partial_{t}^{2}+\beta\
\partial_{r}^{2}\right)\phi-b\int^{\prime}ds\frac{\phi_{r}\phi_{s}}{|r-s|^{2\alpha}}-\frac{\delta\
\phi^{2}}{2}+\frac{\phi^{4}}{4}\Bigg{]},$ (76)

The $\phi$ equation of motion is given by,

$\displaystyle\partial_{t}^{2}\phi_{r}+\beta\partial_{r}^{2}\phi_{r}=-\delta\phi_{r}+\phi_{r}^{3}-\int^{\prime}ds\phi_{s}\frac{1}{|r-s|^{2\alpha}}.$
(77)

####  IV.2.1 Total Entropy of the maximally mixed initial state

In this case, the saddle point configuration will be independent of space, and
the equation of motion can be simplified to be,

$\displaystyle\partial_{t}^{2}\phi=-\delta_{\alpha}\phi+\phi^{3},$ (78)
$\displaystyle\text{where,
}\delta_{\alpha}=\delta+\int^{\prime}ds\frac{1}{|r-s|^{2\alpha}}.$ (79)

The phase transition is effected by taking $\delta_{\alpha}\to 0$. The static
solutions of this equation is,

$\displaystyle\phi=\pm\sqrt{\delta_{\alpha}},$ (80)

and the instanton solution is,

$\displaystyle\phi^{*}=\sqrt{\delta_{\alpha}}\tanh{\sqrt{\frac{\delta_{\alpha}}{2}}(t-t_{0})}.$
(81)

As before, the entropy of the whole system will be given by,

$\displaystyle\hat{S}^{(2)}_{Q}\sim
NL\delta_{\alpha}^{3/2}-\log{\frac{T}{\sqrt{L}}}+\text{const.}$ (82)

####  IV.2.2 Subsystem entropy

For general long-range interactions, there can’t be a simple capillary wave
picture, as the gradient expansion doesn’t converge for $2\alpha<3$. For the
effectively short-ranged case $2\alpha>3$, the story is the same as for the
nearest neighbor model.

In absence of a controlled analytical calculation, we can make reasonable
estimates of the energy of a domain wall pinned to the future boundary at a
subsystem $A$. First we estimate the energy of a flat domain wall of length A
and width w, for a microscopic model with the same low-energy action as the
long range $\phi^{4}$ model. One such model is a classical Ising model in 2D,
with long range interaction along one direction (space) and short range
interaction along the other direction (time), with a Hamiltonian,

$\displaystyle H=-\sum_{t}\sum_{r}JS_{r,t}S_{r,t+1}-\sum_{t}\sum_{r\neq
s}\frac{K}{|r-s|^{2\alpha}}S_{r,t}S_{s,t}\text{ with }J,K>0.$ (83)

The energy of a flat domain wall of length $|A|$ and width w is given by,

$\displaystyle E_{\text{dw}}$ $\displaystyle=|A|J+wK\left(\sum_{r\in L,s\in
L}-\sum_{r\in A,s\in A}-\sum_{r\in A^{c},s\in A^{c}}+2\sum_{r\in A,s\in
A^{c}}\right)\frac{1}{|r-s|^{2\alpha}}$ (84)

We can change the sums into integrals with a regulator $\varepsilon$, and this
is given by,

$\displaystyle E_{\text{dw}}$ $\displaystyle=|A|J+wK\int^{\prime}_{r\in A,s\in
A^{c}}drds\frac{1}{|r-s|^{2\alpha}}+\text{const.}\sim|A|J+wK\left(|A|^{2-2\alpha}-\varepsilon^{2-2\alpha}\right)+\text{const.}$
(85)

Here we have assumed $L>>|A|>>\varepsilon$.

In this simple estimate we can’t estimate the optimal width of the domain
wall, $w$. However this already suggests that $w$ will not scale with $|A|$,
and should instead be given by the time-like correlation length in this model,
$w\sim\xi_{t}$. Hence for $2\alpha\leq 2$, we get a power-law correction to
the volume law entropy, which is distinct from the logarithmic correction that
one obtained for the nearest neighbor / short-ranged case. In the large-N
model, the interactions $J,K$ scale with $N$. The entropy of a small subregion
is hence given by,

$\displaystyle\hat{S}^{(2)}_{A}\sim
N|A|\delta_{\alpha}^{3/2}+\xi_{t}N|A|^{2-2\alpha}+\mathcal{O}(1/N).$ (86)

Any corrections due to the fluctuations of the domain wall will be 1/N
suppressed as in the local case.

##  V Brownian SYK details

In this section we derive the Landau-Ginzburg effective action for the
monitored Brownian SYK model. The Brownian Gaussian random couplings in the
unitary part is defined by the parameters,

$\displaystyle\mathbb{E}\left[J_{a,ij}^{r_{1}r_{2}}(t_{1})J_{a^{\prime},ij}^{r^{\prime}_{1}r^{\prime}_{2}}(t_{2})\right]=\frac{J_{rr^{\prime}}}{2N}\frac{\delta_{t_{1}t_{2}}}{\Delta
t}\delta_{aa^{\prime}}\delta^{r_{1},r_{1}^{\prime}}\delta^{r_{2},r_{2}^{\prime}},$
(87)
$\displaystyle\mathbb{E}\left[U_{a,j_{1}...j_{q}}^{r}(t_{1})U_{a^{\prime},j_{1}...j_{q}}^{r^{\prime}}(t_{2})\right]=\frac{2^{q-2}(q-1)!U}{N^{q-1}}\frac{\delta_{t_{1}t_{2}}}{\Delta
t}\delta_{aa^{\prime}}\delta^{r,r^{\prime}}.$

.

The measurement part for the flavor $i$ at the site $x$ can be cast into,

$\displaystyle\sum_{\nu=1}^{2}M_{\nu}^{x,i}\otimes
M_{\nu}^{x,i{\dagger}}\otimes M_{\nu}^{x,i}\otimes M_{\nu}^{x,i{\dagger}}$
$\displaystyle=$
$\displaystyle\left(1-\frac{s^{2}}{2}\sum_{u=1}^{4}\pi^{+,u}_{x,i}+s^{4}\otimes_{u=1}^{4}\pi^{+,u}_{x,i}\right)$
(88) $\displaystyle\approx$
$\displaystyle\exp\left(-\frac{s^{2}}{2}\sum_{u}\pi^{+,u}_{x,i}\right)$ (89)
$\displaystyle=$ $\displaystyle\exp\frac{\delta
t\gamma}{2}\sum_{u}i\psi_{x,L,i}^{u}\psi_{x,R,i}^{u},$ (90)

where we have used the relation $\pi^{+}_{x,a,j}+\pi^{-}_{x,a,j}=1$ and also
introduced $u=1,...,4$ to denote the four copies of the tensor product. To
derive the above equation, we assume $s\ll 1$ and keep orders up to
$O(s^{2})$. In the last line we introduce $\gamma=s^{2}/\delta t$, and when
the continuum limit is taken, $\gamma$ is kept fixed. All the constants are
neglected because they will not affect the dynamics. The effect of monitoring
every Majorana species $i$ at every site $x$ is described by

$\displaystyle\exp\left(\frac{\gamma}{2}\int
dt\sum_{x,\alpha,i}i\psi_{x,L,i}^{\alpha}\psi_{x,R,i}^{\alpha}\right),$ (91)

where we implicitly sum over all infinitesimal time steps to arrive at the
time integral for a time evolution.

Due to the large-$N$ structure in both Hamiltonian in Eq. 10 in the main text
and monitoring part (91), we can introduce the bilocal field,

$\displaystyle
G_{ab,r}^{uv}(t,t^{\prime})=\frac{1}{N}\sum_{j}\psi_{r,a,j}^{u}(t)\psi_{r,b,j}^{v}(t^{\prime}),$
(92)

to rewrite the Majorana field with the help of the following identity

$\displaystyle 1$ $\displaystyle=$ $\displaystyle\int d\Sigma\exp\int
dt_{1}dt_{2}\Big{[}-\frac{N}{2}\Sigma^{uv}_{r}(t_{1},t_{2})\Big{(}G_{r}^{uv}(t_{1},t_{2})-\frac{1}{N}\sum_{i}\psi_{r,i}^{u}(t_{1})\psi_{r,i}^{v}(t_{2})\Big{)}\Big{]},$
(93)

where $\Sigma_{ab,r}^{uv}(t,t^{\prime})$ is the self-energy. It is a standard
approach for the SYK model [5], which is then generalized to the four contours
with monitoring part [6]. With a slight modification that replace the nearest-
neighbor hopping to a power-law hopping, the large-$N$ action in the replica
space reads

$\displaystyle-\frac{I}{N}$ $\displaystyle=$
$\displaystyle\sum_{r}\Big{[}\frac{1}{2}\text{Tr}\log\left((-1)^{u+1}\partial_{t}-\Sigma_{r}\right)-\frac{1}{2}\int_{t,t^{\prime}}\Sigma_{ab,r}^{uv}G_{ab,r}^{uv}$
(94)
$\displaystyle+\int_{t,t^{\prime}}\delta(t-t^{\prime})\Big{[}\frac{(-1)^{u+v+1}}{4}\delta_{ab}\Big{(}\sum_{r^{\prime}}J^{rr^{\prime}}G_{ab,r}^{uv}G_{ab,r^{\prime}}^{uv}+\frac{U}{2q}(2G_{ab,r}^{uv})^{q}\Big{)}+\frac{i\gamma}{2}G_{LR,r}^{uu}\Big{]}\Big{]},$

where $u,v=1,...,4$ denote the four contours, and
$\int_{t,t^{\prime}}\equiv\int dtdt^{\prime}$. The summations over $a,b$ and
$u,v$ are implicit. Note that the model can be generalized to arbitrary graph
with a modification on the hopping term in the second line of (94).

Saddle-point analysis can be straightforwardly applied to the large-$N$
action. The Schwinger-Dyson equation resulted from (94) reads

$\displaystyle[G_{r}^{-1}]^{uv}_{ab}$ $\displaystyle=$
$\displaystyle(-1)^{u+1}\delta^{uv}\delta_{ab}\partial_{t}-\Sigma_{ab,r}^{uv},$
(95) $\displaystyle\Sigma_{ab,r}^{uv}$ $\displaystyle=$
$\displaystyle\delta(t-t^{\prime})\Big{[}\frac{(-1)^{u+v+1}\delta_{ab}}{2}\Big{(}\sum_{r^{\prime}}J_{rr^{\prime}}2G_{ab,r^{\prime}}^{uv}+U(2G_{ab,r}^{uv})^{q-1}\Big{)}+i\gamma\delta^{uv}\frac{\delta_{aL}\delta_{bR}-\delta_{aR}\delta_{bL}}{2}\Big{]}.$
(96)

For a homogeneous solution in real space, i.e.,
$G_{ab,r}^{uv}=\bar{G}_{ab}^{uv}$ and
$\Sigma_{ab,r}^{uv}=\bar{\Sigma}_{ab}^{uv}$, the Schwinger-Dyson equation is
simplified to be

$\displaystyle[\bar{G}^{-1}]^{uv}_{ab}$ $\displaystyle=$
$\displaystyle(-1)^{\alpha+1}\delta^{uv}\delta_{ab}\partial_{t}-\bar{\Sigma}_{ab}^{uv},$
(97) $\displaystyle\bar{\Sigma}_{ab}^{uv}$ $\displaystyle=$
$\displaystyle\delta(t-t^{\prime})\Big{[}\frac{(-1)^{u+v+1}\delta_{ab}}{2}\Big{(}\hat{J}2\bar{G}_{ab}^{uv}+U(2\bar{G}_{ab}^{uv})^{q-1}\Big{)}+i\gamma\delta^{uv}\frac{\delta_{aL}\delta_{bR}-\delta_{aR}\delta_{bL}}{2}\Big{]},$
(98)

where $\hat{J}=J\zeta(2\alpha)$, and
$\zeta(\alpha)\equiv\sum_{r=1}^{\infty}\frac{1}{r^{\alpha}}$.

To get the solution, we focus on two contours, $u,v=1,2$, because the boundary
condition in $\text{Tr}(\rho)^{2}$ is to connect 1 to 2 and connect 3 to 4
separately. According to Ref. [6], the saddle point solution can be obtained
by replacing $J$ to $\hat{J}$,

$\displaystyle\bar{G}(t_{1},t_{2})=\begin{cases}\frac{e^{-\frac{\hat{J}+U\lambda^{q-2}}{2}|t_{12}|}}{2}\Big{[}\text{sgn}(t_{12})\sigma^{z}-\lambda
i\sigma^{y}+\frac{\tilde{\gamma}\tau^{y}}{1+\tilde{U}\lambda^{q-2}}\Big{]},&\tilde{\gamma}<1,\\\
\\\
\frac{e^{-\frac{\gamma|t_{12}|}{2}}}{2}\left(\text{sgn}(t_{12})\sigma^{z}+\tau^{y}\right)&\tilde{\gamma}\geq
1\end{cases},$ (99)

where $t_{12}\equiv t_{1}-t_{2}$ is the time difference,
$\tilde{\gamma}\equiv\gamma/\hat{J}$, $\tilde{U}\equiv U/\hat{J}$ and Pauli
matrix $\sigma$ ($\tau$) acts on 1 and 2 contours ($L$ and $R$ chains). The
parameter $\lambda$ is given by

$\displaystyle(1-\lambda^{2})(1+\tilde{U}\lambda^{q-2})^{2}=\tilde{\gamma}^{2}.$
(100)

The solution on $3,4$ contours is the same, consistent with the boundary
condition without twist operators. We will discuss the saddle-point solution
in more details in the next section.

###  V.1 Landau-Ginzburg effective action and Rényi entropy

The symmetry of the theory is not affected by the power-law hopping. In the
following, we discuss the $O(2)$ symmetry and the $C_{4}$ symmetry of the
noninteracting and interacting SYK models, separately.

####  V.1.1 Noninteracting SYK2 model

For Brownian randomness Eq. 11 in the main text, it is legitimate to assume
the Green functions are strictly local $G_{ab,r}^{uv}(t,t)\equiv
G_{ab,r}^{uv}(t)$ and antisymmetric $G^{uv}_{ab,r}(t)=-G^{vu}_{ba,r}(t)$ [7],
so for $U=0$ the action can be written as

$\displaystyle-\frac{I}{N}=\frac{1}{2}\text{Tr}\log\left(S\partial_{t}+\Sigma_{r}\right)+\int\frac{1}{2}\text{Tr}\Big{[}\Sigma_{ab,x}G_{ba,x}+\sum_{r^{\prime}}\frac{J_{rr^{\prime}}}{4}G_{ab,r}SG_{ba,r^{\prime}}S+i\frac{\gamma}{2}G_{LR,r}\Big{]},$
(101)

where $S^{uv}=(-1)^{u}\delta^{uv}$ and the trace in the second line is over
the contour indices. The theory is invariant under $O(2)\times O(2)$
transformation, i.e.,

$\displaystyle G_{ab,x}\rightarrow O^{-1}G_{ab,x}O,\quad O^{T}O=1,\quad
O^{T}SO=S,$ (102)

where $O$ acts identically on the left and right chains. (Without the coupling
between the left and the right chains, $\mu=0$ , the action is invariant under
two $O(2)\times O(2)$ for the left and the right chains, respectively,
$G_{ab,x}\rightarrow O^{-1}_{a}G_{ab,x}O_{b}$, where
$O^{T}_{a}O_{a}=1,O^{T}_{a}SO_{a}=1$.) The rotational symmetry is generated by
$\gamma_{(13)}$ and $\gamma_{(24)}$,
$\gamma_{(ij)}^{uv}=\delta^{iu}\delta^{jv}-\delta^{ju}\delta^{iv}$, i.e.,

$\displaystyle O=e^{\theta_{13}\gamma_{(13)}+\theta_{24}\gamma_{(24)}},$ (103)

where $\theta_{13}$ denotes the rotation angle between the contour $1$ and
$3$, and $\theta_{24}$ denotes the rotation angle between the contour $2$ and
$4$.

The saddle point solution (99) for $U=0$ and $\tilde{\gamma}<1$ spontaneously
breaks the relative rotational symmetry. For the noninteracting case, $U=0$,
$\lambda=\sqrt{1-\tilde{\gamma}^{2}}$. There is one Goldstone mode generated
by applying the broken-symmetry generator
$\gamma_{-}\equiv\gamma_{(13)}-\gamma_{(24)}$, i.e.,

$\displaystyle\delta G_{aa,r}(t)$ $\displaystyle=$ $\displaystyle
e^{-\theta_{r}(t)\gamma_{-}}\bar{G}_{aa}(t,t)e^{\theta_{r}(t)\gamma_{-}}-\bar{G}_{aa}(t,t)\approx\sqrt{1-\tilde{\gamma}^{2}}\theta_{r}(t)(\gamma_{(14)}+\gamma_{(23)}),$
(104)

where $\theta_{r}(t)$ denotes the Goldstone mode. In contrast, when
$\tilde{\gamma}>1$, this $O(2)$ symmetry is unbroken and the replicated theory
is in the gapped phase.

Again, with a slight modification the effective theory for the Goldstone mode
reads [6],

$\displaystyle\frac{I_{\text{eff}}}{N}=\frac{\rho}{2}\sum_{k}\int_{\Omega}\left(\frac{\Omega^{2}}{\gamma^{2}}+(1-\epsilon_{k})\right)|\theta_{k}(\Omega)|^{2},$
(105)

where $\theta_{k}=\frac{1}{\sqrt{L}}\sum_{r}\theta_{r}e^{-ikr}$ is the Fourier
transform of the lattice site, and

$\displaystyle\epsilon_{k}\equiv\frac{1}{\zeta(2\alpha)}\sum_{r=1}^{\infty}\frac{\cos
kr}{r^{2\alpha}},$ (106)

resulted from the power-law hopping. Similar results have been obtained in
Ref. [8]. Notice that $\epsilon_{k=0}=1$ consistent with $\theta_{k}(\Omega)$
being a Goldstone mode. The stiffness $\rho=\hat{J}(1-\tilde{\gamma}^{2})$
vanishes at $\tilde{\gamma}=1$, indicating that the transition occurs at
$\gamma=\hat{J}$.

####  V.1.2 Interacting SYK4 model

Now we discuss the interacting case $\tilde{U}>0$. The parameter $\lambda$ is
determined by (100). For small $\tilde{U}$,
$\lambda=\sqrt{1-\tilde{\gamma}^{2}}[1+\tilde{\gamma}^{2}(1-\tilde{\gamma}^{2})^{q/2-2}\tilde{U}+O(\tilde{U}^{2})]$
is well defined when $\tilde{\gamma}<1$, and vanishes continuous as
$\tilde{\gamma}\rightarrow 1$. In the following we will focus on $q=4$, while
our results are true for generic $q$. At the critical point,

$\displaystyle\lambda^{2}\big{(}(2\tilde{U}-1)+(\tilde{U}^{2}-2\tilde{U})\lambda^{2}-\tilde{U}^{2}\lambda^{4}\big{)}=0,$
(107)

which shows that for $\tilde{U}>1/2$ there are two degenerate distinct
physical solutions indicating a discontinuous jump. Thus, the condition for a
continuous transition is $2U<\hat{J}$. On the other hand, when
$\gamma\geq\hat{J}$, the solution is same as the noninteracting case (99) at
$\tilde{\gamma}\geq 1$.

What symmetry out of $O(2)\times O(2)$ is preserved for $\tilde{U}>0$? It is
easy to show that the symmetry reduces to $C_{4}\times C_{4}$, satisfying the
condition
$((O^{-1})^{uu^{\prime}})^{q/2}S^{u^{\prime}v^{\prime}}(O^{v^{\prime}v})^{q/2}=S^{uv}$.
The generator is still given by $\gamma_{(13)}$ and $\gamma_{(24)}$ but the
rotation angle is restricted to multiples of $\pi/2$. The relative rotation
symmetry is spontaneously broken by nonzero $\lambda$ in (99) when $\gamma<J$.
Namely, $\lambda$ serves as an order parameter of the $C_{4}$ symmetry
breaking transition. With a slight modification that replaces $\cos k$ to
$\epsilon_{k}$ and $J$ to $\hat{J}$, the Landau-Ginzburg effective theory
reads

$\displaystyle\frac{I_{\text{eff}}}{N}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1,2;k}\int_{\Omega}\left(\frac{\Omega^{2}}{\gamma}+\hat{J}(1-\epsilon_{k})\right)|\phi_{i,k}(\Omega)|^{2}+\sum_{r}\int_{t}\left(\frac{\gamma-\hat{J}}{2}\vec{\phi}_{r}^{2}+\frac{\gamma}{8}\vec{\phi}_{r}^{4}-\frac{U}{4}(\phi_{1,r}^{4}+\phi_{2,r}^{4})\right),$
(108)

where $\phi_{1}=\delta G^{12}+\delta G^{34}$ and $\phi_{2}=\delta
G^{14}+\delta G^{23}$ transform like a vector under the relative $C_{4}$
rotation. (In this case we have $\delta G^{12}_{RR}=\delta G^{12}_{LL}$,
$\delta G^{34}_{RR}=\delta G^{34}_{LL}$, $\delta G^{14}_{RR}=\delta
G^{14}_{LL}$, $\delta G^{23}_{RR}=\delta G^{23}_{LL}$. So we omit the
subscript of the left and right chains.) This theory features a second order
transition if $2U<\gamma$, and a first order one if $2U>\gamma$, consistent
with the analysis (107) of the saddle-point solution at the transition
$\gamma=\hat{J}$.

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