# Entanglement phases, localization and multifractality of monitored free
fermions in two dimensions

K. Chahine  Institut für Theoretische Physik, Universität zu Köln, D-50937
Cologne, Germany  M. Buchhold  Institut für Theoretische Physik, Universität
zu Köln, D-50937 Cologne, Germany

(August 28, 2024)

###### Abstract

We explore the entanglement structure and wave function properties of
continuously monitored free fermions with $U(1)$-symmetry in two spatial
dimensions (2D). Deriving the fermion replica-Keldysh field theory, and a
bosonic effective long-wavelength action, we discuss similarities and
differences between entanglement phase transitions of monitored fermions in
two dimensions and Anderson-type localization transitions in three dimensions.
We then establish the phenomenology of entanglement transitions of monitored
fermions in 2D by extracting the entanglement entropy, mutual information, and
wave function inverse participation ratio from exact numerical simulations. At
weak monitoring, a characteristic $L\log L$ entanglement growth and a
multifractal dimension $D_{q}=2$ are reminiscent of a metallic Fermi liquid.
For strong monitoring, exponentially localized wave functions yield a
saturation towards area law entanglement. In between, the critical point
displays both an entanglement scaling consistent with an emergent conformal
invariance and strong multifractality. The numerical results are in good
agreement with a mean-field analysis and a one-loop renormalization group
treatment of the field theory. This shapes the picture of a monitoring induced
metal-to-insulator transition in the entanglement content and establishes 2D
monitored fermions as a novel arena to explore the link between non-unitary
quantum dynamics in $D$ dimensions and quantum statistical mechanics in $D+1$
dimensions.

_Introduction._ – The advances in realizing quantum devices with high fidelity
unitary evolution and mid-circuit measurements have put focus on a novel type
of quantum dynamics: the competition between scrambling and localization of
quantum information. Here the competition roots in the non-commutivity of
unitary, i.e., Hamiltonian or gates, and non-unitary, i.e., measurements,
dynamics and it is of genuine quantum mechanical origin. In this new setting,
two types of evolution protocols have crystallized: _quantum circuits_
composed either of discrete unitary gates and projective measurements Skinner
_et al._ (2019); Li  _et al._ (2018, 2019); Gullans and Huse (2020a); Choi
_et al._ (2020); Jian  _et al._ (2020); Fan  _et al._ (2021); Li and Fisher
(2021); Nahum  _et al._ (2021); Jian  _et al._ (2021); Bao  _et al._ (2021);
Turkeshi  _et al._ (2021); Ivanov  _et al._ (2020); Buonaiuto  _et al._
(2021); Zabalo  _et al._ (2020); Jian  _et al._ (2020); Ippoliti  _et al._
(2022); Fisher  _et al._ (2023); Szyniszewski  _et al._ (2020, 2019); Turkeshi
(2022); Pöpperl  _et al._ (2023); Carollo and Alba (2022) or measurements-only
Sriram  _et al._ (2023); Zhu  _et al._ (2022, 2023); Ippoliti  _et al._
(2021); Klocke and Buchhold (2022, 2023); Lavasani  _et al._ (2023); Morral-
Yepes  _et al._ (2023) and _monitored Hamiltonians_ , featuring a continuous
unitary evolution subject to measurements Cao  _et al._ (2019); Alberton  _et
al._ (2021); Buchhold  _et al._ (2021); Poboiko  _et al._ (2023); Ladewig  _et
al._ (2022); Minato  _et al._ (2022); Szyniszewski  _et al._ (2022); Kells
_et al._ (2023); Doggen  _et al._ (2022); Tsitsishvili  _et al._ (2023);
Turkeshi  _et al._ (2022); Piccitto  _et al._ (2022); Müller  _et al._ (2022);
Oshima and Fuji (2023); Paviglianiti and Silva (2023); Minoguchi  _et al._
(2022); Tirrito  _et al._ (2023); Xing  _et al._ (2023); Turkeshi and Schiró
(2023); Fuji and Ashida (2020); Altland  _et al._ (2022); Turkeshi  _et al._
(2023a). The unitary evolution leads to scrambling of quantum information,
e.g., by delocalizing particles or qubits, while measurements yield
localization, causing a quantum phase transition in the entanglement entropy –
the measure for the locality of quantum information.

Scrambling versus localization of quantum information in a $D$-dimensional
monitored system has been linked to aspects of localization-delocalization
transitions in $D+1$-dimensional statistical mechanics of disordered quantum
systems, both for circuits Zabalo  _et al._ (2022); Sierant and Turkeshi
(2022); Klocke and Buchhold (2023); Bao  _et al._ (2020) and Hamiltonian
setups Buchhold  _et al._ (2021); Poboiko  _et al._ (2023); Szyniszewski  _et
al._ (2022); Boorman  _et al._ (2022); Yamamoto and Hamazaki (2023); Jian  _et
al._ (2023); Fava  _et al._ (2023). In particular, for a monitored
$D$-dimensional Hamiltonian evolution, this viewpoint interprets random, non-
unitary measurements similarly to a static disorder potential in $D+1$
dimensions. However, the majority of works so far has focused on dynamics in
one spatial dimension; and for the particular case of fermions with
$U(1)$-symmetry has reported results that are partially at variance Cao  _et
al._ (2019); Alberton  _et al._ (2021); Buchhold  _et al._ (2021);
Szyniszewski  _et al._ (2022); Minato  _et al._ (2022). This is a challenging
limit: the quantum dynamics in $D=1$ spatial dimension displays peculiar
properties Haldane (1981); Witten (1984); just like the statistical mechanics
of disordered systems at the lower critical dimension $D+1=2$ Wegner (1989);
Hikami (1981); Finkel’stein (1984).

Figure 1:  (a) Free fermions hopping on an $L\times L$ square lattice subject
to continuous monitoring with rate $\gamma$. (b) Sketch of the time evolution.
Blue sheets represent snapshots of trajectories in which observables are
computed. The half-system entanglement entropy is obtained from strips
$A=L\times L/2$ shown in yellow. (c) Schematic phase diagram showing
observables and their behavior in each regime. (d) The entanglement entropy
$S(A)$ shows the onset of an area law for strong monitoring and logarithmic
violations for weak monitoring. (d) Multifractal exponent $D_{q}$ of the
inverse participation ratio $P_{q}$: At weak monitoring (blue) the system is
metallic with $D_{q}=2$ for all $q$. At the critical point (pink), the wave
functions exhibit multifractality with $D_{2}=1.80(9)$, $D_{3}=1.1(5)$ and
$D_{4}=0.7(4)$.

Here we explore the link between monitored quantum systems and quantum
disordered ground states in a new arena, which displays an even richer
phenomenology and is less ambiguous. We study the dynamics of weakly
monitored, $U(1)$-symmetric free fermions in $D=2$ spatial dimensions in a
two-thronged approach, combining analytical tools and large scale numerical
simulations. A strong link is provided by fermion Keldysh replica field theory
– its structure is reminiscent of the field theory for free disordered
fermions, i.e., it contains a dissipative four-fermion vertex Kamenev (2011).
However, we also point out clear differences: the vertex is local in space and
time and it has a reduced rotation symmetry in Keldysh coordinates. This
yields a modified long-wavelength theory – a SU$(R)$-symmetric nonlinear sigma
model with a space-time rotation symmetry and dynamical critical exponent
$z=1$, corresponding to the chiral unitary class AIII in three dimensions.

We reveal the phenomenology of a monitoring-induced entanglement phase
transition (MIPT), which is reminiscent of disorder-induced localization
transitions: at weak monitoring, the half-system entanglement entropy displays
a characteristic $L\log L$-growth, which saturates towards an area law at
strong monitoring. Determining the inverse participation ratio Sierant and
Turkeshi (2022); Zabalo  _et al._ (2022); Turkeshi  _et al._ (2023b); Niroula
_et al._ (2023) of the monitored wave functions – a key observable in the
context of Anderson-type localization Mildenberger  _et al._ (2002); Evers and
Mirlin (2008) – we demonstrate multifractal behavior at the transition point.
For weak monitoring, instead, the wave functions obey a behavior familiar from
metallic ground states.

A further link to localization transitions is established through the mutual
information $\mathcal{I}(A,B)$ between two subregions $A,B$, separated by a
distance $d_{AB}$. It shows a scale invariant decay at weak monitoring,
consistent with metallic wave functions, and the onset of exponential
localization at strong monitoring. Resolving exact localization for strong
monitoring turns out to be challenging due to an apparently large localization
length. We thus map out the localization transition by two different scaling
approaches; and by revealing an emergent conformal invariance at the critical
point.

A peculiar role is played by purification of an initial mixed state. We
numerically show that the purification time scale directly reveals the
multifractal exponent. This is particularly relevant: for free fermions the
subsystem entanglement is exclusively determined by particle number
fluctuations, equating the purification transition with an MIPT _and_ a charge
sharpening transition Agrawal  _et al._ (2022); Majidy  _et al._ (2023);
Barratt  _et al._ (2022) with multifractal behavior.

Microscopic model. – We consider fermions on a half-filled 2D $L\times L$
square lattice, described by creation and annihilation operators
$\\{\hat{c}_{\ell}^{\phantom{{}^{\dagger}}}$,
$\hat{c}_{\ell^{\prime}}^{\dagger}\\}=\delta_{\ell,\ell^{\prime}}$ on lattice
sites $\ell,\ell^{\prime}$. The fermions undergo coherent nearest-neighbour
hopping while the particle number $\hat{n}_{\ell}$ at each site is
continuously monitored. In the quantum state diffusion protocol they follow
the stochastic Schrödinger equation (SSE) Alberton  _et al._ (2021); Cao  _et
al._ (2019) with monitoring rate $\gamma$,

$\displaystyle d\ket{\psi_{t}}$
$\displaystyle=\Big{[}-i(\hat{H}-i\hat{D})dt+\sum_{\ell}\xi_{\ell,t}(\hat{n}_{\ell}-\langle\hat{n}_{\ell}\rangle_{t})\Big{]}\ket{\psi_{t}},$
(1) $\displaystyle\hat{H}$
$\displaystyle=-\sum_{\langle\ell,m\rangle}\hat{c}_{m}^{\dagger}\hat{c}_{\ell}^{\phantom{{}^{\dagger}}}+\hat{c}_{\ell}^{\dagger}\hat{c}_{m}^{\phantom{{}^{\dagger}}},\
\hat{D}=\tfrac{\gamma}{2}\sum_{\ell}(\hat{n}_{\ell}-\langle\hat{n}_{\ell}\rangle_{t})^{2}.$

Here $\xi_{\ell,t}$ is a Gaussian white noise with zero mean
$\overline{\xi_{\ell}}=0$ and short-ranged correlations
$\overline{\xi_{\ell,t}\xi_{\ell^{\prime},t^{\prime}}}=\gamma
dt\delta_{\ell,\ell^{\prime}}\delta(t-t^{\prime})$. The overbar denotes the
trajectory average.

Replica master equation. – An analytical treatment of the SSE (1) demands a
proper average over the random measurement outcomes, which is done in a
replica framework for the unnormalized wave function
$|\tilde{\psi}_{t}\rangle$. It is convenient to express the measurement update
via the generalized projector

$\displaystyle\hat{M}(\\{J_{\ell,t}\\})=\prod_{\ell}[\tfrac{2\gamma
dt}{\pi}]^{\frac{1}{4}}\exp[-\gamma dt(J_{\ell,t}-\hat{n}_{\ell})^{2}].$ (2)

After an outcome $J_{\ell,t}\in\mathds{R}$ was recorded in a weak measurement
of $\hat{n}_{\ell}$ at time $t$, it evolves the wave function
$|\tilde{\psi}_{t}\rangle\to|\tilde{\psi}_{t+dt}\rangle=\text{exp}(-i\hat{H}dt)\hat{M}_{\ell}(\\{J_{\ell,t}\\})|\tilde{\psi}_{t}\rangle$.
The Born probability to detect outcome $\\{J_{\ell,t}\\}$ is
$p(\\{J_{\ell,t}\\})=\langle\tilde{\psi}_{t}|\hat{M}_{\ell}(\\{J_{\ell,t}\\})^{2}|\tilde{\psi}_{t}\rangle\langle\tilde{\psi}_{t}|\tilde{\psi}_{t}\rangle^{-1}=\langle\tilde{\psi}_{t+dt}|\tilde{\psi}_{t+dt}\rangle\langle\tilde{\psi}_{t}|\tilde{\psi}_{t}\rangle^{-1}$.
This formulation is equivalent to the SSE (1) in the limit of infinitesimal
time steps $dt\to 0^{+}$. Then the Born probabilities are implicitly
implemented by replacing
$J_{\ell,t}\to\langle\hat{n}_{\ell}\rangle_{t}+\tfrac{\xi_{\ell,t}}{2\gamma
dt}$ with the Gaussian white noise $\xi_{\ell,t}$ defined above sup ;
Minoguchi  _et al._ (2022).

Now we introduce $R$ replicas of the fermion Hilbert space, labeled by
$r=1,...,R$ and fermion operators acting on Hilbert space $r$:
$\hat{c}^{\dagger(r)}_{\ell},\hat{c}^{(r)}_{\ell}$ and
$\hat{n}_{\ell}^{(r)}=\hat{c}^{\dagger(r)}_{\ell}\hat{c}^{(r)}_{\ell}$. For
fixed $R,M>0$, we define the measurement-averaged density matrix

$\displaystyle\rho_{R,M}$
$\displaystyle=\overline{p(\\{J_{\ell,t}\\})^{R}\otimes_{r=1}^{M}|\psi_{t}\rangle\langle\psi_{t}|}=\overline{\Tr(|\tilde{\psi}_{t}\rangle\langle\tilde{\psi}_{t}|)^{R-M}\otimes_{r=1}^{M}|\tilde{\psi}_{t}\rangle\langle\tilde{\psi}_{t}|}$
$\displaystyle=\Tr_{r>M}\tilde{\rho},\text{ with }\ \
\tilde{\rho}=\overline{\otimes_{r=1}^{R}|\tilde{\psi}_{t}\rangle\langle\tilde{\psi}_{t}|}.$
(3)

Here, the trajectory average $\overline{.\phantom{a}.\phantom{a}.}$
corresponds to integrating over all possible outcomes
$J_{\ell,t}\in\mathds{R}$. The equation relates the _nonlinear average_ over
an $M$-replicated, normalized wave function, weighted with Born probability
$p(\\{J_{\ell,t}\\})^{R}$ to a _linear average_ over $R$-replicated,
unnormalized wave functions. It is well-defined for $R\geq M$. For $M>1$, the
physically meaningful case $R=1$ has to be obtained via analytic continuation.

We consider the evolution of $\tilde{\rho}$. Taking
$\partial_{t}\tilde{\rho}$, performing the measurement average and expanding
the result up to $\mathcal{O}(dt)$ yields the replica quantum master equation
(rQME) sup

$\displaystyle\partial_{t}\tilde{\rho}=$ $\displaystyle
i\big{[}\tilde{\rho},\hat{H}_{R}\big{]}-\tfrac{\gamma}{R}\sum_{\ell}\left(\\{\hat{N}_{\ell}(R-\hat{N}_{\ell}),\tilde{\rho}\\}+\tfrac{1}{2}\big{[}\hat{N}_{\ell},\big{[}\hat{N}_{\ell},\tilde{\rho}\big{]}\big{]}\right).$
(4)

Here, $\hat{H}_{R}=\sum_{r}\hat{H}^{(r)}$ and
$\hat{N}_{\ell}=\sum_{r}\hat{n}_{\ell}^{(r)}$ are the sum over each
$r$-replicated Hamiltonian and number operator. Both are quadratic in fermions
and thus invariant under replica-rotations. A symmetry, which is inherited by
$\tilde{\rho}$. The evolution features a competition between $\hat{H}_{R}$ and
the second term $\sim\hat{N}_{\ell}$, which induces the entanglement phase
transition: the latter implements a non-Hermitian evolution towards a replica-
aligned local particle density $\hat{N}_{\ell}=0,R$. The Hamiltonian instead
pushes $\hat{N}_{\ell}\to\tfrac{R}{2}$, aiming to maximize the kinetic energy.

Replica field theory. – The fermion rQME is equivalent to a Keldysh path
integral Sieberer  _et al._ (2016); Kamenev (2011), where the fermion
operators acting at time $t$ and site $\ell$ are replaced by Grassmann
variables
$(\hat{c}^{\dagger(r)}_{\ell},\hat{c}^{(r)}_{\ell})\to(\bar{\psi}^{(r)}_{\alpha,X},\psi^{(r)}_{\alpha,X})$.
Each Grassmann variable acquires an additional Keldysh index $\alpha=1,2$,
distinguishing retarded/advanced and Keldysh sectors, and we draw the
continuum limit of the space-time lattice $(t,\ell)\to(t,\mathbf{x})\equiv
X\in\mathds{R}^{3}$. Following standard procedure Sieberer  _et al._ (2016);
Kamenev (2011); sup , the rQME yields a partition function
$Z=\int\mathcal{D}[\\{\psi,\bar{\psi}\\}]\text{ exp}(iS_{\psi})$ with the
action

$\displaystyle
S_{\psi}=\int_{X}\left\\{\bar{\psi}_{X}G_{0}^{-1}\psi_{X}-\tfrac{i\gamma}{2R}\Tr[(\sigma^{K}_{x}\psi_{X}\bar{\psi}_{X})^{2}]\right\\}.$
(5)

Here $\psi_{X}=\\{\psi^{(r)}_{\alpha,X}\\}$ is a $2R$-Grassmann vector and the
trace runs over replica and Keldysh index; the Pauli matrix $\sigma^{K}_{x}$
acts on the Keldysh index $\alpha$. At half-filling, the bare Keldysh Green’s
function in momentum space $P\equiv(\omega,\mathbf{p})$ is
$G_{0}(P)=\delta_{r,r^{\prime}}[(\omega-\epsilon_{\mathbf{p}})\mathds{1}-i0^{+}\sigma_{z}^{K}]^{-1}$
with dispersion $\epsilon_{\mathbf{q}}=2\text{cos}(p_{x})+2\text{cos}(p_{y})$.

The action $S_{\psi}$ is reminiscent of the Keldysh action for disordered
fermions with $U(1)$-symmetry Kamenev (2011). The quartic fermion vertex,
however, displays two crucial differences compared to static disorder, which
are intrinsic to a monitored, i.e., dynamic, theory. Firstly, the structure in
Keldysh space is modified by $\sigma_{x}^{K}$ (disorder:
$\sigma_{x}^{K}\to\mathds{1}$). This reduces the symmetry of rotations in
Keldysh-replica space: it gaps out $R^{2}-1$ rotation modes, which can be
treated perturbatively. The result is an _emergent space-time invariance_ at
long wavelengths and a dynamical critical exponent $z=1$ sup . This enables an
emergent conformal invariance at the critical point. Secondly, the vertex is
space-time local: measurements, unlike disorder, vary in _time and space_.
This eliminates the possibility of fermions to interfere with their time-
reversed partners and gives rise to _two local_ long-wavelength modes.

In 2D, fermions do not separate into left- and right-movers Buchhold  _et al._
(2021) and bosonization amounts to a Hubbard-Stratonovich decoupling
$\psi_{X}\bar{\psi}_{X}\to\tfrac{1}{2}Q_{X}\in\mathds{C}^{2R\times 2R}$. This
yields

$\displaystyle iS_{Q}=$ $\displaystyle
iS_{0}+\tfrac{\gamma}{8R}\int_{X}\Big{\\{}\left[\Tr(\sigma^{K}_{x}Q_{X})\right]^{2}-\Tr\left[(\sigma^{K}_{x}Q_{X})^{2}\right]\Big{\\}}.$
(6)

with $iS_{0}=\int_{X}\Tr\ln(G_{0}^{-1}+i\tfrac{\gamma}{2R}Q_{X})$. Each
$Q_{X}$ is a local $2R\times 2R$ matrix in Keldysh and replica index. The
replica-diagonals
$Q_{\alpha\beta,X}^{(rr)}=\psi^{(r)}_{\alpha,X}\bar{\psi}_{\beta,X}^{(r)}$
represent the physical fermion bilinears, such as, e.g., the fermion density
$n_{X}^{(r)}=\tfrac{1}{2}(Q_{12,X}^{(rr)}+Q_{21,X}^{(rr)})$. This provides
access to the connected correlation function
$C(\mathbf{x}-\mathbf{x}^{\prime},t)=\langle
n_{\mathbf{x},t}^{(r)}[n_{\mathbf{x}^{\prime},t}^{(r)}-n_{\mathbf{x}^{\prime},t}^{(r^{\prime})}]\rangle=\overline{\langle\hat{n}_{\mathbf{x}}\hat{n}_{\mathbf{x}^{\prime}}\rangle-\langle\hat{n}_{\mathbf{x}}\rangle\langle\hat{n}_{\mathbf{x}^{\prime}}\rangle}$
in the field theory framework Poboiko  _et al._ (2023). For free fermions this
further enables to compute the entanglement entropy
$S_{A}=2\zeta(2)\mathcal{C}_{A}^{(2)}+\mathcal{O}(C_{A}^{(2n\geq 4)})$, where
$C_{A}^{(2n)}$ is the $2n$-th order cummulant of the particle number
$\hat{N}_{A}=\sum_{\ell\in A}\hat{n}_{\ell}$ in subregion $A$ Klich and
Levitov (2009); Song  _et al._ (2012). In particular,
$C_{A}^{(2)}=\int_{\mathbf{x},\mathbf{x}^{\prime}\in
A}C(\mathbf{x}-\mathbf{x}^{\prime},t)$, which has to be computed with
‘absorbing’ temporal boundary conditions Poboiko  _et al._ (2023); sup .

We analytically compute the entanglement in a Gaussian approximation: we
determine the saddle-point of $S_{Q}$ – equivalent to the self-consistent Born
approximation with $Q_{X}=\delta_{r,r^{\prime}}\sigma_{z}$ and the fermion
retarded Green’s function
$g^{R}(P)=(\omega-\epsilon_{\mathbf{p}}+i\tfrac{\gamma}{2R})^{-1}$ – then we
expand around the saddle up to quadratic order in $Q_{X}$. Replica diagonal
and off-diagonal terms decouple and for $|\mathbf{p}|\ll\pi$ and $A=L/2\times
L$ we find

$\displaystyle
C(P)=\tfrac{\mathbf{p}^{2}}{\gamma(\omega^{2}+4\mathbf{p}^{2})}\text{ and
}S_{A}=\tfrac{\pi}{3\gamma}\ln(\pi L/2)L.$ (7)

Figure 2:  Entanglement phase transition. (a) A sharp crossing of $\gamma
c(\gamma)$ locates the critical point at $\gamma_{c}=2.15$. (b) Finite size
scaling collapse of the entanglement entropy for $\gamma_{c}=2.15$. A
different exponent is found for the metallic ($\nu=1$) and the localized phase
($\nu=1.12$). (c) Entropy density scaling with strip size $A=L\times\ell_{A}$
compared with with Eq. (12). Deviations are visible for all curves except for
$\gamma=\gamma_{c}$ (see sup  for original data). (d) A scaling collapse of
the mutual information as a function of the chord distance $\tilde{d}$ is only
possible in the metallic phase. The inset shows the unrescaled data in a semi-
log scale, highlighting the exponential decay of the mutual information for
$\gamma\gg\gamma_{c}$.

_Renormalization group analysis._ – In order to incorporate long-wavelength
fluctuations at the phase transition, we parameterize each matrix
$Q_{X}=\mathcal{R}_{X}\Lambda\mathcal{R}_{X}^{-1}$ in terms of rotations
$\mathcal{R}_{X}$ around the saddle point
$\Lambda=\delta_{r,r^{\prime}}\sigma_{z}^{K}$. For the $2R\times 2R$ matrix
$Q_{X}$, this yields $4R^{2}$ generators of rotations,
$\sigma_{\alpha}^{K}\otimes\Theta_{\alpha}$, $\alpha\in\\{0,x,y,z\\}$, which
are compatible with the $U(2R)$-symmetry. Each $\Theta_{\alpha}$ is a $R\times
R$ hermitian matrix and $\sigma_{\alpha}^{K}$ are Pauli matrices in Keldysh
space with $\sigma_{0}^{K}=\mathds{1}_{2\times 2}$. The saddle point $\Lambda$
is not rotated by the $2R^{2}$ generators $\sim\sigma_{0},\sigma_{z}$, leaving
$2R^{2}$ possible generators and a $U(2R)/U(R)\times U(R)$ manifold. The
quartic part of the action $S_{Q}$ additionally commutes with $\sigma^{K}_{x}$
but not with $\sigma^{K}_{y}$. Thus rotations generated by the former (latter)
are gapless (gapped), except for the trace of $\Theta_{y}$.

Performing the common derivation of the nonlinear sigma model Kamenev (2011);
sup , and eliminating the gapped $\Theta_{y}$-modes in a Gaussian
approximation yields the action

$\displaystyle
iS_{U}=-\frac{g}{2}\int_{X}\left[\partial_{t}U\partial_{t}U^{-1}+\nabla
U\nabla U^{-1}\right]$ (8)

with $U\in\text{SU}(R)$ and $g=(32\gamma^{2})^{-\frac{1}{2}}$. This sigma
model also emerges from a disordered Hamiltonian in the chiral unitary class
AIII Evers and Mirlin (2008). In the replica limit $R\to 1$ and in $2+1$
dimensions, the one-loop renormalization group (RG) flow for the dimensionless
coupling $\tilde{g}=gl$ is (RG scale $l$) Wegner (1989); Hikami (1981); Gade
and Wegner (1991)

$\displaystyle\partial\tilde{g}(l)/\partial\ln(l)=\tilde{g}(l)-\tfrac{1}{4\pi}\
\Rightarrow\
\tilde{g}(l)=\tfrac{1}{4\pi}+(\tilde{g}(l_{0})-\tfrac{1}{4\pi})\tfrac{l}{l_{0}}.$
(9)

Here, we have introduced the UV-scale $l_{0}=1$ (dimensionless lattice
spacing) and $\tilde{g}(l_{0})=g$. The flow equation predicts a monitoring-
induced phase transition at a _critical monitoring rate_
$\gamma_{c,\text{th}}=\tfrac{\pi}{\sqrt{2}}\approx 2.22$. For
$\gamma<\gamma_{c,\text{th}}$ ($\gamma>\gamma_{c,\text{th}}$), the theory
flows to weak (strong) coupling, i.e., $\gamma\to 0$ ($\gamma\to\infty$).

_Numerical simulations._ – The SSE (1) is quadratic in
$\hat{c}_{\ell},\hat{c}_{\ell}^{\dagger}$ and number conserving. It is exactly
and efficiently simulated with Gaussian wave functions Cao  _et al._ (2019);
Alberton  _et al._ (2021) of the form:

$\ket{\psi_{t}}=\prod_{1\leq s\leq L^{2}/2}c^{\dagger}_{s}\ket{0},\quad
c^{\dagger}_{s}=\sum_{1\leq\ell\leq
L^{2}}\psi^{s}_{\ell,t}c^{\dagger}_{\ell}.$ (10)

$\psi^{s}_{\ell,t}\in\mathds{C}$ is the single-particle wave function of
fermion $s$ at site $\ell$. Both the state $\ket{\psi_{t}}$ and the wave
functions $\psi^{s}_{\ell,t}$ at time $t$ implicitly depend on the history of
noise events $\\{\xi_{\ell,t^{\prime}}\\}$ with $t^{\prime}<t$. We initialize
each $\ket{\psi_{t=0}}$ in a random state and evolve it until observables
reach stationary values.

The observables here are the von Neumann entanglement entropy $S(A)$, the
mutual information $\mathcal{I}(A,B)$ and the instantaneous single-particle
wave functions $\psi_{\ell}^{s}$. The trajectory average of $S(A)$ and
$\mathcal{I}(A,B)$ for 2D subsystems $A,B$ are

$\displaystyle S(A)=-\Tr(\overline{\rho_{A}\ln\rho_{A}}),\
\mathcal{I}(A,B)=S(A)+S(B)-S(A\cup B).$

The reduced density matrix
$\rho_{A}=\Tr_{\bar{A}}|\psi_{t}\rangle\langle\psi_{t}|$ is obtained by
tracing over the complement of $A$ sup . Here, we take each subsystem as a
strip of size $A=L\times l_{A}$, see Fig. 1(b). The wave functions
$\psi_{\ell}^{s}$ are characterized by their inverse participation ratio (IPR)
$P_{q}$, its variance $\sigma_{q}$ and anomalous dimension $D_{q}$,

$P_{q}=2L^{-2}\sum_{s,\ell}|\psi^{s}_{\ell,t}|^{2q}\sim L^{-D_{q}(q-1)},\
\sigma_{q}^{2}=\text{var}(\ln P_{q}).$ (11)

Entanglement phase transition. – Numerical simulations underpin two different
entanglement regimes: for weak measurements the entanglement entropy for
$A=L\times L/2$ grows as $S(A)=c(\gamma)L\ln(L)+s(\gamma)L$, which is
reminiscent of a 2D metallic state and consistent with the field theory result
in Eq. (7). The prefactor of the logarithmic growth coincides well with the
predicted value $c(\gamma)\sim\tfrac{\pi}{3\gamma}$, shown in Fig. 2(a). For
strong measurements one observes the onset of a saturation towards an area law
$S(A)\to s(\gamma)L$, visible in Fig. 1(d). However, even up to linear sizes
of $L=80$ and up to $\gamma\sim 4.5$ a faster than linear growth of $S(A)$ is
observed. In order to detect the measurement-induced entanglement phase
transition, we extract the prefactor $c(\gamma)$. It is shown in Fig. 2(a) and
it displays a sharp crossing at $\gamma=2.15$. This indicates a phase
transition close to the theory prediction $\gamma_{c,\text{th}}=2.22$.

We further confirm the entanglement transition and its scaling in two steps.
(i) We perform a scaling analysis of the entanglement entropy according to the
one-loop RG result for the renormalized coupling $\tilde{g}(l)$ in Eq. (9). In
lowest order approximation $\tilde{g}(l)$ replaces the prefactor
$\sim\gamma^{-1}$ in the entanglement scaling in Eq. (7). This predicts a
deviation
$|S(\gamma)-S(\gamma_{c})|\sim|1-\tfrac{\gamma_{c}}{\gamma}|^{\nu}L\ln L$ with
one-loop critical exponent $\nu=1$. For $\gamma_{c}=2.15$, we show the scaling
collapse in Fig. 2(b). It confirms the functional form $\sim L\ln(L)$ but
predicts different scaling exponents for the metallic ($\nu=1$) and the
localized ($\nu=1.12$) side of the transition. This may be attributed to a
strong renormalization of the Gaussian prediction in Eq. (7) in the localized
phase.

(ii) We consider the scaling of the entropy line density $s(A)=S(A)/L$ for
fixed system size $L=64$ and variable strip size $A=L\times l_{A}$ in Fig.
2(c). We compare it with the formula

$s(A)=c(\gamma)\ln\left[\tfrac{L}{\pi}\sin\left(\tfrac{\pi
l_{A}}{L}\right)\right]+s_{0}(\gamma),$ (12)

which we expect to hold for a quasi-one-dimensional system with conformal
symmetry Calabrese and Cardy (2004); Casini and Huerta (2009). At
$\gamma=\gamma_{c}$, we observe a perfect match with Eq. (12), while the
entanglement curve is flatter (sharper) for $\gamma>\gamma_{c}$
($\gamma<\gamma_{c}$), indicating a higher symmetry at the critical value,
i.e., confirming a critical point.

Figure 3:  Multifractality and purification. (a) Multifractal exponent $D_{q}$
as a function of $\gamma$ for different values of $q$ and obtained by two
different methods. (b) The variance $\sigma_{q}$ of the distribution
$\mathcal{P}(\ln P_{q})$ both for the monitored fermions (blue crosses) and
the analytical prediction from the three-dimensional Anderson model
Mildenberger  _et al._ (2002) (orange and green lines). (c) Finite size
scaling for the ancilla purification. The purification rate
$\tau_{\text{pur}}\sim L^{\alpha}$ distinguishes strong (red, $\alpha=0$) and
weak (blue $\alpha=2$) monitoring. (d) Comparison of the purification rate
exponent $\alpha$ and the multifractal exponent $D_{2}$ for different $\gamma$
(dashed lines are guides to the eye). Inset: sketch of the purification setup
with ancilla (yellow).

Mutual Information. – The mutual information $\mathcal{I}(A,B)$ represents an
upper bound for the correlations between two disjoint subsystems $A,B$ Wolf
_et al._ (2008). For free fermions with $U(1)$ symmetry, it is related to
particle number fluctuations between $A$ and $B$ via the entanglement entropy
Klich and Levitov (2009); Song  _et al._ (2012). To leading order, one finds
$\mathcal{I}(A,B)=4\zeta(2)\int_{\mathbf{x}\in A,\mathbf{x}^{\prime}\in
B}C(\mathbf{x}-\mathbf{x}^{\prime},t)$. For two strips of size $L\times 1$,
separated by a distance $d_{AB}$, see Fig. 2(d), Eq. 7 predicts
$\mathcal{I}(A,B)\sim d_{AB}^{-2}/\gamma$. This is confirmed by a scaling
collapse of $\mathcal{I}(A,B)$ for $\gamma\leq\gamma_{c}$ in Fig. 2(d). For
$\gamma\gg\gamma_{c}$, one instead observes an exponential decay
$\log(\mathcal{I}(A,B))\sim-d_{AB}$, consistent with an area law, localized
state. We note that an apparent algebraic decay is still found for relatively
large $\gamma>\gamma_{c}$. The scaling collapse, however, is absent. We
interpret this as a signature for a large correlation length in the area law
phase.

Multifractality. – A striking feature of Anderson-type localization
transitions is multifractality at the critical point Evers and Mirlin (2008).
It describes strong fluctuations of single-particle wave functions in terms of
the so-called multifractal exponent $D_{q}$ of the IPR, defined in Eq. (11).
It distinguishes a metallic phase, with $D_{q}=D$ the spatial dimension, from
a localized, area law phase where $D_{q}\to 0$, both independent of $q$. At a
multifractal critical point $0<D_{q}<2$ becomes a non-trivial function of $q$.

We observe clear signatures of multifractality, demonstrated by the scaling of
$P_{q}$ at the critical point in Fig. 1(e). In Fig. 3(a) we show $D_{q}$ for a
range of $q$ and $\gamma$ values, which witnesses an extended region of
multifractal behaviour. We compute $D_{q}$ in two different ways, from a fit
of $\text{ln}(P_{q})/(1-q)$ versus $\text{ln}(L)$ for $L=10-64$ and directly
through $(1-q)D_{q}=\text{ln}(P_{q})/\text{ln}(L)$ at $L=64$. $D_{q}$ shows a
sigmoid behaviour: it saturates to $D_{q}\to 2$ for $\gamma\to 0$ for all $q$
as expected and slowly decays with increasing monitoring strength. We note
that $D_{q}$ approaches the value $D_{q}\to 0$ only very slowly, i.e., for
both $L$ and $\gamma$ large, consistent with a large localization length.

A second key observable characterizing multifractality is the distribution
function of the IPR, $\mathcal{P}(\ln P_{q})$, in particular its variance
$\sigma_{q}$. We show $\sigma_{q}$ at the critical point $\gamma_{c}$ in Fig.
3(b) and compare it to the analytical prediction for the orthogonal class A in
$D=3$ spatial dimensions. It predicts Mildenberger  _et al._ (2002)

$\sigma_{q}=\left\\{\begin{array}[]{ll}2\pi b|q(q-1)|&\text{ for }|q|<q_{+}\\\
q/q_{+}&\text{ for }|q|>q_{+}\end{array}\right.,$ (13)

with $b\simeq 0.088$ and $q_{+}\simeq\sqrt{2}$. Despite the different symmetry
class and replica limit ($R\to 1$), we find good agreement between our data
and Eq. (13). This further underpins the link between monitored fermions and
Anderson-type localization.

Purification. – MIPTs can be revealed from the purification of an initial
mixed state Gullans and Huse (2020b, a); Gopalakrishnan and Gullans (2021);
Lóio  _et al._ (2023). Here we extract the purification time scale
$\tau_{\text{pur}}\sim L^{\alpha}$ in a modified setup. In an initial stage
($t<0$), the $L\times L$ lattice is coupled to a $L\times 1$ ancilla lattice
via coherent hopping ($\gamma=0$). It entangles the system with the ancilla by
bringing fermions into a superposition between both, see Fig. 3(d). At time
$t=0$, the ancilla is decoupled and only the $L\times L$ lattice undergoes
monitoring described by the SSE (1). The purity is provided by the entropy
density of the ancilla $s_{\text{anc}}\equiv S(L\times
1)/L\sim\text{exp}(-t/\tau_{\text{pur}})$.

For $\gamma<\gamma_{c}$, we observe that the purification exponent $\alpha$
closely resembles the multifractal exponent $D_{2}$. For $\gamma>\gamma_{c}$,
it rapidly approaches $\alpha\to 0$. The relation between purification and
multifractality appears intuitive: the system purifies when particles (or
holes) that were initially shared by system and ancilla are localized in the
system by measurement. The probability of finding such particle in one
measurement step will depend on how its wave function is scrambled, i.e., on
its multifractal structure. For free fermions, purification can thus be used
to probe the multifractal behavior at the MIPT. Due to the equivalence of
particle number fluctuations and entanglement entropy, purification, including
multifractal behavior, is identical to charge sharpening.

Outlook.– Continuously monitored, free fermions in 2D undergo a MIPT, whose
effective field theory and phenomenology are strongly reminiscent of an
Anderson-type localization transition in 3D. However, Anderson transitions are
quantum phase transitions – they vitally depend on wave function interference.
This raises the question of how strong the link between monitored fermions in
$D$ dimensions and Anderson localization in $D+1$ dimensions is – and whether
monitored fermions represent a new universality class, for which the
$D$-dimensional quantum dynamics maps to a $D+1$ _quantum_ statistical
mechanics. The fact that both the entanglement transition and the peculiar
phenomenon of multifractality are observable via purification makes 2D
monitored fermions an ideal test bed to explore these questions. An
interesting future direction would then be to devise experimental schemes to
explore the transition Noel  _et al._ (2022), for instance through appropriate
feedback protocols O’Dea  _et al._ (2022); Iadecola  _et al._ (2023); Buchhold
_et al._ (2022); Piroli  _et al._ (2023); Sierant and Turkeshi (2023).

###### Acknowledgements.

## Acknowledgements

We thank S. Diehl, C. M. Jiang, K. Klocke, I. Poboiko, M. Szyniszewski, X.
Turkeshi and J. H. Wilson for fruitful discussions. We acknowledge support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy Cluster of Excellence Matter and Light for
Quantum Computing (ML4Q) EXC 2004/1 390534769, and by the DFG Collaborative
Research Center (CRC) 183 Project No. 277101999 - project B02. The code for
our numerical computations was implemented in Julia Bezanson  _et al._ (2017).

_Note added_ – During the completion of the manuscript, we learned about
related works on monitored fermions in $D\geq 2$ that appeared simultaneously
Poboiko  _et al._ (2023); Jin and Martin (2023).

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##  Appendix A Replica Master Equation

_Derivation of the rQME._ – Consider the unnormalized, trajectory averaged
density matrix
$\tilde{\rho}=\overline{\otimes_{r=1}^{R}|\tilde{\psi}_{t}\rangle\langle\tilde{\psi}_{t}|}$.
In an infinitesimal time step $t\to t+dt$, for each replica one unitary
operator $\exp{-i\hat{H}dt}$ and one generalized projector
$\hat{M}(\\{J_{\ell,t})$ from Eq. (2) act from the left and from the right on
the density matrix. We attribute each operator a replica label $r$ as in the
main text, indicating the replica Hilbert space, and an additional sign
$\sigma=\pm$ indicating whether it acts from the left or from the right onto
$\tilde{\rho}$, i.e., $\hat{c}_{\ell}\rightarrow\hat{c}_{\ell,\sigma}^{(r)}$.
This simplifies the notation: each pair of fermions
$\hat{c}_{\ell,\sigma}^{(r)},\hat{c}_{\ell^{\prime},\sigma^{\prime}}^{(r^{\prime})}$
is independent if any of the labels differ from each other. The evolution is
then $\tilde{\rho}_{t+dt}=U^{\otimes
2R}\overline{\hat{M}(\\{J_{\ell,t}\\})^{\otimes 2R}}\tilde{\rho}_{t}$, with
$U^{\otimes 2R}=\mathds{1}-idt\sum_{r=1}^{R}(H^{(r)}_{+}-H^{(r)}_{-})$ and the
averaged replica projector. Before the average,

$\displaystyle\hat{M}(\\{J_{\ell,t}\\})^{\otimes
2R}=\prod_{\ell}\left[\frac{2\gamma
dt}{\pi}\right]^{\frac{R}{2}}\exp\left[-\gamma
dt\sum_{\sigma=\pm,r}(J_{\ell,t}-\hat{n}_{\ell,\sigma}^{(r)})^{2}\right]$
$\displaystyle=\prod_{\ell}\left[\frac{2\gamma
dt}{\pi}\right]^{\frac{R}{2}}e^{-\gamma
dtR(2\tilde{J}_{\ell,t}^{2}+L^{2})}\exp{\tfrac{\gamma
dt}{2R}\left(\hat{N}_{\ell,+}+\hat{N}_{\ell,-}\right)^{2}}.$ (14)

Here we have introduced
$\hat{N}_{\ell,\sigma}=\sum_{r=1}^{R}\hat{n}_{\ell,\sigma}^{(r)}$ and
$\tilde{J}_{\ell,t}=J_{\ell,t}-\frac{1}{2R}\left(\hat{N}_{\ell,+}+\hat{N}_{\ell,-}\right)$
and we have used the fermion property that
$(\hat{n}_{\ell,\sigma}^{(r)})^{2}=\hat{n}_{\ell,\sigma}^{(r)}$ and that
$\sum_{\ell}\hat{n}_{\ell,\sigma}^{(r)}=L^{2}/2$ is an exact identity for a
$U(1)$ symmetric, half-filled lattice. Taking the trajectory average over all
possible measurement outcomes amounts to integrating over all possible values
for each $J_{\ell,t}$. Since $J_{\ell,t}$ for different time steps and lattice
sites are independent, we can perform this integration for each infinitesimal
evolution operator individually. The Born probabilities are implemented at the
end by the correct replica limit. Thus, performing the average on
$\hat{M}(\\{J_{l,t}\\})^{\otimes 2R}$ amounts to a Gaussian integral, yielding

$\displaystyle\overline{\hat{M}(\\{J_{l,t}\\})^{\otimes 2R}}$
$\displaystyle=\exp{-\gamma dtRL^{2}+\frac{\gamma
dt}{2R}\left(\hat{N}_{\ell,+}+\hat{N}_{\ell,-}\right)^{2}}$
$\displaystyle\overset{dt\to 0^{+}}{=}\mathds{1}-\frac{\gamma
dt}{R}\sum_{\ell}\left[R^{2}-\tfrac{1}{2}\left(\hat{N}_{\ell,+}+\hat{N}_{\ell,-}\right)^{2}\right].$
(15)

The rQME in Eq. (4) is readily obtained acting the operators $\hat{N}_{\ell}$
on $\tilde{\rho}_{t}$ as indicated by the labels $\sigma=\pm$.

_Stochastic Schrödinger Equation._ – The SSE (1) is derived from the
generalized projector formalism in the limit $dt\to 0^{+}$. For a single wave
function, the evolution of the normalized wave function
$|\psi_{t}\rangle\to|\psi_{t+dt}\rangle$ in an infinitesimal time step is
composed of three steps. First, the measurement outcomes $J_{\ell,t}$ are
drawn from the Born probabilities $p(\\{J_{\ell,t}\\})$. Second, the
unnormalized wave function is computed
$|\tilde{\psi}_{t+dt}\rangle=\text{exp}(-i\hat{H}dt)\hat{M}_{\ell}(\\{J_{\ell,t}\\})|\psi_{t}\rangle$
from the normalized one. Then
$|\psi_{t+dt}\rangle=|\tilde{\psi}_{t+dt}\rangle|||\tilde{\psi}_{t+dt}\rangle||^{-\frac{1}{2}}$
normalization is performed. For infinitesimal time steps, this process can be
made much more efficient by realizing that in the limit $dt\to 0^{+}$, the
variables $J_{\ell,t}$ become Gaussian distributed with mean
$\overline{J_{\ell,t}}=\langle\hat{n}_{\ell}\rangle_{t}$ and variance
$\text{var}(J_{\ell,t})=\tfrac{1}{4\gamma dt}$. This is readily verified by
taking the $m$-th moment of $J_{\ell,t}$,

$\displaystyle\overline{J_{\ell,t}^{m}}=\int_{-\infty}^{\infty}dJ_{\ell,t}\
J_{\ell,t}^{m}p(\\{J_{\ell,t}\\})$ (16)

and the definition of the Born probabilities in the main text. Thus only the
mean of $J_{\ell,t}$ depends on the state but not its variance. This is
implemented by replacing
$J_{\ell,t}\to\langle\hat{n}_{\ell}\rangle_{t}+\tfrac{\xi_{\ell,t}}{2\gamma
dt}$ with $\xi_{\ell,t}$ being the Gaussian white noise defined in the main
text. Inserting the replacement into the generalized projector yields (2):

$\displaystyle\hat{M}(\\{J_{\ell,t}\\})$
$\displaystyle=\prod_{\ell}[\tfrac{2\gamma dt}{\pi}]^{\frac{1}{4}}\exp[-\gamma
dt(\langle\hat{n}_{\ell}\rangle_{t}-\hat{n}_{\ell}+\tfrac{\xi_{\ell,t}}{2\gamma
dt})^{2}]$ (17) $\displaystyle=\prod_{\ell}[\tfrac{2\gamma
dt}{\pi}]^{\frac{1}{4}}\exp[-\gamma
dt(\langle\hat{n}_{\ell}\rangle_{t}-\hat{n}_{\ell})^{2}-\xi_{\ell,t}(\langle\hat{n}_{\ell}\rangle_{t}-\hat{n}_{\ell})].$

To obtain the SSE (1), we expand the exponential to first order in $dt$,
keeping in mind that $\xi_{\ell,t}^{2}=\gamma dt$. It yields

$\displaystyle\hat{M}(\\{J_{\ell,t}\\})\approx\frac{1}{\mathcal{N}}\left[1-\sum_{\ell}\frac{\gamma
dt}{2}(\hat{n}_{\ell}-\langle\hat{n}_{\ell}\rangle_{t})^{2}-\xi_{\ell,t}(\hat{n}_{\ell}-\langle\hat{n}_{\ell}\rangle_{t})\right],$

where $\mathcal{N}$ is a constant prefactor. Applying this expansion to the
wave function and subsequently normalizing it, shows that $\mathcal{N}$ drops
out and one obtains the SSE in the main text.

##  Appendix B Replica field theory

_Fermion Keldysh action._ – The fermion Keldysh action in Eq. (5) is derived
in the common way Kamenev (2011); Sieberer  _et al._ (2016) from the rQME in
Eq. (4): at each infinitesimal time step $t\to t+dt$ fermion coherent states
are inserted on both sides of the density matrix $\tilde{\rho}_{t}$. Each
fermion operator acting on coherent states at time $t$, is then replaced by
Grassmann variables
$c_{\ell,\sigma}^{(r)\dagger},c_{\ell,\sigma}^{(r)}\to\bar{\psi}_{\ell,t,\sigma}^{(r)},\psi_{\ell,t,\sigma}^{(r)}$.
Here $\sigma$ again denotes whether the operator acts from the left
($\sigma=+$) or from the right ($\sigma=-$) onto the density matrix Sieberer
_et al._ (2016). For the monitoring part of the action, this can be readily
inferred from Eq. (14), yielding

$\displaystyle iS_{\text{meas}}=\int
dt\sum_{\ell}\left[\sum_{r=1}^{R}(\bar{\psi}_{\ell,t,+}^{(r)}\psi_{\ell,t,+}^{(r)}+\bar{\psi}_{\ell,t,-}^{(r)}\psi_{\ell,t,-}^{(r)})\right]^{2}.$
(18)

Now, one performs a canonical rotation into the fermion Keldysh basis via
Kamenev (2011)
$\psi_{\ell,t,\alpha}^{(r)}=(\psi_{\ell,t,+}^{(r)}-(-1)^{\alpha}\psi_{\ell,t,-}^{(r)})/\sqrt{2}$
and
$\bar{\psi}_{\ell,t,\alpha}^{(r)}=(\bar{\psi}_{\ell,t,+}^{(r)}+(-1)^{\alpha}\bar{\psi}_{\ell,t,-}^{(r)})/\sqrt{2}$,
where $\alpha=1,2$ is the Keldysh index. Introducing the vector
$\psi_{t,\ell}=\\{\psi_{\alpha,t,\ell}^{(r)}\\}$ and similarly
$\bar{\psi}_{t,\ell}$ and drawing the spatial continuum limit
$(t,\ell)\to(t,\mathbf{x})=X$ the measurement action $S_{\text{meas}}$ readily
obtains the form of the quartic vertex in Eq. (5).

The free part of the action $\sim\int_{X}\bar{\psi}_{X}G_{0}^{-1}\psi_{X}$
with the free Green’s function in Keldysh structure $G_{0}^{-1}$ is familiar
from textbooks, see, e.g., Ref. Kamenev (2011).

_Bosonic action._ – In order to derive the bosonic action $S_{Q}$ in the main
text, we multiply the partition function
$Z=\int\mathcal{D}(\\{\psi_{X}\\})e^{iS_{\psi}}$ by the identity

$\displaystyle
1=\int\mathcal{D}(\\{Q_{X}\\})\delta(Q_{X}-2\psi_{X}\bar{\psi}_{X}).$ (19)

Here $Q_{X}\in\mathds{C}^{2R\times 2R}$ is a $2R\times 2R$ hermitian matrix,
while $\psi_{X}\bar{\psi}_{X}$ is a $2R\times 2R$ matrix of Grassmann
bilinears and $\delta(...)$ is the Dirac $\delta$-function. Shifting this
identity into the Grassmann integration, we use the $\delta$-function to
replace the fermion bilinear matrices $\psi_{X}\bar{\psi}_{X}\to Q_{X}$ in the
quartic term. For the fermion bilinears, one has the identity

$\displaystyle\Tr\left[(\sigma_{x}\psi_{X}\bar{\psi}_{X})^{2}\right]=-\left(\Tr\left[\sigma_{x}\psi_{X}\bar{\psi}_{X}\right]\right)^{2},$
(20)

which has no counterpart for a general $Q_{X}$-matrix structure. In terms of
$Q_{X}$, both terms contribute relevant, i.e., slow, long-wavelength
fluctuations due to the spacetime-local nature of the matrix $Q_{X}$. This is
in contrast to conventional disordered problems Kamenev (2011). We therefore
choose a symmetric decoupling of both channels in this work.

We thus have
$Z=\int\mathcal{D}[\\{\psi_{X},Q_{X}\\}]\delta(Q_{X}-2\psi_{X}\bar{\psi}_{X})e^{iS_{\psi-Q}}$
with

$\displaystyle
S_{\psi-Q}=\int_{X}\left\\{\bar{\psi}_{X}G_{0}^{-1}\psi_{X}-\tfrac{i\gamma}{8R}\left(\Tr[(\sigma_{x}Q_{X})^{2}]-(\Tr[\sigma_{x}Q_{X}])^{2}\right)\right\\}.$
(21)

In order to implement the $\delta$-function, we use its integral
representation in terms of a $2R\times 2R$ Hermitian matrix $\Xi_{X}$,

$\displaystyle\delta(Q_{X}-2\psi_{X}\bar{\psi}_{X})\sim\int\mathcal{D}[\\{\Xi_{X}\\}]e^{\frac{i}{2}\Tr[\Xi_{X}(Q_{X}-2\psi_{X}\bar{\psi}_{X})]}.$
(22)

As common for nonlinear sigma model approaches, we replace the hard constraint
resulting from an integration over $\Xi$ by a soft constraint Sachdev (2011):
we replace $\Xi$ by its saddle point value, obtained by taking the variation
with respect to $Q_{X}$. The remaining action is quadratic in Grassmann
variables. Integrating them out and replacing $\Xi$ by the saddle point value
yields $S_{Q}$ in the main text.

_Saddle point and Gaussian fluctuations._ – The saddle point of the action
$S_{Q}$ in the main text is obtained by taking the variation with respect to
$Q_{X}$. The locality of $Q_{X}$ in space-time makes the variation simpler
compared to disordered fermions and yields the common result Kamenev (2011)

$\displaystyle 0\overset{!}{=}\tfrac{\delta S_{Q}}{\delta
Q_{X}}=\tfrac{i\gamma}{2R}\left[\left(G_{0}^{-1}+i\tfrac{\gamma}{2R}Q\right)_{X,X}^{-1}-\tfrac{1}{2}\sigma_{x}Q_{X}\sigma_{x}+\tfrac{1}{2}\sigma_{x}\Tr(\sigma_{x}Q)\right].$

The equation is solved by $Q_{X}=\Lambda=\delta_{r,r^{\prime}}\sigma_{z}$,
which also implements causality for the saddle point fermion Green’s function
$G^{-1}_{\text{sp}}\equiv\left(G_{0}^{-1}+i\tfrac{\gamma}{2R}\Lambda\right)_{X,X}^{-1}\delta_{r,r^{\prime}}$.
It also implements a half-filled Keldysh Green’s function and fulfills the
nonlinear constraint $Q_{X}^{2}=\mathds{1}$. The result amounts to the self-
consistent Born approximation Kamenev (2011) with an elastic mean-free time
$\tau_{\text{el}}=R/\gamma$.

In order to extract density correlation functions we expand the action around
the saddle point by setting $Q_{X}=\Lambda+\delta Q_{X}$ and then expanding
$S_{Q}$ to quadratic order in $\delta Q_{X}$. The zeroth order term is a
constant and the first order term vanishes, leaving us with

$\displaystyle S_{\delta Q}^{(2)}=\tfrac{\gamma^{2}}{8R^{2}}$
$\displaystyle\int_{X,X^{\prime}}\Tr(G_{\text{sp}}(X-X^{\prime})\delta
Q_{X^{\prime}}G_{\text{sp}}(X^{\prime}-X)\delta Q_{X})$
$\displaystyle-\tfrac{\gamma}{8R}\int_{X}\left\\{\Tr\left[(\sigma_{x}\delta
Q_{X})^{2}\right]-\left[\Tr(\sigma_{x}\delta Q_{X})\right]^{2}\right\\}.$ (23)

This action can be significantly simplified: the Green’s functions
$G_{\text{sg}}$ are diagonal in replica- and Keldysh index. Furthermore,
integrals over $G_{R}(X)G_{R}(-X)$ and $G_{A}(X)G_{A}(-X)$ are zero due to
causality. We thus end up with a separation of the action into replica
diagonal modes $\Phi^{r}_{X}=\delta Q_{X}^{rr}$ and off-diagonal modes
$\delta\tilde{Q}_{X}^{r,r^{\prime}}$. Furthermore, one finds a pairing of
$\Phi^{12}$ and $\Phi^{21}$ in Keldysh index. It is convenient to introduce
_classical_ and _quantum_ fields $\Phi^{q,r}_{X}=\delta Q^{21,rr}_{X}$ and
$\Phi^{c,r}_{X}=\delta Q^{21,rr}_{X}$. In Fourier space, we find

$\displaystyle iS_{\Phi}=$
$\displaystyle-\tfrac{1}{8}\int_{K}(\Phi^{c,r}_{K},\Phi^{q,r}_{K})\left(\begin{array}[]{cc}0&\mathcal{G}^{-1}(K)\\\
\mathcal{G}^{-1}(-K)&0\end{array}\right)\left(\begin{array}[]{c}\Phi^{c,r}_{-K}\\\
\Phi^{q,r}_{-K}\end{array}\right)$ (28)
$\displaystyle+\tfrac{\gamma(1-\delta_{r,r^{\prime}})}{8R}\int_{K}(\Phi^{c,r}_{K},\Phi^{q,r}_{K})\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\left(\begin{array}[]{c}\Phi^{c,r^{\prime}}_{-K}\\\
\Phi^{q,r^{\prime}}_{-K}\end{array}\right).$ (33)

The inverse propagator here is defined via the integral

$\displaystyle\mathcal{G}^{-1}(K)=[\tfrac{\gamma^{2}}{R^{2}}\int_{Q}g^{R}(Q+K)g^{A}(Q)]-\tfrac{\gamma}{R},$
(34)

which for $\omega,\mathbf{k}^{2}\ll\gamma$ and $R\to 1$ yields
$\mathcal{G}^{-1}(K)\approx-i\omega+\tfrac{2}{\gamma}\mathbf{k}^{2}$.
Computing the particle number fluctuations as defined in the main text thus
yields

$\displaystyle
C(K)=\frac{\text{Re}(\mathcal{G}(K))}{2+4\gamma\text{Re}(\mathcal{G}(K))}.$
(35)

When computing the Fourier transform of this equation, one needs to consider
that the observable is measured at the temporal boundary of the
$2+1$-dimensional evolution. This amounts to integral boundaries
$t\in[0,\infty)$ instead of the usual $t\in(-\infty,\infty)$. Without this
boundary, the prefactor for $S_{A}$ would be $\frac{\pi}{6\gamma}$. The
correct expression is a factor of $2$ larger Poboiko  _et al._ (2023), which
yields Eq. (7).

_Nonlinear sigma model and renormalization group treatment._ – The long-
wavelength fluctuations of the bosonic matrix $Q_{X}$ are typically
parametrized in terms of symmetry-compatible rotations
$Q_{X}=\mathcal{R}_{X}\Lambda\mathcal{R}_{X}^{-1}$ around the saddle point
solution $\Lambda$. Each $\mathcal{R}_{X}$ performs local rotations in
$2R\times 2R$-dimensional replica and Keldysh space compatible with a
$U(2R)$-symmetry. Since $\Lambda\sim\sigma_{z}$ in Keldysh space, $2R^{2}$
generators are trivial, i.e., those $\sim\mathds{1},\sigma_{z}$ in Keldysh
space. The resulting rotations act in $U(2R)/U(R)\times U(R)$ symmetric space
and are parametrized by generators
$\sigma_{x}\Theta_{x},\sigma_{y}\Theta_{y}$, where $\Theta_{x,y}$ are
symmetric $R\times R$ matrices acting on replica coordinates. We set
$\mathcal{R}_{X}=\mathcal{R}_{x,X}\mathcal{R}_{y,X}$ with
$\mathcal{R}_{\eta,X}=\exp(\tfrac{i}{2}\sigma_{\eta}\Theta_{\eta})$.

Inserting this parameterization into the measurement part and using identities
for Pauli matrices to compute the trace in Keldysh indices, we find

$\displaystyle\Tr[(\sigma_{x}Q_{X})^{2}]$
$\displaystyle=-2\Tr_{R}\cos(2\Theta_{y,X})\approx
4\Tr_{R}(\Theta_{y,X}^{2}),$ $\displaystyle(\Tr[\sigma_{x}Q_{X}])^{2}$
$\displaystyle=(2\Tr_{R}[\sin(\Theta_{y,X})])^{2}\approx
4(\Tr_{R}\Theta_{y,X})^{2}.$ (36)

Here $\Tr_{R}$ denotes the trace over the remaining replica indices. We have
expanded the nonlinear functions since the action does not vanish for the
traceless part of $\Theta_{y,X}$, i.e., for
$\tilde{\Theta}_{y,X}=\Theta_{y,X}-\frac{1}{R}\Tr_{R}\Theta_{y,X}$, which thus
remains gapped.

The logarithm in the action $S_{Q}$ is rewritten Kamenev (2011)

$\displaystyle\Tr\left[\ln(G_{0}^{-1}+i\tfrac{\gamma}{2R}Q_{X})\right]$
$\displaystyle=\Tr\left[\ln(\mathcal{R}_{X}^{-1}G_{0}^{-1}\mathcal{R}_{X}+i\tfrac{\gamma}{2R}\Lambda)\right]$
$\displaystyle=\Tr\left[\ln(\mathds{1}+G_{\text{sp}}\mathcal{R}_{X}^{-1}[G_{0}^{-1},\mathcal{R}_{X}])\right],$
(37)

with $G_{\text{sp}}=\left(G_{0}^{-1}+i\tfrac{\gamma}{2R}\Lambda\right)^{-1}$.
The logarithm is expanded to leading order in derivatives. The temporal
derivative yields

$\displaystyle\Tr\left[G\mathcal{R}_{X}^{-1}i\partial_{t}\mathcal{R}_{X}\right]$
$\displaystyle=-\tfrac{1}{2}\Tr\left[\sigma_{x}^{K}\sin(\Theta_{y,X})\mathcal{R}_{x,X}^{-1}\partial_{t}\mathcal{R}_{x,X}\right]$
(38)
$\displaystyle\approx-\tfrac{1}{2}\Tr_{R}\left[\Theta_{y,X}e^{-\frac{i}{2}\Theta_{x,X}}(\partial_{t}e^{i\Theta_{x,X}})e^{-\frac{i}{2}\Theta_{x,X}}\right].$

The spatial derivative appears only in quadratic order due to inversion
symmetry

$\displaystyle\Tr((\nabla Q_{X})^{2})=2\Tr_{R}[(\nabla
e^{i\Theta_{x,X}})(\nabla e^{-i\Theta_{x,X}})].$ (39)

Integrating out the gapped relative replica fluctuations
$\tilde{\Theta}_{y,X}$ in the Gaussian approximation and rescaling time by a
factor $\sqrt{2}$ yields Eq. (8) in the main text with $U=e^{i\Theta_{x,X}}$.

##  Appendix C Numerical implementation and observables

_Gaussian state updates._ – Here, we provide details on the numerical
implementation of the trajectory evolution and the numerical analysis behind
some of our results. The time evolution, i.e., the trajectory at each time
$t$, is represented by a Gaussian state $|\psi_{t}\rangle$. It is parametrized
by an $L^{2}\times N$ matrix $\psi_{t}$,

$\ket{\psi_{t}}=\prod_{s=1}^{N}\left(\sum_{\ell=1}^{L^{2}}\psi^{s}_{\ell,t}(t)c_{\ell}^{\dagger}\right)\ket{0}.$

Normalization of the wave functions implies
$\sum_{\ell}(\psi_{\ell,t}^{s})^{*}\psi_{\ell,t}^{s^{\prime}}=\delta_{s,s^{\prime}}$,
i.e., $\psi_{t}^{\dagger}\psi_{t}^{\phantom{\dagger}}=\mathds{1}$. The
trajectory evolution thus amounts to updating $\psi_{t}$, which is done by a
Suzuki-Trotter decomposition of Eq. (1) Cao  _et al._ (2019); Alberton  _et
al._ (2021). Up to first order in $dt$ and up to an overall normalization,
this leads to the matrix update

$\psi_{t+dt}=\text{diag}\left(e^{\xi_{1,t}+\frac{\gamma
dt}{2}(2\langle\hat{n}_{1}\rangle_{t}-1)},\dots,e^{\xi_{N,t}+\frac{\gamma
dt}{2}(2\langle\hat{n}_{N}\rangle_{t}-1)}\right)e^{-ihdt}\psi_{t}.$

Here, $h_{\ell,\ell^{\prime}}=-\delta_{\ell\text{ n.n.}\ell^{\prime}}$ is the
nearest-neighbor hopping matrix. Normalization is obtained by performing a
QR-, i.e., Gram-Schmidt, decomposition $\psi=QR$ and redefining $\psi\equiv
Q$.

_Purification._ – The purification of the monitored system is implemented by
coupling the system initially to an ancilla. While the system is entangled
with the ancilla and thus itself in a mixed state, the global state of the
system _and_ ancilla is in a Gaussian pure state. The global system can thus
be numerically simulated as outlined above. The purification time scales of
the system, however, are typically much longer than the time needed for other
observables to reach stationary values. The purification simulations are thus
computationally more costly and only smaller system sizes up to linear
dimension $L=16$ are performed. To determine the scaling of the purification
rate $\tau_{\text{pur}}$ with the system size $L$, we performed a finite-size
scaling analysis. This was done by rescaling time as $t\to t/L^{\alpha}$ and
the ancilla entanglement density as $s_{\text{anc}}\to
s_{\text{anc}}/L^{\beta}$. Then, we found the $\alpha$ and $\beta$ values
which would collapse the curves for different $L$ values, as shown in figure
4.

Figure 4:  Unrescaled (a) and rescaled (b) purification dynamics of an
$L\times 1$ ancilla for $\gamma=0.5$ and different $L$ values.

_Entanglement and mutual information._ – The entanglement and mutual
information in this work are exclusively computed from the von Neumann
entanglement entropy. For a Gaussian state, it is computed from the
correlation matrix
$\langle\psi_{t}|c^{\dagger}_{\ell}c^{\phantom{\dagger}}_{m}|\psi_{t}\rangle=(\psi_{t}^{\phantom{\dagger}}\psi_{t}^{\dagger})_{\ell,m}$.
In order to compute the von Neumann entanglement of a subsystem $A$, one
considers the correlation matrix
$(\psi_{t}^{\phantom{\dagger}}\psi_{t}^{\dagger})|_{A}$ restricted to
subsystem $A$ and computes its eigenvalues $\\{\lambda_{\alpha}\\}|_{A}$. The
von Neumann entanglement entropy for the subsystem is then

$S(A)=-\sum_{\alpha}\lambda_{\alpha}\log\lambda_{\alpha}+(1-\lambda_{\alpha})\log(1-\lambda_{\alpha}).$

In the metallic phase, the entanglement entropy obeys a logarithmic violation
of the area law, such that for $A=L/2\times L$ one finds a growth law
$S(A)/L=c(\gamma)\ln L+s(\gamma)$, with $\gamma=2.15$ yielding a perfect match
with Eq. (12) (Fig. 5). In order to extract the prefactor of the sub-extensive
growth (shown in Fig. 2(a)), we perform a linear fit of $S(A)/L$ as a function
of $\ln L$. This was done for a range of $\gamma$ values and for increasingly
large $L$ intervals ranging from $L=2$ up to $L=50$.

Figure 5:  Entropy density scaling with strip size $A=L\times\ell_{A}$
compared with with Eq. (12). An excellent match is visible for $\gamma=2.15$
only.

_Inverse participation ratio._ – The inverse participation ratio $P_{q}$ is
computed from the single particle wave functions $\psi_{t}$ using Eq. (10).
The multifractal exponent $D_{q}$ is then extracted in two ways: (a) a linear
fit of $\ln P_{q}/(1-q)$ as a function of $\ln L$ or (b) through the
expression $D_{q}\approx\ln P_{q}/(1-q)\ln L$. We find qualitative agreement
between both methods.