# Effective Theory for the Measurement-Induced Phase Transition of Dirac
Fermions

M. Buchhold  Institut für Theoretische Physik, Universität zu Köln, D-50937
Cologne, Germany  Y. Minoguchi  Vienna Center for Quantum Science and
Technology, Atominstitut, TU Wien, 1040 Vienna, Austria  A. Altland  Institut
für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany  S.
Diehl  Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne,
Germany

(August 27, 2024)

###### Abstract

A wave function  subject to unitary time evolution and exposed to measurements
undergoes pure state dynamics, with deterministic unitary and probabilistic
measurement-induced state updates, defining a quantum trajectory. For many-
particle systems, the competition of these different elements of dynamics can
give rise to a scenario similar to quantum phase transitions. To access  this
competition despite the randomness of single quantum trajectories, we
construct an $n$-replica Keldysh field theory for the ensemble average of the
$n$-th moment of the trajectory projector. A key finding is that this field
theory decouples into one set of degrees of freedom that heats up
indefinitely, while $n-1$ others can be cast into the form of pure state
evolutions generated by an effective non-Hermitian Hamiltonian. This
decoupling is exact for free theories, and useful for interacting ones. In
particular, we study locally measured Dirac fermions in $(1+1)$ dimensions,
which can be bosonized to a monitored interacting Luttinger liquid at long
wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a
quantum Sine-Gordon model with complex coefficients. A renormalization group
analysis reveals a gapless critical phase with logarithmic entanglement
entropy growth, and a gapped area law phase, separated by a Berezinskii-
Kosterlitz-Thouless transition. The physical picture emerging here is a
measurement-induced pinning of the trajectory wave function into eigenstates
of the measurement operators, which succeeds upon increasing the monitoring
rate across a critical threshold.

##  I Introduction

In quantum mechanics, there are two fundamentally distinct dynamical
evolutions: First, a pure state can evolve deterministically according to the
Schrödinger equation, with dynamics generated by a Hamiltonian operator
$\hat{H}$. Second however, it can be updated in a stochastic fashion, when the
quantum system is subject to observation. In the case of a strong, projective
measurement of an observable $\hat{M}$, the wave function abruptly collapses
into one of the eigenstates $\ket{m}$ of $\hat{M}$, with a probability
determined by the overlap of the state before the measurement with $\ket{m}$.
If the Hamiltonian commutes with the measurement operator, after a single
collapse into a certain state $\ket{m}$ the system will be confined to it
indefinitely. In contrast, if $[\hat{H},\hat{M}]\neq 0$, generically the
competition of $\hat{H}$ and $\hat{M}$ will not allow the evolution to come to
rest.

In a many-body context, such a competition of two operators can give rise to
fundamental macroscopic phenomena such as quantum phase transitions. In this
case, two non-commuting terms, e.g. kinetic and potential energy, each
separately stabilize ground states with macroscopically distinct properties.
While finite systems can only undergo a gradual change of properties upon
tuning the ratio of competing energy scales, in the thermodynamic limit a
phase transition separating qualitatively distinct phases of matter will
occur. In this light, it is a natural question whether a many-body system
undergoing competing Hamiltonian and measurement dynamics likewise may undergo
an abrupt change in behavior, and which quantities may host this information.
This question was answered in the affirmative in Skinner  _et al._ (2019); Li
_et al._ (2018a); Jian  _et al._ (2020a); Bao  _et al._ (2020) for
projectively measured random unitary circuits. In these setups, the dynamics
of a one-dimensional spin chain is generated either by randomly selected
quasilocal entangling unitary gates, or by quasilocal measurements. Since the
sets of operators of the entangling gates and the measurements do not commute,
a competition is realized, with a strength tuneable via the ratio of applied
unitaries per time unit vs. applied measurements per time unit.

For Haar random unitaries, moderate size numerics Chan  _et al._ (2019);
Skinner  _et al._ (2019); Choi  _et al._ (2020) and analytical studies Jian
_et al._ (2020a); Bao  _et al._ (2020) have been conducted. Choosing time
evolution randomly from the Clifford- and the measurement operators from the
Pauli-group, efficient numerical analysis of even large systems is possible Li
_et al._ (2018b, 2019); Ippoliti  _et al._ (2021). The two extreme cases are
clearly distinct in their dynamics: The unitarily evolving circuit is
characterized by unbounded growth of the entanglement entropy with system size
(volume law) Calabrese and Cardy (2005); Hosur  _et al._ (2016); Nahum  _et
al._ (2017); Zhou and Nahum (2019); Kim and Huse (2013); Jonay  _et al._
(2018); Akhtar and You (2020); Bertini and Piroli (2020). On the other hand,
for any random initial state local measurements of the spins e.g. in the
$z$-basis will collapse the state into a pure product state of some
configuration of the $z$ projection of the spins, thus characterized by a
saturation of the entanglement entropy to an area law behavior. It was shown
in Refs. Nahum  _et al._ (2017); Li  _et al._ (2018a) that the respective
entanglement growth averaged over the ensemble of trajectories is a good
witness for the phase transition between volume and area law growth at a
finite competition ratio between unitary and measurement dynamics.

This discovery has sparked significant research on the nature of this
transition, its proper description, and its generality in terms of models
hosting such behavior Zabalo  _et al._ (2020); Zhang  _et al._ (2020); Biella
and Schiró (2021); Tang and Zhu (2020); Fuji and Ashida (2020). The
entanglement growth quantifier suggests a physical picture in terms of
information scrambling vs. information localization. This has been made more
precise in Choi  _et al._ (2020); Fan  _et al._ (2021); Fidkowski  _et al._
(2021); Li and Fisher (2021), which explain the robustness of the volume law
phase within a quantum error correction picture, where the fast information
spreading is protected from errors realized as the readout of the
measurements. Giving up the focus on pure state evolution Ref. Gullans and
Huse (2020) characterizes combined unitary and measurement dynamics by its
potential to purify a maximally mixed initial state. Alternatively to random
circuits, models of fermions hopping on a one-dimensional lattice and exposed
to local density measurements have been suggested Cao  _et al._ (2019). These
do not show a volume to area law transition Cao  _et al._ (2019), but rather a
transition from a critical phase with a logarithmic scaling of the
entanglement entropy to an area law Alberton  _et al._ (2021) (the transition
can also be found in non-unitary circuit models Chen  _et al._ (2020); Li  _et
al._ (2020); Jian  _et al._ (2020b)). For free fermion models, a general
correspondence between non-unitary circuit dynamics and unitary but random
Hamiltonian dynamics in $(d+1)$ dimensions has been shown to enable a
classification of the measurement dynamics in terms of symmetries Jian  _et
al._ (2020b). These results suggest a finer structure in the phenomenology of
measurement-induced phase transitions.

In this work, we approach the measurement-induced phase transition from the
perspective of non-equilibrium quantum statistical mechanics, asking in
particular what the proper degrees of freedom are to capture it.  Once this is
done, we discuss how the previously established phenomenology of the
transition, e.g., the behavior of the entanglement entropy and the
purification dynamics, can be understood from a statistical mechanics
viewpoint. To this end, we develop a replica field theory approach for a
minimal model of Dirac fermions in one spatial dimension, undergoing
continuous measurements. This model enables an alternative representation via
bosonization at long wavelength, amounting to a measured non-linear Luttinger
liquid. In terms of complexity, the model is thus comparable to free fermion
problems with disorder; upon integrating out the measurement noise, the
problem becomes formally, and for all practical means, an interacting one. In
our analysis, we are guided by the physical picture of a pinning or
localization transition upon increasing the measurement strength: On the level
of a single, pure state quantum trajectory, the delocalization due to kinetic
energy competes with pinning of fermions due to the measurement-induced
collapse of the wave function.

Care has to be exercised when taking the ensemble average. The binary
measurement outcomes on each point in space lead to an extensive
configurational entropy. The averages that are usually considered, which are
linear in the state, correspond at long times to infinite temperature states
and mask the transition in the trajectory ensemble.  In contrast, averages of
non-linear functions of the state, such as the celebrated entanglement entropy
or connected correlation functions $C_{O_{i},O_{j}}=\langle
O_{i}O_{j}\rangle-\langle O_{i}\rangle\langle O_{j}\rangle$ for some
observables $O_{i,j}$ (the second contribution is nonlinear in the state), do
witness a transition. We demonstrate this on our concrete model: The
trajectory averaged $n$-replica state hosts one structureless ’hot’ mode, but
also $n-1$ ’cold’ modes, which undergo a quantum phase transition in (1+1)
dimensions. The hot mode absorbs all of the configurational entropy, and
exclusively determines any linear average, such as $\langle
O_{i}O_{j}\rangle$. All linear averages are therefore featureless. Nonlinear
averages, however, such as $C_{O_{i},O_{j}}$, also depend on the cold modes,
which are free of configurational entropy, and thus reveal the features of the
measurement-induced evolution. We expect the structures revealed here to be
useful for even broader classes of measurement-induced phase transitions.

Applied to the problem at hand, it allows us to make progress in terms of the
bosonized theory. In particular, we distill an effective non-Hermitian sine-
Gordon model for density fluctuations, and show that it undergoes a
Berezinskii-Kosterlitz-Thouless (BKT) pinning transition. Its gapless phase is
protected by current conservation. The pinning induced by measurements, into
the eigenstates of the measurement operators, drives a gap opening. As
witnesses of the transition, we compute the 2-replica correlation functions
showing algebraic and exponential behavior in the gapless and in the gapped
phases, respectively. Furthermore, we compute the entanglement entropy in the
weakly and strongly monitored regimes, showing logarithmic growth and
saturation with system size $L$, respectively. A measurement rate dependent
effective central charge, appearing as the coefficient of the $\log(L)$ growth
term, is found in the gapless phase. This scenario is consistent with previous
numerical results for a related model of monitored lattice fermions Cao  _et
al._ (2019); Chen  _et al._ (2020); Alberton  _et al._ (2021); Bao  _et al._
(2021). A power law decay of the effective central charge
$c(\gamma)\sim\gamma^{-\kappa}$ is confirmed. The value $\kappa\geq 0.5$ is
bounded from below but non-universal, continuously varying with system
parameters. We believe that such a pinning mechanism, and the BKT universality
class itself, could underlie broader classes of measurement-induced phase
transitions in $1+1$ dimensions.

Our field theory approach complements previous quantum information theoretic
approaches to measurement-induced phase transitions discussed above. We also
note that mappings to statistical mechanics models have been proposed
previously based on a tensor network approaches to the dynamics Bao  _et al._
(2020); Jian  _et al._ (2020a); Skinner  _et al._ (2019); Nahum  _et al._
(2021). What sets our approach apart from these works is that we directly
identify the physically relevant degrees of freedom – the above mentioned
’cold’ modes made of monitored fermions – which are at the root of the
transition. The behavior of entanglement entropies then constitutes an
important hallmark of it, but is not the only signature – as exemplified by
our analysis of replica correlation functions, likewise serving to detect the
transition.

We identify several observables that have been determined in previous works,
and which confirm our findings. In addition, we provide potential quantifiers,
which establish a relation to other setups, including relaxation dynamics and
a purification time scale Gullans and Huse (2020).

Structure of the paper: In the following section, we introduce our setup and
provide a summary of our main results. In Section III, we discuss general
properties of the measurement model and draw the temporal and spatial
continuum limit, introducing a field theory for continuously monitored Dirac
fermions. In Section IV, a two-replica field theory for a general continuous
monitoring setup is derived and then applied to our concrete setup. It
provides the basis for the identification of the relevant degrees of freedom
at the measurement-induced phase transition. Then, in Section V, we perform a
renormalization group analysis of the phase transition. In Section VI, the
replica theory is extended to an arbitrary number of replicas $n$ in a Keldysh
real-time path integral framework. This serves as a basis for the computation
of higher order correlation functions, and we include a discussion of the
Rényi entropy and the von Neumann entanglement entropy. We conclude in Section
VII. Several technical details of our analysis are presented in the appendix.

##  II Extended Synopsis and Key Results

Setup – Our model for the measurement dynamics in one spatial dimension is
based on the stochastic Schrödinger equation for the trajectory wave function
$\ket{\psi_{t}}$ Dalibard  _et al._ (1992a); Dum  _et al._ (1992); Dalibard
_et al._ (1992b), or alternatively the conditioned trajectory projector
$\hat{\rho}^{(c)}_{t}=\ket{\psi_{t}}\bra{\psi_{t}}$ Belavkin (1989);
Carmichael (1993); Jacobs and Steck (2006), which amounts to a continuous time
description of measurements.  Throughout this work, we focus on continuous-
time measurements. The phenomenology of a measurement-induced phase
transition, which we establish for such continuous measurements, however,
includes the case of projective measurements. In that case, the continuous-
time evolution can be understood as an effective, coarse grained description,
which is valid on time scales on which many projective measurements have been
performed. We discuss this aspect in more detail below.

The competition between unitary dynamics and measurements is set by the
dimensionless parameter

$g=\gamma/J,\text{ with }\left\\{\begin{array}[]{cl}J&\text{: kinetic
energy}\\\ \gamma&\text{: measurement rate}\end{array}\right..$ (1)

Here $J$ describes the kinetic energy of the Hamiltonian, and $\gamma$ the
rate at which measurements are performed. This formulation is able to capture
the collapse of the wave function characteristic of physical measurements.
Previous work on measurement-induced phase transitions has mainly focused on
the case of projective (discrete) measurements Skinner  _et al._ (2019); Li
_et al._ (2018a); Nahum  _et al._ (2021); Bao  _et al._ (2020, 2021); however,
the continuous measurement scenario is smoothly connected to the latter.

Phase transitions generally are rooted in a competition of incompatible terms.
In the context of the measurement-induced phase transitions, the competition
is realized by the entangling unitary dynamics, and a disentangling
measurement dynamics. In contrast,  both continuous as well as discrete
(projective) measurements of the local density lead to the same long time
limit. These two types of measurement dynamics can be smoothly interpolated to
one another Jacobs and Steck (2006); Jacobs (2014), but since there is no
competition realized along this interpolation axis, one should expect no
additional phase transition. Indeed, recent works Szyniszewski  _et al._
(2019, 2020) have established that the transition persists upon interpolating
between the limiting cases of strong projective and weak continuous
measurements in random circuits. For our purpose of constructing an
analytically tractable model for the competing dynamics, the continuous time
formulation proves advantageous.

Quantifying the phase transition –  As one may expect quite naturally,
continuously measuring a set of local, mutually commuting operators
$\hat{O}_{j}$, e.g., the local particle density on some lattice site $j$, pins
the system onto their eigenstates. Competing with this is the unitary
dynamics, which leads to the spreading of the local operators via particle
propagation, and therefore to a depinning from eigenstates of the measurement
operators.  The crux of the matter is what type of behavior follows from this
competition and when are the two limiting cases of eigenstate (de-)pinning
observed? The competition of two non-commuting operators driving a phase
transition must thus be encoded in the evolution of the pure state wave
function, similar to a quantum phase transition in a ground state problem.
Importantly however, the pure state considered here corresponds to a random
variable, due to the stochastic nature of measurements. To acquire information
independent of this randomness, suitable ensemble averages need to be
considered. Taking the statistical average over the trajectory ensemble (which
we denote with $\overline{.\phantom{l}.\phantom{l}.}$) in the way familiar
from quantum statistical physics for instance yields

$\displaystyle\overline{\braket{\psi_{t}}{\hat{O}_{i}\hat{O}_{j}}{\psi_{t}}}=\overline{\operatorname{tr}[\hat{O}_{i}\hat{O}_{j}\hat{\rho}_{t}^{(c)}]}=\operatorname{tr}[\hat{O}_{i}\hat{O}_{j}\overline{\hat{\rho}_{t}^{(c)}]},$
(2)

where $\overline{\hat{\rho}_{t}^{(c)}}$ is the trajectory averaged density
matrix. For a measurement dynamics, however,
$\overline{\hat{\rho}_{t}^{(c)}}=\frac{1}{\mathcal{N}}\mathds{1}$, which
corresponds to evaluating the product of the measured operators
$\hat{O}_{i}\hat{O}_{j}$ in an infinite temperature state – thus the
correlations will be trivial except for $g=0$, where there is no averaging.
The masking effect of such trajectory ensemble average can be mitigated by
considering correlation functions which are non-linear in the state – such as,
for example, the equal-time product
$\braket{\psi_{t}}{\hat{O}_{i}}{\psi_{t}}\times\braket{\psi_{t}}{\hat{O}_{j}}{\psi_{t}}$
for the same stochastic wave function $|\psi_{t}\rangle$ in both quantum
mechanical expectation values. We note that the entanglement entropy falls
into this class of quantifiers which are non-linear in the state, involving
even infinite powers in the case of the von Neumann entropy.

In this work, we focus on the non-linear connected covariance matrix for the
measured operators

$\displaystyle
C_{ij}(t)=\overline{\braket{\psi_{t}}{\hat{O}_{i}\hat{O}_{j}}{\psi_{t}}}-\overline{\braket{\psi_{t}}{\hat{O}_{i}}{\psi_{t}}\braket{\psi_{t}}{\hat{O}_{j}}{\psi_{t}}}$
(3)

in the limit $t\to\infty$. Here, the second part
$\sim\overline{\braket{\psi_{t}}{\hat{O}_{i}}{\psi_{t}}\braket{\psi_{t}}{\hat{O}_{j}}{\psi_{t}}}$
is nonlinear in $\hat{\rho}^{(c)}$ and therefore does not correspond to an
infinite temperature average. It is particularly well suited to distinguish
between strong monitoring, where the system remains close to an eigenstate of
the operators $\hat{O}_{j}$ (such that $C_{ij}$ is short-ranged or even zero
for $i\neq j$), and weak monitoring, where longer-ranged correlations between
the measured operators can be established by the Hamiltonian. Indeed, such a
scenario is realized in our model. For $g\ll 1$, the problem becomes linear in
the bosonic formulation. The correlator can be computed analytically,
demonstrating an algebraic decay $C_{ij}(t)\sim|i-j|^{-2}$. In the opposite
regime $g^{-1}\ll 1$, the pinning of particle density due to the measurements
cuts off this correlation function more severely. In the measurement-only
limit $g^{-1}=0$, the wave function evolves into a product state of onsite
occupations $0,1$ of the Dirac fermions with autocorrelations only. In the
bosonized language, this leads to a gap opening, giving rise to an effective
massive bosonic theory with exponentially decaying correlations. This behavior
is backed up by a complementary perturbative calculation for $g^{-1}\ll 1$ of
the correlation function in a fermionic lattice model.

The two opposite regimes thus show behaviors which are qualitatively different
from each other, pointing at the existence of a phase transition at a finite
competition $g_{c}$.

Replica approach – The ability to capture the qualitative behavior for finite
values of $g$ motivates us to study the bosonic non-linear Luttinger model in
more detail. To this end, we find it convenient to work in a replica
formalism. We introduce a product of replicated, conditioned density matrices
$\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}$, which allows us to rewrite

$\displaystyle C_{ij}(t)$ $\displaystyle=$
$\displaystyle\overline{\text{Tr}(\hat{O}_{i}\hat{O}_{j}\hat{\rho}_{t}^{(c)})\text{Tr}\hat{\rho}^{(c)}_{t}}-\overline{\text{Tr}(\hat{O}_{i}\hat{\rho}^{(c)}_{t})\text{Tr}(\hat{O}_{j}\hat{\rho}^{(c)}_{t})}$
(4) $\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left[(\hat{O}_{i}^{(1)}-\hat{O}_{i}^{(2)})(\hat{O}_{j}^{(1)}-\hat{O}_{j}^{(2)})\overline{\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}\right]$

with $\hat{O}_{j}^{(1)}=\hat{O}_{j}\otimes\mathds{1}$ and
$\hat{O}_{j}^{(2)}=\mathds{1}\otimes\hat{O}_{j}$. This formulation offers two
key advantages: First, it encodes the correlation information in an
expectation value linear in the trajectory averaged, replicated state. We then
construct a quantum master equation directly for
$\hat{\rho}^{(R_{2})}_{t}\equiv\overline{\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}$.
It features a non-linear state dependence, which however becomes unimportant
at late times. As a net result, we obtain an evolution linear in the
replicated, averaged state $\hat{\rho}^{(R_{2})}_{t}$. We emphasize that we do
not consider quenched disorder in our system, and statistical averages are
exclusively reserved for taking the average over random measurement
trajectories.

Second, it allows us to introduce new degrees of freedom which capture the
physics of the transition more transparently. In the bosonic model, the
replicas are acted up on by the measured operators
$\hat{O}^{(l)}=f(\hat{\phi}_{x}^{(l)})$ with $l=1,2$ and $f$ either a linear
or a nonlinear function of the bosonic operator $\hat{\phi}_{x}^{(l)}$, which
is associated to density fluctuations. For a linear problem (the problem can
be linearized close to $g,g^{-1}=0$), we find that the replicas can be
decoupled in terms of center-of-mass (or average) and relative coordinates
${\hat{\phi}_{x}^{(a,r)}=\tfrac{1}{\sqrt{2}}}(\hat{\phi}_{x}^{(1)}\pm\hat{\phi}_{x}^{(2)})$.
The center-of-mass coordinates follow an evolution which heats them up
indefinitely, while the relative ones obey a Schrödinger-type evolution under
an effective non-Hermitian Hamiltonian,
$\ket{\psi^{(r)}_{t}}=e^{-i\hat{H}_{\text{eff}}t}\ket{\psi_{0}^{(r)}}$. The
physical reason for this behavior is that all the replicas experience the same
noise realization; therefore, the sum variable (the center-of-mass) collects
all the noise-induced fluctuations, while the noise cancels in the difference
variable.

Including the non-linear terms, the decoupling is no longer exact. However,
assuming that the center-of-mass degrees of freedom still heat up to infinity
irrespectively of the behavior of the relative ones, the center-of-mass
degrees of freedom can be traced out. The remaining relative coordinate
evolution still takes the above form, and can be written as a path integral
governed by the action of a non-Hermitian sine-Gordon model,

$\displaystyle
S=\int_{X}\frac{1}{16\pi}[(\partial_{t}\phi^{(r)}_{X})^{2}-\eta^{2}(\partial_{x}\phi^{(r)}_{X})^{2}]-i\lambda\cos\phi^{(r)}_{X}],$
(5)

where we combined $X=(x,t)$, and with $\eta$ being complex and $\lambda$ real.
We derive the flow equations for this model, and find that it exhibits a BKT
type phase transition as a function of varying the competition ratio $g$,
which enters the coupling constants $\eta(g),\lambda(g)$. This places the
measurement-induced phase transition into the BKT universality class.

This result may be rationalized by the fact that a gapless phase in $(1+1)$
dimensions quite generically is left via the BKT mechanism by the generation
of a mass Kaplan  _et al._ (2009). Here the gapless phase is protected by
current conservation. Indeed, $\hat{\phi}$ in the original model describes
density fluctuations, resulting in gapless relative modes $\hat{\phi}^{(r)}$
at weak monitoring. Upon increasing the measurement strength,  i.e., the rate
at which measurement readouts are performed and the amount of information that
is extracted per readout, these modes generate a mass, and thus get pinned.

We also study the generalization from two to $n$ replicas within a
multicontour Keldysh approach, representing
$Z(n)=\overline{\operatorname{tr}\hat{\rho}^{(c)n}}$ in terms of a functional
integral with $2n$ contours – two for each copy of the state, as in the usual
Keldysh path integral ($n=1$). This is the generating functional for all
correlation functions involving $n$ powers of the quantum trajectory
projector. The noise average then generates correlations between all the
copies of the state. In this framework, and for general Gaussian problems in
the long time limit, we find that there is one center-of-mass mode (the
symmetric superposition, or Fourier mode $k=0$) which heats up indefinitely,
while there are $n-1$ orthogonal ones which do not couple the contours within
one copy, and evolve towads a pure state, generalizing the relative coordinate
identified above. We then discuss how the phase transition found at $n=2$ may
be generalized to arbitrary $n$ within an Abelian bosonization framework.

Entanglement entropy in $n$-replica Keldysh approach – Finally, we embark on a
calculation of the entanglement entropy. To this end, we consider a
bipartition of the system into a subsystem $A$ of length $L$ and
$\mathds{R}\setminus A$ which is traced out, and decompose the partition
function of the subsystem
$Z_{A}(n)=\overline{\operatorname{tr}\hat{\rho}_{A}^{(c)n}}$ into a noiseless
and a noisy contribution. For free theories, including the Gaussian fixed
points of our measurement theory, we show that the noiseless contribution to
entanglement entropy exclusively determines the scaling with the bipartition
size, while the noisy contribution is independent of the bipartition and
amounts at most to a size independent constant. In the gapless regime we find

$\displaystyle S_{\text{vN}}(L)=\tfrac{1}{3}c(\gamma)\log(L),$ (6)

with a $\gamma$-dependent effective central charge $c(\gamma)$. It depends on
the renormalization group flow of the non-Hermitian sine-Gordon theory. In the
non-monitored limit, $c(\gamma\to 0)\to 1$. This reproduces the central charge
of a ground state of free massless Dirac fermions. For weak measurements,
$c(\gamma)$ decays monotonously, and drops to zero at the transition point. In
contrast, in the massive regime $g^{-1}\ll 1$, the entanglement entropy
saturates to an area law. Only for $\gamma=0$ a thermal initial state produces
a volume law entanglement entropy.

##  III Measurement model

In this section, we construct a model for measurements in many-body systems in
the temporal and spatial continuum limit. This model captures the essence of
measurement-induced phase transitions, realizing non-commuting operators by
means of a kinetic energy Hamiltonian competing with exactly local
measurements, and lends itself to a field theory analysis.

###  III.1 Temporal continuum limit: Continuous measurements in quantum state
diffusion approach

_Projective vs. continuous measurements_ – Here we describe the framework of
continuous measurements that we utilize to model the monitored fermion and
boson dynamics. Compared to discrete projective measurements, the continuous
measurements are often referred to as weak measurements. Broadly speaking,
continuous measurements correspond to the limit where the amount of
information extracted per measurement (in the sense of an incomplete readout),
and the duration of each measurement, are reduced to zero simultaneously, in a
way that their ratio $\gamma$ is kept fixed Jacobs and Steck (2006); Wiseman
and Milburn (1993, 2009). This comes with the advantage of enabling us to
straightforwardly take the continuum limit in time, which in turn is useful
for mapping the problem into a field theory in Sec. IV. In the following we
refer to this type of measurement exclusively as ’continuous measurement’ or
’monitoring’ and reserve the terms ’weak’ and ’strong’ measurements (or
monitoring) to situations where $\gamma$ is small or large compared to the
scale of the Hamiltonian. As already pointed out above, continuous
measurements are expected to yield the same coarse grained dynamics as their
projective counterparts on the time scales of many measurements. Especially
they do not exclude the existence of a phase transition Szyniszewski  _et al._
(2019, 2020). In fact, we can replace in the competition ratio from the
framework of discrete unitary circuits and projective measurements

$\displaystyle g=\frac{\\#\text{projective
measurements/time}}{\\#\text{unitaries/time}}\to\frac{\gamma}{J},$ (7)

where $\gamma$ is the time scale of the monitoring and $J$ the scale of the
Hamiltonian, which can take arbitrary values. Accordingly, we will distinguish
weak monitoring $\gamma\ll J$ and strong monitoring $\gamma\gg J$. Indeed,
Ref. Szyniszewski  _et al._ (2019) has demonstrated that the existence of a
phase transition in random circuits upon increasing $g$ is preserved all along
the way from strong projective to weak continuous measurements.

_Stochastic Schrödinger equation_ – The starting point of our approach is a
stochastic Schrödinger equation in the quantum state diffusion framework
Wiseman and Milburn (2009); Carmichael (1993); Gisin and Percival (1992);
Wiseman and Milburn (1993). It corresponds to a monitoring protocol in which
the expectation value of an operator $\langle\hat{O}_{l}\rangle_{t}$ is
continuously measured, for instance a homodyne detection scheme Yang  _et al._
(2018a, b); Wiseman and Milburn (1993). In the quantum state diffusion, the
evolution equation $d\ket{\psi_{t}}=|\psi_{t+dt}\rangle-|\psi_{t}\rangle$ for
the pure state, normalized quantum trajectory wave function $|\psi_{t}\rangle$
is,

$\displaystyle d|\psi_{t}\rangle$ $\displaystyle=$ $\displaystyle-
idt[\hat{H}-i\tfrac{\gamma}{2}\sum_{i}\hat{M}^{2}_{i,t}]|\psi_{t}\rangle+\sum_{i}dW_{i}\hat{M}_{i,t}|\psi_{t}\rangle,$
(8)

for which we define the measurement operators

$\displaystyle\hat{M}_{i,t}=$
$\displaystyle\hat{O}_{i}-\langle\hat{O}_{i}\rangle_{t}$ (9)

for a general set of local measured operators $\hat{O}_{i}$, labeled by an
index $i$ (e.g. a lattice site index). The dynamical update of the state
$\ket{\psi_{t}}$ for an infinitesimal time interval $dt$ is generated by (i) a
deterministic contribution due to a non-Hermitian Hamiltonian
$\hat{H}_{\text{nH}}=\hat{H}-i\tfrac{\gamma}{2}\sum_{i}\hat{M}^{2}_{i,t}$
involving both the Hamiltonian $\hat{H}$ and the measurement operators. These
contributions are associated with the scale $J$ for the Hamiltonian as
anticipated, and $\gamma$ for the measurements, which is implemented by the
measurement operators $\hat{M}_{i,t}$. In addition, (ii), there is a
stochastic contribution: The Wiener process $dW_{i}$ is a local, real valued
Gaussian noise increment, with zero mean $\overline{dW_{i}}=0$ and variance
$\overline{dW_{i}dW_{j}}=\gamma dt\delta_{ij}$ Wiseman and Milburn (1993);
Gisin and Percival (1992); Jacobs and Steck (2006).

The measurement operators $\hat{M}_{i,t}$ involve the quantum mechanical
expectation value
$\langle\hat{O}_{i}\rangle_{t}\equiv\bra{\psi_{t}}\hat{O}_{i}\ket{\psi_{t}}$
evaluated at the time before the update, describing the measurement feedback.
This term makes the quantum trajectory evolution non-linear in the state of
the system. It ensures preservation of the norm of the trajectory wave
function: It is easily verified that all moments of the norm
$\mathcal{N}(n)\equiv\overline{\langle\psi|\psi\rangle^{n}}$ are conserved,
$\partial_{t}\mathcal{N}(n)=0$ for arbitrary $n$. This is an important
condition for describing a physical measurement. For example, the ’raw’
quantum state diffusion protocol, obtained by the replacement
$\hat{M}_{i,t}\to\hat{O}_{i}$ discarding the feedback term, preserves the norm
only on average, $\partial_{t}\mathcal{N}(n)\sim\gamma n(n-1)\mathcal{N}(n)$,
i.e. for $n=1$, and does not qualify for the description of a physical quantum
trajectory undergoing measurements.

A useful alternative representation of the stochastic Schrödinger equation
utilizes the conditioned projector
${\hat{\rho}_{t}^{(c)}=\ket{\psi_{t}}\bra{\psi_{t}}}$, which evolves according
to (cf. Eq. (28) and Refs. Jacobs and Steck (2006); Jacobs (2014) for details)

$\displaystyle d\hat{\rho}^{(c)}_{t}=$ $\displaystyle
dt(-i[\hat{H},\hat{\rho}^{(c)}_{t}]-\frac{\gamma}{2}\sum_{i}[\hat{O}_{i},[\hat{O}_{i},\hat{\rho}^{(c)}_{t}]])$
(10) $\displaystyle+\sum_{i}dW_{i}\\{\hat{M}_{i,t},\hat{\rho}^{(c)}_{t}\\},$

and where we used the anti-commutator
$\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$.

_Connection to projective measurements_ – This becomes particularly
transparent when focusing on the stochastic weak measurement evolution alone
($\hat{H}=0$ for the moment). Consider the long time limit $t\to\infty$ where
$dW_{i}$ acts as a multiplicative noise (multiplied with the state dependent
term $\hat{M}_{i,t}$), and becomes inactive only once the condition

$\displaystyle\hat{O}_{i}\ket{\psi_{t}}=\braket{\hat{O}_{i}}_{t}\ket{\psi_{t}}=O_{i}^{(\alpha)}\ket{\psi_{t}}$
(11)

is fulfilled – in other words, once the system has reached an eigenstate of
the measurement operator $\hat{O}_{i}$, and the instantaneous expectation
value $\braket{\hat{O}_{i}}_{t}$ is equal to one of the eigenvalues
$O_{i}^{(\alpha)}$ of $\hat{O}_{i}$ (please note that any eigenstate
$|\psi_{t}\rangle$ of $\hat{O}_{i}$ fulfills this condition). In this case,
also the double commutator in Eq. (10) vanishes. Thus, this evolution
describes a ‘continuous collapse’ of an initial wavefunction into an
eigenstate of the measurement operator, with the stochastic element provided
by the different noise realizations. We will refer to these eigenstates of the
measurement operators in a quantum optics language as dark states, since the
measurement evolution stops once the system has reached such state, and the
instantaneous expectation values $\langle\hat{O}_{i}\rangle_{t}$ become time-
independent.

_Dark state configurational entropy_ – Measurements come with a large
configurational entropy (as opposed to the entanglement entropy, building up
due to the entangling operations along each pure state trajectory) related to
the spectrum of eigenstates of the measurement operators and the their random
realization in the measurement process. This is best illustrated by means of a
concrete lattice model of $N$ continuously monitored spinless fermions.
Consider an entangling tight-binding hopping Hamiltonian

$\displaystyle\hat{H}=-J\sum_{l}c^{\dagger}_{i}c_{i+1}+\text{H.c.},$ (12)

where $c^{\dagger}_{i},c_{i}$ are fermionic creation and annihilation
operators on a lattice with sites $i=1,...,L$, and a continuous measurement
implemented by monitoring the local fermion densities

$\displaystyle\hat{O}_{i}=\hat{n}_{i}=c^{\dagger}_{i}c_{i}.$ (13)

For each of the local fermionic density measurements, there are two dark
eigenstates $\ket{\sigma_{i}}$ with $\sigma_{i}=0,1$. A random initial state
will ultimately collapse into a pure product state
$\ket{\\{\sigma_{i}\\}}=\prod_{i}\ket{\sigma_{i}}_{i}$, where the
configuration $\\{\sigma_{i}\\}$ is random, subject only to the constraint
$\sum_{i}\sigma_{i}=N$, the total number of fermions in the system. However,
for large systems there are exponentially many such pure states, e.g.
$\left(\begin{array}[]{c}2N\\\ N\end{array}\right)$ at half filling. Of
course, this reasoning generalizes to the quantum non-demolition case, where
$[\hat{H},\hat{M}_{i}]=0$ for all $i$; however, the non-commuting hopping
Hamiltonian will generate competition between measurement and Hamiltonian
evolution, and generically prevent the system from reaching an eigenstate of
the measurement operators. Indeed, this model exhibits a measurement-induced
phase transition, established in Ref. Alberton  _et al._ (2021) based on
numerical simulations.

The large configurational entropy is seen even more directly upon taking the
statistical average over the noise realizations. According to the Îto
calculus, the noise $dW_{i}$ at time $t$ is uncorrelated with the state
$\hat{\rho}^{(c)}_{t}$, such that the noise average of Eq. (10) yields the
Lindblad quantum master equation, i.e. the evolution equation for the
unconditioned density matrix
$\hat{\rho}_{t}\equiv\overline{\hat{\rho}^{(c)}_{t}}$,

$\displaystyle
d\hat{\rho}_{t}=dt(-i[\hat{H},\hat{\rho}_{t}]-\frac{\gamma}{2}\sum_{i}[\hat{n}_{i},[\hat{n}_{i},\hat{\rho}_{t}]]).$
(14)

The right hand side indeed describes a completely positive Lindblad operator,
for the special case of Hermitian Lindblad jump operators $\hat{n}_{l}$
(generalized measurements using non-Hermitian operators, and subsequently
general Lindblad equations, can be described in the POVM (positive operator
valued measurement) framework, cf. Nielsen and Chuang (2000)). Clearly, while
$\hat{\rho}^{(c)}_{t}$ describes a pure state, $\hat{\rho}_{t}$ represents a
mixed state ensemble of pure states. Equation (14) is a deterministic equation
which is linear in the state $\hat{\rho}_{t}$, as required by the foundations
of probability theory for the generator of motion for any complete statistical
representation of a system Primas (1990).

In terms of a physical interpretation, the Lindblad equation corresponds to
unread (or averaged over) measurements. This averaging over measurement
outcomes introduces the extensive configurational entropy, e.g.
$S=\log_{2}(\frac{2N!}{N!^{2}})\to 2N$ for a large, half filled system. For
$H\neq 0$, the Lindblad equation (14) is solved by a totally mixed, infinite
temperature state $\hat{\rho}\sim\mathbb{1}$ (here $\mathbb{1}$ is the
identity in the Hilbert space of fixed total particle number $N$). In fact,
this indefinite heating directed towards a stable, totally mixed dynamical
fixed point is generic for Hermitian measurement operators competing with a
Hamiltonian. This can be illustrated by considering the effect of the
Hamiltonian and the Lindblad operators separately. The Hermitian Lindblad
operators evolve the system towards a density matrix, which is diagonal in
Fock space (or diagonal in the basis of eigenstates of general Hermitian
measurement operators $\hat{O}_{l}$). The Hamiltonian, due to the non-
commutativity $[H,\hat{n}_{l}]\neq 0$, induces transitions between different
eigenstates and mixes different matrix elements until a totally mixed state
$\sim\mathds{1}$ is reached.

Figure 1: Illustration of competing Hamilton and measurement dynamics in the
two-site toy model. (a),(b) Single trajectory dynamics: (a) For strong
monitoring $g=10$ the system is pinned to dark states for most of the time,
with weak, short-time fluctuations. (b) A dominant Hamilton dynamics ($g=0.5$)
leads to depinning, with almost no time spent in the dark states. (c)
Trajectory averaged correlation functions: Quantities linear in the state are
insensitive to the competition ratio, while the correlation function nonlinear
in the state witnesses a monitoring dependence, which develops a non-analytic
behavior in the thermodynamic limit.

_Two-site toy model_ – Before delving into the analysis of the many-body
problem, we illustrate the competition between Hamiltonian and measurement
operators in a minimal toy model, a single fermion on two sites, $L=2,N=1$.
The problem is equivalent to a monitored two-level system, with its Hilbert
space spanned by the two states
$\\{\ket{\uparrow}=\ket{0,1},\ket{\downarrow}=\ket{1,0}\\}$, and
$\hat{H}=\sigma_{x}$, and the operator valued part of $\hat{M}_{i,t}$ given by
$(\mathds{1}_{2\times 2}\mp\sigma_{z})/2$ for $i=1,2$, respectively. For
$g^{-1}=0$ it continuously collapses into the dark states
$\ket{\uparrow},\ket{\downarrow}$, while for $g=0$, it describes Rabi
oscillations. In Fig. 1(a,b) we plot the trajectory expectation value of the
fermion occupation on site $i=1$. For strong monitoring $g^{-1}\ll 1$, the
system is pinned for long times in the dark states, with rare jumps into the
other dark state induced by the Hamiltonian, and small fluctuations in the
rest periods Zoller  _et al._ (1987); Sauter  _et al._ (1987). Conversely, for
weak monitoring $g\ll 1$, the evolution is still closely reminiscent of Rabi
oscillations, and there is a vanishing amount of time spent in the dark
states. We will be guided by this picture of depinning from the dark states
induced by the Hamiltonian dynamics in our analysis of the many-body problem.

The toy model also illustrates the importance of a suitable choice of
trajectory averaged quantities to detect the changes in the trajectories upon
varying the competition ratio $g$. Fig. 1(c) plots averages evaluated after
long evolution time. The observables $\overline{\langle\hat{n}_{1}\rangle}$
and $\overline{\langle\hat{n}_{1}\hat{n}_{2}\rangle}$ are linear in the
conditional state $\hat{\rho}_{t}^{(c)}$: Due to the indefinite heating, they
are not sensitive to the value of $g$. In contrast, the trajectory averaged
covariance matrix for the occupations

$\displaystyle
C_{12}=\overline{\langle\hat{M}_{1,t}\hat{M}_{2,t}\rangle_{t}}=\overline{\langle\hat{n}_{1}\hat{n}_{2}\rangle_{t}}-\overline{\langle\hat{n}_{1}\rangle_{t}\langle\hat{n}_{2}\rangle_{t}}$
(15)

is non-linear in the state and tracks a quantitative change as function of
$g$, rooted in the non-commuting nature of the Hamilton operator and the
measurement operators. In the small Hilbert space considered here, such
changes are only of a quantitative nature, i.e., one observes a crossover
instead of a sharp transition. In Sec. III.3, we will show that correlators of
this type may behave qualitatively differently in the weak and strong
monitoring regimes in a many-body system, which provides a clear indication of
a phase transition at a finite competition ratio $g_{c}$.

###  III.2 Spatial continuum limit: Measured Dirac fermions and bosonization

_Measured Dirac fermions_ – The above microscopic model of measured fermions
offers a measurement-induced phase transition, however in light of the heating
for the average measurement a direct low energy reduction to a continuum limit
may not be fully justified. Here we take a more conservative approach, and
formulate our starting point directly in the spatial continuum in terms of
(1+1) dimensional monitored massless Dirac model. The key ingredient shared
with the lattice model is a competition between a kinetic Hamiltonian giving
rise to delocalization of fermions and creating spatial entanglement, and
local, mutually commuting measurements, which localize or pin the fermions,
and by themselves drive the system into dark eigenstates of product form.

The massless Dirac Hamiltonian reads in terms of the spinor
$\hat{\Psi}_{x}=(\hat{\psi}_{R,x},\hat{\psi}_{L,x})^{T}$

$\displaystyle\hat{H}=iv\int_{x}\hat{\Psi}^{\dagger}_{x}\sigma_{z}\partial_{x}\hat{\Psi}_{x},$
(16)

where $\sigma_{z}$ is the Pauli matrix. We choose two distinct local
measurement operators $\hat{O}_{1,x},\hat{O}_{2,x}$,

$\displaystyle\hat{O}_{1,x}=\Psi^{\dagger}_{x}\Psi_{x}=\hat{J}^{(0)}_{x},\quad\hat{O}_{2,x}=\Psi^{\dagger}_{x}\sigma_{x}\Psi_{x},$
(17)

which are both measured independently but with the same rate $\gamma$. The
measurement operators commute with each other, but they do not commute with
the Hamiltonian, thus realizing the desired competition. They are local, and
stabilize product dark states
$\ket{\sigma_{a,x}}=\prod_{a,x}\ket{\sigma_{a,x}}_{x}$ with $a=\pm$, and
$\sigma_{a,x}=0,1$ the occupations in the basis where $\sigma_{x}$ is
diagonal.

The associated stochastic Schrödinger equation then features a non-Hermitian
Hamiltonian,

$\displaystyle\hat{H}_{\text{nH}}=\hat{H}-i\tfrac{\gamma}{2}\sum_{s=1,2}\int_{x}\hat{M}^{2}_{s,x,t},$
(18)

with $\hat{M}_{s,x,t}=\hat{O}_{s,x}-\braket{\hat{O}_{s,x}}_{t}$ and
uncorrelated noise increments
$\overline{dW_{s,x}dW_{s^{\prime},x^{\prime}}}=\delta_{s,s^{\prime}}\delta(x-x^{\prime})$
for both measurements. In order to characterize the transition we consider the
correlation functions

$\displaystyle C_{y}=\overline{\langle\hat{M}_{x,t}\hat{M}_{x+y,t}\rangle},\
\hat{M}_{x,t}=\hat{M}_{1,x,t}+\hat{M}_{2,x,t}$ (19)

with the symmetric sum of the measurement operators.

We note that this Hamiltonian and the measurement operators would obtain in a
naive low energy approach (at half filling), where the continuum limit is
performed for the lattice model defined with Eqs. (12,13), and the continuum
fermion field is decomposed as
$\hat{\psi}_{x}=\hat{\psi}_{R,x}e^{i2\pi\rho_{0}x}+\hat{\psi}_{L,x}e^{-i2\pi\rho_{0}x}$,
where $\rho_{0}$ here indicates the average fermion density. At half-filling,
$\rho_{0}=0.5$, the sum $\hat{M}_{x,t}$ in Eq. (19) thus corresponds to the
local fermion density $\hat{\psi}^{\dagger}_{x}\hat{\psi}_{x}$. For the main
part in this paper, we will not rely on this connection, and consider the
continuum model in autonomy. Yet, we will discuss striking phenomenological
parallels to lattice models with continuous and stroboscopic measurements in
Sec. VI.4.

_Bosonization_ – The monitored Dirac model lends itself to bosonization, where
the fermion bilinears are replaced by boson fields. The resulting bosonic
model is well suited for a further practical evaluation. The Hamiltonian maps
to the Luttinger Liquid Hamiltonian

$\displaystyle\hat{H}=\frac{v}{2\pi}\int_{x}[(\partial_{x}\hat{\theta}_{x})^{2}+(\partial_{x}\hat{\phi}_{x})^{2}].$
(20)

It is quadratic in the Hermitian operator $\hat{\phi}_{x}$ associated to
density fluctuations, and its conjugate $\hat{\theta}_{x}$ connected to phase
fluctuations. These operators fulfill the canonical commutation relations
$[\partial_{x}\hat{\theta}_{x},\hat{\phi}_{x^{\prime}}]=-i\pi\delta(x-x^{\prime})$.
The measurement operators transform into

$\displaystyle\hat{O}_{1,x}=-\frac{1}{\pi}\partial_{x}\hat{\phi}_{x},\quad\hat{O}_{2,x}=m\cos(2\hat{\phi}_{x})$
(21)

with a ’mass’ $m=\mathcal{O}(1)$ that depends on the normal ordering
prescription. Clearly, also in bosonized language $\hat{O}_{s,x}$ do not
commute with $\hat{H}$. The measurement operators both are functions of the
operator $\hat{\phi}_{x}$ and therefore pairwise commuting
$[\hat{O}_{\alpha,x},\hat{O}_{\beta,x^{\prime}}]=0$ for
$\alpha,\beta\in\\{1,2\\}$, and thus stabilize the eigenstates of
$\hat{\phi}_{x}$. In the measurement-only limit (i.e., for $\hat{H}=0$) this
gives rise to a set of dark states $\hat{M}_{s,x,t}\ket{\Psi_{D}}=0$, which
are eigenstates of the field operator
$\hat{\phi}_{x}\ket{\Psi_{D}}=\phi_{x}\ket{\Psi_{D}}$ for all positions $x$.
The only restriction to the set of real valued eigenvalues $\\{\phi_{x}\\}$ is
to match the condition for a fixed total particle number, which for periodic
boundary conditions reads as $\int_{x}\cos(2\pi\rho_{0}x+\phi_{x})=0$. This
gives rise to an exponentially growing number of dark states with pinned field
operator expectation values $\langle\hat{\phi}_{x}\rangle_{t}=\phi_{x}$,
analogous to the lattice fermion dark states with a pinned fermion particle
number.

As soon as the Hamiltonian is switched on ($\hat{H}\neq 0$), a competition
arises between the measurement operator $\hat{M}_{2,x,t}$, which tends to push
the system towards one of the measurement dark states, and the conjugate field
$\hat{\theta}_{x}$ in the Hamiltonian, which favors a depinning of
$\hat{\phi}_{x}$. We refer to the linear measurement operators
$\hat{M}_{1,x.t}\sim\partial_{x}(\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t})$
as gapless, due to their vanishing in the long wavelength limit (taking
$q\rightarrow 0$ in momentum space). They turn out to be compatible with the
quadratic Hamiltonian, i.e. they do not drive a phase transition. Instead,
their presence gives rise to a unique and well-defined stationary state with a
scale invariant connected correlation function
$\overline{\langle\hat{M}_{x,t}\hat{M}_{x+y,t}\rangle}\sim|y|^{-2}$.

What drives the competition between the pinning and depinning of the field
operator $\hat{\phi}_{x}$ is the interplay between the local, nonlinear
appearance of $\hat{\phi}_{x}$ in the measurement operators $\hat{M}_{2,x}$
and the conjugate fields $\partial_{x}\hat{\theta}_{x}$ in the Hamiltonian. As
we detail below in Eq. (22), for strong monitoring, the measurement operators
can be linearized around the dark state, yielding a measurement operator
$\hat{M}_{2,x,t}\sim m(\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t})$,
which does not vanish in the long wavelength limit. It causes exponentially
decaying correlations $C_{y}$, with a correlation length that is comparable to
the strong-monitoring correlation length in the fermion lattice model Eq. (A).
We therefore denote the linear approximation of $M_{2,x,t}$ as a gapped
measurement operator.

###  III.3 Limits of weak and strong monitoring: Riccati approach

We proceed by analyzing the connected correlation functions in two limiting
cases, where either $M_{2,x,t}$ is irrelevant and can be neglected, or
$M_{2,x,t}$ is dominating and obtains the gapped form $\hat{M}_{2,x,t}\sim
m(\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t})$. In both cases the
measurement operators are linear in the fields $\hat{\phi}_{x}$. For linear
measurement operators, the correlation function (19) affords an analytical
solution. This allows us to demonstrate that indeed for a gapless measurement
operator, the correlations functions obtain a scale-invariant form, while, in
contrast, correlations will not be scale invariant if the measurement
operators are gapped.

We identify the limit in which $M_{2,x,t}$ is irrelevant with the limit of
weak monitoring, i.e., when the Hamiltonian dominates the dynamics $\gamma\ll
v$. Then, the pinning of $\hat{\phi}_{x}$ induced by the nonlinearity in
$\hat{M}_{2,x,t}$ cannot overcome the depinning effect from the Hamiltonian in
Eq. (18). The field operators $\hat{\phi}_{x}$ will explore an extended region
of phase space, and the bounded nonlinear terms are irrelevant and average to
zero. We therefore drop $M_{2,x,t}$ from the evolution.

In the opposite limit of strong measurements, i.e., $v\ll\gamma$, the state of
the system will remain pinned in a measurement dark state $|\Psi_{D}\rangle$
similar to Fig. 1, where this is shown for the fermion density. At random
times, it may jump from the vicinity of one dark state to another dark state
with a different eigenvalue. These jumps are, however, uncorrelated and
exponentially rare in time in the limit of dominant measurements. We take the
pinning into account by assuming that the state $|\psi_{t}\rangle$ remains
close to a measurement dark state $|\Psi_{D}\rangle$ with
$\hat{\phi}_{x}|\Psi_{D}\rangle=\phi_{x}|\Psi_{D}\rangle$. For instance one
may assume $|\psi_{t}\rangle=|\Psi_{D}\rangle+|\delta\psi_{t}\rangle$ with
$\langle\delta\psi_{t}|(\hat{\phi}_{x}-\phi_{x})^{2}|\delta\psi_{t}\rangle/\langle\delta\psi_{t}|\delta\psi_{t}\rangle\ll
1$. We then shift the operator
$\hat{\phi}_{x}\rightarrow\phi_{x}+\hat{\phi}_{x}$ with respect to the
eigenvalue $\phi_{x}$ in the dark state and expand the measurement operators
$\hat{M}_{2,x,t}$ linearly in the small deviation $\hat{\phi}_{x}$, yielding

$\displaystyle\hat{M}_{2,x,t}\sim(\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t}).$
(22)

The more accurate proportionality constant in Eq. (22) is $2m\sin(2\phi_{x})$
and depends on the eigenvalues $\phi_{x}$. We neglect it here (see Appendix B
for more details, including the case of inhomogeneously distributed
$\phi_{x}$).

We consider now the dynamics with the quadratic Hamiltonian (20) and one
dominant measurement operator $\hat{M}_{x,t}$ which is linear in
$\hat{\phi}_{x}$, i.e., either
$M_{x,t}=\partial_{x}(\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t})$ or
$\hat{M}_{x,t}=\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t}$, reflecting
the two different regimes. This yields the stochastic Schrödinger equation

$\displaystyle
d|\psi_{t}\rangle=-dt\Big{[}i\hat{H}+\frac{\gamma}{2}\int_{x}\left(\hat{M}_{x,t}\right)^{2}\Big{]}|\psi_{t}\rangle+\int_{x}dW_{x}\hat{M}_{x,t}|\psi_{t}\rangle.$
(23)

The connected two-point correlation function for linear measurement problems
can be solved exactly by methods developed in the context of Kalman filtering
Kalman and Bucy (1961) in classical control theory Maybeck (2017); Brunton and
Kutz (2019), also used in quantum optics Doherty and Jacobs (1999); Wiseman
and Milburn (2009); Jacobs (2014). The problem maps to a Riccati equation,
which we solve in the stationary state in both limits (see Appendix B and Ref.
Minoguchi  _et al._ (2021) for details).

In the weak monitoring limit, this yields correlation functions in real and
momentum space

$\displaystyle C_{y}=\frac{C_{0}}{|y|^{2}},\ \ C_{k}=\frac{C_{0}|k|}{2}.$ (24)

Here
$C_{0}=\frac{v}{4\gamma}\Big{[}\Big{(}\frac{64\gamma^{2}}{\pi^{2}v^{2}}+4\Big{)}^{\frac{1}{2}}-2\big{]}^{\frac{1}{2}}$.
In the limit $\gamma\rightarrow 0$ this reduces to the ground state
correlations of a Luttinger Liquid with $C_{0}=1/\pi$. In the opposite limit
$\gamma\gg v$ it assumes the asymptotic form $C_{0}=\sqrt{v/2\gamma\pi}$.

This result shows that linear measurement operators
$\hat{M}_{x,t}\sim\partial_{x}\hat{\phi}_{x}-\langle\partial_{x}\hat{\phi}_{x}\rangle_{t}$
stabilize a scale invariant covariance matrix with a $1/|y|^{2}$ scaling
behavior. Precisely this scaling was also detected numerically in the case of
measured lattice fermions Alberton  _et al._ (2021). The only assumption
leading to this result in the present framework is that the nonlinear terms in
the measurement operators are negligible (or technically, $m=0$).

In the strong monitoring limit,
$M_{x,t}=\hat{\phi}_{x}-\langle\hat{\phi}_{x}\rangle_{t}$ yields

$\displaystyle C_{y}\sim\left(\frac{v}{2\pi\gamma}\right)^{|y|}=e^{-|y|/\xi}.$
(25)

This correlation function decays exponentially with the distance and with a
correlation length $\xi^{-1}=\log(2\pi\gamma/v)$. This indicates that the
extreme limit $v=0$, which is uncorrelated except for $y=0$, extends smoothly
to a short range correlated state at non-zero $v$. To further back up these
result beyond the linear approximation performed here, we have computed the
correlation length in the related fermionic lattice model defined by Eqs.
(12,13) perturbatively in Appendix A. This yields a similar correlation length
of $\xi^{-1}\sim\log(\gamma/\nu)$.

To summarize, we obtain a scale invariant correlation function with
characteristic $1/|y|^{2}$ decay at weak monitoring, versus exponentially
decaying correlations with a logarithmically growing correlation length at
strong monitoring. Strictly speaking, to obtain the first result, we needed to
set $\hat{M}_{2,x,t}=0$, and to perform linearization of $\hat{M}_{2,x,t}$ in
the opposite limit. The viability of this procedure reduces to the question
whether or not the nonlinearity in the measurement operator is relevant or
irrelevant. This question will be addressed in the next section.

##  IV Replica field theory for monitored Luttinger Liquids

In this section, we will provide a theoretical description, which is not
restricted to the two limiting cases and also incorporates intermediate
monitoring strength. To this end, we develop a field theory description of a
two-replica setup, which provides a tool to analyze whether an initial
nonlinearity in the measurement operators becomes relevant or irrelevant at
large distances. The replica approach also grants insights into the relevant
degrees of freedom in the measurement evolution, which turn out to be the
relative replica fluctuations.

###  IV.1 Structure of measurement operator correlation functions

In order to motivate the replica approach, we revisit the measurement operator
correlation function on the lattice
$C_{ij}(t)=\overline{\langle\hat{M}_{i,t}\hat{M}_{j,t}\rangle_{t}}$, where
again $\hat{M}_{i,t}=\hat{O}_{i}-\langle\hat{O}_{i}\rangle_{t}$ for the
measured operator $\hat{O}_{i}$. In Eq. (4), we expressed the correlation
function $C_{ij}$ in terms of the conditioned projector $\hat{\rho}^{(c)}_{t}$
and the trajectory average of its product
$\hat{\rho}^{(R_{2})}_{t}=\overline{\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}$
as

$\displaystyle
C_{ij}(t)=\frac{1}{2}\text{Tr}\left[(\hat{O}_{i}^{(1)}-\hat{O}_{i}^{(2)})(\hat{O}_{j}^{(1)}-\hat{O}_{j}^{(2)})\hat{\rho}^{(R_{2})}\right].$
(26)

The density matrix $\hat{\rho}^{(R_{2})}_{t}$ can no longer be written as a
product of pure states and contains inter-replica correlations, and therefore
$C_{ij}(t)$ can behave nontrivially.

Equation (26) shows that (i) the measurement correlation function is the
product of two relative observables $\hat{O}^{(1)}-\hat{O}^{(2)}$ (or their
fluctuation) in the replicated space and (ii) that this correlation function
must be encoded in the linear statistical average $\hat{\rho}^{(R_{2})}_{t}$.
For this and the following section, we introduce a shorthand notation for
expectation values as in Eq. (26). We set

$\displaystyle\langle\langle...\rangle\rangle=\text{Tr}[...\
\hat{\rho}^{(R_{2})}],$ (27)

which is the expectation value of an operator (or operator product) $...$ with
respect to $\hat{\rho}^{(R_{2})}$. It combines the quantum mechanical average
as well as the trajectory average from the previous sections.

###  IV.2 Two-replica master equation

The two-replica density matrix $\hat{\rho}^{(R_{2})}_{t}$ is no longer a
stochastic object but evolves according to a deterministic evolution equation.
The infinitesimal increment $d\hat{\rho}^{(R_{2})}_{t}$ is obtained from
taking the statistical average of the product increment

$\displaystyle d\hat{\rho}^{(R_{2})}_{t}$
$\displaystyle=\hat{\rho}^{(R_{2})}_{t+dt}-\hat{\rho}^{(R_{2})}_{t}=\overline{\hat{\rho}^{(c)}_{t+dt}\otimes\hat{\rho}^{(c)}_{t+dt}}-\overline{\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}$
$\displaystyle=\overline{d\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}+\overline{\hat{\rho}^{(c)}_{t}\otimes
d\hat{\rho}^{(c)}_{t}}+\overline{d\hat{\rho}^{(c)}_{t}\otimes
d\hat{\rho}^{(c)}_{t}}.$ (28)

The stochastic increments $\\{dW_{i}\\}$ at time-step $t\rightarrow t+dt$ are
not correlated with the increments at earlier times. Therefore the trajectory
average in Eq. (28) sets any increments which are only linear in $dW$ to zero
and replaces averages over terms quadratic in $dW$ by
$\overline{(...)dW_{i}dW_{j}}=\overline{(...)}\delta_{ij}\gamma dt$. Here
$(...)$ represents any other stochastic variable, such as
$\langle\hat{O}_{i}\rangle_{t}$ which does not depend on $dW$ but on
stochastic increments at earlier times.

The first and the second term in Eq. (28) only include the increment of one
single density matrix and are thus identical to taking the statistical average
over a single-replica master equation for $d\hat{\rho}^{(c)}$, i.e.,

$\displaystyle\overline{d\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}=$
$\displaystyle
dt\overline{\mathcal{L}^{(1)}\left(\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right)}+\sum_{i}\overline{dW_{i}\left\\{\hat{M}_{i,t}^{(1)},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}}$
$\displaystyle=$ $\displaystyle
dt\mathcal{L}^{(1)}\left(\overline{\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}\right)=dt\mathcal{L}^{(1)}\hat{\rho}^{(R_{2})}_{t}.$
(29)

Here, we have defined the intra-replica Liouvillian
$\mathcal{L}^{(l)}=i[\hat{H}^{(l)},\cdot]-\frac{\gamma}{2}\sum_{i}[\hat{O}_{i},[\hat{O}_{i},\cdot]]$
and measurement operator
$\hat{M}^{(l)}_{i,t}=\hat{O}^{(l)}-\langle\hat{O}_{i}\rangle_{t}$. The
conditioned expectation value
$\langle\hat{O}_{i}\rangle_{t}=\text{Tr}\left[\hat{O}_{i}\hat{\rho}^{(c)}_{t}\right]$
is still a stochastic object, which does, however, not depend on the noise
$dW$ at time $t$, as mentioned above. This yields the familiar Lindblad form
as in Eq. (14), acting on each single replica separately.

The third term, however, builds up correlations between both replicas and
yields an unconventional contribution to the evolution of
$\hat{\rho}^{(R_{2})}_{t}$. The replica-coupling evolution term is

$\displaystyle\overline{d\hat{\rho}^{(c)}_{t}\otimes d\hat{\rho}^{(c)}_{t}}=$
$\displaystyle\sum_{i,j}\overline{\left\\{\hat{M}_{i,t}^{(1)},\left\\{\hat{M}_{j,t}^{(2)},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}\right\\}dW_{i}dW_{j}}$
$\displaystyle=$ $\displaystyle\gamma
dt\sum_{i}\overline{\left\\{\hat{M}_{i,t}^{(1)},\left\\{\hat{M}_{i,t}^{(2)},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}\right\\}}.$
(30)

It contains three different classes of couplings, two of which we have to
consider with care. The first class are terms composed only of operators,
i.e., the ones summarized by

$\displaystyle\overline{\left\\{\hat{O}_{i}^{(1)},\left\\{\hat{O}_{i}^{(2)},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}\right\\}}=\left\\{\hat{O}_{i}^{(1)},\left\\{\hat{O}_{i}^{(2)},\hat{\rho}^{(R_{2})}_{t}\right\\}\right\\}.$
(31)

Here the stochastic average acts only on the density matrix and does not
affect the operators at all. The second class of increments is

$\displaystyle\overline{\left\\{\langle\hat{O}_{i}\rangle_{t},\left\\{\langle\hat{O}_{i}\rangle_{t},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}\right\\}}=4\overline{\langle\hat{O}_{i}\rangle_{t}^{2}\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}.$
(32)

Here we face a problem with computing the stochastic average, since the two
replica density matrix $\hat{\rho}^{(R_{2})}_{t}$ does not give access to the
individual trajectories of $\langle\hat{O}_{i}\rangle_{t}$. It only allows us
to compute the stochastic averages
$\langle\langle\hat{O}^{(l)}\rangle\rangle=\overline{\langle\hat{O}_{i}\rangle_{t}}$
or
$\langle\langle\hat{O}^{(1)}_{j}\hat{O}^{(2)}_{i}\rangle\rangle=\overline{\langle\hat{O}_{i}\rangle_{t}\langle\hat{O}_{j}\rangle_{t}}$,
but this does not include information on the statistical correlations between
$\langle\hat{O}_{i}\rangle_{t}$ and
$\hat{\rho}^{c}_{t}\otimes\hat{\rho}^{c}_{t}$, which are required in the
average in Eq. (32). This also holds for the third type of increment, which is

$\displaystyle\overline{\left\\{\hat{O}_{i}^{(l)},\left\\{\langle\hat{O}_{i}\rangle_{t},\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}\right\\}\right\\}}=2\left\\{\hat{O}_{i}^{(l)},\overline{\langle\hat{O}_{i}\rangle_{t}\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}\right\\}.$
(33)

This problem indicates the initial point of an infinite coupled hierarchy of
replica correlation functions: Indeed, the statistical averages in Eqs. (33)
and (32) can again be formulated as linear problems in a three- or four-
replica framework, respectively (e.g.,
$\overline{\langle\hat{O}_{i}\rangle_{t}\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}=\text{Tr}_{(3)}[\hat{O}^{(3)}_{i}\hat{\rho}^{(R_{3})}_{t}]$
can be written as a partial trace over the third replica in a three-replica
density matrix $\hat{\rho}^{(R_{3})}_{t}$). This illustrates that once the
realm of a single replica description of the problem is left, an infinite
hierarchy of replica equations emerges. In order to find a solution for
$\hat{\rho}^{(R_{2})}$, this hierarchy has to be truncated at finite order.

Here we perform the truncation on the level of the two-replica density matrix,
which amounts to treating the stochastic correlations between
$\langle\hat{O}_{i}\rangle_{t}$ and
$\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}$ in a mean-field type
decoupling by setting

$\displaystyle\overline{\langle\hat{O}_{i}\rangle_{t}\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}$
$\displaystyle\equiv\langle\langle\hat{O}^{(1,2)}_{i}\rangle\rangle_{t}\hat{\rho}^{(R_{2})}_{t},$
(34)
$\displaystyle\overline{\langle\hat{O}_{i}\rangle_{t}^{2}\hat{\rho}^{(c)}_{t}\otimes\hat{\rho}^{(c)}_{t}}$
$\displaystyle\equiv\langle\langle\hat{O}^{(1)}_{i}\hat{O}^{(2)}_{i}\rangle\rangle_{t}\hat{\rho}^{(R_{2})}_{t}.$
(35)

We will see later that if the operators $\hat{O}_{l}$ are linear functions of
boson or fermion fields (as it is the case for the operators $\hat{M}_{1,x,t}$
in the bosonized framework), then the mean-field decoupling in Eqs. (34), (35)
has no effect on the relative correlation function $C_{ij}$, i.e., the mean
field approximation is exact for these correlators. The measurement
expectation values then only show up in a center-of-mass coordinate
$\hat{O}_{i}^{(1)}+\hat{O}_{i}^{(2)}$, which completely decouples from
relative coordinates $\hat{O}^{(1)}_{i}-\hat{O}^{(2)}_{i}$ and effectively
heats up to infinite temperature. This heating of the center-of-mass
coordinate remains operational when the linear operator $\hat{M}_{1,x,t}$ and
the non-linear operator $\hat{M}_{2,x,t}$ are measured simultaneously; in turn
this justifies the mean-field approximation also in this case. The Hermitian
nature of the measurement operators, and the subsequent heating, simplifies
the problem significantly here. For a general set of only nonlinear and non-
Hermitian measurement operators, including nonlinear feedback operations, the
mean-field decoupling is, however, not immediately justified.

After performing the statistical average including the mean-field decoupling,
one obtains a deterministic evolution equation for $\hat{\rho}^{(R_{2})}$, the
two-replica master equation

$\displaystyle\partial_{t}\hat{\rho}^{(R_{2})}_{t}=$
$\displaystyle\mathcal{L}^{(1)}\hat{\rho}^{(R_{2})}_{t}+\mathcal{L}^{(2)}\hat{\rho}^{(R_{2})}_{t}-8\gamma\hat{\rho}^{(R_{2})}\tilde{C}(t)$
(36) $\displaystyle+2\gamma\sum_{l}\left\\{\hat{O}^{(2)}_{i}-\langle\langle
O^{(2)}_{i}\rangle\rangle,\left\\{\hat{O}^{(1)}_{i}-\langle\langle\hat{O}^{(1)}_{i}\rangle\rangle,\hat{\rho}^{(R_{2})}_{t}\right\\}\right\\}.$

Here, the normalization function is
$\tilde{C}_{t}=\sum_{i}\left(\langle\langle\hat{O}^{(1)}_{i}\hat{O}^{(2)}_{i}\rangle\rangle_{t}-\langle\langle\hat{O}^{(1)}_{i}\rangle\rangle_{t}\langle\langle\hat{O}^{(2)}_{i}\rangle\rangle_{t}\right)$.
The first line is the regular master equation for two uncoupled replicas
evolving under the Liouvillians $\mathcal{L}^{(l)}$. If no additional terms
were present, the Liouvillians would heat up each replica towards an infinite
temperature state.

The second line in Eq. (36) is what distinguishes the two-replica setup from a
single replica, which would only heat up to infinite temperature. It leads to
the build-up of inter-replica correlations, and transforms the replica master
equation into a non-Lindblad form. This equation is still non-linear in the
state due to the expectation values $\langle\hat{O}_{i}\rangle_{t}$, contained
in the operators $\hat{M}_{i,t}$ and in $\tilde{C}_{t}$. However, the equation
is deterministic, and the expectation values are no longer stochastic
variables. In the stationary state, they can be replaced by a number
$\langle\hat{O}_{i}\rangle_{t}=o_{i}$. We will analyze the consequences of
this evolution equation in the following section.

###  IV.3 Two-replica master equation in bosonized framework

We now formulate the bosonized measurement setup introduced in Sec. III.2 in
terms of the replica master equation (36). This will allow us to establish the
connection between the generator of dynamics, and the correlation functions.
To this end, we extend the bosonized Hamiltonian and the measurement operators
to replica space by introducing

$\displaystyle\hat{H}^{(l)}$
$\displaystyle=\frac{v}{2\pi}\int_{x}(\partial_{x}\hat{\phi}_{x}^{(l)})^{2}+(\partial_{x}\hat{\theta}_{x}^{(l)})^{2},$
(37) $\displaystyle\hat{M}_{1,x,t}^{(l)}$
$\displaystyle=-\tfrac{1}{\pi}\partial_{x}(\hat{\phi}_{x}^{(l)}-\langle\hat{\phi}_{x}^{(l)}\rangle_{t}),$
(38) $\displaystyle\hat{M}_{2,x,t}^{(l)}$
$\displaystyle=2m(\cos(2\hat{\phi}_{x}^{(l)})-\langle\cos(2\hat{\phi}_{x}^{(l)})\rangle_{t}),$
(39)

where again the label $l$ indicates how the operator acts in replica space
(see the definition above Eq. (26)). The two-replica Hamiltonian is
$\hat{H}=\hat{H}^{(1)}+\hat{H}^{(2)}$. Due to independent noise increments for
$\hat{M}_{1,x,t}$ and $\hat{M}_{2,x,t}$, the two different measurements are
uncorrelated and contribute independently to the replica master equation. This
can be implemented in the two-replica master equation by replacing the index
$i\rightarrow(s,x)$ with $s=1,2$ and $x$ the continuous position.

To gain a better understanding of the two-replica dynamics, we again start by
considering only measurements of the linear operator $\hat{M}_{1,x,t}$ without
measuring $\hat{M}_{2,x,t}$. Apart from the constant terms, this yields the
two-replica master equation

$\displaystyle\partial_{t}\hat{\rho}^{(R_{2})}=$ $\displaystyle
i\left[\hat{\rho}^{(R_{2})},\hat{H}\right]-\frac{\gamma}{2\pi}\sum_{l=1,2}\int_{x}\left[\partial_{x}\hat{\phi}^{(l)}_{x},\left[\partial_{x}\hat{\phi}^{(l)}_{x},\hat{\rho}^{(R_{2})}\right]\right]$
(40)
$\displaystyle+\frac{\gamma}{\pi}\int_{x}\left\\{\partial_{x}\hat{\phi}^{(2)}_{x}-\langle\langle\partial_{x}\hat{\phi}_{x}^{(2)}\rangle\rangle,\left\\{\partial_{x}\hat{\phi}^{(1)}_{x}-\langle\langle\partial_{x}\hat{\phi}_{x}^{(1)}\rangle\rangle,\hat{\rho}^{(R_{2})}\right\\}\right\\}.$

The master equation is quadratic in the field operators, but couples operators
with different replica index $l$. The terms can be decoupled by performing a
coordinate transformation into a replica center-of-mass field
$\hat{\phi}^{(a)}_{x},\hat{\theta}^{(a)}_{x}$ and a relative field
$\hat{\phi}^{(r)}_{x},\hat{\theta}^{(r)}_{x}$ (describing inter-replica
fluctuations) according to the unitary transformation
${\hat{\phi}^{(a,r)}_{x}=(\hat{\phi}^{(1)}_{x}\pm\hat{\phi}^{(2)}_{x})/\sqrt{2}}$,
${\hat{\theta}^{(a,r)}_{x}=(\hat{\theta}^{(1)}_{x}\pm\hat{\theta}^{(2)}_{x})/\sqrt{2}}$.
It preserves the bosonic commutation relations between the operators
$[\partial_{x}\hat{\theta}^{(l)}_{x},\hat{\phi}^{(l^{\prime})}_{y}]=-i\delta_{l,l^{\prime}}\delta(x-y)$
for $l,l^{\prime}=a,r$.

The quadratic Hamiltonian $\hat{H}$ remains a sum
$\hat{H}=\hat{H}^{(r)}+\hat{H}^{(a)}$ with $\hat{H}^{(l)}$ as defined in Eq.
(37), but with center-of-mass and relative operators. The measurement part of
the evolution, however, is transformed such that the relative and center-of-
mass decouple in the new basis. The master equation for $\hat{\rho}^{(R_{2})}$
is therefore separable, which suggests a product ansatz
$\hat{\rho}^{(R_{2})}=\hat{\rho}^{(r)}\otimes\hat{\rho}^{(a)}$. Inserting this
ansatz into Eq. (40) yields

$\displaystyle\partial_{t}\hat{\rho}^{(r)}=$ $\displaystyle
i[\hat{\rho}^{(r)},\hat{H}^{(r)}]-\frac{\gamma}{\pi^{2}}\int_{x}\left\\{(\partial_{x}\hat{\phi}^{(r)}_{x})^{2},\hat{\rho}^{(r)}\right\\},$
(41) $\displaystyle\partial_{t}\hat{\rho}^{(a)}=$ $\displaystyle
i[\hat{\rho}^{(a)},\hat{H}^{(a)}]$ (42)
$\displaystyle+\frac{2\gamma}{\pi^{2}}\int_{x}\left(\partial_{x}\hat{\phi}^{(a)}-\langle\langle\partial_{x}\hat{\phi}^{(a)}\rangle\rangle\right)\hat{\rho}^{(a)}\left(\partial_{x}\hat{\phi}^{(a)}-\langle\langle\partial_{x}\hat{\phi}^{(a)}\rangle\rangle\right).$

This decomposition is a remarkable result, which gives the two replica master
equation a new interpretation. The density matrix $\hat{\rho}^{(r)}$ of the
fluctuations evolves according to a non-Hermitian Schrödinger equation with an
effective Hamiltonian

$\displaystyle\hat{H}_{\text{eff}}^{(r)}=\frac{v}{2\pi}\int_{x}(\partial_{x}\hat{\theta}_{x}^{(r)})^{2}+(1-i\frac{2\gamma}{v\pi})(\partial_{x}\hat{\phi}_{x}^{(r)})^{2},$
(43)

but without feedback from a contour-coupling term (i.e., no coupling between
the left and the right side of $\hat{\rho}^{(R_{2})}$). The density matrix of
the relative degrees is thus evolving towards a stationary state
$\hat{\rho}^{(r)}_{t}=e^{-i\hat{H}^{(r)}_{\text{eff}}t}\hat{\rho}^{(r)}_{0}e^{i\hat{H}^{(r)}_{\text{eff}}t}\rightarrow|\psi^{(r)}_{D}\rangle\langle\psi^{(r)}_{D}|$
which approaches a dark state of the effective Hamiltonian
$\hat{H}_{\text{eff}}^{(r)}|\psi^{(r)}_{D}\rangle=0$. In Appendix E, we
demonstrate that $H_{\text{eff}}^{(r)}$ indeed has one unique dark state,
which is reached from any initial state during the time evolution. We show
that the correlation functions of the replica field operators in the dark
state,
$\langle\psi^{(r)}_{D}|\partial_{x}\hat{\phi}^{(r)}_{x}\partial_{y}\hat{\phi}^{(r)}_{y}|\psi^{(r)}_{D}\rangle$
are identical to the correlation functions (24) obtained from the Riccati
approach, which provides the exact solution for the covariance matrix.

This observation is independent of the mean-field decoupling of the
statistical fluctuations in Eqs. (34), (35). The reason is that due to the
replica symmetry (i.e. invariance of the theory under exchange
$\hat{\phi}^{(1)}\leftrightarrow\hat{\phi}^{(2)}$), independently of the noise
realization we have
$\langle\hat{\phi}^{(1)}\rangle=\langle\hat{\phi}^{(2)}\rangle$. This implies
$\sqrt{2}\langle\hat{\phi}^{(r)}\rangle=\langle\hat{\phi}^{(1)}-\hat{\phi}^{(2)}\rangle=0$.
Therefore, the neglected noise correlations are only affecting
$\hat{\rho}^{(a)}$, but not the density matrix of the fluctuations
$\hat{\rho}^{(r)}$. The dark state therefore describes the exact steady state
of the replica fluctuations.

We now turn to the measurement correlation function Eq. (26), here under the
assumption that the nonlinear terms are irrelevant, $\hat{M}_{2,x,t}=0$. The
relative measurement operator is
$\hat{M}^{(1)}_{x,t}-\hat{M}^{(2)}_{x,t}=\frac{\sqrt{2}}{\pi}\,\partial_{x}\hat{\phi}^{(r)}_{x}$,
yielding the measurement correlation function
$C_{y}=\frac{1}{\pi^{2}}\langle\psi^{(r)}_{D}|\partial_{x}\hat{\phi}^{(r)}\partial_{x}\hat{\phi}_{x+y}^{(r)}|\psi^{(r)}_{D}\rangle$,
the expectation value of
$\partial_{x}\hat{\phi}^{(r)}\partial_{x}\hat{\phi}_{x+y}^{(r)}$ in the dark
state. The scale invariance of the effective Hamiltonian
$\hat{H}_{\text{eff}}^{(r)}$ implies that also the measurement correlation
functions in this limit become scale invariant and algebraically decaying,
confirming the observation from Sec. III.3.

We can also infer what happens in the limit when $\hat{M}_{2,x,t}$ is measured
with a large mass $\gamma m\gg v$, where the linearization
$\hat{M}_{2,x,t}\rightarrow\hat{\phi}^{(l)}_{x}-\langle\hat{\phi}^{(l)}_{x}\rangle_{t}$
is justified. Using the same arguments as for the case $m=0$, this yields a
separable evolution equation for $l=r,a$ with a modified effective Hamiltonian
in Eq. (41) $\hat{H}^{(r)}_{\text{eff}}\rightarrow\hat{H}^{(r)}_{\text{mass}}$
and

$\displaystyle\hat{H}_{\text{mass}}^{(r)}=\hat{H}^{(r)}_{\text{eff}}-i\gamma\int_{x}(\hat{\phi}_{x}^{(r)})^{2}.$
(44)

In $\hat{H}_{\text{mass}}^{(r)}$ the additional dissipative mass term
$\sim\gamma$ introduces a length scale. The measurement correlations are
therefore still described by a pure quantum state but now correlations
functions in the state $|\tilde{\psi}^{(r)}_{D}\rangle$ are exponentially
decaying in space.

The effective cooling and the evolution towards a single dark state in the
relative degrees of freedom $\hat{\theta}^{(r)}_{x},\hat{\phi}^{(r)}_{x}$ have
to be contrasted with the behavior of the center-of-mass fields
$\hat{\theta}^{(a)}_{x},\hat{\phi}^{(a)}_{x}$ described by Eq. (42). Their
evolution does not follow an effective Schrödinger equation but instead
displays enhanced statistical fluctuations. In order to determine the fate of
the local correlation functions for the center-of-mass degrees of freedom, we
consider their evolution equation

$\displaystyle\partial_{t}\langle\langle\hat{\phi}_{x}^{(a)}\hat{\phi}_{x}^{(a)}\rangle\rangle=$
$\displaystyle\text{Tr}\left[(\hat{\phi}_{x}^{(a)})^{2}\partial_{t}\hat{\rho}^{(a)}\right]=-i\langle\langle[\hat{H}^{(a)},\hat{\phi}_{x}^{(a)}\hat{\phi}_{x}^{(a)}]\rangle\rangle$
$\displaystyle+\tfrac{2\gamma}{\pi^{2}}\int_{x^{\prime}}\langle\langle(\partial_{x}\hat{\phi}^{(a)}_{x^{\prime}}-\langle\langle\partial_{x}\hat{\phi}^{(a)}_{x^{\prime}}\rangle\rangle)^{2}(\hat{\phi}_{x}^{(a)})^{2}\rangle\rangle.$
(45)

Due to the quadratic evolution equation for $\hat{\rho}^{(a)}_{t}$, higher
order correlation functions such as in second line of Eq. (IV.3) can be
decoupled via Wick’s theorem. This yields a nonlinear, strictly positive
growth of
$\langle\langle\hat{\phi}_{x}^{(a)}\hat{\phi}_{x}^{(a)}\rangle\rangle$ in
time. The commutator with the Hamiltonian yields a term proportional to
$\langle\langle\hat{\phi}_{x}^{(a)}\hat{\theta}_{x}^{(a)}\rangle\rangle$,
which, however, has a similar non-linear evolution equation (just like
$\langle\langle\hat{\theta}_{x}^{(a)}\hat{\theta}_{x}^{(a)}\rangle\rangle$).
In consequence this yields an asymptotic value
$\langle\langle\hat{\phi}^{(a)}_{x}\hat{\phi}^{(a)}_{x}\rangle\rangle\rightarrow\infty$,
which is consistent with an infinite temperature state in the center-of-mass
degrees of freedom.

###  IV.4 Nonlinear master equation for the replica fluctuations

Now, we will focus on the general case in which the measurement operator
$\hat{M}_{2,x,t}$ cannot be neglected and is a nonlinear function of
$\hat{\phi}^{(l)}_{x}$. We will use the insights from the previous discussion
to obtain the master equation for the replica fluctuations. We demonstrate
that also in this case the time evolution of the relative degrees of freedom
is described by an effective Hamiltonian. It is a non-Hermitian analogue of
the sine-Gordon Hamiltonian.

We start with the full two-replica master equation in the basis of relative
and center-of-mass degrees of freedom

$\displaystyle\partial_{t}\hat{\rho}^{(R_{2})}=$
$\displaystyle-i[\hat{H}^{(r)}_{\text{eff}}\hat{\rho}^{(R_{2})}-\hat{\rho}^{(R_{2})}(\hat{H}^{(r)}_{\text{eff}})^{\dagger}]-i[\hat{H}^{(a)},\hat{\rho}^{(R_{2})}]+\frac{2\gamma}{\pi^{2}}\int_{x}\left(\partial_{x}\hat{\phi}^{(a)}-\langle\langle\partial_{x}\hat{\phi}^{(a)}_{x}\rangle\rangle\right)\hat{\rho}^{(R_{2})}\left(\partial_{x}\hat{\phi}^{(a)}_{x}-\langle\langle\partial_{x}\hat{\phi}^{(a)}_{x}\rangle\rangle\right)$
(46) $\displaystyle-\gamma
m^{2}\int_{x}\sum_{\sigma=\pm}\left[\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}+\sigma\hat{\phi}_{x}^{(r)})\right),\left[\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}+\sigma\hat{\phi}_{x}^{(r)})\right),\hat{\rho}^{(R_{2})}\right]\right]$
$\displaystyle+2\gamma
m^{2}\left\\{\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}+\hat{\phi}_{x}^{(r)})\right)-\langle\langle\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}+\hat{\phi}_{x}^{(r)})\right)\rangle\rangle,\left\\{\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}-\hat{\phi}_{x}^{(r)})\right)-\langle\langle\cos\left(\sqrt{2}(\hat{\phi}_{x}^{(a)}-\hat{\phi}_{x}^{(r)})\right)\rangle\rangle,\hat{\rho}^{(R_{2})}\right\\}\right\\}.$

This equation may look overwhelming at first sight, but we will now show how
its complexity can be drastically reduced by making use of the insights from
the previous section. We will eliminate the coupling of the relative degrees
of freedom to the center-of-mass modes, to obtain an effective master equation
for $\hat{\rho}^{(r)}$.

In order to do this, we trace out the center-of-mass degrees of freedom in Eq.
(46), i.e., we set
$\partial_{t}\hat{\rho}^{(r)}\equiv\text{tr}_{(a)}\partial_{t}\hat{\rho}^{(R_{2})}$,
where $\text{tr}_{(a)}$ denotes the trace over the center-of-mass modes. We
also set
$\langle\langle...\rangle\rangle_{a}=\text{tr}_{(a)}(...\rho^{(R_{2})})$. In
order to integrate out the center-of-mass modes, we have to consider that they
couple to the relative modes nonlinearly. However, we have seen above that the
quadratic part of Eq. (46) is separable, i.e.,
$\rho^{(R_{2})}=\rho^{(r)}\otimes\rho^{(a)}$, and pushes the correlation
functions towards
$\langle\langle\hat{\phi}^{a}_{x}\hat{\phi}^{a}_{x}\rangle\rangle_{a}\rightarrow\infty$,
corresponding to a density matrix $\hat{\rho}^{(a)}$ in an infinite
temperature state.

The bounded vertex operators $\exp(i\sqrt{2}\hat{\phi}^{(a)}_{x})$ alone do
not push the center-of-mass mode away from an infinite temperature state. They
rather become irrelevant in this limit, which follows from
$\langle\langle\exp(i\sqrt{2}\hat{\phi}^{(a)}_{x})\rangle\rangle_{a}=\exp(-\langle\langle\hat{\phi}^{(a)}_{x}\hat{\phi}^{(a)}_{x}\rangle\rangle_{a})\rightarrow
0$, where we assumed that we can apply Wick’s theorem. This also indicates
that the coupling between the relative and the center-of-mass modes will not
modify the correlation functions of the latter. While in general the precise
form of $\hat{\rho}^{(R_{2})}$ may be unknown, this suggests very weak
correlations between the two sets of modes. We therefore approximate the
density matrix by a product
$\hat{\rho}^{(R_{2})}\sim\hat{\rho}^{(r)}\otimes\hat{\rho}^{(a)}$. The the
only assumption we need regarding $\hat{\rho}^{(a)}$ is that it is in a
Gaussian state with
$\langle\langle\hat{\phi}^{(a)}_{x}\hat{\phi}^{(a)}_{x}\rangle\rangle_{a}\rightarrow\infty$
111It is also possible to perturbatively incorporate correlations between the
center-of-mass and the relative degrees of freedom beyond a product state
assumption by applying time-dependent perturbation theory in the Mori-Zwanzig
formalism Zwanzig (2001). However, these corrections are always less relevant
than the first order correction used here..

In this case, tracing out the center-of-mass modes from Eq. (46) can be
performed analytically. First, the dependence on the expectation values
$\langle\langle\cos\left(\sqrt{2}(\hat{\phi}^{(a)}_{x}\pm\hat{\phi}^{(r)}_{x})\right)\rangle\rangle=0$
vanishes. Second, only non-linear vertex operators $\sim\exp(i...)$, which are
independent of $\hat{\phi}^{(a)}_{x}$, persist upon taking the trace with
respect to the average degrees of freedom. Thus only the terms in the third
line of Eq. (46) yield a non-trivial contribution to the evolution equation of
$\rho^{(r)}$. We illustrate this by tracing out the center-of-mass mode in the
cross term

$\displaystyle\langle\langle\cos\left(\sqrt{2}(\hat{\phi}^{(a)}_{x}+\hat{\phi}^{(r)}_{x})\right)\cos\left(\sqrt{2}(\hat{\phi}^{(a)}_{x}-\hat{\phi}^{(r)}_{x})\right)\rangle\rangle_{a}=\frac{1}{2}\langle\langle\cos\left(\sqrt{8}\hat{\phi}^{(a)}_{x}\right)+\cos\left(\sqrt{8}\hat{\phi}^{(r)}_{x}\right)\rangle\rangle_{a}=\frac{1}{2}\cos\left(\sqrt{8}\hat{\phi}^{(r)}_{x}\right).$
(47)

From this procedure, we obtain the evolution equation for the relative replica
fluctuations. In the following we will only focus on this degree of freedom
and drop the label $(r)$ 222The trace has to be evaluated with care. Due to
the noise average there is no straightforward cyclic permutation rule for
operators in the absolute and relative basis.. After keeping the most relevant
nonlinear terms only, the master equation for the fluctuations is

$\displaystyle\partial_{t}\hat{\rho}=$ $\displaystyle
i(\hat{\rho}\hat{H}^{\dagger}_{sg}-\hat{H}_{sg}\hat{\rho}),$ (48)

where $\lambda=\gamma m^{2}$, and the non-Hermitian sine-Gordon Hamiltonian
reads

$\displaystyle\hat{H}_{sg}=$
$\displaystyle\frac{v}{2\pi}\int_{x}\left[(\partial_{x}\hat{\theta}_{x})^{2}+\left(1-\tfrac{2i\gamma}{v\pi}\right)(\partial_{x}\hat{\phi}_{x})^{2}\right]$
(49) $\displaystyle-i\lambda\int_{x}(\cos(\sqrt{8}\hat{\phi}_{x})-1).$

Again we encounter a non-Hermitian Hamiltonian evolution for the relative
degrees of freedom.

On large length- and long time-scales, the irrelevance of higher order
perturbative corrections (scaling $\sim\cos(\zeta\sqrt{8}\hat{\phi})$ at
$\zeta$-th order) again leads to an effective cooling of the replica
fluctuations towards the dark state of the non-Hermitian Hamiltonian
$\hat{H}_{sg}$. Depending on the strength of the nonlinearity and the
imaginary part in the kinetic energy term, this dark state is scale invariant
at large distances, or has built in a length-scale yielding either
algebraically or exponentially decaying correlations. For a Hermitian sine-
Gordon Hamiltonian, a renormalization group treatment predicts a phase
transition between a scale invariant ground state and a gapped ground state
according to the Berezinskii-Kosterlitz-Thouless paradigm Zinn-Justin (1989);
Amit  _et al._ (1980); Kosterlitz and Thouless (1973), and we may conjecture
that a similar scenario applies to the dark state of the non-Hermitian
Hamiltonian. We will examine the nature of the transition in terms of a
renormalization group analysis of $\hat{H}_{sg}$ in the following section.

We conclude this section with an additional remark. The effective Hamiltonian
$\hat{H}_{sg}$ is not PT symmetric 333Both $(\partial_{x}\hat{\phi}_{x})^{2}$
and $\cos(\sqrt{8}\hat{\phi}_{x})$ are PT symmetric and therefore multiplying
them with an imaginary unit generates PT non-symmetric terms. and therefore
its eigenvalues are generally complex. However, the imaginary part of the
eigenvalues is always negative, which guarantees dynamical stability of the
time-evolution. An evolution with a non-Hermitian Hamiltonian with complex
eigenvalues, however, does not preserve the norm of the state. This is
possible because there is no constraint that ensures the norm of
$\hat{\rho}^{(r)}$ and $\hat{\rho}^{(a)}$ to be conserved individually. The
norm of $\hat{\rho}^{(R_{2})}$ is preserved as argued in Sec. III.

##  V Renormalization group approach to the non-Hermitian sine-Gordon theory

The sine-Gordon Hamiltonian in Eq. (49) describes the long-time and large-
distance behavior of the replica fluctuations. In order to determine whether
or not the $\cos$-nonlinearity is relevant or irrelevant at large distances,
i.e., whether it generates a mass term $\sim m^{2}\hat{\phi}_{x}^{2}$ or
vanishes and restores a scale-invariant action, we perform a renormalization
group analysis. While the renormalization group (RG) flow of the sine-Gordon
model is well established in thermal equilibrium (i.e., for a Hermitian
Hamiltonian), only few non-Hermitian situations have been considered
previously Fendley  _et al._ (1993); Ashida  _et al._ (2017); Sarkar (2021).

A convenient starting point for a renormalization group treatment is the non-
Hermitian sine-Gordon action $S_{sg}$. It is derived by writing the propagator
${U(t,0)=\langle\\{\phi_{x,f}\\}|e^{-i\hat{H}_{sg}(t_{f}-t_{i})}|\\{\phi_{x,i}\\}\rangle}$
in terms of a path integral

$\displaystyle
U(t,0)=\int_{\phi_{x,0}=\phi_{x,i}}^{\phi_{x,t}=\phi_{x,f}}\mathcal{D}[\\{\phi_{x,t}\\}]e^{iS(\\{\phi_{x,t}\\})}$
(50)

over the real fields $\\{\phi_{x,t}\\}$ by the conventional Trotterization
procedure, e.g., see Appendix C.

The real-time action for the non-Hermitian sine-Gordon Hamiltonian (49) can be
brought into the Lagrangian form (with $X=(x,t)$)

$\displaystyle
S=\int_{X}\left\\{\frac{K}{16\pi}\left[\frac{1}{\eta}(\partial_{t}\phi_{X})^{2}-\eta(\partial_{x}\phi_{X})^{2}\right]-i\lambda\cos(\phi_{X})\right\\}.$
(51)

Here, $K,\lambda$ are the flow parameters for the renormalization group
treatment with the microscopic value of $K=\eta$ and
$\eta^{2}=1-\frac{2i\gamma}{v\pi}$. The parameter $\eta$ can been eliminated
from the action (51) by a complex Wick rotation
$(x,t)\rightarrow(\eta^{\frac{1}{2}}x,i\eta^{-\frac{1}{2}}t)$. This yields an
imaginary time path integral, which is well-defined for the here realized case
$\text{Im}(\eta)<0$, i.e., when there is no dynamical instability.

Figure 2: (a) Phase diagram of the non-Hermitian sine-Gordon field theory,
obtained from numerical integration of the flow equations (52), (53). In the
grey region the theory flows towards the Gaussian fixed point located at the
origin. The phase boundary is signalled by the black dotted line. For
comparison we also show the transition line for the Hermitian (thermal
equilibrium) theory. Figures (b) and (c) display the renormalization group
flow towards the strong coupling fixed point (b) and the Gaussian fixed point
(c) for the non-Hermitian sine-Gordon theory and the corresponding Hermitian
sine-Gordon theory (with comparably larger starting value of $K$). The initial
parameters of the curves (i) and (iii)-(v) are displayed in (a), the remaining
curves correspond to $|K|=1.3$ for (ii) and $|K|=3/4$ in (vi). All curves
start at $\lambda/A=3$.

We already note that the $\cos$-nonlinearity here becomes irrelevant in the
limit $\gamma\rightarrow 0$ (corresponding to $K,\eta\rightarrow 1$), i.e., in
the limit of zero measurement. This is due to the small prefactor of the
Gaussian part $\sim\frac{1}{16\pi}$, which yields strong fluctuations of the
field $\phi_{X}$. This prefactor arises from rescaling the fields with the
large factor $\phi_{X}\to\phi_{X}/\sqrt{8}$ to eliminate the numerical
prefactor of $\phi_{X}$ in the $\cos(\sqrt{8}\phi_{X})$-term in the sine-
Gordon Hamiltonian Eq. (49). It guarantees that the Gaussian fixed point is
robust against an infinitesimal measurement rate.

The renormalization group flow equations for the couplings $K,\lambda$ are
derived in the conventional way, but here we consider the parameters
$K,\lambda,\eta$ to be arbitrary complex numbers. The fields
$\phi_{X}=\phi_{X}^{(<)}+\phi_{X}^{(>)}$ are decomposed into long-distance
($\phi_{X}^{(>)}$) and short-distance ($\phi_{X}^{(<)}$) fluctuations, where
the long-distance fluctuations correspond to momenta $|k|<\Lambda/\xi$ and the
short distance fluctuations correspond to momenta $\Lambda/\xi<|k|<\Lambda$.
Here $\Lambda$ is a short-distance cutoff ($\Lambda=\pi$ in units of the
lattice spacing) and $\xi=e^{s}$, where $s$ is the rescaling parameter
controlling the renormalization group flow. The renormalization group flow is
then obtained by integrating out the short-distance modes perturbatively in
$\lambda$ and rescaling $x\rightarrow x/\xi$. This yields the flow equations

$\displaystyle\partial_{s}\lambda=$
$\displaystyle\Big{(}2-\frac{2}{K}\Big{)}\lambda,$ (52)
$\displaystyle\partial_{s}K=$ $\displaystyle-\lambda^{2}A,$ (53)

where $A$ is a positive number of order $\mathcal{O}(1)$ determined by the
propagator of the Gaussian theory Amit  _et al._ (1980); José  _et al._ (1977)
(see App. D).

The perturbative flow equations for the non-Hermitian sine-Gordon model in Eq.
(49) take the same form as those in the BKT scenario, but are more general due
to the generally complex nature of the couplings $K,\lambda$. They include
thermal equilibrium and some established non-Hermitian theories in particular
limits, which can be immediately reproduced: If $K$ is real and $\lambda$ is
purely imaginary, i.e., $\lambda=i|\lambda|$, then the nonlinearity $\lambda$
is generally relevant for an initial $K>1$, and $K$ is monotonously growing
under coarse graining, reproducing the conventional KT-flow diagram known from
thermal equilibrium Amit  _et al._ (1980); José  _et al._ (1977). On the other
hand, if $K$ and $\lambda$ are both real (corresponding to a purely imaginary
nonlinearity but real Gaussian part), then $K$ is continuously decreasing,
such that the only asymptotic fixed point of the renormalization group flow is
the Gaussian one Fendley  _et al._ (1993).

In our situation, where the parameter of the Gaussian theory $K$ is generally
complex, both $K,\lambda$ will not only experience a flow of their magnitude,
but also of their complex phase. In this case, the magnitude of the parameter
$K$ in Eq. (53) initially experiences both periods of growth and periods of
shrinking during the RG flow, depending on the complex phase of $\lambda$.
Since $\lambda$ is real initially, one starts with an RG flow that reduces
$K$, shifting the critical point at which $\lambda$ becomes relevant to larger
values of $K$ compared to the corresponding equilibrium RG flow. Figure 2(a)
displays the phase diagram obtained from solving Eqs. (52), (53) numerically
for the initial condition $K=\eta$.

The analysis shows that, while the initial, short distance behavior of the RG
is affected by the complex valued nature of the coefficients, the asymptotic
flow channels in on the one familiar from the Hermitian case. In consequence,
the phase boundary is deformed compared to the latter, but the two familiar
phases and their large distance behavior is reproduced, cf. Fig. 2. In more
detail, after an initial period of rapid phase oscillations, $\lambda$ either
flows to zero, or the complex phases $\varphi_{K},\varphi_{\lambda}$ of
$K=|K|e^{i\varphi_{K}},\lambda=|\lambda|^{i\varphi_{\lambda}}$ get pinned with
respect to each other, fulfilling the relation
$2\varphi_{\lambda}-\varphi_{K}=\pm\pi$. This happens typically at
$s=\mathcal{O}(1)$ and from then on, the flow equations (52), (53) describe
the familiar thermal KT flow. See Fig. 2(b) for examples of RG flow curves
towards the strong coupling fixed point and Fig. 2(c) for examples flowing
towards the Gaussian fixed point.

##  VI n-replica Keldysh construction

Above we have studied two replicas, identifying center-of-mass and relative
coordinates as useful degrees of freedom to capture the transition: In
particular, for Gaussian problems, these coordinates decouple exactly. The
center-of-mass coordinate undergoes indefinite heating, the relative
coordinate is governed by a non-Hermitian Hamiltonian. In this section, we
address the question how this decomposition generalizes to $n$ replicas.
Extending the theory to $n$ replicas enables us to compute higher order
correlation functions, which are of $n$-th order in the conditioned density
matrix $\rho^{(c)}$. In particular, it gives access to the computation of
Rényi entropies and the von Neumann entanglement entropy, which we will
discuss in this section. The construction is done within a replicated Keldysh
functional integral formalism Aleiner  _et al._ (2016); Tsuji  _et al._
(2017); Shenker and Stanford (2015), and yields two key results: First, we
show that the transition established previously in the $2$-replica setup
extends to the $n$ replica scenario. Second, we find that for Gaussian
problems there continues to be an exact decoupling into one mode analogous to
the center-of-mass coordinate, and $n-1$ modes analogous to the relative one;
non-linear problems can then be treated on similar grounds as detailed above
for the two-replica case.

###  VI.1 General construction

Starting point is again Eq. (10) for the conditioned trajectory projector. We
are then interested in the update (see Eq. (28))

$\displaystyle
d\hat{\rho}^{(R_{n})}=\overline{d\Big{[}\otimes_{l=1}^{n}\hat{\rho}^{(c)}\Big{]}}$
(54)

of the trajectory average over $n$ identical copies of the system. We can
iterate this update in time, and represent the process in terms of a path
integral by inserting coherent state resolutions of identity after each time
step in the usual way. The resulting path integral is still conditioned on the
noise realization, we finally take the noise average, yielding the $n$-replica
partition function $Z(n)=\overline{Z(n,\\{dW\\})}$. This bears similarities to
the Keldysh double- or multi-contour constructions, which have been introduced
to study out-of-time ordered correlation functions (OTOCs), with the major
difference that the noise average introduces additional contour couplings
compared to a Hermitian or purely Lindbladian evolution Aleiner  _et al._
(2016); Ansari and Nazarov (2016); Tsuji  _et al._ (2017); Shenker and
Stanford (2015). The procedure is illustrated in Fig. 3.

We consider again the general scenario of a continuously monitored quantum
system, whose wave function evolution is described by the stochastic
Schrödinger equation (8). The present construction is general, and encompasses
fermions or bosons alike; for the sake of concreteness and notation, we first
focus on a (1+1)-dimensional fermion system in spatial continuum, with a
normal ordered Hamiltonian
$\hat{H}=\hat{H}[\hat{\psi}^{\dagger}_{x},\hat{\psi}_{x}]$ and local,
Hermitian measurement operators
$\hat{M}_{s,x,t}=\hat{O}_{s,x}-\langle\hat{O}_{s,x}\rangle_{t}$, where
$\hat{O}_{s,x}=\hat{O}_{s,x}[\hat{\psi}^{\dagger}_{x},\hat{\psi}_{x}]$. Here
$x$ is the position and the index $s$ distinguishes potentially different
types of local measurements. We leave implicit internal degrees of freedom of
the fermion field, like $L,R$ for the Dirac fermions considered above.

We illustrate the construction of the $n$-replica path integral step by step.
For the sake of clarity, we will drop the spatial and measurement indices
$x,s$ for now, set $\gamma=1$, and restore these quantities at the end. The
evolution operator $\hat{V}_{dt}$ evolving the state
$|\psi_{t}\rangle\rightarrow|\psi_{t+dt}\rangle=\hat{V}_{dt}|\psi_{t}\rangle$
then is

$\displaystyle\hat{V}_{dt}=\exp\left[-(i\hat{H}+\hat{M}_{t}^{2})dt+dW\hat{M}_{t}\right],$
(55)

which when expanded up to second order (since $dW^{2}=dt$) yields the
stochastic Schrödinger equation for the increment
$d|\psi_{t}\rangle=|\psi_{t+dt}\rangle-|\psi_{t}\rangle$.

The single replica partition function for the conditioned projector
$Z(1,\\{dW\\})\overset{t\rightarrow\infty}{=}\text{Tr}(\hat{\rho}^{(c)}_{t})\overset{t\rightarrow\infty}{=}\text{Tr}(|\psi_{t}\rangle\langle\psi_{t}|)$
then requires two time strings or contours, corresponding to acting
$\hat{V}_{dt}$ from the left ($+$ contour) and $\hat{V}_{dt}^{\dagger}$ from
the right ($-$ contour) onto $\hat{\rho}^{(c)}_{t}$ at each time step. This
can be expressed via a two-contour path integral over the (Grassmann valued,
for fermions) fields $\psi_{\sigma},\bar{\psi}_{\sigma}$, which carry a
contour index $\sigma=\pm$.

Figure 3: (a) Keldysh representation of a single replica. Time evolution of
the noise averaged density matrix is captured by two Keldysh contours running
horizontally; the spatial dimension is represented by the vertical lines. The
trace operation is represented by a black line connecting final times $t_{f}$,
for a selected position $x$. The noise average (lower blue line) yields the
usual path integral representation of a Lindblad equation, with characteristic
contour coupling terms. (b) Keldysh representation of $n$ replicas. In
addition to the coupling of contours within each replica, the average over the
stochastic term couples all replicas (upper red lines). The trace operation
acts on each replica individually, and is indicated by black lines.

The $n$-replica partition function
$Z(n,\\{dW\\})=\text{Tr}[\otimes_{n=1}^{n}\hat{\rho}^{(c)}_{t}]$ is obtained
from the product over $n$ independent trajectory projectors
$\hat{\rho}^{(c)}$, yielding the partition function
$Z(n,\\{dW\\})=[Z(1,\\{dW\\})]^{n}$. In order to express this via a single
path integral, we add a replica index $l$ to the fields
$\psi_{\sigma}^{(l)},\bar{\psi}_{\sigma}^{(l)}$, accounting for each
individual copy $\hat{\rho}^{(c)}$. The path integral expression for the
stochastic $n$-replica partition function then is

$\displaystyle
Z(n,\\{dW\\})=\int\mathcal{D}[\Psi]\exp\left[i(S_{n,H}[\Psi]+S_{n,dW}[\Psi])\right].$
(56)

The action is composed of a Hamiltonian part $S_{n,H}[\Psi]$ and a measurement
part $S_{n,dW}[\Psi]$,

$\displaystyle S_{n,H}[\Psi]=$
$\displaystyle\sum_{\sigma=\pm}\sum_{l=1}^{n}\sigma\int_{t}\left(\bar{\psi}_{\sigma,t}^{(l)}i\partial_{t}\psi_{\sigma,t}^{(l)}-H[\bar{\psi}_{\sigma,t}^{(l)},\psi_{\sigma,t}^{(l)}]\right),$
(57) $\displaystyle S_{n,dW}[\Psi]=$ $\displaystyle
i\sum_{\sigma=\pm}\sum_{l=1}^{n}\int_{t}\left([M_{\sigma,t}^{(l)}]^{2}-dW_{t}M_{\sigma,t}^{(l)}\right).$
(58)

Here we have bestowed $dW_{t}$ a temporal index, since increments at different
times are uncorrelated and the averaging prescription in the operator
formalism is replaced by
$\overline{dW_{t}dW_{t^{\prime}}}=\delta(t-t^{\prime})$. For fermions, the
functional integral has anti-periodic boundary conditions in time, which we do
not make explicit in the notation here.

The partition function $Z(n,\\{dW\\})$ is the product of $2n$ independent path
integrals ($n$ forward and $n$ backward contours). They share in common that
all contours couple to the same noise increment $dW_{t}$, which acts as a
contour-independent source term. Performing the noise average via integration
over the Gaussian distributed increments $dW_{t}$, the partition function
$Z(n)=\overline{Z(n,\\{dW\\})}$ results, which is no longer of product
structure. The corresponding path integral is

$\displaystyle Z(n)=\int\mathcal{D}\Psi\
\exp\left[i(S_{n,H}[\Psi]+S_{n,M}[\Psi])\right],$ (59)

where the unitary part $S_{n,H}[\Psi]$ (57) remains unchanged by the noise
average. The measurement action $S_{n,M}[\Psi]$ will now be discussed for the
cases $n=1$ and $n>1$ separately.

For the single replica, taking the average over the stochastic increment
couples the $(\pm)$-contours, as illustrated in Fig. 3(a), and yields

$\displaystyle S_{1,M}[\Psi]=$ $\displaystyle
i\int_{t}M_{+,t}^{2}+M_{-,t}^{2}-\frac{1}{2}(M_{+,t}+M_{-,t})^{2}$
$\displaystyle=$ $\displaystyle\frac{i}{2}\int_{t}(O_{+,t}-O_{-,t})^{2}.$ (60)

It therefore removes any state-dependent term
($\sim\langle\hat{O}_{s,x}\rangle_{t}$), and makes the noise averaged single
replica evolution linear in the state. This is then equivalent to the path
integral representation of a Lindblad equation. In particular, for the
Hermitian Lindblad operators considered here, the stationary state described
by it is at infinite temperature.

Next we consider the case $n>1$, which is illustrated in Fig. 3(b). In
addition to intra-replica contour couplings, the noise average produces
additional inter-replica couplings (red lines in Fig. 3(b)). Here, the
measurement expectation values generally do not drop out, and we have to
perform the average in the mean-field decoupling approximation outlined in
Sec. IV.2. In the path integral description this amounts to the approximation
that $M_{\sigma,t}^{(l)}$ is independent of the history of $dW_{t}$.
Importantly, for the same reason as in the two-replica operator formulation,
the thus obtained action functional features linear, noise averaged trajectory
expectation values, such as $\overline{\langle\hat{O}_{s,x}\rangle_{t}}$.
These are now averages obtained from the path integral and, focusing on the
long time limit, we replace them directly by their stationary values, e.g.
$\overline{\langle\hat{O}_{s,x}\rangle_{t}}\to o_{s}$. The action obtained
from this average is

$\displaystyle S_{n,M}[\Psi]=$ $\displaystyle
i\int_{t}\sum_{l=1}^{n}\left([M_{+,t}^{(l)}]^{2}+[M_{+,t}^{(l)}]^{2}\right)-\frac{1}{2}\left(\sum_{l=1}^{n}M_{+,t}^{(l)}+M_{-,t}^{(l)}\right)^{2}.$
(61)

Restoring the spatial and measurement indices in
$\hat{O}_{s,x},\hat{\psi}^{\dagger}_{x},\hat{\psi}_{x}$ and considering
arbitrary $\gamma$, the measurement-induced action is

$\displaystyle S_{n,M}[\Psi]$ $\displaystyle=$
$\displaystyle-i\gamma\int_{X}\left(\sum_{l=1}^{n}\sum_{s}\left[(O^{(l)}_{s,+,X})^{2}+(O^{(l)}_{s,-,X})^{2}\right]-2(\Omega_{s,+,X}+\Omega_{s,-,X})o_{s}-\tfrac{1}{2}\left[\Omega_{s,+,X}+\Omega_{s,-,X}-2n\cdot
o_{s}\right]^{2}\right),$ (62)

with $\Omega_{s,\pm,X}=\sum_{l=1}^{n}O^{(l)}_{s,\pm,X}$. Eq. (62) yields an
interesting structure. While the first term is of the form of an effective
non-Hermitian Hamiltonian acting on each replica individually, the second term
features inter-contour couplings only between two collective degrees of
freedom $\Omega_{s,\pm,X}$. For $n>1$ this potentially enables degrees of
freedom orthogonal to $\Omega_{s,\pm,X}$ (see Sec. VI.2 below), which are free
of contour coupling terms and only evolve according to an effective
Hamiltonian.

###  VI.2 Bosonization of the multi-replica theory and decoupling of the
Gaussian theory

We apply this setup to the continuum measurement model presented in Sec.
III.2. In the fermionic formulation defined by Eqs. (16,17), we first have the
massless Dirac action associated to unitary evolution

$\displaystyle
S_{n,H}[\Psi]=\sum_{l=1}^{n}\sum_{\sigma=\pm}\sigma\int_{X}\bar{\Psi}^{(l)}_{\sigma,X}i(\partial_{t}-v\sigma_{z}\partial_{x})\Psi^{(l)}_{\sigma,X}.$
(63)

We consider two independent sets of measurement operators

$\displaystyle
O^{(l)}_{1,\sigma,X}=\bar{\Psi}^{(l)}_{\sigma,X}\Psi^{(l)}_{\sigma,X},\quad
O^{(l)}_{2,\sigma,X}=\bar{\Psi}^{(l)}_{\sigma,X}\sigma_{x}\Psi^{(l)}_{\sigma,X},$
(64)

each measured with rate $\gamma$. In the bosonized formulation, Eqs. (20,21),
the action is defined with the expressions

$\displaystyle S_{n,H}[\phi]=$
$\displaystyle-\frac{1}{2\pi}\sum_{l=1}^{n}\sum_{\sigma=\pm}\sigma\int_{X}\phi^{(l)}_{\sigma,X}\partial^{2}\phi^{(l)}_{\sigma,X},$
$\displaystyle\hat{O}_{1,\sigma,X}[\phi]=$
$\displaystyle-\frac{1}{\pi}\partial_{x}\phi^{(l)}_{\sigma,X},\quad\hat{O}_{2,\sigma,X}=m\cos(2\phi^{(l)}_{\sigma,X}),$
(65)

where $\partial^{2}\equiv\partial_{t}^{2}-\partial_{x}^{2}$.

Before we discuss the general, nonlinear theory, we now focus on the Gaussian
bosonic setting, relevant to the cases of weak and strong monitoring in the
above model. It is defined with a quadratic Hamiltonian and measurement
operators that are linear functions of Hermitian (bosonic) field operators,
i.e., $O^{(l)}_{\sigma,X}=D\phi^{(l)}_{\sigma,X}$, where
$D=m\pi,i\partial_{x}$ can be either a mass or a derivative. We then find a
decoupling analogous to the one in the 2-replica case discussed in Sec. IV.3.

Figure 4: Illustration of the coupling of the fields $\phi_{X}$ to the
measurement noise at a fixed position $x$ and for different times $t$: (a) In
the replica basis $\\{\phi^{(l)}_{\sigma,X}\\}$, each field couples to the the
same (spatiotemporal) measurement noise $dW_{t}$ (the individual fields are
given an offset with respect to each other for better visibility). (b) In the
Fourier basis $\\{\phi_{\sigma,X}^{(k)}\\}$, defined in Eq. (67), the $k=0$ or
center-of-mass mode couples to an enhanced noise $\sqrt{n}dW_{t}$ while all
the relative modes $(k>0)$ decouple from $dW_{t}$ and are noise-free.

In order to motivate this decoupling on an intuitive level, we start from the
action $S=S_{n,H}+S_{n,dW}$ in Eq. (56), for which the noise has not yet been
integrated out. In the bosonized framework, the action of the Hamiltonian part
$S_{n,H}$ is the described by Eq. (65), and the measurement part $S_{n,dW}$ by
Eq. (58) with the boson measurement operators $\hat{O}_{\alpha,\sigma,X}$
being linearized. One can then derive the equation of motion for the field
$\phi_{+,X}^{(l)}$ of the $l$-th replica from the saddle point condition
$\frac{\delta S}{\delta\phi_{+,X}^{(l)}}$. This yields

$\displaystyle\partial_{t}^{2}\phi^{(l)}_{+,X}=(\partial_{x}^{2}-\frac{\gamma
i}{\pi v}D^{2})\phi^{(l)}_{+,X}+dW_{t}.$ (66)

This equation is linear in both the fields $\phi_{+,X}^{(l)}$ and the noise
$dW_{t}$, and each replica $l$ experiences the same noise realization (see
Fig. 4). Therefore, it is possible to introduce generalized a center-of-mass
modes and relative modes, where the first absorbs all the noise, while it
cancels for the latter. To this end, we Fourier expand for each $l$ in the
equation of motion

$\displaystyle\phi^{(l)}_{\sigma,X}=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}e^{-i\frac{2\pi
kl}{n}}\phi^{(k)}_{\sigma,X},$ (67)

and use $dW_{t}=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}e^{-i\frac{2\pi
kl}{n}}\delta_{k,0}(\sqrt{n}dW_{t})$, to compare coefficients over $k$.
Indeed, all relative modes $k\neq 0$ are then noiseless, while the center-of-
mass mode $k=0$ is affected by a noise of strength $\sqrt{n}dW_{t}$,

$\displaystyle\partial_{t}^{2}\phi^{(k)}_{+,X}=(\partial_{x}^{2}-\frac{\gamma
i}{\pi v}D^{2})\phi^{(k)}_{+,X}+\sqrt{n}dW_{t}\delta_{k,0}.$ (68)

In order to proceed, the noise average is performed, which then yields the
averaged $n$-replica action $S_{n}[\phi]=S_{n,H}+S_{n,M}$, where $S_{n,H}$ is
again provided by Eq. (65) and the averaged measurement action $S_{n,M}$
corresponds to Eq. (62) with the corresponding boson measurement operators,
i.e., in terms of bosonic fields
$\phi=(\phi^{(0)},...,\phi^{(n-1)})^{T},\phi_{X}^{(k)}=(\phi_{+,X}^{(k)},\phi_{-,X}^{(k)})$,
and again linearized. This yields the sum

$\displaystyle S_{n}[\phi]=\sum_{k=0}^{n-1}S^{(k)}[\phi^{(k)}].$ (69)

The above action has one center-of-mass or Fourier $k=0$ mode
$\phi^{(k=0)}=\frac{1}{\sqrt{n}}\sum_{l}\phi^{(l)}$, which is described by

$\displaystyle S^{(0)}[\phi^{(0)}]=-\frac{1}{2\pi}\int_{X}\Big{\\{}$
$\displaystyle\sum_{\sigma=\pm}[\phi_{\sigma,X}^{(0)}(\sigma\partial^{2}-\frac{\gamma
i}{\pi v}D^{2})\phi_{\sigma,X}^{(0)}]$ (70) $\displaystyle+\frac{n\gamma
i}{\pi v}\phi_{c,X}^{(0)}D^{2}\phi_{c,X}^{(0)}\Big{\\}},$

where $\phi_{c,X}^{(l)}=\frac{\phi_{+,X}^{(0)}+\phi_{-,X}^{(0)}}{\sqrt{2}}$.
Due to the intra-replica contour coupling term (last contribution in Eq. (62))
there is heating to an infinite temperature state. This can be inferred from
the Green’s function of the average mode in Fourier space ($D=m\pi,q$)

$\displaystyle(G^{(0)}_{q,\omega})^{-1}=\sigma_{z}(\omega^{2}-q^{2})+\mathds{1}_{2\times
2}\frac{i\gamma(n-2)}{2v\pi}D^{2}+i\frac{\gamma n}{2v\pi}D^{2}\sigma_{x}.$
(71)

It has four poles at frequencies

$\displaystyle\omega_{\pm,\pm}=\pm\sqrt{q^{2}\pm D^{2}\frac{\gamma}{\pi
v}\sqrt{n-1}}.$ (72)

For $n>1$, this implies that at least two of the poles lie on the real axis
444This behavior persists under introducing a regularization: The latter would
enter Eq. (71) as an additional matrix $i\epsilon\omega\sigma_{z}$, the effect
of which is overwritten by any non-zero measurement rate $\gamma$, and does
not shift the poles away from the real axis.. They cause all the matrix
elements of the equal-time correlation function for a given momentum mode $q$
to diverge

$\displaystyle
G^{(0)}(q,t=0)=\int_{\omega}G^{(0)}(q,\omega)\rightarrow\infty.$ (73)

This indicates unbounded fluctuations of $\phi^{(0)}_{X}$, characteristic of
the infinite temperature state, and in line with the findings discussed around
Eq. (IV.3).

In addition, the action $S_{n}[\phi]$ features $n-1$ orthogonal relative modes
$\phi^{(k)},k=1,...,n-1$. These do not involve intra-replica contour coupling
in their action ($k>0$)

$\displaystyle
S^{(k)}[\phi^{(k)}]=-\frac{1}{2\pi}\int_{X}\sum_{\sigma=\pm}\left[\phi_{\sigma,X}^{(k)}(\sigma\partial^{2}-\frac{\gamma
i}{\pi v}D^{2})\tilde{\phi}_{\sigma,X}^{(k)}\right],$ (74)

where we have introduced the shortcut
$\tilde{\phi}^{(k)}_{\sigma,X}=\phi^{(n-k)}_{\sigma,X}$.

Each set of relative fields $\phi^{(k>0)}_{X}$ evolves according to a non-
Hermitian action, yielding a Keldysh Green’s function

$\displaystyle
iG^{K(k>0)}(Q)=\braket{\phi_{c,Q}^{(k>0)}\tilde{\phi}_{c,-Q}^{(k>0)}}=\frac{-i\tilde{\gamma}D^{2}}{(q^{2}-\omega^{2})^{2}+D^{4}\tilde{\gamma}^{2}}.$
(75)

The corresponding equal-time correlation functions

$\displaystyle
G^{K(k>0)}(q,t=0)=2\text{Im}\Big{[}\big{(}i\tilde{\gamma}D^{2}-q^{2}\big{)}^{-\frac{1}{2}}\Big{]}$
(76)

show the same scaling behavior as for a Luttinger Liquid in the ground state,
i.e., $G^{K(k>0)}(q,t=0)\sim|q|$ for $D\sim q$, but with a different,
$\gamma$-dependent amplitude.

As a consequence of the decoupling, the $n$-replica partition function for
linear measurement operators $O_{X}^{(l)}\sim D\phi_{X}^{(l)}$ and a quadratic
Hamiltonian factorizes into $n$ independent products. This remains true when
coupling the replica fields $\phi_{X}$ linearly to local sources $h_{X}$. For
later purposes we introduce such coupling to source terms via
$\int_{X}h_{X}^{T}\phi_{X}$, with
$h=(h^{(0)},...,h^{(n-1)})^{T},h_{X}^{(k)}=(h_{c,X}^{(k)},h_{q,X}^{(k)})^{T}$.
The sourced $n$-replica partition function then obtains by Gaussian
integration and reads

$\displaystyle Z(n)[h]$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{n-1}Z^{(k)}[h],$ (77) $\displaystyle Z^{(k)}[h]$
$\displaystyle=$
$\displaystyle\braket{e^{i\int_{X}h_{X}^{T}\phi_{X}}}=Z^{(k)}[0]\exp\left[-\tfrac{1}{2}\int_{X,X^{\prime}}h^{(k)}_{\sigma,X}\langle\phi^{(k)}_{\sigma,X}\tilde{\phi}^{(k)}_{\sigma^{\prime},X^{\prime}}\rangle\tilde{h}^{(k)}_{\sigma^{\prime}X^{\prime}}\right].$

This allows us to efficiently compute $n$-replica correlation functions of
arbitrary power in the fields $\phi_{X}$, including the $n$-th order Rényi
entropies, which we will discuss in the following section.

Based on the bosonic formulation, and our understanding of the Gaussian
theory, now we are in the position to discuss the measurement-induced phase
transition imposed by a nonlinear measurement operator $O_{2,\sigma,X}$. For
$n=2$ replicas, this problem is equivalent to the boson model that we analyzed
in Secs. III.3-V. For $n>2$, we can proceed in the same manner and bring the
Gaussian part of the action, i.e., the sum $S_{n,H}+S^{(0)}_{n,M}$, into a
replica diagonal form by applying the replica-Fourier transform described in
Eq. (67). The ’hot’ or infinite-temperature mode $\phi^{(k=0)}_{\sigma,X}$ can
then again be integrated out, yielding the remaining action for the $n-1$
equivalent relative modes with $\phi^{(k>0)}_{\sigma,X}$. These modes are
coupled by the $\cos$-nonlinearities, which contain their sum as arguments,
e.g., the term
$\cos(\sqrt{2}\hat{\phi}^{(1)}_{X})\rightarrow\cos(\sum_{k>0}a_{k}^{(1)}\hat{\phi}_{X}^{(k)})$
after integrating out the $k=0$ mode and with the Fourier coefficients
$a^{(l)}_{k}=\sqrt{\frac{2}{n}}e^{i2\pi lk/n}\in\mathds{C}$. The Gaussian part
of the action for $k>0$, however, does not couple different replica sectors
$k,k^{\prime}$ and furthermore is identically the same for each $k>0$.
Therefore the common BKT perturbative renormalization group scheme boils down
to the individual but identical, multiplicative renormalization of the factors
$e^{\pm i\phi^{(k)}_{\sigma,X}}$ for each $k>0$. While deriving the set of
flow equations for $n>2$ may be tedious, the details outlined above ensure
that in the Gaussian sector no coupling terms between different replica
indices $k,k^{\prime}$ can be generated. This in turn ensures that the
Gaussian fixed point for $n>2$ is of the same form as for $n=2$, describing
$n-1$ decoupled free boson theories.

###  VI.3 Entanglement transition

In this section, we discuss how the measurement-induced phase transition
modifies the structure of the entanglement entropy. On both sides of the
transition, the effective theory at long wavelengths is a theory of free
bosons. The entanglement entropy of the bosons can thus be determined from
conventional approaches for free theories, e.g., from the boson covariance
matrix Vidal  _et al._ (2003); Peschel (2003); Cramer  _et al._ (2006), which
is discussed in Appendix G. Here, however, one is mainly interested in the
fermion entanglement entropy, i.e., in the entanglement entropy of the
original, microscopic fermion degrees of freedom.

In order to capture the entanglement entropy of the free Dirac fermions 555The
compactification radius for the Dirac fermions subject to measurements is the
same as for noninteracting fermions, i.e., $K=1$., we implement the approach
outlined in Refs. Casini  _et al._ (2005); Casini and Huerta (2009). Starting
with the definition of the Rényi entropies, this approach then predicts that
the fermion entropies are exclusively determined from the boson  correlation
functions of the relative replica modes $k>0$. The structural difference
between the $n-1$ replica fluctuations modes at the weak and strong
measurement fixed points then already give a hint on the expected scaling
behavior of the entanglement entropy. In the weakly and strongly monitored
limits, it exhibits subextensive logarithmic growth or area law saturation,
respectively, confirming a phase transition in the entanglement entropy. We
use RG arguments to interpolate between these regimes. We also determine the
effective central charge in the weak monitoring regime and find that it
behaves non-universally. Our approach then confirms an algebraic decrease of
the effective central charge $c(\gamma)\sim\gamma^{-\kappa}$ as a function of
the measurement strength $\gamma$ in the weak monitoring regime, and a sharp
drop of $c(\gamma)$ at the measurement-induced transition, both of which was
reported for monitored lattice fermions in Refs. Alberton  _et al._ (2021);
Chen  _et al._ (2020).

General construction – To compute the entanglement entropy in the measurement
problem, we leverage the techniques developed in Ref. Casini  _et al._ (2005)
for free relativistic fermions in $(1+1)$ dimensions to our replica field
theory (Rényi entropies for quantum heat engines have been obtained in a
similar multicontour formalism in Ansari and Nazarov (2016)). More precisely,
we first derive a general expression for the $n$-th Rényi entropy and for the
von Neumann entanglement entropy for the monitored fermions. Then we show that
the fermion entropies are directly related to the correlation functions of the
bosonized model.

For a general bipartition of the one-dimensional system into two disjoint
subsystems $A=[x_{0},x_{0}+L]$ and $B=\mathds{R}\setminus A$, the $n$-th Rényi
entropy for the measured system is defined as

$\displaystyle S_{n}=\tfrac{1}{1-n}\ \overline{\log Z_{A}(n,\\{dW\\})},\
Z_{A}(n,\\{dW\\})\equiv\text{tr}[(\hat{\rho}_{A}^{(c)})^{n}].\ \ \ $ (78)

Here, $Z_{A}(n,\\{dW\\})$ is the reduced partition function for the subset
$A$, which is obtained by first performing a partial trace over $B$ to obtain
$\hat{\rho}^{(c)}_{A}=\text{tr}_{B}\hat{\rho}^{(c)}$, and then taking the
trace of the $n$-replica matrix product $(\hat{\rho}_{A}^{(c)})^{n}$.

The reduced partition function $Z_{A}(n,\\{dW\\})$ can computed from the real-
time path integral for the $n$-replica partition function Eq. (56). In order
to do so, the partial trace and the matrix multiplication in Eq. (78) have to
be implemented by a set of additional boundary conditions, which we discuss in
detail in Appendix F (see also Refs. Casini  _et al._ (2005); Casini and
Huerta (2009)). The key requirement here is that the boundary conditions can
be imposed via a gauge potential, which only modifies derivatives of the
fields. After bosonization, this yields the reduced partition function

$\displaystyle
Z_{A}(n,\\{dW\\})=\prod_{k}\langle\exp\left(-\sqrt{2}i\frac{k}{n}(\phi^{(k)}_{c,x_{0}}-\phi^{(k)}_{c,x_{0}+L})\right)\rangle.$
(79)

Here $\phi^{(k)}_{c,X}$ is the replica Fourier transform of
$\phi^{(l)}_{c,X}=\frac{1}{\sqrt{2}}(\phi^{(l)}_{+,X}+\phi^{(l)}_{-,X})$,
defined in Eq. (67), and the average is performed in the steady state, i.e.,
with respect to the bosonic measurement action discussed in Sec. VI.2.
Crucially, the center-of-mass mode with $k=0$ only yields a trivial
contribution to Eq. (79). The particular values of the noise $\sim dW$
therefore do not modify the reduced $n$-replica partition function, i.e.,
$Z_{A}(n,\\{dW\\})=Z_{A}(n,0)$. This decoupling of the reduced partition
function, and therefore of the Rényi entropies, from the noise is a
consequence of the decoupling of all the $k>0$ modes from the noise. It should
therefore be a general property of Gaussian theories, and it is also found for
the boson entropies in Appendix G. Our interpretation of this decoupling is
that each individual wave function, while showing random fluctuations induced
by the local measurements on the basis of local observables, has the same
global entanglement pattern.

The result in Eq. (79) may be viewed as the bosonic field $\phi^{(k)}_{c,X}$
being coupled to two point-like, opposite charges located at the beginning
$x_{0}$ and the end $x_{0}+L$ of the non-traced-out interval $A$. A similar
relation has been found in Refs. Bao  _et al._ (2021); Zhang  _et al._ (2021)
for monitored Majorana modes, for which the full charges are replaced by half-
charges 666In the framework of a classical XY model, the charges appear as
(half-) vortices, matching the terminology in the references.. In the limit of
either weak or strong monitoring, the bosonic theory can be linearized and we
can use Eqs. (77)-(79) to get explicit results for the Rényi entropies. Apart
from constant contributions $\sim Z^{(k)}[0]$, we obtain

$\displaystyle
S_{n}=\frac{1}{1-n}\sum_{k=1}^{n-1}\frac{2k^{2}}{n^{2}}\langle\phi_{c,x_{0}}^{(k)}\phi_{c,x_{0}}^{(k)}-\phi_{c,x_{0}}^{(k)}\phi^{(k)}_{c,x_{0}+L}\rangle.$
(80)

The correlation functions of the replica Fourier modes $k>0$ are all
independent of $k$ and identical. The summation over $k$ can therefore be
performed, and we can infer the von Neumann entanglement entropy
$S=\lim_{n\rightarrow 1}S_{n}$, yielding

$\displaystyle
S=\frac{2}{3}\langle\phi_{c,x_{0}}^{(k>0)}\phi^{(k>0)}_{c,x_{0}+L}\rangle.$
(81)

Weak monitoring entropy, $\gamma\ll v$ – If the nonlinearity is irrelevant,
the renormalization group flow approaches a Gaussian fixed point at which
$\lambda=0$. The theory is then characterized by a single quantity, i.e., the
fixed point value of $K$. The correlation function at this fixed point is

$\displaystyle\langle\phi^{(k>0)}_{c;q}\phi^{(k>0)}_{c;-q}\rangle=\text{Re}\Big{[}\frac{\pi}{K|q|}\Big{]}\Rightarrow\langle\phi^{(k>0)}_{c,x_{0}}\phi^{(k>0)}_{c,x_{0}+L}\rangle=2c(\gamma)\log(L).$

In this case, Eq. (81) predicts the von Neumann entanglement entropy and the
effective central charge

$\displaystyle S=\frac{1}{3}c(\gamma)\log(L),\quad c(\gamma)=\text{Re}(1/K).$
(82)

For $\lambda=0$ but $\gamma>0$, the system is already initialized at a
Gaussian fixed point with $K=\sqrt{1-i\frac{2\gamma}{\pi v}}$. Then the
prefactor of the $\log$-scaling behaves $\sim 1/\sqrt{\gamma}$ for
sufficiently large $\gamma$. A suppression of the effective central charge
with the measurement strength has been observed to be a general feature of
monitored free fermions, including away from the critical point Alberton  _et
al._ (2021); Chen  _et al._ (2020).

For the general case $\gamma>0,\lambda>0$, the nonlinear terms lead to a
renormalization of the Gaussian part of the theory. In particular, $K$
receives renormalization corrections, and should be evaluated at the fixed
point for a description of the asymptotic long distance physics. Generally, we
observe a growth of the parameter $K$ during the renormalization group flow.
Inserting the fixed point values of $K$ into the above formula then yields a
slightly enhanced suppression of the boson correlation function and of the
effective central charge, followed by a sudden jump to $c(\gamma)\rightarrow
0$ at the transition, which is illustrated in Fig. 5.

Figure 5: Prefactor of the logarithmic scaling function (effective central
charge) $c(\gamma)$ obtained from integrating the RG equation (53) for the
parameter $K$. For $\lambda=0$, we observe the scaling
$c(\gamma)\sim\gamma^{-0.5}$ (red line and inset). For $\lambda\neq 0$, a
transition into the area law phase happens at a critical measurement strength
$\gamma_{c}(\lambda)$, where $c(\gamma)$ drops to zero. In the weak
measurement regime, $c(\gamma)$ increases with $\lambda$. The lines correspond
to $\lambda=0.2\gamma$ (light gray, bold), $\lambda=0.25\gamma$ (dark gray,
dashed) and $\lambda=0.3\gamma$ (black, dotted).

Strong measurement entropy, $\gamma\gg v$ – For strong measurements, the
problem is gapped. Repeating the calculation from above but with non-zero mass
$m$ yields, for a bipartition of the system with subsystem sizes $L\gg
m^{-1}$,

$\displaystyle
S_{\text{vN}}(L)=\frac{c(\gamma)}{3}\log\left(\frac{L}{\sqrt{1+m^{2}L^{2}}}\right)\approx\frac{c(\gamma)}{3}\log
m^{-1}.$ (83)

The entanglement entropy saturates and does no longer depend on $L$. The
overall prefactor of the entropy decays again monotonously with $\gamma$.
Here, however, the inverse mass $m^{-1}$ is a measurement-induced correlation
length. It is also $\gamma$-dependent, approaching the short-distance cutoff
($=1$ on the lattice) of the theory in the limit $\gamma\rightarrow\infty$.

Zero monitoring, $\gamma=0$ – We have observed that in the limit
$\gamma\rightarrow 0$ the entropy approaches the behavior of a Luttinger
Liquid theory in the ground state with $c(0)=1$. However, this has to be taken
with care, since it involves two non-commuting limits. Setting $\gamma=0$
first, i.e., on the level of the microscopic theory, then the state of the
system after finite time always depends on the initial state of the system.
However, when evaluating the entanglement entropy for $\gamma>0$ first and
then taking the limit $\gamma\rightarrow 0$, any initial state dependence
drops out. This is the consequence of the ‘cooling’ effect for the relative
modes with $k\neq 0$, which govern the entanglement entropy, cf. Eq. (80).

For instance, for a thermal state at temperature $T$, the initial condition in
the Keldysh framework is usually implemented via an infinitesimal contribution
to the quantum-quantum sector Kamenev (2011)

$\displaystyle
S^{qq}=\int_{P}2i\epsilon\omega\coth(\frac{\omega}{2T})\phi_{q,-P}\phi_{q,P},$
(84)

with $\epsilon=0^{+}$ infinitesimal, and
$\phi_{q,P}=\frac{\phi_{+,P}-\phi_{-,P}}{\sqrt{2}}$. This initial contribution
appears in the quadratic sector and is identical for each replica, such that
it is directly carried over to the relative and center-of-mass coordinates.
For any non-vanishing measurement rate $\gamma>0$ the infinitesimal
contribution $\sim\epsilon\omega\coth(\frac{\omega}{2T})$, however, is
overwritten by a finite imaginary contribution $\sim\gamma$ in Eq. (75). Only
at the isolated point $\gamma=0$, the initial state matters, and yields the
correlation functions of a Luttinger Liquid at finite temperature
$\langle\phi^{(r)}_{\sigma,x,t}\phi^{(r)}_{\sigma,y,t}\rangle\sim|x-y|$ on
distances $|x-y|>v/T$. For an initial non-zero temperature (or energy
density), we thus recover the expected volume law of the entanglement entropy.
This is in line with numerical results, which show that the volume law is only
asymptotically stable in the absence of monitoring Cao  _et al._ (2019), and
is replaced by a logarithmic scaling at any non-zero monitoring rate Alberton
_et al._ (2021).

Discussion – The entanglement entropy for the fermions in our setup reflects
both the scaling behavior with system size as well as an monotonously
decreasing effective central charge, both of which was numerically observed
for monitored lattice fermions Alberton  _et al._ (2021); Chen  _et al._
(2020). The scaling behavior also includes the isolated point of zero
monitoring.

As already anticipated above, there exists an alternative entanglement measure
in this setup, namely the entanglement entropy of the boson degrees of
freedom, which is discussed in Appendix G. In equilibrium, the ground state
entanglement entropy of the fermions and the bosons coincides Calabrese and
Cardy (2009, 2009). Here, indeed we find that both display the same system-
size scaling behavior of the entanglement entropy, i.e., volume, logarithmic
and area law in the corresponding phases. However, the numerical prefactors of
the entanglement entropy, e.g., the effective central charge $c(\gamma)$, do
not coincide for fermions and bosons.

The comparison with related numerical works Alberton  _et al._ (2021); Bao
_et al._ (2021); Müller  _et al._ (2021); Minato  _et al._ (2021) confirms the