# Charge and Entanglement Criticality in a U(1)-Symmetric Hybrid Circuit of
Qubits

Ahana Chakraborty<EMAIL_ADDRESS>Department of Physics and
Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ
08854, USA  Center for Computational Quantum Physics, Flatiron Institute, 162
5th Avenue, New York, NY 10010, USA  Kun Chen<EMAIL_ADDRESS>Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue,
New York, NY 10010, USA  Aidan Zabalo<EMAIL_ADDRESS>Department
of Physics and Astronomy, Center for Materials Theory, Rutgers University,
Piscataway, NJ 08854, USA  Justin H. Wilson<EMAIL_ADDRESS>Department of
Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana
70803, USA  Center for Computation and Technology, Louisiana State University,
Baton Rouge, Louisiana 70803, USA  J. H. Pixley
<EMAIL_ADDRESS>Department of Physics and Astronomy, Center for
Materials Theory, Rutgers University, Piscataway, NJ 08854, USA  Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA

###### Abstract

We study critical properties of the entanglement and charge-sharpening
measurement-induced phase transitions in a non-unitary quantum circuit
evolving with a U(1) conserved charge. Many critical properties appear
distinct from the generic non-conserving case and percolation; however, upon
interpreting the critical features as mixtures of both entanglement and
charge-sharpening transitions, many critical features are brought within range
of the generic case. Nonetheless, the multifractal properties of the
entanglement transition remain distinct from the generic case without any
symmetry, indicating a unique universality class due to the U(1) symmetry. We
compute entanglement critical exponents and correlation functions via various
ancilla measures, use a transfer matrix for multifractality, and compute
correlators associated with charge sharpening to explain these findings.
Through these correlators, we also find evidence consistent with the charge-
sharpening transition being of the Berezinskii-Kosterlitz-Thouless type
(including the predicted “jump” in stiffness), which simultaneously argues for
a broad critical fan for this transition. As a result, attempts to measure
critical properties in this system will see anomalously large exponents
consistent with overlapping criticality.

##  I Introduction:

Exploring the properties of far-from-equilibrium, non-unitary dynamics
provides a theoretical challenge that links the theory of open quantum systems
and quantum information while also serving as a crucial bridge to
understanding the limits of quantum information processing, including quantum
error correcting codes. Moreover, recently developed noisy intermediate scale
quantum (NISQ) devices Preskill (2018) can directly probe theoretical
predictions regarding the nature and universality of quantum dynamic
protocols. In fact, NISQ devices are poised to probe the interplay of
entangling unitary dynamics and disentangling projective measurements Arute
_et al._ (2019); Noel  _et al._ (2022); Hoke  _et al._ (2023); Crain  _et al._
(2014) thanks to technical developments allowing for the possibility of local,
mid-circuit measurements Koh  _et al._ (2023). Exemplars of this paradigm are
measurement-induced phase transitions (MIPTs) where the rate of mid-circuit
measurements competing with the rate of entangling unitary dynamics (embodied
through the quantum circuit model in Fig. 1(a)) drives a transition in the
dynamics of the system’s entanglement Skinner  _et al._ (2019); Li  _et al._
(2018, 2019); Choi  _et al._ (2020); Chan  _et al._ (2019); Gullans and Huse
(2020a); Jian  _et al._ (2020); Fisher  _et al._ (2023); Czischek  _et al._
(2021). These transitions are analytically tractable in special limits Skinner
_et al._ (2019); Jian  _et al._ (2020); Choi  _et al._ (2020); Vasseur  _et
al._ (2019), but numerical evidence for the generic dynamics involving qubits
indicate a distinguished universality class characterized by a multifractal
logarithmic conformal field theory Zabalo  _et al._ (2022); Lykke Jacobsen and
Cardy (1998); Ludwig (1990); Ludwig and Cardy (1987); Jian  _et al._ (2020);
Bao  _et al._ (2020); Vasseur  _et al._ (2019); Li  _et al._ (2021).

A class of MIPTs were described in Ref. Gullans and Huse (2020b) as
“purification transitions,” (for many cases, these coincide with the
aforementioned entanglement transitions). Information theoretically, the
“mixed phase” can robustly encode quantum information (which could be decoded)
while the “pure phase” destroys any encoded information (i.e., it is
theoretically impossible to decode). Investigating the transition then not
only tells you that the entanglement obeys a universal description, but also
how quantum information is being destroyed. This aspect can change when the
dynamics are enriched with a symmetry Sang and Hsieh (2021); in particular,
with a continuous global U(1) (i.e., charge conserving) symmetry Agrawal  _et
al._ (2022); Barratt  _et al._ (2022); Oshima and Fuji (2023), a new
transition can appear where the system can undergo “charge sharpening” at a
measurement rate $p_{\\#}$ slower than the rate at which it purifies $p_{c}$.
Namely, an initial state that has equal weight in all charge sectors will
collapse into a single sector on short (i.e., an $\sim O(1)$) time scale for
$p>p_{\\#}$ while still being volume law entangled. In contrast, in the charge
fuzzy phase this collapse to a single sector occurs on an $\sim O(L)$ time
scale for a linear system size $L$. In these dynamics, the charge sector that
is labeled by charge density $\mathcal{Q}=Q/L$, where $Q$ is the total
conserved charge, is a (classical) bit of information that can be lost prior
to any quantum information. For the qubit system and $\mathcal{Q}=1/2$, these
transitions appear to be close to one another (Ref. Agrawal  _et al._ (2022)
estimates $p_{\\#}\approx 0.094(4)$ and $p_{c}\approx 0.110(3)$), and
therefore any NISQ device which probes the loss of this quantum information at
the entanglement transition $p_{c}$ may also probe some of the features of the
nearby sharpening transition as the available system sizes may not be larger
than the two correlation lengths ($\xi$ and $\xi_{\\#}$) of each respective
transition. Elucidating and understanding the critical properties of the MIPT
that one can discern in this setup is one of the major achievements of our
present work.

Figure 1: Model, ancilla qubit, and sector resolved phase diagram. (a) shows
brick-layer structure of the monitored quantum circuit consisting of U(1)
symmetric Haar random gates (the blue bricks) and projective $\sigma^{z}$
measurements (the red crosses) at a rate $p$ on the 1D chain of qubits. The
U(1) symmetric gate given in Eq.1 acting on two neighbouring qubits
($\sigma_{1}$ and $\sigma_{2}$)is also shown. (b) An ancilla ($A$) qubit is
maximally entangled to two orthogonal states,
$|\psi_{0};j=\uparrow\rangle_{\mathcal{Q}}=P_{j}^{\uparrow}|\psi_{0}\rangle_{\mathcal{Q}}$
and
$|\psi_{0};j=\downarrow\rangle_{\mathcal{Q}}=P_{j}^{\downarrow}|\psi_{0}\rangle_{\mathcal{Q}}$
by projecting a single site $j$ of the state $|\psi_{0}\rangle_{\mathcal{Q}}$
to up and down state respectively. These states belong to same global charge
sector $\mathcal{Q}$ and the entanglement entropy of $A$ serves as the order
parameter of the entanglement transition within a sector $\mathcal{Q}$. (c)
shows the critical measurement rates, $p_{c}$ (red circles) for the
entanglement transition and $p_{\\#}$ (blue cross) for the charge-sharpening
transition, for different charge densities $\mathcal{Q}$; $p_{\\#}<p_{c}$ for
all charge densities $\mathcal{Q}$. $p_{c}$ is obtained from the finite-size
scaling of the ancilla entanglement entropy when $A$ is coupled in one charge
sector while for $p_{\\#}$, $A$ is coupled across sectors ($\mathcal{Q}$ and
$\mathcal{Q}+1$). We fit $p_{c}(\mathcal{Q})$ to obtain
$p_{c}(\mathcal{Q})=0.44\mathcal{Q}(1-\mathcal{Q})$ shown by red solid line
while the blue solid line is simply a guide to eye.

The generic MIPT enriched by a global U(1) symmetry and resolved into
individual charge sectors $\mathcal{Q}$ is shown in Fig. 1 (c). Here, we say
$p<p_{\\#}(\mathcal{Q})$ is defined as _charge fuzzy_ while
$p>p_{\\#}(\mathcal{Q})$ is _charge sharp_ and the transition occurs within
the volume-law entangled phase of the system. At larger measurement rates
$p>p_{c}(\mathcal{Q})$ the model undergoes an MIPT to an area-law entangled
phase. Thanks to the perspective of a purification transition, we can
putatively probe the sharpening transition and the MIPT separately. In
particular, by coupling an auxiliary ancilla qubit into the system that either
couples across charge sectors (e.g. between $\mathcal{Q}$ and $\mathcal{Q}+1$)
or within a given sector (e.g. between two distinct orthogonal spin states
with equivalent $\mathcal{Q}$) as depicted in Fig 1(b), we should in principle
be able to examine the sharpening transition and the entanglement transition
separately.

However, the nature of the critical properties of the sharpening transition
may have a significant impact on these conclusions. In particular, in Ref.
Barratt  _et al._ (2022) it was shown that a modified version of these
dynamics (in the limit of an infinite onsite Hilbert space dimension) has a
charge sharpening transition which falls into the
Berezinskii–Kosterlitz–Thouless (BKT)Berezinsky (1971); Kosterlitz and
Thouless (1973) universality class but has no effect on the entanglement
transition which remains controlled by percolation. There are several
important implications of this field theoretic prediction. Firstly, the
critical nature of the BKT transition, makes the entire charge fuzzy phase
“critical” in a similar fashion that terminates at the BKT transition Barratt
_et al._ (2022). Second, this critical dynamics could in principle be seen in
a single sector at late times, as probed through correlation functions (not
entanglement measures) averaged over measurement outcomes Barratt  _et al._
(2022). Third, the finite size cross-over regime of the critical fan, defined
by the correlation length ($\xi_{\\#}$) for a BKT transition is known to be
broader than a second order transition ($\xi$), see Fig. 2. As a result, small
system size calculations probing the MIPT could receive a contribution from
the sharpening critical point (and vice versa).

Figure 2: Schematic finite-size cross over diagram showing the two nearby
transitions. The system undergoes a charge sharpening transition from the
charge fuzzy to sharp phase at the measurement rate $p=p_{\\#}$. This is
followed by the usual measurement induced phase transition from volume-law
entanglement to area-law phase at $p=p_{c}$. These three phases, namely the
entangled-fuzzy, entangled-sharp and the disentangled-sharp, are marked in the
figure. The red and the blue dashed lines show the correlation lengths $\xi$
and $\xi_{\\#}$ corresponding to the MIPT and the BKT-sharpening transition
respectively. The latter has a broad critical fan, intertwining these two
transitions at smaller system sizes accessible in finite-size numerics.

In this work, we study the nature of the universality class of the MIPT in the
presence of a continuous U(1) symmetry and the charge sharpening transition in
a qubit chain. First, focusing on the middle charge sector $\mathcal{Q}=1/2$,
we obtain anomalously large (typical) critical exponents and effective central
charge of the log-CFT. We argue that these values are offset by the nearby
sharpening transition, which by assuming it’s a BKT transition allows us to
obtain similar in magnitude (within our numerical accuracy) critical
properties to the MIPT without any symmetry. Moreover, by studying correlation
functions of the conserved charge, we show that critical sharpening
fluctuations are present within a single sector, and these modes can still
look critical at our estimate of the MIPT measurement rate for the sizes we
probe the problem at. This provides a consistent picture to our
interpretation, while at the same time providing evidence that the nature of
the sharpening transition is BKT like. Thanks to the multifractal nature of
the MIPT, we are able to ascertain critical properties that are expected to be
unaffected by the nearby BKT transition (as it is known to not be a
multifractal CFT), which we compute in order to show that the MIPT in the
presence of a U(1) symmetry belongs to its own distinct universality class.

##  II Summary of the main results:

In this manuscript, we present a sector-resolved analysis of the universality
class of the entanglement transition in a U(1) symmetry preserving random-Haar
circuit with projective measurements (shown in Fig. 1(a)). In addition, we
present a detailed study of the MIPT and the critical sharpening properties in
the middle sector. We adopted an efficient method to couple an ancilla to the
charge-conserving circuit (shown in Fig. 1(b)). This allows us to study the
critical properties in different charge sectors, even those which are far off
from half-filling, allowing our finite-size numerics to access larger system
sizes thanks to the constrained many-body Hilbert space. The main findings of
our work are summarized below:

  1. 1.

We compute the sector-resolved phase diagram in Fig. 1(c) which shows that the
system undergoes a charge-sharpening transition at the measurement rate
$p_{\\#}(\mathcal{Q})$ followed by the volume-law to area-law entanglement
transition at $p_{c}(\mathcal{Q})>p_{\\#}(\mathcal{Q})$ in all charge-sectors.
We find the functional form of
$p_{c}(\mathcal{Q})\propto\mathcal{Q}(1-\mathcal{Q})$. Each sector is found to
have a Lorentz invariant critical point at the entanglement transition.

  2. 2.

We present numerical results showing that the estimated critical exponents of
the entanglement transition are anomalously large in the U(1) symmetric
circuit compared to those without any symmetry Zabalo  _et al._ (2020, 2022)
or percolation. To show this, we compute the anomalous scaling dimension
exponent, $\eta$ using the mutual information of a pair of ancillas in section
V.1. We also study the non-unitary log-CFT governing the entanglement
transition in section V.2 and obtain the effective central charge
($c_{\rm{eff}}$) of the log-CFT from the free energy, the typical scaling
dimension $x_{1}^{\rm{typ}}=\eta/2$ (finding excellent agreement with the
ancilla computation) and the higher cumulants (e.g., the $2$nd cumulant
$x_{1}^{(2)}$) that probes the multi-fractal nature of the correlation
functions. These critical exponents are listed in table 1.

| $U(1)$ | No-symmetry | BKT | Subtracting BKT
---|---|---|---|---
| symmetry | ($H$) | ($\\#$) | values ($E$)
$c_{\rm{eff}}$ | 1.27(1) | 0.25(3) | 1 | 0.27(1)
$x_{1}^{\mathrm{typ}}=\eta/2$ | 0.28(2) | 0.14(2) | 0.125 | 0.16(2)
$x_{1}^{(2)}$ | 0.65(2) | 0.15(2) | NA | 0.65(2)
Table 1: A comparison of the critical exponents governing the entanglement
transition with and without U(1) symmetry is shown. No-symmetry values taken
from Ref. Zabalo  _et al._ (2022). Their known values in the BKT universality
class are also listed. The values of $c_{\rm{eff},E}$ and $\eta_{E}$ (in the
last column) obtained after subtracting the corresponding BKT values are close
to the corresponding no-symmetry values, while the $2$nd cumulant
$x_{1}^{(2)}$ is large compared to that without symmetry. This suggests that
the symmetry constraint changes the universality class. Figure 3: Entanglement
Transition. Entanglement entropy $S_{E}$ of the ancilla serves as the order
parameter of the MIPT with $p$. (a) shows the crossing of $S_{E}$ vs $p$ for
different $L=12$ to $48$ with total charge $Q=L/12$ taken at time $t=2L$. This
yield $p_{c}=0.031(4)$ and $\nu=2.68(31)$ from the finite-size scaling
collapse shown in the inset. (b) shows the correlation of the critical expo
nents $\nu$ and $\nu_{\\#}$ at $p_{c}$ (red circles) and $p_{\\#}$ (blue
cross) respectively; this data was obtained from the finite-size scaling of
$S_{E}$ and $S_{\\#}$ (shown in appendix A and B) for different $\mathcal{Q}$
following Eqs. (4) and (5) respectively. (c) shows the collapse of $S_{E}$ vs
$t/L^{z(\mathcal{Q})}$ for different $L=12$ to $48$ at $\mathcal{Q}=12$. This
gives $z(\mathcal{Q}=1/12)=0.97$. $z\approx 1$ for all sectors as shown in the
inset demonstrating each transition is Lorentz invariant.

  3. 3.

The critical sharpening properties are presented in a single, fixed charge
sector, not only across sectors in section VI. These correlations show a BKT-
like scaling behaviour with the expected quantized jump in the stiffness at
the transition. Moreover, they remain critical at the MIPT within the
accessible system sizes available due to a broad finite-size critical fan as
one would expect from a BKT transition (see Fig. 2).

  4. 4.

The close proximity of the BKT-sharpening criticality and the entanglement
transition leads us to interpret the entanglement critical exponents listed in
table 1 as receiving a contribution from the critical sharpening contribution.
In particular, the finite-size estimation of the critical exponents can be
reinterpreted as receiving a combined contribution from the two transitions:
their known values in the BKT universality class (denoted by the subscript
$\\#$) and the MIPT with a U(1) symmetry (denoted by the subscript $E$) and we
compare these values (in magnitude) to the MIPT transition without any
symmetry (denoted by the subscript $H$). In table 1, we presented the values
of these exponents (denoted by a subscript $E$) once we subtracted their known
BKT values. $c_{\rm{eff},E}$ and $\eta_{E}$ closely support the above
perspective while also adding support to the sharpening transition being BKT-
like.

  5. 5.

To carefully separate out the effect of the sharpening transition we look into
the multifractal properties of the log-CFT, which are assumed to not be
contaminated by the BKT transition. As a result, we are able to firmly show
that the presence of a U(1) symmetry enhances the multifractality of the log-
CFT, thus providing numerical evidence that the MIPT in the presence of a U(1)
symmetry belongs to its own distinct universality class.

##  III Model and Ancilla Probes

The quantum circuit consists of a 1D chain of qubits with local charge
$q_{i}=(\sigma_{i}^{z}+1)/2$. The time-evolution of the circuit repeats the
brick-layer structure, shown in Fig. 1(a), consisting of (i) the entangling
unitary gates $U_{i,i+1}$ acting on the bond of the nearest-neighbor sites $i$
and $i+1$. $U_{i,i+1}$ conserve the total charge $q\in 0,1,2$ on the bond and
are chosen from the set of generic Haar-random unitary gates of the form

$\displaystyle U=\begin{pmatrix}e^{i\phi_{0}}&0&0\\\ 0&\boxed{U_{2\times
2}}&0\\\ 0&0&e^{i\phi_{1}}\\\ \end{pmatrix}$ (1)

where $U_{2\times 2}$ is a $2\times 2$ Haar-random matrix and $\phi_{0}$ and
$\phi_{1}$ are random phases. (ii) each site is subjected to projective
$\sigma_{i}^{z}$ measurement with a rate $p$. The non-unitary dynamics
conserve the global U(1) charge $Q=\sum_{i=1}^{L}q_{i}$. This constrains the
dimension of the effective Hilbert space accessible during the dynamics if the
initial condition is chosen from a fixed $\mathcal{Q}=Q/L$ sector. This

allows us to explore larger finite system sizes for the sectors away from
half-filling. We use periodic boundary conditions unless mentioned otherwise.

###  III.1 Ancilla to study the entanglement transition

In our present work, we aim to extract critical exponents that correspond to a
power law decay of the correlation functions of the CFT, via mutual
information between two ancilla qubits. To couple an ancilla to the U(1)
symmetric circuit in a fixed charge sector, previous work in Ref. Agrawal  _et
al._ (2022) utilized a single ancilla coupled to a “bond” between two
neighboring qubits that requires post-selecting to couple into only the
$\lvert\uparrow\downarrow\rangle$ and $\lvert\downarrow\uparrow\rangle$
states. Extending this approach to two ancillas as we need to compute their
mutual information suffers from low statistics due to post-selecting now on
two bonds.

To circumvent this issue, we implement an approach (that utilizes a “single-
site ancilla” protocol schematically shown in Fig. 1(b)) by entangling each
ancilla qubit to a single site of the system, yet conserving the global
charge. We initialize the system in a random Haar state
$\lvert\psi_{0}\rangle_{\mathcal{Q}}$ in the global charge sector
$\mathcal{Q}$. We then create two orthogonal states
$\lvert\psi_{0};j=\uparrow\rangle_{\mathcal{Q}}=P_{j}^{\uparrow}\lvert\psi_{0}\rangle_{\mathcal{Q}}$
and
$\lvert\psi_{0};j=\downarrow\rangle_{\mathcal{Q}}=P_{j}^{\downarrow}\lvert\psi_{0}\rangle_{\mathcal{Q}}$
by projecting a single site $j$ of the system to up and down state
respectively. We note that the states
$\lvert\psi_{0};j=\uparrow\rangle_{\mathcal{Q}}$ and
$\lvert\psi_{0};j=\downarrow\rangle_{\mathcal{Q}}$ remain in the same global
charge sector $\mathcal{Q}$. The ancilla qubit with two orthogonal states
$\lvert\Uparrow\rangle$ and $\lvert\Downarrow\rangle$ is now maximally
entangled to the site $j$ as,

$\lvert\Psi\rangle=\frac{1}{\sqrt{2}}\Big{[}\frac{\lvert\psi_{0};j=\uparrow\rangle_{\mathcal{Q}}}{\|\lvert\psi_{0};j=\uparrow\rangle\|}\lvert\Downarrow\rangle+\frac{\lvert\psi_{0};j=\downarrow\rangle_{\mathcal{Q}}}{\|\lvert\psi_{0};j=\downarrow\rangle\|}\lvert\Uparrow\rangle\Big{]}.$
(2)

This single-site ancilla method allows us to get around the post-selection
problem as we increase the system size and hence can be used to probe the
entanglement transition at different global conserved charge sectors,
including those which are away from the middle sector ($Q=L/2$ or
$\mathcal{Q}=1/2$).

###  III.2 Ancilla to study the charge-sharpening transition

To probe the charge-sharpening (CS) transition, we couple the ancilla to two
neighboring sectors ($\mathcal{Q}$ and $\mathcal{Q}+1$) Agrawal  _et al._
(2022),

$\lvert\Psi\rangle_{\\#}=\frac{1}{\sqrt{2}}\left[\frac{\lvert\psi_{0}\rangle_{\mathcal{Q}}}{\|\lvert\psi_{0}\rangle_{\mathcal{Q}}\|}\lvert\Uparrow\rangle+\frac{\lvert\psi_{0}\rangle_{\mathcal{Q}+1}}{\|\lvert\psi_{0}\rangle_{\mathcal{Q}+1}\|}\lvert\Downarrow\rangle\right].$
(3)

Here $|\psi_{0}\rangle_{Q}$ and $|\psi_{0}\rangle_{Q+1}$ are two initial
states belonging to two different charge sectors $\mathcal{Q}$ and
$\mathcal{Q}+1$ which maximally couple to two orthogonal states of the ancilla
$\Uparrow$ and $\Downarrow$ respectively.

##  IV Phase Diagram:

The model introduced in the previous section exhibits a phase diagram Agrawal
_et al._ (2022); Oshima and Fuji (2023); Barratt  _et al._ (2022) that depends
on the conserving charge sector as we show in Fig. 1(c). Focusing on the
middle sector or coherent superpositions over all charge sectors, with
increasing $p$ the model exhibits two distinct phase transitions: an
entanglement phase transition from the volume-law to area-law phase at the
critical measurement rate $p_{c}$ which is preceded by a charge sharpening
transition at the critical measurement rate $p_{\\#}<p_{c}$ in the volume-law
phase.

###  IV.1 Entanglement Transition

We calculate the entanglement entropy $S_{E}$ of the ancilla with the system
in the steady state which serves as the order parameter of the entanglement
transition. We study variation of $S_{E}$ with $p$ for different system sizes
$L$ commensurate with the global charge Q. We perform a finite size scaling
(FSS) analysis of the order parameter at the entanglement transition point,

$S_{E}(t,L;\mathcal{Q})\sim
h_{\mathcal{Q}}[(p-p_{c})L^{1/\nu(\mathcal{Q})},t/L^{z(\mathcal{Q})}]$ (4)

where $h_{\mathcal{Q}}(x,y)$ is a two-parameter universal scaling function
with a correlation length exponent $\nu(\mathcal{Q})$ and dynamic exponent
$z(\mathcal{Q})$ for different $\mathcal{Q}$ sectors. $\nu(\mathcal{Q})$
governs the divergence of the correlation length $\xi\sim(p-p_{c})^{-\nu}$ at
the entanglement transition.

Following previous work Agrawal  _et al._ (2022), we examine the $p$
dependence of $S_{E}$ by fixing the aspect ratio of time and length of the
system ($t/L=2$) and ensure our crossing and collapse is unaffected by this
choice. At the middle sector ($\mathcal{Q}=1/2)$, our scaling analysis yields
$p_{c}=0.110(3)$ and $\nu=1.65(16)$ (see appendix A) which are in agreement
with the “bond-ancilla” method results Agrawal  _et al._ (2022) and thus
establishes the consistency of the “single-site ancilla” method in the
conserving circuit. In Fig. 3(a) we show $S_{E}$ vs $p$ at $Q=L/12$ for $L=12$
to $48$ and the inset shows the scaling collapse with $p_{c}=0.031(4)$ and
$\nu=2.68(31)$. We summarize the dependence of the global conserved charge
density $\mathcal{Q}=Q/L$ on the entanglement critical point
$p_{c}(\mathcal{Q})$ in Fig. 1(c) (with red circles) and the corresponding
correlation length exponent $\nu(\mathcal{Q})$ in Fig. 3(b) (with red circles)
respectively. Fig. 1(c) shows that changing the value of the conserved charge
density $\mathcal{Q}$ from the half-filling condition weakens the entangling
unitary gates and $p_{c}$ decreases. $p_{c}(\mathcal{Q})$ fits well with the
variance of the Binomial distribution
$p_{c}(\mathcal{Q})=0.44\mathcal{Q}(1-\mathcal{Q})$ shown by red solid line.

Fig. 3(b) shows that $\nu$ increases as $\mathcal{Q}$ deviates from the half-
filling, but with large statistical error at $\mathcal{Q}$ away from half-
filling.

To this end, at the critical point $p_{c}(\mathcal{Q})$, we collapse
$S_{E}(t)$ with $t/L^{z(\mathcal{Q})}$ which gives an estimation of
$z(\mathcal{Q})$ for each charge sector. Fig. 3(c) shows the collapsed data at
$\mathcal{Q}=1/12$ for $L=12,24,36$ and $48$. Our numerical analysis finds
$z(\mathcal{Q})\approx 1$ for all $\mathcal{Q}$ sectors as shown in the inset
of Fig. 3(c). This establishes conformal invariance across all charge sectors
we have considered.

###  IV.2 Sharpening Transition

The entanglement entropy $S_{\\#}$ of the previously mentioned ancilla probes
the charge-shaprening transition point $p_{\\#}$ with the corresponding
correlation length exponent $\nu_{\\#}$ through the following equation,

$S_{\\#}(L;\mathcal{Q})\sim
g_{\mathcal{Q}}[(p-p_{\\#}(\mathcal{Q}))L^{1/\nu_{\\#}(\mathcal{Q})}]$ (5)

where $g_{\mathcal{Q}}(x,y)$ is a two-parameter universal scaling function at
the charge-sharpening transition for charge sectors $\mathcal{Q}$. The
dependence of $p_{\\#}(\mathcal{Q})$ and $\nu_{\\#}(\mathcal{Q})$ on global
conserved charge density $\mathcal{Q}=Q/L$ are shown in Fig. 1(c) (with blue
cross) and in Fig. 3(b) (with blue cross) respectively (data collapse not
shown). For all $\mathcal{Q}$ sectors, the charge-sharpening transition
precedes the entanglement transition,
$p_{\\#}(\mathcal{Q})<p_{c}(\mathcal{Q})$.

Interestingly, for $\mathcal{Q}$ sectors away from half-filling, the two
critical points are approaching each other within their error bars. Moreover,
our estimate of the critical exponents $\nu$ and $\nu_{\\#}$ become strongly
dependent on the sector and we cannot discern them at the small system sizes
we can reach. As the error bars on $\nu$ and $\nu_{\\#}$ are growing as we
move away from the middle sector and the largest deviation from the middle
sector to $\mathcal{Q}=1/12$ differ by at most ‘two $\sigma$’, we cannot rule
out the possibility that these exponents do not depend on the sector and the
trend we witness is a finite size effect. With the recent prediction of the
charge sharpening transition being a BKT transition in the large Hilbert space
limit, this large $\nu_{\\#}$ at the small charge density sectors suggests
that our scaling ansatz could be incorrect and instead is suggestive that the
power law should be replaced with the logarithmic dependence due to the nature
of the BKT correlation length (see appendix B).

##  V Critical Properties of the Entanglement Transition

###  V.1 Mutual Information Probe of the Correlation Functions

We use the “single-site ancilla” method explained in the previous section to
compute the correlation function at the critical measurement rate of the
entanglement transition. We initialize the spin-chain in a fixed global charge
sector $\mathcal{Q}$ and evolve it under U(1) symmetric monitored dynamics up
to late time $t_{0}=2L$. In the steady state, we couple two ancillas $A$ and
$B$ to the circuit at $(r_{1},t_{1})$ and $(r_{2},t_{2})$ and calculate their
mutual information given by, $I_{n}(A,B)=S_{n}(A)+S_{n}(B)-S_{n}(A\cup B)$,
where $S_{n}$ denote the Renyi entropy of order $n$. To compute the order
parameter correlation, we set $t_{1}=t_{2}=t_{0}$ and fix $|r_{1}-r_{2}|=bL$
with $b$ being a constant. The correlation function obeys a one-parameter
finite-size scaling ansatz with time $t-t_{0}$ as,

$C(t-t_{0},L/2,Q)\sim\frac{1}{L^{\eta(Q)}}k_{Q}\Bigg{(}\frac{t-t_{0}}{L}\Bigg{)}.$
(6)

.

Figure 4: Depiction of geometries for the pair of ancilla qubits used in the
computation of $\eta$. Respectively, (a) $\eta$, (b) $\eta_{\perp}$, and (c)
$\eta_{\parallel}$. Figure 5: Probing $\eta$. Mutual information $C(t-t_{0})$
between two ancilla qubits coupled to the monitored U(1) symmetric circuit in
the same charge sector $\mathcal{Q}=Q/L$. (a) shows the scaling collapse of
$C(t-t_{0})$ following Eq. (6) for different system sizes from $L=8$ to $20$
at $\mathcal{Q}=1/4$ for different geometries of the circuit: $\eta=0.45(2)$
for periodic BC at $|r_{1}-r_{2}|=L/2$, $\eta_{\parallel}=1.09(4)$ for open BC
at $|r_{1}-r_{2}|=L$ and $\eta_{\perp}=0.62(2)$ for open BC at
$|r_{1}-r_{2}|=L/2$. Geometries follow Fig. 4, and we use the von Neumann
entropy for this plot. (b) shows the dependence of the bulk critical exponent
$\eta$ on the charge sector with fixed density $\mathcal{Q}$ (at
$n\rightarrow\infty$). $\eta$ exponent remains stable across all $\mathcal{Q}$
sectors by blue triangles. We also show the values $\eta_{E}$ once we subtract
the known BKT values $\eta_{\\#}=0.25$ by yellow circles. The blue dashed line
show the value $\eta_{H}=0.26(1)$ in absence of symmetry in the circuit
obtained from Ref. Zabalo  _et al._ (2020).

Here, we calculate three different anomalous dimension exponent $\eta$
depending on the spatial separation and the boundary conditions (BC) in the
circuit: (a) $\eta$ for $|r_{1}-r_{2}|=L/2$ with periodic boundary condition,
(b) $\eta_{\perp}$ for $|r_{1}-r_{2}|=L/2$ with open boundary condition and
(c) $\eta_{\parallel}$ for $|r_{1}-r_{2}|=L$ with open boundary conditions and
geometries given in Fig. 4. The scaling collapse of these three cases is shown
in Fig. 5(a) at $Q=L/4$ for $L=8$ to $20$ for $n=1$. In Table 2 we show the
dependence of the three anomalous dimension exponents
$\eta,\eta_{\perp},\eta_{\parallel}$ on the Renyi index $n$. With increasing
$n$, the three exponents saturate to the values (e.g., $\eta=0.57(1)$,
$\eta_{\parallel}=1.27(4)$, $\eta_{\perp}=0.77(3)$ at $Q=L/4$) which are
distinct from their corresponding value in the monitored Haar-random circuit
without the U(1) symmetry constraint presented in Ref. Zabalo  _et al._ ,
2020.

n $\eta$ $\eta_{\parallel}$ $\eta_{\perp}$ $\mathcal{Q}=1/2$ $1/4$ $1/2$ $1/4$
$1/2$ $1/4$ 1 0.46(2) 0.45(2) 0.97(2) 1.09(4) 0.61(1) 0.62(2) 2 0.55(3)
0.54(1) 1.13(3) 1.27(4) 0.72(3) 0.71(3) 5 0.57(3) 0.57(1) 1.17(3) 1.27(3)
0.75(2) 0.77(3) $\infty$ 0.57(3) 0.57(1) 1.17(3) 1.27(4) 0.76(2) 0.77(3) No-
symmetry 0.26(1) 0.49(2) 0.34(1)

Table 2: We tabulate the anomalous scaling dimension exponents $\eta$ together
with the surface critical exponents $\eta_{\parallel}$ and $\eta_{\perp}$
using open BC with $|r_{1}-r_{2}|=L$ and $|r_{1}-r_{2}|=L/2$ respectively for
two values of $\mathcal{Q}=1/2$ and $1/4$. For Renyi indices $n>1$, all the
exponents converge to values that are distinct from those without any symmetry
in the circuit.

In Fig. 5(b) we show the variation of the exponent $\eta$, with the global
charge $\mathcal{Q}$. The blue solid line with triangles shows that the $\eta$
exponent (for $n=\infty$) remains same over different charge sectors (see
appendix C for more details) but is different from value $\eta_{H}=0.26(1)$
without the symmetry shown by blue dashed line.

###  V.2 Probing the log-CFT

In this section, we will study the non-unitary CFT that governs the MIPT in
the dynamics of the circuit. We compute its critical properties ranging from
the effective central charge, typical anomalous scaling dimension, and the
higher cumulants to study the multi-fractal nature of the correlation
functions at the transition. Our numerical results indicate that U(1) symmetry
changes the universality class, yielding a distinct multi-fractal scaling at
the entanglement transition.

####  V.2.1 Free Energy

The non-unitary time-evolution of a monitored random circuit can be described
by the transfer-matrix method within the paradigm of statistical mechanics Li
_et al._ (2021); Zabalo  _et al._ (2022); Jian  _et al._ (2020). At late time,
the dynamics is governed by the leading Lyapunov exponents
$\lambda_{0},\lambda_{1},\dots$ of the transfer matrix. Using the conformal
invariance at the entanglement transition point, the circuit dynamics can be
mapped to $(1+1)d$ non-unitary conformal field theory characterized by a
universal number called the effective central charge $c_{\rm{eff}}$. It can be
related to the free-energy density $f=F/A$ obtained from the leading Lyapunov
exponent $\lambda_{0}$ as Zabalo  _et al._ (2022),

$\displaystyle\frac{F(L)}{A}=-\frac{\lambda_{0}}{\alpha
L}=f(L\rightarrow\infty)-\pi\frac{c_{\rm{eff}}}{6L^{2}}+\dots,$ (7)

where $A=\alpha Lt$ is the area of the cylinder of the CFT defined from a
circuit of spatial and temporal length $L$ and $t$ respectively with a space-
time anisotropy factor $\alpha$ to be computed numerically (see Appendix D for
details). Due to the conservation law, the nature of the initial state has a
strong dependence on free energy. As $F=-\ln Z$ where $Z$ is the partition
function of the statistical mechanics model Zabalo  _et al._ (2022) that
involves a trace over all states, we begin by considering initial states that
are equal superpositions over all charge sectors and discuss the effects of
projecting into a given charge sector afterward.

Figure 6: Scaling of the Free energy of the log-CFT. At
$p=p_{c}(\mathcal{Q}=1/2)=0.11$ of the monitored Haar-random circuit with U(1)
symmetry, both (a) free energy density $f(L)$ and (c) difference between
generalized free energies, $f_{1}(L)-f(L)$ show a $1/L^{2}$ scaling with
system size. The slope of the former gives the effective central charge
$c_{\rm{eff}}=1.27(1)$, while that of the latter gives the scaling dimension
of the most relevant operator $x_{1}^{\rm{typ}}=2\eta=0.27(2)$ matching with
that shown Fig. 5. (b) shows the rescaled probability distribution
$P(\mathcal{Q},L)=L\times P(Q,L)$ vs $\mathcal{Q}$ that an initial state with
unrestricted global charge ends up in sector $\mathcal{Q}$ at late time
($t\gtrsim 4L$), which follows a binomial distribution dominated by the middle
sector in $L\rightarrow\infty$ limit. The lines are not fits but simply plots
of the binomial distribution. The initial conditions in (a) and (c) are
randomly chosen from all $\mathcal{Q}$ sectors, except the orange squares in
(a) and (c) respectively are the results selecting only those trajectories
with $Q=L/2$ charge in the steady state. The green triangles in (a) correspond
to the trajectories initialized and hence remained in $\mathcal{Q}=1/2$ sector
throughout the evolution.

In the quantum circuit, each trajectory is defined by a particular initial
condition, a set of unitary gates and a set of measurement outcomes indexed by
$\vec{m}$. $F$ is calculated from the Born probabilities, $p_{\vec{m}}$ of
each measurement outcome in a trajectory, $F=-\sum_{\vec{m}}p_{\vec{m}}\ln
p_{\vec{m}}$ and then finally averaging over random initial conditions,
unitary gates, and measurement outcomes Zabalo  _et al._ (2022).

We first numerically compute the free energy density at
$p=p_{c}(\mathcal{Q}=1/2)$ choosing the initial conditions (a product initial
state or a Haar initial state) randomly with unconstrained total charge. As
time evolves, we calculate free energy in each trajectory from the Born
probabilities of the measurement events. Averaging over all such trajectories
at a long time gives the total free energy $F(t)$ from all accessible
trajectories evolved under the constrained dynamics. $F(t)$ grows linearly
with time $t$ at late times with the slope $\lambda_{0}$ (we waited till time
$t\sim 6L$ to reach the steady state and extract $\lambda_{0}$). The
corresponding free energy density $f(L)$ decays as $1/L^{2}$ (consistent with
the CFT prediction) as shown in Fig. 6(a) by red circles. We compute the slope
of this curve $m_{0}(L_{\rm{min}})$ by systematically eliminating the smallest
system sizes and considering data only from $L=L_{\rm{min}}$ to $L=18$. These
fits are shown by dashed blue lines where the darker color corresponds to
larger $L_{\rm{min}}$. This yields
$m_{0}(L_{\rm{min}})=m_{0}(\infty)+b/L_{\rm{min}}^{2}$. The effective central
charge is computed from $c_{\rm{eff}}=-6m_{0}(\infty)/\pi$. Our numerics
predicts $c_{\rm{eff}}=1.27(1)$ which is significantly different from the
result $c_{\rm{eff}}=0.25(3)$ without U(1) symmetry in the circuit Zabalo  _et
al._ (2022).

We now bin the free energy into each charge sector $F^{Q}$ that it is
projected into in the late time limit ($t\gg L$). We see that the probability
that a given state ends up in a state with total charge $Q$ is precisely given
by the binomial distribution $P(Q,L)=\binom{L}{Q}/2^{L}$. We plot it as
$P(\mathcal{Q},L)=L\times P(Q,L)$ vs $\mathcal{Q}$ in Fig. 6(b),. Importantly,
this shows that in the thermodynamic limit average quantities over initial
states that consider all conserving sectors eventually converge to the middle
sector. To relate the free energy over all sectors (i.e., the grand potential)
and the free energy in the middle sector (i.e., the Gibbs free energy)
requires introducing a chemical potential $\mu$, such that
$F^{\mathcal{Q}}=F+\mu(Qt)$, where $Qt$ is the total charge in space-time.
This implies that in addition to the $1/L^{2}$ dependence expected in Eq. (7)
there is a contribution from $\mu~{}\sim L^{-z}=1/L$ to all of the Lyapunov
exponents in a fixed sector. Focusing on $f^{\mathcal{Q}}=F^{Q}/(\alpha Lt)$
for $\mathcal{Q}=1/2$ as shown in Fig. 6(a) open orange squares, we find that
the data is well fit using the modified scaling form

$f^{\mathcal{Q}}=f^{\mathcal{Q}}(L\rightarrow\infty)+a\mathcal{Q}/L-\pi\frac{c_{\rm{eff}}}{6L^{2}},$
(8)

but with the same effective central charge, which provides evidence for our
relation between the two free energies. We show this fit by green dashed lines
for three different values of $L_{\rm{min}}=8,10$ and $12$ where the darker
color corresponds to larger $L_{\rm{min}}$, by fitting the data from
$L=L_{\rm{min}}$ to $L=18$. Moreover, in the thermodynamic limit
($L\rightarrow\infty$), the Gibbs free energy
($f^{\mathcal{Q}}(L\rightarrow\infty)=0.118$) is approaching the free energy
over all sectors ($f(L\rightarrow\infty)=0.119(1)$), allowing us to conclude
that $f^{\mathcal{Q}}(L\rightarrow\infty)=f(L\rightarrow\infty)$ as expected.
Lastly, the solid green triangles in Fig. 6(a) show the data when the system
is initialized in $\mathcal{Q}=1/2$ sector and hence the circuit is restricted
to $\mathcal{Q}=1/2$ throughout its evolution. These data points closely
overlap with $f^{\mathcal{Q}=1/2}$ where the trajectories are binned to the
middle sector at late times, establishing that the time-evolution is ergodic
as we expect.

####  V.2.2 Leading Scaling Dimension

We now turn to probing the scaling dimensions at the critical point. To do
this, we compute the next leading Lyapunov exponent $\lambda_{1}$ which is
related to the higher generalized free energy as
$f_{1}=F_{1}/A=\lambda_{1}/\alpha L$. Following the procedure prescribed in
Ref.Zabalo  _et al._ (2022) to compute $\lambda_{1}$ (in addition to
$\lambda_{0}$), we create a pair of orthoganonal states $|v_{0}\rangle$ and
$|v_{1}\rangle$. As time evolves, in each time-step of a trajectory,
$|v_{0}\rangle$ and $|v_{1}\rangle$ are subjected to identical $U$ gates and
measurements (locations and at which state it will be projected to) governed
by the Born probabilities $p_{\vec{m}}$ of $|v_{0}\rangle$. After each time-
step, the pair of vectors need to be reorthogonalized. $F_{1}$ is calculated
from the Born probabilities $\tilde{p}_{\vec{m}}$ of each measurement outcomes
in the evolution of $|v_{1}\rangle$,
$F_{1}=\sum_{\vec{m}}\tilde{p}_{\vec{m}}\ln\tilde{p}_{\vec{m}}$. The
difference between the two leading Lyapunov exponents grows as $1/L^{2}$,

$f_{1}(L)-f(L)=\frac{\lambda_{1}-\lambda_{0}}{\alpha L}=\frac{2\pi
x_{1}^{\rm{typ}}}{L^{2}},$ (9)

shown in Fig. 6(c) by the red circles. This is used to calculate the scaling
dimension of the most relevant operator $x_{1}^{\rm{typ}}$. We numerically
estimate $x_{1}^{\rm{typ}}=0.27(2)$ (obtained from the slope $m_{1}(L_{min})$
shown in the inset by eliminating the smallest $L$s in the fit shown by dashed
blue lines in Fig. 6(c)). $x_{1}^{\rm{typ}}$ is related to the decay of the
bulk correlation function through the exponent
$\eta=2x_{1}^{\rm{typ}}=0.55(3)$. This gives an independent estimate of $\eta$
from CFT and importantly matches with our previous result from the order-
parameter correlation function shown in Fig. 5(b) (solid blue line with
triangles).

Lastly, we considered probing the free energy differences after they are
projected into the middle sector
$f_{1}^{\mathcal{Q}=1/2}(L)-f(L)^{\mathcal{Q}=1/2}$ shown in Fig. 6(c) by
orange squares and find that the chemical potential contribution, $\mu$ in Eq.
(8), cancels precisely leaving us with an additional estimate of
$\eta=2x_{1}^{\rm{typ}}=0.51(4)$ (using the same fitting form given in Eq.
(9)) that is in good agreement with that obtained from $f_{1}(L)-f(L)$ with
$\eta=2x_{1}^{\rm{typ}}=0.55(3)$ sampled over all charge sectors (see appendix
E for more details).

####  V.2.3 Multi-fractality

Figure 7: Multifractal scaling of the log-CFT. Evidences of multifractality in
(a) the 2nd cumulant $k_{2}$ vs $1/L^{2}$ giving a large $x_{1}^{(2)}=0.65(2)$
and (b) collapse of distribution of $Y(t)$ onto a single universal curve
$H(s)$ for different values of $L$ and $t$ ($L=8$ to $18$ at $t=16L$ by red
lines and $t=11L$ to $27L$ for $L=18$ by blue lines). The universal curve
$H(s)$ without U(1) symmetry, extracted from Ref. Zabalo  _et al._ (2022), is
shown by the black dashed curve. The $x$-axis is plotted as $s-s_{\rm{min}}$
where the minimal value $s_{\rm{min}}=0.27$ for U(1) symmetric case and
$s_{\rm{min}}=0.14$ without U(1) symmetry. The initial conditions are randomly
chosen from all $\mathcal{Q}$ sectors.

In this subsection, we will probe the multifractal properties of the critical
correlation function at the entanglement transition.

We know that at a generic critical point, where the system is scale-invariant,
all moments of the correlation function vanish as a power law,

$\mathcal{E}[C^{n}(r)]\sim\frac{B_{n}}{r^{2x_{1}(n)}},$ (10)

where $\mathcal{E}[\dots]$ denotes averaging over different random samples. If
the scaling exponent $x_{1}(n)$ of the $n$th moment is a linear function of
$n$, then the system is self-averaging, otherwise, it shows a multifractal
scaling of the correlators. $x_{1}(n)$ can be obtained from the cumulant
expansion,

$\displaystyle\log\mathcal{E}[C^{n}(r)]=$ $\displaystyle n\mathcal{E}[\log
C(r)]$ $\displaystyle+\frac{n^{2}}{2!}\mathcal{E}[\\{\log
C(r)-\mathcal{E}[\log C(r)]\\}^{2}]+\dots,$

which yields,

$x_{1}(n)=nx_{1}^{\rm{typ}}+\frac{n^{2}}{2!}x_{1}^{(2)}+\dots.$ (12)

Here, the coefficient of the linear term $x_{1}^{\rm{typ}}$ governs the decay
of the average typical correlation function and finite higher cumulants
$x_{1}^{(n)}$ represent the multifractal nature of the correlator. We computed
$x_{1}^{\rm{typ}}$ from the difference between the average (over different
trajectories) Lyapunov exponents $\lambda_{1}$ and $\lambda_{0}$ in the
previous subsection. Here we compute the 2nd cumulant $k_{2}$ defined by,

$\displaystyle
k_{2}=\frac{\mathcal{E}[{\left(\lambda_{1}^{\vec{m}}-\lambda_{0}^{\vec{m}}\right)^{2}}]-\left(\mathcal{E}[{\lambda_{1}^{\vec{m}}-\lambda_{0}^{\vec{m}}}]\right)^{2}}{\alpha
L}=2\pi\frac{x_{1}^{(2)}}{L^{2}}.$ (13)

Here, $\lambda_{0}^{\vec{m}}$ and $\lambda_{1}^{\vec{m}}$ are the Lyapunov
exponents from each trajectory $\vec{m}$ in the circuit. We numerically
estimate $x_{1}^{(2)}=0.65(2)$ from $k_{2}(L)$ vs $1/L^{2}$ shown in Fig.
7(a). A large value of $x_{1}^{(2)}$ suggests the presence of strong
multifractality.

This evidence is further verified by collapsing the distribution of
$Y(t)=\alpha t(\lambda_{1}^{\vec{m}}-\lambda_{0}^{\vec{m}})$ onto the
universal curve $H(s)$ which, satisfies (up to a universal prefactor) Zabalo
_et al._ (2022); Chatelain and Berche (2000); Ludwig (1990),

$P[Y(t)]\sim\sqrt{\left(\frac{L}{2\pi\alpha
t}\right)}\exp\left[-\frac{2\pi\alpha t}{L}H\left(\frac{2\pi\alpha
t}{L}Y(t)\right)\right].$ (14)

Fig. 7(b) shows the data collapse onto a single curve $H(s)$ vs $s$ for
different values of $L$ and $t$ ($L=8$ to $18$ at $t=16L$ by red lines and
$t=11L$ to $27L$ for $L=18$ by blue lines). This data collapse of $P[Y(t)]$
using a single curve $H(s)$ establishes multifractal scaling of correlation
functions. $H(s)$ shows a minimum at $s=s_{\rm{min}}$ given by,

$s_{\rm{min}}=\frac{dx_{1}(n)}{dn}|_{n=0}=x_{1}^{\rm{typ}},$ (15)

In Fig. 7(b), we shifted the x-axis to $s-s_{\rm{min}}$ where $H(s)$ shows a
minimum near the typical scaling dimension
$s_{\rm{min}}=x_{1}^{\rm{typ}}=0.27(2)$. The broadening of $H(s)$ is set by
the variance $x_{1}^{(2)}=0.65(2)$.

We compare the multifractal spectrum $H(s)$ that we have obtained with the
MIPT without any symmetry in Fig. 7 (b) (shown as a black dashed line). After
shifting each curve by the location of the minimum in $H(s)$, $s_{\rm{min}}$,
we are able to compare their shapes, which demonstrates that the higher
average moments all differ as well. Thus providing strong evidence that the
U(1) symmetry has modified the universallity class of the MIPT.

##  VI Critical Properties of the Sharpening Transition

Figure 8: Charge sharpening from correlations: (a) shows the spin-spin
connected correlation function $C_{z}(r)$ vs $r$ for different $p$ values in a
log-log plot. The decay is power-law for $p\lesssim 0.14$. For large $p$, the
decay becomes faster than an algebraic decay. The inset shows the power-law
slope, $e_{z}$ which remains $2$ for $p\lesssim 0.14$. (b) shows the power-law
decay of the string correlator $C_{W}(r)\sim 1/|r|^{e_{W}}$ for smaller values
of $p$ in a log-log plot. For large $p$, $C_{W}(r)$ becomes a constant with
$e_{W}\rightarrow 0$ as shown in the inset. These features are consistent with
a BKT criticality. (c) shows a finite-size estimate of $\rho_{s}$ extracted
from $C_{W}$ with $p$ for different $L$. A finite-size BKT scaling collapse
following Eq.19, shown in the inset as a function of $p_{BKT}=(p-p_{\\#})(\log
L/a)^{2}$, yields $p_{\\#}=0.093(7)$. Last, we note that the stiffness
$\tilde{\rho}_{s}$ appears to cross close to where the predicted universal
value at the transition ought to be $\rho_{s}=\frac{1}{\pi}$ Barratt  _et al._
(2022).

Motivated by the description of the charge sharpening transition in the
infinite Hilbert space limit we analyze the sharpening transition in terms of
observables that were shown to exhibit BKT scaling Barratt  _et al._ (2022).
In the large Hilbert space limit this is facilitated by mapping onto a weakly
entangled model that can be simulated at large sizes with matrix product
states. Here, however, we do not have such a mapping for qubit chains and are
restricted to small system sizes where clearly identifying a BKT scaling
unambiguously is close to impossible. Instead, here our goal is to ascertain
if the BKT sharpening diagnostics are both consistent with our numerical
results on small sizes and look critical in a window around the sharpening
transition to see how it could affect the computed properties of the MIPT.

Therefore, we turn to numerically probe the charge-sharpening transition based
on our finite-size numerics following Ref.Barratt  _et al._ (2022). We first
compute the average spin-spin connected correlation function,

$C_{z}(r)=\mathcal{E}\left[\langle\sigma^{z}_{r}\sigma^{z}_{0}\rangle-\langle\sigma^{z}_{r}\rangle\langle\sigma^{z}_{0}\rangle\right].$
(16)

Here, $\langle\dots\rangle$ denotes the expectation value of the operator in
the quantum state of a trajectory and $\mathcal{E}[\dots]$ denotes averaging
over different trajectories constituting the quantum circuit. The correlation
function is expected Barratt  _et al._ (2022) to go like
$C_{z}(r)\sim\rho_{s}/r^{2}$ for $p\leq p_{\\#}$ where $\rho_{s}$ is the
superfluid stiffness and decay exponentially for $p>p_{\\#}$.

We check for this behavior by numerically calculating $C_{z}(r=L/2)$ for
different system sizes $L=8$ to $22$ starting from initial conditions in
$\mathcal{Q}=1/2$ sector. Fig. 8(a) shows $C_{z}(r)$ vs $r$ in a log-log plot
for different values of $0.06\leq p\leq 0.24$. For small values of $p\lesssim
0.14$, $C_{z}(r)$ shows a power-law decay while for larger values of $p\sim
0.24$, $C_{z}(r)$ decays faster than an algebraic decay and resembles an
exponential decay. A power-law fit $C_{z}(r)\approx 1/r^{e_{z}}$ from the
largest $4$ system sizes is shown by dashed lines and the slope $e_{z}$ vs $p$
is shown in the inset. The power-law function fits the data well for smaller
$p$ values indicated by smaller error bars, while for larger values of $p$,
the error-bar increases and the algebraic fit breaks down. Moreover,
$e_{z}\approx 2$ over a range of values, $p\lesssim 0.14$ showing signatures
of the BKT prediction of a critical charge fuzzy phase and a critical regime
extending into the vicinity of our estimate of the MIPT. Hence, this analysis
strongly suggests that the critical fluctuations of the charge sharpening
transition still look critical at the measurement-induced transition $p_{c}$
on our available system sizes.

Next, we consider the string correlation function $C_{W}(r)$ Barratt  _et al._
(2022) that is defined in terms of the string operator $W_{[0,r]}$, namely,

$\displaystyle W_{[0,r]}=\prod_{0<i<r}\sigma^{z}_{i},$ $\displaystyle
C_{W}(r)=\mathcal{E}\left[\langle W_{[0,r]}\rangle^{2}\right].$ (17)

$C_{W}(r)$ is expected to show a power-law decay in the charge fuzzy phase
$C_{W}(r)\sim 1/|r|^{2\pi\rho_{s}}$ and become independent of $r$ above the
sharpening transition. We calculate the string correlator $C_{W}(r=L/2)$ for
$L=8$ to $22$ at different values of $p$ shown in Fig. 8(b). $C_{W}(r)$ shows
a nice power-law decay for small $p$ values and becomes a constant for larger
$p$ values. Hence, this numerical result is qualitatively in good agreement
with the expectation from the BKT prediction. A power law fit $C_{W}(r)\sim
1/|r|^{e_{W}}$ (shown by the dashed lines) yields the slope $e_{W}$ which is
shown as a function of $p$ in the inset. The fitted power-law $e_{W}$
monotonically decreases with increasing $p$ as expected in any finite-size
analysis. This prediction also suggests that these string correlations remain
critical up to our estimate of the entanglement transition as they appear
within a given sector.

In order to provide an additional test of the BKT transition, we introduce a
finite size scaling function for the stiffness $\tilde{\rho}_{s}(p,L)$. From
the expectation $C_{W}(r)\sim 1/|r|^{2\pi\rho_{s}}$ we construct,

$\displaystyle\tilde{\rho}_{s}(p,L)=-\frac{\log C_{W}(L/2)}{2\pi\log(L/2)},$
(18)

which shows a crossing for different system sizes $L=8$ to $20$ as a function
of $p$ as depicted in Fig. 8(c). From the crossing we estimate the location of
the sharpening transition $p_{\\#}$ and perform a finite-size BKT scaling
analysis from the ansatz,

$\tilde{\rho}_{s}\sim h_{s}[(p-p_{\\#})(\log L/a)^{2}],$ (19)

where $h_{s}$ is an unkown scaling function with $p_{\\#}$ and $a$ as the
fitting parameter shown in the inset. Our analysis gives $p_{\\#}=0.093(7)$,
which is very close to our previous estimates of the sharpening transition
(e.g. in Fig. 1). Moreover, the value of the stiffness is expected to undergo
a quantized jump if the sharpening transition is of the predicted BKT type of
size $1/\pi$ Barratt  _et al._ (2022). Based on our estimate of the crossing
in $\tilde{\rho}_{s}$ at $p_{\\#}$ we find good agreement with the stiffness
jumping by $1/\pi$ within the error bars of our estimate of the sharpening
transition. All of this together provides strong evidence that the sharpening
transition in qubit chains is of the BKT type.

In the following subsection we discuss a possible interpretation of the
entanglement critical properties that we have found with values that are
anomalously large relative to measurement-induced criticality without any
symmetry. Following the perspective that our finite size estimates of the
critical properties are going to “feel” the sharpening transition on finite
size simulations, due to its close proximity as depicted in the finite size
cross-over diagram in Fig. 2, we are able to provide a clear explanation of
our results as being shifted by the critical properties of a BKT sharpening
transition.

##  VII Interpretation of the estimated critical entanglement properties

Our numerical results presented in Sec.IV suggest that the two critical
points, the charge-sharpening and the entanglement transition, are approaching
to each other as we go to $\mathcal{Q}$ sectors away from half-filling. Hence,
the entanglement critical properties, which are of our main interests in the
present work, may be influenced by the nearby charge-sharpening transition. To
ascertain this more carefully, in the previous section we showed that on our
finite size simulation the charge sharpening correlations remain critical at
the entanglement transition. A scenario for how this fits into the finite size
scaling regime we access is depicted in the schematic diagram for the two
transitions in Fig. 2. Here we pose a plausible reinterpretation of our above
estimates of the critical exponents considering the nearby sharpening
criticality.

First, we discuss if the presence of the nearby sharpening transition affects
our estimate of the exponent $\eta$. Due to the close proximity of the charge-
sharpening transition with the entanglement transition and the fact that the
sharpening fluctuations can be critical within a given sector, the scaling
dimension we pick up should come from the combination of the measurement-
induced transition and the sharpening transition, i.e.
$\eta=\eta_{\\#}+\eta_{E}$. Here $\eta_{\\#}=1/4$ is assumed for the BKT
charge sharpening transition and $\eta_{E}$ is the critical exponent of the
U(1) symmetric entanglement transition once we subtracted the BKT values from
the finite-size estimation $\eta$. As such, this would provide the estimate of
$\eta_{E}=0.32(3)$, which is close to $\eta_{H}=0.26(1)$ from the Haar random
problem without symmetry.

Second, we turn to our estimate of the effective central charge
$c_{\mathrm{eff}}$. The BKT sharpening transition is also a CFT (but it is not
multifractal) and has a well-defined central charge $c_{\\#}=1$. Our analysis
of the Free energy will then acquire a shift from these critical modes and we
therefore expect that our estimate of the effective central charge is offset
$c_{\mathrm{eff}}=c_{\\#}+c_{\mathrm{eff},E}$, which yields a
$c_{\mathrm{E,eff}}=0.27(1)$ that is quite close to the transition without
symmetry $c_{\mathrm{eff},H}=0.25(2)$. Thus, the interpretation of our
estimate of the effective central charge being contaminated the nearby
sharpening transition provides a natural explanation of our large estimate.
Conversely, it also provides additional evidence that the sharpening
transition is indeed a BKT transition.

Last, we turn to the multifractal properties of the transition. As the BKT
transition is not multifractal, it ensures that its contribution to $x_{1}(n)$
in Eq. (12) is linear in $n$. Combining this with Eq. (12), we see that the
only possible effect of the nearby BKT transition on the multifractal scaling
dimension is to shift the location of the minimum of $H(s)$ to
$s_{\rm{min}}=\frac{\eta_{E}+\eta_{\\#}}{2}.$ Using our numerical estimate of
$s_{\rm{min}}\approx 0.27(2)$ from Fig. 6, we get an estimate
$\eta_{E}=0.3(3)$ This is consistent with our understanding from section VI.
Moreover, all the higher cumulants i.e., $x_{1}^{(n)}$ are unaffected by the
BKT criticality and hence solely characterize the entanglement universality
class. Our observation of $x_{1}^{(2)}=0.65(2)$ being significantly different
from that of the MIPT without symmetry, $x_{1}^{(2)}=0.15(2)$, thus strongly
suggests that the U(1) MIPT is characterized by a unique multifractal log-CFT
that is distinct from its non-symmetric counterpart.

##  VIII Conclusion

In conclusion, this study has provided a comprehensive analysis of the
universality class of the measurement-induced phase transition (MIPT) in U(1)
conserving hybrid quantum circuits. Our sector-resolved approach enabled us to
investigate charges far from the middle sector and access system sizes up to
$L=48$ for a conserved charge density of $\mathcal{Q}=Q/L=1/12$. Our findings
demonstrate that each sector exhibits Lorentz invariance with a phase boundary
that follows the variance of the binomial distribution
$p_{c}(\mathcal{Q})\propto\mathcal{Q}(1-\mathcal{Q})$. The MIPT is
characterized by a logarithmic conformal field theory, which we quantified
through the examination of the transfer matrix and its Lyapunov spectrum
governing the quantum evolution. The effective central charge
($c_{\mathrm{eff}}$), the anomalous scaling dimension ($\eta$), and the
multifractal scaling of correlation functions in the conformal field theory
were all found to be distinctly different from the MIPT without a conservation
law. We then analyzed the charge sharpening physics and found it to be
consistent with a BKT transition though our small system size study is not
conclusive. Importantly, we also found that even within a single charge
sector, critical sharpening physics can still appear in the dynamics of
correlation functions averaged over measurement outcomes and these still look
critical up to the MIPT. Thus, we provided a natural interpretation to our
results that they have been shifted by BKT-related critical exponents. A
separate scenario, which seems less likely, is that the sharpening transition
and the entanglement transition coincide. However, this is at odds with
controlled calculations in the limit of an infinite onsite Hilbert space
dimension and we expect this can be resolved in future studies that find
models that “pull apart” the locations of the sharpening and entanglement
transitions. Nonetheless, regardless of this, by studying the multifractal
spectrum of the log-CFT at the entanglement transition our work offers strong
numerical evidence that the U(1) conservation law alters the universal nature
of the MIPT.

###### Acknowledgements.

We thank Srivatsan Chakram, David Huse, and Matthew Fisher for insightful
discussions as well as Utkarsh Agrawal, Sarang Gopalakrishnan, Andrew Potter,
and Romain Vasseur for discussions and collaborations on related work. This
work was partially supported by the Abrahams Postdoctoral Fellowship at the
Center for Materials Theory Rutgers (A.C.), the Army Research Office Grant No.
79849-PE-H (J.H.P.) and a Sloan Research Fellowship (J.H.P.). J.H.W.
acknowledges financial support from NSF grant DMR-2238895. This work was
performed in part at the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-2210452 (J.H.W., J.H.P.) The
computations were performed using the Beowulf cluster at the Department of
Physics and Astronomy of Rutgers University; and the Office of Advanced
Research Computing (OARC) at Rutgers, The State University of New Jersey
(http://oarc.rutgers.edu), for providing access to the Amarel cluster. The
Flatiron Institute is a division of the Simons Foundation.

Figure 9: Entanglement entropy of the ancilla $S_{E}$ vs $p$ at (a) $Q=L/2$
for $L=8$ to $24$, (b) $Q=L/4$ for $L=8$ to $24$ and (c) $Q=L/8$ for $L=8$ to
$40$. $S_{E}$ is collapsed with the scaling form Eq. (20) (and also consistent
with Eq. (22)) to obtain $p_{c}(\mathcal{Q})$ and $\nu(\mathcal{Q})$ shown in
the insets.

##  Appendix A Critical exponents from power-law scaling of the entangle order
parameter

To probe the entanglement critical point, we use a “single-site Ancilla”
method to calculate the order parameter $S_{E}$. After coupling the ancilla at
$t=0$ in a fixed charge sector $\mathcal{Q}=Q/L$, we evolve the U(1) symmetric
circuit with measurements upto $t=2L$ to reach the steady state. We then
compute the entanglement entropy $S_{E}$ of the ancilla for different
measurement rates $p$ and different system sizes $L$ commensurate with Q. Fig.
9(a)-(c) shows the variation of $S_{E}$ with $p$ in three different charge
sectors (for $n=1$), $\mathcal{Q}=1/2,1/4$ and $1/8$. We have shown $S_{E}$ vs
$p$ in the main text for $\mathcal{Q}=1/12$. In each cases, with increasing
$p$, $S_{E}$ decreases from a volume law phase to an area law phase crossing
at $p=p_{c}$ between different $L$s. The critical point is obtained using a
finite size scaling analysis with two different protocols: (i) using the
standard finite-size scaling ansatz for a second order transition,

$S_{E}(L;\mathcal{Q})\sim\tilde{h}_{\mathcal{Q}}[(p-p_{c}(\mathcal{Q}))L^{1/\nu(\mathcal{Q})}]$
(20)

and collapsing the data onto a universal curve $\tilde{h}_{\mathcal{Q}}$ with
two parameters $p_{c}(\mathcal{Q})$ and $\nu(\mathcal{Q})$ in each
$\mathcal{Q}$ sector. (ii) We use the ansatz proposed in Ref.Agrawal  _et al._
, 2022,

$S_{E}(L;\mathcal{Q})\sim\bar{h}_{\mathcal{Q}}[(p-p_{c}(\mathcal{Q}))L^{1/\nu(\mathcal{Q})},y(\mathcal{Q})L^{-\omega(\mathcal{Q})}],$
(21)

where the leading irrelevant scaling variable $y$ is incorporated to take into
account the shift of the crossing points with increasing system size.
$\bar{h}_{\mathcal{Q}}$ can be expanded in Taylor series around $p=p_{c}$ as,

$S_{E}(L;\mathcal{Q})=a+b(p-p_{c}(\mathcal{Q}))L^{1/\nu(\mathcal{Q})}+c(p-p_{c}(\mathcal{Q}))^{2}L^{2/\nu(\mathcal{Q})}+d/L^{\omega(\mathcal{Q})}.$
(22)

We numerically find the exponents $p_{c}(\mathcal{Q})$ and $\nu(\mathcal{Q})$
and the fitting parameters $a,b,c,d$ (ommited the $\mathcal{Q}$ dependence for
simplicity of the notation) by non-linear fitting from the data. The values
$p_{c}(\mathcal{Q})$ and $\nu(\mathcal{Q})$ shown in the main text in Figs.
1(c) and 3 (b) as well as the insets of Fig. 9 are consistent with both the
scaling anstaz within errorbars.

We next show how we extract the dynamical critical exponent $Z$. We collapse
$S_{E}$ with $t/L^{z(\mathcal{Q})}$ for different $L$s in a fixed
$\mathcal{Q}$ sector. Fig. 10(a)-(c) show the data collapse for
$\mathcal{Q}=1/2,1/4$ and $1/8$ ($\mathcal{Q}=1/12$ is shown in the main
text). This shows $z\approx 1$ across $\mathcal{Q}$ sectors as shown in
Fig2(c) of the main text. This justifies our choice of the aspect ratio
$t/L=2$ (i.e. an $\mathcal{O}(1)$ number) in finding $p_{c}$ and $\nu$.

Figure 10: Entanglement entropy of the ancilla $S_{E}$ is collapsed with
recaled time $t/L^{z(\mathcal{Q})}$ follwoing Eq.4 at (a) $Q=L/2$ for $L=8$ to
$24$, (b) $Q=L/4$ for $L=8$ to $24$ and (c) $Q=L/8$ for $L=8$ to $40$. This
gives the dynamical exponent $z\approx 1$ for all charge sectors.

##  Appendix B BKT finite-size scaling analysis of the charge-sharpening phase
transition

Figure 11: Charge-sharpening order parameter $S_{\\#}$ vs $p$. The panels (a),
(b), (c), (d) are collapsed with the BKT scaling ansatz Eq. (24) at
$\mathcal{Q}=1/2,1/4,1/6$ and $1/12$, respectively. The panels (e), (b), (c),
(d) are collapsed with the second-order phase transition ansatz Eq. (22) at
$\mathcal{Q}=1/2,1/4,1/6$ and $1/12$, respectively. The fitted
$p_{c}(\mathcal{Q})$ and $\nu(\mathcal{Q})$ shown in the insets.

To understand the nature of the charge-sharpening phase transition in a
quantum circuit system, we employed the ancilla method as outlined in
Ref.Agrawal  _et al._ (2022). This method involves coupling an ancilla to two
distinct charge sectors, represented as
$|\Psi\rangle=\left|\psi_{Q}\right\rangle|0\rangle+$
$\left|\psi_{Q-1}\right\rangle|1\rangle$; where $\left|\psi_{Q}\right\rangle$
symbolizes states within the charge sector $\mathcal{Q}$, while $|1\rangle$
and $|0\rangle$ represent ancilla states. The order parameter $S_{\\#}$,
signifying the von Neumann entanglement entropy of the reduced density matrix
${\operatorname{Tr}}_{\text{ancilla}}|\Psi\rangle\langle\Psi|$, is used to
probe the charge-sharpening phase transition. To compute the entropy
$S_{\\#}$, we adjusted the measurement rates $p$ and system sizes $L$ to be
commensurate with Q. Fig. 11 illustrates the variation of $S_{\\#}$ with $p$
for different charge sectors, $\mathcal{Q}=1/2,1/4,1/6$ and $1/12$.

We used finite-size scaling laws of quantum criticality to understand the
behavior near the charge-sharpening phase transition. For our analysis, we
tested two possible types of transitions: a second-order phase transition and
a BKT transition. For the former, the scaling ansatz matched that for the
entanglement phase transition as seen in Eq. (22). On the other hand, for the
BKT transition, we employed a different scaling ansatz:

$S_{\\#}(L;\mathcal{Q})\sim g_{\mathcal{Q}}[(p-p_{\\#}(\mathcal{Q}))(\log
L/a)^{2},y_{\\#}(\mathcal{Q})L^{-\omega_{\\#}(\mathcal{Q})}],$ (23)

Here, $a$ is the ultraviolet scale of BKT criticality, $y_{\\#}$ signifies the
amplitude of the subleading term with the scaling dimension
$\omega_{\\#}(\mathcal{Q})$, and $L$ represents the system size. The choice of
logarithmic scaling for $L$ is guided by the BKT scaling law for the
correlation length $\xi/a\sim\exp(1/\sqrt{|p-p_{c}|})$. In the vicinity of the
critical point, $g_{\mathcal{Q}}$ can be perturbatively expanded,

$S_{\\#}(L;\mathcal{Q})\sim a_{\\#}+b_{\\#}(p-p_{\\#}(\mathcal{Q}))(\log
L/a)^{2}+c_{\\#}(p-p_{\\#}(\mathcal{Q}))^{2}(\log
L/a)^{4}+d_{\\#}L^{-\omega(\mathcal{Q})},$ (24)

where the fitting parameters are $p_{\\#}$, $a$ and
$a_{\\#},b_{\\#},c_{\\#},d_{\\#}$. We tested a wide range of the exponent
$\omega_{\\#}$ to establish the error bars.

##  Appendix C Two methods to calculate mutual information between a pair of
ancilla

In the calculation of the mutual information between a pair of ancillas, here
we compare the newly developed “single-site ancilla” method explained in
Sec.III.1 of the main text with the previously used “bond-ancilla” method (to
find $p_{C}$ and $\nu$) in Ref.Agrawal  _et al._ , 2022. In the “bond-ancilla”
method, to probe the entanglement phase transition in a fixed global charge
sector $\mathcal{Q}$, an ancilla is entangled in a Bell state with a bond of
the adjacent sites $j$ and $j+1$ in the system yielding the wave function
$|\tilde{\Psi}\rangle=(1/\sqrt{2})\big{[}|\dots\uparrow\downarrow\dots\rangle|\Uparrow\rangle+|\dots\downarrow\uparrow\dots\rangle|\Downarrow\rangle\big{]}$.
Here the states $|\dots\uparrow\downarrow\dots\rangle$ and
$|\dots\downarrow\uparrow\dots\rangle$ belong to same global charge sector
$\mathcal{Q}$ and are orthogonal due to the orthogonal configuration at the
bond. Although this method of entangling ancilla accurately predicts the
entanglement critical point ($p_{c}$ and $\nu$), it is difficult to extend
this method to calculate mutual information between a pair of ancillas and
associated $\eta$ exponents at $p_{c}$. As we know, correlation function
Zabalo  _et al._ (2020) is obtained from the mutual information of the two
ancilla entangled to the system. Entangling two ancillas each coupled to a
bond requires $4$ projection operations on the wave function (ex:
$|\dots\uparrow\downarrow\dots\downarrow\uparrow\dots\rangle|\Uparrow\Downarrow\rangle$
) and makes this method numerically challenging due to collapse of the
wavefunction. Moreover, if we go to the charge sectors away from the middle
sectors, the dimension of the constrained Hilbert-space (due to U(1) symmetry)
reduces and it becomes increasingly difficult to couple two ancillas to two
orthogonal states without collapsing the system wavefunction.

To circumvent these issues, we used the efficient “single-site ancilla” method
to couple the ancilla and calculated correlation functions at different charge
sectors. We compare the two methods (the “single-site ancilla” and the “bond-
ancilla”) in Fig. 12 (a)-(b)at the middle sector where the $\eta$ exponent
(for $n=1$) extracted from both the methods are in good agreement. This
establishes the consistency of the new method. However, the “bond-ancilla”
method requires running $\sim 1.5$ times more samples compared to the “single-
site ancilla” method to obtain similar size of the statistical ensemble over
which we averaged the correlation function. Moreover, the “bond-ancilla”
method required us to wait longer to reach the steady state before coupling
the ancillas ($t_{0}\sim L^{2}$), while for the “single-site ancilla” method
it is enough to wait till $t_{0}\sim 2L$. In Fig. 12, we showed both (a) and
(b) at $t_{0}=2L^{2}$ for the purpose of comparison.

Next, we use the “single-site ancilla” method to calculate the correlation
function at $\mathcal{Q}=1/8$. Fig. 12(c) shows the scaling collapse with
$\eta=0.69(5)$. We note that, even after using the efficient method at
$\mathcal{Q}=1/8$, a large number of samples collapsed due to the above-
mentioned post-selection problem and the size of the statistical ensemble we
averaged, is only $\sim 1/4$ compared to that at the middle-sector. This leads
to poor quality of the scaling collapse with large error bars at
$\mathcal{Q}=1/8$ compared to that at half-filling (Fig. 12(b)).

Figure 12: The scaling collapse of the correlation function with time using
Eq.6: the two ancillas are coupled using the “bond-ancilla” method in (a) and
using the “single-site ancilla” method in (b). The critical exponent $\eta$
extracted using these two methods are $\eta=0.41(4)$ from the “bond-ancilla”
method and $\eta=0.44(2)$ from the “single-site ancilla” method at
$p_{c}=0.110$ and $\mathcal{Q}=1/2$. These values are in good agreement with
each other, establishing consistency between these two methods. (c) shows the
scaling collapse of the mutual information at $\mathcal{Q}=1/8$ (using the
“single-site ancilla” method) and $p_{c}=0.048$. This gives $\eta=0.69(5)$,
but with a poorer quality of collapse due to averaging over smaller number of
samples. We use $n=1$ for this plot.

##  Appendix D Computation of the anisotropy parameter $\alpha$

To compute the anisotropy parameter $\alpha$ in the area of the cylinder of
the CFT $A=\alpha Lt$, we followed the procedure prescribed in Ref.Zabalo  _et
al._ (2022). We compute two kinds of correlation functions: (i) the “space-
like” correlation function $I_{\rm{space}}=I_{n=1}(A,B:\delta r=L/2,\delta
t=0)$ from the mutual information of the two ancillas $A$ and $B$, coupled in
the steady state of the circuit at the same instant of time
$t_{1}=t_{2}=t_{0}(=2L)$, at a spatial separation $|r_{2}-r_{1}|=L/2$. (ii)
the “time-like” correlation function $I_{\rm{time}}(\delta
t)=I_{n=1}(A,B:\delta r=0,\delta t)$ from the mutual information of the two
ancillas, where the first ancilla is coupled at $t_{1}=t_{0}=(2L)$ and the
second one is coupled at a separation in time $t_{2}-t_{1}=\delta t$, at the
same site in the system. After coulping the two ancillas in the circuit, we
calculate the time evolution of the correlation functions $I_{n=1}(A,B)$ with
time $t-t_{2}$. This is plotted in Fig. 13(a) where the initial conditions are
chosen only from the middle sector $\mathcal{Q}=1/2$ for a $L=16$ site system.
We set the measurement rate at $p=p_{c}=0.110$ at half-filling. Here,
$I_{\rm{space}}$ is shown by the yellow line while $I_{\rm{time}}$ is shown
for two values of $\delta t=8$ and $9$ which cross $I_{\rm{space}}$.

Figure 13: Computation of the anisotropy paremeter $\alpha$: at
$p=p_{c}(\mathcal{Q}=1/2)=0.11$, we plot the “space-like” correlator
$I_{\rm{space}}$ with yellow color and the “time-like” correlators
$I_{\rm{time}}(\delta t=8)$ (grey) and $I_{\rm{time}}(\delta t=9)$ (red) for
two values of $\delta t$ which cross $I_{\rm{space}}$ at late times. We obtain
$\alpha=0.51(3)$ in case (a), when we restrict to only $\mathcal{Q}=1/2$
trajectories and this matches with the case (b) with $\alpha=0.50(1)$ where
the initial conditions are randomly chosen over all charge sectors.

From this, we have to find out the value of $\delta t=t_{*}$ at which the two
types of the correlations match, $I_{\rm{space}}=I_{\rm{time}}(t_{*})$. Since,
numerically we can only calculate $I_{\rm{time}}(\delta t)$ on a grid of
$\delta t$ with grid size$=1$, we interpolate to obtain,

$t_{*}=8+\frac{I_{\rm{space}}-I_{\rm{time}}(8)}{I_{\rm{time}}(9)-I_{\rm{time}}(8)}(9-8).$
(25)

This gives $t_{*}=8.768$. We compute the value of $\alpha$ from
$\alpha=\log(1+\sqrt{2})L/(\pi t_{*})$ derived in Ref.Zabalo  _et al._ (2022).
This gives $\alpha=0.51(3)$ at half-filling. We use this value of $\alpha$ for
numerical results in Sec.V.2.

Furthermore, we also compute $\alpha$ when the system is initialized to states
superposing with all charge sectors. This is shown in Fig. 13(b) at
$p=p_{c}=0.110$ for $L=16$. Following similar procedure explained above, we
obtain $t_{*}=8.912$ which yields $\alpha=0.50(1)$. Hence the estimates of
$\alpha$ restricting to middle sectors and that from all charge sectors are in
good agreement. This is also consistent with the fact that for large $L$ (in
thermodynamic limit), the middle sectors ($\mathcal{Q}=1/2$) dominates over
all other charge sectors as shown in Fig. 6(b).

##  Appendix E Dependence of initial conditions on $f_{1}(L)$

In section V.2 of the main text, we showed the dependence on the initial
conditions of the free energy $f(L)$ and the difference between the
generalized free energies $f_{1}(L)-f(L)$. We discussed that when we restrict
to the contribution only from a fixed charge sector, the free energy
$f^{\mathcal{Q}}(L)$ picks up a leading $1/L$ dependence (in addition to the
usual $1/L^{2}$ scaling). Here we show that the generalized free energy
$f_{1}(L)$ shows a similar $1/L$ scaling when we compute
$f_{1}^{\mathcal{Q}}(L)$ from the trajectories projected to $\mathcal{Q}$
sector at late times. This is shown in Fig. 14 by blue triangles at
$\mathcal{Q}=1/2$ and $p=p_{c}=0.11$. $f_{1}^{\mathcal{Q}=1/2}(L)$ can be fit
well using a form,

Figure 14: The generalized free energy $f_{1}(L)$ is shown as a function of
$1/L^{2}$ for two different initial conditions: the blue triangles
$f_{1}^{\mathcal{Q}=1/2}$ correspond to restricting to the trajectories to
$\mathcal{Q}=1/2$ charge sector while the red circles correspond to the
contributions from all trajectories with unrestricted charge.
$f_{1}^{\mathcal{Q}=1/2}$ can be fit well by adding a $1/L$ dependence to the
usual $1/L^{2}$ scaling form that applies to the unrestricted charge case.

$f_{1}^{\mathcal{Q}}=f_{1}^{\mathcal{Q}}(L\rightarrow\infty)+a^{\prime}\mathcal{Q}/L+\frac{a^{\prime\prime}}{L^{2}}.$
(26)

This $1/L$ dependence coming from the chemical potential fixing the density to
that charge sector is cancelled in the difference
$f_{1}^{\mathcal{Q}}(L)-f^{\mathcal{Q}}(L)$ from each trajectory with charge
$Q$. This is verified in Fig. 6(b) of the main text. We also plot $f_{1}(L)$
from all trajectories with unrestricted charge in Fig. 14 by the red circles.
In this case, absence of chemical potential leads to the usual $1/L^{2}$
scaling.

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