###### Abstract

In recent years, remarkable progress has been achieved in developing novel
non-perturbative techniques for constructing valence space shell model
Hamiltonians from realistic internucleon interactions. One of these methods is
based on the Okubo–Lee–Suzuki (OLS) unitary transformation applied to no-core
shell model (NCSM) solutions. In the present work, we implement the
corresponding approach to solve for valence space effective electromagnetic
operators. To this end, we use the NCSM results for $A=16-18$, obtained at
$N_{\rm max}=4$, to derive a charge-dependent version of the effective
interaction for the $sd$ shell, which allows us to exactly reproduce selected
NCSM spectra of 18O, 18F and 18Ne within the two valence nucleon space. We
then deduce effective single-particle matrix elements of electric quadrupole
($E2$) and magnetic dipole ($M1$) operators by matching them to the
electromagnetic transitions and moments for 17O and 17F from the NCSM at
$N_{\rm max}=4$. Thus, effective $E2$ and $M1$ operators are obtained as sets
of single-particle matrix elements for the valence space ($sd$ shell) which
allow us to reproduce the NCSM results for $A=17$ exactly. Systematic
comparison of a large set of $sd$ shell results on quadrupole and magnetic
dipole moments and transitions for $A=18$ using effective $E2$ and $M1$
operators that we derive from the full NCSM calculations demonstrates a
remarkable agreement.

##  Chapter 0 Effective operators for valence space calculations
from the ab initio No-Core Shell Model

###  1 Preamble

The authors wish to express their appreciation for the opportunity to
participate in this honorary volume for Prof. Akito Arima. Indeed, Prof. Arima
led many pioneering works in the microscopic theory of nuclear forces and
effective interactions. Several of the authors had the great pleasure to
discuss these fundamental physics topics with him and to receive his sage
advice. With this contribution, we propose extensions to microscopic theory of
valence interactions including electromagnetic effective operators that follow
lines of thinking that can trace their origin to the fundamental works of
Prof. Akito Arima.

###  2 Introduction

Outstanding progress in nuclear many-body computations and in construction of
highly accurate realistic internucleon interactions during the last couple of
decades has made it possible to describe light nuclei and intermediate-mass
nuclei near closed shells from first principles [1, 2, 3, 4, 5, 6, 7]. Among
all these ab initio approaches, the no-core shell model [1] (NCSM), being a
full configuration-interaction type method, solves the eigenvalue problem by
exact diagonalization of the Hamiltonian matrix constructed in a large-
dimensional many-body harmonic-oscillator basis. The latest state-of-the-art
computations, such as those made possible by the Many-fermion dynamics code
for nuclei (MFDn) [8, 9, 10, 11, 12], are capable of providing converged or
nearly-converged solutions for nuclei up to $A\sim 18$, setting benchmarks for
other nuclear structure models [7, 13, 14]. However, for heavier nuclei the
eigenproblem has still to be solved in a more restricted model space,
typically for a few valence nucleons occupying one or two harmonic oscillator
shells beyond a closed-shell core. This approach with an assumed core of
passive nucleons is known as a traditional interacting shell-model (see, e.
g., Refs. [15, 16, 17]). Severe truncation of the model space requires
construction of an appropriate effective Hamiltonian and effective electroweak
operators. From the 1960’s, for more than fifty years, the many-body
perturbation theory (see, e. g., Refs. [18, 19] and references therein) was
the dominant approach to deal with the microscopic derivation of effective
Hamiltonians. It was, however, noticed that when derived from two-nucleon
interaction, an effective valence-space Hamiltonian could not compete in
precision with empirically obtained effective Hamiltonians [20, 21], mainly
because of its deficient monopole part [22]. One of possible reasons was
identified as the absence of three-nucleon forces [23].

New non-perturbative techniques for deriving effective valence-space
interactions have been developed during the last decade, such as similarity-
renormalization group (SRG) inspired methods [24], or application of
Okubo–Lee–Suzuki (OLS) unitary transformation [25, 26, 27] to either a NCSM
solution [28, 29, 30] or to a solution found within the coupled cluster theory
[31, 32]. Recent inclusion of three-nucleon forces via different approaches
[33, 34, 31, 24, 35, 19] seems to cure the deficiency of the monopole part of
the effective interactions and brings encouraging improvement to the
theoretical description.

Construction of appropriate effective electroweak operators to be used in a
valence space has been another long-standing challenge. Since the earlier work
by Arima and collaborators [36, 37], the importance of accounting for the
restricted model space has been realized. Many efforts were devoted to the
construction of effective electromagnetic and Gamow–Teller operators both
microscopically [38, 39, 40, 41, 42, 43] or empirically by a fit of the
optimum form of a corresponding operator to the data [44, 45, 20, 46, 47].

In this work, we use the NCSM solutions for $A=16{-}18$ nuclei in order to
construct an effective interaction and effective $E2$ and $M1$ operators for
the valence $sd$-shell model space. We construct here a charge-dependent
version of the effective interaction presented earlier [30] based on the
Daejeon16 nucleon-nucleon ($NN$) potential [48], using 18O, 18F and 18Ne to
derive neutron-neutron, neutron-proton and proton-proton two-body matrix
elements (TBMEs). Effective $E2$ and $M1$ single-particle matrix elements are
obtained by matching them to the corresponding NCSM results for 17O and 17F
obtained with basis spaces including all many-body excitation oscillator
quanta up to $N_{\rm max}=4$. These constructed effective operators are tested
on a large number of transitions and moments for $A=18$ nuclei.

###  3 Microscopic effective interaction from the NCSM

The derivation of valence-space effective interactions from the NCSM is
described in detail in Refs. [29, 30] and references therein. We start with a
translationally-invariant non-relativistic Hamiltonian for $A$ point-like
nucleons, containing relative kinetic energies and $NN$ interactions:

$H=\sum_{i<j}^{A}\frac{(\vec{p_{i}}-\vec{p_{j}})^{2}}{2mA}+\sum_{i<j}^{A}V_{ij}^{NN}.$
(1)

Here $m$ is the nucleon mass (approximated here as the average of the neutron
and proton mass), $\vec{p}_{i}$ are nucleonic momenta and $V_{ij}^{NN}$
denotes the bare $NN$ interaction. To our chosen Daejeon16 $NN$ interactions
we have added the two-body Coulomb interaction between the protons.

The eigenproblem for $H$ is solved by diagonalizing the Hamiltonian matrix,
constructed in a many-body spherical harmonic-oscillator basis, using the MFDn
code [8, 9, 10, 11, 12]. All nucleons are considered as active particles and
are treated equally. The model space is defined by two parameters: (i) by a
given oscillator quantum, $\hbar\Omega$, and (ii) by the maximum total many-
body harmonic-oscillator excitation quanta, $N_{\rm max}$. This means that we
retain only many-body configurations satisfying the condition
$\sum_{i=1}^{A}(2n_{i}+l_{i})\leq N_{\rm min}+N_{\rm max}$, where $n_{i}$ is
the single-particle radial harmonic-oscillator quantum number, $l_{i}$ is the
single-particle orbital angular momentum quantum number, while $N_{\rm min}$
is the minimal many-body oscillator quanta satisfying the Pauli principle for
the chosen nucleus. The use of the harmonic-oscillator basis allows us to
remove states with center-of-mass excitations by using the Lawson method [49].

In the present study, we select $\hbar\Omega=14$ MeV, which is close to the
empirical value [50] in the vicinity of $A=16$, and which was also adopted in
our previous work [30].

In principle, one may apply the OLS transformation [25, 26, 27] to a “cluster”
of nucleons to develop an effective interaction for the chosen NCSM space (the
“$P$-space”)[1]. This is referred to as the cluster approximation. A compact
presentation of the OLS method is presented in Ref. [51]. The goal is to
achieve an effective interaction at fixed cluster size (conventionally two-
nucleon or three-nucleon) for which the NCSM calculations converge more
rapidly with increasing $P$-space than using the original (non-transformed)
interaction.

In this work we adopt the Daejeon16 $NN$ interaction [48] which is based on a
chiral N3LO interaction [52, 53] and which has been softened already by an SRG
transformation [54, 55] that improves convergence. Hence, following our
procedure [30] we elect to use Daejeon16 directly, without an additional OLS
transformation for our NCSM calculations.

For this work, we adopt the NCSM calculations at $N_{\rm max}$ = 4 for the
$A=16$, 17 and 18F. In addition, to obtain a charge-dependent valence
effective interaction for the first time, we perform new NCSM calculations at
$N_{\rm max}$ = 4 for 18O and 18Ne.

The resulting NCSM eigenvalues and eigenvectors are then employed to define a
new effective interaction for the $N_{\rm max}=0$ space (the
“$P^{\prime}$-space”) through the application of the OLS method. The NCSM
results dominated by $sd$-shell configurations are appropriate for adoption to
define the $P^{\prime}$-space. That is, from the NCSM results for $A=16{-}18$
we apply the OLS method to obtain an effective valence space ($sd$-shell)
Hamiltonian consisting of a core, a single-particle and a two-particle
interaction [29, 30]. This effective interaction is guaranteed to reproduce
the NCSM results for the 16O core and the states of $A=17$ and 18 selected to
be included in the definition of the $P^{\prime}$-space.

From the $A=18$ NCSM results we select the number of eigenpairs (eigenvalues
and eigenvectors) for each $J$ governed by the number of needed two-body
matrix elements. Let us label the total number of eigenpairs (adding up the
dimensions for each $J$) as $d_{nn}$, $d_{pp}$ and $d_{np}$ for valence space
configurations of 18O, 18Ne and 18F respectively. These are the states,
selected from among the NCSM candidates, which have the largest valence space
($0\hbar\Omega$ in the present study) probability. It should be noted that
these are not necessarily the states with the lowest NCSM eigenvalues. The
exact dimensions are $d_{nn}=14$ for 18O, $d_{pp}=14$ for 18Ne, and
$d_{np}=28$ for 18F, since it includes both $T=0$ and $T=1$ states.

Neutron-neutron, proton-neutron ($T=1$) and proton-proton TBMEs of the derived
effective interaction $\langle ab;J,T=1|V_{\rm eff}|cd;J,T=1\rangle$ in the
$sd$ shell from Daejeon16. Note that the Coulomb interaction is included in
the NCSM calculations. See text for details.  $2j_{a}$ $2j_{b}$ $2j_{c}$
$2j_{d}$ $J$ $T$ $V_{nn}$ $V_{pn}^{T=1}$ $V_{pp}$ 1 1 1 1 0 1 $-2.018$
$-1.989$ $-1.536$ 1 1 3 3 0 1 $-0.713$ $-0.769$ $-0.797$ 1 1 5 5 0 1 $-1.446$
$-1.422$ $-1.344$ 3 3 3 3 0 1 $-1.350$ $-1.317$ $-0.847$ 3 3 5 5 0 1 $-2.377$
$-2.500$ $-2.582$ 5 5 5 5 0 1 $-2.618$ $-2.590$ $-2.046$ 1 3 1 3 1 1 $0.137$
$0.136$ $0.482$ 1 3 3 5 1 1 $-0.151$ $-0.151$ $-0.152$ 3 5 3 5 1 1 $0.069$
$0.062$ $0.441$ 1 3 1 3 2 1 $-0.500$ $-0.500$ $-0.116$ 1 3 1 5 2 1 $-1.519$
$-1.498$ $-1.438$ 1 3 3 3 2 1 $-0.090$ $-0.099$ $-0.089$ 1 3 3 5 2 1 $0.459$
$0.452$ $0.428$ 1 3 5 5 2 1 $-1.183$ $-1.164$ $-1.120$ 1 5 1 5 2 1 $-1.358$
$-1.341$ $-0.908$ 1 5 3 3 2 1 $-0.854$ $-0.853$ $-0.828$ 1 5 3 5 2 1 $0.366$
$0.360$ $0.336$ 1 5 5 5 2 1 $-0.549$ $-0.542$ $-0.509$ 3 3 3 3 2 1 $-0.187$
$-0.185$ $0.180$ 3 3 3 5 2 1 $0.861$ $0.855$ $0.824$ 3 3 5 5 2 1 $-0.818$
$-0.837$ $-0.836$ 3 5 3 5 2 1 $-0.108$ $-0.116$ $0.267$ 3 5 5 5 2 1 $0.252$
$0.259$ $0.263$ 5 5 5 5 2 1 $-1.140$ $-1.132$ $-0.695$ 1 5 1 5 3 1 $0.435$
$0.433$ $0.799$ 1 5 3 5 3 1 $0.310$ $0.302$ $0.289$ 3 5 3 5 3 1 $0.186$
$0.169$ $0.508$ 3 5 3 5 4 1 $-1.628$ $-1.621$ $-1.196$ 3 5 5 5 4 1 $1.388$
$1.371$ $1.323$ 5 5 5 5 4 1 $-0.212$ $-0.211$ $0.184$

Proton-neutron ($T=0$) TBMEs of the derived effective interaction $\langle
ab;J,T=0|V_{\rm eff}|cd;J,T=0\rangle$ in the $sd$ shell from Daejeon16. See
text for details.  $2j_{a}$ $2j_{b}$ $2j_{c}$ $2j_{d}$ $J$ $T$ $V_{pn}^{T=0}$
1 1 1 1 1 0 $-2.909$ 1 1 1 3 1 0 $0.381$ 1 1 3 3 1 0 $0.573$ 1 1 3 5 1 0
$-2.196$ 1 1 5 5 1 0 $-1.224$ 1 3 1 3 1 0 $-3.565$ 1 3 3 3 1 0 $2.176$ 1 3 3 5
1 0 $1.458$ 1 3 5 5 1 0 $-0.761$ 3 3 3 3 1 0 $-1.343$ 3 3 3 5 1 0 $0.305$ 3 3
5 5 1 0 $2.465$ 3 5 3 5 1 0 $-5.702$ 3 5 5 5 1 0 $-3.375$ 5 5 5 5 1 0 $-1.043$
1 3 1 3 2 0 $-1.839$ 1 3 1 5 2 0 $3.045$ 1 3 3 5 2 0 $-2.290$ 1 5 1 5 2 0
$-0.497$ 1 5 3 5 2 0 $1.887$ 3 5 3 5 2 0 $-3.370$ 1 5 1 5 3 0 $-3.935$ 1 5 3 3
3 0 $-0.004$ 1 5 3 5 3 0 $-1.537$ 1 5 5 5 3 0 $-2.011$ 3 3 3 3 3 0 $-3.144$ 3
3 3 5 3 0 $-1.743$ 3 3 5 5 3 0 $0.977$ 3 5 3 5 3 0 $-0.646$ 3 5 5 5 3 0
$-2.071$ 5 5 5 5 3 0 $-0.690$ 3 5 3 5 4 0 $-4.218$ 5 5 5 5 5 0 $-4.448$

The OLS transformation leads to a two-nucleon effective Hamiltonian defined in
the $N_{\rm max}=0$ model space by its TBMEs which in the angular momentum $J$
and isospin $T$ coupled form read $\langle ab;JT|H_{\rm eff}|cd;JT\rangle$,
where $a$, $b$, $c$, $d$ denote a full set of single-particle spherical
quantum numbers which label an orbital, e.g.
$a=(n_{a},l_{a},j_{a},t_{z_{a}})$, with $t_{z}$ distinguishing protons and
neutrons.

To obtain the one-body neutron and proton contributions (single-particle
energies, $\epsilon$), we subtract the ground-state energy of 16O (core
energy, $E_{\rm core}$) from 17O and 17F, respectively. To get the TBMEs of
the effective interaction in the valence space, we have to subtract this core
energy and the one-body contributions from diagonal TBMEs of the OLS-derived
effective valence space Hamiltonian $H_{\rm eff}$,

$\langle ab;JT|V_{\rm eff}|ab;JT\rangle=\langle ab;JT|H_{\rm
eff}|ab;JT\rangle-E_{\rm core}-(\epsilon_{a}+\epsilon_{b}),$ (2)

while for non-diagonal TBMEs, we have

$\langle ab;JT|V_{\rm eff}|cd;JT\rangle=\langle ab;JT|H_{\rm
eff}|cd;JT\rangle\,.$ (3)

In this way, from NCSM calculations for 18O, 18Ne and 18F, we derive a set of
neutron-neutron, proton-proton and proton-neutron TBMEs, respectively, to be
used in the valence space shell-model calculations.

Figure 1: Low-energy theoretical excitation spectra of 18O, 18Ne and the $T=1$
part of the 18F spectrum obtained within the NCSM with Daejeon16 at $N_{\rm
max}=4$ (identical to the $sd$-shell calculation with the proposed effective
valence-space interaction). The USDB $T=1$ spectrum for $A=18$ and low-energy
experimental spectra of positive-parity states [56] are shown for comparison.
The $0^{+}_{2}$ state shown in red in experimental spectrum in 18O is most
probably an intruder state as evidenced by the electromagnetic transition
pattern [56]. Its isobaric analogue states in 18F and 18Ne are also marked in
red, and, therefore, should not be compared with nearby theoretical spectra.

The resulting core energy, proton and neutron single-particles energies, as
well as $T=0,1$ proton-neutron TBMEs obtained from Daejeon16 have been
published in an earlier study (see column “bare” of Table II and column “bare”
of Table XI in Ref. [30]), while the neutron-neutron and proton-proton TBMEs
obtained specifically for this work. For the convenience of the reader, we
present the whole set of TBMEs in Tables 3 and 3. By construction, the
$sd$-shell spectra of 18O, 18Ne and 18F obtained with the derived effective
Hamiltonian exactly reproduce the results of the NCSM for the chosen
$P^{\prime}$-space eigenvalues. Comparison of the low-energy $T=1$ states are
shown in Fig. 1 in comparison with experiment [56] and with the results from a
phenomenological interaction USDB [46]. The latter produces identical $T=1$
spectra for all three $A=18$ isobars, because the USDB Hamiltonian is charge-
independent, including the single-particle energies which are the same for
protons and neutrons.

Since there are only small differences among the corresponding proton-proton,
neutron-neutron and proton-neutron ($T=1$) TBMEs, as seen from Table 3, (aside
from the expected, approximately constant, diagonal Coulomb shift for the
proton-proton TBMEs), the excitation spectra support the approximate validity
of isospin symmetry (see Fig. 1).

The low-lying $T=0$ spectrum of 18F, obtained from the NCSM, contains states
with the same $J^{\pi}$ quantum numbers as the USDB spectrum, however, it is
more extended in energy than the USDB spectrum, with different ordering, and
the lowest $3^{+}$ state, being almost degenerate with $1^{+}$ state, becomes
the ground state (see Fig. 2). Except for the three lowest states, the
experimental $T=0$ spectrum is noticeably more dense and it is challenging to
find theoretical counterparts in the $sd$ shell calculation. This may hint at
a possible intruder character of some of those states. In the NCSM calculation
at $N_{\rm max}=4$, states dominated by $2\hbar\Omega$ configurations are
still located at much higher energy and therefore too far from being converged
to shed light on this intruder state issue. Since the present study is focused
on the construction of effective operators and to the comparison between the
NCSM and valence space results, we do not investigate potential intruder
states in the current work. In future applications with larger $P$-spaces, one
may anticipate, however, that as these intruders fall further into the region
of the valence-space dominated states, the mixing that is likely to occur will
lead to complications in identifying the appropriate eigenpairs for retention
in the $P^{\prime}$ space.

Figure 2: Low-energy excitation spectra of 18F (up to 6.2 MeV): theoretical
spectrum obtained within the NCSM with Daejeon16 at $N_{\rm max}=4$ (identical
to the $sd$-shell calculation with the effective interaction of Ref. [30]), in
comparison with the positive-parity experimental spectrum[56] and the one
obtained from USDB. The $0^{+}_{2}$, $T=1$ state in 18F, analogue to the
intruder state in 18O, is shown in red color.

###  4 Effective electromagnetic operators in the $sd$ shell

Considering nucleons as non-relativistic point-like particles and assuming a
long-wavelength approximation for emitted or absorbed radiation, one can
deduce the following expressions for one-body electric quadrupole and magnetic
dipole operators:

$\hat{O}(E2)=\sum_{k=1}^{A}e_{k}r_{k}^{2}\hat{Y}_{2}(\hat{r}_{k})\,,$ (4)

and

$\hat{O}(M1)=\sum_{k=1}^{A}\mu_{N}\left[g^{s}_{k}\vec{s}_{k}+g^{l}_{k}\vec{l}_{k}\right]\,,$
(5)

where $\hat{Y}_{2}(\hat{r}_{k})$ are spherical harmonics of rank $2$,
$\hat{r}_{k}=(\theta_{k},\phi_{k})$ being a spherical angle, $\vec{s}_{k}$ and
$\vec{l}_{k}$ are spin and orbital angular momenta operators of a nucleon,
respectively, $e_{k}$ denotes the electric charge of the $k$th nucleon, with
the bare values being $e_{p}=e$ for a proton and $e_{n}=0$ for a neutron,
while $g^{s}$ and $g^{l}$ are spin and orbital gyro-magnetic factors
($g$-factors) with bare values being $g^{s}_{p}=5.586$, $g^{l}_{p}=1$,
$g^{s}_{n}=-3.826$, $g^{l}_{n}=0$ for protons and neutrons, respectively. For
valence space calculations, $e_{p}$ and $e_{n}$ are conventionally understood
to be effective proton and neutron electric charges, while $g^{l,s}_{k}$ have
to be effective $g$-factors. In principle one could follow the same procedure
as we use to develop the valence space effective two-nucleon interaction to
develop valence space effective $E2$ and $M1$ operators. The procedures for
doing so would follow those demonstrated in Refs. [51, 57]. The results, for
the $P^{\prime}$-space we define here, would be represented as TBMEs for these
operators. That is, there are, in theory, both one-body and two-body
contributions to such derived effective valence electromagnetic operators.
Here, we will obtain approximate theoretical one-body effective valence space
operators for $E2$ and $M1$. We then test those approximate operators in a
number of ways.

Our starting point is the NCSM calculation of $E2$ and $M1$ transitions and
moments in 17O and 17F at $N_{\rm max}=4$. The three lowest states are
$1/2^{+}$, $5/2^{+}$ and $3/2^{+}$ states which are mapped into single-
particle $1s_{1/2}$, $0d_{5/2}$ and $0d_{3/2}$ states in the $sd$-shell
calculation. Therefore, we require that the matrix elements of the $E2$ and
$M1$ operator with bare electric charges and $g$-factors between the many-body
eigenstates from the NCSM at $N_{\rm max}=4$ be exactly reproduced by
effective single-particle matrix elements in the valence space.

####  1 Effective $E2$ operator

To obtain an effective one-body valence-space $E2$ operator, we use the NCSM
results for $E2$ transitions and quadrupole moments for 17O and 17F. For 17F,
we define a valence-state-dependent proton effective charge such that it
exactly reproduces the NCSM values for transitions and moments. Next, we
develop a state-dependent effective charge for the valence neutron. We note
that the NCSM uses bare electric charges, so only protons contribute to the
matrix elements of the $E2$ operator. In contrast, in the valence space 17O
has only one valence neutron and therefore we require that it carries an
effective charge to be able to reproduce electromagnetic transitions and
moments by the valence space calculations. These rules can be summarized as
follows:

$\begin{array}[]{ll}e_{n}(a,b)\langle
b||r^{2}\hat{Y}_{2}(\hat{r})||a\rangle=\langle
J_{f}||\hat{O}(E2)||J_{i}\rangle&({\rm from}\;^{17}{\rm O})\\\
e_{p}(a,b)\langle b||r^{2}\hat{Y}_{2}(\hat{r})||a\rangle=\langle
J_{f}||\hat{O}(E2)||J_{i}\rangle&({\rm from}\;^{17}{\rm F})\\\ \end{array}$
(6)

where $e_{n}(a,b)$ and $e_{p}(a,b)$ are neutron and proton effective state-
dependent charges, $J_{i}$ and $J_{f}$ are the initial and final state
obtained within the NCSM, which are mapped onto single-particle states $a$ and
$b$, while the $E2$ operator $\hat{O}(E2)$ is defined by Eq. (4). Double bars
in this equation and below refer to the matrix element reduced in angular
momentum.

Figure 3: Comparison of the $sd$-shell results for the absolute values of
reduced $E2$ transition matrix elements between various states in 18O, 18F and
18Ne obtained using effective $E2$ operators within shell model with a core
with those obtained by the NCSM (top); the same in the vicinity of zero in
order to provide details on an enlarged scale for the weaker transitions
(bottom).

Figure 4: Comparison of the $sd$-shell results for electric quadrupole moments
of various states in 18O, 18F and 18Ne obtained using effective $E2$ with
those obtained by the NCSM. An outlier in 18F, which is discussed in the text,
is flagged with its spin and sequence in the spectrum indicated.

Matching the NCSM results for $E2$ transitions and quadrupole moments for 17O
and 17F, we have deduced the values for effective electric charges which are
summarized in Table 1. These effective charges $e_{p}(a,b)$ and $e_{n}(a,b)$
depend on the quantum numbers of the initial and final states, $a$ and $b$, so
the effective $E2$ operator is deduced as a set of individually parameterized
single-particle matrix elements $\langle b|\hat{O}(E2)|a\rangle$. However, the
state dependence of the effective charges is not strong as is seen from Table
1: the difference between neutron effective charges $e_{n}(a,b)$ does not
exceed $\sim 0.12e$ and the maximal difference between various proton
effective charges $e_{p}(a,b)$ is $\sim 0.25e$. The weak state dependence of
effective charges tends to support the use of state-independent effective
neutron and proton charges as is conventionally done in the valence-space
calculations, see, e. g., Ref. [47]. We present in Table 1 average theoretical
effective neutron, $\bar{e}_{n}$, and proton, $\bar{e}_{p}$, charges which are
seen to be smaller than the commonly accepted values [47] for $0\hbar\Omega$
model space calculations, $e_{p}=1.36(5)e$ and $e_{n}=0.45(5)e$.

Effective charges (in units of $e$) and effective $g$-factors in the
$sd$-shell valence space found by the respective NCSM calculations for 17O and
17F. The bare values used in the NCSM calculations are also indicated for
reference. Hermiticity requires symmetry of these effective quantities under
the interchange of $a$ and $b$. See text for details.  $(a,b)$ $e_{n}(a,b)$
$e_{p}(a,b)$ $g^{s}_{n}(a,b)$ $g^{l}_{n}(a,b)$ $g^{s}_{p}(a,b)$
$g^{l}_{p}(a,b)$ bare 0.0 1.0 -3.826 0.0 5.586 1.0 $(0d_{5/2},1s_{1/2})$ 0.181
1.171 $(0d_{5/2},0d_{3/2})$ 0.281 1.236 -3.608 0.020 5.252 0.916
$(1s_{1/2},0d_{3/2})$ 0.168 1.297 $(0d_{5/2},0d_{5/2})$ 0.179 1.060 -3.751
0.026 5.499 0.976 $(0d_{3/2},0d_{3/2})$ 0.172 1.248 -3.690 0.033 5.332 0.957
$(1s_{1/2},1s_{1/2})$ -3.729 5.468 $\overline{e}_{n}$ $\overline{e}_{p}$
$\overline{g}^{s}_{n}$ $\overline{g}^{l}_{n}$ $\overline{g}^{s}_{p}$
$\overline{g}^{l}_{p}$ average 0.196 1.202 -3.695 0.026 5.388 0.950

To assess the quality of the operator constructed above, we calculate reduced
transition matrix elements and quadrupole moments in $A=18$ $sd$-shell nuclei
using both state-dependent and average effective charges. Reduced transition
matrix elements,
$\langle\alpha_{f}J_{f}||\hat{O}(E2)||\alpha_{i}J_{i}\rangle$, are required to
express the corresponding $B(E2)$ values of a transition from an initial state
$|\alpha_{i}J_{i}\rangle$ to the final state $|\alpha_{f}J_{f}\rangle$ as

$B(E2;J_{i}\to
J_{f})=\dfrac{1}{2J_{i}+1}|\langle\alpha_{f}J_{f}||\hat{O}(E2)||\alpha_{i}J_{i}\rangle|^{2},$
(7)

where $J_{i}$, $J_{f}$ are the initial (final) angular momenta and
$\alpha_{i}$ ($\alpha_{f}$) are other quantum numbers of the initial (final)
states. The standard definition of a quadrupole moment in a state $|\alpha
J\rangle$ reads

$Q(J)=\sqrt{\dfrac{16\pi}{5}}\langle\alpha
J,M_{J}=J|\hat{O}(E2,M_{L}=0)|\alpha J,M_{J}=J\rangle.$ (8)

Root-mean-square (rms) differences between the results on $E2$ and $M1$
transitions and moments from valence-space calculations with either the state-
dependent (st-dep) or the average parameterizations of the effective $E2$ or
effective $M1$ operator and the respective NCSM results. See text for details.
18O 18F 18Ne rms difference st-dep average st-dep average st-dep average Red
$E2$ M.E. [$e\cdot\rm fm^{2}$] 0.072 0.168 0.114 0.335 0.221 0.267 $B(E2)$
[$e^{2}\cdot\rm fm^{4}$] 0.017 0.039 0.189 0.240 0.208 0.583 $Q$ [$e\cdot\rm
fm^{2}$] 0.063 0.118 0.370 0.478 0.064 0.443 Red $M1$ m.e. [$\mu_{N}$] 0.070
0.088 0.050 0.071 0.108 0.118 $B(M1)$ [$\mu_{N}^{2}$] 0.063 0.081 0.092 0.106
0.060 0.085 $\mu$ [$\mu_{N}$] 0.023 0.032 0.189 0.182 0.020 0.099

$B(E2)$ values for transitions among the lowest states and electric quadrupole
moments of the lowest states in 18O, 18F and 18Ne: NCSM versus valence-space
results with either state-dependent or average effective charges. The
experimental values are shown when available (only the latest result is
retained in the table below when a few experimental values are given in Ref.
[58]).  Exp [56, 58] NCSM $sd$ shell st-dep average 18O $B(E2;2^{+}_{1}\to
0^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 9.3(3) 0.527 0.535 0.634
$B(E2;2^{+}_{2}\to 0^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 3.6(6) 0.001 0.001
0.001 $B(E2;4^{+}_{1}\to 2^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 3.3(2) 0.339
0.359 0.434 $Q(2^{+}_{1})$ [$e\cdot\rm fm^{2}$] $-3.6(9)$ $-1.40$ $-1.46$
$-1.55$ 18Ne $B(E2;2^{+}_{1}\to 0^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 49.6(50)
21.94 21.76 23.79 $B(E2;2^{+}_{2}\to 0^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$]
1.9(9) 0.07 0.07 0.007 $B(E2;4^{+}_{1}\to 2^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$]
24.9(34) 14.43 14.42 16.07 $Q(2^{+}_{1})$ [$e\cdot\rm fm^{2}$] $-9.16$ $-9.23$
$-9.65$ 18F $B(E2;3^{+}_{1}\to 1^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 16.3(6)
8.29 8.51 9.05 $B(E2;3^{+}_{2}\to 1^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$] 1.9(3)
0.014 0.010 0.030 $B(E2;5^{+}_{1}\to 3^{+}_{1})$ [$e^{2}\cdot\rm fm^{4}$]
16.2(7) 7.81 7.95 8.64 $Q(1^{+}_{1})$ [$e\cdot\rm fm^{2}$] 0.32 0.19 0.28
$Q(3^{+}_{1})$ [$e\cdot\rm fm^{2}$] $-5.75$ $-5.93$ $-6.36$ $Q(5^{+}_{1})$
[$e\cdot\rm fm^{2}$] $\pm$7.7(5) $-7.19$ $-7.34$ $-8.29$

We have calculated 66 $E2$ transitions in 18O, 66 $E2$ transitions in 18Ne and
269 $E2$ transitions in 18F within the NCSM at $N_{\rm max}=4$ using the bare
$E2$ operator and in the $sd$-shell model space using the effective operators.
The results are compared in Figs. 3 and 4 where we present absolute values of
reduced $E2$ matrix elements and signed electric quadrupole moments for
different states obtained in the $sd$-shell valence space calculations with
the effective operators versus those obtained by the NCSM. We observe an
excellent correlation between the NCSM and valence-space values with the root-
mean-square (rms) differences between them being very small, especially for
the state-dependent parameterization of the operator, as can be seen from
Table 1. The $B(E2)$ values, used to calculate the rms difference, have been
obtained for $J_{i}\geq J_{f}$. Table 1 shows $B(E2)$ values for transitions
between the lowest states and electric quadrupole moments for the lowest
$J\neq 0$ states in 18O, 18Ne and 18F in comparison with the experimental
data, when available. We clearly see missing strength in the $E2$ transitions
between the lowest $T=1$ states, which may signify a need for larger $N_{\rm
max}$ values, because of the very slow convergence of $E2$ observables [59].
At the same time, the relative order of magnitude of theoretical $B(E2)$
values qualitatively resemble experimental findings. In particular, our
calculation suggests that the quadrupole moment of the $5^{+}_{1}$ state in
18F is negative.

It is worth noting here that the agreement is worse for a couple of cases. In
particular, one of them is seen in Fig. 4 where the quadrupole moment of the
$1^{+}_{7}$ state in 18F with ${Q=0.78~{}e\cdot\rm fm^{2}}$ from the NCSM and
${Q=2.54~{}e\cdot\rm fm^{2}}$ from the valence-space calculations. The
discrepancy is most likely related to the fact that this state is the highest
lying $1^{+}$ state (at around 25 MeV excitation energy) which has only 30% of
the valence-space component in the NCSM calculations. Without this point, the
rms error for $Q$ in 18F reduces to from 0.370 to $0.114~{}e\cdot\rm fm^{2}$
for the state-dependent and from 0.478 to $0.339~{}e\cdot\rm fm^{2}$ for the
average parameterization of the operator.

####  2 Effective $M1$ operator

We proceed in a like manner for the effective $M1$ operator. From comparison
of the NCSM results on magnetic dipole moments and transitions for $A=17$ at
$N_{\rm max}=4$ with single-particle values, we have deduced state-dependent
matrix elements of the orbital and spin contribution of the $M1$ operator as
well as respective state-dependent effective $g$-factors $g^{j}_{k}(a,b)$,
$k=n,p$ and $j=l,s$. The rules can be summarized as follows:

$\begin{array}[]{ll}g^{s}_{n}(a,b)\langle
b||\mu_{N}\,\vec{s}\,||a\rangle=\langle J_{f}||\hat{O}(M1)_{\rm
spin}||J_{i}\rangle&({\rm from}\;^{17}{\rm O})\\\ g^{l}_{n}(a,b)\langle
b||\mu_{N}\,\vec{l}\,||a\rangle=\langle J_{f}||\hat{O}(M1)_{\rm
orbit}||J_{i}\rangle&({\rm from}\;^{17}{\rm O})\\\ g^{s}_{p}(a,b)\langle
b||\mu_{N}\,\vec{s}\,||a\rangle=\langle J_{f}||\hat{O}(M1)_{\rm
spin}||J_{i}\rangle&({\rm from}\;^{17}{\rm F})\\\ g^{l}_{p}(a,b)\langle
b||\mu_{N}\,\vec{l}\,||a\rangle=\langle J_{f}||\hat{O}(M1)_{\rm
orbit}||J_{i}\rangle&({\rm from}\;^{17}{\rm F})\\\ \end{array}$ (9)

where spin and orbital part are referring to the first and second term of the
$M1$ operator, given by Eq. (5), while $J_{i}$ and $J_{f}$ are the initial and
final states obtained within the NCSM which are mapped onto the single-
particle states $a$ and $b$ in the valence space. Note that these equations
help to exactly reproduce $M1$ matrix elements between the NCSM states for
$A=17$ within the valence space, except for those which involve the $s_{1/2}$
orbital due to the fact that it is characterized by $l=0$ orbital quantum
number and therefore there is no way to reproduce a small, but non-zero value
of $\langle J_{f}||\hat{O}(M1)_{\rm orbit}||J_{i}\rangle$. To overcome this
difficulty, we can require that in the case when $a$ or $b$ refers to the
$s_{1/2}$ single-particle state, the valence space matrix element due to non-
zero spin part would exactly reproduce the NCSM matrix element of the full
$M1$ operator. Because of the smallness of the orbital contribution to the
NCSM results, both definitions lead to almost identical results for magnetic
moments and, therefore, their use is practically equivalent. We choose the
latter method in the present work. The corresponding effective $g$-factors are
summarized in Table 1. We notice that there is a more modest quenching of the
spin operators than the quenching obtained in the previous studies in the
$sd$-shell [41, 42, 60, 47]. We notice also that the modifications to orbital
$g$-factors found (positive for neutrons and negative for protons) are
opposite in sign from the corresponding corrections found in the same cited
works. For reference, the best fitted empirical values found for the same form
of the $M1$ operator using the USDB wave functions in the $sd$ shell are
$g^{s}_{n}=-3.55(10)$, $g^{l}_{n}=-0.09(26)$, $g^{s}_{p}=5.15(9)$,
$g^{l}_{p}=1.159(23)$ [47].

Figure 5: Comparison of the $sd$-shell results for the reduced $M1$ matrix
elements between various states in 18O, 18F and 18Ne obtained using effective
$M1$ operators within shell model with a core with those obtained by the NCSM
(top); the same in the vicinity of zero in order to provide details on an
enlarged scale for the weaker transitions (bottom). Two outliers in 18F are
flagged in the top panel with their state identifications and are discussed in
the text.

Figure 6: Comparison of the $sd$-shell results for magnetic dipole moments of
various states in 18O, 18F and 18Ne obtained using effective $M1$ operators
with those obtained by the NCSM. Two outliers in 18F are flagged with their
state identifications and are discussed in the text.

As a next step, we have calculated reduced $M1$ transition probabilities and
magnetic moments in $A=18$ $sd$-shell nuclei. A reduced $M1$ transition matrix
element, $\langle\alpha_{f}J_{f}||\hat{O}(M1)||\alpha_{i}J_{i}\rangle$, is
required to express the corresponding $B(M1)$ value of a transition from an
initial state $|\alpha_{i}J_{i}\rangle$ to the final state
$|\alpha_{f}J_{f}\rangle$ as

$B(M1;J_{i}\to
J_{f})=\dfrac{1}{2J_{i}+1}|\langle\alpha_{f}J_{f}||\hat{O}(M1)||\alpha_{i}J_{i}\rangle|^{2},$
(10)

while the magnetic moment reads

$\mu(J)=\sqrt{\dfrac{4\pi}{3}}\langle\alpha
J,M_{J}=J|\hat{O}(M1,M_{L}=0)|\alpha J,M_{J}=J\rangle.$ (11)

The absolute values of reduced $M1$ matrix elements of 43 $M1$ transitions in
18O, 43 $M1$ transitions in 18Ne and 212 $M1$ transitions in 18F obtained
within the NCSM at ${N_{\rm max}=4}$ using the bare $M1$ operator, and in the
$sd$ shell model space using the effective operators are compared in Fig. 5.
The magnetic moments of all $J\neq 0$ states (11 states in 18O, 11 states in
18Ne and 25 states in 18F) are compared in Fig. 6. We notice again a
remarkable agreement between the two sets of results. The state-dependent
parameterization of the $M1$ single-particle matrix elements leads to smaller
rms differences as compared to those provided by the average parameterization
with an exception of the 18F magnetic moments where both parameterizations
provide nearly the same rms differences (see Table 1). The $B(M1)$ values,
used to compute the rms difference, have been generated for $J_{i}\geq J_{f}$.

Table 2 shows $B(M1)$ values for transitions between the lowest states and
magnetic dipole moments of the lowest states in 18O, 18Ne and 18F in
comparison with the experimental data, when available. In particular, there is
a good agreement with the measured magnetic moment values. We attribute this
to the fact that the $M1$-observables converge faster in the NCSM
calculations.

Two outlying 18F points are clearly seen in Fig. 6 and these are magnetic
moments of the first two $2^{+}$ states: they are only 130 keV separated in
energy and appear to be mixed in isospin. The $2^{+}_{1}$ state contains 88%
of $T=1$ component and 11% $T=0$ component and vice verse for $2^{+}_{2}$. The
values of the reduced $M1$ transition matrix elements between these two states
are also significantly different between the NCSM (4.258 $\mu_{N}^{2}$) and
valence space calculations (3.297 $\mu_{N}^{2}$) as seen from Fig. 5.

$B(M1)$ values for transitions between the lowest states and magnetic moments
of the lowest states in 18O, 18F and 18Ne: NCSM versus valence-space results
obtained using either state-dependent or average $g$-factors. The experimental
values are shown when available (only the latest result is retained in the
table below when a few experimental values are given in Ref. [58]).  Exp [56,
58] NCSM $sd$ shell st-dep average 18O $B(M1;2^{+}_{2}\to 2^{+}_{1})$
[$\mu_{N}^{2}$] 0.25(4) 0.072 0.072 0.071 $B(M1;1^{+}_{1}\to 0^{+}_{1})$
[$\mu_{N}^{2}$] 0.101 0.119 0.125 $\mu(2^{+}_{1})$ [$\mu_{N}$] $-0.57(3)$
$-0.53$ $-0.54$ $-0.52$ $\mu(4^{+}_{1})$ [$\mu_{N}$] $\pm 2.5(4)$ $-2.41$
$-2.41$ $-2.35$ 18Ne $B(M1;2^{+}_{2}\to 2^{+}_{1})$ [$\mu_{N}^{2}$] 0.092
0.092 0.090 $B(M1;1^{+}_{1}\to 0^{+}_{1})$ [$\mu_{N}^{2}$] 0.079 0.089 0.093
$\mu(2^{+}_{1})$ [$\mu_{N}$] 2.62 2.63 2.55 18F $B(M1;1^{+}_{1}\to 0^{+}_{1})$
[$\mu_{N}^{2}$] 5.428 5.366 5.256 $\mu(3^{+}_{1})$ [$\mu_{N}$] $+1.77(12)$
1.857 1.852 1.803 $\mu(5^{+}_{1})$ [$\mu_{N}$] $+2.86(3)$ 2.878 2.878 2.798

###  5 Conclusions and Prospects

In this contribution we present a novel method for deriving effective one-body
electromagnetic operators for valence space calculations from the NCSM
results. The effective $E2$ and $M1$ one-body operators are obtained as a set
of parameterized effective single-particle matrix elements to be used with
many-body states in the valence space. Comparison of a large number of reduced
$E2$ and $M1$ matrix elements, as well as electric quadrupole and magnetic
dipole moments in $A=18$ $sd$-shell nuclei shows a very good agreement with
the NCSM values. Further exploration of these operators in systems with more
valence particles is in progress. One also anticipates systematically
increasing the space for the NCSM calculations as computational resources
allow.

### Acknowledgments

Z. Li and N. A. Smirnova acknowledge the financial support of CNRS/IN2P3 via
ENFIA Master project, France. The work of A. M. Shirokov is supported by the
Russian Foundation for Basic Research under Grant No. 20-02-00357. The work of
I. J. Shin was supported by the Rare Isotope Science Project of Institute for
Basic Science funded by Ministry of Science and ICT and National Research
Foundation of Korea (2013M7A1A1075764). This work was supported in part by the
US Department of Energy (DOE) under Grant Nos. DE-FG02-87ER40371, DE-SC0018223
(SciDAC-4/NUCLEI). Computational resources were provided by the National
Energy Research Scientific Computing Center (NERSC), which is supported by the
US DOE Office of Science under Contract No. DE-AC02-05CH11231, and also
provided by the National Supercomputing Center, Republic of Korea, including
technical support (KSC-2020-CRE-0027). N. A. Smirnova and A.M. Shirokov thank
the Institute for Basic Science, Daejeon, for a hospitality and financial
support of their visits.

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# Quantum Vicsek Model for Active Matter

Hong Yuan (袁红)  Graduate School of CAEP, Beijing 100193, China  L.X. Cui (崔廉相)
Beijing Computational Science Research Center, Beijing 100193, China  L.T.
Chen (陈乐天)  Department of Mathematics and Centre of Complexity Science,
Imperial College London, South Kensington, London SW7 2BZ, United Kingdom
C.P. Sun (孙昌璞<EMAIL_ADDRESS>Graduate School of CAEP, Beijing 100193,
China

###### Abstract

We propose a quantum analog of the Vicsek model, consisting of an ensemble of
overdamped spin$-1/2$ particles with ferromagnetic couplings, driven by a
uniformly polarized magnetic field. The spontaneous magnetization of the spin
components breaks the $SO(3)$ (or $SO(2)$) symmetry, inducing an ordered phase
of flocking. We derive the hydrodynamic equations, similar to those formulated
by Toner and Tu, by applying a mean-field approximation to the quantum analog
model up to the next leading order. Our investigation not only establishes a
microscopic connection between the Vicsek model and the Toner-Tu hydrodynamics
for active matter, but also aims to inspire further studies of active matter
in the quantum regime.

Introduction.—Recent intensive studies of active matter have revolutionized
our understanding of non-equilibrium systems by revealing the spontaneous
emergence of long-range polar order among locally aligning, self-propelled
particles, especially in two dimensions [1, 2, 3, 4, 5], which are similar to
the flocking behavior observed in bird flocks, fish schools, and even
synthetic micro-swimmers [6, 7]. In a pioneering work, Vicsek et al. [4]
introduced a simple yet powerful model demonstrating that even with minimal
local rules, particles could spontaneously align to form long-range polar
order. Another fundamental breakthrough was made almost simultaneously by
Toner and Tu (TT) in 1995 [8], developing a hydrodynamic theory of active
matter. In TT theory, the scaling exponents, which characterize the temporal
and spatial correlations of fluctuations, could be computed exactly,
contrasting with predictions by the Mermin-Wagner-Hohenberg theorem [9, 10].
This seminal work laid the foundation for understanding the collective
dynamics of active matter [11, 12].

Despite the advantages offered by the Vicsek model and TT theory, subsequent
studies revealed certain limitations [13]. For example, Toner himself later
realized that the original hydrodynamic analysis was incomplete due to the
neglect of certain important terms, undermining most claims of exactness in
the original analysis [14]. There remained hope, however, that renormalization
could render these additional terms irrelevant. This expectation was
challenged by recent large-scale simulations of the Vicsek model [15]. These
simulations exhibited scaling behavior that clearly deviated from the
predictions of Toner and Tu, highlighting the need for a refined theoretical
framework.

Our study aims to bridge this gap by proposing a quantum analog of the Vicsek
model, named the quantum Vicsek model (QVM). We consider an ensemble of
overdamped spin$-1/2$ particles with ferromagnetic couplings, driven by a
radically homogeneous magnetic field. The Heisenberg interactions in this
system typically exhibit a collective behavior-spontaneous magnetization,
breaking the $SO(3)$ (or $SO(2)$ in $2D$) symmetry in $3D$, and inducing
spatial motion into an ordered phase analogous to classical flocking. This can
also be seen as a flying XY model in $2D$ [8].

Our model primarily operates in the quantum regime, but it can be applied to
the classical case by the fluctuation-dissipation relations, ignoring
contributions of $\hbar$ terms [16, 17]. Specifically, by taking the continuum
limit of the QVM, we derive the hydrodynamic equations for active matter using
a mean-field approximation up to the next leading order. The similarity
between our hydrodynamic equation from the quantum Vicsek model and the Toner-
Tu equation is reflected by the detailed calculations of critical scaling
exponents according to the renormalization group. Our methodology for this
continuum treatment incorporates quantum aspects absent in classical models,
and the quantum perspective allows for a deeper understanding of the
microscopic mechanisms driving the collective behavior of active particles,
explaining why the results from the Vicsek Model and the Toner-Tu
hydrodynamics may belong to the same universality class at the microscopic
level.

Quantum Vicsek model.—We consider a system consisting of $N$ spin$-1/2$
particles with coordinate $\vec{r}_{j}$ and momentum $\vec{p}_{j}$. The inter-
spin ferromagnetic couplings with strength $J$ has an interaction range
$V_{d}$. There exists approximately $N^{\prime}=V_{d}\rho$ neighboring
particles interacting with $\vec{S}_{j}$, where $\rho$ is the particle
density. Moreover, the entire system is subjected to a homogeneous polar
magnetic $\vec{B}\sim\vec{r}_{j}$ (see the Fig. 1). The Hamiltonian of the
system reads

$H=\sum_{j=1}^{N}\frac{p_{j}^{2}}{2m}\
-\zeta\sum_{j=1}^{N}\vec{r_{j}}\cdot\vec{S_{j}}\
-J\sum_{i<j}^{N^{\prime}}\vec{S_{i}}\cdot\vec{S_{j}},$ (1)

where $m$ is the mass of each particle, and $\zeta$ a constant characterizing
the strength of the interaction between the spins and the magnetic field.

Figure 1: Spins in a homogeneous polar magnetic field.

We assume an environment coupling with this system of spin$-1/2$ particles.
The Caldeira-Leggett type interactions with the reservoir result in the
dissipate dynamics [17, 16] governed by Eq. (1),

$\displaystyle\ddot{\vec{r}}_{j}=-\gamma\dot{\vec{r}}_{j}+\zeta\vec{S}_{j}+\vec{f}(t),$
(2)
$\displaystyle\dot{\vec{S}}_{j}=-\left(J\sum_{i=1}^{N^{\prime}}\vec{S}_{i}+\zeta\vec{r}_{j}\right)\times\vec{S}_{j}-\gamma_{s}(\vec{S}_{j}-\vec{S}_{j}^{st})+\vec{f}_{s}(t),$
(3)

where $\gamma$ and $\gamma_{s}$ are friction coefficients for spatial and spin
parts, respectively, while $\vec{f}(t)$ and $\vec{f}_{s}(t)$ represent the
relevant fluctuation terms. Both Brownian forces of $\vec{f}(t)$ and
$\vec{f}_{s}(t)$ need to adhere to the fluctuation-dissipation theorem (FDT),
i.e., $\langle
f_{a}(t)f_{b}(t)\rangle_{R}=2\gamma(m\beta)^{-1}\delta_{ab}\delta(t-t^{\prime})$
and $\langle
f_{sa}(t)f_{sb}(t)\rangle_{R}=2\gamma_{s}(m\beta)^{-1}\delta_{ab}\delta(t-t^{\prime})$.
Here, $a,b$ are Cartesian indices for dimensions — $x,y$ in $2D$ (or $x,y,z$
in $3D$), $\beta$ is the inverse temperature of the reservoir, and
$\langle\cdot\rangle_{R}$ denotes averaging over the reservoir.
$\vec{S}_{j}^{st}$ is the stationary value of $\vec{S}_{j}$.

With stronger inter-spin coupling of ferromagnetic type, i.e., a large
positive $J$, the spontaneous breaking of $SO(3)$ symmetry emerges to give a
mean field

$\vec{B}_{MF}=\frac{1}{2}J\sum_{j}^{N^{\prime}}\langle\vec{S}_{j}\rangle\equiv\frac{1}{2}JN^{\prime}\vec{S},$
(4)

where $\langle\cdot\rangle$ means some kind of average over classical or
quantum states of the system. It is emphasized that the fluctuation forces in
Eq. (2) and Eq. (3) are the results in the limit of the high temperature, but
it will be modified to include the term $\hbar$ near zero temperature [16,
17].

In the overdamped regime, the inertial term is ignored so that Eq. (2) and Eq.
(3) become the active dynamic equations

$\displaystyle\dot{\overrightarrow{r_{j}}}=u\vec{S}_{j}+\vec{f}(t),$ (5)
$\displaystyle\dot{\vec{S}}_{j}=-\vec{B}_{MF}\times\vec{S}_{j}-\gamma_{s}(\vec{S}_{j}-\vec{S})+\vec{f}_{s}(t).$
(6)

Here $u\equiv\zeta/m$ and the spatial term $\zeta\vec{r}_{j}$ is regarded as
infinitesimal compared to $\vec{B}_{MF}$ in the thermodynamic limit. As
illustrated in Eq. (5), the spin $\vec{S}_{j}$ acts as an energy pump for the
motion of the center of mass, resulting in a nonzero mean velocity aligned
with $\vec{S}_{j}$ due to spin-orbit coupling induced by the magnetic field
$\vec{B}$. Moreover, ferromagnetic interactions typically align $\vec{S}_{j}$
with surrounding spins, leading to the emergence of an ordered
phase—spontaneous magnetization—that breaks the $SO(3)$ symmetry (or $SO(2)$
in $2D$). As a result, nearly all the spins may point in the same direction
$\vec{S}$. In the overdamped regime, this ferromagnetic coupling also induces
an alignment interaction among velocities, whereby a particle reorients itself
according to the average direction of its neighbors within the interaction
range $V_{d}$. This mechanism could therefore result in an ordered flocking
state for spatial motions of the ensemble, propelling the entire system
towards a specific direction paralleling to $\vec{B}_{MF}$.

Figure 2: Numerical simulations of Quantum Vicsek Model (QVM). (a) The phase
diagram of the order parameter $\varphi=|\langle\vec{S}_{j}\rangle|$ as a
function of the relaxation time $\gamma_{s}^{-1}$ and noise amplitude $\xi$.
The average of the order parameter is taking over $10^{5}$ time steps. (b) The
band phase in QVM. (c) Top view of (b), where the shading indicates particle
density. The simulation is based on Eq. (5) and Eq. (6) with periodic boundary
conditions. Simulation parameters: $\rho=0.5$, $u=0.5$, $r_{c}=1.0$, and the
system size $L=32$. For (b), the noise amplitude $\xi=0.5$ and the relaxation
time $\gamma_{s}^{-1}=1.0$.

It is noteworthy that the dynamic equations Eq. (5) and Eq. (6) are consistent
with the dynamics of the Vicsek model [4], where $u\vec{S}_{j}$ acts as the
self-propulsion term. Our model explicitly clarifies the microscopic origins
of both the self-propulsion and the alignment interactions. Thus, this quantum
framework presented here provides a more fundamental understanding of the
Vicsek model, offering a deep insight into the studies of active matter from a
microscopic perspective.

To provide an intuitive demonstration of the QVM, we perform numerical
simulations of Eq. (5) and Eq. (6) in $3D$ space. The noise term $\vec{f}(t)$
in Eq. (5) is not include in the simulations. The mean-field approximation is
applied to particles within a radius $r_{c}$ around each particle. Fixing the
particle number density $\rho$, the model contains two parameters:
fluctuations $\vec{f}_{s}$ and dissipation terms $\gamma_{s}$ of spin. Spin
dissipation is generally characterized by the relaxation time which is
characterized by $\gamma_{s}^{-1}$, while spin fluctuations are modeled using
vector noise [18] with an amplitude $\xi$. As illustrated in Fig. 2(a), as the
relaxation time decreases and the noise intensity is reduced, the order
parameter $\varphi=|\langle\vec{S}_{j}\rangle_{j}|$, where
$\langle\cdot\rangle_{j}$ represents the average over the whole population,
transitions from a disordered phase to an ordered phase. Furthermore, it is
observed that near the onset of order, a bands phase similar to that in the
Vicsek model [18] also exists, as shown in Fig. 2(b). Particles are ordered
within these bands, while particles remain disordered outside the bands.

Hydrodynamic descriptions.—In order to understand how the Vicsek model relates
to the TT theory at the microscopic level, we further consider the
hydrodynamic descriptions from the QVM. To this end, we derive the
hydrodynamic equation through the kinetic approach [19, 20, 21, 22] to the
QVM.

Under the mean-field approximation, this interacting system is usually
described by a single-body distribution $f(\vec{r},\vec{v},t)$, which defines
the particle number density in single-particle phase space at time $t$ [19,
20]. The expected hydrodynamic variables [19, 20, 21, 22], i.e., the spatial
density and the velocity field, are then respectively given by
$\rho(\vec{r},t)=\int f(\vec{r},\vec{v},t)d^{3}\vec{v}$ and
$\vec{V}(\vec{r},t)=\int\vec{v}f(\vec{r},\vec{v},t)d^{3}\vec{v}$, where
$f(\vec{r},\vec{v},t)$ satisfies the Boltzmann equation

$\partial_{t}f+\vec{v}\cdot\nabla_{r}f+\vec{a}\cdot\nabla_{v}f=I[f],$ (7)

where $\partial_{t}$, $\nabla_{r}$ and $\nabla_{v}$ respectively are the
differential operators with respect to time $t$, $\vec{r}$ and $\vec{v}$.
$I[f]$, following the Fokker-Planck theory [23, 24], is given by
$I[f]=\gamma\nabla_{v}\cdot(\vec{v}-V(\vec{r},t)+2\nabla_{v}/m\beta)f$. The
term $m\vec{a}$ is the mean force acting on each particle.

In the overdamped regime, we could approximately replace $\vec{S}_{j}$ with
$\vec{v}_{j}$ in Hamiltonian Eq. (1). The accuracy of this approximation is
enhanced when the symmetry $SO(3)$ spontaneously breaks in the spin part,
leading to accordingly phase transition in velocities, with
$\vec{V}(\vec{r},t)$ being the order parameter. In the mean-field
approximation, the energy reads

$H_{ov}=E_{0}[N]+\sum_{j=1}^{N}[\frac{mv_{j}^{2}}{2}-\frac{J}{2u^{2}}N^{\prime}\vec{V}\cdot\vec{v}_{j}],$
(8)

with the constraint $|\vec{v}_{j}|=u$. Here the last term results from the
Heisenberg interaction, and $E_{0}[N]$ only depends on the particle numbers.
To apply Eq. (7) to QVM, we need to determine the free energy by calculating
the partition function while taking into account that nearly all the
$\vec{v}_{j}$s are aligned with $\vec{V}$. In the high temperature limit,
namely, when the average velocity of particles is much smaller than that of
the thermal motion, we expand the free energy density to the second order of
$\beta mV^{2}$ as

$F(\vec{r})=\rho(-\lambda|\vec{V}|^{2}+\eta|\vec{V}|^{4}+F_{0}),$ (9)

where $\lambda=3J^{2}N^{\prime 2}/8\tilde{m}u^{4}$, $\eta=2\lambda^{2}\beta/9$
and $F_{0}=(3\ln{\tilde{m}\beta}-\ln{2\pi}+2\ln\rho+2\kappa u^{2})/2\beta$.
Here, $\kappa$ is the Lagrange multiplier resulting from the constraint on
velocities, and the renormalized mass is defined as $\tilde{m}\equiv
m+2\kappa/\beta$. For such $\phi^{4}$ field, the symmetry spontaneously breaks
at $|\vec{V}_{0}|^{2}=9/(4\lambda\beta)$, which is just the order parameter
[25]. Consequently, the mean force acting on each particle is given by
$m\vec{a}=-\nabla_{r}(F/\rho)$.

We then obtain the evolution of $\vec{V}(\vec{r},t)$ straightforwardly [19,
20, 21, 22], namely,

$\displaystyle
m\left(\partial_{t}+\lambda_{1}\vec{V}\cdot\nabla_{r}\right)\vec{V}=\tilde{\lambda}\vec{V}-\tilde{\eta}\lvert\vec{V}\rvert^{2}\vec{V}-\frac{1}{\rho}\nabla_{r}\cdot\hat{P}+\vec{R}_{v}.$
(10)

Here, we have considered that, when $t$ is large, the directions of velocity
and position are approximately the same, i.e., $\vec{V}=\xi\vec{r}$. Here,
$\tilde{\lambda}\equiv 2\xi\lambda$, $\lambda_{1}\equiv 1-2\lambda$ and
$\tilde{\eta}=4\xi\eta$.
$\hat{P}_{ab}=(\rho/\beta)\delta_{ab}-\rho(\gamma\beta)^{-1}(\partial_{ra}V_{b}+\partial_{rb}V_{a})$
are the stress tensor; $\partial_{ra}\equiv\partial/\partial r_{a}$ with $a,b$
are Cartesian indices, $\vec{R}_{v}(\vec{r},t)$ is a random force subjected to
the FDT, i.e., $\langle
R_{va}(\vec{r},t)R_{vb}(\vec{r}^{\prime},t^{\prime})\rangle_{R}=2\gamma
m\rho^{-1}[V_{a}V_{b}+\hat{I}_{ab}(m\beta)^{-1}]\delta_{r_{c}}(\vec{r}-\vec{r}^{\prime})\delta(t-t^{\prime})$.
Here $\delta_{r_{c}}(\vec{r}-\vec{r}^{\prime})$ is a Dirac-$\delta$ like
function implying the existence of the cutoff in the spatial freedom. The
detailed discussion about the derivation of hydrodynamic equations is
illustrated in the Supplementary Materials (SM). A complete hydrodynamic
framework is thus formed by Eq. (10) together with the evolution of $\rho$,
which follows the conservation law of particle number, i.e.,
$\partial_{t}\rho=-\nabla_{r}\cdot(\rho\vec{V})$.

Flocking phase transition.—In order to analyze the symmetry broken phase, we
shift to a comoving coordinate frame moving with $\vec{V}_{0}$,
$\vec{V}^{\prime}=(\vec{V}_{\perp},V_{\parallel}\hat{e}_{\parallel})$ with
$\hat{e}_{\parallel}=\vec{V}_{0}/\lvert\vec{V}_{0}\rvert$. Here,
$\vec{V}_{\perp}(\vec{r},t)$ is the Goldstone mode dominating the critical
behavior of this system flocking. Its hydrodynamic equation then becomes

$\displaystyle\left(\frac{\partial}{\partial
t}+\lambda_{1}\vec{V}_{\perp}\cdot\vec{\partial}_{\perp}\right)\vec{V}_{\perp}$
$\displaystyle=-B\vec{\partial}_{\perp}\delta\rho+D\vec{\partial}_{\perp}\vec{\partial}_{\perp}\cdot\vec{V}_{\perp}+D\nabla_{r}^{2}\vec{V}_{\perp}+\vec{R}_{\perp},$
(11)

where
$\partial_{t}\delta\rho=-A_{I}\vec{\partial}_{\perp}\cdot\vec{V}_{\perp}$,
$\vec{\partial}_{\perp}$ denoting the directional derivative along
$\vec{V}_{\perp}$, $\delta\rho=\rho-\rho_{0}$ with $\rho_{0}$ being the mean
of the local number density and $B\equiv(\rho_{0}\beta)^{-1}$. The diffusion
coefficient $D\equiv(\gamma m\beta)^{-1}$ is also subjected to the FDT.
Notably, $\vec{R}_{\perp}$ is the transformed random force with the covariance
$\langle R_{\perp a}(\vec{r},t)R_{\perp
b}(\vec{r}^{\prime},t^{\prime})\rangle_{R}=2\Delta\delta_{ab}\delta_{r_{c}}(\vec{r}-\vec{r}^{\prime})\delta_{\tau_{c}}(t-t^{\prime})$
with the strength $\Delta\equiv\gamma(\rho\beta)^{-1}$. It is also emphasized
that $A_{I}$ is the renormalized coefficient including the impact of the
higher order (as well as nonlinear) term
$\vec{\partial}_{\perp}(\delta\rho\vec{V}_{\perp})$. This simplification
actually decouples the different wavelength components of $\delta\rho$ and
$\vec{V}_{\perp}$. To find the critical scaling exponents, we next implement
the dynamic renormalization group (DRG) method [26, 27, 28] to Eq. (Quantum
Vicsek Model for Active Matter) (See SM for detailed discussion). The DRG
procedure consists of two steps, i.e., eliminate the short wavelength
components of $\vec{V}_{\perp}$ and then apply scale transformation. Thanks to
the simplification mentioned above, Eq. (Quantum Vicsek Model for Active
Matter) becomes

$\displaystyle[-i\omega+D(k^{2}+\vec{k}_{\perp}\vec{k}_{\perp})+i\frac{A_{I}B}{\omega}\vec{k}_{\perp}\vec{k}_{\perp}]\cdot\vec{V}(\vec{k},\omega)=\vec{R}(\vec{k},\omega)-i\lambda_{1}\int_{q,\Omega}\vec{V}(\vec{k}-\vec{q},\omega-\Omega)\cdot\vec{q}_{\perp}\vec{V}(\vec{q},\Omega),$
(12)

where $\int_{q,\Omega}\equiv(2\pi)^{-4}\int
d^{3}\vec{q}\int_{-\infty}^{+\infty}d\Omega$.Here $\vec{V}(\vec{k},\omega)$
and $\vec{R}(\vec{k},\omega)$ being the Fourier transformation of
$\vec{V}_{\perp}(\vec{r},t)$ and $\vec{R}_{\perp}(\vec{r},t)$. The
corresponding covariance of $\vec{R}(\vec{k},\omega)$ is
$\langle\vec{R}(\vec{k},\omega)\vec{R}(\vec{k}^{\prime},\omega^{\prime})\rangle_{R}=(2\pi)^{4}\Delta\delta(\omega-\omega^{\prime})\delta(\vec{k}-\vec{k}^{\prime})$
with $\Delta=\Delta(1-e^{-k^{2}/\Lambda^{2}})$. Here, $\Lambda$ is the
ultraviolet cutoff of $k$ for the above Dirac-$\delta$ like function. In
general, the nonlinear term
$\lambda_{1}\vec{V}_{\perp}\cdot\vec{\partial}_{\perp}\vec{V}_{\perp}$
redefines the parameters of the reduced hydrodynamic equations containing only
the long wavelength part. As a result, keeping up to
$\mathcal{O}(\lambda_{1}^{2})$, the nonlinear coupling strength $\lambda_{1}$
remains unchanged, $\lambda_{I}=\lambda_{1}$, and $A_{I}B$, as a whole, does
not have correction either. In fact, one can conclude that $B$, which measures
the response of velocity field to changes in density, remains unchanged in the
new reduced hydrodynamic equation, i.e., $B_{I}=B$. Besides, the diffusion
coefficient $D$ and the noise strength $\Delta$ are renormalized in the same
way, as required by FDT, namely,
$D_{I}=D[1+\bar{\lambda}_{1}^{2}F(s;\Lambda,A_{I}B/D)]$ and
$\Delta_{I}=\Delta[1+\bar{\lambda}_{1}^{2}F(s;\Lambda,A_{I}B/D)]$ with
$\bar{\lambda}_{1}^{2}\equiv\lambda_{1}^{2}\Delta D^{-3}$. Here
$F(s;\Lambda,A_{I}B/D)$ is a polynomial function of $s$, which, apart from a
constant, contains only odd powers terms, with the lowest order being $s^{1}$.
$s$ is a rescaling factor of space required in step 2. Namely, we rescale
lengths, time, $\vec{V}_{\perp}$ and $\delta\rho$ according to
$\vec{r}\rightarrow s\vec{r}$, $t\rightarrow s^{z}t$,
$\vec{V}_{\perp}\rightarrow s^{\chi}\vec{V}_{\perp}$ and
$\delta\rho\rightarrow s^{\chi}_{\rho}\delta\rho$ respectively.

To further determine the scaling exponents $z$ and $\chi$, we then take an
infinitesimal transformation, i.e., $s=1+l$ ($l\rightarrow 0$), which leads to
differential recursion relations for the parameters

$\displaystyle dD(l)/dl=D[z-2+F_{31}\bar{\lambda}_{1}^{2}],$ (13)
$\displaystyle d\Delta(l)/dl=\Delta[z-2\chi-3+F_{31}\bar{\lambda}_{1}^{2}],$
$\displaystyle d\lambda_{1}/dl=\lambda_{1}[z+\chi-1],$ $\displaystyle
dB(l)/dl=B[\chi_{\rho}-1+z-\chi].$

The scaling factors $z$, $\chi$ and $\chi_{\rho}$ are chosen to keep $D$,
$\Delta$, $\lambda_{1}$ and $B$ fixed at their initial value, so that there
exist the steady solutions $z=3/2$, $\chi=-1/2$ and $\chi_{\rho}=-1$. The
negative $\chi$ just implies a true long-range order exists. Therefore, as an
illustration, we successfully show that our QVM will emerge flocking, i.e., a
stable order phase, in three dimensions and obtain the corresponding critical
exponents through the hydrodynamic method. Apparently, the hydrodynamic
equation derived from the QVM, as specified in Eq. (10), is much similar to
the $3D$ TT equation, while our model provides explicit formulations for the
coefficients. Therefore, it is convinced that the TT theory indeed can be
regarded as the continuation of the Vicsek model, namely, as they may have the
same quantum origin.

Conclusions.—In conclusion, we have proposed a quantum analog of the Vicsek
model (QVM), offering a novel perspective on the study of active matter. The
original Vicsek model is recovered in the classical limit of the QVM, and the
Toner-Tu-like equations, i.e., hydrodynamic equations, naturally emerge from
its continuous formulation. This study provides insight into why the Toner-Tu
equation may belong to the same universality class as the Vicsek model.
Additionally, we have offered a more microscopic description of emergent
flocking phenomena at the quantum level.

By directly examining the critical behaviors of this quantum model in Eq. (1),
it becomes straightforward to analyze theoretical exponents in the
$2-$dimensional case, which merits further exploration. This work represents a
significant step towards unifying classical and quantum theories of active
matter and opens new avenues for exploring the underlying mechanisms of
flocking order and correlations at the quantum level.

###### Acknowledgements.

This study is supported by the National Natural Science Foundation of China
(NSFC) (Grant No. 12088101) and NSAF (Grant No. U2230402).

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