# Construction of non-Abelian electric strings

Tanmay Vachaspati  ∗Physics Department, Arizona State University, Tempe,
Arizona 85287, USA.

###### Abstract

We detail the construction of electric string solutions in $SU(2)$ Yang-Mills-
Higgs theory with a scalar in the fundamental representation and discuss the
properties of the solution. We show that Schwinger gluon pair production in
the electric string background is absent. A similar construction in other
models, such as with an adjoint scalar field and the electroweak model, does
not yield solutions.

##  I Introduction

A homogeneous electric field in Maxwell electrodynamics corresponds to the
familiar gauge potential

$A^{\mu}=(-Ez,0,0,0).$ (1)

where $E$ is the electric field strength. When coupled to external charges,
the electric field is known to decay by Schwinger pair production Schwinger
(1951). Similarly, an $SU(2)$ non-Abelian electric field111We will explicitly
only consider $SU(2)$ non-Abelian gauge theory. The solutions may be embedded
in theories with larger gauge groups., can be derived from the gauge potential

$A^{\mu a}=(-Ez,0,0,0)\delta^{a3}$ (2)

where $a=1,2,3$ is the group index. Unlike in the Maxwell case, it is not
necessary to introduce “external” charges as even the pure non-Abelian gauge
theory includes charged quanta (“gluons”). Schwinger pair production of gluons
will cause the non-Abelian electric field to decay rapidly Matinyan and
Savvidy (1978); Brown and Weisberger (1979); Yildiz and Cox (1980); Ambjorn
and Hughes (1982a, b); Nayak and van Nieuwenhuizen (2005); Cooper and Nayak
(2006); Cooper et al. (2008); Nair and Yelnikov (2010); Kim and Schubert
(2011); Ilderton (2022); Huet et al. (2014); Ragsdale and Singleton (2017);
Karabali et al. (2019); Cardona and Vachaspati (2021). However the story for
non-Abelian electric fields is more subtle, as embedding the Maxwell gauge
potential into the non-Abelian theory is not the only way to obtain a non-
Abelian electric field. As shown by Brown and Weisberger (BW) Brown and
Weisberger (1979), and described in Sec. II, there is a one parameter set of
gauge inequivalent gauge fields that all lead to the same homogeneous electric
field. An analysis of Schwinger pair production in such gauge field
backgrounds shows that Schwinger gluon production is absent Vachaspati (2022).

An issue with BW gauge fields is that, unlike Eq. 2, they are not classical
solutions of the vacuum equations of non-Abelian gauge theory; instead they
require sources. A possibility is that quantum backreaction on the classical
dynamics effectively provides such sources but this is difficult to show. A
second possibility, one that has been reported in Ref. Vachaspati (2023) and
that we will detail in this paper, is that suitable sources can be provided by
an external classical field, such as a scalar field. Then the BW gauge fields
are solutions of the Yang-Mills-Higgs classical equations of motion, much like
other classical solutions such as strings and magnetic monopoles Vilenkin and
Shellard (2000), but a key difference is that the solution contains a flux of
electric field instead of a magnetic field. Such solutions are called
“electric strings”.

The electric string solution presented in Ref. Vachaspati (2023) was arrived
at by using a certain amount of guesswork. In the present paper, we present
some rationale for the guesses, in addition to exploring certain other issues.
The starting point for our discussions is to consider a homogeneous BW
electric field, discussed in Sec. II. In Sec. III we show that a homogeneous
BW electric field can indeed be sourced by a scalar field that transforms in
the fundamental representation of $SU(2)$. The necessary scalar field
configuration also solves its own equation of motion but only for certain
parameters.

Encouraged by the case of the homogeneous electric field, we turn our
attention to “electric string” solutions in which the electric field is
localized to a tubular region. In Sec. IV we construct such a solution. We
find that the tubular electric field is wrapped by magnetic fields in an
oscillating pattern with a slow asymptotic fall off.

The question of Schwinger gluon production in the background of an electric
string is considered in Sec. V and, as in the homogeneous electric field case,
this process is absent. A classical stability analysis of the electric string
solution is postponed for future work.

The existence of an electric string solution in a Yang-Mills with a
fundamental Higgs naturally raises the question if such strings can arise in
other theories. In Appendix A we examine the case when the scalar field is in
the adjoint representation of $SU(2)$ and show that the solution does not
exist. Similarly in Appendix B we examine if the solution can be found in the
electroweak model that has an additional $U(1)$ hypercharge gauge field, and
there too we find that a solution does not exist. These no-go results though
are based on certain assumptions about the structure of the solutions and it
is possible that a more general analysis might successfully find solutions.

Our conclusions are summarized in Sec. VI and some helpful formulae are listed
in Appendix C.

##  II Gauge fields

###  II.1 BW homogeneous gauge fields

The non-vanishing $SU(2)$ gauge fields that give a homogeneous electric field
are Brown and Weisberger (1979),

$W_{\mu}^{1}=\frac{\Omega}{g}\partial_{\mu}t,\ \
W_{\mu}^{2}=-\frac{E}{\Omega}\partial_{\mu}z,\ \ W_{\mu}^{3}=0.$ (3)

where $\Omega$ is a parameter. We will assume without loss of generality that
$\Omega>0$. The field strength is found from

$W_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}+g\epsilon^{abc}W_{\mu}^{b}W_{\nu}^{c}$
(4)

and the non-vanishing field strength is,

$W_{\mu\nu}^{3}=-E(\partial_{\mu}t\,\partial_{\nu}z-\partial_{\nu}t\,\partial_{\mu}z)$
(5)

and $W_{\mu\nu}^{1}=0=W_{\mu\nu}^{2}$.

The currents are found from the classical equations of motion for the gauge
fields,

$j^{\mu a}=\partial_{\nu}W^{\mu\nu a}+g\epsilon^{abc}W_{\nu}^{b}W^{\mu\nu c},$
(6)

and (in the BW gauge) are given by

$j^{\mu 1}=-g\frac{E^{2}}{\Omega}\partial^{\mu}t,\ \ j^{\mu 2}=-\Omega
E\,\partial^{\mu}z,\ \ j^{\mu 3}=0.$ (7)

###  II.2 Temporal gauge

To bring the BW gauge fields to temporal gauge we perform the gauge
transformation,

$U=e^{i\sigma^{2}\pi/4}e^{i\sigma^{1}\pi/4}e^{-i\sigma^{1}\Omega t/2}$ (8)

where $\sigma^{a}$ are the Pauli spin matrices. Then,

$W_{\mu}\to
W_{\mu}^{\prime}=UW_{\mu}U^{\dagger}+\frac{2i}{g}U\partial_{\mu}U^{\dagger}$
(9)

where $W_{\mu}\equiv W_{\mu}^{a}\sigma^{a}$. This gives the gauge fields for a
homogeneous electric field in the form,

$W^{\pm}_{\mu}=-\frac{\epsilon}{g}e^{\pm i\Omega t}\,\partial_{\mu}z,\ \
W^{3}_{\mu}=0,$ (10)

where

$\epsilon\equiv gE/\Omega.$ (11)

Next we also include a string profile function, $f(r)$, since we eventually
want to discuss electric string configurations. Then the gauge fields we will
consider are

$W^{\pm}_{\mu}=-\frac{\epsilon}{g}e^{\pm i\Omega t}f(r)\,\partial_{\mu}z,\ \
W^{3}_{\mu}=0,$ (12)

where $W^{\pm}_{\mu}\equiv W^{1}_{\mu}\pm iW^{2}_{\mu}$, $r$ refers to the
cylindrical radial coordinate. Alternately we can write,

$\displaystyle W^{1}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\cos(\Omega t)f(r)\,\partial_{\mu}z,$ (13)
$\displaystyle W^{2}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\sin(\Omega t)f(r)\,\partial_{\mu}z.$ (14)

The field strengths are

$\displaystyle W^{1}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\bigl{[}-\Omega\sin(\Omega
t)f(r)(\partial_{\mu}t\,\partial_{\nu}z-\partial_{\nu}t\,\partial_{\mu}z)$
(15) $\displaystyle\hskip 14.22636pt+\cos(\Omega
t)f^{\prime}(r)(\partial_{\mu}r\,\partial_{\nu}z-\partial_{\nu}r\,\partial_{\mu}z)\bigr{]}$
$\displaystyle W^{2}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\bigl{[}\Omega\cos(\Omega
t)f(r)(\partial_{\mu}t\,\partial_{\nu}z-\partial_{\nu}t\,\partial_{\mu}z)$
(16) $\displaystyle\hskip 14.22636pt+\sin(\Omega
t)f^{\prime}(r)(\partial_{\mu}r\,\partial_{\nu}z-\partial_{\nu}r\,\partial_{\mu}z)\bigr{]}$
$\displaystyle W^{3}_{\mu\nu}$ $\displaystyle=$ $\displaystyle 0$ (17)

where prime denotes differentiation with respect to the argument. Note that
the electric field is accompanied by a magnetic field in the azimuthal
direction. To decide if the field configuration is electric or magnetic we
will calculate the gauge and Lorentz invariant Lagrangian density,

${\cal L}_{g}=-\frac{1}{4}W_{\mu\nu}^{a}W^{\mu\nu
a}=\frac{\epsilon^{2}}{2g^{2}}(\Omega^{2}f^{2}-f^{\prime 2})$ (18)

Positive ${\cal L}_{g}$ implies that the field is electric-like while negative
values imply a magnetic-like field.

The $SU(2)$ currents are obtained from (6),

$\displaystyle j_{\mu}^{1}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\cos(\Omega
t)\left[f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}f\right]\partial_{\mu}z$
(19) $\displaystyle j_{\mu}^{2}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{g}\sin(\Omega
t)\left[f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}f\right]\partial_{\mu}z$
(20) $\displaystyle j_{\mu}^{3}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon^{2}}{g}\Omega f^{2}\partial_{\mu}t$ (21)

Note that the $a=1,2$ equations may also be written as,

$j_{\mu}^{a}=\frac{1}{f}\left[f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}f\right]W_{\mu}^{a},\
\ (a=1,2).$ (22)

We are interested in finding if these currents can be sourced by scalar
fields. We will first consider the simpler case of a homogeneous electric
field ($f(r)=1$) and show that it can be sourced by a scalar field in the
fundamental representation. The case of a scalar field in the adjoint
representation is discussed in Appendix A with the conclusion that it cannot
source the electric field.

##  III Homogeneous electric field

In this section $f(r)=1$.

###  III.1 Fundamental scalar

The model now is similar to the electroweak model where the symmetry is
$SU(2)\times U(1)$, except that the $U(1)$ charge, commonly denoted by
$g^{\prime}$, is set to zero. In other words, the $SU(2)$ is a gauged symmetry
while the $U(1)$ is a global symmetry.

We denote the fundamental scalar field by $\Phi$. The Lagrangian for the model
is,

$L=-\frac{1}{4}W_{\mu\nu}^{a}W^{\mu\nu a}+|D_{\mu}\Phi|^{2}-V(\Phi)$ (23)

where

$D_{\mu}\Phi=\partial_{\mu}\Phi-i\frac{g}{2}W_{\mu}^{a}\sigma^{a}\Phi$ (24)
$V(\Phi)=m^{2}|\Phi|^{2}+\lambda|\Phi|^{4}$ (25)

where $m^{2}$ may be negative or positive but $\lambda>0$.

The gauge field equations of motion are

$D_{\nu}W^{\mu\nu
a}=j_{\mu}^{a}=i\frac{g}{2}\left[\Phi^{\dagger}\sigma^{a}D_{\mu}\Phi-h.c.\right],$
(26)

where $h.c.$ stands for Hermitian conjugate.

Using $\sigma^{a}\sigma^{b}=\delta^{ab}+i\epsilon^{abc}\sigma^{c}$ we find

$\displaystyle\hskip-28.45274pt\Phi^{\dagger}\sigma^{a}D_{\mu}\Phi=$
$\displaystyle\Phi^{\dagger}\sigma^{a}\partial_{\mu}\Phi-\frac{ig}{2}|\Phi|^{2}W_{\mu}^{a}+\frac{g}{2}|\Phi|^{2}({\vec{W}}_{\mu}\times{\hat{n}})^{a}$
(27)

where the unit vector ${\hat{n}}^{a}$ is given by

${\hat{n}}^{a}\equiv\frac{\Phi^{\dagger}\sigma^{a}\Phi}{\Phi^{\dagger}\Phi}.$
(28)

Inserting (27) in the expression for the current, we get

$j_{\mu}^{a}=i\frac{g}{2}\left[\Phi^{\dagger}\sigma^{a}\partial_{\mu}\Phi-h.c.\right]+\frac{g^{2}}{2}|\Phi|^{2}W_{\mu}^{a}.$
(29)

We will start by solving (29) for $\Phi$ with $j_{\mu}^{a}$ given in (19)-(21)
with $f=1$. We can write $\Phi$ in the Hopf parametrization,

$\Phi=\eta\begin{pmatrix}\cos\alpha\,e^{i\beta}\\\
\sin\alpha\,e^{i\gamma}\end{pmatrix}.$ (30)

We will assume that $\Phi$ is homogeneous, so $\alpha$, $\beta$ and $\gamma$
are only functions of time, and $\eta$ is a constant.

The $\mu=3$ component of (29) is non-trivial only for $a=1,2$, giving

$j_{z}^{a}=\frac{g^{2}}{2}|\Phi|^{2}W_{z}^{a},\ \ (a=1,2).$ (31)

Comparison with (22) gives,

$\Omega^{2}=\frac{1}{2}g^{2}\eta^{2},\ \ {\rm or},\ \
\eta=\frac{\sqrt{2}\,\Omega}{g}$ (32)

(Recall that we are considering $f=1$.) Next we turn to the $\mu=0$
components. Some algebra (see Appendix C) leads to

$\displaystyle
2{\dot{\alpha}}\sin(\gamma-\beta)+\sin(2\alpha)\cos(\gamma-\beta)({\dot{\beta}}+{\dot{\gamma}})$
$\displaystyle=$ $\displaystyle 0$ (33) $\displaystyle
2{\dot{\alpha}}\cos(\gamma-\beta)-\sin(2\alpha)\sin(\gamma-\beta)({\dot{\beta}}+{\dot{\gamma}})$
$\displaystyle=$ $\displaystyle 0$ (34) $\displaystyle
g\eta^{2}(\cos^{2}\alpha\,{\dot{\beta}}-\sin^{2}\alpha\,{\dot{\gamma}})-\frac{\epsilon^{2}\Omega}{g}$
$\displaystyle=$ $\displaystyle 0$ (35)

The solution to these three equations leads to

$\Phi=\eta\begin{pmatrix}z_{1}e^{+i\omega t}\\\ z_{2}e^{-i\omega
t}\end{pmatrix}$ (36)

where $z_{1},\,z_{2}\in\mathbb{C}$ are constants with
$|z_{1}|^{2}+|z_{2}|^{2}=1$, and

$\omega=\frac{\epsilon^{2}}{2\Omega}$ (37)

Now we have to make sure that $\Phi$ solves its own equation of motion,

$D_{\mu}D^{\mu}\Phi+V^{\prime}(\Phi)=0$ (38)

where the prime denotes derivative with respect to $\Phi^{\dagger}$. For $V$
in (25) and $\Phi$ in (36), we can write

$V^{\prime}(\Phi)=(m^{2}+2\lambda\eta^{2})\Phi.$ (39)

We also evaluate

$D_{\mu}D^{\mu}\Phi=-\left(\omega^{2}-\frac{\epsilon^{2}}{4}\right)\Phi$ (40)

Therefore (38) leads to the equation,

$-\omega^{2}+\frac{\epsilon^{2}}{4}+m^{2}+2\lambda\eta^{2}=0.$ (41)

Consistency with (37) implies

$\omega^{2}-\frac{\omega\Omega}{2}-m^{2}-2\lambda\eta^{2}=0.$ (42)

Therefore

$\omega=\frac{1}{2}\left[\frac{\Omega}{2}\pm\sqrt{\frac{\Omega^{2}}{4}+4\left(m^{2}+\frac{4\lambda}{g^{2}}\Omega^{2}\right)}\,\right].$
(43)

Note that (37) implies that $\omega/\Omega>0$. Depending on the signs and
magnitude of the parameters $m^{2}$, $\lambda/g^{2}$ and $\Omega^{2}$, only
one or both roots in (43) are valid.

If we choose potential parameters such that $m^{2}+2\lambda\eta^{2}=0$, we
obtain simpler expressions. With this choice of parameters Eq. (39) gives
$V^{\prime}(\Phi)=0$, i.e. $\Phi$ is at an extrema of its potential. Then (43)
gives two solutions: $\omega=0$ or $\omega=\Omega/2$. The solution with
$\omega=0$ is trivial for then $\epsilon=0$ because of (37) and the electric
field vanishes. For the non-trivial solution $\omega=\Omega/2$, (37) gives

$\epsilon=\Omega,\ \ (V^{\prime}=0).$ (44)

To summarize these results: we have found a solution of the classical
equations of motion that corresponds to a homogeneous electric field,

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\,\Omega}{g}\begin{pmatrix}z_{1}e^{+i\omega t}\\\
z_{2}e^{-i\omega t}\end{pmatrix}$ (45) $\displaystyle{\vec{W}}_{\mu}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{2\omega\Omega}}{g}\left(\cos(\Omega t),\sin(\Omega
t),0\right)\partial_{\mu}z,$ (46)

where $z_{1},\,z_{2}\in\mathbb{C}$ are constants with
$|z_{1}|^{2}+|z_{2}|^{2}=1$ and $\omega$ is given in terms of $\Omega$ and the
parameters in the scalar potential by (43). The electric field is found from
(11) and (37),

$E=\frac{\Omega\sqrt{2\omega\Omega}}{g}$ (47)

##  IV Electric string ($f(r)\neq 1$)

We now move on from the homogeneous electric field to electric string
solutions. The gauge fields are given by (12), the required currents by
(19)-(21), and the currents that $\Phi$ can source by (29). Hence we must now
find $\Phi$ by solving,

$\displaystyle\hskip-14.22636pt\frac{i}{2}g(\Phi^{\dagger}\sigma^{1}D_{\mu}\Phi-(D_{\mu}\Phi)^{\dagger}\sigma^{1}\Phi)$
$\displaystyle\hskip 28.45274pt=-\frac{\epsilon}{g}\cos(\Omega
t)\left(f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}f\right)\partial_{\mu}z$
(48)
$\displaystyle\hskip-14.22636pt\frac{i}{2}g(\Phi^{\dagger}\sigma^{2}D_{\mu}\Phi-(D_{\mu}\Phi)^{\dagger}\sigma^{2}\Phi)$
$\displaystyle\hskip 28.45274pt=-\frac{\epsilon}{g}\sin(\Omega
t)\left(f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}f\right)\partial_{\mu}z$
(49)
$\displaystyle\hskip-14.22636pt\frac{i}{2}g(\Phi^{\dagger}\sigma^{3}D_{\mu}\Phi-(D_{\mu}\Phi)^{\dagger}\sigma^{3}\Phi)=-\frac{\epsilon^{2}}{g}\Omega
f^{2}\,\partial_{\mu}t$ (50)

Guided by the homogeneous electric field case of Sec. III we try,

$\Phi=\eta h(r)\begin{pmatrix}z_{1}e^{+i\omega t}\\\ z_{2}e^{-i\omega
t}\end{pmatrix}$ (51)

where $h(r)$ is a real profile function that is to be determined.

We first consider $\mu=0$ in (48)-(50). We find that (48) and (49) are
trivially satisfied while Eq. (50) gives

$-g\eta^{2}h^{2}\omega=-\frac{\epsilon^{2}}{g}\Omega f^{2}$ (52)

which implies

$h(r)=\frac{\epsilon}{g\eta}\sqrt{\frac{\Omega}{\omega}}\,f(r)$ (53)

and we should have $\omega/\Omega>0$ since $h$ is real.

The $\mu=1,2$ equations are trivially satisfied, so we now consider $\mu=3$,
i.e. the $\mu=z$ equations. Eqs. (48) and (49) then give

$f^{\prime\prime}+\frac{f^{\prime}}{r}+\left(\Omega^{2}-\frac{g^{2}\eta^{2}}{2}h^{2}\right)f=0$
(54)

and (50) is trivially satisfied.

Inserting (53) in (54) gives

$f^{\prime\prime}+\frac{f^{\prime}}{r}+\Omega^{2}\left(1-\frac{\epsilon^{2}}{2\omega\Omega}f^{2}\right)f=0.$
(55)

Next we consider the $\Phi$ equation of motion in (38) together with (53) to
get

$f^{\prime\prime}+\frac{f^{\prime}}{r}+\left[(\omega^{2}-m^{2})-\left(\frac{\epsilon^{2}}{4}+\frac{2\lambda\epsilon^{2}\Omega}{g^{2}\omega}\right)f^{2}\right]f=0.$
(56)

Consistency with (54) requires

$\displaystyle\Omega^{2}$ $\displaystyle=$ $\displaystyle\omega^{2}-m^{2}$
(57) $\displaystyle\frac{\Omega}{2\omega}$ $\displaystyle=$
$\displaystyle\frac{1}{4}+\frac{2\lambda\Omega}{g^{2}\omega}.$ (58)

Eq. (58) can be written as

$\omega=2\left(1-\frac{4\lambda}{g^{2}}\right)\Omega$ (59)

and since we have already assumed $\omega/\Omega>0$ (see (53)) we should
restrict to $\lambda\leq g^{2}/4$. Also since the potential $V$ should be
bounded from below, we have $0\leq\lambda\leq g^{2}/4$.

For fixed parameters $m^{2}$, $\lambda$, Eqs. (57) and (58) can be solved to
obtain $\Omega$ and $\omega$,

$\displaystyle\Omega^{2}=\frac{m^{2}}{4(1-4\lambda/g^{2})^{2}-1}$ (60)
$\displaystyle\omega^{2}=\frac{4(1-4\lambda/g^{2})^{2}m^{2}}{4(1-4\lambda/g^{2})^{2}-1}$
(61)

This is a valid solution provided

$\frac{m^{2}}{4(1-4\lambda/g^{2})^{2}-1}>0$ (62)

which gives the conditions $0<\lambda<g^{2}/8$ or $\lambda>3g^{2}/8$ if
$m^{2}>0$, and $g^{2}/8<\lambda<3g^{2}/8$ if $m^{2}<0$. Taking into account
the tighter restriction discussed below (59) that $\omega>0$ for $\Omega>0$,
the range of $\lambda$ for $m^{2}<0$ is $g^{2}/8<\lambda<g^{2}/4$.

Figure 1: Constraints on the parameters in the $m^{2}$-$8\lambda/g^{2}$ plane.
The unshaded regions give imaginary $\Omega$ and are not allowed. The solution
is only valid in the regions of parameter space where $\omega>0$.

Hence there is a range of values of the parameters $m^{2}$ and $\lambda$ of
the potential for which (55) and (56) are identical, provided (60) and (61)
hold. In this case, (55) can be solved numerically. It is preferable to
rescale variables,

$R=\Omega r,\ \ F=\frac{\epsilon}{\sqrt{2\omega\Omega}}f$ (63)

and then

$F^{\prime\prime}+\frac{F^{\prime}}{R}+(1-F^{2})F=0.$ (64)

Only the combination $\epsilon f(r)$ appears in the gauge field (see (12)),
and we are free to take $f(0)=1$ or,

$F(0)=\frac{\epsilon}{\sqrt{2\omega\Omega}}.$ (65)

Since the electric field strength is proportional to $F$, we would like to
choose boundary conditions such that $F(0)=F_{0}\neq 0$ and $F(\infty)\to 0$.
Different values of $F_{0}$ correspond to different external charges placed at
$z\to\pm\infty$ that produce the electric field. Smoothness at $R=0$ requires
$F^{\prime}(0)=0$.

There are no solutions that fall-off asymptotically for $F_{0}>1$. For
$F_{0}\ll 1$, the non-linear term with $F^{2}$ is sub-dominant and the
solutions are well approximated by a Bessel function of zero order,

$F(R)\approx F_{0}J_{0}(R).$ (66)

A plot of the numerically evaluated $F(R)$ is shown in Fig. 2. The asymptotic
behavior of $F(R)$ is therefore,

$F(R)\sim F_{0}\sqrt{\frac{2}{\pi R}}\cos\left(R-\frac{\pi}{4}\right).$ (67)

The energy density in all the fields is given by the expression,

${\cal
E}=\frac{1}{2}(W_{0i}^{a})^{2}+\frac{1}{4}(W_{ij}^{a})^{2}+|D_{t}\Phi|^{2}+|D_{i}\Phi|^{2}+V(\Phi)$
(68)

Inserting the expressions for the solution we get

$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}\Omega^{2}}{2g^{2}}f^{2}+\frac{\epsilon^{2}}{2g^{2}}f^{\prime
2}+\frac{\epsilon^{2}\omega\Omega}{g^{2}}f^{2}+\frac{\epsilon^{2}\Omega}{g^{2}\omega}f^{\prime
2}+\frac{\epsilon^{4}\Omega}{4g^{2}\omega}f^{4}$ (69)
$\displaystyle+m^{2}\frac{\epsilon^{2}\Omega}{g^{2}\omega}f^{2}+\lambda\frac{\epsilon^{4}\Omega^{2}}{g^{4}\omega^{2}}f^{4},$

or in terms of rescaled variables,

$\displaystyle{\cal E}^{\prime}\equiv\frac{g^{2}}{2\omega\Omega^{3}}{\cal E}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}+\frac{1}{\kappa}\right)F^{\prime
2}+\left(\frac{1}{2}+2\kappa-\frac{1}{\kappa}\right)F^{2}$ (70)
$\displaystyle+\frac{1}{2}\left(\frac{1}{2}+\frac{1}{\kappa}\right)F^{4}$

where (see (59)),

$\kappa\equiv\frac{\omega}{\Omega}=2\left(1-\frac{4\lambda}{g^{2}}\right).$
(71)

Due to the constraints on $\lambda/g^{2}$, we find $1<\kappa<2$ for $m^{2}>0$
and $0<\kappa<1$ for $m^{2}<0$. In Fig. 3 we show ${\cal E}^{\prime}$ for a
sample value of $\kappa$ and $F_{0}$.

Figure 2: $F(R)$ vs. $R$ for $F_{0}=0.5$. Figure 3: ${\cal E}^{\prime}$ vs.
$R$ for $F_{0}=0.5$ and $\kappa=1.5$.

The energy per unit length along the $z-$direction is defined to be the
tension of the string. Therefore the rescaled tension with a radial cutoff at
$R_{c}$ is,

$\displaystyle\mu(R_{c})$ $\displaystyle=$ $\displaystyle
2\pi\int_{0}^{R_{c}}dR\,R\,{\cal E}^{\prime}$ (72) $\displaystyle\approx$
$\displaystyle\left[\left(\frac{1}{2}+\frac{1}{\kappa}\right)0.54+\left(\frac{1}{2}+2\kappa-\frac{1}{\kappa}\right)0.54\right]R_{c}$
$\displaystyle+\frac{1}{2}\left(\frac{1}{2}+\frac{1}{\kappa}\right)0.09\,\ln(R_{c}).$

Therefore the tension diverges linearly with the radial cutoff. Thus while the
electric field is axisymmetric and concentrated along the central axis, it is
not as sharply localized as in a magnetic Nielsen-Olesen string Nielsen and
Olesen (1973), though it is more localized than the homogeneous configuration.
The situation is similar to that of a global string for which the energy
diverges logarithmically and to global monopoles for which the energy diverges
linearly Vilenkin and Shellard (2000).

For $m^{2}<0$, the potential $V(\Phi)$ has a minimum at $\Phi\neq 0$. However
the solution of (64) implies that $\Phi\to 0$ as $r\to\infty$. Therefore
$\Phi$ is not in its true vacuum asymptotically and there is non-vanishing
vacuum energy at spatial infinity. In this case the electric string is a
cylindrical “bubble” of the true vacuum (with non-vanishing $\Phi$) in a
background of the false vacuum phase with $\Phi=0$. This is different from the
case of the electric string with $m^{2}>0$ for then the true vacuum is at
$\Phi=0$ and the potential energy of $\Phi$ goes to zero in the asymptotic
region.

Finally we return to the question of whether the solution corresponds to an
electric string as azimuthal magnetic fields are also present (see (15)-(17)).
Hence we calculate ${\cal L}_{g}$ using (18) and plot the quantity
$F^{2}-(F^{\prime})^{2}$ in Fig. 4. The behavior at large $R$ can be seen from
the properties of the Bessel functions,

$F^{2}-(F^{\prime})^{2}\sim F_{0}^{2}[J_{0}^{2}(R)-J_{1}^{2}(R)]\to
F_{0}^{2}\frac{2\sin(2R)}{\pi R}$ (73)

where we have used $J_{1}(R)=-J_{0}^{\prime}(R)$ and (67). The gauge field
strength is electric-like where $F^{2}-(F^{\prime})^{2}>0$, otherwise it is
magnetic-like. This shows that the solution has alternating electric and
magnetic fields where the electric field is along the $z-$ direction and the
magnetic field is along the azimuthal direction. So a caricature of the
electric string configuration is a tube of electric field along the $z$
direction, wrapped by weak azimuthal magnetic fields, that are again contained
in a sheath of weaker electric field, ad infinitum (see the sketch in Fig. 5).

Figure 4: $F(R)^{2}-F^{\prime}(R)^{2}$ vs. $R$ for $F_{0}=0.5$. The field
strength is electric-like where $F(R)^{2}-F^{\prime}(R)^{2}$ is positive and
magnetic-like where $F(R)^{2}-F^{\prime}(R)^{2}$ is negative. Figure 5: A
sketch of the electric and magnetic fields in an electric string.

To summarize the main result of this section, we have found the electric
string solution,

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{g}\sqrt{\frac{\Omega}{\omega}}\,f(r)\begin{pmatrix}z_{1}e^{+i\omega
t}\\\ z_{2}e^{-i\omega t}\end{pmatrix}$ (74) $\displaystyle W^{\pm}_{\mu}$
$\displaystyle=$ $\displaystyle-\frac{\epsilon}{g}e^{\pm i\Omega
t}f(r)\,\partial_{\mu}z,\ \ W^{3}_{\mu}=0,$ (75)

where $|z_{1}|^{2}+|z_{2}|^{2}=1$, and $\Omega$ and $\omega$ are given by

$\displaystyle\Omega=\left[\frac{m^{2}}{4(1-4\lambda/g^{2})^{2}-1}\right]^{1/2}$
(76)
$\displaystyle\omega=\left[\frac{4(1-4\lambda/g^{2})^{2}m^{2}}{4(1-4\lambda/g^{2})^{2}-1}\right]^{1/2}$
(77)

and $m^{2}$ and $\lambda$ are parameters of the scalar potential (see (25)).
The solution is valid for $0<\lambda<g^{2}/8$ for $m^{2}>0$ and for
$g^{2}/8<\lambda<g^{2}/4$ for $m^{2}<0$. The profile function, $f(r)$, is
common to both the gauge field and the scalar field and satisfies (55) with
boundary conditions $f(0)=1$, $f^{\prime}(0)=0$. The solution for the profile
is closely approximated by the zeroth order Bessel function, $J_{0}(\Omega
r)$, up to a multiplicative constant that fixes the strength of the electric
field at the origin.

##  V Schwinger gluon production?

The gauge particles (“gluons”) are charged under $SU(2)$, as are excitations
of the scalar field $\Phi$, and there can be Schwinger pair production of both
kinds of excitations. Here we are interested in whether the electric field is
protected from pair production of gluons. Pair production of scalar quanta is
similar to pair production of quarks in QCD that results in string breaking.
The effect can be suppressed by considering large masses of the scalar quanta,
i.e. large positive values of the parameter $m^{2}$.

To see that the electric field is stable to Schwinger gluon production, we
perturb the gauge field,

$W_{\mu}^{\pm}=A_{\mu}^{\pm}+e^{\pm i\Omega t}Q_{\mu}^{\pm},\ \
W_{\mu}^{3}=A_{\mu}^{3}+Q_{\mu}^{3}$ (78)

where $A_{\mu}^{a}$ is the background (see Eq. (12))

$A^{\pm}_{\mu}=-\frac{\epsilon}{g}e^{\pm i\Omega t}f(r)\,\partial_{\mu}z,\ \
A^{3}_{\mu}=0$ (79)

and $Q_{\mu}^{a}$ are the perturbations. The scalar field is left unperturbed,

$\Phi=\eta h(r)\begin{pmatrix}z_{1}e^{+i\omega t}\\\ z_{2}e^{-i\omega
t}\end{pmatrix},$ (80)

since we are interested in gluon production and the mass of the $\Phi$ field
can be taken to be large.

As in Ref. Vachaspati (2022), the expressions in (78) are inserted in the
Lagrangian density for the gauge fields

${\cal L}_{g}=-\frac{1}{4}W_{\mu\nu}^{a}W^{\mu\nu a}$ (81)

to obtain the Lagrangian density for the perturbations $Q_{\mu}^{a}$. The
expressions are lengthy but the important point is that there is no explicit
time-dependence in the Lagrangian even though the background in (79) is time-
dependent. For example, the Lagrangian density to second order in the
perturbations is

$\displaystyle\hskip-21.33955pt{\cal L}_{g}^{(2)}=$ (82)
$\displaystyle\frac{1}{2}({\dot{Q}}^{(1)}_{i}-\Omega
Q^{(2)}_{i})^{2}+\frac{1}{2}({\dot{Q}}^{(2)}_{i}+\Omega
Q^{(1)}_{i})^{2}+\frac{1}{2}({\dot{Q}}^{(3)}_{i})^{2}$ $\displaystyle-$
$\displaystyle\frac{1}{4}(\partial_{i}Q^{(1)}_{j}-\partial_{j}Q^{(1)}_{i})^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{4}(\partial_{i}Q^{(2)}_{j}-\partial_{j}Q^{(2)}_{i}+\epsilon
f({\hat{z}}_{i}Q^{(3)}_{j}-{\hat{z}}_{j}Q^{(3)}_{i}))^{2}$ $\displaystyle-$
$\displaystyle\frac{1}{4}(\partial_{i}Q^{(3)}_{j}-\partial_{j}Q^{(3)}_{i}-\epsilon
f({\hat{z}}_{i}Q^{(2)}_{j}-{\hat{z}}_{j}Q^{(2)}_{i}))^{2}$ $\displaystyle+$
$\displaystyle\epsilon
f^{\prime}({\hat{z}}_{i}{\hat{r}}_{j}-{\hat{z}}_{j}{\hat{r}}_{i})Q^{(2)}_{i}Q^{(3)}_{j}$

where ${\hat{z}}_{i}$, ${\hat{r}}_{i}$ are unit vectors in the $z-$ and $r-$
directions, and the contraction of spatial indices is with the Kronecker
delta, e.g.
$({\dot{Q}}^{(3)}_{i})^{2}={\dot{Q}}^{(3)}_{i}{\dot{Q}}^{(3)}_{i}$. Similar
expressions are obtained at all orders in perturbations and there is no
explicit time-dependence in any of them. If we expand the perturbations in
modes in ${\cal L}_{g}^{(2)}$, the mode coefficients correspond to simple
harmonic oscillators with time-independent frequencies, which implies that
there is no particle production.

In the present analysis, since we include the scalar field $\Phi$, there is an
extra term in $Q_{\mu}^{a}$ coming from the covariant gradient term in the
Lagrangian,

$L_{\Phi}=|D_{\mu}\Phi|^{2}-V(\Phi)\to\ldots+\frac{g^{2}}{4}|\Phi|^{2}Q_{\mu}^{a}Q^{\mu
a}$ (83)

where the $\ldots$ include terms that are zeroth and first order in
$Q_{\mu}^{a}$. Since $|\Phi|^{2}=\eta^{2}h^{2}$ is independent of time, the
last term in (83) is simply a mass term for $Q_{\mu}^{a}$ with a mass that is
independent of time. Once again, the quantum state of the modes of
$Q_{\mu}^{a}$ will correspond to simple harmonic oscillators with time-
independent frequencies. Thus the time dependence of the background gauge
field does not lead to any Schwinger particle production of the gauge
excitations.

##  VI Conclusions

We have first constructed homogeneous electric field solutions in non-Abelian
gauge theories with a scalar field that transforms in the fundamental
representation. This construction paved the way for the construction of
electric string solutions that are summarized in Eqs. (74), (75), (76) and
(77). The solutions describe a flux tube of electric field, wrapped by
azimuthal magnetic fields, followed by a sheath of electric field, which is
again wrapped by azimuthal magnetic field, ad infinitum. The strength of the
electric and magnetic fields falls off with distance as $1/\sqrt{r}$. The slow
fall off implies a linear divergence in the energy per unit length of the
electric string for $m^{2}>0$. In the case where $m^{2}<0$, the electric
solution has $\Phi=0$ asymptotically while the true vacuum has $\Phi\neq 0$.
Hence the electric string for $m^{2}<0$ is like a cylindrical bubble solution
that contains gauge fields and $\Phi\neq 0$, that is immersed in a false
vacuum region with $\Phi=0$.

In Sec. V we have shown that Schwinger gluon production is absent in the
electric string background. This still leaves room for classical instabilities
of the electric string solution, especially since unstable modes are known to
exist in the homogeneous BW gauge field background in pure gauge theory
Pereira and Vachaspati (2022). With the $\Phi$ field included, the main
difference is that there is now an interaction term
$|\Phi|^{2}(W^{a}_{i})^{2}$ in the energy functional. Since $\Phi$ is non-zero
in the electric string solution, the gauge field excitations above the
background are massive. This should suppress instabilities but it is difficult
to say if the suppression is sufficient to eliminate the instabilities. We
plan to perform a classical stability analysis of the electric string solution
in future work.

There is a large body of work on the quantization of classical solutions
Rajaraman (1982) The procedure is to consider fluctuations around the
background solution. In our case, the gauge field with fluctuations can be
written as in Eq. (78) and similarly the scalar field is,

$\Phi=\Phi_{0}+\hat{\Phi}$ (84)

where $\Phi_{0}$ is the classical solution and $\hat{\Phi}$ represent
fluctuations. Assuming weak coupling and that there are no classical
instabilities, the fluctuations can be treated to lowest quadratic order in
the action and their eigenmodes are simple harmonic oscillators that can be
quantized in the standard way. The backreaction of these quantum fluctuations
on the classical background will be small. However, this straightforward
quantization does not hold at strong coupling. In that case, the action cannot
be truncated to quadratic order in the fluctuations and the backreaction may
change the classical solution in a significant way. Then lattice methods seem
to be the only recourse. It would be very interesting if strong coupling
effects could control the asymptotic behavior of the electric string so as to
give a finite string tension.

###### Acknowledgements.

I thank Jude Pereira for comments and for the drawing in Fig. 5. This work was
supported by the U.S. Department of Energy, Office of High Energy Physics,
under Award No. DE-SC0019470.

##  Appendix A Homogeneous electric field and adjoint scalar field

The $SU(2)$ gauge fields (see (13) and (14)) will be written as

${\vec{W}}_{\mu}=-\frac{\epsilon}{g}(\cos(\Omega t),\sin(\Omega
t),0)\,\partial_{\mu}z$ (85)

where the vector sign denotes a vector in internal space. From (19)-(21), the
current is

${\vec{j}}_{\mu}=-\frac{\epsilon\Omega}{g}(\Omega\cos(\Omega
t)\partial_{\mu}z,\Omega\sin(\Omega
t)\partial_{\mu}z,\epsilon\partial_{\mu}t)$ (86)

The adjoint scalar will be denoted by ${\vec{\phi}}$. In terms of
${\vec{\phi}}$ the current is

${\vec{j}}_{\mu}=g\,{\vec{\phi}}\times D_{\mu}{\vec{\phi}}$ (87)

where

$D_{\mu}{\vec{\phi}}=\partial_{\mu}{\vec{\phi}}+g{\vec{W}}_{\mu}\times{\vec{\phi}}$
(88)

Therefore, given ${\vec{j}}_{\mu}$, ${\vec{\phi}}$ must satisfy the constraint

${\vec{\phi}}\cdot{\vec{j}}_{\mu}=0$ (89)

for every $\mu$. Setting $\mu=0$ we obtain the requirement $\phi^{3}=0$. And
setting $\mu=3$ gives

$\cos(\Omega t)\phi^{1}+\sin(\Omega t)\phi^{2}=0$ (90)

Therefore

${\vec{\phi}}=\eta(-\sin(\Omega t),\cos(\Omega t),0)$ (91)

where $\eta$ is some unspecified vacuum expectation value of the scalar. Then

$\partial_{\mu}{\vec{\phi}}=-\eta\Omega\,(\cos(\Omega t),\sin(\Omega
t),0)\partial_{\mu}t$ (92)

and

${\vec{\phi}}\times\partial_{\mu}{\vec{\phi}}=\eta^{2}\Omega\,(0,0,1)\partial_{\mu}t$
(93)
${\vec{\phi}}\times({\vec{W}}_{\mu}\times{\vec{\phi}})=({\vec{\phi}}\cdot{\vec{\phi}}){\vec{W}}_{\mu}-({\vec{\phi}}\cdot{\vec{W}}_{\mu}){\vec{\phi}}=\eta^{2}{\vec{W}}_{\mu},$
(94)

since ${\vec{\phi}}\cdot{\vec{W}}_{\mu}=0$.

Eq. (87) gives

$\displaystyle{\vec{j}}_{\mu}$ $\displaystyle=$ $\displaystyle
g\eta^{2}\left[\Omega{\hat{e}}_{3}\partial_{\mu}t+g{\vec{W}}_{\mu}\right]$
(95) $\displaystyle\hskip-21.33955pt=g\eta^{2}(-\epsilon\cos(\Omega
t)\partial_{\mu}z,-\epsilon\sin(\Omega
t)\partial_{\mu}z,\Omega\partial_{\mu}t)$

Comparison of the $\mu=3$ expressions with the desired currents (86) gives,

$\Omega=g\eta$ (96)

The trouble arises in matching the $\mu=0$ expressions, for then,

$\epsilon^{2}=-g^{2}\eta^{2}$ (97)

and a real solution does not exist.

We conclude that the adjoint scalar ${\vec{\phi}}$ cannot source the initial
gauge field in (85).

##  Appendix B Homogeneous electric field and electroweak model

The electroweak model has the same ingredients as our model with an electric
string solution, except that the $U(1)$ symmetry is gauged with gauge coupling
$g^{\prime}$ and we have an extra gauge field, $Y_{\mu}$, called the
hypercharge gauge field. We will look for an electric string solution of the
same form as in (12) and with $Y_{\mu}=0$.

The $W$ currents are unchanged and we still need to satisfy (48)-(50). In
addition, since $Y_{\mu}=0$, the hypercharge current must also vanish,

$\partial_{\nu}Y^{\mu\nu}=j_{\mu}^{Y}=i\frac{g^{\prime}}{2}\left(\Phi^{\dagger}D_{\mu}\Phi-h.c.\right)=0$
(98)

The form of $\Phi$ and $W_{\mu}^{a}$ is fixed and given in (74) and (75).
Inserting these in (98) gives,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime}\epsilon^{2}}{g^{2}}\Omega
f^{2}(|z_{1}|^{2}-|z_{2}|^{2})\partial_{\mu}t$ (99)
$\displaystyle-\frac{g^{\prime}\epsilon^{3}}{2g^{2}}\frac{\Omega}{\omega}f^{3}\left[z_{1}z_{2}^{*}e^{i(\Omega+2\omega)t}+c.c.\right]\partial_{\mu}z,$

The $\mu=0$ component requires $|z_{1}|^{2}=|z_{2}|^{2}=1/2$, while the
$\mu=z$ component requires,

$z_{1}z_{2}^{*}e^{i(\Omega+2\omega)t}+c.c.=0.$ (100)

This forces $\omega/\Omega=-1/2$ but this is in conflict with the requirement
that $\omega/\Omega>0$ discussed below (53). Hence the electric string
solution with $Y_{\mu}=0$ does not exist in the electroweak model. This does
not exclude the possibility of an electric string solution with $Y_{\mu}\neq
0$.

##  Appendix C Some helpful formulae

If we use the Hopf parametrization to write $\Phi$,

$\Phi=\eta\begin{pmatrix}\cos\alpha\,e^{i\beta}\\\
\sin\alpha\,e^{i\gamma}\end{pmatrix}$ (101)

where $\alpha$, $\beta$, $\gamma$ only depend on time. Then

$\displaystyle\hskip-35.56593pt{\vec{n}}\equiv\Phi^{\dagger}{\vec{\sigma}}\Phi=\eta^{2}(\sin(2\alpha)\cos\theta,\sin(2\alpha)\sin\theta,\cos(2\alpha))$
(102)

where $\theta\equiv\gamma-\beta$. And

$\displaystyle\hskip-28.45274pt\Phi^{\dagger}\sigma^{1}{\dot{\Phi}}-h.c.=i\eta^{2}[2\sin\theta\,{\dot{\alpha}}+\sin
2\alpha\cos\theta({\dot{\beta}}+{\dot{\gamma}})]$ (103)
$\displaystyle\hskip-28.45274pt\Phi^{\dagger}\sigma^{2}{\dot{\Phi}}-h.c.=i\eta^{2}[-2\cos\theta\,{\dot{\alpha}}+\sin
2\alpha\sin\theta({\dot{\beta}}+{\dot{\gamma}})]$ (104)
$\displaystyle\hskip-28.45274pt\Phi^{\dagger}\sigma^{3}{\dot{\Phi}}-h.c.=i\eta^{2}2(\cos^{2}\alpha\,{\dot{\beta}}-\sin^{2}\alpha\,{\dot{\gamma}})$
(105)
$\displaystyle\hskip-28.45274pt\Phi^{\dagger}{\dot{\Phi}}-h.c.=i\eta^{2}2(\cos^{2}\alpha\,{\dot{\beta}}+\sin^{2}\alpha\,{\dot{\gamma}})$
(106)

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: Controllable Part-Based 3D Point Cloud Generation
with Cross Diffusion
Kiyohiro Nakayama^1   Mikaela Angelina Uy^1   Jiahui Huang^2   Shi-Min Hu^2   Ke Li^3   Leonidas Guibas^1

^1Stanford University     ^2Tsinghua University     ^3Simon Fraser University

figure: Our approach tackles the task of controllable part-based point cloud generation, where we are able to generate novel shapes - novel configurations of novel parts. Our probabilistic generative model learns a factorized representation of shapes with a cross diffusion network allowing for control. We demonstrate that  not only enables controllable generation but also various shape editing tasks.

This work is supported by ARL grant W911NF-21-2-0104, a Vannevar Bush Faculty Fellowship, and the Natural Sciences and Engineering Research Council of Canada. We very much appreciate Congyue Deng for her helpful discussion at the early stage of this project. We are also grateful for the advice and help from Colton Stearns and Davis Rempe.

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# Measurement Based Statistical Channel Characterization of Air–to–Ground Path
Loss Model at 446MHz for Narrow–Band Signals in Low Altitude UAVs††thanks:
This paper has been accepted for the presentation in the 2020 IEEE 91st
Vehicular Technology Conference (VTC2020-Spring).

Burak Ede12, Serhan Yarkan13, Ali Rıza Ekti 14, Tunçer Baykaş5, Hakan Ali
Çırpan2, Ali Görçin16
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1
Informatics and Information Security Research Center (BİLGEM), TÜBİTAK,
Kocaeli, Turkey  2 Department of Electronics and Communication Engineering,
Istanbul Technical University, İstanbul, Turkey  3 Department of Electrical
and Electronics Engineering, Istanbul Commerce University, İstanbul, Turkey  4
Department of Electrical and Electronics Engineering, Balıkesir University,
Balıkesir, Turkey  5 Department of Electrical and Electronics Engineering,
Medipol University, İstanbul, Turkey  6 Faculty of Electronics and
Communications Engineering, Yıldız Technical University, İstanbul, Turkey

###### Abstract

Powered by the advances in microelectronics technologies, unmanned aerial
vehicles provide a vast variety of services ranging from surveillance to
delivery in both military and civilian domains. It is clear that a successful
operation in those services relies heavily on wireless communication
technologies. Even though wireless communication techniques could be
considered to reach a certain level of maturity, wireless communication links
including UAVs should be regarded in a different way due to the peculiar
characteristics of UAVs such as agility in $3$D spatial domain and versatility
in modes of operation. Such mobility characteristics in a vast variety of
environmental diversity render links including UAVs different from those in
traditional, terrestrial mobility scenarios. Furthermore, UAVs are critical
instruments for network operators in order to provide basic voice and short
messaging services for narrow band communication in and around disaster areas.
It is obvious that such widespread use of UAVs under different scenarios and
environments requires a better understanding the behavior of the communication
links that include UAVs. Therefore, in this study, details of a measurement
campaign designed to collect data for large-scale propagation characterization
of air–to–ground links operated by UAVs at 446MHz under narrowband assumption
are given. Data collection, post-processing, and measurement results are
provided.

###### Index Terms:

UAV, Channel Modeling, Measurements, Path Loss

##  I Introduction

Ubiquitous access has become an essential part of modern daily life. Services,
applications, devices, and even products are equipped with wireless
communication interfaces in such a way that diverse fields and domains are
connected anywhere, anytime. In parallel with the escalating demand for high
data rates, everywhere connectivity with mobility support becomes an
indispensable design requirement for both contemporary and emerging wireless
technologies. Although, it is expected that these wireless technologies should
support crystal clear audio for voice, high–definition video downstreaming,
very–low latency and $4$K online gaming, low-power consumption transmission,
and so on, it is more important to provide service on mission critical
scenarios. Obviously, such an aggressive set of requirements comes at the
expense of several conflicting list of parameters. Furthermore, realization of
these requirements necessitate utilization of crucial resources such as
bandwidth and power.

The conventional strategy to tackle the aforementioned utilization problem is
reusing of resources as frequent as possible. Cell splitting, antenna
sectoring, and small cell concepts are prominent examples to increase capacity
via effective use of available bandwidth, reduce transmit power and impact of
interference. There are products in the market regarding small cell
(microcell, picocell, femtocell in the sense of metro femtocells, public
access femtocells, enterprise femtocells and class $3$ level femtocells, etc.)
solutions which mainly rely on fixed deployment. However, with the emergence
of UAVs, a paradigm shift has been experienced in wireless communication
communities in many aspects. Considering its agility in $3$D spatial domain
and versatility in modes of operation, UAVs bring about almost a completely
novel perspective in contemporary wireless mobile radio communication systems.
Ranging from emergency communications to routing and relaying, UAVs are
considered to be promising solution candidates in various wireless
communication scenarios. In addition, the emerging concept of Internet of
Things (IoT) mandates extending the wireless radio coverage to diverse (and
relatively harsh propagation) environments. In the presence of such a vast
variety of propagation environments, a successful wireless communication link
depends on transceivers which rely on extended measurements and corresponding
designs [1].

In the literature, there has been a significant attention in UAVs and
measurement campaigns including several modes and scenarios. Considering the
transmission modes akin to terrestrial ones, air–to–ground and air–to–air are
the two prominent classes [2]. Among these two, air–to–air class needs further
investigation due to the aforementioned reasons relevant to UAVs modes of
operations [3, 4]. On the other hand, air-to-ground could be considered to be
a transition class since it contains both terrestrial elements and UAVs
simultaneously. Air-to-ground class consists of line–of–sight (LOS),
nonline–of–sight (NLOS), and obstructed line–of–sight (OLOS) [5]. Of course, a
comprehensive air–to–ground model requires the probabilistic transitional
states for LOS, NLOS, and OLOS cases to be defined as well [6, 7, 8]. It is
obvious that further analysis is required to have an extended model which
takes into account shadowing as an additional parameter [9, 10, 11]. A very
detailed collection of studies present in the literature could be found in
[12]. Beside statistical models which depend heavily on theoretical
derivations [13], there are exact [14] and numerical approaches in determining
the propagation characteristics as well. Especially ray tracing method that
runs in downtown scenarios with the extension of building heights is employed
very frequently [15, 4].

In the mission critical scenarios, where the utilization of UAV base
stations/access points such as disaster/public safety regions, rural areas and
downtown areas where the total failure of communication infrastructure would
lead to catastrophic events in terms of wireless communication, finding a
rapid and cost–effective recovery solution that utilizes narrow band voice
channels will be an important task. There have been some commercial services
offered by cellular operators which claim to employ drones as hovering base
stations in order to provide coverage in emergency scenarios and/or when
disaster strikes. However, one of such operators which claimed to have UAV
base stations could not provide service for a while after an earthquake of
magnitude of 5.8 hit Istanbul, Turkey, on September 27th, 2020 [16]. This
instance revealed that establishing communication via UAVs in a disaster
scenario is a challenging task which has multiple dimensions including careful
propagation channel analysis, detailed network planning, and spectrum
management for first responders [17, 18, 19].

Despite all of the measurement campaigns, results, and theoretical analyses
present in the literature which are mostly focused on 800MHz–850MHz,
960MHz–977MHz and 5030MHz–5091MHz for the UAV communication, a generic and
comprehensive propagation model for emergency case communications focusing
specifically on UAVs operating at UHF bands is still required. Therefore, in
this study, a single-frequency path loss measurement campaign at 446MHz is
proposed. Air-to-ground link is established via a narrow band signal generator
(NBSG) mounted on UAV. Measurements are collected within a suburb district,
which is located close by to an inner sea. Measurement site is of hilly
terrain structure with foliage. Considering the fact that such topographical
layouts are difficult to reach and collect measurement data, this study paves
the way for incorporating various terrain profiles into the channel models and
obtaining a more comprehensive air–-to-–ground propagation link. The
organization of the paper is as follows: General characteristics of
air–to–ground path loss link is presented in Section II. In Section III, the
details of the experimental setup, equipment, and data collection stages are
outlined. Results and relevant discussions are presented in Section IV.
Finally, conclusions are drawn in the last section.

##  II General Large Scale Characteristics of Air–to–Ground Link for UAV

In a general air–to–ground wireless communication link scenario, it is
frequently reported in the literature that the propagation mechanism at large
scale is governed by the following statistical path loss equation:

$\Lambda(d)=\underbrace{20\times\log\frac{4\pi
d_{0}}{\lambda}}_{PL_{0}}+10\eta\log\left(d/d_{0}\right),$ (1)

where $d$ is the transmitter-receiver separation, $\eta$ is the path loss
coefficient which depends on the environment, and $d_{0}$ is the reference
distance for path loss measurements along with $PL_{0}$ being the free–space
offset or intercept. A further improvement is possible by incorporating the
LOS into the model as

$\Lambda(d)=\underbrace{20\times\log\frac{4\pi
d_{0}}{\lambda}}_{PL_{0}}+10\eta\log\left(d/d_{0}\right)+\mu_{LOS},$ (2)

where $\mu_{LOS}$ includes all of the large–scale characteristic losses for
LOS.

##  III Measurement Campaign

###  III-A Measurement Campaign and Measurement Setup

The measurements are taken in the test field which is shown in Fig. 1(a) of
The Scientific and Technological Research Council of Turkey (TUBITAK) in
Gebze. A narrow band signal generator is used to transmit a single tone in
446MHz center frequency on the UAV side. DJI Wind 4 is flied up to obtain
measurement and it is used as a air station since its payload capability and
durability in extreme conditions. It is shown in Fig. 1(a).

(a) Block diagram for measurement setup.

(b) Rohde Schwarz FSW26 as receiver.

Figure 1: Measurement campaign and equipment.

On the other hand, R&S FSW Signal and Spectrum Analyzer in Fig. 1(b), is used
as a ground station to receive the transmitting signals. It works in frequency
range from 2Hz to 26.5GHz and offers up to 500MHz analysis bandwidth for
measuring wideband–modulated or frequency–agile signals. Signals are sampled
by 1MHz and they are recorded with 1sn duration. Furthermore, a Bowtie antenna
which frequency range from 400MHz to 1000MHz is used for ground station to
receive narrow band signals. The antenna has omnidirectional radiation pattern
and it has 0.9dBi antenna gain at 400MHz. After the calculation of received
powers of each point, then path loss model are extracted as it will be
explained in Section IV.

###  III-B Measurement Methodology

All the field measurements are conducted in 446 MHz frequency band by using
narrow band signal generator attached to a commercial UAV and all the signals
are captured by the ground station with 1MHz sample rate during one second.
The measurements are collected from 60m to 80m heights with step size 10m
vertically. Local regulations allow UAV to operate up to 120m in height,
however, due to the topographical conditions and the high wind speed after 80m
in height, measurements are performend for 60m to 80m in height. Moreover, the
only allocated location for UAV flight is, where the measurement is taken, is
shown Fig. 1(a) in TUBITAK, Gebze. Also, for each vertical measurement point,
UAV move between 50m to 200m horizontally. Overall grid consists of total 45
determined locations points with 60m to 80m vertically and 50m to 200m
horizontally. Collected signals are fed into a computer that runs MATLAB
2015b. Raw time data which is given in in–phase/quadrature (I/Q) format are
passed through a 255th order FIR filter to maximize power of narrow band
signal and to get rid of spurious signals from the captured signal. In
addition to this, the center frequency offset is calculated for each point and
shifted by that value. Also, another FIR filter with 127th order is used for
removing noise arising from harmonics. Furthermore, offset in the time domain,
is calculated and eliminated from the processed signal. Then, the signal is
down sampled to avoid complexity of the calculation and the average power with
dB scale is calculated for each data set.

Measurement campaign workflow for the aforementioned methods is shown Fig. 2.

Environmental Setup DJI Wind 4 + NBSG RS FSW26 SSA + Bowtie Antenna
Organization Parameter Settings Post Processing 1\. FIR Filter(255th Order)
Center Frequency Offset Calibration 2\. FIR Filter(127th Order) Time-Domain
Offset Calibration Downsampling Average Power(dB) Calculation 1\. Order
Polynomial Fitting Path Loss Curve Results

Figure 2: Illustration of the measurement set–up and post–processing workflow.

##  IV Measurement Results

In this study, measurement campaign focuses in the following aspect on how the
narrow band signal received power is related to the transmitter–-receiver
separation by considering both horizontal and vertical distance. To observe
this simultaneously, Fig. 3 is plotted. As seen in Fig. 3, the received power
decreases with the transmitter-–receiver separation, as expected. Based on the
measurement data when the least-–squares estimation is applied, the following
path loss models are obtained:

$\text{PL}(\text{dB})=\left\\{\begin{matrix}-2.176\times d-35.99,~{}a=60m\\\
-1.244\times d-55.23,~{}a=70m\\\ -1.473\times
d-49.4,~{}a=80m\end{matrix}\right.$ (3)

where $d$ is the transmitter–receiver separation and $a$ stands for the
altitude of UAV.

For the sake of brevity, the path loss coefficients for specific frequency and
altitudes are listed in Table I. Overall mean path loss exponent is found to
be $\eta=2.176,1.244,1.473$ for the altitudes of 60m, 70m, and 80m,
respectively. Another important observation, which is also in conformity with
the results presented in the literature in different frequency [20], is that
as the UAV moves up in the horizontal direction, the path loss exponent,
$\eta$, values decrease in general. Even though, this result looks surprising
it is still in parallel with the independent measurement in some other studies
in literature [20]. This can be explained as the Tx–Rx communication link will
be less sensitive to horizontal distance at high altitude and the channel
condition is better in higher altitudes. Furthermore, the UAV at high altitude
is most likely be affected less in terms of received power change due to the
fact that it flies in the direction of the non–dominant radiation pattern of
the BS antenna. Thus, not only environment but also height dependent channel
propagation parameters are needed for describing the propagation channel for
UAV to ground station communication links.

TABLE I: Values for $D_{\max}$ and Path Loss Exponent, $\eta$, for 446MHz at different altitudes. Altitude (m) | 60 | 70 | 80
---|---|---|---
PL. Exp. ($\eta$) | 2.176 | 1.244 | 1.473

(a) 60m altitude.

(b) 70m altitude.

(c) 80m altitude.

(d) Comparison of PL curves for different heights.

Figure 3: Measurement results for received power levels at different distance.

##  V Concluding Remarks and Future Directions

In this work, the results of a measurement campaign for air to ground channels
of UAV at 446MHz are presented due to the fact that refarming of UHF band will
enable new opportunities for emergency case situations. Based on the
measurement data, it is shown that altitude along with the environment is an
important factor that needs to be considered. Due to the increase in the
elevation, the channel condition gets better and path loss exponent tends to
decrease in general. Thus, in the future studies, we will extend our results
to multiple frequency ranges by also incorporating environment affect in
different locations and topographies.

## Acknowledgement

This work of Tuncer Baykas was supported by the Scientific and Techno logical
Research Council of Turkey (TUBITAK) under Grant 215E324.

Also, this publication was made possible by NPRP12S-0225-190152 from the Qatar
National Research Fund (a member of The Qatar Foundation). The statements made
herein are solely the responsibility of the author[s].

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  *[UAVs]: unmanned aerial vehicle
  *[IoT]: Internet of Things
  *[LOS]: line–of–sight
  *[NLOS]: nonline–of–sight
  *[OLOS]: obstructed line–of–sight
  *[UAV]: unmanned aerial vehicle
  *[I/Q]: in–phase/quadrature


# The influence of obstacles on collective motion of self-propelled objects.

Horacio Serna  [  Wojciech T. Góźdź<EMAIL_ADDRESS>[

###### Abstract

We investigate the influence of obstacles on the collective motion of self
propelled objects in the framework of the Vicsek model. The obstacles are
arranged in a square lattice and have circular shape. We show that by
increasing the radius of the obstacles the collective motion of the self
propelled object can be altered from super diffusive to diffusive. For
obstacles with small radius, the system is composed of large clusters moving
in one direction, for larger radius, the system is composed of small clusters
moving randomly in different directions.

###### keywords:

American Chemical Society, LaTeX

IChF] Institute of Physical Chemistry Polish Academy of Sciences, Kasprzaka
44/52 01-224 Warsaw, Poland. IChF] Institute of Physical Chemistry Polish
Academy of Sciences, Kasprzaka 44/52 01-224 Warsaw, Poland.

##  1 Introduction

Collective motion of self propelled objects1 is a fasctnating phenomenon . It
takes palce at various time and length scales. For example colonies of
bacteria, swarms of bees, school of fish and flocks of birds can move
collectively. Vicsek model is one of the simplest mathematical models to study
the collective motion of self propelled particles 2. Polar liquid, band and
disordered phases were discovered in Vicsek model for different values of the
noise amplitude 3. Recently, a novel cross sea phase4 was reported for noise
amplitudes between the polar liquid and band phases.

In recent years, many works have been devoted to Vicsek model in either the
presence of obstacles or confined into different geometries, revealing
interesting dynamical behaviour. For instance, it has been determined that in
a continuous version of the Vicsek model, the confinement into one-dimensional
channels, squares and circles can lead to spatial coherence of swarms 5. On
the other hand, an on-lattice Vicsek model was studied in the same confining
geometries, showing alignment of self propelled particles parallel to the
boundaries at low noise amplitudes for channels and circles and trapping at
the corners for squares 6. The influence of randomly distributed obstacles on
the collective motion of particles with aligning interactions has been studied
with simulation and theoretical approaches 7, 8. The effects of fixed and
mobile obstacles on the behaviour of Vicsek disks was considered in a recent
work 9. It was also shown that the presence of a small obstacle at the centre
of a large system of point-like Vicsek particles can reverse the direction of
movement of a polar flock 10. Trapping of self propelled point-like Vicsek
particles by arrays of asymmetric V-shaped obstacles was also recently
investigated 11. The effects of topology and geometry on the collective motion
in Vicsek model was studied by confining the system into different assemblies
of circular obstacles 12. Interestingly, the authors found that the presence
of obstacles not only hinders the alignment of particles within the system but
induces sub-diffusive behaviour.

In this article, we study the collective motion in a square lattice of
circular obstacles. We analyze the effects of both the packing fraction of
obstacles and the separation between them (which is equivalent to varying the
number density of obstacles). In contrast to previous works we keep the noise
amplitude fixed and focus on the influence of different arrangement and size
of obstacles on the collective motion of self propelled objects.

##  2 Model and Methods

The collective motion of self of self-propelled particles is studied within
the framework of the Vicsek model 2. The movement of the particles is governed
by the following equation

$\boldsymbol{r}_{i}(t+\delta t)=\boldsymbol{r}_{i}(t)+\boldsymbol{v}_{i}\delta
t,$ (1)

where $\boldsymbol{r}_{i}(t)$ and $\boldsymbol{v}_{i}$ denote the position and
velocity of a particle $i$ at time $t$. The time step is set to $\delta
t=1.0$. The speed, $v_{0}$, of each particle is constant. The angle formed by
the velocity with the $x$ axis is denoted by $\theta_{i}(t)$. The angle
$\theta_{i}(t+\delta t)$ at the next time step is calculated according to the
following rule

$\theta_{i}(t+\delta t)=\langle\theta(t)\rangle_{r_{c}}+\Delta\theta$ (2)

where $\langle\theta(t)\rangle_{r_{c}}$ is an average angle between all the
velocity vectors with the horizontal axis, contained inside a circle of radius
$r_{c}$ with the center set on the particle $i$ given by its coordinates
$\boldsymbol{r}_{i}$

The term $\Delta\theta$ is a random angle uniformly distributed in the
interval $[-\zeta/2,\zeta/2]$, where $\zeta$ is the noise amplitude. We
consider the particles movement at low speed limit, low noise amplitudes, and
low number density. Thus, we set the model parameters to $v_{0}=0.30$,
$\zeta=1.0rad$ $(\approx 57.3^{0})$, and $\rho=N/L^{2}=0.50$, being $N$ the
number of particles and $L$ the side length of the simulation box. For this
set of parameters, the system exhibits collective motion in the form of big
swarming clusters in bulk (See Supplementary Movie M1).

Figure 1: (a) Scheme of the system. The gray and red circles represent the
obstacles and particles, respectively. $l_{0}$ is the lattice constant of the
square lattice and $R_{O}$ is the radius of the obstacles. (b) Representation
of the elastic collisions of a particle with an obstacle within a time step.

We consider a square lattice of fixed circular obstacles ( See Figure 1a). The
simulations are performed in a periodic square box with $L=300r_{c}$. The
collisions of particles with obstacles are elastic. The velocity after the
collision is calculated according to the following formula
$\boldsymbol{v}_{i}=v_{0}(\boldsymbol{\hat{v}}_{i}-2(\boldsymbol{\hat{v}}_{i}\cdot\boldsymbol{\hat{u}})\boldsymbol{\hat{u}})$,
where $\boldsymbol{\hat{v}}_{i}$ is the unit vector parallel to the velocity
vector of particle $i$ before the collision and $\boldsymbol{\hat{u}}$ is the
unit vector normal to the obstacle’s surface at the contact point. To
calculate the contact point, we calculate the fraction of time step that the
particle requires to arrive to the obstacle’s boundary. In the remaining time
step the particle travels with the velocity $\boldsymbol{v}_{i}(t+\delta t)$
after the collision (see Figure 1b). We investigate arrays with $N_{O}=49$,
$225$, $484$ and $900$ obstacles, corresponding to lattice constants,
$l_{0}=\sqrt{N_{O}}/L=42.86r_{c}$, $20.0r_{c}$,$13.64r_{c}$ and $10.0r_{c}$,
respectively. We study the influence of the packing fraction of obstacles by
gradually increasing their radius, $R_{O}$, for each array.

The structure of the system is monitored during the simulations by calculating
the order parameter which measures the degree of alignment of the velocities
of all particles 13:

$\nu_{a}=\frac{1}{N}\sqrt{\left(\sum_{j=1}^{N}\cos{\theta_{j}}\right)^{2}+\left(\sum_{j=1}^{N}\sin{\theta_{j}}\right)^{2}}.$
(3)

Average value of the order parameter calculated during the simulation
$\langle\nu_{a}\rangle$, is used to identify structural transition induced by
the change of the packing fraction of obstacles, $\eta=\pi
N_{O}R^{2}_{O}/L^{2}$, in a similar way as it was done by increasing the noise
amplitude in the bulk system in the original Vicsek model2 and its
variations3, 14, 13. The simulations are run for $5\times 10^{5}$ time steps,
$10^{5}$ for the system to reach the steady state and the rest for averaging
properties.

To characterize the dynamic behaviour of the system, we compute the mean
square displacement,

$\left\langle\Delta
r(t)^{2}\right\rangle=\left\langle\frac{1}{N}\sum_{i=1}^{N}\left(\boldsymbol{r}_{i}(t)-\boldsymbol{r}_{i}(0)\right)^{2}\right\rangle,$
(4)

and the average number of collisions with obstacles per time step, $\langle
C\rangle$. The averages are taken at each time step during $5\times 10^{5}$
time steps at the steady state for all the particles.

##  3 Results and discussion

For the chosen parameters of the noise $\xi=1$, the speed $v_{0}=0.3$, and the
density $\rho=0.5$, the particles form large clusters moving collectively in a
preferred direction in bulk. For confined systems, we can identify two
structural regimes for all four arrays of obstacles. In the first regime where
the obstacles are relatively small, we observe large clusters moving in one
direction along y or x axis. For larger obstacle smaller clusters are formed,
moving in different directions. These theree states are illustrated in Figure
2.

For small values of the packing fraction, $\eta$, ( small obstacles ) the
structure of the systems is similar to the bulk structure. It is interesting
to note that even very small obstacles enforce the movement of clusters along
the x or y axis. Such behavior results form passing the particles through a
square lattice of obstacles. In this state the system is composed of a few big
clusters moving in the same direction and very small clusters and isolated
particles. The order parameter, $\nu_{a}$, has relative large values for such
structure.

Figure 2:  The snapshots from the simulations representing different states of
the system. The parameters of the model are $v_{0}=0.30$, $\zeta=1.0rad$,
$\rho=0.50$, and the system size $L=300r_{c}$. (a) Large clusters in bulk.
Systems with obstacles are composed of 30 obstacles per edge
($l_{0}=10.0r_{c}$), (b) large clusters passing through small obstacles
($R_{O}=1.0r_{c}$, $\eta=0.0314$ ), (c) small clusters moving between
relatively large obstacles ($R_{O}=2.0r_{c}$, $\eta=0.1257$).

For larger values of the packing fraction, $\eta$, (bigger obstacles) the
clusters become smaller and move in different directions. The order parameter,
$\nu_{a}$, has relative small values for such structures. Short movies of the
cases presented in Figure 2 are available in the Supplementary Information(
Movie 1 (a). Movie 2 (b) and Movie 3 (c)).

Figure 3:  (a)The average order parameter, $\langle\nu_{a}\rangle$, as a
function of the packing fraction of obstacles, $\eta$, for all the arrays
considered. Solid lines connect the data as a guide for the eye. (b) The
fluctuations of order parameter, $\nu_{a}$, as a function of the packing
fraction of the obstacles, $\eta$, for all the arrays considered. Solid lines
represent fit curves to double Gaussian functions.

To investigate the alignment of the velocity of the particles, we calculate
the average order parameter, $\langle\nu_{a}\rangle$, as a function of the
packing fraction, $\eta$, according to Eqn. 3. Additionally, we compute the
fluctuations of the order parameter,
$\Delta\nu_{a}=\langle\nu_{a}^{2}\rangle-\langle\nu_{a}\rangle^{2}$. The
results are presented in Figure 3. We observe that for all studied systems,
upon increasing the packing fraction of obstacles, the average order
parameter, $\langle\nu_{a}\rangle$, decreases (See Figure 3a). The decrease of
the order parameter indicates a structural transition between two distinct
regimes characterized by big clusters moving in one direction and small
clusters moving in different directions. The slope of the curve suddenly
change for certain values of $\eta$, indicating the transition from the first
regime to the second regime. The value of the packing fraction at the
inflection point can be used to identify the critical packing fraction for
this transition. The larger the density of the obstacles the smaller the value
of the critical packing fraction (See Figure 3a). The region of the transition
is also characterized by increased values of the fluctuations of the order
parameter, $\Delta\nu_{a}$, as shown in Figure 3b. $\Delta\nu_{a}$ shows a
clear peak that signals a the transition. The data for studied systems can be
fitted with a double Gaussian function as shown in Figure 3b. We take the
position of the peaks as a reference to define the critical packing fraction,
$\eta_{t}$. The values of $\eta_{t}$ and the corresponding
$\langle\nu_{a}\rangle$ are presented in Table 1.

System | $\eta_{t}$ | $R_{t}$ | $\langle\nu_{a}\rangle_{t}$
---|---|---|---
$l_{0}=42.86r_{c}$ | $\approx 0.1395$ | $\approx 9.0310$ | $\approx 0.2735$
$l_{0}=20.00r_{c}$ | $\approx 0.1018$ | $\approx 3.6002$ | $\approx 0.2505$
$l_{0}=13.64r_{c}$ | $\approx 0.0870$ | $\approx 2.2692$ | $\approx 0.2642$
$l_{0}=10.00r_{c}$ | $\approx 0.0718$ | $\approx 1.5117$ | $\approx 0.2496$
Table 1:  The critical packing fraction, $\eta_{t}$, estimated for the studied
systems. $l_{0}$ is the lattice constant of the array of obstacles, $R_{t}$ is
the critical radius of obstacles and $\langle\nu_{a}\rangle_{t}$ is the
critical average order parameter

These results show a new type of the transition in Vicsek-like models, induced
by increasing the packing fraction of obstacles arranged in a square lattice.

Figure 4:  Histograms of the probability distributions of the order parameter,
$P(\nu_{a})$, for different packing fractions of obstacles, $\eta$. The
results are presented for the arrays with $l_{0}=42.86r_{c}$(a),
$l_{0}=20.00r_{c}$(b), $l_{0}=13.64r_{c}$(c) and $l_{0}=10.00r_{c}$(d)

For a better insight into the nature of the transition, we calculate the
probability distribution of $\nu_{a}$ during long simulations of $5\times
10^{5}$ time steps. The results are presented in Figure 4 in the form of
histograms. The histogram for the bulk and for small values of the packing
fractions are narrow and peaked (see orange and blue distributions in Figures
4 a,b,c and d). The histograms calculated for the systems with the packing
fraction of obstacles in the region of the transition are wide and often with
a few distinct peaks (See the yellow histograms in Figure 4 a, b, c and d).
The histograms calculated for the values of the packing fraction of obstacle
after the transition to the states with small alignment of the velocities
become narrower than the histograms at the transition region but wider than in
the region of well aligned velocities (small values of the packing fraction).
We also observe that the larger the lattice constant of the obstacle arrays,
the wider the distribution (See the purple histograms in Figures 4 a,b,c and
d).

Figure 5:  Mean squared displacement, $\left\langle\Delta
r(t)^{2}\right\rangle$ for various packing fractions of obstacles, $\eta$. The
results are presented for different square lattices of obstacles with
$l_{0}=42.86r_{c}$(a), $l_{0}=20.00r_{c}$(b), $l_{0}=13.64r_{c}$(c) and
$l_{0}=10.00r_{c}$(d). The black lines represent the limit of a pure diffusive
regime (solid) and super diffusive regime (dashed)

Finally, to characterize the order-disorder transition from a dynamic point of
view, we calculate the mean squared displacement (MSD) and the average number
of collisions between point-like particles and obstacles per unit time. The
MSD results are presented in Figure 5. For all investigated systems, the
behaviour at long times is super diffusive for small values of $\eta$ which
correspond to the big clusters regime. However, when $\eta$ is approaching
$\eta_{t}$, the MSD of the system starts to show a non-monotonic intermediate
behaviour between super diffusive and diffusive regimes. At packing fractions
about $\eta_{t}$, the MSD plots indicate mixed behavior partly super
diffusive, partly diffusive as shown in Figures 5a, b and c for $\eta=0.1095$,
$0.0830$ and $0.0763$, respectively. We have observed that in this transition
region, big clusters can split into small clusters moving in different
directions and the reverse process also takes place. There is a correlation
between the wide shape of the $\nu_{a}$ histograms (See Figure 4) and the non-
monotonic plots of the MSD. Finally, for $\eta>\eta_{t}$, beyond the
transition region, although the MSD do not show a well developed diffusive
regime, we can state that the systems approach a diffusive behaviour. This
diffusive-like behaviour corresponds to regime of small clusters moving in
different directions.

Figure 6:  The average number of collisions with obstacles per unit time,
$\langle C\rangle$, as a function of the packing fraction of obstacles,
$\eta$. The error bars are comparable to the size of the markers

Figure 6 shows the average number of collisions with obstacles per unit time,
$\langle C\rangle$, as functions of $\eta$ for all studied systems. For small
values of $\eta$, the slope of all curves is high, whereas the slope is
smaller for larger values of $\eta$. Thus the increase of the obstacle’s
radius from the values close to zero causes rapid increase of collision until
some point from which the increase of collisions is substantially slower. Near
this specific point the all the curves bent significantly. We also observe
that the number of collisions for the systems with lower density of obstacles
(larger distance between obstacles) the increase of the collisions is larger
when the packing fraction is increasing (which is equivalent to the increase
of the size of the obstacles). As expected, $\langle C\rangle$ increases
rapidly as $\eta$ increases for the systems with the largest number density of
obstacles ($l_{0}=10.0r_{c}$ and $13.64r_{c}$), whereas for the systems with
the fewer number density of obstacles ($l_{0}=12.86r_{c}$ and $20.00r_{c}$),
it increases slowly. The point where the curves bend is shifted to smaller
values of $\eta$ when the system has smaller density of obstacles but it does
not seem to be related with any structural or dynamical feature of the system
. The location of the transition is not clearly reflected in the graphs of
$\langle C\rangle$, since they are smooth and continuous within the whole
transition region, suggesting a continuous transition.

##  4 Summary and Conclusions

We have investigated the influence of obstacles on the collective motion of
large number of self propelled objects within the framework of Vicsek model.
We have focused on the systems with many obstacles arranged on a square
lattice. We have studied the systems with different packing fraction and
number density of obstacles. We have discovered that by increasing the packing
fraction of obstacles the investigated systems undergo a structural
transition. The structure of the collectively moving objects changes from a
few large clusters moving in a preferred direction to many small clusters
moving in different directions. Such transition takes place for the systems
with different number density of obstacles. The change of the structure of the
systems induced by the change of the packing fractions of obstacles is
reflected in the changes of the order parameter, $\nu_{a}$, (see Figure 3) and
the plots of MSD (see Figure 5). We have determined that the transition is
continuous. We show that this transition is closely related to the dynamic
features of the system. The big clusters regime corresponds to super diffusive
behaviour whereas the small clusters regime is closer to diffusive behaviour.
We also show that the number density of obstacles is increased, the transition
is shifted to smaller values of $\eta$ (See Figure 3a). The transition is also
confirmed by the presence of peaks when the fluctuations of the order
parameter $\Delta\nu_{a}$ are plotted as functions of $\eta$ (see Figure 3b).
From $\Delta\nu_{a}$ we identify the packing fraction at which the transition
occurs, $\eta_{t}$. Interestingly, we have observed non-monotonic behaviour in
the mean square displacement of the systems at the transition region (see
Figure 5). This behaviour results from transitions between two states, one
with a few large clusters moving in a prferred direction and the other
composed of many small clusters moving in different directions. Such behavior
is reflected in the shape of the histograms of the probability of the order
parameter $\nu_{a}$ for $\eta$ close to $\nu_{t}$ (see Figure 4). The
transition described here resembles the transition in the Vicsek model induced
by the increase of the noise parameter.

We would like to acknowledge the support from NCN (grant no
2018/30/Q/ST3/00434).

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