# Classification of Magnetic Vortices
by Angular Momentum Conservation

Kenji Fukushima<EMAIL_ADDRESS>Department of Physics, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Yoshimasa
Hidaka<EMAIL_ADDRESS>Institute of Particle and Nuclear Studies, KEK, 1-1
Oho, Tsukuba, Ibaraki 305-0801 Japan  RIKEN iTHEMS, RIKEN, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Ho-Ung Yee<EMAIL_ADDRESS>Department of Physics,
University of Illinois, Chicago, Illinois 60607, U.S.A.

###### Abstract

Superfluid vortices are quantum excitations carrying quantized amount of
orbital angular momentum in a phase where global symmetry is spontaneously
broken. We address a question of whether magnetic vortices in superconductors
with dynamical gauge fields can carry nonzero orbital angular momentum or not.
We discuss the angular momentum conservation in several distinct classes of
examples from crossdisciplinary fields of physics across condensed matter,
dense nuclear systems, and cosmology. The angular momentum carried by gauge
field configurations around the magnetic vortex plays a crucial role in
satisfying the principle of the conservation law. Based on various ways how
the angular momentum conservation is realized, we provide a general scheme of
classifying magnetic vortices in different phases of matter.

††preprint: KEK-TH-2268, J-PARC-TH-0231

##  I Introduction

When a superfluid where a macroscopic condensate of identical bosons is formed
is under rotation, superfluid vortices emerge and each microscopic constituent
carries the same amount of orbital angular momentum, i.e., an integer multiple
of fundamental quanta, $\hbar$. This is a remarkable way to store macroscopic
amount of angular momentum in a highly coherent quantum state. Not only in
table-top physical systems of superfluids such as 4He, the superfluid vortex
can also be an important constituent in rotating nuclear matter found inside
neutron star, where the extremely high matter density causes a nonzero order
parameter that signifies spontaneous breaking of global baryon number
$U(1)_{\rm B}$ symmetry. In a more interesting scenario of dense quark matter
in quantum chromodynamics (QCD), this order parameter is also responsible for
the superconducting phase of color gauge interactions, most likely the color-
flavor-locked (CFL) superconducting phase Alford  _et al._ (1999). There exist
highly nontrivial vortices in the CFL phase, called non-Abelian CFL vortices
Balachandran  _et al._ (2006), that involve both dynamics of the global baryon
and the local color gauge symmetries; see Ref. Eto  _et al._ (2014) for a
comprehensive review.

As seen in many interesting examples including vortices in CFL quark matter,
some of which we will study in later discussions, the symmetries involved in
vortex contents are entirely or partially gauge symmetries. The prototypical
example is of course the magnetic vortex in Type-II superconductors. In these
cases the vortex profile is fundamentally different from that of the purely
superfluid vortex; a magnetic flux is threaded into the vortex core. Among
many differences between a superfluid vortex and a gauged magnetic vortex, one
may specifically ask about the angular momentum they carry. Surprisingly to
us, we find that this simple question has not been properly addressed in the
literature. As we try to answer the question in various examples across
different fields of physics, we discover surprisingly diverse situations. It
is the purpose of this article to present a compelling list of examples where
the answers are quite different from each other, and also to provide an
overarching physics explanation of why the answers can be so diverse. We will
demonstrate that the angular momentum conservation offers a key guiding
principle to understand the physics origin of the different answers. Our
detailed analysis in the main text shows that the angular momentum carried by
not only the matter sector of the system but also the dynamical gauge fields
surrounding the vortex should be considered in order to fulfill the principle
of angular momentum conservation. Building upon this principle, we attempt a
general classification scheme of magnetic vortices in different phases of
matter, that can hopefully be applied to other physical systems.

A natural starting point of our discussion lies in the vortices in Type-II
superconductors. Quite generally, it is easy to see that the vortex should
carry a nonvanishing orbital angular momentum. Due to one of the Maxwell
equations, $\bm{\nabla}\times\bm{B}=\bm{j}$ (where we chose a natural unit in
which the magnetic constant $\mu_{0}$ is the unity), a smooth and finite
ranged profile of magnetic flux means the existence of azimuthal component of
the current density $\bm{j}$. Under a fairly general assumption that the
charge and momentum carriers are nonrelativistic quasi-electrons in the
conduction band forming the Fermi surface with superconducting gap, the
current and the momentum are linearly related as an operator relation, holding
for all states: $\bm{j}=-{e\over m}\bm{P}$, where $\bm{P}$ is the momentum
density operator, $m$ is the effective mass of conduction electrons, and $-e$
is the charge of electrons. Since $\bm{j}\neq\bm{0}$, we have
$\bm{P}\neq\bm{0}$ and the finite sized vortex should carry a finite angular
momentum by $\bm{L}=\int_{x}\bm{x}\times\bm{P}\neq\bm{0}$.

The linear relation between the current and the momentum for nonrelativistic
electrons is a consequence of Galilean invariance, and is not necessarily
universal. Even though the dispersion relation deviates from the
nonrelativistic Galilean invariant one, the current and the momentum are still
negatively correlated, and there is no reason to exclude nonvanishing angular
momentum. This discussion also implies that the quantization of the angular
momentum in units of $\hbar$ may not be universal. To complicate the situation
more nontrivially, some vortices may also carry a localized electric charge
Khomskii and Freimuth (1995); Clayhold  _et al._ (2003), and the resulting
electromagnetic (EM) field around such a vortex gives rise to a nonvanishing
Poynting vector around the vortex core axis. The total angular momentum should
then include a contribution from the EM field around the vortex.

Although the above features are robust, one can consider the following
thought-experiment, that is somewhat similar to Feynman’s angular momentum
paradox. One places a solenoid below a superconductor sample, and turns on an
external magnetic field to create magnetic vortices piercing the
superconductor. The process can be implemented in azimuthal symmetric way, and
should not change the total angular momentum which is zero initially. Since
the created vortices carry finite amount of angular momentum, where can the
compensating angular momentum be found?

The answer to this question is easy to guess: the background of solid crystal
and the electrons in filled valence bands should carry the compensating
angular momentum. Their inertia is infinitely large and their rotation may not
be detectable, but the torque acting on them during the vortex creation
process should impart to them precisely the negative amount of the angular
momentum of the created vortices. In the next section we will be able to
confirm this quantitatively in a concrete model which is simple and yet
general enough to carry out the analysis of charged magnetic vortices. In this
case the angular momentum carried by EM field also needs to be counted in the
total angular momentum, and the angular momentum conservation holds true quite
nontrivially only after including this EM contribution. We note that the EM
field is localized around and attached to the vortex, so one should think of
it as a part of the vortex profile.

An obvious next question as a continuation of the above thought-experiment of
creating the vortex by a hypothetical solenoid is: what would happen in a
system that has no background matter to absorb the angular momentum? A
concrete example of such system is provided by a relativistic field theory
which is self-consistent by itself without any other degrees of freedom: it
could be identified as the electroweak sector of the Standard Model with Higgs
field condensate, or more simply, a theoretical model by Nielsen and Olesen
Nielsen and Olesen (1973). For this class of examples, our previous argument
of angular momentum conservation becomes powerful enough to dictate that any
magnetic vortices, either charged or not, should carry zero angular momentum.
We call them “spinless vortices.” We will show that this statement is indeed
true for the Nielsen-Olesen model. In showing this for the charged vortex
case, it is again critical to include the EM or gauge contribution to the
total angular momentum. For a similar conclusion for the dyonic solitons, see
Refs. Van der Bij and Radu (2002); van der Bij and Radu (2003); Navarro-Lerida
_et al._ (2014). We make a remark that Appendix C of a renowned paper, Ref.
Julia and Zee (1975), contains an erroneous statement on this for the charged
vortex case, which we will rectify. With this example we showcase the
nontriviality of our argument of angular momentum conservation.

What would happen if a vortex consists of a combination of magnetic vortex of
gauge symmetry and superfluid vortex of global symmetry? To answer this
question, we take an example of the “non-Abelian vortex” Balachandran  _et
al._ (2006); Nakano  _et al._ (2008) in the CFL superconducting phase of dense
quark matter, which may be relevant for the physics of neutron star cores. The
non-Abelian CFL vortices also play a role in the idea of quark-hadron
continuity Schäfer and Wilczek (1999) in the high density region of QCD phase
diagram Hatsuda  _et al._ (2006); Cipriani  _et al._ (2012); Alford  _et al._
(2019); Chatterjee  _et al._ (2019) (see, however, Refs. Cherman  _et al._
(2019); Hirono and Tanizaki (2019); Cherman  _et al._ (2020) for recent
debates on the idea of quark-hadron continuity). In this example, color
symmetry is obviously (non-Abelian) gauged, and the global symmetry is
associated with the $U(1)_{\rm B}$ baryon number. For such an object of
composite nature, one can imagine a creation process by an external magnetic
field for the gauge symmetry together with a physical rotation of the
superfluid for the global symmetry. Our angular momentum conservation argument
then predicts that the total angular momentum should be given only by the
superfluid part of global symmetry without any contribution from the gauged
symmetry part. We will explicitly confirm this expectation in a highly
nontrivial manner.

There is a logical exception to the above argument for spinless vortices in a
system with no background. During the above considered creation process by an
external solenoid, a finite amount of angular momentum may diffuse away to
spatial infinity, resulting in an opposite amount of angular momentum
localized around the vortex. The total angular momentum is conserved and zero,
but the part at infinity is not visible, and should not be thought of as a
contribution to the angular momentum of the localized vortex. The vortex then
carries a left-over angular momentum that is finite. What distinguishes this
case from our first case with background matter is that the opposite angular
momentum to the one carried by the vortex strictly resides at spatial
infinity, or more precisely at the boundary of the system far away from the
vortex. This makes a contrast to the previous case with background matter,
where the bulk of the background absorbs a finite angular momentum.

An instructive example of this class of vortex is provided by the magnetic
vortex on the surface of a time-reversal invariant (i.e., $T$-invariant)
strong topological insulator (TI) in a setup recently studied in Ref. Nogueira
_et al._ (2018). Although the authors of Ref. Nogueira  _et al._ (2018)
considered an interface between TI and a superconductor, we will focus on the
TI part to account clearly for the physics origin of the net fractional (in
units of $\hbar$) angular momentum of the vortex. We will show that the total
angular momentum solely arises from the gauge field configuration surrounding
the vortex on the TI surface, without any TI matter contributions. We will
argue for this peculiar feature that the topological nature of the TI is
responsible for moving apart a finite angular momentum to the (infinitely
separated) boundary, which characterizes this class of example. Ubiquitous
topological vortices with fractional angular momentum in topological phases of
matter as found in Refs. Seiberg and Witten (2016); Seiberg  _et al._ (2016);
Senthil  _et al._ (2019) should belong to this class of magnetic vortex.

The final class of vortex in our classification is delineated by the last
logical possibility: a vortex may not be created by our thought-experiment
with external solenoid in a way that conserves angular momentum, and
additional operations to violate angular momentum conservation must be
performed to create a vortex. This class of vortex is rather exotic and rarely
found in the literature: one example we address in this paper is an object
called the “charged semilocal vortex” of Abraham Abraham (1993). Since this
class of vortex simply falls outside of our principle of angular momentum
conservation, they may or may not carry an angular momentum: in our example
the charged semilocal vortex carries a finite angular momentum. It is an
inhomogeneous profile along the vortex axis that makes it impossible to create
this kind of vortices by simply piercing an external magnetic flux: an
additional “twisting” or “spinning” along the axis is needed to create such a
vortex profile.

In summary, we have the following distinct classes of vortices in regard to
angular momentum conservation and their creation processes:

  * •

Class Ia (spinful vortices): They carry a finite angular momentum due to the
existence of background matter that can absorb angular momentum. Examples are
the vortices in Type-II superconductors.

  * •

Class Ib (topological vortices): They carry a finite angular momentum, but no
background matter exists in the bulk. The angular momentum resides only on the
boundary. The angular momentum carried by the surrounding gauge fields must be
counted for the total angular momentum. Examples are the vortices on the
surface of topological phases of matter.

  * •

Class II (spinless vortices): They do not carry a net angular momentum due to
the angular momentum conservation. The angular momentum carried by the
surrounding gauge fields should be included. Examples are the vortices in
relativistic field theories and cosmology.

  * •

Class III (exotic vortices): They have an inhomogeneous profile along the
vortex axis, so that they cannot be created by a simple procedure of piercing
magnetic flux. They may or may not carry angular momentum. An example is the
charged semilocal vortex.

In the following sections, we present detailed analysis on concrete examples
that belong to each of the above classes, in order.

##  II Class I: Case study of magnetic vortices with nonzero angular momentum

In this section we discuss the case of magnetic vortices that carry a nonzero
angular momentum. Because the angular momentum should be conserved as long as
rotational symmetry is preserved, the angular momentum of magnetic vortices,
if it is nonzero, should be balanced with other contributions. According to
the types of such balancing contributions, we further classify them into two
distinct subclasses; namely, Class Ia and Class Ib.

###  II.1 Class Ia – Incomplete cancellation due to background matter

The most familiar magnetic vortices in Type-II superconductor belong to Class
Ia. The magnetic vortices can carry a nonzero angular momentum but its value
is not quantized in units of $\hbar$, unlike the angular momentum carried by
superfluid vortices. Explicit calculations as shown below make clear where the
difference appears.

For an explicit demonstration we shall consider a relativistic scalar field
theory in the Higgs phase of U(1) symmetry, so that gauged magnetic vortices
emerge. We then take the nonrelativistic reduction and find the equations of
motion that are familiar in condensed matter physics describing magnetic
vortices in Type-II superconductors. The Lagrangian density we study in the
natural unit system ($\hbar=c=1$) reads as

$\mathcal{L}=(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)-U(\phi)+\frac{1}{2}\bm{E}^{2}-\frac{1}{2}\bm{B}^{2}-qA_{0}\,,$
(1)

where $D_{\mu}=\partial_{\mu}+ien_{e}A_{\mu}$ is the covariant derivative,
with $n_{e}$ being the electric charge carried by $\phi$ in units of $e>0$. As
usual, $\bm{E}=-\bm{\nabla}A_{0}-\partial_{0}\bm{A}$ and
$\bm{B}=\bm{\nabla}\times\bm{A}$ are electric and magnetic fields,
respectively. We should choose $n_{e}=-2$ for the Cooper pair in electron
superconductivity. The last term, $qA_{0}$, with a background charge density
$q(x)$, is introduced to keep the total electric charge neutrality, which we
will simply refer to as the “background” in the following. For example, in a
solid with conduction electrons, positively charged ions in the crystal and
other electrons in valence bands neutralize the whole system. We also note
that a finite chemical potential $\mu$ will be introduced by replacing
$ien_{e}A_{0}\to ien_{e}A_{0}-i\mu$. The potential, $U(\phi)$, is chosen to
have a nonzero condensate of $\phi$ in the Higgs phase, the simplest choice of
which would be a polynomial form:

$U(\phi)=-\lambda_{2}|\phi|^{2}+\frac{\lambda_{4}}{2}(|\phi|^{2})^{2}\,.$ (2)

The equations of motion from the Lagrangian are given by

$\displaystyle-(D_{\mu}D^{\mu})\phi+\lambda_{2}\phi-\lambda_{4}|\phi|^{2}\,\phi=0\,,$
(3)
$\displaystyle-\partial_{0}\bm{E}+\bm{\nabla}\times\bm{B}+ien_{e}\bigl{[}(\bm{D}\phi)^{\dagger}\phi-\phi^{\dagger}(\bm{D}\phi)\bigr{]}=0\,,$
(4)
$\displaystyle\bm{\nabla}\cdot\bm{E}+ien_{e}\bigl{[}(D_{0}\phi)^{\dagger}\phi-\phi^{\dagger}D_{0}\phi\bigr{]}-q=0\,,$
(5)

with $(\bm{D})^{i}\equiv D^{i}$, where we note that
$\partial^{i}=\partial/\partial x_{i}=-\partial/\partial x^{i}$ in our metric
convention $(+,-,-,-)$. The magnetic vortices we consider are the static
solutions of the above equations of motion, so we drop the time derivative
terms in the below. Then, Eq. (3) takes a form of

$\bigl{[}(\mu-
en_{e}A^{0})^{2}+\bm{D}^{2}+\lambda_{2}\bigr{]}\phi-\lambda_{4}|\phi|^{2}\phi=0\,.$
(6)

Instead of solving this equation directly, we would like to make the problem
close to a more conventional situation in condensed matter physics, by
considering the nonrelativistic reduction. Because the nonrelativistic energy
is measured from the rest mass energy $m$, we should split the mass term and
rescale the field as

$\mu\to
m+\tilde{\mu}\,,\qquad\lambda_{2}\to-m^{2}+2m\tilde{\lambda}_{2}\,,\qquad\phi\to\frac{\psi}{\sqrt{2m}}\,,$
(7)

where $\tilde{\mu}$ denotes the nonrelativistic chemical potential. Equation
(6) multiplied by $1/\sqrt{2m}$ then becomes

$(\tilde{\lambda}_{2}+\tilde{\mu}-en_{e}A^{0})\psi+\frac{\bm{D}^{2}}{2m}\psi-\frac{\lambda_{4}}{4m^{2}}|\psi|^{2}\psi=0\,,$
(8)

where we have dropped a subleading term proportional to
$(\tilde{\mu}-en_{e}A^{0})^{2}/(2m)$.

We should solve Eq. (8) together with Eqs. (4) and (5) for EM fields. The
Gauss law (5) reads:

$\bm{\nabla}^{2}A^{0}+en_{e}|\psi|^{2}+q=-\frac{en_{e}}{m}(\tilde{\mu}-en_{e}A^{0})|\psi|^{2}\simeq
0\,,$ (9)

where we again drop the last term which is subleading according to the
approximation made in Eq. (8), while we still keep the kinetic term
$\bm{D}^{2}/(2m)$ in Eq. (8). For notational brevity, let us rename our
variables as follows:

$\tilde{\lambda}_{2}+\tilde{\mu}\;\to\;\mu\,,\qquad\frac{\lambda_{4}}{4m^{2}}\;\to\;g\,,\qquad
A^{0}=\frac{\mu}{en_{e}}a\,,\qquad\psi\;\to\;\sqrt{\frac{\mu}{g}}\,\Psi\,.$
(10)

Here, we note that this $\mu$ is different from the original one in Eq. (6).
Together with the Maxwell equation for $\bm{A}$ in Eq. (4), our equations
finally become

$\displaystyle(1-a)\Psi+\frac{1}{m_{H}^{2}}(\bm{\nabla}-ien_{e}\bm{A})^{2}\Psi-|\Psi|^{2}\Psi=0\,,$
(11)
$\displaystyle\bm{\nabla}\times(\bm{\nabla}\times\bm{A})+m_{V}^{2}\biggl{[}\bm{A}|\Psi|^{2}-\frac{i}{2en_{e}}\bigl{(}\Psi\bm{\nabla}\Psi^{\dagger}-\Psi^{\dagger}\bm{\nabla}\Psi\bigr{)}\biggr{]}=0\,,$
(12)
$\displaystyle\bm{\nabla}^{2}a+2m^{2}\frac{m_{V}^{2}}{m_{H}^{2}}(|\Psi|^{2}+\tilde{q})=0\,,$
(13)

where $\tilde{q}\equiv(g/en_{e}\mu)q$, and we also introduce the two typical
mass scales as

$m_{H}^{2}\equiv 2m\mu\,,\qquad m_{V}^{2}\equiv\frac{(en_{e})^{2}\mu}{mg}\,.$
(14)

Physically, $1/m_{H}$ represents the coherent length of the field $\Psi$,
while $1/m_{V}$ represents the penetration length of the magnetic field. If
the penetration length is smaller than the coherent length, $m_{V}>m_{H}$, the
Meissner screening effect is dominant and the phase separation is more
preferred than forming magnetic vortices, which corresponds to Type-I
superconductivity. We are interested in Type-II superconductivity in the
opposite regime with $m_{H}>m_{V}$.

The Ansatz for the vortex solution with the winding number $\nu$ is

$\Psi=f(r)\,e^{i\nu\varphi}\,,\qquad a=a(r)\,,\qquad
A^{i}=-\frac{\nu}{en_{e}}\,\varepsilon^{ij}\,\frac{x^{j}}{r^{2}}\bigl{[}1-h(r)\bigr{]}\,,$
(15)

where $r\equiv\sqrt{x^{2}+y^{2}}$ and $\tan\varphi\equiv y/x$. Introducing a
dimensionless radial coordinate, $\rho\equiv m_{V}\,r$, we can rewrite the
differential equations (with ${}^{\prime}\equiv{d\over d\rho}$) as

$\displaystyle-(\rho\,f^{\prime})^{\prime}+\frac{\nu^{2}h^{2}}{\rho}\,f+\lambda\,\rho\,f(f^{2}-1+a)=0\,,$
(16)
$\displaystyle\rho\biggl{(}\frac{h^{\prime}}{\rho}\biggr{)}^{\prime}-f^{2}h=0\,,$
(17)
$\displaystyle\frac{1}{\rho}(\rho\,a^{\prime})^{\prime}+\frac{2}{\lambda}\,\frac{m^{2}}{m_{V}^{2}}(f^{2}+\tilde{q})=0\,,$
(18)

where $\lambda\equiv m_{H}^{2}/m_{V}^{2}>1$. For the total charge neutrality
condition, we impose the condition,

$\int_{\bm{x}}\tilde{q}=-\int_{\bm{x}}\,f^{2}\,.$ (19)

Here, $\int_{\bm{x}}$ refers to the two-dimensional integration on the plane
perpendicular to the vortex axis. This neutrality condition is demanded by the
fact that the static potential would behave as $a(\rho\gg 1)={Q\over
2\pi}\log\rho$ if the total net charge $Q$ is nonzero. The combination of
$(\mu-en_{e}A^{0})$ appears in the equations of motion and it plays a role of
an effective chemical potential. To have a well-defined effective chemical
potential at spatial infinity, we should impose $Q=0$.

We can numerically solve these differential equations with appropriate
boundary conditions. Let us first consider the conventional “locally neutral”
vortex solution without coupling to electric field, so that $a(r)=0$ simply.
This can be achieved by choosing a space dependent background charge density
$\tilde{q}(x)$ that locally neutralizes the net charge; that is,
$f^{2}+\tilde{q}=0$, leading to $a(r)=0$ from Eq. (18). Most Type-II vortices
behave this way, but there are examples where this does not happen in general;
see Refs. Khomskii and Freimuth (1995); Clayhold  _et al._ (2003). The
regularity of $\Psi$ at $\rho=0$ requires $f(0)=0$, and at infinity it should
approach the vacuum value of $f(\infty)=1$. In the absence of $a$, then the
boundary conditions should be

$f(0)=0\,,\qquad f(\infty)=1\,,\qquad h(0)=1\,,\qquad h(\infty)=0\,.$ (20)

We can easily obtain the numerical solutions using the shooting method to
satisfy these boundary conditions. The left panel of Fig. 1 shows an example
of the profile of the magnetic vortex for $\lambda=1.5$. We see that $h(\rho)$
extends more widely than $f(\rho)$, reflecting $m_{H}>m_{V}$.

Figure 1: (Left panel) Profile of the conventional elementary ($\nu=1$)
magnetic vortex; $f$ and $h$ without coupling to $a$ for $\lambda=1.5$. (Right
panel) Profile of the elementary vortex with the electric field; $f$, $h$, and
$a$ for $\lambda=1.5$ and $m^{2}/m_{V}^{2}=1$.

As a nontrivial example where the local charge density and the electric field
are nonvanishing, let us consider a constant background charge density
$\tilde{q}$, that is determined by the total charge neutrality condition (19)
as

$\tilde{q}=-\frac{1}{S}\int_{\bm{x}}\,f^{2}\,,$ (21)

with $S\equiv\int_{\bm{x}}$ is the transverse area. In the limit of infinitely
large system $\tilde{q}$ would approach the negative unity. In the present
case we should revise the boundary conditions accordingly. That is, $f$ needs
not be unity at large $\rho$, but $f^{2}-1+a$ should be vanishing as $\rho$
gets large. Also, we physically require vanishing electric field at $\rho=0$
and $\rho\to\infty$. Therefore, we impose the following boundary conditions:

$f(0)=0\,,\quad f(\infty)=\sqrt{1-a(\infty)}\,,\quad h(0)=1\,,\quad
h(\infty)=0\,,\quad a^{\prime}(0)=0\,,\quad a^{\prime}(\infty)=0\,.$ (22)

Actually, these boundary conditions are not sufficient to determine the
numerical solution uniquely, but a shift of $a(\rho)\to a(\rho)+c$ with a
constant $c$ still exists. This shift would change the value of $\mu$, and the
magnitude of condensate would also be modified, which would result in a
different value of $\tilde{q}$ in Eq. (21). In other words, we can adjust
$\tilde{q}$ to make a constant shift on $a(\rho)$. To fix this freedom, a
natural condition to impose would be to set $a\to 0$ at large $\rho$, so that
the effective chemical potential at infinity, by definition, remains to be
$\mu$. We choose $\lambda=1.5$ and $m^{2}/m_{V}^{2}=1$ to find that
$a(\infty)\to 0$ is realized with $\tilde{q}\simeq-0.985$. In the right panel
of Fig. 1 we present the numerical solution with these parameters. This
explicitly demonstrates that nontrivial solutions with nonzero local charge
density and electric field certainly exist. We see that the profile of
condensate slightly shrinks as compared to the locally neutral case shown in
the left panel.

Let us now compute the angular momenta carried by the matter and the EM
fields. The matter part of the angular momentum per unit vortex length is

$L^{\rm
matter}_{z}=\int_{\bm{x}}\psi^{\dagger}\Bigl{(}\frac{\hbar}{i}\,D_{\varphi}\Bigr{)}\psi\,,\qquad
D_{\varphi}\equiv\partial_{\varphi}-{ien_{e}\over\hbar}\,A_{\varphi}\,,\qquad
A_{\varphi}\equiv\epsilon^{ij}x^{i}A^{j}=\frac{\nu\hbar}{en_{e}}[1-h(r)]\,,$
(23)

where we reinstate $\hbar$ as a common unit for the angular momentum and also
change the variables back to $r$ and $\varphi$. We note that the boundary
condition (22) guarantees $D_{\varphi}[f(r)e^{i\nu\varphi}]\to 0$ as
$r\to\infty$, and the above integral is convergent. It should be mentioned
that the above form of the angular momentum using $D_{\varphi}$ corresponds to
the kinetic angular momentum, that is the angular momentum carried by matter
alone. We could have defined the canonical angular momentum using
$\partial_{\varphi}$. It is straightforward to find:

$L^{\rm
can,matter}_{z}=\int_{\bm{x}}\psi^{\dagger}\Bigl{(}\frac{\hbar}{i}\,\partial_{\varphi}\Bigr{)}\psi=\nu(2\pi\hbar)\frac{\mu}{g}\int_{0}^{R}dr\,r\,f^{2}(r)=\nu\hbar\,N\,,$
(24)

where $N\equiv\frac{\mu}{g}\int_{\bm{x}}f^{2}$ is the total number of
particles per unit vortex length, and $R$ is the size of the system in radial
direction. This expression is identical to the well-known one for the
quantized angular momentum of a superfluid vortex. In a gauge theory, there is
a contribution from the gauge field:

$L^{\rm
can,gauge}_{z}=\int_{\bm{x}}\bigl{[}\bm{E}\cdot\partial_{\varphi}\bm{A}+(\bm{E}\times\bm{A})_{z}\bigr{]}\,,$
(25)

which vanishes in the vortex configuration (15). The sum of $L^{\rm
can,matter}_{z}$ and $L^{\rm can,gauge}_{z}$ gives the total angular momentum
$L^{\rm can}_{z}$ which is conserved. Alternatively we can consider the
conserved angular momentum as the sum of $L^{\rm matter}_{z}$ and the EM
contribution, $L^{\rm gauge}_{z}$, i.e., $L^{\rm total}_{z}=L^{\rm
matter}_{z}+L^{\rm gauge}_{z}$. Which of $L^{\rm can}_{z}$ or $L^{\rm
total}_{z}$ is the relevant angular momentum depends on the physical setup. In
our present setup we can gradually turn on the magnetic field, so that the
magnetic vortex emerges. In this case, it makes sense to consider $L^{\rm
total}_{z}$, not $L^{\rm can}_{z}$. Although the difference between $L^{\rm
can}_{z}$ and $L^{\rm total}_{z}$ is the only boundary term, it plays an
essential role in the conservation of angular momentum as discussed in Sec.
III.1. For more discussions on the canonical angular momentum, see a concrete
analysis in Ref. Greenshields  _et al._ (2014) and also a general
consideration in Ref. Leader (2016).

With the explicit forms of the vortex profile and the associated vector
potential, the matter part of the angular momentum per unit vortex length
becomes

$L^{\rm
matter}_{z}=\nu(2\pi\hbar)\frac{\mu}{g}\int_{0}^{R}dr\,r\,h(r)\,f^{2}(r)\,.$
(26)

The difference from the canonical expression (24) is the presence of $h(r)$ in
the integrand. Because $h(r)$ decays when $f(r)$ increases as in Fig. 1, we
see that $L^{\rm matter}_{z}$ is smaller than $L^{\rm can}_{z}$. Let us next
consider the EM contribution, i.e.,

$L^{\rm
gauge}_{z}=\int_{\bm{x}}\bigl{[}\bm{x}\times(\bm{E}\times\bm{B})\bigr{]}_{z}\,.$
(27)

Plugging the explicit forms of $\bm{E}$ and $\bm{B}$ into the above, we find
$L^{\rm gauge}_{z}$ as

$L^{\rm
gauge}_{z}=-(2\pi\hbar)\frac{\nu\mu}{(en_{e})^{2}}\int_{0}^{R}dr\,\biggl{[}r\frac{da(r)}{dr}\biggr{]}\,\frac{dh(r)}{dr}\,.$
(28)

In Fig. 2 the integrand corresponding to the local angular momentum density is
plotted, where the variables are made dimensionless again.

Figure 2: The integrand of Eq. (28) in terms of dimensionless variables for
$\lambda=1.5$ and $m^{2}/m_{V}^{2}=1$, which represents the local distribution
of the EM angular momentum.

The angular momentum distribution is peaked around $\rho\sim 1$ and decays at
large $\rho$. We can perform an integration by part and use the equation of
motion to transform the above expression into

$L^{\rm
gauge}_{z}=-(2\pi\hbar)\frac{\nu\mu}{(en_{e})^{2}}\biggl{\\{}r\frac{da(r)}{dr}h(r)\biggl{|}_{0}^{R}+2\,\frac{m^{2}}{\lambda}\int_{0}^{R}dr\,r\,h(r)\bigl{[}f^{2}(r)+\tilde{q}(r)\bigr{]}\biggr{\\}}\,.$
(29)

Because of the boundary conditions (22), the surface contribution vanishes.
Using $\lambda=m_{H}^{2}/m_{V}^{2}$ we can simplify the above expression into

$L^{\rm
gauge}_{z}=-\nu(2\pi\hbar)\frac{\mu}{g}\int_{0}^{R}dr\,r\,h(r)\,\bigl{[}f^{2}(r)+\tilde{q}(r)\bigr{]}\,.$
(30)

Comparing with the matter contribution $L_{z}^{\rm matter}$ in Eq. (26), the
first term is remarkably equal to $-L_{z}^{\rm matter}$, and the total kinetic
angular momentum is thus,

$L^{\rm
total}_{z}=\,-\nu(2\pi\hbar)\frac{\mu}{g}\int_{0}^{R}dr\,r\,h(r)\tilde{q}(r)\,.$
(31)

We see that $L^{\rm total}_{z}$ is proportional to $\tilde{q}$ and this
nonzero value of the total angular momentum is attributed to the presence of
the background. If we had no background, $\tilde{q}=0$, then $L^{\rm
matter}_{z}$ and $L^{\rm gauge}_{z}$ would have perfect cancellation, but we
then allow for a “charged” magnetic vortex. This might be possible due to
finiteness of the system bounded by $R$. A natural realization of this
possibility will be discussed as Class II in the next section.

###  II.2 Class Ib – Incomplete cancellation due to topological boundary

Our next example for incomplete cancellation has been motivated by the
physical setup discussed in Ref. Nogueira  _et al._ (2018) where a fractional
angular momentum in the units of $\hbar$ is found to be carried by the
magnetic vortex at the interface between a superconductor and a $T$-invariant
strong topological insulator (TI)111We note that our result derived in the
following is different by a factor $1/2$ from Ref. Nogueira  _et al._ (2018).
We have identified where this difference stems from, but it is not essential
for our present argument, so we will not go into that detail.. We will
consider a simplified situation that still demonstrates the essential physics
involved; we will show that a nonzero and fractional angular momentum is
localized around a magnetic flux on the boundary surface between a bulk TI and
the vacuum outside. Let us think of this situation from a different
perspective. In the same way we discussed Class Ia in the previous section, we
can imagine a procedure to turn on the magnetic field gradually from zero, and
yet the angular momentum conservation guarantees zero total angular momentum
of the whole system. The only way our result of fractional angular momentum
can be consistent with the angular momentum conservation is that the other
compensating angular momentum should be located in the other part of the TI-
vacuum boundary where the magnetic flux leaves out from the bulk TI. If this
boundary region is far separated from the place where the original incoming
flux enters the TI, we can reasonably neglect this far-away region, and focus
only on the angular momentum localized on the incoming flux alone. This
angular momentum indeed takes a fractional value, as we confirm in the
following discussions. We can say that the fractional angular momentum is
transported from the boundary at infinity to the incoming magnetic flux; this
characterizes the magnetic vortices of Class Ib in our classification. Such
magnetic vortices with fractional angular momenta are not peculiar, but rather
ubiquitous in topological phases of matter; see, for example, Refs. Seiberg
and Witten (2016); Seiberg  _et al._ (2016); Senthil  _et al._ (2019). A
deeper insight to the angular momentum conservation from our discussion should
be useful for better understanding of these systems.

Figure 3: Interface between the TI and the vacuum with a localized flux of
magnetic field $\bm{B}$. The electric charge $Q$ is stored at the interface
which produces the electric field $\bm{E}$.

Let us consider a situation where we have a TI bulk in the $z>0$ region and
the vacuum in $z<0$, with an interface at the $z=0$ surface, as illustrated in
Fig. 3. It is well known that the boundary of TI supports massless surface
states that can be described by a single Dirac fermion field. For our purpose,
let us assume that there are $T$-violating magnetic impurities on the surface,
that opens a mass gap for the surface fermions. Integrating out the massive
surface fermion gives rise to a new term in the effective action in the low
energy limit for the EM fields, which is the Chern-Simons action with a half
integer level, $\nu={1\over 2}$ Qi  _et al._ (2008) (which should not be
confused with the winding number in the previous subsection). To capture the
essential physics of our discussion, we will consider an idealized situation
that this is the only response of the TI surface (with $T$-violating
impurities) to an externally applied EM field. At least in long wavelength and
time limit, the Chern-Simons term becomes dominant over other higher
derivative terms in the action.

From the Chern-Simons action, the charge current in response to an applied EM
field is obtained as

$j^{\mu}=-{\nu\over 2}{e^{2}\over
h}\,\epsilon^{\mu\nu\alpha}F_{\nu\alpha}=-{e^{2}\over
8\pi\hbar}\,\epsilon^{\mu\nu\alpha}F_{\nu\alpha}\,,$ (32)

which in components reads as

$Q={e^{2}\over 4\pi\hbar}B_{z}\,,\qquad j_{x}={e^{2}\over
4\pi\hbar}E_{y}\,,\qquad j_{y}=-{e^{2}\over 4\pi\hbar}E_{x}\,.$ (33)

where $Q$ and $j_{x,y}$ are the charge density and the quantum Hall effect
(QHE) current, respectively. Here, $j_{x,y}$, $B_{z}$, and $E_{x,y}$ in Eq.
(33) represent 3D vector components without distinction between upper and
lower indices.

We consider a magnetic flux that is vertically piercing the interface and is
cylindrically symmetric: $\bm{B}=B_{z}(r)\hat{z}$, where $r$ is the radius in
the $x$-$y$ plane. We further assume that $B_{z}(r)$ is localized for $r\leq
R$, so we can regard it as a flux tube like a magnetic vortex. In fact, we may
realize such a magnetic profile by an external superconducting vortex as
postulated in Ref. Nogueira  _et al._ (2018). As seen from Eq. (33) the
magnetic flux induces a surface charge density $Q$ and this charge gives rise
to a nonzero electric field according to the Gauss law. It is easy to
understand that a nonzero angular momentum emerges from the resulting EM
fields which are indicated by arrows in Fig. 3.

Before going into the computation of the angular momentum carried by the EM
field, we first show that the angular momentum contribution from the TI matter
part at $z>0$ is generally vanishing. The easiest way to confirm this is to
compute the angular momentum that may be transferred to the TI surface states
as we increase the magnetic flux from zero. This is because the TI bulk is
gapped, and only the surface states may carry angular momentum in response to
the applied EM fields in the system. During the process of turning on the
magnetic flux, we have a tangential electric field
$E_{\varphi}=(xE_{y}-yE_{x})/r$ from Faraday’s law,

$2\pi rE_{\varphi}(r)=-2\pi\int_{0}^{r}dr^{\prime}\,r^{\prime}\,{\partial
B_{z}(r^{\prime},t)\over\partial t}\,.$ (34)

Then, according to Eq. (33) in the cylindrical coordinates, we have the QHE
current as $j_{r}=(xj_{x}+yj_{y})/r={e^{2}\over 4\pi\hbar}E_{\varphi}$. From
this, we can compute the torque from the EM force acting on the surface states
along the $\varphi$ direction. The EM force reads,

$F_{\varphi}=QE_{\varphi}-j_{r}\,B_{z}\,,$ (35)

where the second term represents the Lorentz force of $\bm{j}\times\bm{B}$.
Using $Q={e^{2}\over 4\pi\hbar}B_{z}$ and $j_{r}={e^{2}\over
4\pi\hbar}E_{\varphi}$, we see that the force vanishes identically, that is,
the surface states do not experience any tangential force, or torque, by the
Chern-Simons term. In fact, it is easy to verify that this result generally
holds for any geometry. We conclude that no angular momentum is carried by the
TI matter and its surface states. The angular momentum of the whole system
resides solely in the EM sector.

Now let us return back to the angular momentum in the EM sector. For static
fields satisfying $\bm{\nabla}\times\bm{E}=\bm{0}$ and the vector potential
$\bm{A}=A_{\varphi}\hat{\varphi}/r$ satisfying $\bm{\nabla}\cdot\bm{A}=0$, we
can rewrite the angular momentum of EM fields as

$L_{z}=\int_{\bm{x}}\bigl{[}\bm{x}\times(\bm{E}\times\bm{B})\bigr{]}_{z}=\int_{\bm{x}}(\bm{\nabla}\cdot\bm{E})\,A_{\varphi}-\int
A_{\varphi}(\bm{E}\cdot d\bm{S})\,,$ (36)

where the last term is the surface integral on the exterior boundary. Using
the Stokes theorem and the cylindrical symmetry, we find the vector potential
with the boundary condition, $A_{\varphi}(0)=0$, to be

$2\pi
A_{\varphi}(r)=2\pi\int_{0}^{r}dr^{\prime}\,r^{\prime}B_{z}(r^{\prime})\,.$
(37)

Then, with the Gauss law, $\bm{\nabla}\cdot\bm{E}=Q\delta(z)$, the first term
in the above expression of $L_{z}$ becomes

$\int_{\bm{x}}(\bm{\nabla}\cdot\bm{E})\,A_{\varphi}={e^{2}\over
2\hbar}\int_{0}^{\infty}dr\,rB_{z}(r)\int_{0}^{r}dr^{\prime}\,r^{\prime}B_{z}(r^{\prime})={e^{2}\over
4\hbar}\biggl{[}\int_{0}^{\infty}dr\,rB_{z}(r)\biggr{]}^{2}={e^{2}\over
16\pi^{2}\hbar}\Phi_{0}^{2}\,,$ (38)

where $\Phi_{0}$ is the total magnetic flux. For the second term, we consider
a cylindrical boundary at $r=R$, and the Stokes theorem leads to

$2\pi A_{\varphi}(R)=\Phi_{0}\,,$ (39)

which takes a constant value along the boundary. Then, the vector potential
can be taken out from the integrand, which gives

$-\int\,A_{\varphi}(\bm{E}\cdot d\bm{S})=-{\Phi_{0}\over 2\pi}\int\bm{E}\cdot
d\bm{S}=-{\Phi_{0}\over 2\pi}Q_{\rm tot}=-{e^{2}\over
8\pi^{2}\hbar}\Phi_{0}^{2}\,,$ (40)

where $Q_{\rm tot}={e^{2}\over 4\pi\hbar}\Phi_{0}$ from Eq. (33) is used.
Summing the above two terms, we get the total angular momentum as

$L_{z}=-{e^{2}\over 16\pi^{2}\hbar}\Phi_{0}^{2}$ (41)

with the right sign that can easily be confirmed. We note that the original
integral of the angular momentum is convergent by itself as $B_{z}(r)$ is of
finite range, and the above way to split it into two terms is just a
mathematical manipulation for convenience.

We shall suppose that the magnetic flux is quantized as if it were provided by
an adjacent superconducting vortex of the winding number $\nu$ considered in
Ref. Nogueira  _et al._ (2018). We note that the magnetic vortex in
superconductivity does not carry a finite net angular momentum except for the
background contribution, so that the total angular momentum of our interest is
still given by the above formula. The flux quantization gives
${2e\over\hbar}\Phi_{0}=2\pi\nu$, where the factor 2 of $\frac{2e}{\hbar}$
originates from the Cooper pair. This finally leads to

$L_{z}=-{\nu^{2}\over 16}\hbar\,.$ (42)

Therefore, the EM field surrounding the magnetic vortex between a TI and a
superconductor carries a nonzero angular momentum given in Eq. (42).

The conservation of the total angular momentum during the process of turning
on the magnetic flux requires the existence of an opposite and compensating
angular momentum somewhere else. To identify where this compensating component
is, let us consider a global geometry of the bulk TI and its closed boundary.
For simplicity we assume that the bulk TI (which is a blue shaded region in
Fig. 3) is a large ball of radius $R$ and the boundary surface is a sphere of
radius $R$. A localized magnetic tube with a flux $\Phi_{0}$ enters the TI at
$\theta=\pi$, where $\theta$ is the polar angle in 3D spherical coordinates.
The same flux leaves out from the TI at other places of the surface in
cylindrically symmetric (i.e., $\varphi$ independent) way. Let the radial
component of the magnetic field at $r_{\rm 3D}=R$ be $B_{r_{\rm 3D}}(\theta)$
as a function of $\theta$, where $r_{\rm 3D}^{2}=r^{2}+z^{2}$ is the 3D
radius. The flux conservation results in

$\int_{0}^{\pi}d\theta\,\sin\theta\,B_{r_{\rm 3D}}(\theta)=0\,.$ (43)

We consider turning on the magnetic field adiabatically from zero, and the
time-dependent magnetic field gives $E_{\varphi}$ as well as the QHE current
$j_{\theta}$, but the net force on the surface states is vanishing as we saw
before. Therefore, the total angular momentum resides in the EM fields only.

To compute the EM part of the angular momentum, we follow the same steps as
before. Previously we considered only the contribution from the incoming
magnetic flux, but if we perform the same computation including the whole TI
boundary surface, the total $L_{z}$ turns out to be zero as we show in the
following, that is consistent with our angular momentum conservation argument.
From the spherical symmetry of the TI geometry and the cylindrical symmetry of
the vector potential, we have

$2\pi A_{\varphi}(\theta)=2\pi
R^{2}\int_{0}^{\theta}d\theta^{\prime}\,\sin\theta^{\prime}\,B_{r_{\rm
3D}}(\theta^{\prime})\,.$ (44)

The Gauss law gives

$\bm{\nabla}\cdot\bm{E}={e^{2}\over 4\pi\hbar}B_{r_{\rm
3D}}(\theta)\delta(r_{\rm 3D}-R)\,.$ (45)

Then, we find the first term in Eq. (36) to be

$\displaystyle\int_{\bm{x}}(\bm{\nabla}\cdot\bm{E})\,A_{\varphi}$
$\displaystyle=2\pi R^{4}{e^{2}\over
4\pi\hbar}\int_{0}^{\pi}\,d\theta\,\sin\theta\,B_{r_{\rm
3D}}(\theta)\int_{0}^{\theta}\,d\theta^{\prime}\,\sin\theta^{\prime}\,B_{r_{\rm
3D}}(\theta^{\prime})$ $\displaystyle=2\pi R^{4}{e^{2}\over
4\pi\hbar}\,{1\over 2}\biggl{[}\int_{0}^{\pi}\,d\theta\,\sin\theta\,B_{r_{\rm
3D}}(\theta)\biggr{]}^{2}=0\,,$ (46)

using Eq. (43). For the second term in Eq. (36), we can still employ Eq. (40)
with different $Q_{\rm tot}$. Previously we took account of $Q_{\rm tot}$
around the incoming magnetic flux only, but if we sum up all the contributions
from the whole TI surface, it should amount to $Q_{\rm tot}=0$ due to Eq.
(43). In this way we see that the second term is zero as well. We emphasize
that the original expression of the angular momentum is localized in the
region where $\bm{B}\neq\bm{0}$ and $\bm{E}\neq\bm{0}$, that is, it is
localized around the TI boundary where the magnetic flux either enters or
leaves the TI. Therefore, the fractional angular momentum localized around the
magnetic tube at $\theta=\pi$ is compensated by the angular momentum carried
by the outgoing flux in other places of the TI boundary which can be taken
infinitely away.

##  III Class II: Case study of magnetic vortices with zero angular momentum

In this section we consider magnetic vortices in relativistic field theory as
examples of self-consistent systems without any background matter or boundary
that could absorb angular momentum. Such vortices could appear in the Standard
Model and extensions of the Standard Model. They have been considered in the
context of high energy physics and cosmology. A faithful application of our
angular momentum conservation argument to these vortices then dictates that
they should be spinless. We will confirm this claim also in a nontrivial
example where the angular momentum carried by surrounding localized gauge
fields is essential for the cancellation of the total angular momentum. We
emphasize that these localized gauge fields around the vortex core should be
considered as a part of the magnetic vortex configuration under consideration.

###  III.1 Example 1: Relativistic Nielsen-Olesen vortices

Let us illustrate our main points in the simplest example of Nielsen-Olesen
vortices in relativistic scalar theory that we already treated in the previous
section. The formulation of the theory presented below is a standard one, but
we would like to pay a special attention to the cases with nonvanishing charge
density. Consequently, nonzero electric fields accompany the vortices; we then
have a precise description of the charged vortices, taking proper account of
the Gauss law constraint and the Coulomb energy contribution to the Ginzburg-
Landau free energy to be minimized. This endeavor, that we did not find in the
literature in full generality as we present here, turns out to be crucial to
show the exact cancellation of total angular momenta carried by the matter and
the gauge field parts.

This section has some redundancy with our discussions in the previous section,
but to make our analysis as self-contained as possible, let us retain some
calculational details. Showing explicit terms for our later convenience, we
write down the Lagrangian as

$\mathcal{L}=(D_{0}\Phi)^{\dagger}(D_{0}\Phi)-(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)-U(\Phi^{\dagger}\Phi)+{1\over
2}\bm{E}^{2}-{1\over 2}\bm{B}^{2}\,,$ (47)

which is Eq. (1) without background, i.e., $q=0$. Here, we take $n_{e}=1$ and
$D_{\mu}\Phi=(\partial_{\mu}+ieA_{\mu})\Phi$ and, as defined in Sec. II.1, we
adopt a convention of $(\bm{D})^{i}=D^{i}$. The EM fields are
$\bm{E}=-\bm{\nabla}A_{0}-\partial_{0}\bm{A}$ and
$\bm{B}=\bm{\nabla}\times\bm{A}$. We use the unit system with $c=\hbar=1$ in
this section. We also take a conventional form of the potential same as in the
previous section;
$U(\Phi^{\dagger}\Phi)=-\lambda_{2}\Phi^{\dagger}\Phi+{\lambda_{4}\over
2}(\Phi^{\dagger}\Phi)^{2}$. We reproduce the equations of motion and the
Gauss law from this action as

$\displaystyle-
D_{0}^{2}\Phi+\bm{D}^{2}\Phi-U^{\prime}(\Phi^{\dagger}\Phi)\Phi=0\,,$ (48)
$\displaystyle-{\partial_{0}\bm{E}}+\bm{\nabla}\times\bm{B}+ie\bigl{[}(\bm{D}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(\bm{D}\Phi)\bigr{]}=0\,,$
(49)
$\displaystyle\bm{\nabla}\cdot\bm{E}+ie\bigl{[}(D_{0}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(D_{0}\Phi)\bigr{]}=0\,,$
(50)

which are equivalent to Eqs. (3)-(5) with $q=0$ and $n_{e}=1$. We will work in
the Hamiltonian formulation of the theory to look into the dynamics further.

The canonical conjugate field is given by definition as

$\Pi^{\dagger}={\delta{\cal
L}\over\delta\partial_{0}\Phi}=(D_{0}\Phi)^{\dagger}\,.$ (51)

It should be noted that in our convention the above expression defines
$\Pi^{\dagger}$, not $\Pi$. The charge density from the Nöther method is

$Q=-i\bigl{[}(D_{0}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(D_{0}\Phi)\bigr{]}=-i(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)\,,$
(52)

and the Gauss law takes the form of

$\bm{\nabla}\cdot\bm{E}=eQ=-ie(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)\,.$ (53)

The Hamiltonian density from the Legendre transformation (including the EM
sector) is obtained as

$\displaystyle H$
$\displaystyle=\Pi^{\dagger}(\partial_{0}\Phi)+\Pi(\partial_{0}\Phi)^{\dagger}-\bm{E}(\partial_{0}\bm{A})-\mathcal{L}$
$\displaystyle=\Pi^{\dagger}\Pi+(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)+U(\Phi^{\dagger}\Phi)+{1\over
2}(\bm{E}^{2}+\bm{B}^{2})-ieA_{0}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)-A_{0}\bm{\nabla}\cdot\bm{E}$
$\displaystyle=\Pi^{\dagger}\Pi+(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)+U(\Phi^{\dagger}\Phi)+{1\over
2}(\bm{E}^{2}+\bm{B}^{2})\,,$ (54)

where we dropped the total derivative term $\bm{\nabla}\cdot(A_{0}\bm{E})$ in
the second line, and from the second to the last line, we used the Gauss law
to have cancellation between the last two terms. This should be the case since
$A_{0}$ is not a dynamical degrees of freedom in the Hamiltonian formulation
of gauge theory.

For our convenience we introduce a chemical potential $\mu$ via the free
energy to be minimized; $F=H-\mu Q$. This is equivalent to introducing $\mu$
in the covariant derivative, once $F$ in this section is identified as the
Hamiltonian density in the previous section. The free energy is explicitly
given by

$F=H-\mu
Q=\Pi^{\dagger}\Pi+(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)+U(\Phi^{\dagger}\Phi)+{1\over
2}(\bm{E}^{2}+\bm{B}^{2})+i\mu(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)\,.$ (55)

This, together with the Gauss law, constitutes a precise formulation of gauged
Ginzburg-Landau description for the cases with nonzero charge distributions.
From the Gauss law, we see that $\bm{E}$ is not independent but generated
through $\Pi$ and $\Phi$, albeit in a nonlocal way. The variables, $\Pi$,
$\Phi$, and $\bm{A}$, are considered as independent degrees of freedom, with
respect to which the free energy $F$ should be extremized to obtain the
equations of motion.

We are interested in the stationary configurations where magnetic field is
static; $\partial_{0}\bm{B}=0$. In this case, as is familiar in classical
electromagnetism, we can introduce an auxiliary function or static potential
$A_{0}$ such that $\bm{E}=-\bm{\nabla}A_{0}$ and, with a proper boundary
condition at spatial infinity, the Gauss law can be solved nonlocally as

$A_{0}=ie{1\over\bm{\nabla}^{2}}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)\,,\qquad
A_{0}(\infty)=0\,.$ (56)

This boundary condition is necessary, since a nonzero $A_{0}(\infty)$ would
shift our definition of chemical potential $\mu$, that is, the true chemical
potential is $\mu-eA_{0}(\infty)$, as we have already seen in the previous
section. Using this, one of the terms in $F$, that is, the electric field
energy is expressed as

${1\over 2}\bm{E}^{2}={e^{2}\over
2}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi){1\over\bm{\nabla}^{2}}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)\,,$
(57)

which is nothing but the Coulomb energy induced by the charge distributions.
The resulting expression for the free energy $F$ involves only the independent
variables, $\Pi$, $\Phi$, and $\bm{A}$, from which we can proceed to obtain
the equations of motion.

From the variation with respect to $\Pi^{\dagger}$, we get

$\Pi+i\mu\Phi+e^{2}\Phi{1\over\bm{\nabla}^{2}}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)=0\,.$
(58)

Using the expression for $A_{0}$, this can be written as

$\Pi+i\mu\Phi-ieA_{0}\Phi=0\qquad\text{or}\qquad\Pi=-i(\mu-eA_{0})\Phi\,.$
(59)

Recalling the relation, $\Pi=D_{0}\Phi=(\partial_{0}+ieA_{0})\Phi$, this gives
the well-known Josephson relation;

$\partial_{0}\Phi=-i\mu\Phi\,.$ (60)

Since $F$ is quadratic in $\Pi$, one may choose to insert back the solution
for $\Pi$ from Eq. (58) into $F$ to get a more conventional form of the free
energy in terms of $\Phi$ and $\bm{A}$ only. It is explicitly given by

$F=(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)+U(\Phi^{\dagger}\Phi)-\mu(\mu-
eA_{0})\Phi^{\dagger}\Phi+{1\over 2}\bm{B}^{2}\,.$ (61)

One should keep in mind that $A_{0}$ in the above expression is a functional
of $\Phi$ that should be obtained by solving the Gauss law (56) together with
Eq. (58), which is in general nonvanishing for $\mu\neq 0$ corresponding to
nonzero charge distributions in a solution. A more practical way to approach
this problem is indeed what we have described in the preceding paragraphs,
i.e., keeping $\Pi$ as an independent degree of freedom.

The variation of the free energy with respect to $\Phi^{\dagger}$ gives

$-\bm{D}^{2}\Phi+\Phi
U^{\prime}(\Phi^{\dagger}\Phi)-i\mu\Pi-e^{2}\Pi{1\over\bm{\nabla}^{2}}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)=0\,,$
(62)

which, upon using the expression for $A_{0}$, is equivalent to

$-\bm{D}^{2}\Phi+U^{\prime}(\Phi^{\dagger}\Phi)\Phi-i(\mu-eA_{0})\Pi=0\,,$
(63)

and using the solution for $\Pi$, we finally get to

$-\bm{D}^{2}\Phi+U^{\prime}(\Phi^{\dagger}\Phi)\Phi-(\mu-eA_{0})^{2}\Phi=0\,.$
(64)

This, as it should, agrees with the equation of motion for $\Phi$ obtained
from the Lagrangian, as written in Eq. (48), after using the Josephson
relation, $\partial_{0}\Phi=-i\mu\Phi$. However, we again need to recall that
$A_{0}$ is a solution of the Gauss law, which itself involves $\Pi$ and $\Phi$
as

$\bm{\nabla}^{2}A_{0}=ie(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)=-2e(\mu-
eA_{0})\Phi^{\dagger}\Phi\,.$ (65)

Equations (64) and (65) together with the Maxwell equation for $\bm{A}$, i.e.,

$\bm{\nabla}\times(\bm{\nabla}\times\bm{A})+ie\bigl{[}(\bm{D}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(\bm{D}\Phi)\bigr{]}=0\,,$
(66)

constitute our final set of closed equations for $\Phi$, $\bm{A}$ and, $A_{0}$
to be solved for classical configurations in relativistic theory. If $\mu=0$,
then it is consistent with $A_{0}=0$, and there is no electric field. This
situation at $\mu=0$ corresponds to charge neutral vortices in our problem.
For $\mu\neq 0$ there exists nonvanishing charge and the electric field in the
solution, which we will refer to as “charged Nielsen-Olesen vortices.”

We could estimate the angular momentum in the matter part using an expression
like Eq. (23), but here, we shall show an alternative physical approach. To
compute the matter contribution to the angular momentum, we need the linear
momentum density, i.e., $\bm{P}$, obtained from $T^{0i}$ component of the
energy-momentum tensor. From the Nöther method, we see that $T^{0i}$ is given
by

$\bm{P}^{i}=T^{0i}=\bigl{[}(D^{0}\Phi)^{\dagger}(\bm{D}^{i}\Phi)+(\bm{D}^{i}\Phi)^{\dagger}(D^{0}\Phi)\bigr{]}=\bigl{[}\Pi^{\dagger}(\bm{D}^{i}\Phi)+(\bm{D}^{i}\Phi)^{\dagger}\Pi\bigr{]}\,.$
(67)

Using Eq. (58) for $\Pi$, we obtain,

$\bm{P}=-i\bigl{[}(\bm{D}\Phi)^{\dagger}(\mu-eA_{0})\Phi-\Phi^{\dagger}(\mu-
eA_{0})(\bm{D}\Phi)\bigr{]}\,.$ (68)

The kinetic angular momentum carried by the matter part is then

$L_{z}^{\rm matter}=\int_{\bm{x}}\bigl{[}\bm{x}\times\bm{P}\bigr{]}_{z}\,.$
(69)

On the other hand, the gauge field contribution to the angular momentum is
easily found from the Poynting vector, namely from Eq. (27).

The charge neutral case at $\mu=0$ makes the essential difference from the
nonrelativistic case in the previous section. The above equations of motion
for relativistic vortices are mathematically identical, up to trivial scaling
of parameters, to those of “locally” neutral nonrelativistic vortices without
coupling to electric field: both of them are known as Nielsen-Olesen vortices.
For the charge neutral case, it is algebraically trivial to see $\Pi=0$ and
$\bm{P}=\bm{E}=\bm{0}$ from $\mu=A_{0}=0$, and both the matter and gauge field
contributions to the total angular momentum are zero, but this conclusion is
physically nontrivial; vortices are circulating configurations and yet they
have no angular momentum. Intuitively, the absence of matter contribution to
the angular momentum in the relativistic theory can be understood as a
cancellation between particle vortex and anti-particle anti-vortex as follows:
recall that $\Phi\sim a+b^{\dagger}$ and $\Pi\sim i(a-b^{\dagger})$ where $a$
and $b$ are annihilation operators for particle and anti-particle,
respectively. In the superfluid phase of a large occupation number, we can
regard $a$ and $b$ as c-numbers as usual. A vortex profile with winding number
$\nu$ can be viewed as a superposition of a particle vortex of $a\sim
e^{i\nu\varphi}$ and an antiparticle antivortex of $b\sim e^{-i\nu\varphi}$.
In the charge neutral case of $\Pi=0$, their amplitudes are precisely equal,
i.e., $a=b^{\dagger}$, and the antiparticle antivortex contribution to the
angular momentum is precisely opposite to that of the particle vortex. The
absence of antiparticles in the nonrelativistic vortex in the previous section
is the major difference from the relativistic theory discussed in this
section.

Figure 4: Schematic illustration of the charge neutral Nielsen-Olesen vortex
composed from a particle vortex and an antiparticle antivortex.

For the charged case at $\mu\neq 0$, the two systems of equations are
different by terms that we previously neglected in the nonrelativistic
reduction. In addition to this difference for the charged case, a background
charge density that we introduced as $q$ in the previous section is also
absent here. This means that the net charge of a charged Nielsen-Olesen
vortices is not zero, and the electric field grows logarithmically at large
distance in two-dimensional space perpendicular to the vortex string in three
dimensions. This implies that the line density of energy of a charged vortex
is divergent in infinite space, and a sensible solution would exist only in a
finite transverse volume. As we will focus on azimuthally symmetric vortex
configurations to apply our angular momentum conservation argument, we
consider a spatial cutoff in transverse space at a certain distance from the
origin, that is, $r\leq R$ in the radial direction. We will show that the
total angular momentum of a charged vortex within the volume $r\leq R$ is
always zero for any cutoff $R$, when we sum the contributions of both matter
part and the gauge fields.

Our conclusion rectifies a misleading statement in the Appendix C of the well-
known literature, Ref. Julia and Zee (1975), that a charged Nielsen-Olesen
vortex carries a nonzero angular momentum. Our result of vanishing angular
momentum even for charged vortices is independent of the issue of diverging
line energy density in infinite space. Later, we will more precisely point out
where the misleading conclusion in Ref. Julia and Zee (1975) stems from.

The computation in the charged vortex case is more delicate than the neutral
case, and the detailed mechanism for cancellation is similar to that in the
previous section. First of all, since $\Pi\neq 0$, the particle vortex and the
antiparticle antivortex have different amplitudes, and the net matter angular
momentum no longer cancels to be zero. From the radial electric field,
$\bm{E}\neq\bm{0}$, the gauge fields also contribute to the total angular
momentum. We take the following Ansatz,

$\Phi=f(r)\,e^{i\nu\varphi}\,,\qquad A_{0}=a(r)\,,\qquad A_{\varphi}={\nu\over
e}\,\bigl{[}1-h(r)\bigr{]}$ (70)

with the boundary condition for vanishing magnetic flux, i.e., $h(R)=0$ at
sufficiently large boundary $r=R$. Then the equations of motion and the Gauss
law become (with ${}^{\prime}\equiv{d\over dr}$)

$\displaystyle{1\over r}(rf^{\prime})^{\prime}-{\nu^{2}\over
r^{2}}h^{2}f+(\lambda_{2}-\lambda_{4}f^{2})f+(ea-\mu)^{2}f=0\,,$ (71)
$\displaystyle\biggl{(}{h^{\prime}\over r}\biggr{)}^{\prime}-{2e^{2}\over
r}f^{2}h=0\,,$ (72) $\displaystyle{1\over
r}(ra^{\prime})^{\prime}-2e(ea-\mu)f^{2}=0\,.$ (73)

The matter part of the angular momentum is computed from Eq. (69) as

$L_{z}^{\rm
matter}=4\pi\nu\int_{0}^{R}dr\,rh(r)f^{2}(r)\bigl{[}-ea(r)+\mu\bigr{]}$ (74)

and the gauge field contribution from Eq. (27) as

$L_{z}^{\rm gauge}=-{2\pi\nu\over
e}\int_{0}^{R}dr\,ra^{\prime}(r)h^{\prime}(r)={2\pi\nu\over
e}\int_{0}^{R}dr\,[ra^{\prime}(r)]^{\prime}h(r)\,,$ (75)

where in the last equality we performed the integration by part and used the
boundary condition at $r=R$ as in the previous section. Using the Gauss law
(73) to replace $[ra^{\prime}(r)]^{\prime}$, we arrive at

$L_{z}^{\rm
gauge}=-4\pi\nu\int_{0}^{R}dr\,rh(r)f^{2}(r)[-ea(r)+\mu]=-L_{z}^{\rm
matter}\,,$ (76)

which precisely cancels the matter contribution. As a result the total angular
momentum is vanishing. In Appendix C of Ref. Julia and Zee (1975), the surface
term of Eq. (C3) that was neglected is nonzero: this can be seen from the
description of the solution below Eq. (C6) with $Q\neq 0$. It can be shown
that Eq. (C3) precisely cancels Eq. (C4), so that the total angular momentum
is zero. This cancellation has its origin in the angular momentum
conservation, and it holds for any $R$ regardless of an issue of infinite line
energy density of the charged solution.

We finish this subsection by pointing out that our finding of zero angular
momentum for magnetic vortices in the $U(1)$ Abelian Higgs model is consistent
with the particle-vortex duality in 2+1 dimensions Seiberg  _et al._ (2016);
Senthil  _et al._ (2019), where the magnetic vortices in the Abelian Higgs
model are mapped to the elementary excitations of a dual complex scalar field
which are clearly spinless. Checking the spins of other excitations in the web
of dualities Seiberg  _et al._ (2016) would be interesting.

###  III.2 Example 2: Non-Abelian CFL vortices

Figure 5: Schematic illustration of the continuity between the dibaryon vortex
in the hadronic phase and the non-Abelian CFL vortex in CFL quark matter in
QCD.

We can test our assertion in a more non-trivial example of non-Abelian
vortices Balachandran  _et al._ (2006); Nakano  _et al._ (2008) in the CFL
color-superconducting phase of QCD quark matter at high baryon density and low
temperature. The diquark condensates in the CFL phase break both QCD gauge
symmetry and the global $U(1)_{\rm B}$ baryon number symmetry. The non-Abelian
vortices arise from coupled dynamics of color fields and $U(1)_{\rm B}$
superfluidity, and carry fractional winding numbers for both gauge and global
symmetries, such that the total winding number for each color component of the
diquark condensate field is an integer. One might think that the non-Abelian
CFL vortex is peculiar to QCD, but similar structures can also be found in
multi-component superconductivity, see Ref. Babaev (2002) for example. The
minimal non-Abelian vortex carries only $1/2$ of the $U(1)_{\rm B}$ winding
number (that is equivalent to ${1\over 2}\times{2\over 3}={1\over 3}$ winding
number for the diquark field), so that the non-Abelian CFL vortices can be
considered as fractionalized $U(1)_{\rm B}$ vortices. In the hadronic phase,
on the other hand, the minimal dibaryon Cooper-pair superfluid vortex also
carries the same winding number $1\over 2$, so that across the two phases the
dibaryon vortex should transmute to the non-Abelian CFL vortex Alford  _et
al._ (2019), which is schematically illustrated in Fig. 5 (see also Ref.
Chatterjee  _et al._ (2019) for an alternative scenario). Since the angular
momentum must be conserved during this transmutation process, we expect the
angular momenta of the two vortices to be equal. The minimal dibaryon vortex
of $1/2$ of the $U(1)_{\rm B}$ winding number is a usual superfluid vortex and
carries the angular momentum $L_{z}=N_{\rm B}/2$ where $N_{\rm B}$ is the
total baryon number. In contrast, the non-Abelian CFL vortex is also
accompanied by color gauge fields, and in general, its total angular momentum
receives contribution from these localized color fields. It is a nontrivial
check to see that the total angular momentum of the non-Abelian CFL vortex
from both matter part and the gauge fields is indeed $L_{z}=N_{\rm B}/2$,
i.e., the same as in the hadronic phase, as we will show below. Essentially,
this means that the color-magnetic part of the non-Abelian CFL vortex does not
contribute to the angular momentum, and only the $U(1)_{\rm B}$ superfluid
part makes a finite contribution. This situation provides another example of
confirming our assertion that the gauged magnetic vortex does not carry
angular momentum.

The diquark condensate in the CFL phase is described by a $3\times 3$ matrix
field, $\Phi=\Phi^{i\alpha}$, where $i$ and $\alpha$ are color and flavor
indices, respectively. More precisely, there are two such fields for left-
handed and right-handed diquarks, and we assume that they share the same
configuration in a vortex solution. We can always perform suitable color
rotations, such that the profile of the non-Abelian CFL vortex appears only in
the global $U(1)_{\rm B}$ and the eighth component of the color field
$A_{\mu}^{8}$ with the generator
$t^{8}={1\over\sqrt{12}}\mathrm{diag}(-2,1,1)$222This matrix representation is
an unconventional choice; in later discussions we will focus on the $u$-quark
sector and this choice is good for that purpose.. Therefore, we will show
expressions only in these parts in the following. The QCD covariant derivative
with $A_{\mu}^{8}$ only is

$D_{\mu}\Phi=\left(\partial_{\mu}-igA_{\mu}^{8}t^{8}\right)\Phi\,,$ (77)

where $g$ is the QCD coupling constant. We will work with the gauge invariant
Lagrangian in terms of the diquark field given by

$\mathcal{L}={\rm
tr}\bigl{[}(D_{0}\Phi)^{\dagger}(D_{0}\Phi)-(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)\bigr{]}-V\left({\rm
tr}(\Phi^{\dagger}\Phi)\right)+{1\over 2}\bm{E}^{8}\cdot\bm{E}^{8}-{1\over
2}\bm{B}^{8}\cdot\bm{B}^{8}\,,$ (78)

where $\bm{E}^{8}=-\bm{\nabla}A_{0}^{8}-\partial_{0}\bm{A}^{8}$ and
$\bm{B}^{8}=\bm{\nabla}\times\bm{A}^{8}$. The concrete shape of the potential
$V$ is not important for our purpose. The ensuing analysis is very similar to
that in the previous subsection, and we will highlight only the important
differences and the major results.

The chemical potential $\mu_{\text{B}}$ is introduced for the baryon charge
density $Q_{\rm B}$, which is

$Q_{\rm B}=-{2\over 3}i\,{\rm
tr}\bigl{[}(D_{0}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(D_{0}\Phi)\bigr{]}\,.$
(79)

Here the coefficient is understood from the baryon charge $2/3$ carried by the
diquark field $\Phi$. The color charge that appears in the Gauss law
constraint is given by

$\bm{\nabla}\cdot\bm{E}^{8}=ig\,{\rm
tr}\bigl{[}(D_{0}\Phi)^{\dagger}t^{8}\Phi-\Phi^{\dagger}t^{8}(D_{0}\Phi)\bigr{]}\,,$
(80)

which is easily obtained from the equation of motion for $A_{0}^{8}$.
Introducing the canonical conjugate field $\Pi\equiv D_{0}\Phi$ and following
the same steps in the previous section, we can find the Hamiltonian and the
free energy as

$\displaystyle F$ $\displaystyle={\rm
tr}\bigl{[}\Pi^{\dagger}\Pi+(\bm{D}\Phi)^{\dagger}(\bm{D}\Phi)\bigr{]}+V\left({\rm
tr}(\Phi^{\dagger}\Phi)\right)$ $\displaystyle\quad+{g^{2}\over 2}{\rm
tr}(\Pi^{\dagger}t^{8}\Phi-\Phi^{\dagger}t^{8}\Pi){1\over\bm{\nabla}^{2}}{\rm
tr}(\Pi^{\dagger}t^{8}\Phi-\Phi^{\dagger}t^{8}\Pi)$
$\displaystyle\quad+{2\over 3}i\mu_{\text{B}}{\rm
tr}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)+{1\over
2}\bm{B}^{8}\cdot\bm{B}^{8}\,.$ (81)

Recall that the second line arises from the Coulomb energy of color field,
${1\over 2}\bm{E}^{8}\cdot\bm{E}^{8}$ [see Eq. (57)]. By minimizing $F$ with
respect to $\Pi$, $\Phi$, and $\bm{A}^{8}$, we obtain the equations of motion.
It is convenient to introduce an auxiliary variable $A_{0}^{8}$ to render the
nonlocality into local equations of motion, which is defined by
$\bm{E}^{8}=-\bm{\nabla}A_{0}^{8}$ for static configuration, so that the Gauss
law becomes

$\bm{\nabla}^{2}A_{0}^{8}=-ig{\rm
tr}\left(\Pi^{\dagger}t^{8}\Phi-\Phi^{\dagger}t^{8}\Pi\right)\,.$ (82)

Then, the equation of motion for $\Pi$ is easily solved as

$\Pi=-i(gA_{0}^{8}t^{8}+2\mu_{\text{B}}/3)\Phi$ (83)

and the Gauss law becomes

$\bm{\nabla}^{2}A_{0}^{8}=2g\,{\rm
tr}\bigl{[}\Phi^{\dagger}t^{8}(gA_{0}^{8}t^{8}+2\mu_{B}/3)\Phi\bigr{]}\,.$
(84)

In the above the term $\propto\mu_{\text{B}}$ should be understood as the
unity matrix in color space. The other equations of motion are

$\bm{D}^{2}\Phi-\Phi V^{\prime}\left({\rm
tr}(\Phi^{\dagger}\Phi)\right)+\bigl{(}gA_{0}^{8}t^{8}+2\mu_{\text{B}}/3\bigr{)}^{2}\Phi=0\,,$
(85)

and

$\bm{\nabla}\times(\bm{\nabla}\times\bm{A}^{8})=ig{\rm
tr}\bigl{[}(\bm{D}\Phi)^{\dagger}t^{8}\Phi-\Phi^{\dagger}t^{8}(\bm{D}\Phi)\bigr{]}\,.$
(86)

Equations (84), (85), and (86) form a closed set to solve for the vortex
profile of $\Phi$, $\bm{A}^{8}$, and $A_{0}^{8}$. The non-Abelian CFL vortex
solution has the following form Balachandran  _et al._ (2006):

$A_{0}^{8}=a(r)\,,\qquad A^{8}_{\varphi}={\nu\over g}{\sqrt{12}\over
3}\bigl{[}1-h(r)\bigr{]}$ (87)

with $\bm{A}^{8}=A^{8}_{\varphi}\hat{\varphi}/r$ and

$\Phi=\begin{pmatrix}f(r)\,e^{i\nu\varphi}&0&0\\\ 0&b(r)&0\\\
0&0&b(r)\end{pmatrix}\,.$ (88)

The boundary condition is $h(\infty)=0$ which ensures,

$-\bm{D}\Phi=(\bm{\nabla}+ig\bm{A}^{8}t^{8})\Phi~{}~{}\to~{}~{}i{\nu\over
3}\,{\hat{\varphi}\over r}\Phi\quad\text{as }r\to\infty\,.$ (89)

This signifies that the vortex carries a superfluid winding number $\nu/3$
with respect to the diquark global $U(1)$ (which is equivalent to $\nu/2$ with
respect to $U(1)_{\rm B}$ symmetry). To see how the color-magnetic vortex is
embedded in the above solution, we can factorize it as follows,

$\Phi=e^{i{\nu\over 3}\varphi}\begin{pmatrix}e^{i{2\nu\over 3}\varphi}&0&0\\\
0&e^{-i{\nu\over 3}\varphi}&0\\\ 0&0&e^{-i{\nu\over
3}\varphi}\end{pmatrix}\begin{pmatrix}f(r)&0&0\\\ 0&b(r)&0\\\
0&0&b(r)\end{pmatrix}\,,$ (90)

where the overall phase corresponds to the global $U(1)$ and the middle
matrix, $e^{-i\nu{\sqrt{12}\over 3}t^{8}\varphi}$, belongs to $SU(3)$, and
this is why this configuration as implemented in Eq. (88) is called a “non-
Abelian” vortex.

The equations of motion, (84), (85), and (86), become, after some algebra,

$\displaystyle{1\over r}(ra^{\prime})^{\prime}-{g^{2}\over
3}a(2f^{2}+b^{2})-{4g\over 3\sqrt{3}}\mu_{B}(-f^{2}+b^{2})=0\,,$ (91)
$\displaystyle{1\over r}(rf^{\prime})^{\prime}-{\nu^{2}\over
9r^{2}}(1+2h)^{2}f-fV^{\prime}(f^{2}+2b^{2})+\left(-{2g\over\sqrt{12}}a+{2\over
3}\mu_{B}\right)^{2}f=0\,,$ (92) $\displaystyle{1\over
r}(rb^{\prime})^{\prime}-{\nu^{2}\over
9r^{2}}(1-h)^{2}b-bV^{\prime}(f^{2}+2b^{2})+\left({g\over\sqrt{12}}a+{2\over
3}\mu_{B}\right)^{2}b=0\,,$ (93) $\displaystyle\left({1\over
r}h^{\prime}\right)^{\prime}+{g^{2}\over 3}{f^{2}\over r}(1+2h)-{g^{2}\over
3r}(1-h)b^{2}=0\,,$ (94)

where Eq. (91) corresponds to the Gauss law.

To compute the matter part of the angular momentum, we need the momentum
density,

$\bm{P}^{i}=T^{0i}={\rm
tr}\bigl{[}(D_{0}\Phi)^{\dagger}(\bm{D}^{i}\Phi)+(\bm{D}^{i}\Phi)^{\dagger}(D_{0}\Phi)\bigr{]}={\rm
tr}\bigl{[}\Pi^{\dagger}(\bm{D}^{i}\Phi)+(\bm{D}^{i}\Phi)^{\dagger}\Pi\bigr{]}\,.$
(95)

Substituting the solution of $\Pi$ for the above $\bm{P}^{i}$, we obtain,

$\displaystyle\bm{P}$ $\displaystyle=P_{\varphi}\hat{\varphi}=-i\,{\rm
tr}\bigl{[}(\bm{D}\Phi)^{\dagger}(gA_{0}^{8}t^{8}+2\mu_{\text{B}}/3)\Phi-\Phi^{\dagger}(gA_{0}^{8}t^{8}+2\mu_{\text{B}}/3)(\bm{D}\Phi)\bigr{]}$
$\displaystyle=\biggl{[}{\nu\over
3r}(1+2h){f^{2}}\biggl{(}-{4g\over\sqrt{12}}a+{4\over
3}\mu_{\text{B}}\biggr{)}+{4\nu\over
3r}(1-h)b^{2}\biggl{(}{g\over\sqrt{12}}a+{2\over
3}\mu_{\text{B}}\biggr{)}\biggr{]}\hat{\varphi}\,.$ (96)

Thus, the matter part of the angular momentum is

$\displaystyle L_{z}^{\rm matter}$
$\displaystyle=2\pi\int_{0}^{R}dr\,\,r(rP_{\varphi})$
$\displaystyle={2\pi\nu\over
3}\int_{0}^{R}dr\,r\biggl{[}(1+2h){f^{2}}\biggl{(}-{4g\over\sqrt{12}}a+{4\over
3}\mu_{\text{B}}\biggr{)}+{4}(1-h)b^{2}\biggl{(}{g\over\sqrt{12}}a+{2\over
3}\mu_{\text{B}}\biggr{)}\biggr{]}\,.$ (97)

The color gauge field contribution is as before

$L_{z}^{\rm gauge}=-(2\pi){\sqrt{12}\nu\over
3g}\int_{0}^{R}dr\,ra^{\prime}(r)h^{\prime}(r)\,.$ (98)

Integrating by part and using the Gauss law, we have,

$L_{z}^{\rm gauge}=(2\pi\nu){\sqrt{12}\over
3}\int_{0}^{R}dr\,r\biggl{[}{g\over 3}a(2f^{2}+b^{2})+{4\over
3\sqrt{3}}\mu_{\text{B}}(-f^{2}+b^{2})\biggr{]}h\,.$ (99)

Summing up $L_{z}^{\rm matter}$ in Eq. (97) and $L_{z}^{\rm gauge}$ in Eq.
(99), we get the total angular momentum per unit vortex length to be

$L_{z}^{\rm tot}=2\pi\nu\int_{0}^{R}dr\,r\left[f^{2}\left(-{2g\over
3\sqrt{3}}a+{4\over 9}\mu_{\text{B}}\right)+b^{2}\left({2g\over
3\sqrt{3}}a+{8\over 9}\mu_{\text{B}}\right)\right]\,.$ (100)

One might think that the above result is an involved expression, but there is
an elegant physical interpretation. To this end, we shall compute the baryon
charge density as

$\displaystyle Q_{\rm B}$ $\displaystyle=$ $\displaystyle-{2\over 3}i\,{\rm
tr}\bigl{[}(D_{0}\Phi)^{\dagger}\Phi-\Phi^{\dagger}(D_{0}\Phi)\bigr{]}=-{2\over
3}i\,{\rm tr}(\Pi^{\dagger}\Phi-\Phi^{\dagger}\Pi)$ (101) $\displaystyle=$
$\displaystyle{4\over 3}{\rm
tr}\bigl{[}\Phi^{\dagger}(gA_{0}^{8}t^{8}+2\mu_{\text{B}}/3)\Phi\bigr{]}\,,$

from which the total baryon charge per unit vortex length reads:

$N_{\rm B}=2\pi\int_{0}^{R}dr\,r\left[f^{2}\left(-{4g\over 3\sqrt{3}}a+{8\over
9}\mu_{\text{B}}\right)+b^{2}\left({4g\over 3\sqrt{3}}a+{16\over
9}\mu_{\text{B}}\right)\right]\,.$ (102)

Comparing $L_{z}^{\rm tot}$ and $N_{\rm B}$, we see that the following
relation holds:

$L_{z}^{\rm tot}={\nu\over 2}N_{\rm B}\,.$ (103)

This confirms that the total angular momentum of the non-Abelian vortex in the
CFL phase contains only the contribution from the global $U(1)_{\rm B}$
vortex; the total angular momentum (in the unit of $\hbar$) is $\nu$ times the
number of the Cooper pairs. We note that this result completely agrees with
Eq. (14) in Ref. Alford  _et al._ (2019), but in Ref. Alford  _et al._ (2019)
only the $U(1)_{\rm B}$ contribution to the angular momentum was postulated
without rigorous justification.

##  IV Class III: Case study without angular momentum conservation

The last logical possibility in our classification is that magnetic vortices
cannot be created by simply turning on external magnetic flux in an
azimuthally symmetric way. What distinguishes this case from all previous
cases is that the principle of angular momentum conservation does not naïvely
apply in the vortex creation process. The vortices classified in this class
are characterized by inhomogeneous profiles along the vortex axis, which means
that not only the external magnetic flux but also something else are needed to
create the vortices: roughly speaking, a kind of twisting along the axis would
be required. Such vortices do exist as we discuss below, although they seem to
be rare in the literature.

An example that belongs to this class is provided by an object called “charged
semilocal vortex” as constructed by Abraham in Ref. Abraham (1993). This
Abraham vortex is an extension of the semilocal vortex Vachaspati and
Achucarro (1991) that has been discussed in the context of electroweak strings
in cosmology (see Ref. Achucarro and Vachaspati (2000) for a review). They
also appear quite commonly as topological BPS (Bogomol’nyi-Prasad-Sommerfield)
objects in supersymmetric gauge theories. The simplest model of the Abraham
vortex consists of two charged scalar fields, $\Phi_{a}$ $(a=1,2)$, with the
equal charge, and a $U(1)$ gauge field $A_{\mu}$. The Hamiltonian in the
critical limit reads:

$H=\sum_{a=1,2}\left(|D_{0}\Phi_{a}|^{2}+|\bm{D}\Phi_{a}|^{2}\right)+{g^{2}\over
2}\biggl{(}\,\,\sum_{a=1,2}|\Phi_{a}|^{2}-v^{2}\biggr{)}^{2}+{1\over
2}(\bm{E}^{2}+\bm{B}^{2})\,,$ (104)

where $D_{\mu}\Phi_{a}=(\partial_{\mu}-igA_{\mu})\Phi_{a}$, and the Gauss law
is

$\bm{\nabla}\cdot\bm{E}=ig\sum_{a=1,2}\Bigl{[}(D_{0}\Phi_{a})^{\dagger}\Phi_{a}-\Phi_{a}^{\dagger}(D_{0}\Phi_{a})\Bigr{]}\,.$
(105)

The vortex solution relies on the following Bogomol’nyi bound,

$\displaystyle H$ $\displaystyle=\sum_{a=1,2}\bigl{(}|D_{0}\Phi_{a}\pm
D_{3}\Phi_{a}|^{2}+|D_{1}\Phi_{a}\pm iD_{2}\Phi_{a}|^{2}\bigr{)}+{1\over
2}|E_{x}\mp B_{y}|^{2}+{1\over 2}|E_{y}\pm B_{x}|^{2}$
$\displaystyle\quad+{1\over 2}\biggl{[}B_{z}\mp
g\biggl{(}\sum_{a=1,2}|\Phi_{a}|^{2}-v^{2}\biggr{)}\biggr{]}^{2}\mp\alpha
Q_{2}\mp v^{2}gB_{z}\mp\bm{\nabla}\cdot(\bm{E}A_{z})\,.$ (106)

Here, $\bm{A}=(A_{x},A_{y},A_{z})$, $\bm{E}=(E_{x},E_{y},E_{z})$,
$\bm{B}=(B_{x},B_{y},B_{z})$ and we imposed additional conditions that
$\partial_{3}\Phi_{2}=i\alpha\Phi_{2}$ with a constant $\alpha$ and all other
$\partial_{3}$ is vanishing. We defined $Q_{2}$ as

$Q_{2}=i\bigl{[}(D_{0}\Phi_{2})^{\dagger}\Phi_{2}-\Phi_{2}^{\dagger}(D_{0}\Phi_{2})\bigr{]}\,.$
(107)

The equations we obtain from this, for the upper sign, are

$D_{0}\Phi_{a}+D_{3}\Phi_{a}=0\,,\quad D_{1}\Phi_{a}+iD_{2}\Phi_{a}=0\,,\quad
E_{i}=\epsilon_{ij}B_{j}\,,\quad
B_{z}-g\biggl{(}\sum_{a=1,2}|\Phi_{a}|^{2}-v^{2}\biggr{)}=0\,.$ (108)

It can be checked that these solve the original equations of motion. In Ref.
Abraham (1993) it was also shown that these equations admit nice solutions
with zero net gauged $U(1)$ charge but nonzero $Q_{2}$, which are somewhat
misleadingly called “charged” semilocal vortices. These solutions have finite
line energy density, due to the fact that the total $U(1)$ charge is zero. The
solutions are possible only for $\alpha\neq 0$, that corresponds to a
“twisting” along the vortex axis. Due to this, the vortex string carries a net
linear momentum along the axis direction. Although the total $U(1)$ charge is
zero, the charge density profile in space is nonzero, and there exists
nontrivial profile of local electric and magnetic fields. This leads to a
nonvanishing contribution of the electromagnetic fields to the total angular
momentum. As pointed out in Ref. Abraham (1993), the solutions carry nonzero
total angular momentum, but we would not go into technicality here, and the
readers can directly consult Ref. Abraham (1993). The “twisting”, parametrized
by $\alpha$, can be considered as spinning the vortex to give a finite angular
momentum. This is an extra operation that would be needed to create such a
vortex profile by hand, and the angular momentum conservation cannot be
applied to the situation. In other words, in this peculiar system belonging to
this class, the angular momentum conservation is not satisfied by $L_{z}^{\rm
matter}$ or $L_{z}^{\rm gauge}$ or their sum.

##  V Conclusion

In this work, we apply the principle of angular momentum conservation to
understand the origin of angular momentum carried by magnetic vortices in
various physical systems in condensed matter, high energy, nuclear physics,
and cosmology. We find that this simple principle is powerful enough to allow
us an overarching scheme of classifying the known examples, according to how
the principle of angular momentum conservation is satisfied. We find the four
distinct classes of examples in our classification scheme; spinful (class Ia),
topological (class Ib), spinless (class II) and exotic (class III) vortices,
as already summarized in Introduction. We present detailed analyses for these
examples, and emphasize that the angular momentum carried by localized gauge
fields around the vortex core plays a crucial role in satisfying the angular
momentum conservation. We believe that our study gives a clear answer to the
seemingly confusing, but surprisingly rich, question of angular momentum
carried by magnetic vortices that are ubiquitous in many branches of physics.

###### Acknowledgements.

K. F. is supported by Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant Nos. 18H01211 and 19K21874. Y. H. is supported by Japan Society
for the Promotion of Science (JSPS) KAKENHI Grant Nos. 17H06462 and 18H01211.
H.-U. Y. is supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, Grant No. DE-FG0201ER41195, and within the
framework of the Beam Energy Scan Theory (BEST) Topical Collaboration.

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