aainstitutetext: Facultad de Ingeniería, Arquitectura y Diseño, Universidad
San Sebastián, sede Valdivia,
General Lagos 1163, Valdivia 5110693, Chilebbinstitutetext: Centro de Estudios
Científicos (CECS),
Avenida Arturo Prat 514, Valdivia, Chileccinstitutetext: Departamento de
Física, Universidad Nacional de La Plata,
CC67, 1900 La Plata, Argentinaddinstitutetext: Instituto de Física La Plata,
Consejo Nacional de Investigaciones Científicas y Técnicas
Diagonal 113 entre 63 y 64, 1900 La Plata, Argentinaeeinstitutetext:
Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción,
Chile

#  Exact oscillations and chaos on a non-Abelian coil

Fabrizio Canfora c,d Nicolas Grandi e Marcelo Oyarzo e Julio Oliva
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct new exact solutions of the Georgi-Glashow model in $3+1$
dimensions. These configurations are periodic in time but lead to a stationary
energy density and no energy flux. Nevertheless, they possess a characteristic
frequency which manifests itself through non-trivial resonances on test
fields. This allows us to interpret them as non-Abelian self sustained coils.
We show that for larger energies a transition to chaotic behavior takes place,
which we characterize by Poincaré sections, Fourier spectra and exponential
growth of the geodesic deviation in an effective Jacobi metric, the latter
triggered by parametric resonances.

##  1 Introduction

Time-periodic configurations arising in nonlinear hyperbolic problems are
notoriously difficult to construct (see time-periodicYM1 ; time-periodicYM2 ;
time-periodicYM3  and references therein) and, at the same time, extremely
interesting physically (see e.g. time-periodicYM4 ; time-periodicYM5 ; time-
periodicYM6 ). In Euclidean spaces, the relevance of topologically non-trivial
configurations which are periodic in Euclidean time, representing instantons
at finite temperature, is particularly relevant for the analysis of the phase
diagram of gauge theories BalaBook ; manton . The interest in these
configurations arises, in part, from the difficulty to study time dependent
configurations in lattice gauge theories lattice1 ; lattice2 . It also
experienced a remarkable growth in the recent years, due to the intensive
research in out-of-equilibrium physics (see e.g. entanglement5 ; entanglement6
; entanglement7 ; entanglement8 ; entanglement9 ; entanglement9a1 ;
entanglement9a2 ; entanglement9b ; entanglement9c ; entanglement9d ;
entanglement10 ; entanglement11 ; entanglement12  and references therein).

In the present paper we construct new exact, time dependent solutions to the
Einstein-Yang-Mills-Higgs system in $3+1$ dimensions, with quite intriguing
physical properties. These configurations are periodic in real time in such a
way that the energy-density is stationary and their non-Abelian Poynting
vector vanishes, so that there is no energy flux. In spite of this, as we will
show below, they possess a characteristic frequency which manifests itself
through non-trivial resonances of test fields, charged under the non-Abelian
gauge symmetry, which propagate in these backgrounds. These new analytic
solutions possess genuine non-Abelian features as they can be interpreted as
non-Abelian self-sustained coils.

Besides the intrinsic interest to construct analytical time-dependent
configurations, the technical tools allow to discuss very interesting open
questions on the chaotic behavior of Yang-Mills theory. The analysis of chaos
in non-Abelian gauge theories raised huge interest since the early years soon
after the discovery of Yang-Mills theory (see chaosYM1 ; chaosYM2 ; chaosYM3 ;
chaosYM4 ; chaosYM5  and references therein). In recent years, two references
in particular chaosYM5.1 ; chaosYM5.2  triggered a burst of activity on this
topic due to the discovery of novel relations with holography and quantum
chaos (see chaosYM6 ; chaosYM7 ; chaosYM8 ; chaosYM9  and references therein).
The usual starting point of these analyses is a homogeneous Ansatz for the
Yang-Mills-Higgs fields with, very often, the Higgs field in the fundamental
representation, which only depend on time, in such a manner that the
corresponding field equations can be analyzed with the available tools of
chaotic dynamics (see chaosYMreview ). On the other hand, this starting point
prevents, in many situations, to include non-trivial topological fluxes, that
either need some non-trivial dependence on space-like coordinates, or the
presence of the Higgs field in the adjoint representation, in order to get a
gauge-invariant version of the magnetic flux. Therefore, if one is interested
in the analysis of the interplay of topology and chaos, it is important to
generalize a little bit the notion of homogeneous field and to construct an
Ansatz in which the fields depend non-trivially on the spatial coordinates,
keeping alive the topological fluxes, but in such a way that the field
equations reduce to a dynamical system.

An important technical tool to succeed in the aforementioned construction
turns out to be the non-spherical hedgehog Ansatz developed for the Skyrme
model, originally introduced in 56a0 -crystal4 , that allowed to discover the
first analytic and topologically non-trivial solutions in the Skyrme model
which are periodic in time in such a way that the energy-momentum tensor is
static 56a ; 56a1 . As explained below, in a certain sense the results
presented here represent an extension of those in 56a  and 56a1  to the Yang-
Mills-Higgs case, with the Higgs in the adjoint representation of the gauge
group.

At a first glance, the analytic solutions representing non-Abelian self-
sustained coils, to be described in the following sections, could suggest the
appearance of some integrable sector of the theory. In fact, this is not the
case: the chaotic behavior appears anyway. However, in the analysis of the
chaotic regime, the analytic solutions manifest themselves through
“integrability islands” in the corresponding Poincaré sections. One of the
main tools that we will use in the analysis of chaotic dynamics was introduced
in cerruti-pettini  and is based on the Jacobi metric arnold . Our analysis
shows that such a tool, which to the best of our knowledge has not been
employed so far in the analysis of chaos in Yang-Mills theory, is actually
very effective when compared with different techniques.

The paper is organized as follows: in Section 2.1 the conventions and the
Georgi-Glashow model are presented. In Section 2.2 we introduce the time
dependent Ansatz for the Yang-Mills and Higgs fields in flat spacetime. Later,
the new exact solutions of the system are derived, as well as some of their
perturbations. The cases with and without vacuum expectation value are studied
separately in sections 3.1 and 3.2 respectively. In Section 5 we study the
resonance frequencies of the configurations with a quantum scalar field probe
in the fundamental of $SU(2)$. Some remarks and conclusions are given in the
last section.

##  2 Basic setup

In this section, the model and the time-dependent Ansatz are introduced,
together with the corresponding equations of motion and the resulting energy
momentum tensor and non-Abelian Poynting vector.

###  2.1 The model

Our starting point is the Georgi-Glashow model for $SU\left(2\right)$, with
field content given by a Lie algebra valued $1$-form gauge potential $A$ and a
Higgs field $\Phi$ which transforms in the adjoint representation. They are
algebra valued objects

$A=A_{\ \mu}^{a}t_{a}dx^{\mu}\ ,\qquad\qquad\Phi=\Phi^{a}t_{a}\ ,$ (1)

where we consider anti-Hermitian matrices $t_{a}\equiv i\sigma_{a}$, where
$\left\\{\sigma_{a}\ ,\ a=1,2,3\right\\}$ are the Pauli matrices. These
generators fulfill $t_{a}t_{b}=-\delta_{ab}-\varepsilon_{abc}t_{c}$.

The action for the model reads

$I\left[A,\Phi\right]=\int
d^{4}x\sqrt{-g}\left(-\frac{1}{4e^{2}}F^{a\mu\nu}F_{a\mu\nu}-\frac{1}{2e^{2}}D_{\mu}\Phi^{a}D^{\mu}\Phi_{a}-\frac{\lambda}{4}\left(\Phi^{a}\Phi_{a}-\nu^{2}\right)^{2}\right)\
,$ (2)

where $e$ is a positive gauge coupling constant, $\lambda$ is a positive
scalar self coupling, and $\nu$ is the vacuum expectation value of the Higgs
field. As usual, the field strength and the covariant derivative are defined
by

$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\left[A_{\mu},A_{\nu}\right]\
,$ (3) $\displaystyle D_{\mu}\cdot$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\cdot+\left[A_{\mu},\cdot\right]\ .$ (4)

The field equations are obtained by computing the stationary variation with
respect to the fields $A_{\ \mu}^{a}$ and $\Phi^{a}$ which respectively give
the following expressions

$\displaystyle D_{\mu}F^{\mu\nu}-\left[\Phi,D^{\nu}\Phi\right]$
$\displaystyle=$ $\displaystyle 0\ ,$ (5) $\displaystyle
D_{\mu}D^{\mu}\Phi-e^{2}\lambda\left(\Phi^{a}\Phi_{a}-\nu^{2}\right)\Phi$
$\displaystyle=$ $\displaystyle 0\ .$ (6)

The energy momentum tensor of this model is computed by varying the action
with respect to the metric, resulting in

$T_{\mu\nu}=T_{\mu\nu}^{\sf Gauge}+T_{\mu\nu}^{\sf Higgs}\ ,$ (7)

with

$\displaystyle T_{\mu\nu}^{\sf Gauge}=\frac{1}{e^{2}}\left(F_{a\mu\lambda}F_{\
\nu}^{a\ \lambda}-\frac{1}{4}g_{\mu\nu}F^{a\rho\sigma}F_{a\rho\sigma}\right)\
,$ (8) $\displaystyle T_{\mu\nu}^{\sf
Higgs}=\frac{1}{e^{2}}\left(D_{\mu}\Phi^{a}D_{\nu}\Phi_{a}-\frac{1}{2}g_{\mu\nu}D_{\sigma}\Phi^{a}D^{\sigma}\Phi_{a}-g_{\mu\nu}\frac{\lambda
e^{2}}{4}\left(\Phi^{a}\Phi_{a}-\nu^{2}\right)^{2}\right)\ .$ (9)

From now on we set $e=1$ without loss of generality, since the only relevant
combination is $\lambda e^{2}$.

###  2.2 The time dependent Ansatz

In the present section we define an appropriate Ansatz which allows us to
solve the field equations analytically with a time dependent profile.

Let us first fix the geometry considering flat spacetime in cylindric
coordinates

$ds^{2}=-dt^{2}+dz^{2}+d\rho^{2}+\rho^{2}d\varphi^{2}\ .$ (10)

The range of the coordinates are the usual, $\varphi\in\left[0,2\pi\right]$
with $\varphi\sim\varphi+2\pi\ ,\ \rho\in[0,+\infty[$ and $t,z\in\mathbb{R\
}$. In this background we define our Ansatz for the gauge field and Higgs
fields, as

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{W(t)}{\sqrt{2}}\left(t_{1}\,\rho d\varphi-
t_{2}\,d\rho\right)-\frac{1}{2}t_{3}\,d\varphi\ ,$ (11) $\displaystyle\Phi$
$\displaystyle=$ $\displaystyle G(t)\,t_{3}\ .$ (12)

Both the gauge $W(t)$ and Higgs $G(t)$ profiles depend explicitly on time. The
non-Abelian field strength defined in (3) for the Ansatz (11) reads

$F=\frac{\dot{W}}{\sqrt{2}}\left(\,dt\wedge d\rho\,t_{2}-\rho\,dt\wedge
d\varphi\,t_{1}\right)-W^{2}\rho\,d\rho\wedge d\varphi\,t_{3}\ .$ (13)

It has two electric components, one of them is along the second generator of
the gauge group while pointing in the radial spatial direction, while the
other is aligned with the first generator and it points around the cylinder.
The magnetic field is aligned with the third generator and it goes along the
axis of the cylinder.

With the above Ansatz, the energy momentum tensor has a natural cylindrical
symmetry, and it can be written as

$T_{\mu\nu}\,dx^{\mu}\otimes dx^{\nu}=\frac{1}{e^{2}}\left({\cal
E}\,dt^{2}-p_{\perp}(d\rho^{2}+\rho^{2}d\theta^{2})-p_{z}\,dz^{2}\right)\,,$
(14)

with

$\displaystyle\mathcal{E}=\frac{1}{2}\left(\dot{G}^{2}+\dot{W}^{2}\right)+\frac{1}{2}W^{2}\left(4G^{2}+W^{2}\right)+\frac{\lambda}{4}\left(G^{2}-\nu^{2}\right)^{2}\
,$ (15) $\displaystyle
p_{\perp}=-\frac{1}{2}\left(\dot{G}^{2}+W^{4}\right)+\frac{\lambda}{4}\left(G^{2}-\nu^{2}\right)^{2}\,,$
(16) $\displaystyle
p_{z}=p_{\perp}+W^{2}\left(2G^{2}+W^{2}\right)-\frac{1}{2}W^{\prime 2}\,.$
(17)

It is worth to emphasize that, in spite of considering a time dependent
configuration, there are no energy fluxes. This feature can be interpreted as
an interplay between the non-Abelian character of the solution and the time
dependence of the gauge fields, in such a way that the trace in the definition
of the energy momentum tensor cancels out the radiation of the gauge field. We
will discuss this feature in more detail in the forthcoming sections.

To give some physical content to the above construction, let us first recall
one of the most useful features of the Georgi-Glashow model: the presence of a
scalar field in the adjoint representation allows us to construct a gauge
invariant quantity representing the effective Abelian gauge field of the
theory

${F}_{\sf eff}={\rm tr}\left(\Phi F\right)\ .$ (18)

For the configuration (11) and (12), the above projection gives

${F}_{\sf eff}=\rho\,G\,W^{2}d\rho\wedge d\varphi\ .$ (19)

In the present case, the $2$-form (19) corresponds to an effective uniform
Abelian magnetic flux along the $z$-axis. The exact configurations that will
be discussed in the following are periodic in time, hence the effective
Abelian magnetic field will be periodic as well.

Now let us consider a cylinder of radius $R_{0}$ inside of which the fields
are given by the Ansatz (11)-(12), while they vanish outside. In order to
match the fields in the interior of the cylinder with those outside it, we
require the usual Maxwell junction conditions for the corresponding Abelian
part (19). These conditions tell us that the normal component to the interface
of the effective Abelian magnetic field must be continuous, which is satisfied
by $B_{\sf{eff}}=G\left(t\right)W\left(t\right)^{2}\partial_{z}$. Also, since
the Poynting vector is zero everywhere, there is no energy flux outside the
cylinder. Consequently, if we are able to construct explicitly exact solutions
for the gauge and Higgs profiles which are periodic in time, then such
configurations can be interpreted as coils with a self-generated $AC$ current.

A very important property of the ansatz for the gauge and Higgs fields given
in Eqs. (11) and (12) is that it reduces the full coupled system of non-linear
partial differential equations to the following two coupled ordinary
differential equations

$\displaystyle\frac{d^{2}G}{dt^{2}}+4W^{2}G+\lambda
G\left(G^{2}-\nu^{2}\right)=0\ ,$ (20)
$\displaystyle\frac{d^{2}W}{dt^{2}}+4G^{2}W+2W^{3}=0\ .$ (21)

It is conceptually useful to rewrite this system of second order differential
equations as a Newtonian system for the time-dependent variables
$\left(G,W\right)$ in the form

$\frac{d^{2}G}{dt^{2}}=-\frac{\partial}{\partial G}V\left(G,W\right)\
,\qquad\qquad\frac{d^{2}W}{dt^{2}}=-\frac{\partial}{\partial
W}V\left(G,W\right)\ ,$ (22)

in terms of the effective potential

$V(G,W)=2W^{2}G^{2}+\frac{1}{2}W^{4}+\frac{\lambda}{4}\left(G^{2}-\nu^{2}\right)^{2}\
.$ (23)

The additive constant was introduced to set to zero the energy of the
configuration with a non-trivial vacuum expectation value. This potential is
bounded from below and has two global minimum at $G=\pm\nu\ ,\ W=0$ and a
saddle point at $G=0,W=0$. A plot of the level curves of this potential is
shown in Fig. 1.

Figure 1:  Level curves of the effective potential $V$ for the cases $\nu\neq
0$ (left) and $\nu=0$ (right).

As a first interesting result notice that, integrating the system (22)-(23)
once, we recover the conservation of the energy density ${\cal E}$, in spite
of the fact that the field configuration is time dependent. This is consistent
with the absence of energy fluxes in our configuration.

Naively, one could conclude that this configuration is static and hence there
is no a characteristic frequency of the system. Nevertheless, this is not the
case as we will show by computing the time-dependent transition amplitude of a
scalar probe field in the adjoint representation propagating in the exact
solutions of the above form. Such transition amplitude discloses a clear
resonance effect when the frequency of the test field matches the
characteristic frequency of the background solutions. The present situation is
reminiscent of the spin-from-isospin effect for Skyrmions and non-Abelian
monopoles spinfromisospin1 ; spinfromisospin2 ; spinfromisospin3  in which
case the energy-momentum tensor is spherically symmetric and yet these
configurations are not spherically symmetric in the obvious sense as the
angular momentum operator is naturally supplemented by an extra term arising
from the internal symmetry group. This fact is behind the commonly used
statement “gauge field are invariant up to an internal transformation”.

Notice that the equations (20) and (21) have the shift symmetry

$t\to t-t_{0}\ ,$ (24)

which implies that one of the integration constants of the system sets the
zero of the time variable. Moreover, they have the scaling invariance

$\displaystyle(t,W,G,\nu,\lambda)\rightarrow\left(\frac{t}{T},TW,TG,T\nu,\lambda\right)\
,$ (25)

where $T$ is an arbitrary constant. For vanishing $\nu$ this implies that a
second integration constant sets the time scale and the overall scale of the
fields. For $\nu$ finite, these can be fixed by the value of $\nu$.

##  3 Exact solutions

In this section we present our exact solutions and analyze their properties,
studing separately the cases with a without vacuum expectation value. In each
case, we explore the vacuum and perturbative solutions, the pure Yang-Mills
and pure Higgs cases, and the solutions with both fields turned on.

###  3.1 Configurations with non-vanishing vacuum expectation value

In this section we will consider configurations with non-vanishing vacuum
expectation value $\nu\neq 0$.

#### Perturbative solution:

The first trivial observation in this case is that there is a vacuum static
solution in which $W(t)=0$ and $G(t)=\pm\nu$. Such solution can be perturbed
as

$\displaystyle W(t)=\epsilon w(t)\ ,$ (26) $\displaystyle G(t)=\pm\nu+\epsilon
g(t)\ ,$ (27)

where $\epsilon$ is a small parameter and $w(t)$ and $g(t)$ are new unknown
functions. Plugging this back into the equations of motion and expanding to
first order in $\epsilon$, we get a perturbative solution

$\displaystyle W(t)=\epsilon\sin(2\nu(t-t_{0})+\delta)\ ,$ (28) $\displaystyle
G(t)=\pm\nu+\epsilon a\cos(\sqrt{2\lambda}\nu(t-t_{0}))\ ,$ (29)

where $\epsilon$ now becomes a small integration constant, and $a,t_{0}$ and
$\delta$ are integration constants of order one. Notice that these solutions
are periodic only when $\sqrt{\lambda/2}=p/q$ with $p,q\in\mathbb{N}$. The
period then reads

$t\sim t+\frac{\pi}{\nu}\sqrt{\frac{2}{\lambda}}\,p=t+\frac{\pi}{\nu}q$ (30)

#### Pure Yang-Mills solution:

There is a pure Yang-Mills sector of the theory, which is obtained setting
$G(t)=0$. In this case, the field equations (20)-(21) reduce to the equations
of a quartic oscillator, namely

$\frac{d^{2}W}{dt^{2}}+2W^{3}=0\ ,$ (31)

that can be solved in the form

$W(t)=\pm a\ {\rm sn}\\!\left(a\left(t-t_{0}\right),-1\right)\ ,$ (32)

where ${\rm sn}(x,m)$ is the Jacobi elliptic sine function, and $a$ is a
constant of integration. Notice that the same constant sets both the time
scale and the amplitude of the oscillation. This can be traced back to the
scaling symmetry (25), taking into account that the value of $\nu$ does not
enter into the present pure Yang-Mills solution.

We can calculate the energy density of the configuration according to the
expression (15), obtaining

${\cal E}=\frac{1}{4}\left(2a^{4}+\lambda\nu^{4}\right)\ ,$ (33)

where we see that the energy density is conserved.

Solutions (32) are periodic, their period can be obtained from the periodicity
properties of the Jacobi elliptic sine, resulting in the expression

$t\sim t+\frac{2}{a}K_{20}(-1)\ ,$ (34)

where the function $K_{pq}(m)$ has been defined according to

$\displaystyle K_{pq}(m)=p\,K(m)+i\,q\,K(1-m)\ .$ (35)

In this expression, $K$ is the complete elliptic integral of the first kind,
and $p,q\in\mathbb{N}$. Here and in what follows, we are choosing the values
of $p$ and $q$ as the smallest integers that make the resulting period real.

#### Pure Higgs solution:

There is also a pure Higgs configuration, which is obtained by setting
$W(t)=0$ and solving the remaining equation for $G(t)$, resulting in

$G(t)=\pm\nu\sqrt{\frac{2q}{1+q}}\,{\rm
sn}\\!\left(\sqrt{\frac{-\lambda}{1+q}}\,\nu\,(t-t_{0}),q\right)\ ,$ (36)

where $q$ is a constant of integration.

As for the pure Yang-Mills case, this is a periodic solution whose period is
given by that of the Jacobi sine, in the form

$t\sim t+\frac{2}{\nu}\sqrt{\frac{1+q}{-\lambda}}K_{20}(q)\ ,$ (37)

where $K_{pq}$ defined as in equation (35).

Solution (36) is explicitly real for $q<-1$. However, using the definition of
and properties of the Jacobi elliptic functions, it can be analytically
continued to $q\in(-1,0]$ in the form

$G(t)=\pm\nu\sqrt{\frac{-2q}{1+q}}\,{\rm
sc\\!}\left(-\sqrt{\frac{\lambda}{1+q}}\,\nu\,(t-t_{0}),1-q\right)\ ,$ (38)

where $\mbox{sc}(x,m)=i\,\mbox{sn}(-ix,1-m)$ is another Jacobi function.

Expression (38) is again periodic, but in this case the period is written in
the form

$t\sim+\frac{2}{\nu}\sqrt{\frac{1+q}{\lambda}}K_{22}(1-q)\ ,$ (39)

which connects smoothly to (37) as $q\to-1$.

This configuration has an energy density given by

${\cal E}=\frac{\lambda}{4}\left(\frac{1-q}{1+q}\right)^{2}\nu^{4}\ ,$ (40)

which is again conserved.

#### Solution with both fields:

For the generic case with non-vanishing Higgs, the solution reads

$\displaystyle G(t)$ $\displaystyle=$ $\displaystyle\pm_{1}\sqrt{2}\,\nu\,{\rm
dn}\\!\left(\sqrt{8-\lambda}\,\nu\left(t-t_{0}\right),\frac{\lambda}{8-\lambda}\right)\
,$ (41) $\displaystyle W(t)$ $\displaystyle=$
$\displaystyle\pm_{2}\sqrt{\frac{\lambda(\lambda-4)}{8-\lambda}}\,\nu\,{\rm
sn}\\!\left(\sqrt{8-\lambda}\,\nu\left(t-t_{0}\right),\frac{\lambda}{8-\lambda}\right)\
,$ (42)

where ${\rm dn}^{2}(x,m)=1-m\,{\rm sn}^{2}(x,m)$ is another Jacobi elliptic
function. This solution is explicitly real for $\lambda\in[4,8)$. There is no
integration constant controlling the frequency of the oscillation, nor its
amplitude. However, the vacuum expectation value parameter $\nu$ changes the
amplitude and the frequency of the configuration in the same amount, due to
the scaling symmetry discussed in the previous section (25). The period is
given by

$t\sim
t+\frac{2}{\nu\sqrt{8-\lambda}}K_{22}\\!\left(\frac{\lambda}{8-\lambda}\right)\,.$
(43)

Using the identities and the relations between the Jacobi elliptic functions
one can write (41)-(42) in an alternative form which is manifestly real for
$\lambda>8$. In such case we have

$\displaystyle G\left(t\right)$ $\displaystyle=$
$\displaystyle\pm_{1}\sqrt{2}\,\nu\,\text{dc}\\!\left(\sqrt{\lambda-8}\,\nu\,\left(t-t_{0}\right),1-\frac{\lambda}{8-\lambda}\right)\
,$ (44) $\displaystyle W\left(t\right)$ $\displaystyle=$
$\displaystyle\pm_{2}\sqrt{\frac{\lambda\left(\lambda-4\right)}{\lambda-8}}\,\nu\,\text{sc}\\!\left(\sqrt{\lambda-8}\,\nu\left(t-t_{0}\right),1-\frac{\lambda}{8-\lambda}\right)\
.$ (45)

Where $\mbox{dc}(x,m)=\mbox{dn}(-ix,1-m)$ is a further elliptic function. The
period now reads

$t\sim
t+\frac{2}{\nu\sqrt{\lambda-8}}K_{22}\\!\left(1-\frac{\lambda}{8-\lambda}\right)\
.$ (46)

The $\lambda=8$ can be integrated from the equations (20)-(21) and it reads

$\displaystyle G\left(t\right)$ $\displaystyle=$
$\displaystyle\sqrt{2}\,\nu\,\sin\\!\left(2\sqrt{2}\nu\left(t-t_{0}\right)\right)\
,$ (47) $\displaystyle W\left(t\right)$ $\displaystyle=$ $\displaystyle
2\,\nu\,\cos\\!\left(2\sqrt{2}\nu\left(t-t_{0}\right)\right)\ .$ (48)

Here, $t_{0}$ is the only integration constant of the solution. The period of
this solution can be written as

$t\sim t+\frac{\pi}{\sqrt{2}\nu}\ .$ (49)

For any value of the coupling $\lambda$, the energy density of this exact
configuration is given by the expression

${\cal E}=\frac{1}{4}(2\lambda-7)\lambda\nu^{4}\ ,\\\ $ (50)

which behaves smoothly in the $\lambda\to 8$ limit.

In Fig.2 we overlap the solutions we found in this section with the level
curves of the effective potential (23) for $\nu=1,\lambda=6,8,12$.

Figure 2: Solutions with non-vanishing vacuum expectation value $\nu\neq 0$,
for the particular cases $\lambda=6,8,12$ from left to right. In yellow is the
perturbative solution, while in orange is the exact solution. The vertical and
horizontal lines represent the pure Yang-Mills and pure Higgs solutions
respectively, the last one corresponding to the smallest possible amplitude
$q\to-\infty$.

###  3.2 Configurations with vanishing vacuum expectation value

#### Perturbative solution:

With $\nu=0$ we still have a static solution, now at $W(t)=G(t)=0$, that can
perturbed to obtain

$\displaystyle W(t)=\epsilon(t-t_{0})\ ,$ (51) $\displaystyle G(t)=\epsilon
a(t-t_{0})\ .$ (52)

Higher order perturbations result in further corrections to the overall
coefficient of the linear term, up to order $\epsilon^{3}$ at which there is
an additional correction which goes as $(t-t_{0})^{5}$.

#### Pure Yang-Mills solution:

The pure Yang-Mills configuration is the same as in the case with non-
vanishing vacuum expectation value, which is to be expected since the Higgs
field plays no role in it.

#### Pure Higgs solution:

Regarding the pure Higgs configuration, it satisfies the equation of motion

$\frac{d^{2}G}{dt^{2}}+\lambda G^{3}=0\ .$ (53)

This is again a quartic oscillator, with solution

$G(t)=\pm\sqrt{2}a\,{\rm sn}\left(a\sqrt{\lambda}(t-t_{0}),-1\right)\ ,$ (54)

where $a$ is an integration constant. The period takes the form

$t\sim t+\frac{2}{a\sqrt{\lambda}}K_{20}(-1)\ .$ (55)

The energy density on the other hand, reads

$\displaystyle{\cal E}=a^{4}\lambda\ .$ (56)

It is interesting to notice that formulas (54) to (56) can be obtained from
the corresponding equations for the finite vacuum expectation value case, by
taking the limit $\nu\to 0$ and $q\to-1$ with the constraint
$\nu/\sqrt{1+q}=i\,a$.

Figure 3: Solutions with non-vanishing vacuum expectation value $\nu=0$, for
the particular cases $\lambda=1,2,3$ from left to right. In orange is the
exact solution. The vertical and horizontal lines represent the pure Yang-
Mills and pure Higgs solutions respectively.

#### Solution with both fields:

The fact that for the linearly perturbed solution we have a Higgs profile $G$
that is proportional to the Yang-Mills profile $W$, suggests that in the non-
perturbative case we can try to reduce the equations (20)-(21) into a unique
equation, by considering the ansatz

$G(t)=\pm\sqrt{\frac{2}{4-\lambda}}W(t)\ .$ (57)

Here the proportionality factor has been chosen so that the resulting
equations for $G(t)$ and $W(t)$ coincide. Notice that the shape of the
potential provides that $\lambda$ must be positive and the equation (57)
implies that $\lambda<4\ $. The resulting master equation is given by

$\frac{d^{2}W}{dt^{2}}+2\left(\frac{8-\lambda}{4-\lambda}\right)W^{3}=0\ ,$
(58)

which can be solved by

$W(t)=\pm a\ {\rm
sn}\\!\left(a\sqrt{\frac{8-\lambda}{4-\lambda}}\left(t-t_{0}\right),-1\right)\
,$ (59)

Notice that as before the amplitude $a$ is tied to the frequency due to the
scaling symmetry, but now it is an integration constant. Consequently, the
profile for the Higgs field reads

$G(t)=\pm\sqrt{\frac{2}{4-\lambda}}\,a\,{\rm
sn}\\!\left(a\sqrt{\frac{8-\lambda}{4-\lambda}}\left(t-t_{0}\right),-1\right)\
.$ (60)

These solutions are explicitly real for $\lambda<4$ and cannot be extended to
$\lambda>4$.

The energy density of this configuration is

${\cal
E}=\frac{a^{4}\left(\lambda-8\right)\left(\lambda-6\right)}{2\left(\lambda-4\right)^{2}}\
,$ (61)

while the period can be obtained in terms of the complete elliptic integral of
the first kind $K$ as

$t\sim t+\frac{2}{a}\sqrt{\frac{\lambda-4}{\lambda-8}}\,K_{20}(-1)\ .$ (62)

##  4 Chaotic behaviour

At a first glance, one could think that the appearance of the nice analytic
solutions described in the previous sections may hint at the integrability of
the Yang-Mills-Higgs sector described by the Ansatz in Eqs. (11) and (12).
This possibility becomes quite clear taking into account that using the
homogeneous Ansatz which is usually employed in the analysis of chaos in Yang-
Mills theory (see chaosYM6 ; chaosYM7 ; chaosYM8 ; chaosYM9  and references
therein) it has not been possible to find analytic solutions, to the best of
our knowledge. In the following sections we will show that this is not the
case: the chaotic behavior appears nevertheless, if one increases the energy
of the system.

To characterize the chaotic regime, we will use three different and somewhat
complementary techniques:

  1. 1.

Poincaré sections: The phase space of the system is $4$-dimensional and can be
parameterized by the coordinates $(G,W)$ and the canonical momenta
$(p_{G}=\dot{G},p_{W}=\dot{W})$. The conservation of the energy (15) reduces
in one the dimensionality of the space where the trajectories develop.
Poincaré sections are then constructed by performing one further projection
onto the plane $(W,p_{W})$.

Regular trajectories appear in the Poincaré section as sets of points that can
be connected with smooth curves. Chaotic behavior on the other hand,
corresponds to sparse sets that fill the section.

The presence of analytic solutions manifests itself through “integrability
islands”.

  2. 2.

Fourier analysis: Chaos can often be confused with a quasiperiodic behavior, a
combination of linear oscillators with non-commensurable frequencies. In order
to exclude the latter possibility of our analysis, we consider the discrete
Fourier spectrum of one of the canonical variables.

A non-smooth Fourier spectrum is a clear signature of a chaotic regime.

  3. 3.

Geodesic divergence: In classical mechanics, the time evolution of a Newtonian
system of the kind defined by (22)-(23) can be described as a non-affine
parametrization of the geodesic curves on a manifold endowed with the so-
called Jacobi metric arnold ; loris , defined according to

$\displaystyle ds^{2}=g_{ij}\,dq^{i}dq^{j}$ $\displaystyle=2({\cal
E}-V)\left(dW^{2}+dG^{2}\right)\;,$ $\displaystyle=4({\cal E}-V)^{2}\,dt^{2}\
,$ (63)

where $i,j$ run on the independent generalized coordinates $q^{i}=(G,W)$ and
${\cal E}$ is the energy of the system (15).

The relation between the curvature of the manifold and the stability of the
geodesics is expressed in terms of the Jacobi-Levi-Civita equation for the
Jacobi field $\eta^{i}$, measuring the deviation between two infinitesimally
close geodesics

${\nabla_{s}^{2}\eta^{i}}-\mathcal{R}^{i}{}_{jkl}\frac{dq^{j}}{ds}\frac{dq^{k}}{ds}\eta^{l}=0\
.$ (64)

where $\nabla_{s}$ is the covariant derivative. In a two dimensional manifold
the Riemann tensor can be written in terms of the scalar curvature
$\mathcal{R}$ in the form ${\cal R}^{i}{}_{jkl}={\cal
R}(\delta^{i}{}_{k}g_{jl}-\delta^{i}{}_{l}g_{jk})/2$. This implies that

${\nabla^{2}_{s}\eta^{i}}+\frac{\mathcal{R}}{2}\left(\eta^{i}-\frac{dq^{i}}{ds}\frac{dq^{j}}{ds}\eta_{j}\right)=0\
.$ (65)

Where in the second term we used the fact that $s$ is an affine parameter and
the tangent vector is normalized to one. Contracting with $dq_{i}/ds$ and
$\varepsilon_{ij}\,dq^{j}/ds$ (with $\varepsilon_{ij}$ the Levi-Civita tensor)
and taking into account the geodesic equation $\nabla_{s}(dq^{i}/ds)=0$, we
can write

${\frac{d^{2}\eta_{\perp}}{ds^{2}}+\frac{\mathcal{R}}{2}\eta_{\perp}=0\
,\qquad\qquad\quad\nabla_{s}^{2}\eta_{\parallel}}=0\ ,$ (66)

Here we have defined $\eta_{\parallel}=\eta^{i}(dq^{i}/ds)$ and
$\eta_{\perp}=\epsilon_{ij}\eta^{i}(dq^{j}/ds)$.

It is clear that a negative scalar curvature $\mathcal{R}<0$ would lead to
solutions with an exponential grow in time for $\eta_{\perp}$. However, for
our Newtonian system the Ricci scalar $\mathcal{R}$ is given by

$\displaystyle\mathcal{R}$ $\displaystyle=\frac{1}{2\left({\cal
E}-V\right)^{3}}\left[4W^{2}\left(2G^{2}+W^{2}\right)^{2}+G^{2}\left(4W^{2}+\lambda\left(G^{2}-\nu^{2}\right)\right)^{2}\right]+$
$\displaystyle~{}~{}~{}~{}+\frac{1}{2\left({\cal
E}-V\right)^{2}}\left[(4+3\lambda)G^{2}+10W^{2}-\lambda\nu^{2}\right]\ ,$ (67)

Recalling that ${\cal E}-V>0$, the curvature scalar has a chance to be
negative only when $G$ and $W$ are small enough, so that the last term in the
second line superseeds the rest. In most configurations this is not the case,
thus the instabilities we eventually find should come from parametric
resonance, as the scalar ${\cal R}$ is time dependent (for a similar
situation, see cerruti-pettini ).

The equation for the nearby geodesic deviation can be rewritten as

$\frac{d^{2}Y}{dt^{2}}+\Sigma\,Y=0\ ,$ (68)

where $Y=\eta_{\perp}/\sqrt{{\cal E}-V}$ and $\Sigma$ is a function of time
defined as

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle 2{\cal R}({\cal
E}-V)^{2}-\frac{1}{2({\cal E}-V)}\frac{d^{2}V}{dt^{2}}-\frac{3}{4({\cal
E}-V)^{2}}\left(\frac{dV}{dt}\right)^{2}$ (69)

The form of the solution $Y$ as a function of time gives us insight about the
behaviour of the perturbation of the configuration, varying the initial
conditions. If $Y$ is constant, it gives a signal of stability, while if $Y$
grows exponentially in time, the system could develop a chaotic behavior in
that region.

###  4.1 Chaos with vanishing vacuum expectation value

The solution in (59)-(60) provides a uni-parametric family of analytic
solutions with parameter $a$, in terms of which the energy is fixed through
(33). By evaluating the solution at the initial time $t_{0}$, we obtain a set
of initial conditions with periodic evolution. To depart from the analytic
solution, we write one of the canonical variables in terms of the energy, say
$p_{G}^{2}(t_{0})=2({\cal E}-V)-\dot{W}(t_{0})$, and then move the value of
the energy ${\cal E}$ away from (33).

As we increase the energy with a fixed value of the coupling $\lambda$, we
find chaotic behaviour above a critical value, see the figure 4.

Figure 4: Transition to chaos with vanishing vacuum expectation value. From
left to right: Poincaré section, frequency spectrum and the logarithm of the
geodesic deviation $\log|Y|$. We evolved up to $t_{f}=15000$ with $a=0.5$.
Notice that between ${\cal E}=0.127$ and ${\cal E}=0.128$ there is a
transition to chaos.

###  4.2 Chaos with non-vanishing vacuum expectation value

The solution (42) is devoid of integration constants. We will proceed as
before in order to explore the phase space, evaluating the initial conditions
using the solution and deforming one of the fields away from the analytic
regime by varying the energy ${\cal E}$. The results are presented in figure
5.

Figure 5: Transition to chaos with non-vanishing vacuum expectation value.
From left to right: Poincaré section, frequency spectrum and the logarithm of
the geodesic deviation $\log|Y|$. We evolved up to $t_{f}=15000$ with
$\lambda=4.2$. Notice that between ${\cal E}=1.195$ and ${\cal E}=0.179$ there
is a transition to chaos.

We checked the transition to chaos can be observed also on the effective
$U(1)$ field (18). This is interesting since such a non-linear phenomenon is
not expected in the standard linear $U(1)$ gauge dynamics, representing a
genuine non-Abelian effect.

##  5 Probe scalar field

In order to explore the features of these configurations let us consider a
probe scalar field $\psi$ which transforms in the fundamental representation
of $SU(2)$. The covariant derivative is defined as

$D_{\mu}\psi=\partial_{\mu}\psi+A_{\mu}\psi\ ,$ (70)

in such a way that its commutator gives the field strength (3) _i.e_
$\left[D_{\mu},D_{\nu}\right]\psi=F_{\mu\nu}\psi$. The action principle for
the scalar field is given by

$I[\psi,\psi^{\dagger}]=-\int
d^{4}x\sqrt{-g}\left(D_{\mu}\psi\right)^{\dagger}D^{\mu}\psi\ .$ (71)

The equation coming from the variation of this action (71) in the background
(11) expands as

$\displaystyle-\partial_{t}^{2}\psi+\partial_{z}^{2}\psi+\frac{1}{\rho}\partial_{\rho}\psi+\partial_{\rho}^{2}\psi+\frac{1}{\rho^{2}}\partial_{\varphi}^{2}\psi-W^{2}\psi-\frac{1}{4\rho^{2}}\psi+$
(72)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{2}{\rho^{2}}\left(-\frac{W}{\sqrt{2}}\rho
t_{1}-\frac{1}{2}t_{3}\right)\partial_{\varphi}\psi+\frac{1}{\rho}\frac{W}{\sqrt{2}}t_{2}\psi+\frac{2W}{\sqrt{2}}t_{2}\partial_{\rho}\psi\left.=\right.0\
.$

In order to apply time-dependent perturbation theory, to evaluate transitions
amplitudes of the state of the scalar triggered by the interaction with the
background field, it is convenient to separate the above equations into two
terms. The first term corresponds to

$\displaystyle
H_{0}\psi=-\partial_{t}^{2}\psi+\partial_{z}^{2}\psi+\frac{1}{\rho}\partial_{\rho}\psi+\partial_{\rho}^{2}\psi+\frac{1}{\rho^{2}}\partial_{\varphi}^{2}\psi-\frac{1}{4\rho^{2}}\psi-\frac{t_{3}}{\rho^{2}}\partial_{\varphi}\psi\left.\right.\
$

which defines the action of $H_{0}$ on the scalar, while the second term
defines $H_{\sf int}$ by

$H_{\sf
int}\psi=-W^{2}\psi-\frac{\sqrt{2}W}{\rho^{2}}t_{1}\partial_{\varphi}\psi+\frac{1}{\rho}\frac{W}{\sqrt{2}}t_{2}\psi+\frac{2W}{\sqrt{2}}t_{2}\partial_{\rho}\psi\
.$ (73)

This splitting allows to analyze the time-dependent part of the gauge field
with time-dependent perturbation theory taking advantage of the fact that the
“unperturbed Hamiltonian” $H_{0}\psi=0$ can be solved exactly. Hereafter we
proceed in a canonical fashion, and details can be found in the Appendix.

Using the symbols $\uparrow,\downarrow$ to denote the up and down components
of the field $\psi$, and the indices $n,\ell,m$ to identify the longitudinal,
radial and angular modes respectively, we obtain the eigenstates of the free
hamitonian $H_{0}$ as $|\uparrow n\ell m\pm\rangle$ and $|\downarrow n\ell
m\pm\rangle$ where $\pm$ denote left and right movers in the angular
direction. The referred transition amplitude turns out to be given by

$\displaystyle\langle\downarrow\ell^{\prime}m^{\prime}n^{\prime}+|H_{\sf
int}|\uparrow\ell mn-\rangle$ $\displaystyle=$ $\displaystyle-\frac{\pi
L\mathcal{N}_{\ell^{\prime}m^{\prime}n^{\prime}}\mathcal{\bar{N}}_{\ell
mn}}{R_{0}\sqrt{2\omega_{\ell^{\prime}m^{\prime}n^{\prime}}\bar{\omega}_{\ell
mn}}}\delta_{m}^{m^{\prime}}\delta_{\ell}^{\ell^{\prime}}\alpha_{n}^{m-\frac{1}{2}}\int
dt\,W(t)e^{i\left(\omega_{\ell^{\prime}m^{\prime}n^{\prime}}-\bar{\omega}_{\ell
mn}\right)t}\times$ (74) $\displaystyle\ \
\times\int_{0}^{R_{0}}d\rho\,\rho\,J_{m^{\prime}+\frac{1}{2}}\left(\alpha_{n^{\prime}}^{m^{\prime}+\frac{1}{2}}\frac{\rho}{R_{0}}\right)J_{m+\frac{1}{2}}\left(\alpha_{n}^{m-\frac{1}{2}}\frac{\rho}{R_{0}}\right)\
,$

where $J_{\eta}$ are Bessel functions and their zeros are labeled by
$\alpha_{n}^{\eta}$ with $n=1,2,\dots\ $, the constants ${\cal N}_{\ell mn}$
are for normalization in a cylinder of length $L$ and radius $R_{0}$, and
$\omega_{\ell mn}$ denote the eigenfrequencies of the unperturbed Hamiltonian
$H_{0}$.

The above formula for the transition amplitude corresponds to a probe scalar
field, coupled to the time-dependent topologically non-trivial Yang-Mills-
Higgs background, and it is the main technical result of the present section.
In particular, Eq. (74) shows that if the classical background is in its
integrable phase, then as it has been discussed in the analysis of the
Poincaré sections, the Fourier spectrum of the gauge field $W(t)$ has few
relevant peaks. In these cases, the transition amplitude will be different
from zero in such few cases, corresponding to the resonances between
$\omega_{\ell^{\prime}m^{\prime}n^{\prime}}-\bar{\omega}_{\ell mn}$ and the
Fourier components of $W$. On the other hand, in the chaotic regime, the
amplitude will be different from zero in a broad band of values for
$\omega_{\ell^{\prime}m^{\prime}n^{\prime}}-\bar{\omega}_{\ell mn}$.
Therefore, the transition amplitudes of the probe scalar field can detect
whether the non-Abelian coil is in the chaotic or integrable regime.

##  6 Conclusions

In the present paper we have discussed how the chaotic behavior of time-
dependent configurations in the $SU(2)$ Georgi-Glashow model is affected by
the Higgs coupling constant, by the vacuum expectation value as well as by the
presence of topologically non-trivial fluxes, which in the present case
correspond to the flux of the non-Abelian magnetic field projected along the
Higgs field.

There are many intriguing questions which have not been analyzed in detail so
far in the literature. For instance: does the presence of the Higgs potential
and of the vacuum expectation value increase or decrease the chaotic behavior
of the theory? What is the effect of non-trivial topological fluxes? The main
technical problem to solve in order to answer such questions is related to the
construction of a suitable Ansatz for the gauge and Higgs fields. Indeed, one
can easily write down explicit expressions both for the gauge and for the
Higgs fields where all the components depend on time only, as it is usually
done in the literature on the chaotic behavior of Yang-Mills-Higgs theory: see
chaosYM1 -chaosYMreview  and references therein. In this way the field
equations reduce consistently to a dynamical system which can be analyzed
using the known tools of chaotic dynamics. However, if all the fields only
depend on time, then the topological fluxes may vanish. This is the reason why
it is useful to design an Ansatz in such a way that the fields depend in a
non-trivial way also on the spatial coordinates, keeping alive the topological
fluxes, but with the field equations reducing to an autonomous dynamical
system. In the present work we have constructed such an Ansatz.

A byproduct of the analysis is that we have also have identified an integrable
sector where the field equations can be integrated analytically and the
corresponding exact solutions represent the non-Abelian version of self-
sustained alternating current generator. Moreover, the Ansatz has been
constructed in such a way that one can, for instance, increase (or decrease)
the control parameters, such as the Higgs coupling and the vacuum expectation
value, and analyze how this change affects the chaotic properties. This
situation is especially suitable to be studied using the tools introduced by
Casetti, Pettini and Cohen (see cerruti-pettini-cohen  and references therein)
in their geometric approach to the search for the stochasticity threshold in
Hamiltonian dynamics. Using these tools we have shown that as one increases
the energy, integrability is lost. Moreover, we proved that the chaotic
behavior and sensitive dependence on the initial condition shown by the
exponential growth of the geodesic deviation in Jacobi metric, are triggered
by a parametric resonance.

### Acknowledgements

F. C. and J. O. have been funded by Fondecyt Grants 1200022 and 1221504. The
work of M. O. is partially funded by Beca ANID de Doctorado 21222264. N. G.
wants to thank Centro de Estudios Científicos (CECs) and Universidad de
Concepción (UdeC) by hospitality and support during this work. The Centro de
Estudios Científicos (CECs) is funded by the Chilean Government through the
Centers of Excellence Base Financing Program of ANID.

##  Appendix A Perturbation theory

The free Hamiltonian $H_{0}$, can be diagonalized by the following field
configuration that fulfills the equation (72) with $W=0$

$\Phi\left(t,\vec{x}\right)=\bar{\phi}\left(t,\vec{x}\right)+\phi\left(t,\vec{x}\right)\
,$ (75) $\displaystyle\bar{\phi}^{a}\\!\left(t,\vec{x}\right)\\!$
$\displaystyle\equiv$ $\displaystyle\\!\sum_{\ell
mn}^{\text{{\LARGE\\_}}}\\!\frac{\mathcal{\bar{N}}_{\ell
mn}}{\sqrt{2\bar{\omega}_{\ell mn}}}\sin\\!\left(\frac{\pi\ell
z}{L}\right)J_{m-\frac{1}{2}}\\!\\!\left(\frac{\alpha^{m-\frac{1}{2}}_{n}\rho}{R_{0}}\right)\\!\left(\begin{array}[]{c}\boldsymbol{l}_{\uparrow\ell
mn}e^{i\left(m\varphi-\bar{\omega}_{\ell
mn}t\right)}+\tilde{\boldsymbol{l}}_{\uparrow\ell mn}^{\
\dagger}e^{i\left(m\varphi+\bar{\omega}_{\ell mn}t\right)}\\\
\\!\boldsymbol{l}_{\downarrow\ell mn}e^{-i\left(m\varphi+\bar{\omega}_{\ell
mn}t\right)}+\tilde{\boldsymbol{l}}_{\downarrow\ell mn}^{\
\dagger}e^{-i\left(m\varphi-\bar{\omega}_{\ell
mn}t\right)\\!}\end{array}\\!\right)^{\\!\\!a}$
$\displaystyle\phi^{a}\\!\left(t,\vec{x}\right)\\!$ $\displaystyle\equiv\\!$
$\displaystyle\sum_{\ell mn}\\!\frac{\mathcal{N}_{\ell
mn}}{\sqrt{2\omega_{\ell mn}}}\sin\\!\left(\frac{\pi\ell
z}{L}\right)J_{m+\frac{1}{2}}\\!\\!\left(\frac{\alpha^{m+\frac{1}{2}}_{n}\rho}{R_{0}}\right)\\!\left(\begin{array}[]{c}\\!\boldsymbol{r}_{\uparrow\ell
mn}e^{-i\left(m\varphi+\omega_{\ell
mn}t\right)}+\tilde{\boldsymbol{r}}_{\uparrow\ell mn}^{\
\dagger}e^{-i\left(m\varphi-\omega_{\ell mn}t\right)}\\!\\\
\boldsymbol{r}_{\downarrow\ell mn}e^{i\left(m\varphi-\omega_{\ell
mn}t\right)}+\tilde{\boldsymbol{r}}_{\downarrow\ell mn}^{\
\dagger}e^{i\left(m\varphi+\omega_{\ell
mn}^{\prime}t\right)}\end{array}\\!\right)^{\\!\\!a}$

Where $J_{\eta}$ are the Bessel functions and $\alpha^{\eta}_{n}$ is the
$n^{\text{th}}$ zero of the Bessel function $J_{\eta}$. In this expansion we
have eight types of creation and annihilation operators
$(\boldsymbol{l}_{\uparrow},\tilde{\boldsymbol{l}}_{\uparrow},\boldsymbol{l}_{\downarrow},\tilde{\boldsymbol{l}}_{\downarrow},\boldsymbol{r}_{\uparrow},$
$\tilde{\boldsymbol{r}}_{\uparrow},\boldsymbol{r}_{\downarrow},\tilde{\boldsymbol{r}}_{\downarrow})$.
The states that we used in (74) are

$\displaystyle|\uparrow\ell mn-\rangle=\boldsymbol{l}_{\uparrow\ell
mn}^{\dagger}|0\rangle\;,\qquad|\downarrow\ell^{\prime}m^{\prime}n^{\prime}+\rangle=\boldsymbol{r}_{\downarrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}|0\rangle\;,$
(78)

where $|0\rangle$ stands for the vacuum state of the theory. This solution
satisfies the boundary conditions $\Phi|_{\partial M}=0$ where $M$ is the
solid cylinder of radius $R_{0}$ and length $L$. The range of the integer
numbers $\ell,\ m,\ n$, as well as the definition of the summations in (75)
are

$\sum_{\ell
mn}^{\text{{\LARGE\\_}}}\equiv\sum_{\ell=1}^{\infty}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\
,\qquad\qquad\sum_{\ell
mn}\equiv\sum_{\ell=1}^{\infty}\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\ .$ (79)

The normalization constants are

$\mathcal{\bar{N}}_{\ell mn}=\sqrt{\frac{2}{\pi
L}}\frac{1}{R_{0}J_{m+\frac{1}{2}}\left(\alpha^{m-\frac{1}{2}}_{n}\right)}\
,\qquad\quad\mathcal{N}_{\ell mn}=\sqrt{\frac{2}{\pi
L}}\frac{1}{R_{0}J_{m+\frac{3}{2}}\left(\alpha^{m+\frac{1}{2}}_{n}\right)}\ ,$
(80)

and the frequencies that are fixed by the boundary conditions are given by

$\left(\bar{\omega}_{\ell
mn}\right)^{2}=\left(\frac{\alpha^{m-\frac{1}{2}}_{n}}{R_{0}}\right)^{2}+\left(\frac{\pi\ell}{L}\right)^{2}\
,\qquad\qquad\left(\omega_{\ell
mn}\right)^{2}=\left(\frac{\alpha^{m+\frac{1}{2}}_{n}}{R_{0}}\right)^{2}+\left(\frac{\pi\ell}{L}\right)^{2}\
.$ (81)

The state with smallest energy in the system is given by
$\bar{\omega}_{011}=\omega_{001}=(\alpha_{1}^{\frac{1}{2}}/R_{0})^{2}$.

The conjugate momenta for the Lagrangian defined in (71), following the
standard definitions are

$\displaystyle P_{a}$ $\displaystyle=$
$\displaystyle\sqrt{\gamma}\left(\partial_{t}\psi^{\dagger}-\psi^{\dagger}A_{t}\right)_{a}\
,$ (82) $\displaystyle P^{\prime a}$ $\displaystyle=$
$\displaystyle\sqrt{\gamma}\left(\partial_{t}\psi+A_{t}\psi\right)^{a}\ ,$
(83)

here $\gamma$ is the determinant of the induced metric
$\gamma_{\mu\nu}=g_{\mu\nu}+\delta_{\mu}^{t}\delta_{\nu}^{t}$, which is the
spatial section of the metric (10), then $\sqrt{\gamma}=\rho$. The canonical
momenta given in terms of (75) through the above definition, forms a
representation of the canonical algebra

$\left[\Phi^{a}\left(t,\mathbf{x}\right),P_{b}\left(t,\mathbf{y}\right)\right]=i\delta_{b}^{a}\delta\left(\mathbf{x}-\mathbf{y}\right)\
,\quad\left[\Phi_{a}^{\dagger}\left(t,\mathbf{x}\right),P^{\prime
b}\left(t,\mathbf{y}\right)\right]=i\delta_{a}^{b}\delta\left(\mathbf{x}-\mathbf{y}\right)\
,$ (84)

with the following commutation relation for the creation/annihilation
operators

$\displaystyle\left[\boldsymbol{l}_{\uparrow\ell
mn},\boldsymbol{l}_{\uparrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]$
$\displaystyle=$ $\displaystyle\left[\tilde{\boldsymbol{l}}_{\uparrow\ell
mn},\tilde{\boldsymbol{l}}_{\uparrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\left[\boldsymbol{r}_{\uparrow\ell
mn},\boldsymbol{r}_{\uparrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\left[\tilde{\boldsymbol{r}}_{\uparrow\ell
mn},\tilde{\boldsymbol{r}}_{\uparrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\delta_{\ell}^{\ell^{\prime}}\delta_{m}^{m^{\prime}}\delta_{n}^{n^{\prime}}\
,$ $\displaystyle\left[\boldsymbol{l}_{\downarrow\ell
mn},\boldsymbol{l}_{\downarrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]$
$\displaystyle=$ $\displaystyle\left[\tilde{\boldsymbol{l}}_{\downarrow\ell
mn},\tilde{\boldsymbol{l}}_{\downarrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\left[\boldsymbol{r}_{\downarrow\ell
mn},\boldsymbol{r}_{\downarrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\left[\tilde{\boldsymbol{r}}_{\downarrow\ell
mn},\tilde{\boldsymbol{r}}_{\downarrow\ell^{\prime}m^{\prime}n^{\prime}}^{\dagger}\right]=\delta_{\ell}^{\ell^{\prime}}\delta_{m}^{m^{\prime}}\delta_{n}^{n^{\prime}}\
.$

To see how this works, we compute one commutator between $\Phi^{a}$ and
$P_{b}$. The following representations of the Dirac delta will be useful

$\displaystyle\delta\left(\varphi-\varphi^{\prime}\right)$ $\displaystyle=$
$\displaystyle\sum_{m=-\infty}^{\infty}\frac{1}{2\pi}e^{im\left(\varphi-\varphi^{\prime}\right)}\
,$ $\displaystyle\delta\left(\rho-\rho^{\prime}\right)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{2\rho}{R_{0}^{2}J_{\eta+1}\left(\alpha_{n}^{\eta}\right)^{2}}J_{\eta}\left(\frac{\alpha_{n}^{\eta}\rho}{R_{0}}\right)J_{\eta}\left(\frac{\alpha_{n}^{\eta}\rho^{\prime}}{R_{0}}\right)\
,$ (85) $\displaystyle\delta\left(z-z^{\prime}\right)$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\infty}\frac{2}{L}\sin\left(\frac{\pi\ell
z^{\prime}}{L}\right)\sin\left(\frac{\pi\ell z}{L}\right)\ .$

The expression of $P_{b}$ following the definition for our magnetic background
is

$P_{a}\equiv\rho\partial_{t}\Phi^{\dagger}\equiv\bar{p}_{a}+p_{a}\ ,$ (86)

where

$\displaystyle\bar{p}_{a}\\!$ $\displaystyle=$ $\displaystyle\\!\sum_{\ell
mn}^{\text{{\LARGE\\_}}}\rho i\mathcal{\bar{N}}_{\ell
mn}\sqrt{\frac{\bar{\omega}_{\ell mn}}{2}}\sin\\!\left(\frac{\pi\ell
z}{L}\right)J_{m-\frac{1}{2}}\\!\left(\frac{\bar{\chi}_{n}^{m}\rho}{R_{0}}\right)\\!\left(\begin{array}[]{c}\boldsymbol{l}_{\uparrow\ell
mn}^{\dagger}e^{-i\left(m\varphi-\bar{\omega}_{\ell
mn}t\right)}-\tilde{\boldsymbol{l}}_{\uparrow\ell
mn}e^{-i\left(m\varphi+\bar{\omega}_{\ell mn}t\right)}\\\
\boldsymbol{l}_{\downarrow\ell
mn}^{\dagger}e^{i\left(m\varphi+\bar{\omega}_{\ell
mn}t\right)}-\tilde{\boldsymbol{l}}_{\downarrow\ell
mn}e^{i\left(m\varphi-\bar{\omega}_{\ell
mn}t\right)}\end{array}\right)^{\\!\\!T},$ (89) $\displaystyle p_{a}\\!$
$\displaystyle=$ $\displaystyle\\!\sum_{\ell mn}\rho i\mathcal{N}_{\ell
mn}\sqrt{\frac{\omega_{\ell mn}}{2}}\sin\\!\left(\frac{\pi\ell
z}{L}\right)J_{m+\frac{1}{2}}\\!\left(\frac{\chi_{n}^{m}\rho}{R_{0}}\right)\\!\left(\begin{array}[]{c}\boldsymbol{r}_{\uparrow\ell
mn}^{\dagger}e^{i\left(m\varphi+\omega_{\ell
mn}t\right)}-\tilde{\boldsymbol{r}}_{\uparrow\ell
mn}e^{i\left(m\varphi-\omega_{\ell mn}t\right)}\\\
\boldsymbol{r}_{\downarrow\ell mn}^{\dagger}e^{-i\left(m\varphi-\omega_{\ell
mn}t\right)}-\tilde{\boldsymbol{r}}_{\downarrow\ell
mn}e^{-i\left(m\varphi+\omega_{\ell mn}t\right)}\end{array}\right)^{\\!\\!T},$
(92) $\displaystyle\bar{\chi}_{n}^{m}$ $\displaystyle\equiv$
$\displaystyle\alpha_{n}^{m-1}\;,\quad\chi_{n}^{m}\equiv\alpha_{n}^{m+1}\;.$

The commutator can be written as

$\left[\Phi^{a}\left(t,\vec{x}\right),P_{b}\left(t,\vec{x}^{\prime}\right)\right]=\left(\left[\bar{\phi}^{1},\bar{p}_{1}\right]+\left[\phi^{1},p_{1}\right]\right)\delta_{1}^{a}\delta_{b}^{1}+\left(\left[\bar{\phi}^{2},\bar{p}_{2}\right]+\left[\phi^{2},p_{2}\right]\right)\delta_{2}^{a}\delta_{b}^{2}\
.$ (93)

Let us compute explicitly the first parenthesis of (93). The first commutator
is

$\displaystyle\left[\bar{\phi}^{1},\bar{p}_{1}\right]$ $\displaystyle=$
$\displaystyle\sum_{\ell mn}^{\text{{\LARGE\\_}}}i\rho\mathcal{\bar{N}}_{\ell
mn}^{2}\sin\left(\frac{\pi\ell z}{L}\right)\sin\left(\frac{\pi\ell
z^{\prime}}{L}\right)J_{m-\frac{1}{2}}\left(\frac{\bar{\chi}_{n}^{m}\rho}{R_{0}}\right)J_{m-\frac{1}{2}}\left(\frac{\bar{\chi}_{n}^{m}\rho^{\prime}}{R_{0}}\right)e^{im\left(\varphi-\varphi^{\prime}\right)}\
,$ (94) $\displaystyle=$
$\displaystyle\delta\left(z-z^{\prime}\right)\sum_{\ell
mn}^{\text{{\LARGE\\_}}}i\frac{1}{2\pi}\frac{2}{R_{0}^{2}J_{m+\frac{1}{2}}\left(\bar{\chi}_{n}^{m}\right)^{2}}\rho
J_{m-\frac{1}{2}}\left(\frac{\bar{\chi}_{n}^{m}\rho}{R_{0}}\right)J_{m-\frac{1}{2}}\left(\frac{\bar{\chi}_{n}^{m}\rho^{\prime}}{R_{0}}\right)e^{im\left(\varphi-\varphi^{\prime}\right)}\
,$ $\displaystyle=$ $\displaystyle
i\frac{1}{2\pi}\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)\sum_{m=1}^{\infty}e^{im\left(\varphi-\varphi^{\prime}\right)}\
.$

while the second term is

$\displaystyle\left[\phi^{1},p_{1}\right]\\!$ $\displaystyle=$
$\displaystyle\\!\sum_{\ell mn}\frac{2i}{\pi
L}\frac{\rho}{R_{0}^{2}J_{m+\frac{3}{2}}\\!\left(\chi_{n}^{m}\right)^{2}}\sin\\!\left(\frac{\pi\ell
z}{L}\right)\sin\\!\left(\frac{\pi\ell
z^{\prime}}{L}\right)J_{m+\frac{1}{2}}\\!\left(\frac{\chi_{n}^{m}\rho}{R_{0}}\right)J_{m+\frac{1}{2}}\\!\left(\frac{\chi_{n}^{m}\rho^{\prime}}{R_{0}}\right)e^{-im\left(\varphi-\varphi^{\prime}\right)}\
,$ (95) $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}i\frac{1}{2\pi}\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)e^{-im\left(\varphi-\varphi^{\prime}\right)}\
.$

Replacing in the first parenthesis in (93)

$\left[\bar{\phi}^{1},\bar{p}_{1}\right]+\left[\phi^{1},p_{1}\right]=i\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)\frac{1}{2\pi}\left(\sum_{m=1}^{+\infty}e^{im\left(\varphi-\varphi^{\prime}\right)}+\sum_{m=0}^{\infty}e^{-im\left(\varphi-\varphi^{\prime}\right)}\right)\
,$ (96)

changing the sign in the second summation and using the fact that we get the
representation of the delta function, thus

$\left[\bar{\phi}^{1},\bar{p}_{1}\right]+\left[\phi^{1},p_{1}\right]=i\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)\delta\left(\varphi-\varphi^{\prime}\right)\
.$ (97)

One can show that the same mechanism works for the second parenthesis in (93),

$\left[\bar{\phi}^{2},\bar{p}_{2}\right]+\left[\phi^{2},p_{2}\right]=i\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)\delta\left(\varphi-\varphi^{\prime}\right)\
.$ (98)

Replacing back into (93) we find

$\left[\Phi^{a}\left(t,\vec{x}\right),P_{b}\left(t,\vec{x}^{\prime}\right)\right]=i\delta_{b}^{a}\delta\left(z-z^{\prime}\right)\delta\left(\rho-\rho^{\prime}\right)\delta\left(\varphi-\varphi^{\prime}\right)\
,$ (99)

as promised.

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