# The Advective Flux Transport Model: Improving the Far-Side with Active
Regions observed by STEREO 304Å

Lisa A. Upton  Southwest Research Institute, 1050 Walnut St, Suite 300,
Boulder, CO 80302  Space Systems Research Corporation, Alexandria, VA, 22314
Ignacio Ugarte-Urra  Space Science Division, Naval Research Laboratory,
Washington, DC 20375, USA  Harry P. Warren  Space Science Division, Naval
Research Laboratory, Washington, DC 20375, USA  David H. Hathaway  Hansen
Experimental Physics Lab., Stanford University, HEPL-4085, Stanford, CA
94305-408, USA

(Revised ApJ)

###### Abstract

Observations the Sun’s photospheric magnetic field are often confined to the
Sun-Earth line. Surface flux transport (SFT) models, such as the Advective
Flux Transport (AFT) model, simulate the evolution of the photospheric
magnetic field to produce magnetic maps over the entire surface of the Sun.
While these models are able to evolve active regions that transit the near-
side of the Sun, new far-side side flux emergence is typically neglected. We
demonstrate a new method for creating improved maps of magnetic field over the
Sun’s entire photosphere using data obtained by the STEREO mission. The STEREO
He II 304 Å intensity images are used to infer the time, location, and total
unsigned magnetic flux of far-side active regions. We have developed and
automatic detection algorithm for finding and ingesting new far-side active
region emergence into the AFT model. We conduct a series of simulations to
investigate the impact of including active region emergence in AFT, both with
and without data assimilation of magnetograms. We find that while the He II
304 Å can be used to improve surface flux models, but care must taken to
mitigate intensity surges from flaring events. We estimate that during Solar
Cycle 24 maximum (2011-2015), $4-6\times 10^{22}$ Mx of flux is missing from
SFT models that do not include far-side data. We find that while He II 304 Å
data alone can be used to create synchronic maps of photospheric magnetic
field that resemble the observations, it is insufficient to produce a complete
picture without direct magnetic observations from magnetographs.

Sun: magnetic fields

††facilities: SDO (AIA,HMI), STEREO (EUVI)††software: SolarSoft (Freeland &
Handy, 2012), VSO (Hill et al., 2009)

##  1 Introduction

One of the biggest discoveries about the Sun in the last century was its
magnetic nature. Magnetism on the Sun is ubiquitous and plays a fundamental
role in nearly every process occurring on the sun - from the driving the solar
cycles (Babcock, 1961; Leighton, 1969) to eruptive events like flares (Webb &
Howard, 2012), which impact space weather and can produce geomagnetic storms
here on Earth (Baker & Lanzerotti, 2016). Unfortunately, magnetic solar
observations are currently limited to near-side or Earth-facing side of the
Sun. Surface Flux Transport (SFT) models (see review articles Sheeley, 2005;
Jiang et al., 2014; Yeates et al., 2023, and references within) simulate the
evolution of the Sun’s photospheric magnetic field, creating instantaneous
maps of the Sun’s magnetic field over the entire surface of the Sun, e.g.,
’synchronic maps’. The Advective Flux Transport (AFT) (Upton & Hathaway,
2014b, a) model is a state-of-the-art SFT model that uses data assimilation of
magnetograms combined with a convective simulation to produce synchronic maps
with unparalleled realism. AFT has been shown to accurately (within a factor
of 2) reproduce the evolution of the total unsigned flux of simple active
regions over the course of their lifetimes (Ugarte-Urra et al., 2015). AFT,
and other SFT models, help fill in the blank for the far-side of the Sun,
improving our ability to make predictions and space weather forecasts (Arge et
al., 2010; Mackay & Yeates, 2012; Henney et al., 2015; Schrijver et al., 2015,
SFT) (Yardley et al., 2021; Warren et al., 2021; Mackay & Upton, 2022, AFT).

While AFT and other SFT models are able to produce maps that include decaying
active regions on the far-side, unfortunately new or additional active region
emergence that occurs on the far-side ischallenging. The AFT Baseline model
performs data assimilation of SDO/HMI magnetograms (Scherrer et al., 2012) to
incorporate the near side magnetic flux. The AFT Baseline maps are the most
accurate synchronic maps AFT is currently able to produce with near-side data
alone. However, while these maps continue to evolve the flux that has been
observed on the near-side, they still lack new active region emergence that
occurs on the far-side of the Sun. To improve upon the AFT Baseline, we create
an new version of AFT which continues to perform data assimilation of SDO/HMI
magnetograms, but also includes far-side Active Regions.

In the last two decades, several missions to explore and directly observe the
far-side of the Sun have been launched, beginning with the twin Solar
TErrestrial RElations Observatory (Kaiser et al., 2008, STEREO) satellites in
2006. This pair of satellites entered orbits with one moving progressively
ahead of Earth (STEREO A) and one moving progressively behind (STEREO B). Data
obtained by pair with the EUVI/STEREO (Howard et al., 2008) instruments
enabled stereoscopic imaging of the Sun in the extreme ultraviolet (EUV). By
2011, combining STEREO observations with data obtained by the Atmospheric
Imaging Assembly (Lemen et al., 2012, AIA) provided continuous full 360 degree
coverage of the active latitudes for nearly four years and at continuous
partial far-side coverage for well over a decade. In 2020, a newer mission
that directly images the Sun’s far-side was launched: ESA/NASA’s Solar Orbiter
(Müller et al., 2020). Solar Orbiter’s primary mission is not specifically to
observe the far-side of the Sun, but rather to fly close to the Sun in an
inclined orbit, allowing it to take very high resolution measurements of the
Sun and from higher latitudes than can be observed from Earth’s orbit.
However, its unique trajectory enable it to provide valuable data on the Sun’s
far-side during portions of its orbit. Most notably, this includes the first
far-side magnetograms ever obtained (Yang et al., 2023).

In Ugarte-Urra et al. (2015), we showed that 304 Å images can be used as a
proxy for magnetic flux measurements when magnetic field data is not
available. In this paper, we demonstrate how the 304 Å proxy can be used to
systematically identify far-side active regions observed in 304 Å and
incorporate them into AFT in an autonomous way. We begin by discussing some
revisions that we have made to optimize the flux-luminosity relationship for
use with the AFT model. We then describe the process for automatically
detecting active regions in the 304 Å maps and how the far-side active region
emergence is incorporated into the AFT simulations. Finally, we discuss the
impact of the far-side active region on both active region and solar cycle
time scales.

##  2 Revised Flux-Luminosity Relationship

Figure 1: Relationship between the 304 Å photon flux and the total unsigned
magnetic flux in the AFT maps of 500 active regions in the period 2010–2018.
The photon flux is obtained for full Sun synchronic Carrington maps above a
threshold of 500 DN/s/arcsec2. The magnetic flux is calculated for AFT
synchronic maps considering magnetic flux densities in the range 100–900G.

In a previous study Ugarte-Urra et al. (2015) we showed that He II 304 Å
images can be used to infer the total unsigned magnetic flux of a region on
the Sun when measurements of the magnetic field are not available. The proxy
is given by an empirical power-law relationship between the total photon flux
in 304 Å and the total unsigned magnetic flux. This method was shown to be
sufficient to obtain accurate information about active regions emerging in the
far side of the Sun, and was employed to study the continuous long term
evolution of active region magnetic flux (Ugarte-Urra et al., 2015, 2017;
Iglesias et al., 2019).

In the current study we revisit this technique with the objective of
systematizing a processing pipeline that consists in: acquisition, calibration
and manipulation of 304 Å images from EUVI/STEREO and AIA/SDO, followed by the
merger of the images to create a synchronic map of the full Sun that, in its
final product, is corrected for center-to-limb variations, sensitivity
differences between instruments and sensitivity decay as a function of time.
These maps are then used to estimate new empirical relationships for the total
photon flux and the total unsigned magnetic flux under various thresholding
scenarios. The maps and flux-flux relationships then serve as inputs to the
far-side magnetic flux emergence detection algorithm described in Section 4

Several of these steps were already part of the analysis in Ugarte-Urra et al.
(2015). Here we have introduced a correction for the center-to-limb variation
of the He II 304 Å line. We have adopted the parameterization described in
Mango et al. (1978): $I(\mu)=I_{c}[1+a(1-\mu)]$, where $I_{c}$ is the disk-
center intensity, $\mu$ is the cosine of the heliocentric angle, and $a=0.36$
is the relative limb brightening. This factor was obtained from the fits of
that curve to solar disk limb-brightening maps made from the median of the
weakest on disk intensities in a Carrington rotation. $a$ is the average of
the fit coefficient for six Carrington rotations (2055, 2078, 2085, 2095,
2110, 2137). The effect of the center-to-limb variation was also removed in
our previous analyses (Ugarte-Urra et al., 2015; Ugarte-Urra & Warren, 2012),
but was dealt with within a more generic description (”edge effects”) and
corrected using quiet Sun data from the same map. Before combining AIA and
EUVI images into a single map, AIA 304 Å images are scaled to EUVI intensity
levels with a time dependent factor provided in the SolarSoft STEREO beacon
directories (see Ugarte-Urra et al., 2015). This correction takes care of the
sensitivity decay of the AIA instrument. The resulting 304 Å full Sun
synchronic maps (Level 1), with a spatial resolution of 1$\deg$ in longitude
and latitude, constitute the main database and are saved with a cadence of 8h,
i.e. three files per day.

Figure 2: Active Region Detection. The combined STEREO/AIA 304Å map for
January 1st 2013 (top left). The active regions identified in the STEREO/AIA
304 Å map with the automated detection algorithm (top right). The AFT map for
January 1st 2013 (bottom left). The AFT active regions that were identified in
the STEREO/AIA map (bottom right). The centroid positions of the active
regions identified in the 304 Å image are marked with a black/red squares in
the 304 Å (top right) and AFT (bottom right) images.

For the purposes of the emergence detection algorithm, we also make a higher
order (Level 2) processed file (Figure 2a). This map is corrected for the
temporal decay of EUVI sensitivity, computed from the decay in the centroid
value of a histogram of intensities during the years 2007 – 2019. The centroid
for each datapoint per year was taken from the twenty 304 Å synchronic maps in
a five day interval at a 6h cadence. The correction we implement is:
$f=1-1.268\times 10^{-9}\rm dt$ where the corrected intensity is
$I_{c}=I_{304}/f$ and $\rm dt$ is the time in seconds since the launch of
STEREO (Oct 25, 2006). This correction amounts to a $f=0.51$ as of January 1,
2019. In these maps we also set to zero regions close to the limb of any of
the AIA and EUVI images that make the map, defined as the locations that are
beyond 0.9 times the radial distance to the solar disk center. Finally, we
remove from the maps intensity spikes, namely flares, with a despiking
algorithm available in SolarSoft (CLEAN_EXPOSURE.PRO) designed for cosmic ray
despiking in spectra, with the coefficients optimized for our purpose. Pixels
identified as spikes are filled in with the median of the surrounding pixels.
This step is necessary to avoid the effect that large intensity levels from
flares have on the magnetic flux proxy, which relies on ’standard’ 304 Å
active region intensities.

An example of the corrected STEREO/AIA map for January 1, 2013 is shown in the
upper left panel of Figure 2. We use these new corrected maps to compute new
empirical power-law relationships between the total 304 Å photon flux and the
total unsigned magnetic flux of active regions. We calculate them from a
dataset of 500 active regions observed at central meridian, starting with NOAA
11065 (May 1, 2010) and ending with NOAA 12703 (April 4, 2018), which averages
to 1 in 3 NOAA regions in that period. For each region we extract an area of
450$\arcsec$ in Helioprojective Cartesian coordinates ($\sim 30\deg$ in
Carrington Heliographic Coordinates) and integrate the 304 Å intensities above
a predefined threshold. A threshold of 295 DN/s/arcsec2 (units that the maps
are saved in) returns us intensities above 2 times the standard deviation of
the map’s histogram. In Fig. 1 we show the flux-flux relationship above a
threshold of 500 DN/s/arcsec2. The total unsigned magnetic flux in this case
is obtained from the AFT Baseline maps considering only magnetic flux
densities in the range 100 – 900 G (Ugarte-Urra et al., 2017). Near central
meridian those fluxes are basically the assimilated values from HMI
magnetograms. Based on visual inspection of the data, we have discarded data
points with less than $1\times 10^{6}$ DN/s and the corresponding magnetic
flux given by the linear fit to the power law:
$\log_{10}I=C+(\alpha*\log_{10}\Phi)$. This relationship inferred from data
observed on the Earth side, allow us to obtain estimates of magnetic flux for
regions on the far side for which we do not have magnetic field information.

##  3 AFT 304 Automated Detection Algorithm

The corrected STEREO/AIA 304 maps are also used to identify far-side active
region emergence to input into the AFT model. We have developed an automatic
detection algorithm for finding and adding far-side emergence to AFT. The
first step of the emergence detection algorithm is to identify the active
regions. We begin by creating a binary map of active region locations. We zero
out all values in the corrected STEREO/AIA maps where the intensity is less
than 500 DN/s/arcsec2. We then smooth the map with a 7 pixel boxcar average
filter. All values where the intensity is less than 500 DN/s/arcsec2 are then
zeroed out again, producing a binary map of discrete areas that correspond to
the active regions. Multiplying the corrected STEREO/AIA 304 map by the binary
map clearly reveals the STEREO/AIA 304 active regions and their boundaries.
Each discrete active region area is labeled with a unique identifier and a
list is generated.

We then calculate the photon flux (in DN/s) of each active region by
multiplying the corrected STEREO/AIA 304 maps by area per pixel and
integrating. Using the flux-luminosity relationship derived in the previous
section (Figure 1), we calculate the proxy magnetic flux ($\Phi_{304}$) for
each of the potential active regions. We then exclude regions with a proxy
flux less than the threshold of $5.7\times 10^{20}$ Mx, the lower limit of the
flux-luminosity relationship. The remaining detected active regions are shown
in the top right panel of Figure 2. Finally, we calculate the latitude and
longitude of the centroid of each of the remaining active regions (indicated
by the square red dots in the image).

The next step of the automatic detection algorithm is to identify which far-
side active regions have new flux emergence ($\Phi_{new}$). This is done by
comparing the flux in the STEREO/AIA 304 detected active regions
($\Phi_{304}$) with the flux in active regions in the AFT maps ($\Phi_{AFT}$).
We begin by eliminating all near-side active regions (which are presumed to
already be included from the data assimilation of HMI maps) by excluding any
304 active region located closer than $\pm$ 75 degrees from the central
meridian. For the remaining active regions, we calculate the total unsigned
magnetic flux for each active region in the AFT map ($\Phi_{AFT}$), again
using the areas from the binary map above (panel d of Figure 2). We then
compare the total unsigned flux for each active region in the AFT map to the
proxy flux calculated from the corrected STEREO/AIA 304 maps. If the 304 flux
is at least 1.25 times as large as the AFT flux, new flux ($\Phi_{new}$) is
assumed to have emerged on the far-side, where

$\displaystyle\Phi_{new}=\Phi_{AFT}-\Phi_{304}.$ (1)

The final step of the automated algorithm is to add the newly emerged far-side
active region flux into the AFT model. This is done by first calculating the
area of the active region from flux-area relationship (Sheeley, 1966; Mosher,
1977):

$\displaystyle A(\Phi_{304})$ $\displaystyle=7.0\times 10^{19}/\Phi_{304}$ (2)

where $\Phi_{304}$ is the total magnetic flux of the active region in Maxwells
as determined by the 304 proxy and $A$ is the total sunspot area in units of
micro Hemispheres. Using the total area ($A$) and the latitude of the centroid
of the active regions ($\lambda$), we then calculate the bipole tilt angle and
bipole separation distance in terms of the latitudinal and longitudinal
separations ($\Delta\phi$ and $\Delta\theta$).

In this work, we use the longitude separations given by Hathaway & Upton
(2016) For a selection of active regions in SOHO/MDI magnetograms, they
identified the centroid positions of the bipolar components and calculated the
longitudinal separations as a function of the active region area
($\Delta\phi(A)$, where $A$ is in units of millionths of a hemisphere). They
found that a good fit to the data (blue line in Figure 3 ) could be found with
the following equation:

$\displaystyle\Delta\phi(A)$ $\displaystyle=3+8\times\tanh{(A/500)}$ (3)

The active region tilts were derived in a similar manner by Stenflo &
Kosovichev (2012). They found that the tilt as a function of latitude
($\Delta\theta(\lambda)$) could be described by the equation:

$\displaystyle\Delta\theta(\lambda)$
$\displaystyle=\Delta\phi\times\tan(32.1\sin(\lambda))$ (4)

Stenflo & Kosovichev (2012)

We split the new active region flux equally between two opposite polarity
bipoles. The bipoles are inserted into AFT as two Gaussian spots with a peak
field strength of $\pm$ 900G with Hale’s polarity and the latitudinal and
longitudinal separations ($\Delta\phi$ and $\Delta\theta$) centered on the
centroid of the active region. This is repeated for all active regions where
new flux emergence has been identified. AFT is then evolved in time to produce
the map for the next time step.

After the completion of this project, we have conducted a more thorough
analysis of longitude separation of the active regions in SOHO/MDI. For 3031
active regions observed by MDI, we identified the day of maximum area for each
active region. We then determined the centroid of each polarity in the active
region and calculated the longitudinal extent. We divided them into 31 sunspot
area bins and found the average (as shown in Figure 3 with 2 sigma error bars
on the averages). We find that a better fit (red line in Figure 3 ) to the
longitudinal separations as a function of the active region area
($\Delta\phi(A)$) is given by the following equation:

$\displaystyle\Delta\phi(A)$ $\displaystyle=2.2+4.5\times\tanh{(A/300)}$ (5)

We note, that while this equation is a better fit to the MDI observations,
more than 80% of the ARs have maximum area less than 200 and do not expect
that the difference in the fit would cause a significant impact to the results
presented in this paper. However, we do recommend (and plan ourselves) to use
this revised fit to the longitude separation of the active regions moving
forward.

Figure 3: Longitudinal Extent of Active Regions. Black dots represent
individual active regions. Red dots represent the average in 31 sunspot area
bins with 2 sigma error bars on the averages. The blue line shows the fit used
in this work (Equation 3), while the red line shows the recommended fit
(Equation 5)

##  4 Simulations

To investigate the impact of including the STEREO/AIA 304 detected active
regions, we ran a series of AFT simulations. Each AFT simulation was run from
June 1, 2010 through January 1, 2020. In one set of simulations (referred to
as AFT+304), we performed data assimilation of the HMI magnetograms (as is
done in the AFT Baseline) and we incorporated all STEREO/AIA 304 detected far-
side active regions. In the second set of simulations (referred to as AFT 304
only), ALL the STEREO/AIA 304 detected active regions (near-side and far-side)
were incorporated in the AFT model and no data was assimilated from
magnetograms. (Note: aside from the initial condition map, these simulation
are informed exclusively by the He II 304 Å intensity maps.)

Figure 4: Fraction of the Sun observed in the Level1 and Level 2 STERE0/AIA
304 maps as a function of time. The dotted lines highlight when SDO data
became available in May 2010 and when communication with STEREO-B spacecraft
was lost (October 1, 2014). The shaded area indicates the period of superior
solar conjunction when STEREO instruments were turned down. The right panel
shows a close-up of the 2014–2016 period delimited by the dashed lines. Figure
5: Active Region 11499. We plot the total photon flux in the EUVI/AIA 304 maps
for NOAA Active Region 11499 in the top left panel. The bottom left panel
shows total unsigned flux as given by the 304 Å proxy (green circles), the AFT
Baseline (red), and AFT+304 (orange). The gray/white background color
indicates the availability (not available/available) of nearside magnetic
observations. Columns on the right are cutouts of the active region in the AFT
Baseline (left column), AFT+304 (middle column), and EUVI/AIA 304 for select
times during its evolution (marked by the vertical dashed lines in the plots
on the left).

While we used all available STEREO/AIA 304 Level 2 intensity maps, there are
gaps in coverage, beginning the middle of 2014 and lasting through to the end
of 2015, that impact the ability to detect far-side active regions during that
period. Figure 4 illustrates this effect. It shows the fraction of the Sun
observed daily in the Level 1 and Level 2 304 maps. This fraction shows a slow
secular change due the increase/decrease in far-side area coverage as the
STEREO spacecrafts progress in their orbits, interrupted by two sudden events:
a step-function increase in May 2010 when SDO data became available, and a
sudden drop in October 2014 when communications with STEREO B spacecraft were
lost (Cox & Ossing, 2018). In 2014 and 2015 there are many instances where the
daily fraction goes to zero due to data loss. The loss of coverage is most
notably due to the period (shaded area) of superior solar conjunction when the
STEREO spacecrafts were behind the Sun and instruments were powered down, but
extends several months, before and after, to a period of limited science
operations (Ossing et al., 2017). We have also identified several days in 2014
where the gaps in our database do not correlate with gaps in the EUVI
database. The latter contribution is less than 6% of the missing data in that
period.

Preliminary results showed that, while the despiking of the STEREO/AIA 304
maps (as described in 2) significantly reduced the occurrence of large
intensity spikes from flares and false flux deposition, small flares and
intensity fluctuations still resulted in spurious flux being added to the AFT
simulations. In order to mitigate this, we restricted the amount of flux
allowed to emerge per active region per 8h time interval. For each of the two
sets of simulations, (AFT+304 and AFT 304 only), we applied four different
threshold caps: (a) $1\times 10^{20}$ Mx, (b) $5\times 10^{20}$ Mx, (c)
$1\times 10^{21}$ Mx, and (d) $5\times 10^{21}$ Mx, for a total of eight new
simulations.

Figure 6: Evolution of the total unsigned magnetic flux for six active
regions. Each panel shows the total unsigned magnetic flux estimated from the
EUV 304 Å proxy compared to simulations of the AFT Baseline, AFT+304 and AFT
304 only models.

##  5 Results

To analyze the impact of using the the STEREO/AIA 304 data in the AFT
simulations, we investigate the flux evolution on both small and large
temporal and spatial scales. For the small temporal and spatial scales, we
examine the evolution of active regions over a few solar rotations. For large
temporal and spatial scales, we examine the evolution of global solar
properties over several years.

###  5.1 Active Regions

We investigated the detailed evolution of 7 active regions selected during the
period 2011 – 2014 where we have the best data coverage. The regions (11263,
11283, 11339, 11402, 11499, 11575, 12035) were selected for experiencing flux
emergence on the far-side and not too complex in their evolution, although
some of them do exhibit multiple emergence episodes during their lifetime
(e.g. 11499, 11575). Figure 5 shows the sample case of 11499 as a summary of
the analysis performed on each region. First, we computed the total photon
flux above threshold (500 DN/s/arcsec2) in the Level 2 304Å maps as the region
rotates around the Sun. The top left panel of the figure shows the flux
changes observed in active region 11499, with a characteristic intensity
increase corresponding to the active region emergence (April 24, 2012),
followed by an intensity decrease during the active region decay phase.
Interspersed in this secular trend, there are short term variations due to
localized additional flux emergence, as well as flares (May 9 – 11). The 24h
averaged photon flux is then used to estimate the total unsigned magnetic flux
using the Figure 1 proxy. The resulting flux is plotted in the bottom left
panel with green circles. The gray shaded areas in that panel represent the
time the region is in transition over the far-side of the Sun.

At this moment, it is worth stopping to compare the magnetic flux proxy curve
to the curve returned by the AFT Baseline model (red line) with no far-side
information. The AFT Baseline model remains oblivious of the presence of
active region 11499 until it rotates over the East limb and it is assimilated
into the model via the HMI magnetograms. This occurs on May 5, 11 days after
the region emerged on the far-side on April 24. Images for key moments for
that evolution for the 304 Å maps and the model are shown on the first and
third image columns. On April 30, 2012, the region is clearly visible in the
EUV 304 Å image, but it is still absent in the AFT Baseline model. Note that
we show here the magnetic flux curves with an emergence threshold cap of
$5\times 10^{21}$ Mx per time step. The discussion of the results obtained
using different caps is provided at the end of this section.

Figure 7: Evolution of the total unsigned magnetic flux for six active
regions. Each panel shows the total unsigned magnetic flux estimated from the
EUV 304 Å proxy compared to three simulations of the AFT 304 only model with
three different magnetic flux caps: $1\times 10^{20}$, $5\times 10^{20}$,
$1\times 10^{21}$ Mx.

The new AFT+304 simulation, however, which incorporates the far-side
emergences from the EUV proxy, does have a bipole emerged at the location of
the EUV 304 Å active region intensity enhancement (Figure 5, second image
column). The corresponding orange magnetic flux curve of the AFT+304 model,
seen on the lower left panel, is in good agreement with the magnetic flux
proxy discussed earlier. This is an important result because it demonstrates
that it is possible to inform a magnetic flux transport model with an EUV
imaging proxy to incorporate new far-side flux improving the local magnetic
flux fidelity of a purely magnetic model with only near-side information.
While both models track each other well while the region is on the Earth side,
further emergence on its second far-side pass makes the models diverge again
until near-side assimilation makes them converge. In the third far-side pass,
the active region does not experience additional flux emergence and simply
decays, hence performance of the Baseline and AFT+304 models is comparable and
in agreement with the proxy.

To understand the validity of the 304 Å proxy model, we also studied the
evolution of the active region in the AFT 304 only model, where no near-side
data is assimilated. This is shown with the pink curve in the bottom left
panel and the fourth image column in Figure 5. The AFT 304 only simulation is
able to incorporate successfully far-side flux emergence using the EUV proxy,
improving on the Baseline. On the Earth side, the AFT 304 only does not self-
correct with data and can diverge from the AFT+304. Any errors in
incorporating short-term spurious new flux, arising for instance from flare
emission (e.g. May 11), may result in unrealistic accumulation of magnetic
flux. More along these lines will be discussed in Section 5.2 when looking at
the global field. A similar self-correction occurs with the active region
tilt, which in the case of the AFT 304 only simulation follows an estimate
from Joy’s law.

Below we provide a few comments on each of the other six Active Regions
investigated in this study:

11263

Active region 11402, below, provides a good example of a simple emergence and
decay evolution. Active Region 11263 shows that the model is also capable of
dealing with more complex scenarios. The AFT+304 simulation is able to capture
11263 initial far-side emergence, as well as the far-side emergence of a
bipolar region to its SE around August 22-24, 2011 (AR 11279), and a large
region on September 19 (AR 11302). While not evident from the integrated total
unsigned flux curve, the model was able to resolve the different longitudes
and latitudes for these three emerging areas within the area of integration.

11283

In the evolution of this region, it is worth highlighting the impact that
flare emission can have in the magnetic flux proxy and how the magnetic flux
cap mitigates that effect. During the near-side pass, flare activity
(September 8-9, 2011) that survived our cleanup process, produces a magnetic
flux spike. This spike is ignored by the AFT Baseline and AFT+304 simulations
due to the assimilation process, but it has an impact on the AFT 304 only
case, mitigated by the magnetic flux cap that only allows to add flux at a
maximum rate of $5\times 10^{21}$ Mx per time step. As the flare intensity
decays rapidly, the simulation only incorporates a fraction of the peak flux.

11339

As in several other examples, the AFT+304 and 304 only models capture the
emergence process of this region in the far side as it happens, whereas the
Baseline model plays catch up with near side assimilated data 15 days later.
The 304 Å proxy underestimates, however, the magnetic flux observed by HMI on
the near side. There is an intrinsic uncertainty in the proxy prediction,
evident from the spread in Fig. 1, something that we already discussed in
Ugarte-Urra et al. (2015). Active region 11339 is a region with a very large
sunspot area. Sunspots, while providing significant magnetic flux to an active
region, do not correlate well with EUV intensity. While we mitigate this
impact in our analysis by setting a magnetic flux integration range of 100 –
900 G (Ugarte-Urra et al., 2017), this shows that a simple flux-flux
relationship cannot capture the intricacies of all active regions.

11402

The simple evolution of this region is a good representation of the ability of
the Baseline model to capture the overall flux evolution in an active region,
and how the AFT+304 upgrade extends that to the emergence time. It also
reveals the limitations of the 304 only model. In this case, the discrepancy
in the spatial flux distribution compared to the observed one plays a critical
role in the cancellation process leading to the decay (i.e. more concentrated
in the AFT only model, more fragmented in the observations). A limitation that
the near-side assimilation corrects in the AFT+304 case.

11575

In the development of this active region, it is interesting to point at the
flux emergence taking place on the near-side around September 29, 2012. It is
captured both by the assimilated data in the AFT Baseline and AFT+304, but
also the AFT 304 only using the proxy. As the region reaches the West limb,
the Baseline stops assimilating new data of the emergence and flux transport
takes over leading the region into cancellation and decay, while the two other
simulations with far-side information continue adding significant flux and
providing a more comprehensive description of the evolution of that region in
the far-side.

12035

This is another active region with multiple far-side emergences in different
rotations that are well represented by the AFT+304 model. The evolution of
this active region in the AFT 304 only reveals the challenge that a model
without near side data assimilation has over long timescales. Active region
12035 emerges over an already large background flux in AFT 304 only. This is a
consequence of a period of high activity in the belts. A nearby decaying
active region rotates into the field of view of analysis, causing it to appear
as if there is additional growth before the decay phase begins.

We outlined in Section 4 that for each of the two sets of simulations we
applied four different threshold caps for the amount of flux that is allowed
to emerge per active region per 8-hour interval between STEREO/AIA 304 maps.
The purpose of these caps is to limit the effect that large increases in
intensity originating from large flares can have in the magnetic flux proxy.
Figure 7 shows a comparison of the AFT 304 only for the three largest
thresholds ($5\times 10^{20}$, $1\times 10^{21}$, $5\times 10^{21}$ Mx). We
find that a too stringent cap does not allow the model to capture the
emergence of flux sufficiently rapidly, therefore underestimating the maximum
total unsigned flux in the region. It is for that reason and for the sake of
clarity that we do not show the $1\times 10^{20}$ Mx case in Figure 7. Our
analysis indicates that a cap of $5\times 10^{21}$ Mx strikes the best balance
between limiting the effects of large flares while at the same time accurately
capturing the emergence timescales in the active regions’ flux evolution. This
amounts to a flux emergence rate of about $6.25\times 10^{20}$ Mx/hr, which
would be sufficient to capture the largest emergence rates in Norton et al.
(2017) or the $2.5\times 10^{20}$ Mx/hr upper limit we measured for active
regions in Ugarte-Urra & Warren (2012).

###  5.2 Global Field

The active region evolution illustrates the flux evolution on small temporal
and spatial scales, we now turn our attention to the flux evolution on large
temporal and spatial scales. Here we examine the evolution of global solar
properties over several years. We begin with a qualitative comparison by
showing a magnetic butterfly diagram for eight of our simulations (shown in
the top two rows of Figure 8). The magnetic butterfly diagram is calculated by
averaging the magnetic field at the surface over all longitudes and for each
latitude and each Carrington Rotation.

Figure 8: Magnetic Butterfly Diagrams . Magnetic butterfly diagrams for the
AFT+304 and AFT 304 only simulations are shown in the top and bottom rows
respectively. The titles of each panel are color coded based on the flux cap
(1e20, 5e20, 1e21, and 5e21) used.

For the AFT+304 simulations (top row, Figure 8), the butterfly diagrams are
nearly identical, thought some minor differences can be seen with careful
inspection. There are some slight variations in field strength, but
structurally they are essentially the same (e.g., the morphology of the
dappled pattern in the butterfly wings, or the oscillating leading and
following polarity polar streams). The striking similarity in these cases is
due to the fact that the data assimilation process continually corrects the
near-side of the simulation with the observations. The butterfly diagram for
the Baseline simulation is also nearly identical to these three and so is not
shown. Any differences in the strength are due to the additional flux that is
present on the far-side of the simulation, which is now included in the
average for each latitude and each Carrington Rotation.

Differences in the magnetic butterfly diagram are more pronounced for the AFT
304 only cases (bottom row, Figure 9) because there is no data assimilation
process to regulate the simulations. The butterfly diagrams for the for AFT
304 only simulations have clear differences both in the amplitude of the field
strength and also structurally. As the active region flux cap increases, the
strength of the magnetic field in the butterfly wings saturate and the
stronger flux concentrations become wider. This causes the wings of the
butterfly to widen as well. As a consequence, more flux cancellation across
the equator occurs and more following polarity flux fills the polar streams.
This causes the polar fields to reverse more quickly and become stronger by
the end of the cycle.

Comparing the AFT 304 only cases to the Baseline or AFT+304 simulations, we
find that the AFT 304 only simulations have distinct differences in both the
mottled pattern and in the polar streams. This is largely due to the fact that
the Active Regions in the AFT 304 only case are all given the average Joy’s
Law tilt. Another difference is an apparent gap in the emergence of new flux
for the AFT 304 only cases in 2015. This was caused by the previously
mentioned superior solar conjunction. Afterwards, the flux emergence does pick
up again somewhat, but remains notably weak for the remainder of the cycle.
This is caused by the diminishing latitudinal coverage due to both the loss of
the STEREO B spacecraft observations as well as the return of the STEREO A
spacecraft to the near-side of the Sun, as illustrated in Figure 4.

Despite the differences, some structural similarities persist. For instance,
the general outline of the butterfly wings and the latitudes of the polarity
inversion lines are very similar. Additionally, a few of the following
polarity streams are present in all of the simulations (e.g., in the northern
hemisphere from 2011-2013 and in the southern hemisphere from 2014-2016).
Overall, the AFT 304 only are able to reproduce the butterfly diagram
reasonable well considering no direct magnetic date (other than the initial
map) were used, particularly for the first 4-5 years when nearly full coverage
was available.

Next we quantify the differences between the AFT simulations. We begin by
calculating the Total Unsigned Flux (TUF) over the entire Sun, both near-side
and far-side. The TUF for the AFT+304 simulations (using four different flux
caps) are shown by the orange/brown lines in the top left panel of Figure 9.
For reference, the TUF measured in the Baseline is shown in black. The TUF for
the AFT+304 simulations seems to be fairly consistent (i.e., matching the
overall evolution) with the AFT Baseline) and across all four simulations. In
each case, the higher flux cap increases the TUF by a fraction of the Baseline
values, with the biggest increase (about $10\%$) seen during solar maximum. In
order to better visualize this, we also plot the difference between the TUF in
each simulation and the AFT Baseline) TUF (middle panels) from 2011-2015. The
flux difference calculated by subtracting the TUF from the AFT Baseline from
each of the the AFT+304 simulations (top middle panel) represents an estimate
of how much flux is missing from the AFT Baseline due to new far-side active
region growth and emergence. The analysis of the active region evolution in
the previous section showed that flux cap of $5\times 10^{21}$ Mx strikes the
best balance between capturing the active region emergence while limiting the
effects of large flares. For this time period, the missing far-side flux is
typically on the order of $4-6\times 10^{22}$ Mx, or about the size of a
fairly large active region. This is an estimate during the time leading up to
and including Solar Cycle 24 maximum (April 2014), and it should be noted that
Cycle 24 was considered a very small cycle (the smallest observed in 100
years). Larger cycles would be expected to have even more far-side emergence
and this should be considered a lower estimate of the amount of flux missing
from SFT models during solar maximum when only near-side data is included.

Figure 9: Global Simulation Properties. Plots of the Total Unsigned Flux
(left), Flux difference (relative to the Baseline, middle), and the Axial
Dipole (right) are shown in the top (AFT+304 simulations) and bottom (AFT 304
only simulations) rows. The plots are color coded based on the flux threshold
(1e20, 5e20, 1e21, and 5e21) used. For reference, the measured Baseline TUF
and Axial Dipole values are shown in black

Similarly, the TUF for the AFT 304 only simulations is shown by the
pink/purple lines in the bottom left panel of Figure 9. We note that flux
emergence is underestimated prior to 2012 and again after the 2nd half of 2014
for all but the strongest flux cap used ($5\times 10^{21}$ Mx). This is due to
the lack of 304 coverage as the STEREO spacecrafts progress in their orbit
(leading to 2012, and after 2015) and during the superior solar conjunction
(2014-2015). During the superior solar conjunction in 2014-2015, the TUF drops
well below the AFT Baseline in all four cases. In the later half of 2015, as
the STEREO A spacecraft emerges from the conjunction, the largest threshold
cases again significantly exceed the AFT Baseline, while the mid threshold
cases ($5\times 10^{20}$ Mx and $1\times 10^{21}$ Mx) reach similar values and
the $1\times 10^{2}$ Mx threshold case falls well below the AFT Baseline TUF.
In 2017, the TUF in all these cases begin a progressive decline below the AFT
Baseline as STEREO A progresses towards the Earth vantage and far-side
coverage is lost.

The flux difference calculated by subtracting the TUF from the AFT Baseline
from the AFT 304 only simulations (bottom middle panel) far exceeds the
estimate obtained from the AFT+304 simulations in the $1\times 10^{21}$ Mx and
$5\times 10^{21}$ Mx threshold cases. On global time scales, these thresholds
are not consistent with the Sun. The amplitude of the $1\times 10^{20}$ Mx and
$5\times 10^{20}$ Mx threshold cases are more inline with the AFT Baseline and
the AFT 304 only results, respectively. However, the structure in the TUF
evolution is not well captured. This is due to a steady accumulation of flux
from flaring regions that are not being corrected by the data assimilation of
magnetograms.

Another quantitative metric to consider is the progression of the Sun’s axial
dipole moment. This quantity measures the polar field evolution - i.e., the
decay, reversal, and build up of magnetic flux in the polar regions. The
ability to reproduce the evolution of the polar fields is essential for SFT
models, as the polar field at the time of solar cycle minimum is a key
predictor of the strength of the next cycle (Schatten & Sofia, 1987; Hathaway,
2015; Upton & Hathaway, 2014b). The axial dipole strength is given as

$D(t)=(3/4\pi)\int B_{r}(\theta,\phi,t)\>cos\theta\>d\Omega,$ (6)

where $\,B_{r}(\theta,\phi,t)\,$ is the radial photospheric magnetic field at
latitudes $theta$, longitudes $\phi$, and time $t$ integrated over the solid
angle $\Omega$ (Wang & Sheeley, 1991). The axial dipole moment from each of
the simulations is shown in the right column of Figure 9, with the AFT+304
simulations shown in the top right panel and the AFT 304 only simulations
shown in the bottom right panel. In both cases, the axial dipole moment from
the AFT Baseline is shown in black.

The axial dipole moment from all of the AFT+304 simulations are nearly
identical to the AFT Baseline, once again showing that the data assimilation
process is able to correct for inaccuracies in the far-side emergence. For the
AFT 304 only simulations, again, the $1\times 10^{20}$ Mx and $5\times
10^{20}$ Mx threshold cases do the best at reproducing the axial dipole
evolution until 2015. During this same time period, the $1\times 10^{21}$ Mx,
and $5\times 10^{21}$ Mx threshold cases have too much flux reaching the pole,
causing the axial dipole to reverse too quickly. During 2014, following the
superior solar conjunction, the dipole evolution falls briefly in all cases.
While this is followed by a short period of growth, they all begin a steady
decline in 2016 . This occurs because the loss of He II 304 Å coverage results
in an apparent gap in active region emergence, which allows leading polarity
flux to be carried to the pole (see Figure 9, bottom row) during the
conjunction. Afterwards, active region emergence picks up briefly before the
combined effects of the declining cycle and declining far-side coverage result
in only a small amount of new flux is being added, primarily close to the
equator. Since most of the flux cancellation occurs at the lowest latitudes,
very little new flux reaches the poles. The flux that is transported at this
stage is primary leading polarity flux, causing the weak decline in the polar
fields.

Determining the best threshold case for the global parameters in the AFT 304
only simulations is not straight forward. The $1\times 10^{21}$ Mx threshold
case is most consistent with the TUF in the best case from the AFT+304
results, but does not do well at reproducing the polar field evolution. The
$1\times 10^{20}$ Mx case performs remarkably well at reproducing the polar
field evolution, but underestimates the TUF. Choosing the best case for the
AFT 304 only simulations is thus dependent on the end-use application.

##  6 Discussion

###  6.1 AR Detection

We have developed an automated detection algorithm to identify active regions
in He II 304 Å intensity images and incorporate them into the AFT SFT model.
For this purpose, we created higher order (Level 2) processed files which
included several corrections, including despiking the intensity to mitigate
against flaring events. We found that the power-law relationship is sensitive
to how the boundary of the active region is defined and to the magnetic flux
density thresholds and have computed a new empirical relationship tailored to
the AFT Baseline maps. Our automatic detection algorithm identifies the
location and infers flux in active regions from the He II 304 Å intensity
images using this relationship. The algorithm them compares the active regions
to the AFT maps to determine if new active region growth has occurred. When
new growth occurs, idealized bipoles are added to the AFT maps.

Despite despiking the Level 2 He II 304 Å intensity images, we found that
flaring events still occurred frequently. To further mitigate against this, we
implemented a cap on how much a given active region was allowed to grow in the
8 hour interval between Level 2 He II 304 Å intensity images. We then conduced
a series of simulations to investigate the performance of the automated
detection algorithm with several flux cap thresholds: $1\times 10^{20}$ Mx,
$5\times 10^{20}$ Mx, $1\times 10^{21}$ Mx, and 5$\times 10^{21}$ Mx. For each
of these flux caps, we ran AFT with and without data assimilation, AFT+304 and
AFT 304 only respectively. The latter of which is informed exclusively by the
He II 304 Å intensity maps.

###  6.2 AFT Simulations

Results from the AFT+304 and AFT 304 only simulations showed that the $5\times
10^{21}$ Mx threshold case (corresponding to as active region growth rate of
about $6.25\times 10^{20}$ Mx/hr) performed the best at capturing the Active
Region growth. However, the global parameters in the AFT 304 only simulations
did not fare as well: the TUF was much too high and the axial dipole reversed
polarity much too quickly, suggesting too much flux is being transported to
the poles. Figure 7 shows that while this threshold does well at capturing the
active region growth phase, the active region decay phase is not as well
reproduced. This can be attributed to a few different factors, including
residual flaring events, active region properties, and small scale flux
interactions. Flaring events will cause excess flux to be deposited into the
active region, directly resulting in more residual flux to be transported
poleward. The active regions in these simulations are incorporated into AFT as
idealized Gaussian spots with tilts and separation distances determined from
statistical properties. Differences between the simulated and observed tilts
and separations will change the axial dipole moment of the active regions,
which could also result in more flux being transported to the poles. Another
possibility is that interactions with nearby small scale flux, not included in
the AFT idealized active regions, may play a non-trivial role in the flux
cancellation process. Yeates (2020) showed that idealized bipoles, as opposed
to observed complex morphologies, could also lead to an overestimation of the
axial dipole. Ugarte-Urra et al. (2015) found that AFT was able to reproduce
the evolution of the active regions to within a factor of 2 when starting with
the observed active region morphology. We find that the active regions in the
AFT+304 performed even better (see Figure 7), suggesting that the idealized
active region are the most likely source of this discrepancy. These results
demonstrates that while He II 304 Å does remarkably well at driving the SFT
model, these data alone are insufficient.

We find that the AFT+304 simulations fared much better. This configuration
provides the most accurate representation of the flux distribution on the
entire surface of the Sun by combining the benefits of incorporate far-side
emergence while using data assimilation of magnetograms to correct for
inconsistencies in the implementation of the far-side Active Regions. Using
the $5\times 10^{21}$ Mx threshold case, we estimate that for Cycle 24 surface
flux transport models are missing approximately $4-6\times 10^{22}$ Mx from
far-side Active Regions. While not implemented here, we intend to use these
maps as the inner boundary condition for various types of atmospheric and
coronal modeling in order to determine the impact of including this missing
flux in making space weather predictions. He II 304 Å images can be used as a
supplemental proxy for magnetic flux measurements when direct magnetic field
observations are not complete.

###  6.3 Future Work

While STEREO is currently positioned close to the sun-earth line, which limits
the utility of the results for new applications, in a few years STEREO A will
begin returning far-side images. Studies that use the older STEREO data, such
as this one, are essential for determining how we can best use the far-side
EUV observations when they become available again. In addition, these
methodologies and results can be used to validate other far-side proxies of
the far-side, such as those obtained by helioseismic and Machine
Learning/Artificial Intelligence (ML/AI) methodologies, as well as adapted for
implementation with far-side observations from newer missions.

While helioseismic results show considerable promise, they are less reliable
than direct imaging. Direct imaging in the EUV can precisely confirm the
existence, location, and relative size of active regions. Helioseismic
results, on the contrary, are still plagued with false positives and false
negatives. Even far-side active region detections that are confirmed by direct
far-side imaging or subsequent rotation on the near-side, possess large
uncertainties in relevant parameters such as active region size, strength,
location. The EUV automated far-side detection process described here can be
used to create reliable active region catalogs. These catalogs can then be
used to assess the uncertainty in far-side helioseismology active region
detections and aid in making decisions on how best to include far-side results
in SFT models.

The advent of ML/AI is providing new avenues for advancing helioseismic
techniques and improving the accuracy. Both HMI and GONG helioseismology teams
and we are currently exploring ML/AI methodologies to provide improved data
products, including helioseismic inferred far-side magnetic maps (Chen et al.,
2022; Creelman et al., 2024). We are currently working with both of these
teams to adapt the techniques described in this paper for use with the
helioseismic data, which we plan to address in follow-on papers. In addition
to the advances ML/AI techniques offer to heliosesmology, these techniques are
also being used to improve other existing data products. For example, ML/AI
groups training on combined AIA/STEREO EUV images to create AI-generated
magnetograms are consistent with HMI magnetograms (Kim et al., 2019; Jeong et
al., 2022). Using our automated detection codes with these data to populate
AFT with far-side active regions may prove more accurate than with the EUV
observations directly. Another group is using ML/AI to cross-calibrate
Hinode/SOT-SP data (Lites et al., 2013) with HMI data in order to create
improved vector magnetograms(Fouhey et al., 2023; Wang et al., 2024) . These
new vector magnetograms are expected to minimize issues with the 180 dis-
ambiguity as well as limb effects, problems which have long stood in the way
of making vector date viable for assimilation into SFT models

Finally, the methodologies developed here may be adapted for and improved by
new direct far-side observations. For example, Solar Orbiter captures high-
resolution EUV images as well as magnetograms. This data combination provide a
unique opportunity to perform additional studies like this one. Coordinated
observations can be used to cross-calibrate observations made by the two
instruments and develop the ability to create proxies that can be used when
one or the other is not observing the same target. These data, when available,
could then be incorporated in SFT models to provide supplementary knowledge of
far-side active region evolution and to access helioseismic results(Yang et
al., 2023) .

We would like to thank the anonymous referee for valuable comments and
suggestions that improved the quality of the paper. AIA and HMI data are
courtesy of NASA/SDO and the AIA and HMI science teams. L.A.U. was supported
by NASA Heliophysics Guest Investigator NNH15ZDA001N-HGI, NASA Heliophysics
Living With a Star grants NNH16ZDA010N-LWS and NNH18ZDA001N-LWS, and by NASA
grant NNH18ZDA001N-DRIVE to the COFFIES DRIVE Center managed by Stanford
University. I.U.U. and H.P.W. were supported by the NASA Heliophysics Guest
Investigator and Living With a Star programs and the Office of Naval Research.
D.H.H. was supported by NASA contract NAS5-02139 (HMI) to Stanford University.

## References

  * Arge et al. (2010) Arge, C. N., Henney, C. J., Koller, J., et al. 2010, in American Institute of Physics Conference Series, Vol. 1216, Twelfth International Solar Wind Conference, ed. M. Maksimovic, K. Issautier, N. Meyer-Vernet, M. Moncuquet, & F. Pantellini, 343–346
  * Babcock (1961) Babcock, H. W. 1961, The Astrophysical Journal, 133, 572
  * Baker & Lanzerotti (2016) Baker, D. N., & Lanzerotti, L. J. 2016, American Journal of Physics, 84, 166
  * Chen et al. (2022) Chen, R., Zhao, J., Hess Webber, S., et al. 2022, ApJ, 941, 197
  * Cox & Ossing (2018) Cox, M. W., & Ossing, D. A. 2018, The Fall and Rise of STEREO Behind
  * Creelman et al. (2024) Creelman, M., Jain, K., Oien, N., & Wentzel, T. M. 2024, arXiv e-prints, arXiv:2402.15768
  * Fouhey et al. (2023) Fouhey, D. F., Higgins, R. E. L., Antiochos, S. K., et al. 2023, ApJS, 264, 49
  * Freeland & Handy (2012) Freeland, S. L., & Handy, B. N. 2012, SolarSoft: Programming and data analysis environment for solar physics, Astrophysics Source Code Library, record ascl:1208.013, , , ascl:1208.013
  * Hathaway (2015) Hathaway, D. H. 2015, Living Reviews in Solar Physics, 12, 4
  * Hathaway & Upton (2016) Hathaway, D. H., & Upton, L. A. 2016, Journal of Geophysical Research (Space Physics), 121, 10,744
  * Henney et al. (2015) Henney, C. J., Hock, R. A., Schooley, A. K., et al. 2015, Space Weather, 13, 141
  * Hill et al. (2009) Hill, F., Martens, P., Yoshimura, K., et al. 2009, Earth Moon and Planets, 104, 315
  * Howard et al. (2008) Howard, R. A., Moses, J. D., Vourlidas, A., et al. 2008, Space Sci. Rev., 136, 67
  * Iglesias et al. (2019) Iglesias, F. A., Cremades, H., Merenda, L. A., et al. 2019, Advances in Space Research, in press
  * Jeong et al. (2022) Jeong, H.-J., Moon, Y.-J., Park, E., Lee, H., & Baek, J.-H. 2022, ApJS, 262, 50
  * Jiang et al. (2014) Jiang, J., Hathaway, D. H., Cameron, R. H., et al. 2014, Space Science Reviews, 186, 491
  * Kaiser et al. (2008) Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, Space Sci. Rev., 136, 5
  * Kim et al. (2019) Kim, T., Park, E., Lee, H., et al. 2019, Nature Astronomy, 3, 397
  * Leighton (1969) Leighton, R. B. 1969, ApJ, 156, 1
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17
  * Lites et al. (2013) Lites, B. W., Akin, D. L., Card, G., et al. 2013, Sol. Phys., 283, 579
  * Mackay & Upton (2022) Mackay, D. H., & Upton, L. A. 2022, ApJ, 939, 9
  * Mackay & Yeates (2012) Mackay, D. H., & Yeates, A. R. 2012, Living Reviews in Solar Physics, 9, 6
  * Mango et al. (1978) Mango, S. A., Bohlin, J. D., Glackin, D. L., & Linsky, J. L. 1978, ApJ, 220, 683
  * Mosher (1977) Mosher, J. M. 1977, PhD thesis, California Institute of Technology, Pasadena.
  * Müller et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1
  * Norton et al. (2017) Norton, A. A., Jones, E. H., Linton, M. G., & Leake, J. E. 2017, ApJ, 842, 3
  * Ossing et al. (2017) Ossing, D. A., Wilson, D., Balon, K., et al. 2017, in 2017 IEEE Aerospace Conference, 1–20
  * Schatten & Sofia (1987) Schatten, K. H., & Sofia, S. 1987, Geophys. Res. Lett., 14, 632
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207
  * Schrijver et al. (2015) Schrijver, C. J., Kauristie, K., Aylward, A. D., et al. 2015, Advances in Space Research, 55, 2745
  * Sheeley (1966) Sheeley, Jr., N. R. 1966, ApJ, 144, 723
  * Sheeley (2005) —. 2005, Living Reviews in Solar Physics, 2, 5
  * Stenflo & Kosovichev (2012) Stenflo, J. O., & Kosovichev, A. G. 2012, ApJ, 745, 129
  * Ugarte-Urra et al. (2015) Ugarte-Urra, I., Upton, L., Warren, H. P., & Hathaway, D. H. 2015, ApJ, 815, 90
  * Ugarte-Urra & Warren (2012) Ugarte-Urra, I., & Warren, H. P. 2012, ApJ, 761, 21
  * Ugarte-Urra et al. (2017) Ugarte-Urra, I., Warren, H. P., Upton, L. A., & Young, P. R. 2017, ApJ, 846, 165
  * Upton & Hathaway (2014a) Upton, L., & Hathaway, D. H. 2014a, ApJ, 792, 142
  * Upton & Hathaway (2014b) —. 2014b, ApJ, 780, 5
  * Wang et al. (2024) Wang, R., Fouhey, D., Higgins, R., et al. 2024, ApJ, submitted
  * Wang & Sheeley (1991) Wang, Y.-M., & Sheeley, Jr., N. R. 1991, ApJ, 375, 761
  * Warren et al. (2021) Warren, H. P., Floyd, L. E., & Upton, L. A. 2021, Space Weather, 19, e02860
  * Webb & Howard (2012) Webb, D. F., & Howard, T. A. 2012, Living Reviews in Solar Physics, 9, 3
  * Yang et al. (2023) Yang, D., Gizon, L., Barucq, H., et al. 2023, A&A, 674, A183
  * Yardley et al. (2021) Yardley, S. L., Pagano, P., Mackay, D. H., & Upton, L. A. 2021, A&A, 652, A160
  * Yeates (2020) Yeates, A. R. 2020, Sol. Phys., 295, 119
  * Yeates et al. (2023) Yeates, A. R., Cheung, M. C. M., Jiang, J., Petrovay, K., & Wang, Y.-M. 2023, Space Sci. Rev., 219, 31


# Quantum Holonomy based in a Kaluza-Klein description for defects in $C_{60}$
fullerenes

Everton Cavalcante  Electronic address<EMAIL_ADDRESS>Centro de
Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba, Patos,
PB, Brazil  Claudio Furtado  Electronic address<EMAIL_ADDRESS>Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970, João Pessoa, PB, Brazil

###### Abstract

In this paper, we discuss a new way to get a quantum holonomy around
topological defects in $C_{60}$ fullerenes. For this, we use a Kaluza-Klein
extra dimension approach. Furthermore, we discuss how an extra dimension could
promote the formation of new freedom degrees which would open a discussion
about a possible qubits computation.

A,B,C

###### pacs:

F,G,H

##  I Introduction

Nowadays, great efforts in theoretical physics are devoted to study models
with extra dimensions. Extra dimensions concept have a long story that began
by the decade of 1920 when Kaluza [1] and Klein [2] proposed the first of
these kind of models. Within the Kaluza-Klein (KK) approach, the metric tensor
is represented by a function of an additional vector and scalar fields,
$g_{\mu\nu}=g_{\mu\nu}\left(A_{\mu}(x),\phi(x)\right)$. This change in the
metric not only reproduces the field equations for $g_{\mu\nu}(x)$, but also
recovers correctly the Maxwell theory for $A_{\mu}(x)$ and the massless Klein-
Gordon equation for $\phi(x)$. Thus, a new relationship between
electromagnetic systems and general relativity in five dimensions has been
introduced, and a new branch of research has been founded (for a review on
Kaluza-Klein theory see f.e. [3, 4, 5]). All of these efforts, as we already
know, have not yet been able to unite gravity theories and quantum phenomena.
However, such efforts have been elucidating possibilities of research in a
many branches of the physics. One of these possibilities has proposed in this
work.

The quantum dynamics for fermions in the presence of topological defects in
solids is described by local reference frames, and the spinors transform under
infinitesimal Lorentz transformations [6]. In this way, to preserve a local
Lorentz group, we introduce a covariant derivative
$\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}(x)$. This one allow to describe a
geometric phase on spinor as a function of $\Gamma_{\mu}(x)$ [7]. On the other
hand, a way to implement a holonomic quantum computation (HQC) was proposed by
Zanardi and Rasetti [8]. Such an idea is based on non-Abelian geometric phases
called holonomies [9, 7]. One of the interesting characteristics of HQC is its
stability [10], which inspired a lot of works dedicated to their applications
in different quantum systems, such as ion traps [11, 12], Josephson-junction
devices [13], neutral atoms [14], optical computer [15], conical defects in
graphene layers [16], nuclear magnetic resonance [17] and nuclear Zeeman-
perturbed systems [18]. Besides this, the quantum gates have been realized in
terms of operations possessing a purely geometrical nature, with the stability
of quantum gates against local perturbations must be emphasized especially
[19]. Clearly, in HQC, the states of a quantum system are used for encoding
information by means of a holonomy (unitary evolution) for a final state where
the solution of a given computational problem can be decoded. This state can
be presented as

$|\psi\rangle=U(\alpha)|\psi_{0}\rangle=e^{-i\int_{0}^{t}E(t^{\prime})dt^{\prime}}\Gamma_{A}(\alpha)|\psi_{0}\rangle$
(1)

The exponential of the integral here is the dynamical phase, which can be
omitted after redefining the zero energy level as $E(0)=0$, the
$\Gamma_{A}(\alpha)=\exp\int_{c}A(\alpha)d\alpha$ is the holonomy. The
quantity $A(\alpha)$ corresponds to the Berry potential with components
$A^{\mu\nu}=\langle\psi^{\mu}(\alpha)|\frac{\partial}{\partial\alpha}|\psi^{\nu}(\alpha)\rangle$
used to describe arising of geometric phases in any non-adiabatic cyclic
evolution of a quantum system [20]. The most known examples of non-adiabatic
cyclic evolutions are the Aharonov-Bohm effect [21], the scalar Aharonov-Bohm
effect [22], the Aharonov-Casher effect [23], the He-McKellar-Wilkens effect
[24, 25], and the dual Aharonov-Bohm effect [26].

Recently, two interesting ideas about graphene layers were proposed. One of
these ideas is the possible implementing a quantum holonomy associated with a
defect structure in a conical graphene [27, 28], where the geometric
description for a defects in solids proposed for Katanaev and Volovich was
used [29]. It was demonstrated there how to construct quantum gates depending
only of the number of sectors ($n_{\Omega}$) in the graphene layer removed in
order to build a particular conical configuration described by a two-
dimensional metric. However, in [30], it has been shown that it is possible to
extend the usual two-dimensional metric describing a topological defect in a
graphene layer to a three-dimensional metric by adding an Kaluza-Klein extra
dimension to introduce the Fermi-point degree of freedom within the
geometrical phase acquired by wave function of the fermions which go around
the apex of a defect in a graphene layer.

In other work developed by us [31], a geometrical model of a fullerene
molecule with a $Ih$ symmetry was described. For this, we have combined the
well known non-Abelian monopole approach [32] and the geometric theory of
defects [29]. Thus, we have applied a continuum formulation to describe this
spherical geometry with a bi-dimensional spherical metric. Our aim in this
work is to discuss a way to get a holonomy matrix in the $C_{60}$ molecule
through a Dirac phase factor method within an extra dimension approach.

This paper is organized as follows. In section II, we give a brief review of
the geometrical approach for a $Ih$ symmetrical fullerene molecule. In section
III, we introduce the extra-dimensional approach for topological defects.
After that, in section IV, we study the geometrical phases for massive and
massless modes of fermions and discuss a new way of obtaining quantum
holonomies for the system. Finally, in section V, we present our conclusions.

##  II Geometric approach for topological defects in $C_{60}$ fullerenes

In 1985 the molecule of fullerene was discovered by Kroto, Curl, and Smalley
[33]. In this molecule, the carbon atoms are arranged in a closed spherical
shell. This allotrope became known as fullerene in tribute to the architect
Fuller who designed intriguing constructions known as shaped polyhedrons. The
fullerene, or $C_{60}$, is formed by twelve pentagonal rings combined with
twenty hexagon rings of the carbon. It is well known in the theoretical and
experimental setting [34, 35], that the X-ray diffraction pattern of $C_{60}$
is well fitted when suggesting that the molecule shape is a truncated
icosahedron of the point-group symmetry Ih. The carbon atoms are arranged in
20 six-membered and 12 five-membered rings. The C-C bonds separating two
hexagons are double bonds ($h$) while the pentagon C-C bonds ($p$) are single
[35]. As in the case of graphene, the conduction band of $C_{60}$ fullerene
can be described by a tight-binding model [32, 36, 37]. Therefore, the Fermi
surface reduces to two $K$-points located in the Brillouin zone.

It is well known that the existence of pentagonal rings in $C_{60}$ fullerene
can be interpreted as a disclination in the spherical lattice formed and
described by a ”cut and glue” Volterra process [38]. And these topological
defects are associated with the symmetries during the generation of the
molecule. The disclination becomes especially relevant in two-dimensional
solids, being responsible for the value of the strength, and for the arising
of curvature in the lattice. The geometric approach to its description is more
powerful, because, in this theory, the disclinations are well described by a
curvature of the lattice. The curvature of the molecule induces an effective
gauge field that is obtained via the parallel transport of the reference
frames in a closed-circuit around the defect [36, 37, 39, 40, 41, 42, 43]:

$\oint\omega_{\mu}dx^{\mu}=-\frac{\pi}{6}\sigma^{3},$ (2)

which enforces a mixture of the Fermi points ($\textbf{K}_{\pm}$) and
generates a non-Abelian gauge field (or a K-spin flux) as demonstrated in [36,
37], offsetting the discontinuity of the Bravais labels (A/B):

$\oint A_{\mu}dx^{\mu}=\frac{\pi}{2}\tau^{2},$ (3)

where $\tau^{2}$ is the second Pauli matrix mixing up the $\textbf{K}_{+}$ and
$\textbf{K}_{-}$ components of the spinor on the $\textbf{K}_{\pm}$ space. In
other words, the mixture of Fermi points induces an effective field arising
due to a fictitious magnetic monopole in the center of the fullerene by
replacing the fields of 12 disclinations.

The first geometric continuous model using the Dirac equation to describe low-
energy electrons in the $C_{60}$ fullerenes was proposed by Gonzales, Guinea
and Vozmediano [32]. Within this model, the fullerene molecule is described by
a non-Abelian magnetic monopole introduced in the centre of a two-dimensional
sphere. The field produced by this monopole represents the fields of twelve
disclinations presenting in the $Ih$ (Icosahedron) symmetry. This field theory
model was used by Kolesnikov and Osipov [44], where pentagonal rings in
fullerene are simulated by two kinds of gauge fields. One of these gauge
fields describes the elastic properties of disclinations, while the second one
is a non-Abelian gauge field describes the K-spin fluxes.

To study the properties of electrons taking place due to the presence of a
topological defect in graphene and $Ih$ fullerenes, it is natural to formulate
the Dirac equation in a curved space induced by this defect. However, here we
will introduce Kaluza-Klein extension in the spherical metric in the next
section.

##  III The extra dimension approach for $C_{60}$ fullerenes

Since the quantum flux in (2) can be obtained within the geometrical approach
[39], a holonomy matrix can be calculated when the spinor suffers parallel
transport around the apex of a graphitic cone. The effect of the parallel
transport of a spinor around the apex of the graphitic cone is analogous to
the Aharonov–Bohm effect [45]. Recently, the geometrical description for a
second quantum flux in graphene cones (3) was proposed in [30], where the
Fermi point degree of freedom, or K-spin flux, is described via a Kaluza-Klein
theory. The line element in the presence of a non-Abelian gauge field
($B_{\mu}$) is given in the form [5]

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+(dy+\kappa B_{\mu}dx^{\mu})^{2},$ (4)

where $g_{\mu\nu}(x)$ is the metric tensor in (2 + 1) dimensions, and $y$
corresponds to the coordinate of the extra dimension ($\kappa$ is a constant
called the Kaluza constant). Following the prescription of Refs. [5, 31, 30]
for a two-dimensional spherical geometry, which describes the $C_{60}$
fullerene, we have:

$ds^{2}=V_{f}^{2}dt^{2}-R^{2}d\theta^{2}-\alpha^{2}R^{2}\sin^{2}\theta
d\phi^{2}-\bigg{(}dy+\frac{n_{\Omega}}{2}R\sin\theta d\phi\bigg{)}^{2}$ (5)

where $\alpha=1-\frac{n_{\Omega}}{6}$ is related to the angular sector removed
from a spherical sheet in order to form the conical singularities, $V_{f}$ is
the Fermi velocity of order $10^{6}$ m/s, and $R$ is the radius of fullerene,
of the order of nanometers. This effective geometry is considered by the fact
pointed by Kolesnikov and Osipov [44]: eigenfunctions of the low-energy levels
oscillate relatively slowly with a distance. In this way, we can find the
impact of the twelve pentagons considering the two-dimensional spherical
metric near the defect. Without the extra dimension, the element (5) describes
a sphere in (2 + 1) dimensions.

The bases of this space-time are known as tetrads (${e^{a}}_{\mu}(x)$), which
are defined at each point in space-time by a local reference frame
$g_{\mu\nu}(x)=\eta_{ab}{e^{a}}_{\mu}{e^{b}}_{\nu}$ [40]. The tetrad and its
inverse, ($e^{\mu}_{a}=\eta_{ab}g^{\mu\nu}e^{b}_{\nu}$), satisfy the
orthogonal relationships: $e^{a}_{\mu}e^{b\mu}=\eta^{ab}$,
$e^{a}_{\mu}e^{\mu}_{b}=\delta^{a}_{b}$,
$e^{\mu}_{a}e^{a}_{\nu}=\delta^{\mu}_{\nu}$. The Greek indices ($\mu,\nu$) for
space-time frame coordinates, and the Latin indices ($a,b$) for local frame
coordinates. For the local reference frame
($\theta^{a}=e^{a}_{\mu}(x)dx^{\mu}$), we choose the tetrads:

$\displaystyle e^{a}_{\ \mu}(x)=\left(\begin{array}[]{cccc}V_{f}&0&0&0\\\
0&R&0&0\\\ 0&0&\alpha R\sin\theta&0\\\
0&0&\frac{n_{\Omega}}{2}R\sin\theta&1\end{array}\right);\ e^{\mu}_{\
a}(x)=\left(\begin{array}[]{cccc}\frac{1}{V_{f}}&0&0&0\\\ 0&\frac{1}{R}&0&0\\\
0&0&\frac{1}{\alpha R\sin\theta}&0\\\
0&0&-\frac{n_{\Omega}}{2\alpha}&1\end{array}\right).$ (14)

In a space with a curvature, the components of covariant derivative are given
by

$\nabla_{\mu}=\partial_{\mu}+\frac{i}{4}\omega_{\mu ab}(x)\Sigma^{ab}$ (15)

where $\Sigma^{ab}=\frac{i}{2}[\gamma^{a},\gamma^{b}]$, and the $\gamma^{a}$
matrices are defined in the local reference frame and are identical to the
Dirac matrices in the Minkowski spacetime in Weyl (or chiral) representation
[30, 46].

The beautiful way to obtain the one-form connection (${{\omega}^{a}}_{b}$) is
based on the first Maurer-Cartan structure equation:

$d\theta^{a}+{\omega^{a}}_{b}\wedge\theta^{b}=0\mbox{.}$ (16)

where the non-zero components are given by:

$\displaystyle\omega^{\ 1}_{\phi\ 2}(x)=-\omega^{\ 2}_{\phi\
1}(x)=-\alpha\cos\theta;$ (17) $\displaystyle\omega^{\ 1}_{\phi\
3}(x)=-\omega^{\ 3}_{\phi\ 1}(x)=-\frac{n_{\Omega}}{2}\cos\theta.$ (18)

In this way, we can write the Dirac equation for a massless fermion in this
background as

$i\frac{\gamma^{0}}{V_{f}}\frac{\partial\psi}{\partial
t}+i\frac{\gamma^{1}}{R}\bigg{(}\frac{\partial}{\partial\theta}+\frac{\cot\theta}{2}\bigg{)}\psi+i\frac{\gamma^{2}}{\alpha
R\sin\theta}\frac{\partial\psi}{\partial\phi}-i\gamma^{2}\frac{n_{\Omega}}{2\alpha}\frac{\partial\psi}{\partial
y}+i\gamma^{3}\frac{\partial\psi}{\partial y}-\frac{n_{\Omega}}{4\alpha
R}\cot\theta\gamma^{0}\gamma^{5}\psi=0.$ (19)

Since the extra dimension is periodic, we can expand the Dirac spinor in the
Fourier series with respect to this dimension:

$\psi(t,\theta,\phi,y)=\sum_{l=-\infty}^{\infty}\exp{\bigg{(}2\pi
il\frac{y}{L}\bigg{)}}\psi_{l}(t,\theta,\phi)$ (20)

where the scale of the extra dimension is denoted by $L$. Now, the Dirac
equation (19) takes the form

$i\frac{\gamma^{0}}{V_{f}}\frac{\partial\psi_{l}}{\partial
t}+i\frac{\gamma^{1}}{R}\bigg{(}\frac{\partial}{\partial\theta}+\frac{\cot\theta}{2}\bigg{)}\psi_{l}+i\frac{\gamma^{2}}{\alpha
R\sin\theta}\frac{\partial\psi_{l}}{\partial\phi}-\frac{n_{\Omega}}{4\alpha
R}\cot\theta\gamma^{0}\gamma^{5}\psi_{l}+\frac{2\pi
l}{L}\frac{n_{\Omega}}{2\alpha}\gamma^{2}\psi_{l}-\frac{2\pi
l}{L}\gamma^{3}\psi_{l}=0.$ (21)

Note that the last two terms in (21) correspond to the massive modes created
by the extra dimension of the KK theory. Obviously, we can apply the Dirac
phase factor method in (21). We will do that in the next section.

##  IV Quantum Holonomy

The possibility to realize a quantum gate based by a geometric description
(Eq. 5) establishes a problem of obtaining the geometrical phase ($\Phi$) of
the system in (21). To do this, we must use the Dirac phase factor method,
where we suppose that the Dirac spinor is written as

$\psi_{l}(t,\theta,\phi)=e^{i\Phi}\psi_{l}^{0}(t,\theta,\phi)=\exp{\bigg{(}-\int\Gamma_{\mu}(x)dx^{\mu}\bigg{)}}\psi_{l}^{0}(t,\theta,\phi)$
(22)

which $\Gamma_{\mu}(x)=\frac{i}{4}\omega_{\mu ab}(x)\Sigma^{ab}$ being the
spin connection. As we found, we can calculate the spinorial connection by
defining a local reference frame at each point along the closed curse around
the defect. Thus the geometrical phase $\Phi$ defines the holonomy matrix:
$U(\alpha,\theta)=e^{i\Phi}$, which stands here for a parallel transport of a
spinor along a path around the defect are respectively

$\Phi=\int\bigg{(}\frac{\alpha\cos\theta}{2}\Sigma^{3}-\frac{n_{\Omega}}{4}\cos\theta\Sigma^{2}\bigg{)}d\phi=\pi\alpha\cos\theta\Sigma^{3}-3\pi(1-\alpha)\cos\theta\Sigma^{2}$
(23)

and

$U(\alpha,\theta)=e^{i\pi\alpha\cos\theta\Sigma^{3}-3i\pi(1-\alpha)\cos\theta\Sigma^{2}}.$
(24)

Here it is important to note that we can reduce the holonomy matrix (24) to a
single parameter $\alpha$ if we assume that $\theta$ is small, so,
$\cos\theta\approx 1$. Remembering that parameter
$\alpha=1-\frac{n_{\Omega}}{6}$ is related to the number of sectors removed
($n_{\Omega}$) in the creation of the defect in $C_{60}$ molecule. More about
the synthesis of defects in graphene allotropes are found in [47, 48, 49].

Since equation (23) was obtained without adiabatic approximation, it’s an
Aharonov-Anandan phase. So that, in (24) we have

$U(\alpha)=e^{i\pi\alpha\Sigma^{3}-3i\pi(1-\alpha)\Sigma^{2}}.$ (25)

This approximation requires to use $\rho=R\theta$ in the metric (5). For the
fullerene molecule, we consider $\rho\ll R$, where $\rho$ is the dimension of
the local conical defect, and $R$ is the sphere radius. That way we have:
$\sin\theta\approx\frac{\rho}{R}$ and
$\cos\theta=\sqrt{1-\frac{\rho^{2}}{R^{2}}}\approx 1$.

Since the holonomy transformation (25) involves the sum of two non-commuting
matrices in the argument of the exponential function, we have that
$e^{A+B}\neq e^{A}e^{B}$. However, using the Hausdorff formula
$e^{A+B}=e^{A}e^{B}e^{-\frac{1}{2}[A,B]}\dots$, and defining
$\zeta=3(1-\alpha)$, we find

$U(\alpha)\approx
e^{i\pi\alpha\Sigma^{3}}e^{-i\pi\zeta\Sigma^{2}}e^{i\pi^{2}\alpha\zeta\Sigma^{1}}.$
(26)

By using the definition of the exponential of a matrix, that is,
$e^{A}=\sum\limits_{i=0}^{\infty}\frac{A^{n}}{n!}$, the expression (26) can be
rewritten in the form:

$U(\alpha)\approx q_{0}I+q_{1}i\Sigma^{1}-q_{2}i\Sigma^{2}+q_{3}i\Sigma^{3},$
(27)

where the parameters $q_{i}$ are defined as

$\displaystyle\begin{array}[]{ll}q_{0}=\cos(\pi\alpha)\cos(\pi\zeta)\cos(\pi^{2}\alpha\zeta)+\sin(\pi\alpha)\sin(\pi\zeta)\sin(\pi^{2}\alpha\zeta),\\\
q_{1}=\cos(\pi\alpha)\cos(\pi\zeta)\sin(\pi^{2}\alpha\zeta)-\sin(\pi\alpha)\sin(\pi\zeta)\cos(\pi^{2}\alpha\zeta),\\\
q_{2}=\cos(\pi\alpha)\sin(\pi\zeta)\cos(\pi^{2}\alpha\zeta)+\sin(\pi\alpha)\cos(\pi\zeta)\sin(\pi^{2}\alpha\zeta),\\\
q_{3}=\sin(\pi\alpha)\cos(\pi\zeta)\cos(\pi^{2}\alpha\zeta)-\cos(\pi\alpha)\sin(\pi\zeta)\sin(\pi^{2}\alpha\zeta).\\\
\end{array}$ (32)

In this way the holonomy matrix can be described by an unitary matrix:

$U(\alpha)\approx\left(\begin{array}[]{cccc}q_{0}+iq_{3}&-\left(q_{2}+iq_{1}\right)^{*}&0&0\\\
q_{2}+iq_{1}&\left(q_{0}+iq_{3}\right)^{*}&0&0\\\
0&0&q_{0}+iq_{3}&-\left(q_{2}+iq_{1}\right)^{*}\\\
0&0&q_{2}+iq_{1}&\left(q_{0}+iq_{3}\right)^{*}\\\ \end{array}\right)\mbox{.}$
(33)

It is also possible to show that $\sum q_{n}^{2}=1$. Also, the result we
obtained in eq. (33) is very important because, as is well known, quantum
gates can be represented by unitary transformations applied at defined moments
of time throughout the execution of the algorithm. So, any unitary
transformation can be written as a combination of the elementary quantum
logical keys [50, 51].

It is important to emphasize that continuum models elucidate a long-range
physics at the same time as they are restricted to a few dozens of electron-
volts around the Fermi point. Concurrent to this, the electronic density of
states decrease in the neighborhood of the defect [36, 37]. Besides that, if
we consider the $C_{60}$ molecule at low temperatures, we can reduce the
thermal noise and the density of electrons on the conduction band. In this
way, the quantum holonomy could mixed up the $K_{\pm}$ Fermi points, and the
electron location on sublattices $A$ and $B$. For this, we assumed that the
sublattices were mixed when the control parameter changes into the range
$0<\alpha<1$, and the gauge transformation on K-spin part is a rotation the
components of a possible two-qubits spinor in the next step of the quantum
computation issue.

##  V Conclusion

Let us discuss our results. We have proposed a discussion about a quantum
holonomy matrix based on the introduction of Kaluza-Klein extra dimension as
an alternative manner to explain the mixture of Fermi points in the $C_{60}$
fullerene molecule with icosahedral ($Ih$) symmetry. We have shown that it is
possible to describe the mix of the Fermi points ($K_{\pm}$) within the
$(4+1)$-dimensional geometrical approach. This result is important since it
shows that the implementation of quantum gates by an extra dimension approach
is an acceptable approximation, at least for the low-energy states, where the
eigenfunctions of the quasiparticles inside the molecule do not oscillate too
rapidly with distance. Besides that, the adding of an extra dimension promote
more freedom degrees on a possible two-qubits logical basis. That can be an
EPR pair of entangled electrons, a pair electron-hole, or another one.

It is hard to achieve the implementation of the quantum computation associated
with topological defects in solids, even more with the use of extra
dimensions. However, we wish to open a discussion about a new possibility to
implement it.

##  VI ACKNOWLEDGMENTS

We thank CAPES , CNPQ and FAPESQ-PB for financial support.

## References

  * [1] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss., Phys. Math. K1, 966 (1921).
  * [2] O. Klein, Z. Phys. 37, 895 (1926).
  * [3] A. Salam, J. H. Strathdee, Ann. Phys. 141, 316 (1982).
  * [4] M. J. Duff, Kaluza-Klein in perspective, hep-th/9410046.
  * [5] D. Bailin, A. Love, Rep. Prog. Phys. 50, 1087 (1987).
  * [6] Kleinert H., Gauge Fields in Condensed Matter, Vol. 2, World Scientific (1989).
  * [7] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, The Geometric Phase in Quantum Systems: Foundation, Mathematical Concepts and Applications in Molecular and Condensed Matter Physics, Springer-Verlag, New-York (2003).
  * [8] P. Zanardi, M. Rasetti, Phys. Lett. A 264, 94 (1999).
  * [9] F. Wilczek, A. Zee, Phys. Rev. Lett. 52, 2111 (1984).
  * [10] V. I. Kuvshinov, A. V. Kuzmin, Phys. Lett. A 316, 391 (2003).
  * [11] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, D. J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).
  * [12] J. Pachos, Phys. Rev. A 66, 042318 (2002).
  * [13] M. Cholascinski, Phys. Rev. B 69, 134516 (2004).
  * [14] A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, P. Zoller, Phys. Rev. A 66, 032309 (2002).
  * [15] J. Pachos, S. Chountasis, Phys. Rev. A 62, 052318 (2000).
  * [16] K. Bakke, C. Furtado, S. Sergeenkov, EPL 87, 30002 (2009).
  * [17] J. A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Nature 403, 869 (2000).
  * [18] J. Teles, et. al., Quantum Inf. Process 14, 1889 (2015).
  * [19] J. Pachos, P. Zanardi, Int. J. of Mod. Phys. B 15, 1257 (2001).
  * [20] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).
  * [21] Y. Aharonov, D. Bohm, Phys. Rev. 155, 485 (1959).
  * [22] B. E. Allmam, et. al., Phys. Rev. A 48, 1799 (1993).
  * [23] Y. Aharonov, A. Casher, Phys. Rev. Lett. 53, 319 (1984).
  * [24] X.-G. He, B. H. J. McKellar, Phys. Rev. A 47, 3424 (1993).
  * [25] M. Wilkens, Phys. Rev. Lett. 72, 5 (1994).
  * [26] C. Furtado, G. Duarte, Phys. Scr. 71, 7 (2005).
  * [27] K. Bakke, C. Furtado, Eur. Phys. Lett. 87, 30002 (2009).
  * [28] K. Bakke, C. Furtado, Quant. Inf. Proc. 12, 119 (2013).
  * [29] M. O. Katanaev, I. V. Volovich, Ann. Phys. (N.Y.) 216, 1 (1992).
  * [30] K. Bakke, A. Yu. Petrov, C. Furtado, Annals of Physics, 327, 2946 (2012).
  * [31] E. Cavalcante, C. Furtado, Journal of Phys. and Chem. of Solids, 75, 1265 (2014).
  * [32] J. Gonzalez, F. Guinea, M. A. H. Vozmediano, Nucl. Phys. B 406, 771 (1993).
  * [33] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E. Smalley, Nature 318, 162 (1985).
  * [34] K. Hedberg, L. Hedberg, D. S. Bethune, C. A. Brown, M. de Vries, H. C. Dorn, R. D. Johnson, Science 254, 410 (1991).
  * [35] E. F. Sheka, B. S. Razbirin, E. A. Nikitina and D. K. Nelson, J. Phys. Chem. A 115 3480 (2011).
  * [36] P. E. Lammert and V. H. Crespi, Phys. Rev. Lett., 85, 5190 (2000).
  * [37] P. E. Lammert and V. H. Crespi, Phys. Rev. B 69, 035406 (2004).
  * [38] M. Kleman, Points, Lignes, Parois: dans les fluides anisotropes et les solides cristallins, Editions de Physique, France, 1977.
  * [39] C. Furtado, F. Moraes, A. M. de M. Carvalho, Phys. Lett. A 372, 5368 (2008).
  * [40] N. D. Birrell, and P. C. W. Davies Quantum fields in curved spaces, Cambridge (1982).
  * [41] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, K. Geim, Rev. Mod. Phys. 81 (2009).
  * [42] M. A. H. Vozmediano, M. I. Katsnelson, F. Guinea, Phys. Rep. 496, 109 (2010).
  * [43] J. K. Pachos, Contemp. Phys. 50, 375 (2009).
  * [44] D. V. Kolesnikov and V. A. Osipov, Eur. Phys. J. B 49, 465 (2006).
  * [45] Y. Aharonov and D. Bohm, Eur. Phys. Rev. 115, 485 (1959).
  * [46] V. P. Gusynin, S. G. Sharapov, J. P. Carbote, Int. J. Mod. Phys. B 21, 4611 (2007).
  * [47] A. Krishnan, E. Dujardin, M. M. J. Treacy, et. al. Nature 388, 451 (1997).
  * [48] H. Terrones, T. Hayashi, M. Muñoz-Navia, et. al. Chem. Phys. Lett. 343, 241 (2001).
  * [49] M. Ge, K. Sattler, Chem. Phys. Lett. 220, 192 (1994).
  * [50] I. S. Oliveira, Introdução a Física do Estado Sólido, Ed. Livraria da Física (2011).
  * [51] M. Nielsen, I. Chuang, Quantum Information and Quantum Computation, Cambridge Press (2000).