# Three Dimensional Simulations of Solar Wind Preconditioning and the 23 July
2012 Interplanetary Coronal Mass Ejection
Solar Physics

Ravindra T. Desai  Han Zhang  Emma E. Davies  Julia E. Stawarz  Joan Mico-
Gomez  Pilar Ivá$\mathbf{\tilde{n}}$ez-Ballesteros  Blackett Laboratory,
Imperial College London, London, UK  Department of Physics, University of
Durham, Durham, UK  Universitat de Valéncia, Valéncia, Spain  Universidad
Autónoma de Madrid, Madrid, Spain

###### keywords:

Interplanetary Coronal Mass Ejections, Magnetohydrodynamics, Solar Wind,
Disturbances, Magnetohydrodynamics, Space Weather

##  1 Introduction

S-Introduction Coronal mass ejections (CMEs) are characterised by the large-
scale eruption of plasma from the solar corona and release copious amounts of
energy over incredibly short time periods. Upon reaching interplanetary space,
CMEs, hereafter referred to as interplanetary CMEs (ICMEs), can propagate at
velocities significantly greater than the ambient solar wind and will
therefore be decelerated due to drag forces (Cargill et al., 1996). In recent
years, the phenomenon of an initial ICME clearing the path for a successive
eruption has been implicated as a significant causal mechanism for producing
some of the most severe space weather events on record (Liu et al., 2019).
There is therefore a current need to understand how solar wind preconditioning
can affect an observed CME which poses a threat to operational spacecraft and
ground-based infrastructure, if Earth-directed. (SWPS, 2015).

The most extreme ICMEs are believed to occur along a power law distribution
(Riley, 2012) and can possess velocities $>$2000 km s-1, the most famous of
which is the Carrington Event of September 1859 (Carrington, 1859). This ICME
propagated Sun-to-Earth in just 17.6 hours and induced the largest geomagnetic
storm on record (Tsurutani et al., 2003). On 23 July 2012, however, the Solar
Terrestrial Relations Observatory-Ahead (STEREO-A) spacecraft, orbiting at
0.96 au, experienced a non-Earth directed ICME (Dryer et al., 2012) of a
magnitude comparable to the 1859 Carrington event (Baker et al., 2013) which
arrived 18.6 hours after its eruption from the solar corona with a velocity of
$\approx$2250 km s-1 (Russell et al., 2013).

Two distinct reasons have been put forth to explain the rapid transit time and
extreme characteristics of the 23 July 2012 event. The first reason is the CME
was produced by two separate eruptions spaced 10-15 minutes apart which
appeared to merge within the solar corona (Liu et al., 2014a). This idea was
further supported by the presence of two flux ropes within the in-situ
magnetic field measurements at STEREO-A where the ICME magnetic flux
consequently reached a record 109 nT (Russell et al., 2013). The second reason
is that the solar wind conditions ahead of the ICME had been significantly
affected by events prior to 23 July (Wu et al., 2014), the most significant of
which Liu et al. (2014a) identify as a CME which erupted from the same active
region at 05:30 UT on 19 July. This propagated along a similar path and is
thought to have removed existing solar wind plasma and non-radially directed
magnetic fields which resulted in minimal slowdown for the larger subsequent
eruption (Temmer and Nitta, 2015). It has however been highlighted that large
ICMEs are significantly less affected by the ambient conditions (Wu et al.,
2005; Liou et al., 2014; Cash et al., 2015) which leads to the question as to
if, and how, the 23 July 2012 ICME was, or could have been, influenced by
prior preconditioning events?

The complex interaction between ICMEs and the solar wind have been analysed
within a number of simulation studies (e.g. Cargill et al., 1996; Odstrcil and
Pizzo, 1999; Riley et al., 2003; Lugaz, Manchester, and Gombosi, 2005), and
Manchester et al. (2008) and Werner et al. (2019) include the effects of solar
wind preconditioning when simulating the solar storms of 28 October 2003 and
6-9 September 2017, respectively. This study focuses on solar wind
preconditioning with respect to the Carrington-scale event observed on 23 July
2012.

This article is organised as follows: Section PLUTO describes the simulation
code, PLUTO, the governing equations and implementation. Section 23july2012
builds upon these concepts and describes a simulation of the conditions
leading up to and including the extreme ICME of 23 July 2012. Section
preconditioning then uses this simulation as a baseline to study solar wind
preconditioning for events of this magnitude. Section summary then concludes
with a discussion on the implications for understanding and forecasting these
extreme events.

##  2 Simulation Development

PLUTO

###  2.1 Governing Equations

The simulation code, PLUTO, is an open-source parallel modular multiphysics
code developed for the study of astrophysical plasmas (Mignone et al., 2007).
It is written to solve partial differential equations deriving from
conservation laws of the form,

$\frac{\partial\mathbf{U}}{\partial t}=-\nabla\cdot\mathbf{T(U)+S(U)},$ (1)

where U is the state vector of conservative quantities, T is a tensor denoting
the flux of each component of the state vector and S defines the source terms.
This general integration sequence does not require the explicit forms of U, T
and S, and this flexibility allows the implementation of different physical
equations. PLUTO has the capability to solve non-relativistic or relativistic
hydrodynamic (HD) or magnetohydrodynamic (MHD) systems of equations on either
cartesian, cylindrical or spherical geometries.

PLUTO has been validated against a number of test-problems (e.g. Mignone et
al., 2007) and extensively applied to the study of astrophysical plasmas,
including; stellar and extragalactic jets (e.g. Bodo et al., 2003; Mignone et
al., 2013), the solar corona (e.g. Reale et al., 2016; Petralia, Reale, and
Testa, 2018), stellar winds (e.g. Alvarado-Gómez et al., 2016; Pantolmos and
Matt, 2017), and 1D simulations of the fast-forward shock associated with the
23 July 2012 event (Riley et al., 2016).

A detailed description of the numerics and physics capabilities are described
by Mignone et al. (2007) and for completeness the system of equations solved
in this study are described below. The MHD conservation equations are written
as:

$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf{v}=0,$ (2)
$\frac{\partial\mathbf{m}}{\partial
t}+\nabla\cdot\left[\mathbf{mv}-\mathbf{BB}+\mathbf{I}(p+\mathbf{B}^{2})\right]^{T}=-\rho\nabla\mathbf{\Phi}+\rho\mathbf{g},$
(3) $\frac{\partial\mathbf{B}}{\partial
t}+\nabla\times(\mathbf{v}\times\mathbf{B})=0,$ (4)

$\frac{\partial(E_{t}+\rho\mathbf{\Phi})}{\partial
t}+\nabla\cdot\left[(\frac{\rho\mathbf{v}^{2}}{2}+\rho
e+p+p\mathbf{\Phi})\mathbf{v}+(\mathbf{v}\times\mathbf{B})\times\mathbf{B}\right]=\mathbf{m}\cdot\mathbf{g},$
(5)

where $\mathbf{v}$ is the gas velocity in an inertial reference frame, $\rho$
is the gas mass density, and $p$ its thermal pressure. $\mathbf{m}$ represents
the momentum density term equal to $\rho\mathbf{v}$, $\mathbf{I}$ is the unit
tensor, and $\mathbf{\Phi}$ and $\mathbf{g}$ are the potential and vector part
of the body force which are set to zero in this study. The total number
density $n$ is defined by $\rho=\mu H$, where $\mu$ is the mean molecular
weight in units of the mass of the hydrogen atom $H$. $\mathbf{B}$ is the
magnetic field and the total energy density, $E_{t}$, is defined as,

$E_{t}=\rho e+\frac{\mathbf{m}^{2}}{2\rho}+\frac{\mathbf{B}^{2}}{2}.$ (6)

In Equations 4 and 5 the electric field $\mathbf{E}$ is provided through the
generalised Ohm’s law with the resistive and Hall terms neglected, leaving an
ideal MHD description. A closure is provided to these systems using an
equation of state $\rho\hskip 2.84526pte=\rho\hskip 2.84526pte(p,\rho)$, which
is set to represent an ideal gas.

To evolve these equations in time, the code uses volume averages determined by
piece-wise monotonic interpolation inside each grid cell. This generic
implementation allows the code to easily switch between a number of different
solvers and, in this study, the Roe solver of Cargo and Gallice (1997) is
utilised. The Linearized Roe Riemann solver is based on characteristic
decomposition of the Roe matrix which approximates the solutions to the
discontinuous left and right Riemann solutions at the cell interfaces (Roe,
1981). The MHD formulation evolves $\rho$, $\mathbf{v}$, $T$ and $\mathbf{B}$
in time and requires an additional step to enforce the solenoidal constraint
on the divergence of the magnetic field. The eight-wave divergence cleaning
formulation of Powell (1994) and (Powell et al., 1999) is used, with the
constrained transport scheme of Balsara and Spicer (1999) and Londrillo and
del Zanna (2004), to add an additional corrective source term. This utilises
the upwinded fluxes in the higher-order Godunov scheme combined with the
electric field to maintain a divergence-free magnetic field to machine
precision.

The heliospheric simulations can be implemented in one, two or three spatial
dimensions with either one, two or three components of the vector fields. In
this study, a spherical coordinate system is used where $r$, $\theta$ and
$\phi$ represent the radial, polar and azimuthal directions respectively, in a
heliocentric rotating or inertial reference frame. The mean molecular weight
is set to $\mu$=0.6 to represent a fully ionised electron-proton plasma with
small amounts of heavier species. The domain is specified to capture the solar
wind expanding outward from an inner boundary specified at an arbitrary
distance from the solar surface. The simulations are driven by updating this
inner inflow boundary at each time-step with commensurate solar wind values.
For idealised scenarios the inputs are theoretically derived, or, to represent
real time periods, the values at the inner boundary are updated to match the
output of coronal models. The outer boundaries are represented by Von-Neumann
boundary conditions for continuous outflow, and the simulation domain thus
represents the continuous evolution of the solar wind within a predefined
region of the heliosphere.

Figure 1.: Initialisation phase of a three dimensional MHD simulation of the
solar wind. The simulations are shown as the fast solar wind first reaches the
outer boundary while the slow solar wind has not yet done so. The left-hand
and centre panels (a–b) show the plasma number density normalised by the
radial distance squared in au from above and at the equatorial plane,
respectively. The right-hand panel (c) shows the heliospheric current sheet
which is revealed by masking magnetic field values above an arbitrarily small
value. Note that 214.8 Rs $\simeq$ 1 au solarwind

###  2.2 The Solar Wind

setup

To simulate the nascent solar wind, a three-dimensional MHD simulation is
initialised to reproduce solar wind conditions starting from 21 June 2012. The
simulations are driven using inflow coronal boundary conditions generated by
CORHEL (Riley, Linker, and Mikić, 2001; Linker, 1999) from the archived
simulation run Camilla_Scolini_012516_SH_1 , downloaded from NASA’s Community
Coordinated Modeling Center (CCMC). This CORHEL coronal boundary condition
uses the magnetic field structure from the Magnetohydrodynamics Algorithm
outside a Sphere (MAS) polytropic coronal model (Linker, 1999) with
empirically derived velocity, density and temperature variables (Riley,
Linker, and Mikić, 2001). The input data is available at a 6 hour cadence and
a resolution of $\theta=180$ and $\Phi=360$ points. A cubic spline
interpolation sequence is used to resolve this to the variable simulation
timestep of the heliospheric grid which is specified at a lower resolution of
$\theta=90$ and $\Phi=180$ points with a radial extent of 192 points from 30
Rs out to beyond 1 au.

Figure solarwind shows the initialisation stage of this simulation where the
solar wind expands outward into a low density background. The plasma density
across the simulation domain is initially specified to fall off as a function
of $r^{-2}$, but the simulations are only judged to have reached a steady-
state when solar wind plasma from the inner boundary has reached the outer
boundary and asymptoted to a steady state solution. The left and centre panels
show two projections of the normalised density where the fast solar wind has
reached the outer boundary but the slow solar wind has not yet done so. The
propagation of fast and slow solar wind and stream interaction regions (SIRs)
are visible as well as the rotation of the Sun which wraps these streams along
an Archmidean spiral (Parker, 1958). The right-hand panel shows the
heliospheric current sheet (HCS), calculated by masking values above an
arbitrarily small value. The HCS is produced at the Sun’s magnetic equator and
maintains its tilted orientation with respect to the ecliptic plane as it
evolves outward.

In this study a polytropic index of $\gamma=1.5$ is used to capture the
measured polytropic index of the free-streaming solar wind (Totten, Freeman,
and Arya, 1995). This still, however, doesn’t fully capture the kinetic
heating of the solar wind via a variety of electron and ion-scale
instabilities (e.g. Stawarz et al., 2009; Goldstein et al., 2015) and the MHD
simulations therefore result in a steeper radial temperature profile than that
observed (e.g. Cranmer et al., 2009).

###  2.3 Interplanetary Coronal Mass Ejections

ICME

CMEs are launched along a specific direction and are then able to deflect and
rotate as they erupt outward (Gopalswamy et al., 2003; Cremades, Bothmer, and
Tripathi, 2006; Gui et al., 2011; Vourlidas et al., 2011) as they interact
with the solar wind (Wang et al., 2004; Isavnin, Vourlidas, and Kilpua, 2013)
or through interactions with other ICMEs (Lugaz et al., 2012; Shen et al.,
2012). This can cause limb ICMEs to strike Earth (Wang et al., 2004) and
30-40% of Halo ICMEs to not strike Earth (Yermolaev et al., 2005; Shen et al.,
2014).

To model and capture these three dimensional processes, a pulse is injected on
the inner boundary by varying the solar wind density and velocity according to
the relation:

$\lambda(t,\theta,\phi)=\lambda_{0}+\lambda_{p}sin^{2}(\pi\frac{t-t_{0}}{t_{p}})cos(\frac{\pi}{2}\frac{\theta-\theta_{0}}{\Theta})cos(\frac{\pi}{2}\frac{\phi-\phi_{0}}{\Phi}),$
(7)

within the limits:

$t_{0}<t_{p}<t_{0}+T,\hskip
14.22636pt\theta_{0}-\Theta<\theta<\theta_{0}+\Theta,\hskip
14.22636pt\phi_{0}-\Phi<\phi<\phi_{0}+\Phi,$ (8)

where $\lambda$ represents the variable of interest, and $\lambda_{p}$ its
peak increase. $\theta_{0}$, $\phi_{0}$ and $t_{0}$ are the latitudinal and
azimuthal launch position and start time, respectively, and $\Phi=\Theta$
which represents the half-angle subtended by the pulse and $t_{p}$ the pulse
duration.

Gaussian and sinusoidal pulses, and variations thereof, have been commonly
used to represent ICMEs (Odstrcil and Pizzo, 1999; Pomoell and Poedts, 2018;
Liou et al., 2014; Riley et al., 2016) and, as these pulses propagate outward,
the outer regions expand faster than the inner regions and produce a
characteristic cone shape with associated MHD forward and reverse shocks.
These pulses do not strictly conserve the Rankine-Huguenot jump conditions but
are of sufficient smoothness that MHD solvers are able to relax these into a
structure which has been shown to well-represent an ICME (e.g. Chané et al.,
2006), with the exception of the internal flux rope magnetic field.

8

Figure 2.: Three dimensional MHD simulations of the 19 July ICME in upper
panels (a–b) and 23 July ICME in lower panels (c–d). The left-hand panels (a
and c) show the radial velocity and the right-hand panels (b and d) show the
scaled plasma number density. Note that 214.8 Rs $\simeq$ 1 au. mainevent

##  3 Simulating the 23 July 2012 Event

23july2012

In this study the preconditioning events leading up to 23 July and the 23 July
ICME are approximated by two distinct CMEs erupting on 19 July and 23 July.
The background solar wind is modelled for the time period encompassing this
event from 15 July, thus allowing the simulated heliosphere to reach steady-
state before the passage of the ICMEs. The simulations are driven using two
publically available simulations (Anthony_Mannucci_022515_SH_1, ;
Camilla_Scolini_012516_SH_1, ) produced by the MAS polytropic coronal model
(Linker et al., 2016) which cover Carrington rotation 2125 from 21 June until
18 July, and Carrington rotation 2126 from 18 July until 16 August. These
coronal simulations are driven by GONG synotoptic magnetogram data. To avoid
numerical instability, the final outputs of Carrington rotation 2125 are
omitted and the model inputs are interpolated over 24 hours, which ensures a
smooth transition between changes in the locations of the coronal hole
boundary between the two inputs.

Figure 3.: Time series output from the MHD simulations presented in Figure
mainevent at the location of STEREO-A compared to the STEREO-A timeseries from
Liu et al. (2014a, see Figure 5 therein). (a) Plasma density, (b) bulk
velocity, (c) temperature and (d) magnitude of the magnetic field. Note, the
total measured density (a) (blue) were calculated through extrapolation from
the low energy electron measurements (Liu et al., 2014a). stereoA

The ICMEs are initialised in-line with measurements (Cash et al., 2015; Liou
et al., 2014), see Table 1, and the simulations run until the final eruption
has reached the outer boundary on 24 July 2012. Estimates used for the initial
speed of the 23 July ICME range from 2500$\pm$500 km s-1 to $>$3000 km s-1
(Baker et al., 2013; Liu et al., 2014a; Intriligator et al., 2015; Liou et
al., 2014; Intriligator et al., 2015; Riley et al., 2016; Liu et al., 2014b),
the upper range of which exceeds the theorised $\approx$ 3000 km/s limit
available from the energy budget of solar active regions (Gopalswamy et al.,
2005; Yashiro et al., 2004). In this study a launch velocity of 2800 km/s is
found to produce a reasonable match.

Figure mainevent shows the simulation of the solar wind and 19 July and 23
July ICMEs. The 19 July ICME is visibly smaller than the 23 July ICME and the
initial eruption produces a density cavity in its wake. The ICMEs propagate
through various solar wind conditions; the 19 July ICME encountering a denser
slower solar wind stream, which cause significant azimuthal variation across
solar longitude, whilst the 23 July ICME propagates from a fast solar wind
stream into a slow solar wind stream. The limitations in using two distinct
coronal models inputs become clear as in certain locations, the changes in the
coronal hole boundaries between the two inputs causes fast solar wind to run
into slower solar wind which results in some visible discontinuities. Further
interpolation routines might remove this and indeed Cash et al. (2015)
implement a variety of different solar wind backgrounds from either before or
after the event and finally a hydrodynamic solar wind. It is not necessarily
clear, however, which selection is correct due to the lack of continuous
magnetogram coverage during this period. These effects are also minimal in the
sector where the 19 and 23 July ICMEs propagate, and the ICME propagation
along the Sun-STEREO-A line is not significantly affected.

Figure stereoA shows the time series from a probe within the simulation at the
location of STEREO-A which is compared to the STEREO-A in-situ measurements.
The time series is also shown for a simulation of the background solar wind
without the ICMEs. Two initial ICMEs are visible in the STEREO-A data on 18
and 19 July which in this study are represented by the single and larger 19
July ICME. The 19 July ICME appears to feature a second peak in density which
is attributed to the dense solar wind stream encountered en-route which is
likely to have been cleared by the first initial ICME visible in the STEREO-A
time series, but which is not simulated here. The low density cavity ahead of
the 19 July ICME is well reproduced with the background plasma density dipping
to comparable values to the observed $\approx$1 cm-3. The velocity profile of
the 23 July ICME is subsequently a close match with the simulated ICME
arriving close in time to the observed ICME, and initially dropping of at a
similar rate. A second peak in velocity is visible in Figure mainevent, which
represents a reverse shock, although this feature is only marginally visible
within the measurements. The density peaks at a similar time and amplitude
and, with the velocity profile, indicates that the simulations have reproduced
an ICME of comparable magnitude to the observed event.

Notable differences do however exist. The initial ramp-up in density and
velocity for the few hours prior to the 23 July ICME is not captured, some of
which Russell et al. (2013) tentatively attribute to initial disturbances from
the 23 July event itself. The simulated ICME density is also narrower in time,
reaching higher values but decreasing more rapidly too. The simulated magnetic
field reaches $>$100 nT, as in the STEREO-A time-series, but this is due to
compressed pre-existing solar wind fields rather than an intrinsic field and
therefore decreases rapidly too. These discrepancies are attributed to the
approximation of a complex multi flux rope event by a single smooth pulse. The
simulated 23 July ICME also reveals a distinct temperature spike not captured
within the in-situ measurements.

Further efforts to optimise the input parameters to match the observations
could decrease the discrepancies but might well result in over-fitting due to
the various error sources deriving from the choice in background solar wind
conditions and the observational constraints on the initial and final
conditions. Given these, the concept of a high-speed ICME travelling through a
low density interplanetary medium is judged to be well captured and the
simulations presented in Figures mainevent and stereoA are thus taken as a
reasonable reproduction of the 23 July 2012 event, and indeed of a
Carringtons-scale event, from which physical processes can be inferred.

Figure 4.: Normalised plasma number density for four different launch times
from 20 – 23 July. Upper panels (a–d) show the case of a single ICME without
solar wind preconditioning and lower panels (e–h) show the case of a preceding
ICME which produces a large density cavity in front of the subsequent event.
The right-hand lower panel (h) corresponds to the simulations shown in Figures
mainevent and stereoA. Note that 214.8 Rs $\simeq$ 1 au. 8plot

##  4 Preconditioning Study

preconditioning

Figure 5.: Parametric study showing the variations in radial velocity in left-
hand panel (a), and transit time in right-hand panel (b) resulting from the 23
July ICME launched at different times behind the 19 July ICME (orange solid
line) and also in its absence (blue dashed line). The actual velocity observed
at STEREO-A and transit time are marked by annotated dashed black lines.
launchtime

The simulated times series in Figure stereoA show the solar wind density ahead
of the main event to increase towards their nominal levels faster than the
observations do although they still have not fully recovered by 23 July. This
faster recovery appears to be due to the higher simulated solar wind speeds
ahead of the 23 July ICME of $\approx$550 km s-1 which gives a ballistic
refilling timescale of $\approx$3.1 days for fresh solar wind to propagate
into the depleted sector, following the passage of the 19 July ICME. The
STEREO-A observations, however, show a lower velocity of $\approx$440 km s-1,
which gives a ballistic refilling timescale of $\approx$3.9 days which extends
up until the 23 July ICME. The ballistic refilling argument does, however,
only provide a minimum timescale for the refilling as it does not account for
the density cavity ahead of the fresh solar wind and how the remaining
rarefied solar wind has been impacted by the initial event. Temmer et al.
(2017) indeed find that the interplanetary medium remains disturbed for
$\approx$ 3 – 6 days based on statistical comparisons between observed ICMEs
and modelled background solar wind.

To further study the concept of an initial ICME clearing the path for a second
ICME, the simulation presented in Figures mainevent and stereoA are used as a
baseline and the launch time of the 23 July ICME is varied with respect to the
19 July ICME, from 20 July until 24 July. The simulations are also run in the
absence of an initial event to fully isolate the effects of solar wind
preconditioning. From an operational forecasting perspective this approach
allows one to examine the amount of time after an initially observed ICME that
subsequent ICMEs will have an enhanced geophysical impact. Figure 8plot shows
four of these permutations, with and without preconditioning, which reveals
the significant density cavity ahead of the subsequent ICME and how an earlier
launch time results in the subsequent ICME propagating through this cavity.

The peak velocity at the location of STEREO-A is shown in Figure 5 with the
simulation time corresponding to the actual launch marked. The peak velocity,
together with the southward component of the magnetic field, provides an
indicator of the geoeffectiveness of the event (Burton, McPherron, and
Russell, 1975; Wu and Lepping, 2005), while the transit time is also shown
which provides the lead time that is required to accurately predict a given
CMEs evolution. The results show that the initial simulated 19 July ICME
induced a speed increase in the subsequent event right up until 24 July,
although, on and after 23 July this effect appears minimal as the simulations
with and without the 19 July ICME converge. As the simulated 23 July ICME is
launched earlier in time, the peak velocities increase as the 23 July ICME
spends more and more time in the wake of the 19 July ICME and eventually runs
into its back if launched before 21 July, a phenomenon (Lugaz, Manchester, and
Gombosi, 2005) not examined in this study . The velocity increase appears the
most significant when launched on 21 July.

The preconditioning study results, combined with further analysis of the
STEREO-A data suggest that the preconditioning for the 23 July 2012 event was
marginal and therefore this event was produced without the aid of solar wind
preconditioning. The trends do, however, indicate that solar wind
preconditioning can be significant for an event of this magnitude and that
this event therefore could have been even more severe if launched directly
into the depleted solar wind observed by STEREO-A two days prior.

##  5 Discussion and Conclusionssec:Conclusion-and-Future

summary

This study describes 3D MHD PLUTO simulations of the inner heliosphere leading
up to and including the 19 and 23 July ICMEs, the latter of which has been
identified as a Carrington-scale event. The simulations were driven by CORHEL
coronal boundary conditions for Carrington rotation 2125 and 2126, which
highlighted the relevance of continuously derived inputs with respect to
observing and modelling the solar corona. The simulations produced an event of
comparable magnitude to the 23 July 2012 event with time-series output from
the location of STEREO-A matching the observations. It should, however, be
noted that the simulations of the 23 July 2012 event are constrained around
the STEREO-A observations at 0.96 au. It is therefore likely that multiple
initial conditions exist that produce an ICME at 1 au that matches those
observed, as indeed is highlighted by Cash et al. (2015). The uncertainties in
the near-Sun properties of the CME do not allow for this event to be
constrained further and simulations of this events should therefore be
interpreted as physically realistic events of comparable magnitude to the 23
July 2012 event rather than exact replicas.

The 23 July 2012 simulation was then used to further examine and quantify the
impact of solar wind preconditioning. The launch time of the 23 July ICME was
varied with respect to a 19 July preconditioning ICME and also in its absence
to constrain the phenomenon of one ICME clearing the path for another. The
simulations were able to reproduce the observed depleted solar wind densities
ahead of the 23 July ICME and the simulated 23 July ICME reached a maximum
velocity of $>$2750 km s-1 when launched on 21 July but tended towards 2000 km
s-1 when launched after 23 July.

Several studies have indicated that large ICMEs are significantly less
affected by the ambient solar wind (Wu et al., 2005; Cash et al., 2015). The
simulations of the 23 July ICME presented here are, however, towards the
larger end of those considered, see Table 1 and Cash et al. (2015, Table 2.
therein). While the precise numbers presented in this parametric
preconditioning study might well differ depending on the solar wind
backgrounds and ICME parameters used, the depletion behind the 19 July ICME
shown in Figure 8plot is consistent with the large-scale interaction between
ICMEs and the ambient solar wind and also comparable to those produced by
flux-rope ICME simulations (Shen et al., 2014). The trends presented in Figure
launchtime therefore indicate that solar wind preconditioning can be a
significant factor for ICMEs of comparable magnitude to the 23 July event and,
therefore, Carrington-scale events. The simulations also reveal that the 23
July event itself likely occurred when the solar wind had nearly fully
recovered and it can be inferred that the 23 July would have reached even
higher velocities at 1 au if launched closer in time to the 19 July
preconditioning ICME.

The preconditioning window identified in Figure launchtime, although focussed
on a particular event, can provide context for historical events, such as the
August 1972 ICME which reached Earth in a record 14.6 hours (Knipp et al.,
2018) and the Carrington event of September 1859 which reached Earth in 17.6
hours (Tsurutani et al., 2003), which have been implicated as involving
multiple ICME eruptions. In the absence of in-situ measurements, the trends
presented here indicate that preconditioning variations in the ambient solar
wind could explain the difference in these arrival times. The significant
increase in ICME velocity and decrease in transit time due to solar wind
preconditioning could also plausibly enhance the severity of a range of CMEs
which do not initially present themselves as Carrington-scale within the solar
corona. This study also did not, however, consider the effects of solar wind
preconditioning inside the solar corona. A comprehensive parametric
preconditioning study, with regards to the numerous ICME input parameters that
can be varied, and also with a flux rope CME description, is therefore left
for a future endeavour.

The preconditioning window induced by an initial ICME, and the decrease in its
significance over time, presents a quantifiable prediction that can be used by
current space weather forecasting efforts to estimate the increased likelihood
of a severe space weather event occurring.

ICME |  | Launch Time
---
(UT)
| Radial Velocity
---
(km s-1)
| Mass
---
(g)
| Longitude
---
(∘)
| Latitude
---
(∘)
| Half Width
---
(∘)
| Duration
---
(hrs)
19 July | 05:30 | 1500 | 1.0e+16 | 134 | 0 | 45 | 3.5
23 July | 04:30 | 2800 | 2.0e+16 | 134 | 0 | 45 | 3.5
Table 1.: Input data for the 19 July and 23 July 2012 ICMEs corresponding the
simulations presented in Figure mainevent and stereoA. The ICMEs are launched
at the inner boundary of 30 RS and further input values are taken from Cash et
al. (2015, see Tables 1 & 2 therein). inputtable

#### Acknowledgments

The authors would like to extend their gratitude to Robert Forsyth, Jeremy
Chittenden, Jonathan Eastwood, Timothy Horbury, Peter Riley and Jon Linker for
useful discussions, and Ying Liu for providing updated STEREO-A time-series
estimates. RTD acknowledges funding from the NERC grant NE/P017347/1 (Rad-
Sat). EED acknowledges STFC studentship ST/N504336/1. JES acknowledges funding
from the UKRI STFC grant ST/S000364/1. PI and JM acknowledge the Erasmus
programme. The CORHEL coronal boundary conditions were downloaded from the
Coordinated Community Modelling Centre. This study used the Imperial College
High Performance Computing Service (doi: 10.14469/hpc/2232).

Disclosure of Potential Conflicts of Interest: The authors declare that they
have no conflicts of interest.

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# Real-space and reciprocal-space topology in the Eu(Ga1-xAlx)4 square net
system

Jaime M. Moya  Applied Physics Graduate Program, Rice University, Houston, TX,
77005 USA  Department of Physics and Astronomy, Rice University, Houston, TX,
77005 USA  Jianwei Huang  Department of Physics and Astronomy, Rice
University, Houston, TX, 77005 USA  Shiming Lei  Department of Physics and
Astronomy, Rice University, Houston, TX, 77005 USA  Kevin Allen  Department of
Physics and Astronomy, Rice University, Houston, TX, 77005 USA  Yuxiang Gao
Department of Physics and Astronomy, Rice University, Houston, TX, 77005 USA
Yan Sun  Shenyang National Laboratory for Materials Science,Institute of Metal
Research, Chinese Academy of Sciences  Ming Yi  Department of Physics and
Astronomy, Rice University, Houston, TX, 77005 USA  E. Morosan<EMAIL_ADDRESS>Department of Physics and Astronomy, Rice University, Houston, TX, 77005 USA

###### Abstract

Magnetotransport measurements on the centrosymetric square-net Eu(Ga1-xAlx)4
compounds reveal evidence for both reciprocal- and real-space topology. For
compositions $0.50\leq x\leq 0.90$, several intermediate field phases are
found by magnetization measurements when $H\parallel c$, where a maximum in
the topological Hall effect (THE) is observed, pointing to the existence of
topological (real-space topology) or non-coplanar spin textures. For $0.25\leq
x\leq 0.39$, magnetization measurements reveal an intermediate field state,
but no transition is visible in the Hall measurements. For $x=0.15$, only one
magnetic transition occurs below the Néel temperature $T_{N}$, and no
intermediate field spin reorientations are observed. The Hall effect varies
smoothly before the spin-polarized (SP) state. However, in the SP state, Hall
measurements reveal a large anomalous Hall effect (AHE) for all compositions,
a consequence of reciprocal-space topology. Density functional theory
calculations in the paramagnetic state indeed reveal a Dirac point that lies
very near the Fermi energy, which is expected to split into Weyl nodes in the
SP state, thereby explaining the observed AHE. These results suggest the
Eu(Ga1-xAlx)4 family is a rare material platform where real- and reciprocal-
space topology exist in a single material platform.

##  I Introduction

The advent of topological materials has generated much interest not only from
a fundamental physics perspective, but also due to their potential
applications in revolutionary electronic devices. For example, the theoretical
prediction of the quantum anomalous Hall effect (QAHE) [1, 2], a consequence
of Berry curvature [3] in reciprocal space, promises the possibility of
quantized, chiral, dissipation-free electron transport without magnetic
fields, ideal for energy efficient electronic devices. The reality of such a
device was brought one step closer with the experimental realization of the
QAHE in the topological insulator Cr-doped Bi(Sb)2Te3 [4].

The notions of topology in condensed matter systems also extends to real space
via topological spin textures which are particle-like, non-coplanar spin
configurations characterized by a topological charge [5]. Topological spin
textures have been proposed for applications in next-generation memory, logic,
spintronic and neuromorphic computing devices [6, 7, 8, 9, 10]. Furthermore,
itinerant electrons coupled to topological spin textures acquire Berry phase
when traversing the non-coplanar spin textures. Therefore, the topological
spin textures can be regarded as real-space sources of Berry curvature,
resulting in topological the Hall effect (THE) [5].

Combining reciprocal- and real-space topology extends the phase space of
approaches for future spintronic applications. For example, proof-of-principle
experiments based on topological insulator/magnetic heterostructures exhibit
both THE and QAHE: this approach takes advantage of THE to read out the spin-
state information, which, in turn, can be transmitted without dissipation
using the QAHE chiral edge states [11, 12, 13, 14, 15, 16]. Compared to their
heterostructure counterparts, bulk magnetic topological materials offer the
chance for stronger coupling between magnetism and itinerant electrons.
However, the concurrence of reciprocal- and real-space topology has not yet
been observed in a bulk system.

The Eu(Ga1-xAlx)4 series can be considered an ideal platform for such
reciprocal- and real-space topology coexistence. The $x$ = 0 compound EuGa4
orders magnetically into a simple A-type antiferromagnetic (AFM) structure
[17]. No intermediate phases are observed below the Néel temperature in the
magnetic field - temperature ($H~{}-~{}T$) phase diagram [18]. However, above
$T_{N}$, angle-resolved photoemission spectroscopy (ARPES) data show the
existence of four-fold degenerate spinless nodal rings (NR) near the Fermi-
level in EuGa4 [18]. With the application of a magnetic field
$H~{}\parallel~{}c$, when the magnetization is saturated in the spin-polarized
(SP) state, the spin-degeneracy is lifted, and two topological Weyl-nodal
rings (NRs) are realized, protected by mirror symmetry [18]. The topological
NRs, whose signatures are observed in magneto-transport measurements [18], are
responsible for the large quantum mobility and large, unsaturated
magnetoresistance (MR) reaching a value of 200,000% at $\mu_{0}H~{}=~{}14~{}$T
and $T~{}=~{}2~{}$K [18, 19]. The $x$ = 1 compound EuAl4 [20, 21, 22, 23, 24,
25] has a complex $H-T$ phase diagram with several spin reorientation
transitions for $H\parallel c$ [23, 22, 21]. A non-zero THE in select regions
of the $H~{}-~{}T$ phase diagram led to the proposal of topological spin-
textures in EuAl4 [22]. Later, the existence of a skyrmion lattice was
confirmed via small-angle nuetron scattering measurements [21]. The evidence
for the reciprocal-space topology in EuGa4 and real-space topology in EuAl4
prompts the search for the coexistence of the two types of topological states
across the Eu(Ga1-xAlx)4 series.

Structurally, Eu(Ga1-xAlx)4crystallizes in the tetragonal space group
$I4/mmm$, which hosts two crystollographic sites for Al or Ga [20]. For $x$ =
0.5, the Al and Ga preferentially occupy the two different sites, leading to
an ordered structure of EuGa2Al2 [20]. EuGa2Al2 has also been shown to have a
complex $H-T$ phase diagram for $H\parallel c$, with a non-zero THE maximum
centered around an intermediate-field phase, pointing to the existence of
either a topological spin texture or another non-coplanar spin texture [26].
The exact nature of this intermediate field state is yet to be determined.

The existence of reciprocal- and real-space topology for the two end members
EuGa4 and EuAl4, respectively, and the persistence of the THE in EuGa2Al2,
motivate a detailed study on the Eu(Ga1-xAlx)4 series, with a goal of
identifying compositions where both topological phenomena might coexist. To
this end, Eu(Ga1-xAlx)4 single crystals with $x$ = 0.15, 0.24, 0.31, 0.39,
0.50, 0.58, 0.71 and 0.90 were synthesized. In the magnetically ordered state,
the THE for $x\geq 0.50$ with $H\parallel c$ is observed, pointing to the
existence of topological or, more generally, non-coplanar spin textures.
Additionally, a large intrinsic AHE is registered in the field-induced spin-
polarized (SP) state when the magnetization is saturated for compositions 0.24
$\leq$ x $\leq$ 0.71, suggesting reciprocal-space band topology. Density
functional theory (DFT) calculations indicate the existence of a Dirac point
near the Fermi energy in EuGa2Al2. The Dirac point is expected to split into
Weyl nodes in the SP state, responsible for the large AHE in Eu(Ga1-xAlx)4.

##  II Experimental Methods

Single crystals of Eu(Ga1-xAlx)4 were synthesized using a self-flux method
described in Ref. [20]. Powder X-ray diffraction measurements were collected
with a Bruker D8 Advance diffractometer with Cu Kα radiation. Rietveld
refinements were done using Fullprof software [27] and the obtained lattice
parameters are consistent with the previous results [20]. Quantitative
elemental analysis by Wavelength Dispersive Spectrometry (WDS) of
Eu(Ga1-xAlx)4 phase was performed using the EPMA (Electron Probe Micro-
Analyzer) instrument at Rice University, with a JEOL JXA 8530F Hyperprobe
equipped with a field emission (Schottky) emitter and five WDS spectrometers.
The analytical conditions used were 15 kV accelerating voltage, 20 nA beam
current and a spot beam size of $\sim$ 300 nm. The standards used for
composition calibration were synthetic in-house produced stoichiometric
compounds EuGa4 and EuAl4. Careful background offsets were manually selected
for each element to avoid interferences with higher order X-rays during peak
and background measurement. Each element (Ga, Al and Eu) was simultaneously
analyzed on two spectrometers, in order to improve the statistics on the
standard deviation and detection limit calculation for each measurement. The
reproducibility of the standards was accurate and precise, with an error below
1% for each element. ZAF matrix correction was employed for quantification.

Magnetization measurements were done using a Quantum Design (QD) Dynacool
system equipped with a vibrating sample magnetometer. Four-probe resistivity
measurements were made using the electrical transport option in the same
system with the typical applied current $j~{}=~{}5~{}$mA and frequency
$f~{}=~{}9.15~{}$Hz. Measurements of the longitudinal resistivity as a
function of magnetic field $\mu_{0}H$, $\rho_{xx}(\mu_{0}H)$, and Hall
measurements, $\rho_{yx}(H)$, were measured in a complete field-sweep loop
with four quadrants: Quadrant I from $\mu_{0}H~{}>~{}0$ to $\mu_{0}H~{}=~{}0$,
Quadrant II from $\mu_{0}H~{}=~{}0$ to $\mu_{0}H~{}<~{}0$, Quadrant III from
$\mu_{0}H~{}<~{}0$ to $\mu_{0}H~{}=~{}0$ and Quadrant IV from
$\mu_{0}H~{}=~{}0$ to $\mu_{0}H~{}>~{}0$. The subsequent measurements were
symmetrized or antisymmetrized, respectively. No hysteresis was observed for
any composition.

The electronic band structure was calculated based on density functional
theory by using the code of Vienna Ab initio Simulation Package (VASP) [28]
with projected augmented wave potential. The exchange and correlation energies
were considered at the level of the generalized gradient approximation (GGA),
following the Perdew–Burke–Ernzerhof parametrization scheme [29]. The energy
cut-off was set as 500 eV. The calculations have dealt with f-electrons as the
valance states. To calculate the surface state, we projected the Bloch
wavefunctions into maximally localized Wannier functions (MLWFs) [30] derived
from the Eu-5d, Eu-6s, Eu-6p, Ga-4s, Ga-4p, Al-3s, and Al-3p orbitals. The
tight-binding model Hamiltonian was constructed from the MLWF overlap matrix.
Based on the tight-binding Hamiltonian, the surface state was considered under
open boundary conditions with the half-infinite two-dimensional model using
the iterative Green’s-function method [31, 32].

Figure 1:  Magnetization $M$ (red squares, left axis), resistivity $\rho_{xx}$
(gold triangles, inner right axis), and Hall resistivity $\rho_{yx}$ (blue
circles, outer right axis) as a function of magnetic field $\mu_{0}H$ measured
with magnetic field $H~{}\parallel~{}c$ at temperature $T~{}=~{}2.5~{}$K for
Eu(Ga1-xAlx)4 (a) $x$ = 0.15, (b) $x$ = 0.24, (c) $x$ = 0.31, (d) $x$ = 0.39,
(e) $x$ = 0.50, (f) $x$ = 0.58, (g) $x$ = 0.71 and (h) $x$ = 0.90. The
transport measurements are measured with the current $j~{}\parallel~{}a$.
Phase boundaries are marked with a dashed line.

##  III Results and Discussion

Hall effect measurements have proven a powerful tool to identify systems with
large sources of Berry curvature $\mathbf{\Omega_{k}}$ [33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 22, 47, 5, 48]. Systems with non-zero
$\mathbf{\Omega_{k}}$ have an additional term in the Hall conductivity
$\sigma_{xy}$ besides the normal metal component, and this is the anomalous
Hall conductivity $\sigma_{xy}^{A}$ that depends on $\mathbf{\Omega_{k}}$ as
$\sigma_{xy}^{A}~{}=~{}-(e^{2}/h)\int d^{3}k\Omega_{k}^{z}/(2\pi)^{3}$ [47].
Berry curvature can either be generated via the Karpus-Luttinger-type (KL)
mechanism [49], first discussed in the context of ferromagnets, or the scalar
spin chirality (SSC) [50, 51, 52] mechanism. In the former, spin-orbit
coupling (SOC) has always been emphasized as a key ingredient while the
derivation of the quantum Hall conductivity by Thouless et al. [53] and
subsequent interpretation of the non-quantized anomalous Hall conductivity by
Haldane [54] made clear the relationship between reciprocal-space topology and
the anomalous Hall effect. In the latter, when the SSC defined as
$\chi_{ijk}~{}=~{}\mathbf{S}_{i}\cdot(\mathbf{S}_{j}\times\mathbf{S}_{k})$
(where $\mathbf{S}_{i,j,k}$ are the spins of three adjacent sites in a
lattice) is non-zero, the real-space non-coplanar spin textures act as sources
of Berry curvature. In the literature the Hall response contributed from the
SSC-type mechanism is often referred to as the topological Hall effect. Since
topological spin textures are non-coplanar, they should therefore exhibit a
topological Hall effect if they are metallic [5, 38, 55].

###  III.1 Topological Hall Effect

Since both types of Hall effects are related to the magnetic properties,
field-dependent magnetization data $M(H)$ for Eu(Ga1-xAlx)4 (blue squares,
Fig. 1) are compared with resistivity measurements $\rho_{xx}$ (gold
triangles) and Hall resistivity $\rho_{yx}$ (blue circles), measured at
$T=2.5~{}K~{}<~{}T_{N}$, with $H\parallel c$, and current $j\parallel a$.
Signatures of field-induced spin reorientation are clearly observed by
anomalies in both $M$ and $\rho_{xx}$, which are marked by the vertical dashed
lines in Fig. 1. Qualitatively, the behavior of $\rho_{yx}$ below the fields
$\mu_{0}H_{c}$ where $M$ saturates can be grouped into two categories
depending on composition, as discussed below.

Figure 2:  Topological Hall effect for Eu(Ga1-xAlx)4with $x=~{}0.71$. (a)
Isothermal magnetization $M$, (b) longitudinal resistivity $\rho_{xx}$, and
(c) Hall resistivity $\rho_{yx}$ measured as a function of magnetic field
$\mu_{0}H$ with field $H~{}\parallel~{}c$ at temperatures
$2.5~{}\text{K}~{}\leq~{}T~{}\leq~{}20.5~{}$K. Transport measurements are
measured with current $j~{}\parallel~{}a$. (d) $\rho_{yx}$ at
$T~{}=~{}2.5~{}$K (blue circles) and $T~{}=~{}20.5~{}$K (yellow circles) with
fits to Eq. 4 (cyan lines) and Eq. 5 (pink lines). (e) and (f) The difference,
$\Delta\rho_{yx}$, between $\rho_{yx}$ and fits to Eq. 4 and Eq. 5,
respectively.

For $0.15\leq x\leq 0.39$ (Fig. 1a-d), $\rho_{yx}$ is concave down below
$\mu_{0}H_{c}$ with no obvious discontinuities even though there are clear
magnetic transition features in both $M$ and $\rho_{xx}$ curves. For $x=0.15$
(Fig. 1a), only one magnetic phase transition is observed, which is similar to
the behavior of the end-compound EuGa4, where the field induces a magnetic
transition from the AFM ground state to the fully SP state [56]. It is
therefore unlikely for this composition to host topological spin textures. For
$0.24\leq x\leq 0.39$ (Fig. 1b-d), the magnetic field drives the system
through multiple magnetic phases, which are clearly captured in $M$ and
$\rho_{xx}$, but not very evident in the $\rho_{yx}$ curves.

For $0.50\leq x\leq 0.90$ (Fig. 1e-h), $\rho_{yx}$ shows discontinuities
corresponding to some of the magnetic transitions that are also revealed in
$M$ and $\rho_{xx}$. The discontinuity in $\rho_{yx}$ suggests the existence
of the THE, which is observed in many well established skyrmion hosting
materials, such as MnSi [57, 46], Gd2PdSi3 [38], Gd3Ru4Al${}_{1}2$ [55], and
GdRu2Si2 [58]. Such a discontinuity is associated with the first-order nature
of the field-induced magnetic transition. Therefore it is concluded that the
Eu(Ga1-xAlx)4 compounds with 0.5 $\leq~{}x~{}\leq$ 0.90 are promising
candidates to host real-space topological spin textures. Next the THE for
these compositions is evaluated.

Figure 3:  (a) The Hall resistivity $\rho_{yx}$ measured with magnetic field
$H~{}\parallel~{}c$ and current $j~{}\parallel~{}a$ for fields
$0~{}\text{T}~{}\leq~{}\mu_{0}H~{}\leq~{}6~{}$T and temperatures
$2.5~{}\text{K}~{}\leq~{}T~{}\leq 20.5~{}$K for Eu(Ga1-xAlx)4 $x$ = 0.90.
(b)$\rho_{yx}$ at $T~{}=~{}2.5~{}$K (blue symbols) and $\rho_{yx}$ at
$T~{}=~{}20.5~{}$K (yellow symbols) scaled to the high-field regime of of the
$T~{}=~{}2.5~{}$K data. (c) The difference between the lower temperature
$\rho_{yx}$ and scaled $T~{}=~{}20.5~{}$K data, $\Delta\rho_{yx}$ for
$2.5~{}\text{K}~{}\leq~{}T~{}\leq 19.5~{}$K. Figure 4:  Magnetic field -
temperature $H~{}-~{}T$ phase diagrams with magnetic field $H~{}\parallel~{}c$
for Eu(Ga1-xAlx)4 (a) $x$ = 0.50, (b) $x$ = 0.58, (c) $x$ = 0.71 and (d) $x$ =
0.90. Blue squares are determined from maxima in d$M$/d$H$ while purple
hexagons are determined from d$(MT)$/d$T$. The color of the contour maps
correspond to the contribution to the topological Hall resistivity
$\Delta\rho_{yx}$.

The measured Hall resistivity $\rho_{yx}$ for a metallic magnet can have
multiple contributions:

$\rho_{yx}=R_{0}\mu_{0}H+\rho_{yx}^{A}+\Delta\rho_{yx}.$ (1)

Here, the first term $R_{0}\mu_{0}H$ is the normal Hall resistivity arising
from the Lorentz force the charge carriers experience as they traverse a
perpendicular magnetic field $\mu_{0}H$, where $R_{0}$ is the Hall
coefficient. When the system has one dominant carrier, the normal Hall effect
is typically linear in field.

The second term $\rho_{yx}^{A}$ is the anomalous Hall resistivity which scales
with $M$. $\rho_{yx}^{A}$ can also have multiple contributions; however,
depending on the magnitude of the conductivity
$\sigma_{xx}~{}=~{}\rho_{xx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$, one contribution
usually dominates. Empirically, for highly conductive systems i.e.
$\sigma_{xx}~{}>~{}10^{6}~{}(\Omega\text{cm})^{-1}$, the anomalous Hall
conductivity
$\sigma_{xy}^{A}~{}=~{}\rho_{yx}^{A}/(\rho_{xx}^{2}+\rho_{yx}^{2})$ is
dominated by skew-scattering and varies linearly with $\sigma_{xx}$ [47].
Assuming $\rho_{yx}~{}<<~{}\rho_{xx}$, $\rho_{yx}^{A}~{}=~{}S_{H}M\rho_{xx}$,
when skew scattering dominates, where $S_{H}$ is the anomalous Hall
coefficient.

The so-called intrinsic regime, where the AHE is dominated by the KL mechanism
[49], is empirically found to be in the range
$10^{4}~{}(\Omega\text{cm})^{-1}~{}<~{}\sigma_{xx}~{}<~{}10^{6}~{}(\Omega\text{cm})^{-1}$
[47]. In this regime, $\sigma_{xy}^{A}$ is roughly independent of
$\sigma_{xx}$, or $\sigma_{xy}^{A}~{}\sim~{}constant$ [47] and
$\rho_{yx}^{A}~{}=~{}S_{H}^{\prime}M\rho_{xx}^{2}$ [40]. In the bad metal
regime, $\sigma_{xx}~{}<~{}10^{4}~{}(\Omega\text{cm})^{-1}$,
$\sigma_{yx}^{A}~{}\propto~{}\sigma_{xx}^{n}$ where $n~{}\sim 1.6$ [47, 59].
All samples studied in this work display large conductivities
($\sigma_{xx}~{}>~{}10^{4}~{}(\Omega\text{cm})^{-1}$), therefore the remainder
of this manuscript is concerned with the highly conductive and intrinsic
regimes.

The third term in Eq. 1, $\Delta\rho_{yx}$, is the THE and is due to the non-
zero SSC generated by non-coplanar spin textures. Such a contribution has been
observed in both trivial non-coplanar spin textures [60, 61], as well as
topological spin textures [5, 38, 55, 46].

To parse out the different contributions to the Hall effect in Eu(Ga1-xAlx)4,
two different methods are used. The first method is valid when
$\rho_{yx}<<\rho_{xx}$ and assumes a single dominant carrier type. This method
is applicable where both topological Hall and anomalous Hall contributions to
$\rho_{yx}$ are expected, and when $\rho_{yx}(H>H_{c})$ is linear. Here, Eq. 1
can be written as

$\rho_{yx}=R_{0}\mu_{0}H+S_{H}\rho_{xx}M+\Delta\rho_{yx}$ (2)

assuming skew scattering as the dominant scattering mechanism, or

$\rho_{yx}=R_{0}\mu_{0}H+S_{H}^{\prime}\rho_{xx}^{2}M+\Delta\rho_{yx}$ (3)

assuming the intrinsic KL-type scattering mechanism [47].

For fields greater than $\mu_{0}H_{c}$, $\Delta\rho_{yx}$ is necessarily zero
since all spins are aligned in the SP state, and Eqs. 2 and 3 become

$\frac{\rho_{yx}}{\mu_{0}H}=R_{0}+S_{H}\frac{\rho_{xx}M}{\mu_{0}H}$ (4)

and

$\frac{\rho_{yx}}{\mu_{0}H}=R_{0}+S_{H}^{\prime}\frac{\rho_{xx}^{2}M}{\mu_{0}H},$
(5)

respectively. $R_{0}$ and $S_{H}$ or $S_{H}^{\prime}$ can thus be extracted as
the intercept and slope of the corresponding line generated by plotting
$\rho_{yx}/\mu_{0}H$ vs. $\rho_{xx}M/\mu_{0}H$ or vs.
$\rho_{xx}^{2}M/\mu_{0}H)$. $\Delta\rho_{yx}$ is then estimated below
$\mu_{0}H_{c}$ as the difference

$\Delta\rho_{yx}=\rho_{yx}-R_{0}\mu_{0}H-\rho_{yx}^{A}.$ (6)

This analysis is demonstrated in Fig. 2 for Eu(Ga1-xAlx)4 with $x~{}=~{}0.71$,
where the temperature dependence of $M(H)$, $\rho_{xx}(H)$ and $\rho_{yx}(H)$
are shown for $H~{}\parallel~{}c$ and $j~{}\parallel~{}a$ (panels a-c). Fits
of $\rho_{yx}(H)$ at $T~{}=~{}2.5~{}$K (blue) and $T~{}=~{}20.5~{}$K (yellow)
to Eq. 4 (cyan line) and Eq. 5 (pink line) are presented in Fig. 2d. For
$T~{}=~{}2.5~{}$K, both equations capture the behavior of $\rho_{yx}(H)$ for
$\mu_{0}H~{}>~{}\mu_{0}H_{c}$, while for $\mu_{0}H~{}<~{}\mu_{0}H_{c}$, there
are clear deviations of the measured $\rho_{yx}(H)$ compared to the fits,
indicating an additional contribution $\Delta\rho_{yx}$. When
$T~{}=~{}20.5~{}$K, both equations capture all features of $\rho_{yx}(H)$ such
that $\Delta\rho_{yx}~{}=~{}0$.

The temperature and field dependence of $\Delta\rho_{yx}$ obtained after
subtracting the the normal and anomalous contributions from the measured
$\rho_{yx}$ (Eq. 6) are shown in Fig. 2e,f evaluated from fits to Eq. 4 and 5,
respectively. Qualitatively, both methods for estimating $\Delta\rho_{yx}$
result in similar temperature and field dependencies. However, Eq. 5 does a
slightly better job of minimizing $\Delta\rho_{yx}$, consistent with Ref.
[26]. A similar analysis was done for Eu(Ga1-xAlx)4 for x = 0.50 and 0.58 (the
Supplementary Materials Fig. S1 and S2).

This method works for $x$ = 0.50, 0.58, and 0.71. However, a different method
needs to be used to evaluate the THE for $x=0.90$ because $\rho_{yx}$ above
$\mu_{0}H_{c}$ shows a nonlinear field dependence (Fig. 3) for this
composition, which requires a multi-band description for the normal Hall
effect, instead of a single band analysis.

To extract $\Delta\rho_{yx}$ for Eu(Ga1-xAlx)4 with $x$ = 0.90, we follow the
procedure done in Ref. [34]. Assuming there is only a weak temperature
dependence of $R_{0}$ and $\rho_{yx}^{A}$, the scaled $\rho_{yx}$ above
$T_{N}$ is used to extract the THE at low temperatures. Here, the
$T~{}=~{}20.5~{}$K (20.5 K$~{}>~{}T_{N}$) data is scaled to the lower
temperature data, such that the high field data collapse onto each other. Fig.
3b shows this analysis where the $T~{}=~{}20.5~{}$K $\rho_{yx}$ data (yellow)
has been scaled to the $T~{}=~{}2.5~{}$K $\rho_{yx}$ data (blue).
$\Delta\rho_{yx}$ shown in Fig. 3c for all measured $T$ is obtained by
subtracting the scaled high temperature from the lower temperature data,
showing a clear THE. Such treatment precludes the analysis of $\rho_{yx}^{A}$.

As a result of this analysis, a contour map of $\Delta\rho_{yx}$ is produced,
shown in Fig. 4(a-d) for Eu(Ga1-xAlx)4 with $x$ = 0.50, 0.58, 0.71 and 0.90.
The $H-T$ phase diagrams determined from isothermal magnetization (blue
squares) and temperature-dependent magnetic susceptibility measurements
(purple hexagons)[26, 20] are shown together with the contour plots .

Overall, non-zero $\Delta\rho_{yx}$ can be observed in certain phase region
for all four compositions. For $x=0.5$ (Fig. 4a), both the $H-T$ phase diagram
and $\Delta\rho_{yx}$ map are consistent with previous results [26]. Here, the
AFM1 phase for H = 0 was determined to be an incommensurate helical structure
with a magnetic propagation vector $q_{mag}$ along the crystallographic $a$
axis, and magnetic moments rotating in in the $ab$ plane. The ground state,
AFM3, was determined to be a cycloidal state again with $q_{mag}$ along the
crystollographic $a$ axis, but with moments rotating in the $bc$ plane. The
phase AFM2 that separates AFM1 and AFM3 on cooling was determined to have
mixed magnetic propagation vectors. With the application of magnetic field
$H\parallel c$, AFM3 and AFM1 are separated by the intermediate field phase (A
phase), where $\Delta\rho_{yx}$ is centered around, indicative of a
topological spin texture or a more generic non-coplanar spin texture.

For Eu(Ga1-xAlx)4with $x$ = 0.58, 0.71 and 0.90 (Fig. 4b-d), the maximum THE
appears in the regions I, II, and II, respectively. We focus on Eu(Ga1-xAlx)4
$x$= 0.9, the compound closest in composition to the confirmed skyrmion host
EuAl4 [21]. Despite the existence of the THE in both compounds (the THE in
EuAl4 was reported in Ref. [22]), the phase diagrams are strikingly different.
For comparison, the $H~{}-~{}T$ phase diagram ($H\parallel c$) determined by
magnetization measurements for EuAl4 is presented in Fig. 5 which is
consistent with previous reports [21]. EuAl4 possesses four different
magnetically ordered phases on zero-field cooling, labeled IV, VI, V and I in
Fig. 5. This is in contrast to Eu(Ga1-xAlx)4 with $x$ = 0.9, which only has
two phases below $T_{N}$, labeled I and III in Fig. 4d. Also, with the
application of magnetic field $H\parallel c$, the zero-field phases in EuAl4
(Fig. 5) are separated by two additional phases, phase II (rhombic skyrmion
lattice) and III (square skyrmion lattice) [21], compared to only one
intermediate field state in Eu(Ga1-xAlx)4 $x$= 0.9 (phase II in Fig. 4d) where
the THE reaches a maximum. The different $H-T$ phase diagrams reflect the
ability to fine-tune the magnetic interactions in the two compounds using
chemical substitution, possibly favoring one type of skyrmion lattice over the
other.

The Dzyaloshinskii-Moriya (DM) interaction [62, 63] has been emphasized as a
key ingredient in stabilizing topological spin textures in non-centrosymetric
crystals. However, the Eu(Ga1-xAlx)4 crystal structure is centrosymmetric and
therefore DM interactions should be absent [5]. Instead, recent theoretical
studies [64, 65, 66, 67] suggested that the interplay between the Ruderman-
Kittel-Kasuya-Yosida (RKKY) interaction [68, 69, 70] and the four-spin
interaction could stabilize topological spin-textures in metallic
centrosymmetric magnets. In these theories [64, 65, 66, 67], the RKKY
interaction stabilizes $q_{mag}$ set by the Fermi surface nesting vector,
while the four-spin interaction, which depends on the in-plane bond-dependent
anisotropy and easy axis anisotropy, increases the propensity towards
multi-$q$ order. Being able to tune these interactions in centrosymmetric
compounds offers the opportunity for tunable topological spin textures, which
is in contrast to their non-centrosymmetric counter parts, where the type of
topological spin texture is set by the crystalline symmetry [5]. Future
experimental and theoretical works on Eu(Ga1-xAlx)4 will elucidate the
influence of chemical substitution on these tuning parameters, which may, in
turn, enable targeted engineering of topological spin textures.

###  III.2 Anomalous Hall effect (AHE)

Figure 5:  Magnetic field - temperature phase diagram of EuAl4 (Eu(Ga1-xAlx)4
$x$ = 1) for $H\parallel c$. The phase boundaries are determined by maxima in
d$M$/d$H$ (blue squares) and d$(MT)$/d$T$ (purple hexagons).  Figure 6:
Magnetic field dependence of the Hall resistivity $\rho_{yx}$ (blue symbols)
for Eu(Ga1-xAlx)4 ($x$ = 0.15, 0.24, 0.31, 0.39, 0.50, 0.58, 0.71, 0.90)
measured at $T=2.5$ K with $H\parallel c$ and $j\parallel a$ for
$0~{}\text{T}\leq\mu_{0}H\leq 14~{}\text{T}$. Data are offset such that
$\rho_{yx}(H=0)=0$. Pink lines are fits to Eq. 5 in the field-polarized state
up to $\mu_{0}H=9$ T, the highest field we are able to measure magnetization.
The grey lines are linear extrapolations of the pink lines. Figure 7:  The
anomalous Hall conductivity $\sigma_{xy}^{A}$ plotted as a function of
longitudinal conductivity $\sigma_{xx}$ for Eu(Ga1-xAlx)4 $x$ = 0.24, 0.31,
0.39, 0.50, 0.58 and 0.71 extracted from the field polarized regime
($\mu_{0}H=9$T) at temperature $T~{}=~{}2.5~{}$K with fits to Eq. 5 (large,
closed symbols) compared to other metallic magnets (small open symbols) taken
from Ref. [71, 35, 72, 36, 73, 34]. Figure 8:  The electronic structure
calculated along certain high symmetry lines for Eu(Ga1-xAlx)4, $x$ = 0.5,
corresponding to the ordered structure EuGa2Al2, in the paramagnetic state.
The yellow and magenta lines mark out the bands that form a Dirac point. The
red dashed line indicates the Fermi level determined by the ARPES experiments
[74].

We now turn to the AHE in the SP state above $\mu_{0}H_{c}$ (where $M$ is
fully saturated at $T=2.5$ K). Extended measurements of $\rho_{yx}(H)$,
$\rho_{xx}(H)$, and $M(H)$ up to $\mu_{0}H$ = 9 T are shown in Supplementary
Materials Fig. S3. Eq. 5 is used to extract the anomalous Hall resistivity as
$\rho_{yx}^{A}=S_{H}^{\prime}\frac{\rho_{xx}^{2}M}{\mu_{0}H}$. Such an
analysis is valid only for the single-band Hall effect so the analysis is
restricted to Eu(Ga1-xAlx)4 with $0.24~{}\leq~{}~{}x~{}\leq~{}0.71$. Fits to
the $T=2.5$ K data are shown in Fig. 6 as pink lines for
$\mu_{0}H_{c}\leq\mu_{0}H\leq 9$ T. Next, $\rho_{yx}^{A}$ is converted to
anomalous Hall conductivity $\sigma_{xy}^{A}$ using the tensor relation
$\sigma_{xy}^{A}~{}=~{}\rho_{yx}^{A}/(\rho_{xx}^{2}+\rho_{yx}^{2})$. In Fig. 7
$\sigma_{xy}^{A}(\mu_{0}H=9\text{T},T=2.5\text{K})$ is plotted for each
composition compared to $\sigma_{xx}(\mu_{0}H=0~{}\text{T},T=2.5~{}\text{K})$,
which serves as a measure of disorder and naturally varies with $x$.

For $x=0.24,~{}0.31,~{}0.39,\text{and}~{}0.71$, $\sigma_{xy}^{A}$ is nearly
independent of $\sigma_{xx}$, consistent with the intrinsic KL mechanism. The
resulting $\sigma_{xy}^{A}~{}=~{}600-1000~{}(\Omega~{}\text{cm})^{-1}$, is
also comparable with the theoretical limit of $10^{2}\sim
10^{3}~{}(\Omega\text{cm})^{-1}$ when $\sigma_{xy}^{A}$ is generated by
reciprocal-space Berry curvature [39]. The compounds with $x$ = 0.50 and 0.58
are much more conductive, entering the regime where $\sigma_{xy}^{A}$ is
dominated by the skew-scattering mechanism. For these compositions,
$\sigma_{xy}^{A}$ increases with increasing $\sigma_{xx}$, which appears to be
in line with the skew scattering mechanism. However, the Hall angle
$\Theta_{H}~{}=~{}\tan^{-1}(\sigma_{xy}^{A}/\sigma_{xx})$ of typical metals
where skew scattering dominates is usually less than 1% [39]. $\Theta_{H}$
measured for Eu(Ga1-xAlx)4 with x = 0.50 is $\sim$ 7% indicating that another
mechanism (possibly intrinsic KL) is also contributing to the large
$\sigma_{xy}^{A}$, a consequence of reciprocal space Berry curvature.

First-principles calculations are used to further study the topological
electronic structure, which is the origin of the reciprocal-space Berry
curvature for the Eu(Ga1-xAlx)4 series. The band structure of EuGa2Al2 in the
paramagnetic state obtained by density functional theory (DFT) calculations is
presented in Fig 8, which shows a Dirac point along the $\Gamma-Z$ path. The
Dirac point is protected by the four-fold rotational symmetry with respect to
the $z$ axis in this tetragonal system: the two bands highlighted with magenta
and yellow lines have different irreducible representations of the $C_{4v}$
point group and cross each other along $\Gamma-Z$ without opening a gap when
spin-orbit coupling is considered. Note that this type of Dirac crossing is
sometimes also called semi-Dirac or anisotropic Dirac crossing because the
band dispersion is linear along one axis ($k_{z}$), while being quadratic
along other axes ($k_{x}$ and $k_{y}$). Such a crossing has been discussed in
the study on the non-magnetic isostructural analogue BaAl4 [75]. However, the
Dirac crossing is located $\sim$0.4 eV above the Fermi level in BaAl4.

The existence of the Dirac point in EuGa2Al2 is verified by ARPES measurements
[74]. In the Brillouin zone (BZ) $Z$ plane, there is a hole band (yellow
dashed line) and an electron band (magenta dashed line) which intersect each
other around the BZ center. The ARPES measurements [74] agree well with the
DFT calculations and confirm the existence of the Dirac point along the
$\Gamma-Z$ path. More importantly, the Dirac point in EuGa2Al2 is found to be
located close to the Fermi level by angle-resolved photomession spectroscopy
measurements [74], in contrast to the case of BaAl4 [75]. In the SP state, the
spin-degeneracy of the bands will be lifted, which could lead to the formation
of Weyl points. The mechanism is similar to that of GdPtBi [76, 77, 78], where
the gapless quadratic band touching in the electronic structure sets the stage
for the field induced Weyl physics and the AHE.

Finally, it is noted that the energy of the Dirac point in Eu(Ga1-xAlx)4
series may slightly vary as Al/Ga ratio changes. In EuGa4, for example, the
Dirac point was determined to be above the Fermi level [18]. The origin is
likely related to the difference in electronegativity between Al and Ga.
Consequently, a change in Al/Ga ratio in Eu(Ga1-xAlx)4 also provides an
opportunity to change the intrinsic anomalous Hall conductivity.

##  IV Conclusions

In conclusion, magnetotransport measurements on Eu(Ga1-xAlx)4 single crystals
with $0.15~{}\leq~{}x~{}\leq~{}0.9$ reveal evidence for a THE for
$x~{}\geq~{}0.5$, pointing to the existence of non-coplanar spin textures. For
intermediate composition $0.24~{}<~{}x~{}<~{}0.39$, the Hall resistivity
varies smoothly with magnetic field, although multiple magnetic ordered phases
are clearly revealed in the isothermal magnetization and longitudinal
resistivity measurements. Our measurements suggest a weak THE contribution, if
any of the phase regimes host topological spin textures. At the lowest doping
$x$ = 0.15, only one magnetic ordered phase can be observed before reaching
the SP state, similar to the end compound EuGa4. Therefore, no topological
spin texture is expected. In the SP state, evidence for a large AHE in
Eu(Ga1-xAlx)4 is found, which is attributed to the appearance of Weyl nodes,
generated after spin-splitting of a Dirac point. The existence of the Dirac
point near the Fermi level in the paramagnetic state in EuGa2Al2 is confirmed
by DFT calculations which has been verified by ARPES measurements. Therefore
Eu(Ga1-xAlx)4 for $x~{}\geq~{}0.5$ is established as a rare platform to study
the field-tunable phenomena associated with reciprocal- and real-space
topology.

###### Acknowledgements.

YG, SL and EM acknowledge partial support from AFOSR Grant no.
FA9550-21-1-0343. JMM has been supported by the NSF Graduate Research
Fellowship Program under Grant no. DGE-1842494 and National Science Foundation
(NSF) DMR Grant no. 1903741. KA and EM have been partially supported by the
Robert A. Welch Foundation grant C-2114. The use of the EPMA facility at the
Department of Earth Science, Rice University, Houston, Texas, is kindly
acknowledged. The authors thank Gelu Costin for help with EPMA measurements.

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Supplemental Materials:
Signatures of real-space and reciprocal-space topology in the Eu(Ga1-xAlx)4
system

Figure S1:  (a) Isothermal magnetization $M$, (b) longitudinal resistivity
$\rho_{xx}$, and (c) Hall resistivity $\rho_{yx}$ measured as a function of
magnetic field $\mu_{0}H$ with field $H~{}\parallel~{}c$ at temperatures
$2.5~{}\text{K}~{}\leq~{}T~{}\leq~{}20.5~{}$K for Eu(Ga1-xAlx)4 x = 0.50.
Transport measurements are measured with current $j~{}\parallel~{}a$. (d)
$\rho_{yx}$ at $T~{}=~{}2.5~{}$K (blue symbols) and $T~{}=~{}20.5~{}$K
(yellow) with fits to Eq. 4 (cyan lines) and Eq. 5 (pink lines). The
difference between $\rho_{yx}$ and fits to (e) Eq. 4 or (f) Eq. 5 resulting in
$\Delta\rho_{yx}$ for $2.5~{}\text{K}~{}\leq~{}T~{}\leq~{}20.5~{}$K. Figure
S2:  (a) Isothermal magnetization $M$, (b) longitudinal resistivity
$\rho_{xx}$, and (c) Hall resistivity $\rho_{yx}$ measured as a function of
magnetic field $\mu_{0}H$ with field $H~{}\parallel~{}c$ at temperatures
$2.5~{}\text{K}~{}\leq~{}T~{}\leq~{}20.5~{}$K for Eu(Ga1-xAlx)4 x = 0.58.
Transport measurements are measured with current $j~{}\parallel~{}a$. (d)
$\rho_{yx}$ at $T~{}=~{}2.5~{}$K (blue symbols) and $T~{}=~{}20.5~{}$K
(yellow) with fits to Eq. 4 (cyan lines) and Eq. 5 (pink lines). The
difference between $\rho_{yx}$ and fits to (e) Eq. 4 or (f) Eq. 5 resulting in
$\Delta\rho_{yx}$ for $2.5~{}\text{K}~{}\leq~{}T~{}\leq~{}20.5~{}$K. Figure
S3:  Magnetization $M$ (red squares, left axis), resistivity $\rho_{xx}$ (gold
triangles, inner right axis), and Hall resitivity $\rho_{yx}$ (blue circles,
outer right axis) as a function of magnetic field $\mu_{0}H$ measured with
magnetic field $H~{}\parallel~{}c$ up to 9 T at temperature $T~{}=~{}2.5~{}$K
for Eu(Ga1-xAlx)4 (a) $x$ = 0.15, (b) $x$ = 0.24, (c) $x$ = 0.31, (d) $x$ =
0.39, (e) $x$ = 0.50, (f) $x$ = 0.58, (g) $x$ = 0.71 and (h) $x$ = 0.90. The
transport measurements are measured with the current $j~{}\parallel~{}a$.