z

# Solar-wind electron precipitation on weakly magnetized bodies:
the planet Mercury

Federico Lavorenti<EMAIL_ADDRESS>Laboratoire Lagrange,
Observatoire de la Côte d’Azur, UCA, CNRS, Nice, France  Dipartimento di
Fisica “E. Fermi”, Università di Pisa, Pisa, Italy  Pierre Henri  Laboratoire
Lagrange, Observatoire de la Côte d’Azur, UCA, CNRS, Nice, France  LPC2E,
CNRS, Univ. d’Orléans, OSUC, CNES, Orléans, France  Francesco Califano
Dipartimento di Fisica “E. Fermi”, Università di Pisa, Pisa, Italy  Jan Deca
LASP, University of Colorado Boulder, Boulder, CO, USA  Institute for Modeling
Plasma, Atmospheres and Cosmic Dust, NASA/SSERVI, Silicon Valley, CA, USA
LATMOS, Université de Versailles à Saint Quentin, Guyancourt, France  Simon
Lindsay  School of Physics & Astronomy, University of Leicester, Leicester, UK
Sae Aizawa  IRAP, CNRS-CNES-UPS, Toulouse, France  Dipartimento di Fisica ”E.
Fermi”, Università di Pisa, Pisa, Italy  Japan Aerospace Exploration Agency,
Sagamihara, Kanagawa, Japan  Johannes Benkhoff  ESA/ESTEC, Keplerlaan 1, 2200
AG Noordwijk, The Netherlands

###### Abstract

Mercury is the archetype of a weakly magnetized, airless, telluric body
immersed in the solar wind. Due to the lack of any substantial atmosphere, the
solar wind directly precipitates on Mercury’s surface. Using a 3D fully-
kinetic self-consistent plasma model, we show for the first time that solar-
wind electron precipitation drives (i) efficient ionization of multiple
species (H, He, O and Mn) in Mercury’s neutral exosphere and (ii) emission of
X-rays from the planet’s surface. This is the first, independent evidence of
X-ray auroras on Mercury using a numerical approach.

††preprint: APS/123-QED Box dimensions | $(L_{x},L_{y},L_{z})$ | (-9:+6,$\pm$6,$\pm$6) $R$
---|---|---
Number of cells | $(N_{x},N_{y},N_{z})$ | (960,768,768)
Spatial resolution | $(dx,dy,dz)$ | 0.015 $R$
Time resolution | $dt$ | 1.4 ms
Macro-particles per cell111Total number of macro-particles, ions plus electrons. | PPC | 128
Solar-wind number flux | $\mathcal{F}_{{}_{SW}}$ |  $1.2\cdot 10^{9}$ 1/cm2s
Solar-wind energy flux | $\mathcal{E}_{{}_{SW}}$ |  $2.4\cdot 10^{10}$ eV/cm2s
Ion-to-electron mass ratio | $m_{i}/m_{e}$ | 100
Light-to-Alfvén speed ratio | $c/V_{A,i}$ | 178
Planet-to-gyro radius ratio | $R/\rho_{i}$ | 10
Table 1: Common numerical parameters of RunN and RunS with purely northward
and southwards IMF, respectively.

The planet Mercury is our neighbor example of a rocky, weakly magnetized body
immersed in the solar-wind plasma. Mercury’s environment presents us the ideal
scenario to better understand the physics governing the interaction between
rocky bodies (such as planets, moons, asteroids and comets) and the solar
wind. Mercury’s complexity is due to the strong coupling between the solar-
wind plasma and the planet’s magnetosphere, exosphere and surface. The
physical processes controlling the electron interactions in the system are a
scientific enigma with underlying roots ranging from plasma to solid-state
physics [27]. As an example, electron acceleration in the magnetosphere is
thought to be at the origin of X-ray aurora-like emission from the surface of
Mercury. However, on the one hand, this hypothesis remains to be confirmed; on
the other hand, the dependence of such process on solar-wind conditions
remains unknown.

The Sun acts as an external energy driver, sustaining the dynamics at Mercury.
Solar radiation and particles are the source (via sputtering and desorption)
and sink (via radiation pressure and ionization) of the neutral exosphere
surrounding Mercury. Moreover, the shape and dynamics of Mercury’s magnetic
field is strongly altered by the solar wind. Mercury’s intrinsic magnetic
field generates a scaled-down, Earth-like magnetosphere able to – partially –
shield the planet’s surface from the impinging solar wind. Part of the solar
wind enters the magnetosphere, interacts with Mercury’s magnetic field and,
given the absence of an atmosphere, precipitates down to the planet’s surface.
The study of such plasma precipitation is a key point to understand the strong
coupling between Mercury’s magnetosphere, exosphere and surface, and around
weakly magnetized bodies in general.

To date, only two missions (Mariner 10 and MESSENGER) have been devoted to the
exploration of Mercury’s environment. Mariner 10 provided the first electron
observations showing electron fluxes in the range $\sim$20-600 eV throughout
the magnetosphere [29]. Sporadic bursts above tens of keV were also detected
[38, and references therein]. MESSENGER [33] could not measure the core of the
electron distribution function, but it provided direct observations of high-
energy electrons above 35 keV [11, 12] and indirect observations of
suprathermal electrons in the range $\sim$1-10 keV [21, 2, 7, 13]. MESSENGER
also reported the first evidence of electron-induced X-ray emissions from the
surface [22, 23]. These missions provided a novel – but still fragmented –
picture of the solar-wind electron interaction with Mercury and weakly
magnetized bodies, in general. In the near future, the joint ESA/JAXA space
mission BepiColombo [3] will revolutionize our understanding of Mercury’s
environment thanks to (i) its two-satellite composition and (ii) its
instrumental payload with resolution down to electron kinetic scales [28].
BepiColombo will observe the whole electron [31, 15] and X-ray spectrum [5]
with unprecedented resolution. However, the complexity of these measurements
is such that only through global models, including the electron dynamics, the
true potential of these measurements can be unveiled. Our work presents such a
model to interpret the fragmented picture left by past MESSENGER observations,
while paving the way for the future BepiColombo ones.

In the past, global numerical models of Mercury’s interaction with the solar
wind have been focused on the ion dynamics [17, 36, 30, 9]. Such models
included self-consistently the ion kinetic physics, but neglected the kinetic
physics of electrons (treated as a massless neutralizing fluid). Those models
neglect electron acceleration processes and so their feedback on the
magnetosphere and surface. Previous studies that did model electron
trajectories, then, prescribed constant electromagnetic fields (i.e. a test-
particle approach [32, 37]). Such a method completely neglects the kinetic
electron dynamics and their feedback on the large-scale global evolution.
Schriver et al. [32] provided a first estimate of electron precipitation maps
at Mercury, but their results did not address the energy distribution of
electrons at the surface (due to the small statistical sample of test
electrons). To overcome these limitations, in this work we study the
precipitation of electrons on Mercury-like bodies using a global fully-kinetic
model. Our model includes the electron dynamics self-consistently from the
large, planetary scale down to the electron gyro-radius.

We assess electron precipitation at the surface of Mercury under northward and
southward solar-wind conditions. Our numerical results are then used to
compute (i) the electron impact ionization rates in Mercury’s low altitude
exosphere, and (ii) the X-ray photon emission profiles from Mercury’s surface.
This novel, self-consistent approach provides (i) the first estimate of the
efficiency of electron impact ionization – a process usually neglected in
exosphere models – for multiple exospheric species, and (ii) the first
estimate of X-ray luminosity of a rocky, weakly magnetized body driven by
solar-wind electron precipitation.

Figure 1: Electron precipitation maps from our fully-kinetic simulations using
a purely northward (top panels) or southward (bottom panels) IMF. Panels a-b:
number fluxes in units of electrons/cm2s. Panels c-d: energy fluxes in units
of eV/cm2s. All maps use a Mollweide projection of the surface of Mercury.
Vertical axis corresponds to geographical latitude. Horizontal axis
corresponds to local time (LT, as indicated in panel a), where LT 12 is the
subsolar longitude. Dashed black lines indicate the boundary between open and
closed magnetic field lines.

Figure 2: Electron impact ionization (EII) rates computed in our two
simulations. Panel a: surface maps of EII rates for selected exospheric
species (H, He, O and Mn). Maps in the same format as Fig 1. Panel b: box-plot
of the distribution of EII rates for all exospheric species considered. EII
rates are normalized to photoionization rates. Horizontal green lines shows
the median, limits of the boxes are the first and third quartile, the whiskers
are the maxima of the data. In panel b, EII rates obtained from the two runs
(RunN and RunS) are merged. The red box in panel b highlights the region of
non-negligible EII (i.e. where the EII-to-photoionization frequency ratio is
between 0.1 and 1).

We use the semi-implicit, particle-in-cell code iPIC3D that solves the Vlasov-
Maxwell system of equations in a three-dimensional Cartesian box by
discretizing the ion and electron distribution function [25]. The simulation
setup includes (i) a uniform solar-wind plasma injected from the sunward side
of the box and (ii) a scaled-down model of the planet Mercury with radius $R$
= 230 km (radius reduced by around a factor 10 from its real value) and
magnetic dipole moment 200 nT$/R^{3}$. The dipole field is shifted northward
by 0.2 $R$ in agreement with the MESSENGER magnetic field observations [1]. A
scaled-down setup enables us to run a global fully-kinetic simulation on
present state-of-the-art computing facilities. Scaling-down the planet, but
keeping the good ordering of physical spatial and temporal scales, preserves
the global magnetosphere structure and dynamics [20]. At the box boundaries,
the outermost cells are populated with solar-wind plasma and the
electromagnetic fields are linearly smoothed to their solar-wind values.
Inside the planet, macro-particles are removed using a charge-balanced scheme
[20]. We initialize the simulations with a solar-wind density $n_{{}_{SW}}$ =
30 cm-3, speed $V_{{}_{SW}}$ = 400 km/s in the sun-planet direction, a
magnetic field amplitude $B_{{}_{SW}}$ = 20 nT and temperature $T_{i,_{SW}}$ =
$T_{e,_{SW}}$ = 21.5 eV. Two different simulation setups, with purely
northward (RunN) or southward (RunS) interplanetary magnetic field (IMF) are
used. These two limit cases target the role of magnetic reconnection in the
magnetosphere (inhibited in RunN and enhanced in RunS). This simulation setup
has been validated in Lavorenti et al. [20] studying the large-scale structure
of Mercury’s magnetosphere. The numerical parameters employed in our two runs
are reported in Tab. 1.

In both simulations, the global system reaches a quasi-steady state after a
time $T\approx$ 10 s. Starting from this time $T$, we integrate the plasma
precipitation on the planet’s surface for a time interval $\Delta t$ = 50 ms,
corresponding to about two electron gyro-periods. From the precipitated
plasma, we compute the electron precipitation maps, as shown in Fig. 1
222Solar-wind ion precipitation maps in the same format as Fig. 1 can be found
in the Supplementary Materials.. The precipitation maps are obtained using
$\sim 10^{6}$ macro-particles, enabling a good representation of the electron
distribution function at the surface.

The electron precipitation maps in Fig. 1 show significant spatial
inhomogeneities for both explored IMF configurations. In the case of northward
IMF (RunN), the solar-wind electrons are (i) energized up to energies of a few
keV and (ii) concentrated onto the northern and southern cusps. In RunN, the
northern cusp extends down to latitude $\sim+60^{\circ}$ while the southern
cusp goes up to $\sim-30^{\circ}$. Energy fluxes in the cusps reach values of
$\sim 10^{12}$ eV/cm2s, two orders of magnitude higher than in the pristine
solar-wind (Tab. 1). In the case of southward IMF (RunS), high-energy
electrons (up to few keV) precipitate at low latitudes close to the magnetic
equator (from -50∘ to +60∘) and mainly at the nightside. This energy flux is
higher in the dawn side (LT 0-6 h) as compared to the dusk side (LT 18-24 h).
Such dawn-dusk asymmetry is a consequence of the dawnward drift of electrons
injected from the neutral line in the tail towards the planet [6, 8, 20]. Such
dawnward enhancements can also be due to a dawn-dusk asymmetry in the plasma
sheet thickness [34, Chap. 4 therein]. In RunS, low-energy electrons (around
tens of eV) precipitate around the northern and southern poles. These
electrons precipitate directly from the solar wind onto the surface without
crossing the reconnection region.

We find that 1.5 times more electrons precipitate on the surface in RunS as
compared to RunN. The rate of precipitating electrons is $1.7\cdot 10^{25}$
s-1 ($2.6\cdot 10^{25}$ s-1) in RunN (RunS), corresponding to an effective
area of 2% (3%) of the total planet surface area. The rates, fluxes and
energies reported here are in agreement with the findings of Schriver et al.
[32]. For unmagnetized bodies (such as Mars, Venus, comets or the Moon), the
effective area is 50% of the planet surface area. In those cases,
precipitation is much higher as compared to Mercury, but the solar wind does
not suffer acceleration in the magnetosphere [18]. A weak magnetic field, like
the one of Mercury, thus (i) filters the solar-wind in precise regions of the
surface and (ii) accelerates the incoming particles by around a factor 100 in
energy. Both effects (filtering and acceleration) are not possible around
unmagnetized objects.

Figure 3: X-ray emission from the surface of Mercury due to electron
precipitation. Panel a: maps showing the X-ray photon flux emitted from the
surface of Mercury in our two runs. Maps in same format as Fig. 1-2a. Panel b:
X-ray photon flux integrated over surface area as a function of photon energy.
Contributions from different surface atoms are shown. To better visualize the
lines, we use a gaussian line profile with width $\sigma=5$ eV.

Before hitting the rocky surface of the planet, electrons interact with the
exosphere. We address the efficiency of electron impact ionization (hereafter
EII) of multiple exosphere species, using the electron energy distribution
from our simulations, because we want to quantitatively estimate the
efficiency of EII (usually considered a secondary, negligible effect) in
comparison to photoionization (that is the primary ionization process). For
details on the method used, see the Appendix. The distribution of EII rates
over the planet’s surface is shown in Fig. 2a for hydrogen (H), helium (He),
oxygen (O) and manganese (Mn) for both simulation runs 333A more complete set
of maps for all species considered can be found in the Supplementary
Materials.. These four species have the highest EII-to-photoionization
frequency ratio, as shown in Fig. 2b (photoionization rates $\nu_{ph}$ are
taken from Huebner and Mukherjee [14] for quiet sun conditions and rescaled to
Mercury’s aphelion). For H, He, O and Mn, EII is relevant (i) in the dayside,
where locally $\nu_{EII}\approx\nu_{ph}$ and (ii) in the nightside, where
ionization of neutrals is dominated by EII with typical rates of $\sim$ 0.1
$\nu_{ph}$. The main difference between photoionization and EII remains the
strongly inhomogeneous and time-dependent character of the latter. Variations
in the IMF direction, such as moving from northward to southward IMF, induce
strong variations in EII rates locally at the surface. We also show that EII
of sodium (Na), magnesium (Mg), aluminum (Al), silicon (Si), potassium (K) and
calcium (Ca) is negligible, as shown in Fig. 2b. This result support the
common assumption of negligible EII for the Na exosphere [35, 16].
Nevertheless, compared to photoionization, EII should not be considered a
secondary process for the H, He, O and Mn exosphere.

When hitting the surface, electrons induce the emission of photons from the
surface atoms via X-ray fluorescence (hereafter XRF). XRF is driven by
electrons with energies above a few hundreds eV [5, Tab. 5 therein]. From the
electron energy distribution at the surface obtained from our simulations, we
compute the flux of emitted X-rays, as shown in Fig. 3a. We include XRF
emission from surface O, Na, Mg, Al, Si, S, Ca and Fe. In Fig. 3a the photon
flux is integrated over these species. Fig. 3b, instead, shows the relative
emission from each surface atoms as a function of photon energy, integrated
over the planet surface. The spatial distribution of surface atoms is assumed
uniform, with density $n_{s}$. The details on the method is described in the
Appendix. We find that auroral X-ray emissions from the surface of Mercury
present strong spatial inhomogeneities dependent upon the upstream IMF
conditions. Regions of strongest X-ray emission correspond to regions of high-
energy electron precipitation, namely, the poles (in the case of northward
IMF) and the low latitude dawn-midnight sector (in the case of southward IMF).
In these regions, the emitted X-ray flux reaches values of the order of 107
photons/cm2s (mostly coming from the O-K$\alpha$ line). MESSENGER/XRS
instrument was able to measure X-rays from Si-K$\alpha$ and Ca-K$\alpha$ lines
at Mercury’s nightside [22], and partially at the dayside [23] during periods
of low solar activity. Our results corroborate the idea that Mercury’s X-ray
auroras are due to high-energy electron precipitation at the surface.
Moreover, our modeled X-ray flux provides a new mean to interpret and plan
future in-situ observations by the BepiColombo/MIXS instrument [5]. The
integrated X-ray luminosity of Mercury from electron-induced XRF is
$\mathcal{L}\approx 10^{26}$ photons/s. Such luminosity is comparable (or even
higher) to that of other solar system bodies shining in X-rays [4], such as
comets, Jupiter and Saturn. Nonetheless, remote observations from Earth of
Mercury’s X-ray aurora remains challenging due to the strong background of
X-ray photons coming from the Sun.

To conclude, using a novel 3D fully-kinetic approach, we investigated the
properties of solar-wind electron precipitation on the surface of Mercury-like
bodies. The magnetosphere of those bodies acts as (i) a shield, allowing only
few percents of the solar wind to precipitate onto the surface, and (ii) an
accelerator, increasing electron energies by a factor $\sim$100\. Using such
precipitation pattern as the input to exosphere and surface impact processes,
we find two main results. First, electron impact ionization of exospheric H,
He, O and Mn is efficient, with rates comparable to photoionization. This is
particularly important on the nightside where photoionization is inhibited.
The rates provided here are crucial to complement fluid and hybrid models of
Mercury’s environment, that cannot model electron acceleration processes.
Second, electron accelerated in the magnetosphere induce X-ray emission from
the surface with fluxes of the order of $10^{7}$ photons/cm2/s. This result
corroborate and provide the physical origin of the X-ray auroras observations
by MESSENGER/XRS instrument; it also paves the way for the future planning of
BepiColombo/MIXS observations.

###### Acknowledgements.

This work was granted access to HPC resources at TGCC under the allocation
A0100412428 and A0120412428 made by GENCI via the DARI procedure. We
acknowledge the CINECA award under the ISCRA initiative, for the availability
of high performance computing resources and support for the project IsC93. We
acknowledge the support of CNES for the BepiColombo mission. Part of this work
was inspired by discussions within International Team 525: “Modelling
Mercury’s Dynamic Magnetosphere in Anticipation of BepiColombo” at the
International Space Science Institute, Bern, Switzerland. We acknowledge
support by ESA within the PhD project “Global modelling of Mercury’s outer
environment to prepare BepiColombo”. JD gratefully acknowledges support from
NASA’s Solar System Exploration Research Virtual Institute (SSERVI): Institute
for Modeling Plasmas, Atmosphere, and Cosmic Dust (IMPACT), the NASA High-End
Computing (HEC) Program through the NASA Advanced Supercomputing (NAS)
Division at Ames Research Center, and NASA’s Rosetta Data Analysis Program,
Grant No. 80NSSC19K1305.

*

##  Appendix A Methods to compute EII rates and XRF emissions

To compute electron impact ionization (EII) rates and X-ray fluorescence (XRF)
emissions, we employ the electron energy distribution at the surface computed
from our fully-kinetic simulations. We also use the cross sections from Kim,
Y.-K. et al. [19], Zatsarinny et al. [39], Golyatina and Maiorov [10] for EII,
and from Llovet et al. [24] for XRF. For a given process (EII or XRF with a
given atomic species), the rate of interaction is:

$\nu_{X}(\phi,\theta)=\int^{\infty}_{0}f_{e}(\phi,\theta,E)\sigma_{X}(E)\sqrt{\frac{2E}{m_{e}}}dE$
(1)

where $f_{e}$ is the distribution of electrons at the surface as a function of
longitude $\phi$, latitude $\theta$ and energy $E$ and $\sigma_{X}$ is the
cross section of the process under consideration. Eq. 1 is used to compute the
EII rates in Fig. 2. We use the rate $\nu_{EII}$ to estimate the efficiency of
EII with a given exosphere species because (i) it is independent on the
density of target atoms and (ii) it is directly comparable to the
photoionization rate $\nu_{ph}$, a value commonly used and well-known for
multiple atomic species, e.g. Huebner and Mukherjee [14].

To compute the X-ray flux emitted via particle-induced XRF from the surface,
information on surface density and composition are needed. We employ the
Mercury’s geochemical surface composition of McCoy et al. [26, Tab. 7.1]
derived from MESSENGER observations. Using a mean surface mass density
$n_{mean}=3~{}\text{g/cm}^{3}$, we obtain the following densities for Si
($5~{}10^{22}\text{cm}^{-3}$), O ($4.5~{}10^{22}\text{cm}^{-3}$), Mg
($2.5~{}10^{22}\text{cm}^{-3}$), Al ($1.3~{}10^{22}\text{cm}^{-3}$), Na
($6.5~{}10^{21}\text{cm}^{-3}$), Ca ($4.6~{}10^{21}\text{cm}^{-3}$), S
($2.3~{}10^{21}\text{cm}^{-3}$) and Fe ($5~{}10^{20}\text{cm}^{-3}$). We also
assume an electron penetration depth of $\delta$ = 1 $\mu$m. The total X-ray
photon flux is:

$\mathcal{F}_{XRF}(\phi,\theta)=\delta\sum_{s}\nu_{{}_{XRF},s}(\phi,\theta)n_{s}$
(2)

where $n_{s}$ is the density of a given surface species. The species-
dependent, spatially integrated X-ray photon flux is:

$F_{XRF,s}=\delta\iint\nu_{{}_{XRF},s}(\phi,\theta)n_{s}\sin(\theta)d\theta
d\phi$ (3)

Eq. 2 is used to compute the flux in Fig. 3a. Eq. 3 is used to compute the
flux in Fig. 3b.

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# Phase transition dynamics in the three-dimensional field-free $\pm$J Ising
model

Ozan S. Sarıyer<EMAIL_ADDRESS>Pîrî Reis University, School of
Arts and Sciences, Istanbul 34940 Turkey

###### Abstract

By using frustration-preserving hard-spin mean-field theory, we investigated
the phase transition dynamics in the three-dimensional field-free $\pm J$
Ising spin glass model. As the temperature $T$ is decreased from paramagnetic
phase at high temperatures, with a rate $\omega=-dT/dt$ in time $t$, the
critical temperature depends on the cooling rate through a clear power-law
$\omega^{a}$. With increasing antiferromagnetic bond fraction $p$, the
exponent $a$ increases for the transition into the ferromagnetic case for
$p<p_{\text{c}}$, and decreases for the transition into the spin glass phase
for $p>p_{\text{c}}$, signaling the ferromagnetic-spin glass phase transition
at $p_{\text{c}}\approx 0.22$. The relaxation time is also investigated, at
adiabatic case $\omega=0$, and it is found that the dynamic exponent $z\nu$
increases with increasing $p$.

##  I Introduction

Spin glass systems exhibit aging, rejuvenation and memory effects owing to
their long relaxation times. Dupuis  _et al._ (2001) Better understanding of
the spin dynamics in such systems is a key for applications such as
information storage, magnetic resonance imaging, biomarking, biosensing, and
targeted drug delivery such as hyperthermia in cancer treatment. Perovic  _et
al._ (2013) Materials such as Fe2TiO5, YEr (2%) and Fe0.5Mn0.5TiO3 are
classified as _Ising spin glasses_. Svedlindh  _et al._ (1987a)

In this paper, we study the dynamic effects on phase transitions in the $\pm
J$ random-bond Ising spin glass model in zero external magnetic field that is
defined on a lattice by the dimensionless Hamiltonian Toulouse (1977)

$\mathcal{H}=-\sum_{\langle ij\rangle}J_{ij}s_{i}s_{j}.$ (1)

Here, the sum is over $ij$-bonds, $s_{i}=\left\\{+1,-1\right\\}$ is the Ising
spin at $i$-site, and $J_{ij}=\left\\{+J,-J\right\\}$ is the $ij$-bond
interaction dealt quench-randomly, with an antiferromagnetic ($-J$)
probability $p$. A dimensionless temperature for the system can be defined as
$T=1/J$.

The hard-spin mean-field theory (HSMFT) is defined by the set of self-
consistent equations

$m_{i}=\sum_{\left\\{s_{j}\right\\}}\left\\{\left[\prod_{j}\frac{1+m_{j}s_{j}}{2}\right]~{}\tanh\left(\sum_{j}J_{ij}s_{j}\right)\right\\}$
(2)

for the local magnetization $m_{i}$ at each $i$-site, with nearest-neighbors
labeled by $j$. For it preserves the local frustration, it is useful in
studying spin glass systems. Banavar  _et al._ (1991); Netz and Berker (1991a,
b, c); Netz (1992, 1993); Ames and McKay (1994); Berker  _et al._ (1994);
Kabakçıoğlu  _et al._ (1994); Akgüç and Yalabık (1995); Tesiero and McKay
(1996); Monroe (1997); Pelizzola and Pretti (1999); Kabakçıoğlu (2000); Kaya
and Berker (2000); Yücesoy and Berker (2007); Robinson  _et al._ (2011);
Çağlar and Berker (2011, 2015); Sarıyer  _et al._ (2012)

Previously, we studied the $\pm J$ model on a simple cubic lattice under time-
dependent external magnetic field by HSMFT, and obtained the phase diagram
(see Fig. 1) from distinctive behavior of hysteresis loops. Sarıyer  _et al._
(2012) At temperatures above a critical $T_{\text{c}}^{0}$ that depends on
$p$, system is in paramagnetic (PM) phase. (In general, the critical
temperature $T_{\text{c}}(\omega)$ depends on the cooling rate $\omega$, and
$T_{\text{c}}^{0}\equiv T_{\text{c}}(\omega=0)$ denotes the non-dynamic
critical temperature at infinitely-slow cooling rate.) At low temperatures,
$T<T_{\text{c}}^{0}(p)$, there exist a ferromagnetic (FM) phase for
$p<p_{\text{c}}$, and a spin glass (SG) phase for $p>p_{\text{c}}$. The FM-SG
phase boundary is at $p_{\text{c}}\approx 0.22$. The phase diagram of Fig. 1
is consistent both with theoretical Ozeki and Nishimori (1987) and
experimental Binder and Young (1986) results, and is mirror symmetric about
$p=0.5$, with antiferromagnetic (AFM) phase replacing the FM phase for
$p>1-p_{\text{c}}$. We can omit the $p>0.5$ subspace in general, due to the
FM$\leftrightarrow$AFM symmetry of classical Ising spins on bipartite
lattices.

Figure 1:  Zero-field phase diagram of the $\pm J$ model on a simple cubic
lattice. PM, FM, and SG respectively denote paramagnetic, ferromagnetic, and
spin glass phases.

In the same previous work, dynamic effect of the magnetic field sweep rate on
the dissipative loss in a hysteresis cycle was also studied Sarıyer  _et al._
(2012), while in the present article, we consider the dynamics of the model by
means of (_i_) adiabatic relaxation time behavior for infinitely-slow cooling
rate $\omega=0$, and (_ii_) the effects of nonzero cooling rate $\omega$ on
the critical temperature $T_{\text{c}}(\omega)$.

To this end, we consider an $L\times L\times L$-site simple cubic lattice with
periodic boundary conditions. HSMFT results become independent of the lattice
size for $L>15$ for this system Yücesoy and Berker (2007); Sarıyer  _et al._
(2012), and we work with $L=20$ here.

##  II Relaxation time

A particular realization at a given $(T,p)$ is generated by the assignment of
the quenched-random interactions $J_{ij}=\pm 1/T$ with a probability $p$ of
AFM bonds $-1/T$, and initially, an unbiased random assignment of spins in the
range $s_{i}\in\left[-1,+1\right]$. A time step $dt$ corresponds to successive
iterations of Eq. (2) on $L^{3}$ randomly chosen sites. The relaxation time
$\tau$ corresponds to number of such time steps required for the system to
converge from an initial random seed to a self-consistent solution that is
determined when the mean-square deviation in magnetization, $\Delta
m^{2}=L^{-3}\sum_{i}\left[m_{i}(\tau)-m_{i}(\tau-1)\right]^{2}$, drops below a
predetermined threshold, _i.e._ , when $\Delta m^{2}\leq 10^{-12}$.

Figure 2:  Relaxation time $\tau$ as a function of absolute reduced
temperature $|\tilde{T}|=|T-T_{\text{c}}^{0}|/T_{\text{c}}^{0}$ for $p=0.25$.
Faded markers show the results for $20$ distinct realizations of AFM bond
distributions, while the dark markers show averages over those random
replicas. Blue and red show the data below and above the critical temperature,
$\tilde{T}<0$ and $\tilde{T}>0$ respectively. Line is the best fit to the
power-law Eq. (4), for $\tilde{T}\geq 0.1$. Figure 3:  Dynamic critical
exponent $z\nu$ as a function of AFM bond fraction $p$. Error bars show the
standard deviation of the fit procedure to Eq. (4). Vertical line shows the
critical $p_{\text{c}}\approx 0.22$ between the ordered FM and SG phases (see
Fig. 1).

A characteristic signature of spin glass systems is that the relaxation time
$\tau$ diverges as $T\to{T_{\text{c}}^{0}}^{+}$, and remains infinite for
$T<T_{\text{c}}^{0}$. Fig. 2 is a log-log plot of the relaxation time $\tau$
as a function of $|\tilde{T}|$, for a representative value $p=0.25$, while we
obtain the same behavior for other $p$ values. Here,

$\tilde{T}\equiv\frac{T-T_{\text{c}}^{0}}{T_{\text{c}}^{0}}$ (3)

is the reduced temperature that measures the distantness to critical
temperature. Faded markers show the results for $20$ distinct realizations of
AFM bond distributions, while the dark markers show averages over such random
realizations. We see that the relaxation time $\tau$ averaged over replicas,
peaks as $\tilde{T}\to 0^{+}$ and stays at large values for $\tilde{T}<0$ as
expected. Deviation from a true divergence at $\tilde{T}\leq 0$ is due to
finite-size effects.

The divergence of $\tau$ as $\tilde{T}\to 0^{+}$, referred as the critical
slowing-down, is due to the divergence of the correlation length $\xi$. A
scaling relation between the relaxation time and the correlation length
$\tau\sim\xi^{z}$, and another one between the correlation length and the
reduced temperature, $\xi\sim\tilde{T}^{-\nu}$, yield to the power-law Binder
and Young (1986)

$\tau\sim\tilde{T}^{-z\nu}.$ (4)

Fitting the average relaxation time to Eq. (4) in the range
$0.1\leq\tilde{T}\leq 1$, we obtain the line in Fig. 2. Following the same
procedure for several $p$ values, we obtain the dynamic critical exponent
$z\nu$ as a function of AFM bond concentration $p$, as presented in Fig. 3. We
observe that $z\nu$ increases with increasing $p$.

Figure 4: Reduced difference
$(T_{\text{c}}^{0}-T_{\text{c}})/T_{\text{c}}^{0}$ between the apparent
dynamic ($T_{\text{c}}$) and the true non-dynamic ($T_{\text{c}}^{0}$)
critical temperatures, as a function of cooling rate $\omega$, for $p=0.25$.
Faded markers show the results for $20$ distinct realizations of AFM bond
distributions, while the dark markers show averages over those random
replicas. The line shows the best fit to the power-law Eq. (6). The average
magnetization, $M=L^{-3}\sum_{i}m_{i}$ and the SG order parameter
$Q=L^{-3}\sum_{i}m_{i}^{2}$ are shown in the bottom-right and top-left insets
as functions of $-\tilde{T}$ for all $20$ realizations for $\omega=0.0025$.

##  III Finite temperature sweep rate

To determine the effect of finite cooling rate

$\omega=-\frac{dT}{dt}\neq 0$ (5)

on the critical temperature $T_{\text{c}}(\omega)$, we start with a fully
relaxed realization of the system at a fixed AFM bond distribution at a high
temperature, $T=100$. The system is then cooled with a rate of change in
temperature $-dT$ per time step. As defined above, one time step $dt$
corresponds to updating $L^{3}$ randomly chosen sites according to Eq. (2). We
repeat the procedure for $\omega$ spanning two orders of magnitude in the
range $[0.00025,0.025]$, and for various $p$. For each set of values for
$\omega$ and $p$, we generate $20$ distinct realizations of AFM bond
distributions and work with the averages over those replicas.

For finite $\omega\neq 0$, the system will not find enough time to fully relax
to an ordered state when it reaches at the true non-dynamic critical
temperature $T_{\text{c}}^{0}=T_{\text{c}}(\omega=0)$. And when it reaches an
ordered state, the temperature will be already cooled below
$T_{\text{c}}^{0}$. Thus, the apparent critical temperature
$T_{\text{c}}(\omega)$ is expected to be lower than $T_{\text{c}}^{0}$, and to
shift down with increasing cooling rate $\omega$.

The reduced difference $(T_{\text{c}}^{0}-T_{\text{c}})/T_{\text{c}}^{0}$
between the apparent dynamic $T_{\text{c}}$ and the true non-dynamic
$T_{\text{c}}^{0}$ is plotted in Fig. 4 in log-log axes as a function of
$\omega$, for a representative value $p=0.25$, while we obtain the same
behavior for other $p$ values. The average magnetization,
$M=L^{-3}\sum_{i}m_{i}$ (shown in the bottom-right inset for $20$ distinct
realizations for $p=0.25$ and $\omega=0.0025$), is not suitable to determine
$T_{\text{c}}$ for PM-SG phase transition at $p>p_{\text{c}}$. Instead, the SG
order parameter $Q=L^{-3}\sum_{i}m_{i}^{2}$ (shown in the top-left inset for
$20$ distinct realizations for $p=0.25$ and $\omega=0.0025$) is used for the
whole range of $p$, implementing the fact that $Q=0$ for $T>T_{\text{c}}$ and
$Q>0$ for $T<T_{\text{c}}$. We checked that for $\omega=0$, this condition
yields the same equilibrium PM phase boundary as obtained from hysteretic
behavior (see Fig. 1).

Figure 5:  Temperature sweep rate exponent $a$ of Eq. (6) as a function of AFM
bond fraction $p$. Error bars show the standard deviation of the fit procedure
to Eq. (6). Vertical line shows the critical $p_{\text{c}}\approx 0.22$
between the ordered FM and SG phases (see Fig. 1).

Particularly at higher $\omega$, the reduced difference in critical
temperatures shows a clear power-law behavior,

$\frac{T_{\text{c}}^{0}-T_{\text{c}}}{T_{\text{c}}^{0}}\sim\omega^{a},$ (6)

as shown by the fit of average data to Eq. (6), resulting in the line in Fig.
4. Such is the case for all other $p$ values, and the resulting exponents are
shown in Fig. 5 for various $p$. We clearly observe the peak in the exponent
$a$ at the FM-SG phase transition at $p_{\text{c}}\approx 0.22$.

##  IV Conclusion

We simulated the phase transition dynamics of $\pm J$ Ising spin glass model
on a simple cubic lattice by the frustration preserving hard-spin mean-field
theory. We obtained the dynamic exponents $z\nu$ for the relaxation time, and
$a$ for the cooling rate dependence of critical temperature,
$T_{\textit{c}}(\omega)$.

We found that the dynamic exponent $z\nu$ varies in the range of about $0.5$
to $0.7$, which is an order of magnitude smaller than the values reported by
AC magnetometry experiments Svedlindh  _et al._ (1987a, b); Mauger  _et al._
(1988); Gunnarsson  _et al._ (1988); Nam  _et al._ (2000); Quilliam  _et al._
(2008); Wang  _et al._ (2010); Perovic  _et al._ (2011, 2013); Scholz and
Dronskowski (2016) and stochastic simulations Ogielski and Morgenstern (1985);
Ogielski (1985). Such is the pitfall of mean-field theories in general.
Neglecting the long range correlations close to critical points where
correlation length diverges, mean-field theories exhibit inadequate values for
critical temperatures, amplitudes and exponents. However, they offer
qualitative pictures of phase transitions. In this particular case, although
our values for the dynamic exponent $z\nu$ do not agree with previous results,
the qualitative feature that it increases with increasing AFM bond ratio $p$
is acceptable. This can be understood on the basis of increasing relaxation
time with increasing frustration (increasing raggedness and number of local
minima in the free energy landscape) with increasing $p$. This may also
explain the various experimental values for $z\nu$ ranging in the range from
$5.6\pm 0.1$ for Sn0.9Fe3.1N Scholz and Dronskowski (2016) to $10.4\pm 1.0$
for Fe0.5Mn0.5TiO3 Svedlindh  _et al._ (1987b).

Similarly, although the values might be incorrect, the cooling rate exponent
$a$ shows a peak at $p=p_{\text{c}}\approx 0.22$, which clearly indicates the
FM-SG transition. In these simulations, we initialized the cooling from a high
temperature $T=100$. We observed that starting the cooling from $T=10$,
changes our results only within the standard deviations even for the highest
cooling rates, and hence, we infer that $T=100$ is a safe high-temperature
starting point. Nonetheless, the effect of initial temperature on the dynamic
properties, is another interesting direction to cover. We expect that as the
starting temperature gets smaller, the apparent dynamic critical temperatures
will decrease further. This is because the system will find even smaller times
to relax to an ordered state, and by the time it relaxes, the temperature will
be even more smaller than the true non-dynamic $T_{\text{c}}^{0}$.

###### Acknowledgements.

I thank Dr. Aykut Erbaş for valuable discussions and for careful reading of
the manuscript. Numerical calculations were run by a machine partially
supported by the 2232 TÜBİTAK Reintegration Grant of project #115C135.

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