# Multiscale Modelling of the Antiferromagnet Mn2Au: From ab-initio to
Micromagnetics

Joel Hirst  Materials & Engineering Research Institute, Sheffield Hallam
University, Howard Street, Sheffield S1 1WB, United Kingdom Unai Atxitia
Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie
Universitat Berlin, 14195 Berlin, Germany  Instituto de Ciencia de Materiales
de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain  Sergiu Ruta  Materials &
Engineering Research Institute, Sheffield Hallam University, Howard Street,
Sheffield S1 1WB, United Kingdom Jerome Jackson  Scientific Computing
Department, STFC Daresbury Laboratory, Warrington WA4 4AD,United Kingdom  Leon
Petit  Scientific Computing Department, STFC Daresbury Laboratory, Warrington
WA4 4AD,United Kingdom  Thomas Ostler  Department of Physics & Mathematics,
University of Hull, Hull HU6 7RX, UK  Materials & Engineering Research
Institute, Sheffield Hallam University, Howard Street, Sheffield S1 1WB,
United Kingdom Department of Engineering and Mathematics, Sheffield Hallam
University, Howard Street, Sheffield S1 1WB, United Kingdom

###### Abstract

Antiferromagnets (AFMs) are strong candidates for the future spintronic and
memory applications largely because of their inherently fast dynamics and lack
of stray fields, with Mn2Au being one of the most promising. For the numerical
modelling of magnetic material properties, it is common to use ab-initio
methods, atomistic models and micromagnetics. However, each method alone
describes the physics within certain limits. Multiscale methods bridging the
gap between these three approaches have been already proposed for
ferromagnetic materials. Here, we present a complete multiscale model of the
AFM Mn2Au as an exemplar material, starting with results from ab-initio
methods going via atomistic spin dynamics (ASD) to an AFM Landau-Lifshitz-
Bloch (AFM-LLB) model. Firstly, bulk Mn2Au is modelled using a classical spin
Hamiltonian constructed based on earlier first-principles calculations.
Secondly, this spin model is used in the stochastic Landau-Lifshitz-Gilbert
(LLG) to calculate temperature-dependent equilibrium properties, such as
magnetization and magnetic susceptibilities. Thirdly, the temperature
dependent micromagnetic parameters are used in the AFM-LLB. We validate our
approach by comparing the ASD and AFM-LLB models for three paradigmatic cases;
(i) Damped magnetic oscillations, (ii) magnetization dynamics following a heat
pulse resembling pump-probe experiments, (iii) magnetic domain wall motion
under thermal gradients.

##  I  Introduction

The interest in antiferromagnetic (AFM) materials has increased in recent
years for several reasons. From fundamental questions regarding the effects of
atomic-scale spin interactions on the macroscale magnetic configurations [1,
2, 3, 4], to contenders for driving future technologies of smaller, faster,
and more energy efficient devices [5, 6, 7, 8, 9, 10]. Possible applications
are in the fields of terahertz (THz) science, ultrafast spin dynamics and
spin-caloritronics – combined spin transport and heat.

AFMs are robust against strong external magnetic fields and have intrinsically
fast THz magnetization dynamics compared to the (gigahertz) GHz dynamics found
in ferromagnets. It has been known since the late 1940’s that the resonant
modes of certain AFMs can reach the THz range [11, 12]. This frequency
enhancement in AFMs emerges at the macroscale from the interplay of the
magnetic anisotropy and the atomic-scale antiferromagnetic spin exchange
interaction. If antiferromagnets are to be used in THz emitters and spintronic
devices [13, 14, 15, 16], it becomes essential to accurately calculate, from
first-principles, the nature and values of the anisotropy and exchange
interactions between spins. Microscopic interactions also play a key role in
the field of ultrafast spin dynamics [17]. Experiments have demonstrated an
ultrafast and more energy efficient magnetic order quenching in AFM than in
FMs [18]. Further understanding and control of the ultrafast spin dynamics in
AFMs requires not only accurate determination of the exchange interactions,
but also how the magnetic system reacts to an ultrafast load of heat from
femtosecond laser pulses.

Metallic AFMs are convenient for all of those aforementioned applications. For
instance, the antiferromagnets CuMnAs and Mn2Au have been proposed [19, 20,
21, 22, 23, 24] for several device applications because their magnetic state
can be controlled via electric currents, and they can be read out via their
magnetoresistive properties [25, 26]. They have relatively high critical
temperature with experimental measurements placing the Néel temperature of
Mn2Au between 1300 and 1600 K [27] and around 480 K for CuMnAs [28], making
them ideal for memory and spintronics applications. While initial studies
focused on CuMnAs rather than Mn2Au, it has been demonstrated that the Joule
heating created by electric currents puts the system too close to the critical
temperature, so that undesired thermal fluctuations strongly influence the
dynamics. In this regard, the much higher critical temperature of Mn2Au is
preferred.

Established approaches to calculate the magnetic properties include ab-initio
methods for the electronic structure at the atomic scale and generally at zero
temperature [29, 30, 31, 32, 33], atomistic spin models coupled to a heat-bath
for the calculation of thermal properties at the [34, 35, 36], and
micromagnetics for the calculation of the magnetic distribution at the
micrometer scale [37, 38, 39]. Ab-initio methods and atomistic spin models
naturally describe AFMs. Modeling AFMs with micromagnetism is more problematic
since the micromagnetic models are based on small variations of the
magnetization at the nanoscale, while in AFMs the spin orientation varies
strongly at the atomic scale. While micromagnetic modelling of AFMs remains a
challenge, ferromagnetic models based on the Landau-Lifshitz-Gilbert (LLG)
equation are now well established and used extensively in fundamental and
device research. Several codes are available with a plethora of features and
are implemented on various high performance computing platforms [37, 40, 38].
These micromagnetic models are particularly useful when attempting to model
scenarios in which the dynamics occur across $\mu$m distances and where high
frequency spinwaves do not contribute to the dynamics, for example, when
modelling domain structures [41] or vortex core dynamics [42]. One of the main
drawbacks of conventional micromagnetic modeling is that the simulations
remain at fixed temperature, rendering LLG based micromagnetics obsolete in
scenarios where the temperature changes dynamically or when it varies
spatially as transient changes in the macrosopin magnetisation are not
naturally taken into account.

It is possible to use atomistic spin dynamics (ASD) in scenarios where thermal
effects play an important role. In ASD, each magnetic lattice site has a
magnetic moment, the dynamics of which are simulated with the stochastic LLG
equation [43, 44]. The magnetic atoms interacts with neighbouring moments
which can be written as an extended Heisenberg Hamiltonian [34, 45]. Atomistic
modelling has become an essential tool for modelling ultrafast magnetization
dynamics [46, 47] and temperature dependent effects such as laser pump-probe
experiments [48, 49, 50]. However, simulations of systems on $\mu$m
lengthscales and $\mu$s timescales (such as the magnetic grains found in novel
magnetic recording media) become computationally expensive, and a finite
temperature micromagnetic approach valid for AFMs is needed. Along this line,
extensive work already exists using the so-called Landau-Lifshitz-Bloch (LLB)
finite temperature micromagnetic formalism to model ferro and ferrimagnets,
including for FePt [51, 52], GdFeCo [53] and permalloy [54]. The LLB framework
has been used to model magnetic systems on device-level length and timescales.
However, the LLB formalism for AFMs is incomplete. First attempts have been
made to calculate the domain wall velocity due to a thermal gradient for a
generic AFM using an LLB model [55]. However, the proposed model for AFM-LLB
lacks a fundamental aspect of the magnetic properties at finite temperatures,
the relaxation of the magnetization length. The correct description of the
magnetization quenching dynamics (aka longitudinal relaxation dynamics) is
essential for the modelling of ultrafast magnetization dynamics and switching.
A revised version of the AFM-LLB that includes the recently derived exchange
enhanced longitudinal effective damping [56] is necessary in order to extend
the LLB framework to AFMs.

Here we present a complete multiscale model of the AFM Mn2Au. The model starts
with parameters from Density Functional Theory (DFT) calculations [57], then
going via atomistic spin dynamics to a newly developed LLB equation for AFMs.
We calculate the thermal equilibrium properties using ASD and use the results
as input into an AFM-LLB model, where instead of having a spin at each
atomistic site we simulate a collection of spins, known as a macrospin (in
much the same way as is done in traditional LLG based micromagnetics). This
approach, whereby one starts with results from first-principle calculations
and end up with a description on micromagnetic length scales, is known as
multiscale modelling, and this is the first of its kind for AFMs where both
the transverse and longitudinal processes have been accurately within an LLB
framework.

The work is broken down in the following way. We begin with an introduction of
the atomistic spin model with the DFT parameters that are used to parameterise
the atomistic Hamiltonian. We then present the transverse terms of the AFM LLB
model and present results for the Antiferromagnetic Resonance (AFMR) and
compare to ASD and analytical expressions. We then introduce the longitudinal
term in the LLB model and compare the dynamics from ASD and LLB simulations
for step changes in temperature as well as for ultrafast laser heating using a
two-temperature model. Finally, as an example where both the longitudinal and
transverse components of the AFM-LLB are important (and where ASD simulations
take orders of magnitude more computational time than an LLB approach) we
present a comparison between LLB, ASD and analytical expressions for the the
domain wall (DW) motion due to a thermal gradient.

##  II Atomistic Model

The dynamics of each atomistic spin is governed by the stochastic Landau-
Lifshitz-Gilbert equation (s-LLG) equation:

$\frac{\partial\mathbf{S}_{i}}{\partial
t}=-\frac{\gamma_{i}}{\left(1+\lambda_{i}^{2}\right)\mu_{i}}\left[\mathbf{S}_{i}\times\mathbf{H}_{i}+\lambda_{i}\mathbf{S}_{i}\times\left(\mathbf{S}_{i}\times\mathbf{H}_{i}\right)\right]$
(1)

where $\mathbf{S}_{i}$ is a normalised unit vector of the spin at site $i$,
$\lambda_{i}$ is the effective damping parameter, $\gamma_{i}$ is the
gyromagnetic ratio, $\mu_{i}$ is the atomic magnetic moment and
$\mathbf{H}_{i}$ is the effective field acting on the spin at site $i$. The
effective field is the sum of a field-like stochastic term and the negative
derivative of the spin Hamiltonian:

$\mathbf{H}_{i}=\bm{\zeta}_{i}(t)-\frac{\partial\mathcal{H}}{\partial\mathbf{S}_{i}}$
(2)

where the Hamiltonian given by:

$\mathcal{H}=-\sum_{\langle
ij\rangle}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-d_{\eta}\sum_{i}\left(S_{i}^{\eta}\right)^{2}-\mu_{s}\mathbf{B}\cdot\sum_{i}\mathbf{S}_{i}$
(3)

where $J_{ij}$ is the exchange interaction between site $i$ and site $j$,
$\mathbf{B}$ is the Zeeman field and $d_{\eta}$ is the uniaxial anisotropy
constant with $\eta=x,y,z$. The stochastic term $\bm{\zeta}_{i}(t)$ in Eq. (2)
describes the coupling of the spin system to an external thermal bath and
accounts for the thermal fluctuations. The noise processes are governed by the
equations:

$\begin{gathered}\left\langle\zeta_{i}^{a}(t)\zeta_{j}^{b}(t)\right\rangle=2\delta_{ij}\delta_{ab}\delta\left(t-t^{\prime}\right)\frac{\mu_{i}\lambda_{i}k_{B}T}{\gamma_{i}}\\\
\left\langle\zeta_{i}^{a}(t)\right\rangle=0\end{gathered}$ (4)

Where $a$ and $b$ are the Cartesian components and $T$ is the temperature of
the system. The integration of the s-LLG equation is completed using the Heun
integration method. The simulations are performed on an in-house code capable
of running on graphical processing units (GPUs).

Figure 1: The Mn2Au unit cell. The exchange interactions used in this work are labelled $J_{1}$ to $J_{4}$. The values can be found in Table 1 Constant | Value | Unit
---|---|---
Lattice constant, $a$ | 3.330 | Å
Lattice constant, $c$ | 8.537 | Å
Magnetic Moment, $\mu_{s}$ | 3.8663 | $\mu_{B}$
Exchange constant, $J_{1}$ | -5.3422 | mRy
Exchange constant, $J_{2}$ | 0.6484 | mRy
Exchange constant, $J_{3}$ | -0.6341 | mRy
Exchange constant, $J_{4}$ | -6.8986 | mRy
Sum of FM exchange, $J_{0,\nu\nu}$ | 2.5934 | mRy
Sum of AFM exchange, $J_{0,\nu\kappa}$ | -30.8040 | mRy
Uniaxial Anisotropy in $z-$direction, $d_{||}$ | -0.0090 | mRy
Uniaxial Strain Anisotropy in $z-$direction, $d^{*}_{||}$ | 0.0004 | mRy
Table 1: Parameters used in the Hamiltonian for the atomistic modelling of
Mn2Au.

For the atomistic simulations of Mn2Au, we take parameters for the exchange
constants, magnetic moment and anisotropy from previous work by Ruta et al
[57]. Using the LMTO-ASA method [58], they calculate a strong negative
uniaxial anisotropy constant along the $z-$direction meaning the Néel vector
has no preferential direction in the plane. We opt to use an additional
uniaxial anisotropy term along the [100] direction induced by in-plane strain
with a value of 0.0004 mRyd/Mn as calculated by Shick et al [59]. In such a
way, the system has a single favoured orientation of the Néel vector. We use
the first first four nearest neighbour interactions between Mn sites as
illustrated in Fig. 1. The parameters used in the atomistic Hamiltonian can be
found in Table 1. For the exchange constant $J_{1},J_{2}$ and $J_{3}$ there
are four interactions for each atomic site, and for $J_{4}$ there is just one.

Figure 2: The sublattice magnetization of Mn2Au as a function of temperature.
The solid line is a fit to the expression $M/M_{S}=m_{e}(T)=(1-T/T_{N})^{b}$.
The circular points show the longitudinal susceptibility,
$\widetilde{\chi}_{\|}(T)$, calculated from ASD and the dashed line shows
$\widetilde{\chi}_{\|}(T)$ taken from MFA, which is discussed in Sec. IV.

##  III Transverse Dynamics of the Antiferromagnetic Landau-Lifshitz-Bloch
Equation

The LLB model captures the dynamics of a collection of exchange coupled
atomistic spins, namely the dynamics of thermal averaged
$\mathbf{m}_{i}=\langle\mathbf{S}_{i}\rangle$. It was first derived by Garanin
[60] within a mean-field approximation for ferromagnets, which implies
$\mathbf{m}_{i}=\mathbf{m}$ for lattice sites $i$. Extensions of the model for
non-homogeneous magnetization states exist within micromagnetic computational
approaches [51]. More recently, the LLB model was extended to ferrimagnets by
Atxitia et al [53]. The LLB model contains both transverse and longitudinal
relaxation terms and interpolates between Landau-Lifshitz-Gilbert equation at
low temperatures (transverse dynamics) and the Bloch equation at high
temperatures (longitudinal dynamics). To start with, a comparison of the
transverse dynamics between the AFM-LLB and atomistic models is given. We
begin with an LLB equation that neglects changes in the magnetization length,
and purely describes the transverse dynamics,

$\frac{{d\mathbf{m}}_{\nu}}{dt}=-\gamma\left[\mathbf{m}_{\nu}\times\mathbf{H}_{\text{eff},{\nu}}\right]-\gamma\alpha_{\perp}\frac{\left[\mathbf{m}_{\nu}\times\left[\mathbf{m}_{\nu}\times\mathbf{H}_{\mathrm{eff},{\nu}}\right]\right]}{m^{2}_{\nu}}$
(5)

The above looks remarkably similar to the LLG equation in that we have a
precessional and relaxation term in both models. In the above,
$\mathbf{m}_{\nu}$ is the macrospin magnetization in sublattice $\nu$. It is
not of constant length and its equilibrium value, $m_{e}(T)$ is temperature
dependent. The dimensionless transverse damping parameter is given by,

$\alpha_{\perp}=\begin{cases}\lambda\Big{(}1-\frac{T}{3T_{N}}\Big{)},&T<T_{N}\\\
\\\ \lambda\frac{2T}{3T_{N}},&T>T_{N}\end{cases}$ (6)

Figure 3: The $m_{z}$ motion following a rotation of of both sublattices by 20
degrees out of the easy plane at (a) $T=300$ K, (b) $T=1000$ K and (c)
$T=1200$ K. Solid line are the AFM-LLB model and points are ASD. The dotted
line shows the relaxation of the transverse dynamics. The relaxation time can
be related to $\alpha_{\perp}$ through with Eq. (10).

It is worth noting that for situations where the magnetization length is
fairly constant, $m_{\nu}=m_{e,\nu}$, Eq. (5) can be written in terms of a
normalized vector, $\mathbf{n}_{\nu}=\mathbf{m}_{\nu}/m_{e,\nu}$.

$\frac{{d\mathbf{n}}_{\nu}}{dt}=-\gamma\left[\mathbf{n}_{\nu}\times\mathbf{H}_{\text{eff},{\nu}}\right]-\gamma\frac{\alpha_{\perp}}{m_{e,\nu}}\left[\mathbf{n}_{\nu}\times\left[\mathbf{n}_{\nu}\times\mathbf{H}_{\mathrm{eff},{\nu}}\right]\right],$
(7)

which is identical to the LLG equation [43] with the identification
$\alpha_{\rm{LLG}}=\alpha_{\perp}/m_{e}$.

The Néel temperature, $T_{N}$, is taken from atomistic simulations of the
equilibrium magnetization, $m_{e}(T)$. For these calculations, we simulate
$30\times 30\times 30$ unit cells with periodic boundary conditions. The
atomistic damping in Eq. (1), $\lambda_{i}$, is set 1.0 for both sublattices
to allow for a faster relaxation of the magnetization length. The equilibrium
magnetization as a function of temperature can be found in Fig. 2. The fit
function, $m_{e}(T)=(1-T/T_{N})^{b}$ shown by the solid black line in Fig. 2,
has free fitting parameters $T_{N}$ and $b$ which provide a good estimate of
the transition temperature for an infinite system. Fitting to the equilibrium
magnetization from ASD yields $T_{N}=$ 1335 K and $b=0.34$. A more accurate
way to determine the critical temperature is through calculations of the
parallel susceptibility, $\widetilde{\chi}_{\|}(T)$. In an atomistic spin
model, the parallel susceptibility is found by measuring the fluctuations of
the magnetization

$\widetilde{\chi}_{\|}=\frac{\mu_{s}N}{k_{\rm{B}}T}\left(\left\langle
S_{\|}^{2}\right\rangle-\left\langle S_{\|}\right\rangle^{2}\right)$ (8)

where $\langle...\rangle$ is an ensemble average and $k_{\rm{B}}T$ is the
thermal energy. Due to the strong uniaxial easy-plane anisotropy in Mn2Au, the
parallel component is taken as the average of the $x$ and $y$ components of
the sublattice magnetization. It is worth noting that the thermal equilibrium
parameters are the same for both sublattices. A value of $T_{N}=1333$ K was
extracted from fitting to $\widetilde{\chi}_{\|}(T)$ calculated from atomistic
spin dynamics.

Moving away from atomistic spin dynamics and returning to the discussion of
the AFM-LLB model – for a single macrospin containing two sublattices with the
magnetization length initialised at $m_{\nu}=m_{e}(T)$, the effective field,
$\mathbf{H}_{\text{eff},{\nu}}$, in Eq. (5) is given by

$\mathbf{H}_{\mathrm{eff},\nu}=\mathbf{B}+\mathbf{H}_{a,\nu}+\frac{J_{0,\nu\kappa}}{\mu}\mathbf{\Pi}_{\kappa}$
(9)

where $\mathbf{B}$ is the applied magnetic field, $\mathbf{H}_{a,\nu}$ is the
anisotropy field, $J_{0,v\kappa}$ is the sum of the inter-lattice exchange
(see Table 1), and $\mathbf{\Pi}_{\kappa}$ is given by,
$\mathbf{\Pi}_{\kappa}=-\left[\mathbf{m}_{\nu}\times\left[\mathbf{m}_{\nu}\times\mathbf{m}_{\kappa}\right]\right]/m_{\nu}^{2}$
where $\mathbf{m}_{\kappa}$ is the magnetization vector of the second
sublattice. The anisotropy field in the macrospin model, $\mathbf{H}_{a,\nu}$,
is defined as $\mathbf{H}_{a,\nu}=\frac{2K(T)}{M_{S}}$ where the temperature
dependence of the anisotropy constant is governed by Callen-Callen scaling;
$K(T)=K(0)m^{3}_{e}(T)$ [61].

Figure 4: The transverse relaxation time, $\tau_{\perp}$ as function of
temperature. Empty points represent ASD simulations with values of
$\lambda=0.05$ (square), $0.01$ (circle) and $0.001$ (triangles). Solid
circles are the AFM-LLB model. The solid line is given by Eq. (10).

To validate the temperature-dependent transverse dynamics described by the LLB
model, we present simulation results showing the temperature dependence of the
AFMR for Mn2Au using both atomistic s-LLG and AFM-LLB models, as well as a
comparison to the Kittel equation for AFMR [11, 12]. Magnetic oscillations in
AFMs have a resonance frequency in the THz range, orders of magnitude faster
than the GHz range observed in ferromagnets. As a result of the complexity in
current THz-signal generation methods [62, 63, 64], AFM oscillators have been
proposed for use in adjustable room-temperature THz-frequency signal devices
[13, 14, 15]. While attempts to experimentally measure the in-plane AFMR in
Mn2Au driven by Néel spin orbit torques have previously failed [65], it piqued
the magnetism community’s interest, highlighting the importance of not only
measuring, but also calculating, the resonant frequency in AFMs. Since the
AFMR sets the speed limit of the AFM dynamics, it is highly relevant for the
development of ultrafast memory applications. For the atomistic simulations of
the AFMR, the system is initially set to relax to its equilibrium
magnetization for each given temperature. Once relaxed, all spins are rotated
by 20 degrees out of the easy plane inducing a torque and thereby exciting the
AFMR mode. The relaxation time of the transverse motion, $\tau_{\perp}$, can
be related to the effective field and transverse damping via

$\tau_{\perp}=\frac{m_{e}}{\alpha_{\perp}\omega_{\mathrm{ex}}}$ (10)

where $\omega_{\mathrm{ex}}=\gamma
H_{E}=\gamma\frac{J_{0,\nu\kappa}}{\mu_{s}}m_{e}$. An example of the $m_{z}$
dynamics at $T=0$ K can be seen in Fig. 3. The value of $\tau_{\perp}$ was
found by fitting the sublattice magnetization to the equation
$m_{z}(t)/m_{e}=\cos(\omega_{\rm{AFMR}}t)\exp(-t/\tau_{\perp})$.

A comparison of the temperature dependence of $\tau_{\perp}$ can be found in
Fig. 4. We calculate $\tau_{\perp}$ atomistically for values of
$\lambda=0.05,0.01$ and $0.001$. Note for lower values of $\lambda$ we see a
reduction in the relaxation time as we approach $T_{N}$. This effect is
explained by the fact at lower damping we are further from equilibrium with
the thermal bath and energy is predominantly transferred between the
sublattices. The temperature dependence of $\tau_{\perp}$ in the LLB model is
independent of $\lambda$. The exact form of the damping dependence on
$\tau_{\perp}$ remains an open question.

The AFMR frequency of the dynamics was extracted via Fast Fourier Transform
(FFT) of $m_{z}$ of a single sublattice. The results of this frequency
analysis can be found in Fig. 5 showing excellent agreement between the models
and the analytical expression. The analytic expression for the AFMR, shown by
the solid line in Fig. 5, is given by:

$f=\frac{\gamma}{2\pi}\sqrt{H_{A}(H_{A}+2H_{E})}$ (11)

where $H_{A}$ and $H_{E}$ are the anisotropy and exchange fields respectively.

Figure 5: AFMR frequency of Mn2Au . The solid line is the result from Eq.
(11), square points are from ASD, and circular points are the LLB simulations
respectively.

##  IV Longitudinal Relaxation

It has been demonstrated that a femtosecond laser-pulse can induce ultrafast
magnetization dynamics in ferro- and ferri-magnets [17, 47, 46]. Whilst
mechanisms leading to the ultrafast demagnetization in magnetic materials
broadly are still under debate, in certain ferrimagnets, a single laser-pulse
can induced switching of the magnetization [46]. The magnetic order dynamics
in AFMs, i.e. Néel vector length dynamics, is less known. In order to describe
the magnetic order quenching in Mn2Au, we introduce a term in the LLB equation
that accounts for changes in the magnetization length. Here we are only
interested in the longitudinal dynamics – parallel to the magnetization
direction – and can therefore ignore the transverse terms.

$\frac{d\mathbf{m}_{\nu}}{dt}=\gamma\alpha^{\mathrm{AF}}_{\|}\frac{\left(\mathbf{m}_{\nu}\cdot\mathbf{H}_{\mathrm{eff},\nu}\right)}{m_{\nu}^{2}}\mathbf{m}_{\nu}$
(12)

The effective field for a purely longitudinal relaxation (where one can ignore
any external, anisotropy or inter-macrospin exchange fields) is given by

$\mathbf{H}_{\mathrm{eff},\nu}^{\|}=\begin{cases}\frac{1}{2\widetilde{\chi}_{\|}}\left(1-\frac{m^{2}_{\nu}}{m_{\mathrm{e}}^{2}}\right)\mathbf{m}_{\nu},&T\lesssim
T_{N}\\\ \\\
-\frac{1}{\widetilde{\chi}_{\|}}\left(1+\frac{3}{5}\frac{T_{N}}{T-T_{N}}m^{2}_{\nu}\right)\mathbf{m}_{\nu},&T\gtrsim
T_{\mathrm{c}}\end{cases}$ (13)

where $\widetilde{\chi}_{\|}$ is the reduced longitudinal susceptibility. In
this work we opt to use the susceptibility taken from a MFA [66]. We find that
the agreement in the longitudinal relaxation between ASD and LLB when using
$\widetilde{\chi}_{\|}$ taken from ASD simulations gave large discrepancies
between the two models at lower temperature while using
$\widetilde{\chi}_{\|}$ taken from a MFA gave excellent agreement for all
temperatures. This problem has been noted elsewhere - Vogler et al mention
that calculations of $\widetilde{\chi}_{\|}$ only work for hard magnetic
materials with strong uniaxial anisotropy [67] with similar problems having
been identified in ferrimagnets [68]. The reason for the difference is
unclear, and should be investigated as part of future work. The method of
calculating the MFA longitudinal susceptibility follows directly from [69].
Using the same value for the exchange constants in ASD and MFA approaches
yields a higher critical temperature compared to the atomistic results
($T_{N}^{\text{MFA}}>T_{N}^{\text{ASD}}$). Therefore, we rescale the exchange
constant to match the value for the Néel temperature extracted from Fig. 2.
The process of using a scaled MFA susceptibility has been seen previously in
atomistic modelling of GdFeCo [70].

Figure 6: The damping dependence of $\beta$. The MFA LLB equation [60] is
fitted to the longitudinal dynamics from atomistic simulations following a
step change in temperature from $T=0$ K to $T>T_{N}$. All other parameters in
the LLB besides $\beta$ remain fixed.

Returning to the discussion of the longitudinal relaxation in the LLB
equation, assuming that the macrospin magnetization remains along single axis,
$\mathbf{m}_{\nu}=[m_{\nu},0,0]$, Eq. (12) simplifies to

$\frac{dm_{\nu}}{dt}=\gamma\alpha^{\mathrm{AF}}_{\|}H^{\|}_{\mathrm{eff},\nu}.$
(14)

Experimental works studying the ultrafast magnetic order dynamics in the FM
and AFM phases of Dy have shown that the AFM phase exhibits faster dynamics
than the FM phase [18]. This speed up is attributed to the exchange of angular
momentum between sublattices. For Mn2Au , it will be difficult to conduct a
similar comparative study due to the absence of a FM phase. However, a recent
theoretical work by Jakobs & Atxitia [56] has concluded that the speed up of
the AFM dynamics comes from the exchange-enhancement of the effective damping
parameter

$\alpha^{\mathrm{AF}}_{\|}=\alpha^{\mathrm{FM}}_{\|}\left(1+\frac{2}{z\left|m_{\nu}\right|^{\beta}}\right)$
(15)

where, $\alpha^{\mathrm{FM}}_{\|}$ is the ferromagnetic longitudinal damping
parameter (where the relaxation is solely due to the dissipation of angular
momentum to the heat bath) and is defined by

$\alpha^{\mathrm{FM}}_{\|}=\lambda\frac{2T}{3T_{N}}$ (16)

In Eq. (15), theory predicts that $z$ is the number of nearest neighbours
antiferromagnetically coupled to a given spin. While for a simple cubic with
only nearest neighbours this number is 6, for Mn2Au this would correspond to
5, as sketched in Fig. 1. Our simulations show that $z\approx 6.0$. The
exponent $\beta$ in Eq. (15) is a phenomenological parameter, necessary for
the description of high non-equilibrium situations where the temperature goes
well above the critical temperature. While for small deviations from the
equilibrium of the magnetic order parameter, $\beta=1$, for larger deviations,
for example when the temperature of the system changes from $T=0$ K to
$T=2T_{N}$, the exponent takes a value of around 2. We also find that
alongside the temperature dependence of the $\beta$ exponent, it also exhibits
a damping dependence, which can be found in Fig. 6. To determine the damping
dependence of $\beta$, numeric solutions of the LLB equation after step change
in temperature from $T=0$ K to $T>T_{N}$ were fitted against the corresponding
atomistic simulations. For the fitting, we use a MFA form of the effective
field that is not dependent on any atomistically derived parameters. Instead
of using Eq. (13), the effective field is given by [53]

$H_{\mathrm{eff},\nu}^{\|}=\frac{\left(m_{\nu}-m_{0,\nu}\right)}{\mu_{s}\beta
L^{\prime}\left(\xi_{\nu}\right)}$ (17)

in the above, $m_{0,\nu}$ is not the equilibrium magnetization and is given by
$m_{0,\nu}=L\left(\xi_{\nu}\right)$ where
$L\left(\xi_{\nu}\right)=\operatorname{coth}(\xi_{\nu})-1/\xi_{\nu}$ is the
Langevin function and $L^{\prime}(\xi_{\nu})=dL/d\xi_{\nu}$ with
$\xi_{\nu}=\beta\mu_{s}H_{\nu}^{\mathrm{MFA}}$ and $\beta=1/k_{\mathrm{B}}T$.
The MFA field is defined as $H^{\mathrm{MFA}}_{\nu}=3k_{B}T_{N}m_{\kappa}$
[60].

Figure 7: A comparison at five different temperatures between the AFM-LLB and
the atomistic modelling. Symbols represent the atomistic spin dynamics results
and the solid lines correspond to the AFM-LLB. No stochastic noise was present
in the AFM-LLB simulations.

Using the results of our determination of $\beta$ for a value of
$\lambda=0.01$ we have furthermore carried out simulations of the longitudinal
dynamics after a temperature step (Fig. 7). In these temperature step
simulations, we begin with a perfectly ordered AFM configuration of Mn2Au at
$T=0$ K, we then apply a temperature step and record the relaxation to the
equilibrium magnetization. Fig. 7 shows the longitudinal relaxation for 5
different temperatures (with $\lambda=0.01$ for all simulations), the points
are simulations using the stochastic LLG equation and the solid lines are from
the LLB model. The agreement at lower temperatures is excellent. At
temperatures well above $T_{N}$, finite size effects lead to a longer
relaxation time in the atomistic model. It is also worth stressing that
instantaneous step changes in temperature from 0 Kelvin to near $T_{N}$ are
radical, and some disagreement would be expected.

To highlight the importance of the $\beta$ exponent in Eq. (15), we performed
ASD and LLB simulations with values for $\lambda=0.00005$ for a step change to
above the critical temperature, as shown in Fig. 8. We also include ASD
results for an entirely ferromagnetic exchange to show the difference in the
relaxation rates between FM and AFM. We find that relaxation dynamics in AFMs
is defined by two distinct processes, (i) a exponential decay due to
dissipation of angular momentum to the heat-bath and (ii) a power-law decay
due to angular momentum exchange between sublattices. While for the FM version
of Mn2Au the relaxation dynamics is described by an exponential decay, for
AFMs, the exponential decay only dominates for values of the order parameter
$n>1/3$. As the magnetic order reduces to small values, the rate of angular
momentum dissipation remains constant,
$(\mu_{\rm{at}}/\gamma)\dot{n}\approx(2/3)\lambda k_{B}T/n^{\beta-1}$, leading
to a power-law decay.

Figure 8: The longitudinal relaxation following a step change in temperature
from 0 K to 1400 K. The Gilbert damping was set to 0.00005. Square points are
from ASD simulations of the ground state AFM configuration of mn2Au. Circular
points are from ASD simulations representing an FM configuration of Mn2Au
where the absolute values of $J_{1}$ to $J_{4}$ have been used. The solid line
shows the AFM-LLB.

Although in the simulations a step-like increase of temperature is easy to
achieve, experimentally, a strong and rapid heating of the system is possible
by applying, for example, a femtosecond laser pulse. In these scenarios the
temperature of the system is difficult to determine, however, given semi-
classical considerations [71] one can define a temperature of an electron and
a phonon bath. A laser pulse will couple more strongly to the electron system
giving large and rapid temperature increases, in hundreds of femtoseconds, to
a temperature above $T_{N}$, however, the electron system also quickly cools
downs on the picosecond time scale by transferring energy to the phonon system
via electron-phonon coupling. The electronic temperature is calculated using a
two-temperature model (TTM) [72]. Fig. 9 shows the longitudinal magnetization
dynamics for a transient change in temperature following heating from a laser
pulse. We begin at $T=300$ K then heat to just below and above $T_{N}$, shown
as TTM 1 and TTM 2 in Fig. 9. What is also worth noting is that the AFM-LLB
accurately captures both remagnetization and demagnetization processes.

These results show that the exchange-enhanced damping derived by Jakobs &
Atxitia is successful in describing the faster relaxation times within the LLB
model, especially at lower temperatures. It also shows good agreement when
using a TTM, which is highly applicable to simulations of all-optical
switching in AFMs.

Figure 9: (Top) The electronic temperature dynamics using a TTM. Dotted line
shows the Néel temperature. (Bottom) Magnetization dynamics from atomistic
simulations (black squares) and the LLB equation using a ferromagnetic (dotted
lines) and ferrimagnetic (solid lines).

##  V Domain Wall Motion

The manipulation of domain walls (DW) have been proposed for use in the next
generation of logic and memory devices [73, 74]. The motion of the DWs can be
modulated using spinwaves [75, 76], spin currents [77, 78, 79] or external
magnetic fields [80, 81], for example. Another possibility is to drive DW
motion by thermal gradients [82, 83]. Under thermal gradient, motion of the
domain wall is induced towards the hotter end of the nanowire due to a
reduction in the free energy, $\Delta F(T)$. For large thermal gradients in
ferromagnets, there is a precession of the internal magnetization, which leads
to a reduction in the domain wall velocity, known as the Walker breakdown
[80]. In previous ASD simulations of AFMs, it has been shown that there is no
breakdown in the DW velocity due to symmetry of the torques acting on the DW
[84], meaning DWs in AFMs are able to reach higher speeds.

Conventional micromagnetic methods have limitations in the modelling of
thermally driven DWs, as they cannot account for the dynamic changes in the
magnetization length. It has been shown experimentally that ultrashort laser
pulses are able to drive DW motion [83]. In these scenarios, heating and
cooling will play a role, and the effects of this cannot be captured using an
LLG based micromagnetic model. This could be simulated using an atomistic
approach, but the calculations are expensive and require large ensembles to
reduce statistical noise. The LLB model overcomes these issues, making it an
important tool for temperature dependant calculations of topological
structures such as domain-wall, spin-spirals and skyrmions. As an example of
simulations that require both accurate descriptions of the longitudinal and
transverse dynamics, we present a comparison of the DW motion under a thermal
gradient using both atomistic and AFM-LLB models and compare to previously
derived analytical expressions for the DW velocity. As we are considering both
longitudinal and transverse dynamics, the LLB equation now reads

$\displaystyle\frac{{d\mathbf{m}}_{\nu}}{dt}=\gamma\left[\mathbf{m}_{\nu}\times\mathbf{H}_{\text{eff},\nu}\right]-\gamma\alpha_{\|}^{\nu}\frac{\left(\mathbf{m}_{\nu}\cdot\mathbf{H}_{\text{eff},\nu}\right)}{m_{\nu}^{2}}\mathbf{m}_{\nu}$
$\displaystyle-\gamma\alpha_{\perp}^{\nu}\frac{\left[\mathbf{m}_{\nu}\times\left[\mathbf{m}_{\nu}\times\mathbf{H}_{\text{eff},\nu}\right]\right]}{m_{\nu}^{2}}$
(18)

To be able to describe DWs within an LLB framework, we introduce a term into
the LLB Hamiltonian that describes the exchange coupling between neighbouring
macrospins that is given by:

$\mathbf{H}_{\mathrm{ex}}^{v,i}=\frac{2A(T)}{d^{2}M_{S}m_{e}^{2}}\sum_{j}\left(\mathbf{m}_{v,j}-\mathbf{m}_{v,i}\right)$
(19)

Where $A(T)$ is the exchange stiffness, $d^{2}$ is the surface area between
neighbouring macrospins, and $M_{S}$ is the saturation magnetization. The
value and temperature dependence of the exchange stiffness is calculated using
atomistic simulations of the DW width, $\delta_{0}$. The relation that links
the DW width to the exchange stiffness is given by

$\delta_{0}(T)=\sqrt{\frac{A(T)}{K(T)}}$ (20)

where $K(T)$ is the anisotropy energy where again we use Callen-Callen scaling
[61], $K(T)=Km^{3}_{e}(T)$. The exchange stiffness, $A(T)$ scales with
$m_{e}^{2}$ within a MFA. Fig. 10 shows the exchange stiffness as a function
of temperature with points representing atomistic simulation results, and the
dotted line showing the scaling law $m_{e}^{2}$.

Figure 10: Exchange stiffness as a function of temperature from atomistic
simulations. Dotted line shows the proportionality to the equilibrium
magnetization with $A(0)$ taken from Eq. (22).

For the atomistic simulations of the domain wall width, we ensure the length
of the system in the $x$-direction is much wider than the zero Kelvin domain
wall width. The Mn sites at the left hand boundary of the slab remain fixed in
an anti-parallel alignment to the spins at the right hand boundary. The
remainder of the spins are allowed to relax to form a domain wall profile. The
reduced magnetization, $m(x)$, is then fitted to a hyperbolic tangent function
to find the width parameter:

Figure 11: The domain wall width at 0 K, 600 K and 1200 K. Solid lines are
from LLB simulations and points are from ASD. The Néel domain-wall width is
given by $\delta(T)=\pi\delta_{0}(T)$. The dotted lines show the width of the
domain wall at 0 K.

$m(x)=m_{e}\tanh\Big{(}\frac{x-b}{\delta_{0}}\Big{)}$ (21)

where $b$ is the central position of the DW, and $\delta_{0}$ is the wall-
width parameter. A comparison of the DW profile at $T=$0 K can be found in
Fig. 11. The Néel wall width can be related to the wall width parameter
through $\delta=\pi\delta_{0}$. We calculate $\delta(0)=31.2$ nm from
atomistic simulations and see good agreement between ASD and LLB in Fig. 11
for intermediate and high temperatures.

It is possible to relate the zero temperature exchange stiffness $A(0)$
directly to the $J_{ij}$ values and lattice constants found in Table 1 through
the equation [85]

$A(0)=\frac{1}{V_{0}}\sum_{i,v}\frac{\left|J_{i,v}\right|}{\left(a_{i,v}\right)^{2}}$
(22)

where $V_{0}$ is the unit cell volume, $v$ represents all the neighbours for
site $i$ where $x$ or $y$ are non-zero as the domain wall forms in the
$xy$-plane, and $a_{v}$ is the absolute distance between site $i$ and site
$v$. In total, there are 12 interactions that contribute to $A(0)$. Referring
to Fig. 1, the only interaction that doesn’t contribute is $J_{4}$. Using the
values for $J_{1}$ to $J_{3}$ in Table 1 yields $A(T=0\ \mathrm{K})=2.34\times
10^{-11}$ J/m, which agrees well with calculations of the exchange stiffness
through atomistic simulations of the DW width, as shown in Fig. 10 We should
emphasize here that we use a different equation for the exchange coupling to
Chen et al [55]. They use an exchange term that sums over the opposite
sublattice in the neighbouring macrospins. This term would be non-zero when
sublattices in neighbouring macrospins are parallel, leading to a disagreement
between LLB and ASD simulations of the AFMR (see Sec. II).

For the atomistic calculations of the DW velocity, we begin with a DW
initialised at 0 Kelvin along the $x-$direction, we then apply a linear
thermal gradient in the direction of the DW. It has been shown that the DW
velocity induced by a thermal gradient in an AFM can be approximated by [86]

$v_{\mathrm{DW}}^{\mathrm{LLB}}=\frac{2\gamma}{M_{s}\alpha_{\perp}}\frac{\partial
T}{\partial z}\frac{\partial A}{\partial T}$ (23)

As seen in previous analyses of domain walls [86, 84], the weak temperature
dependence of $\alpha_{\perp}$ is neglected and the exchange stiffness is
linearized to $dA/dT\approx A(0)/T_{C}$. Fig. 12 shows the domain wall
velocity for Mn2Au in atomistic and LLB simulations due to a thermal gradient,
showing good agreement with Eq. (23). The velocity is calculated by, once
again, fitting the DW profile to Eq. (21) to find the centre of the domain
wall, then tracking the movement of this central position in time. While the
relation appears linear in Fig. 12, in reality the velocity will reach a
saturation point governed by the magnon group velocity [87]. In the atomistic
modelling of thermally induced DW motion, Selzer et al [84] state that there
is no acceleration phase, and the DW moves with constant velocity, which is
what we observe in Mn2Au in both LLG and LLB simulations.

Figure 12: Domain Wall velocity as a function of the temperature gradient.
$\lambda$ was set to 0.01 for the simulations. The solid line is given by Eq.
(23). The inset shows an example of the domain wall at three different
timesteps. The wall is moving from left to right in the figure.

##  VI Summary

Using a multiscale approach, it has been demonstrated that antiferromagnets
such as Mn2Au can be modelled micromagnetically on micrometer length scales
using an AFM-LLB model. We started with parameters from previous work for the
exchange, anisotropy and magnetic moment which were fed into an atomistic
model of Mn2Au. Then, we calculated the temperature dependent parameters
$m_{e}(T)$, $A(T)$, $\widetilde{\chi}_{\|}(T)$ using atomistic spin dynamics
before using these as input into the AFM-LLB.

To verify the dynamics described by the proposed LLB model for Mn2Au, first we
presented consistent results between ASD, LLB and the Kittel relation for the
antiferromagnetic resonance frequency. As well as serving as a validation of
the transverse dynamics, we have revealed the first estimate of the in-plane
AFMR frequency in Mn2Au. The fact the resonant frequency sits within the THz
range opens the possibility sub-picosecond switching and generation of THz
electromagnetic signals at ambient temperatures. Second, we compared ASD
simulations results to the AFM-LLB model for the longitudinal dynamics
following step changes in temperature and laser pulse heating - both of which
give excellent agreement between the models. Finally, we provided a comparison
of the domain wall motion due to a thermal gradient as an example of how the
LLB model for an AFM can be utilised on $\mu$m length and $\mu$s timescales -
opening the door to micromagnetic simulations of AFM materials for use in
realistic spintronic devices. While questions such as the damping dependence
of the transverse relaxation time, and the poor agreement for the longitudinal
relaxation between ASD and LLB with $\widetilde{\chi}$ from ASD remain
unanswered, the AFM-LLB opens the possibility for the description laser
induced local thermal gradients on length-scales and the benchmark of other
thermodynamic effects in AFMs on micrometer length-scales.

##  VII Acknowledgements

This work was supported by the EPSRC TERASWITCH project (project ID
EP/T027916/1). Simulations were undertaken using the HPC cluster at Sheffield
Hallam University and resources provided by the Cambridge Tier-2 system
operated by the University of Cambridge Research Computing Service
(www.hpc.cam.ac.uk) funded by EPSRC Tier-2 capital grant EP/P020259/1. U. A.
gratefully acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation)—Project-ID 328545488—TRR 227, Project No. A08. J.
H. and U.A. acknowledge support from COST-Action CA17123.

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# Quantum non-locality:
from denigration to the Nobel prize, via quantum cryptography

Nicolas Gisin  Group of Applied Physics, University of Geneva, 1211 Geneva 4,
Switzerland
Constructor University, Geneva, Switzerland

In the late 1960s, a young physicist was sailing along the coast of California
towards Berkeley, where he got a post-doc position in astronomy. But his real
goal was not astronomy, at least not immediately. First, John Clauser eagerly
wanted to test some predictions of quantum theory that were at odds with a
then recent and mostly ignored result by an Irish physicist John Stewart Bell,
working at the celebrated CERN near Geneva. Bell, inspired by David Bohm’s
hidden variable model of quantum theory, proved that all possible correlations
that can be described by local variables necessarily satisfy some
inequalities, today known as Bell inequalities. These inequalities are
mathematically quite trivial. However, quantum theory predicts that they can
be violated even when the correlation is between outcomes of far distant
measurements. Denote $a$ and $b$ the measurement outcomes and $x$ and $y$ the
measurement settings (e.g. polarizers’ orientations), and denote $\lambda$ the
hypothetical hidden local variables. Accordingly, the entire statistics of the
experiment is captured by the so-called “correlation” - strictly speaking,
conditional probability distribution - $p(a,b|x,y,\lambda)$. The $\lambda$’s
are hidden in the sense that they are not part of quantum theory, though the
usual quantum state $\psi$ could well be a part of $\lambda$. Here, local - or
Bell-local - refers to the assumption that the correlation factorizes in two
parts, one for each of the distant sides of the experiment:

$p(a,b|x,y,\lambda)=p(a|x,\lambda)\cdot p(b|y,\lambda)$ (1)

That’s the only assumption necessary to derive Bell inequalities. The
$\lambda$’s denote the state of the system as described by any possible future
physical theory (except that the settings $x$ and $y$ are assumed to be
independent of $\lambda$). In this sense, Bell inequalities go way beyond
quantum theory: a violation of a Bell inequality proves that no future theory
can satisfy the locality condition (1).

John Clauser, Abner Shimony, Michael Horne and Richard Holt were among the
very few who understood this in the 1960s and all wanted to test Bell
inequalities, Clauser to prove quantum theory wrong, Holt, a young student at
Harvard, to prove the Bell-locality assumption (1) wrong. Clauser was in a
good position thanks to existing equipment at Berkeley. Indeed, Carl Kocher
had done a similar experiment in 1967, though for other purposes.
Unfortunately, Kocher, and even earlier Chien-Shiung Wu, had only measured the
correlations when the polarizers were either parallel or orthogonal, while a
proper violation of Bell inequality requires intermediate orientations. Note
that assuming that polarization is a 2-dimensional quantum system, a qubit as
one says today, correlations at $45^{o}$ can be derived from the parallel and
orthogonal correlations assuming no-signaling [1]:
$E_{45}=(E_{\parallel}+E_{\perp})/\sqrt{2}$. That wasn’t known at the time.
But regardless, the visibilities measured by Kocher and Wu were below 50%,
while a proper violation requires visibilities larger than 71%. Hence the race
was on. Clauser got there first, confirming quantum predictions, against his
expectation. But then Holt obtained his own result, confirming the inequality,
against his expectation. Somehow, the score was one to one.

At that time, these fascinating and intriguing results interested almost no
one, except some hippies who could later claim to have saved physics [2].
Clauser had long discussions with them, though the last time I met him he had
turned into a loud climate skeptic.

In the 1970’s, my friend Alain Aspect was doing his French civil service in
Africa, reading physics, as we all do. When he hit on Bell inequalities, it
was love at first sight: “I want to work on that”. Back in Paris, he traveled
to Geneva to meet John Bell and told him about his plans. Bell replied: “Do
you have a permanent position?”. Indeed, in those times, working on - or even
just showing interest in - Bell inequalities was a kind of scientific suicide.
Bohr had it all solved, went the dogma. Looking back, it is difficult to
appreciate how deeply denigrated was all research around Bell inequalities and
entanglement - the quantum resource necessary to violate them. At the time,
French had no agreed-upon translation of entanglement, some used
“enchevêtrement”, others “intrication” (the latter has by now been officially
recognized by the French academy).

Fortunately, the French system allowed young physicists such as Aspect to hold
permanent positions, so he decided to score the winning goal. Crucially, he
planned to add fast switches that would allow one to chose the measurement
settings $x$ and $y$ while the photons were already too far away to possibly
influence the other side. Aspect was able to achieve this using newly
developed lasers to pump his entanglement source, while Clauser and Holt had
to use flash lamps. In a series of three beautiful experiments in the early
1980’s, Aspect settled the dispute in favor of quantum theory. Accordingly, no
future theory will ever satisfy the locality condition (1). Today, this is
often expressed by the short expression non-local, which really means not-
Bell-local.

Despite these beautiful experiments and the intellectually fascinating
discoveries, Bell inequalities remained dismissed and poorly understood. Even
to this day, the clear terminology non-local (equivalently, not-Bell-local) is
too often blurred as not satisfying “local-realism”, as if non-realism was a
way out [3, 4]. The fact is that assumption (1) is no longer tenable. As an
example, consider the scientific background provided by the Nobel Committee
[5]. A few lines after correctly presenting Bohm’s non-local hidden variable
model, one reads that Bell inequality violation shows “that no hidden variable
theory would be able to reproduce all the results of quantum mechanics”,
contradicting the just cited Bohm model (which does predict violation of Bell
inequalities). The correct statement is that no _local_ variable theory is
able to reproduce all results of quantum mechanics. And a few lines further,
locality is defined as no-signaling - no communication without any physical
object carrying the information, despite the fact that one of the main
contribution of quantum information to the foundations of physics is a clear
distinction between these two concepts. Next, realism is defined as
determinism, even though Bell inequalities also hold in all stochastic
theories satisfying (1). All this illustrates that Bell inequalities are still
poorly understood by the general physics community. The 2022 Nobel Prize in
physics allows one to hope that henceforth Bell inequalities will be part of
all physics cursus.

One major step towards a better appreciation of Bell inequalities came from a
young Polish PhD student at Oxford University, Artur Ekert. In 1991, he
realized that non-local quantum correlations are nothing but cryptographic
keys! Indeed, in both cases, the correlation is private and, after some error
correction, the bits on both sides are identical. This proposal to exploit
non-local correlations for cryptographic applications changed everything
(though it took several years to prove Ekert’s intuition correct [6, 7]).
Moreover, just a few years later, Peter Shor showed how one can exploit
entanglement to break the commonly used public key cryptography system RSA.
Thus, in the 1990’s, non-locality and entanglement were in the spotlight, at
last.

But that would not have sufficed. The entanglement source used so far was too
complex. Leonard Mandel, at Rochester University, realized that a humble non-
linear crystal could provide highly entangled photons when pumped by a simple
diode. Moreover, the entangled photons could easily be coupled into optical
fibers, opening thus the road to quantum cryptography using existing
infrastructure, e.g. our demonstrations of quantum non-locality over the
Swisscom network and quantum cryptography under Lake Geneva [8], illustrated
in Fig. 1.

Figure 1: Quantum cryptography under Lake Geneva was the first quantum
experiment requiring a satellite photo to illustrate it. Nowadays in
commercial use [9].

The focus thus changed from foundations of quantum physics to quantum
information science and technologies. New ideas emerged, like quantum
teleportation and quantum error correction, in addition to fast experimental
progress. Anton Zeilinger had been interested in foundations since his early
days as a physicist in neutron interferometry. He quickly joined the quantum
information community and became a leading figure. His demonstration of
quantum teleportation, immediately after the one in Rome, by De Martini and
Popescu, attracted enormous attention, both within the scientific community
and from the public at large. Soon thereafter, Zeilinger went further and
demonstrated the teleportation of entanglement. Generally, quantum
teleportation is the resource behind quantum computation and long-distance
quantum communication. Zeilinger also improved on Aspect’s experiments in fast
choices of the measurement settings (though with low detection efficiencies,
hence much simpler experiments were possible [10]). Next, the so-called
detection loophole was closed in an optical experiment by Paul Kwiat’s group
at Ilinois University [11], before a series of “final” loophole-free
experiments, one of which was carried out by Zeilinger’s group. Zeilinger
continued with a long series of remarkable experiments, including dense
coding, and the demonstration of 3 and 4 photon entanglement, culminating with
the long-distance free-space communication in the Canary Islands, making him a
clear leader of the new field of experimental quantum information.

The next step was the understanding that entanglement is actually not
necessary for point-to-point quantum cryptography, though it remains essential
for the security proofs. Indeed, while in tests of Bell inequalities one sets
the source about half way between the measurement devices, in applications it
is much more practical to put it on one side. Hence, a mere single photon
travels to the receiver. This, and a few additional tricks - especially the
move to wavelengths compatible with standard telecom optical fibers, which we
initiated in Geneva with the development of specific single-photon detectors,
made it possible to not only demonstrate quantum cryptography, but to
industrialize and commercialize it. Today, there are many small companies
selling quantum cryptography equipments, some quite advanced as illustrated in
Fig. 2. Development efforts will continue, but no longer with the aim of
excluding local variables satisfying (1); the goals now are to make the
equipments cheaper, faster and able to cover longer distances, probably by
exploiting quantum teleportation.

On the conceptual side, the violation of Bell inequalities dramatically
revolutionized our world-view. Interestingly, Newton’s theory of gravity was
also non-local, even signaling. But Einstein improved on it, making gravity
local. It is thus not surprising that he strongly objected to quantum non-
locality, not fully appreciating that it is of a very different sort: without
any action at a distance, just non-local randomness without any possibility to
use it for signaling [12, 13]. In contrast to Newton’s non-locality, quantum
non-locality is here to stay; the experimental evidence is clear on that
point. Today, non-local quantum correlations are explored for device-
independent quantum information processing [7], in particular device-
independent quantum cryptography, a truly fascinating research field
unthinkable before Bell’s work. Another timely and exciting conceptual goal is
to take non-locality beyond the simple Bell scenario and place it in the
context of quantum networks with several independent sources of entanglement
[14]; this already led to the remarkable result that some quantum networks
can’t be described using only real-number Hilbert spaces [15].

Sincere congratulations John, Alain and Anton, you made me so happy.
Congratulations also to the Nobel Committee for recognizing, finally, the
game-changing findings of the late John Stewart Bell, with whom, along with
his wife Mary, I had the pleasure of sharing several cheese raclettes in
downtown Geneva.

Figure 2: A picture of Aspect’s lab around 1981, followed by a commercial
quantum cryptography equipment [9], with the entanglement source used by
Zeilinger in the middle (photo by Paul Kwiat and Michael Reck).

## Acknowledgement

Many thanks to Benjamin Feddersen for polishing my English.

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