# Magnetically induced polarization in centrosymmetric bonds

Igor Solovyev<EMAIL_ADDRESS>National Institute for Materials
Science, MANA, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan  Department of
Theoretical Physics and Applied Mathematics, Ural Federal University, Mira
str. 19, 620002 Ekaterinburg, Russia  Institute of Metal Physics, S.
Kovalevskaya str. 18, 620108 Ekaterinburg, Russia  Ryota Ono  Graduate School
of Science and Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-
shi 265-8522, Japan  Sergey Nikolaev  National Institute for Materials
Science, MANA, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan  Tokyo Tech World
Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo
Institute of Technology, 4259 Nagatsuta, Midori-Ku, Yokohama, Kanagawa,
226-8503, Japan  Laboratory for Materials and Structures, Tokyo Institute of
Technology, 4259 Nagatsuta, Midori-Ku, Yokohama, Kanagawa, 226-8503, Japan

###### Abstract

We reveal the microscopic origin of electric polarization $\vec{P}$ induced by
noncollinear magnetic order. We show that in Mott insulators, such $\vec{P}$
is given by all possible combinations of position operators
$\hat{\vec{r}}_{ij}=(\vec{r}_{ij}^{\,0},\vec{\boldsymbol{r}}_{ij}^{\phantom{0}})$
and transfer integrals
$\hat{t}_{ij}=(t_{ij}^{0},\boldsymbol{t}_{ij}^{\phantom{0}})$ in the bonds,
where $\vec{r}_{ij}^{\,0}$ and $t_{ij}^{0}$ are spin-independent contributions
in the basis of Kramers doublet states, while
$\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$ and
$\boldsymbol{t}_{ij}^{\phantom{0}}$ stem solely from the spin-orbit
interaction. Among them, the combination
$t_{ij}^{0}\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$, which couples to the spin
current, remains finite in the centrosymmetric bonds, thus yielding finite
$\vec{P}$ in the case of noncollinear arrangement of spins. The form of the
magnetoelectric coupling, which is controlled by
$\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$, appears to be rich and is not
limited to the phenomenological law
$\vec{P}\sim\boldsymbol{\epsilon}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$
with $\boldsymbol{\epsilon}_{ij}$ being the bond vector connecting the spins
$\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{j}$. Using density-functional
theory, we illustrate how the proposed mechanism work in the spiral magnets
CuCl2, CuBr2, CuO, and $\alpha$-Li2IrO3, providing consistent explanation to
available experimental data.

_Introduction_. In order to make a material ferroelectric, it is essential to
break the inversion symmetry. Canonically, this implied some crystallographic
instability towards polar atomic displacements [1]. Nevertheless, there is a
very special class of materials, called multiferroics, where inversion
symmetry can be broken by magnetic means in an otherwise perfect
crystallographically centrosymmetric lattice [2]. The microscopic origin of
multiferroicity is the fundamental physical problem and its practical
realization is an important step towards mutual control of electric
polarization and magnetism in novel electronic devices.

There can be various scenarios of the magnetic inversion symmetry breaking. In
certain multiferroic materials, the inversion symmetry is microscopically
broken by local distortions, so that individual bonds can be formally
associated with some polarization vectors. When arranged in an
antiferroelectric manner, these bonds result in zero net polarization.
Nevertheless, if some of them become inequivalent owing to complex magnetic
order, the perfect cancellation of polarization vectors does not occur and the
system becomes ferroelectric.

However, what if the bond itself is centrosymmetric? Can it become
electrically polarized by magnetic means? An affirmative answer to these
questions was given by Katsura, Nagaosa, and Balatsky (KNB) [3], who
considered a very special microscopic model and argued that the noncollinear
alignment of spins can induce the polarization
$\vec{P}\sim\boldsymbol{\epsilon}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$,
which lies in the plane of spins $\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{j}$
and is perpendicular to the bond vector $\boldsymbol{\epsilon}_{ij}$. This
finding was supported by phenomenological considerations [4] and the proposed
mechanism was called the “spin-current mechanism”, which is widely used for
the analysis of magnetoelectric (ME) coupling in spiral magnets [5], typically
in combination with two other mechanisms – the “exchange striction” [6, 7] and
“spin-dependent $p$-$d$ hybridization” [8].

Nevertheless, the analysis remains largely phenomenological. First, the
properties of all spiral multiferroics are usually discussed from the
viewpoint of the spin-current model [9, 10, 11, 12, 13, 14]. However, there
are only few materials, such as CuCl2 [9] and CuBr2 [10], consisting solely of
the centrosymmetric bonds. In other materials, the situation is not so
straightforward [11, 12, 13, 14]: as the symmetry is low, the bonds are not
necessarily centrosymmetric, thus allowing for alternative explanations [15,
16, 17]. Then, if KNB model fails to explain the properties of spiral magnets,
it typically causes some confusion with identifying the problem and choosing a
suitable alternative [18]. Although density-functional theory (DFT) provides a
powerful tool for calculating the polarization [19, 20], the formal mapping of
DFT results on a specifically selected model [21] does not shed light on
microscopic mechanisms underlying this model.

In this Letter, we formulate a transparent microscopic theory of electric
polarization induced by noncollinear magnetic order. We explicitly show how
and why the noncollinear arrangement of spins gives rise to electric
polarization even in centrosymmetric bonds. We relate the ME coupling to
fundamental symmetry properties of the position operator and transfer
integrals in the basis of Kramers states and argue that the paradigm of the
spin-current induced polarization appears to be much richer and goes beyond
the phenomenological low
$\vec{P}\sim\boldsymbol{\epsilon}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$
[17, 18, 21]. We evaluate all relevant parameters on the basis of DFT and show
how they are manifested in the properties of real spiral magnets.

_Basic theory_. The simplest toy model, which captures the physics, is the
1-orbital Hubbard model with spin-orbit interaction (SOI),

$\hat{\cal{H}}=\sum\limits_{ij}\sum_{\sigma\sigma^{\prime}}t_{ij}^{\sigma\sigma^{\prime}}\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\phantom{\dagger}}_{j\sigma^{\prime}}+U\sum\limits_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow},$
(1)

where $\hat{c}^{\dagger}_{i\sigma}$ ($\hat{c}^{\phantom{\dagger}}_{i\sigma}$)
creates (annihilates) a hole with pseudospin $\sigma$$=$ $+$ or $-$ at site
$i$,
$\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\phantom{\dagger}}_{i\sigma}$,
$\hat{t}_{ij}=[t_{ij}^{\sigma\sigma^{\prime}}]$ are the transfer integrals,
and $U$ is the on-site Coulomb repulsion. The Wannier functions,
$|i\sigma\rangle=\hat{c}^{\dagger}_{i\sigma}|0\rangle$, can be chosen as the
Kramers pairs, transforming under the time reversal as
$\hat{T}|i$$\mp\rangle=\pm|i$$\pm\rangle$. All model parameters are derived
from DFT [22], in the subspace of Wannier states, which are primarily
responsible for the magnetism [20, 23]. The Coulomb $U$ is evaluated within
constrained random-phase approximation [24].

Since the polarization in metals is screened by free electrons, the
multiferroicity is the property of insulating state. Then, the problem can be
solved in the spirit of superexchange (SE) theory, by treating $\hat{t}_{ij}$
as a perturbation [25]. Let
$|\alpha_{i}^{o}\rangle=\cos\frac{\theta_{i}}{2}|i$$+\rangle-\sin\frac{\theta_{i}}{2}\,e^{-i\phi_{i}}|i$$-\rangle$
be the occupied orbital in the limit $\hat{t}_{ij}$$=$$0$, where $\theta_{i}$
and $\phi_{i}$ specify the direction
$\boldsymbol{e}_{i}=(\cos\phi_{i}\sin\theta_{i},\sin\phi_{i}\sin\theta_{i},\cos\theta_{i})$
of spin, and
$|\alpha_{i}^{u}\rangle=\sin\frac{\theta_{i}}{2}\,e^{i\phi_{i}}|i$$+\rangle+\cos\frac{\theta_{i}}{2}|i$$-\rangle$
is the unoccupied orbital. To the first order in $\frac{\hat{t}_{ij}}{U}$,
$|\alpha_{i}^{o}\rangle$ will transform to the Wannier function
$|w_{i}\rangle=|\alpha_{i}^{o}\rangle+\sum_{j}|\alpha_{i\to j}^{o}\rangle$,
acquiring the tails

$|\alpha_{i\to
j}^{o}\rangle=-\frac{1}{U}|\alpha_{j}^{u}\rangle\langle\alpha_{j}^{u}|\hat{t}_{ji}|\alpha_{i}^{o}\rangle$
(2)

on surrounding sites $j$. Considering expectation values of the kinetic energy
$\langle w_{i}|\hat{t}_{ij}|w_{i}\rangle$, one can readily formulate the spin
model

$\mathcal{E}=\sum\limits_{\langle
ij\rangle}\left(-J_{ij}\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}+\boldsymbol{D}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]+\boldsymbol{e}_{i}\cdot\tensor{\Gamma}_{ij}\boldsymbol{e}_{j}\right),$
(3)

describing the energy change in terms of isotropic ($J_{ij}$), Dzyaloshinskii-
Moriya (DM, $\boldsymbol{D}_{ij}$), and symmetric anisotropic
($\tensor{\Gamma}_{ij}$) interactions. A similar model can be formulated for
polarization [26]:

$\vec{P}=\sum\limits_{\langle
ij\rangle}\left(\vec{\mathsf{P}}_{ij}\,\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}+\vec{\boldsymbol{\mathcal{P}}}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]+\boldsymbol{e}_{i}\cdot\vec{\boldsymbol{\Pi}}_{ij}\boldsymbol{e}_{j}\right),$
(4)

in terms of the vector $\vec{\mathsf{P}}_{ij}\equiv[\mathsf{P}_{ij}^{v}]$,
rank-2 tensor
$\vec{\boldsymbol{\mathcal{P}}}_{ij}\equiv[\mathcal{P}_{ij}^{v,c}]$, and
rank-3 tensor $\vec{\boldsymbol{\Pi}}_{ij}\equiv[{\Pi}_{ij}^{v,ab}]$ [27]. The
model parameters can be obtained from matrix elements of the position operator
in the framework of general theory for polarization in periodic systems [19,
20]:

$\vec{P}=-\frac{e}{V}\sum\limits_{i}\langle w_{i}|\vec{r}\,|w_{i}\rangle$ (5)

(where $V$ is the volume and $-$$e$ is the electron charge). Then, the only
matrix elements that contribute to the magnetic dependence of $\vec{P}$ are of
the type $\langle\alpha_{i}^{o}|\vec{r}\,|\alpha_{i\to j}^{o}\rangle$. Other
contributions, such as $\langle\alpha_{i\to j}^{o}|\vec{r}\,|\alpha_{i\to
j}^{o}\rangle$ or single-ion anisotropy of $\vec{P}$, vanish in the 1-orbital
model [17, 18].

The $2$$\times$$2$ matrices $\hat{t}_{ij}$ and $\hat{\vec{r}}_{ij}$ can be
decomposed in terms of the unity $\hat{\mathbb{1}}$ and the vector
$\hat{\boldsymbol{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$
of Pauli matrices as
$\hat{t}_{ij}=t_{ij}^{0}\hat{\mathbb{1}}+i\boldsymbol{t}_{ij}^{\phantom{0}}\hat{\boldsymbol{\sigma}}$
and
$\hat{\vec{r}}_{ij}=\vec{r}_{ij}^{\,0}\hat{\mathbb{1}}+i\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}\hat{\boldsymbol{\sigma}}$
with the real coefficients $(t_{ij}^{0},\boldsymbol{t}_{ij}^{\phantom{0}})$
and $(\vec{r}_{ij}^{\,0},\vec{\boldsymbol{r}}_{ij}^{\phantom{0}})$.
Furthermore, the hermiticity yields $t_{ji}^{0}=t_{ij}^{0}$,
$\boldsymbol{t}_{ji}=-\boldsymbol{t}_{ij}$,
$\vec{r}_{ji}^{\,0}=\vec{r}_{ij}^{\,0}$, and
$\vec{\boldsymbol{r}}_{ji}=-\vec{\boldsymbol{r}}_{ij}$ [28]. The corresponding
spin model parameters are summarized in Table 1 [29].

Table 1: Isotropic ($J_{ij}$ and $\vec{\mathsf{P}}_{ij}$), antisymmetric ($\boldsymbol{D}_{ij}$ and $\vec{\boldsymbol{\mathcal{P}}}_{ij}$), and anisotropic symmetric ($\boldsymbol{\Gamma}_{ij}=\boldsymbol{\Gamma}_{ij}^{\prime}-\frac{1}{2}{\rm Tr}\boldsymbol{\Gamma}_{ij}^{\prime}$ and $\vec{\boldsymbol{\Pi}}_{ij}=\vec{\boldsymbol{\Pi}}_{ij}^{\prime}-\frac{1}{2}{\rm Tr}\vec{\boldsymbol{\Pi}}_{ij}^{\prime}$) parameters of exchange interactions and polarization. $\otimes$ denotes the direct product.  exchange | polarization
---|---
$J_{ij}=-\frac{(t_{ij}^{0})^{2}}{U}$ | $\vec{\mathsf{P}}_{ij}=-\frac{2e}{V}\frac{\vec{r}_{ij}^{\,0}t_{ij}^{0}}{U}$
$\boldsymbol{D}_{ij}=\frac{2t_{ij}^{0}\boldsymbol{t}_{ij}^{\phantom{0}}}{U}$ | $\vec{\boldsymbol{\mathcal{P}}}_{ij}=-\frac{2e}{V}\frac{\vec{r}_{ij}^{\,0}\boldsymbol{t}_{ij}^{\phantom{0}}+t_{ij}^{0}\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}}{U}$
$\boldsymbol{\Gamma}_{ij}^{\prime}=\frac{2\boldsymbol{t}_{ij}\otimes\boldsymbol{t}_{ij}}{U}$ | $\vec{\boldsymbol{\Pi}}_{ij}^{\prime}=-\frac{e}{V}\frac{\vec{\boldsymbol{r}}_{ij}\otimes\boldsymbol{t}_{ij}+\boldsymbol{t}_{ij}\otimes\vec{\boldsymbol{r}}_{ij}}{U}$

A very special case is when two sites $i$ and $j$ are connected by spacial
inversion. Since $\hat{t}_{ij}$ is a scalar, but $\hat{\vec{r}}_{ij}$ is a
(true) vector, inversion symmetry requires that $\hat{t}_{ji}=\hat{t}_{ij}$,
but $\hat{\vec{r}}_{ji}=-\hat{\vec{r}}_{ij}$. In combination with hermiticity,
we have $\boldsymbol{t}_{ij}=0$ and $\vec{r}_{ij}^{\,0}=0$. Then, the DM
interaction $\boldsymbol{D}_{ij}$ vanishes, which is the general property of
centrosymmetric (c) bonds, and so does $\boldsymbol{\Gamma}_{ij}$, being not
independent in the 1-orbital model [30]. Thus, the only interaction in the
c-bonds will be $J_{ij}$. The behavior of electric polarization is different:
$\vec{\mathsf{P}}_{ij}=0$ and $\vec{\boldsymbol{\Pi}}_{ij}=0$, while
$\vec{\boldsymbol{\mathcal{P}}}_{ij}=-\frac{2e}{V}\frac{t_{ij}^{0}\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}}{U}\equiv\vec{\boldsymbol{\mathcal{C}}}_{ij}$
can persist even in the c-bond (in agreement with symmetry arguments [21]),
and the corresponding polarization,
$\vec{\boldsymbol{\mathcal{C}}}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$,
is induced solely by the noncollinear arrangement of spins.

In the noncentrosymmetric (nc) bonds, $\boldsymbol{t}_{ij}$ and
$\vec{r}_{ij}^{\,0}$ are finite, resulting in nonzero $\boldsymbol{D}_{ij}$,
$\boldsymbol{\Gamma}_{ij}$, $\vec{\mathsf{P}}_{ij}$, and
$\vec{\boldsymbol{\Pi}}_{ij}$. Moreover, $\vec{\boldsymbol{\mathcal{P}}}_{ij}$
acquires an additional asymmetric contribution
$\vec{\boldsymbol{\mathcal{A}}}_{ij}=-\frac{2e}{V}\frac{\vec{r}_{ij}^{\,0}\boldsymbol{t}_{ij}^{\phantom{0}}}{U}$.
$\vec{\mathsf{P}}_{ij}$ and $\vec{\boldsymbol{\mathcal{A}}}_{ij}$ can be
viewed as a regular polarization of the nc-bond
($-\frac{e\vec{r}_{ij}^{\,0}}{V}$), which is additionally modulated by the
spin texture. If this texture results from the competition of
$\boldsymbol{D}_{ij}$ and $J_{ij}$, a similar competition takes place between
$\vec{\boldsymbol{\mathcal{A}}}_{ij}$ and $\vec{\mathsf{P}}_{ij}$, as it
depends on the same ratio
$\frac{\boldsymbol{t}_{ij}^{\phantom{0}}}{t_{ij}^{0}}$. Therefore, there will
be a _cancellation_ of contributions associated with
$\vec{\boldsymbol{\mathcal{A}}}_{ij}$ and $\vec{\mathsf{P}}_{ij}$ [18].

The 1-orbital model is subjected to hidden symmetries, that allow to fully
eliminate $\boldsymbol{t}_{ij}^{\phantom{0}}$ by rotating the spins at the
sites $i$ and $j$ [30]. In such local coordinate frame, the bond is solely
described by the transfer integral
$\tilde{t}_{ij}^{0}=\sqrt{(t_{ij}^{0})^{2}+\boldsymbol{t}_{ij}^{\phantom{0}}\cdot\boldsymbol{t}_{ij}^{\phantom{0}}}$,
exchange interactions – by $J_{ij}$, and the electric polarization – by the
competition of $\vec{\mathsf{P}}_{ij}$ and
$\vec{\boldsymbol{\mathcal{C}}}_{ij}$ (with $\tilde{t}_{ij}^{0}$ instead of
$t_{ij}^{0}$). Using
$|\alpha^{u}_{j}\rangle\langle\alpha^{u}_{j}|=\hat{\mathbb{1}}-|\alpha^{o}_{j}\rangle\langle\alpha^{o}_{j}|$
in Eq. (2), each term in the spin model can be further expressed via the
expectation value of some quantity in the ground state. Since
$[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ is related to the spin current
$\hat{\boldsymbol{j}}_{ij}^{s}=\frac{i\tilde{t}_{ij}^{0}}{2}\big{(}\hat{\tilde{c}}^{\dagger}_{i\sigma}\hat{\boldsymbol{\sigma}}_{\sigma\sigma^{\prime}}\hat{\tilde{c}}^{\phantom{\dagger}}_{j\sigma^{\prime}}-\hat{\tilde{c}}^{\dagger}_{j\sigma}\hat{\boldsymbol{\sigma}}_{\sigma\sigma^{\prime}}\hat{\tilde{c}}^{\phantom{\dagger}}_{i\sigma^{\prime}}\big{)}$
[31], one can find that
$\vec{P}\sim\vec{\boldsymbol{\mathcal{C}}}_{ij}\cdot\langle\boldsymbol{j}^{S}_{ij}\rangle$
[29], in analogy with the DM interactions [32].

The $3$$\times$$3$ tensor
$\vec{\boldsymbol{\mathcal{C}}}_{ij}=[\mathcal{C}_{ij}^{v,c}]$ can be
generally decomposed into symmetric and antisymmetric parts,
$\vec{\boldsymbol{\mathcal{C}}}_{ij}=\vec{\boldsymbol{\mathcal{C}}}_{ij}^{S}+\vec{\boldsymbol{\mathcal{C}}}_{ij}^{A}$.
The latter is expressed as $\varepsilon_{vca}\pi_{ij}^{a}$, in terms of the
vector $\boldsymbol{\pi}_{ij}=[\pi_{ij}^{a}]$ and Levi-Civita symbol
$\varepsilon_{vca}$. The corresponding
$\vec{P}=\boldsymbol{\pi}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$
reminds of the KNB expression [3, 4]. Nevertheless, $\boldsymbol{\pi}_{ij}$ is
not necessarily parallel to $\boldsymbol{\epsilon}_{ij}$, and
$\vec{\boldsymbol{\mathcal{C}}}$ can include
$\vec{\boldsymbol{\mathcal{C}}}_{ij}^{S}$. Thus, the proposed spin-current
theory is not limited to the conventional law
$\vec{P}\sim\boldsymbol{\epsilon}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$
and includes other interesting options. Below, we consider examples of how it
works for different symmetries.

_Relativistic $j$$=$$\frac{1}{2}$ manifold of $t_{2g}$ states_. In the cubic
environment, the sixfold degenerate $t_{2g}$ states are split by SOI into
fourfold degenerate $\Gamma_{8}$ states and the Kramers pair of $\Gamma_{7}$
states
$|+\rangle=\frac{1}{\sqrt{3}}(|xy$$\downarrow$$\rangle-|yz$$\uparrow$$\rangle+i|zx$$\uparrow$$\rangle)$
and
$|-\rangle=\frac{1}{\sqrt{3}}(|xy$$\uparrow$$\rangle+|yz$$\downarrow$$\rangle+i|zx$$\downarrow$$\rangle)$.
The latter can be viewed as the effective $j=\frac{1}{2}$ pseudospin states,
which play a key role in the physics of spin-orbit Mott insulators realized in
$5d$ Ir oxides [33]. Since the KNB expression was derived assuming this
symmetry of states [3], we start our analysis with this example and consider a
perfect bond obeying the $C_{\infty}^{z}$ symmetry along $z$. It is
straightforward to show that the only nonzero elements of
$\vec{\boldsymbol{r}}_{ij}=[r_{ij}^{v,c}]$ will be
$r_{ij}^{y,x}=-r_{ij}^{x,y}=\frac{1}{3}\,(\langle
xy_{i}|y|zx_{j}\rangle-\langle zx_{i}|y|xy_{j}\rangle)$ [29]. The
_antisymmetric_ tensor can be presented as $\varepsilon_{abz}\pi_{ij}^{z}$,
where $ab=$ $xy$ or $yx$, and the vector
$\boldsymbol{\pi}_{ij}=(0,0,\pi_{ij}^{z})$ is indeed parallel to the bond.
Thus, we do recover the KNB expression
$\vec{P}\sim\boldsymbol{\epsilon}_{ij}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$
[3]. Nevertheless, this form of $\vec{P}$ is the consequence of the
_particular symmetry of undistorted $t_{2g}$ states_. Other symmetries may
yield different $\vec{P}$.

_$z^{2}$ states with SOI_. The simplest example, illustrating this idea is the
$z^{2}$ states, which are stabilized by the crystal field and mixed with the
$yz$ and $zx$ states by SOI. The Kramers states are [34]:
$|+\rangle\propto|z^{2}$$\downarrow$$\rangle+\xi(i|yz$$\uparrow$$\rangle-|zx$$\uparrow$$\rangle)$
and
$|-\rangle\propto|z^{2}$$\uparrow$$\rangle+\xi(i|yz$$\downarrow$$\rangle+|zx$$\downarrow$$\rangle)$
($\xi$ being the ratio of SOI to the crystal field splitting), and the non-
vanishing elements $r_{ij}^{v,c}$ are:
$r_{ij}^{y,x}=r_{ij}^{x,y}\propto\langle z^{2}_{i}|x|zx_{j}\rangle-\langle
zx_{i}|x|z^{2}_{j}\rangle$ [29]. Thus, the tensor $[r_{ij}^{v,c}]$ is
_symmetric_ , meaning that for two noncollinear spins
$\boldsymbol{e}_{1,2}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\pm\cos\theta)$,
the polarization $\vec{P}\sim(\cos\varphi,-\sin\varphi,0)$ is still
perpendicular to the bond, but does not necessarily lie in the spin plane.

_$\alpha$ -Li2IrO3_. As the first realistic example, it is instructive to
consider the spin-orbit Mott insulator $\alpha$-Li2IrO3, which attracted a
great deal of attention as a possible material for realizing the Kitaev spin
liquid state [35]. In this monoclinic compound (space group $C2/m$), there are
two types of nearest c-bonds: the strongest one ($01$ in Fig. 1a) along the
monoclinic $b$ axis, which is transformed to itself by the twofold rotation
about $b$; and two rotationally non-invariant weak bonds ($02$ and $03$),
which are connected by the twofold rotation.

Figure 1:  (a) Tensor $\vec{\boldsymbol{\mathcal{C}}}_{ij}$ in the nearest
c-bonds of $\alpha$-Li2IrO3 (in $\mu$C/m2), in the local coordinate frames
denoted by $x$, $y$ and $z$. (b) Double-$\boldsymbol{q}$ magnetic structure
realized in $\alpha$-Li2IrO3. Inversion center is denoted by $*$. (c)
Magnetically induced polarization in the chains with $+\boldsymbol{q}$ and
$-\boldsymbol{q}$ depending on the rotation of the spin-spiral plane about the
monoclinic $b$ axis, as explained in (d) in the planes $ab$ and $bc$. $\theta$
is the angle formed by the spin-spiral plane and the monoclinic $a$ axis.

The minimal model was constructed for the magnetic $j$$=$$\frac{1}{2}$ bands
[29]. In the local coordinate frame, where the $z$ axis is along the bond, the
tensor $\vec{\boldsymbol{\mathcal{C}}}_{01}$ has only $x$ and $y$ components
(see Fig. 1a). Hence, the polarization is perpendicular to the bond and,
considering only the antisymmetric part of
$\vec{\boldsymbol{\mathcal{C}}}_{01}$, we would indeed obtain
$\vec{P}\sim\boldsymbol{\epsilon}_{01}\times[\boldsymbol{e}_{0}\times\boldsymbol{e}_{1}]$
[3, 4]. However, besides the antisymmetric contribution,
$\vec{\boldsymbol{\mathcal{C}}}_{01}$ clearly shows a strong symmetric one,
which is expected for this type of symmetry [36]. In the low-symmetry bonds
$02$ and $03$, $\vec{\boldsymbol{\mathcal{C}}}_{ij}$ is even more complex: all
elements are finite and inequivalent, so that the polarization can be
perpendicular as well as parallel to the bond.

Since $\alpha$-Li2IrO3 is a noncollinear magnet with
$\boldsymbol{q}\approx(\frac{1}{3},0,0)$ [37], it is interesting to ask
whether it can become multiferroic. The magnetic texture can be viewed as the
double-$\boldsymbol{q}$ spiral, propagating in alternating zigzag chains with
$+\boldsymbol{q}$ and $-\boldsymbol{q}$ (Fig. 1b). The spiral order induces an
appreciable polarization in each of the chains (Fig. 1c). However, the chains
are connected by the inversion operation, the contributions with
$+\boldsymbol{q}$ and $-\boldsymbol{q}$ cancel each other, and
$\alpha$-Li2IrO3 remains antiferroelectric.

_Copper dihalides_. CuCl2 and CuBr2 form the chain-like monoclinic structure
(space group $C2/m$), where all Cu-Cu bonds are centrosymmetric. Below the
Néel temperature ($T_{\rm N}=24$ and $74$ K for CuCl2 and CuBr2,
respectively), they develop a cycloidal magnetic order with
$\boldsymbol{q}\approx(1,\frac{1}{4},\frac{1}{2})$ (Fig. 2a), which coincides
with the onset of ferroelectricity [9, 10].

Figure 2:  (a) Crystal and magnetic structure of CrCl2 in the $ab$ plane. (b)
Rotations of the spin spiral plane about the axis $b$. (c) Angles $\theta$ and
$\theta_{P}=\tan^{-1}\frac{P^{z}}{P^{x}}$ specifying the spiral plane and
electric polarization, respectively. (d) and (e) Angle dependence of electric
polarization in CuCl2 and CuBr2.

The minimal model was constructed for the magnetic “Cu $x^{2}$-$y^{2}$” bands
[29]. The obtained parameters $J_{ij}$ nicely reproduce the experimental
cycloidal order [38]. Polarization induced by the cycloidal order is related
to $\vec{\boldsymbol{\mathcal{C}}}_{ij}$. Since twofold rotations about $b$
transform the chains to themselves, the symmetry properties of
$\vec{\boldsymbol{\mathcal{C}}}_{ij}$ in these chains are similar to
$\vec{\boldsymbol{\mathcal{C}}}_{01}$ in $\alpha$-Li2IrO3 (Fig. 1a) and the
polarization is expected to be perpendicular to $b$. In the nearest bonds $\pm
b$, the antisymmetric part of $\vec{\boldsymbol{\mathcal{C}}}_{\pm b}$ is
given by $\boldsymbol{\pi}_{\pm b}=(0,\pm 0.5,0)$ and $(0,\pm 8,0)$ $\mu$C/m2
for CuCl2 and CuBr2, respectively, which is too small to account for the total
$\vec{P}$, and a substantially larger contribution stems from the symmetric
part. The behavior of $\vec{P}$ (after summation over all bonds) is summarized
in Fig. 2d and e. For CuCl2, we note a good agreement with the experiment [9],
including deviation of $\vec{P}$ from the spin-spiral plane by the angle
$\theta+\theta_{P}$. Similar behavior is expected for CuBr2 with somewhat
larger $T_{\rm N}$ and $\vec{P}$ due to stronger SOI and hybridization
mediated by Br $4p$ states.

_Cupric oxide_. CuO with the centrosymmetric monoclinic structure (space group
$C2/c$) has attracted much attention as a simple binary material that becomes
multiferroic at exceptionally high temperature [13]. Ferroelectricity is
driven by a spiral magnetic order with
$\boldsymbol{q}\approx(\frac{1}{2},0,-\frac{1}{2})$ (see Fig. 3a) emerging
between $213$ and $230$ K [39] (and is expected even at room temperature under
hydrostatic pressure [40]).

Figure 3:  (a) Spiral magnetic order in CuO. (b)-(c) Nearest c\- and (d) nc-
bonds in the monoclinic planes $ab$ and $bc$ with the vectors of induced
polarization. (e) Electric polarization (total and centrosymmetric part
described by $\vec{\boldsymbol{\mathcal{C}}}_{ij}$) as the function of angle
$\theta$ formed by the spins in the $ac$ plane with the axis $a$. Here, $x$
and $y$ are chosen along $a$ and $b$, respectively, and $z$ is perpendicular
to $a$ and $b$. (f) Electric polarization perpendicular ($P^{\perp}$) and
parallel ($P^{\parallel}$) to the bond, calculated for the nearest c\- and nc-
bonds as a function of $\theta$.

Construction of the minimal model is similar to Cu dihalides [29]. Although
the crystal structure of CuO includes both c\- and nc-bonds (Fig. 3), the
parameters associated with the latter can be largely eliminated [41], while
$\vec{\boldsymbol{\mathcal{C}}}_{ij}$ is responsible for 90 % of $\vec{P}$
(Fig. 3e).

Since the symmetry is low, the form of $\vec{\boldsymbol{\mathcal{C}}}_{ij}$
is complex [29], and the polarization vectors in individual bonds are
specified by all three projections, as shown in Fig. 3. Notably, there can be
appreciable components along the bonds (Fig. 3f). Nevertheless, due to the
twofold rotations about $y$ ($b$), only the $y$ component of $\vec{P}$ will
survive after summation over all equivalent bonds. This is an important point
where the symmetry comes into play: the experimental polarization is induced
along the $y$ axis not because of the phenomenological rule
$\boldsymbol{q}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ [3, 4].
Quite the contrary, it is a consequence of the particular $C2/c$ symmetry of
CuO. Total $P^{b}\approx 55$ $\mu$C/m2 is comparable with the experimental
$P^{b}\sim 150$ $\mu$C/m2 [13, 42].

_Conclusion_. We have presented a toy theory revealing the fundamental origin
of the magnetic inversion symmetry breaking in centrosymmetric systems. Due to
intrinsic symmetries of the transfer integrals and position operator in the
basis of Kramers states, the combination
$t_{ij}^{0}\vec{\boldsymbol{r}}_{ij}^{\phantom{0}}$ remains finite, yielding
finite polarization for noncollinear spins. This polarization depends on the
symmetry of Kramers states, providing new alternatives beyond phenomenological
spin-current model. The abilities of proposed theory are illustrated on spiral
magnets CuCl2, CuBr2, CuO, and $\alpha$-Li2IrO3.

_Acknowledgement_. We are grateful to Anonymous Referee of Ref. [18] for
drawing our attention to intersite matrix elements of $\vec{r}$. I.S. was
supported by program AAAA-A18-118020190095-4 (Quantum).

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CPS

    Cyber-Physical System
NCS

    Networked Control System
LAN

    Local-Area Network
WLAN

    Wireless Local-Area Network
KPI

    Key Performance Indicator
WPAN

    Wireless Personal Area Network
CSMA/CD

    Carrier-Sense Multiple Access with Collision Detection
CLEAVE

    ControL bEnchmArking serVice on the Edge
OS

    Operating System
UDP

    User Datagram Protocol
TCP

    Transmission Control Protocol
RMS

    Root Mean Square
RTT

    Round-Trip Time
CI

    Confidence Interval
AP

    Access Point
API

    Application Programming Interface
SSF

    Swedish Foundation for Strategic Research
TECoSA

    Trustworthy Edge Computing Systems and Applications
ABC

    Abstract Base Class
URL

    Uniform Record Locator
AWS

    Amazon Web Services
EC2

    Elastic Compute 2
AMI

    Amazon Machine Image
SSH

    Secure Shell
IP

    Internet Protocol
VPN

    Virtual Private Network
NAT

    Network Address Translation
NTP

    Network Time Protocol
SDR

    Software-Defined Radio
VLAN

    Virtual Local-Area Network
APT

    Advaced Packaging Tool
LTE

    Long-Term Evolution
SSID

    Service Set Identifier
EPC

    Evolved Packet Core
UE

    User Equipment
eNodeB

    Evolved Node B
DNS

    Domain Name System
MNIST

    Modified National Institute of Standards and Technology
HTTP

    Hyper-Text Transfer Protocol
YAML

    YAML Ain’t Markup Language
O-RAN

    Open Radio Access Network
FOSS

    Free and Open Source Software
COSMOS

    Cloud enhanced Open Software defined MObile wireless testbed for city-Scale deployment
OMF

    ORBIT Management Framework
POWDER

    Platform for Open Wireless Data-driven Experimental Research
COTS

    Commercial Off-The-Shelves
RF

    Radio-Frequency
VM

    Virtual Machine
REST

    REpresentational Sate Transfer
OS

    Operating System
MaaS

    Metal-as-a-Service
OEDL

     ORBIT Management Framework (OMF) Experiment Description Language
RSpec

    Resource Specification
LXC

    LinuX Containers
MEC

    Mobile Edge Computing
USRP

    Universal Software Radio Peripheral
OAI

    OpenAirInterface
NR

    New Radio
FPGA

    Field-Programmable Gate Array

# Ainur: A Framework for Repeatable End-to-End Wireless Edge Computing Testbed
Research

Manuel Olguín Muñoz1 , Seyed Samie Mostafavi2 , Vishnu N. Moothedath3 , James
Gross4  0000-0002-3383-2335 0000-0001-9316-0414 0000-0002-2739-5060
0000-0001-6682-6559 School of Electrical Engineering & Computer Science
KTH Royal Institute of Technology, Sweden
Email: {1molguin, 2ssmos, 3vnmo<EMAIL_ADDRESS>
###### Abstract

Experimental research on wireless networking in combination with edge and
cloud computing has been the subject of explosive interest in the last decade.
This development has been driven by the increasing complexity of modern
wireless technologies and the extensive softwarization of these through
projects such as a Open Radio Access Network (O-RAN). In this context, a
number of small- to mid-scale testbeds have emerged, employing a variety of
technologies to target a wide array of use-cases and scenarios in the context
of novel mobile communication technologies such as 5G and beyond-5G. Little
work, however, has yet been devoted to developing a standard framework for
wireless testbed automation which is hardware-agnostic and compatible with
edge- and cloud-native technologies. Such a solution would simplify the
development of new testbeds by completely or partially removing the
requirement for custom management and orchestration software.

In this paper, we present the first such mostly hardware-agnostic wireless
testbed automation framework, _Ainur_. It is designed to configure, manage,
orchestrate, and deploy workloads from an end-to-end perspective. Ainur is
built on top of cloud-native technologies such as Docker, and is provided as
FOSS to the community through the KTH-EXPECA/Ainur repository on GitHub. We
demonstrate the utility of the platform with a series of scenarios, showcasing
in particular its flexibility with respect to physical link definition,
computation placement, and automation of arbitrarily complex experimental
scenarios.

##  I Introduction

Experimental research in the area of wireless networking has received ever
increasing attention over the last years, driven, on the one hand, by the
complexity of modern networked systems and corresponding applications. On the
other, networked systems are more and more based on software instead of
dedicated hardware, which allows experimental testbeds to be rededicated
simply through an update as system versions evolve — in contrast to the
redeployment of hardware necessitated $1015$ years ago. The complexity of
these systems, as well as their softwarization are expected to continue
growing, driving in turn an expanding interest in testbed-based experimental
research in wireless systems.

Over the last years, several small- to mid-scale testbeds have emerged that
leverage a large degree of freedom with respect to hardware and software, for
example the  1. COSMOS 2. POWDER 3. Drexel Grid Software-Defined Radio (SDR)
testbeds. COSMOS (_Cloud enhanced Open Software defined MObile wireless
testbed for city-Scale deployment_) is a testbed spanning an area of roughly
$1$ square mile ($2.6\text{\,}{\mathrm{km}}^{2}$) featuring SDRs,
$\mathrm{mm}$-wave equipment, optical fibers, cloud integration, and compute
for core network functionality and application data processing [1, 2]. It
contains over $200$ rooftop, intermediate, and mobile nodes, and is controlled
and managed by a central node. Cloud enhanced Open Software defined MObile
wireless testbed for city-Scale deployment (COSMOS) relies on the OMF
(originally developed for ORBIT [3]), and employs the  Experiment Description
Language (OEDL), a domain-specific imperative language based on Ruby, for
experiment development and definition.

POWDER (_Platform for Open Wireless Data-driven Experimental Research_)
promises research on wireless and mobile networks with a level of
programmability down to the waveform [4]. The testbed spans a
$15\text{\,}{\mathrm{km}}^{2}$ area and features about $15$ fixed programmable
radio nodes, based on off-the-shelves SDRs and featuring edge-like compute
capabilities and integration with cloud resources, which interact with $50$
mobile nodes. Platform for Open Wireless Data-driven Experimental Research
(POWDER) experiments are defined and developed in _profiles_ , which
correspond to Virtual Machine (VM) images containing the necessary software
and configurations. These profiles are defined through using Resource
Specification
(RSpec)111https://groups.geni.net/geni/wiki/GENIExperimenter/RSpecs documents.

Finally, the _Drexel Grid SDR Testbed_ features SDRs that connect either over-
the-air, through a channel emulator, or over a combination of the two, to
facilitate realistic and reproducible experimentation [5]. Primarily intended
for SDR-centric research, it does not integrate any core, cloud or edge
components. However, the testbed extensively employs the LinuX Containers
(LXC) runtime for the deployment of both experimental code and SDR software,
which affords users great freedom when it comes to development of experiments.

Experimentation is key to to fully understanding the implications of next-
generation wireless systems, cloud, and edge computing paradigms, and thus
more of these testbeds are sure to emerge in the near future. Yet, little work
has so far been devoted to general-purpose, hardware-agnostic software
frameworks for the management and automation of such systems. Existing
platforms implement their own, ad-hoc software solutions which are not
compatible with other testbeds, and in many cases are not even compatible with
reigning cloud-native standards. This is, for instance, the case with COSMOS
and POWDER; their reliance on domain-specific languages limits their
integration with cloud-native solutions, which generally build upon general-
purpose languages such as Python and Go. These testbeds further leverage
virtualization technology based on VMs instead of more lightweight and edge-
compatible solutions such as containers.

To the best of our knowledge, CloudRAFT [6] is the only work to tackle (to a
certain extent) this challenge. CloudRAFT corresponds to a cloud-based
framework for mobile network experimentation, with a focus on simplifying the
management of testbed resources. The goal of this project is to integrate,
coordinate, share, and improve upon existing testbeds, and employs pre-built
VMs containing the necessary software for experiments. Although it provides
some automation for testbed resource provisioning and experiment execution,
its focus is largely rather on the sharing and partitioning of testbed
systems. Testbeds currently working with CloudRAFT include a variety of
domain-specific setups, including SDR SDR-based testbed as a well as a ground
vehicular robot for mobility-related experimentation.

In this work, we present our solution to the challenge of testbed automation:
Ainur, a framework for wireless testbed automation with a specific focus on
end-to-end experimental research in the context of edge-computing using cloud-
and edge-native technologies. Ainur is designed to deploy experimental runs
from a workload perspective by configuring the physical testbed, initializing
all involved software components, deploying and executing the experimental
workload, collecting logs and data, and finally gracefully degrading the
system. The framework allows for dynamic, software-definition of physical and
logical links, network topology, cloud and edge computing resources, as well
as experimental workload deployment and orchestration. It heavily leverages
cloud-native technologies, such as Docker containers, in order to supports a
wide variety of different testbed hardware setups and experimental
configurations and workloads, as well as to be as easily extendable as
possible. Furthermore, we make Ainur available to the community as Free and
Open Source Software (FOSS). It can be obtained from the KTH-EXPECA/Ainur
repository on GitHub [7], released under an Apache version $2.0$ license.

The rest of this paper is structured as follows. Section II presents the
framework as well as the key concepts and technologies supporting its design
and architecture. This section also discusses briefly the assumptions made
about the underlying hardware on which the framework is set to run, and we
present an overview of our experimental testbed. Next, Section III describes
two demonstration procedures through which we will showcase the flexibility
and potential of this tool for the automatic, repeatable, end-to-end
experimentation in the context of wireless testbeds. Finally, in Section IV we
summarize our contributions and conclude this paper.

##  II The Ainur Framework

Figure 1: Layered structure of an Ainur experiment Figure 2: Lifecycle of an
experimental run in Ainur. Blocks in orange are optional. Figure 3: Network
structure assumed by Ainur.

Ainur is designed as an end-to-end wireless network testbed management
framework, fully flexible in terms of the communication network stack,
computation hosts, and the distributed application deployed on top. The core
goal of the framework is to facilitate the creation of the desired
communication and computation elements to discover their implications to the
performance of end-to-end distributed applications. Ainur achieves this by
offering a Python Application Programming Interface (API) which is used to
describe an end-to-end experiment in a procedural manner, and we plan to
eventually provide toolkits for declarative configuration of experiments.

Conceptually, in Ainur, an experiment is decomposed into a layered structure,
consisting of  1. the distributed application (workload)  2. computation hosts
3. communication networks connecting the hosts  . In Figure 1, the workload is
represented by a client-server process pair. These could correspond, for
instance, to the emulation of an inverted pendulum and a matching controller.
However, Ainur makes no assumptions about the nature of the workloads deployed
on the framework, and virtually any process or combination of processes can be
used.

Hosts correspond to either bare-metal machines or cloud instances on Amazon
Web Services (AWS) Elastic Compute 2 (EC2). Users are free to combine and
interconnect these in any configuration. Ainur can provision wired networks
over Ethernet, and supports a number of software-defined wireless
communication stacks, currently including WiFi, 4G Long-Term Evolution (LTE),
and 5G. To realize these various wireless communication protocols, it is
assumed that the testbed is equipped with Universal Software Radio Peripheral
(USRP) SDRs and some computation hosts dedicated to signal processing and/or
radio management.

To collect data from workloads, Ainur configures both a shared, distributed,
storage location which all processes can reach, as well as a logging service
which automatically captures structured and unstructured data from the
standard output of workload processes. The logging service additionally
collects data from the other two layers, and the resulting dataset could, for
instance, be analyzed to relate the performance of the control loop to the
quality of the wireless link.

###  II-A Ainur Software Stack

To realize our vision for an automated, flexible, cloud-native, and workload-
agnostic framework for cloud and edge computing experimentation, we built
Ainur on top of a combination of well-established tools and frameworks. Below
we briefly touch on the most important of these:

missingfnum@@desciitemPython:

Ainur is built on-top of Python 3.8. We chose this language for its
flexibility, ease of prototyping, and for the extensive ecosystem of third-
party cloud-native frameworks and libraries.

missingfnum@@desciitemAnsible:

One of the “cloud-native frameworks” mentioned above, Red Hat Ansible is a
powerful Python framework for bare-metal configuration, provisioning, and
automation. We use it extensively in Ainur for configuration of network
interfaces and services on managed hosts.

missingfnum@@desciitemContainers:

We leverage Docker containers to great extent for the virtualization and
orchestration of workloads, as well as the encapsulation and management of
complex network configurations. See Section II-C for more details.

missingfnum@@desciitemAWS and boto3:

Ainur employs the Amazon boto3 library for Python to directly interface with
AWS EC2 and deploy, configure, and manage remote cloud instances.

missingfnum@@desciitem OpenAirInterface (OAI):

OAI is an open-source project that implements 3GPP technology on general
purpose x86 computing hardware and off-the-shelf SDRs like the USRP [8]. Ainur
can deploy, configure, and manage OAI software components that implement 4G
LTE and 5G New Radio (NR).

missingfnum@@desciitemMango Communications 802.11:

Finally, the Mango Communications project implements real-time 802.11 (WiFi)
MAC and PHY in Xilinx Field-Programmable Gate Arrays. It can be used on a
variety of hardware platforms including USRP SDRs, and is employed in Ainur to
provision end-to-end WiFi links.

###  II-B Main Software Components

Ainur follows a layered architecture which closely mimics the conceptual
layers of the TCP/IP stack. Components are deployed in bottom-up order,
starting with the establishment of physical links, through the establishment
of Internet Protocol (IP) connectivity and deployment of links to the cloud,
and ending with the distribution and initialization of workloads on top of a
container orchestration layer. The architectural modules of the framework can
broadly be classified according to the below categories:

missingfnum@@desciitemConfiguration Layer:

The lowest layer of Ainur, it handles parsing of configuration files
describing experimental scenarios. Optional, as it is only needed when Ainur
is running experiments described in a declarative manner.

missingfnum@@desciitemLogging layer:

This layer handles logging across all layers of the system to a central
repository. Concretely, it manages the configuration and lifecycle of a
Fluentd server to which any component in the network can send logs. Currently,
the two main components relying on this scaffolding are the physical layer and
the workloads.

missingfnum@@desciitemPhysical Layer:

Creates and deploys the underlying physical connections of the workload data
network. This layer interacts with hardware such as managed switches and SDRs
(and their associated computation hosts) to create links between the desired
workload hosts. These links correspond to ethernet, WiFi, and even an entire
software-defined 4G LTE and 5G networks.

missingfnum@@desciitemCloud Layer:

Handles integration with cloud services (currently, AWS EC2). Only deployed in
the case of an experiment requiring cloud instances, it manages the
instantiation and configuration of remote cloud resources.

missingfnum@@desciitemIP Layer:

Configures and establishes IP layer connectivity of packets between hosts in
the experimental setup, both local and cloud. This includes assigning valid IP
addresses to hosts, establishing Virtual Private Network (VPN) routes to cloud
instances through a pair of pre-configured VPN gateways, and configuring
routing tables to ensure any two hosts in the workload network can communicate
with each other.

missingfnum@@desciitemWorkload Layer:

Finally, this layer deploys, scales, and orchestrates containerized workloads,
as well as configures a shared, distributed storage for workloads to store
data. This components leverages Docker Swarm to spin up and manage workload
containers on desired hosts. It also establishes overlay networks abstracting
away the physical topology and allowing containers to interconnect through
dynamically assigned hostnames.

###  II-C Containerization

Containers are a key cloud-native technology for the virtualization and
sandboxing of arbitrary processes [9]. They allow for easy packaging of
software in predefined, consistent, and conflict-free execution environments,
including all necessary dependencies. Containers are lightweight and portable
compared to VM-based virtualization, making them an ideal solution for the
distribution, deployment, and orchestration of software in distributed
computing environments. They deploy quickly and are very configurable, while
abstracting away the complexities and delays that come with in creating,
managing, and moving (potentially huge) VM disk images.

Ainur leverages containerization extensively across the framework, in order to
1. deploy and orchestrate workloads  2. support a wide spectrum of different
physical layer configurations out-of-the-box, as well as allow for easy
extension to new ones  3. support automated collection of logs  . Workloads in
the framework are deployed packaged in Docker containers, orchestrated through
docker-compose and Docker Swarm. This allows the framework to remain mostly
agnostic to the nature of the workloads, and thus support a wide ranged of
different applications and system architectures. It also allows for easy
deployment to different compute nodes in the network, without having to take
into consideration details such as required libraries and versions.

As expressed above, Ainur also leverages containers for the communication
stack. With the advent of Open Radio Access Network (O-RAN) and SDRs, all
components of the wireless network stack can run on general-purpose
processors. They can thus be containerized and distributed across multiple
hosts.

As a final point, the automation of logging is another advantage of using
containers and an orchestration framework. Ainur employs the Fluentd unified
layer for log collection. It natively integrates with Docker containers and
allows for the automatic collection of text data from the standard output and
error streams of processes executing inside containers. It automatically
decouples data sources from containers running on different systems and allows
Ainur to collect and classify the logs from any components of the network,
whether on the wireless stack or workload.

###  II-D General assumptions

Ainur makes a number of assumptions about its execution environment. Apart
from generic ones regarding the presence of necessary authentication
information for remote access to local and remote hosts and services, the
framework assumes a split network architecture such as the one depicted in
Figure 3. Ainur runs in a dedicated control host and the management plane
resides in a physically distinct network from the workload data. This is a key
requirement to be able to reconfigure the physical links in the workload data
network without disrupting management traffic. Also assumed is the existence
of VPN gateways to the cloud, time synchronization, and configured hostname
lookups (possibly using a Domain Name System (DNS) service).

###  II-E Obtaining Ainur

The framework is released as Free and Open Source Software under a permissive
Apache license. It can be downloaded from the Ainur repository [7] of the KTH-
EXPECA organization on GitHub. Given the software’s complexity, the repository
also includes detailed documentation on its requirements, configuration,
deployment, and execution, as well as links to external resource containing
information and guides about related concepts and services.

Figure 4: The possible workload offloading setups and physical layer
configurations used in the demo. Not pictured, in Phy Option 2:  1.
SDRprocessing hosts  2. additional base-station host for the Core Network and
Evolved Node B (eNodeB)  .

(a) Workload 1: MNIST image classifier

(b) Workload 2: inverted pendulum NCS

Figure 5: Demonstration workloads

##  III Demo

In this demo, we will show the flexibility of Ainur for running end-to-end
experimental workloads on both the Edge and the Cloud. This will be done
interactively, and members of the audience will be invited to propose testbed
configurations.

Figure 4 illustrates the possible configurations for the testbed. Workloads
are deployed on client hosts, and computation is offloaded either to an edge
server (cloudlet), or to AWS EC2 instances on the cloud. Communication between
the client- and server-sides of the workload will occur over one of three
possible physical link layer setups:

missingfnum@@desciitemWiFi:

SDR SDR is configured as an IEEE 802.11n access point, and client hosts
connect to it using on-board WiFi.

missingfnum@@desciitem4G LTE:

SDR SDR is configured as an 4G LTE base station, and another is configured as
an 4G LTE User Equipment (UE). Together these radios act as an LTE bridge
between the client-side and the server-side of the network, and all client-
server traffic is routed through them.

missingfnum@@desciitemEthernet:

connects everything through plain ethernet.

The number of workload instances (and therefore client hosts) deployed in each
execution of this demonstration will range from $110$. Workloads deployed to
the cloud will be able to target any of AWS’s datacenters.

We will employ two different workloads for this demonstration, illustrated in
Figure 5. The first of these (Figure 5(a)) consists of a proof-of-concept web
application which implements a simple classifier to identify hand-drawn digits
from the widely-used Modified National Institute of Standards and Technology
(MNIST) dataset [10]. The application consists of a client with a web
interface, and HTTP Hyper-Text Transfer Protocol (HTTP) server which responds
to requests from the client and performs the actual image recognition. This
workload is intended to showcase in an interactive manner the effects of
placing computation at different points of the network, and how Ainur
simplifies these deployments.

The second workload (Figure 5(b)) corresponds to NCS Networked Control System
(NCS) balancing an inverted pendulum, implemented on a software framework for
the emulation of NCSs using cloud-native technologies [11]. It consists of an
emulation of the physical inverted pendulum system plant and a software-
implemented proportional-differential controller. These components communicate
with each other over the User Datagram Protocol (UDP). This workload will be
used to showcase the utility of Ainur for automating the execution of batches
of experiments potentially including multiple different clients, servers, and
physical layers.

###  III-A Testbed Setup

This demonstration will be performed on a testbed consisting of  1.
$10$Raspberry Pi 4 Model B boards, acting as client-side workload hosts  2. a
Raspberry Pi 4 Model B acting as VPN gateway for the workload network  3. a
Raspberry Pi 4 Model B acting as VPN gateway for the management network, as
well as hosting the necessary Network Time Protocol (NTP) and DNS server
software  4. a i386 workstation, acting as an edge-side workload host  5. a
configurable number of AWS EC2 cloud instances acting as cloud servers  6. a
separate i386 workstation hosting the Fluent server and on which Ainur is
deployed as well.  These nodes are interconnected using a combination of
managed switches, SDRs, and VPN gateways; please refer to Figure 3 for an
architectural overview of this setup.

###  III-B Demo Procedure

####  III-B1 Workload 1

Figure 6: Screenshot of the user interface of demo workload 1. After deciding
on placement of the compute backend and the type of physical layer connecting
client and backend, participants will use this interface to interact with the
system.

Participants will be asked to choose  1. a location to offload computation to
(local (i.e. no offloading), edge, or any AWS datacenter)  2. a physical layer
for the first hop of the network (WiFi, 4G LTE, or Ethernet).  Through the
Ainur command line, we will deploy a single client-server pair according to
the specifications. Next, participants will be able to access the client web
interface and interact with the server by requesting classification of images
from the MNIST database. The server will respond to these, and the assigned
labels will be visible on the web interface together with timing statistics
for each request, see Figure 6.

####  III-B2 Workload 2

Participants will be asked to specify the full, arbitrarily complex,
experimental scenario, including the number of clients, physical layers
(single or multiple, and which clients are on each physical link), and
offloading configuration (number of instances on the edge vs. on the cloud,
which AWS datacenters to deploy to). Configuration will be specified through a
YAML file, which will then be parsed by Ainur for the automatic execution of
the scenario.

##  IV Conclusion

In this paper, we have introduced the Ainur framework for repeatable end-to-
end testbed automation in the context of wireless networking and edge
computing research. The framework simplifies the execution and verification of
end-to-end experimentation in these testbeds by automating the  1.
establishment of physical links between hosts, including the configuration of
complex wireless systems such as 4G LTE and 5G  2. provisioning of and
connection to remote cloud instances  3. initialization of IP layer
connectivity between hosts  4. collection of logs and data  5. deployment,
scaling, and lifecycle management of containerized processes  . We have
described its general architecture, which follows a layered design mimicking
the network stack layers the framework directly interacts with, as well as the
underlying assumptions about its deployment environment and specific
requirements for its deployment.

Finally, we have outlined a demonstration which showcases the flexibility and
power of the framework by deploying two different workloads to our testbed. We
show how Ainur deployment of these to different points in the network, as well
as how the framework makes possible repeatable, large-scale experimentation in
distributed systems.

We believe our framework represents an important step towards repeatable,
replicable, yet low-access barrier end-to-end wireless testbed
experimentation. It has been released as FOSS and can be found on GitHub [7].

## References

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  * [2] Jiakai Yu et al.  “COSMOS: Optical Architecture and Prototyping”  In _2019 Optical Fiber Communications Conference and Exhibition (OFC)_ , 2019, pp. 1–3
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  * [7] “Ainur GitHub Repository”  In _GitHub_ EXPECA group at KTH  URL: https://github.com/KTH-EXPECA/Ainur
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  * [10] Li Deng  “The MNIST Database of Handwritten Digit Images for Machine Learning Research [Best of the Web]”  In _IEEE Signal Processing Magazine_ 29.6, 2012, pp. 141–142  DOI: 10.1109/MSP.2012.2211477
  * [11] Manuel Olguín Muñoz et al.  “CLEAVE: Scalable and Edge-Native Benchmarking of Networked Control Systems”  In _Proceedings of the 5th International Workshop on Edge Systems, Analytics and Networking_ , EdgeSys ’22  Rennes, France: Association for Computing Machinery, 2022, pp. 37–42  DOI: 10.1145/3517206.3526272

  *[OMF]: ORBIT Management Framework
  *[O-RAN]: Open Radio Access Network
  *[FOSS]: Free and Open Source Software
  *[COSMOS]: Cloud enhanced Open Software defined MObile wireless testbed for city-Scale deployment
  *[POWDER]: Platform for Open Wireless Data-driven Experimental Research
  *[SDR]: Software-Defined Radio
  *[SDRs]: Software-Defined Radio
  *[]:
  *[OEDL]:   () Experiment Description Language
  *[VM]: Virtual Machine
  *[RSpec]: Resource Specification
  *[ SDR]: Software-Defined Radio
  *[LXC]: LinuX Containers
  *[VMs]: Virtual Machine
  *[API]: Application Programming Interface
  *[AWS]: Amazon Web Services
  *[EC2]: Elastic Compute 2
  *[LTE]: Long-Term Evolution
  *[USRP]: Universal Software Radio Peripheral
  *[OAI]: OpenAirInterface
  *[NR]: New Radio
  *[TCP]: Transmission Control Protocol
  *[IP]: Internet Protocol
  *[VPN]: Virtual Private Network
  *[DNS]: Domain Name System
  *[eNodeB]: Evolved Node B
  *[MNIST]: Modified National Institute of Standards and Technology
  *[NCS]: Networked Control System
  *[UE]: User Equipment
  *[HTTP]: Hyper-Text Transfer Protocol
  *[NCSs]: Networked Control System
  *[UDP]: User Datagram Protocol
  *[NTP]: Network Time Protocol
  *[YAML]: YAML Ain’t Markup Language