# Mixed topology ring states for Hall effect and orbital magnetism
in skyrmions of Weyl semimetals

M. Redies<EMAIL_ADDRESS>Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Department of Physics, RWTH Aachen University, 52056 Aachen, Germany  F. R.
Lux  Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  Department of
Physics, RWTH Aachen University, 52056 Aachen, Germany  J.-P. Hanke  Institute
of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany  P. M.
Buhl  Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz,
Germany  S. Blügel  Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  Y.
Mokrousov<EMAIL_ADDRESS>Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz,
Germany

###### Abstract

As skyrmion lattices are attracting increasing attention owing to their
properties driven by real-space topology, properties of magnetic Weyl
semimetals with complex $k$-space topology are drawing increasing attention.
We consider Hall transport properties and orbital magnetism of skyrmion
lattices imprinted in topological semimetals, by employing a minimal model of
a mixed Weyl semimetal which, as a function of the magnetization direction,
exhibits two Chern insulator phases separated by a Weyl state. We find that
while the orbital magnetization is topologically robust and Hall transport
properties exhibit a behavior consistent with that expected for the recently
discovered chiral Hall effect Lux  _et al._ (2020), their evolution in the
region of the Chern insulator gap is largely determined by the properties of
the so-called mixed topology ring states, emerging in domain walls that
separate the skyrmion core from the ferromagnetic background. In particular,
we show that these localized ring states possess a robust orbital chirality
which reverses sign as a function of the skyrmion radius, thereby mediating a
smooth switching dynamics of the orbital magnetization. We speculate that
while the emergent ring states can possibly play a role in the physics of
Majorana states, probing their properties experimentally can provide insights
into the details of skyrmionic spin structures.

##  I Introduction

Topological chiral spin textures such as magnetic skyrmions have established
themselves as an exciting platform for the realization of novel physical
effects and for innovative ideas in the realm of practical applications of
magnetic systems Lai  _et al._ (2017); Zhang  _et al._ (2015a); Tomasello  _et
al._ (2015); Zhang  _et al._ (2020a); Zázvorka  _et al._ (2019); Bourianoff
_et al._ (2018); Pinna  _et al._ (2019); Zhang  _et al._ (2015b); Luo  _et
al._ (2018); Zhang  _et al._ (2020a). On the other side of the topology scale,
magnetic materials which exhibit non-trivial $k$-space topology in the
electronic structure $-$ such as quantum anomalous Hall insulators Deng  _et
al._ (2020); Chang  _et al._ (2013), or antiferromagnetic topological
insulators Otrokov  _et al._ (2019); Niu  _et al._ (2020) $-$ are at the heart
of current research in solid state physics and spintronics. Bringing together
the benefits of skyrmions, such as their efficient dynamics and stability,
with the advantages of $k$-topological materials, normally associated with
dissipationless transport and topological protection, appears to be an
exciting avenue to pursue. To date, however, skyrmions have been mostly
realized in metallic ferromagnetic materials that do not exhibit a distinct
non-trivial $k$-space topology, such as metallic ferromagnetic materials Miao
_et al._ (2014); Münzer  _et al._ (2010). And while the interest in skyrmions
realized in insulating materials is rising Zhang  _et al._ (2020b); Kong and
Zang (2013); Onose  _et al._ (2012), the emergence of global Chern insulating
states in skyrmion lattices has been recently shown theoretically Lado and
Fernández-Rossier (2015); Hamamoto  _et al._ (2015); Göbel  _et al._ (2018,
2017, 2019).

At the same time, another flavor of $k$-space topology exhibited by magnetic
Weyl semimetals is gaining increasing attention, and the number of specific
material candidates which exhibit distinct topological band crossings in their
electronic structure is constantly growing Wang  _et al._ (2017); Liu  _et
al._ (2019). In three dimensions, Weyl semimetals host bands crossings known
as Weyl nodes, that appear in pairs and carry an opposite non-zero topological
charge. Recently, it was suggested that in two dimensions (2D) it is natural
to interpret the emergence of topological band crossings in the context of
mixed Weyl points, emphasizing that their topology and properties can be
classified most optimally by including the magnetization direction
$\hat{\mathbf{m}}$ and the mixed components of the Berry curvature tensor into
the topological analysis Hanke  _et al._ (2017a); Niu  _et al._ (2019). The 2D
mixed Weyl semimetals thus behave in many aspects similarly to the 3D Weyl
semimetals in $(\mathbf{k},\hat{\mathbf{m}})$-space. Recently, it was realized
that the presence of Weyl points in spin textures in 2D or 3D Weyl semimetals
can have a drastic effect on their magneto-electric properties, orbital
magnetism and dynamics, see e.g. Refs. [Araki, 2020; Araki and Nomura, 2018;
Lux  _et al._ , 2018; Hanke  _et al._ , 2017a; Niu  _et al._ , 2019].

In this work, by performing explicit tight-binding calculations of skyrmion
lattices imprinted into a 2D Weyl semimetal, we attempt to understand the role
that the complex mixed topology can have on the Hall transport properties and
orbital magnetism in these systems. Our main finding is the demonstration that
properties of skyrmions of mixed Weyl semimetals, whose chemical potential
resides in the vicinity of Weyl points, are largely determined by so-called
ring states, which are localized at the skyrmion boundary and carry an orbital
moment of a specific orbital chirality. While we discuss how the transport
properties and orbital magnetization in these systems can be used to get
insights into the details of spin distribution, we also consider a range of
phenomena where ring states can be utilized for shaping skyrmion dynamics and
for mediating the emergence of novel topological phases. This article is
structured as follows. In Sec. II, we describe the tight-binding model and
parameters used in this paper. The details of the setup and transport
calculations are given in Sec. III. In Sec. IV we present and discuss the
results of our calculations, providing an outlook in Sec. V.

##  II Model

In order to assess the transport properties of 2D skyrmions in mixed Weyl
semimetals we choose the tight-binding model such that the underlying
electronic structure of the ferromagnetic host exhibits a mixed Weyl point in
an extended phase space of the magnetization direction and $k$-space. A
similar model has been used to study the topological properties of
ferromagnetic mixed Weyl semimetals in the past Hanke  _et al._ (2017a). The
tight-binding Hamiltonian of this model on a honeycomb lattice with two
structurally inequivalent atoms per unit cell and two spin-split orbitals per
site reads:

$\begin{split}H&=\,\lambda\sum_{i\epsilon\zeta}\left(\hat{\mathbf{m}_{i}}\cdot\boldsymbol{\sigma}\right)c_{i\epsilon}^{\dagger}c_{i\zeta}\\\
-t\sum_{\left<ij\right>\epsilon}c_{i\epsilon}^{\dagger}c_{j\epsilon}&+it_{\text{so}}\sum_{\left<ij\right>\epsilon\zeta}\hat{\mathbf{e}}_{z}\cdot\left(\boldsymbol{\sigma}\times\hat{\mathbf{d}}_{ij}\right)c_{i\epsilon}^{\dagger}c_{j\zeta}\end{split}$
(1)

The properties of this model of magnetic graphene with Rashba spin-orbit
interaction have been extensively studied in the past for collinear
ferromagnetic and antiferromagnetic cases Castro  _et al._ (2008); Matte  _et
al._ (2009); Qiao  _et al._ (2010). In Eq. (1), $t$ is the magnitude of the
nearest neighbour hopping, $t_{\text{so}}$ is the magnitude of Rasba-like
spin-orbit coupling Bychkov and Rashba (1984) , and $\lambda$ characterizes
the magnitude of the Stoner exchange splitting Stoner (1936, 1938). In the
above expression the indices $i$ and $j$ run over nearest-neighbor atoms,
while $\epsilon$ and $\zeta$ mark the spin channel. Further,
$\hat{\mathbf{e}}_{z}$ is the unit vector in the $z$-direction (out of the two
dimensional plane), while $\hat{\mathbf{d}}_{ij}$ is the unit vector
connecting atom sites $i$ and $j$. The operators $c_{i\epsilon}^{\dagger}$ and
$c_{i\epsilon}$ are the creation and annihilation operators of an electron on
site $i$ with spin $\epsilon$. The vector of Pauli matrices is denoted as
$\boldsymbol{\sigma}$, while $\hat{\mathbf{m}_{i}}$ is the unit vector of the
magnetization direction at atom $i$, which generally varies in real space when
a spin texture is present.

Throughout this work, we choose $t=-1.0$ eV, $t_{so}=~{}0.4$ eV and
$\lambda=1.4$ eV in order to realize a mixed Weyl point in the electronic
structure for the ferromagnetic case when all $\hat{\mathbf{m}_{i}}$ are
aligned along a single direction $\hat{\mathbf{m}}$. The bandstructure of the
model for the out-of-plane direction of the ferromagnetic magnetization with
these parameters is shown in Fig. 1(a). The emergence of the mixed Weyl point
upon varying the magnetization direction is shown in Fig. 1(c). The
topologically non-trivial character of this crossing point can be shown by
computing the flux of the Berry curvature tensor with components in $k$\- and
$\theta$-space, which quantifies the mixed topological charge of the mixed
Weyl point to be $+2$ for $\theta=\pi/2$, and $-2$ for $\theta=-\pi/2$ Niu
_et al._ (2019).

Figure 1: (a) The bandstructure of the ferromagnetic model with an out-of-
plane magnetization. The green dashed line marks the region $[-0.47,-0.46]$
eV. (b) The bandstructure of a skyrmion with the parameterization described in
the text. The red-shaded area in (a) and (b) indicates the band gap region of
about 0.6 eV of the ferromagnetic model. (c) A section of the bandstructure is
shown for different angles $\theta$ that the magnetization in the
ferromagnetic model makes with the $z$-axis. (d) The magnetization is
characterized by the azimuthal angle $\theta$ and the polar angle $\phi$. The
skyrmion is placed in the $xy$-plane. (e,f) The Hall conductance (e) and
orbital magnetization (f) of the ferromagnetic model with an out-of-plane
magnetization (along $+z$) as a function of band filling. Red shaded areas
coincide with the band gap marked in (a).  All plots except for (b) and (d)
are for $\phi=0$.

##  III Computational approach

Similarly to our previous work on transport properties of chiral bobbers
Redies  _et al._ (2019) we calculate the Hall conductance and orbital
magnetization of the system by employing the $k$-space Berry curvature
formalism, with the only non-vanishing component of the Berry curvature tensor
in $k$-space being

$\Omega_{xy}^{n}=-2\imaginary\innerproduct{\frac{\partial
u_{n\bm{k}}}{\partial k_{x}}}{\frac{\partial u_{n\bm{k}}}{\partial k_{y}}},$
(2)

where $u_{n\bm{k}}$ is the lattice-periodic part of the Bloch wave function of
the band $n$ for a Bloch vector $\bm{k}$. The intrinsic Hall conductance (in
units of $e^{2}/h$) is then evaluated as the Brillouin zone integral of the
Berry curvature of all occupied states Blügel  _et al._ (2017)

$\sigma_{xy}=\sum_{n}^{occ}\int\limits_{\rm
BZ}\Omega_{xy}^{n}(\bm{k})d\bm{k}.$ (3)

For an insulator the Hall conductance is quantized and it is proportional to
the value of an integer Chern number
$\mathcal{C}:=\int_{S}\Omega_{xy}dS/2\pi=-\sigma_{xy}/2\pi$. We calculate the
out-of-plane component of the orbital magnetization (OM) in the system,
$M_{orb}$, employing the modern theory of orbital magnetism Thonhauser  _et
al._ (2005); Xiao  _et al._ (2005); Thonhauser (2011), according to which

$\begin{split}M_{orb}=&M^{LC}_{orb}+M^{IC}_{orb}=\frac{e}{2\hbar
c}\imaginary\int\limits_{\rm BZ}\frac{d\bm{k}}{(2\pi)^{2}}\\\
&\sum_{n}^{occ.}\matrixelement{\frac{\partial u_{n\bm{k}}}{\partial
k_{x}}}{\times\left(H_{\bm{k}}+\varepsilon_{n\bm{k}}-2\mu\right)}{\frac{\partial
u_{n\bm{k}}}{\partial k_{y}}},\end{split}$ (4)

where $M^{LC}_{orb}$ and $M^{IC}_{orb}$ are the local circulation and
itinerant circulation parts of the OM, respectively, $\varepsilon_{n\bm{k}}$
is the energy of the $n$-th Bloch state at $\bm{k}$, and $H_{\bm{k}}$ is the
lattice-periodic part of the Hamiltionian
$H_{\bm{k}}:=e^{-i\bm{k}\bm{r}}He^{i\bm{k}\bm{r}}$. In order to compute the
derivatives of the Bloch states in the tight-binding basis we employ first-
order perturbation theory:

$\ket{\frac{\partial u_{n\bm{k}}}{\partial k_{i}}}=\sum_{m\neq
n}\frac{\matrixelement{u_{m\bm{k}}}{\nicefrac{{\partial H_{\bm{k}}}}{{\partial
k_{i}}}}{u_{n\bm{k}}}}{\varepsilon_{n\bm{k}}-\varepsilon_{m\bm{k}}+i\eta}\ket{u_{m\bm{k}}},$
(5)

which results in the well-known gauge-invariant expression for the Berry
curvature and OM Yao  _et al._ (2004); Thonhauser (2011). In order to avoid
divergences for (nearly) degenerate states a broadening $\eta=10^{-8}\mbox{
Hartree}$ is introduced.

The calculated Hall conductance (HC) and OM of the model as a function of band
filling for the out-of-plane magnetization are shown in Fig. 1(e) and (f),
respectively. As evident from these plots, at half filling the system is a
Chern insulator with the Chern number of $-2$. Therefore, the variation of the
OM in the topologically non-trivial gap is linear with the chemical potential
$\mu$, according to the relation valid for insulators Thonhauser (2011):

$\frac{dM_{orb}}{d\mu}=\frac{e}{(2\pi)^{2}\hbar c}\,\sigma_{xy},$ (6)

where $e$ is the electron charge and $c$ is the speed of light. The emergence
of the metallic point in the spectrum of the model is due to the change in the
Chern number from $-2$ to $+2$ at half filling upon reversing the direction of
$\hat{\mathbf{m}}$.

Based on the model (1) we imprint the skyrmion lattice by varying the
direction of $\hat{\mathbf{m}}_{i}$ in real space according to the
parametrization of the skyrmions discussed below. We consider a hexagonal
lattice of skyrmions where in the center of each unit cell we place a Néel
skyrmion.  Skyrmions of this type can be stabilized experimentally in the
presence of a small external magnetic field, the effect of which on the
electronic properties we do not take into account. The chemical unit cell
contains two atoms, which are located at $(0,\pm a,0)^{T}$. The lattice
vectors of the chemical unit cell are $(l_{\text{chem}},0,0)^{T}$ and
$l_{\text{chem}}\cdot(\frac{1}{2},\frac{\sqrt{3}}{2},0)^{T}$, where
$l_{\text{chem}}=\sqrt{3}a$ and $a$ is the distance between the inequivalent
atoms in the chemical unit cell. The unit cell of the skyrmion lattice is
hexagonal and contains 1568 atoms. The lattice vectors of the supercell are
$(l_{\text{mag}},0,0)^{T}$ and
$l_{\text{mag}}\cdot(\frac{1}{2},\frac{\sqrt{3}}{2},0)^{T}$, where
$l_{\text{mag}}=28\,l_{\text{chem}}$. This corresponds to 28 atoms in between
the centers of neighboring skyrmions and 1568 atoms in the unit cell. We use
an adaptive integration scheme to evaluate the integrals in Eqs. (3) and (4).
This leads to the convergence of the results presented in Figs. 3 and 4 with
6144 $k$-points, and results presented in Fig. 1(e,f) with 16042 $k$-points.
The full source code is available as open source Redies (2020).

##  IV Results

###  IV.1 Emergence of mixed topology ring states in skyrmions of mixed Weyl
semimetals

Here, we analyze the electronic structure of a skyrmion lattice of a mixed
Weyl semimetal at half filling, and in the following the zero of energy is
associated with the position of Fermi energy for the case when exactly two
electronic states per structural unit cell are occupied. For the ferromagnetic
system this corresponds to the position of the metallic Weyl point in the
spectrum when the magnetization is in-plane, see Fig. 1(c). Correspondingly,
in the limit of very large skyrmions, when the regions with homogeneous out-of
plane magnetization in the skyrmion center and in-between the skyrmions are
large, we expect an emergence of electronic states around the zero energy.
This state is expected to be localized to the region where the magnetization
lies in-plane, i.e. within the domain wall separating the skyrmion center with
the outside region.

Figure 2:  Emergence of mixed topology ring states (MTRS) in a skyrmion
lattice of a mixed Weyl semimetal with skyrmion radius of
$r_{\text{max}}=\frac{1}{2}\,l_{mag}$. (a-c) The local density of states
(LDOS) in real space integrated over the energy range from $-$0.3 to $+$0.3
eV, corresponding to the gap of the ferromagnetic system in Fig. 1(a,b). The
magnitude of the integrated LDOS is indicated in the color bar. The arrows to
the right indicate the spin direction along the orange line in the
$(xy)$-plane. Here the dark red (blue) arrows indicate a
$\mathbf{m}=(-)\hat{e}_{z}$ magnetization direction. In (a) $d=0.2$, in (b)
$d=0.5$ and in (c) $d=0.8$. The LDOS marks the formation of MTRS. (d) The LDOS
integrated in the energy range $[-0.47,-0.46]\mbox{ eV}$, away from the gap,
marked green in Fig 1(a). The magnetic structure is identical to (c). In this
energy region the MTRS are absent. In the center of the skyrmion
$\mathbf{m}=+\mathbf{e}_{z}$, while $\mathbf{m}=-\mathbf{e}_{z}$ at the edges
of the unit cell.

We verify this by performing explicit calculations of the electronic structure
of the skyrmion lattice with the model described above. The Néel skyrmions of
radius $R$ (with the maximal value of it being $R_{max}=l_{\rm mag}/2$), are
parameterized by the angle $\alpha$ that a spin at the distance $r$ from the
center of the skyrmion makes with the $z$-axis, while the parameter $d$ is
introduced to tune the width of the domain wall region where the spins are
lying in-plane  so that larger $d$ results in a larger region of “flat”
magnetization (see examples in Fig 2(a-c) for different values of $d$). Note
that the rate of change of the magnetization outside of the in-plane regions
does not change. We use this parametrization to artificially tune the internal
width of the states which emerge in the “flat” region. Below we consider the
case of $R=R_{max}/2$ when introducing the parameterization in terms of $r$
and $d$ as follows:
$\alpha(r,d)=\pi\frac{\beta(r/R_{max},d)-\beta(1,d)}{\beta(0,d)-\beta(1,d)}$,
with
$\beta(x,d)=\operatorname{atan}\left(R_{max}(6x-3d-3)\right)+\operatorname{atan}\left(R_{max}(6x+3d-3)\right)$.
As we show below, this parameterization allows us to manipulate the spatial
spread of states associated with the transition between the two out-of-plane
domains.

In order to analyze the spatial localization of the electronic states, we
compute the space-resolved local density of states (LDOS), presenting the
results in Fig. 2. Namely, we look at the LDOS of states which are appearing
in the electronic structure of the skyrmion in the gap of the out-of-plane
ferromagnetic system between $-$0.3 and $+$0.3 eV, see Fig. 1(a,b). In Fig. 2
(a-c) the spatial distribution of the skyrmion’s LDOS, which has been
integrated in this energy region, is shown for different widths of the domain
wall as controlled by the parameter $d$. Quite remarkably, our calculations
show that, irrespective of the domain wall width, the LDOS is vanishing
exactly outside of the domain wall $-$ i.e., in the center of the skyrmion,
and in the region between the skyrmions. This is in sharp contrast to all
other states of the skyrmion lattice which lie outside of the gap in the
“metallic” region: as an example, in Fig. 2(d) the LDOS of the system, which
has been integrated over the energy region $[-0.47,-0.46]$ eV (i.e. outside of
the gap as marked with a green dashed line in Fig. 1(a)), is finite at any
point in the unit cell which marks the delocalized character of the
constituting states.

The precise localization of the gap states in the domain that we observe
irrespective of the domain wall width allows us to directly associate them
with the emergence of the band crossing in the electronic structure of the
model ferromagnetic system for the in-plane magnetization. Recently it was
shown that this band crossing acquires a non-trivial topological character
once the magnetization direction $\hat{\mathbf{m}}$ is included into the
topological analysis. In this sense the ferromagnetic 2D model that we study
is an example of what is called a mixed Weyl semimetal Hanke  _et al._
(2017a); Niu  _et al._ (2019) $-$ a term motivated by necessity of including
the so-called mixed Berry curvature into the analysis of the band topology.
Since the gap states that we observe arise as a result of the non-trivial
mixed topology of our model, we refer to them as mixed topology ring states
(MTRS). It has been shown, that, depending on the symmetry of the model, the
mixed Weyl points can be associated with a transition between the Chern
insulating phases with different Chern number arising for different directions
of $\hat{\mathbf{m}}$ Hanke  _et al._ (2017a); Niu  _et al._ (2019). This is
exactly the case for our model: as the direction of $\hat{\mathbf{m}}$ is
changed from along the $z$-axis to the opposite to it, the Chern number at
half-filling changes sign from $+2$ to $-2$. The skyrmion lattice that we
consider thus presents a lattice of domains with an opposite Chern number
separated by the domain walls, and the MTRS can be naturally interpreted as
the topological edge states localized at the boundary separating the two
domains.

###  IV.2 Hall conductance and orbital magnetization in skyrmions of a mixed
Weyl semimetal

In this section we investigate the possible influence of MTRS on the Hall
transport properties and orbital magnetization exhibited by the skyrmion
lattices of mixed Weyl semimetals. In order to do that, we parametrize the
magnetization distribution within the skyrmion of radius $R$ in a way that
$\alpha(r)=\pi\left(1-\frac{\delta(r)-\delta(0)}{\delta(R_{\text{max}})-\delta(0)}\right)$
with $\delta(x)=-\operatorname{atan}\left(a\cdot(r-R)\right)+\frac{\pi}{2}$.
Unlike the previous parameterization used in section IV.A, which exhibits a
pronounced flat region in the middle of the wall, the parametrization we use
from here on is designed to model most closely the skyrmions in systems which
favor an out-of-plane magnetization direction, which leads to domain walls of
roughly constant magnetization gradient across the wall.  When compared to the
approach used in the preceding subsection, the parametrization employed from
now on is very close to that routinely used to model skyrmion profiles in
micromagnetic studies Romming  _et al._ (2015); foo . It is important to note
that the MTRS emerge for both types of the profile parametrization.

We first consider the case of $R=R_{max}/2$, and show the skyrmion profile
along the path between the skyrmion centers for various values of $a$ in Fig.
3(a). We present the results of our calculations of the Hall conductance and
orbital magnetization in the system in Fig. 3(b,c) as a function of band
filling and parameter $a$, of which the latter controls the domain wall width
and correspondingly the MTRS spread in real space. Concerning the overall
energy dependence for all values of $a$, while HC exhibits a symmetric
structure with respect to the middle of the bulk gap at $E_{F}=0$ eV, the OM
is antisymmetric, which originates in the symmetry of the ferromagnetic band
structure and the properties of the OM around the mixed Weyl points Niu  _et
al._ (2019). Overall $-$ excluding the region of the bulk gap, marked with a
shaded area in Fig. 3(b,c), which will be considered in detail below $-$ the
computed HC appears to be extremely sensitive to the domain wall width. This
manifestly “non-topological” behavior of the HC emerges in contrast to the
expectations of the topological Hall effect in this system, insensitive to the
details of the spin distribution but determined by the overall topological
charge, with the latter remaining constant in our calculations.

The observed sensitivity of the HC to the domain wall width, i.e. to the
magnitude of the magnetization gradient in it, can be best understood by
referring to the novel phenomenon of the chiral Hall effect Lux  _et al._
(2020), which has been recently shown to be prominent in chiral skyrmions with
interfacial spin-orbit coupling, and which is originated in the change of the
magnetization within the walls  as given by their chirality along the line
which passes through the center of the skyrmion. Within the theory of the
chiral Hall effect, emerging already for spin-spiral solutions Lux  _et al._
(2020), the chiral signal is proportional to the sense of the spin chirality
among the neighboring spins $\mathbf{S}_{i}\times\mathbf{S}_{j}$, and thus the
change in sign of the Hall conductivity upon changing the sign of the spin
chirality serves as one of the trademarks of the chiral Hall effect Kipp  _et
al._ (2020). In Fig. 3(d) we present the results of the HC calculations as a
function of the domain wall width, but assuming that the domain wall has an
opposite sense of spin rotation from that shown in Fig. 3(a). While the
topological charge of the skyrmion does not depend on the chosen chirality,
the sign of the HC is opposite for two opposite chiralities for almost all
values of $a$ and majority of energies. This underpins the origin of the
observed HC in the chiral Hall effect originated in the domain walls of
skyrmions. A perfect reversal of the HC at a given energy with chirality (i.e.
reversal in sign but not in magnitude) is not expected given very strong spin-
orbit interaction strength which makes the electronic structure of two flavors
of skyrmions different.

On the other hand, the overall value of the orbital magnetization is
remarkably insensitive to the domain wall width. While the emergence of the
chiral orbital magnetization, arising in analogy to the chiral Hall effect,
would be also expected for interfacial skyrmions Lux  _et al._ (2020), for our
particular choice of the model and symmetric parametrization of the skyrmion
profile the chiral part of the OM vanishes from symmetry arguments, thus
reducing to the well-known effect of topological orbital magnetization
Hoffmann  _et al._ (2015); Hanke  _et al._ (2016); dos Santos Dias  _et al._
(2016); Hanke  _et al._ (2017b); Lux  _et al._ (2018).  The topological OM
remains extremely robust with respect to the changes in the details of spin
distribution, as visible in Fig. 3(c), and its behavior is manifestly not very
sensitive to the change in the spin chirality of the skyrmion (not shown).

Figure 3: Transport properties of a skyrmion lattice of a mixed Weyl semimetal
with $R=R_{max}/2$. (a) The real-space cut through the skyrmion profile along
the path connecting two skyrmion centers for different values of parameter $a$
(see text for more details). (b,c) The dependence of the Hall conductance
$\sigma_{xy}$ (b), and orbital magnetization $M$ (c), on the band filling and
parameter $a$. The inset in (b) zooms into the region of quantization where
MTRS reside.  (d) Displays the HC of the skyrmions with profile shown in (a)
but with reversed sense of the spin rotation in the walls (for impression see
the sketch shown as an inset).  In all plots the color indicates the value of
$a$ in accordance to the color scale shown in (a).

In the following, we focus on the region in energy which corresponds to the
position of the bulk gap of the model for the out-of-plane magnetization,
marked with a shaded area in Fig. 3(b-c), and zoomed into in the inset of Fig.
3(b). The behavior of the HC, shown in the inset, displays isolated spikes on
the background of a plateau value of $-2$ $\frac{e^{2}}{h}$.  The latter value
can be traced back to the quantized value of the HC for the out-of-plane
ferromagnetic model, see Fig. 1(e), as for the given value of $R$ the overall
out-of-plane magnetization in the unit cell is positive. On the other hand,
the spikes at isolated positions correspond to the contributions of ring
states to the HC. These contributions become somewhat more pronounced as the
width of the domain wall increases, owing to the increased spread of the ring
states and increased probability of electron hopping among the ring states
positioned in different unit cells. In order to investigate this effect in
more detail, we study the dependence of the HC and OM on the radius of the
skyrmion $R$, while assuming a very large value of parameter $a$, thus
localizing the ring states in a narrow region of the domain wall, see sketches
in the right column of Fig. 5. The overall qualitative behavior of the HC and
OM with $R$, discussed below, does not qualitatively depend on the value of
parameter $a$ chosen.

Figure 4: Transport properties of a skyrmion lattice of a mixed Weyl semimetal
in the region of the bulk gap as a function of relative radius $R/R_{max}$ and
a value of parameter $a$ of 105. (a) The energy dependence of the orbital
magnetization as a function of $R/R_{max}$. The color of the line corresponds
to the magnitude of the relative radius according to the color bar. (b) The
energy dependence of the Hall conductance as a function of $R/R_{max}$. The HC
of the systems with $R/R_{max}\leq 0.7$ are plotted with a solid line “$-$”,
with $R/R_{max}=0.8$ is plotted with a dashed line “$---$”, with
$R/R_{max}=0.9$ is plotted with a dashed-dotted line “$-\cdot-$”, and with
$R/R_{max}=1$ is plotted with a dotted line “$\cdot\cdot\cdot$”.  Figure 5:
The evolution of the bandstructure around the Fermi energy in the region of
MTRS (in green, scale on the left) and of the orbital magnetization $M$ (in
blue, scale on top, as a function of band filling) with respect to
$R/R_{max}$. The positions of the jumps in the saw-tooth pattern of the
orbital magnetization coincide with the positions of the MTRSs. The sketches
on the left mark the division of the unit cell into regions with roughly
magnetization up (tainted in blue), and roughly magnetization down (tainted in
red), with $R/R_{max}=0.3$ in (d), in $R/R_{max}=0.8$ in (e), and
$R/R_{max}=1$ in (f).

We first discuss the evolution of the HC with the relative radius $R/R_{max}$,
presented in Fig. 4(b). Over a very large range of radii $R<0.8R_{max}$ the
conductance is ideally quantized to $-2$ $\frac{e^{2}}{h}$ in the gap, despite
a large number of MTRS, see Fig. 5(a-b). In this case the Hall effect is
dominated by the ferromagnetic background in between the skyrmions with
$\hat{\mathbf{m}}$ along $+z$ which provides a Chern number of $-2$. The ring
states are strongly localized in the wall region (which explains the flat
dispersion of the ring states visible in Fig. 5(a)), do not hybridize with
each other across the unit cell boundaries, and do not contribute to the HC.
In the other limit of $R=R_{max}$, see Fig. 5(f) for a sketch, the film
comprises mainly areas of the texture with $\hat{\mathbf{m}}$ along $-z$,
which explains mostly quantized value of the HC of $+2$ $\frac{e^{2}}{h}$.
The crossover point of the HC from $\sigma_{xy}=-2\,\frac{e^{2}}{h}$ to
$\sigma_{xy}=2\,\frac{e^{2}}{h}$ positioned between $R/R_{\text{max}}=0.7$ and
$R/R_{\text{max}}=0.8$ coincides with the point where the integrated over the
unit cell out-of-plane component of the magnetization changes sign as a result
of the competition between domains with $\hat{\mathbf{m}}$ along $+z$ and
domains with $\hat{\mathbf{m}}$ along $-z$. The region of $0.8R_{max}\leq
R\leq R_{max}$ is the region of where the MTRS interact with each other
strongly across the unit cell boundaries, exhibit a strong dispersion (see
Fig. 5(b)) and give rise to a drastic variation in the HC. In the limit of
$R=R_{max}$ the MTRS are localized in the unit cell corners, thus interacting
noticeably along the unit cell boundaries, which gives rise to a small but
finite dispersion of the bands (see Fig. 5(c)) and peak-like contributions to
the HC.

We come now to the discussion of the OM of the skyrmion lattice exhibited in
response to the variation of the radius. This behavior, presented in Fig.
4(a), is distinctly different from that exhibited by the Hall conductance.
Clearly, in the limiting cases of $R=0$ and $R=R_{max}$ the slope of the OM as
a function of energy in the gap has to be opposite, as follows from Eq. (6),
however, while in the case of the HC the transition between the two Chern
insulator cases is relatively abrupt, the same transition in terms of the OM
seems to be quite smooth, with the overall slope of the OM changing
continuously as a function of $R$, and the OM curves being quite smooth as a
function of energy, when compared to the case of the HC. This can be
understood by looking in detail at the contributions of the MTRS to the OM
variation in the gap, Fig. 5(a-c). Here, we observe that although the slope of
the OM in between the groups of MTRS is strictly constant and consistent with
the sign of the quantized value of the HC in the same region, Fig. 5(b), the
overall curve of the OM for all values of $R$ displays a characteristic saw-
tooth shape, where the overall slope is “renormalized” by the contributions
from MTRS. Interestingly, these contributions are consistent in sign among all
MTRS for a given $R$: they are positive in the region of $R<0.8R_{max}$, and
negative for $R\geq 0.8R_{max}$. This “coherence” of MTRS is the reason that
the slope of OM is getting consistenly and gradually increased as $R$ goes
from zero to $R_{max}$.

According to our calculations, the OM of MTRS is predominantly originated in
the local circulation part, which is consistent with the strong localization
of the ring states. In a sense, the ring states can be envisaged as a group of
localized electronic states which have a common unique sense of bound orbital
current circulation around the skyrmion center $-$ which we can naturally
refer to as having a specific sense of orbital chirality. And while in the
case of the Hall effect it is the Hall conductance which changes abruptly at
the point of phase transition between two areas of out-of-plane magnetized
domains, the corresponding marker in terms of orbital magnetism is the orbital
chirality of the ring states which exhibits a sharp transition.  Manifestly,
as follows from our calculations of the OM with reversed spin chirality of
skyrmions, while the number and exact energetic positions of the states depend
on the width of the domain wall for both spin chiralities, the orbital
chirality of MTRS is remains a robust quantity even under the change in the
skyrmion chirality.

##  V Discussion

In our work, we have considered the intrinsic Hall effect and orbital
magnetism exhibited by skyrmion lattices of mixed Weyl semimetals by referring
to the Berry phase framework. Our main finding is the discovery of the special
type of ring states which form as a result of complex mixed topology in real-
space textures of these materials. These electronic states reside at the
domain wall region, which serves as the boundary between the two Chern
insulator phases and separates the skyrmion cores from the ferromagnetic
background. We have shown that the degree of the localization of the ring
states and the strength of their inter-cell interaction can be controlled by
the parameters which determine the texture details. In turn, the ring states
mediate the transition between the two Chern insulating phases occurring upon
switching of the magnetization direction from out-of-plane to the opposite,
when e.g. an external magnetic field changes its strength and sign in
experiment. This concerns the Hall conductance, but mostly the orbital
magnetization of the samples, as the ring states carry a specific orbital
chirality and orbital momentum.

The scaling of the Hall conductance as a function of the skyrmions domain wall
width is a strong indicator of a prominent chiral Hall effect, where such a
behavior has recently been predicted and which could have possible
ramifications for the unambiguous electrical detection of magnetic skyrmions
Lux  _et al._ (2020). In simplified model systems, the emergence of this
effect can be understood from the interplay of spin-orbit interaction and
exchange coupling which gives rise to an effective magnetic field Nakabayashi
and Tatara (2014). This phenomenology is similar to the topological Hall
effect (THE) Bruno  _et al._ (2004), but in contrast, it materializes already
at the leading order perturbation theory whereas the THE is subleading Lux
_et al._ (2020). From the viewpoint of Weyl semimetals, the coaction of chiral
magnetism and spin-orbit coupling (which is responsible for the chiral Hall
effect) is directly linked to an emergent chiral anomaly Ilan  _et al._
(2020). Our model elevates this phenomenological interpretation to a more
realistic setting. To which degree this explanation can be uphold will be the
subject of future investigations.

Ultimately, the ring states arise as a result of a strong variation in the
electronic structure around a specific magnetization direction, and thus it is
rewarding to explore in the future what impact can the ring states have on the
local real-space variation of the spin-orbit torques and the Dzyaloshinskii-
Moriya interaction, as well as current-induced skyrmion dynamics Hanke  _et
al._ (2017a); Niu  _et al._ (2019); Araki (2020); Hanke  _et al._ (2020); Liu
_et al._ (2013); Kurebayashi and Nomura (2019); Kurebayashi and Nagaosa
(2019), where the ring states can play a role of local “hooks” strongly
coupling skyrmions to applied currents. On the other hand, the unique orbital
properties of the ring states lend themselves as a possible platform for
obtaining a detailed information on the skyrmion parameters with e.g. XMCD
type of techniques. Additionally, peculiar topological nature of these one-
dimensional edge states emerging in skyrmions of Weyl semimetals hints at
exceptional possibilities that the ring states can play in mediating the
physics of Majorana states emerging upon deposition of skyrmions on
superconductors Pershoguba  _et al._ (2016); Güngördü  _et al._ (2018);
Palacio-Morales  _et al._ (2019); Yang  _et al._ (2016); Rex  _et al._ (2019);
Palacio-Morales  _et al._ (2019), which presents an exciting future direction
to pursue.

##  VI Acknowledgements

We gratefully acknowledge computing time on the supercomputers JUQUEEN and
JURECA at Jülich Supercomputing Center, and at the JARA-HPC cluster of RWTH
Aachen. We acknowledge funding under SPP 2137 “Skyrmionics” of Deutsche
Forschungsgemeinschaft (DFG), DARPA TEE program through grant MIPR#
HR0011831554 from DOI. We gratefully acknowledge financial support from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 856538, project ”3D MAGiC”).

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# Dual-Arm Adversarial Robot Learning

Elie Aljalbout
Technical University of Munich
<EMAIL_ADDRESS>

###### Abstract

Robot learning is a very promising topic for the future of automation and
machine intelligence. Future robots should be able to autonomously acquire
skills, learn to represent their environment, and interact with it. While
these topics have been explored in simulation, real-world robot learning
research seems to be still limited. This is due to the additional challenges
encountered in the real-world, such as noisy sensors and actuators, safe
exploration, non-stationary dynamics, autonomous environment resetting as well
as the cost of running experiments for long periods of time. Unless we develop
scalable solutions to these problems, learning complex tasks involving hand-
eye coordination and rich contacts will remain an untouched vision that is
only feasible in controlled lab environments. We propose dual-arm settings as
platforms for robot learning. Such settings enable safe data collection for
acquiring manipulation skills as well as training perception modules in a
robot-supervised manner. They also ease the processes of resetting the
environment. Furthermore, adversarial learning could potentially boost the
generalization capability of robot learning methods by maximizing the
exploration based on game-theoretic objectives while ensuring safety based on
collaborative task spaces. In this paper, we will discuss the potential
benefits of this setup as well as the challenges and research directions that
can be pursued.

> Keywords: Adversarial Learning, Robotic Manipulation, Learning Platforms

##  1 Introduction

With the recent progress in robot learning, many robotic tasks became possible
to learn either via imitation or trial-and-error. These tasks range from
classical robotics problems such as reaching [1, 2] and collision avoidance
[3, 4] to more complex tasks involving locomotion [5, 6, 7] and manipulation
[8, 9, 10, 11]. However, most methods that solve these tasks are either tested
in simulation, assume to have access to a perfect estimate of the environment’
state, or strongly restrict the interaction between the robot and its
surrounding based on predefined heuristics or physical constraints. The main
reason for these restrictions is the sample inefficiency of modern robot
learning approaches, as well as the difficulty of safe environment exploration
and resetting, without a human in the loop. Sample inefficiency is most
commonly attributed to the complexity and noise encountered in sensory
information processing. Solutions to the problem range from including
pretrained perception modules [12] in the learning pipeline to integrating
self-supervised state representation learning objectives into task learning
[13, 10, 4, 14]. Safe exploration and environment resetting are rarely
mentioned in publications and temporary solutions include engineering the
environment or having a human manually stop the robot in dangerous situations
and reset the environment at the end of each trial. While these approaches may
work for simplified lab experiments, there is a clear need for scalable
solutions in order to bring robots into real-world environments and human
surroundings. In this paper we propose a simple paradigm to approach some of
these problems. Namely, we propose dual-arm adversarial robot learning (DAARL)
as a paradigm to automate robot learning experiments and ease the data
collection. Besides the obvious benefits for dual-arm manipulation tasks,
using a second hand makes it simpler to randomize single arm manipulation
experiments, e.g. for the classical peg-in-hole task, a second robot can be
used to hold the hole at different poses, allowing the learned policy to
generalize to different settings. Additionally, for tasks requiring some sort
of remote sensing such as vision, a second arm can be used to collect data for
training perception modules. This data can either be used in a supervised
fashion to train separate perception modules or even to integrate perception
in the decision making (for instance by including state representation
learning objectives in the task learning process). Furthermore, with the
mutual awareness of the two robots, it is possible to design simple policies
to solve the task at hand, and use these simplistic policies to sample data
for training a more complex one. In this paper, we will introduce this
paradigm (section 2), discuss its potential gains (section 3) and challenges
(section 4), and discuss the general picture (section 5).

##  2 Methodology

Figure 1: Illustration of DAARL: each robot is either a trainer or a trainee
with its own policy. The trainee is the main learner. Its policy is trained to
solve the task. The trainer policy and robot are responsible for resetting and
randomizing the environment between episodes. It can also be used to generate
adversarial actions to make the trainee policy more robust.

Figure 1 illustrates the general idea behind DAARL. Each robot is either a
trainer or a trainee. This condition can be relaxed for certain dual-arm
manipulation tasks, but we would still use this naming convention for the sake
of distinguishing the two robots. For single-arm tasks the trainer robot is
only used to reset and randomize the environment for the task at hand, e.g.
randomize the pose of the hole for single-arm peg-in-hole. The two robots are
controlled by two separate policies, $\pi_{tr}$ controls the trainer and
$\pi_{te}$ controls the trainee. The latter is the one performing the task
while the trainer is the one responsible for resetting the environment and
maximizing exploration. In addition, the trainer could also be used to
generate disturbances during the task execution by the trainee. This behavior
can also be adversarial similar to the work in [15]. It is important to note
that the two policies don’t have to run at the same frequency. For instance,
if the trainer is only taking care of the environment reset of a single-arm
task, it would only be used at the end of each episode, while the trainee
policy is used at a higher frequency during the task execution. Considering a
Markov Decision Process (MDP) as a tuple
$\mathcal{M}=\\{\mathcal{S},\mathcal{A}_{f},\mathcal{A}_{l},T,\rho_{0},r,\gamma\\}$
, where $\mathcal{S}$ denotes the state space, $\mathcal{A}_{f}$ and
$\mathcal{A}_{l}$ the action spaces of the trainer and trainee policies. $T$
describes the system dynamics. $\rho_{0}$ defines the initial state
distribution.
$r:\mathcal{S}\times\mathcal{A}_{f}\times\mathcal{A}_{l}\to\mathbb{R}$ is the
reward function , and $\gamma$ is a parameter for discounting future rewards.
The overall objective for this problem can be formulated as follows:

$\min_{\pi_{tr}}\max_{\pi_{te}}\mathbb{E}_{s_{0}\sim\rho_{0},a_{tr}\sim\pi_{tr},a_{te}\sim\pi_{te}}\left[\sum_{t=0}^{H}\gamma^{t}r(s,a_{tr},a_{te})\right]$
(1)

As the solution to such problems can be complex [16], it might be desirable to
train the two policies in an alternating fashion as described in [15]. This
also makes it easier to control the degree of exploration by weighting the
rewards for the trainer and trainee policies differently: $r_{tr}=\alpha
r(s,a_{tr},a_{te})$ and $r_{te}=\beta r(s,a_{tr},a_{te})$. For dual-arm
manipulation tasks, each robot would play the role of the trainer and the
trainee, purely attempting to solve the tasks at times and tricking the other
robot at other times. This would then require to have for each robot two
policies, which are trained in an alternating fashion.

##  3 Potential Gains

Safety during training. This aspect is especially relevant for single-arm
tasks where the usual setup might include static (task-relevant) objects on
surfaces (e.g. table) or objects in a human’s hand. Due to the access to the
trainer robot’s state and hence the object’s, it is possible to ensure safety
by embedding this knowledge in the controllers and action spaces of both
robots. Namely, it is possible to predict and prevent robot-robot collisions,
and subsequently avoid collisions altogether since these are mostly probable
to occur around the object.

Autonomous Randomized Experiments. When learning via trial and error, it is
important to be able to reset the environment, without having a human in the
loop. For instance, if the task is ”object pushing”, it is necessary to reset
the pose of the object after every training episode. While this is very simple
in simulation, in real-world experiments, many problems can be faced such as
objects out of reach, collisions, and having to switch/modify the end-effector
to reset the environment. Having a second robot, makes it possible to fetch
objects from a larger workspace and resetting them to an initial pose.
Additionally, it is often desirable to randomize the initial state of the
environment. Randomizing the initial state improves the generalization
capability of the learned policy [17]. Besides reducing the dependence on a
human to reset the environment, a second robot makes it possible to have more
diverse initial states, since the objects to be manipulated can be placed in
the trainer’s hand which can be anywhere within this robot’s workspace. Unlike
the case where the object to be manipulated is placed in predefined positions
on certain surfaces. Taking the example of a ”pick and place” task, the
trainer robot can hold the objects in hand (with the hand wide open) and the
trainee attempts to pick the object. Of course, this has its limitations. For
instance, the objects might fall outside of the workspace of both robots, a
human is then required to return it into the scene. Finally, a dual-arm setup
can also be beneficial for reward computation. For instance, if the object to
be manipulated is rigid and always in the trainer’s hand, it is possible to
have access to its state without having to use any kind of external state
estimation. This makes it easier to compute the reward.

Human-Robot Interaction (HRI). Another benefit of having a second robot, is to
simulate a human for HRI tasks during training. Namely, the trainer robot can
imitate the motion of a human that might be encountered during the task. By
exposing the trainee to human-like motions during training, it is possible to
train it for HRI tasks without a human in the loop. The trainer could also be
designed to act in human-like ways. It is also possible to enforce certain
properties on the trainer (e.g. time optimality) which are hard to impose on a
human training the robot (for instance due to lack of motivation).

Data Collection. In Dual-arm settings, each robot can easily collect labeled
data about the objects in the other robot’s end effector, or ones previously
moved by the other robot. The robots need to be calibrated with respect to
each other and their kinematics should be known. For instance, for vision-
based tasks, it is possible to collect datasets of images and object locations
and use that later on for training segmentation, object detection, or tracking
modules. This can also be done with a single-arm setting. However, the second
arm makes the data collection faster, more flexible and enables collecting
data about dynamic objects. Even for unsupervised/self-supervised state
representation learning, collecting observations of the environment can be way
better in the presence of an active agent (second robot) randomizing the state
of the environment, and diversifying the observations. In both supervised and
unsupervised cases, the obtained perception modules can later on be used for a
variety of tasks even single-arm tasks. Another major benefit is to collect
observation-action trajectories using simple (local) hand-designed teacher
policies. The teacher policies could take advantage of the calibration between
the two robots and generate actions based on the obtained knowledge concerning
the accurate whereabouts of the task-relevant objects. Such policies can be
hand-designed and sub-optimal. They can then be used to train a global policy
as in [18]. More interestingly, the teacher policies could be fully blind
(i.e. do not use vision and rely on true state information), but later on be
used to collect data with images as part of the observation to train vision-
based policies. Chen et al. [19] proposed a similar concept for training
autonomous driving agents.

Guided Exploration. In robot learning, especially reinforcement learning-based
methods, sample efficiency is a major problem. For that, a good balance
between exploration and exploitation is very important to ensure
generalization and escaping bad local optimas while also being sample
efficient. However, both exploration and exploitation can be challenging. For
exploration, a uniform random policy is already better than the task policy at
collecting diverse samples. However, even a random policy can collect similar
state-action trajectories, and hence more aggressive exploration is needed.
The adversarial side of DAARL (section 2) help collecting samples where the
current policy still struggles, instead of ones which are random but
potentially very similar to the ones already encountered. Even for
exploitation, most reinforcement learning methods need a lot of trials to
learn a successful policy that can be exploited. As explained in the previous
paragraph, DAARL could accelerate that by simplifying the process of
programming teacher policies based on the mutual awareness of the two robots.

##  4 Challenges & Limitations

Like any other paradigm, building DAARL-based pipelines and applications can
be challenging. We discuss major challenges and limitations in this section:

Increased Complexity: First, by using a second robot for learning single-arm
tasks, some extra engineering effort is needed. As previously discussed, the
transformation between the two robots needs to be identified and a coordinated
motion planning and control architecture is needed to ensure safety and smooth
experiments111These components are required anyway for dual-arm tasks. In
addition, every design and instrumentation choice needs to be made twice. For
instance, the choice of action spaces and controllers used for every policy
and robot are different. Additionally, for single-arm tasks, the trainer robot
needs to be equipped with basic skills to be able to do its job on certain
tasks. For example, if the role of the trainer is to reset and randomize the
environment for a ”pick and place” task involving multiple objects, it needs
to be able to pick the objects itself and place it. At first, it might sound
as if DAARL defies its purpose. However, the ”pick and place” skill doesn’t
need to be perfect for the trainer but just good enough to serve its purpose,
i.e. it could be hard-coded, sub-optimal, or based on object-specific
assumptions. Nonetheless, that’s an extra effort needed to get started. This
motivates adopting a developmental approach to learning skills with DAARL:
starting with basic atomic skills that require the least expert knowledge, and
using the learned skills later-on while learning different skills and solving
different tasks. Early tasks could even be supervised by a human.

Extra Cost: Besides the additional engineering efforts required to get started
with DAARL, there’s an extra cost needed to own the second robot, run it and
maintain it. However, one of the main motivations for robot learning research
is to enable robots to perform multiple tasks in unstructured and unknown
environments such as households. We believe the actual robots deployed for
such environments won’t be single-arm robots attached to a static table, but
rather something more dexterous and mobile like a humanoid. We might as well
start training robots in these settings, solve some of the challenges that are
faced in such settings, and take advantage of all the information that can be
gained.

Limitations: As previously mentioned, DAARL doesn’t completely remove the need
for humans in the loop during training. We hope that it at least reduces the
need for human supervision, and potentially eliminate it for certain tasks.
Furthermore, the adversarial part of DAARL introduces additional
hyperparameters and implementation details that need to be well chosen. For
instance, in section 2, we suggested an alternating training procedure to
solve the adversarial problem. Our choice was mostly based on our own previous
experiences in similar optimization problems as well as related work [15].
Other alternatives might improve the training, reduce the number of
hyperparameters and potentially strike a better balance between exploration
and exploitation.

##  5 Discussion

In this paper, we propose dual-arm adversarial robot learning as a new
paradigm for automating real-world robot learning experiments, reducing the
dependence on humans in the loop, and boosting the exploration and
generalization capabilities of such methods. Our main aim is to propose a
feasible alternative to training in simulation, while reducing the chance of
falling in common pitfalls of real-world robotic experiments, especially ones
involving randomness. We also discuss other benefits of DAARL such as data
collection, human-robot interaction, and safety-enabling potentials. We also
discussed challenges and limitations of DAARL-like systems. Most importantly,
we note that some of the majors advantages of DAARL, such as requiring less
human interventions, are only limited to certain tasks. For that we briefly
propose a developmental skill and task learning approach based on sorting the
dependencies of tasks and skills on each others. Future work could attempt to
smartly apply DAARL to various tasks, solve some of the mentioned challenges,
improve the concept further, and even propose other smart mechanisms and
platforms for real-world robot learning.

### Acknowledgments

We would like to thank the reviewers for their valuable feedback and comments.

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