# Chiral response of spin spiral-states as the origin of chiral transport
fingerprints of spin textures

Jonathan Kipp  Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  Department of
Physics, RWTH Aachen University, 52056 Aachen, Germany  Fabian R. Lux
Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz,
Germany  Yuriy Mokrousov  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

The transport properties of non-trivial spin textures are coming under closer
scrutiny as the amount of experimental data and theoretical simulations is
increasing. To extend the commonly accepted yet simplifying and approximate
picture of transport effects taking place in systems with spatially varying
magnetization, it is important to understand the transport properties of
building blocks for spin textures $-$ the homochiral spin-spiral states. In
this work, by referring to phenomenological symmetry arguments based on the
gradient expansion, and explicit calculations within the Kubo framework, we
study the transport properties of various types of spin-spirals in a two-
dimensional model with strong spin-orbit interaction. In particular, we focus
on the contributions to the magnetoconductivity, the planar Hall effect and
the anomalous Hall effect, which are sensitive to the sense of chirality of
the spiral states. We analyze the emergence, symmetry, and microscopic
properties of the resulting chiral magnetoconductivity, chiral planar Hall
effect, and chiral Hall effect in terms of spin-spiral propagation direction,
cone angle, spiral pitch, and disorder strength. Our findings suggest that the
presence of spin-spiral states in magnets can be readily detected in various
types of magnetotransport setups. Moreover, the sizable magnitude of chiral
contributions to the conductivity of skyrmions estimated from homochiral
spirals implies that chiral, as opposed to topological, magnetotransport can
play a prominent role for the detection of non-trivial spin textures.

##  I Introduction

Figure 1: Parametrization of Néel, Bloch, and cone-type spirals. Parts (a)-(c)
show the parametrization of Néel (a), Bloch (b), and cone-type (c) spin-
spirals on a sphere spanned by magnetic moments. The spiral propagates along a
direction $\mathbf{q}$, and the magnetization rotates around an axis
$\mathbf{e}_{r}$, which encloses a cone angle $\alpha$ with the initial
magnetic moment $\hat{\boldsymbol{n}}_{0}$. For the Néel spiral
$\mathbf{e}_{r}\bot\mathbf{q}$, while for the Bloch spiral,
$\mathbf{e}_{r}\parallel\mathbf{q}$. In this work, the plane of the honeycomb
lattice is the $xy$-plane (axis $\hat{\boldsymbol{z}}$ is perpendicular to it)
which contains $\mathbf{q}$. Plots (d)-(f) show the real-space distribution
(as seen from the top) of corresponding magnetization textures for
$\mathbf{q}\parallel\hat{\boldsymbol{x}}$.

The electron dynamics taking place in diverse magnetization textures is one of
the most intensively pursued areas of solid state physics at the crossroads
between topology, transport and physics of non-uniform media. In particular,
the transport manifestations of electron dynamics in spin textures exposed to
external electric fields have come to occupy a very important role in modern
skyrmionics and spintronics due to numerous implications for practical
implementations of skyrmion-based design philosophy. Since the early days of
the field, a commonly accepted approach for treating the transverse transport
of complex spin-textures in two dimensions, such as skyrmions, has relied
heavily on the presence of the so-called emergent field, coupling spatial
gradients $\partial_{x}\hat{\mathbf{n}}$ and $\partial_{y}\hat{\mathbf{n}}$.
Since the seminal paper of Bruno and co-workers [1], the validity of the
ansatz for the transverse Hall conductivity as an object directly proportional
to the emergent field has been tested and validated in many cases from
effective models [2, 3, 4], microscopic calculations [5, 6] and direct
comparison to experiments [7, 8], while ”topological” contributions have also
been reported for magnetoconductivity and the planar Hall effect [9, 10, 11,
12, 13]. However, in recent years, evidence has started to accumulate that the
simple picture of the emergent field and the corresponding topological Hall
effect often does not suffice to explain results from experimental transport
measurements. The community has become more aware of the fact that the
emergent field picture has to be radically extended [14, 15, 16], particularly
in strongly spin-orbit coupled systems. On the one hand, recent works have
shown that in spin-orbit coupled systems the second order in magnetization
gradients terms, which go beyond the conventional topological-like signal, can
contribute significantly to the orbital magnetism and Hall effect exhibited by
the textures [17, 16]. On the other hand, it has been demonstrated that an
effect that is much more pronounced in strongly spin-orbit coupled systems,
since it is linear in the gradients of $\hat{\mathbf{n}}$, can emerge in the
context of Hall currents and orbital magnetism in generic spin textures [18].
The corresponding chiral Hall effect (CHE) was shown to be directly sensitive
to the sense of local spin chirality and, in contrast to the topological Hall
effect, to fine details of spin distribution [18, 19], which potentially makes
it a powerful tool for tracking texture dynamics with magnetotransport means.
Recently, it was also demonstrated from model considerations and first-
principles calculations that the CHE can be prominent in canted spin-orbit
coupled ferromagnets and antiferromagnets [20]. For textures, a signal
consistent with such chiral contributions has been reported experimentally for
the anomalous Hall effect (AHE) [21], the planar Hall effect (PHE) [9], and
magnetoconductivity (MC), although additional analysis is necessary to
unambiguously pin down the exact microscopic origin of the observed signal.

In the context of chiral contributions to the magnetotransport of complex spin
textures, understanding chiral transport properties of elemental one-
dimensional spin-spiral states presents an important milestone and a pivotal
starting point. Spin-spirals ahve emerged as the ground state of various
transition-metal compounds in bulk [7], at surfaces [22], and in one-
dimensional systems [23] as a result of complex exchange interactions.
Further, spin-spirals serve as natural building blocks for more complex spin
structures such as multi-$q$ states [24], domain walls [25] and skyrmions [7,
26], while their treatment often provides a key to modeling fluctuating
magnets [27, 28]. Proper theoretical understanding of symmetry and microscopic
mechanisms behind the transport properties of spin-spiral states thus provides
a necessary foundation for building a coherent picture of transport phenomena
in diverse spin textures. A distinct transport signal of spiral states has
been observed in experiments lately [21], and a recent theoretical work based
on an effective Rashba model predicted the emergence of the chiral Hall effect
for specific types of spin-spiral states [29]. At this point, a comprehensive
picture of magnetotransport of spin-spirals grounded in a realistic electronic
structure on a specific lattice has emerged as a necessary next step.
Technically, computing transport properties of extended spin-spiral states is
very challenging. The spin-spiral order in combination with spin-orbit
interaction is inconsistent with the generalized Bloch theorem [30, 31], which
necessitates the use of very large unit cells containing thousands of atoms.
The formidable computational effort explains the noticeable lack of expansive
studies in this direction (see e.g. [32, 25]).

In our work, we study in detail the magnetotransport properties of spin-spiral
states in two-dimensional magnets, and we make predictions concerning the
impact of these properties on chiral transport fingerprints of large
skyrmions. We refer to an effective tight-binding model of electrons on a
honeycomb lattice for explicit calculations of transport properties from the
electronic structure of spin-spiral states in the Kubo linear-response
formalism. Inspecting components of the conductivity tensor (anti-)symmetrized
with respect to chirality, we uncover chiral contributions to anisotropic
magnetoconductivity (denoted below as MC), the planar Hall effect, and the
anomalous Hall effect. Specifically, we find perfect agreement between
numerical results from the Kubo formalism and symmetry-based predictions from
the gradient expansion for the existence of chiral contributions as a function
of the spin-spiral propagation direction and the type of spiral. We address
intrinsic and extrinsic origins of the considered effects, uncovering a non-
trivial competition of disorder and Berry phase effects on the chiral Hall
effect exhibited by spin-spirals and skyrmions. The strong and extremely non-
trivial response of the chiral transport to the pitch of the spirals found
here suggests that chiral magnetotransport serves as a unique marker of the
fine details of the spin distribution in spin textures. Finally, we show how
by extracting effective parameters from explicit calculations of spin-spirals
the chiral transport properties of large-scale skyrmions can be predicted.

##  II Approach

###  II.1 Model

We investigate the existence and properties of chiral magnetotransport effects
on a bipartite honeycomb lattice of magnetic spins. To model the electronic
structure, we employ an effective two-dimensional lattice tight-binding
Hamiltonian (in the $xy$-plane) which reads:

$\begin{split}H=-t\sum\limits_{\langle
ij\rangle\alpha}c_{i\alpha}^{\dagger}c_{j\alpha}^{\phantom{\dagger}}&+i\alpha_{\rm
R}\sum\limits_{\langle
ij\rangle\alpha\beta}\hat{\mathbf{e}}_{z}\cdot(\boldsymbol{\sigma}\times{\mathbf{d}}_{ij})_{\alpha\beta}\,c_{i\alpha}^{\dagger}c_{j\beta}^{\phantom{\dagger}}\\\
&+\lambda_{\rm
ex}\sum_{i\alpha\beta}(\hat{\mathbf{s}}_{i}\cdot\boldsymbol{\sigma})_{\alpha\beta}\,c_{i\alpha}^{\dagger}c_{i\beta}^{\phantom{\dagger}},\end{split}$
(1)

where $c_{i\alpha}^{\dagger}$ ($c_{i\alpha}^{\vphantom{\dagger}}$) denotes the
creation (annihilation) of an electron with spin $\alpha$ at site $i$,
$\langle...\rangle$ restricts the sums to nearest neighbors, the unit vector
$\mathbf{d}_{ij}$ points from $j$ to $i$, and $\boldsymbol{\sigma}$ stands for
the vector of Pauli matrices. Besides the hopping with amplitude $t$, Eq. 1
contains the Rashba spin-orbit coupling of strength $\alpha_{\text{R}}$
originating for example in the surface potential gradient perpendicular to the
plane (i.e. along $\hat{\boldsymbol{z}}$). The remaining term in Eq. 1 is the
local exchange term with $\lambda_{\rm ex}$ characterizing the strength of
exchange splitting and $\hat{\mathbf{s}}_{i}$ stands for the direction of spin
on site $i$. Here, we work with the following parameters of the model: $t=1.0$
eV, $\alpha_{\rm R}=0.4$ eV, and $\lambda_{\rm ex}=1.4$ eV, which corresponds
to the case of a strongly spin-orbit coupled magnet.

###  II.2 Parametrization of spin spirals

In this work, we impose magnetic spiral distribution of spins
$\hat{\mathbf{s}}_{i}$ in the Hamiltonian (1), and we study the transport
properties of such states. We proceed by first defining a continuous,
normalized vector field $\hat{\mathbf{n}}(\mathbf{x})$ which is discretized on
the lattice as

$\displaystyle\hat{\boldsymbol{s}}_{i}\equiv\hat{\mathbf{n}}(\mathbf{R}_{i}-\mathbf{R}_{0}),$
(2)

where $\mathbf{R}_{i}$ corresponds to the real-space position of lattice site
$i$, and $\mathbf{R}_{0}$ represents a choice for the origin of the continuous
coordinate system. General spin-spirals are one-dimensional, periodic patterns
of spins, whose modulation is characterized by a single phase factor
$\Psi(\mathbf{x})=\mathbf{q}\cdot\mathbf{x}$. Explicitly, we first construct
such a spin-spiral in the $x$-direction followed by a rotation
$R^{\mathbf{e}_{z}}_{\phi_{q}}$ around the $z$-axis into the actual direction
of the $\mathbf{q}$-vector, as characterized by its polar coordinate
$\phi_{q}$:

$\hat{\mathbf{n}}=R^{\mathbf{e}_{z}}_{\phi_{q}}R^{\mathbf{e}_{r}}_{\Psi}\hat{\mathbf{n}}_{0}.$
(3)

Here, $\hat{\mathbf{n}}_{0}$ defines the initial orientation of
$\hat{\mathbf{n}}$ for $\Psi=0$. Following the direction of $\mathbf{q}$ in
real space, the pattern describes a rotational motion around the axis defined
by $\mathbf{e}_{r}$. With $\hat{\mathbf{n}}_{0}=\mathbf{e}_{z}$ fixed, three
different cases are considered here, see Fig. 1: the Bloch spiral
$\mathbf{e}_{r}=\mathbf{e}_{x}$, the Néel spiral
$\mathbf{e}_{r}=\mathbf{e}_{y}$ and a tilted conical Néel phase
$\mathbf{e}_{r}=\sin\alpha~{}\mathbf{e}_{y}+\cos\alpha~{}\mathbf{e}_{z}$ with
cone angle $0\leq\alpha\leq\pi/2$, interpolating between the ferromagnetic
phase for $\alpha=0$ and the Néel spiral for $\alpha=\pi/2$. All different
cases are contained in the parametrization:

$\mathbf{e}_{r}=\sin\beta\sin\alpha~{}\mathbf{e}_{x}+\cos\beta\sin\alpha~{}\mathbf{e}_{y}+\cos\alpha~{}\mathbf{e}_{z},$
(4)

where the angle $0\leq\beta\leq\pi/2$ can rotate from the tilted Néel phase
($\beta=0$) to a tilted Bloch phase ($\beta=\pi/2$).

The wavevector pitch $q=\|\mathbf{q}\|$ is the main tuning knob to adjust the
magnetization pattern in this work. The limiting cases are the ferromagnetic
(FM) pattern for $q=0$ and the antiferromagnetic pattern for $q=\pi$. The
$q=0$ limit corresponds to the regular out of plane (along the $z$-axis) FM
case. However, depending on the orientation of the rotation axis
$\mathbf{e}_{r}$ in real space, the two spins in the AFM pattern are not
antiparallel, but enclose a polar angle of $2\cdot\alpha$, twice the cone
angle that the rotation axis $\mathbf{e}_{r}$ encloses with the $z$-axis. This
can be understood by realizing that a phase $\pi$ between spins on neighboring
atomic sites corresponds to opposite positions on the cone around the rotation
axis $\mathbf{e}_{r}$, which has an opening angle of $2\cdot\alpha$.

###  II.3 Gradient expansion

Within the linear response theory, the conductivity tensor
$\sigma_{\alpha\beta}$ describes the electric current response in the system
which is linear in electric field $\mathbf{E}$ as
$j^{\alpha}=\sigma_{\alpha\beta}E^{\beta}$. In order to arrive at a way to
categorize different physical effects that are encapsulated in the
conductivity tensor, we start by writing an asymptotic expansion in the
gradients $\partial_{i}n_{j}$ for a smooth magnetization texture, i.e.,

$\sigma_{\alpha\beta}[\hat{\mathbf{n}}]\\!=\\!\braket{\sigma_{\alpha\beta}^{\mathrm{col}}(\hat{\mathbf{n}})+\sigma_{\alpha\beta\gamma\delta}^{\chi}(\hat{\mathbf{n}})\partial_{\gamma}n_{\delta}+\mathcal{O}(\partial^{2})}.$
(5)

Here, the bracket $\braket{\bullet}$ indicates the real-space integral
$\int\mathrm{d}\mathbf{x}/\mathcal{V}$, where $\mathcal{V}$ is the real-space
volume of the system, $\mathbf{x}=(x,y)$ is the position vector, and notation
$[\hat{\mathbf{n}}]$ implies the functional dependence on the overall texture
as given by the distribution of $\hat{\mathbf{n}}$ in real space. The
conductivity tensor itself can be decomposed into symmetric and antisymmetric
components with respect to the interchange of indices $\alpha$ and $\beta$:
$\sigma_{\alpha\beta}[\hat{\mathbf{n}}]=\sigma_{[\alpha\beta]}[\hat{\mathbf{n}}]+\sigma_{(\alpha\beta)}[\hat{\mathbf{n}}]$.
Here $[\alpha\beta]$ indicates the antisymmetrization, whereas $(\alpha\beta)$
represents a symmetrization of indices. Via the Onsager reciprocity relations,
$\sigma_{[\alpha\beta]}[\hat{\mathbf{n}}]=-\sigma_{[\alpha\beta]}[-\hat{\mathbf{n}}]$
and
$\sigma_{(\alpha\beta)}[\hat{\mathbf{n}}]=\sigma_{(\alpha\beta)}[-\hat{\mathbf{n}}]$.
The symmetric tensor then describes magnetoconductivity, anisotropic
magnetoconductivity and planar Hall effects, while its antisymmetric
counterpart captures the anomalous Hall effect. The superscript ”col”
indicates that
$\sigma_{\alpha\beta}^{\mathrm{col}}(\hat{\mathbf{n}}(\mathbf{x}))$ fully
describes the linear response of a collinear magnetized state. For slowly
varying textures, the next-to-leading-order term
$\sigma_{\alpha\beta\gamma\delta}^{\chi}(\hat{\mathbf{n}}(\mathbf{x}))$
couples to the first-order gradients of the magnetization texture and is
therefore sensitive to the chirality of $\hat{\mathbf{n}}$, thereby providing
the chiral part of the overall conductivity tensor. When the texture
constitutes a spin-spiral with a wave-vector $\mathbf{q}$, as introduced
above, the corresponding chiral corrections to the conductivity are fully
antisymmetric in $\mathbf{q}$.

Table 1: Character table of $C_{6v}$. Shown are the characters of each irreducible representation for each conjugacy class, alongside linear and quadratic basis functions which generate the respective representations. $C_{6v}$ | $E$ | 2$C_{6}\text{(z)}$ | $2C_{3}\text{(z)}$ | $C_{2}\text{(z)}$ | 3$\sigma_{v}$ | 3$\sigma_{d}$ | linear | quadratic
---|---|---|---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | $z$ | $x^{2}+y^{2},z^{2}$
$A_{2}$ | 1 | 1 | 1 | 1 | -1 | -1 | $n_{z}$ |
$B_{1}$ | 1 | -1 | 1 | -1 | 1 | -1 |  |
$B_{2}$ | 1 | -1 | 1 | -1 | -1 | 1 |  |
$E_{1}$ | 2 | 1 | -1 | -2 | 0 | 0 | $(x,y)$ | $(xz,yz)$
$E_{2}$ | 2 | -1 | -1 | 2 | 0 | 0 |  | $(x^{2}-y^{2},xy)$
Table 2: Kronecker products in $C_{6v}$. Shown is the reduction of all possible combinations of Kronecker products among the irreducible representations of $C_{6v}$.  $\otimes$ | $A_{1}$ | $A_{2}$ | $B_{1}$ | $B_{2}$ | $E_{1}$ | $E_{2}$
---|---|---|---|---|---|---
$A_{1}$ | $A_{1}$ | $A_{2}$ | $B_{1}$ | $B_{2}$ | $E_{1}$ | $E_{2}$
$A_{2}$ | $A_{2}$ | $A_{1}$ | $B_{2}$ | $B_{1}$ | $E_{1}$ | $E_{2}$
$B_{1}$ | $B_{1}$ | $B_{2}$ | $A_{1}$ | $A_{2}$ | $E_{2}$ | $E_{1}$
$B_{2}$ | $B_{2}$ | $B_{1}$ | $A_{2}$ | $A_{1}$ | $E_{2}$ | $E_{1}$
$E_{1}$ | $E_{1}$ | $E_{1}$ | $E_{2}$ | $E_{2}$ | $A_{1}+A_{2}+E_{2}$ | $B_{1}+B_{2}+E_{1}$
$E_{2}$ | $E_{2}$ | $E_{2}$ | $E_{1}$ | $E_{1}$ | $B_{1}+B_{2}+E_{1}$ | $A_{1}+A_{2}+E_{2}$

Table 3: Classification of electric transport effects in a noncollinear magnet. For the symmetry group $C_{6v}$, the conductivity tensor in the $x$-$y$–plane factors into the irreducible representations $\sigma=E_{1}\otimes E_{1}=A_{1}+A_{2}+E_{2}$. Each representation is generated from specific combinations of the components $\sigma_{\alpha\beta}$ which we refer to as channels. Within each channel, different kinds of physical effects can appear which are sensitive to different aspects of the underlying magnetization texture and which we roughly divide into those effects which are already present in a collinear ferromagnet and those which require a finite, first-order gradient $\partial_{i}n_{j}$.  Irrep | Channel |  Name |  Collinear effects $\mathcal{O}(\partial^{0})$ |  Chiral effects $\mathcal{O}(\partial^{1})$
---|---|---|---|---
$A_{1}$ | $(\sigma_{(xx)}+\sigma_{(yy)})/2$ |  isotropic longitudinal |  longitudinal conductivity (LC), magnetoconductivity (MC), anisotropic magnetoconductivity (AMC) |  chiral magnetoconductivity (CMC)
$A_{2}$ | $\sigma_{[xy]}$ |  antisymmetric transverse |  anomalous Hall effect (AHE) |  chiral Hall effect (CHE)
$(E_{2})_{1}$ | $(\sigma_{(xx)}-\sigma_{(yy)})/2$ |  anisotropic longitudinal |  longitudinal planar Hall effect (LPHE) |  chiral longitudinal planar Hall effect (CLPHE)
$(E_{2})_{2}$ | $\sigma_{(xy)}$ |  symmetric transverse |  planar Hall effect (PHE) |  chiral planar Hall effect (CPHE)

Following [33], one can expand $\sigma^{\mathrm{col}}$ and $\sigma^{\chi}$
into powers of the magnetization vector $\hat{\mathbf{n}}$ and reduce the
number of possible coupling terms using the restrictions imposed by the
crystallographic point group of the nonmagnetic lattice. For example, this
method has been successfully applied recently to study the chiral corrections
to the spin-Hall magnetoresistance in the noncollinear magnet Cu2OSeO3 [34].
Below, we demonstrate the way that it can be done for the partial case of
$C_{6v}$ symmetry of the Hamiltonian as given by Eq. (1). A systematic way to
perform this reduction is guided by the representation theory of $C_{6v}$ as
summarized by the character table in Tab. 1. Accordingly, the gradient
operator $\nabla=(\partial_{x},\partial_{y})$ generates the representations
$\Gamma_{\mathbf{x}}=\Gamma_{\nabla}=E_{1}$. The magnetization
$\hat{\mathbf{n}}$ – as an axial vector – decomposes into
$\hat{\mathbf{n}}_{\parallel}=(n_{x},n_{y},0)$ belonging to $E_{1}$ and
$\hat{\mathbf{n}}_{\perp}=(0,0,n_{z})$, belonging to $A_{2}$, i.e.,
$\Gamma_{\hat{\mathbf{n}}}=A_{2}+E_{1}$. Using the Schur orthogonality
relations [35, 36], one can derive the decomposition of the Kronecker products
$\Gamma_{i}\otimes\Gamma_{j}=\bigoplus_{k}\lambda_{k}\Gamma_{k}$ among any of
the irreducible representations (irreps) $\Gamma_{i}$, where $\lambda_{k}$ are
positive integers. Summarized in Tab. 2, the result of this procedure can be
used to study the symmetry-allowed couplings in the expansion of
$\sigma_{\alpha\beta}$: since it couples two polar vectors in the
$x$-$y$–plane, it carries the representation $\sigma=E_{1}\otimes
E_{1}=A_{1}+A_{2}+E_{2}$. Different irreps therefore distinguish different
categories of physical effects which are summarized in Table 3 and which will
be introduced in the following. Due to the Onsager relations, the $A_{1}$ and
$E_{2}$ contributions to $\sigma_{\alpha\beta}$ have to be even in $n_{i}$
while the $A_{2}$ term is odd. In the absence of magnetism or possible
external fields, all components that do not belong to the totally symmetric
representation $A_{1}$ have to vanish by Neumann’s symmetry principle [37,
38]. The $A_{1}$ term $(\sigma_{(xx)}+\sigma_{(yy)})/2$ then captures the
isotropic contributions to the longitudinal conductivity. In the presence of a
finite magnetization, the components of $\hat{\mathbf{n}}$ and its gradients
can form a basis for the irreducible representations $A_{2}$ and $E_{2}$,
thereby leading to a finite anomalous Hall effect in $\sigma_{[xy]}$ and a
finite planar Hall effect in $\sigma_{(xy)}$ for example. Using representation
theory, one can study what different terms appear in the gradient expansion of
Eq. (5) if each of the tensors is expanded as a power series in $n_{i}$. For
the collinear case, one finds that the magnetization carries the following
irreducible representations

$\displaystyle A_{1}\colon$ $\displaystyle n_{\parallel}^{2},n_{\perp}^{2}$
(6) $\displaystyle A_{2}\colon$ $\displaystyle n_{z}$ (7) $\displaystyle
E_{1}\colon$ $\displaystyle(n_{x}n_{z},n_{y}n_{z})$ (8) $\displaystyle
E_{2}\colon$ $\displaystyle\left((n_{x}^{2}-n_{y}^{2})/2,n_{x}n_{y}\right),$
(9)

up to second order in $n_{i}$, where $n_{\perp}^{2}=n_{z}^{2}$ and
$n_{\parallel}^{2}=n_{x}^{2}+n_{y}^{2}$. Here, one has to take into account
that $\hat{\mathbf{n}}$ transforms as a pseudovector. At the given order in
$n_{i}$, there are thus $5$ couplings to the collinear magnetic state:

$\displaystyle\sigma_{A_{1}}^{\mathrm{col}}(\hat{\mathbf{n}})$
$\displaystyle=\gamma_{\text{\tiny LC}}+\gamma_{\text{\tiny
MC}}\|\hat{\mathbf{n}}\|^{2}+\gamma_{\text{\tiny
AMC}}(n_{\perp}^{2}-n_{\parallel}^{2})+\mathcal{O}(n_{i}^{4})$ (10)
$\displaystyle\sigma_{A_{2}}^{\mathrm{col}}(\hat{\mathbf{n}})$
$\displaystyle=\gamma_{\text{\tiny AHE}}n_{z}+\mathcal{O}(n_{i}^{3})$ (11)
$\displaystyle\sigma_{E_{2}}^{\mathrm{col}}(\hat{\mathbf{n}})$
$\displaystyle=\gamma_{\text{\tiny
PHE}}((n_{x}^{2}-n_{y}^{2})/2,n_{x}n_{y})+\mathcal{O}(n_{i}^{4}),$ (12)

which we refer to as the longitudinal conductivity (LC), the
magnetoconductivity (MC), the anisotropic magnetoconductivity (AMC), the
anomalous Hall effect (AHE) and the planar Hall effect (PHE). The PHE is
commonly understood as the off-diagonal components $\sigma_{(xy)}$ of the
symmetrized conductivity tensor. Due to the peculiarity of the $C_{6v}$ \-
symmetry, the same coefficient controls anisotropic contributions to the
longitudinal conductivity $(\sigma_{(xx)}-\sigma_{(yy)})/2$. In Table 3, this
latter term is therefore listed as the longitudinal planar Hall effect (LPHE).

Moving on to the gradient-induced effects in two dimensions, one can deduce
from Tab. 1 and Tab. 2 that $\Gamma_{\nabla\otimes\hat{\mathbf{n}}}\equiv
E_{1}\otimes(A_{2}+E_{2})=A_{1}+A_{2}+E_{1}+E_{2}$. The relevant irrep is
$A_{2}$ (due to the Onsager relations), and it occurs only once:

$\displaystyle A_{1}\colon$ $\displaystyle(\nabla\times\hat{\mathbf{n}})_{z}$
(13) $\displaystyle A_{2}\colon$ $\displaystyle\nabla\cdot\hat{\mathbf{n}}.$
(14) $\displaystyle E_{1}\colon$
$\displaystyle(\partial_{x}n_{z},\partial_{y}n_{z})$ (15) $\displaystyle
E_{2}\colon$
$\displaystyle(\partial_{x}n_{x}-\partial_{y}n_{y},\partial_{x}n_{y}+\partial_{y}n_{x}).$
(16)

We assume that partial integration can be performed under the real-space
integral in Eq. (5) which renders the $A_{2}$ contribution integrate to zero.
The next order in $n_{i}$ can be obtained by reducing the tensor product
$\Gamma_{\nabla\otimes\hat{\mathbf{n}}}\otimes\Gamma_{\hat{\mathbf{n}}}=2A_{1}+2A_{2}+B_{1}+B_{2}+4E_{1}+2E_{2}$:

$\displaystyle A_{2}\otimes A_{2}\to A_{1}\colon$
$\displaystyle(\nabla\cdot\hat{\mathbf{n}})n_{z}$ (17) $\displaystyle
E_{1}\otimes A_{2}\to A_{1}\colon$
$\displaystyle(\hat{\mathbf{n}}\cdot\nabla)n_{z}$ (18) $\displaystyle
E_{2}\otimes A_{2}\to E_{2}\colon$ $\displaystyle
n_{z}(\partial_{x}n_{x}-\partial_{y}n_{y},\partial_{x}n_{y}+\partial_{y}n_{x}),$
(19) $\displaystyle E_{1}\otimes E_{1}\to E_{2}\colon$
$\displaystyle(n_{x}\partial_{x}n_{z}-n_{y}\partial_{y}n_{z},n_{x}\partial_{y}n_{z}+n_{y}\partial_{x}n_{z}).$
(20)

Under partial integration, one recognizes that the $A_{1}$ contributions
combine into the Lifshitz invariant
$(\nabla\cdot\hat{\mathbf{n}})n_{z}-(\hat{\mathbf{n}}\cdot\nabla)n_{z}$ which
is also responsible for the DMI interaction under $C_{6v}$ symmetry [39, 40].
The two $E_{2}$ contributions are the same under partial integration as well.
Combined, this leads to

$\displaystyle\sigma_{A_{1},\alpha\beta}^{\chi}(\hat{\mathbf{n}})\partial_{\alpha}n_{\beta}=$
$\displaystyle\gamma_{\text{\tiny
CMC}}(\nabla\cdot\hat{\mathbf{n}}-\hat{\mathbf{n}}\cdot\nabla)n_{z}+\mathcal{O}(n_{i}^{4}),$
(21)
$\displaystyle\sigma_{E_{2},\alpha\beta}^{\chi}(\hat{\mathbf{n}})\partial_{\alpha}n_{\beta}=$
$\displaystyle\gamma_{\text{\tiny
CPHE}}~{}n_{z}(\partial_{x}n_{x}-\partial_{y}n_{y},\partial_{x}n_{y}+\partial_{y}n_{x})$
$\displaystyle+\mathcal{O}(n_{i}^{4}),$ (22)

describing the chiral magnetoconductivity (CMC) and the chiral planar Hall
effect (CPHE).

The symmetrized tensor product
$\mathrm{Sym}(\hat{\mathbf{n}}\otimes\hat{\mathbf{n}})$ carries the
representation
$\Gamma_{\mathrm{Sym}(\hat{\mathbf{n}}\otimes\hat{\mathbf{n}})}=2A_{1}+E_{1}+E_{2}$.
For the next order in $n_{i}$, one therefore finds
$\Gamma_{\nabla\otimes\hat{\mathbf{n}}}\otimes\Gamma_{\mathrm{Sym}(\hat{\mathbf{n}}\otimes\hat{\mathbf{n}})}=4A_{1}+4A_{2}+2B_{1}+2B_{2}+6E_{1}+6E_{2}$.
Since this order is odd under time reversal, we are only interested in the
four $A_{2}$ representations given by

$\displaystyle A_{2}\otimes A_{1}\to 2A_{2}\colon$
$\displaystyle(\nabla\cdot\hat{\mathbf{n}})n_{\perp}^{2},(\nabla\cdot\hat{\mathbf{n}})n_{\parallel}^{2},$
(23) $\displaystyle E_{1}\otimes E_{1}\to A_{2}\colon$
$\displaystyle(\hat{\mathbf{n}}\cdot\nabla)n_{\perp}^{2}/2$ (24)
$\displaystyle E_{2}\otimes E_{2}\to A_{2}\colon$ $\displaystyle
n_{x}n_{y}(\partial_{x}n_{y}+\partial_{y}n_{x})$ (25)
$\displaystyle+(n_{x}^{2}-n_{y}^{2})(\partial_{x}n_{x}-\partial_{y}n_{y})/2.$
(26)

The last term may be recombined with other $A_{2}$ representations of this
order such that it can be written as
$(\hat{\mathbf{n}}\cdot\nabla)n_{\parallel}^{2}/2$. And therefore

$\displaystyle\sigma_{A_{2},\alpha\beta}^{\chi}(\hat{\mathbf{n}})\partial_{\alpha}n_{\beta}=$
$\displaystyle\gamma_{\text{\tiny
CHE}}~{}(\nabla\cdot\hat{\mathbf{n}})(n_{\perp}^{2}-n_{\parallel}^{2})+\mathcal{O}(n_{i}^{5}),$
(27)

is the only coupling that describes the chiral Hall effect (CHE) in $C_{6v}$
symmetric systems. In summary, we find that

$\displaystyle\sigma_{A_{1}}[\hat{\mathbf{n}}]=$
$\displaystyle(\gamma_{\text{\tiny LC}}+\gamma_{\text{\tiny
MC}})+\gamma_{\text{\tiny AMC}}\braket{n_{\perp}^{2}-n_{\parallel}^{2}}$
$\displaystyle+\gamma_{\text{\tiny
CMC}}\braket{(\nabla\cdot\hat{\mathbf{n}}-\hat{\mathbf{n}}\cdot\nabla)n_{z}}+\mathcal{O}(n_{i}^{4})$
(28) $\displaystyle\sigma_{A_{2}}[\hat{\mathbf{n}}]=$
$\displaystyle\gamma_{\text{\tiny AHE}}\braket{n_{z}}+\gamma_{\text{\tiny
CHE}}\braket{(\nabla\cdot\hat{\mathbf{n}})(n_{\perp}^{2}-n_{\parallel}^{2})}+\mathcal{O}(n_{i}^{5})$
(29) $\displaystyle\sigma_{E_{2}}[\hat{\mathbf{n}}]=$
$\displaystyle\gamma_{\text{\tiny
PHE}}\braket{((n_{x}^{2}-n_{y}^{2})/2,n_{x}n_{y})}$
$\displaystyle+\gamma_{\text{\tiny
CPHE}}\braket{n_{z}(\partial_{x}n_{x}-\partial_{y}n_{y},\partial_{x}n_{y}+\partial_{y}n_{x})}$
$\displaystyle+\mathcal{O}(n_{i}^{4}),$ (30)

where the factors $\gamma_{i}$ are temperature-dependent material constants
that can be extracted from an underlying microscopic model. Inserting the
spin-spiral defined by the combination of Eq. (3) and Eq. (4), one arrives at

$\displaystyle\sigma_{A_{1}}[\hat{\mathbf{n}}]=$
$\displaystyle(\gamma_{\text{\tiny LC}}+\gamma_{\text{\tiny
MC}})+\frac{1}{2}\gamma_{\text{\tiny AMC}}\cos^{2}(\alpha)(3\cos(2\alpha)-1)$
$\displaystyle+\gamma_{\text{\tiny
CMC}}q\cos\beta\sin^{3}\alpha+\mathcal{O}(n_{i}^{4})$ (31)
$\displaystyle\sigma_{A_{2}}[\hat{\mathbf{n}}]=$
$\displaystyle\gamma_{\text{\tiny
AHE}}\cos^{2}\alpha+\frac{1}{2}\gamma_{\text{\tiny
CHE}}q\cos\beta\sin(\alpha)\sin^{2}(2\alpha)$
$\displaystyle+\mathcal{O}(n_{i}^{5})$ (32)
$\displaystyle\sigma_{E_{2}}[\hat{\mathbf{n}}]=$
$\displaystyle-\frac{1}{8}\gamma_{\text{\tiny
PHE}}\sin^{2}(\alpha)(3\cos(2\alpha)+1)\boldsymbol{\vartheta}_{2}$
$\displaystyle+\frac{1}{2}\gamma_{\text{\tiny
CPHE}}q\sin^{3}(\alpha)\boldsymbol{\vartheta}_{1}+\mathcal{O}(n_{i}^{4}),$
(33)

where
$\boldsymbol{\vartheta}_{n}=(\cos(n\beta-2\phi_{q}),-\sin(n\beta-2\phi_{q}))$.

Table 4: The existence of chiral magnetotransport of spin-spiral states. For each type of the spiral state (Néel, Bloch, cone) the emergence of the corresponding effect [non-chiral (AHE) and chiral (CHE) anomalous Hall effect, non-chiral (MC) and chiral (CMC) magnetoconductivity, non-chiral (PHE) and chiral (CPHE) chiral planar Hall effect] is marked with the sign “$\checkmark$” when it is allowed by symmetry arguments of the gradient expansion and confirmed by explicit Kubo transport calculations, while its absence is marked with an empty entry. Crystallographic directions $[100]$, $[0\overline{1}1]$ and $[110]$ mark the direction of the $\mathbf{q}$-vector, which correspond to the value of $\phi_{q}$ of 0, $-\pi/2$ and $\pi/3$, respectively. Type | AHE | CHE | MC | CMC | PHE | CPHE
---|---|---|---|---|---|---
Néel | [100] |  |  | ✓ | ✓ |  |
[0$\overline{1}$1] |  |  | ✓ | ✓ |  |
[110] |  |  | ✓ | ✓ | ✓ | ✓
Bloch | [100] |  |  | ✓ |  |  | ✓
[0$\overline{1}$1] |  |  | ✓ |  |  | ✓
[110] |  |  | ✓ | ✓ | ✓ | ✓
Cone | [100] | ✓ | ✓ | ✓ | ✓ |  |
[0$\overline{1}$1] | ✓ | ✓ | ✓ | ✓ |  |
[110] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

Figure 2: Electronic structure of Néel, Bloch and cone-type spirals.
Bandstructures of Néel (a) and Bloch (b) spin-spirals in the energy interval
$[-1,1]$ eV for $\mathbf{q}=\pm 0.31a_{0}^{-1}\cdot\hat{\boldsymbol{x}}$ (blue
and red). (c-d) Bandstructures of a cone-type spiral with the cone angle
$\alpha=$0.6\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$\approx$$$ in the energy
range of $[-1,1]$ eV for opposite orientations of the wavevector
$\mathbf{q}=\pm 0.31a_{0}^{-1}\cdot\hat{\boldsymbol{x}}$, reflecting the
changes in the electronic structure due to opposite sense of chirality. The
color of the bands in (c-d) indicates the value of the band-resolved Berry
curvature $\Omega_{xy}(\mathbf{k})$. (e-h) The real-space magnetic texture
corresponding to band structures in the respective columns. For more details
see main text.

###  II.4 Kubo Formalism

Given specific electronic structure, we calculate the transverse and diagonal
conductivity at zero temperature using the Kubo formalism, which allows us to
take into account the effect of disorder in the system on the conductivity
tensor. In order to do so, we replace the retarded and advanced Green
functions $G_{0}$ of the perfect crystal by the full Green function
$G=\frac{1}{G_{0}^{-1}-\Sigma}$, where $\Sigma(E,\mathbf{k})$ is the self-
energy representing the effect of disorder. Within this work, we are using a
constant broadening model such that
$\Sigma(E,\mathbf{k})=-i\Gamma\cdot\mathcal{I}$. With the constant broadening
$\Gamma$ we obtain a Green function diagonal in the eigenspace of the
Hamiltonian:

$\displaystyle
G^{R/A}(E,\mathbf{k})_{mn}=\frac{\delta_{mn}}{E-\epsilon_{n\mathbf{k}}\pm
i\Gamma},$ (34)

where $\epsilon_{n\mathbf{k}}$ are the single-electron eigenenergies. The
antisymmetric part of the conductivity tensor, which can be expressed in terms
of $G^{R/A}$ [41], splits into two contributions:

$\displaystyle\sigma_{[\alpha\beta]}^{I}=$
$\displaystyle-\frac{1}{2\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\begin{subarray}{c}mn\\\
m\neq
n\end{subarray}}\Im\\{v_{mn}^{\alpha}(\mathbf{k})v_{nm}^{\beta}(\mathbf{k})\\}$
(35)
$\displaystyle\times\frac{(\epsilon_{m\mathbf{k}}-\epsilon_{n\mathbf{k}})\Gamma}{((E_{F}-\epsilon_{m\mathbf{k}})^{2}+\Gamma^{2})((E_{F}-\epsilon_{n\mathbf{k}})^{2}+\Gamma^{2})}$

and

$\displaystyle\sigma_{[\alpha\beta]}^{II}=$
$\displaystyle-\frac{1}{\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\begin{subarray}{c}mn\\\
m\neq
n\end{subarray}}\Im\\{v_{mn}^{\alpha}(\mathbf{k})v_{nm}^{\beta}(\mathbf{k})\\}$
(36)
$\displaystyle\times\frac{\Gamma}{(\epsilon_{m\mathbf{k}}-\epsilon_{n\mathbf{k}})((E_{F}-\epsilon_{m\mathbf{k}})^{2}+\Gamma^{2})}$
$\displaystyle-\frac{1}{(\epsilon_{m\mathbf{k}}-\epsilon_{n\mathbf{k}})^{2}}\Im\left\\{\ln{\left(\frac{E_{F}-\epsilon_{m\mathbf{k}}+i\Gamma}{E_{F}-\epsilon_{m\mathbf{k}}+i\Gamma}\right)}\right\\},$

where $\alpha$ and $\beta$ are the Cartesian indices and $E_{F}$ is the Fermi
energy. We refer to $\sigma_{\alpha\beta}^{I}$ as the Fermi-surface term,
since it only picks up contributions from the Fermi surface. The term
$\sigma_{\alpha\beta}^{II}$ collects terms from all occupied states up to the
Fermi level and is therefore referred to as the Fermi-sea term. The symmetric
part of the conductivity tensor is given by [42]

$\displaystyle\sigma_{(\alpha\beta)}=$
$\displaystyle\frac{1}{\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{mn}\Re\\{v_{mn}^{\alpha}(\mathbf{k})v_{nm}^{\beta}(\mathbf{k})\\}$
(37)
$\displaystyle\times\frac{\Gamma^{2}}{((E_{F}-\epsilon_{m\mathbf{k}})^{2}+\Gamma^{2})((E_{F}-\epsilon_{n\mathbf{k}})^{2}+\Gamma^{2})}.$

In evaluating the Kubo expressions for the conductivity of spin-spiral states
we have considered systems treated in a super-cell with up to 1600 atoms in
the unit cell, using up to $2\cdot 10^{5}$ $k$-points for performing Brillouin
zone integrations.

##  III Chiral transport properties of spin-spirals

Below we compare the results of the gradient expansion performed up to linear
order for the considered model to the explicit tight-binding calculations of
the conductivity tensor by using the Kubo formalism. From explicit
calculations of the conductivity for the system in a spin-spiral state
$\hat{\mathbf{n}}_{\mathrm{cone}}$ (defined in Eq. (3) and Eq. (4)), we
extract the contributions to the conductivity tensor which are even (non-
chiral, $\sigma^{nc}$) and odd (chiral, $\sigma^{c}$) in spiral wave-vector
$\mathbf{q}$ by performing the corresponding decomposition:

$\displaystyle\sigma_{\alpha\beta}^{c(nc)}=\frac{\sigma_{\alpha\beta}(\mathbf{q})\mp\sigma_{\alpha\beta}(-\mathbf{q})}{2}.$
(38)

Using this definition, we can make a connection to the gradient expansion in
the long-wavelength limit. Here, one finds the asymptotic relationships

$\displaystyle\sigma_{\alpha\beta}^{nc}$
$\displaystyle\sim\braket{\sigma^{\mathrm{col}}_{\alpha\beta}(\hat{\mathbf{n}})},~{}\text{for}~{}q\to
0,$ (39) $\displaystyle\sigma_{\alpha\beta}^{c}$
$\displaystyle\sim\braket{\sigma^{\chi}_{\alpha\beta\gamma\delta}(\hat{\mathbf{n}})\partial_{\gamma}n_{\delta}},~{}\text{for}~{}q\to
0.$ (40)

We thus scrutinize the existence of chiral contributions to the
magnetoconductivity, $\sigma^{c}_{(\alpha\alpha)}$ ($\mathbf{q}$-chiral part
of the diagonal components of the conductivity tensor
$\sigma_{\alpha\alpha}$), chiral planar Hall effect,
$\sigma^{c}_{(\alpha\beta)}$ ($\mathbf{q}$-chiral part of the
$\alpha\leftrightarrow\beta$ symmetric off-diagonal components of the
conductivity tensor $\sigma_{\alpha\beta}$) and chiral Hall effect,
$\sigma^{c}_{[\alpha\beta]}$ ($\mathbf{q}$-chiral part of the
$\alpha\leftrightarrow\beta$ antisymmetric off-diagonal components of the
conductivity tensor $\sigma_{\alpha\beta}$). The predictions of the explicit
calculations concerning the existence of MC and CMC, PHE and CPHE, AHE and
CHE, are presented in Table 4. They are entirely consistent with the symmetry
analysis of the gradient expansion also beyond the long-wavelength limit:
whenever an empty instance is met in the table, the gradient expansion
predicts a vanishing contribution for a given direction of the spin-spiral and
its type, while explicit calculations provide negligible values of the
conductivity. Below, we discuss in detail the emergence of chiral
contributions to the MC, PHE and AHE.

Figure 3: Chiral magnetoconductivity (CMC), chiral planar Hall effect (CPHE)
and chiral Hall effect (CHE). (a,d): chiral magnetoconductivity (CMC),
$\sigma^{c}_{xx}$, calculated for a Néel-type spiral with $\mathbf{q}=\pm
0.47a_{0}^{-1}\cdot\hat{\boldsymbol{x}}$ as a function of Fermi energy $E_{f}$
and broadening $\Gamma$, (a), and as a function of $E_{f}$ and $q$ (along
$\hat{\boldsymbol{x}}$) at $\Gamma$ of 100 meV, (d). The insets depict the
corresponding behavior of the non-chiral part of the conductivity tensor
element, $\sigma^{nc}_{xx}$. (b,e): chiral planar Hall effect (CPHE),
$\sigma^{c}_{(xy)}$, calculated for a Bloch spiral with $\mathbf{q}=\pm
0.47a_{0}^{-1}\cdot\hat{\boldsymbol{x}}$ as a function of $E_{f}$ and
$\Gamma$, (b), and as a function of $E_{f}$ and $q$ (along
$\hat{\boldsymbol{x}}$) at $\Gamma$ of 100 meV, (e). The corresponding non-
chiral part of the conductivity tensor element, $\sigma^{nc}_{(xx)}$, vanishes
in this case, see Table 4. (c,f): chiral Hall effect (CHE),
$\sigma^{c}_{[xy]}$, calculated for a cone-type spiral with $\mathbf{q}=\pm
0.47a_{0}^{-1}\cdot\hat{\boldsymbol{x}}$ and a cone angle of
$\alpha=$0.6\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$\approx$$$ as a function
of $E_{f}$ and $\Gamma$, (c), and as a function of $E_{f}$ and $q$ (along
$\hat{\boldsymbol{x}}$) at $\Gamma$ of 100 meV, (f). The insets depict the
corresponding behavior of the non-chiral part of the conductivity tensor
element, $\sigma^{nc}_{[xy]}$.

###  III.1 Longitudinal chiral conductivity

The discussion of chiral effects in the longitudinal conductivity falls into
two categories: the isotropic contributions $(\sigma_{(xx)}+\sigma_{(yy)})/2$
from the totally symmetric irrep $A_{1}$, which we defined as CMC, and
anisotropic contributions $(\sigma_{(xx)}-\sigma_{(yy)})/2$ from irrep
$E_{2}$, which we referred to as CLPHE. For the spiral, the two effects
evaluate to

$\displaystyle\gamma_{\text{\tiny
CMC}}\braket{(\nabla\cdot\hat{\mathbf{n}}-\hat{\mathbf{n}}\cdot\nabla)n_{z}}=$
$\displaystyle\gamma_{\text{\tiny CMC}}q\cos\beta\sin^{3}\alpha,$ (41)
$\displaystyle\gamma_{\text{\tiny
CPHE}}\braket{n_{z}(\partial_{x}n_{x}-\partial_{y}n_{y})}=$
$\displaystyle\frac{1}{2}\gamma_{\text{\tiny
CPHE}}q\sin^{3}(\alpha)\cos(\beta-2\phi_{q}).$ (42)

For a Bloch-type spiral, $\beta=\pi/2$, the CMC is zero and the CLPHE terms
only exist along specific directions: $\cos(\pi/2-2\phi_{q})=\sin(2\phi_{q})$.
From this argument it is clear that a Bloch spiral along the $x$-direction
will not display any chiral longitudinal conductivity. Consequently, we take
these longitudinal effects under closer inspection for a Néel-type spiral,
$\beta=0$. Inspecting Eq. 41 we realize that

$\sigma_{(xx)}^{c}=\sigma_{A_{1}}+\sigma_{E_{2}}^{1}$ (43)

contains both isotropic and anisotropic components. However, both isotropic
and anisotropic components follow the angular dependence dictated by symmetry

$\sigma_{(xx)}^{c}(\alpha)=\sin^{3}\alpha\left.\sigma_{(xx)}^{c}\right|_{\alpha=\pi/2}$
(44)

in the long-wavelength limit, independent of the type of spiral.

The bandstructure of the model for the Néel spiral with $\pm\mathbf{q}$ along
$x$ and the magnitude of $|\mathbf{q}|=0.31\,a_{0}^{-1}$, presented in Fig. 2
a), shows a strong chiral asymmetry. Drastic modifications brought to the
energetic positions of the states as well as their relative splittings give
rise to a pronounced chiral contribution to the longitudinal conductivity. In
Fig. 3 (a) we show the CMC-signal given by $\sigma_{xx}^{c}$ as a function of
Fermi energy $E_{f}$ and broadening $\Gamma$, taking a specific value of
$|\mathbf{q}|=0.47\,a_{0}^{-1}$ for a Néel spiral propagating along the
$x$-axis. For comparison, we show the non-chiral MC in an inset. We observe
the particle-hole symmetry of the model which is evident from the symmetry of
$\sigma_{xx}^{c}$ with respect to Fermi energy $E_{f}$. This symmetry is
fulfilled by all quantities displayed in Fig. 3 (a-f). Strong contributions to
CMC, shown in Fig. 3 (a), arise from certain specific energies (e.g.
$\pm$1.3\text{\,}\mathrm{e}\mathrm{V}$$ and broader humps around
$\pm$2.0\text{\,}\mathrm{e}\mathrm{V}$$), corresponding to the features in the
electronic structure which are most affected by chirality. In contrast, non-
chiral MC given by $\sigma_{xx}^{nc}$, is less sensitive to the Fermi energy.
The magnitude of the computed CMC signal reaches as much as 30% of the non-
chiral part of the conductivity. Clearly, the largest values of CMC are
reached for the smallest values of $\Gamma$, and the rapid decay of
$\sigma_{xx}^{c}$ with $\Gamma$ is evident. A careful analysis of the CMC
scaling with $\Gamma$ for very small values of $\Gamma$ reveals an expected
$1/\Gamma$ behavior.

In Equation 44 we uncovered a direct connection between the CMC for a Néel
spiral and that of the cone-type spiral, valid in the limit of small spin
orbit coupling (SOC). This connection is intuitive in the sense that the Néel
spiral is the limiting case of $\beta=0$ for the cone-type spiral, where the
cone angle approaches $\alpha=\pi/2$. We examine the prediction of Equation 44
made from the relation between the Néel and the cone-type spiral in Figure 4
a), comparing the scaling of the CMC-signal for the cone-type spiral,
$\sigma_{xx,\text{Cone}}^{c}$ (blue), to the function
$\sin^{3}(\alpha)\cdot\sigma_{xx,\text{Neel}}^{c}$ (red). For SOC strength of
up to $\alpha_{R}=$0.04\text{\,}\mathrm{e}\mathrm{V}$$ we observe a perfect
match where the cone-type spiral is equivalent to a Néel spiral (left side,
$\alpha=0$) up to cone angles as large as $\pi/4$. For even larger cone
angles, the CMC signals deviate slightly (about 2-3 %) due to the influence of
higher order terms included in the numerical calculation, which are neglected
in the gradient expansion. We can conclude here that the first nontrivial
order in the gradient expansion reproduces the numerical calculations
exceptionally well, when SOC is only of moderate strength.

The dependence of the CMC signal on the magnitude of $q$ is shown in Figure 3
d) for a fixed value of $\Gamma$. Clearly, the spin-spiral pitch influences
the magnitude of CMC in the most drastic way. The data reveals that CMC
vanishes in the collinear ferromagnetic (FM) limit, $q=0$, and then quickly
increases in magnitude over the whole energy range, as $q$ is increased, with
the effect being particularly prominent at around $\pm 2$ eV. Although the
qualitative distinction between $\sim q$ and $\sim q^{2}$ contributions to the
longitudinal conductivity of domain walls has been discussed in the past based
on model arguments and material-specific calculations [32, 25, 43, 44], the
true chiral nature of the CMC manifests in the dependence of the overall
signal on the sense of chirality of the spiral states observed here.
Remarkably, the chiral magnetoconductivity may change sign several times for
different values of $q$ at a fixed value of the Fermi energy. Such complex
behavior finds its roots in a strong influence of the spin-spiral vector on
the electronic structure of the system, with the corresponding shifts of the
bands occurring over the range of eVs. CMC therefore directly stems from the
presence of strong spin-orbit coupling, tying the rearrangements of bands to
the sense of chirality as it is visible in Fig. 2. The extreme sensitivity of
CMC to wavevector $q$ suggests that for spin-textures hosted in materials with
strong spin-orbit interaction, the measured longitudinal conductivity can
fluctuate dramatically even upon small variations in spin distribution,
brought about, for example, by an external magnetic field. Naively, such
changes in measured magnetotransport can be erroneously interpreted as arising
from qualitative modifications in texture properties.

###  III.2 Chiral planar Hall effect

Traditionally, the planar Hall effect is understood as the $\sigma_{(xy)}$
entry of the conductivity tensor, which we find in the second component of
irrep $E_{2}$, $\sigma_{(xy)}=\sigma_{E_{2,2}}[\hat{\mathbf{n}}]$. For the
case of $C_{6v}$ symmetry and the cone spiral as defined before, one finds the
chiral contribution as a sum of two terms

$\displaystyle\sigma_{E_{2,2}}[\hat{\mathbf{n}}]=$
$\displaystyle\frac{1}{8}\gamma_{\text{\tiny
PHE}}\sin^{2}(\alpha)(3\cos(2\alpha)+1)\sin(2\beta-2\phi_{q})$
$\displaystyle-\frac{1}{2}\gamma_{\text{\tiny
CPHE}}q\sin^{3}(\alpha)\sin(\beta-2\phi_{q}).$ (45)

The non-chiral contribution vanishes exactly. For tilted conical Néel spirals,
$\beta=0$, the angular dependence is $\sin(2\phi_{q})$ for both terms. As a
direct consequence the tensor component describing the CPHE for Néel spirals
is only nonzero for $\phi_{q}\neq 0$, i.e. when the wavevector $\mathbf{q}$ is
not parallel to the $x$-axis. In contrast, the term in
$\sigma_{E_{2,2}}[\hat{\mathbf{n}}]$ proportional to $\gamma_{\text{\tiny
CPHE}}$ survives for the corresponding spirals of Bloch-type along the
$x$-axis, $\beta=\pi/2$ and $\phi_{q}=0$. This can be understood by realizing
that for $\beta=\pi/2$, the term linear in $\gamma_{\text{\tiny CPHE}}$ is
proportional to $\cos(2\phi_{q})$, while the term linear in
$\gamma_{\text{\tiny PHE}}$ is proportional to $\sin(2\phi_{q})$. This
effective phase shift of $\pi/2$ results in a non-zero contribution to the
symmetric transverse conductivity $\sigma_{(xy)}$. Therefore, we investigate
$\sigma_{(xy)}$ for the Bloch spiral along the $x$-axis, which has anisotropic
contributions from irrep $E_{2}$ proportional to $\gamma_{\text{\tiny CPHE}}$
only. This is in contrast to the case of $\sigma_{xx}$ calculated for Néel
spirals along $x$, where both anisotropic and isotropic components from irreps
$A_{1}$ and $E_{2}$ contribute.

The band structure of the model for a Bloch spiral with $\pm\mathbf{q}$ along
$x$ and the magnitude of $|\mathbf{q}|=0.31\,a_{0}^{-1}$, presented in Figure
2 b), exhibits certain chiral asymmetry of the bands. In comparison to the
band structure of the Néel spiral shown in Figure 2 a), the rearrangement of
bands is not as drastic for the Bloch spiral. Nonetheless, the chiral PHE is
very prominent, as can be observed for a Bloch spiral propagating along the
$x$-axis with $|\mathbf{q}|=0.47\,a_{0}^{-1}$ in Fig. 3 b). Similar to the
CMC-signal shown in Fig. 3 a), most pronounced features in the CPHE originate
from the energy regions around $\pm$1.3\text{\,}\mathrm{e}\mathrm{V}$$ and
around $\pm$2.0\text{\,}\mathrm{e}\mathrm{V}$$. While these features are
generally broader in energy and similar in magnitude to CMC, strong
fingerprints of CPHE appear also around $0\text{\,}\mathrm{e}\mathrm{V}$. In
comparison to CMC, chiral PHE is not as rapidly decreasing with increasing
$\Gamma$, although the overall trend remains.

Moving on to Fig. 3 d), we inspect CPHE as a function of $q$ for a fixed value
of $\Gamma$. We observe that the CPHE signal vanishes for the FM limit and
then increases in magnitude, but not as rapidly as the CMC signal does.
Moreover, at fixed $E_{f}$, sign changes with $q$ occur only at very specific
energies from the range around $\pm$1\text{\,}\mathrm{e}\mathrm{V}$$ for the
CPHE, and not over wide ranges of energies as is the case for CMC. This is a
consequence of an overall smoother distribution of CPHE as a function of
$E_{f}$ and $q$. We expect that CMC and CPHE are clearly distinguishable
experimentally in materials displaying spiral textures, since they display
opposite signs over a wide range of spiral pitch $q$. One of the most
interesting features of the planar Hall effect for Bloch-type spirals
propagating along the $x$-axis is suppressed non-chiral PHE, which is given by
$\sigma_{(xy)}^{nc}$. This numerical result is in perfect agreement with the
predictions from the gradient expansion, see Table 4. The non-chiral PHE
vanishes completely, while we observe a planar Hall effect, comparable in
magnitude to AMC, which is purely chiral in nature.

###  III.3 Chiral Hall effect

We finally discuss chiral corrections to the AHE exhibited by spin-spiral
states. The AHE is described by the antisymmetric conductivity
$\sigma_{[xy]}$, where we find the chiral term

$\gamma_{\text{\tiny
CHE}}\braket{(\nabla\cdot\hat{\mathbf{n}})(n_{\perp}^{2}-n_{\parallel}^{2})}=\frac{1}{2}\gamma_{\text{\tiny
CHE}}q\cos\beta\sin(\alpha)\sin^{2}(2\alpha),$ (46)

for the cone spiral as introduced before. Correspondingly, this effect is
referred to as the chiral Hall effect (CHE). It is zero for $\alpha=$$$ and
$\alpha=$$$, and it attains a maximum value at $\alpha\approx$$$. We inspect
the CHE-signal for the cone-type spiral at the intermediate value
$\alpha=$0.6\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$\approx$$$.

In Fig. 3 c) we present the calculated CHE-signal, as given by
$\sigma_{[xy]}^{c}$, for the cone-type spiral propagating along the $x$-axis
with $|\mathbf{q}|=0.47\,a_{0}^{-1}$ and a cone angle
$\alpha=$0.6\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$$ as a function of Fermi
energy $E_{f}$ and broadening $\Gamma$. The overall observed structure of the
CHE as a function of the Fermi energy is much richer than that exhibited by
CMC and CPHE, as well as the non-chiral part of the AHE, $\sigma_{[xy]}^{nc}$
(shown in the inset). Consequently, CHE is characterized by a stronger
sensitivity to finer details of the electronic structure mediated by the sense
of chirality. In comparison to CMC and CPHE, the most remarkable feature of
both chiral and non-chiral parts of the AHE is a qualitatively different
behavior with $\Gamma$: in contrast to the data shown in Fig. 3 a)-b) for CMC
and CPHE, the CHE-signal does not scale as dramatically with the broadening.
Especially the most pronounced features of the CHE located in the energy
region $[$-1\text{\,}\mathrm{e}\mathrm{V}$,$1\text{\,}\mathrm{e}\mathrm{V}$]$
are least sensitive to $\Gamma$ variation. This behavior can be explained by
the robustness of intrinsic contributions to the CHE. In fact, with decreasing
disorder the values of the CHE converge to the clean limit values, which are
given by the integrated values of Berry curvature $\Omega_{xy}$
antisymmetrized with respect to $\mathbf{q}$ [20]. The rise of non-vanishing
intrinsic chiral AHE becomes apparent from looking at Fig. 2 c)-d), where the
strongly chirality-dependent electronic structure and Berry curvature of the
states is visible. On the other hand, by comparing the chiral and non-chiral
parts of the AHE, we observe that the disorder corrections to the intrinsic
conductivity are much more important for the CHE than for the conventional
AHE: the convergence to the clean limit values is quite slow and the
saturation does not occur up to broadening $\Gamma$ on the scale of
$0.01\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. This points to possibly very
prominent extrinsic contributions to the chiral Hall effect in realistic spin
textures.

Figure 4: Scaling with the cone angle and spin-spiral pitch. (a) Chiral
magnetoconductivity (CMC) for a cone-type spiral (blue) calculated for cone
angles in the range $\alpha\in[0,\pi/2]$ and CMC for a Néel-type spiral (red)
calculated for cone angle $\alpha=0$ and extrapolated to larger cone angles by
multiplying with $\sin(\alpha)^{3}$, see Equation 44. The relationship between
CMC for cone-type and Néel spirals obtained from the gradient expansion is
reproduced with explicit tight-binding calculations. The calculation was done
for spin-orbit coupling $\alpha_{R}=$0.04\text{\,}\mathrm{e}\mathrm{V}$$ at a
spiral pitch $q=0.031\,a_{0}^{-1}$. (b) Ferromagnetic limit of the chiral Hall
effect (CHE) for a cone-type spiral. Surface term (orange), sea term (red) and
the sum of the two (blue) yielding the CHE for non zero broadening $\Gamma$.
(c) The CHE signal near the FM limit of $q=0$. The linear slope
$\gamma\approx$-10.752\text{\,}\mathrm{e}^{2}\mathrm{a}_{\mathrm{0}}\mathrm{/}\mathrm{h}$$
in the limit of $q\rightarrow 0$, participating in gradient expansion
prediction Equation 51, can be estimated from calculations. For all plots, the
data were calculated for the values of
$E_{f}=$-0.625\text{\,}\mathrm{e}\mathrm{V}$$ and
$\Gamma=$0.2\text{\,}\mathrm{e}\mathrm{V}$$. A value of
$\alpha_{R}=$0.4\text{\,}\mathrm{e}\mathrm{V}$$ was used in (b) and (c).

In Fig. 3 f) we analyze the behavior of the CHE signal with the magnitude of
the wave-vector $q$. What can be concluded from comparing all the effects
among each other is that while the magnitude of the AHE and CHE is expectedly
significantly smaller than that of the CMC and CPHE (at the same value of
$\Gamma$), the magnitudes of the chiral and non-chiral AHE are comparable.
Among the latter two, the CHE is much more sensitive to $q$, which is
especially visible for smaller values of $q$, where the structure of the
chiral signal is especially non-trivial with respect to the Fermi energy.
Here, small deviations from the zero-$q$ limit may result in sign changes in
CHE, which bares relevance for magneto-transport detection of textures (see
discussion of CMC). By inspecting the $q=0$ limit more closely in Fig. 4 b),
we find that the data correctly reproduces the ferromagnetic limit, where the
CHE is vanishing. The data presented in Fig. 4 c) also illustrates the linear
scaling behavior of the CHE signal $\sigma_{[xy]}^{a}$ with increasing $q$
away from the ferromagnetic limit. This stands in agreement with the gradient
expansion prediction, and confirms that higher-order beyond-linear corrections
to the chirality-sensitive signal are negligible in the limit of vanishing
$q$.

The limit on the validity of the gradient expansion technique is apparent from
the calculations: deviations from linear-in-$q$ behavior visible above
$q\approx 0.03\cdot$\text{\,}\mathrm{a}_{\mathrm{0}}^{-1}$$ $-$ a value which
depends significantly on the parameters of the model. Importantly, in this
range of the $q$-vectors we observe that the non-chiral corrections to the
$q=0$ values of the AHE are significantly smaller in magnitude than the chiral
signal. This bares consequences for our estimates of the qualitative
importance of the linear versus higher-order in the magnetization gradients
contributions to the transport properties of the large-scale textures, as
discussed below.

##  IV Impact on transport properties of skyrmions

While super cell calculations for one-dimensional textures are already
computationally demanding, treating two-dimensional textures such as a lattice
of skyrmions requires even more resources. One can however deduce certain
transport properties in general slowly varying noncollinear textures from the
properties of the spiral phase. For example, we consider a skyrmion texture
defined by $\theta(r)=\pi(1-r/w)$, and
$\hat{\mathbf{n}}_{\mathrm{sk}}=(\sin(\theta)\cos(\varphi+\eta),\sin\theta\sin(\varphi+\eta),\cos(\theta))^{T}$
where $(r,\varphi)$ are polar coordinates in the plane. We imagine that the
skyrmion $\hat{\mathbf{n}}_{\mathrm{sk}}$ is embedded in a ferromagnetic host
$\hat{\mathbf{n}}_{\mathrm{fm}}=\mathbf{e}_{z}$ and integrate the real-space
averages in Eq’s. (28)-(30) within the volume of the skyrmion $\mathcal{V}=\pi
w^{2}$. The helicity $\eta$ interpolates between the Néel skyrmion for
$\eta=0$ and the Bloch skyrmion with $\eta=\pi/2$. We can then estimate the
change in conductivity which is induced by the skyrmion:

$\displaystyle\sigma_{A_{1}}^{\mathrm{sk}}-\sigma_{A_{1}}^{\mathrm{fm}}$
$\displaystyle=-\frac{\pi}{w}\gamma_{\text{\tiny CMC}}\cos\eta$ (47)
$\displaystyle\sigma_{A_{2}}^{\mathrm{sk}}-\sigma_{A_{2}}^{\mathrm{fm}}$
$\displaystyle=\left(\frac{4}{\pi^{2}}-1\right)\gamma_{\text{\tiny
AHE}}-\frac{32}{9\pi w}\gamma_{\text{\tiny CHE}}\cos\eta$ (48)
$\displaystyle\sigma_{E_{2}}^{\mathrm{sk}}-\sigma_{E_{2}}^{\mathrm{fm}}$
$\displaystyle=0.$ (49)

The explicit helicity dependence is one of the hallmarks of “chiral” transport
effects [19]. In contrast to the topological Hall effect, the CHE is further
characterized by its $1/w$ scaling dependence in the long-wavelength limit. In
the case of the THE, a $1/w^{2}$-scaling would be expected, originating from
the emergent magnetic field
$B_{\mathrm{eff}}\sim\hat{\mathbf{n}}\cdot(\partial_{x}\hat{\mathbf{n}}\times\partial_{y}\hat{\mathbf{n}})$
and which, for large $w$, will eventually be the subdominant contribution of
the two different effects. Concerning practical computations, the coefficient
$\gamma_{\text{\tiny AHE}}$ can be simply extracted from a calculation of a
collinear state, while the material parameters $\gamma_{\text{\tiny CMC}}$ and
$\gamma_{\text{\tiny CHE}}$ could be estimated from the spiral calculations as
presented before via

$\displaystyle\gamma_{\text{\tiny CMC}}$
$\displaystyle\approx\lim\limits_{q\to
0}\partial_{q}\frac{\sigma_{A_{1}}[\hat{\mathbf{n}}]}{\cos\beta\sin^{3}\alpha},$
(50) $\displaystyle\gamma_{\text{\tiny CHE}}$
$\displaystyle\approx\lim\limits_{q\to
0}\partial_{q}\frac{\sigma_{A_{2}}[\hat{\mathbf{n}}]}{\cos\beta\sin\alpha\sin^{2}(2\alpha)/2}.$
(51)

The relation is approximate since strong spin-orbit coupling would lead to
higher-order corrections in the $\alpha$ and $\beta$ dependence which could
not be factored out. Irrespective of this, the large effective values of
$\gamma$ correspond to the observation that CHE reaches values comparable in
magnitude to the collinear AHE already for small values of pitch $q$. This
speaks to the fact that the chiral transport signal should be the dominant
source of texture-driven conductivity for wide regimes of texture profiles.

##  V Conclusions

In this work, we have pursued the idea that homogeneous spin-spiral states can
exhibit macroscopic transport properties that are sensitive to their sense of
winding, or, chirality. Inspired by previous effective analysis, we referred
to an explicit two-dimensional electronic model and rigorously demonstrated
the emergence of chiral magnetoconductivity, the chiral planar Hall effect and
the chiral Hall effect. Based on Kubo and gradient expansion techniques we
showed that the corresponding chiral signal can be very prominent, depending
on the type of the spiral state. It also follows from our analysis that in
addition to the high sensitivity of the effects to the electronic structure,
the length-scale of the one-dimensional spin texture has a profound effect on
the magnitude and sign of the chiral signal. This can prove extremely useful
in dealing with aspects such as transport signatures of texture dynamics or
magnetic phase transitions. Besides bringing fundamental novel insights into
the interplay between spin chirality and magnetotransport, we show how an
ability to predict the underlying properties of simpler spiral states paves
the way to understanding and educated engineering of transport fingerprints
for more complex textures such as magnetic skyrmions.

## Acknowledgements

We acknowledge funding under SPP 2137-”Skyrmionics” of the DFG (German
Research Foundation). We also gratefully acknowledge the Jülich Supercomputing
Centre and RWTH Aachen University for providing computational resources under
project Nos. jiff40 and jpgi11. The work was also supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) $-$ TRR 173 $-$
268565370 (project A11), TRR 288 – 422213477 (project B06), and project MO
1731/10-1 of the DFG. J.K. acknowledges funding under HGF-RSF Joint Research
Group “TOPOMANN”, Grant No. DB001827. The authors express their gratitude to
Prof. Stefan Blügel and Dr. Juba Bouaziz for fruitful discussion.

## References

  * Bruno  _et al._ [2004] P. Bruno, V. K. Dugaev, and M. Taillefumier, Topological hall effect and berry phase in magnetic nanostructures, Phys. Rev. Lett. 93, 096806 (2004).
  * Tatara and Kawamura [2002] G. Tatara and H. Kawamura, Chirality-driven anomalous hall effect in weak coupling regime, Journal of the Physical Society of Japan 71, 2613 (2002).
  * Onoda  _et al._ [2004] M. Onoda, G. Tatara, and N. Nagaosa, Anomalous hall effect and skyrmion number in real and momentum spaces, Journal of the Physical Society of Japan 73, 2624–2627 (2004).
  * Nakazawa  _et al._ [2018] K. Nakazawa, M. Bibes, and H. Kohno, Topological hall effect from strong to weak coupling, J. Phys. Soc. Jpn. 87, 033705 (2018).
  * Franz  _et al._ [2014] C. Franz, F. Freimuth, A. Bauer, R. Ritz, C. Schnarr, C. Duvinage, T. Adams, S. Blügel, A. Rosch, Y. Mokrousov, and C. Pfleiderer, Real-space and reciprocal-space berry phases in the hall effect of ${\mathrm{mn}}_{1-x}{\mathrm{fe}}_{x}\mathrm{Si}$, Phys. Rev. Lett. 112, 186601 (2014).
  * Spencer  _et al._ [2018] C. S. Spencer, J. Gayles, N. A. Porter, S. Sugimoto, Z. Aslam, C. J. Kinane, T. R. Charlton, F. Freimuth, S. Chadov, S. Langridge, J. Sinova, C. Felser, S. Blügel, Y. Mokrousov, and C. H. Marrows, Helical magnetic structure and the anomalous and topological hall effects in epitaxial b20 ${\mathrm{fe}}_{1-y}{\mathrm{co}}_{y}\mathrm{Ge}$ films, Phys. Rev. B 97, 214406 (2018).
  * Neubauer  _et al._ [2009] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Topological hall effect in the $a$ phase of mnsi, Phys. Rev. Lett. 102, 186602 (2009).
  * Ritz  _et al._ [2013] R. Ritz, M. Halder, C. Franz, A. Bauer, M. Wagner, R. Bamler, A. Rosch, and C. Pfleiderer, Giant generic topological hall resistivity of mnsi under pressure, Phys. Rev. B 87, 134424 (2013).
  * Hirschberger  _et al._ [2020] M. Hirschberger, T. Nakajima, M. Kriener, T. Kurumaji, L. Spitz, S. Gao, A. Kikkawa, Y. Yamasaki, H. Sagayama, H. Nakao, S. Ohira-Kawamura, Y. Taguchi, T.-h. Arima, and Y. Tokura, High-field depinned phase and planar hall effect in the skyrmion host ${\mathrm{gd}}_{2}{\mathrm{pdsi}}_{3}$, Phys. Rev. B 101, 220401 (2020).
  * Vir  _et al._ [2019] P. Vir, J. Gayles, A. S. Sukhanov, N. Kumar, F. m. c. Damay, Y. Sun, J. Kübler, C. Shekhar, and C. Felser, Anisotropic topological hall effect with real and momentum space berry curvature in the antiskrymion-hosting heusler compound ${\mathrm{mn}}_{1.4}\mathrm{PtSn}$, Phys. Rev. B 99, 140406 (2019).
  * Kumar  _et al._ [2020] V. Kumar, N. Kumar, M. Reehuis, J. Gayles, A. S. Sukhanov, A. Hoser, F. m. c. Damay, C. Shekhar, P. Adler, and C. Felser, Detection of antiskyrmions by topological hall effect in heusler compounds, Phys. Rev. B 101, 014424 (2020).
  * Swekis  _et al._ [2021] P. Swekis, J. Gayles, D. Kriegner, G. H. Fecher, Y. Sun, S. T. B. Goennenwein, C. Felser, and A. Markou, Role of magnetic exchange interactions in chiral-type hall effects of epitaxial mnxptsn films, ACS Applied Electronic Materials 3, 1323 (2021).
  * Göbel  _et al._ [2021] B. Göbel, I. Mertig, and O. A. Tretiakov, Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles, Physics Reports 895, 1 (2021), beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles.
  * Yamaguchi and Yamakage [2021] T. Yamaguchi and A. Yamakage, Theory of magnetic-texture-induced anomalous hall effect on the surface of topological insulators, Journal of the Physical Society of Japan 90, 063703 (2021).
  * Hirschberger  _et al._ [2021] M. Hirschberger, Y. Nomura, H. Mitamura, A. Miyake, T. Koretsune, Y. Kaneko, L. Spitz, Y. Taguchi, A. Matsuo, K. Kindo, R. Arita, M. Tokunaga, and Y. Tokura, Geometrical hall effect and momentum-space berry curvature from spin-reversed band pairs, Phys. Rev. B 103, L041111 (2021).
  * Bouaziz  _et al._ [2021] J. Bouaziz, H. Ishida, S. Lounis, and S. Blügel, Transverse transport in two-dimensional relativistic systems with nontrivial spin textures, Phys. Rev. Lett. 126, 147203 (2021).
  * Lux  _et al._ [2018] F. R. Lux, F. Freimuth, S. Blügel, and Y. Mokrousov, Engineering chiral and topological orbital magnetism of domain walls and skyrmions, Commun. Phys. 1, 60 (2018).
  * Lux  _et al._ [2020a] F. R. Lux, F. Freimuth, S. Blügel, and Y. Mokrousov, Chiral hall effect in noncollinear magnets from a cyclic cohomology approach, Phys. Rev. Lett. 124, 096602 (2020a).
  * Redies  _et al._ [2020] M. Redies, F. R. Lux, J.-P. Hanke, P. M. Buhl, S. Blügel, and Y. Mokrousov, Mixed topology ring states for hall effect and orbital magnetism in skyrmions of weyl semimetals, Phys. Rev. B 102, 184407 (2020).
  * Kipp  _et al._ [2021] J. Kipp, K. Samanta, F. R. Lux, M. Merte, J. P. Hanke, M. Redies, F. Freimuth, S. Blügel, M. Ležaić, and Y. Mokrousov, The chiral hall effect in canted ferromagnets and antiferromagnets, Communications Physics 4, 99 (2021).
  * Meynell  _et al._ [2014] S. A. Meynell, M. N. Wilson, J. C. Loudon, A. Spitzig, F. N. Rybakov, M. B. Johnson, and T. L. Monchesky, Hall effect and transmission electron microscopy of epitaxial mnsi thin films, Phys. Rev. B 90, 224419 (2014).
  * Bode  _et al._ [2007] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, and R. Wiesendanger, Chiral magnetic order at surfaces driven by inversion asymmetry, Nature 447, 190 (2007).
  * Menzel  _et al._ [2012] M. Menzel, Y. Mokrousov, R. Wieser, J. E. Bickel, E. Vedmedenko, S. Blügel, S. Heinze, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Information transfer by vector spin chirality in finite magnetic chains, Phys. Rev. Lett. 108, 197204 (2012).
  * Spethmann  _et al._ [2020] J. Spethmann, S. Meyer, K. von Bergmann, R. Wiesendanger, S. Heinze, and A. Kubetzka, Discovery of magnetic single- and triple-$\mathbf{q}$ states in $\mathrm{Mn}/\mathrm{Re}(0001)$, Phys. Rev. Lett. 124, 227203 (2020).
  * Seemann  _et al._ [2012] K. M. Seemann, F. Garcia-Sanchez, F. Kronast, J. Miguel, A. Kákay, C. M. Schneider, R. Hertel, F. Freimuth, Y. Mokrousov, and S. Blügel, Disentangling the physical contributions to the electrical resistance in magnetic domain walls: A multiscale study, Phys. Rev. Lett. 108, 077201 (2012).
  * Heinze  _et al._ [2011] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nature Physics 7, 713 (2011).
  * Freimuth  _et al._ [2020] F. Freimuth, S. Blügel, and Y. Mokrousov, Anisotropy and temperature-dependence of spin-orbit torques: Role of magnons (2020), arXiv:2003.05028 [cond-mat.mes-hall] .
  * Zhang  _et al._ [2020] L.-c. Zhang, D. Go, J.-P. Hanke, P. M. Buhl, S. Grytsiuk, F. R. Lux, S. Blügel, and Y. Mokrousov, Imprinting and driving electronic orbital magnetism using magnons, Communications Physics 3, 227 (2020).
  * Lux  _et al._ [2020b] F. R. Lux, P. Prass, S. Blügel, and Y. Mokrousov, Effective seiberg-witten gauge theory of noncollinear magnetism (2020b), arXiv:2005.12629 [cond-mat.mes-hall] .
  * Sandratskii [1986] L. M. Sandratskii, Energy band structure calculations for crystals with spiral magnetic structure, physica status solidi (b) 136, 167 (1986).
  * Sandratskii [1991] L. M. Sandratskii, Symmetry analysis of electronic states for crystals with spiral magnetic order. i. general properties, Journal of Physics: Condensed Matter 3, 8565 (1991).
  * Yuan  _et al._ [2012] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and A. Brataas, Spin-orbit-coupling-induced domain-wall resistance in diffusive ferromagnets, Phys. Rev. Lett. 109, 267201 (2012).
  * Birss [1964] R. R. Birss,  _Symmetry and magnetism_ (North-Holland Publishing Company, 1964).
  * Aqeel  _et al._ [2021] A. Aqeel, M. Azhar, N. Vlietstra, A. Pozzi, J. Sahliger, H. Huebl, T. T. M. Palstra, C. H. Back, and M. Mostovoy, All-electrical detection of skyrmion lattice state and chiral surface twists, Phys. Rev. B 103, L100410 (2021).
  * Dresselhaus  _et al._ [2007] M. Dresselhaus, G. Dresselhaus, and A. Jorio,  _Group Theory: Application to the Physics of Condensed Matter_ (Springer, 2007).
  * Hamermesh [2012] M. Hamermesh,  _Group Theory and Its Application to Physical Problems_ , Dover Books on Physics (Dover Publications, 2012).
  * Neumann [1885] F. E. Neumann,  _Vorlesungen über die Theorie der Elasticität der festen Körper und des Lichtäthers: gehalten an der Universität Königsberg_ , Vol. 4 (BG Teubner, 1885).
  * Nowick [2005] A. S. Nowick,  _Crystal properties via group theory_ (Cambridge university press, 2005).
  * Dzyaloshinskii [1964] I. Dzyaloshinskii, Theory of helicoidal structures in antiferromagnets. i. nonmetals, Sov. Phys. JETP 19, 960 (1964).
  * Bogdanov and Yablonskii [1989] A. N. Bogdanov and D. Yablonskii, Thermodynamically stable “vortices” in magnetically ordered crystals. the mixed state of magnets, Zh. Eksp. Teor. Fiz 95, 178 (1989).
  * Czaja  _et al._ [2014] P. Czaja, F. Freimuth, J. Weischenberg, S. Blügel, and Y. Mokrousov, Anomalous hall effect in ferromagnets with gaussian disorder, Phys. Rev. B 89, 014411 (2014).
  * Czaja [2013] P. Czaja,  _Anomalous Hall Effect and Orbital Magnetism in disordered Ferromagnets from Ab Initio_ , Master’s thesis, RWTH Aachen, Department of Physics, RWTH Aachen University, 52056 Aachen, Germany (2013).
  * Yavorsky  _et al._ [2002] B. Y. Yavorsky, I. Mertig, A. Y. Perlov, A. N. Yaresko, and V. N. Antonov, Giant magnetoresistance due to a domain wall in fe: Ab initio study, Phys. Rev. B 66, 174422 (2002).
  * van Hoof  _et al._ [1999] J. B. A. N. van Hoof, K. M. Schep, A. Brataas, G. E. W. Bauer, and P. J. Kelly, Ballistic electron transport through magnetic domain walls, Phys. Rev. B 59, 138 (1999).