††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Unified Theory of the Anomalous and Topological Hall Effects with Phase
Space Berry Curvatures

Nishchhal Verma  Department of Physics, The Ohio State University, Columbus,
OH 43210, USA  Zachariah Addison  Department of Physics, The Ohio State
University, Columbus, OH 43210, USA  Mohit Randeria  Department of Physics,
The Ohio State University, Columbus, OH 43210, USA

###### Abstract

Hall experiments in chiral magnets are often analyzed as the sum of an
anomalous Hall effect, dominated by momentum-space Berry curvature, and a
topological Hall effect, arising from the real-space Berry curvature in the
presence of skyrmions, in addition to the ordinary Hall resistivity. This
raises the questions of how one can incorporate, on an equal footing, the
effects of the anomalous velocity and the real space winding of the
magnetization, and when such a decomposition of the resistivity is justified.
We provide definitive answers to these questions by including the effects of
all phase-space Berry curvatures in a semi-classical approach and by solving
the Boltzmann equation in a weak spin-orbit coupling regime when the
magnetization texture varies slowly on the scale of the mean free path. We
show that the Hall resistivity is then just the sum of the anomalous and
topological contributions, with negligible corrections from Berry curvature-
independent and mixed curvature terms. We also use an exact Kubo formalism to
numerically investigate the opposite limit of infinite mean path, and show
that the results are similar to the semi-classical results.

The Hall effect in magnetic materials has a long history Hall (1879). One
might be tempted to think the primary explanation for the effect is that the
magnetization ${\bf M}$ can exert a Lorentz force on the materials electrons,
however, this effect is negligible for the non-relativistic charge carriers in
metals Wannier (1947). Karplus and Luttinger Karplus and Luttinger (1954)
correctly identified the importance of spin-orbit coupling (SOC) in the
anomalous Hall effect (AHE); their analysis is now best understood in terms of
the momentum-space Berry curvature of the electron Nagaosa (2006); Nagaosa
_et al._ (2010); Xiao  _et al._ (2010). Scattering in the presence of SOC also
makes extrinsic contributions Smit (1955); Berger (1964) to the AHE, but the
anomalous velocity that arises from Berry curvature effects is an intrinsic
effect that dominates in many experiments Nagaosa  _et al._ (2010); Yao  _et
al._ (2004), and will be our focus here.

Understanding the Hall effect with spatially varying magnetic textures poses
further challenges. In addition to the AHE, experiments see a topological Hall
effect (THE) in a variety of chiral magnetic materials that harbor skyrmions,
including B20 crystals Lee  _et al._ (2009); Neubauer  _et al._ (2009);
Kanazawa  _et al._ (2011) and thin films Li  _et al._ (2013); Gallagher  _et
al._ (2017); Ahmed  _et al._ (2018), and heavy metal/magnetic insulator
bilayers Ahmed  _et al._ (2019); Shao  _et al._ (2019). Skyrmions give rise to
an emergent magnetic field that derives from the real-space Berry curvature,
resulting in a THE proportional to their topological charge density
$n_{\text{sk}}$ Ye  _et al._ (1999); Tatara and Kawamura (2002); Bruno  _et
al._ (2004); Onoda  _et al._ (2004); Nagaosa  _et al._ (2012); Nagaosa and
Tokura (2013); Hamamoto  _et al._ (2015); Nakazawa  _et al._ (2018); Ishizuka
and Nagaosa (2018).

Theories of the anomalous and topological Hall effect have for the most part
been distinct and, despite important recent progress Kim  _et al._ (2013);
Akosa  _et al._ (2018); Lux  _et al._ (2018); Akosa  _et al._ (2019); Zhang
_et al._ (2020); Lux  _et al._ (2020); Bouaziz  _et al._ (2021) on electrons
with SOC interacting with skyrmions, a single theory that incorporates both
real and momentum space Berry curvature effects on an equal footing to
calculate electronic transport has remained elusive. The experiments Lee  _et
al._ (2009); Neubauer  _et al._ (2009); Kanazawa  _et al._ (2011); Li  _et
al._ (2013); Gallagher  _et al._ (2017); Ahmed  _et al._ (2018, 2019); Shao
_et al._ (2019), on the other hand, are routinely interpreted as a sum of an
anomalous and topological Hall resistivity, in addition to the ordinary Hall
effect proportional to the magnetic field.

Figure 1:  The semi-classical wave-packet follows the texture and is
influenced by real-space Berry curvature arising from the presence of
skyrmions, in addition to the anomalous velocity that it acquires from an
external electric field and momentum space Berry curvature. Our results are
obtained in the regime where spin texture length scale $L_{s}\gg$ mean free
path $\ell\gg a$, the lattice spacing, and weak spin-orbit coupling
$\lambda\ll E_{F}$, the Fermi energy. The table summarizes the three
contributions to $\rho_{xy}$, their scaling with these parameters, their
dependence on the magnetic texture $\hat{\bf m}({\bf r})$, and their relation
to Berry curvatures. Mixed momentum- and real-space curvatures contribute to
the Hall resistivity at higher order in $(\lambda/E_{F})$ and $(a/L_{s})$.

In this paper, we demonstrate within a semi-classical theory that the Hall
response (excluding the ordinary Hall effect) is just the sum of two terms,
the AHE and the THE. The semi-classical approach Xiao  _et al._ (2010);
Freimuth  _et al._ (2013) is a natural avenue to study the effects of all
phase space Berry curvatures, ${\bf r}$-space, ${\bf k}$-space, and mixed, on
an equal footing in the regime where the length scale $L_{s}$ on which the
spin texture varies and the mean free path $\ell$ from impurity scattering are
both much larger than the microscopic scales of the average inter-particle
spacing $k_{F}^{-1}$ or the lattice spacing $a$. We are also interested in the
regime of weak SOC $\lambda\ll E_{F}$, the Fermi energy.

To determine the Hall resistivity we solve the Boltzmann equation to linear
order in the electric field in the presence of all phase-space curvatures and
real and momentum space derivatives of the semi-classical energy eigenvalues.
Systematically classifying the resulting array of terms in powers of the small
parameters $\lambda/E_{F}$ and $\ell/L_{s}$, and extracting the leading
contributions in the regime $L_{s}\gg\ell\gg k_{F}^{-1}\simeq a$ and
$\lambda\ll E_{F}$, we find that

$\rho_{xy}=\rho_{xy}^{\text{AHE}}+\rho_{xy}^{\text{THE}}+\delta\rho_{xy}.$ (1)

Our results are summarized in the table in Fig. 1, where we show how each term
depends (i) on the small parameters that control our calculation, (ii) on the
spatially varying magnetization ${\bf M}=M_{s}\,\hat{\bf m}({\bf r})$, and
(iii) on the Berry curvatures. While the first two terms represent the AHE and
the THE respectively, the correction term $\delta\rho_{xy}$ is a curvature-
independent boundary contribution proportional to the vorticity of the local
electronic velocity field. It vanishes when the spin texture is periodic,
e.g., a skyrmion crystal, and is negligible for a disordered skyrmion array in
the thermodynamic limit. We show that the mixed curvatures contribute to the
Hall resistivity at higher order in the small parameters $(\lambda/E_{F})$ and
$(a/L_{s})$ than the terms shown in Fig. 1.

Finally, we also present results using the Kubo formula in the opposite regime
where $\ell\gg L_{s}\gtrsim k_{F}^{-1}\simeq a$. We focus on a disorder free
system with $\ell=\infty$, use exact diagonalization in the magnetic unit cell
of a skyrmion crystal, and compute the total Hall conductivity using the TKNN
formula Thouless  _et al._ (1982) in the magnetic Brillouin zone, which
includes the effects of both the anomalous velocity and of the skyrmion
topological charge density. We show how the semi-classical results allow us to
qualitatively understand all of the non-trivial parameter dependencies of the
Hall response including the dependence on the density, the SOC, and the
exchange coupling between the charge carriers and the spin.

Model: We analyze a minimal Hamiltonian for studying the confluence of
anomalous and topological Hall effects. It can arise either from a “s-d model”
of itinerant electrons interacting with local moments in a metallic magnet
with Rashba SOC, or alternatively, it can be used to model the conduction
electrons in a metal proximate to a magnetic insulator where broken inversion
symmetry at the interface induces a Rashba SOC.

We consider a 2D Hamiltonian

$\widehat{\mathcal{H}}=\dfrac{\widehat{{\bf
p}}^{2}}{2m}+\dfrac{a\lambda}{\hbar}\left(\widehat{{\bf p}}\times\hat{{\bf
z}}\right)\cdot\boldsymbol{\sigma}-J\;\hat{{\bf m}}(\widehat{{\bf
r}})\cdot\boldsymbol{\sigma}+\widehat{\mathcal{H}}_{\text{imp}}$ (2)

which describes itinerant electrons of mass $m$ and Rashba SOC $\lambda$ whose
spin $\boldsymbol{\sigma}$ is coupled to a magnetic texture ${\bf
M}=M_{s}\,\hat{{\bf m}}({\bf r})$ via an exchange interaction $J$. Elastic
scattering of electrons off a disorder potential is described by
$\widehat{\mathcal{H}}_{\text{imp}}$ and leads to a mean free path $\ell\gg
k_{F}^{-1}$. The small hats denote unit vectors and the wide hats denote
quantum mechanical operators. Based on the separation of time-scales
associated with the itinerant electrons and the dynamics of spins in the
texture, we assume that the texture is static. The model has three energy
scales: the Fermi energy $E_{F}$, SOC $\lambda$, and exchange coupling $J$,
and three length scales: the inter-particle spacing $k_{F}^{-1}$ ($\approx a$,
the lattice spacing), the mean-free path $\ell$, and the length scale $L_{s}$
associated to the spatial variations of the magnetic texture. We will focus on
the weak SOC regime $\lambda\ll J,E_{F}$, relevant for experiments.

Semi-classical Equations of Motion: Let us focus on the semi-classical regime
$L_{s}\gg k_{F}^{-1}$. To analyze the dynamics of electron wave packets in
phase space $\boldsymbol{\xi}=(x,y,k_{x},k_{y})$, we follow the standard
prescription Xiao  _et al._ (2010) to construct the semi-classical Hamiltonian

$\mathcal{H}(\boldsymbol{\xi})=\dfrac{\hbar^{2}{\bf k}^{2}}{2m}+{\bf
d}(\boldsymbol{\xi})\cdot{\boldsymbol{\sigma}}$ (3)

where ${\bf d}(\boldsymbol{\xi})=a\lambda({\bf k}\times\hat{{\bf
z}})-J\hat{{\bf m}}({\bf r})$ captures the quantum mechanical nature of the
spin. The semi-classical eigenenergies are
$\mathcal{E}_{\pm}(\boldsymbol{\xi})=\hbar^{2}{\bf k}^{2}/2m\pm|{\bf
d}(\boldsymbol{\xi})|$. The corresponding wavefunctions posses non-trivial
phase space geometry encoded in the Berry curvatures

$\displaystyle\Omega^{\pm}_{\alpha,\beta}(\boldsymbol{\xi})$ $\displaystyle=$
$\displaystyle\pm\dfrac{1}{2}\hat{{\bf
d}}(\boldsymbol{\xi})\cdot\left(\partial_{\alpha}\hat{\bf
d}(\boldsymbol{\xi})\times\partial_{\beta}\hat{{\bf
d}}(\boldsymbol{\xi})\right)$ (4)

each corresponding to one of the six orthogonal planes in the 4D phase space
spanned by $\boldsymbol{\xi}$. The dynamics of the semi-classical theory
describe intra-band processes such that each electronic band may be treated
independently, and we will suppress the band index unless necessary.

The curvatures modify the equations of motion as well as the invariant measure
in phase space. To simplify notation, we introduce a $4\times 4$ matrix,

$[\Gamma(\boldsymbol{\xi})]_{\alpha,\beta}=\Omega_{\alpha,\beta}(\boldsymbol{\xi})-[i\sigma_{y}\otimes\mathds{1}]_{\alpha,\beta}$
(5)

to write the equations of motion

$\dot{\xi}_{\alpha}(\boldsymbol{\xi})=[\Gamma^{-1}(\boldsymbol{\xi})]_{\alpha\beta}\;\big{(}\partial_{\beta}\widetilde{\mathcal{E}}(\boldsymbol{\xi})+eE\;\delta_{\beta,y}\big{)}/\hbar$
(6)

where $E$ is the external electric field along the $\hat{{\bf y}}$ direction
and the electron charge is $(-e)$. Here
$\widetilde{\mathcal{E}}(\boldsymbol{\xi})\simeq\mathcal{E}(\boldsymbol{\xi})$
up to corrections of order $(\lambda/E_{F})(a/L_{s})$ that can be ignored in
the regime of interest. Our compact notation hides all the familiar terms,
including the anomalous velocity, inside $\Gamma^{-1}$; see appendix A for
more details

The combination of a spatially varying magnetic texture and SOC leads to
finite real-space, momentum-space and mixed real-momentum space curvatures.
The electrons acquire an anomalous velocity proportional to the momentum-space
Berry curvature $\Omega_{k_{x},k_{y}}$, an “anomalous force” proportional to
the real-space Berry curvature $\Omega_{x,y}$ and corrections to the group
velocity and generalized force proportional to the mixed real-momentum-space
Berry curvatures.

Crucially, in addition to the equations of motion, the curvatures also modify
the volume element that remains invariant under phase-space flows. Thus to
satisfy Liouville’s theorem, one must use the integration measure Xiao  _et
al._ (2010); Addison  _et al._
$dV_{\boldsymbol{\xi}}=\sqrt{\det[\Gamma(\boldsymbol{\xi})]}\,d^{4}\boldsymbol{\xi}/(2\pi)^{2}V$,
where $V$ is the volume of the system. We note that in the presence of an
external magnetic field $B_{z}\hat{{\bf z}}$,
$\sqrt{\det[\Gamma(\boldsymbol{\xi})]}$ reduces to the well-known factor of
$(1+e\Omega_{k_{x},k_{y}}B_{z}/\hbar)$ when only the momentum-space curvature
is present, however, we will need the more general result here.

Hall Conductivity: With electric field applied along $\hat{y}$, we must
calculate the transverse current along $\hat{x}$:

$j_{x}=-e\int
dV_{\boldsymbol{\xi}}\;\dot{x}(\boldsymbol{\xi})\;f(\boldsymbol{\xi})$ (7)

where $f(\boldsymbol{\xi})$ is the electronic distribution function. The
distribution function reduces to the equilibrium Fermi-Dirac function
$f^{0}[\mathcal{E}(\boldsymbol{\xi})]$ in the absence of the external electric
field. The goal is to find contributions that are linear order in $E$ to
calculate the electric conductivity.

The anomalous Hall contribution to the current derives from the intrinsic
anomalous velocity and couples to the equilibrium distribution function
$f^{0}[\mathcal{E}({\bf\xi})]$. We isolate the terms in $\dot{x}$ linear in
$E$ to find

$\sigma^{\text{AHE}}_{xy}=-\dfrac{e^{2}}{\hbar}\sum_{l=\pm}\int\dfrac{d^{2}{\bf
r}\;d^{2}{\bf
k}}{(2\pi)^{2}V}\;\Omega^{l}_{k_{x},k_{y}}(\boldsymbol{\xi})\;f^{0}_{l}[\mathcal{E}_{l}(\boldsymbol{\xi})]$
(8)

where $l=\pm$ indexes the two bands. We emphasize that
$\sqrt{\det[\Gamma(\boldsymbol{\xi})]}$ in the measure exactly cancels the
determinant factor in $\Gamma^{-1}(\boldsymbol{\xi})$ so that the final answer
depends only on the momentum-space Berry curvature. We further expand
$\Omega_{k_{x},k_{y}}(\boldsymbol{\xi})$ to lowest order in $\lambda/J$ to
find

$\displaystyle\sigma_{xy}^{\text{AHE}}$
$\displaystyle\approx-\dfrac{e^{2}a^{2}}{2\hbar}\,\overline{m}_{z}\left({\lambda}/{J}\right)^{2}\sum_{l=\pm}\,l\,n_{l}$
(9)

where $\overline{m}_{z}=\int d^{2}{\bf r}\;\hat{{\bf m}}_{z}({\bf r})/V$ is
the average out-of-plane magnetization and the band-resolved density
$n_{l}=\int d^{2}{\bf k}\;f^{0}[\mathcal{E}_{l}({\bf k})]/(2\pi)^{2}$ with
$\mathcal{E}_{l}({\bf k})=\mathcal{E}_{l}({\boldsymbol{\xi}};\lambda=0)$.

The corresponding resistivity is found from the conductivity via
$\rho_{xy}=-\sigma_{xy}/(\sigma^{2}_{xx}+\sigma^{2}_{xy})$ where
$\sigma_{xy}\ll\sigma_{xx}=(e^{2}/h)k_{F}\ell$. This relationship will be used
to convert conductivities to resistivities for each contribution to the Hall
effect. For the AHE this leads to the scaling relation
$\rho_{xy}^{\text{AHE}}\sim(\lambda/E_{F})^{2}(a/\ell)^{2}$.

Figure 2: The Hall conductivity calculated from the Kubo formula. (a) The blue
curve is the THE calculated at $\lambda=0$ with $J/t=10$. The red curve shows
$\mathcal{K}(n)=\mathcal{K}_{+}+\mathcal{K}_{-}$ [eq. (14)] which describes
how the band strucure controls the semi-classical THE result (13). Despite
their different regimes of validity, both the Kubo $\sigma_{xy}$ and
$\mathcal{K}(\mu)$ have the same sign for all densities and vanish at the Van-
Hove filling where the Fermi surface undergoes a Lifshitz transition. (b) The
THE conductivity ($\lambda=0$) shows a crossover from a linear regime at small
$J$ to saturation at large $J$. The slope is independent of density $n$, while
the saturating value increases with $n$. Both these behaviors can be
qualitatively explained by analyzing $\mathcal{K}(n)$ as a function of $J$
(see appendix B). (c) Variation of $\sigma_{xy}$ with spin-orbit coupling
$\lambda$ at fixed $J/t=10$ and $n=0.2$. With increasing $L_{s}/a$, the
results rapidly converge to a finite value that is very weakly
$\lambda$-dependent. Thus there is no linear in $\lambda$ contribution in the
Kubo result in agreement with the semi-classical analysis.

All other contributions to the Hall response involve the electric field
induced perturbations to the distribution function determined by solving the
Boltzmann equation. We expand the distribution function to linear order in the
electric field, $f=f^{0}+g+\mathcal{O}(E^{2})$ and substitute it into the
Boltzmann equation with a relaxation time $\tau=\ell/v_{F}$ to find the
equation for $g$:

$\displaystyle\left(1+\tau\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\right)g(\boldsymbol{\xi})=-\tau\;\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]$
(10)

where $\dot{\boldsymbol{\xi}}^{(I)}$ and $\dot{\boldsymbol{\xi}}^{(D)}$ are
the electric field independent and dependent parts of $\dot{\boldsymbol{\xi}}$
in eq. (6). We now take advantage of the fact that
$\tau\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\sim(\ell/L_{s})(a/L_{s})\ll
1$ when $\ell/L_{s}\ll 1$ to invert the operator on the left hand side and
solve for $g(\boldsymbol{\xi})$. This is analogous to the Zener-Jones
calculation Ziman (2007) of the Hall conductivity in the weak field regime
$\omega_{c}\tau\ll 1$. Solving the Boltzmann equation for $L_{s}\ll\ell$ is
technically much harder. We will investigate aspects of this regime using the
Kubo formalism below.

The term $g^{(1)}(\boldsymbol{\xi})$ linear in $\tau$ does not contribute to
the Hall conductivity and the leading order contribution proportional to
$\tau^{2}$ is

$g^{(2)}(\boldsymbol{\xi})=\tau^{2}\;\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\Big{(}\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]\Big{)}.$
(11)

We emphasize that this equation involves all six curvatures along with mixed
derivatives of the semi-classical eigenenergies. Combining
$g^{(2)}(\boldsymbol{\xi})$ with eq. (7) we calculate the current which is
linear in $E$:

$\displaystyle j_{x}^{(2)}$ $\displaystyle=-e\tau^{2}\int
dV_{\boldsymbol{\xi}}\;\dot{x}^{(I)}(\boldsymbol{\xi})\;\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\Big{(}\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]\Big{)}.$
(12)

We organize the calculation of the conductivity by classifying the various
terms in eq. (12) in powers of the small parameters $\lambda/E_{F}$ and
$a/L_{s}$; see appendix B for details. Here we discuss the leading order
contributions.

We first focus on the zeroth order term in $(\lambda/E_{F})$. Without SOC, all
curvatures vanish except the real-space curvature which leads to the
topological Hall contribution

$\sigma_{xy}^{\text{THE}}=\dfrac{e^{2}\tau^{2}}{\hbar^{3}}\,n_{\text{sk}}\;\sum_{l=\pm}\mathcal{K}_{l}(\mu)\bigg{|}_{\lambda=0}.$
(13)

Here $n_{\text{sk}}=\int d^{2}{\bf r}\,\hat{{\bf m}}\cdot(\partial_{x}\hat{\bf
m}\times\partial_{y}\hat{{\bf m}})/(4\pi V)$ is the skyrmion density and

$\mathcal{K}_{\pm}(\mu)=\mp\hbar^{4}\int\dfrac{d^{2}{\bf
k}}{(4\pi)}\left(\dfrac{\partial f^{0}_{\pm}}{\partial\mathcal{E}}\right){\bf
v}^{T}(\mathbb{M}^{-1}-\text{Tr}\mathbb{M}^{-1}){\bf v}$ (14)

is a Fermi surface integral that depends on the chemical potential $\mu$ (or
filling $n$) and the band index. Here ${\bf v}=\boldsymbol{\nabla}_{{\bf
k}}\mathcal{E}(\boldsymbol{\xi})/\hbar$ is the band velocity vector and
$\mathbb{M}^{-1}_{\mu\nu}=\partial_{k_{\mu},k_{\nu}}\mathcal{E}(\boldsymbol{\xi})/\hbar^{2}$
is the inverse mass tensor. The semi-classical theory illuminates the
relationship between the real-space Berry curvature which is a property of the
spatial evolution of the semi-classical Bloch eigenstates and the skyrmion
density which is a property of the spatial evolution of the magnetization
vector. In the absence of spin-orbit coupling $\Omega_{x,y}^{\pm}=\mp\hat{{\bf
m}}\cdot(\partial_{x}\hat{\bf m}\times\partial_{y}\hat{{\bf m}})/2$. The
result of eq. (13) bears a striking resemblance to the canonical solution
Ziman (2007) for the semi-classical Hall conductivity with the real space
Berry curvature $\Omega_{x,y}$ playing the role of an external magnetic field,
in agreement with the intuitive picture behind the THE. The corresponding
resistivity is independent of $\tau$ and scales as
$\rho_{xy}^{\text{THE}}\sim(a/L_{s})^{2}$.

Next we focus on terms linear in $(\lambda/E_{F})$. Even though there are
several terms, there is only one that is linear in $(a/L_{s})$. It originates
from mixed spatial and momentum space derivatives of the semi-classical
energies $\mathcal{E}(\boldsymbol{\xi})$ and is independent of all Berry
curvatures:

$\delta\sigma_{xy}=-\dfrac{e^{2}\tau^{2}}{2m}\sum_{l=\pm}\,\omega_{l}\,n_{l}$
(15)

where $\omega_{l}=1/V\int d^{2}{\bf r}\,\,\hat{{\bf
z}}\cdot(\boldsymbol{\nabla}_{r}\times{\bf v}_{l}({\bf r}))$ is the average
“vorticity” of electrons in band $l$ with velocity ${\bf v}_{l}({\bf r})$ that
is linear in $\lambda$ (see appendix C for details) and $n_{l}$ is the band-
resolved density defined below eq. (9). The intuition behind this term is that
real-space gradients of the magnetic texture can lead to orbital electronic
motion akin to the dynamics induced by an external magnetic field. For the
Rashba SOC considered here, the vorticity simplifies to $\sim\int d{\bf
r}\,\,\boldsymbol{\nabla}_{r}\cdot\hat{{\bf m}}({\bf r})$. This term has been
discussed in the literature Kim  _et al._ (2013); Akosa  _et al._ (2018,
2019); Zhang  _et al._ (2020) as a ${\cal O}(\lambda)$ correction to the
emergent magnetic field arising from skyrmions. Here this contribution arises
not from SOC corrections to the real-space Berry curvature, but instead from
mixed momentum and real space derivatives of the semi-classical eigenvalues.
Like the THE the corresponding resistivity is independent of $\tau$, but
instead scales as $\delta\rho_{xy}\sim(a/L_{s})(\lambda/E_{F})$. We note,
however, that $\delta\rho_{xy}$ vanishes identically for any periodic spin
texture, like a skyrmion crystal. More generally, for any smooth texture for
which ${\bf v}({\bf r})$ has continuous first order partial derivatives, we
can use Stokes’ theorem and show that the vorticity leads only to a boundary
term that is negligible in the thermodynamic limit.

All other contributions to $\rho_{xy}$, including the mixed curvature terms,
are higher order in either $(\lambda/E_{F})$ (which is not relevant for
experiments) or in $(a/L_{s})$ at which point the semi-classical analysis
presented here is itself not applicable. Thus we have used the semi-classical
approach that treats all curvatures on equal footing to conclude that AHE and
THE resistivities are indeed additive and the largest contribution to the Hall
effect for $L_{s}\gg\ell$.

Kubo formula analysis: We next turn to the opposite limit of small skyrmions
such that $a\approx k_{F}^{-1}\lesssim L_{s}\ll\ell$. We in fact set the mean-
free path to infinity and use an exact Kubo formula to numerically calculate
the Hall conductivity for a lattice model of itenerant electrons in the
presence of a skyrmion crystal; see appendix D for details. The starting
Hamiltonian is a tight-binding generalization of eq.(2) describing electrons
on a lattice with nearest neighbor hopping $t$ and Rashba SOC $\lambda$,
coupled to a background spin texture described by local moments ${\bf m}_{\bf
i}$ at each lattice site ${\bf i}$. The skyrmion crystal defines an enlarged
$N_{s}\times N_{s}$ unit cell, where $N_{s}=L_{s}/a$, and results in a
magnetic Brillouin zone with $N_{\text{b}}=2N_{s}^{2}$ bands. We present here
results for a triangle lattice, but as we show in appendix D our results are
independent of the lattice for low densities.

We use exact diagonalization to compute the energy eigenvalues and
eigenfunctions of our lattice Hamiltonian and then use the TKNN formula
Thouless  _et al._ (1982) to determine the Hall conductivity in terms of the
momentum-space Berry curvature in the magnetic Brillouin zone. Note that this
numerically exact procedure includes all the effects of the anomalous velocity
as well as the real-space Berry curvature arising from the skyrmions, however,
unlike the semi-classical theory it is hard to decompose the final result into
AHE and THE contributions. We thus proceed as follows. We first show that in
various limits one obtains just the AHE (in a ferromagnetic background), or
just the THE (in a skyrmion crystal with $\lambda=0$) . Finally, we consider
the full problem and gain qualitative insights into the numerical results by
comparing them with the semi-classical results described above.

First, consider the simplest ferromagnetic case with uniform magnetization
$\hat{{\bf m}}_{\bf i}=\hat{\bf z}$ (independent of ${\bf i}$). which is just
the lattice version of the continuum model analyzed in ref Xiao  _et al._
(2010) with their $\Delta\sigma_{z}$ corresponding to our $J\sigma_{z}$. An
AHE is seen in this case provided both $\lambda$ and $\Delta$ are non-zero.
The SOC $\lambda$ breaks the two-fold spin degeneracy of the bands everywhere
except at the time-reversal invariant momenta (TRIM) where time reversal (TR)
enforces a Kramers degeneracy. A non-zero $\Delta$ destroys TR symmetry,
causes band inversion, and creates Berry curvature hotspots at TRIMs which
then lead to an enhancement of the AHE conductivity whenever the Fermi level
falls near the TRIM points.

We next look at a skyrmion crystal, but set $\lambda=0$ so that there is no
AHE (even though the net $M_{z}$ is non-zero). The Fourier modes of the
periodic texture cause scattering between momentum eigenstates and lead to
band folding. At strong coupling $J/t\gg 1$, the bands separate into two
sectors with the spins aligned/anti-aligned with the local magnetic texture.
The corresponding Hall conductivity is the THE arising from non-zero skyrmion
number. It shows a non-trivial dependence on the band filling as seen in Fig.
2(a) (blue curve). Comparing this with the semi-classical THE prediction of
eq. (13) (red curve) we see that these results, though obtained in very
different regimes, share some qualitative features. Both have the same sign at
each density and vanish at the van Hove filling where the Fermi surface
undergoes a Lifshitz transition.

Next consider the $J/t$ dependence of the numerical results for the THE shown
in Fig. 2(b). We see a linear regime at small $J$ crossing over to saturation
at large $J$. We can gain insight into these results by analyzing the $J$
dependence of the semi-classical THE of eq. (13), which predicts an initial
slope independent of density $n$ and a saturation value that increases with
$n$ (see appendix B)

Finally, we turn to the Kubo results for a skyrmion crystal with non-zero SOC.
In Fig. 2(c) we plot these results at strong coupling $J/t=10$ and find that
in general the Hall response depends on the SOC $\lambda$. We see that with
increasing $L_{s}/a$, the results converge to a non-zero value which is very
weakly dependent on $\lambda$. The large $L_{s}/a$ limit allows us to make
contact with the semi-classical results, where we showed above that there is
no linear in $\lambda$ contribution to the Hall conductivity. For the
parameters considered here, the AHE contribution that scales like
$(\lambda/J)^{2}\sim 10^{-4}$ is also negligible.

Discussion: We have presented a complete semi-classical analysis in the weak
SOC $\lambda\ll E_{F}$ regime for $a\ll\ell\ll L_{s}$ and demonstrated that
the Hall resistivity the sum of an anomalous Hall contribution, arising from
the momentum space Berry curvature and proportional to the average out-of-
plane magnetization, and a topological Hall contribution, arising from the
real-space Berry curvature and proportional to the skyrmion density. All
corrections were explicitly shown to be higher order in the small parameters.
The semi-classical results are valid for any spin texture without any
assumption about its periodicity. In the opposite limit $L_{s}\ll\ell=\infty$
(zero disorder) we have presented exact Kubo formula results for skyrmion
crystals.

We conclude by noting effects that we have not included and questions for
further study. We focussed on the intrinsic anomalous Hall effect, arising for
momentum space Berry curvature, often the dominant contribution Nagaosa  _et
al._ (2010) to the AHE, but did not consider extrinsic effects such as skew
and side jump scattering. We have also not analyzed non-periodic spin textures
which vary on a length scale $L_{s}\lesssim\ell$. Such a regime has been
analyzed Bouaziz  _et al._ (2021) in the context of electrons scattering off a
single skrymion with the prediction of a novel non-collinear Hall effect
linear in the SOC. It would be interesting to extend our semi-classical
analysis to this regime.

Finally, in the semiclassical regime that we have examined in detail, with
$L_{s}\gg\ell$, there is a novel vorticity term [eq. (15)] that is linear in
$\lambda$, but we were able to use Stokes’ theorem to reduce it to a boundary
term that vanishes for periodic textures. An interesting question Addison  _et
al._ for further study is the fate of this term in the presence of
singularities, such as Bloch points, that may act as obstructions to the use
of Stokes theorem.

Acknowledgements: This work was supported by NSF Materials Research Science
and Engineering Center Grant DMR-2011876. Z.A. was also supported by the Ohio
State University President’s Postdoctoral Scholars Program. We gratefully
acknowledge Roland Kawakami, Siddharth Seetharaman, Po-Kuan Wu, and Fengyuan
Yang for insightful discussions.

## References

  * Hall (1879) E. H. Hall, American Journal of Mathematics 2, 287 (1879).
  * Wannier (1947) G. H. Wannier, Phys. Rev. 72, 304 (1947).
  * Karplus and Luttinger (1954) R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954).
  * Nagaosa (2006) N. Nagaosa, Journal of the Physical Society of Japan 75, 042001 (2006), https://doi.org/10.1143/JPSJ.75.042001 .
  * Nagaosa  _et al._ (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Reviews of Modern Physics 82, 1959 (2010).
  * Smit (1955) J. Smit, Physica 21, 877 (1955).
  * Berger (1964) L. Berger, Physica 30, 1141 (1964).
  * Yao  _et al._ (2004) Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004).
  * Lee  _et al._ (2009) M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, Phys. Rev. Lett. 102, 186601 (2009).
  * Neubauer  _et al._ (2009) A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Phys. Rev. Lett. 102, 186602 (2009).
  * Kanazawa  _et al._ (2011) N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. 106, 156603 (2011).
  * Li  _et al._ (2013) Y. Li, N. Kanazawa, X. Z. Yu, A. Tsukazaki, M. Kawasaki, M. Ichikawa, X. F. Jin, F. Kagawa, and Y. Tokura, Phys. Rev. Lett. 110, 117202 (2013).
  * Gallagher  _et al._ (2017) J. C. Gallagher, K. Y. Meng, J. T. Brangham, H. L. Wang, B. D. Esser, D. W. McComb, and F. Y. Yang, Phys. Rev. Lett. 118, 027201 (2017).
  * Ahmed  _et al._ (2018) A. S. Ahmed, J. Rowland, B. D. Esser, S. R. Dunsiger, D. W. McComb, M. Randeria, and R. K. Kawakami, Phys. Rev. Materials 2, 041401 (2018).
  * Ahmed  _et al._ (2019) A. S. Ahmed, A. J. Lee, N. Bagués, B. A. McCullian, A. M. Thabt, A. Perrine, P. K. Wu, J. R. Rowland, M. Randeria, P. C. Hammel, D. W. McComb, and F. Yang, Nano Letters 19, 5683 (2019), arXiv:1905.03650 .
  * Shao  _et al._ (2019) Q. Shao, Y. Liu, G. Yu, S. K. Kim, X. Che, C. Tang, Q. L. He, Y. Tserkovnyak, J. Shi, and K. L. Wang, Nature Electronics 2, 182 (2019), arXiv:1904.07107 .
  * Ye  _et al._ (1999) J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, and Z. Tešanović, Phys. Rev. Lett. 83, 3737 (1999).
  * Tatara and Kawamura (2002) G. Tatara and H. Kawamura, Journal of the Physical Society of Japan 71, 2613 (2002).
  * Bruno  _et al._ (2004) P. Bruno, V. K. Dugaev, and M. Taillefumier, Physical Review Letters 93, 096806 (2004).
  * Onoda  _et al._ (2004) M. Onoda, G. Tatara, and N. Nagaosa, Journal of the Physical Society of Japan 73, 2624 (2004).
  * Nagaosa  _et al._ (2012) N. Nagaosa, X. Z. Yu, and Y. Tokura, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 5806 (2012).
  * Nagaosa and Tokura (2013) N. Nagaosa and Y. Tokura, Nature Nanotechnology 8, 899 (2013).
  * Hamamoto  _et al._ (2015) K. Hamamoto, M. Ezawa, and N. Nagaosa, Phys. Rev. B 92, 115417 (2015).
  * Nakazawa  _et al._ (2018) K. Nakazawa, M. Bibes, and H. Kohno, Journal of the Physical Society of Japan 87, 033705 (2018).
  * Ishizuka and Nagaosa (2018) H. Ishizuka and N. Nagaosa, Science Advances 4, eaap9962 (2018).
  * Kim  _et al._ (2013) K. W. Kim, H. W. Lee, K. J. Lee, and M. D. Stiles, Physical Review Letters 111 (2013), 10.1103/PhysRevLett.111.216601.
  * Akosa  _et al._ (2018) C. A. Akosa, A. Takeuchi, Z. Yuan, and G. Tatara, Physical Review B 98, 184424 (2018).
  * Lux  _et al._ (2018) F. R. Lux, F. Freimuth, S. Blügel, and Y. Mokrousov, Communications Physics 1, 60 (2018).
  * Akosa  _et al._ (2019) C. A. Akosa, H. Li, G. Tatara, and O. A. Tretiakov, Physical Review Applied 12, 54032 (2019).
  * Zhang  _et al._ (2020) S. S. Zhang, H. Ishizuka, H. Zhang, G. B. Halász, and C. D. Batista, Physical Review B 101, 024420 (2020).
  * Lux  _et al._ (2020) F. R. Lux, F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev. Lett. 124, 096602 (2020).
  * Bouaziz  _et al._ (2021) J. Bouaziz, H. Ishida, S. Lounis, and S. Blügel, Phys. Rev. Lett. 126, 147203 (2021).
  * Freimuth  _et al._ (2013) F. Freimuth, R. Bamler, Y. Mokrousov, and A. Rosch, Phys. Rev. B 88, 214409 (2013).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * (36) Z. Addison, N. Verma, S. Seetharaman, and M. Randeria, In Preparation .
  * Ziman (2007) J. Ziman,  _Electrons and Phonons_ (Oxford University Press, 2007).

APPENDICES

##  Appendix A Semi-classical Equations of Motion with Phase Space Berry
Curvatures

Semi-classical theory describes transport in terms of electron wave-packets
whose width is larger than microscopic lattice scale $a$ but much smaller than
mean-free path $\ell$ so that the average position ${\bf r}$ and average
momentum ${\bf k}$ of the wavepacket are well-defined simultaneously. This is
in addition to their time evolution which is governed by a semi-classical
Hamiltonian. The magnetic texture presents a new length-scale related to its
size $L_{s}$. The construction now requires a gradient expansion which
introduces an additional constraint that the width is smaller than $L_{s}$.
The Hamiltonian thus obtained is a function of phase-space variables
$\boldsymbol{\xi}=(x,y,k_{x},k_{y})$ :

$\mathcal{H}(\boldsymbol{\xi})=\dfrac{\hbar^{2}{\bf k}^{2}}{2m}+{\bf
d}(\boldsymbol{\xi})\cdot\boldsymbol{\sigma},\quad{\bf
d}(\boldsymbol{\xi})=a\lambda({\bf k}\times\hat{{\bf z}})-J\hat{{\bf m}}({\bf
r}),\quad\mathcal{E}_{\pm}(\boldsymbol{\xi})=\dfrac{\hbar^{2}{\bf
k}^{2}}{2m}\pm|{\bf d}(\boldsymbol{\xi})|$ (16)

and hosts six types of Berry curvatures, each corresponding to a plane in the
4D phase space

$\displaystyle\Omega^{\pm}_{\alpha,\beta}(\boldsymbol{\xi})$ $\displaystyle=$
$\displaystyle\pm\dfrac{1}{2}\hat{{\bf
d}}(\boldsymbol{\xi})\cdot(\partial_{\alpha}\hat{\bf
d}(\boldsymbol{\xi})\times\partial_{\beta}\hat{{\bf d}}(\boldsymbol{\xi}))$
(17)

where $\pm$ label the two bands. The curvatures introduce non-trivial Poisson
bracket relations between the phase space variables that lead to corrections
in the equations of motion and the invariant measure. Both these quantities
are captured by the completely anti-symmetric matrix
$[\Gamma(\boldsymbol{\xi})]_{\alpha,\beta}=\Omega_{\alpha,\beta}(\boldsymbol{\xi})-[i\sigma_{y}\otimes\mathds{1}]_{\alpha,\beta}$
as defined in the main text. Here we explicitly write the expression for
completeness:

$\displaystyle\hbar\begin{pmatrix}\dot{x}\\\ \dot{y}\\\ \dot{k_{x}}\\\
\dot{k_{y}}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{\text{det}[\Gamma(\boldsymbol{\xi})]}}\left[\begin{pmatrix}0&\Omega_{k_{x},k_{y}}&-\Omega_{y,k_{y}}&\Omega_{y,k_{x}}\\\
-\Omega_{k_{x},k_{y}}&0&\Omega_{x,k_{y}}&-\Omega_{x,k_{x}}\\\
\Omega_{y,k_{y}}&-\Omega_{x,k_{y}}&0&\Omega_{x,y}\\\
-\Omega_{y,k_{x}}&-\Omega_{x,k_{x}}&-\Omega_{x,y}&0\end{pmatrix}-\begin{pmatrix}0&0&-1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\\\
\end{pmatrix}\right]\begin{pmatrix}\partial_{x}\widetilde{\mathcal{E}}(\boldsymbol{\xi})\\\
\partial_{y}\widetilde{\mathcal{E}}(\boldsymbol{\xi})+eE\\\
\partial_{k_{x}}\widetilde{\mathcal{E}}(\boldsymbol{\xi})\\\
\partial_{k_{y}}\widetilde{\mathcal{E}}(\boldsymbol{\xi})\end{pmatrix}$ (18)
$\displaystyle dV_{\boldsymbol{\xi}}$ $\displaystyle=$
$\displaystyle\dfrac{d^{2}{\bf r}d^{2}{\bf
k}}{(2\pi)^{2}V}\sqrt{\text{det}[\Gamma(\boldsymbol{\xi})]}$ (19)

There is one additional change in the equations. The non-trivial spatial and
momentum variation in the eigenfunction of the semi-classical Bloch
Hamiltonian $|u(\boldsymbol{\xi})\rangle$ leads to a shift in the energy
$\widetilde{\mathcal{E}}(\boldsymbol{\xi})=\mathcal{E}(\boldsymbol{\xi})+\delta\mathcal{E}(\boldsymbol{\xi})$
with

$\delta\mathcal{E}(\boldsymbol{\xi})=-\sum_{i=x,y}\text{Im}\bigg{[}\bigg{(}\partial_{r_{i}}\langle{u(\boldsymbol{\xi})}|\bigg{)}(\mathcal{E}(\boldsymbol{\xi})-\mathcal{H}(\boldsymbol{\xi}))\bigg{(}\partial_{k_{i}}|{u(\boldsymbol{\xi})}\rangle\bigg{)}\bigg{]}.$
(20)

We can ignore $\delta\mathcal{E}(\boldsymbol{\xi})$ in our calculation because
it scales as $(\lambda/E_{F})(a/L_{s})$ and thus leads to higher order
corrections to the Hall effect not considered here.

The matrix representation in eq. (18) contains all contributions of the
curvatures. In particular, the anomalous velocity can be extracted from the
electric field dependent part of the velocity in $+x$ direction

$\dot{x}^{(D)}(\boldsymbol{\xi})=\dfrac{e}{\hbar}\dfrac{\Omega_{k_{x},k_{y}}(\boldsymbol{\xi})}{\sqrt{\text{det}[\Gamma(\boldsymbol{\xi})]}}E.$
(21)

The determinant factor in the denominator may seem unfamiliar but is
absolutely crucial for calculating the correct intrinsic anomalous Hall
response. There is a complete cancellation of the phase-space measure factors
in the Hall current so that the anomalous Hall conductivity only depends on
the momentum-space Berry curvature:

$j_{x}=e\int\dfrac{d^{2}{\bf r}d^{2}{\bf
k}}{(2\pi)^{2}V}\sqrt{\text{det}[\Gamma(\boldsymbol{\xi})]}\;\dot{x}^{(D)}(\boldsymbol{\xi})\;f^{0}(\boldsymbol{\xi})=\left(\dfrac{e^{2}}{\hbar}\int\dfrac{d^{2}{\bf
r}d^{2}{\bf
k}}{(2\pi)^{2}V}\;\Omega_{k_{x},k_{y}}(\boldsymbol{\xi})\;f^{0}(\boldsymbol{\xi})\right)E$
(22)

The quantity within brackets is $\sigma_{xy}^{\text{AHE}}$. Even though the
expression contains only the momentum-space Berry curvature
$\Omega_{k_{x},k_{y}}(\boldsymbol{\xi})$, we must keep in mind that
$\Omega_{k_{x},k_{y}}(\boldsymbol{\xi})$ is a function of momentum and real
space and the two integrals are not separable:

$\sigma_{xy}^{\text{AHE}}=-\sum_{l=\pm}l\dfrac{e^{2}}{\hbar}\int\dfrac{d^{2}{\bf
r}\;d^{2}{\bf k}}{(2\pi)^{2}V}\;\left[\dfrac{a^{2}\lambda^{2}Jm_{z}({\bf
r})}{2|{\bf
d}(\boldsymbol{\xi})|}\right]\;f^{0}[\mathcal{E}_{l}(\boldsymbol{\xi})].$ (23)

Since there is already an explicit $\lambda^{2}$ and $\lambda/E_{F}$ is a
small parameter, we can set $\lambda=0$ in the rest of the expression to find
the leading contribution. The spatial dependence in the semi-classical
eigenenergies drops out when $\lambda=0$, that is
$(\mathcal{E}_{l}(\boldsymbol{\xi};\lambda=0)=\mathcal{E}_{l}({\bf k}))$ and
the spatial and momentum integrals become separable

$\displaystyle\sigma_{xy}^{\text{AHE}}$
$\displaystyle\approx-\sum_{l=\pm}l\dfrac{e^{2}a^{2}}{2\hbar}\bigg{(}\dfrac{\lambda}{J}\bigg{)}^{2}\left(\int\dfrac{d^{2}{\bf
r}}{V}\;m_{z}({\bf r})\right)\left(\int\dfrac{d^{2}{\bf
k}}{(2\pi)^{2}}f^{0}[\mathcal{E}_{l}({\bf k})]\right).$ (24)

We thus find that the intrinsic contribution only probes the net out-of-plane
magnetization even for spatially varying textures.

##  Appendix B Solution to the Boltzmann Equation

Focussing on contributions that come from electric field induced perturbations
to the distribution function, we write the full distribution function in the
presence of electric field as $f=f^{0}+g$ where $g$ is linear order in the
field. We then use the relaxation time approximation to write the Boltzmann
equation as

$\displaystyle\dot{\boldsymbol{\xi}}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\left(f^{0}(\boldsymbol{\xi})+g(\boldsymbol{\xi})\right)=-\dfrac{g(\boldsymbol{\xi})}{\tau}.$
(25)

We write
$\dot{\boldsymbol{\xi}}=\dot{\boldsymbol{\xi}}^{(I)}+\dot{\boldsymbol{\xi}}^{(D)}$
where $I$ and $D$ refer to electric field dependent and independent components
to find an equation for $g(\boldsymbol{\xi})$

$\left(1+\tau\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\right)g(\boldsymbol{\xi})=-\tau\;\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})].$
(26)

The differential operator on the left has a particular scaling. With
$g(\boldsymbol{\xi})=g[\mathcal{E}(\boldsymbol{\xi})]$, we can infer that
$\tau\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\mathcal{E}\sim(\ell/L_{s})(a/L_{s})$.
Now since $a\ll\ell\ll L_{s}$, both these ratios are small and hence, we can
invert the operator to find

$g(\boldsymbol{\xi})=-\tau\;\left(1-\tau\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\right)\;\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]=g^{(1)}(\boldsymbol{\xi})+g^{(2)}(\boldsymbol{\xi})$
(27)

where the superscripts label the order in $\tau$. The first order term
$g^{(1)}(\boldsymbol{\xi})$ does not result in any Hall conductivity. While we
find that at the end of a long calculation, we can use time-reversal (TR)
symmetry to understand why it vanishes. Onsager’s reciprocity relation forces
Hall conductivity to be odd under TR. The conductivity arising from
$g^{(1)}(\boldsymbol{\xi})$ doesn’t have this property:

$\sigma_{xy}\sim\dfrac{e}{V}\int\dfrac{d^{2}{\bf r}d^{2}{\bf
k}}{(2\pi)^{2}}\underbrace{\sqrt{\text{det}[\Gamma(\boldsymbol{\xi})]}}_{\text{TR
even}}\;\underbrace{\dot{x}(\boldsymbol{\xi})}_{\text{TR
odd}}\;\Big{(}-\tau\underbrace{\dot{\boldsymbol{\xi}}^{(D)}}_{\text{TR
odd}}\cdot\underbrace{\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]}_{\text{TR
even}}\Big{)}.$ (28)

It is clearly even under time-reversal and hence must vanish. The
contributions from $g^{(2)}(\boldsymbol{\xi})$ survive this argument.

The calculation for Hall conductivity involves combining the distribution
function with velocity and the appropriate phase space volume factor. The
algebra is tedious but there are a few simplifying factors. Products and
derivatives of the curvatures can be excluded as they are all higher order in
$(a/L_{s})$. There will still be many terms and hence we need to introduce a
classification scheme for bookkeeping

$\prod\limits_{i=1}^{m}\prod\limits_{j=1}^{n}\partial_{k_{i}}\partial_{r_{j}}(\cdot)\longrightarrow(m,n).$
(29)

Here $(m,n)$ labels expressions that have $m$ momentum derivatives and $n$
spatial derivatives. These numbers count both: derivatives with respect to the
semi-classical energies and the implicit derivatives hidden inside the
curvatures.

With these labels, the Hall conductivity is

$\sigma_{xy}\;\sim\;\int\dfrac{d^{2}{\bf r}d^{2}{\bf
k}}{(2\pi)^{2}V}\;\underbrace{\phantom{\Big{(}}\text{Phase space
volume}\phantom{\Big{)}}}_{(0,0)+(1,1)}\;\times\;\underbrace{\phantom{\Big{(}}\text{Velocity}\phantom{\Big{)}}}_{(1,0)+{\color[rgb]{0,0,0}(2,1)}}\;\times\;\underbrace{\phantom{\Big{(}}\dot{\boldsymbol{\xi}}^{(I)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}\phantom{\Big{)}}}_{(1,1)+{\color[rgb]{0,0,0}(2,2)}}\;\times\underbrace{\Big{(}\dot{\boldsymbol{\xi}}^{(D)}\cdot\boldsymbol{\nabla}_{\boldsymbol{\xi}}f^{0}[\mathcal{E}(\boldsymbol{\xi})]\Big{)}}_{(1,0)+{\color[rgb]{0,0,0}(2,1)}}.$
(30)

The first tuple indexes derivatives acting on the semi-classical
eigenenergies, while the second tuple indexes derivatives deriving from the
Berry curvatures. Since we are focussing on contributions that involve at most
one curvature, there are only two broad categories: no curvature $(3,1)$ and
one curvature $(4,2)$. The number of terms inside each category is still quite
large.

We now turn to energy scaling relations and take advantage of the fact that
$\lambda/E_{F}$ is a small parameter. To the leading order, we find that
$\partial_{k_{x}}\mathcal{E}\sim\lambda^{0}$, $\Omega_{x,y}\sim\lambda^{0}$,
$\partial_{x}\mathcal{E}\sim\lambda$, , $\Omega_{x,k_{y}}\sim\lambda$ and
$\Omega_{k_{x},k_{y}}\sim\lambda^{2}$. We are now ready to calculate the
contributions order by order in $\lambda/E_{F}$ and $a/L_{s}$:

  * •

Zeroth order in $\lambda/E_{F}$

The equations of motion are quite simple since all spatial derivatives vanish.
There are no $(3,1)$ terms and the only non-zero $(4,2)$ term has both spatial
derivatives coming from the real-space Berry curvature. The resulting
contribution is the Topological Hall response $\sigma_{xy}^{\text{THE}}$.
Simplifying eq. (13) we find

$\sigma_{xy}^{\text{THE}}=-\dfrac{e^{2}\tau^{2}}{\hbar^{3}}n_{\text{sk}}\dfrac{2h^{2}}{m}\begin{cases}\dfrac{\pi\hbar
n}{m},&\text{for $0<n<\dfrac{mJ}{\pi\hbar}$}\\\ J,&\text{for
$\dfrac{mJ}{\pi\hbar}<n$}\end{cases}$ (31)

where $n$ is the electron density. Hence as $J/E_{F}$ is tuned, $\sigma_{xy}$
crosses over from a linear in $J$ regime to a saturating value that is
independent of $J$ but increases with density. Here $n_{\text{sk}}$ is the
skyrmion density

$n_{\text{sk}}=\dfrac{1}{V}\int\dfrac{d^{2}{\bf r}}{4\pi}\,\,\hat{{\bf
m}}({\bf r})\cdot(\partial_{x}\hat{{\bf m}}({\bf
r})\times\partial_{y}\hat{{\bf m}}({\bf r}))$ (32)

For small densities $n<\dfrac{mJ}{\pi\hbar}$ this can be written as

$\sigma_{xy}^{\text{THE}}=\dfrac{ne\tau}{m}\bigg{(}\dfrac{\tau
eB_{\text{eff}}}{m}\bigg{)}$ (33)

with $eB_{\text{eff}}=-2\pi n_{\text{sk}}$ acting like an effective magnetic
field induced by the pressence of the spatially dispersive magnetic texture.

  * •

First order in $\lambda/E_{F}$

There are both $(3,1)$ and $(4,2)$ type of contributions. We leave $(3,1)$ to
the next section since it has a rather interesting origin, and focus on
$(4,2)$, which has two possible origins.

The first involves $\Omega_{x,y}$ multiplied with four momentum derivatives of
energies. We will now show that the resultant Hall conductivity is even in
$\lambda$ and hence is either zeroth order (discussed above) or second order
(can be ignored). It can be checked that the semi-classical eigenenergies and
the real-space curvature satisfy the relations

$\mathcal{E}({\bf r},{\bf k},\lambda)=\mathcal{E}({\bf r},-{\bf
k},-\lambda);\quad\Omega_{x,y}({\bf r},{\bf k},\lambda)=\Omega_{x,y}({\bf
r},-{\bf k},-\lambda).$ (34)

As a result, the integrand in phase space will switch $\lambda$ upon flipping
the momentum ${\bf k}\rightarrow-{\bf k}$. The resulting Hall conductivity
changes $\sigma(\lambda)\rightarrow\sigma(-\lambda)$ under the flip. However,
since ${\bf k}$ is a dummy variable that is being integrated over, Hall
conductivity must satisfy $\sigma(\lambda)=\sigma(-\lambda)$ and is hence even
in $\lambda$. There cannot be any first order corrections.

The other possibility both involves mixed curvatures, which as we showed, are
at least linear in $\lambda/E_{F}$. Since the overall type has to be $(4,2)$,
the pre-factors that come with mixed curvature should be of the type $(3,1)$.
That is, there will be an additional spatial derivative in the full
expression. It can either come from a different mixed curvature piece or from
a first order spatial derivative of the semi-classical eigenenergies. It is
easy to see that both these situations lead to second order contributions.

In sum, the only linear order contribution in SOC is of type $(3,1)$. It is
the subject of the next section.

##  Appendix C Hall conductivity independent of Curvatures

There are many simplifications when the curvatures are absent. We therefore
find it instructive to present the full derivation, starting from the fact
that the semi-classical energy is a function of both real space and momentum,
$\mathcal{E}({\bf r},{\bf k})$. The derivation also appeals to the generality
of the result and that it may apply to systems beyond the model Hamiltonian
that we have considered in this paper.

With external Electric field, ${\bf E}$, the dynamics of the wave-packet is
governed by the equations :

$\begin{pmatrix}\dot{{\bf r}}\\\ \dot{{\bf
k}}\end{pmatrix}=\begin{pmatrix}0&\mathds{1}\\\
-\mathds{1}&0\end{pmatrix}\begin{pmatrix}\boldsymbol{\nabla}_{r}\mathcal{E}/\hbar+e{\bf
E}/\hbar\\\ \boldsymbol{\nabla}_{k}\mathcal{E}/\hbar\end{pmatrix}.$ (35)

that lead to the following second-order shift in the distribution function

$g^{(2)}(\boldsymbol{\xi})=-\dfrac{e\tau^{2}}{\hbar^{2}}\left(\dfrac{\partial
f^{0}}{\partial\mathcal{E}}\right)\big{[}\boldsymbol{\nabla}_{k}\mathcal{E}\cdot\boldsymbol{\nabla}_{r}-\boldsymbol{\nabla}_{r}\mathcal{E}\cdot\boldsymbol{\nabla}_{k}\big{]}{\bf
E}\cdot\boldsymbol{\nabla}_{k}\mathcal{E}$ (36)

and a Hall conductivity

$\sigma_{\alpha\beta}=\dfrac{e^{2}\tau^{2}}{\hbar^{3}}\dfrac{1}{V}\int\dfrac{d^{2}{\bf
r}\;d^{2}{\bf q}}{(2\pi)^{2}}\;\left(\dfrac{\partial
f^{0}}{\partial\mathcal{E}}\right)\;(\partial_{k_{\alpha}}\mathcal{E})\big{[}\boldsymbol{\nabla}_{k}\mathcal{E}\cdot\boldsymbol{\nabla}_{r}-\boldsymbol{\nabla}_{r}\mathcal{E}\cdot\boldsymbol{\nabla}_{k}\big{]}(\partial_{k_{\beta}}\mathcal{E}).$
(37)

where we have suppressed the sum over the band index for brevity.

It is not obvious from the expression, as it stands, to see that the anti-
symmetric response, $\sigma_{xy}-\sigma_{yx}$, is finite. Therefore, we next
use integration by parts to rewrite the tensor as

$\sigma_{\alpha\beta}=-\dfrac{e^{2}\tau^{2}}{\hbar^{3}}\dfrac{1}{V}\int d{\bf
r}\;d{\bf
q}\;f^{0}[\mathcal{E}]\;\big{[}\boldsymbol{\nabla}_{k}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{r}-\boldsymbol{\nabla}_{r}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{k}\big{]}(\partial_{k_{\beta}}\mathcal{E})+\mathcal{S}_{\alpha\beta}$
(38)

where $\mathcal{S}$ is a symmetric tensor,
$\mathcal{S}_{\alpha\beta}=\mathcal{S}_{\beta\alpha}$, and the integrand is
explicitly anti-symmetric

$\big{[}\boldsymbol{\nabla}_{k}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{r}-\boldsymbol{\nabla}_{r}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{k}\big{]}(\partial_{k_{\beta}}\mathcal{E})=\boldsymbol{\nabla}_{k}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{r}\partial_{k_{\beta}}\mathcal{E}-\boldsymbol{\nabla}_{r}\partial_{k_{\alpha}}\mathcal{E}\cdot\boldsymbol{\nabla}_{k}\partial_{k_{\beta}}\mathcal{E}.$
(39)

Thus, the net anti-symmetric part can survive.

Back to our model Hamiltonian, we see that this effect cannot be described as
an anomalous or topological Hall response. It survives in the absence of both
Berry curvatures. As we will show now, its origin lies in vorticity of the
local electronic velocity field. We expand the integrand

$\partial_{k_{x}}^{2}\mathcal{E}\partial_{x,k_{y}}\mathcal{E}+\partial_{k_{x},k_{y}}\mathcal{E}\partial_{y,k_{y}}\mathcal{E}-\partial_{x,k_{x}}\mathcal{E}\partial_{k_{x},k_{y}}\mathcal{E}-\partial_{y,k_{x}}\mathcal{E}\partial_{k_{y}}^{2}\mathcal{E}$
(40)

and use the fact that
$\partial_{k_{\alpha},k_{\beta}}\mathcal{E}=\delta_{\alpha,\beta}\;\hbar^{2}/m+\mathcal{O}(\lambda^{2})$
to ignore the middle two terms when $\lambda/E_{F}$ is small. The other two
terms to first order can be written as

$\dfrac{\hbar^{2}}{m}\left(\partial_{x,k_{y}}\mathcal{E}-\partial_{y,k_{x}}\mathcal{E}\right)=\dfrac{\hbar^{2}}{m}\left({\bf
z}\cdot\boldsymbol{\nabla}_{\bf r}\times(\boldsymbol{\nabla}_{\bf
k}\mathcal{E})\right)=\dfrac{\hbar^{3}}{m}\left({\bf
z}\cdot\boldsymbol{\nabla}_{\bf r}\times{\bf v}({\bf r})\right).$ (41)

An intuitive picture behind ordinary Hall effect is that electrons undertake
cyclotron orbits under the action of the magnetic field. This results in
electron velocity field forming vortices. This contribution, on the other
hand, doesn’t require an external magnetic field and instead uses the
underlying magnetic texture to mimic vortices. The explicit connection to the
texture is

$\displaystyle\partial_{x,k_{y}}\mathcal{E}_{\pm}-\partial_{y,k_{x}}\mathcal{E}_{\pm}=\mp
a\lambda\left(\partial_{x}m_{x}+\partial_{y}m_{y}\right)=\mp
a\lambda\boldsymbol{\nabla}\cdot\hat{m}({\bf r})$ (42)

which has been reported elsewhere in the literature Freimuth  _et al._ (2013);
Akosa  _et al._ (2018, 2019); Zhang  _et al._ (2020) as a correction to the
effective magnetic field in the presence of SOC. The resulting Hall
conductivity for small densities $n<mJ/\pi\hbar$ is

$\delta\sigma_{xy}=-\dfrac{ne^{2}\tau}{m}\left(\dfrac{\tau\lambda
a}{\hbar}\int\dfrac{d^{2}{\bf
r}}{V}\;\dfrac{\boldsymbol{\nabla}_{r}\cdot\hat{{\bf m}}({\bf r})}{2}\right).$
(43)

and can be interpreted as arrising from an effective magnetic field
$\sim\lambda\boldsymbol{\nabla}_{r}\cdot\hat{{\bf m}}({\bf r})$. Lastly, we
note that this integral is a boundary term. Therefore unless there are
singular features in the semi-classical velocity, the integral has to vanish.
That being said the general result in eq. (37) may still be finite for systems
with alternative kinetic dispersion relations $\mathcal{E}(\boldsymbol{\xi})$.

##  Appendix D Kubo Formula Calculation

Figure A1: Hall conductivity from exact diagonalization for a skyrmion crystal
with (a) triangle and (b) square lattice with strong coupling $J/t=10$. The
first panel shows the bands for a skyrmion unit cell with $L_{s}/a=6$. The
skyrmion potential causes band folding between momentum eigenstates. The
number of bands increases on increasing the lattice resolution of the
skyrmion, as seen in the second panel with $L_{s}/a=14$. The resulting DoS and
Hall conductivity are shown in the last two panels.

While the semi-classical calculation produces intuitive results, the algebra
is only controlled when $\lambda/E_{F}$, $(a/L_{s})$, and $\ell/L_{s}$ are
small parameters. In this section, we will discuss the opposite limit with
$a,L_{s}\ll\ell=\infty$. We use a tight binding model with magnetic unit cell
area $\sim(L_{s}/a)^{2}$ and calculate the Hall conductance using the TKNN
Kubo formula. The results of the calculation are exact and contain information
deriving from all the types of contributions to the Hall effect. Guided by the
semi-classical calculation here we discuss certain limiting cases.

We consider a tight-binding version of the continuum model

$\mathcal{H}=-t\sum\limits_{\langle{\bf i},{\bf
j}\rangle,\sigma}c^{\dagger}_{{\bf i}\sigma}c^{\phantom{{\dagger}}}_{{\bf
j}\sigma}-i\lambda\sum\limits_{\langle{\bf i},{\bf
j}\rangle,\sigma,\sigma^{\prime}}c^{\dagger}_{{\bf i}\sigma}\left[{\bf
r}_{ij}\times{\bf
z}\cdot\boldsymbol{\sigma}\right]_{\sigma\sigma^{\prime}}c^{\phantom{{\dagger}}}_{{\bf
j}\sigma^{\prime}}-J\sum\limits_{{\bf
i},\sigma,\sigma^{\prime}}c^{\dagger}_{{\bf i}\sigma}\left[\hat{{\bf
m}}_{i}\cdot\boldsymbol{\sigma}\right]_{\sigma\sigma^{\prime}}c^{\phantom{{\dagger}}}_{{\bf
i}\sigma^{\prime}}$ (44)

where the vector field $\hat{{\bf m}}_{\bf i}$ models a discrete version of a
skyrmion

${\bf m}_{{\bf i}}=\begin{pmatrix}\sin\left(2\pi{\bf i}\cdot{\bf
a}_{1}\right)\\\ \sin\left(2\pi{\bf i}\cdot{\bf a}_{2}\right)\\\
\cos\left(2\pi{\bf i}\cdot{\bf a}_{1}\right)+\cos\left(2\pi{\bf i}\cdot{\bf
a}_{2}\right)+1\end{pmatrix},\quad\hat{{\bf m}}_{\bf i}=\dfrac{{\bf m}_{{\bf
i}}}{\sqrt{{\bf m}_{{\bf i}}\cdot{\bf m}_{{\bf i}}}}$ (45)

with winding number $+1$. Here ${\bf a}_{1}$ and ${\bf a}_{2}$ are the lattice
vectors for the skyrmion lattice and ${\bf i}$ labels a position inside the
skyrmion unit cell. The corresponding magnetic Brillouin Zone (MBZ) is spanned
by vectors ${\bf b}_{1}$ and ${\bf b}_{2}$ that satisfy ${\bf a}_{i}\cdot{\bf
b}_{j}=2\pi\delta_{i,j}$. These vectors permit a momentum representation

$c^{\phantom{{\dagger}}}_{{\bf
k}\sigma}=\dfrac{1}{\sqrt{N_{uc}}}\sum\limits_{{\bf k}\in\text{MBZ}}e^{i{\bf
k}\cdot{\bf i}}c^{\phantom{{\dagger}}}_{{\bf i}\sigma}$ (46)

with ${\bf k}$ taken from a $N_{k}\times N_{k}$ BZ mesh. The Bloch Hamiltonian
which is then diagonalized to find the energies and wavefunctions

$\mathcal{H}({\bf k})|u_{n,{\bf k}}\rangle=\epsilon_{n}({\bf k})|u_{n,{\bf
k}}\rangle.$ (47)

The wave-functions lead to the Berry curvature

$\Omega_{n}({\bf k})=-2\text{Im}\langle\partial_{k_{x}}u_{n,{\bf
k}}|\partial_{k_{y}}u_{n,{\bf k}}\rangle$ (48)

that is then combined with the TKNN formula to calculate the Hall conductivity

$\sigma_{xy}=-\dfrac{e^{2}}{\hbar}\dfrac{1}{V}\int\limits_{\text{MBZ}}\dfrac{d^{2}{\bf
k}}{(2\pi)^{2}}\sum\limits_{n=1}^{N_{b}}\Omega_{n}({\bf
k})\;\Theta(\mu-\epsilon_{n}({\bf k})).$ (49)

Finally, we replace the integral by a discrete sum

$\dfrac{1}{V}\int\limits_{\text{MBZ}}\dfrac{d^{2}{\bf
k}}{(2\pi)^{2}}\longrightarrow\dfrac{1}{\mathcal{V}N_{k}^{2}}\sum\limits_{{\bf
k}\in\text{MBZ}}.$ (50)

where $\mathcal{V}=|{\bf a}_{1}\times{\bf a}_{2}|$ is the area of the unit
cell. We chose the normalization so that the density of electrons per unit
cell

$n=\dfrac{1}{N_{s}^{2}N_{k}^{2}}\sum\limits_{n,{\bf
k}\in\text{MBZ}}\Theta(\mu-\epsilon_{n}({\bf k})).$ (51)

goes from 0 (empty) to 2 (filled) as the chemical potential $\mu$ is varied
across the spectrum (see Fig. A1).

The resolution of the skyrmion within the unit cell is controlled by
$N_{s}=L_{s}/a$. Larger $N_{s}$ lead to a better real-space mesh but also give
rise to a larger Bloch Hamiltonian with $2N_{s}^{2}$ bands. The bottleneck in
our code is the matrix diagonalization step whose complexity is
$\mathcal{O}(n^{3})$ where $n=N_{s}^{2}$ is the size of the matrix. Since this
step has to be repeated $N_{k}^{2}$ times, the overall complexity is
$\mathcal{O}(N_{k}^{2}N_{s}^{6})$ and hence the continuum limit
$(N_{s}\rightarrow\infty$ is more difficult than thermodynamic limit
$N_{k}\rightarrow\infty$. We find that $N_{k}\rightarrow\infty$ and
$N_{s}\rightarrow\infty$ limits can be different, especially for the vorticity
correction term which is sensitive to $N_{s}$.


# Cospectral lifts of graphs

F. Ramezani1
Department of Mathematics, Faculty of Science,
K.N.Toosi University of Technology, Tehran, Iran,
P.O. Box 16315-16181

###### Abstract

We prove that for a pair of cospectral graphs $G$ and $H$, there exist their
non trivial lifts $G^{\prime}$ and $H^{\prime}$ which are cospectral. More
over for a pair of cospectral graphs on $6$ vertices, we find some cospectral
lifts of them.

AMS Subject Classification: 05C50.
Keywords: lifts of graphs, eigenvalues, cospectral graphs.

11footnotetext: Corresponding Author, Email address<EMAIL_ADDRESS>
##  1 Introduction

Let $G=(V,E)$ be a simple graph on the vertex set
$V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and edge set $E$. The adjacency matrix
of $G$ is an $n$ by $n$ matrix $A(G)$ whose $(i,j)$-th entry is $1$ if
vertices $v_{i}$ and $v_{j}$ are adjacent and $0$, otherwise. The spectrum of
$G$ is the multi-set of eigenvalues of $A(G)$. Two graphs $G$ and $G^{\prime}$
are called cospectral if they share the same spectrum. We say $G$ is
determined by spectrum (DS for short) if it has no non-isomorphic cospectral
mate.

The problem of constructing cospectral graphs, has been investigated by some
authors. For a survey of results on this area we refer the reader to [2, 3,
6]. In [4] the authors have used the concept $m$-cospectrality to construct
new cospectral graphs. Haemers et all in [1] have considered Godsil-McKay
switching method to construct non-isomorphic cospectral graphs see the paper
for more details. In this article we use the concept lift of graphs to
construct new non-isomorphic cospectral graphs from given small cospectral
pairs of graphs.

##  2 Preliminaries

In this section we mention some basic definitions and results which will be
used during the paper. We denote by $\dot{E}$ the set of all ordered pairs
$\\{(i,j)|\textrm{ }i<j,\textrm{ }\\{v_{i},v_{j}\\}\in E\\}$. For an Abelian
group of order $k$, say $Gr$, a $k-$Abelian signature $s$ of the graph $G$ is
a map $s:\dot{E}\longrightarrow Gr$. A $k$-Abelian lift of the graph $G$,
associated with the signature $s$, which is denoted by $G(s)$, is a graph on
the vertex set $V(G)\times[k]$ ($[k]=\\{0,1,\ldots,k-1\\}$,
$Gr=(\\{g_{0},g_{1},\ldots,g_{k-1}\\},*$)), where for any $(i,j)\in\dot{E}$
and $a,b\in[k]$ there is an edge between $(v_{i},a)$ and $(v_{j},b)$ if and
only if $s(i,j)*g_{a}=g_{b}$. Note that in the graph $G(s)$, for any
$(i,j)\in\dot{E}$, there is a matching between the vertex sets
$V_{i}=\\{v_{i}\\}\times[k]$ and $V_{j}=\\{v_{j}\\}\times[k]$. If a graph have
$m$ edges there may be $k^{m}$ different $k$-Abelian lifts of $G$, since the
sets $V_{i},V_{j}$ are matched in $k$ different ways. If the signature $s$
maps all pairs to the same element $g\in Gr$, then we denote the corresponding
lift $G(s)$ with $G_{g}$. We illustrate the definition of the $k$-lifts of a
graph in the following figure. In the following graph the graph $G$ is the
cycle $C_{4},$ and the corresponding signature is
$s:\dot{E}\longrightarrow\mathbb{Z}_{2}$, with
$s(1,3)=0,s(1,4)=0,s(2,3)=1,s(2,4)=0.$

⋮⋮⋮⋮.4132$\longrightarrow$

$G\hskip 99.58464ptG(s)$

Figure1. 2-lift of G corresponding to the signature s

Let $Gr=(\\{g_{1}=1,g_{2},\ldots,g_{n}\\},*)$ be a group of order $n$. For any
group element say $g\in Gr$ there is an $n\times n$ permutation matrix $P_{g}$
in correspondence, which is defined bellow,

$P_{g}(i,j)=\begin{cases}1&\text{ {if} }g_{i}*g=g_{j},\\\
0&\text{otherwise}.\end{cases}$

###### Lemma 1

The function $\phi:Gr\rightarrow SL(n,\mathbb{R})$, where $SL(n,\mathbb{R})$
is the set of $n\times n$ real non-singular matrices and $\phi(g)=P_{g}$, is a
group homomorphism.

The eigenvalues of the graph $G(s)$ has been studied in the literature. For
instance in the following theorem from [5] the authors have obtained the
eigenvalues of Abelian $t$-lifts. See [5] for more details and the notations.

###### Theorem 1

Let $G$ be a multigraph and $\phi$ be a signature assignment to an Abelian
group. Let $\beta$ be a common basis of eigenvectors of the permutation
matrices in the image of $\phi$. For every $\mathbf{x}\in\beta$, let
$A_{\mathbf{x}}$ be the matrix obtained from the adjacency matrix of $G$ by
replacing any $(u,v)$-entry of $A(G)$ by
$\sum_{(e,u,v)\in\overrightarrow{E}(G)}\lambda_{\mathbf{x}}(\phi(e,u,v))$.
Then the spectrum of the $t$-lift $G(\phi)$ of $G$ is the multiset union of
the spectra of the matrices $A_{\mathbf{x}}(\mathbf{x}\in\beta)$.

##  3 Main result

Our main problem here is ”for given pair of cospectral graphs $G$ and $H$, is
there $k$-Abelian signatures $s,s^{\prime}$ which $G(s)$ and $H(s^{\prime})$
are cospectral?”. We look for general answers of this question.

It is known that for $l,l^{\prime}\in Gr$ the permutation matrices
$P_{l},P_{l^{\prime}}$ commute, so they have common basis of eigenvectors. The
following theorem is a straight consequence of Theorem 1.

###### Theorem 2

Let G be a graph and $s$ be a $k$-cyclic signature of $G$. Let $\beta$ be a
common basis of eigenvectors of the permutation matrices in the image of $s$.
For every $\mathbf{x}\in\beta$, let $A_{\mathbf{x}}$ be the matrix defined
bellow

$A_{\mathbf{x}}(i,j)=\begin{cases}\lambda_{\mathbf{x}}(P_{s(i,j)})&i<j,\\\
0&i=j,\\\ \lambda_{\mathbf{x}}^{-1}(P_{s(i,j)})&i>j.\end{cases}$

Then the spectrum of $G(s)$ is the multi-set union of the spectra of the
matrices $A_{\mathbf{x}}(\mathbf{x}\in\beta)$.

###### Lemma 2

Let $Gr$ be a group of order $n$. For any $g\in Gr$, any eigenvalues of the
permutation matrix $P_{g}$ is an $n$’th root of unity.

###### Proof.

The assertion follows by the fact that the order of any element in the group
divides the order of group. Hence $g^{n}=1$, and therefore $P_{g}^{n}=I_{n}.$
Hence the minimal polynomial of $P_{g}$, say $m(P_{g},x)$ divides the
polynomial $x^{n}-1,$ thus the assertion follows.  $\Box$

###### Lemma 3

If $G$ and $H$ are cospectral graphs on $n$ verices and $Gr$ be a finite group
of order $t$. If for $g\in Gr$ the matrix $P_{g}$ is symmetric, then the
graphs $G_{g}$ and $H_{g}$ are cospectral.

###### Proof.

Since the signature corresponds the fixed element $g$ to all the edges of the
graph $G$, and $P_{g}^{-1}=P_{g},$ then by Theorem 1, the eigenvalues of the
graph $G_{g}$ are the multi-set union of the matrices $\omega_{i}A(G)$, where
$\omega_{i}$’s are the eigenvalues of $P_{g}$ for $i=1,2,\ldots,n.$ On the
other hand the eigenvalues of $\omega_{i}A(G)$ are $\omega_{i}\lambda_{j}$
where $\lambda_{j}$ is the $j$’th eigenvalue of $G$. Hence the spectrum of
$G_{g}$ and $H_{g}$ are the multi-set
$\\{\omega_{i}\lambda_{j}\\}_{i=1,\ldots,t}^{j=1,\ldots,n}.$

$\Box$

###  3.1 Examples

We consider two cospectral graphs $G$ and $H$, shown in the Figure 1. We try
to find possible Abelian lifts of them say $G(s)$ and $H(s^{\prime})$ which
are also cospectral.

$\textrm{{Figure1.} Two cospectral graphs }G\textrm{ and }H$

We first consider all possible $t$-Abelian signatures of the graphs $G$ and
$H$, suppose that the matrices $A(G)_{\mathbf{x}},A(H)_{\mathbf{x}}$,
corresponding to the graphs $G$, $H$ and their prescribed signatures, which is
introduced in Theorem 1 are of the following general forms,

$A(G)_{\mathbf{x}}=\begin{pmatrix}0&u&0&0&0&0\\\ u^{-1}&0&v&w&0&0\\\
0&v^{-1}&0&x&y&0\\\ 0&w^{-1}&x^{-1}&0&z&0\\\ 0&0&y^{-1}&z^{-1}&0&r\\\
0&0&0&0&r^{-1}&0\end{pmatrix}\textrm{,
}A(H)_{\mathbf{x}}=\begin{pmatrix}0&u_{1}&v_{1}&0&0&0\\\
u^{-1}_{1}&0&w_{1}&0&0&0\\\ v^{-1}_{1}&w^{-1}_{1}&0&x_{1}&y_{1}&z_{1}\\\
0&0&x^{-1}_{1}&0&0&0\\\ 0&0&y^{-1}_{1}&0&0&r_{1}\\\
0&0&z^{-1}_{1}&0&r^{-1}_{1}&0\end{pmatrix}.$

note that $r,u,v,\ldots,z,r_{1},u_{1},v_{1},\ldots,z_{1}$ are the complex
variables and stand for the eigenvalues of the permutation matrices
corresponding to each edge. Using Theorem 2 we find sufficient conditions on
the signatures such that the corresponding lifts become cospectral.

###### Theorem 3

Let $s,s^{\prime}$ be $k$-Abelian lifts on the graphs $G,H$ respectively. If
the following situations hold, the graphs $G(s)$ and $H(s^{\prime})$ are
cospectral.

  * •

$\frac{w}{v}=\frac{y}{z}$

  * •

$2(\frac{xv}{w}+\frac{w}{xv})=\frac{y_{1}r_{1}}{z_{1}}+\frac{z_{1}}{y_{1}r_{1}}+\frac{u_{1}w_{1}}{v_{1}}+\frac{v_{1}}{u_{1}w_{1}}.$

###### Proof.

We consider all posibilities for the matrices
$A(G)_{\mathbf{x}},A(H)_{\mathbf{x}}$ corresponding to the graphs $G,H$.
Comparing the coefficients of
$\chi(A(G)_{\mathbf{x}},t),\chi(A(H)_{\mathbf{x}},t)$, the equality holds if
and only if

$2=\frac{wz}{vy}+\frac{vy}{wz},\hskip 56.9055pt(1)$
$\frac{xz}{y}+\frac{y}{xz}+\frac{xv}{w}+\frac{w}{xv}=\frac{y_{1}r_{1}}{z_{1}}+\frac{z_{1}}{y_{1}r_{1}}+\frac{u_{1}w_{1}}{v_{1}}+\frac{v_{1}}{u_{1}w_{1}}.\hskip
56.9055pt(2)$

The first equation follows by comparing the coefficients of $t^{2}$ and the
second one follows by comparing the coefficients of $t,t^{3}$. Consider the
first equality above, note the variables are the $n$’th roots of unity hence
the equality $(1)$ holds if and only if $wz=vy$, hence the first assertion of
the statement follows. The second assertion follows by the Equation (2) and
the first equality. Hence the graphs according to the mentioned signatures are
cospectral. $\Box$

###### Corollary 1

The following constraints on the variables
$r,u,v,\ldots,z,r_{1},u_{1},v_{1},\ldots,z_{1}$ will give cospectral lifts of
the graphs $G$ and $H$.

$z=\frac{vy}{w},\textrm{ }u_{1}=y_{1}=x,\textrm{ }w_{1}=r_{1}=v,\textrm{
}z_{1}=w$

###### Example 1

In the graphs $G$ and $H$ the following signatures have the condition stated
in Theorem 3. Hence the graphs $G(s)$ and $H(s^{\prime})$ are cospectral. The
group members are denoted in the cyclic representation.

$s(1,2)=(1,2,3),s(2,3)=(1,3,2),s(2,4)=\textrm{id},$
$s(3,4)=(1,3,2),s(3,5)=(1,3,2),s(4,5)=(1,2,3),s(5,6)=(1,2)$
$s^{\prime}(1,2)=(1,3,2),s^{\prime}(1,3)=\textrm{id},s^{\prime}(2,3)=(1,3,2),$
$s^{\prime}(3,4)=\textrm{id},s^{\prime}(3,5)=(1,3,2),s^{\prime}(3,6)=\textrm{id},s^{\prime}(5,6)=(1,3,2)$

The adjacency matrices of the graphs $G(s)$ and $H(s^{\prime})$ are of the
following forms.

$A(G(s))=\begin{pmatrix}0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0\\\
0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0\\\ 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0\\\
0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0\\\ 1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0\\\
0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0\\\ 0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0\\\
0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0\\\ 0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0\\\
0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0\\\ 0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0\\\
0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0\\\ 0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0\\\
0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0\\\ 0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1\\\
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0\\\ 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0\\\
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0\end{pmatrix},A(H(s^{\prime}))=\begin{pmatrix}0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0\\\
0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0\\\ 0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0\\\
0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0\\\ 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0\\\
1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0\\\ 0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,1\\\
0,0,1,1,0,0,0,0,0,0,1,0,0,1,1,0,0,0\\\ 1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0\\\
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0\\\ 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0\\\
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0\\\ 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1\\\
0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0\\\ 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0\\\
0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0\\\ 0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0\\\
0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0\end{pmatrix}.$

## References

  * [1] A. Abiad, A.E. Brouwer and W. H. Haemers, Godsil-McKay switching and isomorphism, Electronic Journal of Linear Algebra 28 (2015), 4–11.
  * [2] E. R. van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241–272.
  * [3] E. R. van Dam and W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Math., 309 (2009), 576 -586.
  * [4] N. Ghareghani, F. Ramezani, B. Tayfeh-Rezaei, Graphs cospectral to starlike trees, Linear Algebra Appl., 429 (2008), 2691 -2701.
  * [5] B. Mohar and B. Tayfeh-Rezaie, Median eigenvalues of bipartite graphs, J. Algebraic Combin. 41 (2015), 899–909.
  * [6] F. Ramezani, On the signed graphs with two distinct eigenvalues, arXiv:1511.03511v2.