  * November 2022

# Coherent backscattering in the topological Hall effect

Hong Liu1,2,3, Rhonald Burgos Atencia3,4,5, Nikhil Medhekar1,2 and Dimitrie
Culcer3,4 1 School of Physics and Astronomy, Monash University, Victoria 3800,
Australia  2 ARC Centre of Excellence in Future Low-Energy Electronics
Technologies, Monash University, Victoria 3800, Australia  3School of Physics,
The University of New South Wales, Sydney 2052, Australia  4ARC Centre of
Excellence in Low-Energy Electronics Technologies, UNSW Node, The University
of New South Wales, Sydney 2052, Australia  5 Facultad de Ingenierías,
Departamento de Ciencias Básicas, Universidad del Sinú, Cra.1w No.38-153,
4536534, Montería, Córdoba 230002, Colombia<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

The mutual interplay between electron transport and magnetism has attracted
considerable attention in recent years, primarily motivated by strategies to
manipulate magnetic degrees of freedom electrically, such as spin-orbit
torques and domain wall motion. Within this field the topological Hall effect,
which originates from scalar spin chirality, is an example of inter-band
quantum coherence induced by real-space inhomogeneous magnetic textures, and
its magnitude depends on the winding number and chiral spin features that
establish the total topological charge of the system. Remarkably, in the two
decades since its discovery, there has been no research on the quantum
correction to the topological Hall effect. Here we will show that, unlike the
ordinary Hall effect, the inhomogeneous magnetization arising from the spin
texture will give additional scattering terms in the kinetic equation, which
result in a quantum correction to the topological Hall resistivity. We focus
on 2D systems, where weak localization is strongest, and determine the
complicated gradient corrections to the Cooperon and kinetic equation. Whereas
the weak localization correction to the topological Hall effect is not large
in currently known materials, we show that it is experimentally observable in
dilute magnetic semiconductors. Our theoretical results will stimulate
experiments on the topological Hall effect and fill the theoretical knowledge
gap on weak localization corrections to transverse transport.

††: Materials for Quantum Technology

Keywords: Topological Hall effect, weak localization, inhomogenous
magnetization, chiral spin texture

##  1 Introduction

Classical information technology relies heavily on electron transport and
magnetism, with semiconductors and ferromagnets playing complementary roles in
information processing and storage [1]. Magnetism has been most successful in
high-density memory applications [2], such as hard disks [3, 4], while
considerable research is being devoted to switching magnetic memory components
by electrical currents via spin-orbit torques[5, 6, 7, 8, 9, 10, 11, 12]. The
latter effort has been galvanised by the rise of topological materials, where
spin-orbit interactions are often strong, and room-temperature magnetization
switching has been accomplished by means of an electrical current [13]. At the
same time the discovery of topological magnetic materials has generated
considerable interest in topologically protected spin textures, including
vortices, merons and skyrmions with potential application in next-generation
logic and memory devices [14, 15].

Electrical transport in media hosting skyrmions and chiral spin textures is
influenced by the exchange interaction between itinerant carriers and
localized spins forming such textures. A famous example of this interplay is
the topological Hall effect [16, 17, 18, 19, 20, 21, 22, 23, 24]. Unlike the
anomalous Hall effect, which relies on the spin-orbit interaction [9], the
topological Hall effect stems from the inhomogeneous spin texture that yields
a fictitious magnetic field, not requiring any spin–orbit coupling.
Stabilization and dynamics of different chiral spin textures depend strongly
on their topological properties [25, 26, 27]. The effect was first discussed
in two-dimensional diluted magnetic semiconductors [16] but has since grown
into a distinct research area in its own right. Then topological Hall effect
has been studied theoretically in references [28, 29, 30, 31, 32, 33, 34, 35,
36], where spin-flip scattering [32, 35] and the evolution from strong
exchange coupling to weak exchange coupling [37] are explicitly considered. To
date the topological Hall effect has been experimentally observed in a vast
range of materials [38, 39, 40, 41, 42, 43, 44], summarized in a recent review
[45]. They include ferromagnetic semiconductors, magnetic Kagome lattices,
transition-metal oxides, SrRuO3-based materials, transition metal compounds,
Heusler compounds, and magnetic topological insulators, among others [46, 47,
48, 49]. Topological spin textures that might produce topological Hall effect
include Bloch-type skyrmions induced by bulk Dzyaloshinskii–Moriya interaction
(DMI), Néel-type skyrmions mainly associated with interfacial DMI, scalar spin
chirality generally originated from geometrical frustration interaction, and
noncollinear magnetism and bubble caused by the competition of various
magnetic interactions including magnetocrystalline anisotropy, demagnetization
and exchange interaction [45].

Transport studies have overwhelmingly focused on the classical regime.
Theoretical studies to date have primarily relied on classical transport
theory ($\hbar/\epsilon_{\rm F}\tau\ll 1$ with $\epsilon_{\rm F}$ Fermi
energy, $\tau$ momentum relaxation time and $\hbar$ Planck constant).
Nevertheless, besides classical transport, quantum coherence modifies
electronic transport properties significantly. The weak localization
correction to the conductivity arises as a result of the quantum interference
between closed, time-reversed loops that circle regions in which one or more
impurities are present, and is noticeable when the electron mean free path is
much shorter than the phase coherence length. Weak localization is strongly
affected both by spin-orbit interactions in the band structure and by the form
of the impurity potential, as well as by external inhomogeneous magnetic
fields and by domain walls [50, 51, 52, 53]. In this context, the effect on
weak localization of inhomogeneous magnetic structures in real space has
received almost no attention, despite the inexorable rise of magnetic systems
with topological spin textures such as skyrmions. One notable exception is a
recent study of weak localization in longitudinal transport due to scattering
off chiral spin textures [54]. To date no studies have considered weak
localization in the Hall conductivity of topological spin textures, presumably
because the effect is expected to vanish in ordinary Hall transport [55],
though it is believed to play a part in the anomalous Hall effect [56]. The
fundamental scientific questions in this context are: What is the effect of
inhomogeneous topological magnetic textures on coherent backscattering? Can
there be a weak localization correction to Hall transport? The resolution of
these questions requires a complex kinetic equation, in which the transport
theory needs to be augmented by a series of non-trivial gradient corrections.

We seek to answer these fundamental questions in the present work by
investigating theoretically the weak localization correction to the
topological Hall effect in 2D systems with chiral spin textures. We derive a
generalized quantum kinetic equation that captures the effect of inhomogeneous
magnetic textures and provide a complete quantum kinetic framework for
describing the topological Hall effect including corrections up to the second
order in the spatial gradient. We show that the topological Hall effect is a
manifestation of inter-band quantum coherence induced by a real-space magnetic
texture [57], and accounting for the interplay between diagonal and off-
diagonal density matrix response is vital in order to capture the underlying
physics and determine the correct result both in the Born approximation and
for coherent backscattering. We restrict ourselves to scalar impurity
scattering, noting that spin-flip scattering can straightforwardly be included
along the lines of reference [54]. This however will only increase the
complexity of the theory by modifying the Cooperon quantitatively without
bringing about any qualitative changes in our results.

We concentrate on a simple model that captures inter-band coherence between
spin up and spin down bands, namely that of a two-dimensional electron gas
with an inhomogenous magnetization. Due to the interplay of magnetic
inhomogeneity and inter-band coherence we find a series of additional
scattering terms in the kinetic equation stemming from the gradient of the
magnetization, which give rise to a weak localization correction in the
transverse transport direction. Since the interference effects leading to weak
localization disappear in weak external magnetic fields, these corrections can
typically be identified straightforwardly in experiment, and are frequently
used to characterize samples, in particular transport in novel materials. They
provide valuable information about the system, such as symmetries of the
system, the phase coherence length, and the quasiparticle mass [58]. The weak
localization correction is somewhat smaller than the topological Hall effect
itself, but it is observable in systems with sizable topological Hall
resistivities, hence we expect our result to stimulate experiments.

Our paper is organized as follows. Section 2 presents a comprehensive
discussion of our method. We derive a general quantum kinetic equation that
accounts both homogeneous and inhomogenous magnetization in section 2.1. Then
the gradient correction to the scattering term arising from the inhomogenous
magnetization is shown in section 2.2. In sections 2.3 and 2.4, we solve the
quantum kinetic equation without gradient correction in scattering term, which
will reproduce the topological Hall effect in transverse direction and the
weak localization in the longitudinal direction. In section 2.5, we calculated
the weak localization correction to the topological Hall effect by taking the
additional gradient scattering term in the quantum kinetic equation. In
section 3, we provide a brief overview and physical explanation of our main
results, then apply our model to various skyrmion systems and discuss
experimental observation. In section 4 we summarize our findings and outlook.

##  2 Theoretical Formalism

###  2.1 Quantum kinetic equation

The total Hamiltonian of the system is

$\displaystyle H=H_{0}+U({\bm{r}})+H_{\bm{E}}.$ (1)

Here $H_{0}$ is the band Hamiltonian describing a two dimensional electron
system with an inhomogeneous magnetization. The band Hamiltonian can be
further decomposed into $H_{0}=h_{\bm{k}}+H_{\rm M}({\bm{r}})$, where the
kinetic energy $h_{\bm{k}}=\frac{\hbar^{2}k^{2}}{2m}$, with $m$ effective mass
and $\hbar$ the Planck constant. The spatially varying magnetization gives
rise to a Zeeman-like splitting $H_{\rm
M}({\bm{r}})={\bm{\sigma}}\cdot{\bm{M}}({\bm{r}})$, where $M$ has units of
energy. For simplicity, we assume the local magnetization
${\bm{M}}({\bm{r}})=M{\bm{n}}({\bm{r}})$, and that the 3D unit vector
${\bm{n}}({\bm{r}})=(\sin\theta\cos\phi_{r},\sin\theta\sin\phi_{r},\cos\theta)$
is a slowly varying function of the spatial coordinates, where
$\cos\theta=\epsilon_{k}/\sqrt{\epsilon^{2}_{k}+M^{2}}$ and
$\sin\theta=M/\sqrt{\epsilon^{2}_{k}+M^{2}}$. The phase $\phi_{r}$ is
dependent on location. The conduction electrons are considered to be non-
interacting, but are scattered by spin-independent impurities. The disorder
potential consists of uncorrelated short-range scalar impurities. The total
impurity potential is $U({\bm{r}})=\sum_{I}U({\bm{r}}-{\bm{R}}_{I})$, where
$\sum_{I}$ is a summation over all impurities and ${\bm{R}}_{I}$ is the
coordinate of impurity $I$. The reciprocal-space matrix element of a short-
range impurity potential matrix element is
$U_{{\bm{k}}{\bm{k}}^{\prime}}\equiv u$. The customary solution to dealing
with disorder is performing an average over all the possible impurity
configurations, whereupon the total impurity potential in reciprocal space
$U_{{\bm{k}}{\bm{k}}^{\prime}}$ becomes $\langle
U_{{\bm{k}}{\bm{k}}^{\prime}}\rangle=0$ and $\langle
U_{{\bm{k}}{\bm{k}}^{\prime}}U_{{\bm{k}}_{1}{\bm{k}}^{\prime}_{1}}\rangle=n_{\rm
i}u^{2}$. $n_{\rm i}$ represents the impurity concentration. $H_{\bm{E}}$ is
the external electric driving field.

Our discussion is based on the density matrix transport theory, where central
quantity is density matrix $\hat{\rho}$ in the quantum Liouville equation

$\displaystyle\frac{\partial\hat{\rho}}{{\rm d}t}+\frac{\rm
i}{\hbar}[H,\hat{\rho}]=0.$ (2)

We decompose the density matrix $\hat{\rho}=f+g$, where $f$ is disorder-
averaged density matrix and $g$ is the fluctuating part, which is integrated
over within a chosen approximation scheme that determines the effective
scattering term, discussed in detail below. We obtain the kinetic equation for
the disorder-averaged density matrix as [57]

$\displaystyle\frac{\partial f}{{\rm d}t}+[H,f]+J(f)=0,$ (3)

where $J(f)$ is the scattering term with impurities. $J(f)$ will be calculated
in section 2.2. Taking the inhomogenous magnetization into account in the
Hamiltonian, a rigorous definition of the distribution function follows from
the one-particle density matrix $f_{{\bm{r}}_{1}{\bm{r}}_{2}}(t)$. The aim of
this subsection is to derive the equation of motion for a semiclassical analog
of the quantum mechanical density matrix, which is provided by the Wigner
representation. The Wigner representation for the density matrix can be
expressed as

$\displaystyle f_{{\bm{k}},{\bm{r}}}=\int\frac{{\rm
d}{\bm{q}}}{(2\pi)^{d}}{\rm
e}^{i{\bm{q}}\cdot{\bm{r}}}f_{{\bm{k}}_{+}{\bm{k}}_{-}},$ (4)

where
$f_{{\bm{k}}_{1}{\bm{k}}_{2}}=f_{{\bm{k}}_{+}{\bm{k}}_{-}}\equiv\langle{\bm{k}}+{\bm{q}}/2|f|{\bm{k}}-{\bm{q}}/2\rangle$
with ${\bm{k}}=({\bm{k}}_{1}+{\bm{k}}_{2})/2$ and
${\bm{q}}={\bm{k}}_{1}-{\bm{k}}_{2}$. The inverse transformation of the
density matrix is

$\displaystyle f_{{\bm{k}},{\bm{r}}}=\int\frac{{\rm
d}{\bm{s}}}{(2\pi)^{d}}{\rm
e}^{-i{\bm{k}}\cdot{\bm{s}}}f_{{\bm{r}}_{+}{\bm{r}}_{-}},$ (5)

where
$f_{{\bm{r}}_{1}{\bm{r}}_{2}}=f_{{\bm{r}}_{+}{\bm{r}}_{-}}\equiv\langle{\bm{r}}+{\bm{s}}/2|f|{\bm{r}}-{\bm{s}}/2\rangle$
with ${\bm{r}}=({\bm{r}}_{1}+{\bm{r}}_{2})/2$ and
${\bm{s}}={\bm{r}}_{1}-{\bm{r}}_{2}$. $d$ is the dimension of the system.
Using equations (4) and (5), we transform the equation (3) into the quantum
kinetic equation that captures the physics of an inhomogeneous magnetization

$\displaystyle\frac{\partial f_{{\bm{k}},{\bm{r}}}}{\partial t}+\frac{{\rm
i}}{\hbar}[H_{\rm
M}({\bm{r}}),f_{{\bm{k}},{\bm{r}}}]+\frac{1}{2\hbar}\bigg{\\{}\frac{\partial
h_{\bm{k}}}{\partial{\bm{k}}}\nabla
f_{{\bm{k}},{\bm{r}}}\bigg{\\}}-\frac{1}{2\hbar}\bigg{\\{}\nabla H_{\rm
M}({\bm{r}})\frac{\partial
f_{{\bm{k}},{\bm{r}}}}{\partial{\bm{k}}}\bigg{\\}}+J(f_{{\bm{k}},{\bm{r}}})=\mathcal{D}_{\bm{E}},$
(6)

where $\\{{\bm{a}}{\bm{b}}\\}={\bm{a}}\cdot{\bm{b}}+{\bm{b}}\cdot{\bm{a}}$,
and the driving term due to the external electric field is

$\displaystyle\mathcal{D}_{\bm{E}}=\frac{e}{\hbar}{\bm{E}}\cdot\frac{\partial
f_{{\bm{k}},{\bm{r}}}}{\partial{\bm{k}}}.$ (7)

###  2.2 The scattering integral with gradient corrections

Figure 1: The diagrams labeled by (a) (b) (c) (d) are corresponding to the
scattering terms $J^{(1)}(f)$,$J^{(2)}(f)$,$J^{(3)}(f)$,$J^{(4)}(f)$ in
equations (9),(10),(11) and (12). The dashed red lines represent impurity
lines. Solid right and left arrow lines represent Green’s function $G^{\rm R}$
and $G^{\rm A}$, respectively. For simplicity, $G^{\rm R,A}(\epsilon)\equiv
G^{\rm R,A}$. Red dot with two legs represents averaged density matrix $f$.
The Cooperon operator $C$ is represented by orange rectangular box.

In section 2.1, we obtain the general form of the kinetic equation for Wigner
distribution function suitable for describing transport in inhomogeneous
systems. To calculate the weak localization correction to transport we require
the Wigner transformation of the scattering term $J(f)$ in equation (3) that
yields $J(f_{{\bm{k}},{\bm{r}}})$. The operator form of scattering term can be
expressed as

$\displaystyle J(f)=\frac{1}{2\pi\hbar}\int^{\infty}_{\infty}{\rm
d}\epsilon\langle[\hat{U},G^{\rm R}(\epsilon)[\hat{U},f]G^{\rm A}(\epsilon)],$
(8)

which is corresponding to four elements as follows

$\displaystyle J^{(1)}(f)=\frac{1}{2\pi\hbar}\int^{+\infty}_{-\infty}{\rm
d}\epsilon\langle\hat{U}G^{\rm R}(\epsilon)\hat{U}fG^{\rm
A}(\epsilon)\rangle,$ (9) $\displaystyle
J^{(2)}(f)=\frac{1}{2\pi\hbar}\int^{+\infty}_{-\infty}{\rm
d}\epsilon\langle\hat{U}G^{\rm R}(\epsilon)f\hat{U}G^{\rm
A}(\epsilon)\rangle,$ (10) $\displaystyle
J^{(3)}(f)=\frac{1}{2\pi\hbar}\int^{+\infty}_{-\infty}{\rm d}\epsilon\langle
G^{\rm R}(\epsilon)\hat{U}fG^{\rm A}(\epsilon)\hat{U}\rangle,$ (11)
$\displaystyle J^{(4)}(f)=\frac{1}{2\pi\hbar}\int^{+\infty}_{-\infty}{\rm
d}\epsilon\langle G^{\rm R}(\epsilon)f\hat{U}G^{\rm
A}(\epsilon)\hat{U}\rangle.$ (12)

$G^{\rm R,A}(\epsilon)$ are retarded and advanced Green’s function. The
$J^{(1)}(f)$ and $J^{(4)}(f)$ are complex conjugates and contribute to the
dressed Cooperon in the weak localization. $J^{(2)}(f)$ and $J^{(3)}(f)$ are
complex conjugate and contribute to the bare Cooperon in the weak
localization. The diagrams corresponding to the four scattering terms are
shown in figure 1. In the Born approximation, the disorder-averaged retarded
Green’s functions in the eigenstate basis for a 2D electron gas with an
inhomogeneous magnetization is

$\displaystyle G^{\rm
R}_{\bm{k}}(\epsilon)=\left(\begin{array}[]{cc}\frac{1}{\epsilon-\epsilon_{{\bm{k}},+}+\frac{{\rm
i}\hbar}{2\tau}}&0\\\ 0&\frac{1}{\epsilon-\epsilon_{{\bm{k}},-}+\frac{{\rm
i}\hbar}{2\tau}}\end{array}\right).$ (15)

When the spin polarization due to the interaction with the magnetization is
taken into account, the lifetime $\tau$ becomes spin-dependent. For the most
part of this paper, we neglect this spin-dependence, to avoid unnecessary
complication, but we will restore it at the end. To determine the real space
dependence of the Green’s function, we express the Green’s function in the
Pauli spin eigenstate basis as $G^{\rm
R}_{{\bm{k}},{\bm{r}}}(\epsilon)=TG^{\rm R}_{\bm{k}}(\epsilon)T^{\dagger}$
with with

$\displaystyle T=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}{\rm
e}^{-\frac{{\rm i}\phi_{r}}{2}}&-\sin\frac{\theta}{2}{\rm e}^{-\frac{{\rm
i}\phi_{r}}{2}}\\\ \sin\frac{\theta}{2}{\rm e}^{\frac{{\rm
i}\phi_{r}}{2}}&\cos\frac{\theta}{2}{\rm e}^{\frac{{\rm
i}\phi_{r}}{2}}\end{array}\right).$ (18)

For the short-range scalar impurity, the four constituents of the Born-
approximation scattering term in the Wigner representation are

$\displaystyle J^{(1)}(f_{{\bm{k}},{\bm{r}}})=\frac{2^{d}\times
2^{d}}{2\pi\hbar}\int^{\infty}_{\infty}{\rm d}\epsilon\int\frac{{\rm
d}{\bm{k}}_{1}}{(2\pi)^{d}}\int{\rm d}\Delta{\bm{r}}_{2}\int{\rm
d}\Delta{\bm{r}}_{3}\int\frac{{\rm
d}\Delta{\bm{k}}_{2}}{(2\pi)^{d}}\int\frac{{\rm
d}\Delta{\bm{k}}_{3}}{(2\pi)^{d}}{\rm e}^{-2{\rm
i}\Delta{\bm{k}}_{2}\cdot\Delta{\bm{r}}_{3}}$ (19) $\displaystyle\times{\rm
e}^{2{\rm i}\Delta{\bm{k}}_{3}\cdot\Delta{\bm{r}}_{2}}\langle
U_{{\bm{k}}{\bm{k}}_{1}}G^{\rm
R}_{{\bm{k}}_{1},{\bm{r}}+\Delta{\bm{r}}_{2}-\Delta{\bm{r}}_{3}}(\epsilon)U_{{\bm{k}}_{1}{\bm{k}}}f_{{\bm{k}}+\Delta{\bm{k}}_{2},{\bm{r}}+\Delta{\bm{r}}_{2}}G^{\rm
A}_{{\bm{k}}+\Delta{\bm{k}}_{3},{\bm{r}}+\Delta{\bm{r}}_{3}}(\epsilon)\rangle,$
(20) $\displaystyle
J^{(2)}(f_{{\bm{k}},{\bm{r}}})=\frac{2^{d}}{2\pi\hbar}\int^{\infty}_{\infty}{\rm
d}\epsilon\int\frac{{\rm d}{\bm{k}}_{1}}{(2\pi)^{d}}\int{\rm
d}\Delta{\bm{r}}_{2}\int\frac{{\rm d}\Delta{\bm{k}}_{3}}{(2\pi)^{d}}{\rm
e}^{2{\rm i}\Delta{\bm{k}}_{3}\cdot\Delta{\bm{r}}_{2}}\langle
U_{{\bm{k}}{\bm{k}}_{1}}G^{\rm
R}_{{\bm{k}}_{1},{\bm{r}}+\Delta{\bm{r}}_{2}}(\epsilon)$ (21)
$\displaystyle\times
f_{{\bm{k}}_{1}+\Delta{\bm{k}}_{3},{\bm{r}}+\Delta{\bm{r}}_{2}}U_{{\bm{k}}_{1}{\bm{k}}}G^{\rm
A}_{{\bm{k}}+\Delta{\bm{k}}_{3},{\bm{r}}}(\epsilon)\rangle,$ (22)
$\displaystyle
J^{(3)}(f_{{\bm{k}},{\bm{r}}})=\frac{2^{d}}{2\pi\hbar}\int^{\infty}_{\infty}{\rm
d}\epsilon\int\frac{{\rm d}{\bm{k}}_{3}}{(2\pi)^{d}}\int{\rm
d}\Delta{\bm{r}}_{2}\int\frac{{\rm d}\Delta{\bm{k}}_{1}}{(2\pi)^{d}}{\rm
e}^{-2{\rm i}\Delta{\bm{k}}_{1}\cdot\Delta{\bm{r}}_{2}}\langle G^{\rm
R}_{{\bm{k}}+\Delta{\bm{k}}_{1},{\bm{r}}}U_{{\bm{k}}{\bm{k}}_{3}}(\epsilon)$
(23) $\displaystyle\times
f_{{\bm{k}}_{3}+\Delta{\bm{k}}_{1},{\bm{r}}+\Delta{\bm{r}}_{2}}G^{\rm
A}_{{\bm{k}}_{3},{\bm{r}}+\Delta{\bm{r}}_{2}}(\epsilon)U_{{\bm{k}}_{3}{\bm{k}}}\rangle,$
(24) $\displaystyle J^{(4)}(f_{{\bm{k}},{\bm{r}}})=\frac{2^{d}\times
2^{d}}{2\pi\hbar}\int^{\infty}_{\infty}{\rm d}\epsilon\int\frac{{\rm
d}{\bm{k}}_{3}}{(2\pi)^{d}}\int{\rm d}\Delta{\bm{r}}_{1}\int{\rm
d}\Delta{\bm{r}}_{2}\int\frac{{\rm
d}\Delta{\bm{k}}_{1}}{(2\pi)^{d}}\int\frac{{\rm
d}\Delta{\bm{k}}_{2}}{(2\pi)^{d}}{\rm e}^{-2{\rm
i}\Delta{\bm{k}}_{1}\cdot\Delta{\bm{r}}_{2}}$ (25) $\displaystyle\times{\rm
e}^{2{\rm i}\Delta{\bm{k}}_{2}\cdot\Delta{\bm{r}}_{1}}\langle G^{\rm
R}_{{\bm{k}}+\Delta{\bm{k}}_{1},{\bm{r}}+\Delta{\bm{r}}_{1}}(\epsilon)f_{{\bm{k}}+\Delta{\bm{k}}_{2},{\bm{r}}+\Delta{\bm{r}}_{2}}U_{{\bm{k}}{\bm{k}}_{3}}G^{\rm
A}_{{\bm{k}}_{3},{\bm{r}}+\Delta{\bm{r}}_{2}-\Delta{\bm{r}}_{1}}(\epsilon)U_{{\bm{k}}_{3}{\bm{k}}}\rangle.$
(26)

The $2^{d}$ comes from Jacobian transformation in the integral. The density
matrix and Green’s function in equations above are $2\times 2$ matrices. The
gradient correction to the scattering then can be calculated by expanding the
Green’s function and density matrix in small quantities. Taking
$J^{(2)}(f_{{\bm{k}},{\bm{r}}})$ as an example [see figure 2(d)],

$\displaystyle G^{\rm
R}_{{\bm{k}}_{1},{\bm{r}}+\Delta{\bm{r}}_{2}}(\epsilon)\approx G^{\rm
R}_{{\bm{k}}_{1},{\bm{r}}}(\epsilon)+\Delta{\bm{r}}_{2}\frac{\partial G^{\rm
R}_{{\bm{k}}_{1},{\bm{r}}}(\epsilon)}{\partial\bm{r}},$ (27) $\displaystyle
f_{{\bm{k}}_{1}+\Delta{\bm{k}}_{3},{\bm{r}}+\Delta{\bm{r}}_{2}}\approx
f_{{\bm{k}}_{1},{\bm{r}}}+\Delta{\bm{r}}_{2}\frac{\partial
f_{{\bm{r}},{\bm{k}}_{1}}}{\partial\bm{r}}+\Delta{\bm{k}}_{3}\frac{\partial
f_{{\bm{k}}_{1},{\bm{r}}}}{\partial{\bm{k}}_{1}},$ (28) $\displaystyle G^{\rm
A}_{{\bm{k}}+\Delta{\bm{k}}_{3},{\bm{r}}}(\epsilon)\approx G^{\rm
A}_{{\bm{k}},{\bm{r}}}(\epsilon)+\Delta{\bm{k}}_{3}\frac{\partial G^{\rm
A}_{{\bm{k}},{\bm{r}}}(\epsilon)}{\partial\bm{k}},$ (29) $\displaystyle
C(\Delta{\bm{k}}_{3}\\!+\\!{\bm{k}},\Delta{\bm{r}}_{2}\\!+\\!{\bm{r}},{\bm{Q}})=C({\bm{Q}},{\bm{r}})\\!+\\!\Delta{\bm{k}}_{3}\frac{\partial
C({\bm{Q}},{\bm{r}})}{\partial\bm{k}}+\Delta{\bm{r}}_{2}\frac{\partial
C({\bm{Q}},{\bm{r}})}{\partial\bm{r}}.$ (30)

Then we can integrate over the small quantities using

$\displaystyle\int{\rm d}\Delta{\bm{r}}_{2}\int\frac{{\rm
d}\Delta{\bm{k}}_{3}}{(2\pi)^{d}}{\rm e}^{2{\rm
i}\Delta{\bm{k}}_{3}\cdot\Delta{\bm{r}}_{2}}\Delta{\bm{r}}_{2}\Delta{\bm{k}}_{3}=-\frac{1}{2{\rm
i}}\frac{1}{2^{d}}.$ (31)

At this stage we are ready to find the gradient correction to the Born-
approximation scattering term as well as the maximally crossed diagrams. In
the Pauli basis

$\displaystyle[J^{0,(2)}_{\rm
DB}(f_{{\bm{k}},{\bm{r}}})]^{\alpha\delta}=\frac{1}{2\pi\hbar}\sum_{\beta\gamma\rho\nu}\int{\rm
d}\epsilon\int\frac{{\rm d}{\bm{k}}_{1}}{(2\pi)^{d}}\langle
U^{{\bm{k}}{\bm{k}}_{1}}_{\alpha\beta}G^{\rm
R}_{\beta\gamma}({\bm{k}}_{1},{\bm{r}})f_{\gamma\rho}({\bm{k}}_{1},{\bm{r}})U^{{\bm{k}}_{1}{\bm{k}}}_{\rho\nu}G^{\rm
A}_{\nu\delta}({\bm{k}},{\bm{r}})\rangle,$ (32) $\displaystyle[J^{{\rm
g},(2)}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})]^{\alpha\delta}=-\frac{1}{2{\rm
i}}\frac{1}{2\pi\hbar}\sum_{\beta\gamma\rho\nu}\int{\rm
d}\epsilon\int\frac{{\rm d}{\bm{k}}_{1}}{(2\pi)^{d}}\Big{[}\langle
U^{{\bm{k}}{\bm{k}}_{1}}_{\alpha\beta}\frac{\partial G^{\rm
R}_{\beta\gamma}({\bm{k}}_{1},{\bm{r}})f_{\gamma\rho}({\bm{k}}_{1},{\bm{r}})}{\partial\bm{r}}$
(33) $\displaystyle\times U^{{\bm{k}}_{1}{\bm{k}}}_{\rho\nu}\frac{\partial
G^{\rm A}_{\nu\delta}({\bm{k}},{\bm{r}})}{\partial\bm{k}}\rangle+\langle
U^{{\bm{k}}{\bm{k}}_{1}}_{\alpha\beta}\frac{\partial G^{\rm
R}_{\beta\gamma}({\bm{k}}_{1},{\bm{r}})}{\partial\bm{r}}\frac{\partial
f_{\gamma\rho}({\bm{k}}_{1},{\bm{r}})}{\partial{\bm{k}}_{1}}U^{{\bm{k}}_{1}{\bm{k}}}_{\rho\nu}G^{\rm
A}_{\nu\delta}({\bm{k}},{\bm{r}})\rangle\Big{]},$ (34)
$\displaystyle[J^{0,(2)}_{\rm
WL}(f_{{\bm{k}},{\bm{r}}})]^{\alpha\delta}=\sum_{\beta\gamma\rho\nu}\frac{1}{2\pi\hbar}\int\\!\int\frac{{\rm
d}\epsilon{\rm d}{\bm{Q}}}{(2\pi)^{d}}G^{\rm
R}_{\beta\gamma}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})G^{\rm
A}_{\nu\delta}({\bm{k}},{\bm{r}}),$ (35) $\displaystyle[J^{{\rm g},(2)}_{\rm
WL}(f_{{\bm{k}},{\bm{r}}})]^{\alpha\delta}=-\frac{1}{2{\rm
i}}\sum_{\beta\gamma\rho\nu}\frac{1}{2\pi\hbar}\int{\rm
d}\epsilon\int\frac{{\rm d}{\bm{Q}}}{(2\pi)^{d}}$ (36)
$\displaystyle\times\Big{[}\frac{\partial G^{\rm
R}_{\beta\gamma}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})}{\partial{\bm{r}}}\big{[}\frac{\partial
C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})}{\partial\bm{Q}}f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})+C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})\frac{\partial
f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})}{\partial({\bm{Q}}\\!-\\!{\bm{k}})}\big{]}G^{A}_{\nu\delta}({\bm{k}},{\bm{r}})$
(37)
$\displaystyle+\frac{\partial}{\partial\bm{r}}\big{[}C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})G^{\rm
R}{\beta\gamma}({\bm{Q}}-{\bm{k}},{\bm{r}})f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})\big{]}\frac{\partial
G^{\rm A}_{\nu\delta}({\bm{k}},{\bm{r}})}{\partial{\bm{k}}}$ (38)
$\displaystyle+\big{[}\frac{\partial
C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})}{\partial\bm{r}}\frac{\partial
f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})}{\partial({\bm{Q}}\\!-\\!{\bm{k}})}\\!+\\!\frac{\partial
C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})}{\partial\bm{Q}}\frac{\partial
f_{\gamma\rho}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})}{\partial\bm{r}}\big{]}G^{\rm
R}_{\beta\gamma}({\bm{Q}}\\!-\\!{\bm{k}},{\bm{r}})G^{A}_{\nu\delta}({\bm{k}},{\bm{r}})\Big{]}.$
(39)

Here ${\bm{Q}}$ is the momentum change after a multiple scattering process
with disorder. $C({\bm{Q}},{\bm{r}})$ is the Cooperon, which is dependence
both on momenta ${\bm{Q}}$ and real space location ${\bm{r}}$. Summing over
all the terms in four scattering elements, we get
x$J(f_{{\bm{k}},{\bm{r}}})=J^{0}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})+J^{0}_{\rm
WL}(f_{{\bm{k}},{\bm{r}}})+J^{\rm g}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})+J^{\rm
g}_{\rm WL}(f_{{\bm{k}},{\bm{r}}})$. $J^{0}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})$
and $J^{0}_{\rm WL}(f_{{\bm{k}},{\bm{r}}})$ are is the Born-approximation and
weak localization correction to the scattering term without gradient
correction, respectively. $J^{\rm g}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})$ and
$J^{\rm g}_{\rm WL}(f_{{\bm{k}},{\bm{r}}})$ are the gradient correction to the
Born-approximation scattering and to the weak localization scattering term,
respectively.

Figure 2:  (a) Definition of the Green’s functions as arrowed solid lines in
which Greek letters are spin indices. (b) Definition of dashed lines:
impurities lines expressed in both retarded and advanced cases. (c) Definition
of the Wigner distribution function. (d) Indices label of
$J^{(2)}(f_{{\bm{k}},{\bm{r}}})$ as an example to calculate the gradient
correction in scattering term.

###  2.3 Recovering the topological Hall effect

In this section we solve Eq. (6) and demonstrate that the original topological
Hall effect formula of Ref. [16] can be reproduced by including the scattering
term $J^{0}_{\rm DB}(f_{{\bm{k}},{\bm{r}}})$. The calculation of the
topological Hall effect in the density matrix formalism is decomposed into
part for the charge density part and a part for the spin density. The solution
of equation (6) in equilibrium is separated into
$f_{0}({\bm{k}},{\bm{r}})=n_{0}({\bm{k}})+{S}_{0}({\bm{k}},{\bm{r}})$ with
$n_{0}({\bm{k}})=\frac{1}{2}(f_{0,+}+f_{0,-})$ the charge part and
${S}_{0}({\bm{k}},{\bm{r}})=\frac{1}{2}(f_{0,+}-f_{0,-}){\bm{\sigma}}\cdot{\bm{n}}({\bm{r}})$
the spin-part, while $f_{0,\pm}=f_{0}(\epsilon_{k}\pm M)$ is the Fermi-Dirac
distribution function for the conduction band and valence band resepctively.
The spin direction can be mapped within the frame
${\bm{n}}({\bm{r}}),\partial_{i}{\bm{n}}({\bm{r}}),\partial_{i}{\bm{n}}({\bm{r}})\times{\bm{n}}({\bm{r}})$,
where $\sigma_{\parallel}={\bm{\sigma}}\cdot{\bm{n}}({\bm{r}})$ and
$\sigma_{\perp}={\bm{\sigma}}\cdot\partial_{i}{\bm{n}}({\bm{r}})+{\bm{\sigma}}\cdot\partial_{i}{\bm{n}}({\bm{r}})\times{\bm{n}}({\bm{r}})$.
Then we can decompose the spin density matrix into parallel and perpendicular
parts with
${S}({\bm{k}},{\bm{r}})=S_{\parallel}({\bm{k}},{\bm{r}})+S_{\perp}({\bm{k}},{\bm{r}})$.
In matrix language, $S_{\parallel}({\bm{k}},{\bm{r}})$ is parallel to the
Hamiltonian $H_{\rm M}({\bm{r}})$ i.e. it commutes with the Hamiltonian, while
the remainder $S_{\perp}({\bm{k}},{\bm{r}})$ is orthogonal to the Hamiltonian
$H_{\rm M}({\bm{r}})$. Because $[H_{\rm
M}({\bm{r}}),S_{\parallel}({\bm{k}},{\bm{r}})]=0$, the parallel part
$S_{\parallel}({\bm{k}},{\bm{r}})$ does not change in time under the action of
the time evolution operator $e^{iH_{0}t/\hbar}$. Therefore
$S_{\parallel}({\bm{k}},{\bm{r}})$ represents the conserved spin fraction and
$S_{\perp}({\bm{k}},{\bm{r}})$ is the precessing spin fraction.

To find the density matrix out of equilibrium, we consider an electric field
${\bm{E}}$ applied in the $\hat{\bm{x}}$ and obtain the driving term. We solve
the kinetic equation in Pauli basis, the solution of the kinetic equation is
$f_{\bm{E}}({\bm{k}},{\bm{r}})=n_{\bm{E}}({\bm{k}},{\bm{r}})+S_{\bm{E}}({\bm{k}},{\bm{r}})$
can be found using the blueprint of reference [57].

$\displaystyle\begin{array}[]{rl}n_{\bm{E}}({\bm{k}},{\bm{r}})=&\displaystyle\frac{e{\bm{E}}\tau}{2\hbar}\cdot\frac{\partial}{\partial{\bm{k}}}(f_{0,+}+f_{0,-}),\\\\[12.91663pt]
S_{\bm{E}}({\bm{k}},{\bm{r}})=&\displaystyle\frac{e{\bm{E}}\tau}{2\hbar}\cdot\frac{\partial}{\partial{\bm{k}}}\Big{[}(f_{0,+}-f_{0,-}){\bm{\sigma}}\cdot{\bm{n}}({\bm{r}})\Big{]},\end{array}$
(42)

where $1/\tau=2\pi n_{\rm i}u^{2}\hbar/\rho(\epsilon_{\rm F})$ is the momentum
relaxation time and $\epsilon_{\rm F}$ is the Fermi energy.
$\rho(\epsilon_{\rm F})=m/(2\pi\hbar^{2})$ is the density of state for two
dimensional electron gas.

To calculate the topological Hall effect, we regard the gradient terms in the
quantum kinetic equation as perturbation, and fed
$f_{\bm{E}}({\bm{k}},{\bm{r}})$ into the kinetic equation to get the effective
driving term.

$\displaystyle\mathcal{D}^{(1)}_{\nabla}=-\frac{\hbar{\bm{k}}}{m}\cdot\nabla
f_{\bm{E}}({\bm{k}},{\bm{r}})+\frac{1}{2\hbar}\Big{\\{}\nabla H_{\rm
M}({\bm{r}})\frac{\partial
f_{\bm{E}}({\bm{k}},{\bm{r}})}{\partial\bm{k}}\Big{\\}},$ (43)

where we have
$\mathcal{D}^{(1)}_{\nabla}\propto\hat{k}_{i}{\bm{\sigma}}\cdot\partial_{i}{\bm{n}}({\bm{r}})$
which contributes to the perpendicular part of density matrix
$S_{\perp}({\bm{k}},{\bm{r}})$. Solving the kinetic equation as follows

$\displaystyle\frac{\partial S^{(1)}_{\nabla,\perp}}{\partial t}+\frac{\rm
i}{\hbar}[H_{\rm M}({\bm{r}}),S^{(1)}_{\nabla,\perp}]+J^{0}_{\rm
DB}(S^{(1)}_{\nabla,\perp})=\mathcal{D}^{(1)}_{\nabla},$ (44)

we will get the perpendicular spin density matrix in first-order gradient
written as $S^{(1)}_{\nabla,\perp}$, and the solution is

$\displaystyle S^{(1)}_{\nabla,\perp}=\lim_{\eta\rightarrow
0}\int^{\infty}_{0}dte^{-\eta
t}e^{-iH_{M}t/\hbar}\mathcal{D}^{(1)}_{\nabla}e^{iH_{M}t/\hbar}=\frac{\hbar}{2}\hat{k}_{i}{\bm{\sigma}}\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times{\bm{n}}({\bm{r}})]\frac{e{\bm{E}}\cdot\hat{\bm{k}}\tau}{\hbar^{2}k}\frac{\partial
f_{0}}{\partial k}.$ (45)

Similarly, the driving term for second-order gradient can be expressed as

$\displaystyle\mathcal{D}^{(2)}_{\nabla}=-\frac{\hbar{\bm{k}}}{m}\cdot\nabla
S^{(1)}_{\nabla,\perp}+\frac{1}{2\hbar}\Big{\\{}\nabla H_{\rm
M}({\bm{r}})\frac{\partial S^{(1)}_{\nabla,\perp}}{\partial\bm{k}}\Big{\\}}.$
(46)

With

$\displaystyle\hat{\bm{k}}\cdot\nabla\Big{[}{\bm{\sigma}}\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times{\bm{n}}({\bm{r}})]\Big{]}\approx\hat{k}_{j}{\bm{\sigma}}\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})],$
(47)
$\displaystyle\Big{\\{}\hat{\bm{k}}\cdot\nabla[{\bm{\sigma}}\cdot{\bm{n}}({\bm{r}})]\Big{[}{\bm{\sigma}}\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times{\bm{n}}({\bm{r}})]\Big{]}\Big{\\}}=2{\bm{n}}({\bm{r}})\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})],$
(48)

we have $\mathcal{D}^{(2)}_{\nabla}$ contributes to the charge density matrix
and parallel part of spin density matrix. Solving $J^{0}_{\rm
DB}(f^{(2)}_{\nabla})=\mathcal{D}^{(2)}_{\nabla}$, we get
$f^{(2)}_{\nabla}=n^{(2)}_{\nabla}+S^{(2)}_{\nabla,\parallel}$.

$\displaystyle
n^{(2)}_{\nabla}=M\hat{k}_{i}\hat{k}_{j}\frac{\partial}{\partial
k}\Big{[}\frac{e{\bm{E}}\cdot\hat{\bm{k}}\tau^{2}}{\hbar^{2}k}\Big{]}{\bm{n}}({\bm{r}})\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})],$
(49) $\displaystyle
S^{(2)}_{\nabla,\parallel}=\hat{k}_{i}\hat{k}_{j}{\bm{\sigma}}\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})]\Big{[}\frac{e{\bm{E}}\cdot\hat{\bm{k}}\tau^{2}}{2m}\frac{\partial
f_{0}}{\partial k}\Big{]}.$ (50)

The topological Hall current can be written as
$j^{y}=\Tr[\hat{v}_{y}n^{(2)}_{\nabla}]$, where $\hat{v}_{y}$ is the velocity
operator in the transverse direction. Then we have the topological Hall
conductivity as

$\displaystyle\begin{array}[]{rl}&\displaystyle\sigma^{\rm
T}_{yx}=-\frac{e^{2}}{h}\frac{e\tau}{m}\frac{M\tau}{\hbar}\frac{h}{4\pi
e}{\bm{n}}({\bm{r}})\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})],\end{array}$
(52)

where
$B_{t}=\frac{\phi_{0}}{4\pi}{\bm{n}}({\bm{r}})\cdot\big{[}\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})\big{]}$
and $\phi_{0}=\frac{hc}{e}$ is the magnetic flux quantum. The topological Hall
conductivity obtained here is in exact agreement with reference [16], where
the total $\sigma^{\rm T}_{yx}=\sum_{s}\sigma^{s}_{yx}$ with
$s=\uparrow,\downarrow$.

###  2.4 Weak localization correction to the longitudinal conductivity

The weak localization correction to the longitudinal current can be calculated
by including $J^{0}_{\rm WL}(f_{{\bm{k}},{\bm{r}}})$. In this paper, we
consider scalar disorder scattering, therefore the impurity line in Fig. 2(b)
is reduced to $n_{\rm i}u^{2}$. The calculation of Cooperon is standard. For
system with strong exchange interaction, the twisted Cooperon $\tilde{C}(Q)$
in the eigenstate basis takes the form

$\displaystyle\begin{array}[]{rl}&\displaystyle\tilde{C}(Q)\approx\frac{\hbar}{2\pi\tau\rho(\epsilon_{\rm
F})}\frac{2}{\tau^{2}v^{2}_{\rm F}Q^{2}}\left(\begin{array}[]{c c c
c}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\end{array}\right).\end{array}$ (58)

Solving the kinetic equation $J^{0}_{\rm DB}(f^{(0)}_{\rm WL})=-J^{0}_{\rm
WL}[f_{\bm{E}}({\bm{k}},{\bm{r}})]$, we find the correction to the density
matrix

$\displaystyle f^{(0)}_{\rm WL}=-\frac{2}{h}\sum_{\bm{Q}}\frac{1}{\tau
v^{2}_{\rm F}Q^{2}\rho(\epsilon_{\rm F})}f_{\bm{E}}({\bm{k}},{\bm{r}}).$ (59)

The weak localization correction to the longitudinal current is calculated as
$\delta j^{x}=\Tr[\hat{v}_{x}f^{(0)}_{\rm WL}]$. This yields the standard
result $\delta\sigma_{xx}=-\frac{e^{2}}{h\pi}\ln\frac{l_{\phi}}{l}$, where
$l_{\phi},l$ are the phase coherence length and mean free path respectively.

###  2.5 Weak localization correction to the topological Hall effect

In sections 2.3 and 2.4, we have solved the density matrix pertubatively
without the gradient scattering term. In this section, we calculate the
density matrix with the full $J(f_{{\bm{k}},{\bm{r}}})$. The usual weak
localization term in the Drude conductivity does not enter the gradient
correction due to an inhomogenous magnetization, while $J^{\rm g}_{\rm
DB}(f_{{\bm{k}},{\bm{r}}})$ and $J^{\rm g}_{\rm WL}(f_{{\bm{k}},{\bm{r}}})$
must be included. The kinetic equation with an additional gradient scattering
term will give rise to a weak localization correction of topological Hall
effect. Solving the kinetic equation to determine the density matrix
$f_{\nabla,{\rm WL}}$ with both gradient and weak localization corrections is
key to our approach. We write the kinetic equation explicitly as follows

$\displaystyle\frac{\partial f_{\nabla,{\rm WL}}}{\partial t}+\frac{\rm
i}{\hbar}[H_{\rm M}({\bm{r}}),f_{\nabla,{\rm WL}}]+J^{0}_{\rm
DB}(f_{\nabla,{\rm WL}})$ (60)
$\displaystyle=-\frac{\hbar{\bm{k}}}{m}\cdot\nabla f^{(0)}_{\rm
WL}+\frac{1}{2\hbar}\Big{\\{}\nabla H_{\rm M}({\bm{r}})\frac{\partial
f^{(0)}_{\rm WL}}{\partial\bm{k}}\Big{\\}}-J^{0}_{\rm
WL}(f^{(2)}_{\nabla})-J^{\rm g}_{\rm DB}(f^{(0)}_{\rm WL})-J^{\rm g}_{\rm
WL}[f_{\bm{E}}({\bm{k}},{\bm{r}})],$ (61)

We found that the contributions from $-J^{0}_{\rm WL}(f^{(2)}_{\nabla})$ and
$-\frac{\hbar{\bm{k}}}{m}\cdot\nabla f^{(0)}_{\rm
WL}+\frac{1}{2\hbar}\Big{\\{}\nabla H_{\rm M}({\bm{r}})\frac{\partial
f^{(0)}_{\rm WL}}{\partial\bm{k}}\Big{\\}}$ cancel each other. The
contribution from $J^{\rm g}_{\rm DB}(f^{(0)}_{\rm WL})=0$ in the system we
consider here. To calculate the contribution from $J^{\rm g}_{\rm
WL}[f_{\bm{E}}({\bm{k}},{\bm{r}})]$, we need to transform the twisted Cooperon
$\tilde{C}({\bm{Q}})$ to the Pauli basis using
$\tilde{C}({\bm{Q}},{\bm{r}})=T\otimes T\tilde{C}({\bm{Q}})T^{\dagger}\otimes
T^{\dagger}$ and exchange the spin indices to yield the Cooperon
$C({\bm{Q}},{\bm{r}})$. Feeding the Cooperon matrix element
$C^{\alpha\beta}_{\nu\rho}({\bm{Q}},{\bm{r}})$, the density matrix element
$f_{\bm{E}}({\bm{k}},{\bm{r}})$ as well as the Green’s function matrix element
$G^{\rm R,A}_{\alpha\beta}({\bm{k}},{\bm{r}})$ into $J^{\rm g}_{\rm
WL}[f_{\bm{E}}({\bm{k}},{\bm{r}})]$, we obtain the density matrix contribution
to first order in the gradient $f^{(1)}_{\nabla,{\rm WL}}$ as the solution of
$J^{0}_{\rm DB}(f^{(1)}_{\nabla,{\rm WL}})=-J^{\rm g}_{\rm
WL}[f_{\bm{E}}({\bm{k}},{\bm{r}})]$. To find the density matrix
$f^{(2)}_{\nabla,{\rm WL}}$ to second order in the gradient, we go through the
same process as that in the topological Hall effect calculation with

$\displaystyle\mathcal{D}^{(2)}_{\nabla,{\rm
WL}}=-\frac{\hbar{\bm{k}}}{m}\cdot\nabla f^{(1)}_{\nabla,{\rm
WL}}+\frac{1}{2\hbar}\Big{\\{}\nabla H_{\rm M}({\bm{r}})\frac{\partial
f^{(1)}_{\nabla,{\rm WL}}}{\partial\bm{k}}\Big{\\}}.$ (62)

Finally, we get the density matrix contributing to the charge current in the
transverse direction

$\displaystyle\qquad n^{(2)}_{\nabla,{\rm
WL}}=\sum_{\bm{Q}}\frac{4\pi}{M}\frac{1}{\rho(\epsilon_{\rm
F})}\frac{1}{\tau^{2}v^{2}_{\rm
F}Q^{2}}\frac{\partial^{2}n_{\bm{E}}({\bm{k}},{\bm{r}})}{\partial
k^{2}}\hat{k}_{i}\hat{k}_{j}{\bm{n}}({\bm{r}})\cdot[\partial_{i}{\bm{n}}({\bm{r}})\times\partial_{j}{\bm{n}}({\bm{r}})].$
(63)

After taking trace $\delta j^{y}=\Tr[\hat{v}_{y}n^{(2)}_{\nabla,{\rm WL}}]$,
we get the gradient correction to the topological Hall conductivity

$\displaystyle\delta\sigma^{\rm
g}_{yx}=\frac{e^{2}}{\hbar}\frac{1}{4\pi^{2}}\frac{\hbar}{\epsilon_{\rm
F}\tau}\frac{\hbar}{M\tau}\frac{e\tau B_{t}}{m}\ln\frac{l_{\phi}}{l}.$ (64)

##  3 Results and Discussion

The primary result of the paper is the weak localization correction in the
absence of an external magnetic field, which is equation (64). The averaged
component of the topological magnetic field perpendicular to the 2D plane is
given by $B_{t}\equiv Q\phi_{0}/\mathcal{A}$, where $Q$ is an integer number
and $\mathcal{A}$ is an area of the chiral spin texture which is also
proportional to the density of isolated chiral spin texture. The ration
between correction to topological Hall resistivity and topological Hall
resistivity is

$\displaystyle\frac{\delta\rho^{\rm g}_{yx}}{\rho^{\rm
T}_{yx}}=-\frac{1}{2\pi}\frac{\hbar}{\epsilon_{\rm
F}\tau}\bigg{(}\frac{\hbar}{M\tau}\bigg{)}^{2}\ln\frac{l_{\phi}}{l},$ (65)

which is around $10^{-3}$ order of topological Hall resistivity for systems
such as dilute magnetic semiconductor. For our theory to be applicable, we
require $\hbar/\epsilon_{\rm F}\tau<1$. The $<$ sign means the Fermi wave
vector is greater than the inverse of the mean free path, but not much
greater, so that disorder effects are observable. Conventionally, when one
restricts one’s attention to the weak momentum scattering regime where
$\hbar/\epsilon_{\rm F}\tau\ll 1$, yet, in this case, weak
localization/antilocalization corrections tend to be negligible. In this
paper, we consider the strong exchange interaction case with adiabatic
parameter $M\tau/\hbar>1$.

All electrical measurements of magnetic skyrmions have been based on the
interpretation that the topological Hall resistivity is proportional to the
number of skyrmions multiplied by the magnetic flux quantum [16]. Significant
progress has been achieved due to recent advances in imaging techniques that
have led to the discovery of different material platforms hosting skyrmions
with size ranging from sub 100 nm [59, 60, 61, 62] down to sub 10 nm [63, 64].
These sizes correspond to emergent magnetic fields ranging from 1 to 4000 T.
In the Berry phase picture, spin of conduction electrons must fully align to
the local magnetic moment during their movement, an adiabatic process is
needed, which is only satisfied for strong coupling. So far, most experimental
works on the topological Hall effect belong to this case. In weak coupling
case, especially in the diffusive regime, the condition of adiabatic picture
fails, and nonadiabatic contribution needs to be taken into account. Theory
predictions of nonadiabatic effect have been made [33, 37, 28], while
experimental confirmation is still lack[45]. In this paper, our theory has
been simplified and based on strong exchange interaction where the emergent
topological field is an averaged component perpendicular to the 2D plane, and
the dispersion of conduction electron is chosen to be quadratic. Therefore,
our theory is applicable to most current topological Hall experiments.

With this in mind, it is important to note the principal difference in the
topological Hall effect behavior for dense and dilute skyrmion systems. The
first case corresponds to the dense skyrmionic system with the distance
between the skyrmions $d_{\rm sk}$ of the same order as their size $a$. The
second case is a dilute system with skyrmions randomly distributed with large
distances between each other, so that $d_{\rm sk}\ll a$. A consistent theory
of the topological Hall effect has to take into account the evolution of a
skyrmion arrangement upon changing the external conditions. Especially, the
topological Hall resistivity is usually measured under the external magnetic
field. The skymion size decreases with increasing external magnetic field. The
skyrmion sheet density, in principle, can depend on the magnetic field thus
leading to the transition from dense to dilute regime [60, 34, 59]. Skyrmions
size in ferromagnetic layer composition Ir/Fe/Co/Pt multi-layers and their
sheet density can be varied over a wide range by changing the composition of
the structure and by tuning the external parameters [34, 62]. Smallest
skyrmion of few-nm size was detected in a strong magnetic field in a PdFe
bilayer on Ir(111) surface using spin-polarized scanning tunneling microscopy.
Fe/Ir(111)-PdFe/Ir(111) became the model systems for study of different
regimes of magnetic ordering from single skyrmions to skyrmion lattices. The
considered theory of weak localization correction to the topological Hall
effect in a dilute array of chiral spin textures is valid when the mean free
path $l$ is larger than skymions size $a$. We have considered the dilute
regime, when the scattering rate on spin textures is much smaller than that on
nonmagnetic impurities $\tau_{s}=\tau$. Neglecting thermal effects [65], we
assume that the scattering on nonmagnetic impurities gives the dominating
contribution to the transport scattering time and determines the mean free
path.

For instance, scattering off nonmagnetic impurities gives the dominant
contribution to the transport scattering time of nanoscale skyrmions observed
in atomically thin magnetic layers, such as PdFe/Ir(111) [66]. In thin
ferromagnetic films PdFe/Ir, the conduction electrons in the magnetic film
interact with skyrmions as well as with nonmagnetic impurities. The estimated
parameters in the Tr/Fe/Co/Pt multilayers are $M=0.6$ $eV$ and $\epsilon_{\rm
F}=5$ $eV$. With effective in-plane mass $m=m_{0}$. If we estimate the
momentum relaxation time $\tau=a/v_{\rm F}$ with $a=50$ nm, we have
$M\tau/\hbar\approx 30$. In addition, the large Fermi energy in the system
limit the regime in weak momentum scattering $\hbar/\epsilon_{\rm F}\tau\ll
1$, therefore the weak localization correction to the topological Hall
resistivity in this system is negligible.

Taking dilute magnetic semiconductor as example, a 2D layer containing
randomly distributed magnetic skyrmions with sheet density $n_{\rm sk}$ is
much lower than the concentration of background nonmagnetic impurities. In the
dilute regime of low skyrmion surface density considered, the total transport
scattering time is independent of the carrier spin, being determined by
scattering on host nonmagnetic impurities[35]. The substantial advantage of
dilute magnetic semiconductors is that both the Fermi energy and the exchange
interaction strength can be tuned, allowing us to control the adiabatic
parameter in a wide range covering the weak coupling and adiabatic regimes of
topological Hall effect. For electrons, the exchange interaction $M=2$ $meV$,
the concentration $n=10^{11}$ ${\rm cm}^{-2}$, $m=0.22$ $m_{0}$, the momentum
relaxation time $\tau=1$ ps, we can get the adiabatic parameter
$M\tau/\hbar=3$ and the momentum scattering regime $\hbar/\epsilon_{\rm
F}\tau=0.6$. The condition of adiabaticity is fulfilled. Here we plot the
carrier density and band splitting dependencies of ${\rm
Ratio}=\delta\rho^{\rm g}_{yx}/\rho^{\rm T}_{yx}$ in figure 3(a) and (b). The
ratio between the weak localization correction to the topological Hall
resistivity and topological Hall resistivity is around $10^{-3}$, but up to
$10^{-2}$ for small $M=1$ $meV$ in diluted magnetic semiconductors. In
addition, typical measured value of the giant topological Hall resistivity in
FeGe thin films is 80 n$\Omega$ cm to 160 n$\Omega$ cm [40, 67]. Therefore,
the weak localization correction to topological Hall resistivity is expected
to be observable in these systems.

Figure 3: (a) The carrier density dependence of ${\rm Ratio}=\delta\rho^{\rm
g}_{yx}/\rho^{\rm T}_{yx}$ with $M=2$meV. (b) The magnetization magnitude $M$
dependence of ${\rm Ratio}=\delta\rho^{\rm g}_{yx}/\rho^{\rm T}_{yx}$ with
$n=10^{12}{\rm cm}^{-2}$. Both (a) and (b) take the momentum relaxation time
is $\tau=1$ ps, effective electron mass $m=0.22m_{0}$. We have set
$l_{\phi}/l=5,10$ in agreement with what is measured in experiment.

In addition to the above, a large topological Hall effect occurs in single
layer SrRuO3 [68] and interface-driven SrRuO3-SrIrO3 superlattice [69]. The
maximum of the topological Hall resistivity in these materials can be 0.4
$\mu\Omega{\rm cm}$. Topological insulators interfaced with a magnetic
insulators have also reported a topological Hall resistivity around 0.2
$\mu\Omega{\rm cm}$ [70], with 1.5 $\mu\Omega{\rm cm}$ in CrTe2/Bi2Te3 [44].
In addition, in the 2D antiferromagnetic Topological Insulator MnBi4Te7, the
topological Hall resistivity reaches a giant value of 7 $\mu\Omega{\rm cm}$
[71]. The Fermi level and band gap can be tuned for MnBi4Te7 by defect
engineering and chemical doping in experiment [72]. In light of the above we
expect an observable value for the weak localization correction to the
topological Hall resistivity in experimental measurements of the 2D
antiferromagnetic topological insulator MnBi4Te7.

In contrast to the dilute arrays of spin textures mentioned above, the theory
is also applicable to skyrmion crystals, which are characterized by a higher
density of magnetic vortices $n_{\rm sk}a^{2}<1$ with $n_{\rm sk}$ the
skymions density. This assumption is typical when considering electron
transport in real skyrmion crystals. An electron spin state is co-aligned
everywhere with a local magnetization and skyrmions do not lead to spin-flip
scattering, satisfying the strong exchange interaction we consider in the
paper. Nevertheless, further experimental data on these materials is needed
before a reliable estimate of weak localization in Hall transport can be
produced.

##  4 Conclusion

We have demonstrated that a quantum correction to the topological Hall effect
existst in general and we have provided a comprehensive theoretical blueprint
for evaluating it by developing a full theory of the quantum interference-
induced corrections to transverse charge transport. Our theory is applicable
to a number of systems that host inhomogenous chiral spin textures, and we
have estimated the correction for the different materials with large
topological Hall resistivity. The theoretical results discussed in this paper
will stimulate experiments on topological Hall effect measurement and fill the
knowledge gap on the weak localization correction to the transverse transport.
Further attention must be awarded to distinguishing the topological Hall
effects from other effects that may be present in experiment. From an
experimental point of view it is still challenging to differentiate the
topological contribution from anomalous Hall effect as along with the
topological Hall effect the formation of magnetic skyrmions also reduces an
average magnetization and, hence, the anomalous Hall effect [73].

## Acknowledgement

This work is supported by the Australian Research Council Centre of Excellence
in Future Low-Energy Electronics Technologies, project number CE170100039.

## Data availability statement

All data that support the findings of this study are included within the
article (and any supplementary files).

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