#  Spectral features of magnetic domain walls on surface of 3D topological
insulators.

I. P. Rusinov<EMAIL_ADDRESS>Tomsk State University, Tomsk, 634050
Russia  St. Petersburg State University, 199034 St. Petersburg, Russia  V. N.
Men’shov  NRC Kurchatov Institute, Kurchatov Sqr. 1, 123182 Moscow, Russia
Tomsk State University, Tomsk, 634050 Russia  St. Petersburg State University,
199034 St. Petersburg, Russia  E. V. Chulkov  St. Petersburg State University,
199034 St. Petersburg, Russia  Donostia International Physics Center (DIPC),
20018 San Sebastián/Donostia, Spain  Departamento de Polímeros y Materiales
Avanzados: Física, Química y Tecnología, Facultad de Ciencias Químicas,
Universidad del País Vasco UPV/EHU, 20080 San Sebastián/Donostia, Basque
Country, Spain

(August 27, 2024)

###### Abstract

We present a theoretical investigation of electron states hosted by magnetic
domain walls on the 3D topological insulator surface. The consideration
includes the domain walls with distinct vectorial and spatial textures. The
study is carried out on the basis of the Hamiltonian for quasi-relativistic
fermions by using a continual approach and tight-binding calculations. We
derive the spectral characteristics and spatial localization of the the one-
dimensional low-energy states appearing at the domain walls. The antiphase
domain walls are shown to generate the topologically protected chiral states
with linear dispersion, the group velocity and spin-polarization direction of
which depend on an easy axis orientation. In the case of an easy plane
anisotropy, we predict a realization of a dispersionless state, flat band in
the energy spectrum, that is spin-polarized along the surface normal.
Modification of the surface states in the multi-domain case, which is
approximated by a periodic set of domain walls, is described as well. We find
that the magnetic domain walls with complex internal texture, such as Néel-
like or Bloch-like walls, also host the topological states, although their
spectrum and spin structure can be changed compared with the sharp wall case.

magnetic topological insulators, band topology

##  I INTRODUCTION

The effect of quantized conductivity without an external magnetic field was
revealed and studied in topological insulators (TIs) that are time-reversal
invariant semiconductors with strong spin-orbit coupling [1, 2, 3]. However,
namely the breaking of time-reversal symmetry (TRS) through the introduction
of a magnetic order in TIs provides conducive ground for an emergence of
fascinating phenomena such as quantum anomalous Hall effect (QAHE) [4, 5, 6,
7, 8], axion insulator state [8], and Majorana fermions [9], which would allow
extending potential of spintronic applications. At present, there are several
viable approaches for creating a magnetic order in TIs on the basis of
tetradymite-like semiconductors [10]. Accordingly, five alternative platforms
suitable for realization of phenomena associated with quantized transverse
conductivity can be outlined as follows. 1) QAHE was first detected in thin
films composed of a few quintuple-layers (QLs) of TIs Crx(Bi,Sb)2-xTe3 [11]
and Vx(Bi,Sb)2-xTe3 [12], where randomly dissolved moments of transition metal
atoms form ferromagnetic (FM) long-range order. 2) By using magnetic
modulation doping, when the rich-Cr/V-doped thin layers are inserted near both
the surfaces of (Bi,Sb)2Te3 films, experimentalists have succeeded in
observing QAHE [13, 14, 15] and axion insulator state [16, 17, 15]. 3) The
magnetic-proximity-effect-induced QAHE has been implemented in the
(Zn,Cr)Te/(Bi,Sb)2Te3/(Zn,Cr)Te heterostructure due to a fine tuning of the
composition of FM insulator interfaced with TI [18]. 4) The giant exchange gap
at the Dirac point of the surface state could be achieved due to a magnetic
extension effect, when a thin layer of FM insulator is deposited on the
surface of a nonmagnetic TI, which are both structurally and compositionally
compatible with each other [19, 20]. Perhaps this effect is responsible for
anomalous Hall regime at temperatures of several Kelvin observed in the
magnetic topological bulk crystals in which Mn ions self-organize into a
periodic MnBi2Te4/Bi2Te3 superlattice [21]. 5) Recently, the existence of
quantized Hall conductivity, accompanied by zero longitudinal resistance, was
experimentally demonstrated in the thin flakes of an intrinsic
antiferromagnetic (AFM) TI MnBi2Te4 with odd numbers of septuple-layers (SLs),
concretely 5 SLs, under zero extrinsic magnetic field [22]. This is a typical
behavior of QAHE, theoretically predicted for intrinsic AFM TI in Ref. 23. In
turn, the flakes of MnBi2Te4 with even number of SLs, namely 6 SLs, have
exhibited axion insulator state at zero magnetic field [24].

Despite rapid progress in this field, the exploration of magnetic TIs with in-
plane easy axis [25, 26, 27] or with magnetic moment canted towards the
surface [28, 29] remains scarce. The recent study [30] has predicted the in-
plane sublattice magnetization in vanadium-based family of AFM TIs
V(Bi,Sb)2(Se,Te)4, which could significantly broaden the base for the search
of exotic fermion states at the TI surface.

It should be stressed that, in the magnetic TI samples prepared for spin-
dependent transport measurements, the physical boundaries such as surfaces,
interfaces and side faces play a particular role [8, 10]. Conceptually, given
that the surface/interface states are fully gapped at the Dirac point due to
the spontaneous magnetization normal to the surface, the topologically
protected edge chiral channels running along the side faces are responsible
for the quantized Hall conductivity [31, 5, 6, 7, 8, 10]. In reality, the
magnetization distribution at the surface can be highly distinct from the bulk
magnetic order. It is thought that this distinction can be caused by various
factors. One of them are thermal magnetization fluctuations which, as a rule,
are enhanced in the surface region of a magnetic [32]. At the same time, the
magnetic fluctuations are frozen by static symmetry-reducing imperfections
[33]. The structural defects and compositional disorder inevitably occur
during the exfoliation or the epitaxial growth of the TI samples [34, 35, 36].
For instance, the surface of an exfoliated MnBi2Te4 flake can even be
subjected to strong chemical reconstruction, so that the topmost SL becomes
rather as a Mn-doped Bi2Te3 and a MnxBiyTe double layer with a clear van der
Waals gap in between [37]. These and other symmetry-reducing imperfections can
dominate locally the exchange and dipolar interactions between moments as well
as magnetic anisotropy at the surface. In turn, an interplay between the
exchange coupling and anisotropy drives nucleation and energetic stability of
the magnetic landscape at the terminating surface including complex space-
varying textures of magnetization. It is also important to emphasize that, in
tetradymite TI materials, the topologically protected surface states are
mostly localized inside a few terminating QLs/SLs, i.e., roughly speaking,
inside the same region where the magnetic order is sensitive to the
perturbations and therefore can be remarkably modified compared with the bulk
magnetic order.

The existence of magnetic DWs on the surface of magnetic TIs and the 1D
electron states associated with them is experimentally confirmed [36, 38, 39].
Using magnetic force microscopy, Sass and coauthors presented microscopic
evidence of the DWs on the as-grown (0001)-surface of MnBi2Te4, which is
consistent with opposite surface magnetizations of antiphase domains or
terraces separated by SL steps [36]. The emergence of multidomain states at
topological phase transitions under external field sweep was found in various
magnetic systems based on TIs [38, 40, 17, 39, 15, 41, 42]. The inconsistency
of spectroscopy (ARPES) results [43, 44, 45, 46] with the earlier ones [47,
48, 29, 49] indicates the possibility of a spatially inhomogeneous structure
of the surface magnetization in MnBi2Te4. In addition, the magnetic DWs on the
AFM TI surface could also be induced intentionally using a magnetic force
microscope tip [50] or by spatially modulated external magnetic field due to
Meissner repulsion from a bulk superconductor, as it has been realized in Cr-
doped TI (Bi,Sb)2Te3 [51]. On the theoretical side, the issue on particular
quasiparticle states on the magnetic DWs of TIs was raised previously in Refs.
52, 53, 54, 55, 56, 30.

In this work, we study how the magnetic inhomogeneities such as DWs can modify
a picture of the topological surface states. The DW spatial textures are
modelled as static one-dimensional boundaries between magnetic domains of
different orientations on the TI surface. We discuss the properties of the DW
induced fermion states in the context of the real space localization and the
spin-resolved spectral function. We demonstrate that such DWs host
topologically protected chiral fermion states, the energy spectrum of which
can evolve from a linear dispersion to a flat band depending on the easy axis
orientation. The results are obtained by means of two complementary methods,
one based on the effective low-energy Dirac equation description and the other
based on a numerical tight-binding approach.

The rest of the paper is organized as follows. In Sec. II we present the
microscopic model of DWs on the magnetic TI surface and outline methodological
details of the description of the electron states bound to DWs. In Sec. III,
we successively investigate the characteristics of the low-energy surface
states generated by a single magnetic antiphase DWs under different directions
of easy axis, a pair of the antiphase DWs, and the multi-domain periodic
configurations of magnetization. Here we also study how the noncollinear and
noncoplanar textures of the magnetization modify the spectral properties of
the surface states. Finally, in Sec. IV, we summarize the obtained results and
discuss possible manifestations of DWs in spectroscopy and transport
properties of TIs.

##  II MODEL AND METHODS

Figure 1:  Energy dispersion in the momentum space $(k_{x},k_{y})$ for
topological surface state subjected to the uniform surface magnetization
oriented along axes the $z$-axis (left panel), the $x$-axis (central panel)
and the $y$-axis (right panel).

The essential low-energy physics on the 3D TI surface can be described by
massless two-component fermions with the spin-momentum locking [1, 2, 3]. In
the presence of the exchange field at the surface, the motion of the fermions
can be modelled with effective 2D Hamiltonian:

$H(\mathbf{k})=-\nu[\mathbf{k}\times\boldsymbol{\sigma}]_{z}+J(\mathbf{M}\cdot\boldsymbol{\sigma}).$
(1)

Here, the first term is the lowest-order expansion of the Dirac quasiparticle
energy in the small in-plane momentum $\mathbf{k}=(k_{x},k_{y})$ around the
$\overline{\Gamma}$-point in the surface Brillouin zone (BZ), $\nu$ is the
Fermi velocity (one assumes $\hbar=1$), $\boldsymbol{\sigma}$ is trio of Pauli
matrices $\sigma_{i}$ ($i=x,y,z$) for the spin degree of freedom. In the
tetradymite TI systems, both in the case of randomly distributed _3d_
transition metal dopants and in the case of regularly aligned layers of
magnetic ions, the exchange coupling is mediated by _p_ -orbitals of the TI
host. It can lead to appearance of a magnetic ordering and hence a surface
magnetization $\mathbf{M}(x,y)$. The latter causes a spin polarization of the
the Dirac fermions owing to an effective interaction $J$. We present the
corresponding exchange energy by the second term in Eq. (1) that allows us to
capture the DW fingerprints in the spectral properties of the surface
electrons in any of the five platforms for magnetic ordering in TI listed
above. It should be noted that the interaction strength $J$ depends profoundly
on strategy of the formation magnetic order in the system. We also suppose
that the interaction $J$ is the same in the in-plane and out-of-plane
direction (for certainty $J>0$) and ignore the particle–hole asymmetry in the
surface bands. We suggest that the surface states modelled with Eq. (1) reside
in the projected bulk band gap. It should be noted, we do not include possible
hexagonal warping term [57, 58], which could be avoided at small momentum near
the BZ center.

If the surface magnetization is uniform, the energy spectrum of the
Hamiltonian (1) is given by the relation $E^{2}=(\nu k_{x}-JM_{y})^{2}+(\nu
k_{y}+JM_{x})^{2}+J^{2}M^{2}_{z}$. Thus, when the vector $\mathbf{M}$ is
oriented in the surface plane, the 2D electron states show a gapless cone
spectrum with the Dirac point shifted from the BZ origin in the direction
perpendicular to $\mathbf{M}$. In the case of out-of-plane easy axis, the
surface states are gapped. The corresponding 2D spectral relations for the
three orientations of the uniform surface magnetization are illustrated in
Fig. 1. However, for topological surface states, an interplay of the band
topology and magnetic ordering is not reduced to these spectral features.
Indeed, the topological indices of the surface states of a magnetic TI might
be closely linked with the sign of either the exchange energy gap or exchange
shift of the Dirac cone in 2D BZ, depending on the magnetic easy axis
direction. In the other words, the fermions moving on the magnetically
inhomogeneous surface can acquire the spatially varying topological indices.
Thus, DW, across which the surface magnetization changes its sign, is also a
border separating domains with different topological indices. Thereby, such a
border is expected to host peculiar topologically protected 1D state.

As emphasized above, despite the presence of magnetic order in the TI bulk,
the terminating surface can display rather considerable local variations of
the magnetization $\mathbf{M}(x,y)$, including DWs with various textures. The
magnetic DWs of certain types can bind the Dirac fermions. In order to
elucidate this issue, we address a particular class of magnetic configurations
in which the $\mathbf{M}$, magnitude is fixed, $|\mathbf{M}(x,y)|=M_{0}$,
while the vector $\mathbf{M}(x,y)/M_{0}=\mathbf{m}(x,y)$ is a function of the
position at the surface plane. Furthermore, we restrict our study to a 1D
model in the sense that the magnetization undergoes large-scale modulation in
only one spatial direction, for definiteness
$\mathbf{m}(x,y)\to\mathbf{m}(x)$, while retaining lattice periodicity in the
orthogonal direction and, consequently, conserving momentum $k_{y}$. We
consider several representative orientation configurations of the surface
magnetization. We do not analyze here the energy of these configurations, but
make use of them as a playground for the exploration of substantive
characteristics of the topological surface states. At first, we will address
isolated antiphase DWs of zero width such as
$\mathbf{m}(x)=\mathbf{m}_{0}\operatorname{sgn}(x)$, which may differ in the
$\mathbf{m}_{0}$-direction with respect to the crystallographic axes of the
system. If the distance between the neighboring DWs is not too large, the DW
induced bound states overlap and hybridize. This aspect is explored by using
configurations of a pair of DWs and a 1D periodic domain lattice. We also
consider non-collinear magnetic DWs at the TI surface and study the bound
states appearing at the isolated 180∘ Néel-type and Bloch-type DWs, which
profiles are simulated by piece wise unit vectors $\mathbf{m}$.

Our $k$-linear model approach, Eq. (1), has the advantage that it allows us to
readily explain main trends in modification of the topological surface states
caused by perturbations of magnetization that are not easily accounted within
_ab initio_ simulations. We employ both the continuum model analysis and the
tight-binding calculations. The former is used to find explicit low-energy
solutions for the Dirac-like equation. The latter are based on a lattice
regularization of the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian (1) via the
substitution $k_{x,y}\to(1/a)\sin(k_{x,y}a)$ where $a$ is 2D square lattice
constant. The tight-binding simulation is featured by the dimensionless
parameter $JM_{0}a/\nu$. To avoid an appearance of false Dirac points at the
boundary of BZ, known as the double-fermion problem [59], we include the
Willson mass term $\frac{2w\sigma_{z}}{a^{2}}[2-\cos(k_{x}a)-\cos(k_{y}a)]$,
which does not significantly affect the surface state behavior at small
momenta. The tight-binding calculations are performed using recursive
technique for the Green functions [60, 61]. Having obtained the retarded Green
function ($G^{R}(E,k_{y})$) for a given magnetization distribution, we can
analyze further the corresponding momentum-resolved one-particle spectral
function $\rho(E,k_{y})=(-1/\pi)\imaginary\Tr G^{R}(E,k_{y})$ and that for
each spin polarization
$S_{i}(E,k_{y})=(-1/\pi)\imaginary\Tr\sigma_{i}G^{R}(E,k_{y})$. Furthermore,
we can obtain the total density of states (DOS)
$\rho(E)=(2\pi/a)\int_{-\pi/a}^{\pi/a}dk_{y}\rho(E,k_{y})$, and the net spin
polarization $S_{i}(E)=(2\pi/a)\int_{-\pi/a}^{\pi/a}dk_{y}S_{i}(E,k_{y})$ as
function of the state energy, $E$. The spatial distribution of the surface
electron state is evaluated by the local density of states
$\rho(E,x,k_{y})=(-1/\pi)\imaginary[E+i\epsilon-H(x,k_{y})]^{-1}$ at energy
$E$, where $i\epsilon$ is a small broadening width. The results of the tight-
binding numerical simulations are presented in Figs. 2-9.

##  III THE MAIN RESULTS

###  III.1 Single antiphase domain wall

Figure 2:  Bound electron states induced by single sharp antiphase DW on the
magnetic TI surface for increasing angle $\theta$. Panels (a,b), (c,d), (e,f),
(g,h) correspond to $\theta=0$, $\pi/6$, $\pi/3$, $\pi/2$, respectively.
Projected topological surface bands associated with left and right magnetic
domains are in blue and green, respectively. In the panel (a) these bands
coincide. Dependence $E(k_{y})$ for the DW induced bound state obtained in
terms of spectral function is in red. Total DOS, $\rho(E)$, and the spin
polarization, $S_{i}(E)$, are shown as well. The top of the panels (a), (c),
(e), (g) contains the schematic image of magnetic texture of antiphase DW
indicated by arrows. Figure 3:  Spatial profiles of the probability density of
the planar DW bound state for distinct momenta $k_{y}$ indicated with small
color squares on the dispersion curves in Fig. 2 (g).

We begin by considering the electron state bound to a single collinear DW,
when the two its sides differ in sign of the magnetization. For definiteness,
DW is assumed to extend along the $y$-direction being centered at $x=0$. The
spatial configuration of such an antiphase DW is merely given by a step
function profile,
$\mathbf{m}(x,y)=\mathbf{m}(x)=\mathbf{m}_{0}\operatorname{sgn}(x)$, i.e., the
DW magnetization changes sharply its direction to just the opposite one under
crossing over the boundary $(x=0,y)$, while the magnetization within the
domains stays uniform. We describe the unit vector
$\mathbf{m}_{0}=(\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$
with spherical coordinates, where the polar angle $\theta$ and azimuthal angle
$\phi$ set the magnetization orientation with respect to the $z$-axis and in
the surface plane $(x,y)$, respectively. The vector $\mathbf{m}_{0}$ is
typically aligned with the magnetic easy axis. The infinitely thin DW
approximation is justified, provided that the DW width is significantly
shorter than the localization scale of the electron state.

The system homogeneity along the $y$-direction implies that the Bloch
excitation with wave vector $k_{y}$ propagates along DW. Therefore, upon
replacement $k_{x}\to-i\partial_{x}$, the problem is reduced to the 1D real-
space Dirac equation $H(x,k_{y})\Theta(x,k_{y})=E(k_{y})\Theta(x,k_{y})$ for
the bispinor envelope function
$\Theta(x,k_{y})=(\varphi(x,k_{y}),\chi(x,k_{y}))^{\mathrm{T}}$, which must
decay into the domain regions, i.e. $\Theta(|x|\to\infty,k_{y})=0$. On the
other hand, the function $\Theta(x,k_{y})$ satisfies the boundary conditions
that ensure its continuity at $x=0$ and the derivative jump which reads

$i\sigma_{y}[\partial_{x}\Theta(x,k_{y})\rvert_{0^{+}}-\partial_{x}\Theta(x,k_{y})\rvert_{0^{-}}]=\\\
=2k_{0}(\mathbf{m}_{0}\cdot\boldsymbol{\sigma})\Theta(0,k_{y}),$ (2)

where $k_{0}=JM_{0}/\nu$ is the characteristic momentum.

As a prime example of the eigen-problem we address the case when the
magnetization is tilted in the $(x,z)$ plane by angle $\theta$ and the $m_{y}$
component is absent, i.e. $\sin(\phi)=0$. The <<tail-to-tail>> DW, the two
sides of which differ by angle $\pi$ (for definiteness $0\leq\theta\leq\pi/2$)
harbors the bound state featured by the envelope function

$\Theta(x,k_{y})=\Theta_{0}\left(\begin{array}[]{c}\alpha\\\
1\end{array}\right)\sum_{\pm}h(\pm x)\exp(\mp p_{\pm}x)$ (3)

and the energy dispersion

$E(k_{y})=\nu k_{y}\cos(\theta).$ (4)

Here $h(x)$ is the Heviside function, the momenta $p_{\pm}=k_{0}\pm
k_{y}\sin(\theta)$ determine the localization length of the bound state,
$\Theta_{0}$ is a normalization constant, $\alpha=\tan((\pi/2-\theta)/2)$. The
spectral branch (4) exists within the momentum interval
$|k_{y}\sin(\theta))|<k_{0}$, where $p_{\pm}>0$, in other words, the 1D
linearly dispersive mode (4) meets the 2D Dirac cones when
$|k_{y}\sin(\theta))|=k_{0}$.

Thus, we find that at the TI surface there is only one chiral bound state per
single magnetic DW, propagating along DW with group velocity
$\nu_{*}=\nu\cos(\theta)$. The properties of the DW induced state (3)–(4)
alter as one changes the tilt angle $\theta$. When the domain magnetization is
parallel to the surface normal, the emergent state is massless linearly
dispersing and completely spin-polarized along the $x$-axis fermion with
energy $E=\nu k_{y}$ and spinor construction $\alpha=1$. On the contrary, when
the vector $\mathbf{m}$ is parallel to the $x$-axis, a heavy fermion with
$E=0$ and spin polarization along the $z$-axis ($\alpha=0$) appears. In any
case, i.e. at arbitrary angle $\theta$, one can see that the spin polarization
of the bound state (3)–(4) is exactly perpendicular to both the DW propagation
direction and the domain magnetization,
$(\mathbf{m}_{0}\cdot\langle\boldsymbol{\sigma}\rangle)=0$. In this sense,
besides spin-momentum coupling, the fermion state possesses an peculiar sort
of chirality when the fermion spin localized at DW is tightly locked to the
surface magnetization. The density of states (DOS) for the solution (3)–(4) is
given by

$\rho(E)=\frac{a}{2\pi\nu\cos(\theta)}h(\nu k_{0}\cot(\theta)-|E|).$ (5)

The DOS (5) is transformed from the constant value $\rho(E)=a/(2\pi\nu)$ for
$\theta=0$ to the utterly narrow peak $\rho(E)=(ak_{0}/\pi)\delta(E)$ for
$\theta=\pi/2$, where $\delta(E)$ is delta-function.

We have also numerically verified the existence and peculiarities of the DW
induced bound states using the lattice approximation. In the tight-binding
calculations, the model dimensionless parameter is chosen as $JM_{0}a/\nu=0.1$
only for convenience, in order to facilitate the visualization of the results.
One can easily verify that the same qualitative results would be observed for
any value of this parameter. In Fig. 2, we plot representative pictures of the
spectral dependencies, $E(k_{y})$, the total DOS, $\rho(E)$, and the spin
polarization, $S_{i}(E)$, for four selected values of the angle $\theta$;
besides, the corresponding DW configurations are schematically depicted. What
we are most interested in is the low-lying states in the middle of the
projected 1D BZ $|k_{y}|<\pi/a$ with the momenta that are small compared to
momentum of the order of the reciprocal lattice length. First, the energy
spectrum of these states comprises pronounced projections of the two Dirac
cones, originated from the corresponding magnetic domains, separated by
momentum $2k_{0}\sin(\theta)$ and gapped by energy $2JM_{0}\cos(\theta)$.
Second, and most important, there is a 1D massless linearly dispersing in-gap
chiral fermion branch (at $\cos(\theta)\neq 0$) associated with the DW induced
bound state. The properties of the emergent DW states controlled by the polar
angle are drastically different in the cases when the easy axis is normal to
the surface or lies in its plane. As seen in Fig. 2 (a,b), at
$\cos(\theta)=1$, the fully spin-polarized along the $x$-axis dispersionless
mode with constant DOS spans the exchange gap of the size $2JM_{0}$ in the
Dirac cone. Away from the angle $\theta=0$, the state gradually changes its
character: the group velocity, $\nu_{*}=\nu\cos(\theta)$, of the bound state
reduces linearly with the $m_{z}$-component, the spin-polarization deviates
from the $x$-axis in the $(x,z)$ plane, and DOS as a function of energy
concentrates around $E=0$. Particular cases are demonstrated in Fig. 2 for
$\theta=\pi/6$ (c,d panels) and for $\theta=\pi/3$ (e,f panels). Eventually,
at $\cos(\theta)=0$, one observes in Fig. 2(g) a perfectly flat band with
energy $E=0$ that bridges the two the Dirac cones with nodes at momenta $\pm
k_{0}$. The corresponding heavy fermion state, fully spin-polarized along the
$z$-axis, manifests in a dramatic enhancement of DOS (Fig. 2 (h). In Fig. 3 we
depict the local density profile of the flat band state for three momenta
$k_{y}$.

Thus, combining analytic model and tight-binding numerical simulations in the
long wave-length limit we consistently demonstrate that the antiphase DW at
any polar orientation $\theta$ supports the in-gap chiral degenerate bound
state. The chirality consists in locking of three vectors that are mutually
orthogonal to each other, that is to say they constitute the orthogonal trio:
the domain polarization $\mathbf{m}_{0}$ (oriented as a rule along a favoring
axis of a magnetocrystalline anisotropy in the surface plane), the state spin
polarization $\langle\boldsymbol{\sigma}\rangle$, and the propagation
direction, i.e. the wave-vector $\mathbf{y}k_{y}$. The inversion of the DW
magnetization $\mathbf{m}(x)$, such as <<tail-to-
tail>>$\leftrightarrow$<<head-to-head>>, entails the alteration of the vectors
$\langle\boldsymbol{\sigma}\rangle$ and $\mathbf{y}k_{y}$ to just the opposite
ones. In other words, the fermions moving on the <<tail-to-tail>> and <<head-
to-head>> DWs have the opposite chiralities.

The spectral and spin features of the DW induced state are highly sensitive to
the orientation of $\mathbf{m}_{0}$ with respect to the surface normal.
Notably also, the existence of the eigen localized solution (3)–(4) is
guaranteed by the antiphase DW texture, which allows us to consider such a
magnetic defect as a topological boundary and, consequently, the emergent in-
gap quasiparticle state as topologically protected one. When the magnetization
points perpendicular to the TI surface, an exchange interaction induced
magnetization opens up the gap of the size $2JM_{0}$. At the boundary between
up and down out-of-plane magnetic domains, according to the known arguments
[62, 63], a chiral linearly dispersive state appears [Fig. 2 (a)] because of
the inversion of the gap from $+2JM_{0}$ to $-2JM_{0}$. To provide a
compelling evidence that the DW induced state (3)–(4) has a topological origin
at any polar angle we use the reasoning similar to that given in Ref. [30,
64]. In the general case, which is addressed here, the role of the <<effective
gap>> could be played by the momentum dependent energy parameter
$\Delta(x,k_{y})=2\nu[p_{+}h(x)-p_{-}h(-x)]=2[JM_{0}\operatorname{sgn}(x)+\nu
k_{y}\sin(\theta)]$. Provided that $k_{y}$ is restricted to the realm of the
existence of the bound state (3)–(4) in the 1D BZ,
$|k_{y}\sin(\theta)|<k_{0}$, the parameter $\Delta(x,k_{y})$ changes its sign
just at the magnetic DW. On the contrary, if $|k_{y}\sin(\theta)|>k_{0}$, the
parameter $\Delta(x,k_{y})$ has the same sign on both sides of DW. Thus, the
1D gapless surface bound state (3)–(4) appears due to the <<gap>>
$\Delta(x,k_{y})$ closing at DW, which is guaranteed by the texture of the
antiphase DW at any polar orientation $\theta$. In the partial case of the
easy plane, $\theta=\pi/2$, the linearly dispersing branch of the surface
spectrum, Fig. 2 (a,c,e), transforms into the flat-band branch with energy
$E=0$ connecting the Dirac points at $k_{y}=\pm k_{0}$, Fig. 2 (g). It should
be acknowledged that the existence of a 2D dispersionless state, so-called
<<heavy fermion>>, was predicted by Volkov and Pankratov in the case of a
supersymmetric DW in a ferroelectric semiconductor A4B6 [65].

When the surface magnetization remains in the $(y,z)$ plane, the antiphase DW
featured by the vector $\mathbf{m}_{0}=(0,\sin(\theta),\cos(\theta))$ hosts
massless linearly dispersing chiral fermion for any angle
$0\leq\theta\leq\pi/2$. In according to the boundary conditions of Eq. (2), it
is described by the envelope function
$\Theta(x,k_{y})\sim\exp(-k_{0}e^{i\theta}|x|)$ and spectrum $E(k_{y})=\nu
k_{y}$ and is fully-spin-polarized along the $x$-axis. In the case of the
magnetization is oriented along DW, i.e. $\theta=\pi/2$ and $\varphi=\pi/2$,
the bound state disappears.

Figure 4:  Bound electron states induced by a pair of single sharp antiphase
DWs on the magnetic TI surface for out-of-plane easy axis. Dependence
$E(k_{y})$ for the induced bound state obtained in terms of the spectral
function is present in panels (a,b) in red for two different distances between
DWs, $L=50a$ and $L=10a$. Total DOS, $\rho(E)$, and the spin polarization,
$S_{i}(E)$, are shown in panel (c). The top of panels (a) and (b) contains the
schematic image of out-of-plane antiphase magnetic texture, where the domain
magnetizations are indicated by arrows. Figure 5:  The same as Fig. 4, but for
the case of the in-plane easy axis.

###  III.2  The flat band state for a single domain wall with in-plane
magnetization

Now we turn to consideration of the magnetic TI surface where domains are
oriented in the basic plane due to preferable easy-plane anisotropy. In order
to understand what happens with the surface state we set the single azimuthal
DW that separates two regions with differing magnetization orientations
specified by angles $\phi_{+}$ and $\phi_{-}$. The magnetization spatial
profile reads:
$\mathbf{m}(x)=(\cos(\phi_{+}),\sin(\phi_{+}),0)h(x)+(\cos(\phi_{-}),\sin(\phi_{-}),0)h(-x)$.
We find readily the DW induced state pinned to energy $E=0$ and express its
envelope function as

$\displaystyle\begin{split}\Theta(x)&=\left(\begin{array}[]{c}\varphi_{0}\\\
0\end{array}\right)\sum_{\pm}h(\pm x)\exp(q_{\pm}x),\\\ &\quad-
k_{0}\cos(\phi_{-})<k_{y}<-k_{0}\cos(\phi_{+});\\\
\Theta(x)&=\left(\begin{array}[]{c}0\\\
\chi_{0}\end{array}\right)\sum_{\pm}h(\pm x)\exp(-q^{*}_{\pm}x),\\\ &\quad-
k_{0}\cos(\phi_{+})<k_{y}<-k_{0}\cos(\phi_{-})\\\ \end{split}$ (6)

where the momenta $q_{\pm}=k_{y}+k_{0}\exp(i\phi_{\pm})$ determine the
localization length of the bound state, $\varphi_{0}$ and $\chi_{0}$ are
normalization constants. From here we conclude that the azimuthal DW generates
the dispersionless bound state with perfect spin polarization aligned
perpendicular to the surface plane unless only the $M_{x}$ components
coincide. The realm of the flat band connecting the projections of the two
Dirac cones in the 1D BZ is determined by angles $\phi_{+}$ and $\phi_{-}$.
The condition $\phi_{+}=-\phi_{-}$ or $\phi_{+},\phi_{-}=\pm\pi/2$ means
disappearing of the bound state. On the contrary, in the case of the antiphase
DW, when either $\phi_{+}=0$, $\phi_{-}=\pi$ or $\phi_{+}=\pi$, $\phi_{-}=0$,
the state occupies the widest possible realm in the momentum space,
$|k_{y}|<k_{0}$, Fig. 2 (g). The DOS for the flat band state of Eq. (6) is
given by
$\rho(E)=[ak_{0}/(2\pi)]\left|\cos(\phi_{+})-\cos(\phi_{-})\right|\delta(E)$.
Our consideration demonstrates that the 1D flat band state is robust even when
the orientations of the magnetic domains are rotated with respect to each
other in the basic plane.

###  III.3 Pair of antiphase domain walls

If the distance between two DWs is not too large, the DW induced bound states
can overlap and hybridize, effectively leading to a deformation of their
spectra. To analyze this modification we describe the surface magnetization
distribution as composed of three different regions divided by the sharp DWs
fixed at $x=\pm L/2$. In the case of the out-of-plane anisotropy, the up-
magnetized stripe domain of a finite width $L$ is placed between the two half-
infinite down-magnetized domains, which can be represented as
$\mathbf{m}(x)=(0,0,-1)h(|x|-L/2)+(0,0,1)h(L/2-|x|)$. When the overlap of the
tails of the states originating from the DWs is small, two 1D linear spectral
branches appear within the exchange gap $2JM_{0}$, which are slightly gapped
$\sim JM_{0}\exp(-k_{0}L)$ at $k_{y}=0$ due to the states hybridization, Fig.
4 (a). If the DWs are relatively close to each other, $Lk_{0}\ll 1$, the large
overlap leads to formation of a pair of weakly localized states with
parabolic-like bands (Fig. 4 (b)), whose edges are near the the exchange gap
edges, $JM_{0}-|E|\sim JM_{0}(k_{0}L)^{2}$ at $k_{y}=0$. The DOS and spin
polarization of these states are depicted in Fig. 4 (c).

In the case of the in-plane anisotropy, the magnetization configuration
includes the <<tail-to-tail>> and <<head-to-head>> DWs. For the sake of
definiteness, we assume that the side domain regions magnetized along the
$x$-axis are separated by the intermediate region with the opposite
magnetization direction, which is described by
$\mathbf{m}(x)=(-1,0,0)h(|x|-L/2)+(1,0,0)h(L/2-|x|)$. In the spectral picture
of Fig. 5 (a) one can see that, at large enough width $L$, one of the cones,
associated with the middle domain, is weakly pronounced, while the heavy
fermion spectral branch acquires weak dispersion given by
$E(k_{y})\approx\sqrt{2JM_{0}\nu(k_{0}-k_{y})}\exp(-(k_{0}-k_{y})L)$. When
$Lk_{0}\ll 1$, both this cone and the bound state disappear, Fig. 5 (b). The
respective DOS and spin polarization are depicted in Fig. 5 (c). Our analysis
shows that the DW induced flat band state is highly sensitive to the DW-to-DW
distance.

###  III.4 Periodical array of domain walls

Figure 6:  Electron states induced by periodical array of sharp antiphase DWs
on the magnetic TI surface with the out-of-plane easy axis. The spin-resolved
spectral function is calculated for texture with the period $2L$, where
$L=30a$. The amplitude and sign of the spin polarization $S_{x}(E)$ of the
energy states $E(k_{y})$ are coded in red-blue color scheme. The top of the
panel contains the schematic image of the corresponding magnetic texture,
where the domain magnetizations are indicated by arrows. Total DOS, $\rho(E)$,
is shown as well. Figure 7:  Electron states induced by periodical array of
sharp antiphase DWs on the magnetic TI surface for the in-plane easy axis. The
calculated spin-resolved spectral function is shown for the texture with
period $2L$, where $L=100a$, the panel (a), and $L=30a$ in the panel (c). In
the latter case, the amplitude and sign of the spin polarization
$S_{x}(E,k_{y})$ of the energy states $E(k_{y})$ are coded in red-blue color
scheme. The top of the panels (a) and (c) contains the schematic image of the
corresponding magnetic texture, where the domain magnetizations are indicated
by arrows. Total DOSs, $\rho(E)$, are shown as well. The panels (b) and (d)
present spatial profiles of the probability density of the states with
distinct momenta $k_{y}$ indicated with small squares of the corresponding
color on the dispersion curves in panels (a) and (c), respectively.

In low-temperature phase, the surface of an as-grown magnetic material sample
should host a set of domains with different magnetization directions [33, 66,
67]. Here the question arises on how such a multi-domain magnetic texture, in
particular the domain length-scale, affects the electronic and spin structure
of the TI surface states. To figure that out, we address the 1D periodically
modulated collinear textures of the surface magnetization which consist of
alternating magnetization stripes aligned in the opposite directions:
$\mathbf{m}(x)=\mathbf{m}_{0}\operatorname{sgn}\left[\sin(\pi x/L)\right]$,
where $2L$ is the texture period. The domain stripes have finite width along
the $x$-axis and infinite length along the $y$-axis and are separated from one
another by sharp antiphase DWs. The net surface magnetization is supposed to
be vanished. The domain polarization $\mathbf{m}_{0}$ is fixed to an easy
axis. To be more specific, we study the modification of the topological
surface states under the magnetic texture $\mathbf{m}(x)$ for two cases: the
out-of-plane anisotropy, when $\mathbf{m}_{0}=(0,0,1)$, and the in-plane
anisotropy, when $\mathbf{m}_{0}=(1,0,0)$.

The tight-binding calculations are performed using an enlarged unit cell,
which is the stripe of width $2L=Na$ in the $x$-direction, where $N$ is the
number of sites per the unit cell. The latter contains two anti-aligned
domains. As we have already clarified, due to the local variation of exchange
field, an isolated antiphase DW supports the bound state. As the magnetization
forms a multi-domain texture, the surface states stemming from the neighboring
DWs hybridize. Hence, it is natural to expect that the surface Dirac-like
spectrum rebuilds owing to the formation of the Bloch-like subbands
$E_{n}(k_{x},k_{y})$ in 2D BZ restricted to the realm $|k_{y}|<\pi/a$,
$|k_{x}|<\pi/(2L)$.

In the case of the $z$-easy axis, the 2D Dirac surface states are gapped out
by the domain magnetizations pointing outwards and inwards from the sample
surface, whereas the single DW produces the 1D in-gap state. As the DWs are
arranged regularly, the low-energy spectrum can be expressed in the form
$E^{\pm}_{0}(k_{x},k_{y})=\pm\sqrt{\nu^{2}k_{y}^{2}+\xi^{2}(k_{x})+\Delta^{2}}$.
In the regime where the typical localization length $k^{-1}_{0}$ is smaller
than the inter-DW distance, the dispersion in $k_{x}$ appears due to the weak
overlap of the envelope function tails, $\xi(k_{x})\sim\nu
k_{0}\exp(-k_{0}L)\sin(k_{x}L)$, the tiny hybridization gap $2\Delta$ between
the branches $E^{+}(k_{x},k_{y})$ and $E^{-}(k_{x},k_{y})$ is of the next
order of magnitude in the overlap. The situation when the inter-DW distance is
comparable to the typical localization length $k^{-1}_{0}$ is illustrated by
the spectral picture in Fig. 6 for $L=30a$. Moreover, at the BZ boundary
$|k_{x}|=\pi/(2L)$, the remarkable gap opens up in the spectral density of the
propagating states about energy $|E|\approx 0.16\nu/a$ at $k_{y}\approx 0$.
The $x$-component of spin polarization $S_{x}(E,k_{y})$ is depicted in Fig. 6
by means of color intensity of the spectral branches. This image tells about
the spin-momentum locking of the surface states, excluding the narrow realm in
the vicinity $k_{y}\approx 0$.

More complicated situation arises in the case of the magnetization lying in
the surface plane. When the adjacent DWs lie at quite some distance from one
another, $L\gg k^{-1}_{0}$ (Fig. 7 (a) for $L=100a$), the flatness and spin
polarization of the low-energy states near $E=0$ changes very little as
compared with the single-DW case. There occur the dispersion
$\xi(k_{x})\sim\nu k_{0}\exp(-k_{0}L)\sin(k_{x}L)$ along $k_{x}$ and the tiny
gap $2\Delta$ between the branches $E_{0}^{+}(k_{x},|k_{y}|<k_{0})$ and
$E_{0}^{-}(k_{x},|k_{y}|<k_{0})$. As follows from Fig. 7 (b), the spatial
profile of the probability density of the low-energy state at $k_{y}\approx 0$
is significantly different from that at $|k_{y}|\approx k_{0}$. With regard to
the delocalized states, as seen in Fig. 7 (a) for the spectral density
projected onto $k_{y}$, the continual cone spectrum is split into a set of
subbands, $E_{n}(|k_{y}|=k_{0})\approx\pi\nu n/(2L)$, $n=1,2,3...$. As the
inter-DW distance is decreased to intermediate value $L\sim k_{0}^{-1}$, we
find that the low-energy spectrum loses the flat behavior becoming parabolic
along momentum $k_{y}$ which can be seen in Fig. 7 (c) for $L=30a$.
Interestingly, while the bands at zero energy disappear, two new flat-like
bands at the finite energy $|E|\approx\pm 0.13\nu/a$ emerge due to a
rebuilding of the cone states in the exchange field of the periodic texture
with $L=30a$, Fig. 7 (c). Correspondingly, two sharp peaks are present in the
full DOS. One also notes that the component $S_{x}(E,k_{y})$ reduces near
$k_{y}\approx 0$, where the electron spin density is polarized normal to the
surface. The real space behavior of some states with different eigen-values is
shown in Fig. 7 (d).

Our analysis shows that the spectral characteristics of the low-energy states
in the case of the $z$-easy axis are generally stable in relation to varying
inter-DW distance. On the contrary, in the case of the in-plane anisotropy,
the dispersionless states bound to DW are highly sensitive to change of the
distance between neighboring DWs. When $L\lesssim k_{0}^{-1}$, these states
are interfering with each other shifting the flat band level from zero energy.
As a result, the spectrum acquires the curvature and the gap. Moreover, the
drastic redistribution of DOS of the 2D cone states takes place under the
magnetization spatial modulation.

###  III.5  Noncollinear and noncoplanar domain walls

Figure 8:  Bound electron states induced by the Néel-like texture on the
magnetic TI surface with the easy in-plane anisotropy. The spin-resolved
spectral function is calculated for the texture with the middle region width
$2l$, where $2l=50a$ in the panel (a) and $2l=20a$ in the panel (b). The top
of the panels (a) and (c) contains the schematic image of the corresponding
magnetic texture, where the right and left side regions are in green and blue,
respectively, the middle region is in brown, the arrows indicate the moments
directions. The projected surface band structure associated with the left and
right sides of the texture is in blue and green, respectively. The
dependencies $E(k_{y})$ for the DW induced bound states are associated with
red curves. The total DOS, $\rho(E)$, and the spin polarization, $S_{z}(E)$,
are shown as well. Figure 9:  The same as Fig. 8, but for the case of polar
Néel-like texture.

Above, we have considered the antiphase DW described by the collinear
magnetization texture $\mathbf{m}(x)=\mathbf{m}_{0}\operatorname{sgn}(x)$.
Such a sharp profile is believed to be a reasonable approximation if the DW
size is much smaller than the fermion localization length $\sim k_{0}^{-1}$,
in this case, an internal structure of the DW is not important. Now we turn
attention to more complex DW textures displaying noncollinear and noncoplanar
arrangement of the surface magnetization varying over finite scale. To explore
the properties of the bound electron states induced by such textures we
address several typical examples of 1D 180∘ noncollinear and noncoplanar
configurations, where the magnetization orientation varies only along the
$x$-axis,
$\mathbf{m}(x)=(\mathbf{m}_{x}(x),\mathbf{m}_{y}(x),\mathbf{m}_{z}(x))$, and
the magnitude is fixed, $|\mathbf{m}(x)|=1$. In order to avoid time consuming
calculations, we employ the approach, where the spatial profile
$\mathbf{m}(x)$ is composed of three differently oriented regions:
$\mathbf{m}(x)=\mathbf{m}_{\text{s}}h(|x|-l)\operatorname{sgn}(x)+\mathbf{m}_{\text{m}}h(l-|x|)$,
$\mathbf{m}_{\text{s}}$ and $\mathbf{m}_{\text{m}}$ are independent of the
coordinate, the boundaries between the regions are perpendicular to the
$x$-axis. The magnetizations of the side regions, $\pm\mathbf{m}_{\text{s}}$,
are antiparallel to each other and orthogonal to that of the middle region,
$\mathbf{m}_{\text{m}}$, i.e.
$(\mathbf{m}_{\text{s}}\cdot\mathbf{m}_{\text{m}})=0$. The approximation
simulates a helical rotation or cycloidal one of the magnetization vector
about the DW normal moving from one side to another for the Bloch-like texture
or the Néel-like one, respectively. In the framework of our approach, the
bound states created by these textures are described by solution of the
eigenvalue problem
$H(x,k_{y})\Theta_{n}(x,k_{y})=E_{n}(k_{y})\Theta_{n}(x,k_{y})$ with the
Hamiltonian (1) under boundary conditions at $x=l$ and $x=-l$ presented in Eq.
(2).

In the present approach, the coplanar vectors $\mathbf{m}_{\text{s}}=(1,0,0)$
and $\mathbf{m}_{\text{m}}=(0,1,0)$ set the Néel-like DW on the TI surface.
The schematic of the magnetization profiles along the DW can be seen at the
top of Fig 8. A hallmark of this Néel-like DW is an existence of the
dispersionless and strongly spin-polarized state with energy $E_{0}(k_{y})=0$
within the momentum interval $|k_{y}|<k_{0}$, at any width $2l$ of the middle
region. When compared with the case of the sharp DW
$\mathbf{m}=(1,0,0)\operatorname{sgn}(x)$ referred to Eq. 3 at $\theta=\pi/2$,
the presence of the middle region is reflected in the phase shift of the
envelope function away from the DW core:
$\Theta_{0}(x\to\pm\infty,k_{y})\sim\exp(\mp|k_{0}+k_{y}|x\pm ik_{0}l)$. The
spectral characteristics of the surface electrons are illustrated by the
tight-binding simulations displayed in Fig. 8. They confirm clearly the
existence of the flat band state and indicate a large sharp peak in both full
DOS $\rho(E)$ and the spin polarization $S_{z}(E)$ near zero-energy. It should
be noted that in the case of a broad middle region, $l>k_{0}^{-1}$, the DW
also supports the spin-degenerate gapped states with quasi-parabolic bands
$E_{n}(k_{y})=E_{n}(-k_{y})$ within the local gap in the 2D projected bands
(Fig. 8), here $n=\pm 1,\pm 2,...\pm N$. The eigenstates have opposite
energies with respect to the zero energy, $E_{n}(k_{y})=-E_{-n}(k_{y})$. With
increasing the width, $2l$, a number of the gapped states, $2N$, grows. The
bound states with $n\neq 0$ manifest in modification of DOS at the edges of
the quasi-parabolic bands. In the opposite case of the relatively sharp DW,
when $l<k_{0}^{-1}$, the additional states with $n\neq 0$ do not occur. The
specified above trends can be observed in Fig. 8 by comparing the spectral
features for $2l=50a$ and $2l=20a$.

We performed a similar analysis of the bound states for the noncoplanar
surface magnetic texture defined by the vectors
$\mathbf{m}_{\text{s}}=(0,0,1)$ and $\mathbf{m}_{\text{m}}=(0,1,0)$, which is
here referred to as the Bloch-like DW. We argue that, at any width of the
middle region, the Bloch-like DW hosts the strongly spin-polarized state with
linear momentum-energy relationship $E_{0}(k_{y})=\nu k_{y}$ spanning the
exchange gap. The asymptotic behavior of the envelope function of the state
away from the DW core, given by the relation
$\Theta_{0}(x\to\pm\infty)\sim\exp(\mp k_{0}(x-il))$ shows that the existence
of the middle region results in a phase shift of the function. Furthermore, if
the middle region is so wide that $l>k_{0}^{-1}$, the additional bound gapped
states with quasi-parabolic dispersion, $E_{n}(k_{y})=E_{n}(-k_{y})$ ($n=\pm
1,\pm 2,...\pm N$), appear in the gap. These states are doubly degenerate in
spin, their spectrum is mirrored with respect to zero energy,
$E_{n}(k_{y})=-E_{-n}(k_{y})$. With increasing $2l$, a number of the gapped
states, $2N$, increases. In the case, when $l<k_{0}^{-1}$, the additional
states with $n\neq 0$ are absent.

Let us consider two the polar Néel-like textures, where the surface
magnetization is confined in the $(x,z)$-plane. Fig. 9 displays the spectral
picture of the surface electron states for magnetic configuration given by
$\mathbf{m}_{\text{s}}$=(1,0,0) and $\mathbf{m}_{\text{m}}$=(0,0,1). As seen
in Fig. 9, in the case of the relatively wide middle region of $2l=50a$, a
pair of quasi-linear bands appears within the local gap starting from the
Dirac points at $k_{y}=\pm k_{0}$ and crossing at $k_{y}=0$. This band
crossing is accompanied by an opening of a tiny hybridization gap
$\sim\exp(-2k_{0}l)$, which divides the states with opposite spatial parity,
$E_{0}(k_{y})$ and $E_{-1}(k_{y})$. As it can be inferred by comparing the
spectra at different values of the width $2l$ (see, for example, the plots for
$2l=50a$ and $2l=20a$ in Fig. 9), when the middle region is narrowed the gap
$|E_{0}(0)-E_{-1}(0)|$ enlarges. At the same time, the low-energy dispersive
state $E_{0}(k_{y})$ sinks deeper into the local gap, and eventually in the
small $2l$ limit, it takes the form of the perfect flat band close to zero-
energy connecting the two Dirac points, $E_{0}(k_{y})\approx-2\nu k_{0}^{2}l$
at $k_{0}l\to 0$. Note also, that the state $E_{0}(k_{y})$ is partially spin-
polarized at the finite width $2l$, however, it becomes fully spin-polarized
state if the DW middle region is completely shrinked. In turn, the state
$E_{-1}(k_{y})$ moves towards the band continuum edge and leaves the local gap
below a certain value of the width $2l=k_{0}^{-1}$ ($E_{-1}(k_{y})=-\nu k_{0}$
at $2k_{0}l=1$).

One can also show that the two in-gap bound states are hosted by the Néel-like
DW whose texture is defined by the vectors $\mathbf{m}_{\text{s}}=(0,0,1)$ and
$\mathbf{m}_{\text{m}}=(1,0,0)$. Provided that the middle region is narrow
enough, $k_{0}l\ll 1$, the one of these states leaves the exchange gap for the
band continuum, while the other remains within the gap as the chiral fermion
with the perfect linear dispersion $E_{0}(k_{y})=\nu k_{y}$. It is evident
that a single 90∘ DW, which contains the $(0,\pm 1,0)$-oriented domain, does
not support an in-gap bound state. However, noncollinear and noncoplanar 180∘
textures, in which the middle region magnetization $\mathbf{m}_{\text{m}}$ is
aligned along $y$-axis, can generate from one to several in-gap states. In any
case, the 180∘ Néel- or Bloch-type DWs investigated certainly provides the
existence of the topologically protected chiral state $E_{0}(k_{y})$ inside
the exchange gap.

##  IV DISCUSSION AND SUMMARY

We have shown that the real-space magnetic textures, such as surface DWs, are
capable of affecting substantially the momentum-space behavior of low-energy
quasiparticles in TIs. Thus, any particular magnetization texture
$\mathbf{M}(x,y)$ should provide its own spectral portrait for the surface
electron states in the surface exchange gap. The averaged spectral function
$A(E,\mathbf{k})=-(1/\pi)\imaginary\Tr\langle
G^{R}(E,\mathbf{k}|\mathbf{M}(x,y))\rangle$, in which the contributions of all
the textures are collected accordingly to their statistic weight, is thought
to be observed in the spectroscopy measurements. Such an assumption may
qualitatively explain controversial ARPES results for the surface spectra of
intrinsic AFM TI MnBi2Te4 with out-of-plane easy axis obtained by various
experimental groups [47, 35, 44, 43, 45]. In the seminal work [47] a clear
exchange gap of about 70 meV at the $\overline{\Gamma}$-point separating the
upper and lower parts of the surface cone was revealed, which is close to the
theoretically predicted value 88 meV. However, Shikin et al. [35] have found
in the ARPES spectra both large (60–70 meV) and reduced (<20 meV) surface gaps
at the Dirac point. Some other ARPES measurements [44, 43, 45] have detected
the gapless Dirac states at the surface of MnBi2Te4 below the Néel temperature
instead of the expected exchange gapping of the states. Within the developed
theory, we speculate that in the experimental findings cited here above the
ARPES surface spectra may contain a superposition of the DW-induced spectral
features like the ones presented in Figs. 2, 4, 6 averaged spatially over
multiple DWs caught out in the light spot of a micron scale. If the
concentration of the DWs is relatively low, they can hardly ensure a visible
spectral signal inside the large exchange gap, which is associated with almost
homogeneous out-of-plane magnetization of the broad domains. With the
increasing concentration of DWs, one can note the appearance of the Dirac-like
gapless dispersion crossing the exchange energy gap. Under the higher
concentration, the Dirac-like spectrum becomes gapped (i.e., with the
<<reduced>> gap in the terms of Ref. [35] at the $\overline{\Gamma}$-point
owing to the hybridization of the states bound to the neighboring DWs (see
Fig. 4 and the corresponding comments). From that standpoint, the disagreement
of the ARPES findings for the surface spectra of intrinsic AFM TI MnBi2Te4 can
be attributed to the growth conditions and magnetic treatment of tested
samples.

The disagreement between some spectroscopic probes indicating a relatively
large exchange gap at the TI surface and the low temperature range of the QAHE
realization remains an important matter [68]. The quantized conductivity
effects in real systems are affected by a complex network of topologically
non-trivial and trivial conducting channels running along boundaries,
separating regions of distinct topological phases [69]. Our results
demonstrate the presence of the DW-induced states embodying the conducting
channels at the magnetic surface of TI and describe their characteristics. In
magneto-transport measurements, when the chemical potential is fixed within
the energy gap, the conductivity is realized by means of both a carrier
propagation along the edge channels of the sample and a percolation through 1D
channels caused by the network of the magnetic DWs [70, 42]. Therefore the
measured conductivity can display remarkable deviation from the expected
quantized value.

We have revealed a unique possibility to tune the group velocity and spin-
polarization orientation of the DW-induced states by reorienting the easy axis
direction, which could be useful to design new devices with controlled
propagation velocity and chirality for the robust excitations responsible for
quantized dissipationless conductivity. We have predicted that, when the
moments lie within the easy-plane parallel to the TI surface plane, both the
head-to-head and tail-to-tail planar textures generate the chiral zero-energy
flat band state in the single-particle spectrum. Due to the flatness of the
energy dispersion and respectively the large peak of DOS this state could
attract the particular attention from the perspective of enhanced electron
correlations. Thereby, the surface of the planar magnetic TI carrying the DWs
could provide unique platform for realizing phase transitions driven by
instabilities of the unusual quasiparticles — the heavy chiral fermions with
nontrivial topology. In particular, in such a system, it would be interesting
to seek for the unconventional superconductivity with spin triplet pairing
[71] or triplet excitonic insulator [72].

To summarize: We have provided the physical picture to understand the nature
and properties of the DW-induced states at the TI magnetic surface. Implying a
rich domain texture of the surface we have shown the existence of various
topologically protected states stemming from the DWs. The relevance of our
results is based on the fact that the properties of the DW-induced states are
explained in terms of general approach, and their description is material
independent. It can be applied to the surfaces of intrinsic magnetic TIs and
the interfaces in hetrostructures with induced magnetization under any
direction of anisotropy axis. Further investigations of the interplay between
domain texture and the surface fermion states in the magnetic TIs, especially
via different experimental techniques, are strongly desired to open the way to
realize novel quantum phenomena.

##  V ACKNOWLEDGMENTS

This work was supported by the Saint Petersburg State University (project ID
No. 73028629) and Russian Science Foundation (Grant No. 18-12-00169).

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