# Orbital Hall effect and topology on a two-dimensional triangular lattice:
from bulk to edge

Anderson L. R. Barbosa<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal Rural de Pernambuco, 52171-900, Recife, PE, Brazil  Luis
M. Canonico  Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC
and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  Jose H. García
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,
Campus UAB, Bellaterra, 08193 Barcelona, Spain  Tatiana G. Rappoport
<EMAIL_ADDRESS>Centro de Física das Universidade do Minho e do
Porto (CFUMUP) e Departamento de Física, Universidade do Minho, P-4710-057
Braga, Portugal  Instituto de Física, Universidade Federal do Rio de Janeiro
C.P. 68528, 21941-972 Rio de Janeiro RJ, Brazil

###### Abstract

We investigate a generalized multi-orbital tight-binding model on a triangular
lattice, a system prevalent in a wide range of two-dimensional materials, and
particularly relevant for simulating transition metal dichalcogenide
monolayers. We show that the interplay between spin-orbit coupling and
different symmetry-breaking mechanisms leads to the emergence of four distinct
topological phases [Eck, P., et al., Phys. Rev. B, 107 (11), 115130 (2023)].
Remarkably, this interplay also triggers the orbital Hall effect with
distinguished characteristics. Furthermore, by employing the Landauer-Büttiker
formula, we establish that in the orbital Hall insulating phase, the orbital
angular momentum is carried by edge states present in nanoribbons with
specific terminations. We also show that, as expected, they do not have
topological protection against the disorder of the edge states belonging to a
first-order topological insulator.

The past few decades have seen a growing interest in manipulating the quantum
degrees of freedom for developing new quantum devices. The emergence of two-
dimensional materials and the tunability of their properties added to their
capability to sustain highly crystalline interfaces reignited the excitement
in spintronics and provided a platform for tackling long-standing challenges
in the field while paving the way for groundbreaking technological
applications like spin-orbit-based memory systems and ultra-compact devices
[2, 3, 4]. The tunability of the properties of the 2D materials also enabled
the control of the spin-orbit coupling, which is essential for the electrical
manipulation of the spins. Nonetheless, due to its electrostatic origin, this
interaction tends to appear on heavy metals, which are scarce, and their usage
poses environmental risks.

Recently, researchers have turned their attention to the manipulation of the
orbital degrees of freedom and exploiting novel phenomena such as the orbital
Hall effect [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and the charge-to-orbital
conversion via orbital Rashba-Edelstein effect [16, 17, 18, 19]. In analogy
with their spin counterparts, these effects enable the electrical manipulation
of the orbital degrees of freedom without a mediating interaction like the
spin-orbit coupling, opening the possibility of using light elements in novel
sustainable devices [20] and reigniting the interest in orbitronics [21, 22].
Novel experimental studies have demonstrated that the orbital Hall effect
(OHE) can arise in three-dimensional (3D) systems even if the orbital angular
momentum of the system vanishes in equilibrium [23, 24] and the orbital
currents may have very large decay lengths [25, 26].

Furthermore, it has been shown that the OHE can induce magnetization dynamics
via the orbital torque in magnetic hetero-bilayers [27, 28, 29, 30]. Also, the
reciprocal version of this effect, the orbital pumping, where an oscillating
magnetic moment creates an orbital current, was recently confirmed [31, 32].
In the case of two-dimensional (2D) materials, the interplay between their
band structure, orbital degrees of freedom, and the presence of sizable Berry
curvature pockets in reciprocal space has garnered significant attention.
Theoretical works have shown that, like their 3D counterparts, multi-orbital
2D materials can host orbital textures that trigger sizable OHE [33].
Additionally, several theoretical works have predicted that monolayers [34,
35, 36] and bilayers [37, 38] of transition metal dichalcogenides (TMDs)
exhibit OHE within their energy gap and are characterized by an orbital Chern
number. Furthermore, these materials exhibit orbital currents that flow
through their edge states, which were previously considered trivial from the
$\mathbb{Z}_{2}$ perspective [39, 40].

Recently, a theoretical work by Costa et al. [41] demonstrated the connection
between the orbital Hall insulating phase of certain transition metal
dichalcogenides (TMD) monolayers and their high-order topological insulating
phases [42, 43, 44], hinting a connection between the non-topological edge
states and the orbital Hall conductivity plateaus [34, 37, 38, 35], and
evinced the existence of the OHE in centrosymmetric systems. They showed that
the accumulation of orbital angular momentum can occur at the edges of ribbons
of these materials due to the presence of metallic states that depend on the
ribbon orientation. However, despite this array of theoretical predictions,
direct experimental evidence of the transport and manipulation of the atomic
orbital angular momentum remains elusive.

The development of orbitronic devices based on 2D materials requires a robust
criterion to identify orbital-Hall-capable materials. Recent studies by Han et
al. have taken a significant step toward this goal [45, 46]. The authors
conducted a detailed analysis of the microscopic origin of the orbital
textures and uncovered the interplay between crystalline and orbital
symmetries that promote hybridizations leading to the emergence of these
textures. From another perspective, Eck et al. [1] recently presented the
conditions that enable the appearance of 2D high-order topological insulators
(HOTI) on the triangular lattice from the interplay between the crystalline
and orbital symmetries. Our current study further explores the correlation
between orbital hybridization, orbital symmetries, and the OHE to gain
insights into its connection with various topological phases.

In this work, building on the previous work of Eck et al., we explored how
symmetry reduction can trigger different spin and orbital topological phases.
To this end, we focus on the $d_{z^{2}}$, $d_{xy}$, and $d_{x^{2}-y^{2}}$
orbitals within the $d$-subshell, residing on a triangular lattice; this is a
well-established model for characterizing TMD monolayers, where the symmetries
can be controlled by removing specific hopping elements. This flexibility
allowed us to explore various topological phases, which are subsequently
probed through quantum transport calculations of the orbital and spin
conductivities using the Kubo and Landauer-Büttiker formalisms. Combining
these methods, we identified the distinctive orbital-Hall conductivity
plateaus within the insulating gap of high-order topological insulators [42,
43, 44], determined whether the metallic edge states emerge within the energy
gap for specific ribbon orientations, and studied the impact of the disorder
on the transport of orbital angular momentum in device geometries.

##  I The model

We begin with the three-band tight-binding model, which is commonly used to
describe the low-energy behavior of H-TMDs. In this model, a triangular
lattice accommodates the $d_{z^{2}}$, $d_{xy}$, and $d_{x^{2}-y^{2}}$
orbitals. To simplify the analysis, we adopt the same tight-binding parameters
as in Ref. 47 for MoS2, but with a spin-orbit coupling strength that is $15$
times larger than the original value. This allows us to transition between
different phases of the system by simply removing some of the original hopping
elements, which modify the symmetries of the model without altering the values
of the other parameters. Using the basis
$\\{d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}\\}$, the Hamiltonian of the system can be
written as $\mathcal{H}=\sigma_{0}\bigotimes H_{0}+H_{SOC}$, where

$H_{0}=\begin{bmatrix}h_{0}&h_{1}&h_{2}\\\ h_{1}^{*}&h_{11}&h_{12}\\\
h_{2}^{*}&h_{12}^{*}&h_{22}\end{bmatrix},$ (1)

with

$\displaystyle h_{0}=2t_{0}(\cos 2\alpha+2\cos\alpha\cos\beta)+\epsilon_{1},$
$\displaystyle h_{1}=-2\sqrt{3}t_{2}\sin\alpha\sin\beta+2it_{1}(\sin
2\alpha+\sin\alpha\cos\beta)$ $\displaystyle h_{2}=2t_{2}(\cos
2\alpha-\cos\alpha\cos\beta)+2\sqrt{3}it_{1}\cos\alpha\sin\beta,$
$\displaystyle h_{11}=2t_{11}\cos
2\alpha+(t_{11}+3t_{22})\cos\alpha\cos\beta+\epsilon_{2},$ $\displaystyle
h_{22}=2t_{22}\cos 2\alpha+(3t_{11}+t_{22})\cos\alpha\cos\beta+\epsilon_{2},$
$\displaystyle
h_{12}=\sqrt{3}(t_{22}-t_{11})\sin\alpha\sin\beta+4it_{12}\sin\alpha(\cos\alpha-\cos\beta),$
$\displaystyle\text{and }\alpha=\frac{1}{2}k_{x}a\mbox{ and
}\beta=\frac{\sqrt{3}}{2}k_{y}a.$ (2)

This is a symmetry-based tight-binding Hamiltonian that does not originate
from a Slater-Koster approach, as the hopping elements between the $d$
orbitals are mediated by the $p$ orbitals of the chalcogenes in 2H-TMDs. By
examining the TMD Hamiltonian, it becomes apparent that, apart from the term
proportional to $t_{2}$, Eq. 1 is formally equivalent to a Slater-Koster
Hamiltonian utilizing $p_{x}$, $p_{y}$, and $p_{z}$ orbitals in a two-
dimensional triangular lattice where the inversion and mirror symmetries of
the lattice are broken (see Appendix A).

In this map between the two Hamiltonians, $t_{0}$, $t_{11}$ and $t_{22}$ are
Slater-Koster hopping elements for $\sigma$ and $\pi$ bonds for $p_{z}$,
$p_{x}$ and $p_{y}$ orbitals that respect all the symmetries of a two-
dimensional triangular lattice. The terms proportional to $t_{0}$, $t_{11}$
and $t_{22}$ describe the nearest-neighbor hopping in a triangular lattice of
layer group $p6/mmm$, which is generated by a six-fold rotation, three
vertical reflection planes $\sigma_{v}$, three diagonal reflection planes
$\sigma_{d}$ and one horizontal reflection plane $\sigma_{h}$ [1].

If inversion symmetry is broken, the six-fold rotation ($C_{6}$) reduces to
$C_{3}$, and the Hamiltonian presents the term proportional to $t_{12}$. On
the other hand, if only the mirror symmetry with respect to the plane is
broken, it is possible to have hybridization between $p_{z}$ and the in-plane
orbitals $p_{x}$ and $p_{y}$, which gives rise to the term proportional to
$t_{1}$. The extra term proportional to $t_{2}$ is the only one that cannot be
mapped into a Slater-Koster Hamiltonian. It is important to notice, however,
that although we can formally map one Hamiltonian into another, the meaning of
the different terms is not the same for the $d$ orbitals. For instance,
$t_{1}$ hybridizes orbitals belonging to different irreducible
representations, as in the case of the $p$ orbitals, but does not break the
same mirror symmetry, which is preserved in 2H TMDs monolayers.

The spin-orbit coupling is written as

$H_{\rm SOC}=\sigma_{z}\frac{\lambda}{2}\begin{bmatrix}0&0&0\\\ 0&0&2i\\\
0&-2i&0\end{bmatrix}.$ (3)

Here, $\lambda$ is the strength of the spin-orbit coupling, and $\sigma_{0}$
and $\sigma_{z}$ are the $2\times 2$ identity matrix and the $z$ Pauli
matrices, respectively. To investigate the connection between topology and the
orbital Hall plateau, we analyze four different phases. To consider the
different phases without modifying the spin-orbit coupling or the strength of
the other hoppings, we take $\lambda$ as fifteen times stronger than the SOC
of MoS2. The band structures of the four phases are shown in figure 1. For
phase I, $t_{1}=t_{2}=t_{12}=0$, which, in the case of $p$ orbitals, is
equivalent to a system where all the symmetries of the triangular lattices are
kept. Fig. 1(a) shows the band structure of the phase. For this case, the
subspace for $m_{l}=0$ is separated from the subspace where $m_{l}=\pm l$.
Because of the inversion symmetry, all bands are spin degenerate. The system
has a lower band with $m_{l}=0$. The two other bands that belong to the
$m_{l}\pm l$ sub-space are split by the spin-orbit coupling.

Figure 1: Bulk energy bands of the four topological phases of the tree bands
model for various terms in the Hamiltonian. Phase I (a):
$t_{1}=t_{2}=t_{12}=0$. Phase II (b): $t_{2}=t_{12}=0,t_{1}\neq 0$. Phase III
(c): $t_{1}=t_{2}=0,t_{12}\neq 0$. Phase IV (d): $t_{1}\neq t_{2}\neq
t_{12}\neq 0$.

For phase II, in the case of $p$ orbitals, it indicates that the mirror
symmetry is broken with $t_{1}=t_{2}=0$ and $t_{12}\neq 0$, while preserving
the inversion symmetry, and consequently, the spin-degenerate energy bands.
However, there is a finite hybridization between the bands belonging to the
$m_{l}=\pm l$ subspace and the one for $m_{l}=0$. As a result, the band
structure of Fig. 1(b) presents a gap at the band crossing of Fig. 1(a).

In the case of phase III, the subspaces of $m_{l}=\pm l$ and $m_{l}=0$ are
decoupled but the inversion symmetry is broken. As a result, the bands in Fig.
1(c) can present spin and orbital angular momentum textures. This breaks the
spin degeneracy of the bands belonging to the $m_{l}=\pm l$ subspace. Phase IV
presents all the parameters of a three-band model for MoS2 so that all hopping
of the Hamiltonian are finite and a much stronger spin-orbit coupling that
produces a large splitting of the conduction band states, as seen in Fig. 1(d)
but still keeps the spin and orbital character of the TMDs in the vicinity of
the band gap.

##  II Topology and the spin and orbital angular momentum transport

To explore the topology of the different phases, we use a two-fold approach:
we calculate the band structure of a nanoribbon and analyze the edge states
for the four different phases, while comparing it with the spin and orbital
Hall conductivities of the bulk system. For the computation of the spin and
orbital Hall conductivities, we used the Kubo-Bastin formula [48, 49, 50]:

$\sigma_{\alpha\beta}^{\mathcal{O}}(\mu,T)=ie\hbar\int_{-\infty}^{\infty}d\varepsilon
f(\varepsilon,\mu,T)\times\\\ Im\left(Tr\langle
J_{\alpha}^{\mathcal{O}}\delta(\varepsilon-H)v_{\beta}\partial_{\varepsilon}G^{+}(\varepsilon)\rangle\right),$
(4)

where $J_{\alpha}^{\mathcal{O}}=\frac{1}{2\Omega}\\{v_{\alpha},\mathcal{O}\\}$
is the angular momentum current density operator with $\mathcal{O}$ being
either the $S_{z}$ spin operator or the $L_{z}$ orbital angular momentum
operators, where the later is evaluated within the atom-centered
approximation, $v_{\alpha}$ is the $\alpha$ component of the velocity operator
in the $v_{\alpha}=-\frac{i}{\hbar}\left[H,R_{\alpha}\right]$ with
$R_{\alpha}$ being the projection of along the $\alpha$ direction of the
position operator, $H$ is the Hamiltonian of the system and $\Omega$ the area
of the sample. $G^{+}(\varepsilon)$ and $\delta(\varepsilon-H)$ are the
retarded Green’s and spectral functions, respectively. These are approximated
using the kernel-polynomial method [51, 52] as implemented in the LSQUANT
toolkit [53]. For all of our transport calculations, we used $1024$ Chebyshev
moments with the Jackson kernel to obtain an energy resolution of
$\delta\approx 3$ meV, while assuming periodicity allowed us to consider a
grid of $512\times 512$ points in reciprocal space.

Starting with phase I, Fig. 2(a) and Fig. 2(b) show the band structure of a
nanoribbon with the color code of panels Fig. 2(a) and panels Fig. 2(b)
indicating the $S_{z}$ and $L_{z}$ projections respectively. Because of the
spin degeneracy, we project the spin-up states. The band structure of the
nanoribbon does not have in-gap edge states, signaling that the system lies in
a topologically trivial state. Additionally, from the spin and orbital
projection of the energy states of the nanoribbon, it is clear that the
vanishing contribution from hoppings $t_{1},t_{2}$ and $t_{12}$ forbids the
hybridization between the orbitals with $d_{z^{2}}$, and
$d_{x^{2}-y^{2}},d_{xy}$ and makes the system inversion symmetric.
Consequently, the SOC enforces the energy states formed by linear combinations
of orbitals $d_{x^{2}-y^{2}}$ and $d_{xy}$ to have a well-defined atomic
angular momentum character. Thus making the system a trivial SOC insulator, as
the null orbital and spin Hall conductivities presented in Fig. 2(c) confirm.

Figure 2: Spin (a) and orbital (b) character of the spin up energy bands of a
nanoribbon with breadth $W=50\vec{a}_{2}$ for a system in the phase I. (c)
Spin (orange) and orbital (green) Hall conductivities for a system in phase I.

For phase II, the band structure of the nanoribbon in Fig. 3(a) and Fig. 3(b)
show that after enabling the hybridization between the orbitals $d_{z^{2}}$,
$d_{xy}$, and $d_{x^{2}-y^{2}}$ through $t_{1}$ while maintaining
$t_{2}=t_{12}=0$, the system exhibits inversion symmetry with two pairs of in-
gap spin-polarized edge states. Interestingly, the orbital-angular momentum
character of the spin-up states (Fig. 3(b)) unveils that the combined action
of the SOC and orbital hybridization results in the mixing of $d_{z^{2}}$
states and $m_{-l}=\frac{1}{\sqrt{2}}(d_{x^{2}-y^{2}}-id_{xy})$ states, in
resemblance with the BHZ model of quantum spin Hall insulators, and
consequently the edge states of this system carry opposite spin and orbital
angular momentum. The quantized spin Hall conductivity characterizes the
topological insulator and the opposite bulk spin and orbital Hall
conductivities shown in Fig. 3(c) confirms the relation with the BHZ model.

Figure 3: Spin (a) and orbital (b) character of the spin-up energy bands of a
nanoribbon with breadth $W=50\vec{a}_{2}$ for a system in phase II. (c) Spin
(orange) and orbital (green) Hall conductivities for a system in phase II.

To further inquire into the effects of the orbital hybridization on the
appearance of topologically non-trivial phases, in phase III, we follow a
similar analysis as we did for phase II. ./ Fig. 4(a) and Fig. 4(b) show the
band structure and the respective spin and orbital character of the states of
the nanoribbon. Here, we forbid the hybridization of the orbitals $d_{z^{2}}$
with $d_{x^{2}-y^{2}},d_{xy}$ by setting the hoppings $t_{1}=t_{2}=0$, and we
break the inversion symmetry by allowing $t_{12}\neq 0$. Analyzing the orbital
and spin character of the edge states of the system, we can see that the
$d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals are responsible for the non-trivial
topological properties of the system, whereas the states related to
$d_{z^{2}}$ are effectively decoupled. In contrast with phase II, the
hybridization and the inversion symmetry breaking favor the appearance of
pairs of edge states with different spin and orbital characters at opposite
sides of the ribbon Brillouin zone, in connection with a spin- and orbital-
valley locking. Similarly, as before, these edge states are endowed with
sizable orbital angular momentum, providing these states with additional
topological protection since any backscattering event would require not only a
change of the spin character but also the orbital part. The spin and orbital
Hall conductivities shown in Fig. 3(c) confirm the topologically non-trivial
nature of this phase and the spin and orbital Hall plateaus have the same
sign, in contrast with phase II

Figure 4: Spin (a) and Orbital (b) character of the energy bands of a
nanoribbon with breadth $W=50\vec{a}_{2}$ for a system in phase III. (c) Spin
(orange) and orbital (green) Hall conductivities for a system in phase III.

Finally, for phase IV, we allow all the hopping terms, and essentially the
model reduces to the three-band model for 2H-TMDs presented by Liu et al. [47]
but with an enhanced spin-orbit coupling. From analyzing the spin and orbital
characters of the energy states of the nanoribbon shown in Fig. 5(a) and Fig.
5(b), respectively, we found that the system presents edge states within its
main energy gap. They are similar to the edge states presented in MoS2 [34,
37] but with large splitting from the strong spin-orbit coupling. There are
four pairs of edge states so that the system is topologically trivial from the
$\mathbb{Z}_{2}$ classification and exhibits a behavior that is drastically
different from the previous phases.

Figure 5: Spin (a) and orbital (b) character of the energy bands of a
nanoribbon with breadth $W=50\vec{a}_{2}$ for a system in phase IV. (c) Spin
(orange) and orbital (green) conductivity for a system in phase IV.

One can notice that the spin character of the edge states in a single side of
the ribbon Brillouin zone contains opposite spin polarizations in contrast
with the energy states shown in Fig. 4(a). However, when focusing on the
orbital character of the edge states of this system, we found that they retain
a similar orbital character as shown in Fig. 4(b). These properties correlate
with the absence of spin Hall conductivity plateaus within the energy gap but
the presence of the orbital Hall insulating phase as shown in Fig. 5(c) which
is characterized by an orbital Chern number = 1 [37]. The even number of edge
state pairs in the case of flat termination together with the absence of edge
states for zigzag edges is consistent with a higher-order topological phase,
which is present in this case [54].

##  III The orbital Hall conductivity of nanoribbons: Landauer-Büttiker
calculations

We proceed to further analyze the higher-order topological phase and check
whether the edge states possess the capability to carry orbital angular
momentum. To accomplish this, we have designed an OHE setup consisting of a
mesoscopic device connected to four semi-infinite terminals subjected to
voltage biases $V_{i}$. In our calculations, we considered two distinct device
geometries. Fig. 6(a) showcases a configuration with all flat edges, whereas
Fig. 6(b) presents a conventional Hall-bar geometry characterized by a
combination of flat and zigzag edges.

Figure 6: (a) Triangular lattice device with edges that either preserve or
break the rotational symmetry of the original lattice. (b) Triangular lattice
device with edges that break the rotational symmetry of the original lattice.
The device (blue) is connected to four leads (red).

In the Landauer-Büttiker formalism, the orbital (spin) projected current
through the $i$th terminal in the linear regime at low temperature is given by

$I^{\mathcal{O}}_{i,\eta}=\frac{e^{2}}{h}\sum_{j}\tau_{ij,\eta}^{\mathcal{O}}\left(V_{i}-V_{j}\right),$
(5)

with $\mathcal{O}$ being one of the $L_{z}$ orbital angular momentum or
$S_{z}$ spin states ($\circlearrowleft(\uparrow)$ or
$\circlearrowright(\downarrow)$). The orbital (spin) transmission coefficient
is calculated from the scattering matrix
$\mathcal{S}=\left[\mathcal{S}_{ij}\right]_{i,j=1,\dots,4}$ as

$\tau_{ij,\eta}^{\mathcal{O}}=\textbf{Tr}\left[\left(\mathcal{S}_{ij}\right)^{\dagger}\mathcal{P}^{\mathcal{O}}_{\eta}\mathcal{S}_{ij}\right],$
(6)

where the matrices
$\displaystyle\mathcal{P}^{L_{\eta}}_{\eta}=\mathbb{1}_{N}\otimes
l^{\eta}\otimes\sigma^{0}$ and
$\mathcal{P}^{S_{\eta}}_{\eta}=\mathbb{1}_{N}\otimes
l^{0}\otimes\sigma^{\eta}$ are a projector, $\mathbb{1}_{N}$ is an identity
matrix with dimension $N\times N$. The dimensionless integer $N$ refers to the
number of propagating wave modes in the terminals, proportional to the
terminal width ($W$) and the Fermi vector ($k_{F}$), computed through the
equation $N=k_{F}W/\pi$. The matrices $l^{\eta}$ and $\sigma^{\eta}$ with
$\eta=\\{x,y,z\\}$ are the orbital angular momentum and spin matrices,
respectively, and the cases with $\eta=0$ refer to the identity matrices in
the orbital and spin subspaces. Thus, by setting either $\eta=0$ or
$\eta=\\{x,y,z\\}$, the charge and orbital (spin) can be respectively
addressed.

The pure OHC (SHC)
$I^{L_{z}(S_{z})}_{i,z}=\frac{\hbar}{e}(I^{\circlearrowleft(\uparrow)}_{i}-I^{\circlearrowright(\downarrow)}_{i}$),
$i=3,4$ can be obtained by assuming that the charge current vanishes in the
transverse terminals,
$I^{c}_{i,0}=I^{\circlearrowleft(\uparrow)}_{i}+I^{\circlearrowright(\downarrow)}_{i}=0$,
while the charge current is conserved in the longitudinal terminals,
$I^{c}_{1,0}=-I^{c}_{2,0}=I^{c}$ [55, 56, 57]. By applying these conditions to
Eq.(5), we obtain [11]

$\displaystyle
I^{\mathcal{O}}_{i,\eta}=\frac{e}{2\pi}\left[\left(\tau_{i2,\eta}^{\mathcal{O}}-\tau_{i1,\eta}^{\mathcal{O}}\right)\frac{V}{2}-\tau_{i3,\eta}^{\mathcal{O}}V_{3}+\tau_{i4,\eta}^{\mathcal{O}}V_{4}\right],$
(7)

for $i=3,4$, where $V$ is a constant potential difference between longitudinal
terminals, and $V_{3,4}$ is the transversal terminal voltage.

Figure 7: Band structure (a), charge current (b) and orbital and spin Hall
current (c) of a mesoscopic device with the dimensions and design of Fig.
6(a). Band structure (d) charge current (e) and orbital and spin Hall current
(f) of a mesoscopic device a mesoscopic device with the dimensions and design
of Fig. 6(b). (c) and (f): Spin (orange) and orbital (green) conductance for a
Mo$S_{2}$ monolayer, which is in phase IV.

We begin our investigation by examining the influence of ribbon termination on
the orbital Hall conductance in a higher-order topological insulator. To
facilitate a clearer visualization of the role of in-gap edge states in
orbital transport, we utilize a triangular lattice with parameters akin to
MoS2, which belongs to the same topological phase as phase IV. The numerical
calculations of this section were implemented in the KWANT software [58].

It is essential to note that to establish a connection between the edge states
of the original honeycomb lattice of MoS2 and the triangular lattice of $d$
orbitals, the zigzag edge of the honeycomb lattice corresponds to the flat
edge of the triangular lattice, which consists of one of the sublattices.
Simultaneously, the armchair edge of the honeycomb lattice is associated with
the zigzag edge of the triangular lattice.

Fig. 7(a) presents the band structure of the type of device sketched in Fig.
6(a) for the parameters of MoS2. In that device, all edges break the three-
fold rotation symmetry of the triangular lattice. The band structure of the
system is very similar to the band structure of a MoS2 nanoribbon in the
three-orbitals approximation [41]. Fig. 7(b) presents the longitudinal charge
current of the device, where it is clear that the in-gap edges are metallic
and can conduct charge. Fig. 7(c) depicts the transverse spin and orbital
currents. While the spin current is zero inside the gap, we can see a finite
orbital current in the same energy window of the in-gap edge states. In
contrast, if the device also presents some edges that preserve the three-fold
rotation symmetry of the triangular lattice, as sketched in figure Fig. 6(b),
the band structure remains similar, as shown in panel Fig. 7(d) but the charge
current is suppressed inside the gap (panel e) and the system does not present
either spin or orbital currents, as shown in Fig. 7(f).

Figure 8: Band structure (a) charge current (b) and orbital and spin Hall
current (c) of a mesoscopic device a mesoscopic device with the dimensions and
design of Fig. 6(a). Band structure (d) charge current (e) and orbital and
spin Hall current (f) of a mesoscopic device a mesoscopic device with the
dimensions and design of Fig. 6(b). (c) and (f): Spin (orange) and orbital
(green) current, which is in phase II.

This can be compared with the characteristics of a first-order topological
insulator, as the one in phase II. Fig. 8(a) presents the band structure for
the device sketched in Fig. 6(a) for a topological insulator in phase II,
discussed previously. Fig. 8(b) presents the longitudinal charge current of
the device, where it is clear that the in-gap edges are metallic and can
conduct charge. Fig. 8(c) depicts the transverse spin and orbital currents.
Both spin and orbital currents are finite with opposite signs. While the spin
current has a clear plateau inside the gap, the orbital current does not seem
to have a quantized value. ./ 8d, e, and f present the same results but for
the device sketched in figure Fig. 6(b). Differently from the previous cases,
the results are robust independent of the type of edges the device presents,
underlying the importance of topological protection by a global symmetry.

Figure 9: Band structure (a) and orbital and spin Hall current with disorder
strengths for $U=0$ (b), $U=0.1$ eV (c), $U=0.2$ eV (d) of a mesoscopic device
of Mo$S_{2}$ monolayer, which is in phase IV, with dimensions ten times bigger
than as shown Fig. 6(a). Band structure (e) and orbital and spin Hall current
with disorder strengths for $U=0$ (f), $U=0.1$ eV (g), $U=0.2$ eV (h) of a
mesoscopic device, which is in phase II, with dimensions ten times bigger than
as shown in Fig. 6(a).

Finally, to study the robustness of the orbital and spin Hall currents against
scalar disorder, we included an Anderson disorder term that varies randomly
from site to site according to a uniform distribution in the interval
$(-U/2,U/2)$, with $U$ being the disorder strength. Fig. 9(b), Fig. 9(c), and
Fig. 9(d) present both the orbital and spin Hall currents of MoS2 for $U=0$,
$U=0.1$ eV, $U=0.2$ eV, respectively. From these ./, one can notice that the
orbital Hall currents are not protected against disorder. Although the plateau
observed in bulk calculations survives with increasing disorder [14], our
results show a decrease in the orbital Hall current. For the case of the
topological insulator of phase II, panels Fig. 9(f), Fig. 9(g), and Fig. 9(h)
show that the spin currents are not affected by increment of the disorder, as
one would expect for the topologically protected states. In contrast, the
orbital Hall currents are affected by the disorder.

##  IV Conclusion

To summarize, our study focuses on a generalized multi-orbital tight-binding
model on a triangular lattice. This system was recently used to obtain higher-
order topological phases in two-dimensional materials but also is especially
relevant for modeling transition metal dichalcogenide monolayers. By modifying
the hopping terms of the model, we changed the hybridization between orbitals
and uncovered the intricate interplay between spin-orbit coupling and various
symmetry-breaking mechanisms, which has led to the identification of four
distinct topological phases [1]. Notably, this interplay has also resulted in
the emergence of an orbital Hall effect with unique characteristics. For the
two first-order topological phases, the quantized spin Hall conductivity is
accompanied by a non-quantized orbital Hall plateau, while the higher-order
topological phase presents the orbital Hall effect in the absence of the spin
Hall effect.

Furthermore, we have applied the Landauer-Büttiker formula to establish that
in the orbital Hall insulating phase, the orbital angular momentum is carried
by edge states that exist in nanoribbons with flat terminations. Importantly,
we have demonstrated that these edge states lack the topological protection
against disorder observed in the edge states associated with a first-order
topological insulator. These findings provide valuable insights into the
behavior of edge states in systems with higher-order topology and their role
in the onset of the orbital Hall effect in these systems.

##  V ACKNOWLEDGMENTS

TGR acknowledges funding from FCT-Portugal through Grant No.
CEECIND/07471/2022. L.M.C. acknowledges funding from Ministerio de Ciencia e
Innovación de España under grant No. PID2019-106684GB-I00 / AEI /
10.13039/501100011033, FJC2021-047300-I, financiado por
MCIN/AEI/10.13039/501100011033 and the European Union
“NextGenerationEU”/PRTR.". J.H.G. acknowledge funding from MCIN/AEI
/10.13039/501100011033 and European Union "NextGenerationEU/PRTR” under grant
PCI2021-122035-2A-2a and funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 881603. ICN2 is
funded by the CERCA Programme/Generalitat de Catalunya and supported by the
Severo Ochoa Centres of Excellence programme, Grant CEX2021-001214-S, funded
by MCIN/AEI/10.13039.501100011033. ALRB acknowledges financial support from
Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPQ (Grant No
309457/2021-1).

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##  Appendix A Slater-Koster for $p$ orbitals in a triangular lattice

To calculate the first-neighbors hoppings of a hamiltonian of $p$ orbitals in
a triangular lattice, we define $n_{z}=\cos{\theta}$,
$n_{x}=\cos{\phi}\sin{\theta}$ and $n_{y}=\sin{\phi}\sin{\theta}$.

For the hopping between $p$ orbitals one has

$\displaystyle t_{p_{i},p_{i}}=n_{i}^{2}V_{pp\sigma}+(1-n_{i}^{2})V_{pp\pi}$
(8) $\displaystyle t_{p_{i},p_{j}}=n_{i}n_{j}(V_{pp\sigma}-V_{pp\pi})$ (9)

with $i=x,y,z$ and $i\neq j$.

For two-dimensional lattice, we define $n_{z}=\cos{\theta=0}=1$,
$n_{x}=\cos{\phi}$ and $n_{y}=\sin{\phi}$

and $t_{p_{z},p_{z}}=n_{i}^{2}V^{z}_{pp\sigma}$. Using
$\cos(\pi/3)=\cos(-\pi/3)=-\cos(2\pi/3)=-\cos(-2\pi/3)=\frac{1}{2}$,
$\sin(\pi/3)=-\cos(-\pi/3)=\sin(2\pi/3)=-\sin(-2\pi/3)=\frac{\sqrt{3}}{2}$,
$\cos(0)=-\cos(\pi)=1$ and $\sin{0}=\sin(\pi)=0$

one can construct the hopping elements that preserve all the symmetries of the
lattice

$\displaystyle h_{p_{x},p_{x}}(k_{x},k_{y})$
$\displaystyle=\frac{1}{4}(V_{pp\sigma}+3V_{pp\pi})(e^{i(\alpha+\beta)}+e^{-i(\alpha+\beta)a}+e^{i(\alpha-\beta)}+e^{-i(\alpha-\beta)})$
(10) $\displaystyle+V_{pp\sigma}(e^{2i\alpha}+e^{-2i\alpha})$ (11)
$\displaystyle h_{p_{x},p_{x}}(k_{x},k_{y})$
$\displaystyle=(V_{pp\sigma}+3V_{pp\pi})\cos\alpha\cos\beta+2V_{pp\sigma}\cos{2\alpha}$
(12) $\displaystyle h_{p_{y},p_{y}}(k_{x},k_{y})$
$\displaystyle=(3V_{pp\sigma}+V_{pp\pi})\cos\alpha\cos\beta+2V_{pp\pi}\cos{2\alpha}$
(13)

using the expressions above, one has
$\cos(\pi/3)\sin(\pi/3)=-\cos(-\pi/3)\sin(-\pi/3)=\cos(-2\pi/3)\sin(-2\pi/3)=-\cos(2\pi/3)\sin(2\pi/3)=\frac{\sqrt{3}}{4}$
and $\cos(0)\sin(0)=\cos(\pi)\sin(\pi)=0$

$\displaystyle h_{p_{x},p_{y}}(k_{x},k_{y})$
$\displaystyle=\frac{\sqrt{3}}{4}(V_{pp\sigma}-V_{pp\pi})(e^{i(\alpha+\beta)}+e^{-i(\alpha+\beta)}-e^{i(\alpha-\beta)}-e^{-i(\alpha-\beta)})$
(14) $\displaystyle h_{p_{x},p_{y}}(k_{x},k_{y})$
$\displaystyle=-\sqrt{3}(V_{pp\sigma}-V_{pp\pi})\sin\alpha\sin\beta.$ (15)

As can be seen, if all the symmetries are preserved, the subspace of the
$p_{z}$ orbital is separated from the subspace of $p_{x}$ and $p_{y}$
orbitals. However, if the vertical reflection symmetry is broken, it allows
for the hybridization between them, which can be given by
$t_{p_{i}p_{z}}=\lambda_{h}n_{i}=-t_{p_{z}p_{i}}$

$\displaystyle h_{p_{x},p_{z}}(k_{x},k_{y})=$
$\displaystyle\frac{1}{2}\lambda_{h}(e^{i(\alpha+\beta)}-e^{-i(\alpha+\beta)}+e^{i(\alpha-\beta)}-e^{-i(\alpha-\beta)}+2e^{2i\alpha}-2e^{-2i\alpha})$
(16) $\displaystyle h_{p_{x},p_{z}}(k_{x},k_{y})=$
$\displaystyle-2i\lambda_{h}(\sin{2\alpha}+\sin\alpha\cos\beta)$ (17)
$\displaystyle h_{p_{y},p_{z}}(k_{x},k_{y})=$
$\displaystyle\frac{\sqrt{3}}{2}\lambda_{h}(e^{i(\alpha+\beta)}-e^{-i(\alpha+\beta)}-e^{i(\alpha-\beta)}+e^{-i(\alpha-\beta)})$
(18) $\displaystyle h_{p_{y},p_{z}}(k_{x},k_{y})=$ $\displaystyle
2\sqrt{3}i\lambda_{h}\cos\alpha\sin\beta.$ (19)

We can also break inversion symmetry by breaking $\sigma_{v}$ while still
respecting the three-fold rotation symmetry. For this purpose, we can
introduce a hopping given by
$t^{v}_{p_{y},p_{x}}=\lambda_{v}\cos(3\phi)=-t^{v}_{p_{x},p_{y}}$, so that

$\displaystyle h^{v}_{p_{x},p_{y}}(k_{x},k_{y})=$
$\displaystyle\lambda_{v}(-e^{i(\alpha+\beta)}+e^{-i(\alpha+\beta)}-e^{i(\alpha-\beta)}+e^{-i(\alpha-\beta)}+e^{2i\alpha}-e^{-2i\alpha})$
(20) $\displaystyle h^{v}_{p_{x},p_{y}}(k_{x},k_{y})=$
$\displaystyle\lambda_{v}(2i\sin{2\alpha}-4i\sin\alpha\cos\beta)$ (21)
$\displaystyle h^{v}_{p_{x},p_{y}}(k_{x},k_{y})=$ $\displaystyle
4i\lambda_{v}\sin{\alpha}(\cos\alpha-\cos\beta).$ (22)

##  Appendix B Low Energy Hamiltonian

To better understand how phase IV is related to the appearance of an orbital
Hall plateau in the absence of spin Hall conductivity, it is helpful to look
at a simplified low-energy Hamiltonian. Since the original Hamiltonian,
$H_{0}$, is written in a basis that spans states with orbital angular momentum
quantum numbers $m_{l}$ of $-l$, $0$, and $l$, we can map it to a basis of $p$
orbitals, and rewrite it in terms of the orbital angular momentum operators
$L_{x}$, $L_{y}$, and $L_{z}$. We can then expand this Hamiltonian around the
Dirac points, which is relevant for phase IV:

The Hamiltonian for each valley $H_{K(K^{\prime})}(q_{x},q_{y})$, with
$\vec{q}=\vec{k}-\vec{K}(\vec{K}^{\prime})$ reads as

$\displaystyle H_{K}(q_{x},q_{y})=$
$\displaystyle\left(-\frac{3}{2}(t_{11}+t_{22})+\epsilon_{2}\right)L_{z}^{2}-3\sqrt{3}t_{12}\tau_{z}L_{z}+\lambda
s_{z}L_{z}+(-3t_{0}+\epsilon_{1})\left(\frac{L^{2}}{l(l+1)}-L_{z}^{2}\right)+\frac{3\sqrt{3}}{2}t_{2}\tau_{z}(q_{x}L_{x}-q_{y}L_{y})$
$\displaystyle+\frac{3}{2}t_{1}\left\\{L_{z},(q_{x}L_{x}-q_{y}L_{y})\right\\}-\frac{3\sqrt{3}}{4}(t_{11}-t_{22})\left(L_{x}(q_{x}L_{x}+q_{y}L_{y})+L_{y}(q_{x}L_{y}-q_{y}L_{x})\right)$
(23)

where $\tau_{z}=\pm 1$ represents the different valleys and $s_{z}$ is the
spin operator. From the equation above, it is clear that up to first order in
$q_{x}$ and $q_{y}$ the spin-orbit coupling does not produce any spin texture.
On the other hand, the last term of the Hamiltonian, proportional to the
difference $t_{11}-t_{22}$, is connected to an orbital Rashba coupling while
the terms proportional to $t_{1}$ and $t_{2}$ incorporate an orbital
Dresselhaus coupling. Furthermore, $t_{12}$ is responsible for a strong
orbital-valley locking when it dominates over other contributions. It becomes
clear that the spin-valley coupling that is seeing in TMDs is mediated by an
orbital-valley coupling

Figure 10: Spin (a) and orbital (b) character of the energy bands of a
nanoribbon with breadth $W=50\vec{a}_{2}$ for a system in the phase IV. (c)
Spin (orange) and orbital (green) conductivity for a system in the phase IV.

To test the new low-energy Hamiltonian, we first present the band structure
for the parameters of the three bands model of MoS2 in figure 10 in the
vicinity of $K$ (a) and $K^{\prime}$ Dirac points (b). The band structure is
consistent with both the tight-binding and the two-bands low energy model,
keeping the spin character of the bands in the two Dirac points while
explicitly taking into account the orbital characters of the three states in
$K$ and $K^{\prime}$. figure 10c-d show the band structure for the parameters
used in Phase IV and it is able to also provide a consistent description of
the bands.