# Layer dependent topological phases and transitions in TaRhTe4: From
monolayer and bilayer to bulk

Xiao Zhang<EMAIL_ADDRESS>Leibniz IFW Dresden, Helmholtzstr. 20,
01069 Dresden, Germany  Institute of Theoretical Physics and Würzburg-Dresden
Cluster of Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden,
Germany  Ning Mao  Max Planck Institute for Chemical Physics of Solids,
D-01187 Dresden, Germany  Oleg Janson  Leibniz IFW Dresden, Helmholtzstr. 20,
01069 Dresden, Germany  Jeroen van den Brink  Leibniz IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany  Institute of Theoretical Physics and
Würzburg-Dresden Cluster of Excellence ct.qmat, Technische Universität
Dresden, 01062 Dresden, Germany  Dresden Center for Computational Materials
Science (DCMS), TU Dresden, 01062 Dresden, Germany  Rajyavardhan Ray
<EMAIL_ADDRESS>Leibniz IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany  Dresden Center for Computational Materials Science (DCMS), TU
Dresden, 01062 Dresden, Germany  Department of Physics, Birla Institute of
Technology Mesra, Ranchi, Jharkhand, India - 835215

###### Abstract

The recently synthesized ternary quasi-2D material TaRhTe4 is a bulk Weyl
semimetal with an intrinsically layered structure, which poses the question
how the topology of its electronic structure depends on layers separations.
Experimentally these separations may be changed for instance by intercalation
of the bulk, or by exfoliation to reach monolayer or few-layer structures.
Here we show that in the monolayer limit a quantum spin Hall insulator (QSHI)
state emerges, employing density functional calculations as well as a minimal
four-orbital tight-binding model that we develop. Even for weak spin-orbit
couplings the QSHI is present, which has an interesting edge state that
features Rashba-split bands with quadratic band minima. Further we find that a
weak topological insulator (WTI) manifests in the bilayer system due to
sizable intralayer hopping, contrary to the common lore that only weak
interlayer interactions between stacked QSHIs lead to WTIs. Stacked bilayers
give rise to a phase diagram as function of the interlayer separation that
comprises a Weyl semimetal, WTI and normal insulator phases. These insights on
the evolution of topology with dimension can be transferred to the family of
layered ternary transition metal tellurides.

##  I Introduction

Topological insulators (TIs) and topological semimetals are new classes of
quantum materials [1, 2, 3, 4, 5, 6] with potential applications in
spintronics and quantum computation [7, 8, 9]. TIs feature a nontrivial bulk
insulating gap accompanied by topologically robust gapless edges for two-
dimensional (2D) TIs or surface states in three-dimensional (3D) TIs with
linearly dispersing double degenerate bands meeting at Dirac points [10, 11,
12, 13]. 2D TIs are characterized by a $\mathcal{Z}_{2}$ topological index and
are also known as quantum spin Hall insulators (QSHI) [14, 15, 16, 17]. For
3DTIs, however, four indices $\mathcal{Z}_{2}=(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$
are required based on which they can be classified as strong ($\nu_{0}=1$) or
weak ($\nu_{0}=0$; $\nu_{i}=1$ for any $i$) [18, 19]. Correspondingly, strong
(weak) TIs possess surface states with an odd (even) number of Dirac bands on
all (selected) facets [20, 21, 22]. Arguably, the simplest weak TI (WTI) is
obtained by stacking of QSHIs whereby the facets perpendicular to the stacking
direction are free of Dirac nodes (trivial) [11, 23].

Weyl semimetals (WSMs), on the other hand, host pairs of topologically
protected linear band crossings, called Weyl points (WPs) [4, 5, 24]. Each WP
is characterized by a topological charge, chirality $\chi=\pm 1$ [25, 26, 27].
The surface states of WSMs are Fermi arcs that terminate at the projections of
bulk WPs of opposite chirality [28, 29, 30].

In recent years, a large number of TIs and WSMs have been experimentally
realized, for example, monolayer $1T^{\prime}$-WTe2 as QSHI [31, 17]; Bi2Se3
[32, 33] and Bi2Te3 [34, 35] as 3D strong TIs; ${\rm Bi_{14}Rh_{3}I_{9}}$ [36]
and ${\rm Bi_{12}Rh_{3}Cu_{2}I_{5}}$ [37] as WTIs; TaAs, [38, 39] WTe2 [40,
41], MoTe2 [42, 43] and TaIrTe4 [44, 45] as as WSMs. Many such compounds are
layered materials with van der Waals (vdW) type interlayer interactions and
exhibit different properties as a function of layer thickness [46, 47, 48, 49,
50]. Of particular relevance are binary and ternary tellurides known to be WSM
in the bulk while the monolayers are QSHIs. From the practical angle, QSHIs
with large bulk topological gap for room temperature applications and/or WSMs
with minimal symmetry-allowed WPs near the Fermi energy are desired. At the
same time, lack of a clear understanding of the topological phase transitions
as a function of number of layers is detrimental to possible application in
development of efficient quantum devices.

Figure 1: Crystal structures of bulk and monolayer TaRhTe4. (a) Crystal
structure of bulk and (b) monolayer TaRhTe4 (top view) where the unit cell is
shown in solid black lines. In the latter, the sublattices A and B are also
shown. The orbitals of Ta(Te)-atoms marked by 1A and 1B (2A and 2B) mainly
contribute to minimal tight-binding model. The effective inversion center,
marked by a cross, lies at the center of this Ta-Te plaquette. (c) Brillouin
zone (BZ) of monolayer TaRhTe4 along with the high symmetry points. (d)
Reduced four-site lattice for the tight-binding model with sublattices A and
B. The arrows depict the hopping terms ($s_{0}$) and the (spin-orbit coupling)
SOC terms ($s_{z}$) of the model.

Here, we focus on the recently synthesized layered ternary quasi-2D material
TaRhTe4 [51, 52, 53] and elucidate the topological electronic properties of
different layered structures, from monolayer and bilayer to bulk using density
functional theory (DFT). TaRhTe4 is isostructural to TaIrTe4 [45, 44, 54] and
both share the same nonsymmorphic space group as WTe2 [24, 40, 55], all of
which are type-II WSMs in bulk. In comparison, the monolayers of TaIrTe4 and
$1T^{{}^{\prime}}$-WTe2 are QSHIs with time-reversal protected edge states
[56, 57, 58, 31]. Interestingly, the bilayer TaIrTe4, owing to its relatively
small interlayer separation, has been predicted to be a QSHI as well [50].
With increasing interlayer separation, there is a phase transition to a
trivial normal insulator (NI) phase. Although remarkable, the QSHI ground
state in the bilayer TaIrTe4 is at odds with the common understanding that
interlayer coupling between stacked QSHIs gaps out the edge states, leading to
a trivial NI phase [25]. Furthermore, it is unknown which topological phases
transitions (TPTs) occur while increasing the layer thickness from the
monolayers and bilayers to the bulk in any layered chalcogenide, which can be
important to bring to fruition the envisioned technological applications of
these materials.

In this work, we establish that the ambient strucutres of monolayer as well as
bilayer TaRhTe4 are QSHI. The monolayer has a bulk gap of $\sim 65$ meV,
indicative of a largely insulating behavior at room temperature, while the
bilayer is a zero gap semimetal with an average direct gap of $\sim 50$ meV
around $\Gamma$. We obtain a minimal 4-orbital tight binding (TB) model for
the monolayer which reveals that the QSHI phase is driven by hybridization
between Ta-$5d_{x^{2}-y^{2}}$ and Te-$5p_{x}$ orbitals. Our analysis of the
interlayer and intralayer hoppings for the bilayer QSHI elucidates the
importance of interlayer hoppings in the stabilization of the QSHI phase,
suggesting that the bilayer TaRhTe4 behaves effectively as a single electronic
entity despite vdW separated physical layers in the unit cell.

Finally, the bulk TaRhTe4 is confirmed to be a WSM with three quartet WPs
close to Fermi energy arising from band crossings between the top of the
valence band and the bottom of the conduction band (bands $N,N+1$ where $N$ is
the number of valence electrons). Topological phase transitions between WSM,
WTI, and (NI) phase are realized by calibrating the intra- and inter-bilayer
distances, culminating in a rich phase diagram. The topological phase
transition of WTI-NI is the same as that in the bilayer TaRhTe4. Phase
transitions of WSM-WTI and WSM-NI open up a band gap near the Fermi level and
are accompanied by creation and annihilation of WPs. Interestingly, we observe
that the WPs remain confined in the $k_{z}=0$ plane for WSM-NI as well as WSM-
WTI transitions.

##  II Monolayer system

###  II.1 Crystal structure and symmetries

The three-dimensional bulk TaRhTe4 crystallizes in noncentrosymmetric
$Pmn2_{1}$ (31) space group, shown in Fig. 1(a), similar to WTe2 and TaIrTe4.
Each unit cell consists of two AB-stacked TaRhTe4 formula units and possess
mirror $M_{x}$, glide mirror
$\widetilde{M}_{y}=t[(\mathbf{e}_{x}+\mathbf{e}_{z})/2]M_{y}$, and two fold
screw $\widetilde{C}_{2z}=t(\mathbf{e}_{z}/2)C_{2z}$ symmetries. Here,
$t(\mathbf{e}_{\mu})$ represents tranlations by a lattice constant along
$\mathbf{e}_{\mu}$, $M_{x}$ represents a mirror plane with normal along
$\mathbf{e}_{x}$ direction while $C_{2z}$ represents 2-fold rotational
symmetry about the $\mathbf{e}_{z}$ axis.

The 2D monolayer TaRhTe4 was constructed out of the bulk crystal structure by
inserting large vacuum ($\gtrsim 20$ Å) between the AB-stacked layers of the
bulk unit cell (see Methods for details). Figure 1(b) shows the top view of
monolayer TaRhTe4, Ta and Rh atoms form zigzag chains along the $a$ axis. The
resulting structure is dynamically stable [see Supplemental Information (SI)
[59] for the phonon dispersion]. In comparison to bulk, an effective inversion
symmetry can be found in the monolayer TaRhTe4. Apart from the time reversal
symmetry $\mathcal{T}$ and lattice translations
$\\{t(\mathbf{e}_{x}),t(\mathbf{e}_{y})\\}$, the monolayer structure also has
a glide reflection $\widetilde{M}_{x}=t(\mathbf{e}_{y}/2)M_{x}$ and a two-fold
screw rotation $\widetilde{C}_{2x}=t(\mathbf{e}_{x}/2)C_{2x}$. The product of
these two generators is exactly the spatial inversion symmetry
$\widetilde{M}_{x}\widetilde{C}_{2x}=\mathcal{I}$ [58, 31, 60, 45]. The
corresponding inversion center lies at the center of the line joining the Ta
atoms. Consequently, the monolayer unit cell can be thought of as composed of
sublattices A and B atoms in the unit cell related to each other via the
effective inversion symmetry, as shown in Fig. 1(b). The corresponding BZ
along with the high symmetry points are shown in Fig. 1(c).

###  II.2 Electronic properties

In Fig. 2(a), the band structure, obtained within scalar relativistic
approximations (‘no SOC’), of monolayer TaRhTe4 along a high-symmetry path in
BZ is shown. It is a Dirac semimetal with two Dirac nodes located at
$\Sigma=(\pm 0.0833,0.253)\,$Å-1 along Y-S path, as expected for a non-
symmorphic crystal structure [31]. By analyzing the density of states (not
shown), we found that Ta-$5d$ and Te-$5p$ orbitals dominate around the Fermi
level, whereas the contribution of Rh-$4d$ orbitals is negligible. We thus
show the band-weight contributions from Te-$5p_{x}$ and Ta-$5d_{x^{2}-y^{2}}$
orbitals in Fig. 2(a). The overlap of orange and blue dots implies a $p$-$d$
hybridization around the Dirac cone, which is analogous to the case in
monolayer TaIrTe4 [50]. It is important to note that band inversion is present
even without SOC, similar to the case of $1T^{{}^{\prime}}$-WTe2 [58, 31, 60].

Figure 2: Electronic properties of monolayer TaRhTe4. Band structure and
orbital contributions close to the Fermi energy $\epsilon_{\rm F}$ in
monolayer TaRhTe4 (a) without (‘no SOC’) and (b) with SOC. The inset shows a
zoom into the Dirac cone. (c) Wilson loop spectrum for the full DFT band
structure when the BZ loop is chosen along $k_{x}$ direction. (d) Surface
spectral function for the termination shown in Fig. 1(b). The edge states are
marked by green arrows.

As shown in Fig. 2(b), SOC lifts the degeneracy of the Dirac point and opens
up a gap $E_{g}\sim 65$ meV which is sufficiently large for room-temperature
QSHI applications. Due to the effective inversion symmetry and time reversal
symmetry, each band is doubly degenerate. Further, the Wilson loop spectrum
[61] in Fig. 2(c) indicates that the bands are topologically nontrivial with
the topological invariant $\mathcal{Z}_{2}=1$, implying a QSHI ground state
characterized by presence of helical edge states protected by the time-
reversal symmetry $\mathcal{T}$.

Figure 2(d) displays an edge state spectrum of monolayer TaRhTe4 in a
nanoribbon geometry for the edge terminations marked by arrows in Fig. 1(b).
Indeed, a pair of counter-propagating edge states cross at $\bar{\Gamma}$ and
carry opposite spin polarization. It is interesting to note that the
topological edge Dirac cone lies within the bulk gap and resembles Rashba
splitting of edge states with parabolic band minima akin to the nearly free
electron dispersion. However, the energy position of the Dirac point is
sensitive to the nanoribbon/edge termination [see SI [59] for details] as well
as edge disorder [57].

###  II.3 Four-band minimal tight-binding model

Given the effective band inversion in monolayer TaRhTe4 even without SOC, the
role of SOC in the realization of the QSHI state is a priori not clear. To
further understand the electronic properties of monolayer TaRhTe4, especially
the role of SOC, we obtain a minimal tight-binding (TB) model. A minimal TB
model is also helpful in investigations of transport properties, studies of
disorder effects among others [62, 57]. As shown in the blow-up of Fig. 2(a),
the orbital contributions to states close to the Fermi energy are
predominantly from Ta-$5d_{x^{2}-y^{2}}$ orbitals and $5p_{x}$ orbitals
centered at one of the Te atom marked by atoms 1 and 2 in Fig. 2(a) and their
sublattice symmetry partners ($A/B$). We, therefore, construct a maximally
projected Wannier model using a 4-orbital basis set consisting of these
orbitals (see Methods for details).

Figure 3: TB model for monolayer TaRhTe4. Band structures obtained from the
4-orbital TB model for (a) without and (b) with SOC along with the orbital
contributions. The DFT bands are shown with black dashed lines for comparison.
(c) Wilson loop spectra for the TB model (blue curve), and for the model where
the SOC terms are scaled down to 10% of the real values (purple color).

Figure 3(a) shows the orbital-resolved TB band structure without SOC terms.
For comparison, the corresponding DFT band structure is also shown (dashed
lines). The Wannier TB model successfully reproduces the bulk Dirac cones as
well as the band inversion in the vicinity of the Fermi level, that are the
two characteristic features of the Generalized Gradient Approximation (GGA)
band structure within the scalar relativistic approximation (‘no SOC’). To
reproduce the the Dirac node, however, we need to include hopping terms up to
the third-nearest neighbors (up to $\sim 20$Å). The dominant hopping terms are
shown in Fig. 1(d). We note that, compared to the DFT band structure, the
location of the Dirac point in BZ is shifted towards (but not exactly at) the
Y point regardless of the number of neighbors considered in the model. The
details of the Hamiltonian and the corresponding TB parameters are provided in
Methods.

Upon inclusion of SOC terms which preserve the time reversal symmetry
$\mathcal{T}$, glide mirror symmetry $\widetilde{M}_{x}$, glide screw
$\widetilde{C}_{2x}$ symmetry, and therefore the effective inversion symmetry
$\mathcal{I}$, the resulting TB Hamiltonian matrix is $8\times 8$ and consists
of three additional terms (see Methods). The corresponding TB band structure
along with the DFT band structure (with SOC) is shown in Fig. 3(b). In the TB
and the DFT band structure alike, each band is doubly degenerate. The
fundamental gap of minimal model is $\sim 53$ meV, somewhat smaller than the
DFT result. However, the band inversion around Y point is reproduced
reasonably well. Most importantly, the Wilson loop spectrum in Fig. 3(c)
indicates that the minimal TB model captures the QSHI ground state.

In order to clarify the role of SOC in driving the QSHI phase in monolayer
TaRhTe4, we numerically/artificially tune the SOC strength in the 4-orbital
minimal TB model. Specifically, we considered the situation when the effective
strength of SOC, $\lambda$, is only 5% and 10% of the value in the material-
based minimal model, $\lambda_{0}$ (equivalently, in the minimal TB model with
parameters tabulated in Table 2). Figure 3(d) shows the Wilson loop
calculations for the latter case, $\lambda/\lambda_{0}=0.1$. Remarkably, we
find $\mathcal{Z}_{2}=1$ even in this case, implying that the SOC plays a
secondary role in driving the QSHI phase in the monolayer TaRhTe4.

##  III Bilayer system

Figure 4: Structure and electronic properties of bilayer TaRhTe4. (a) Side
view of AB stacking TaRhTe4. For ease of discussion, all the atoms have been
numbered. The interlayer distance is defined as $d$, where the equilibrium
interlayer distance $d_{0}=6.58$ Å. (b) Layer resolved band structure of
bilayer TaRhTe4 without SOC. (c) Energy bands of bilayer TaRhTe4 with SOC. (d)
Topological $\mathcal{Z}_{2}$ invariant as a function of the interlayer
distance $\Delta d$.

A bilayer TaRhTe4 system was constructed from the bulk unit cell in a fashion
similar to the monolayer (see Methods for details). The unit cell consists of
two AB-stacked layers. The structure is shown in Fig. 4(a). Compared to the
monolayer, the bilayer breaks the screw symmetry due to the displacement
between the layers in the $\mathbf{e}_{x}$-$\mathbf{e}_{y}$ plane. Thus, the
effective inversion symmetry is also broken and the space group consists only
of mirror reflection $M_{x}$.

###  III.1 Role of interlayer separation

Figure 4(b) shows the band structure of bilayer TaRhTe4 with layer-resolved
orbital contributions along a high symmetry path for the scalar relativistic
case (no SOC). The highest-lying valence band $N$, where $N$ is the number of
valence electrons, and lowest-lying conduction band $N+1$ are quite close
along $\Gamma$-X and Y-S but do not cross.

When SOC is included, bands are no longer doubly degenerate, shown in Fig.
4(c). The ground state is a zero-gap semimetal with direct band gaps around
the $\Gamma$ point. The average value of the direct gap is $\sim 50$ meV (min:
$\sim 30$ meV, max: $\sim 100$ meV). In contrast to the monolayer case. the
band degeneracy is lifted due to the lack of effective inversion symmetry.
Yet, the bilayer TaRhTe4 also hosts a nontrivial topological ground state. As
the system is quasi-2D but contains two vdW separated layers, the topological
index $\mathcal{Z}_{2}=(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ based on the Wannier
center evolution [63] was used (see Methods). $\mathcal{Z}_{2}=(0;001)$,
obtained for the bilayer TaRhTe4, indicates the presence of edge states on the
facets whose normal is perpendicular to the layer direction and corresponds to
a QSHI ground state. Strictly speaking, the ground state is a WTI, but
corresponds to a 2D QSHI due to the quasi-2D structure of the bilayer. In SI,
we provide an edge state spectrum of bilayer TaRhTe4 in a nanoribbon geometry.

With increasing interlayer separation $d=d_{0}+\Delta d$, where $d_{0}=6.58$ Å
is the interlayer separation in the bulk 3D compound, a topological phase
transition to a trivial insulator occurs at $\Delta d=0.23$ Å ($\Delta
d/d_{0}=3.5\%;d=6.81$Å); see Fig. 4(d)). This behavior is similar to the Ir
compound, TaIrTe4, and has been attributed to the strong interlayer coupling
[50].

Interestingly, in TaRhTe4, the ground state is insensitive to the stacking
pattern of the layer in the bilayer unit cell. We find that the AA stacked
bilayer with no relative displacement in the $\mathbf{e}_{x}$-$\mathbf{e}_{y}$
plane between the layers, and thus retains the screw symmetry and inversion
symmetry, also hosts a QSH ground state. The AA stacked TaRhTe4 transitions to
a trivial insulator when the interlayer distance is larger than $6.42$ Å.
Although the AA stacked structures are deemed unstable [31], efforts are being
made to manipulate the interlayer stacking in some transition metal
dichalcogenides [64]. This is especially relevant since ternary composition
offers a wider possibility to manipulate the composition and structure.
Additionally, this also allows access to the QSH state in bilayer systems for
different interlayer interactions [65].

###  III.2 Intralayer and interlayer hoppings

In order to gain further insights into the topological phase transition in
bilayer TaRhTe4, we analyze the dependence of intra- and interlayer hopping
parameters on the interlayer distance based on a highly accurate $44$-orbital
Wannier model. (see Methods for details). Figure 5(a) shows selected dominant
intralayer hoppings between Ta-Ta and Ta-Te atoms (see SI [59] for further
details). The hoppings between Ta-Te atoms are typically $\sim 1$ eV while Ta-
Ta hoppings are an order of magnitude smaller. As could be expected, the
intralayer hoppings are almost unchanged with increasing the interlayer
separation $d$.

Figure 5: Intra- and interlayer hopping strengths in bilayer TaRhTe4. Dominant
hopping strengths as a function of interlayer distance: (a) intralayer
hoppings (b) interlayer hoppings. The atoms in the layer-1 and layer-2 of
bilayer structure are marked by numbers in Fig. 4(a). The vertical black line
represents the topological phase transition boundary between QSHI and NI
phases.

In comparison, the dominant interlayer hoppings range between $\sim 10$ and
$\sim 100$ meV , as shown in Fig. 5(b). With increasing interlayer separation
$d$, the orbital overlaps decrease and eventually become zero as $\Delta
d/d_{0}\gg 1$. Some of these terms, in fact, vanish even at $\Delta d\sim 1.5$
Å ($\Delta d/d_{0}\sim 18.6\%$). Evidently, the interlayer couplings are
reasonably strong to drive the system into a QSHI phase, and decay smoothly
across the topological phase boundary.

The interlayer hopping driven QSHI ground state in bilayer TaRhTe4 is somewhat
counter-intuitive as coupling between the Dirac cones from each (mono)layer
are expected to gap them out rendering a trivial ground state. The large
interlayer hoppings are a result of enhanced bandwidths of the monolayer
densities of states. The QSHI phase is, however, stabilized by the presence of
additional band inversion between the layer-resolved orbital contributions at
around Y point, see Fig. 4(c). As shown in the Supplementary Fig. S4, with
increasing interlayer separation, the contribution by layer-1 to the topmost
valence band decreases and the contribution to the lowest valence band
increases, while the behavior of layer-2 is just the opposite. Across the
phase transition point, the energy gap near the Y point closes and reopens,
with significant modifications in the band weight contributions from layer-1
and layer-2. After crossing the phase boundary, the contribution of both the
van der Waals separated layers tend to equate in the conduction and valence
band edges.

Furthermore, it would be expected that systems with odd number of layers would
be QSHIs akin to the monolayer TaRhTe4. Remarkably, the QSHI bilayer systems
implies that few-layer systems with an even number of layers per unit cell
would likely also host a QSHI ground state.

Owing to presence of a robust QSHI ground state in monolayer and bilayer
systems, we obtain the exfoliation energies for both of them. It is found that
the exfoliation energy for monolayer TaRhTe4 is $2.85$ meV/Å2, while the
energy for bilayer system is $1.75$ meV/Å2 (see SI). For comparison, we also
computed the exfoliation energy for graphene in the same way as mentioned in
the Methods for reference, and find it to be $1.3$ meV/Å2 comparable to the
experimental data [66]. The exfoliation energy of bilayer TaRhTe4 is
comparable to that of graphene, while the energy of monolayer system is about
2 times larger than that of graphene, rendering these systems particularly
appealing for possible technological applications.

##  IV Topological phase transitions: from bilayer to bulk

###  IV.1 Weyl semimetal phase in 3D TaRhTe4

The unit cell of the 3D bulk TaRhTe4 is structurally equivalent to stacked
bilayers as shown in Fig. 1(a). We define the inter-bilayer distance as $c$
and intra-bilayer distance as $d$. The space group and symmetry of the 3D
structure have been discussed earlier. The band structure with SOC effects
included along high symmetry paths is shown in the Supplementary Fig. S7. The
top of the occupied band (band $N$ where $N$ is the number of valence
electrons in the system, blue curve) leads to three-dimensional hole pockets
along $\Gamma$-X and $\Gamma$-Y while band $N+1$ (red curve) for electron
pocket around $\Gamma$.

The DFT calculations confirm that 3D TaRhTe4 is a Weyl semimetal [67]. Within
GGA, we obtain three quartets of Weyl points (WPs) between the highest
occupied band (blue curve) and the lowest unoccupied band (red curve). All the
WPs lie in the $k_{z}=0$ plane and are 4-fold degenerate due to symmetry.
Table 1 lists the position and energy of these WPs alongside their chirality.

Table 1: Positions, Chirality ($\chi$), and energies of the Weyl points formed between bands $N$ and $N+1$ in bulk TaRhTe4, where $N$ is the number of valence electrons. All WPs are four-fold degenerate, related by the symmetries $C_{2}(z)$ and $M_{x}$. | $(k_{x},k_{y},k_{z})\frac{2\pi}{a}$ | $\chi$ |  $E$ (meV)
---|---|---|---
$W_{1}$ | $(0.150553,0.105029,0)$ | $-1$ | $125.2$
$W_{2}$ | $(0.019475,0.098853,0)$ | $+1$ | $-43.0$
$W_{3}$ | $(0.025053,0.066748,0)$ | $+1$ | $-63.7$

###  IV.2 Phase diagram with intra- and inter-bilayer distance

We now turn our attention to possible topological phase transitions from a
monolayer and bilayer QSHI to the 3D WSM phase in TaRhTe4. This is achieved by
tuning the intra-bilayer ($d=d_{0}+\Delta d$) and inter-bilayer distance
($c=c_{0}+\Delta c$), where $c_{0}=d_{0}$ represents the interlayer separation
for the bulk 3D TaRhTe4 [see the inset of Fig. 6(a)]. Several interesting
possibilities emerge: (i) In the limit of $c,d\rightarrow\infty$, one would
expect to recover the topology of the monolayer, while (ii) $c=d=c_{0}$
represents the bulk 3D system. (iii) $\Delta d=0$, $\Delta c=\infty$
corresponds to a WTI since it can be understood as a stacking of bilayer
TaRhTe4 which is a QSHI, along $z$ axis. (iv) $\Delta c\neq\Delta d\neq 0$
represents a generic situation. Due to the large layer separation, the
un-/weakly-coupled layers contained in an unit cell are likely to lead to a
topological trivial phase.

We consider a sizable range for the inter- and intralayer separation, up to
$\Delta c/c_{0}=\Delta d/d_{0}=51.7\%$. It is important to note that while not
all the points in the phase diagram may not be experimentally achievable, a
sizable region of the phase diagram is relevant for intercalated and/or
exfoliated systems for which the TPTs and their boundaries are not known.
Additionally, the phase diagram must be symmetric about the $c=d$ diagonal and
computation of the ground state properties in the bottom triangular region of
the $c-d$ phase digram would suffice.

Figure 6: Phase diagram: Topological phase transitions from bilayer to bulk
TaRhTe4 as a function inter- and intra-bilayer separations. (a) Left panel:
Phase diagram of 3D TaRhTe4 in the $c-d$ plane. Ground state at each point
(filled circles) is obtained from DFT calculation. The black dashed curves are
a guide to the eye for the phase boundaries. Paths I, II and III highlight the
WSM-WTI, WSM-NI, and WTI-NI phase boundaries, respectively. Right panel: side
view of TaRhTe4with two unit cells, where the inter-bilayer distance
$c=c_{0}+\Delta c$ and the intra-bilayer distance $d=d_{0}+\Delta d$. (b) The
number of WPs as a function of $\Delta s$ along Path II and Path III, where
$\Delta s=\sqrt{{\Delta c}^{2}+{\Delta d}^{2}}$. (c) The coordinates of WPs in
BZ selected from Path III of phase diagram. The corresponding chosen points
are marked in (b). The red crosses represent the positions in the BZ where the
WPs annihilate.

The resulting phase diagram, shown in Fig. 6(a), is quite rich and consists of
topological insulator (QSHI/WTI), WSM and NI regions. Computations were
carried out only in the bottom triangle of phase diagram, because $\Delta c$
and $\Delta d$ are symmetric along the layered direction. All the structures
computed are represented by filled circles. The phase boundaries are only a
guide to the eye and encompass the phase transitions discussed above.

To begin with, for $\Delta c/c_{0}=\Delta d/d_{0}\gg 1$, the unit cell is
equivalent to well-separated non-interacting monolayers. The expected ground
state is thus a QSHI. From computations of the 3D $\mathcal{Z}_{2}$ invariant
of the considered largest value of $\Delta c/c_{0}=\Delta d/d_{0}=51.7\%$,
however, we find a trivial NI ground state. On the other hand, for $\Delta
d=0$ and $\Delta c/c_{0}\gg 1$ corresponds to a bilayer system and the ground
state is a WTI as discussed earlier. For such a suitable but fixed $\Delta
c/c_{0}$, varying $\Delta d$ (Path I in Fig. 6(a)) leads to a topological
phase transition between the WTI and the NI phase, i.e. from
$\mathcal{Z}_{2}=(0;001)$ to $\mathcal{Z}_{2}=(0;000)$ for $\Delta c=2.4$ Å
and $\Delta d=0.23$ Å, i.e., $\Delta c/c_{0}=36.5\%$ and $\Delta
d/d_{0}=3.5\%$, and was discussed above in detail.

Similarly, for $\Delta c=\Delta d=0$, corresponding to the bulk 3D TaRhTe4,
the ground state is a WSM. Keeping $\Delta d=0$ fixed, and increasing $\Delta
c/c_{0}$ corresponds to a structural phase transition from the bulk 3D
structure to the AB-stacked bilayer TaRhTe4. By the same token, a topological
phase transition from the WSM phase to the WTI ground state is found, as shown
in Path II along $\Delta d=0$.

The WSM phase extends to regions of small $\Delta c$ and $\Delta d$, whereas
WTI is located in the area with large $\Delta c$ but small $\Delta d$. With
large $\Delta c/c_{0}$ and $\Delta d/d_{0}$, the TaRhTe4 system becomes a NI.
Naturally, band gap opens when the phase transitions of WSM-WTI and WSM-NI
occur. At the same time, the inter- and intra-bilayer separation induces
creation and annihilation of WPs in the WSM phase.

To further illustrate these topological phase transitions, we consider paths
II and III which, respectively, highlight the WSM-WTI and WSM-NI topological
phase transitions. Figure 6(b) shows the number of Weyl points as a function
of $\Delta s$ along these paths. Because of the time-reversal symmetry, Weyl
nodes are always created and annihilated as quartets. Along Path II, the
number of WPs in TaRhTe4 decreases to 4 before the phase transition, and
completely annihilated at around $\Delta s=\sqrt{\Delta c^{2}+\Delta
d^{2}}\sim 2\,$Å.

Similar behavior can be observed along Path III. A band gap opens up already
at around $\Delta s\sim 1.7$ Å. Due to $c=d$, all symmetries in equilibrium 3D
structure of bulk TaRhTe4 are preserved. Therefore, all the Weyl points are
still constrained in $k_{z}=0$ plane. In Fig. 6(c), we show the evolution of
the WPs along path III. For brevity, we depict the coordinates of WPs for
selected values of $\Delta s$. the three quartet of WPs found in the bulk
TaRhTe4, for $\Delta s=0$, are denoted by orange dots. Their positions in the
BZ are listed in Table 1. As $\Delta s$ increases, the quartet $W_{1}$ moves
continuously closer to the origin and eventually annihilate each other on the
$k_{x}$ axis. Concurrently, the WPs $W_{2}$ and $W_{3}$ move along the $k_{y}$
axis increasing their relative separations. At $\Delta s=0.5\,\text{\AA}$,
only two quartets ($W_{2}$ and $W_{3}$) survive and their positions in the BZ
is shown by green diamonds. Further increasing $\Delta s$ creates further WPs
[see Fig. 6(b)] close to the $k_{x}$ axis which annihilate rather quickly with
small changes in $\Delta s$ (not shown). Simultaneously, both quartet $W_{2}$
and $W_{3}$ also get annihilated. Increasing $\Delta s$ even further to
$\Delta s=1.7\,\text{\AA}$ creates a new quartet of WPs close to $k_{x}=0$
axis, shown by purple triangles which eventually annihilate each other at the
phase boundary between the WSM and the NI states.

##  V Conclusion & Outlook

Our investigation of the monolayer, bilayer and bulk TaRhTe4 systems using DFT
brings forth several interesting and important aspects of layered vdW
tellurides in general and TaRhTe4 in particular. First, monolayer TaRhTe4 is a
QSHI with 65 meV bulk gap, suggesting viability for room temperature
applications. At the same time, the minimal 4-orbital tight binding model
presented here will enable further investigations of disorder and geometric
effects for realistic samples.

Second, our analysis of the inter- and intralayer TB parameters clarifies
that, even if vdW separated, the bilayer effectively behaves as a single
electronic unit and hosts a QSHI ground state. Presence of QSHI ground state
for both monolayer and bilayer, as well as the insensitivity of the QSHI
ground state to the stacking pattern in the bilayer, suggests that even few
layer TaRhTe4 systems would also be QSHIs, and may also have interesting
consequences for several related compositions and compounds.

Third, and perhaps the most important, a rich phase diagram containing
QSHI/WTI, WSM and NI phases is obtained by varying the interlayer and inter-
bilayer separations, comprehensively cover a vast majority of possible
realistic situations under different synthesis conditions.

These insights are easily extendable not only to multi-/few-layer TaRhTe4 but
to binary and ternary tellurides in general, and intercalated and exfoliated
tellurides in particular. Moreover, the higher degree of control offered by
ternary compounds in terms of their composition and structure compared to
their binary counterparts suggest that these our findings will likely impact
the field of 2D chalcogenides significantly and drive not only future
theoretical and experimental investigations, but also open up new avenues for
quantum devices.

##  VI Methods

###  VI.1 Crystal structures

The bulk crystal structure consists of four formula units and two vdW
separated layers per unit cell. The lattice constants are $a=3.76$ Å,
$b=12.55$ Å, and $c=13.17$ Å [52]. The interlayer separation is
$c_{0}=6.58\,\text{\AA}$. The optimized atomic positions were used while the
lattice constants were kept fixed at experimental values [52].

Monolayer TaRhTe4 was constructed from the bulk unit cell by inserting a large
vacuum between the two layers in the bulk unit cell. It was found that a
vacuum $\gtrsim 15\,\text{\AA}$ does not influence the electronic properties
of the monolayer. The atomic positions of monolayer structure were first
optimized under $Pmn2_{1}$ symmetry such that the force on each atoms is less
than 1 meV/Å. Then the lattice constants were tuned to minimize the total
energy of the system. At last, the atomic positions were optimized again under
the new lattice constants. The in-plane lattice parameters are $a=3.72$ Å and
$b=12.42$ Å, which are slightly smaller than bulk values. The electronic bands
of final structure matches the results by using the structure from Materials
Cloud [68]. For comparison, the energy bands with optimized atomic positions
but unoptimized lattice constants are presented in SI [59].

Similarly, bilayer TaRhTe4 systems were prepared by inserting large vacuum
between the bilayers in the bulk unit cell. For the bilayer system with intra-
bilayer distance equal to the interlayer separation in the bulk 3D system, but
large inter-bilayer distance, the atomic positions were optimized with a force
threshold of 1 meV/Å for each atom. Other bilayer structures with different
intra-bilayer separations realized by inserting the vacuum directly in the
optimized structure. The resulting crystal structures for monolayer as well as
bilayer systems are presented in SI [59].

###  VI.2 DFT calculations

The DFT calculations were carried out for monolayer, bilayer and bulk TaRhTe4
systems using the Full-Potential Local-Orbital (FPLO) code [69], version 18.57
[70]. The Perdew-Burke-Ernzerhof implementation [71] of the GGA was employed
in all cases. For the numerical integration of monolayers and bilayers, we use
a $k$ mesh with ($20\times 10\times 1$) intervals in the BZ, and a $k$ mesh
with ($20\times 10\times 10$) intervals for bulk TaRhTe4. The linear
tetrahedron method was used for the numerical integration in the Brillouin
zone. As applicable, the scalar relativistic correction (‘no SOC’) was used,
while the SOC effects (‘with SOC’) were included via the 4-spinor formalism as
implemented in the FPLO code.

###  VI.3 Topological electronic properties

For the monolayer insulators, a highly accurate maximally projected Wannier
model was constructed which was used for the computation of the topological
properties. It was constructed by projecting the band structure onto a subset
of atomic-like basis functions (Wannier functions) within a given energy
window. We focused on a set of isolated bands across the Fermi energy, lying
between $-6.5$ eV and $+2.3$ eV, which have dominant contributions from the
Ta-$5d$ orbitals, Rh-$4d$ orbitals, and Te-$5p$ orbitals.

The topology of the band structure was determined by calculating the
$\mathcal{Z}_{2}$ index with via the Wilson loop method (own code) [61].
Sensitivity of the ground state to the Te positions was explicitly checked by
considering an inversion symmetric unit cell with relaxed atomic positions for
which the $\mathcal{Z}_{2}$ index matches with the corresponding value
obtained via the method of Wannier center evolution provided by FPLO,
suggesting the topology is robust against perturbations to Te positions.

The edge state spectra were obtained by computing the bulk projected bands
using nanoribbon geometry as implemented in pyfplo module of the FPLO code.
Two blocks of unit cell are used as ribbon width in all semi-slab
calculations. 200 points were sampled across the energy window spanning
$[-0.15,0.15]$ eV.

For the bilayers, with direct gap throughout the BZ, the topological
properties were ascertained from the
$\mathcal{Z}_{2}=(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ index for the 3D systems,
obtained via the Wannier center evolution algorithm [63] as implemented in the
FPLO code.

The Wannier tight-binding model comprises the Ta-$5d$ orbitals, Rh-$4d$
orbitals, and Te-$5p$ orbitals with energy window lying between $-7.0$ eV and
$+2.5$ eV. The basis set for the Wannier projections consists of 88 orbitals
for the scalar relativistic case (‘no SOC’). Correspondingly, the basis of
Wannier functions involved 176 orbitals when spin-orbit interactions were
considered. Because varying the interlayer separations does not significantly
change the distribution of bands, the energy window was kept between $-7.0$ eV
and $+2.5$ eV in all bilayer computations.

For the bulk TaRhTe4, on the other hand, the ground state is semimetallic. To
study its underlying topology of the electronic structure, we constructed a
Wannier model projecting the band structure onto Wannier functions for the
isolated manifold of bands between $-6.5$ eV and $+4.3$ eV, which have
dominant contributions from the Ta-$5d$, Rh-$4d$, $5s$, and Te-$5p$ orbitals.
Therefore, the basis set for the Wannier projections consists of 92 orbitals
for the scalar relativistic case. Correspondingly, the basis of Wannier
functions involved 184 orbitals when spin-orbit interactions were considered.
The WPs were obtained and confirmed by computing the Chern numbers as
implemented in pyfplo module of the FPLO code, and outlined in Ref. [44].

###  VI.4 Exfoliation energy

In order to estimate the energy cost of exfoliating monolayer and bilayer
TaRhTe4 from a real bulk system, we calculated their exfoliation energies. We
define exfoliation energy as the energy needed to completely detach all the
layers in the bulk unit cell. It is worth noting that in certain studies [72,
73], exfoliation energy is alternatively defined as the energy expended in
removing the top layer from the surface of a bulk crystal. Thereby, we compute
the total energy of the bulk TaRhTe4 as a function of the intra-bilayer
distance $d$ and inter-bilayer distance $c$, and obtain the difference from
the energy of the equilibrium structure. For bilayer system, we vary the
interlayer distance $c$ only. To arrive at monolayer limit, we vary both the
intra-bilayer and inter-bilayer distances at the same time. For these
calculations, we use a $k$ mesh with ($20\times 20\times 10$) intervals in the
BZ. Because there are 2 layers per unit cell, we divide the energy by 4 times
surface area ($3.76\times 12.55\;\mathrm{\AA}^{2}$) of unit cell as the final
monolayer exfoliation energy, whereas divide 2 times surface area as the final
bilayer exfoliation energy. As a benchmark, we also obtain the exfoliation
energy for graphite. The situation of graphene is same as that of the
monolayer, i.e. we divide the energy by by 4 times surface area to obtain the
exfoliation energy.

Table 2: Tight-binding model parameters of the 4-orbital minimal model for monolayer TaRhTe4. The model parameters correspond to Eq. (1) and Eq. (9) and are shown in Fig. 1(d). Parameter | Value (meV) | Parameter | Value (meV)
---|---|---|---
A. TB model without SOC $H_{0}$
$\mu_{p}$ | $590\,$ | $t_{dABx1}$ | $30\,$
$\mu_{d}$ | $-2070\,$ | $t_{dABy}$ | $-26\,$
$t_{px}$ | $940\,$ | $t_{dABy1}$ | $-11\,$
$t_{dx}$ | $-290\,$ | $t_{dABy2}$ | $-15\,$
$t_{pAB}$ | $240\,$ | $t_{0AB}$ | $-120\,$
$t_{pABx}$ | $-38\,$ | $t_{0ABx}$ | $-81\,$
$t_{dAB}$ | $-110\,$ | $t_{0ABx1}$ | $-26\,$
$t_{dABx}$ | $27\,$ | $t_{0x}$ | $57\,$
B. SOC terms $H_{\rm SOC}$
$\lambda_{0AB}^{z}$ | $16.8\,$ | $\lambda_{px}^{z}$ | $7.5\,$
$\lambda_{dx}^{z}$ | $7.1\,$ | $\lambda_{0}^{z}$ | $9.8\,$

###  VI.5 Phonon spectrum

We calculate the phonon spectrum of monolayer TaRhTe4 by using Vienna ab
initio Simulation Package (VASP) [74], version 5.4.2, and phonopy package
[75], version 2.20.0. The second-order harmonic interatomic force constants
are calculated by the density-functional perturbation theory with a ($2\times
2\times 1$) supercell and ($5\times 2\times 1$) $k$ mesh, employing projector
augmented-wave pesudopotential [76], where the cutoff energy is set as $500$
eV. In SI, we show the phonon spectrum of monolayer TaRhTe4.

###  VI.6 Minimal TB model for monolayer TaRhTe4

For the monolayer, a 4-orbital TB model was also obtained. Starting from the
symmetry based Hamiltonian in Ref. 57 and considering long range hopping
terms, up to $\sim 20$Å, the Hamiltonian without SOC is written as:

$\displaystyle\begin{split}H_{0}=&\left[\frac{\mu_{p}}{2}+t_{px}\cos(k_{x}a)\right]\Gamma_{1}^{-}+\left[\frac{\mu_{d}}{2}+t_{dx}\cos(k_{x}a)\right]\Gamma_{1}^{+}\\\
&+\left[t_{dAB}(1+e^{ik_{x}a})+t_{dABx}(e^{-ik_{x}a}+e^{2ik_{x}a})\right.\\\
&+t_{dABx1}(e^{-2ik_{x}a}+e^{3ik_{x}a})+t_{dABy}e^{-ik_{y}b}(1+e^{ik_{x}a})\\\
&+t_{dABy1}e^{-ik_{y}b}(e^{-ik_{x}a}+e^{2ik_{x}a})\\\
&\left.+t_{dABy2}e^{-ik_{y}b}(e^{-2ik_{x}a}+e^{3ik_{x}a})\right]e^{i\mathbf{k}\cdot\Delta_{1}}\Gamma^{+}_{2}\\\
&+\left[t_{pAB}(1+e^{ik_{x}a})+t_{pABx}(e^{-ik_{x}a}+e^{2ik_{x}a})\right]e^{i\mathbf{k}\cdot\Delta_{2}}\Gamma_{2}^{-}\\\
&+\left[t_{0AB}(1-e^{ik_{x}a})+t_{0ABx}(e^{-ik_{x}a}-e^{2ik_{x}a})\right.\\\
&\left.+t_{0ABx1}(e^{-2ik_{x}a}-e^{3ik_{x}a})\right]e^{i\mathbf{k}\cdot\Delta_{3}}\Gamma_{3}\\\
&-2it_{0x}\sin(k_{x}a)\left[e^{i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{-}+e^{-i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{+}\right]+\text{H.c.}\end{split}$
(1)

These hopping terms are illustrated in Fig. 1(d). The $4\times 4$ matrices
$\Gamma_{i}$ are linear combinations of products $\tau_{i}\sigma_{i}$ of Pauli
matrices and $\Delta_{i}$ are vectors corresponding to interatomic distances
and is defined below.

Specifically, the matrices $\Gamma_{i}$ have the form:

$\displaystyle\Gamma_{0}$ $\displaystyle=$ $\displaystyle\tau_{0}\sigma_{0},$
(2) $\displaystyle\Gamma_{1}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{\tau_{0}}{2}(\sigma_{0}\pm\sigma_{3}),$ (3)
$\displaystyle\Gamma_{2}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\tau_{1}+i\tau_{2})(\sigma_{0}\pm\sigma_{3}),$ (4)
$\displaystyle\Gamma_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\tau_{1}+i\tau_{2})i\sigma_{2},$ (5)
$\displaystyle\Gamma_{4}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\tau_{0}\pm\tau_{3})(\sigma_{1}+i\sigma_{2}),$ (6)
$\displaystyle\Gamma_{5}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{\tau_{3}}{2}(\sigma_{0}\pm\sigma_{3}),$ (7)
$\displaystyle\Gamma_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\tau_{1}+i\tau_{2})\sigma_{1},$ (8)

where the matrices $\tau_{j}\sigma_{i}$ are products of Pauli matrices acting
in orbital space with respect to the basis
$\\{d_{\mathbf{k}Ads},d_{\mathbf{k}Aps},d_{\mathbf{k}Bds},d_{\mathbf{k}Bps}\\}$,
where $d_{\mathbf{k}cls}$ annihilates an electron with momentum $\mathbf{k}$,
spin-$S_{z}$ eigenvalue $s=\uparrow,\downarrow$ and orbital $l=p,d$ (Te,Ta) in
sublattice $c=A,B$. In other words, $\tau_{j}$ acts on the sublattice degree
of freedom and $\sigma_{i}$ acts on the orbital degree of freedom.

The vectors in unit cell are defined as
$\Delta_{1}=\mathbf{r}_{Ad}-\mathbf{r}_{Bd}$,
$\Delta_{2}=\mathbf{r}_{Ap}-\mathbf{r}_{Bp}$,
$\Delta_{3}=\mathbf{r}_{Ad}-\mathbf{r}_{Bp}$, and
$\Delta_{4}=\mathbf{r}_{Ad}-\mathbf{r}_{Ap}$, where the vector
$\mathbf{r}_{cl}$ denotes the position of the corresponding lattice site,
associated with an orbital $l$ in sublattice $c$, in the unit cell. We list
the coordinates of vectors as $\mathbf{r}_{Ad}=(-0.25a,0.16b)$,
$\mathbf{r}_{Bp}=(0.25a,0.044b)$, $\mathbf{r}_{Ap}=(-0.25a,-0.044b)$, and
$\mathbf{r}_{Bd}=(0.25a,-0.16b)$.

When taking into account SOC, the minimal model is constructed by localized
Wannier orbitals of the Ta-$d$ and Te-$p$ orbitals along with the spin degrees
of freedom, which gives rise to an eight-orbital model. Based on the minimal
tight-binding model without SOC, we include spin-orbit coupling terms that
preserve: time reversal symmetry $\mathcal{T}$, glide mirror symmetry
$\widetilde{M}_{x}$, glide screw $\widetilde{C}_{2x}$ symmetry, and inversion
$\mathcal{I}$ symmetry. In the basis of the Hamiltonian in Eq. (1), the matrix
representations of the symmetry operations are
$\hat{\mathcal{T}}=is_{y}\Gamma_{0}K$ with complex conjugation $K$,
$\hat{\mathcal{I}}=s_{0}\tau_{1}\sigma_{3}$, and
$\hat{M}_{x}=is_{x}\tau_{0}\sigma_{3}$ for the nontranslational component of
$\widetilde{M}_{x}$. By taking the hopping parameters ($\geqslant 0.014$) eV
from Wannier fit result, the Bloch Hamiltonian of the final tight-binding
model with SOC is
$H(\mathbf{k})=s_{0}H_{0}(\mathbf{k})+H_{\text{SOC}}(\mathbf{k})$, with

$\displaystyle\begin{split}H_{\text{SOC}}=&\lambda_{dx}^{z}s_{z}\sin(ak_{x})\Gamma_{5}^{+}+\lambda_{px}^{z}s_{z}\sin(ak_{x})\Gamma_{5}^{-}\\\
&-i\lambda_{0AB}^{z}s_{z}(1+e^{iak_{x}})e^{i\mathbf{k}\cdot\Delta_{3}}\Gamma_{6}\\\
&-i\lambda_{0}^{z}s_{z}(e^{i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{+}-e^{-i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{-})+\text{H.c.}\end{split}$
(9)

where $s_{i}$ are Pauli matrices acting in spin space and $s_{0}$ is the
$2\times 2$ identity matrix. The full Hamiltonian is given by $H=H_{0}+H_{\rm
SOC}$. Parameter values in Eqs. (1) and (9) are listed in Table 2.

###### Acknowledgements.

The authors thank Hui Liu and Ion Cosma Fulga for helpful discussions and
Ulrike Nitzche for technical assistance. RR acknowledges helpful discussions
with Anupam Roy. We acknowledge financial support from German
Forschungsgemeinschaft (DFG, German Research Foundation) via SFB1143 Project
No. A05 and under Germany’s Excellence Strategy through Würzburg-Dresden
Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat
(EXC 2147, Project No. 390858490).

## References

  * Haldane [2017] F. D. M. Haldane, “Nobel lecture: Topological quantum matter,” Rev. Mod. Phys. 89, 040502 (2017).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
  * Moore [2010] J. E. Moore, “The birth of topological insulators,” Nature (London) 464, 194 (2010).
  * Wan  _et al._ [2011] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,” Phys. Rev. B 83, 205101 (2011).
  * Weng  _et al._ [2015] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, “Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides,” Phys. Rev. X 5, 011029 (2015).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057 (2011).
  * Tsai  _et al._ [2013] W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, “Gated silicene as a tunable source of nearly 100% spin-polarized electrons,” Nat. Commun. 4, 1500 (2013).
  * de Juan  _et al._ [2017] F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, “Quantized circular photogalvanic effect in weyl semimetals,” Nat. Commun. 8, 15995 (2017).
  * He  _et al._ [2022] Q. L. He, T. L. Hughes, N. P. Armitage, Y. Tokura, and K. L. Wang, “Topological spintronics and magnetoelectronics,” Nat. Mater. 21, 15 (2022).
  * Fukui and Hatsugai [2007] T. Fukui and Y. Hatsugai, “Topological aspects of the quantum spin-hall effect in graphene: ${\mathrm{z}}_{2}$ topological order and spin chern number,” Phys. Rev. B 75, 121403 (2007).
  * Fu and Kane [2007] L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
  * Fu and Kane [2006] L. Fu and C. L. Kane, “Time reversal polarization and a ${Z}_{2}$ adiabatic spin pump,” Phys. Rev. B 74, 195312 (2006).
  * Kane and Mele [2005a] C. L. Kane and E. J. Mele, “Quantum spin hall effect in graphene,” Phys. Rev. Lett. 95, 226801 (2005a).
  * Kane and Mele [2005b] C. L. Kane and E. J. Mele, “${Z}_{2}$ topological order and the quantum spin hall effect,” Phys. Rev. Lett. 95, 146802 (2005b).
  * Roy [2009] R. Roy, “${Z}_{2}$ classification of quantum spin hall systems: An approach using time-reversal invariance,” Phys. Rev. B 79, 195321 (2009).
  * Qian  _et al._ [2014] X. Qian, J. Liu, L. Fu, and J. Li, “Quantum spin hall effect in two-dimensional transition metal dichalcogenides,” Science 346, 1344 (2014).
  * Fu  _et al._ [2007] L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Phys. Rev. Lett. 98, 106803 (2007).
  * Moore and Balents [2007] J. E. Moore and L. Balents, “Topological invariants of time-reversal-invariant band structures,” Phys. Rev. B 75, 121306 (2007).
  * Hsieh  _et al._ [2008] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological dirac insulator in a quantum spin hall phase,” Nature (London) 452, 970 (2008).
  * Ringel  _et al._ [2012] Z. Ringel, Y. E. Kraus, and A. Stern, “Strong side of weak topological insulators,” Phys. Rev. B 86, 045102 (2012).
  * Fan  _et al._ [2017] Z. Fan, Q.-F. Liang, Y. B. Chen, S.-H. Yao, and J. Zhou, “Transition between strong and weak topological insulator in zrte5 and hfte5,” Sci. Rep. 7, 45667 (2017).
  * Liu  _et al._ [2012a] C.-X. Liu, X.-L. Qi, and S.-C. Zhang, “Half quantum spin hall effect on the surface of weak topological insulators,” Physica E 44, 906 (2012a), sI:Topological Insulators.
  * Soluyanov  _et al._ [2015] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-ii weyl semimetals,” Nature (London) 527, 495 (2015).
  * Burkov and Balents [2011] A. A. Burkov and L. Balents, “Weyl semimetal in a topological insulator multilayer,” Phys. Rev. Lett. 107, 127205 (2011).
  * Lv  _et al._ [2021] B. Q. Lv, T. Qian, and H. Ding, “Experimental perspective on three-dimensional topological semimetals,” Rev. Mod. Phys. 93, 025002 (2021).
  * Armitage  _et al._ [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).
  * Jia  _et al._ [2016] S. Jia, S.-Y. Xu, and M. Z. Hasan, “Weyl semimetals, fermi arcs and chiral anomalies,” Nat. Mater. 15, 1140 (2016).
  * Xu  _et al._ [2015] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y. Huang, W.-F. Tsai, H. Lin, P. P. Shibayev, F.-C. Chou, R. J. Cava, and M. Z. Hasan, “Observation of fermi arc surface states in a topological metal,” Science 347, 294 (2015).
  * Lv  _et al._ [2015a] B. Q. Lv, S. Muff, T. Qian, Z. D. Song, S. M. Nie, N. Xu, P. Richard, C. E. Matt, N. C. Plumb, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, J. H. Dil, J. Mesot, M. Shi, H. M. Weng, and H. Ding, “Observation of fermi-arc spin texture in taas,” Phys. Rev. Lett. 115, 217601 (2015a).
  * Muechler  _et al._ [2016] L. Muechler, A. Alexandradinata, T. Neupert, and R. Car, “Topological nonsymmorphic metals from band inversion,” Phys. Rev. X 6, 041069 (2016).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5, 438 (2009).
  * Xia  _et al._ [2009] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single dirac cone on the surface,” Nat. Phys. 5, 398 (2009).
  * Hsieh  _et al._ [2009] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A tunable topological insulator in the spin helical dirac transport regime,” Nature 460, 1101 (2009).
  * Chen  _et al._ [2009] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, “Experimental realization of a three-dimensional topological insulator, bi2te3,” Science 325, 178 (2009).
  * Rasche  _et al._ [2013] B. Rasche, A. Isaeva, M. Ruck, S. Borisenko, V. Zabolotnyy, B. Büchner, K. Koepernik, C. Ortix, M. Richter, and J. van den Brink, “Stacked topological insulator built from bismuth-based graphene sheet analogues,” Nature Materials 12, 422 (2013).
  * Carrillo-Aravena  _et al._ [2022] E. Carrillo-Aravena, K. Finzel, R. Ray, M. Richter, T. Heider, I. Cojocariu, D. Baranowski, V. Feyer, L. Plucinski, M. Gruschwitz, C. Tegenkamp, and M. Ruck, “Bi12rh3cu2i5: A 3d weak topological insulator with monolayer spacers and independent transport channels,” physica status solidi (b) 259, 2100447 (2022).
  * Lv  _et al._ [2015b] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of weyl semimetal taas,” Phys. Rev. X 5, 031013 (2015b).
  * Lv  _et al._ [2015c] B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of weyl nodes in taas,” Nat. Phys. 11, 724 (2015c).
  * Li  _et al._ [2017] P. Li, Y. Wen, X. He, Q. Zhang, C. Xia, Z.-M. Yu, S. A. Yang, Z. Zhu, H. N. Alshareef, and X.-X. Zhang, “Evidence for topological type-ii weyl semimetal wte2,” Nat. Commun. 8, 2150 (2017).
  * Wu  _et al._ [2016] Y. Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, “Observation of fermi arcs in the type-II weyl semimetal candidate WTe2,” Phys. Rev. B 94, 121113 (2016).
  * Deng  _et al._ [2016] K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, Z. Xu, J. Denlinger, A. Fedorov, H. Yang, W. Duan, H. Yao, Y. Wu, S. Fan, H. Zhang, X. Chen, and S. Zhou, “Experimental observation of topological fermi arcs in type-II weyl semimetal MoTe2,” Nat. Phys. 12, 1105 (2016).
  * Jiang  _et al._ [2017] J. Jiang, Z. K. Liu, Y. Sun, H. F. Yang, C. R. Rajamathi, Y. P. Qi, L. X. Yang, C. Chen, H. Peng, C.-C. Hwang, S. Z. Sun, S.-K. Mo, I. Vobornik, J. Fujii, S. S. P. Parkin, C. Felser, B. H. Yan, and Y. L. Chen, “Signature of type-II Weyl semimetal phase in MoTe2,” Nat. Commun. 8, 13973 (2017).
  * Koepernik  _et al._ [2016] K. Koepernik, D. Kasinathan, D. V. Efremov, S. Khim, S. Borisenko, B. Büchner, and J. van den Brink, “TaIrTe4: A ternary type-II Weyl semimetal,” Phys. Rev. B 93, 201101 (2016).
  * Haubold  _et al._ [2017] E. Haubold, K. Koepernik, D. Efremov, S. Khim, A. Fedorov, Y. Kushnirenko, J. van den Brink, S. Wurmehl, B. Büchner, T. K. Kim, M. Hoesch, K. Sumida, K. Taguchi, T. Yoshikawa, A. Kimura, T. Okuda, and S. V. Borisenko, “Experimental realization of type-II Weyl state in noncentrosymmetric TaIrTe4,” Phys. Rev. B 95, 241108 (2017).
  * Otrokov  _et al._ [2019] M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau, and E. V. Chulkov, “Unique thickness-dependent properties of the van der waals interlayer antiferromagnet ${\mathrm{mnbi}}_{2}{\mathrm{te}}_{4}$ films,” Phys. Rev. Lett. 122, 107202 (2019).
  * Huang  _et al._ [2017] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, “Layer-dependent ferromagnetism in a van der waals crystal down to the monolayer limit,” Nature 546, 270 (2017).
  * Li  _et al._ [2020] Z. Li, J. Li, K. He, X. Wan, W. Duan, and Y. Xu, “Tunable interlayer magnetism and band topology in van der waals heterostructures of $\mathrm{Mn}{\mathrm{bi}}_{2}{\mathrm{te}}_{4}$-family materials,” Phys. Rev. B 102, 081107 (2020).
  * Mahatara and Kiefer [2021] S. Mahatara and B. Kiefer, “Layer dependent magnetism and topology in monolayer and bilayers Re$X$ 3 ($X$ = Br, I),” J. Phys.: Condens. Matter 33, 455801 (2021).
  * Guo  _et al._ [2020] P.-J. Guo, X.-Q. Lu, W. Ji, K. Liu, and Z.-Y. Lu, “Quantum spin hall effect in monolayer and bilayer ${\mathrm{tairte}}_{4}$,” Phys. Rev. B 102, 041109 (2020).
  * Sufyan  _et al._ [2021] A. Sufyan, G. Macam, C.-H. Hsu, Z.-Q. Huang, S.-M. Huang, H. Lin, and F.-C. Chuang, “Theoretical prediction of topological insulators in two-dimensional ternary transition metal chalcogenides (MM’X4, M=Ta, Nb, or V; M’= Ir, Rh, or Co; X=Se or Te),” Chin. J. Phys. 73, 95 (2021).
  * Shipunov  _et al._ [2021] G. Shipunov, B. R. Piening, C. Wuttke, T. A. Romanova, A. V. Sadakov, O. A. Sobolevskiy, E. Y. Guzovsky, A. S. Usoltsev, V. M. Pudalov, D. V. Efremov, S. Subakti, D. Wolf, A. Lubk, B. Kiefer, and S. Aswartham, “Layered van der Waals topological metals of TaTMTe4 (TM = Ir, Rh, Ru) family,” J. Phys. Chem. Lett. 12, 6730 (2021).
  * Lv  _et al._ [2022] M. Lv, G. Yu, and W. Chen, “2d nbirte4 and tarhte4 monolayers: two fascinating topological insulators as electrocatalysts for oxygen reduction,” Inorg. Chem. Front. 9, 6133 (2022).
  * Kumar  _et al._ [2021] D. Kumar, C.-H. Hsu, R. Sharma, T.-R. Chang, P. Yu, J. Wang, G. Eda, G. Liang, and H. Yang, “Room-temperature nonlinear hall effect and wireless radiofrequency rectification in weyl semimetal tairte4,” Nat. Nanotechnol. 16, 421 (2021).
  * Wang  _et al._ [2016] C. Wang, Y. Zhang, J. Huang, S. Nie, G. Liu, A. Liang, Y. Zhang, B. Shen, J. Liu, C. Hu, Y. Ding, D. Liu, Y. Hu, S. He, L. Zhao, L. Yu, J. Hu, J. Wei, Z. Mao, Y. Shi, X. Jia, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng, H. Weng, X. Dai, Z. Fang, Z. Xu, C. Chen, and X. J. Zhou, “Observation of fermi arc and its connection with bulk states in the candidate type-ii weyl semimetal ${\mathrm{wte}}_{2}$,” Phys. Rev. B 94, 241119 (2016).
  * Shi  _et al._ [2019] Y. Shi, J. Kahn, B. Niu, Z. Fei, B. Sun, X. Cai, B. A. Francisco, D. Wu, Z.-X. Shen, X. Xu, D. H. Cobden, and Y.-T. Cui, “Imaging quantum spin hall edges in monolayer wte2,” Sci. Adv. 5, eaat8799 (2019).
  * Lau  _et al._ [2019] A. Lau, R. Ray, D. Varjas, and A. R. Akhmerov, “Influence of lattice termination on the edge states of the quantum spin hall insulator monolayer $1{T}^{{}^{\prime}}\text{$-$}{\mathrm{wte}}_{2}$,” Phys. Rev. Materials 3, 054206 (2019).
  * Ok  _et al._ [2019] S. Ok, L. Muechler, D. Di Sante, G. Sangiovanni, R. Thomale, and T. Neupert, “Custodial glide symmetry of quantum spin hall edge modes in monolayer ${\mathrm{wte}}_{2}$,” Phys. Rev. B 99, 121105 (2019).
  * [59] Supplemental material contains further information on (i) crystal structures used, (ii) surface spectral function for different edge terminations in monolayer TaRhTe4, and (iii) the DFT bandstructure for bulk TaRhTe4.
  * Hu  _et al._ [2021] M. Hu, G. Ma, C. Y. Wan, and J. Liu, “Realistic tight-binding model for monolayer transition metal dichalcogenides of $1{T}^{{}^{\prime}}$ structure,” Phys. Rev. B 104, 035156 (2021).
  * Yu  _et al._ [2011] R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, “Equivalent expression of $z_{2}$ topological invariant for band insulators using the non-abelian berry connection,” Phys. Rev. B 84, 075119 (2011).
  * Culcer  _et al._ [2020] D. Culcer, A. C. Keser, Y. Li, and G. Tkachov, “Transport in two-dimensional topological materials: recent developments in experiment and theory,” 2D Materials 7, 022007 (2020).
  * Soluyanov and Vanderbilt [2011] A. A. Soluyanov and D. Vanderbilt, “Computing topological invariants without inversion symmetry,” Phys. Rev. B 83, 235401 (2011).
  * Shinde  _et al._ [2018] S. M. Shinde, K. P. Dhakal, X. Chen, W. S. Yun, J. Lee, H. Kim, and J.-H. Ahn, “Stacking-controllable interlayer coupling and symmetric configuration of multilayered MoS2,” NPG Asia Materials 10, e468 (2018).
  * Wu  _et al._ [2019] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, “Topological insulators in twisted transition metal dichalcogenide homobilayers,” Phys. Rev. Lett. 122, 086402 (2019).
  * Jung  _et al._ [2018] J. H. Jung, C.-H. Park, and J. Ihm, “A rigorous method of calculating exfoliation energies from first principles,” Nano Lett. 18, 2759 (2018).
  * Liu  _et al._ [2016] J. Liu, H. Wang, C. Fang, L. Fu, and X. Qian, “van der waals stacking-induced topological phase transition in layered ternary transition metal chalcogenides,” Nano Letters 17, 467 (2016).
  * [68] https://www.materialscloud.org/discover/2dtopo/details/RhTaTe4.
  * Koepernik and Eschrig [1999] K. Koepernik and H. Eschrig, “Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme,” Phys. Rev. B 59, 1743 (1999).
  * [70] https://www.fplo.de.
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865 (1996).
  * Wang  _et al._ [2015] W. Wang, S. Dai, X. Li, J. Yang, D. J. Srolovitz, and Q. Zheng, “Measurement of the cleavage energy of graphite,” Nat. Commun. 6, 7853 (2015).
  * Liu  _et al._ [2012b] Z. Liu, J. Z. Liu, Y. Cheng, Z. Li, L. Wang, and Q. Zheng, “Interlayer binding energy of graphite: A mesoscopic determination from deformation,” Phys. Rev. B 85, 205418 (2012b).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169 (1996).
  * Togo  _et al._ [2008] A. Togo, F. Oba, and I. Tanaka, “First-principles calculations of the ferroelastic transition between rutile-type and ${\text{cacl}}_{2}$-type ${\text{sio}}_{2}$ at high pressures,” Phys. Rev. B 78, 134106 (2008).
  * Blöchl [1994] P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B 50, 17953 (1994).

## Supplemental Information

In Supplementay A, we provide the details of the relaxed crystal structures
for monolayer, bilayer and bulk TaRhTe4 used in this study. Supplementay B
illustrates the DFT band structure of monolayer TaRhTe4 which lattice
constants are same to the bulk. Supplementay C shows the exfoliation energy of
monolayer and bilayer TaRhTe4. In Supplementay D, the phonon spectrum of
monolayer TaRhTe4 is provided. Supplementay E gives the layer-resolved band
weight of bilayer TaRhTe4. Supplementay F presents the surface spectral
function for different edge terminations in the monolayer TaRhTe4.
Supplementay G shows bandstructure and Wilson loop of AA-stacked bilayer
TaRhTe4, while Supplementay H shows the DFT bandstructure for bulk TaRhTe4.

##  A Crystal structures of monolayer, bilayer, and bulk TaRhTe4.

Table S1: Crystal structure of monolayer TaRhTe4. It was constructed from the
bulk unit cell by inserting a large vacuum between the two layer in the bulk
unit cell. The atom positions and lattice constants are optimized under
$Pmn2_{1}$ symmetry. A. Lattice parameters
---
Basis vector |  $x$ [Å] |  $y$ [Å] |  $z$ [Å]
$\vec{a}$ |  $3.71915$ |  $0.00000$ |  $0.00000$
$\vec{b}$ |  $0.00000$ |  $12.4221$ |  $0.00000$
$\vec{c}$ |  $0.00000$ |  $0.00000$ |  $23.9814$
Axis angle |  $\alpha$ [∘] |  $\beta$ [∘] |  $\gamma$ [∘]
|  $90$ |  $90$ |  $90$
B. Atomic Wyckoff positions
Ta |  $0$ |  $-0.05312$ |  $0.08669$
Ta |  $1/2$ |  $0.26344$ |  $0.07862$
Rh |  $0$ |  $0.46000$ |  $0.08401$
Rh |  $1/2$ |  $-0.24965$ |  $0.08126$
Te |  $1/2$ |  $0.06155$ |  $0.02046$
Te |  $0$ |  $-0.19740$ |  $0.00513$
Te |  $0$ |  $-0.35181$ |  $0.13854$
Te |  $1/2$ |  $0.40777$ |  $0.16015$
Te |  $1/2$ |  $-0.43785$ |  $0.02673$
Te |  $0$ |  $0.31959$ |  $-0.00074$
Te |  $0$ |  $0.14877$ |  $0.14484$
Te |  $1/2$ |  $-0.10929$ |  $0.16604$

Table S2: Crystal structure of bilayer TaRhTe4. It was constructed from the
bulk unit cell by inserting a large vacuum between the bilayer in the bulk
unit cell. The atom positions and lattice constants are not optimized. A.
Lattice parameters
---
Basis vector |  $x$ [Å] |  $y$ [Å] |  $z$ [Å]
$\vec{a}$ |  $3.75672$ |  $0.00000$ |  $0.00000$
$\vec{b}$ |  $0.00000$ |  $12.5476$ |  $0.00000$
$\vec{c}$ |  $0.00000$ |  $0.00000$ |  $30.6581$
Axis angle |  $\alpha$ [∘] |  $\beta$ [∘] |  $\gamma$ [∘]
|  $90$ |  $90$ |  $90$
B. Atomic Wyckoff positions
Ta |  $0$ |  $0.05312$ |  $0.28409$
Ta |  $1/2$ |  $-0.05312$ |  $0.06936$
Ta |  $0$ |  $0.27076$ |  $0.06376$
Ta |  $1/2$ |  $-0.27076$ |  $0.27848$
Rh |  $0$ |  $-0.46534$ |  $0.28204$
Rh |  $1/2$ |  $0.46534$ |  $0.06731$
Rh |  $0$ |  $-0.24412$ |  $0.06376$
Rh |  $1/2$ |  $0.24412$ |  $0.27848$
Te |  $0$ |  $0.06713$ |  $0.01782$
Te |  $1/2$ |  $-0.06694$ |  $0.23429$
Te |  $0$ |  $0.19176$ |  $0.21742$
Te |  $1/2$ |  $-0.19081$ |  $0.00238$
Te |  $0$ |  $0.34590$ |  $0.32485$
Te |  $1/2$ |  $-0.34578$ |  $0.109175$
Te |  $0$ |  $0.41368$ |  $0.12874$
Te |  $1/2$ |  $-0.41280$ |  $0.34376$
Te |  $0$ |  $-0.43343$ |  $0.02092$
Te |  $1/2$ |  $0.43304$ |  $0.23636$
Te |  $0$ |  $-0.32326$ |  $0.21514$
Te |  $1/2$ |  $0.32340$ |  $0$
Te |  $0$ |  $-0.15175$ |  $0.33065$
Te |  $1/2$ |  $0.15042$ |  $0.11414$
Te |  $0$ |  $-0.10495$ |  $0.13257$
Te |  $1/2$ |  $0.10627$ |  $0.34764$

Table S3: Crystal structure of bulk TaRhTe4 A. Lattice parameters
---
Basis vector |  $x$ [Å] |  $y$ [Å] |  $z$ [Å]
$\vec{a}$ |  $3.75672$ |  $0.00000$ |  $0.00000$
$\vec{b}$ |  $0.00000$ |  $12.5476$ |  $0.00000$
$\vec{c}$ |  $0.00000$ |  $0.00000$ |  $13.1660$
Axis angle |  $\alpha$ [∘] |  $\beta$ [∘] |  $\gamma$ [∘]
|  $90$ |  $90$ |  $90$
B. Atomic Wyckoff positions
Ta |  $0$ |  $0.05311$ |  $-0.19200$
Ta |  $0$ |  $0.27076$ |  $0.29495$
Rh |  $0$ |  $-0.46534$ |  $-0.19677$
Rh |  $0$ |  $-0.24412$ |  $0.29496$
Te |  $0$ |  $0.06743$ |  $0.19535$
Te |  $0$ |  $0.19140$ |  $-0.34448$
Te |  $0$ |  $0.34526$ |  $-0.10208$
Te |  $0$ |  $0.41389$ |  $0.44296$
Te |  $0$ |  $-0.43341$ |  $0.19958$
Te |  $0$ |  $-0.32390$ |  $-0.34990$
Te |  $0$ |  $-0.15087$ |  $-0.08964$
Te |  $0$ |  $-0.10633$ |  $0.45250$

##  B Band structure of monolayer TaRhTe4 with bulk lattice constants.

In Supplementary Fig. S1(a), we show the band structure of monolayer TaRhTe4
without SOC. The Dirac cone can be found along path Y-S akin to the case with
optimized lattice constants, but the band along $\Gamma$-X touches Fermi level
showing the difference. With SOC, in Fig. S1(b), the band gap $E_{g}=76$ meV
is larger than the optimized results ($65$ meV).

Figure S1: Band structure of monolayer TaRhTe4 (a) without SOC and (b) with
SOC.

##  C Exfoliation energy of monolayer and bilayer TaRhTe4.

Since each unit cell of the bulk TaRhTe4 has two layers, we increase the
interlayer distance between unit cells to approach the bilayer limit. To
arrive at the monolayer limit, it is necessary to increase both the interlayer
distance between unit cells and the interlayer distance within unit cells. The
exfoliation energy is defined as the total energy after changing the distance
between layers minus the total energy of the bulk in the equilibrium state. In
Fig. S2, we show the exfoliation energy as a function of $\Delta d$, where
$\Delta d=d-d_{0}$ denotes the difference of interlayer distance and
equilibrium distance.

The exfoliation energy for monolayer TaRhTe4 is $28.5$ meV/Å2, while the
energy for bilayer TaRhTe4 is $17.5$ meV/Å2. In addition, we exfoliate
graphene from graphite in the same way for reference. We obtain the
exfoliation energy of graphene to be $13$ meV/Å2, which is slightly smaller
than the result ($21$ meV/Å2) in another calculation.[66]

Figure S2: Exfoliation energy of monolayer and bilayer TaRhTe4.

##  D Phonon spectrum of monolayer TaRhTe4.

Figure S3 serves as confirmation of the dynamic stability of monolayer
TaRhTe4. It displays the phonon spectrum of monolayer structure. The acoustic
branches and the other optical branches are clearly separated from each other
in the vicinity of the $\Gamma$ point.[53] There are phonon branches observed
below zero frequency through the BZ.

Figure S3: Phonon dispersion spectrum of monolayer TaRhTe4.

##  E Layer-resolved orbital contributions to the bandstructure in bilayer
TaRhTe4.

In order to understand the topological phase transtion in the bilayer TaRhTe4,
we present the band weight of bilayer system projected to orbitals from
different layers. Figure S4(b) is the band weight of equilibrium bilayer
structure. We find that the layer-1 dominates valence band edge (near the Y
point) whereas conduction band edge are governed by layer-2. Approaching the
TPT point, the band gap is closing near the Y point as shown in Fig. S4(c).
After crossing the phase boundary, in Fig. S4(d), the contribution of both the
van der Waals separated layers tend to equate in the conduction and valence
band edges.

Figure S4: Layer resolved band weight of bilayer TaRhTe4 with different layer
displacements.

##  F Edge states of monolayer and bilayer TaRhTe4.

In Supplementary Fig. S5, we show the spectra of edge states of monolayer
TaRhTe4 with different terminations. In Supplementary Fig. S6, we show the
spectra of edge states of bilayer TaRhTe4.

Figure S5: Edge state spectra of monolayer TaRhTe4 with different
terminations.  Figure S6: Edge state spectra of bilayer TaRhTe4 and
corresponding terminations.

##  G Bandstructure and Wilson loop of AA-stacked bilayer TaRhTe4.

In Supplementary Fig. S7, we show the bandstructure and Wilson loop of AA-
stacked TaRhTe4. The interlayer distance (vertical distance of Rh atoms from
different layers) is $6.358$ Å.

Figure S7: (a) Band structure of AA-stacked bilayer TaRhTe4 along high
symmetry points with SOC. (b) Wislon loop for AA-stacked bilayer TaRhTe4.

##  H Bandstructure of bulk 3D TaRhTe4.

Supplementary Figure S8 shows the BZ and DFT bandstructure for bulk (3D)
TaRhTe4. The top of the valence band (band $N$) and the bottom of the
conduction band (band $N+1$), used for obtaining the WPs, are highlighed in
blue and red color, respectively.

Figure S8: (a) Brillouin zone of bulk TaRhTe4along with the high symmetry
points. (b) Band structure of 3D TaRhTe4 along high symmetry points with SOC.
The blue curve represents the highest valence band, while the red curve is the
lowest conduction band.