#  Distinguishing nodal and nonunitary superconductivity in quasiparticle
interference of an Ising superconductor with Rashba spin-orbit coupling: an
example of NbSe2

Jozef Haniš<EMAIL_ADDRESS>Institute of Physics, Pavol Jozef
Šafárik University in Košice, 04001 Košice, Slovakia  Marko Milivojević
Institute of Informatics, Slovak Academy of Sciences, 84507 Bratislava,
Slovakia  Faculty of Physics, University of Belgrade, 11001 Belgrade, Serbia
Martin Gmitra<EMAIL_ADDRESS>Institute of Physics, Pavol Jozef Šafárik
University in Košice, 04001 Košice, Slovakia  Institute of Experimental
Physics, Slovak Academy of Sciences, Watsonova 47, 04001 Košice, Slovakia

(August 28, 2024)

###### Abstract

The NbSe2 monolayer with Rashba spin-orbit coupling represents a paradigmatic
example of an Ising superconductor on a substrate. Using a single-band model
and symmetry analysis, we present general superconducting pairing functions
beyond the nearest-neighbor approximation, uncovering new types of gap
functions, including the nodal singlet gap function and the triplet non-
unitary pairing function that breaks time-reversal symmetry. The non-unitarity
builts in the asymmetrical band dispersion in the superconducting
quasiparticle energy spectra. Performing exact T-matrix calculations of
quasiparticle interference due to a single scalar impurity scattering, we
found that the interference patterns possess characteristic features
distinguishing the type of pairing and possible nematic and chiral symmetry
violations.

quasiparticle interference, impurity scattering, spin-orbit coupling, NbSe2
monolayer.

###### pacs:

##  I Introduction

Since its initial discovery [1], transition-metal dichalcogenides (TMDC) have
attracted lots of attention due to their intriguing electronic structure [2,
3] and potential application in many subfields of solid state physics such as
spintronics [4, 5, 6, 7, 8, 9], valleytronic [10], optotronics [11], and
superconductivity [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. TMDC
crystalizing in several structural polytypes. Trigonal prismatic polytype is a
stable structure of NbSe2 exhibiting superconductivity in its bulk form [23,
24, 25, 26] as well as down to a single layer limit [27, 28]. Monolayer NbSe2,
unlike the bulk structure, lacks inversion symmetry, which leads to uniquely
resolved large spin split bands due to spin-orbit coupling near the K points
[29]. Due to time-reversal symmetry, the spins are locked to momentum with
opposite directions in K and K′ points and ${\bf D}_{3{\rm h}}$ symmetry,
restricting the spins’ orientation to the out-of-plane direction. Formed
Cooper pairs break their rotational invariance in spin space leading to the
novel pairing dubbed as Ising superconductivity [30, 27]. A key consequence of
the Ising superconductivity is robustness to the in-plane magnetic fields
exceeding considerably the Pauli limit [27, 28]. The Ising superconductivity
is attracting significant attention [31, 32, 33, 34, 35, 36] as it is
considered as a principal mechanism for the giant in-plane upper critical
magnetic field also observed in the bulk layered misfit structures [37, 38,
39].

Misfit layered compounds are a class of heterostructured materials that
consist of two structurally different materials forming an ordered
superstructure [40, 41, 42]. (LaSe)1.14(NbSe2)x, $x=1,2$ are examples in which
the alternating LaSe and NbSe2 layers form a slab stacked along their $c$
direction with a mismatch along the $a$ direction [43, 44]. Electronic band
structure near the Fermi level reminds Nb $d$-band with an offset due to
strong electron doping, from 0.5 to 0.6 electron per Nb [45, 46]. The misfit
structures, however, represent an unprecedented platform for charge transfer
control [47] providing access to topological superconductivity [13] due to the
high tunability of chemical potential that can also be achieved by engineering
La vacancies [38] or alloying. The Ising superconductivity has been observed
in the misfit structures [37, 39, 45] suggesting that the spins near the K
valleys still possess significant out-of-plane components.

Classification of the possible superconducting states according to the
irreducible representations (IR) of a given symmetry group [48, 49] can be
performed without entering into the microscopical details of the origin of the
superconductivity [50, 51, 52]. Although there are some analyses of the
superconducting TMDC systems with broken horizontal mirror plane symmetry
[53], the full classification of the possible pairing functions is still
missing. Doping control in the misfit structures [45], prediction that
superconductivity in gated MoS2 [54, 55] can possess exotic topological
pairing [53], as well as the recent experiment [56] reporting competition
between the nodal and nematic superconductivity in a monolayer NbSe2 on a
substrate motivate us to study superconducting pairing in the reduced ${\bf
C}_{\rm 3v}$ symmetry case. In such cases, both inversion and horizontal
mirror plane symmetry are broken, giving rise to the Rashba spin-orbit
coupling that tilts the spins to in-plane directions.

Electronic states of TMDC materials can be well described near the K and
$\Gamma$ valleys using a single-band effective tight-binding model [57]. We
extend the model considering spin-flip Rashba spin-orbit coupling term
modeling effect of a substrate, gating or electron doping of the NbSe2
monolayer in the reduced ${\bf C}_{\rm 3v}$ symmetry and constructed all
possible superconducting pairing functions classified according to the
irreducible representations (IRs) [58, 59, 60, 61].

The benefit of the proposed general approach is that it allows us to go beyond
the nearest-neighbor approximation for the pairing function. Consequently, we
found novel exotic types of gap functions such as the nodal singlet and non-
unitary triplet functions.

As quasiparticle interference (QPI) has been used to investigate
superconducting gap function [62, 63, 64, 65, 66], we calculate QPI for each
of the superconducting pairing functions. The obtained QPI patterns have
characteristic features that allow distinguishing between the different types
of pairing.

This paper is organized as follows. In Section II we study all possible types
of a superconducting pairing function in a system with ${\bf C}_{3{\rm v}}$
symmetry. In Section III, we present the numerical method for QPI calculation
and define the effective quasiparticle BdG model for NbSe2 using the single-
band model of the NbSe2 monolayer with Rashba spin-orbit coupling described in
Appendix A.

In Section IV, by assuming a single scalar impurity scattering, we analyze QPI
patterns in the first Brillouin zone for different types of gaps. Final
remarks are given in the Conclusions section, where we also summarize the main
consequences of our results.

##  II Gap functions classification according to irreducible representations

In this Section, we will focus on the construction of the gap function for a
system with the point group symmetry ${\bf C}_{3{\rm v}}$. The considered
geometry of the system is shown in the Fig. 1. The group contains 6 elements
and two generators: three-fold rotation $C_{3}$ and vertical mirror plane
$\sigma_{\rm v}$. The superconducting pairing can be transformed according to
the one-dimensional (1D) irreducible representations (IR) $A_{1}$ or $A_{2}$,
and two-dimensional (2D) IR $E$, see Table 1.

Table 1: Table of one-dimensional and two-dimensional IRs of the group ${\bm{C}}_{3{\rm v}}$, where $s=0,1,2$ represents different elements of the subgroup ${\bm{C}}_{3}$. IR | $C_{3}^{s}$ | $\sigma_{\rm v}C_{3}^{s}$
---|---|---
$A_{1}$ | 1 | 1
$A_{2}$ | 1 | -1
$E$ | $\begin{pmatrix}{\rm e}^{{\rm i}\frac{2\pi}{3}s}&0\\\ 0&{\rm e}^{-{\rm i}\frac{2\pi}{3}s}\end{pmatrix}$ | $\begin{pmatrix}0&{\rm e}^{-{\rm i}\frac{2\pi}{3}s}\\\ {\rm e}^{{\rm i}\frac{2\pi}{3}s}&0\end{pmatrix}$

Antisymmetry condition on the gap parameter due to the Pauli principle implies
that in the single band case, the orbital part of the superconducting gap must
be

symmetric/antisymmetric to the momentum change (${\bf k}\rightarrow{-\bf k}$)
in the singlet/triplet case. The straightforward way to implement such a rule
is to construct the order parameter using the even/odd function. Here, we will
use the trigonometric $\cos/\sin$ function in such a way that the translation
invariance of the system is preserved, with an argument that is a scalar
function of the type ${\bf k}\cdot{\bf r}$.

The general formulation of the singlet and triplet gap functions can be
written as follows

$\displaystyle\Delta_{\mu,m}^{\rm s}$ $\displaystyle=$
$\displaystyle\sum_{g\in{\bf C}_{3{\rm
v}}}{\Gamma}_{mm}^{(\mu)*}(g)\cos{\big{(}{\bf
k}\cdot\mathcal{D}^{+}(g^{-1}{\bf r})\big{)}}{\bf d}_{0},$ (1)
$\displaystyle\Delta_{\mu,m}^{{\rm t},z}$ $\displaystyle=$
$\displaystyle\sum_{g\in{\bf C}_{3{\rm
v}}}\Gamma_{mm}^{(\mu)*}(g)\sin{\big{(}{\bf k}\cdot\mathcal{D}^{+}(g^{-1}{\bf
r})\big{)}}\mathcal{D}^{-}(g^{-1}){\bf d}_{z},$
$\displaystyle\Delta_{\mu,m}^{{\rm t},xy}$ $\displaystyle=$
$\displaystyle\sum_{g\in{\bf C}_{3{\rm
v}}}\Gamma_{mm}^{(\mu)*}(g)\sin{\big{(}{\bf k}\cdot\mathcal{D}^{+}(g^{-1}{\bf
r})\big{)}}\mathcal{D}^{-}(g^{-1}){\bf d}_{1},$

where $\mu=A_{1},A_{2},E$ defines the IR, $m=1,\ldots,|\mu|$, where $|\mu|$ is
the dimension of the IR $\mu$, and $\Gamma_{mm}^{(\mu)*}(g)$ is the conjugated
matrix element of the IR $\mu$ for the given group element $g$. The spin space
is spanned conventionally by the matrices $d_{l}={\rm i}\sigma_{l}\sigma_{y}$
forming a pseudovector under rotation in spin space, where $\sigma_{l}$ are
Pauli matrices for $l=x,y,z$, and $\sigma_{0}$ is the unit matrix. The
component with $l=0$ corresponds to the antisymmetric part of the singlet
pairing function, while $l=x,y,z$ components constitute the symmetric parts of
the triplet pairing functions. Whereas the singlet gap function can be
constructed using $d_{0}$ solely, in the case of the triplet pairing the gap
function should be composed using the pseudovector ${\bf
d}=(d_{x},d_{y},d_{z})$. Due to the ${\bf C}_{3{\rm v}}$ symmetry, it is
possible to decouple the $d_{z}$ component and the $(d_{x},d_{y})$ multiplet,
leading to two types of triplet gap functions that can be constructed, see Eq.
(1). This is the reason for choosing the ${\bf d}_{z}$ and ${\bf d}_{1}$
vectors, being equal to $(0,0,d_{z})$ and $(d_{x},0,0)$, and belonging to the
orthogonal subspaces of the pseudovector space.

The components of the normal vector and pseudovector transform according to
the matrix representations equal to

$\displaystyle\mathcal{D}^{\pm}(C_{3})=\begin{pmatrix}\cos{\frac{2\pi}{3}}&-\sin{\frac{2\pi}{3}}&0\\\
\sin{\frac{2\pi}{3}}&\cos{\frac{2\pi}{3}}&0\\\
0&0&1\end{pmatrix},\mathcal{D}^{\pm}(\sigma_{\rm
v})=\pm\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}.$

Representation for the other group elements can be obtained using the
multiplication rule
$\mathcal{D}(g_{1})\mathcal{D}(g_{2})=\mathcal{D}(g_{1}g_{2})$.

###  II.1 Singlet gaps

Here we analyze the contribution of the first neighbors and start the
construction of the gap functions using ${\bf r}_{1}=a{\bf e}_{x}$ as the
initial coordinate that coincides with one of the Bravais lattice vectors
(${\bf a}_{1}={\bf r}_{1}$ and ${\bf a}_{2}={\bf r}_{2}$). The other vectors
are obtained by using the action of the elements of the ${\bf C}_{3{\rm v}}$
group: ${\bf r}_{2}=\mathcal{D}^{+}(C_{3}){\bf r}_{1}$, ${\bf
r}_{3}=\mathcal{D}^{+}(C_{3}^{2}){\bf r}_{1}$, ${\bf
r}_{4}=\mathcal{D}^{+}(\sigma_{\rm v}){\bf r}_{1}$, ${\bf
r}_{5}=\mathcal{D}^{+}(\sigma_{\rm v}C_{3}){\bf r}_{1}$, ${\bf
r}_{6}=\mathcal{D}^{+}(\sigma_{\rm v}C_{3}^{2}){\bf r}_{1}$. The $k$-dependent
functions that can be used to construct the gap function are $v_{i}={\bf
k}\cdot{\bf r}_{i}$, $(i=1,\ldots,6)$. Note that under an element $g$ of the
point group, ${\bf k}\rightarrow g{\bf k}$ and $f({\bf k}\cdot{\bf r})$ is
mapped into $f((g{\bf k})\cdot{\bf r})=f({\bf k}\cdot(g^{-1}{\bf r}))$.

Using the symmetrization procedure in Eq. (1), we can construct the order
parameter function with $\cos$ having $v_{i}$ (or some linear combination) as
an argument. The gap function in the IR $A_{1}$ has the form

$\Delta_{A_{1}}^{\rm
s}=\Big{[}2\cos{(k_{x}a)}+4\cos{\frac{k_{x}a}{2}}\cos{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{0}.$
(2)

Note that, around the $\Gamma$ point, $\Delta_{A_{1}}^{\rm s}\approx d_{0}$,
resembling the well known $s$-wave gap.

Figure 1: Perspective view of the NbSe2 monolayer crystal structure with the
considered lattice vectors ${\bf a}_{1}$ and ${\bf a}_{2}$, vertical mirror
plane $\sigma_{\rm v}$ that coincides with the $yz$-plane, and the main
$C_{3}$ axis.

The IR $E$ is a two-dimensional representation for which we construct the
order parameter in the two-component form $\Delta_{E}^{\rm
s}=(\Delta_{E,1}^{\rm s},\Delta_{E,2}^{\rm s})$. The components are connected
by the vertical mirror plane symmetry $\sigma_{\rm v}$ as $\sigma_{\rm
v}\Delta_{E,1}^{\rm s}=\Delta_{E,2}^{\rm s}$ and read

$\displaystyle\Delta_{E,1/2}^{\rm s}$ $\displaystyle=$
$\displaystyle\Big{[}\cos{(k_{x}a)}-\cos{\frac{k_{x}a}{2}}\cos{\frac{\sqrt{3}k_{y}a}{2}}$
(3) $\displaystyle\pm{\rm
i}\sqrt{3}\sin{\frac{k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{0}.$

Close to the $\Gamma$ point, $\Delta_{E,1/2}^{\rm
s}\approx(k_{x}^{2}-k_{y}^{2})\pm{\rm i}k_{x}k_{y}$, resembling the
$d+\mathrm{i}d$ type of a superconducting gap.

In the first neighbor approximation, the order parameter that transforms
according to the IR $A_{2}$ is zero. To obtain a nonzero gap function one
needs to consider for the symmetrization procedure the initial vector ${\bf
r}\rightarrow 2{\bf a}_{1}-{\bf a}_{2}$ which corresponds to the third
neighbor. The gap function then reads

$\displaystyle\Delta_{A_{2}}^{\rm s}$ $\displaystyle=$ $\displaystyle
2\Big{[}\sin{\frac{k_{x}a}{2}}\sin{\frac{3\sqrt{3}k_{y}a}{2}}-\sin{(2k_{x}a)}$
$\displaystyle\times\sin{(\sqrt{3}k_{y}a)}+\sin{\frac{5k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{0}.$

The leading polynomial around the $\Gamma$ is of the sixth order and equals
$k_{x}^{5}k_{y}-10/3k_{x}^{3}k_{y}^{3}+k_{x}k_{y}^{5}$. If we express this
term in polar coordinates, we find that $\Delta_{A_{2}}^{\rm
s}\approx\sin{(6\varphi)}d_{0}$, has six nodal lines.

###  II.2 Triplet gaps

In the case of the triplet pairing, the order parameter is composed using the
pseudovector ${\bf d}=(d_{x},d_{y},d_{z})$, with components in the matrix form
equal to

$d_{x}=\begin{pmatrix}-1&0\\\ 0&1\\\ \end{pmatrix},d_{y}=\begin{pmatrix}{\rm
i}&0\\\ 0&{\rm i}\\\ \end{pmatrix},d_{z}=\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix}.$ (4)

To construct an irreducible pseudovector, we will use the $\sin$ functions
(since the gap must be antisymmetric under the change ${\bf k}\rightarrow{-\bf
k}$), while we can independently use the multiplet of $(d_{x},d_{y})$ that
transform according to the IR $E$ and the pseudocomponent $d_{z}$,
transforming according to the representation $A_{2}$. Due to their different
symmetry properties, we construct the gap function using independently $d_{z}$
and the $(d_{x},d_{y})$ multiplet.

####  II.2.1 Triplets gap functions with $d_{z}$ pseudovector component

To construct triplet gap functions for the $d_{z}$ pseudovector component
solely, i.e., ${\bf d}_{z}=(0,0,d_{z})$, we consider for the initial
coordinate the vector ${\bf r}={\bf a}_{1}$, meaning that the nearest-neighbor
approximation is employed. The gap function for the $A_{1}$ representations
equals

$\displaystyle\Delta_{A_{1}}^{{\rm t},z}$ $\displaystyle=$
$\displaystyle\Big{[}\Big{(}\cos{\frac{k_{x}a}{2}}-\cos{\frac{\sqrt{3}k_{y}a}{2}}\Big{)}\sin{\frac{k_{x}a}{2}}\Big{]}d_{z}.$
(5)

Around the $\Gamma$ point, this gap function can be approximated as
$k_{x}(3k_{y}^{2}-k_{x}^{2})d_{z}$, representing the $f$-wave gap.

In the case of the $A_{2}$ IR, we are unable to obtain the gap function
considering the nearest neighbors, but in the case of second-neighbors, for
the initial coordinate ${\bf r}\to{\bf a}_{1}-{\bf a}_{2}$, the nonzero gap of
the form

$\displaystyle\Delta_{A_{2}}^{{\rm
t},z}=\Big{[}\sin{(\sqrt{3}k_{y}a)}-2\sin{\frac{\sqrt{3}k_{y}a}{2}}\cos{\frac{3k_{x}a}{2}}\Big{]}d_{z}$
(6)

is constructed. Around the $\Gamma$ point, the $\Delta_{A_{2}}^{{\rm t},z}$
gap function can be approximated as $k_{y}(k_{y}^{2}-3k_{x}^{2})d_{z}$,
suggesting the $f$-wave character of superconducting order parameter.

For the gap function within the two-dimensional IR $E$ considering the first
neighbors we obtain

$\displaystyle\Delta_{E,1/2}^{{\rm t},z}$ $\displaystyle=$
$\displaystyle\Big{[}\sin{(k_{x}a)}+\sin{\frac{k_{x}a}{2}}\cos{\frac{\sqrt{3}k_{y}a}{2}}$
(7) $\displaystyle\pm{\rm
i}\sqrt{3}\cos{\frac{k_{x}a}{2}\sin{\frac{\sqrt{3}k_{y}a}{2}}}\Big{]}d_{z}.$

We have additionally checked that the relation $\sigma_{\rm
v}\Delta_{E,1}^{{\rm t},z}=\Delta_{E,2}^{{\rm t},z}$ is satisfied, meaning
that the constructed gap fulfills the required symmetry. In the vicinity of
the zone center, the gap can be approximated as $(k_{x}\pm{\rm i}k_{y})d_{z}$,
resembling the $p+\mathrm{i}p$ gap.

####  II.2.2 Triplets gap functions with $d_{x}$ and $d_{y}$ pseudovector
components

We now construct triplet gaps using the doublet $(d_{x},d_{y})$. The gap
function transforming according to the IR $A_{1}$ is zero for nearest neighbor
coordinates. The non-zero contribution we get for ${\bf r}\to{\bf a}_{1}+{\bf
a}_{2}$ reading

$\displaystyle\Delta_{A_{1}}^{{\rm t},xy}$ $\displaystyle=$
$\displaystyle\Big{[}3\cos{\frac{k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{x}-\Big{[}\sqrt{3}\Big{(}2\cos{\frac{k_{x}a}{2}}$
(8)
$\displaystyle+\cos{\frac{\sqrt{3}k_{y}a}{2}}\Big{)}\sin{\frac{k_{x}a}{2}}\Big{]}d_{y}.$

Around the $\Gamma$ point, the gap function can be approximated as
$k_{y}d_{x}-k_{x}d_{y}$, consistent with the previous results based on the
method of invariants [67].

In the case of the IR $A_{2}$ representation, we can obtain a nonzero gap
function considering nearest-neighbors in the form

$\displaystyle\Delta_{A_{2}}^{{\rm t},xy}$ $\displaystyle=$
$\displaystyle\Big{[}\Big{(}2\cos{\frac{k_{x}a}{2}}+\cos{\frac{\sqrt{3}k_{y}a}{2}}\Big{)}\sin{\frac{k_{x}a}{2}}\Big{]}d_{x}$
(9)
$\displaystyle+\Big{[}\sqrt{3}\cos{\frac{k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{y}.$

Around the $\Gamma$ point, the gap function $\Delta_{A_{2}}^{{\rm t},xy}$ can
be approximated as $k_{x}d_{x}+k_{y}d_{y}$, as shown in [67]. Finally, for the
two-dimensional IR $E$ we were unable to construct the gap function using the
initial coordinate ${\bf r}=a{\bf e}_{x}$ for nearest-neighbors.

The relation $\sigma_{\rm v}\Delta_{E,1}^{{\rm t},xy}=\Delta_{E,2}^{{\rm
t},xy}$ is not satisfied, but rather $\sigma_{\rm v}\Delta_{E,1}^{{\rm
t},xy}=-\Delta_{E,2}^{{\rm t},xy}$. The solution $\Delta_{E}^{{\rm
t},xy}=(\Delta_{E,1}^{{\rm t},xy},\Delta_{E,2}^{{\rm t},xy})$ that has the
correct symmetry properties with respect to the vertical mirror symmetry can
be constructed using the initial vector ${\bf r}\to{\bf a}_{1}+2{\bf a}_{2}$

$\displaystyle\Delta_{E,1/2}^{{\rm t},xy}$ $\displaystyle=$
$\displaystyle\Big{[}2\sin{(\sqrt{3}k_{y}a)}-\cos{\frac{3k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}$
(10) $\displaystyle\pm{\rm
i}\sqrt{3}\sin{\frac{3k_{x}a}{2}}\cos{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{x}$
$\displaystyle+\Big{[}\sqrt{3}\sin{\frac{3k_{x}a}{2}}\cos{\frac{\sqrt{3}k_{y}a}{2}}$
$\displaystyle\mp 3{\rm
i}\cos{\frac{3k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}\Big{]}d_{y}.$

Around the $\Gamma$ point, the gap function can be approximated as
$\Delta_{E,1/2}^{{\rm t},xy}\approx(k_{x}d_{y}+k_{y}d_{x})\pm{\rm
i}(k_{x}d_{x}-k_{y}d_{y})$. Furthermore, it can be shown that the gap
functions $\Delta_{E,1/2}^{{\rm t},xy}$ are non-unitary. According to [68],
the unitary gap function $\Delta({\bf k})$ satisfies the equation

$\Delta({\bf k})\Delta^{{\dagger}}({\bf k})=\Delta_{U}^{2}({\bf
k})\mathcal{I}_{2},$ (11)

where $\Delta_{U}^{2}({\bf k})$ is the norm of the unitary gap, while
$\mathcal{I}_{2}$ is the identity $2\times 2$ matrix. By analyzing the
equation

$\Delta_{E,1/2}^{{\rm t},xy}(\Delta_{E,1/2}^{{\rm
t},xy})^{{\dagger}}=4\begin{pmatrix}A^{2}({\bf k})&0\\\ 0&B^{2}({\bf
k})\end{pmatrix},$ (12)

where

$\displaystyle A^{2}({\bf k})$ $\displaystyle=$
$\displaystyle\Big{[}-2\cos{\frac{3k_{x}a}{2}}\sin{\frac{\sqrt{3}k_{y}a}{2}}+\sin{(\sqrt{3}k_{y}a)}\Big{]}^{2},$
$\displaystyle B^{2}({\bf k})$ $\displaystyle=$ $\displaystyle
3\sin^{2}{(\frac{3k_{x}a}{2})}\cos^{2}{(\frac{\sqrt{3}k_{y}a}{2})}+\big{(}\cos{\frac{3k_{x}a}{2}}$
(13)
$\displaystyle\times\sin{\frac{\sqrt{3}k_{y}a}{2}}+\sin{(\sqrt{3}k_{y}a)}\big{)}^{2},$

one could conclude that the gap functions $\Delta_{E,1/2}^{{\rm t},xy}$ are
non-unitary, since the functions $A^{2}({\bf k})$ and $B^{2}({\bf k})$ are not
identically equal. As an illustration, we expand the functions $A^{2}({\bf
k})$ and $B^{2}({\bf k})$ around the $\Gamma$ point up to the ${\bf k}^{2}$
term and get $A^{2}({\bf k})\approx 0$, $B^{2}({\bf k})\approx
27/4(k_{x}^{2}+k_{y}^{2})a^{2}$. We note that the gap functions
$\Delta_{E,1}^{{\rm t},xy}$ and $\Delta_{E,2}^{{\rm t},xy}$ could realize a
certain linear combination $c_{1}\Delta_{E,1}^{{\rm
t},xy}+c_{2}\Delta_{E,2}^{{\rm t},xy}$. Assuming that $c_{1}$ and $c_{2}$ are
arbitrary complex numbers, the unitarity given by Eq. (11) is restored if and
only if the relation $|c_{1}|^{2}=|c_{2}|^{2}$ is satisfied.

##  III Effective quasiparticle BdG model for NbSe2

We are interested in studying the effects of electron superconducting pairing
in monolayer NbSe2 at a low-temperature regime, neglecting superconducting
phase fluctuation [69]. We model the superconductivity by Bogoliubov de-Gennes
(BdG) formalism with Hamiltonian

$\mathcal{H}_{\mathrm{BdG}}=\frac{1}{2}\sum_{\mathbf{k}}\Psi^{\dagger}_{\mathbf{k}}\mathcal{H}_{\mathbf{k}}^{\mathrm{BdG}}\Psi_{\mathbf{k}},$
(14)

considering the Nambu spinor,
$\Psi^{\dagger}_{\mathbf{k}}=[c^{\dagger}_{\mathbf{k}\uparrow},c^{\dagger}_{\mathbf{k}\downarrow},c_{\mathbf{-k}\uparrow},c_{\mathbf{-k}\downarrow}]$
with fermionic creation $c^{\dagger}_{\mathbf{k}\uparrow}$ and annihilation
$c_{-\mathbf{k}\uparrow}$ operators. The $4\times 4$ BdG Hamiltonian in
reciprocal space $\mathcal{H}_{\mathbf{k}}^{\mathrm{BdG}}$ reads [70]

$\mathcal{H}_{\mathbf{k}}^{\mathrm{BdG}}=\begin{pmatrix}H_{\mathrm{e}}(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{{\dagger}}(\mathbf{k})&-H_{\mathrm{e}}^{\mathrm{T}}(-\mathbf{k})\end{pmatrix},$
(15)

where the electron-like Hamiltonian $H_{\mathrm{e}}(\mathbf{k})$ is the
effective tight-binding Hamiltonian

$H_{\mathrm{e}}(\mathbf{k})=\mathcal{H}_{\mathrm{orb}}(\mathbf{k})+\mathcal{H}_{\mathrm{I}}(\mathbf{k})+\mathcal{H}_{\mathrm{R}}(\mathbf{k}),$
(16)

with the orbital part $\mathcal{H}_{\mathrm{orb}}$ describing the dispersion
of the Nb $d$-band in the vicinity of the Fermi level, intrinsic
$\mathcal{H}_{\mathrm{I}}(\mathbf{k})$ and Rashba spin-orbit coupling
$\mathcal{H}_{\mathrm{R}}(\mathbf{k})$ contributions. More details of the
Hamiltonian are discussed in Appendix A.

The superconducting order parameter is equal to
$\Delta_{\mathbf{k}}=\Delta_{0}\Delta^{\mathcal{P}}_{r}(\mathbf{k})$, where
$\Delta_{0}$ is the gap amplitude, while
$\Delta^{\mathcal{P}}_{r}(\mathbf{k})$ is the $\mathbf{k}$-dependent part of
the gap function respecting the point group symmetry representations,
$r=\\{A_{1},A_{2},E_{1/2}\\}$, for singlet and triplet pairings,
$\mathcal{P}=\\{{\rm s},{\rm t}\\}$, as discussed in Section II.
Superconducting critical temperature of monolayer 1H-NbSe2 is $T_{c}\approx
2\leavevmode\nobreak\ \mathrm{K}$ [71, 72, 25]. This yields from BCS theory
for superconducting gap $\Delta(0\leavevmode\nobreak\ \mathrm{K})\approx 0.3$
meV. For simplicity, in what follows, we will use $\Delta_{0}=1$ meV,
approaching the same order of magnitude as the extracted superconducting gap.

###  III.1 QPI & Spectral functions

QPI is extracted from scanning tunneling spectroscopy measurements by
examining the Fourier transform of $\mathrm{d}I/\mathrm{d}V$ maps of the local
density of states (LDOS). It is a quite powerful qualitative tool and
represents a unique probe of wavelengths of LDOS-modulated oscillations caused
by impurities present in the system, which in turn contains information on the
electronic structure of the pure metallic [73, 74], semiconducting [75, 76]
and superconducting [77, 78] systems. Calculated QPI patterns are similar to
those experimentally observed [79] and have been used for identification of
the disorder type [80], superconducting gap features such as phase structure,
sign [81, 82] and orbital order [83]. It has been shown that quasiparticle
responses of many impurities possess identical poles as a single impurity case
[84, 85], therefore, we approximate the description of the elastic scattering
process on a spin-conserving single impurity. Such impurity potential can be
written in real Nambu space as

$V=v\tau_{3}\otimes\sigma_{0},$ (17)

where $v$ is the impurity strength, $\tau_{3}$ is the Pauli matrix for
electron-holes and $\sigma_{0}$ is the identity operator in the spin space.
Considering the potential $V$ as a perturbation, the LDOS can be calculated
using the T-matrix approach via Green’s function written as

$G(\mathbf{k},\mathbf{k}^{\prime};\omega)=G_{0}(\mathbf{k},\mathbf{k}^{\prime};\omega)+\delta
G(\mathbf{k},\mathbf{k}^{\prime};\omega)$ (18)

with

$\delta
G(\mathbf{k},\mathbf{k}^{\prime};\omega)=\sum_{\mathbf{k}_{1},\mathbf{k}_{2}}G_{0}(\mathbf{k},\mathbf{k}_{1};\omega)T(\mathbf{k}_{1},\mathbf{k}_{2};\omega)G_{0}(\mathbf{k}_{2},\mathbf{k}^{\prime};\omega),$
(19)

where $G_{0}(\mathbf{k},\mathbf{k}^{\prime};\omega)$ is retarded bare Green’s
function and $T(\mathbf{k},\mathbf{k}^{\prime};\omega)$ is the T-matrix,
represent the solution of the scattering problem for single scalar impurity
[86], and $\omega=\epsilon+\mathrm{i}\delta$ is the quasiparticle energy with
small lifetime broadening ($\delta\approx 0.1$ meV). The spin-conserving
scalar impurity $V$ leads to a T-matrix independent of momentum. Hence, the
T-matrix reads

$T(\omega)=V\cdot[\mathbb{1}-V\cdot G_{0}(\omega)]^{-1}$ (20)

where $G_{0}(\omega)=1/\Omega\int[\omega-\mathcal{H}_{0}(\mathbf{k})]^{-1}$ is
the integrated bare Green’s function within the first Brillouin zone (BZ). We
assume in normal phase $\mathcal{H}_{0}(\mathbf{k})=H_{\rm e}(\mathbf{k}),$
and for superconducting phase
$\mathcal{H}_{0}(\mathbf{k})=\mathcal{H}_{\mathbf{k}}^{\rm BdG}.$ As QPI
describes LDOS of scattered quasiparticle process with momentum
$\mathbf{k}\rightarrow\mathbf{k}+\mathbf{q}$, we substitute momentum
$\mathbf{k}^{\prime}$ with $\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{q}$ and
integrate the Green’s function (18) over all possible $\mathbf{k}$ points in
the first BZ. After the integration, the scattered quasiparticle amplitude can
be expressed as follows

$\rho(\mathbf{q};\omega)=-\frac{1}{\pi}\mathrm{Im}\left\\{\frac{1}{\Omega}\int\mathrm{d}\mathbf{k}\,\delta
G(\mathbf{k},\mathbf{k}+\mathbf{q};\omega)\right\\}.$ (21)

The QPI $\rho(\mathbf{q};\omega)$ can be evaluated via convolution theorem
[87], which reduces computational complexity allowing to study of fine grid
maps with small broadening.

For calculating the spectral characteristics of the band structure we use an
imaginary part of the spectral function defined as

$\mathcal{A}(\mathbf{k};\omega)=-\frac{1}{\pi}\mathrm{Im}\\{\mathrm{Tr}[G_{0}(\mathbf{k};\omega)\cdot(\mathbb{1}-V\cdot
G_{0}(\omega))^{-1}]\\},$ (22)

obtained from Eq. (18) taking $\mathbf{k}^{\prime}=\mathbf{k}$. The Bloch
spectral (density) function has interpretation as $\mathbf{k}$-resolved DOS
[88].

##  IV QPI patterns

In this section, we discuss calculated QPI maps separately for each derived
superconducting gap function $\Delta_{r}^{p}(\mathbf{k})$ within the
$\mathbf{C}_{3{\rm v}}$ symmetry. The QPIs were calculated for the Fermi
energy of the single-band electron Hamiltonian $H_{\rm e}(\mathbf{k})$, see
Appendix A, that effectively describes doped NbSe2 by 0.56 electrons. Such
doping corresponds to a rigid shift of the Nb $d$-band as reported for
(LaSe)1.14(NbSe2)2 misfit using angle-resolved photoemission spectroscopy,
scanning tunneling microscopy, and quasiparticle interference measurements
[45]. Similar electron doping was reported also for intercalated bulk NbSe2 by
imidazole cations [89].

In Fig. 2(a) we show spectral function $\mathcal{A}(\mathbf{k})$ for the doped
NbSe2 monolayer at the Fermi level with the Fermi pockets around the $\Gamma$
and K points. The QPI map for the scalar impurity with the potential of
$v=-0.1$ eV is shown in Fig. 2(b). For small $\mathbf{q}$ momenta centered
around the $\Gamma$ point there are three distinct contrasts forming contours
that correspond to $\Gamma$-intravalley and two K-intravalley scattering
processes, see schematically sketched $\mathbf{q}$ vectors in the Fig. 2(a).
The two distinct K-intravalley contours reflect strong spin-orbit split bands
at the K pockets. For large scattering $\mathbf{q}$ we identified K-point
centered patterns: intervalley ${\rm K-K^{\prime}}$ scattering, and two
contours corresponding to the $\Gamma-{\rm K}$ intervalley scattering implying
the effect of the spin-orbit split bands around the K point.

Figure 2: Calculated spectral characteristics for doped NbSe2 monolayer by
0.56 electrons in normal state. (a) spectral function
$\mathcal{A}(\mathbf{k})$ at the Fermi level; (b) QPI map for the scalar
impurity with the potential $v=-0.1$ eV.

Bogoliubov quasiparticles can be viewed as Bloch states with modulated
dispersion for which an elastic scattering can also result in modulation of
the QPI signal. As the single-particle bandstructure is known in our case,
from the contours at the Fermi energy, see Fig. 2, $\mathbf{k}$-dependence of
the gap function, and density of states (DOS), see Appendix B, we can identify
the origin of the most relevant scattering channels. We start by examining QPI
map for the singlet gap functions. In Fig. 3(a) we show a calculated QPI map
for $A_{1}$ representation. The finite smearing of the entirely gaped system,
see Fig. 7(a), leads to a QPI signal that preserves contour features discussed
for the normal case.

For $A_{2}$ representation the $\Gamma$-intravalley and intervalley $\Gamma-$K
scatterings maintain significant contributions. The nodal lines connecting
$\Gamma-$K, $\Gamma-$M, and K$-$K’ points lead to a rich QPI pattern
resembling a dahlia flower, see Fig. 3(b). The in-gap states for the
$\Delta^{\rm s}_{A_{2}}$ gap function, see the v-shaped DOS dependence shown
in Fig. 7(b), lead to the small $q$ momenta scattering events well-visible
near the $\Gamma$ point as an inner corolla. It combines the small momentum
scattering near the $\Gamma$ and K pockets resulting in 18 petals. 12 petals
point to regions in between the nodal lines, and six petals point along the
$\Gamma-$M lines that contain the $\sigma_{\rm v}$ plane. The dahlia-like QPI
pattern near the $\Gamma$ point follows the six-fold symmetry of the
$\Delta^{\rm s}_{A_{2}}\approx\sin(6\varphi)d_{0}$, see the zone center zoom
in Fig. 3(b).

In the case of $E$ representations, for the 2D multiplet
$(\Delta_{E,1},\Delta_{E,2})$ we will restrict our analysis to the first
component $\Delta_{E,1}$ only, representing the energetically most favorable
situation [60]. The QPI pattern for the singlet pairing shows similar contours
as the normal phase. The gap function $\Delta_{E,1}^{\rm s}$ opens a global
gap at the Fermi level, see Fig. 7(c). For the $\Gamma$ pocket, the gap
function amplitude is smaller than for the K pockets. This leads within the
finite broadening to the enhanced $\Gamma$ intravalley contour, and $\Gamma-$K
intervalley contours. The finite broadening and reduced gap amplitude is also
responsible for the enhanced signal at the zone center.

Figure 3: Calculated QPI patterns in the superconducting state for singlet
order parameters for (a) $A_{1}$, (b) $A_{2}$, and (c) $E$ representations.
Plots in the bottom row are the zone center zooms for the corresponding top
row. The inset shows the first Brillouin zone with nodal lines. For
calculations superconducting pairing amplitude $\Delta_{0}=1$ meV, the real
part of the quasiparticle energy $\omega$ equal to Fermi energy, and the
amplitude of the scalar impurity potential $v=-0.1$ meV was used.

We now move on to the case of the triplet pairing with $d_{z}$ pseudovector
component. In Fig. 4 we plot QPI map for $A_{1}$, $A_{2}$ and $E$
representations. In the case of the $A_{1}$ representation the K-intravalley
and K$-$K’ intervalley scatterings are suppressed. Instead, we observe a
significant signal enhancement close to the $\Gamma$ and K points, see Fig.
4(a). The enhancement is related to the presence of the nodal lines connecting
$\Gamma-$M points. The $\Gamma$ pocket serves 6-fold symmetric in-gap states.
As the quasiparticles can scatter preferentially along the nodal lines with
small momentum due to finite lifetime, the QPI signal forms a daisy-like
flower pattern with six petals along the nodal lines, see the zoom near the
zone center and inset of the first BZ showing the nodal lines. The fine
structure around the K point has 3-fold symmetry and the sharp QPI signal is
governed by the umklapp processed combining quasiparticles scattered off the
nodal line in-gap states within the $\Gamma$ pocket and the states within the
K pocket. The K pocket states are gaped, residing within the coherent peaks,
but due to a finite broadening they offer a significant number of states to
absorb the scattering. The gap function for the $A_{2}$ representation
possesses six-fold nodal symmetry at the $\Gamma$ and K points. The nodal
lines connect $\Gamma-$K and K$-$K’ points. The nodal quasiparticles modulate
the QPI map such that it resembles a daisy flower pattern, see Fig. 4(b). The
pattern is rotated by 30 degrees compared to the $A_{1}$ representation. The
nodal character of the gap function $\Delta_{A_{2}}^{{\rm t},z}$ enhances the
QPI signal on the apexes of the petals pointing along the nodal lines. The
nodal character of the gap function also contributes to the small $\mathbf{q}$
momenta scattering forming characteristic inner corolla near the $\Gamma$
point. Close to the K point the 3-fold inner pattern is rotated by 60 degrees
compared to $A_{1}$ representation signaling the contribution from the K$-$K’
scatterings. We note that the K$-$K’ processes can be distinguished as they
form a clear 3-fold symmetric pattern while the $\Gamma-$K processes capture
6-fold symmetry. We note that the nodal gap functions for $A_{1}$ and $A_{2}$
representations result in v-shaped DOS, see Fig. 7(d,e). In Fig. 4(c) we show
the QPI map for the $E$ representation. The gap function $\Delta_{E,1}^{{\rm
t},z}$ is non-zero for the considered Fermi energy, and the global gap is
opened, see Fig. 7(f). The Bogoliubov quasiparticles are gaped, however, the
amplitude of the gap for the $\Gamma$ pocket is larger than for the K pockets.
We note that the gap amplitude drops to zero at the K point, therefore the
inner K pocket is less gapped than the outer one. Due to the finite broadening
the K intravalley QPI contour is enhanced as well as the K$-$K’ intervalley
contours in comparison to the processes involving the $\Gamma$ pocket.

Figure 4: Calculated QPI patterns in the superconducting state for triplet
order parameters with pseudovector component $d_{z}$ for (a) $A_{1}$, (b)
$A_{2}$, and (c) $E$ representations. Plots in the bottom row correspond to
the zone center zoom. The insets show the first BZ with corresponding nodal
lines. The other parameters are as in Fig. 3.

Finally, we analyze the triplet pairing constructed using the multiplet
$(d_{x},d_{y})$. The QPI patterns for the $A_{1}$, $A_{2}$, and $E$
representations are similar, containing significant contour traces of the K
intravalley and all the intervalley scattering processes except the $\Gamma$
intervalley scattering which is suppressed, see Fig. 5. The suppression is due
to the gap opened for the $\Gamma$ pocket for $A_{1}$ and $A_{2}$
representations, while for the $E$ representation the gap has a node along the
$\Gamma$-K lines. In Fig. 7(g,h) we plot DOS for the $A_{1}$ and $A_{2}$
representations. The main coherence peaks enclose the gap opened for the
$\Gamma$ pocket with a constant density corresponding to linearly dispersing
in gap states around the K valleys forming the narrow v-shaped dependence near
the zero energy. DOS for $E$ representation shows a broad finite valued
v-shaped dependence, see Fig. 7, due to the gapless spectrum around K valleys
and nodal features on the $\Gamma$ pocket along the $\Gamma$-K directions.
Nematicity of the singlet and triplet with $d_{z}$ gap functions for the two-
dimensional representation $E$, as well as the chiral symmetry violation of
the triplet with $(d_{x},d_{y})$ multiple, can be detected with QPI. We
discuss it in Appendix C and D.

Figure 5: Calculated QPI patterns in the superconducting state for triplet
order parameters with pseudovector $(d_{x},d_{y})$ multiplet for (a) $A_{1}$,
(b) $A_{2}$, and (c) $E$ representations. Plots in the bottom row are the zone
center zooms for the corresponding top row. The other parameters are as in
Fig. 3.

##  V Conclusion

For a single-band model of NbSe2 monolayer, Ising superconductor with Rashba
spin-orbit coupling, we constructed using a group theoretical approach all
possible superconducting gap functions beyond the nearest-neighbor
approximation for all irreducible representations of ${\bf C}_{3{\rm v}}$
symmetry. We found the nodal gap function for the singlet pairing in $A_{2}$
representation and non-unitary triplet gap functions for two-dimensional $E$
representation. Our analysis indicates that the nodal and nematic
superconducting pairing, recently observed experimentally can be connected to
the nodal and two-dimensional superconducting pairing functions. Breaking the
unitarity is associated with the asymmetrical band dispersion in the
superconducting energy spectra. Calculated QPI for single scalar impurity
revealed characteristic patterns for the pairing functions. The symmetry-based
approach of combining group theoretical analysis and the QPI technique paves
the way toward understanding exotic superconductivity. We showed that QPI can
capture nematic response for the two-dimensional representation that breaks
time-reversal symmetry for singlet and triplet with $d_{z}$, and chiral
symmetry violation for the triplet with $(d_{x},d_{y})$ multiplet. Our
approach to superconducting gap function construction can be extended to the
case of an arbitrary class of the point group symmetry [90] and the case of
multiorbital superconductivity.

###### Acknowledgements.

J.H. acknowledges the financial support provided by the Ministry of Education,
Science, Research and Sport of the Slovak Republic. M.M. acknowledges the
financial support provided by the Ministry of Education, Science, and
Technological Development of the Republic of Serbia. This project has received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme under the Programme SASPRO 2 COFUND Marie Sklodowska-Curie grant
agreement No. 945478. M.G. acknowledges financial support from Slovak Research
and Development Agency provided under Contract No. APVV-20-0425, and Slovak
Academy of Sciences project IMPULZ IM-2021-42 and project FLAG ERA JTC 2021
2DSOTECH.

##  Appendix A Model Hamitonian

###  A.1 Effective tight-binding model for NbSe2

To describe efficiently electronic states of NbSe2 monolayer we consider an
effective single-orbital tight-binding model Hamiltonian [57]. The orbital
part of the Bloch Hamiltonian up to the seventh Nb neighbor atoms reads

$\displaystyle{\cal H}_{\rm orb}({\bf
k})=\varepsilon_{0}+2t_{1}(\cos{2\alpha}+2\cos{\alpha}\cos{\beta})+2t_{2}(\cos{2\beta}$
$\displaystyle+$ $\displaystyle
2\cos{3\alpha}\cos{\beta})+2t_{3}(\cos{4\alpha}+2\cos{2\alpha}\cos{2\beta})$
$\displaystyle+$ $\displaystyle
4t_{4}(\cos{\alpha}\cos{3\beta}+\cos{4\alpha}\cos{2\beta}+\cos{5\alpha}\cos{\beta})+2t_{5}\times$
$\displaystyle(\cos{6\alpha}+2\cos{3\alpha}\cos{3\beta})+2t_{6}(\cos{2\beta}+2\cos{3\alpha}\cos{\beta})$
$\displaystyle+$ $\displaystyle
4t_{7}(\cos{7\alpha}\cos{\beta}+\cos{5\alpha}\cos{3\beta}+\cos{2\alpha}\cos{4\beta}),$
(23)

where $\alpha=k_{x}a/2$, $\beta=\sqrt{3}k_{y}a/2$, $a$ is the lattice
constant, $k_{x}$ and $k_{y}$ are components of the wave vectors in Cartesian
frame, $\varepsilon_{0}$ is the energy offset of the corresponding to the
chemical potential of the electron-doped NbSe2 layer with respect to the
isolated NbSe2 layer, and $t_{i}$, $i=1,...,7$ are the real hopping
parameters. The model describes Nb $d$-band close to the Fermi level. By
including intrinsic SOC in accordance with the ${\bf D}_{3{\rm h}}$ point
group of the free-standing NbSe2 monolayer, the effective $d$-band split and
the horizontal mirror plane $\sigma_{\rm h}$ constrains spins in the out-of-
plane direction. The intrinsic spin-orbit coupling Hamiltonian $H_{\rm I}$
[57] up to the third neighbor equals to

$\displaystyle{\cal H}_{\rm I}({\bf k})=2\sigma_{z}\Big{(}$
$\displaystyle\lambda_{\rm I}^{(1)}$
$\displaystyle(\sin{2\alpha}-2\sin{\alpha}\cos{\beta})$ $\displaystyle+$
$\displaystyle\lambda_{\rm I}^{(3)}$
$\displaystyle(\sin{4\alpha}-2\sin{2\alpha}\cos{2\beta})\Big{)}\,,$ (24)

where $\sigma_{z}$ is the Pauli matrix, and the nonzero real parameters
$\lambda_{\rm I}^{(1)}$ and $\lambda_{\rm I}^{(3)}$ quantify the strength of
the interaction in the first and third neighbor approximation. The parameter
$\lambda_{\rm I}^{(2)}=0$ due to vertical mirror plane symmetry.

When a monolayer of NbSe2 is placed on a substrate, the horizontal mirror
plane symmetry $\sigma_{\rm h}$ is broken, triggering the appearance of Rashba
spin-orbit due to a proximity effect by virtue of an electrical field
perpendicular to the plane. This leads to a reduction of the ${\bf D}_{3{\rm
h}}$ symmetry to ${\bf C}_{3{\rm v}}$. The effective Rashba spin-orbit
coupling can be derived similarly as the intrinsic counterpart. In real space,
the Rashba Hamiltonian has the following form

$H_{\rm R}={\rm i}\sum_{l,i}\lambda_{\rm R}^{(l)}({\bf e}_{z}\times{\bf
e}_{i}^{(l)})\cdot{\bm{\sigma}},$ (25)

where $\lambda_{\rm R}^{(l)}$ are the real-valued Rashba parameters for the
$l$-th order, ${\bf e}_{z}$ is the unit vector in the $z$-direction, and ${\bf
e}_{i}^{(l)}={\bf d}_{i}^{(l)}/|{\bf d}_{i}^{(l)}|$ is the normalized distance
of each Nb atom from the centered one. By performing the Fourier transform of
the Rashba Hamiltonian, we obtain the following form in the $k$-space

${\cal H}_{\rm R}({\bf k})={\rm i}\sum_{l,i}\lambda_{\rm R}^{(l)}{\rm e}^{{\rm
i}{\bf k}\cdot{\bf d}_{i}^{(l)}}({\bf e}_{z}\times{\bf
e}_{i}^{(l)})\cdot{\bm{\sigma}}.$ (26)

Next, we consider contributions up to the third neighbors, obtaining

$\displaystyle{\cal H}_{\rm R}({\bf k})=2\lambda_{\rm
R}^{(1)}\Big{(}\sqrt{3}\cos{\alpha}\sin{\beta}\sigma_{x}-f(\alpha,\beta)\sigma_{y}\Big{)}$
$\displaystyle+2\lambda_{\rm
R}^{(2)}\Big{(}(\sin{2\beta}+\cos{3\alpha}\sin{\beta})\sigma_{x}-\sqrt{3}\sin{3\alpha}\cos{\beta}\sigma_{y}\Big{)}$
$\displaystyle+2\lambda_{\rm
R}^{(3)}\Big{(}\sqrt{3}\cos{2\alpha}\sin{2\beta}\sigma_{x}-f(2\alpha,2\beta)\sigma_{y}\Big{)}.$
(27)

where $f(\alpha,\beta)=(\sin{2\alpha}+\sin{\alpha}\cos{\beta})$, $\sigma_{x}$
and $\sigma_{y}$ are Pauli matrices, while $\lambda_{\rm R}^{(1)}$,
$\lambda_{\rm R}^{(2)}$, $\lambda_{\rm R}^{(3)}$ are parameters to be
determined by fitting the DFT data to the model Hamiltonian ${\cal H}_{\rm
orb}({\bf k})+{\cal H}_{\rm I}({\bf k})+{\cal H}_{\rm R}({\bf k})$. In
addition to the Rashba parameters, the orbital $\varepsilon_{0}$, $t_{i}$,
$i=1,...,7$ and intrinsic $\lambda_{\rm I}^{(1)/(3)}$ parameters need to be
determined.

Table 2: Fitting parameters of the single-orbital band model of NbSe2. Besides the energy offset $\varepsilon_{0}$, seven hopping parameters $t_{i}$, $i=1,...,7$, model the spin-independent band structure. Spin-orbit properties are described using the parameters $\lambda_{\rm I}^{(1)/(2)}$ and $\lambda_{\rm R}^{(1)/(2)/(3)}$, representing the interaction strengths of the intrinsic and Rashba spin-orbit Hamiltonian. $\varepsilon_{0}$ [meV] | -346.26
---|---
$t_{1}$ [meV] | 33.52
$t_{2}$ [meV] | 97.26
$t_{3}$ [meV] | -2.11
$t_{4}$ [meV] | -13.53
$t_{5}$ [meV] | -10.30
$t_{6}$ [meV] | 3.48
$t_{7}$ [meV] | 1.69
$\lambda_{1}$ [meV] | 13.27
$\lambda_{2}$ [meV] | -1.94
$\lambda_{\rm R}^{1}\,[\mu{\rm eV}]$ | -9.60
$\lambda_{\rm R}^{2}\,[\mu{\rm eV}]$ | -0.29
$\lambda_{\rm R}^{3}\,[\mu{\rm eV}]$ | -4.70

###  A.2 Fitting the model to the DFT data

To obtain relevant parameters for the single-band model, which is, besides the
superconducting gap function, one of the inputs in BdG Hamiltonian, we
performed DFT calculations of electronic structure for the NbSe2 monolayer.
This was done using the plane wave DFT suite QUANTUM ESPRESSO (QE) package
[91, 92] using the full relativistic SG15 Optimized Norm-Conserving Vanderbilt
(ONCV) pseudopotentials [93, 94, 95], with the kinetic energy cut-offs for the
wave function and charge density 45 Ry and 180 Ry, respectively. For the
Brillouin zone sampling, a $12\times 12\times 1$ $k$-points mesh was
considered using the Monkhorst-Pack scheme. The energy convergence threshold
for self-consistent calculation, including the spin-orbit coupling, was sent
to $10^{-10}$ Ry/bohr. A vacuum of 15 ${\rm\AA}$ in the $z$-direction was
used.

Figure 6: Calculated energy bands dispersion close to the Fermi level along
high symmetry lines in the first Brillouin zone for NbSe2 monolayer in the
electric field of 0.1 V/nm perpendicular to the monolayer plane, and the
corresponding spin expectation values. The circles are DFT date, and the solid
lines are effective single orbital tight-binding model.

We obtained the orbital and spin-orbital hopping parameters by fitting the DFT
data for the NbSe2 monolayer in a perpendicular field of 0.1 V/nm. The
parameters are gathered in Table 2. A comparison between the numerical band
structure and spin expectation values and our model is shown in Fig. 6.

##  Appendix B Density of states

In Fig. 7 we plot the density of states (DOS) as a function of the
quasiparticle energy for the BdG Hamiltonian considering superconducting gap
functions derived in Sec. II. The energy dependencies were calculated using
the linear tetrahedron method with $8192\times 8192$ $k$-point sampling of the
first BZ. We consider $\Delta_{0}=10$ meV in order to reduce the computational
complexity in terms of $k$-sampling. We note that the overall qualitative
dependencies of the DOS are unaffected when compared to $\Delta_{0}=1$ meV.
For the singlet pairing the gap has finite value for $A_{1}$ and $E$
representations, see Fig. 7(a,c). The K pocket states form the gap coherence
peaks for the $A_{1}$ representation, while for the $E$ the peaks originate
from the $\Gamma$ pocket. For the $A_{2}$ representation the nodal line
character of the gap function results in a v-shape dependence, see Fig. 7(b).

For the triplet pairing with $d_{z}$ pseudovector component the gap functions
for $A_{1}$ and $A_{2}$ representations show the v-shape gap, see Fig. 7(d,e).
For the $E$ representation the global gap is open with coherence peaks
originating from the K pocket.

The triplet pairing with $(d_{x},d_{y})$ multiplet shows similar v-shaped DOS
for the gap functions for the $A_{1}$ and $A_{2}$, see Fig. 7(g,h). In the
case of the $E$ representation, see Fig. 7(i), the DOS is finite with a rather
broad v-shape dependence around the zero energy.

Figure 7: Calculated density of states for NbSe2 monolayer in the electric
field of 0.1 V/nm perpendicular to the monolayer plane for singlet gap
functions (a) for $A_{1}$ representation; (b) for $A_{2}$ representation; (c)
for $E$ representation. Density of states for triplet gap functions with
pseudovector component $d_{z}$ (d) for $A_{1}$ representation; (e) $A_{2}$
representation; (f) for $E$ representation; and for pseudovector multiplet
$(d_{x},d_{y})$ (g) $A_{1}$ representation; (h) for $A_{2}$ representation;
(i) for $E$ representation. For calculations the amplitude $\Delta_{0}=10$ meV
was used.

##  Appendix C Nematicity response in QPI

The gap functions for the two-dimensional representation $E$ in singlet and
triplet pairing lead to a nematic phase with a global gap. This pairing type
belongs to the chiral topological class [96]. Any combination of the form
$\Delta_{E}^{\mathcal{P}}(\varphi)=\cos(\varphi)\Delta_{E,1}^{\mathcal{P}}+\sin(\varphi)\Delta_{E,2}^{\mathcal{P}}$
follows the irreducible representation and can be detected by the QPI. The gap
functions in the lowest order in momentum in the vicinity of the Brillouin
zone center are listed in Table 3. The $\varphi$ rotates the gap functions
where for the singlet pairing one recognizes the rotated $d+\mathrm{i}d$ gap,
while for the triplet with $d_{z}$ pseudovector, it rotates the $p+{\rm i}p$
type gap. In Fig. 8 we plot the QPI patterns for the singlet and triplet with
$d_{z}$, $\mathcal{P}=\\{s,({\rm t},z)\\}$ and phase angle
$\varphi=\\{\pi/4,\pi/2,3\pi/4\\}$. In case of $\Delta_{E}^{{\rm t},z}(\pi/4)$
the QPI pattern resembles $p_{y}$-like orbital for the small $\mathbf{q}$.
This can be understood as the gap function having reduced amplitude for
$k_{x}\simeq 0$ allowing due to the finite lifetime of quasiparticles an
enhanced scattering along the $q_{y}$. Similarly one explains other calculated
QPI patterns.

$\varphi$ |  $\pi/4$ |  $\pi/2$ | $3\pi/4$
---|---|---|---
$\Delta_{E}^{\rm s}(\varphi)/d_{0}$ |  $k_{x}^{2}-k_{y}^{2}$ |  $(k_{x}-ik_{y})^{2}$ | $ik_{x}k_{y}$
$\Delta_{E}^{{\rm t},z}(\varphi)/d_{z}$ |  $k_{x}$ |  $(k_{x}-ik_{y})$ | $ik_{y}$
Table 3: Nematic gap functions dependencies near the Brillouin zone center for
different phase angles. Figure 8: Calculated QPI patterns in the
superconducting state for $E$ representation with $(d_{x},d_{y})$ multiplet
for different phase angles. Singlet pairing (a) for $\varphi=\pi/4$, (b)
$\varphi=\pi/2$, (c) $\varphi=3\pi/4$. Triplet pairing with pseudovector
$d_{z}$ (d) for $\varphi=\pi/4$, (e) $\varphi=\pi/2$, (f) $\varphi=3\pi/4$.
The other parameters are as in Fig. 3.

##  Appendix D Chiral symmetry response in QPI

As discussed in Sec. II.2.2 the triplet $\Delta_{E}^{{\rm t},xy}$ gap function
is non-unitary except for the special phase angles $\varphi$. In Table 4 we
list the symmetry properties of the $\Delta_{E}^{{\rm t},xy}$ gap function.
The ${\bf C}_{\rm 3v}$ symmetry is reduced to ${\bf C}_{1}$ for $\varphi\in
3\pi/4+n\pi$ and to ${\bf C}_{\rm 1v}$ for $\varphi\in\pi/4+n\pi$, where $n$
is the integer. Along with the unitarity of the $\Delta_{E}^{{\rm t},xy}$ gap
function, the chiral symmetry of the BdG quasiparticle spectra is recovered
for the above special angles. In Fig. 9 we plot differences of the QPI signals
$\Delta\rho(\mathbf{q},\omega)=\rho(\mathbf{q},\omega)-\rho(\mathbf{q},-\omega)$
for $\Delta_{E}^{{\rm t},xy}$ and energy $\omega=0.07$ eV that falls to the
spectral gap opened between electron and hole spin down quasiparticle bands
near the K points. The non-zero signal of the $\Delta\rho(\mathbf{q},\omega)$
demonstrates the chiral symmetry violation of the BdG quasiparticle spectra.
For the case of $\varphi=0$, Fig. 9(a), the signal preserves the threefold
symmetry, while for the $\varphi=\pi/3$, Fig. 9(b), the pattern breaks the
threefold symmetry.

$\Delta^{{\rm t},xy}_{E}$ | unitary | chiral
---|---|---
${\bf C}_{\rm 3v}\xrightarrow{\varphi}{\bf C}_{1}$ | 0 | 0
${\bf C}_{\rm 3v}\xrightarrow{\varphi\in(3\pi/4+n\pi)}{\bf C}_{1}$ | 1 | 1
${\bf C}_{\rm 3v}\xrightarrow{\varphi\in(\pi/4+n\pi)}{\bf C}_{1{\rm v}}$ | 1 | 1
Table 4: Symmetries of the $\Delta_{E}^{{\rm t},xy}$ gap function for
different phase angles $\varphi$. Figure 9: Calculated difference of the QPI
patterns
$\Delta\rho(\mathbf{q},\omega)=\rho(\mathbf{q},\omega)-\rho(\mathbf{q},-\omega)$
in the superconducting triplet state for $E$ representation with pseudovector
$(d_{x},d_{y})$ multiplet for $\Delta_{0}=10$ meV and $\omega=0.07$ eV. (a)
phase angle $\varphi=0$; and (b) $\varphi=\pi/3$.

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