# Indoor Wireless Signal Modeling with Smooth Surface Diffraction Effects

Ruichen Wang  ECE Department
University of Maryland, College Park
Maryland, United States
<EMAIL_ADDRESS>Samuel Audia  CS Department
University of Maryland, College Park
Maryland, United States
<EMAIL_ADDRESS>Dinesh Manocha  CS and ECE Departments
University of Maryland, College Park
Maryland, United States
<EMAIL_ADDRESS>Ruichen Wang1, Samuel Audia2, Dinesh Manocha12,  1Department
of Electrical and Computer Engineering, University of Maryland, College Park
(UMD), Maryland, United States  2Department of Computer Science, University of
Maryland, College Park (UMD), Maryland, United States  Email<EMAIL_ADDRESS>
###### Abstract

We present a novel algorithm that enhances the accuracy of electromagnetic
field simulations in indoor environments by incorporating the Uniform
Geometrical Theory of Diffraction (UTD) for surface diffraction. This
additional diffraction phenomenology is important for the design of modern
wireless systems and allows us to capture the effects of more complex scene
geometries. Central to our methodology is the Dynamic Coherence-Based EM Ray
Tracing Simulator (DCEM), and we augment that formulation with smooth surface
UTD and present techniques to efficiently compute the ray paths. We validate
our additions by comparing to analytical solutions of a sphere, method of
moments solutions from FEKO, and ray-traced indoor scenes from WinProp. Our
algorithm improves shadow region predicted powers by about $5dB$ compared to
our previous work, and captures nuanced field effects beyond shadow
boundaries. We highlight the performance on different indoor scenes and
observe 60% faster computation time over WinProp.

###### Index Terms:

Geometric Optics, Uniform Geometrical Theory of Diffraction, Wireless
Modeling, Ray Tracing

##  I Introduction

The increased complexity of indoor wireless systems has required the
development of more complex simulation tools to accurately design and test
them. Earlier simulation methods used simple line-of-sight models [1] and
current methods tend to improve the accuracy by using multipath bounces and
diffraction [2]. Typical methods for high-fidelity electromagnetic field
modeling are intensive in both memory and computation time [3]. Large-scale,
complex scenes, such as realistic indoor spaces, use ray tracing solutions to
capture fields in a reasonable amount of time and computation.

Ray optical techniques are limited by the fact that they cannot accurately
capture edge diffraction, smooth edge diffraction, or higher order effects
such as cavity responses due to their assumption that a surface interaction is
local and planar [4]. Edge diffraction is typically computed using the Uniform
Geometrical Theory of Diffraction (UTD) [4] or Physical Theory of Diffraction
(PTD) [5] which superimposes solutions for edge diffraction onto solutions of
lower order effects. The latter method requires the expensive operation of
integrating currents throughout surfaces in the scene and, as such, is not
combined with pure geometric optics solutions. Therefore, we focus on UTD,
which also allows for the accurate capture of smooth edge diffraction on
Perfect Electrical Conductor (PEC) [6] and PEC bodies coated with a thin layer
of dielectric [7] solely by tracing additional rays.

Main Results: We present an approach that augments the Dynamic coherence-based
EM Ray Tracing Simulator (DCEM) [8][9] that can handle complex dynamic scenes
efficiently, with off surface source and receiver smooth surface diffraction
modeled using UTD [10, 11]. This theory has long been understood and used, but
its usage in indoor simulation has been mostly limited to edge diffraction.
Our formulation has been motivated by the importance of smooth edge
diffraction for accurately capturing the real-world geometry of buildings
[12]. Moreover, our approach for smooth edge diffraction can also handle human
obstacles, which can be reasonably approximated by PEC cylinders [13]. The
novel components of our work include:

  1. 1.

Incorporation of smooth surface diffraction effects into a general, scalable,
and dynamic ray tracing simulation offering a more comprehensive tool for
indoor wireless system design and testing.

  2. 2.

Rigorous validation against analytical solutions and commercial tools
demonstrating the simulations marked improvements in accuracy, performance
runtime, and efficiency.

  3. 3.

Our approach can enable efficient dynamic scene modeling in crowded spaces.

We validate our additions by first comparing them to two small objects: a
sphere and a metal tea cup. We compare our solution for a sphere with the
analytical scattering solution, while the teacup is compared to the commercial
tool FEKO’s [14] method of moments simulation. We also compare a room-scale
simulation containing a PEC sphere to show the extension of our work to larger
objects or scenes, and the results show clear improvements in terms of
received powers ($5dB$ on average compared to the original DCEM w/o surface
diffraction predictions) in the shadow region. For the third test, we compare
the heat maps of the room computed by our method to the ones produced by DCEM
and WinProp, where we observe clear received power raised close to shadow
boundaries while maintaining the prediction accuracy outside the shadow
region. We also compare the running time and efficiency and observe
significant saving in running time of around $60\%$ compared to WinProp at
default settings. Additionally, we demonstrate contributions beyond the shadow
boundary on the smooth surface as well as attenuated fields in the lit region,
as described by the UTD theory.

##  II Uniform Geometrical Theory of Diffraction

###  II-A Limitations of Geometric Optics

Geometric optics is an asymptotic high-frequency approximation of the
propagation of electromagnetic energy [5]. The method approximates the path of
travel for electromagnetic fields as a ray representing the shortest path
between two points. Upon intersecting a surface, the ray reflects and
potentially refracts, splitting energy between the two directions. It is
assumed that the ray travels through a homogeneous medium. That medium can,
however, have an associated loss which is typically modeled by an
exponentially decaying extinction term. Surface interactions are further
assumed to be locally flat and infinite. At high frequencies and with smooth
curvatures, this approximation matches well with measured data. Furthermore,
the method provides run times orders of magnitude faster for large problem
sizes compared to full-wave solvers, which require expensive, memory-intensive
computations.

For common frequencies of interest, like those for wireless communication, as
well as realistic, nonsmooth geometry, the geometric optics assumption no
longer holds. To make up for these shortcomings, the geometric optics field is
superimposed with diffracted fields. This was originally proposed by Keller
with the Geometrical Theory of Diffraction (GTD) [15], which accounted for
diffraction effects using an extended ray model. The original GTD formulation
had singularities in the transition between lit and unlit regions known as the
shadow boundary. The Uniform Geometrical Theory of Diffraction [5] was then
developed to account for these singularities and provide smooth, bounded
calculations across the shadow boundary region.

###  II-B Off Surface UTD Surface Diffraction

Geometric optics can be extended by the Uniform Geometrical Theory of
Diffraction to account for scattering along smooth convex surfaces even in
regions that are not directly illuminated. We use the off surface formulation
of the diffraction because we assume that transmitters and receivers are not
located on scattering surfaces. UTD can largely be described by two equations:
one for the lit region and one for the shadowed region. They are as follows
[5]:

$\bar{E}(P)=\bar{E}^{i}(P)U^{i}+\bar{E}^{r}(P)U^{r}$ (1)

and

$\bar{E}(P)=\bar{E}^{d}(P)(1-U).$ (2)

The $U^{i}$ and $U^{r}$ terms represent the Heaviside step function,
signifying a direct path visibility for the incident and multipath reflected
fields, respectively. $\bar{E}^{i}$ is the incident field directly from a
transmitter at a receiver location $P$ while $\bar{E}^{r}(P)$ and
$\bar{E}^{d}(p)$ represent the reflected and diffracted fields, respectively.
They are given by

$\bar{E}^{r}(P)=\bar{E}^{i}(Q)\cdot\bar{\bar{R}}\sqrt{\frac{\rho_{1}^{r}\rho_{2}^{r}}{(\rho_{1}^{r}+s^{r})(\rho_{2}^{r}+s^{r})}}e^{-jks^{r}}$
(3)

and

$\bar{E}^{d}(P)=\bar{E}^{i}(Q_{1})\cdot\bar{\bar{T}}(Q_{1},Q_{2})\sqrt{\frac{\rho^{d}}{s^{d}(\rho^{d}+s^{d})}}e^{-jks^{d}}.$
(4)

The $\rho$ terms account for the surface curvature while $s$ represents the
distance from a scattering point $Q$ on the surface and the receiver.
$\bar{\bar{R}}$ is the UTD reflection coefficient accounting for both parallel
and perpendicularly polarized field components [11]. $\bar{\bar{T}}$ is a
transfer function that describes the amplitude and phase variation of the ray
along geodesic paths from $Q_{1}$ to $Q_{2}$ as well as the diffraction at a
surface point $Q_{2}$ [10]. The square root terms can be interpreted as the
dispersion of the field along the geodesic path, and the exponential accounts
for the phase difference between the scattering point and the receiver.

##  III The Dynamic Coherence-Based EM Ray Tracing Simulation

###  III-A Prior work

In previous works [8][9], DCEM was developed to handle large urban models and
perform accurate computations in dynamic scenes with scalability to many
moving objects. DCEM exploits a coherence-based framework and performs
accurate computations in dynamic scenes. The underlying ray tracing algorithm
models direct rays, reflected rays, diffracted rays, and scattered rays to
perform EM propagation simulations. It uses three main characteristics:
backward ray tracing, bounding volume hierarchy (BVH), and propagation path
caching to accelerate the performance in dynamic environments [8]. DCEM was
further extended to large urban scenes with multiple receivers with spatial
consistency and channel correlation to improve ray tracing simulation’s
efficiency and accuracy in dynamic scenarios and address its scalability [9].

###  III-B Augmentation to DCEM for smooth surface diffraction

In this work, we improve the DCEM’s prediction accuracy in shadow regions by
introducing smooth surface diffraction calculations as described in the
previous section. By adapting how the field is calculated in the lit and unlit
regions, the effects of convex surfaces are accounted for. This will commonly
produce larger fields in shadow regions where there would otherwise be a sharp
shadow boundary as well as interference effects in lit regions from encircling
fields. We include those ray paths that satisfy the generalized Fermat’s
principle into the shadow region to improve the prediction accuracy. This
method was explored in some other literature as [16, 17], but only for small
objects on small scales (a few $cms$).
In this work, when computing diffracted ray paths from backward ray tracing,
we mark the diffraction surfaces, consider the sources that lie in the
diffraction shadow region from the receiver’s perspective by UTD smooth
surface diffraction, and then perform path validation back to the receiver as
with reflection paths. Equation 4 is used to compute the field strength. While
introducing more diffraction rays, this removes an erroneous discontinuity
between the lit and shadowed regions as well as contributes to the field
beyond the curved surface that would have previously been occluded. Path
caching also helps accelerate the simulation process by restoring diffraction
paths and reusing them for calculating other diffracted ray paths in the local
area (adjacent frames). For this paper, we did not perform dynamic
simulations, but we kept the potential for efficient BVH updating in dynamic
scenes for future exploration [8].

In Fig. 1, a typical ray path along which the electrical field propagates from
the source to the field point $P$ is shown. According to GTD, the field can be
expressed in terms of Equation 1 and Equation 2 in the lit and unlit regions,
respectively. More detailed derivations and discussions can be found in [18].

Figure 1: Diffracted ray path geometry illustration for scattering from a
convex surface: some rays cast from the original source reach the convex
surface and travel on the surface to further go beyond shadow boundaries.
Without smooth surface diffraction, the incident rays at $Q^{\prime}$ will be
diffracted there. But now with smooth surface diffraction, the new paths from
equivalent source $Q^{\prime}$ can be generated as $Q^{\prime}QP$.

###  III-C Simulation setup

Our implementation of UTD surface diffraction was verified on small shapes
such as spheres and cylinders because measured data for indoor spaces is
unavailable and the scale makes higher fidelity simulation methods unfeasible.
Analytical solutions were used where possible, such as for a sphere [19], and
Method of Moments [20] provided by the FEKO [14] simulation suite was used for
all other cases. Specifically, we run evaluation tests on a teacup shaped
model and a sphere model, as shown in Fig. 3 and Fig. 4.

We built two room-scale spaces to run comparison simulations in our ray
tracing solver and the industry-standard WinProp. One comparison between DCEM
w/ and w/o smooth surface diffraction calculations took place in Fig. 2(a) to
demonstrate the dynamic movements, and another comparison between DCEM and
WinProp took place in Fig. 2(c) with multiple static objects to show the
performance of accuracy and efficiency. Fig. 2 shows the 2D and 3D views of
the two simulation environments (dimensions of (a)(b): $6m*4.5m*3m$ and
dimensions of (c)(d): $5m*5m*3m$). In both simulation room setups, the
transmitters are placed at $0.2m$, a height similar to that of the metal balls
(on the ground with a $0.2m$ radius). The transmitters are omnidirectional
with the transmitting power of $40dBm$ at $2GHz$ (wavelength $15cm$). As
defined in WinProp, the circles are approximated by different numbers of line
segments. In Fig. 2(a)(b), the balls are approximated by 30 line segments; in
Fig. 2(c)(d) they are approximated by 10. The latter has fewer lines for
approximation to reduce the simulation time in WinProp, with reasonable
prediction outputs. The same materials are used in both simulation setups: the
walls are $10cm$ bricks and the balls have $1mm$ metal surface.

Figure 2: Comparison indoor environments. In (a), the transmitter is at the
bottom left corner and the circles on the right represent metal ball
positions. In (b), the positions of transmitters and spheres are displayed. In
(c), the transmitter is at the center of the room and the four circles around
the transmitter represent metal ball positions. In (d), the positions of
transmitters and spheres are displayed. The simulated room has a floor and
ceiling that was not shown in the screenshot for better illustration purposes.

##  IV Results and discussions

###  IV-A Evaluation results from Analytical, FEKO, and DCEM solutions

In Fig. 3, we show the validation field strength results of a $0.2m$ sphere
computed by analytical solutions, DCEM without smooth surface diffractions,
and our approach that uses DCEM with smooth surface diffractions. We see
significant improvements by introducing smooth surface diffractions to
reproduce the response periodicity. However, the amplitudes of peaks still
show clear differences which partially result from the proximity of the smooth
surface diffraction methods, but they might be aligned better in future
improvements by appropriate parameter tweaks.
In Fig. 4, we show the teacup simulation result comparisons in polar plots.
The field strength around the teacup model aligned very well and we can
clearly identify the responses from the main body (lower half of the plots)
and the handle part (upper half of the plots).

Figure 3: Left: a perfect sphere conductor with a radius of 2 cm[14]. Right:
Field strength for the sphere: the yellow line represents the analytical
solution as ground truth, the red line comes from the simulation of DCEM w/o
smooth surface diffraction, and the blue line from the simulation of DCEM w/
smooth surface diffraction.

Figure 4: Left: a teacup shaped perfect conductor model with 12.5 cm height
and radius of 5 cm [14]. Middle: polar plot from FEKO. Right: polar plot from
our DCEM-based method. We observe that the lower part has almost the same
results as representing the teacup’s cup part, while some minor differences in
the upper part come from the handle part. However, it’s clear that the field
characteristics are perfectly identified and the minor differences might
result from simulation precision settings.

###  IV-B Improvements in DCEM with smooth surface diffraction

In this section, we show the improvements in DCEM simulations with smooth
surface diffraction. The setups are discussed in Section III.C. In Fig. 5, we
show a snapshot of the heatmap focused on Sphere B, where smooth surface
diffractions take place. While there are also diffracted rays around Sphere A,
we selected Sphere B for a clearer and simpler comparison. In Fig. 6, we show
received powers along a typical ray path by smooth surface diffraction. With
smooth surface diffraction, DCEM becomes better at shadow region field
strength calculations.

Figure 5: Heatmap comparison of DCEM without smooth surface diffraction (left)
vs. DCEM with smooth surface diffraction (right) of Sphere B illustrated in
Fig. 2(a). We observe the same reflected rays on the front side (indicated
with a blue circle); the differences mostly lie in the red circled area. This
is as expected for the smooth surface diffractions that take place at the
shadow boundaries and raise the field strength in the shadow region area. We
explicitly show the received power along a typical path into the shadow region
in Fig. 6; more discussions follow.

Figure 6: Received power comparison of DCEM without smooth surface diffraction
vs. DCEM with smooth surface diffraction of Sphere B along a boundary path
into a shadow region (indicated with the blue arrow above the received power
plot). We observe the power strengths are the same at the sphere boundary and
result in a $5dB$ difference at the end of the $0.62m$ path. The smooth
surface diffractions contribute more significantly closer to the diffraction
point on the smooth surface before $0.15m$ and decay afterward. However, as
indicated in the heatmap Fig. 5, the received powers should align again when
the shadow region created by the metal ball blockage ends.

###  IV-C Results comparison between WinProp and DCEM

In this section, we show the simulation results in WinProp and DCEM to show
better shadow region prediction accuracy, and also compare the running times
of WinProp and DCEM under such settings. The WinProp settings are specified
below. We enabled the ray optical propagation model with a 3D ray tracing
algorithm: the max number of transmissions, reflections, and diffractions are
3, 2, and 1, respectively. We used Fresnel Coefficients for transmission and
reflections and GTD/UTD for diffractions. The TX frequency is $2GHz$, and the
resolution is set to $0.02m$. The other setups are the same as discussed in
Section III.C.

In Fig. 7 and Fig. 8, we show heatmap comparisons between WinProp and DCEM at
Sphere positions 1 and 3, as indicated in Fig. 2(c). Since WinProp does not
offer smooth surface diffraction calculations, we observe clear shadow region
boundaries at both Sphere positions. DCEM with smooth surface diffraction
shows more power in the shadow region through diffraction around the
diffraction points. We observed the same differences in the results of the
other two sphere positions, 2 and 4 (not included in this paper). In Fig. 9,
we show the received power along a line across the shadow region in both
WinProp and DCEM to demonstrate how many improvements the smooth prediction
introduced by reducing the shadow region and keeping other simulations
unaffected.

Figure 7: Heatmap comparison of WinProp (left) vs. DCEM (right) of Sphere 1,
illustrated in Fig. 2(c). The propagation paths are the same in the blue
circled area, where most reflections happen. In the red circled area, we
observe a stronger field and a thinner shadow region in the DCEM heatmap (on
the right). We explicitly show the received power along a line across the
shadow region to demonstrate the prediction differences in Fig. 9.

Figure 8: Heatmap comparison of WinProp (left) vs. DCEM (right) of Sphere 3,
illustrated in Fig. 2(c), a confirmation comparison to the observations and
conclusions made in Fig. 7. Again, the blue circled area shows the same
reflection calculations, and the red circled area indicates stronger shadow
region strength predictions in general.

Figure 9: Received power comparison along a line across the shadow region
bonds in WinProp (the left heatmap, the path indicated in a blue dashed line,
results in blue dots,), and DCEM (the right heatmap, the path indicated in a
red dashed line, results in a red line). We see the width of the shadow region
in WinProp simulations (blue dots) is approximately $0.4m$ (bottom to bottom)
while that of the DCEM results (red lines) is around $0.1m$. This is expected
because the smooth surface diffraction enables more illuminated areas around
the original shadow boundary regions, decaying fast after a wavelength
($0.15m$ in our settings). In addition, we see the powers at the diffraction
point and outside of the region align well (shown as the powers at the two
sides of this plot). This also means that the surface diffraction shall not
distort other propagation path simulations.

In the next Table I, we show the running time recorded in three test runs
under the same setups as specified earlier. We see a significant savings in
running time of around $175min$ compared to WinProp at default settings. We
believe that this discrepancy could be explained by redundant rays tracing
repeated paths resulting in a longer running time.

TABLE I: Running time comparison of WinProp and our approach based on DCEM. We highlight the speedups. Simulator | Runtime test1 | Runtime test2 | Runtime test3
---|---|---|---
WinProp | 305min | 300min | 302min
DCEM | 132min | 125min | 124min

##  V Conclusion and Future work

We proposed the addition of UTD Surface Diffraction effects to indoor wireless
characterization. Comparisons of our simulation to WinProp show the inclusion
of additional diffraction effects that more accurately represent the
underlying physics; however, validation against measured data would still be
beneficial. We would also like to explore how our approach scales to larger,
more complex indoor environments and how it could handle dynamic simulations
of crowds using the PEC cylinder approximation. We would also like to evaluate
the accuracy by comparing with real-world measurements.

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# Al2B2 and AlB4 monolayers: emergence of multiple two-dimensional Dirac nodal
line semimetals with novel properties

Saeid Abedi  Department of Physics, Isfahan University of Technology, Isfahan,
84156-83111, Iran  Esmaeil Taghizadeh Sisakht  Department of Physics, Isfahan
University of Technology, Isfahan, 84156-83111, Iran  S. Javad Hashemifar
Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111,
Iran  Nima Ghafari Cherati  Department of Physics, Isfahan University of
Technology, Isfahan, 84156-83111, Iran  Ismaeil Abdolhosseini Sarsari
Department of Physics, Isfahan University of Technology, Isfahan, 84156-83111,
Iran  Francois M. Peeters  Department of Physics, University of Antwerp,
Groenenborgerlaan 171, B-2020 Antwerpen, Belgium

###### Abstract

Topological semimetal phases in two-dimensional (2D) materials have gained
widespread interest due to their potential applications in developing
nanoscale devices. Despite the prediction of the Dirac/Weyl points in a wide
variety of 2D candidates, materials featuring topological nodal lines are
still in great scarcity. Herein, we predict two stable thinnest films of
aluminum diboride with hyper- and hypo-stoichiometries of Al2B2 and AlB4 as
new 2D nonmagnetic Dirac nodal line semimetals (NLSMs) which promise to offer
many novel features. Our elaborate electronic structure calculations combined
with analytical studies reveal that, in addition to the multiple Dirac points,
these 2D configurations host various type-I closed nodal lines (NLs) around
the Fermi level, all of which are semimetal states protected by the time-
reversal and in-plane mirror symmetries. The most intriguing NL in Al2B2
encloses the K point and crosses the Fermi level with a considerable
dispersion, thus providing a fresh playground to explore exotic properties in
dispersive Dirac nodal lines. More strikingly, in the case of 2D
superconductor AlB4 which exhibits a high transition temperature, we provide
the first evidence for a set of 2D nonmagnetic open type-II NLs in weak spin-
orbit coupling limit, coinciding with closed type-I NLs near the Fermi level.
The coexistence of superconductivity and nontrivial band topology in AlB4 not
only makes it a promising material to exhibit novel topological
superconducting phases, but also the rather large energy dispersion of type-II
nodal lines in this configuration, may offer a distinguished platform for
realization of novel topological features in two-dimensional limit.

ZnO clusters, Magic number, GW, Heat capcity, IR

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XXXX XXXX; E-mail<EMAIL_ADDRESS>
Topological classification of quantum states of matter has provided a new
paradigm for the study of modern condensed matter physics. This vibrant area
of research has emerged thanks to the discovery of two- (2D) and three-
dimensional (3D) topological insulators (TIs) Bernevig  _et al._ (2006); König
_et al._ (2007); Fu  _et al._ (2007); Moore (2010); Wang  _et al._ (2013);
Zhao  _et al._ (2011). TIs are insulating in bulk, while host conducting
edge/surface states at the boundary that are protected by the time-reversal
(TR) symmetry. Beyond the topological insulating phase, a new class of
topological materials is realized that exhibit a gapless bulk band structure
and thus called topological semimetals (TSMs) or topological metals Burkov
_et al._ (2011); Burkov (2016); Armitage  _et al._ (2018); Yang and Nagaosa
(2014); Baik  _et al._ (2015).
TSMs are characterized by the key features of the band crossing originated
from the crystal symmetries. Depending on the type of band degeneracy, the
codimension, and the band dispersion, one can categorize TSMs into three
general classes of nodal Dirac/Weyl points Wan  _et al._ (2011); Young  _et
al._ (2012); Liu  _et al._ (2014); Soluyanov  _et al._ (2015); Lv  _et al._
(2015); Xu  _et al._ (2015a); Chen  _et al._ (2015), nodal lines (NLs) Kim
_et al._ (2015); Mullen  _et al._ (2015); Weng  _et al._ (2015); Hirayama  _et
al._ (2017); Bzdusek  _et al._ (2016); Zhang  _et al._ (2017); Feng  _et al._
(2018); Liu  _et al._ (2018); Tian  _et al._ (2020), and nodal surfaces Liang
_et al._ (2016); Zhong  _et al._ (2016); Wu  _et al._ (2018); Xu  _et al._
(2015b); Topp  _et al._ (2017). The first class includes Dirac/Weyl semimetals
where the band crossings are isolated points with four-fold/two-fold
degeneracy and the quasi-particle excitations disperse linearly and obey the
Dirac/Weyl equations. In the the second class, degenerate dispersive lines
appear either as an open or a closed nodal line across the Brillouin Zone (BZ)
and are divided into type-I, type-II, and hybrid topological NLs Wang  _et
al._ (2018). The third class of TSMs involve band crossings that form a nodal
surface in the BZ Liang  _et al._ (2016); Zhong  _et al._ (2016); Wu  _et al._
(2018); Xu  _et al._ (2015b); Topp  _et al._ (2017); Fu  _et al._ (2019).

Recent developments in the realization of 2D topological materials will pave
the way for studying exotic quantum phenomena at the nanoscale that might lead
to novel quantum devices. 2D NLSMs are new members of this family that have
been the focus of attention in recent studies. Besides the important
properties of these systems including non-dispersive Landau energy levels Rhim
and Kim (2015), high-temperature surface superconductivity Kopnin  _et al._
(2011), and specific long-range Coulomb interactions Huh  _et al._ (2016), one
of the most notable characteristics of 2D NLSMs is that their topological
features can be revealed by the angle-resolved photoemission spectroscopy
(ARPES) measurements Zhou  _et al._ (2018). In spite of the large number of
discovered 3D TSMs, there are a limited number of 2D materials exhibiting
NLSMs Jin  _et al._ (2017); Li  _et al._ (2018); Zhou  _et al._ (2018); Zhong
_et al._ (2019); Wu  _et al._ (2019). Some examples include 2D Lieb lattice
Yang  _et al._ (2017); Feng  _et al._ (2020), monolayer borophene Gupta  _et
al._ (2018), honeycomb-Kagome lattice Lu  _et al._ (2017), transition metal
chalcogenide monolayers Jin  _et al._ (2017), Cu2Si Feng  _et al._ (2017), and
CuSe Gao  _et al._ (2018) monolayers, where the two last cases are
experimentally realized. Therefore, the exploration of new 2D NLSM materials
is of great interest.

The bulk structure of AlB2-type materials are known to be 3D TSMs and exhibit
specific properties including superconductivity in MgB2 Nagamatsu  _et al._
(2001) and ZrB2 Gasparov  _et al._ (2001), excellent thermoelectricity in AlB2
Sharma  _et al._ (2011) and MgB2 Putti  _et al._ (2002), and super-hardness in
OsB2 Cumberland  _et al._ (2005). Moreover, novel topological states including
triple point, nexus, and nodal links has been verified experimentally and
theoretically in TiB2 and ZrB2 Zhang  _et al._ (2017); Liu  _et al._ (2018);
Feng  _et al._ (2018); Yi  _et al._ (2018); Lou  _et al._ (2018), where, the
dominant features of the energy bands near the crossing points come from the
Ti- and Zr-3d states and the appeared NLs are protected by mirror reflection
symmetries Zhang  _et al._ (2017); Feng  _et al._ (2018). Alternatively, a
topological Dirac nodal line (DNL), dispersed along the K-H direction and
protected by the combination of inversion and TR symmetries Jin  _et al._
(2019), has been observed in the conventional high-temperature superconductor
MgB2 and its non-superconducting sister AlB2 Jin  _et al._ (2019); Takane  _et
al._ (2018). In contrast to TiB2 and ZrB2, the DNLs in AlB2 and MgB2 originate
from the B-2p electrons. As a result, the spin-orbit coupling (SOC) in these
materials is negligible and consequently the involved topological nodal lines
are experimentally feasible Takane  _et al._ (2018).

It has been shown that the mechanical cleavage of AlB2 flakes may lead to
highly single-crystalline 2D layers of this material Humood  _et al._ (2018).
In addition, recent ARPES measurements on a typical cleaved (001) surface of
AlB2 revealed the feasibility of both B- and Al-terminations Sunko  _et al._
(2020). In a similar work, a B-terminated monolayer of AlB2 was successfully
synthesized on Al(111) via molecular beam epitaxy Geng  _et al._ (2020). These
experimental works, evidence the feasibility of aluminum diboride thin films
with the desired terminations. Herein, we study three atomic layers AlB2 thin
films with the Al- and B- termenations, leading to the Al2B2 and AlB4
stoichiometires, by using the first-principles calculations, group theory
analyses and effective Hamiltonian models (including WTB and continuum
models).

After confirmation of the structural stability of these compounds, we
investigate the topological aspects of the characteristic band structures.
Then, we present a thorough group theory analysis to explain the protection
mechanism behind the emergence of various Dirac points and NLs. Meanwhile, we
also provide effective continuum models using the method of invariants Bir and
Pikus (1974) to systematically reproduce the momentum distribution of DNLs and
responsible low-energy band dispersions around these Dirac nodes. Our results
indicate that there exist several type-I 2D NLs in a relatively small energy
range around the Fermi levels of Al2B2 and AlB4 which enclose high symmetry
points in the BZ. Very interestingly, for the Al2B2 configuration there is an
NL that encloses the K point and crosses the Fermi level with a dispersion of
$\sim$0.6 eV. Most strikingly, we realize the coexistence of an open type-II
2D nodal line with other Dirac nodal features in AlB4 configuration which
arises from the intersection of those energy bands that contributes to
determining the superconducting behavior in this material. Our predictions not
only enrich the family of 2D NLSMs, but also highlight the potential of these
topological materials as promising candidates to explore exotic properties in
dispersive 2D Dirac nodal lines and the interplay of the superconductivity and
topological Dirac nodal line states Campi  _et al._ (2021).

Figure 1:  (a) Top view and BZ of aluminum diboride thin films with lattice
vectors $\bm{a}_{1}$ and $\bm{a}_{2}$. (b) and (c) Side views of thinnest
films of Al-B2-Al and B2-Al-B2 configurations. (d) and (e) Phonon frequencies
of Al2B2 and AlB4.

Results and discussion
Crystal structures and phonon dispersions. Depending on the (001) surface
termination, one may consider two configurations for the thinnest AlB2 films
with the in-plane mirror symmetry, which is crucial for the emergence of nodal
line in the system. Figure 1(a) shows the similar top view of these two
configurations where the lattice vectors $\bm{a}_{1}$ and $\bm{a}_{2}$
indicate their surface unit cells. The first pattern of the atomic layers lies
within the hypo-stoichiometric Al1+xB2 films and consists of a boron layer
intercalated between two aluminum layers with the Al-B2-Al stacking (Fig.
1(b)). The second one is the thinnest hyper-stoichiometric AlB2+x film with a
sandwich B2-Al-B2 configuration (Fig. 1(c)), where the two boron layers form
an AA-stacking order enclosing a triangular aluminum layer. The crystal
structure of our hypo- and hyper-stoichiometric thin films, refered to as
Al2B2 and AlB4, respectively, belongs to the layer group
$P6\texttt{\small/}mmm$ (No.80) with the corresponding point group $D_{6h}$
Litvin and Wike (1991). Our structural optimizations lead to the in-plane
lattice constants 2.994 Å and 2.949 Å for the AlB4 and Al2B2 films,
respectively. To evaluate the dynamic stability of these structures, we
calculated and considered their phonon spectra. The obtained phonon band
structures, presented in Figs.1(d) and (e), clearly indicate absence of any
imaginary phonon mode, thus implying their dynamical stability.
Having established the stability of these configurations, we now turn our
attention to the electronic and topological properties. As both the AlB4 and
Al2B2 structures are composed of light elements, the SOC is quite weak in
these materials and one would expect to observe a definite sign of possible
Dirac features in experimental measurements. Therefore, we shall ignore the
spin of electrons throughout our calculations.
DNLs in Al2B2. We begin by investigating the electronic and topological
properties of Al2B2 configuration. As mentioned before, the point group of
this structure is $D_{6h}$ which contains 24 symmetry elements. We have shown
in Fig. 2(a) those symmetry elements that are important for our purpose. They
are the two different sets of twofold symmetry axes ($3C^{\prime}_{2}$ and
$3C^{\prime\prime}_{2}$), two different sets of vertical mirror planes
($3M_{\sigma}$ and $3M_{\tau})$, and one horizontal symmetry plane ($M_{h}$).
Figure 3(a) depicts the electronic band structure of Al2B2 along the highly
symmetric directions (M-K-$\Gamma$-M) in the BZ.

Figure 2:  Three types of mirror planes $M_{\sigma}$, $M_{\tau}$, and $M_{h}$
for (a) Al2B2 and (b) AlB4 configurations, as indicated by the red, blue, and
green planes, respectively. Two of rotational axes $C^{\prime}_{2}$ and
$C^{\prime\prime}_{2}$ are shown by dashed arrows.

As seen from Figs. 3(b) and (c), the atom-projected and orbital-projected
bands show that the metallicity of Al2B2 comes from the hybridization among
$s$ and $p$ electron valence shells of both Al and B atoms. Interestingly, in
the energy range from -2.4 to 1.2 eV, we find multiple band crossing features
(consisting of Dirac points (DPs) and nodal points (NPs)) that offer a
fruitful line of investigation for 2D topological NLs in this configuration. A
close inspection of the band structure reveals that in this energy window
there exist two Dirac points and five type-I 2D NLs. We mark the two Dirac
points as DP1 and DP2 and indicate the nodal points as NP$i$ and
NP$i^{\prime}$ ($i$ runs from 1 to 5) that are the pertinent points in the
nodal loop NL$i$ (see Fig. 3(a)). The formation of multiple topological nodal
loops and Dirac fermions in this band structure suggests that the
understanding of the mechanism behind it has to be questioned. Therefore, what
follows is a discussion to demystify the reason for the observation of the
introduced 2D NLs and Dirac points utilizing the group theory analysis, TB
approximation, and method of invariants.

Figure 3:  (a) The calculated electronic band structure of Al2B2
configuration. Energy dispersion display multiple band crossing features
including both Dirac and nodal points as presented by magenta circles. Special
bands are marked by Greek letters. (b) and (c) Atom-projected band structures
for Al2B2 contributed by Al and B atoms, respectively. The size of dots is
proportional to the weight of contributed orbitals.

First of all, to gain a better insight into the band structure of monolayer
Al2B2 in the whole BZ, we have constructed a TB Hamiltonian in the basis of
maximally localized Wannier functions. The comparison of the band structures
of Al2B2 as calculated by density functional theory (DFT) and the WTB
Hamiltonian is shown in Fig. 4. One can see that in the mentioned energy range
the Wannier band structure exactly matches the DFT results. Next, we have
diagonalized our Hamiltonian in the whole BZ to provide the 3D energy bands
whose intersections form the mentioned 2D DNLs. Figures 5(a)-(d) depict
zoomed-in band structures in the regions of band crossing near nodal points
NP$i$ and NP$i^{\prime}$ (see top views in Figs. 5(e)-(h)). From the
transparent top views of bands surfaces, it is clear that the two points NP$i$
and NP$i^{\prime}$ are residing on a 2D nodal loop NL$i$. The extensions of
nodal points are displayed throughout 3D energy-momentum spaces which vividly
describe the form of characteristic topology of each NL. As seen, NL1 and NL5
surround the K and M points, respectively and NL2, NL3, and NL4 are three
concentric nodal loops centered around the $\Gamma$ point. These Dirac nodal
loops have dispersion at different energy levels in the BZ such that cover a
large portion of the mentioned energy window.
Let us now turn to the mechanism by which the Dirac points DP1 and DP2 emerge.
As shown in Fig. 3(a), the DP1 point is located along the $\Gamma$-K direction
and 25 meV below the Fermi level. The little group for Al2B2 configuration
along this high symmetry direction is $C_{2v}$ with two perpendicular mirror
reflections planes; $M_{\tau}$ and $M_{h}$ which their intersection introduces
the twofold rotation axis $C_{2}$. Our symmetry analysis shows that the
irreducible representations (IRs) of the crossing bands around the DP1 point
are $\Gamma_{3}$ and $\Gamma_{4}$ (Fig. 4). Since the IRs $\Gamma_{3}$ and
$\Gamma_{4}$ under the $M_{\tau}$ or $C_{2}$ symmetry operation have opposite
parities, the two bands around the DP1 point do not interact. As a result, six
symmetry protected Dirac points DP1 emerge in the whole BZ due to the star of
$k$. The Dirac point DP2 with an n-type character is located at the K point
which stems from the underlying point group $D_{6h}$ of the lattice in a
similar manner to the formation of Dirac cones in graphene.
We now begin to consider the mechanism which generates NL1. An important
feature of this nodal line is that it crosses the Fermi level and disperses
from about $-0.2$ to $+0.4$ eV (see Fig. 3(a)). For more convenience, in the
vicinity of the nodal points NP1 and NP1′ we label the responsible electronic
bands as $\alpha$, $\beta$, and $\gamma$.

Figure 4:  Comparison of the band structures of Al2B2 along highly symmetric
directions as calculated by DFT and the WTB Hamiltonian. The Wannier band
structure exactly matches the DFT results. The IRs of energy bands along these
symmetry paths and at high symmetry points K and $\Gamma$ are also shown.
Figure 5:  (a)-(d) 3D visualizations of energy bands obtained from Al2B2 WTB
Hamiltonian. Bands whose crossings lead to the formation of nodal lines around
the K, $\Gamma$, and M points are shown. The extensions of nodal points can be
seen vividly throughout 3D energy-momentum spaces which form the
characteristic topology of each NL. Color bars represent the energy of
electrons. (e)-(h) The corresponding top views of (a)-(d). The momentum
distributions of gapless nodal points are shown. Note that in Figs.(b) and (f)
some data values were removed for better visualization.  Figure 6:  (a)
Electronic band structure of Al2B2 along the typical low-symmetry path
X-$\Gamma$ in the BZ as shown in (b).

As seen from Figs. 3(b) and (c), the electronic states of these bands near the
Fermi level in the vicinity of the nodal points NP1 and NP1′ are mainly
decided by Al-s and Al-px,y. Similar to the $\Gamma$-K path, the little group
along the K-M direction is also $C_{2v}$ with the mentioned symmetry
operations. Group theory analysis shows that the $\alpha$, $\beta$, and
$\gamma$ bands belong to three different $\Gamma_{1}$, $\Gamma_{2}$ and
$\Gamma_{4}$ IRs, respectively (Fig. 4). Along the $\Gamma$-K direction, the
$\alpha$ and $\gamma$ bands have opposite eigenvalues either with respect to
$C_{2}$ or $M_{h}$ symmetry operations. These opposite parities ensure the
decoupling of the $\alpha$ and $\gamma$ bands leading to the formation of
topologically protected NP1. On the other hand, along the symmetry line K-M,
the opposite parities of $\beta$ and $\gamma$ bands under either $M_{\tau}$ or
$M_{h}$ operations allow the emergence of topological point NP1′. Besides the
justification of the two nodal points NP1 and NP1′ along high-symmetry paths,
we also examine the mechanism of the other touching points in the NL1. Figure
6 illustrates the band structure of Al2B2 along a typical low-symmetry line
X-K where X is a point between the $\Gamma$ and M points. Due to the mirror
reflection symmetry $M_{h}$ of monolayer Al2B2, we found that the little group
of the X-K line is $C_{s}$ and the two crossing bands near the Fermi level
belong to different IRs $\Gamma_{1}$ and $\Gamma_{2}$ as marked in Fig. 6(a).
This is valid for the set of the mentioned typical paths. These different
representations have opposite parities with respect to $M_{h}$, giving rise to
a distribution of touching points that construct the 2D topological NL1 (see
Figs. 5(a) and (e)).
To further analyze the structure of NL1 and the corresponding crossing bands,
we use group theory to derive a matrix Hamiltonian with invariant expansion
method Bir and Pikus (1974) which describes the electron states in the
vicinity of the K point. This method enables us to systematically incorporate
the allowed terms up to any desired order of wave vector $\bm{k}$. The little
group of the wave vector K in Al2B2 is $D_{3h}$. At this point the two
degenerate Bloch states of the $\alpha$ and $\beta$ bands transform according
to the 2D IR $\Gamma_{6}$ as denoted by $\psi^{\alpha}$ and $\psi^{\beta}$,
while the corresponding wave function of the $\gamma$ band ($\psi^{\gamma}$)
transforms as IR $\Gamma_{4}$ . These basis functions allow us to construct
the $3\times 3$ matrix Hamiltonian $\mathcal{H}^{\mathrm{K}}$ around the K
point which falls into four blocks

$\mathcal{H}^{\mathrm{K}}=\left(\begin{array}[]{cc}\mathcal{H}_{11}&\mathcal{H}_{12}\\\
\mathcal{H}_{21}&\mathcal{H}_{22}\end{array}\right)\,,$ (1)

where each block $\mathcal{H}_{ij}$ has a dimension of $n_{i}\times n_{j}$ and
$n_{i}$ ($n_{j}$) denotes the dimension of representation $\Gamma_{i}$
($\Gamma_{j}$). The diagonal blocks $\mathcal{H}_{11}$ ($2\times 2$) and
$\mathcal{H}_{22}$ ($1\times 1$) describe energy bands that transform
according to the IRs $\Gamma_{6}$ and $\Gamma_{4}$, respectively. Each block
$\mathcal{H}_{ij}$ depends on a tensor operator ${\bm{\mathscr{K}}}$ which
represents a function of the components of the wave vector $\bm{k}$. Bir and
Pikus Bir and Pikus (1974) showed that for a symmetry element $g$ with matrix
representation $\mathcal{D}(g)$, the invariance condition

$\mathcal{D}(g)\,\mathcal{H}(g^{-1}{\bm{\mathscr{K}}})\,\mathcal{D}^{-1}(g)=\mathcal{H}({\bm{\mathscr{K}}}),$
(2)

leads to the general form $\mathcal{H}_{ij}$ as

$\mathcal{H}_{ij}({\bm{\mathscr{K}}})=\sum_{\kappa,\,\nu}a^{ij}_{\kappa\nu}\sum_{l=1}^{d_{\kappa}}{X}_{l}^{(\kappa,ij)}\mathscr{K}_{l}^{(\kappa,\nu)\,\ast}\,.$
(3)

The parameters $a^{ij}_{\kappa\nu}$ are material-dependent constants which can
be determined via a fitting procedure. $X_{l}^{(\kappa,ij)}$ are those
linearly independent $n_{i}\times n_{j}$-dimensional basis matrices that
transform as those irreducible representations $\Gamma_{\kappa}$ which occurs
in the product $\Gamma_{i}\times\Gamma_{j}^{\ast}$. Here, $l$ denotes the
$l$th basis matrix and runs up to $d_{\kappa}$, the dimension of
representations $\Gamma_{\kappa}$. Also, $\mathscr{K}_{l}^{(\kappa,\nu)}$ are
the $l$th component of the $\nu$th-order of the tensor operator
${\bm{\mathscr{K}}}$ which transform in a similar way. To construct the basis
matrices $X_{l}^{(\kappa,ij)}$ and an arbitrary order of the tensor operators
$\mathscr{K}_{l}^{(\kappa,\nu)}$, one can utilize Clebsch-Gordan coefficients
which have been tabulated in Ref. Koster  _et al._ (1963). The construction
procedure of these matrices- and tensor-components has been described in Refs.
Geissler  _et al._ (2013); Winkler (2003).
Besides the invariance condition (2), the TR symmetry of the system also
imposes an additional constraint on the diagonal blocks $\mathcal{H}_{ii}$
Winkler (2003). Therefore, when using the invariant method to construct the
effective Hamiltonians, we will additionally consider this constraint to list
the allowed tensor-components $\mathscr{K}_{l}^{(\kappa,\nu)}$.
Let us now proceed by generating the clear form of the Hamiltonian
$\mathcal{H}^{\mathrm{K}}$ up to second order in $\bm{k}$ components using the
introduced method. As mentioned, around the K point the Bloch functions
$\psi^{\alpha}$ and $\psi^{\beta}$ transform according to the IR $\Gamma_{6}$
of $D_{3h}$. Also, the Bloch function $\psi^{\gamma}$ transforms as
representation $\Gamma_{4}$. Therefore, from a group theoretical point of
view, one may adopt $\\{x-iy,x+iy\\}$ and $\\{z\\}$ as basis functions to
obtain the basis matrices $X_{l}^{(\kappa,ij)}$. Choosing this set of basis
functions, we obtained the basis matrices as listed in Table S1 (see
Supplementary Note 1). Likewise, we can obtain the irreducible tensor
components up to $k_{i}k_{j}$ terms that transform accordingly and are allowed
by TR invariance. According to Herring’s rule, considering the TR symmetry may
lead to three different possibilities ’$a$’, ’$b$’, and ’$c$’ for a spinless
system Herring (1937). In case ’$a$’, no additional degeneracies occur by
applying the TR operator. However, when the two inequivalent wave vectors
$\bm{k}$ and $-\bm{k}$ are related via a symmetry element $R$ (known as case
’$a_{2}$’), one needs to consider the additional condition Bir and Pikus
(1974)

$\mathcal{T}^{-1}\mathcal{H}(R^{-1}{\bm{\mathscr{K}}})\mathcal{T}=\mathcal{H}^{\ast}(f{\bm{\mathscr{K}}})=\mathcal{H}^{t}(f{\bm{\mathscr{K}}}),$
(4)

where $f=\pm 1$ is the parity of ${\bm{\mathscr{K}}}$ under the TR symmetry.
In Al2B2 the symmetry element $M_{\sigma}$ maps the two inequivalent points K
and K′ onto each other. Choosing the operator $M_{\sigma}$, the TR operator
have the form of $\mathcal{T}=K$ Winkler and Zülicke (2010), where $K$ is the
complex conjugate operator. Therefore the condition (4) for the K point is
written as

$\mathcal{H}^{\mathrm{K}}(R^{-1}{\bm{\mathscr{K}}})=\mathcal{H}^{\ast}(f{\bm{\mathscr{K}}}).$
(5)

We have listed in Table S1 those tensor components that satisfy the condition
(5). Then, inserting the tabulated symmetrized matrices and tensor components
in Eq. (3) we arrive at

$\displaystyle\mathcal{H}_{11}=[E_{0}+Af_{1}(k)]\openone+B(k_{x}\sigma_{x}+k_{y}\sigma_{y})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+C(f_{2}(k)\sigma_{x}+f_{3}(k)\sigma_{y}),$
(6) $\displaystyle\mathcal{H}_{22}=E^{\prime}_{0}+A^{\prime}f_{1}(k),$ (7)
$\displaystyle\mathcal{H}_{12}=\mathcal{H}_{21}=0,$ (8)

where we have defined functions $f_{1}(k)=k_{x}^{2}+k_{y}^{2}$,
$f_{2}(k)=k_{y}^{2}-k_{x}^{2}$, and $f_{3}(k)=2k_{x}k_{y}$. Therefore, the
diagonalization of parameter-dependent Hamiltonian (1) results in the three-
band energy spectra Eqs. (S1-S2) (see Supplementary Note 1). We then performed
a fitting procedure of these energy bands with first-principle data to obtain
the numerical values of parameters. The comparison between DFT bands and the
bands from our continuous model around the K point is shown in Supplementary
Fig.S1. One can see that around this point the fitted bands are in good
agreement with the first-principles data. Furthermore, they also reproduce
well the band dispersions in 2D BZ (around K point), and therefore the locus
of the touching points between these energy bands leads to the emergence of
NL1 as depicted in Fig. 7(a).
Having confirmed the existence of NL1 in Al2B2, we next study the reasons
behind the formation of NL2, NL3, and NL4 which enclose the $\Gamma$ point. In
Fig. 3(a), we observe that all of these nodal lines are located completely
below the Fermi level in the range of -2.35 to -1.35 eV which can be revealed
by ARPES measurements Zhou  _et al._ (2018). Let us first focus on NL3 and NL4
that have distribution near the the $\Gamma$ point. Here, two of the
responsible bands are hole-like (labeled as $\lambda$ and $\delta$ bands) and
the other one is electron-like (labeled as $\nu$ band).

Figure 7:  Transparent top views of crossing bands obtained by continuum model
around (a) the K and (b) $\Gamma$ points of Al2B2. The momentum distributions
of gapless nodal points (NL1, NL2, and Nl3) are reproduced correctly by the
continuum model.

As seen from Figs. 3(b) and (c), the $\lambda$ and $\delta$ bands are largely
composed of B-px,y orbitals while the $\nu$ band is mainly contributed by Al-
pz and B-pz states. The little group of wave vectors $\bm{k}$ along the
$\Gamma$-M direction with symmetry elements $E$, $C_{2}$, $M_{\sigma}$, and
$M_{h}$ shares the same point group symmetry as $\Gamma$-K and K-M paths.
Therefore, the energy bands of Al2B2 along this direction belong to one of the
irreducible representations of the $C_{2v}$ point group. As shown in Fig. 4, a
symmetry analysis unveils that along $\Gamma$-K ($\Gamma$-M) directions the
$\lambda$, $\delta$, and $\nu$ bands belong to three different 1D irreducible
representations $\Gamma_{1}$($\Gamma_{2}$), $\Gamma_{2}$($\Gamma_{1}$), and
$\Gamma_{4}$, respectively. For both symmetry directions, the corresponding
bands with different IRs $\Gamma_{1}$ and $\Gamma_{4}$ have opposite parities
with respect to either $C_{2}$ or $M_{h}$ symmetry elements. Therefore, they
will not hybridize with each other, allowing them to meet at NP3 and NP4′.
Similarly, for symmetry direction $\Gamma$-K ($\Gamma$-M), the energy bands
with IRs $\Gamma_{2}$ and $\Gamma_{4}$ have opposite parities under mirror
symmetries $M_{h}$ or $M_{\tau}$ ($M_{\sigma}$), leading to the formation of
the nodal point NP4 (NP3′). Here, a similar analysis to what we described for
the formation of NL1 could be applied to confirm that no gap opens at the
crossing between the corresponding bands along a typical low-symmetry line
X-$\Gamma$, leading to the emergence of NL3 and NL4.
We now turn to the method of invariants. This approach also provides a good
description of momentum distributions of these nodal loops. The little point
group at $\Gamma$ point in the Al2B2 is $D_{6h}$. At this point, the Bloch
functions $\\{\psi^{\lambda},\psi^{\delta}\\}$ and $\psi^{\nu}$ transform
according to the IRs $\Gamma_{6}^{+}$ and $\Gamma_{2}^{-}$, respectively.
Taking the set of symmetrical basis functions $\\{x-iy,x+iy\\}$ and {$z$}, we
obtained the basis matrices and the irreducible tensor components that
transform accordingly as listed in Supplementary Table S2. Note that at the
$\Gamma$ point we have $\bm{k}=-\bm{k}=0$. Therefore, according to the Herring
test, these irreducible representations belong to the case ’$a_{1}$’ Bir and
Pikus (1974). On the other hand, odd functions of $k_{i}^{n}k_{j}^{m}$ are not
invariants of $\Gamma_{6}^{+}\otimes\Gamma_{6}^{+\ast}$ and
$\Gamma_{2}^{-}\otimes\Gamma_{2}^{-\ast}$. As a result, TR invariance implies
no additional constraint. Based on these two irreducible representations, we
divide the $3\times 3$ effective Hamiltonian $\mathcal{H}^{\mathrm{\Gamma}}$
into four blocks and obtain each block using the method of invariants as
follows

$\displaystyle\mathcal{H}_{11}=[E_{0}+Af_{1}(k)]\openone+B[f_{2}(k)\sigma_{x}+f_{3}(k)\sigma_{y}]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+C[f_{4}(k)\sigma_{x}+f_{5}(k)\sigma_{y}],$
(9) $\displaystyle\mathcal{H}_{22}=E^{\prime}_{0}+A^{\prime}f_{1}(k),$ (10)
$\displaystyle\mathcal{H}_{12}=\mathcal{H}_{21}=0,$ (11)

where $f_{4}(k)=k_{y}^{4}-6k_{x}^{2}k_{y}^{2}+k_{x}^{4}$, and
$f_{5}(k)=4(k_{x}k_{y}^{3}+k_{x}^{3}k_{y})$. The fitting procedure of the
eigenvalues of this model with the DFT band structure is presented in
Supplementary Fig. S2. Using the fitted values we have shown in Fig. 7(b) a
transparent top view of energy bands corresponding to NL3 and Nl4,
respectively. As seen, our effective Hamiltonian reproduces correctly the
hexagon (NL3) and hexagram (NL4) form of the momentum distribution of gapless
nodal points.
The other nodal line that we shall examine is NL2. Along the $\Gamma$-M
direction, the IRs of touching bands at the NP2′ point are identical to those
of NP4′. Hence, the same group theory analysis is applied to explain the
formation of topological nodal point NP2′. On the other hand, along the
$\Gamma$-K direction, the NP2 nodal point emerges due to the crossing of two
energy bands with different IRs $\Gamma_{1}$ and $\Gamma_{3}$. Here, the
protection of this point is justified by the opposite parity of the two bands
under $M_{h}$ or $M_{\tau}$. It should be noted that the same interpretation
as discussed earlier applies to other touching points of NL2 whose point
groups are $C_{s}$.
The remaining nodal line in Al2B2 is NL5 which encloses the M point. This
nodal loop is located at least 0.48 eV above the Fermi energy and is less
significant. For this nodal line, along both $\Gamma$-M and $\Gamma$-K
directions the corresponding bands of touching points NP5 an NP5′ have the
same IRs $\Gamma_{1}$ and $\Gamma_{4}$. As the wave functions of these bands
have opposite parities under $C_{2}$ and $M_{h}$, they prohibit the
interaction between them which allow the appearance of these nodal points.

DNLs in AlB4. Now that the electronic and topological properties of monolayer
Al2B2 were investigated, we move to examine the case of monolayer AlB4. This
configuration share the same symmetry as Al2B2 with $D_{6h}$ point group.
Figure 2(b) depicts those symmetry elements of AlB4 which we are interested in
the following. The band structure of AlB4 along the highly symmetric paths
(M-$\Gamma$-K-M) is displayed in Fig. 8(a). One can observe from Figs. 8(b)
and (c) that around the $\Gamma$ point and close to the Fermi level energy
bands mainly originate from boron $\sigma$ orbitals (B-s and B-px,y). However,
there exist also pure $\pi$-bonding boron states as well as mixed orbitals
(constructed from B-p, B-s, and Al-s) that

Figure 8:  (a) The calculated electronic band structure of Al2B2
configuration. Energy dispersion display multiple band crossing features
including both Dirac and nodal points as presented by magenta circles. Special
bands are marked by Greek letters. (b) and (c) Atom-projected band structures
for Al2B2 contributed by Al and B atoms, respectively. The size of dots is
proportional to the weight of contributed orbitals.  Figure 9:  Comparison of
the band structures of AlB4 as calculated by DFT and the WTB Hamiltonian. The
Wannier band structure exactly matches the DFT results.  Figure 10:  (a) and
(b) 3D visualizations of energy bands obtained from AlB4 WTB Hamiltonian.
Bands whose crossings lead to the formation of nodal lines around the $\Gamma$
point are shown. The extensions of nodal points can be seen vividly throughout
3D energy-momentum spaces which form the characteristic topology of each NL.
Color bars represent the energy of electrons. (c) and (d) The transparent top
views of (a) and (b). The momentum distributions of gapless nodal points are
shown. Note that in Figs.(b) and (d) some data values were removed for better
visualization.

cross the Fermi level. In a recent study, the role of these low-energy bands
in determining the superconducting behavior of the system has been explored
Zhao  _et al._ (2019). It was revealed that the AlB4 film captures a robust
three-gap superconducting nature with a high critical temperature ($\sim$47 K)
which provides a platform for further investigation of multigap
superconductors. Surprisingly, we found that along with the superconducting
behavior of these energy bands there are linear band crossing features that
emerge as a consequence of lattice symmetries. This led us to investigate the
topological aspects of the AlB4 band structure which contains several
dispersive NLs and Dirac cones.
Let us label the mentioned bands using Greek letters as indicated in Fig.
8(a). In the energy range from -2.2 to 1 eV, the crossing points consist of
either nodal points (NPs) or Dirac points (DPs). There are three Dirac points
as indicated by DP$1$ to DP$3$. Also, the nodal points NP$i$, and
NP$i^{\prime}$ ($i$ runs from 1 to 3) are pertinent points which belong to the
2D nodal loop NL$i$. In addition, there exists a distinct nodal point (NP-II)
that corresponds to an open 2D type-II nodal line (NL4). This is to the best
of our knowledge the first evidence for the existence of a 2D nonmagnetic open
type-II NL in systems at weak SOC limit. Figure 9 shows the comparison of
Wannier band structure of AlB4 with DFT data. Using the obtained WTB
Hamiltonian we provide in Fig. 10 3D energy bands of AlB4 near the Fermi level
in the vicinity of the $\Gamma$ point. An inspection of these 3D bands and the
corresponding transparent top views (Figs. 10(c) and (d)) reveals that the
momentum distribution of type-I nodal loops NL1 and NL3 show a hexagon, while
for the other type-I nodal loop i.e. NL2, it is a hexagram. Besides, one can
see from Figs. 10(a) and (c) that the open type-II nodal lines NL4 are
terminated to the closed nodal hexagram NL2. The coexistence of these two
nodal line types which can be confirmed through ARPES Feng  _et al._ (2017) or
infrared spectroscopy Shao  _et al._ (2020) measurements, may provide an ideal
platform to study many novel physical properties Yu  _et al._ (2016); Udagawa
and Bergholtz (2016); O’Brien  _et al._ (2016) that have been proposed for
NLSMs. For example, it is preferred to examine the signatures of Landau levels
spectrum collapse Yu  _et al._ (2016) on a 2D platform rather than in a 3D
material candidate. Also, it might be beneficial to utilize this topological
material to explore experimentally the anticipated enhanced correlations in
nodal line semimetals Shao  _et al._ (2020). Furthermore, the interplay of the
superconductivity and these topological Dirac nodal line states suggests that
topological superconducting phases Jin  _et al._ (2019) may be realized in
this structure which is a subject for further investigations.
The hexagon and hexagram shapes of NL1 (NL3) and NL2 are due to the fact that
the underlying symmetries and irreducible representations of the corresponding
bands are identical to that of NL4 and NL3 in Al2B2 lattice, respectively.
Therefore, the same group theory and method of invariants analysis applies to
explain the formation mechanism of these NLs, and no further discussion seems
necessary to highlight here.
Before we turn to the open nodal loop NL4, we shall briefly explain the
reasons why Dirac points DP1, DP2, and DP3 occur in the band structure of this
configuration. DP1 and DP2 locate along the $\Gamma$-K and $\Gamma$-M
directions $\sim$1.18 eV and $\sim$1.77 eV below the Fermi level,
respectively. Since the number of such points in the star of $k$ in the BZ is
six, one expect to see the same number of Dirac points. As seen in Fig. 9, the
IRs of the touching bands around both Dirac points DP1 and DP2 are
$\Gamma_{3}$ and $\Gamma_{4}$. Therefore, just similar to the case of DP1 in
Al2B2, the opposite parities of $C_{2}$ rotation or mirror $M_{\sigma}$ (or
$M_{\tau}$) allow the two bands to touch each other at DP1 and DP2. It is
worth mentioning that at first glance, one may expect that similar to a
typical point NP$i$, these points also belong to a nodal line distribution.
However, the situation is different here. Along a typical low symmetry
X-$\Gamma$ direction with point group $C_{s}$, both bands (around DP1 and DP2)
have the same IR $\Gamma_{2}$ which prevents them to touch each other, leading
to the opening of a gap. The other Dirac point that locates at the K point and
$\sim$0.65 eV below the Fermi level is DP3 which has the characteristics of
the Dirac points in graphene.
Now, we will focus on the topological aspects of type-II nodal line NL4. Along
the $\Gamma$-M direction, the two electron-like bands $\beta$ and $\gamma$
(with the same slope sign) meet $\sim$0.6 eV below the Fermi level and show a
band crossing as labeled by NP-II (Fig. 8(a)). Figures 11(a) and (b) display
band structures of AlB4 along two typical low-symmetry lines X-$\Gamma$ where
X is a point along the high-symmetry M-K direction (Fig. 11(c)). As seen, for
a typical line X1-$\Gamma$, where X1 is between K and C, there exists no
crossing between the $\beta$ and $\gamma$ bands. Conversely, when X is between
C and M, these two bands crosses along the corresponding X-$\Gamma$ path
(Fig.11b). Therefore, the crossing point of the $\gamma$ and $\beta$ bands are
only distributed in certain Brillouin zone slices, shown in Fig., to form a
set of open type-II nodal lines (NL4) in the system. Here, C specifies the low
symmetry path wherein NL4 is linked with NL2 (see Fig. 10(c)). The rather
large energy dispersion of these type-II nodal lines ($\sim$1.3 eV) makes AlB4
a distinguished 2D candidate for realization of novel topological properties.

Figure 11:  (a) and (b) Electronic band structures of AlB4 along typical low-
symmetry paths X1-$\Gamma$ and X3-$\Gamma$ in the BZ as shown by rel lines in
(c). The crossing of $\beta$ and $\gamma$ bands exists for a typical path
X-$\Gamma$ where X is located between X2 and M points.

The formation of the topologically protected momentum distribution NL4 is
justified as follow. Along the high symmetry direction $\Gamma$-M with point
group $C_{2v}$, the IRs of $\beta$ and $\gamma$ bands are $\Gamma_{4}$ and
$\Gamma_{2}$, respectively (see Fig. 9). The opposite parities of these two
bands under mirror symmetries $M_{h}$ or $M_{\sigma}$ decouple the associated
electronic states and allow the formation of the nodal point NP-II. Moreover,
along the typical low symmetry line X3-$\Gamma$ with point group $C_{s}$,
these bands have different IRs $\Gamma_{2}$ and $\Gamma_{1}$. The situation is
just similar to the case of NL1 in Al2B2 which permits the appearance of open
type-II NL4.
Eventually, using the method of invariants, we derive the effective
Hamiltonian model which describes quite well those nodal lines of AlB4 that
locate below the Fermi level. For the little group $D_{6h}$ at $\Gamma$ point
the relevant Bloch states belong to the doubly degenerate IRs
$\Gamma_{6}^{-}$, $\Gamma_{6}^{+}$, and nondegenerate IR $\Gamma_{2}^{-}$ as
shown in Fig. 9. Thus, the corresponding Bloch functions allow us to construct
an effective $5\times 5$ block diagonal Hamiltonian
$\mathcal{H}^{\mathrm{\Gamma}}$ around the $\Gamma$ point as

$\mathcal{H}^{\mathrm{\Gamma}}=\left(\begin{array}[]{ccc}\mathcal{H}_{11}&\mathcal{H}_{12}&\mathcal{H}_{13}\\\
\mathcal{H}_{21}&\mathcal{H}_{22}&\mathcal{H}_{23}\\\
\mathcal{H}_{31}&\mathcal{H}_{32}&\mathcal{H}_{33}\end{array}\right)\,.$ (12)

Therefore, using the method of invariants we adopt $\\{x-iy,x+iy\\}$,
$\\{x-iy,x+iy\\}$, and {$z$} as basis functions to obtain the basis matrices
as well as tensor components (see Supplementary Note 3). Inserting the
obtained basis matrices and tensor components (TR adds no additional
constraint as discussed in the case of Al2B2) in each block of
$\mathcal{H}^{\mathrm{\Gamma}}$ we arrive at

$\displaystyle\mathcal{H}_{11}=[E_{0}+Af_{1}(k)]\openone+B[f_{2}(k)\sigma_{x}+f_{3}(k)\sigma_{y}]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+C[f_{4}(k)\sigma_{x}+f_{5}(k)\sigma_{y}],$
(13)
$\displaystyle\mathcal{H}_{22}=[E^{\prime}_{0}+A^{\prime}f_{1}(k)]\openone+B^{\prime}[f_{2}(k)\sigma_{x}+f_{3}(k)\sigma_{y}]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+C^{\prime}[f_{4}(k)\sigma_{x}+f_{5}(k)\sigma_{y}],$
(14)
$\displaystyle\mathcal{H}_{33}=E^{\prime\prime}_{0}+A^{\prime\prime}f_{1}(k),$
(15) $\displaystyle\mathcal{H}_{ij}=0~{}~{}~{}~{}~{}~{}{i\neq j},$ (16)

where we have previously defined $f_{i}(k)$ functions. By fitting the
parameters with DFT results (see Supplementary Fig. S3), our continuous five-
band model describes very well the formation of open type-II nodal line NL4 as
well as NL1 and NL2 around the $\Gamma$ point as shown in Figs.12(a) and (b).

Figure 12:  Transparent top views of crossing bands obtained by continuum
model around the $\Gamma$ point of AlB4. The momentum distributions of gapless
nodal lines (a) NL1 and (b) NL2 and NL4 are reproduced correctly by the
continuum model.

Finally, we make some notes on the freestanding monolayer AlB2. Although,
group theory analysis of the energy bands unveils that there exist some
topologically protected Dirac points along the high symmetry lines in the BZ
of this structure, the absence of the in-plane mirror symmetry reduces the
point group of the system to $C_{6v}$ and thus prevents formation of any nodal
line feature in the momentum space (See Supplementary Note 4). It is worth
mentioning that the monolayer AlB2 with a B-terminated surface has been
recently synthesized on Al(111) Geng  _et al._ (2020) and the existence of the
p-type Dirac Fermions in this configuration was confirmed by the recent
experimental ARPES measurements Geng  _et al._ (2020). This may serve as a
possible way to explore our predicted topological NLSM states in aluminum
diborides thin films with in-plane mirrors.

Summary and conclusions
In summary, using first-principles calculations and group theory analyses, we
accurately investigated the electronic structure of Al2B2 and AlB4 thin films
to reveal the existence of many nontrivial topological features in these
systems. The dynamical stability of these structures was also proven by
calculation of their phonon spectra. Our results indicate that the electronic
band structures of both configurations host multiple topologically protected
band crossings including 2D Dirac nodal lines as well as Dirac points. We then
used symmetry analyses to explain the protection mechanism of the nodal lines
and then applied the method of invariants to construct proper effective
Hamiltonians for systematic investigation of the formation mechanism of these
nodal lines. We demonstrated formation of five and three type-I topological
NLs in the low energy electronic states of the Al2B2 and AlB4 compounds,
respectively. It is argued that all of the nodal line states are protected by
the time reversal and in-plane mirror symmetries. In the case of Al2B2, three
concentric nodal loops sit below the Fermi level and surround the $\Gamma$
point, while an isolated nodal ring appears around the M point above the Fermi
level. Moreover, it hosts a dispersive nodal loop centered around the K point,
appearing as six arcs in the corners of the Brillouin zone. This nodal loop
crosses the Fermi level with a considerable dispersion and thus provide an
excellent platform to study the fascinating characteristics of dispersive
Dirac nodal lines. For the AlB4 thin film, on the other hand, three concentric
nodal loops were evidenced in the low energy electronic states around the
$\Gamma$ point with hexagon or hexagram momentum distributions. More
importantly, we identify the first evidence for the emergence of a set of open
2D nonmagnetic type-II nodal arcs in the AlB4 Brillouin zone, linked with a
type-I nodal line. The rather large energy dispersion ($\sim$1.3 eV) of these
Dirac nodal arcs, their coexistence with multiple type-I NLs, and the high-
temperature superconducting behavior of AlB4 suggest this structure as a
distinguished material for advanced research on 2D topological
superconductors.

Methods
All structural and electronic properties are obtained within the Kohn-Sham DFT
calculations with the full potential linearized augmented plane wave (LAPW)
method as implemented in the computer package WIEN2k Blaha  _et al._ (2018).
The generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof
(PBE) formulation Perdew  _et al._ (1996); Blöchl (1994a) was used for the
exchange-correlation functional. The expansion cutoff
$R_{\mathrm{MT}}K_{\mathrm{max}}$ was set to 8, while a $30\times 30\times 1$
$\Gamma$-centered $k$ mesh was applied to sample the BZ for self-consistent-
field calculations. The crystal structures are fully relaxed with a total
energy and force convergence criteria of $10^{-4}$ Ry and $10^{-3}$ Ry/a.u.,
respectively. A vacuum of more than 20 Å was used to minimize the interactions
between the neighboring replica of the system. The maximally localized Wannier
functions were constructed by employing the Wannier90 package Pizzi  _et al._
(2020). The Al s, p and B s, p orbitals are chosen as initial projections for
WTB model construction.

In order to calculated the phonon spectra, the interatomic force constants
were obtained through a supercell approach by using the HIPHIVE Eriksson  _et
al._ (2019) package. We used $10\times 10\times 1$ supercells and then
generated the reference rattled structures by applying displacements, randomly
sampled from a normal distribution with a standard deviation of 0.01 Å. The
HIPHIVE package employs VASP Kresse and Furthmüller (1996a, b) for the
required DFT calculations, being performed with the projector augmented-wave
(PAW) Blöchl (1994b) pseudopotentials and PBE exchange-correlation functional.
The PAW plane-wave energy cutoff was set to 500 eV. After finding the force
constants, the phonon frequencies were calculated using the PHONOPY package
Togo and Tanaka (2015).

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Author contributions
S. Abedi and E. Taghizadeh Sisakht contributed equally to this work.

Conflicts of interest
There are no conflicts to declare.

## Supplementary Information

###  Supplementary Note 1 Supporting information of the continuum model for
Al2B2 configuration around the K point.

Table S1:  Symmetrized matrices for the invariant expansion of the diagonal
blocks $\mathcal{H}_{ii}$ for the point group $D_{3h}$.

$\begin{array}[]{clclcl}\hline\cr\hline\cr\mbox{Block}&\lx@intercol\mbox{Representations}\hfil\lx@intercol&\lx@intercol\mbox{Symmetrized
matrices}\hfil\lx@intercol&\lx@intercol\mbox{Tensor
components}\hfil\lx@intercol\\\
\hline\cr\mathcal{H}_{11}&\Gamma_{6}\otimes\Gamma_{6}^{\ast}&\Gamma_{1}:&\openone&\Gamma_{1}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}+\Gamma_{2}+\Gamma_{6}&\Gamma_{2}:&\sigma_{z}&\Gamma_{2}:&-\\\
&&\Gamma_{6}:&\sigma_{x},\sigma_{y}&\Gamma_{6}:&(k_{x},k_{y});(k_{y}^{2}-k_{x}^{2},2k_{x}k_{y})\\\\[4.30554pt]
\mathcal{H}_{22}&\Gamma_{4}\otimes\Gamma_{4}^{\ast}&\Gamma_{1}:&\openone&\Gamma_{1}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}\\\ \hline\cr\hline\cr\end{array}$

$E_{1,2}(k_{x},k_{y})={E_{0}}+A({k_{x}}^{2}+{k_{y}}^{2})\pm\sqrt{B^{2}({k_{x}}^{2}+{k_{y}}^{2})+C^{2}({k_{x}}^{2}+{k_{y}}^{2})^{2}-2BC({k_{x}}^{3}-3{k_{x}}{k_{y}}^{2})},$
(S1)
$E_{3}(k_{x},k_{y})={E^{\prime}_{0}}+A^{\prime}({k_{x}}^{2}+{k_{y}}^{2}).$
(S2)

The fitted parameters are $E_{0}=1.19$ eV, $A=-113.16$ eVÅ2, $B=8.78$ eVÅ,
$C=132.10$ eVÅ2, $E^{\prime}_{0}=-0.56$ eV, $A^{\prime}=66.33$ eVÅ2 . The
fitted bands are shown in Supplementary Fig. S1.

Supplementary Fig S1:  Comparison between DFT bands and the bands from the
continuum model near the K point of Al2B2.

###  Supplementary Note 2 Supporting information of the continuum model for
Al2B2 configuration around the $\Gamma$ point.

Table S2:  Symmetrized matrices for the invariant expansion of the diagonal
blocks $\mathcal{H}_{ii}$ for the point group $D_{6h}$.

$\begin{array}[]{clclcl}\hline\cr\hline\cr\mbox{Block}&\lx@intercol\mbox{Representations}\hfil\lx@intercol&\lx@intercol\mbox{Symmetrized
matrices}\hfil\lx@intercol&\lx@intercol\mbox{Tensor
components}\hfil\lx@intercol\\\
\hline\cr\mathcal{H}_{11}&\Gamma_{6}^{+}\otimes\Gamma_{6}^{+\ast}&\Gamma_{1}^{+}:&\openone&\Gamma_{1}^{+}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}^{+}+\Gamma_{2}^{+}+\Gamma_{6}^{+}&\Gamma_{2}^{+}:&\sigma_{z}&\Gamma_{2}^{+}:&-\\\
&&\Gamma_{6}^{+}:&\sigma_{x},\sigma_{y}&\Gamma_{6}^{+}:&(k_{y}^{2}-k_{x}^{2},2k_{x}k_{y});\;(k_{x}^{4}-6k_{x}^{2}k_{y}^{2}+k_{y}^{4},4k_{x}k_{y}^{3}+4k_{x}^{3}k_{y})\\\\[8.61108pt]
\mathcal{H}_{22}&\Gamma_{2}^{-}\otimes\Gamma_{2}^{-\ast}&\Gamma_{1}^{+}:&\openone&\Gamma_{1}^{+}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}^{+}\\\ \hline\cr\hline\cr\end{array}$

$E_{1,2}(k_{x},k_{y})={E_{0}}+A({k_{x}}^{2}+{k_{y}}^{2})\pm\sqrt{B^{2}({k_{x}}^{2}+{k_{y}}^{2})^{2}+C^{2}({k_{x}}^{2}+{k_{y}}^{2})^{4}-2BC({k_{x}}^{6}-15{k_{x}}^{4}{k_{y}}^{2}+15k_{x}^{2}k_{y}^{4}-k_{y}^{6})},$
(S3)
$E_{3}(k_{x},k_{y})={E^{\prime}_{0}}+A^{\prime}({k_{x}}^{2}+{k_{y}}^{2}).$
(S4)

The fitted parameters are $E_{0}=-0.37$ eV, $A=-43.88$ eVÅ2, $B=11.10$
eVÅ2,$C=100.17$eVÅ4, $E^{\prime}_{0}=-2.85$ eV, $A^{\prime}=12.36$ eVÅ2 . The
fitted bands are shown in Supplementary Fig. S2.

Supplementary Fig S2:  Comparison between DFT bands and the bands from the
continuum model near the $\Gamma$ point of Al2B2.

###  Supplementary Note 3 Supporting information of the continuum model for
AlB4 configuration around the $\Gamma$ point.

Table S3:  Symmetrized matrices for the invariant expansion of the diagonal
blocks $\mathcal{H}_{ii}$ for the point group $D_{6h}$.

$\begin{array}[]{clclcl}\hline\cr\hline\cr\mbox{Block}&\lx@intercol\mbox{Representations}\hfil\lx@intercol&\lx@intercol\mbox{Symmetrized
matrices}\hfil\lx@intercol&\lx@intercol\mbox{Tensor
components}\hfil\lx@intercol\\\
\hline\cr\mathcal{H}_{11}&\Gamma_{6}^{-}\otimes\Gamma_{6}^{-\ast}&\Gamma_{1}^{+}:&\openone&\Gamma_{1}^{+}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}^{+}+\Gamma_{2}^{+}+\Gamma_{6}^{+}&\Gamma_{2}^{+}:&\sigma_{z}&\Gamma_{2}^{+}:&-\\\
&&\Gamma_{6}^{+}:&\sigma_{x},\sigma_{y}&\Gamma_{6}^{+}:&(k_{y}^{2}-k_{x}^{2},2k_{x}k_{y});\;(k_{x}^{4}-6k_{x}^{2}k_{y}^{2}+k_{y}^{4},4k_{x}k_{y}^{3}+4k_{x}^{3}k_{y})\\\\[8.61108pt]
\mathcal{H}_{22}&\Gamma_{6}^{+}\otimes\Gamma_{6}^{+\ast}&\Gamma_{1}^{+}:&\openone&\Gamma_{1}^{+}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}^{+}+\Gamma_{2}^{+}+\Gamma_{6}^{+}&\Gamma_{2}^{+}:&\sigma_{z}&\Gamma_{2}^{+}:&-\\\
&&\Gamma_{6}^{+}:&\sigma_{x},\sigma_{y}&\Gamma_{6}^{+}:&(k_{y}^{2}-k_{x}^{2},2k_{x}k_{y});\;(k_{x}^{4}-6k_{x}^{2}k_{y}^{2}+k_{y}^{4},4k_{x}k_{y}^{3}+4k_{x}^{3}k_{y})\\\\[8.61108pt]
\mathcal{H}_{33}&\Gamma_{2}^{-}\otimes\Gamma_{2}^{-\ast}&\Gamma_{1}^{+}:&\openone&\Gamma_{1}^{+}:&1;\;{k_{x}^{2}+k_{y}^{2}}\\\
&=\Gamma_{1}^{+}\\\ \hline\cr\hline\cr\end{array}$

$E_{1,2}(k_{x},k_{y})={E_{0}}+A({k_{x}}^{2}+{k_{y}}^{2})\pm\sqrt{B^{2}({k_{x}}^{2}+{k_{y}}^{2})^{2}+C^{2}({k_{x}}^{2}+{k_{y}}^{2})^{4}-2BC({k_{x}}^{6}-15{k_{x}}^{4}{k_{y}}^{2}+15k_{x}^{2}k_{y}^{4}-k_{y}^{6})},$
(S5)
$E_{3,4}(k_{x},k_{y})={E^{\prime}_{0}}+A^{\prime}({k_{x}}^{2}+{k_{y}}^{2})\pm\sqrt{B^{\prime
2}({k_{x}}^{2}+{k_{y}}^{2})^{2}+C^{\prime
2}({k_{x}}^{2}+{k_{y}}^{2})^{4}-2B^{\prime}C^{\prime}({k_{x}}^{6}-15{k_{x}}^{4}{k_{y}}^{2}+15k_{x}^{2}k_{y}^{4}-k_{y}^{6})},$
(S6)
$E_{5}(k_{x},k_{y})={E^{\prime\prime}_{0}}+A^{\prime\prime}({k_{x}}^{2}+{k_{y}}^{2}).$
(S7)

The fitted parameters are $E_{0}=0.28$ eV, $A=-42.10$ eVÅ2, $B=12.55$
eVÅ2,$C=74.19$eVÅ4, $E^{\prime}_{0}=0.88$ eV, $A^{\prime}=-36.14$ eVÅ2,
$B^{\prime}=8.80$ eVÅ2,$C^{\prime}=58.35$eVÅ4, $E^{\prime\prime}_{0}=-2.72$
eV, $A^{\prime\prime}=11.64$ eVÅ2 . The fitted bands are shown in
Supplementary Fig. S3.

Supplementary Fig S3:  Comparison between DFT bands and the bands from the
continuum model near the $\Gamma$ point of AlB4.

###  Supplementary Note 4 Electronic properties of monolayer AlB2.

The electronic band structure of freestanding monolayer AlB2 is shown in Fig.
S4. Within 2.5 eV of the Fermi level there are several bands which cross to
each other to form four $p-$type Dirac cones below (red circles) and a gapless
$n-$type Dirac cone above (blue circle) the Fermi surface, respectively. We
mark these bands by Greek letters as shown in the Figure. Group theory
analysis of these energy bands shows that along the $\Gamma$-M and K-M
symmetry directions the small groups are $C_{s}$ which has an out of plane
symmetry element. Along these directions $\beta$, $\gamma$ and $\delta$ bands
belong to the $\Gamma_{1}$ IR which have positive mirror parity. Also, the IR
of $\alpha$ and $\nu$ bands are $\Gamma_{2}$ with a negative mirror eigenvalue
Altmann and Herzig (1994). As a result, within the mentioned energy window one
can observe the formation of four Dirac points DP1-DP4 due to the crossing of
bands with opposite mirror parities. The remained Dirac point (DP3) is located
at the K point which arises from the boron hexagonal lattice as the Dirac’s
cones of graphene. Note that due to the absence of in-plane mirror symmetry in
monolayer AlB2, along arbitrary low-symmetry directions X-$\Gamma$ and X′-K
one can see no crossing feature (see Figs. S5(a) and (b)). Therefore, we do
not expect to observe symmetry protected nodal lines in this structure. The
existence of the $p-$type Dirac Fermions in this configuration has been
confirmed by recent experimental ARPES measurements Geng  _et al._ (2020)
which provides an excellent platform for designing new nanoelectronics.

Supplementary Fig S4:  The electronic band structure of monolayer AlB2 along
the high symmetry paths M-$\Gamma$-K-M. The $p-$ and $n-$type Dirac cones are
shown by red and blue circles, respectively.  Supplementary Fig S5:  The
electronic band structure of monolayer AlB2 along typical low-symmetry
directions (a) X-$\Gamma$ and (b) X′-K as shown in (c). The zoomed-in band
structures in the red ellipses are shown in (d) and (e).