# 3D carbon allotropes: Topological quantum materials with obstructed atomic
insulating phases, multiple bulk-boundary correspondences, and real topology

###### Abstract

The study of topological phases with unconventional bulk-boundary
correspondences and nontrivial real Chern number has garnered significant
attention in the topological states of matter. Using the first-principle
calculations and theoretical analysis, we perform a high-throughput material
screening of the 3D obstructed atomic insulators (OAIs) and 3D real Chern
insulators (RCIs) based on the Samara Carbon Allotrope Database (SACADA).
Results show that 422 out of 703 3D carbon allotropes are 3D OAIs with
multiple bulk-boundary correspondences, including 2D obstructed surface states
(OSSs) and 1D hinge states, which are in one dimension and two dimensions
lower than the 3D bulk, respectively. The 2D OSSs in these OAIs can be
modified when subjected to appropriate boundaries, which benefits the
investigation of surface engineering and the development of efficient
topological catalysts. These 422 OAIs, which have 2D and 1D boundary states,
are excellent platforms for multi-dimensional topological boundaries research.
Remarkably, 138 of 422 OAIs are also 3D RCIs, which show a nontrivial real
topology in the protection of spacetime inversion symmetry. Our work not only
provides a comprehensive list of 3D carbon-based OAIs and RCIs, but also
guides their application in various aspects based on multiple bulk-boundary
correspondences and real topological phases.

###### keywords:

3D carbon allotropes, Real Chern insulators, Hinge Fermi arc states,
Obstructed atomic insulators

Jianhua Wang#, Ting-Ting Zhang#, Qianwen Zhang, Xia Cheng, Wenhong Wang,
Shifeng Qian*, Zhenxiang Cheng*, Gang Zhang, Xiaotian Wang*

(#These authors contributed equally to this work.)

J. Wang
School of Material Science and Engineering, Tiangong University, Tianjin
300387, China
Institute for Superconducting and Electronic Materials (ISEM), University of
Wollongong, Wollongong 2500, Australia
T. Zhang
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
Q. Zhang, X. Cheng
School of Physical Science and Technology, Southwest University, Chongqing
400715, China
W. Wang
School of Material Science and Engineering, Tiangong University, Tianjin
300387, China
S. Qian
Anhui Province Key Laboratory of Optoelectric Materials Science and
Technology, Department of Physics, Anhui Normal University, Anhui, Wuhu
241000, China
<EMAIL_ADDRESS>
Z. Cheng
Institute for Superconducting and Electronic Materials (ISEM), University of
Wollongong, Wollongong 2500, Australia
<EMAIL_ADDRESS>
G. Zhang
Institute of High-Performance Computing, Agency for Science, Technology and
Research (A$\\*$STAR), 138632 Singapore, Singapore
X. Wang
Institute for Superconducting and Electronic Materials (ISEM), University of
Wollongong, Wollongong 2500, Australia
School of Physical Science and Technology, Southwest University, Chongqing
400715, China
<EMAIL_ADDRESS>

##  1 Introduction

In condensed-matter physics, band topology [1, 2, 3, 4, 5] has flourished and
developed over the last ten years. Many materials have been predicted as
topological insulators [6, 7, 8, 9, 10] or topological semimetals [11, 12, 13,
14, 15, 16, 17, 18, 19, 20] with topological boundary states. Meanwhile,
topological trivial insulators have lately received attention due to the
advancement of topological quantum chemistry theory [21, 22]. A topological
trivial insulator is a material whose band representations (BRs) of valence
bands can be characterized by a sum of elementary BRs (EBRs) [21], which means
the trivial state can be described by a set of exponentially localized Wannier
functions. Based on the Wyckoff positions where the orbitals responsible for
inducing the band representations (BRs) are situated, topological trivial
insulators can be classified into two categories: obstructed atomic insulators
(OAIs) and atomic insulators [23, 24, 25, 26, 27, 28, 29, 30]. For the atomic
insulators, their BRs are all induced from the orbitals that are located at
atomic-occupied Wyckoff positions (WPs); however, for the OAIs, their BRs come
from the orbitals that are located at atom-occupied and atom-unoccupied WPs
[31, 32]. Such atom-unoccupied WPs in OAIs are named obstructed Wannier charge
centers (OWCCs) [31, 32]. When the cleavage termination cuts through those
OWCCs in a 3D OAI, the 2D obstructed surface states (OSSs) will occur [27, 28,
29]. Unlike topological insulators, their topological surface states are
connected to the bulk states, which generally host small spin-orbit coupling-
induced gaps [6, 7]; for OAIs, their OSSs are usually clean, located between
the large energy gap, and simply to distinguish in the energy spectrum from
the bulk states, which is advantageous for theoretical analysis and
experimental detection [30]. Remarkably, it has been suggested that the 2D
OSSs in a 3D OAI can be used as a descriptor to define catalytic active sites
in 3D inorganic crystalline materials [27]. That is, the crystal surface with
OSSs could be active for molecule bonding and adsorption. Moreover, the
existence of OSSs in SrIn2P2 [30] has been validated by the utilization of
scanning tunneling microscopy and angle-resolved photoemission spectroscopy.

Figure 1: The schematic procedure for discovering the obstructed atomic
insulators and real Chern insulators among 703 3D carbon allotropes based on
the Samara Carbon Allotrope Database (SACADA).

Note that the study of topological matters has recently revolutionized with
the finding of second-order topological phases [33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Second-order topological phases
in 3D induce unconventional bulk-boundary correspondence because they exhibit
boundary states in two dimensions lower than the 3D bulk, as opposed to the
previous first-order topological phases, which contained topological boundary
states in one dimension lower than the 3D bulk. The second-order topological
phases in 3D are manifested in the hinge modes located at the hinges of the
sample materials with a tube geometry. So far, second-order topological phases
in 3D have only been experimentally verified in Bi, BixSb1-x, and Bi2TeI [34,
52, 51], and the scarcity of suitable materials hinders further
investigations. For a 3D OAI, it is also interesting to study the existence of
(3-2)D boundaries, i.e., 1D hinges, besides its (3-1)D boundaries, i.e., 2D
OSSs.

Moreover, as guaranteed by certain symmetry constraints, such as the spacetime
inversion symmetry ($\mathcal{P}\mathcal{T}$) and the absence of spin-orbit
coupling (SOC) effect [53, 54, 55], the band eigenstates are required to be
real [55, 56, 57, 58]. The real Chern insulator state (RCI) state,
characterized by the nontrivial $\mathbb{Z}_{2}$ valued real Chern number
$\nu_{R}$ (also known as the second Stiefel-Whitney number) [56, 57], has been
revealed in a few two-dimensional systems [59]. Notably, the concept can be
extended to three dimensions. In this work, we term the 3D OAIs with a
nontrivial real topology as 3D RCIs, where every 2D horizontal or vertical
slice has a nontrivial $\nu_{R}$.

From a materials science perspective, finding ideal material candidates for 3D
OAIs and 3D RCIs with multiple dimensions of boundaries and real topological
phases continues to be a crucial subject. In this study, we aim to make
advancements in predicting the properties of 3D OAIs and 3D RCIs in all forms
of carbon allotropes by high-throughput material screening. In addition to the
widely recognized carbon allotropes such as graphite [60, 61, 62], diamond
[63, 64], carbon nanotubes [65, 66, 67], graphene [68, 69, 70], and fullerenes
[71, 72], there have been over 700 more carbon allotropes that have been
either theoretically predicted or experimentally synthesized. Through
comprehensive examination utilizing first-principle calculations and
theoretical analysis based on the Samara Carbon Allotrope Database (SACADA)
[73], we have investigated a total of 703 carbon allotropes. Our findings
indicate that among these candidates, 422 candidates are 3D OAIs. In addition,
out of these OAIs, 138 can be categorized as 3D
$\mathcal{P}\mathcal{T}$-symmetric RCIs that possess a nontrivial $\nu_{R}$.
Note that there have been no papers reporting the comprehensive analysis of
all OAIs and RCIs for all forms of carbon allotropes to yet. Using 37-SG.
194-lon (No. 37 in SACADA and space group (SG.) 194) and 35-dia-a (No. 35 in
SACADA and SG. 224) as examples, we will show their OAI state and RCI state,
respectively. Moreover, we shall examine the OSS states in the example of
37-SG. 194-lon by choosing four surface planes, i.e., (001)1,
$(10\overline{1}0)$, (001)2, and (010) planes, as cleavage terminations, and
find there would exist four completely different 2D OSSs on these surface
planes. In other words, the 2D OSSs in 3D OAIs can be modified when subjected
to appropriate boundaries, making them more ideal for on-demand surface
engineering. Besides the 2D OSSs, the 1D hinge states on vertical hinges can
also be found in 37-SG. 194-lon. That is, 2D and 1D boundaries (surface and
hinge) coexisted in a single 37-SG. 194-lon, which provides a platform for
studying boundaries across multiple dimensions. Unlike 37-SG. 194-lon, the
35-SG. 224-dia-a hosts a real Chern insulator state, characterized by a
$\nu_{R}$, which can be viewed as a promising platform for exploring the
fascinating physics of real topological phases.

##  2 Results and discussion

###  2.1 Database screening scheme

The crystallographic data for all carbon allotropes can be acquired from the
SACADA database [73], which provides a comprehensive list of 703 3-periodic
carbon allotropes.

The first step is identifying the 3D OAIs from the 703 3D carbon allotropes
using the theory of elementary band representation (EBR) [21, 22]. The EBRs
correspond to the smallest sets of band structures that can be derived from
maximal localized atomic-like Wannier functions. A sum of EBRs can fully
describe the band structures below the Fermi level of atomic insulators and
OAIs. However, the OAI is the insulator with the EBRs at the WPs that are
unoccupied by atoms. Such atom-unoccupied WPs are known as OWCCs.

To see if the OWCCs are present in the 703 3D carbon allotropes, we estimated
the BRs of occupied bands at high-symmetry points for each 703 carbon
allotropes (see Figure 1). To characterize the symmetry properties, the
concept of symmetry-data vector $B$ is defined [21], and its explicit form is
provided below:

$B=\left\\{n_{1}^{1}G_{K_{1}}^{1}\oplus
n_{2}^{2}G_{K_{1}}^{2}\oplus\ldots,n_{2}^{1}G_{K_{2}}^{1}\oplus
n_{2}^{2}G_{K_{2}}^{2}\oplus\cdots\right\\},$ (1)

the $G_{K_{i}}^{j}$ is the $j$th irrep of the little group at the maximal
momenta Ki, and $n_{i}^{j}$ is the multiplicity of irrep $G_{K_{i}}^{j}$.

$B$ can be decomposed on the basis of EBRs. The decomposition is subjected to

$B=\sum_{i}p_{i}(EBR)_{i}.$ (2)

The coefficient $p_{i}$ can show the topological features of the system [21].
For clarity, we select 37-SG. 194-lon, as an example, the $B$ and
corresponding EBRs for 37-SG. 194-lon are shown in Table 1. It is obvious that
the coefficients $p_{\left(A_{1}^{\prime}\right)_{2c}}\uparrow G=1$ and
$p_{\left(A_{g}\right)_{6g}}\uparrow G=1$, and the others are zero, indicating
the bands below the Fermi level can be expressed as a combination of EBRs:
${\left(A_{1}^{\prime}\right)_{2c}}\uparrow G$ \+
${\left(A_{g}\right)_{6g}}\uparrow G$. Hence, the OWCCs of 37-SG. 194-lon
reside at the Wyckoff positions 2c and 6g, which are mismatching with atomic-
occupied Wyckoff positions 4f (see Figure 2a), reflecting 37-SG. 194-lon is an
OAI. After screening, we found 422 out of 703 carbon allotropes exhibit the
OWCCs, indicating the nature of the OAI (see Table S1 in the Supporting
Information).

Table 1: The EBRs induced by maximal symmetry Wyckoff positions in 37-SG. 194-lon. EBRs | $\Gamma(0,0,0)$ | $M(1/2,0,0)$ | $K(1/3,1/3,0)$ | $A(0,0,1/2)$ | $L(1/2,0,1/2)$ | $H(1/3,1/3,1/2)$
---|---|---|---|---|---|---
$(A^{\prime}_{1})_{2c}\uparrow G$ | $\Gamma_{1}^{+}\oplus\Gamma_{4}^{-}$ | $M_{1}^{+}\oplus M_{4}^{-}$ | $K_{5}$ | $A_{1}$ | $L_{1}$ | $H_{1}$
$(A_{g})_{6g}\uparrow G$ |  $\Gamma_{1}^{+}\oplus\Gamma_{3}^{+}\oplus$ $\Gamma_{5}^{+}\oplus\Gamma_{6}^{+}$ |  $M_{1}^{+}\oplus M_{1}^{-}\oplus M_{2}^{-}\oplus$ $M_{3}^{+}\oplus M_{3}^{-}\oplus M_{4}^{-}$ | $K_{1}\oplus K_{2}\oplus K_{5}\oplus K_{6}$ | $A_{1}\oplus A_{3}$ | $2L_{1}\oplus L_{2}$ | $H_{1}\oplus H_{2}\oplus H_{3}$
Figure 2: (a) Crystal structure of 37-SG. 194-lon. In (a), the OWCCs residing
at the Wyckoff positions 2c and 6g are denoted by small blue balls. Four
surface terminations, including (001)1, $(10\overline{1}0)$, (001)2, and
(010), are highlighted by arrows. (b) The 3D Brillouin zone (BZ) and the
projected (001), $(10\overline{1}0)$, and (010) surfaces. (c) Calculated band
structure for 37-SG. 194-lon along the symmetry paths
$\Gamma$–M–K–$\Gamma$–A–L–H–A.

The second step is identifying the real Chern insulator state based on the
selected 422 OAIs. To confirm the real Chern topologies, we evaluate the
$\nu_{R}$ for 2D $k_{z}=0$ and $k_{z}=\pi$ planes. For these two planes, the
$\nu_{R}$ can be readily extracted from the parity eigenvalues at the time-
reversal invariant momentum (TRIM) points on each plane, with [56, 57]

$(-1)^{\nu_{R}}=\prod_{i}(-1)^{\left\lfloor\left(n_{-}^{\Gamma_{i}}/2\right)\right\rfloor},$
(3)

where $\lfloor\cdots\rfloor$ is the floor function, and $n_{-}^{\Gamma_{i}}$
is the number of occupied bands with negative $\mathcal{P}\mathcal{T}$
eigenvalue at TRIM point ${\Gamma_{i}}$. If one finds that the $\nu_{R}$ is
nontrivial ($\nu_{R}$ = 1) for $k_{z}=0$ and $k_{z}=\pi$ planes, which
confirms the RCI nature for the carbon allotropes (see Figure 1). Finally, we
find 138 candidates are RCIs with nontrivial $\nu_{R}$ (see Table S1 in the
Supporting Information).

Figure 3: (a) Schematic diagrams of the numbers of OWCCs, which are cut
through by (001)2, $(10\overline{1}0)$, (001)1, and (010) crystal planes.
(b)-(e) 2D OSSs on (001)2, $(10\overline{1}0)$, (001)1, and (010) crystal
planes, respectively.  Figure 4:  (a) Calculated hinge spectrum for the 1D
tube geometry of 37-SG. 194-lon sample. The hinge bands are shown in red. (b)
Energy spectrum for 37-SG. 194-lon sample with a 1D tube geometry. (c) Charge
spatial distributions of the two groups of hinge modes marked by triangle and
quadrangle in (b) under different viewpoints.

###  2.2 3D OAIs with 2D OSSs and 1D hinges: 37-SG. 194-lon as an example

Upon conducting an examination to calculate the BRs of the valence bands and
determine the presence of OWCCs among the 703 3D carbon allotropes, it was
discovered that 422 of these allotropes are 3D OAIs, as detailed in the
Supporting Information (see Tables S2-S422 and Figures S1-S420). In this
section, we will focus on an example, 37-SG. 194-lon, to provide a
comprehensive analysis of multiple dimensions of boundaries.

The band structure is shown in Figure 2c, in which the insulating state with
an indirect band gap of 3.3457 eV is obvious. It is important to acknowledge
that the hexagonal wurtzite-like 37-SG. 194-lon is a realistic material,
initially discovered by Frondel and Marvin [74] in the acid-insoluble residue
obtained from small specimens collected from the rim of the crater. The
optimized lattice constants are calculated as $a=b=2.51$ Å and $c=4.18$ Å,
which are in good agreement with the experimental data ($a=b=2.51$ Å and
$c=4.12$ Å) [74].

As shown in Table 1, the BRs are induced by the $A_{1}^{\prime}$ and $A_{g}$
orbitals in the D6h point group at Wyckoff positions 2c and 6g of SG
$\mathrm{P6}_{3}/\mathrm{mmc}$, which are atom-unoccupied positions in 37-SG.
194-lon. That is to say, the Wyckoff positions are 2c and 6g occupied by the
OWCCs (see Figure 2a). Therefore, the outcome clearly indicates that 37-SG.
194-lon is a 3D OAI.

Next, we proceed to examine the 2D boundaries, i.e., OSSs, of 37-SG. 194-lon.
In contrast to the topological insulator, which exhibits topological surface
states that always intersect the bulk energy gap irrespective of the system
boundaries, the OSSs of the OAIs, manifest as floating bands within the bulk
energy gap, exclusively emerge along the boundaries that cut through the
OWCCs. In Figure 3a, we have chosen four surface planes, namely (001)2,
$(10\overline{1}0)$, (001)1, and (010), which cut through 1, 2, 3, and 4
OWCCs, respectively. These surface planes are chosen to investigate the
occurrence of various numbers of OSSs on the associated surface planes. The
calculated OSSs on the four surface planes are shown in Figures 3b-e. As
expected, 1, 2, 3, and 4 OSSs appear on (001)2, $(10\overline{1}0)$, (001)1,
and (010) surface planes, respectively, due to these 2D boundaries cutting
through 1, 2, 3, and 4 OWCCs.

The OSSs in Figures 3b-e typically exhibit cleanliness and are situated within
a large energy gap. Consequently, they can be easily separated in the energy
spectrum from the bulk states, thereby facilitating theoretical research and
experimental detection. The tunability of the OSS numbers can be achieved by
selecting boundaries that cut through the numbers of OWCCs. This tunability is
advantageous for the investigation of surface engineering. Since 2019, carbon-
based catalyst [75, 76] has been found to be an efficient, low-cost, metal-
free alternative to platinum for oxygen reduction in fuel cells. Furthermore,
prior studies have put forth the notion of 2D surface states [27, 77, 78, 79,
80] as a ubiquitous catalytic descriptor in the field of topological
catalysis. It is anticipated that the 422 OAIs with 2D OSSs (listed in Table
S1 of the Supporting Information) may be considered as potential candidates
for the future development of highly efficient carbon-based metal-free
topological catalysts.

In 3D OAIs, the presence of gaps in the 3D bulk states allows for the
existence of boundary states on (3-1)D surfaces. Interestingly, unconventional
bulk-boundary correspondence also appears in 3D OAIs because they host second-
order topological phases. That is, boundary states in two dimensions lower
than the 3D bulk, i.e., (3-2)D hinges also appear in 3D OAIs. In order to
determine the hinge spectrum, a tight-binding model (TB) for 1D tube geometry
was built based on a sample of 37-SG. 194-lon preserving its $C_{3z}$ symmetry
(see Figure 4c), and then the nanotube band calculation was performed using
the Wannier function [81, 82]. Figures 4a and b illustrate the presence of two
groups of threefold degenerate hinge states, denoted by red dots with triangle
and quadrangle. The charge spatial distributions of these two groups of hinge
states are depicted in Figure 4c.

Figure 5: (a) Crystal structure of 35-SG. 224-dia-a. In (a), the OWCCs
residing at the Wyckoff position 4b are denoted by small blue balls. (b) 3D
bulk BZ and the projected of (001) plane. (c) Calculated band structure for
35-SG. 224-dia-a along the symmetry paths $\Gamma$–X–M–$\Gamma$–R–X. (d) 2D
OSSs on (001) plane. (e) Calculated hinge spectrum for the 1D tube geometry of
35-SG. 224-dia-a sample. The three groups of hinge bands are shown in red. (f)
Energy spectrum for 35-SG. 224-dia-a sample with a 1D tube geometry. (g)
Charge spatial distributions of three groups of hinges marked by circle,
triangle, and quadrangle in (f) under different viewpoints. (h) The schematic
diagram for 3D $\mathcal{P}\mathcal{T}$-symmetric RCIs with hinge Fermi arcs,
in which every 2D slice normal to $k_{z}$ has a nontrivial $\nu_{R}$. (i) and
(j) Wilson loop spectrum calculated on $k_{z}=0$ and $k_{z}=\pi$ planes.

3D OAIs containing 2D and 1D boundaries (surface and hinge) can be considered
a useful platform for multi-dimensional boundary research. With the aid of
surface-sensitive probes, such as angle-resolved photoemission spectroscopy or
scanning tunneling spectroscopy, the 2D OSSs can be detected. As demonstrated
in Bi [34], scanning tunneling spectroscopy is a useful method for examining
1D hinge modes.

Table 2: Parity information of the 35-SG. 224-dia-a at the eight TRIM points. |  $k_{z}$=0 | $k_{z}=\pi$
---|---|---
| $\Gamma$ | $X$ | $M$ | $Y$ | $Z$ | $U$ | $R$ | $T$
$n_{+}$ | 10 | 8 | 8 | 8 | 8 | 8 | 10 | 8
$n_{-}$ | 6 | 8 | 8 | 8 | 8 | 8 | 6 | 8
$\nu_{R}$ | 1 | 1

###  2.3 3D RCIs with a real topology: 35-SG. 224-dia-a as an example

The $\mathcal{P}\mathcal{T}$ enforces a real topology for the 3D OAIs, which
can be characterized by the $\nu_{R}$. Through a comprehensive analysis,
$\nu_{R}$ for the $k_{z}=0$ and $k_{z}=\pi$ planes was calculated for the 422
3D OAIs. The findings revealed that among these OAIs, 138 of them displayed
$\nu_{R}=1$, i.e., the nontrivial real topology for both planes. Further
information regarding the selected 138 3D RCIs can be found in Tables S423-559
(Supporting Information).

In this section, we will focus on an example, 35-SG. 224-dia-a, to
comprehensively analyze its real topology, and 2D and 1D boundaries (surfaces
and hinges). 35-SG. 224-dia-a was designed by Li et al. [83] by moving carbon
atoms in T-carbon [84, 85] half of the lattice vector along a-axis and
stacking it with T-carbon, as illustrated in Figure 5a. As demonstrated in
Table S14 of the Supporting Information, the decomposition of the BR of dia-a
into a linear combination of the EBRs in space group SG. 224 reveals the
occurrence of the OWCCs (represented by blue balls in Figure 5a) in 35-SG.
224-dia-a, which are located at the Wyckoff position 4b. The insulating band
structure for 35-SG. 224-dia-a with an indirect bulk gap of 1.0836 eV is
exhibited in Figure 5c. Due to the 2D surface plane (001) cuts through two
OWCCs, two OSSs can be observed, as depicted in Figure 5d. Significantly, the
OSSs depicted in Figure 5d exhibit a gap near the Fermi level. This
characteristic becomes advantageous when observing the hinge Fermi arcs in the
system. Figures 5e and f illustrate the presence of three groups of twofold
degenerate hinge states, denoted by red dots with circle, triangle, and
quadrangle. The charge spatial distributions of the three groups of hinges are
depicted in Figure 5g.

In addition, the 35-SG. 224-dia-a hosts a nontrivial real topology. In order
to validate the real topology, it is convenient to perform calculations of the
$\nu_{R}$ for the $k_{z}=0$ or $k_{z}=\pi$ plane using the parity information
at the eight TRIM points. In Table 2, $n_{+}\left(n_{-}\right)$ shows the
numbers of the valence states with positive (negative)
$\mathcal{P}\mathcal{T}$-eigenvalues for $k_{z}=0$ and $k_{z}=\pi$ planes.
Based on Eq. (3), the calculated $\nu_{R}$ for the $k_{z}=0$ and $k_{z}=\pi$
planes is 1, reflecting the nontrivial real topology for both planes.
Actually, every 2D slice normal to $k_{z}$ has a nontrivial $\nu_{R}$, as
shown in the schematic diagram for 3D RCIs in Figure 5h. Furthermore, the
$\nu_{R}$ can also be evaluated using the Wilson-loop method. We calculate the
Wilson loop spectrum on $k_{z}=0$ and $k_{z}=\pi$ planes for 35-SG. 224-dia-a,
as shown in Figures 5i and j. Only one crossing point at $\theta$ = $\pi$ of
the Wilson loop spectrum in both planes indicates $\nu_{R}$ = 1 for both.
These results are consistent with those obtained by the parity information at
the TRIM points in Table 2.

##  3 Conclusion

In summary, from first-principle calculations and theoretical analysis based
on the SACADA, it has been determined that out of a total of 703 carbon
allotropes, 422 are 3D OAIs. The atom-unoccupied WPs (i.e., OWCCs) in these
422 OAIs are also provided as a reference for subsequent research inquiries.
We have chosen a specific example, 37-SG. 194-lon, to conduct an investigation
on the relationship between the bulk and boundaries in OAIs. Moreover, the
numbers of 2D OSSs can be determined by the numbers of the cut OWCCs by the 2D
boundary. The tunable OSSs in these 422 OAIs can be utilized for the purpose
of exploring surface engineering and developing efficient carbon-based metal-
free topological catalysts in subsequent research endeavors. Moreover, the
37-SG. 194-lon hosts second-order topological phases, which induce
unconventional bulk-boundary correspondence and are visible as hinge Fermi
arcs. The coexistence of 2D OSSs and 1D hinges in 37-SG. 194-lon benefits the
follow-up investigations of multi-dimensional topological boundaries. Further
screening reveals that 138 of 422 OAIs host a real topology, which is
characterized by a nontrivial $\nu_{R}$. These 138
$\mathcal{P}\mathcal{T}$-symmetric 3D RCIs are promising candidates for
exploring the fascinating physics of real topological phases. Our findings
show a complete list of 3D carbon-based OAIs and RCIs, and provide guidance
for their use in a variety of fields, such as surface engineering, metal-free
topological catalysts based on carbon allotropes, multi-dimensional
boundaries, and nontrivial real band topology.

##  4 Calculation methods

We perform the first-principles calculations in the framework of density-
functional theory (DFT) within the Vienna Ab initio simulation package (VASP)
[86, 87]. The exchange-correlation effect is treated in the generalized
gradient approximation (GGA) in the Perdew Burke Ernzerhof (PBE) function
[88]. All calculations are performed with a plane-wave cutoff of 500 eV, and
the convergence criterion for the electronic self-consistence loop was set to
be 10-6 eV, and for the structural relaxation, the Hellmanne-Feynman forces on
each atom were taken to be -0.01 eV/Å. The boundary bands were calculated
based on the iterative Green functions method, realized using the WANNIERTOOLS
package [81]. The irreducible representations of electronic states from DFT
results were calculated using the irvsp code [89].

Acknowledgements

W. Wang thanks the National Key R$\&$D Program of China (Grant no.
22022YFA1402600) for support. X. Wang thanks Australian Research Council
Discovery Early Career Researcher Award (Grant no. DE240100627) for support.
S. Qian thanks the National Natural Science Foundation of China (No. 12174003)
and the Foundation for Distinguished Young Scholars in Higher Education
Institutions of Anhui Provincial (No. 2022AH020019) for support.

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