Higher-order topological Dirac phase in Y3InC: a first-principles study

Higher-order topological insulators hosting intriguing topologically protected hinge or corner states are of significant research interest. However, materials that possess higher-order topological hinge states associated with gapless bulk Dirac phases still need to be explored. Using first-principles calculations with hybrid exchange functional, we explore the electronic structure and topological properties of Y3InC and a few of its sister compounds, totaling 16 bulk materials. A symmetry-protected triple point phase, with dominated d-t 2g character, is observed in Y3InC without spin–orbit coupling (SOC). Interestingly, the SOC induces a twin Dirac node phase in the bulk Y3InC. Furthermore, the computed Z 4 topological invariant reveals the higher-order topological nature of investigated materials. To demonstrate the gapless hinge states, we conduct edge state calculations using a rod-shaped geometry of Y3InC. Remarkably, Y3InC is identified to host multi-Dirac nodes in the bulk and surface phases together with the higher-order hinge states. These results lay the groundwork for further experimental and theoretical investigations into cubic antiperovskite materials for higher-order topological phases.


Introduction
Topological materials have gained enormous research interest due to their peculiar electronic structure properties originating from bulk-boundary correspondence [1][2][3][4][5][6][7].Recently, a novel type of topological material characterized as a higher-order topological insulator (HOTI) with extended bulk-boundary correspondence gathered significant research interest.This extended bulk-boundary correspondence notably facilitates topologically protected lower-dimensional hinge/corner states in a three-dimensional insulator [8][9][10][11][12][13][14][15].Interestingly, for a second-order topological insulator, these symmetry-protected gapless hinge states occur in the energy window where the bulk and surface states are gapped [16][17][18].In contrast, a third-order topological insulator protects zero-dimensional corner states [19].Nonetheless, theoretical studies show that these high-order topological features exist not only in the insulating materials but also in gapless materials [20][21][22][23][24][25][26].The recent extension of this higher-order topology to gapless topological systems such as Dirac materials enables symmetry-protected hinge/corner states to be present in higher-order Dirac material phases (HODM) [20,21,[23][24][25][26]. Unlike HOTIs, the HODM phase allows the gapless bulk, surface, and hinge states to coexist.Yet another intriguing characteristic of HODMs is the higher-order Fermi arcs localized on the hinge edge where two surfaces meet.These protected higher-order Fermi arcs represent the bulk-hinge correspondence [20][21][22][23][24][25][26].Notably, these topological higher-order hinge states are symmetry-mediated.Hence, the crystalline symmetries such as rotation, inversion, and reflections combined with time-reversal symmetry play a crucial role in the origin of the aforementioned higher-order topological and Dirac phases [20].In this regard, several bulk topological indexes facilitate the possible higher-order topology in solid materials [8,13,17].However, when materials possess inversion and mirror symmetries, topological invariants such as Z 4 invariant and mirror Chern numbers are found exceptionally useful [8,13].Due to these symmetry constraints, only a handful of materials, including Cd 3 As 2 , KMgBi, XTe 2 (X: Mo, W), β-CuI and NaCuSe [20,[24][25][26], etc, are found to be Dirac materials with higher-order Fermi arcs in the early stage of the study.Therefore, a search for materials with higher-order Dirac phases for experimental synthesis is still in great demand.
A possible candidate for HODMs is the perovskite family of materials, which has been constantly investigated in condensed matter physics and material science due to its wide variety of structural and electronic properties [27].Cubic antiperovskite (A 3 BX type) compounds are a family of materials with a Pm-3m space group belonging to a similar symmetry as cubic perovskite (ABX 3 type) with an interchange in the atomic positions between A and X elements [28,29].These materials gathered massive attention due to their various properties, such as magnetic, superconducting, magnetoresistance, thermoelectric, topological features, etc [30][31][32][33][34].Moreover, several antiperovskite materials are explored for topological crystalline insulating and Dirac phases [35][36][37][38].Recently, one of the antiperovskite carbides, such as Sc 3 AlC, has shown higher-order topological features stemming from the triple point phase without considering spin-orbit coupling (SOC) [39].However, the role of SOC in these materials is crucial.To explore this material family, we have chosen a few experimentally synthesizable materials for our examination of the higher-order topological phase, and our representative material is Y 3 InC.The bulk Y 3 InC compound was synthesized in early 1995 using powder sintering, fusion, and annealing techniques [40].Further, the sample characterizations were reported as a cubic Pm-3m structure [40].The unexplored electronic and topological features and the clean band structure near the intriguing topological states, such as quadratic and linear Dirac nodes of Y 3 InC, inspired us to choose this material as our representative material.Thus, the detailed topological higher-order features of an experimentally synthesized cubic antiperovskite Y 3 InC and its few sister compounds of A 3 BC-type materials (A = Sc, Y, La, or Lu; B = Al, Ga, In, or Tl), totaling to 16 compounds, were examined in this work.Among these materials, few Y and Sc-based materials have been explored for topological triple and Dirac nodes in some previous studies [41,42].However, a detailed topological study of these materials for HODM phases has not been performed yet.

Computational methods
We performed first-principles calculations within the framework of density functional theory using the projector-augmented-wave method, as implemented in the Vienna ab initio simulation package [43,44], to examine the ground state properties of investigated compounds.The kinetic energy cutoff was set to 500 eV.The atomic positions and lattice constants were relaxed until the residual force met the convergence criteria of 10 −3 eV atom −1 under the Perdew-Burke-Ernzerhof (PBE) [45] functional.A 10 × 10 × 10 Monkhorst-Pack grid was used in our computations.Electronic structure calculations were performed using both PBE as well as Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [46], including SOC.Since the band gap of PBE functional is often underestimated [47], we have attempted to improve the band structure using the hybrid HSE06 functional approach.Thus, both PBE and HSE06 are used to compute the electronic structure calculations.For all the topological properties, such as Z 4 topological invariant, surface states, and surface arc, we have utilized the HSE06 results.Phonon spectra were computed using the density functional perturbation theory method as implemented in the Phonopy code [48].We have used a 2 × 2 × 2 supercell to compute the force constants.The Z 4 topological index was examined from the parity of occupied bands at the time-reversal invariant momenta (TRIM) points [13].Further, topological surface spectra were calculated using the combination of density functional theory methods and maximally localized Wannier functions-based tight-binding models, as implemented in Wannier90 [49][50][51] and WannierTool [52] packages.Hinge states were examined by creating a rod geometry with a finite length along the a direction (11 cell size) and a semi-infinite nature along the b direction while preserving the periodicity along the c direction with the help of PythTB program [53] and WannierTool package [52].The irreducible representations were identified using irvsp code [54].together with time-reversal symmetry (T).The 'A'-type atoms occupy the faces of the cubic structure, 'B'-type atoms are placed at the corner, and the 'C' atom is at the center, enabling an octahedral bonding between C and B-type atoms.The optimized lattice parameters for all the 16 investigated compounds are summarized in supplemental table S1 [55].The optimized lattice parameter for Y 3 InC is 4.92 Å, which is very close to the experimental value 4.90 Å [40].Our results for other investigated compounds also agree with the available experimental and other theoretical reports [56][57][58][59][60][61][62] (see supplemental table S1 [55]).The bulk and (001) surface Brillouin zones of cubic A 3 BC structure are shown in figure 1(c).In the present study, we focus on an interesting candidate, Y 3 InC, as the representative material.Thus, the succeeding discussion will describe the structural, electronic, and topological properties of Y 3 InC.To examine the dynamical stability of Y 3 InC, we performed phonon dispersion calculations.A total of fifteen phonon modes are found, among the lowest, three phonon modes originated from the acoustic branch, and the remaining are optical phonon modes.The absence of imaginary phonon modes confirms the dynamical stability of Y 3 InC, as shown in figure 1(d).The dynamical stability of the remaining compounds is evident from the phonon dispersion plots presented in supplemental figure S1 [55].

Electronic and topological properties
Upon establishing the stability of these compounds, we performed band structure calculations under PBE and HSE06 functionals to achieve a deep understanding of the electronic structure properties.The computed band structures of all the compounds under PBE functional are shown in supplementary figure S2 [55].In figures 2(a) and (d), we provided the band structures of representative bulk Y 3 InC without and with SOC under hybrid HSE06 functional.To get a better understanding of the electronic structure of Y 3 InC in the presence and absence of SOC, we have carried out a detailed orbital analysis of the same.It is evident that the 'd' states of Y and In dominate near the Fermi level.Without SOC, as shown in figure 2 Notably, the bands at Γ high symmetry point possess a triply degenerate phase.Subsequently, the application of SOC further split the threefold d-t 2g bands above the Fermi level (6-fold when spin is considered) into fourfold and twofold bands.The twofold d-t 2g bands are found above the fourfold d-t 2g bands, as shown in figure 2(e).In the upcoming session, we will describe the possible topological features that originated due to crystalline symmetry and SOC.It is well known that crystalline symmetries play a crucial role in characterizing the topological features of materials.Without SOC, the topological characteristic of Y 3 InC belongs to the triple point phase.Triple degenerate points (TP) in semimetals/metals represent unique quasiparticles with no counterparts in higher-energy physics and are crystal symmetry-mediated topological excitations [63].As shown in figure 2(a), the band degeneracy at Γ near the Fermi level involves three bands, stemming from the combination of one non-degenerate and a double degenerate band.The bands involved in the triple-point degeneracy at Γ high symmetry point near the Fermi level are bands 25, 26, and band 27.The irreducible representation that protects this band degeneration is GM 5+ , which exhibits a three-dimensional character.Further, it is noticed that, along Γ − X, these bands split into the double degenerate bands, including bands 25 and 26 with irreducible representation DT 5 and single band 27 with irreducible representation DT 3 .The band dispersion around this triple point is found to be parabolic, which might originate from the localized nature of d-orbitals, enabling a quadratic triple point (QTP) nature at Γ high symmetry point.A three-dimensional band representation of QTP near the Γ high-symmetry point is exhibited in figure 2(c).In contrast, the triple band degeneracy along Γ-X (and other equivalent symmetry directions) exhibits a tilted linear band dispersion (see figure 2(c)).The vital point to note here is that in the absence of SOC, the bulk Y 3 InC hosts a multi-triple point phase (a total of seven) that are symmetry symmetry-mediated.
As we discussed earlier, the band degeneracy of Y 3 InC significantly alters when SOC is considered.At Γ high symmetry point, SOC-induced fourfold degenerate band can be seen with a parabolic curvature enabling a quadratic Dirac point (QDP) near the Fermi level (see figure 2(d)).The observed irreducible representation for constituent bands (band 49 to band 52) is GM 10 , which is four-dimensional, enabling a fourfold degenerate point at Γ.
From the Γ point, these bands split into twofold bands along X direction, with irreducible representation DT 6 around the point (0.25, 0, 0).There exists a small gap (around 0.01 eV) between the fourfold QDP and the above twofold band (see figures 2(d) and (e)).A three-dimensional band representation of QDP near the Γ high-symmetry point is shown in figure 2(f).Notably, SOC causes the origination of twin Dirac nodes along Γ-X high symmetry direction near the Fermi level.Dirac points are marked as DP1(around −0.02 eV) and DP2 (around −0.06 eV) in the inset of figure 2(d).A band dispersion around these Dirac points is exhibited in figure 2(f).The band structures (with SOC under both PBE and HSE06 functionals) for all the remaining compounds are provided in supplemental figures S2 and S3 [55].Except for Sc/Y-based Al/Ga compounds, all the other materials with significant SOC effect host twin Dirac cones.Overall, the SOC effect on Y 3 InC turned the triple point phase into a twin Dirac phase.Also, we noticed twelve linear Dirac points and one Dirac phase with the parabolic feature in bulk.
Subsequently, we identified the topological characterization of investigated compounds by computing the Z 4 topological invariant.We have examined the possibility of higher-order topological phases in Y 3 InC and its sister compounds by computing the Z 4 topological invariant.The centrosymmetric nature of Y3InC allows the characterization of Z 4 topological invariant from the parity of occupied bands [13] using the following equation: where n k+ and n k− represent the number of occupied states with parity ±1 at different TRIM points.The value of the Z 4 topological parameter helps to identify various topological materials, such as strong TI, trivial insulators, and higher-order topological phase.Here, the topological invariant Z 4 = 2 indicates the higher-order topological state [13].In table 1, we summarize the value of Z 4 topological invariants of all the examined compounds.The Z 4 = 2 value for all the studied compounds indicates the higher-order topological hinge states in these materials together with their bulk Dirac cone nature.Our examined bulk topological characterizations of Y 3 InC confirm the existence of topological surface and hinge states.Surface states are one of the vital characteristics of topological materials.In this section, we demonstrate the (001) surface states of Y 3 InC without and with the SOC effect.Firstly, let us describe the surface states without considering the SOC effect.In figures 3(a)-(c), we show the surface states along the, X-Γ-X, Ȳ-Γ-Y and M-Γ-M high symmetry paths on the (001) surface.As discussed earlier, Y 3 InC hosts a triple-point phase consisting of quadratic and linear triple points without SOC.A topological surface state, marked as surface state SS 1 , connects the bulk QTP is seen in figure 3(a) around 0.23 eV above the Fermi level.Notably, the surface state SS 1 is found to start from Γ, pass through the boundary X high-symmetry point, and then return to Γ.A similar surface state can be seen in figure 3(b), as represented by SS 3 .In addition, the surface state marked as SS 2 below the Fermi level in figure 3(a) represents the topological surface state related to the triple point aligned along Γ-X high-symmetry line.As we can see from figure 3(a), the SS 2 surface state connects two high-symmetry points, Γ and X, through the triple point (TP).Similarly, the topological surface state SS 4, marked in figure 3(b), joins the Γ and Y high-symmetry points through the TP located along the Γ-Y path.In contrast, one topological surface state was observed along the M-Γ-M (SS 5 in figure 3(c)) connecting points in the bulk, and the surface state SS 6 stems from the respective crystalline symmetries and is not a topological surface state.Yet another interesting feature related to the surface state is the surface arcs, which represent the projections of these surface states to specific k-planes.Subsequently, we have examined the surface arc on the k x -k y plane at the Fermi level, as shown in figure 3(d).The sharp red contour, as represented by the arc in figure 3(d), exhibits the surface arc due to the topological surface state SS 1 (see figure 3(a)) and SS 2 (see figure 3(a)) near the Fermi level, which is also related to the TP located near the Fermi level.
Next, we discuss the (001) surface states of Y 3 InC in the presence of SOC effect.It is shown in figure 3 that Y 3 InC exhibits a QDP at Γ and twin Dirac nodes along the high-symmetry directions.In figures 3(e)-(g), we provide the (001) surface spectra of Y 3 InC along X-Γ-X, Ȳ-Γ-Y and M-Γ-M high symmetry paths, respectively.Similar to the case without SOC, topological and crystalline symmetry-mediated surface states are observed in the presence of SOC.The computed surface states marked as SS 1 + and SS 1 − in figure 3(e) represent the topological surface states connected to the QDP around 0.25 eV above the Fermi level, which starts from the Γ point and then returns to the same point through the boundary.In figure 3(f), SS 3 + and SS 3 − represent similar topological surface states related to QDP.In contrast, the surface states named SS 2 + and SS 2 − are related to the two closely aligned Dirac nodes in the Γ-X high-symmetry direction.Both of these surface states start from the Γ point, pass through the Dirac points and boundary, and then finally return to Γ. Similarly, figure 3(f) shows the topological surface states for DPs (labeled as SS 4 + and SS 4 − ).Another topological surface state can be seen in figure 3(g) (marked as SS 5 + , SS 5 − ) along M-Γ-M high symmetry path.However, the surface states marked as SS 6 + and SS 6 − in figure 3(g) are crystalline-symmetry mediated.To understand the surface arcs derived from the topological Dirac nodes near the Fermi level, we have computed the k x -k y projection of surface states at the energy level E-E F = −0.05eV, as shown in figure 3(h).Two surface arc contours, marked arc 1 and arc 2 , represent the two surface states due to SOC-derived topological surfaces (SS 2 + and SS 2 − ).Based on these findings, Y 3 InC indeed exhibits rich surface states that connect multiple Dirac nodes on the (001) surface.Supplementary figure S4 [55] summarizes the surface states (with SOC) for the remaining compounds.
It is worth noting that antiperovskite materials might show local structural distortions as a function of temperature, which might alter the electronic structure and, hence, the topological features.Since the topological features in Y 3 InC and sister compounds are symmetry-protected, these structural distortions due to increased temperature might destroy the observed Dirac phases in these materials.

Hinge states
Hinge states, which are key features of higher-order topological materials, are edge states that connect two surfaces.For Y 3 InC, the examined bulk Z 4 topological invariant hints at the possible bulk-hinge correspondence.Here, the [0 0 1] direction can be considered as a hinge direction where two mirror-invariant surfaces (100) and (010) meet [10].To examine the hinge state and relation between the bulk Dirac nodes, we computed the electronic spectra of a rod geometry (see figure 4(a)) with finite size along the a-direction, semi-infinite along the b-direction, and periodic along the c-direction.Here, we expect the hinge states along the period c direction.Further, figure 4(b) shows the hinge spectrum along the Z-Γ-Z direction.From the color bar, it is evident that a significant contribution of the hinge edge state can be seen near the Fermi level.In addition, the hinge correspondence to the bulk Dirac nodes is evident near the Fermi level, as labeled by HS 1 and HS 2 in figure 4(b).It is observed that the existence of twin Dirac nodes is along Γ-Z high symmetry path (which is similar to Γ-X high symmetry line) in the bulk Y 3 InC.Here, the hinge states represented as HS 1 and HS 2 in figure 4(b) are connected to the bulk Dirac nodes located along Γ-Z.Moreover, a closer look at the hinge state HS 1 along Z-Γ exhibits a hinge arc connected to the HS 1 located along Γ-Z.A similar nature is observed in the case of hinge state HS 2 .Our results confirm the presence of a hinge arc that connects the bulk Dirac points and propose Y 3 InC as a fertile candidate for a higher-order Dirac state.Overall, Y 3 InC is a rich topological material with several interesting properties, which are summarized in the schematic diagram in figure 5 highlighting the multiple Dirac nodes in bulk and surface of Y 3 InC along with the hinge states.3 BC (A: Sc, Y, La, Lu; B: Al, Ga, In, Tl) family This section briefly discusses the structural, electronic, and topological properties of the remaining compounds, A 3 BC (A: Sc,Y, La, Lu; B: Al, Ga, In, Tl) in this study.The optimized lattice parameters of all these compounds are provided in supplementary table S1 [55].The electronic structure of Sc, Y, and Lu-based compounds exhibits a similar band profile under HSE06 with SOC, whereas La-based compounds show a slightly different band profile (see figure in S3 [55]).The QDP around the Γ is noticed above the Fermi level for Sc, Y, and Lu-based compounds, whereas for La-based compounds, the fourfold degenerate band is observed below the Fermi level with a slightly different band curvature.Moreover, for all the compounds, we have noticed the presence of two Dirac nodes along the Γ-X high symmetry line.The computed Z 4 topological invariant (shown in table 1) for all these compounds shows the possible higher-order topological features in these materials.To capture the topological features on the surface of these materials, we have computed the surface spectra along X-Γ-M path on the (001) surface (see supplementary figure S4 [55]).Interestingly Sc and Y-based compounds exhibit fertile surface states similar to Y 3 InC, whereas the surface spectra of La and Lu-based compounds are found to be merged into the bulk band states.Overall, all the investigated compounds exhibit a possibility of hosting a higher-order Dirac phase.

Other materials in the A
Finally, we have explored the higher-order topological phase of the antiperovskite compounds within the framework of the first principles calculations.We used parity-based Z 4 bulk topological invariant to predict the possibility of a higher-order phase in these systems since these compounds host inversion symmetry.To elucidate the bulk-boundary correspondence, we constructed a nanoribbon geometry to highlight the hinge states, similar to previous reports [8,20].We have created a nanoribbon geometry (which is one-dimensional) with finite length along the x-and semi-infinite along the y-direction, and infinite along the z-direction, which is parallel to the C 4 symmetry axis.This particular nanoribbon geometry was chosen to observe the one-dimensional states that correspond to the bulk Dirac states noticed along the Γ − Z high symmetry path.In bulk, along Γ-Z high symmetry direction, Y 3 InC exhibits twin Dirac nodes, which are located very close to the Fermi level, with slight differences in energy and k-point position.In figure 4(b), the color code helps to identify the contribution from the atoms located at the hinge of the structure, and the high intensity near the points (we have marked them as hinge states HS 1 and HS 2 ) helped to capture the hinge states of the system.Yet another vital point that helped to confirm the hinge states (HS 1 and HS 2 ) is the arc-like states that, starting from the Dirac point located along the +Z direction, connect the Brillouin zone center and extend to the other Dirac point in the −Z direction.This feature is reported in several previous works [8,20].

Conclusion
To conclude, we demonstrated the topological higher-order Dirac phase in Y 3 InC.Our first-principles calculation revealed the coexistence of gapless bulk, surface, and hinge states in the system.The detailed bulk electronic structure calculations exhibited the SOC driven twin-Dirac nodes in the bulk phase of Y 3 InC.The topological higher-order Dirac phase was confirmed by the computed Z 4 topological invariant.A topological hinge arc that connects bulk Dirac nodes in Y 3 InC was demonstrated using the edge state calculations of a rod geometry.Our results show that the experimentally synthesizable Y 3 InC and fifteen other anti-perovskite carbides are potential candidates for HODMs that will further attract future experimental and theoretical investigations.

3. 1 .
Crystal structure and stabilityThe A 3 BC-type (A = Sc, Y, La, or Lu; B = Al, Ga, In, or Tl) compounds crystallize in cubic antiperovskite structure with space group Pm-3 m (SG No: 221), as shown in figure1(a).A top view of the crystal structure is provided in figure1(b).This structure belongs to the O h point group, which offers multiple rotation symmetries (C 2 , C 4 , C 3 ), two inequivalent mirror symmetries (M z , M xy ), and an inversion symmetry (I)
(b), d-t 2g states, including 'd xy ' , 'd xz ' and 'd yz ' are dominating around Γ high symmetry point, above the Fermi level.The key orbitals below the Fermi level around Γ high symmetry point are d-e g states (d x2− y 2 and d z2 ) of Y and In.

Figure 2 .
Figure 2.For bulk Y3InC (a) Band structure without SOC under HSE06 functional (b) Orbital projected band structure without SOC under HSE06 [blue and cyan color spheres represent d-t2g (dxy, dxz, and dyz) and d-eg (dx 2−y2 and dz2) states respectively].Note that both d-t2g and d-eg states are contributions from Y and In.(c) Three-dimensional representation of band dispersion around quadratic triple point (QTP) and triple point (TP).(d) Band structure with SOC under HSE06 functional, and inset figure represent the zoom in around Γ high symmetry point exhibiting QDP and DPs.[Note that a small gap (around 0.01 eV) exists between the QDP and the above two-fold bands] (e) Orbital projected band structure with SOC under HSE06 [blue and cyan color spheres represent d-t2g (dxy, dxz, and dyz) and d-eg (dx 2−y2 and dz2) states, respectively].Note that both d-t2g and d-eg states are contributions from Y and In.(f) Three-dimensional representation of band dispersion around quadratic Dirac point (QDP) and Dirac points (DP1 and DP2).

Table 1 .
Computed Z4 topological invariant under HSE06 functional with SOC for the bulk A3BC-type materials (A = Sc, Y, La, or Lu; B = Al, Ga, In, or Tl).