Glide Symmetry Protected Higher-Order Topological Insulators from Semimetals with butterfly-like Nodal Lines

Most topological insulators discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the gapped TI phases and the gapless TSMs. Recently, we have proposed a new family of TSMs in time-reversal invariant {\it spinless} systems, which host butterfly-like nodal-lines (NLs) consisting of a pair of identical concentric intersecting coplanar ellipses (CICE). In this Communication, we unveil the intrinsic link between this exotic class of nodal-line semimetals (NLSMs) and a $\mathbb{Z}_{4}$ = 2 topological crystalline insulator (TCI), by including substantial SOC. We demonstrate that in three space groups ({\it i.e.} $Pbam$ (No.55), $P4/mbm$ (No.127) and $P4_2/mbc$ (No.135)), the TCI supports a fourfold Dirac fermion on the (001) surface protected by two glide symmetries, which originates from the intertwined drumhead surface states of the CICE NLs. The higher order topology is further demonstrated by the emergence of one-dimensional helical hinge states, indicating a new higher order topological insulator protected by a glide symmetry.


Abstract
Most topological insulators discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the gapped TI phases and the gapless TSMs. Recently, we have proposed a new family of TSMs in time-reversal invariant spinless systems, which host butterfly-like nodal-lines (NLs) consisting of a pair of identical concentric intersecting coplanar ellipses (CICE). In this Communication, we unveil the intrinsic link between this exotic class of nodal-line semimetals (NLSMs) and a Z 4 = 2 topological crystalline insulator (TCI), by including substantial SOC. We demonstrate that in three space groups (i.e. P bam (No.55), P 4/mbm (No.127) and P 4 2 /mbc (No.135)), the TCI supports a fourfold Dirac fermion on the (001) surface protected by two glide symmetries, which originates from the intertwined drumhead surface states of the CICE NLs. The higher order topology is further demonstrated by the emergence of onedimensional helical hinge states, indicating a new higher order topological insulator protected by a glide symmetry. topological phase and a gapless TSM. Nevertheless, similar studies for NLSMs with complex NL configurations remain deficient. In this work, we unveil the intrinsic link between an exotic class of nodal-line semimetals (NLSMs) and a Z 4 = 2 topological crystalline insulator (TCI). The new type of NLSM has been proposed in Ref. [37] recently in spinless systems, which hosts a butterfly-like NL consisting of a pair of concentric intersecting coplanar ellipses (CICE) residing on a plane in k space as shown in Fig.1a.
In this Communication, we include the P bam (No.55) symmetry-invariant SOC in a minimal tight-binding (TB) model which exhibits CICE in Ref. [37], and (i) demonstrate that the CICE act as the origin of a TCI protected by two glide symmetries. With substantial SOC, as shown in Fig.1b, the CICE become anticrossing, thus driving a phase transition from NLSM to a TCI with Z 4 = 2 [38][39][40][41] due to the fact that the CICE are essentially sets of NLs stemming from the double-band-inversion (DBI). (ii) Consequently, the intertwined drumhead surface states (DSSs) on the (001) surface (wallpaper group (WG) pgg) stemming from the CICE nodal lines, evolve to the topological surface states (TSSs) with a fourfold Dirac fermion [42,43] protected by the two glide symmetries, corresponding to the TCI as shown in Fig.1b and Fig.2d, respectively. (iii) We further uncover the higher-order topology of the system featuring the 1D helical hinge states when the sample geometries are distinctively and properly selected. Moreover, it is the new HOTI protected by a nonsymmorphic glide symmetry.
(1) describe the pair of concentric elliptic NLs, the NL anisotropy, and the angle between the NLs (see details in [37]). Since the CICE are composed of two NLs, it is anticipated to observe a pair of DSS [28] intertwined on the (001) surface.
In order to determine its band topology, we implement the symmetry-indicator theory [38,40,41]. Crystals in the SG P bam (No. 55) are characterized by four symmetry indicators (SIs) [38,40,41], three Z 2 weak TI indices and one Z 4 index. The Z 4 index is defined as, is the number of occupied bands with parity + (−) at the TRIM points K. Due to the nonsymmorphic symmetries, bands are fourfold degenerate at all TRIM points except Γ. Besides, since inversion I anticommutes with the glide, G, or screw, S, symmetry operations (here S includes S y = 2 [010] 1 = {2 010 | 1 2 1 2 0}, and S x = 2 [100] 1 = {2 100 | 1 2 1 2 0}), at X, U, Y, and T, the parity of each four-fold degenerate state must be (+, +, −, −), which does not contribute to Z 4 . By enumerating the parity of the states at other TRIM points, we obtain Z 2,2,2,4 = (0, 0, 0, 2), corresponding to eight possible topological states [41]. To further narrow down the possible phases, we have calculated the mirror Chern numbers of M z and C m 001 0,π (m 001 0,π denotes the k z = 0, π mirror planes), following the method implemented in Ref. [16]. We find that C m 001 0,π = (2, 0). The corresponding Dirac surface states on the (010) surface are shown in Fig. 2b, where the relevant k points and the schematic locations of the Dirac cones are illustrated in Fig. 2a. Therefore, given that C m 001 0,π = (2, 0), there are two possible topological phases, which are listed in Table I = G x = {m 100 | 1 2 1 2 0}) . Note that the nontrivial characteristics of the bands agree with the analysis from the elementary band representations (EBRs) [44][45][46]. The physical EBRs for the 2a Wyckoff position [47][48][49] require that the parity of Γ, Z, S and R has the same sign. However, because of the double band inversion at S guaranteed by the CICE-NL, both valence and conduction bands violate the physical EBRs, suggesting the emergence of nontrivial topology.
To further determine the topological phase uniquely, we have also investigated the (001) surface bands, because the presence of topological surface states on the (001) surface excludes the scenario of S-protected TCI. The (001) surface bands are shown in Fig. 2c, where the (001) surface BZ and the corresponding high symmetry k points are displayed Fig. 2a.
obtain the fourfold Dirac fermions via SOC-induced splitting of the intertwined DSSs of the CICE TSM [37], where the degenerate dispersions alongS −X (Ȳ ) are guaranteed by G.
Consequently, the CICE-NL induced topological phase belongs to the G-protected TCI. In the following, we provide more physical insights on this TCI phase and the fourfold Dirac fermions.
In general, for time reversal symmetric systems, strong topological insulators (STIs) with a single band inversion at one TRIM point can be regarded as an elementary building block of the nontrivial insulating phase [15,40]. For each STI, the topological surface states of the (001) surface can be described by the Hamiltonian, h k = k x s 2 −k y s 1 [40]. The gapless feature of h k is protected by the TRS operator, T = −is 2 K, where K is the complex conjugation operator. For the current case, since there is a DBI at theS point, the induced TCI phase can be viewed as two copies of STIs. Accordingly, for the TSSs of the TCI, the only allowed TR invariant mass term takes the form, M = mµ 2 ⊗ s 3 , where µ 1,2,3 are the Pauli matrices acting on the two copies of h k , and m is constant. If M can be prohibited by any spatial symmetry Q, the anomalous gapless surface states will persist, indicating that the existing topology is protected by Q, which can be G x and G y , as derived below.
AtS, the eigenvalues of G x and G y are ±1. To preserve TRS, the only available representations are G x = µ 2 ⊗ s 1 and G y = µ 2 ⊗ s 2 , which in turn lead to the rotational symmetry about the z-axis C 2z (= G x × G y ) = −iµ 0 ⊗ s 3 . Obviously, M cannot survive with G x and G y , but is allowed by C 2z . Consequently, there exist representations for G x and G y to support the (001) TSSs atS, which is the fourfold Dirac fermion shown in Fig. 2d described by the where all g's and a's are real parameters.
Higher-order topological insulator protected by a glide symmetry. In addition to the topological surface states belonging to the (d − 1) bulk-edge correspondence, i.e., the fourfold Dirac fermion and the M z -protected surface states, the G x , G y and M z symmetries can give rise to higher order (d − 2) bulk-edge correspondence.
We have considered the nanorod geometry, shown in Fig. 3a, with open boundary conditions along the [011] and [011] directions, respectively, and periodic boundary conditions along [100]. We find that the nanorod can support two pairs of hinges modes along the intersection lines between the (011) and (011) surfaces and the (011) and (011) surfaces, respectively. None of the above surfaces hosts gapless surface states, since there is no TCI phase supporting them. As discussed in Ref. [22], the hinge modes formed by the intersection of the (011) and (011) facets are generated via bending the (001) surface along the [001] direction, which, however, preserves the G y symmetry for the hinges and the entire crystal. The original pair of Dirac cones forming the fourfold Dirac fermion on the (001) surface become gapped with opposite mass terms and reside on the (011) and (011) surfaces, respectively (red and black massive Dirac cones in Fig. 3a). Hence, the surface insulating phases that reside on the two facets differ by an odd Z 2 index, leading to the emergence of an odd number of helical hinge modes on the domain wall between the two facets. Similarly, the hinge modes along the intersection of the (011) and (011) facets are generated via bending the (010) surface along the [001] direction, which preserve the M z symmetry. Fig. 3b shows the band structure of the nanorod along the k x symmetry direction where the two pair of hinge modes are denoted in red and cyan, respectively. The distribution in real space of the two types of hinge modes along the [100] direction are displayed in Fig. 3c with the corresponding colors. For the purpose of clarity, on-site potentials V = 0.2 eV are added on the hinges formed by the {(011), (011)} and {(011), (011)} facets to better differentiate the two hinge states in energy. The constraint imposed by the glide symmetry G y allows two possible topologies of hinge bands [52], the hourglass connectivity and the analogue of the quantum spin Hall (QSH) effect (see Fig. 3d). From the the hinge modes in cyan displayed in Fig. 3c, we find that the hinge band connectivity is the analogue of the quantum spin Hall effect (bottom panel of Fig. 3d).
In summary, inclusion of SOC in the model Hamiltonian describing our recently proposed a family of butterfly-like CICE NLs in SG P bam (No. 55) [37] unveils intrinsic connection of the CICE NLSM and the TCI protected by two glide symmetries. The SOC drives the TSM to a Z 4 = 2 TCI with higher order topology, supporting in turn a fourfold Dirac fermion on the (001) surface protected by two coexisting glide symmetries of WG pgg. As a candidate material of this type of TCI, Sr 2 Pb 3 (SG P 4/mbm No.127) has been studied in [42].
Moreover, its higher order topology is corroborated for the first time by the emergence of 1D hinge states protected by glide symmetry. This intriguing TCI phase provides a platform for exploring exotic physics, such as the electron transport and thermoelectric effect on the surfaces/hinges. Our proposed glide-protected HOTI may have important implications on the emergence of Majorana zero modes via proximity of the HOTI to a superconductor [53], which exhibit distinct features compared to those in TI/SC heterostructures. Finally, our The artificial system can be designed as layered structure, where these conditions can be satisfied by tuning the interlayer hopping or SOC parameters, such as γ, λ 10,13 and ζ 233 . [6] Roy, R.
Topological phases and the quantum spin Hall effect in three dimensions.  [011] [011] [100] [010] [001] [011] [011] Hourglass Type Hamiltonian where τ 0 , σ 0 and s 0 are 2 × 2 identity matrices. The matrices τ i act on the p z and d xy orbitals, σ i on the two sublattices and s i on electron spin. Note that the invariance of the hamiltonian under a space-group operation g = {R|v} means that Similarly, the invariance of the hamiltonian under the anti-unitary time-reversal operation T = T K can be expressed as where T = −iτ 0 ⊗ σ 0 ⊗ s 2 and K is ordinary complex conjugation.

FROM ATOMIC ORBITALS TO EBRS
Roughly speaking, a band representation [S1, S2] can be understood as the collection of electronic bands generated by placing a set orbitals at different positions in the unit cell, in such a way that the set is closed under all the symmetries of the crystal. A band representation obtained by placing orbitals at a maximal Wickoff position w that transform according to an irreducible representation τ of the site symmetry group G w is an elementary band representation (EBR) and is denoted by τ ↑ G| w . Any band representation that is not elementary can be written as a sum of EBRs [S3].
As our system respects time reversal symmetry (TRS), we are interested only in real and physically irreducible representations [S4]. Following the Bilbao Crystallographic Server [S5, S6] (BCS) conventions, a physically irreducible representation is denoted as the sum of two irreducible representations, which can be the same or different depending on their reality type [S4], with the + sign between the components of the pair omitted. For instance, Γ + 1 in Eq. (S5) below is a real irreducible representation whereas R + 1 and R + 2 are complex conjugate irreducible representations that together constitute the real, physically irreducible In order to find the EBR contents of the model it is enough to know the transformation properties of the atomic orbitals s and p z . The site-symmery group 2/m has four single-valued real irreducible representations (irreps), A g , A u , B g , B u , and two double-valued u . The spinless orbitals d xy and p z belong the single-valued irreps A g and A u respectively. As a consequence, in the absence of SOC the spectrum of the TB hamiltonian (5) is described by the sum of two EBRs induced from the 2a Wyckoff position, namely Then the application BANDREP at the BCS gives all the irreps at the different points in the Brillouin Zone (BZ). For the high symmetry TRIM points the result is where the numbers in parentheses give the dimensions of the irreps, which coincide with the degeneracies of the corresponding bands. This means, for instance, that the hamiltonian for spinless electrons in Eq. (5) of the paper necessarily has four non-degenerate bands at the Γ- , etc. Note that in order to obtain this information one does not have to diagonalize the hamiltonian H 0 (k), as the result depends only on the orbital contents of the model. Note also that the degeneracies of the bands at the high symmetry points are independent of the parameters of the model, barring accidental degeneracies if some parameters are set to zero or otherwise fine-tuned.
Electron spin and SOC can be easily incorporated by noting that spin transforms according to the physically irrep 1Ē g 2Ē g of the site-symmetry group 2/m [S4]. Then, the double-valued irreps for s and p orbitals are obtained by taking the products of the respec-tive single-valued irreps with the spin representation and this implies that the spectrum of the total hamiltonian H(k) = H 0 (k) + H SOC (k) is described by the sum of two EBRs Then BANDREP immediately gives the irrep contents and the structure of the spectrum for the total hamiltonian H(k). For the high symmetry TRIM points the result is

(S9)
Then any band representation can be characterized by a vector that tells how many times each physically irreducible representation appears at every high symmetry point. For instance, for the two EBRs in Eq. (S8) the vectors are and for the bands of the total hamiltonian H(k) u ↑ G| 2a = (2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2). (S11) The topology of an isolated subset of bands can be characterized by a set of symmetrybased indicators [S7]. In order to determine the indicators, one tries to write the vector giving the irreps for the subset of bands as a linear combination of all the vectors for the EBRs of the space group. There are several possibilities: 1. If the vector for the subset of bands can be written as a linear combination with positive integer coefficients, then all symmetry-based indicators (SI) for the subset vanish. The subset may still be topologically non-trivial, but all the eigenvalue-based topological invariants will be zero. In order to ascertain that the subset is non-trivial, one should use other tools, such as Wilson loops.
2. If the vector for the subset of bands can be written as a linear combination with integer coefficients, but some of the coefficients are necessarily negative, then the subset exhibits non-trivial fragile topology.
Note that EBR is a 12 × 8 non-square matrix, where 8 is the number of EBRs and 12 the number of irreps. Then, for any isolated subset of bands characterized by a 12-component vector B = (B 1 , . . . , B 12 ) giving the numbers of irreps, we will have to solve the linear set of equations where the solution X = (X 1 , . . . , X 8 ) gives the coefficients of the linear combination of the 8 EBRs. The presence of non-integer the coefficients in {X 1 , . . . , X 8 } implies the existence of symmetry-based indicators.
The possible existence of non-integer solutions and hence the existence of SI and nontrivial topological indicators may be predicted by diagonalizing the EBR matrix. As EBR is a non-square matrix with integer coefficients, one has to use a special diagonalization procedure known as Smith decomposition: where L and R are unitary integer square matrices of dimensions 12×12 and 8×8 respectively, and D is the following 12 × 8 diagonal matrix (S15) The number of non-zero diagonal elements is the rank of the group, equal to five in the case of SG55 with SOC and TRS. One can show that all the eigenvalues must be positive integers. If some of the eigenvalues{n 1 , n 2 , · · · } are greater than one, then the group of SI is Z n 1 × Z n 2 × . . .. In this case the SI group is Z 2 × Z 4 , in agreement with (2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2) − (2, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1) = (0, 2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1) (S16) and its SI are Adding the topological invariants of the two complementary bands and remembering that Z n indicators are defined mod n, gives for the total 8-band system as expected. Note that the conduction and valence bands are always complementary.

PHASE DIAGRAM FOR THE TB MODEL
The 18 cases in Table S1  δ 0 , δ ± = α + β ± γ , ξ ± = (λ 10 ± λ 13 ) 2 + ζ 2 233 (S21) according to Table S3. Then, as long as Z 4 is even, it is easy to choose the parameters in such a way that the desired ordering is obtained and the valence and conduction bands are separated by a gap throughout the BZ. For Z = 1 or 3, the bands must cross at a high symmetry line or plane and the phase is semimetallic [S9].  Table S1. The middle bands correspond to case 12 in Table S1 and Z 4 = 3; there is a band crossing at the T Z line and the phase is semimetallic. In the last example, at the bottom of Fig. S1, all the SI vanish and the irrep contents of the valence and conduction bands coincide with the EBRs 1Ē g 2Ē u ↑ G| 2a respectively. There are no band inversions and the system is in an atomic limit. i x G P E T j Z 8 t F 5 + x i F G t G E 2 T L 9 + 9 s x k N j Z m F g M y G n i V n 1 F s P / v F 5 K o 1 s / k 1 G S E k b C R q w 3 S h W j m C 3 6 s q H U K E j N L H C h p d 2 S i Q n X X J C 9 S s n W 9 1 b L r k P 7 q u q 5 V a 9 x X a n d i x G P E T j Z 8 t F 5 + x i F G t G E 2 T L 9 + 9 s x k N j Z m F g M y G n i V n 1 F s P / v F 5 K o 1 s / k 1 G S E k b C R q w 3 S h W j m C 3 6 s q H U K E j N L H C h p d 2 S i Q n X X J C 9 S s n W 9 1 b L r k P 7 q u q 5 V a 9 x X a n d i x G P E T j Z 8 t F 5 + x i F G t G E 2 T L 9 + 9 s x k N j Z m F g M y G n i V n 1 F s P / v F 5 K o 1 s / k 1 G S E k b C R q w 3 S h W j m C 3 6 s q H U K E j N L H C h p d 2 S i Q n X X J C 9 S s n W 9 1 b L r k P 7 q u q 5 V a 9 x X a n d i x G P E T j Z 8 t F 5 + x i F G t G E 2 T L 9 + 9 s x k N j Z m F g M y G n i V n 1 F s P / v F 5 K o 1 s / k 1 G S E k b C R q w 3 S h W j m C 3 6 s q H U K E j N L H C h p d 2 S i Q n X X J C 9 S s n W 9 1 b L r k P 7 q u q 5 V a 9 x X a n d The only band inversion is at point S, Z 4 = 2, case 4 in Table S1. Middle: Band inversions at the R, S and Z points, Z 4 = 3, case 12. This is a semimetallic phase, note the band crossing at the T Z line. Bottom: This phase has no SI and the irrep contents of the valence and conduction bands are those of EBRs, case 1 in Table S1.