Introduction

The discovery of the quantum spin Hall effect (QSHE)1,2,3 and topological insulators (TIs)4,5,6 which are protected by time-reversal symmetry (TRS), has indicated that symmetry plays a crucial role in classifying the topology of free fermion states7,8,9. Subsequently, the concept has been generalized to spatial symmetries in crystalline systems. For instance, topological crystalline insulators (TCIs)10 are protected by other space-group symmetries (\({{{\mathcal{Q}}}}\))11,12, such as mirror13 and rotational symmetries14,15,16. Consequently, such type of systems are known to harbor symmetry-protected topological (SPT) phase17.

In free fermion systems, SPT insulators harbor a central paradigm referred to as the bulk-boundary correspondence18. A d-dimensional bulk with gapped excitations hosts anomalous gapless, topologically nontrivial boundary states in lower (d − 1) dimensions4,19. Recently, a higher-order bulk-boundary correspondence has been uncovered in types of TCI, which exhibit a gapped (d − 1)-dimensional boundary but a gapless (d − 2)-dimensional boundary14,20,21,22. Hence, this type of TCIs are dubbed higher-order topological insulators (HOTIs)14,20,21,22,23,24,25,26,27. In general, an nth order topological insulator describes a TCI in d-dimensions having symmetry protected (d − n)-dimensional gapless boundary states, but gapped otherwise, when the sample geometry is properly selected, being compatible with \({{{\mathcal{Q}}}}\). HOTIs protected by various symmetries have been studied, such as \({{{{\mathcal{C}}}}}_{4}^{z}{{{\mathcal{T}}}}\), the mirror symmetry22, and the inversion symmetry23, respectively.

In contrast to the gapped topological phase, a topological semimetal (TSM) has gapless bulk band structures, which are characterized by the topologically robust band-crossings manifolds between occupied and unoccupied bands in momentum space. Among them, nodal-line semimetals (NLSMs)28,29, which harbor one-dimensional (1D) nodal lines (NLs), possess the highest variability. NLSMs with NLs integrated in various configurations have been studied under the assumption that the spin-orbit coupling (SOC) is negligible or absent, e.g., a chain link30,31,32, a Hopf link31, and a knot33.

However, from another perspective, it is intuitive to raise the question whether additional topology could be unearthed when these intricate degenerate links are gapped out by substantial SOC. The intrinsic link between gapped TIs or TCIs and gapless TSMs is essential to trace the origin of the topology of the insulating phase due to the band inversions and the evolution of the surface states. It has been known that a \({{\mathbb{Z}}}_{2}\) strong TI can be realized from NLSMs with a single nodal ring when SOC is included34,35,36, and a HOTI from NLSMs with monopole nodal lines24, which represent the intrinsic link between a gapped 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 \({{\mathbb{Z}}}_{4}\) = 2 topological crystalline insulator (TCI). This type of NLSM has been discovered and reported 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.

Fig. 1: Phase transition from a NLSM to a TCI, crystal structure, and band structures.
figure 1

a Brillouin zone (BZ) of bulk and the projected (001) surface in SG Pbam (No. 55), with the high-symmetry points. The CICE nodal lines are on the xy-plane centered at S point. b Schematic band structures demonstrating the SOC-driven transition from a NLSM exhibiting the CICE NLs37 (left panel) to a TCI (right panel). Gray denotes the bulk states, and the magenta, red and blue indicate the (001) surface states, i.e., the interwined DSSs of the NLSM (left panel), and the TSSs of the TCI with two intertwined Dirac cones (right panel). c Orthorhombic crystal structure of the lattice model in Eqs. (1) and (2), consisting of a lattice with two sublattices A (in blue) and B (in gray). d Band structure of the model with and without SOC, respectively.

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In this paper, we include the Pbam (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 \({{\mathbb{Z}}}_{4}=2\)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 fermion42,43 protected by the two glide symmetries, corresponding to the TCI as shown in Figs. 1b and 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, which indicates the discovery of the three-dimensional (3D) HOTI protected by a nonsymmorphic glide symmetry.

Fig. 2: Topological surface states and the evolution from DSSs to TSSs.
figure 2

a Brilliouin zones for the bulk, the projected (001) and (010) surfaces, on which the predicted positions of the fourfold Dirac fermion and Dirac cones are shown schematically. Surface band spectrum of the b (010) and c (001) surfaces along the high-symmetry k paths, where the fourfold Dirac fermion emerges at \(\bar{S}\) on the (001) surface and a Dirac cone lies along the \(\tilde{{{\Gamma }}}\) − \(\tilde{X}\) direction of the (010) surface, respectively. d Schematic of the SOC-driven transition of the (001) surface states from the intertwined DSSs stemming from the CICE NLSM37 (left panel) to the fourfold Dirac fermion at \(\bar{S}\) of the TCI (right panel). e The z-directed Wilson loop along the high symmetry directions for occupied states hosting positive (+sector, red) and negative (−sector, blue) surface glide eigenvalues, indicating a bulk topology of (χx, χy) = (2, 2).

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Results

The lattice model

The CICE can be sustained by two glide mirror symmetries and only nine space groups (SGs) are feasible to host it37. The minimal 4-band TB model for the spinless CICE NLSM in SG Pbam (No. 55) is a lattice, consisting of two sublattices denoted by A (gray) and B (blue) which occupy the 2a Wyckoff position at rA = (0, 0, 0) and \({{{{\bf{r}}}}}_{B}=(\frac{1}{2},\frac{1}{2},0)\) in the unit cell (see Fig. 1c for the structure). There are two orbitals, pz and dxy, for each sublattice, described by the Pauli matrix σ and τ for the A and B sublattices, respectively. σ0 and τ0 are identity matrices. The SG Pbam (No.55) can be generated by the mirror symmetry, \({{{{\mathcal{M}}}}}_{z}=\{{m}_{001}| 000\}\), and the two glide-mirror symmetries, \({{{{\mathcal{G}}}}}_{x}=\{{m}_{100}| \frac{1}{2}\frac{1}{2}0\}\) and \({{{{\mathcal{G}}}}}_{y}=\{{m}_{010}| \frac{1}{2}\frac{1}{2}0\}\), normal to the [100] and [010] directions, respectively, accompanied by a translation of \([\frac{1}{2}\frac{1}{2}0]\). For a spinless system, employing the basis \({{\Psi }}={({p}_{z}^{A},{d}_{xy}^{A},{p}_{z}^{B},{d}_{xy}^{B})}^{T}\), the symmetry-constrained TB Hamiltonian is of the form,

$$\begin{array}{ll}{H}_{0}({\bf{k}})=[(\alpha \cos {k}_{x}+\beta \cos {k}_{y}+\gamma \cos {k}_{z})+{\delta }_{0}]{\tau }_{0}{\sigma }_{3}\\ + \cos \frac{{k}_{x}}{2}\cos \frac{{k}_{y}}{2}\cos {k}_{z}({\lambda }_{10}{\tau }_{1}{\sigma }_{0}+{\lambda }_{13}{\tau }_{1}{\sigma }_{3})\\ +\sin {k}_{z}({\lambda }_{32}{\tau }_{3}{\sigma }_{2})+\sin \frac{{k}_{x}}{2}\sin \frac{{k}_{y}}{2}\sin {k}_{z}({\lambda }_{12}{\tau }_{1}{\sigma }_{2}),\end{array}$$
(1)

where α, β, γ, and λij are hopping strength, and δ0 represents the chemical potential. As discussed in detail in ref. 37, the CICE emerge on the mirror plane (gray shaded area in Fig. 1a) centered at the high symmetry k point S = (π, π, 0) [R = (π, π, π)], under the conditions {αδS(R) < 0 ∩ αβ > 0 ∩ α ≠ β}, where δS,R = δ0 − (α + β ∓ γ). The various terms in Eq. (1) describe the pair of concentric elliptic NLs, the NL anisotropy, and the angle between the NLs (see details in ref. 37). Since the CICE are composed of two NLs, it is anticipated to observe a pair of DSS28 intertwined on the (001) surface.

Topological crystalline insulator

We consider the effect of SOC, and the minimal TB model contains eight spinful bands. The minimal SOC Hamiltonian in SG Pbam (No. 55), HSOC(k), to gap out the CICE-NL is of the form,

$$\begin{array}{ll}{H}_{SOC}({{{\bf{k}}}})=&\mathop{\sum }\limits_{i=1}^{3}{\zeta }_{01i}\sin {k}_{i}{{{\Gamma }}}_{01i}\\ &+\,{\zeta }_{233}\cos \frac{{k}_{x}}{2}\cos \frac{{k}_{y}}{2}\cos {k}_{z}{{{\Gamma }}}_{233}+{\zeta }_{111}\sin \frac{{k}_{x}}{2}\cos \frac{{k}_{y}}{2}\cos {k}_{z}{{{\Gamma }}}_{111}\\ &+\,{\zeta }_{112}\cos \frac{{k}_{x}}{2}\sin \frac{{k}_{y}}{2}\cos {k}_{z}{{{\Gamma }}}_{112}+{\zeta }_{223}\sin \frac{{k}_{x}}{2}\sin \frac{{k}_{y}}{2}\sin {k}_{z}{{{\Gamma }}}_{223},\end{array}$$
(2)

where ζijk denotes SOC strength, and Γijk = τiσjsk (i, j, k ∈ {0, 1, 2, 3}). s0,1,2,3 are identity matrix and Pauli matrices operating in spin space, respectively. The band structure of H0 and H = H0 + HSOC are shown in Fig. 1d by the red and blue curves, respectively. In the presence of SOC, the CICE-NL TSM evolves into an insulating phase. The parameters are tuned to allow the system to host a single CICE centered at the S point in the absence of SOC and to have no additional band inversions at other k points including SOC. In addition to {αδS < 0 ∩ αβ > 0 ∩ α ≠ β}, either condition \(\{{\delta }_{S} \,<\, 0\cap {\delta }_{R} \,>\, 0\cap {\delta }_{{{\Gamma }},Z} \,>\, \sqrt{{\lambda }_{+}^{2}+{\zeta }_{233}^{2}}+\sqrt{{\lambda }_{-}^{2}+{\zeta }_{233}^{2}}\}\) or \(\{{\delta }_{S} \,>\, 0\cap {\delta }_{R}\, <\, 0\cap {\delta }_{{{\Gamma }},Z} \, < \, -\!\!\sqrt{{\lambda }_{+}^{2}+{\zeta }_{233}^{2}}-\sqrt{{\lambda }_{-}^{2}+{\zeta }_{233}^{2}}\}\), should be satisfied, where λ± = λ10 ± λ13 and δΓ,Z = δ0 + (α + β ± γ). If additional band inversions emerge beyond the one which gives rise to the CICE, the band topology may be changed in the presence of SOC, and the semimetal may evolve to distinct insulating phases, although the CICE nodal lines may still exist in the absence of SOC.

In order to determine its band topology, we implement the symmetry-indicator theory38,40,41. Crystals in the SG Pbam (No. 55) are characterized by four symmetry indicators (SIs)38,40,41, three \({{\mathbb{Z}}}_{2}\) weak TI indices and one \({{\mathbb{Z}}}_{4}\) index. The \({{\mathbb{Z}}}_{4}\) index is defined as, \({{\mathbb{Z}}}_{4}\equiv \frac{1}{4}{\sum }_{K\in TRIMs}({n}_{K}^{+}-{n}_{K}^{-})\,{{{\rm{mod}}}}\,4\), where \({n}_{K}^{+}\) (\({n}_{K}^{-}\)) is the number of occupied bands with parity +(−) at the TRIM points K. Due to the nonsymmorphic symmetries, bands are four-fold degenerate at all TRIM points except Γ. Besides, since inversion \({{{\mathcal{I}}}}\) anticommutes with the glide, \({{{\mathcal{G}}}}\), or screw, \({{{\mathcal{S}}}}\), symmetry operations (here \({{{\mathcal{S}}}}\) includes \({{{{\mathcal{S}}}}}_{y}={2}_{1}^{[010]}=\{{2}_{010}| \frac{1}{2}\frac{1}{2}0\}\), and \({{{{\mathcal{S}}}}}_{x}={2}_{1}^{[100]}=\{{2}_{100}| \frac{1}{2}\frac{1}{2}0\}\)), at X, U, Y, and T, the parity of each four-fold degenerate state must be (+, +, −, −), which does not contribute to \({{\mathbb{Z}}}_{4}\). By enumerating the parity of the states at other TRIM points, we obtain \({{\mathbb{Z}}}_{2,2,2,4}=(0,0,0,2)\), corresponding to eight possible topological states41. To further narrow down the possible phases, we have calculated the mirror Chern numbers of \({{{{\mathcal{M}}}}}_{z}\) and \({C}_{{m}_{0,\pi }^{001}}\) (\({m}_{0,\pi }^{001}\) denotes the kz = 0, π mirror planes), following the method implemented in ref. 16. We find that \({C}_{{m}_{0,\pi }^{001}}=(2,0)\). 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 \({C}_{{m}_{0,\pi }^{001}}=(2,0)\), there are two possible topological phases, which are listed in Table 1. The first one is \({{{\mathcal{S}}}}\)-protected TCI with \({\nu }_{{2}_{1}^{[010]}}={\nu }_{{2}_{1}^{[100]}}=1\), while the second one is \({{{\mathcal{G}}}}\)-protected TCI with nontrivial \({{\mathbb{Z}}}_{2}\) topological invariants \({\nu }_{{g}_{a}^{(010)}}\) and \({\nu }_{{g}_{b}^{(100)}}\) (where \({g}_{a}^{(010)}={{{{\mathcal{G}}}}}_{y}=\{{m}_{010}| \frac{1}{2}\frac{1}{2}0\}\), and \({g}_{b}^{(100)}={{{{\mathcal{G}}}}}_{x}=\{{m}_{100}| \frac{1}{2}\frac{1}{2}0\}\)). 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 position47,48,49 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.

Table 1 The two possible topological states with SIs \({{\mathbb{Z}}}_{2,2,2,4}=(0,0,0,2)\) and mirror Chern number \({C}_{{m}_{0,\pi }^{(001)}}=(2,0)\) in SG Pbam (No. 55)41. The \({\mathbb{Z}}\) invariants, \({C}_{{m}_{0,\pi }^{(001)}}\), are the mirror Chern numbers for the mirror planes \({{{{\mathcal{M}}}}}_{(001)}\) with kz = 0, π, respectively. All the listed ν’s are \({{\mathbb{Z}}}_{2}\) classified topological invariants. The set (ν0; ν1ν2ν3) are the invariants for 3D \({{\mathbb{Z}}}_{2}\) TIs. \({\nu }_{{g}_{a}^{010}}\) and \({\nu }_{{g}_{b}^{100}}\) represent the invariants for the glide symmetries \({{{{\mathcal{G}}}}}_{y}\) and \({{{{\mathcal{G}}}}}_{x}\), respectively. \({\nu }_{{{{\mathcal{I}}}}}\) is the inversion \({{{\mathcal{I}}}}\) protected TCI index, where \({\nu }_{{{{\mathcal{I}}}}}=1\), features the hinge states in a 3D finite geometry preserving \({{{\mathcal{I}}}}\). \({\nu }_{{2}^{001}}\), \({\nu }_{{2}_{1}^{010}}\) and \({\nu }_{{2}_{1}^{100}}\) denote the invariants for the rotational and screw symmetries 2[001], \({2}_{1}^{[010]}\) and \({2}_{1}^{[100]}\), respectively.
Full size table

To further determine the topological phase certainly, we have also investigated the (001) surface bands, because the presence of topological surface states on the (001) surface excludes the scenario of \({{{\mathcal{S}}}}\)-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. Interestingly, the calculations reveal the emergence of nontrivial surface states around \(\bar{S}\) which are composed of two intertwined surface Dirac cones, shown schematically in Fig. 2d (right panel). We refer to these topological surface states (TSSs) as fourfold Dirac fermions, which are of a particular type of wallpaper fermions42. As Fig. 2d shows, one can also obtain the fourfold Dirac fermions via SOC-induced splitting of the intertwined DSSs of the CICE TSM37, where the degenerate dispersions along \(\bar{S}-\bar{X}\ (\bar{Y})\) are guaranteed by \({{{\mathcal{G}}}}\). Consequently, the CICE-NL induced topological phase belongs to the \({{{\mathcal{G}}}}\)-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 phase15,40. For each STI, the topological surface states of the (001) surface can be described by the Hamiltonian, hk = kxs2 − kys140. The gapless feature of hk is protected by the TRS operator, \({{{\mathcal{T}}}}=-{is}_{2}{{{\mathcal{K}}}}\), where \({{{\mathcal{K}}}}\) is the complex conjugation operator. For the current case, since there is a DBI at the \(\bar{S}\) 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 = mμ2 ⊗ s3, where μ1,2,3 are the Pauli matrices acting on the two copies of hk, and m is constant. If M can be prohibited by any spatial symmetry \({{{\mathcal{Q}}}}\), the anomalous gapless surface states will persist, indicating that the existing topology is protected by \({{{\mathcal{Q}}}}\), which can be \({{{{\mathcal{G}}}}}_{x}\) and \({{{{\mathcal{G}}}}}_{y}\), as derived below.

At \(\bar{S}\), the eigenvalues of \({{{{\mathcal{G}}}}}_{x}\) and \({{{{\mathcal{G}}}}}_{y}\) are ±1. To preserve TRS, the only available representations are \({{{{\mathcal{G}}}}}_{x}={\mu }_{2}\otimes {s}_{1}\) and \({{{{\mathcal{G}}}}}_{y}={\mu }_{2}\otimes {s}_{2}\), which in turn lead to the rotational symmetry about the z-axis \({C}_{2z}(=\!{{{{\mathcal{G}}}}}_{x}\times {{{{\mathcal{G}}}}}_{y})=-{i}{\mu }_{0}\otimes {s}_{3}\). Obviously, M cannot survive with \({{{{\mathcal{G}}}}}_{x}\) and \({{{{\mathcal{G}}}}}_{y}\), but is allowed by C2z. Consequently, there exist representations for \({{{{\mathcal{G}}}}}_{x}\) and \({{{{\mathcal{G}}}}}_{y}\) to support the (001) TSSs at \(\bar{S}\), which is the fourfold Dirac fermion shown in Fig. 2d described by the k ⋅ p Hamiltonian

$$\begin{array}{ll}{H}_{TSS}({{{\bf{q}}}})={g}_{0}({q}_{x}{s}_{2}-{q}_{y}{s}_{1})+\mathop{\sum}\limits_{i=1,3}{g}_{i}{\mu }_{i}\otimes ({q}_{x}{s}_{1}+{q}_{y}{s}_{2})\\ +\,{q}_{x}{q}_{y}({a}_{3}{\mu }_{3}+{a}_{1}{\mu }_{1})+{g}_{23}{\mu }_{2}{s}_{3},\end{array}$$
(3)

where all g’s and a’s are real parameters.

The z-oriented Wilson loop50,51, \({\bar{z}}_{n}^{\pm }({k}_{x},{k}_{y})=\left\langle {W}_{n0}^{\pm }| \hat{z}| {W}_{n0}^{\pm }\right\rangle\), is calculated on the (001) surface, where \(\left|{W}_{n0}^{\pm }\right\rangle\) is the n-th Wannier orbit with glide \({{{{\mathcal{G}}}}}_{x}\) (\({{{{\mathcal{G}}}}}_{y}\)) eigenvalues \(\pm \!{{\rm{e}}}^{-{\rm{i}}{k}_{y}}\) (\(\pm {{\rm{e}}}^{-{\rm{i}}{k}_{x}}\)) in the home unit cell, R = 0 and \(\hat{z}\) is the position operator. Figure 2e shows \({\bar{z}}^{\pm }\) along the high symmetry k directions for occupied states hosting positive (+sector, red) and negative (−sector, blue) surface glide eigenvalues. Employing the analysis for the two \({{\mathbb{Z}}}_{4}\) indexes (χx, χy) derived in ref. 42, the bulk topology of our system is (χx, χy) = (2, 2).

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 \({{{{\mathcal{M}}}}}_{z}\)-protected surface states, the \({{{{\mathcal{G}}}}}_{x}\), \({{{{\mathcal{G}}}}}_{y}\) and \({{{{\mathcal{M}}}}}_{z}\) 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 \([01\bar{1}]\) 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 \((0\bar{1}1)\) surfaces and the (011) and \((01\bar{1})\) 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 \((0\bar{1}1)\) facets are generated via bending the (001) surface along the [001] direction, which, however, preserves the \({{{{\mathcal{G}}}}}_{y}\) 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 \((0\bar{1}1)\) 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 \({{\mathbb{Z}}}_{2}\) 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 \((01\bar{1})\) facets are generated via bending the (010) surface along the [001] direction, which preserve the \({{{{\mathcal{M}}}}}_{z}\) symmetry.

Fig. 3: Bulk-surface-hinge correspondence of the HOTI, and the band connectivity the 1D hinge states.
figure 3

a Geometry of the nanorod, which is periodic along the [100] direction, and finite-size along the [011] and \([01\bar{1}]\) directions. Two pairs of hinge modes (denoted in red and blue) emerge along the intersection lines between the (011) and \((0\bar{1}1)\) facets and the (011) and \((01\bar{1})\) facets, respectively. The first (second) pair of hinge states, denoted by cyan (red), preserves the glide \({{{{\mathcal{G}}}}}_{y}\) (mirror \({{{{\mathcal{M}}}}}_{z}\)) symmetry. b Energy dispersion along kx for the nanorod where the topological hinge modes are denoted with the corresponding cyan and red colors, respectively. c Distribution in real space of the hinge states of the nanorod. For purpose of clarity, on-site potentials V = 0.2 eV are added for the hinges. d Schematic of two symmetry-allowed band connectivities of the hinge states (cyan) protected by glide symmetry \({{{{\mathcal{G}}}}}_{y}\).

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Figure 3b shows the band structure of the nanorod along the kx 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 \(\{(011),(0\bar{1}1)\}\) and \(\{(0\bar{1}\bar{1}),(01\bar{1})\}\) facets to better differentiate the two hinge states in energy. The constraint imposed by the glide symmetry \({{{{\mathcal{G}}}}}_{y}\) allows two possible topologies of hinge bands52, the hourglass connectivity and the analog of the quantum spin Hall (QSH) effect (see Fig. 3d). From the hinge modes in cyan displayed in Fig. 3c, we find that the hinge band connectivity is the analog of the quantum spin Hall effect (bottom panel of Fig. 3d).

Discussion

In ref. 37, we demonstrated that 9 (Pbam (No. 55), Pccn (No. 56), Pnnm (No. 58), Pnma (No.62), P4/mbm (No. 127), P4/mcn (No.128), P42/mbc (No. 135), P42/mnm (No.136), P42/ncm (No. 138)) out of 230 SGs can host CICEs centered at certain TRIM points in the absence of SOC. However, only two wallpaper groups, namely pgg ((001) surface of Pbam (No. 55)) and p4g ((001) surface of P4/mbm (No. 127) and P42/mbc (No. 135)), containing double-glide lines, can support fourfold Dirac fermions when SOC is introduced. Consequently, only 3 (Pbam (No. 55), P4/mbm (No. 127), and P42/mbc (No. 135)) of the 9 SGs can support the fourfold Dirac fermions on their (001) surfaces. In SG Pnma (No.55), the CICE resides on the (010) rather than the (001) plane, and the (010) surface (pg) contains a single glide line. Therefore, the (010) TSSs of the corresponding TCI are hourglass fermions instead. For the rest 5 SGs, due to the lack of glide symmetry preserved on their (001) surfaces, the nonsymmorphic fourfold Dirac or hourglass fermions would not be expected.

In summary, the inclusion of SOC in the model Hamiltonian describing our recently proposed family of butterfly-like CICE NLs in SG Pbam (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 \({{\mathbb{Z}}}_{4}=2\) 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, Sr2Pb3 (SG P4/mbm No. 127) has been studied in42. Moreover, its higher order topology is corroborated by the emergence of 1D hinge states protected by glide symmetry, which has never been found before. 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 superconductor53, which exhibit distinct features compared to those in TI/SC heterostructures. Finally, our EBRs analysis of the tight binding model in Supplementary Discussion demonstrates a rich phase diagram featuring TCI, strong TI, and obstructed atomic insulator (OAI) phases.

As described in detail in the Supplementary Note, we have carried out systematic ab initio electronic structure calculations to identify material candidates which exhibit (i) a single CICE in the absence of SOC and (ii) fourfold-Dirac TCI/HOTI phase in the presence of SOC. Unfortunately, this task has proved immensely challenging and the calculations failed in finding such an ideal material. This is due to the fact that the band structures in real materials are more complex involving multiple orbitals and bands. Nevertheless, these calculations raise the intriguing question of realizing these quantum states by designing new materials according to our simple model, so that the topological nature can be captured and not buried in the complex band structure. As stated above, we have constructed a minimal (8-band in the presence of SOC) tight-binding model with a bipartite lattice in space group Pbam (No. 55), based on the Wyckoff position 2a with site-symmetry group 2/m, and with two orbital per sublattice. Even though we employed the (pz, dxy)-derived orbitals, this choice is not unique and one can equally well use the (pz, s)-derived orbitals. The conditions for the emergence of CICE nodal lines in the absence of SOC are given by Equations (4) and (5) of ref. 37 with the additional proviso that no other band inversions occur. The conditions to avoid additional band inversions in the presence of SOC are given in the paragraph below Eq. (2) of this work. 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, λ10,13, and ζ233.