Novel Family of Topological Semimetals with Butterfly-like Nodal Lines

Topological semimetals (TSMs) Fang et al. (2016); Weng et al. (2016); Armitage et al. (2018)have emerged among the most active frontiers in condensed matter physics in recent years, drawing widespread attention from both the theoretical and the experimental communities. In the noninteracting limit, TSMs describe systems which are characterized by the topologically robust band-crossings manifolds between conduction and valence bands in momentum $k$ space. These mainfolds can be zero-dimensional (0D) nodal points, e.g., three-dimensional (3D) Weyl semimetals (WSMs) Wan et al. (2011); Burkov and Balents (2011); Armitage et al. (2018) and Dirac semimetals (DSMs) Young et al. (2012); Wang et al. (2012); Young and Kane (2015); Chang et al. (2017a); Armitage et al. (2018); Yu et al. (2018), and one-dimensional (1D) nodal lines/loops, e.g., nodal-line semimetals (NLSMs) Burkov et al. (2011); Fang et al. (2016). Around these band-crossings, electron excitations behave drastically differently from the conventional Schrödinger fermions in normal metals. For example, the low-energy electrons in 3D DSMs and WSMs resemble the relativistic Dirac and Weyl fermions, making it possible to mimic high-energy physics phenomena. Meanwhile, TSMs are distinguished from normal semimetals by the accompanying topological indices due to the aforementioned manifolds. Moreover, because of these unique electronic features, TSMs present exotic properties in different ways, such as Fermi arcs Xu et al. (2015) and drumhead surface states (SSs) Bian et al. (2016) on surfaces of WSMs and NLSMs, respectively, and novel transport phenomena e.g., the negative magnetoresistance related to the chiral anomaly in both Weyl and Dirac SMs Burkov (2014); Zhang et al. (2016); Hu et al. (2019).

Among TSMs, NLSMs possess the highest variability. NLs can be integrated in various configurations, e.g., a chain link Bzdušek et al. (2016); Chang et al. (2017b); Yan et al. (2017); Yu et al. (2018), a Hopf link Chang et al. (2017b), and a knot Bi et al. (2017), where each of them carries its unique topology. Since the essential characteristics, band crossings, of various TSMs are mostly protected by crystalline symmetries, a thorough classification of a particular type of TSMs in all space groups can greatly accelerate the experimental discovery. There exist well-established classifications for DSMs and WSMs Yang and Nagaosa (2014); Gao et al. (2016); Wieder et al. (2016), and the triple point semimetals Zhu et al. (2016). However, for most types of NLSMs proposed today, except the chain link Bzdušek et al. (2016) and some types of intersecting rings Gong et al. (2018), the symmetry criteria of the emergence of particular nodal lines remain deficient.

In this Letter, we introduce a new type of NLSM in time-reversal invariant spinless systems, which hosts a butterfly-like nodal-line (NL) consisting of a pair of concentric intersecting coplanar ellipses (CICE) at half-filling residing on a plane in $k$ space, as indicated by the blue, red concentric ellipses in Fig.1(a). Meanwhile, the half-filling CICE consequently guarantees the presence of another pair of CICE formed by band crossings beyond half-filling, which is indicated by the magenta lines in Fig.1(b). We demonstrate that CICE can be sustained by nonsymmorphic crystalline symmetries including two glide symmetries, and only 9 space groups (SGs) are feasible to host it. These SGs are classified into two categories by their point group symmetries, which are $D_{2h}$ and $D_{4h}$. Moreover, we provide a tight-binding model for one of the SGs $Pbam$ (No. 55) which exhibits CICE and hosts exotic intertwined drumhead surface states. In the end, five material candidates from these two categories, hosting the proposed CICE, are suggested for further experimental studies.

Symmetry Criteria and Space Groups– Conceptually, a pair of CICE can be constructed by integrating two NL fermions. As shown in Fig. 1(a), CICE denoted by the intersection of the red and blue ellipses on the $k_{z}=0$ plane can be decomposed into two individual NL fermions.Each NL, the accidental two-fold band-crossings due to the band inversion, is further validated by the inherent crystal symmetry belonging to its parent bands, which is the mirror (or glide) reflection symmetry, $\mathcal{M}_{z}(\mathcal{G}_{z}):(k_{x},k_{y},k_{z})\rightarrow(k_{x},k_{y},-k_{z})$. On the $k_{z}=0$ plane, states carrying different mirror (or glide) eigenvalues forbid their mutual hybridization, thus supporting the NL fermion. Since at each $k$ point $\mathcal{M}_{z}$ ($\mathcal{G}_{z}$) only supplies two different eigenvalues, additional symmetry constrains along $k_{x}$ and $k_{y}$ axes are demanded to sustain the four-fold degenerate points (FDPs) on the CICE, $i.e.,$ the intersecting points of the two NL fermions, marked in Fig. 1(d).

Therefore, the additional symmetries required along $k_{x}$ and $k_{y}$ to guarantee the two-fold degeneracy can be realized by introducing an antiunitary symmetry, $\mathcal{T}\mathcal{Q}$, which combines time-reversal symmetry (TRS) $\mathcal{T}$ and a spatial symmetry $\mathcal{Q}$. Thus, the Kramer-like two-fold degeneracy is enforced at $\mathcal{T}\mathcal{Q}$-invariant points where $(\mathcal{T}\mathcal{Q})^{2}=-1$. For a spinless system, $\mathcal{T}^{2}=+1$, which in turn requires that $\mathcal{Q}$ is a nonsymmorphic symmetry, with eigenvalues of $\pm i$ Young and Kane (2015); Wang et al. (2016) at certain points on the boundaries of BZ. Consequently, the qualified candidates of $\mathcal{Q}$ for ensuring the degeneracy on the $k_{x}$ ($k_{y}$) are, $\mathcal{G}_{x}$ or $\mathcal{S}_{y}$ ($\mathcal{G}_{y}$ or $\mathcal{S}_{x}$), so the CICE should be centered at $(\pi,\pi,0\textrm{ or }\pi)$. Here, $\mathcal{S}_{x(y)}=\{\mathcal{C}_{2x(2y)}|\bm{t}_{x(y)}\}$ denotes a two-fold screw rotation with respect to the $k_{x(y)}$-axis accompanied by a translation $\bm{t}_{x(y)}=\frac{1}{2}\hat{x}(\hat{y})$, and $\mathcal{G}_{x(y)}=\{\mathcal{M}_{x(y)}|\bm{t}_{y(x)}\}$ is a glide symmetry normal to $k_{x(y)}$, which contains a fractional translation $\bm{t}_{y(x)}=\frac{1}{2}\hat{y}(\hat{x})$. To avoid replicas of CICE on other symmetry-related planes, $n$-fold rotation and rotoinversion symmetries with $n>2$ with respect to the $k_{x(y)}$ axes are not allowed. In addition, each Kramer-paired states should carry the same mirror symmetry eigenvalue of $\mathcal{M}_{z}$ (or $\mathcal{G}_{z}$). Furthermore, symmetry-enforced degeneracy are not allowed at any generic point of the $k_{z}=0$ plane other than the $k_{x(y)}$-axis.

In summary, the criteria for generating CICE in a spinless crystal preserving TRS include: (i) the little group of the center of CICE is nonsymmorphic with corresponding point group (PG) $D_{2h}$ or $D_{4h}$; (ii) the crystal contains two glide $\mathcal{G}_{x(y)}$ or screw $\mathcal{S}_{y(x)}$ symmetries with respect to the axes lying on a mirror $\mathcal{M}_{z}$ or a glide $\mathcal{G}_{z}$ plane; (iii) at the center of the CICE, $\mathcal{M}_{z}$ ($\mathcal{G}_{z}$) should commute with other preserved and required symmetries. According to the above criteria, we have exhaustively scanned all 230 SGs, and determined 9 possible SGs to host CICE. The corresponding positions of the center of the CICE and the corresponding axes of $\mathcal{S}$ or $\mathcal{G}$ are listed in Table. 1.

Lattice Model and Surface States– To validate the criteria derived above and explore the underlying topology of CICE, we construct a minimal 4-band tight-binding lattice model for the SG $Pbam$ (No. 55). The minimal required symmetries are $\mathcal{M}_{z}=\{m_{001}|000\}$, $\mathcal{G}_{x}=\{m_{100}|\frac{1}{2}\frac{1}{2}0\}$ and $\mathcal{G}_{y}=\{m_{010}|\frac{1}{2}\frac{1}{2}0\}$ (see Supplementary Material for details). The model is a bipartite lattice, where the sublattices denoted by A (gray) and B (blue) occupy the 2a Wyckoff position at $\mathbf{r}_{A}=(0,0,0)$ and $\mathbf{r}_{B}=(\frac{1}{2},\frac{1}{2},0)$ in a unit cell (see Fig. 2 (a) for the structure). Each sublattice contains two orbitals, $p_{z}$ and $d_{xy}$ described by the Pauli matrix $\bm{\sigma}$, and $\bm{\tau}$ for the A and B sublattices. For a spinless system, employing the basis $\Psi=(p_{z}^{A},d_{xy}^{A},p_{z}^{B},d_{xy}^{B})^{T}$ the symmetry-constrained tight-binding Hamiltonian is of the form,

where $\alpha,\beta,\gamma,\delta_{0},$ and $\lambda_{ij}$ are constants.

Since the CICE can emerge on the mirror plane (gray shaded area in Fig. 2(b)) centered at the high symmetry $k$ point $S=(\pi,\pi,0)$ [$R=(\pi,\pi,\pi)$] (Fig. 2(b)), we derive the effective $k\cdot p$ Hamiltonian around the S(R) point,

At $q_{z}=0$, the Hamiltonian is diagonalized as $E_{S(R)}(q_{x},q_{y},0)=\textrm{diag}(E_{+}^{A+B},E_{-}^{A+B},E_{+}^{A-B},E_{-}^{A-B})$ on the basis $\Psi^{\prime}=(p_{z}^{A+B},d_{xy}^{A+B},p_{z}^{A-B},d_{xy}^{A-B})^{T}$, where $|\varphi^{A\pm B}\rangle=\frac{1}{\sqrt{2}}(|\varphi^{A}\rangle\pm|\varphi^{B}\rangle)$ ($\varphi=p_{z},d_{xy}$) denote the bonding/antibonding states of the relevant orbitals.

If $|\lambda_{13}|>|\lambda_{10}|$, each ellipse in the half-filling CICE is the line crossing between the two bands $E_{+}^{A+B}$ ($E_{+}^{A-B}$) and $E_{-}^{A+B}$ ($E_{-}^{A-B}$) respectively, while the ellipses in the second pair are given by the crossings between $E_{+}^{A+B}$ ($E_{+}^{A-B}$) and $E_{-}^{A-B}$ ($E_{-}^{A+B}$). We refer to this type of CICE as type-I NL. Otherwise, if $|\lambda_{13}|<|\lambda_{10}|$, the first and second pairs of CICE are exchanged and we have type-II NL. Thus, the corresponding NLs for the half-filling CICE can be obtained by solving the equations,

After further analyses, we find that when the condition

is satisfied, where $\delta_{S,R}=\delta_{0}-(\alpha+\beta\mp\gamma)$, the terms in the first line of Eq. (1) describe two concentric elliptic NLs with double band inversions at the S (R) point (Fig. 1(d)). The terms in the second line in Eq. (1) adjust the anisotropy of each NL, resulting in two twisted elliptic NLs (see dispersion along $k_{x}=k_{y}$ in Fig 2(c), where the band width differs in the two original elliptic NLs). The angles of the elliptic NLs with respect to the $k_{x}$-axis are determined via $\theta_{\pm}=\pm\frac{1}{2}\textrm{arctan}\frac{\lambda}{2(\alpha-\beta)}$, where $\lambda=\textrm{max}\{|\lambda_{10}|,|\lambda_{13}|\}$.

To explore the unique topological properties of CICE NL, the model parameters are tuned to allow the system to host CICE centered at the $S$ point and to have no additional band inversions at other time-reversal-invariant momentum points (TRIM). The corresponding band structure is shown in Fig. 2(c), in which the distinctive features of CICE can be recognized by comparing the bands along $S-\Gamma$ and $S-X$ (or $S-Y$). Since CICE are composed by two NLs, and the essential $\mathcal{G}_{x}$ and $\mathcal{G}_{y}$ symmetries are preserved on (001) surface, we anticipate to observe two intertwined drumhead surface states (DSSs) Burkov et al. (2011). The DSS, shown schematically in Fig. 2(d), can be described by the $k\cdot p$ Hamiltonian around the $\bar{S}$ point,

where $\mu_{1,2,3}$ are Pauli matrices acting in orbital space, and $a_{1,3}$ are real constants. Two saddle-like hyperbolic paraboloids (red and gray surfaces) are intertwined with each other, resulting in the flat and doubly degenerate bands along $\bar{S}-\bar{X}$ and $\bar{S}-\bar{Y}$ respectively, which are enforced by $\mathcal{G}_{x}$ and $\mathcal{G}_{y}$ symmetries combined with $\mathcal{T}$. We would like to emphasize that these remarkable features exhibited by the new DSSs allow them to provide a great platform for studies of exotic emergent phenomena.

The calculated $(001)$ surface band structure along $\bar{X}-\bar{S}-\bar{\Gamma}$ for $|\lambda_{10}|<|\lambda_{13}|$ is shown in Fig. 2(e). Intriguingly, we notice that another type of DSSs, shown in Fig. 2(f), can be realized when $|\lambda_{10}|>|\lambda_{13}|$ with all remaining parameters unchanged. We refer to the two different types of DSSs as type-I/type-II for the former/latter case. Type-II DSSs can be comprehended from the way one proceeds to decompose the second pair of CICE into two single NLs. As shown in Fig. 1(b) the NLs (shown by magenta color) are allowed by the same band configurations with swapped conduction bands in comparison to the configurations of Fig. 1 (a). In contrast to the NLs of the half-filling CICE, the NLs in magenta are due to the crossings between two occupied bands (one-quarter filling) and two unoccupied bands (three-quarter filling), respectively, and hence might be irrelevant for electron excitation at half-filling system. However, the CICE TSM introduces another possibility. As one cannot distinguish whether the FDPs of CICE belong to the half-filling CICE or the second pair of CICE, both of them can provide topological DSSs on an equal footing due to the inherent band inversion. Even though both types of NL contribute to the DSS on the (001) surface, $\mathcal{G}_{x}$ and $\mathcal{G}_{y}$ permit solely one pair of DSSs, forcing in turn the other pair merge into the bulk states. Consequently, the DSS of CICE-NL in the spinless case may appear in either way depending on the coupling parameter details.

Material Candidates– We propose a series of compounds as material candidates for the experimental realization of this new type of TSMs that host butterfly-like CICE NLs, such as ZrX₂ (X = As, P) with CICE centered at TRIM point U ($\pi,\pi,0$), as well as GeTe₅Tl₂, CYB₂ and Al₂Y₃ at M ($\pi,\pi,0$), respectively. Here we take the ZrAs₂ and GeTe₅Tl₂ as representatives. The equilibrium lattice constants and electronic structure of both compounds were determined by first principles density functional theory (DFT) calculations using the VASP Kresse and Hafner (1993) and WIEN2k Blaha et al. (2020) packages (CYB₂ and Al₂Y₃ were found using the Advanced Search Tools of https://www.topologicalquantumchemistry.org/Bradlyn et al. (2017); Vergniory et al. (2019), see Supplementary Materials for details).

The crystal structure of ZrAs₂, is orthorhombic with SG $Pnma$ (No. 62) and is displayed in Fig. 3 (a). The calculated lattice parameters $a=6.847$ Å, $b=3.718$ Å  and $c=9.123$ Å are in agreement with the experimental ones Trzebiatowski et al. (1958); Blanchard et al. (2010). The band structure without SOC close to the $U=(\pi,0,\pi)$ point of the BZ is shown in Fig. 3 (b) along two high-symmetry lines ($X-U$ and $U-\Gamma$). The second material, GeTe₅Tl₂, has tetragonal structure with SG $P4/mbm$ (No. 127) and the crystal structure is depicted in Fig. 3 (d). In the DFT calculations the experimental structure of GeTe₅Tl₂ Marsh (1990); Abba Toure et al. (1990) is applied, and the band structures along two essential high symmetry $k$ paths $M\rightarrow X$ and $M\rightarrow\Gamma$ are shown in Fig. 3 (d), which reveal band crossings alike those observed in ZrAs₂.

In order to corroborate the CICE NLs in ZrAs₂ around $U$ we used the Bloch functions obtained with DFT to construct a Wannier-function based model employing the Wannier90 packageMostofi et al. (2008). The model reproduces the bands around the Fermi level, allowing the scan of band crossings in the BZ more efficiently than direct DFT calculations. As shown in Fig. 3 (c) the nodal points around $U$ point form a butterfly-like CICE with a small energy dispersion. In the case of GeTe₅Tl₂, the band crossings, shown in Fig 3 (f), occur closer to the $M$ point, yielding a smaller area enclosed by the CICE and a lower energy dispersion.

Conclusion– In summary, we have proposed a new type of TSMs which unveil intriguing butterfly-like CICE NDLs. We have derived the symmetry criteria to generate the CICE, identified the 9 SGs which can host such complex NDLs and determined the positions of the CICE centers in the BZ for each SG. For one of the SGs $Pbam$ (No. 55) and for spinless fermions we have introduced a model which hosts CICE and supports the intriguing intertwined drumhead surface states. Finally, we have predicted candidate materials which can host such exotic NL landscapes.

Acknowledgments– The work at CSUN was supported by NSF-Partnership in Research and Education in Materials (PREM) Grant No. DMR-1828019. The work of J.L.M. has been supported by Spanish Science Ministry grant PGC2018-094626-B-C21 (MCIU/AEI/FEDER, EU) and Basque Government grant IT979-16. M.G.V. thanks support from DFG INCIEN2019-000356 from Gipuzkoako Foru Aldundia. H.L. acknowledges the support by the Ministry of Science and Technology (MOST) in Taiwan under grant number MOST 109-2112-M-001-014-MY3.