Abstract
Topological metals and semimetals (TMs) have recently drawn significant interest. These materials give rise to condensed matter realizations of many important concepts in highenergy physics, leading to wideranging protected properties in transport and spectroscopic experiments. It has been wellestablished that the known TMs can be classified by the dimensionality of the topologically protected band degeneracies. While Weyl and Dirac semimetals feature zerodimensional points, the band crossing of nodalline semimetals forms a onedimensional closed loop. In this paper, we identify a TM that goes beyond the above paradigms. It shows an exotic configuration of degeneracies without a welldefined dimensionality. Specifically, it consists of 0D nexus with tripledegeneracy that interconnects 1D lines with doubledegeneracy. We show that, because of the novel form of band crossing, the new TM cannot be described by the established results that characterize the topology of the Dirac and Weyl nodes. Moreover, triplydegenerate nodes realize emergent fermionic quasiparticles not present in relativistic quantum field theory. We present materials candidates. Our results open the door for realizing new topological phenomena and fermions including transport anomalies and spectroscopic responses in metallic crystals with nontrivial topology beyond the Weyl/Dirac paradigm.
Introduction
Understanding nontrivial topology in gapless materials including metals and semimetals has recently emerged as one of the most exciting frontiers in the research of condensed matter physics and materials science^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}. Unlike conventional metals, topological metals/semimetals (TMs) are materials whose Fermi surface arises from the degeneracy of conduction and valence bands, which cannot be avoided due to their nontrivial topology. To date, the known TMs include Dirac semimetals, Weyl semimetals, and nodalline semimetals. Dirac or Weyl semimetals have zerodimensional (0D) band crossings, i.e., the Dirac or Weyl nodes and a Fermi surface that consists of isolated 0D points in the bulk Brillouin zone (BZ). By contrast, nodalline semimetals feature onedimensional (1D) band crossings and a Fermi surface that is made up of 1D closed loops in the BZ. Therefore, the band crossings serve as a key signature of nontrivial topology in metals and can be used to classify TMs. More importantly, these band crossings can give rise to fundamentally new physical phenomena. Since lowenergy excitations near the Dirac or Weyl nodes mimic elementary fermions, TMs provide a unique opportunity to study important concepts of highenergy physics such as Dirac fermions, Weyl fermions, and the chiral anomaly in tabletop experiments. The correspondence with highenergy physics, in turn, leads to a cornucopia of topologically protected phenomena. The resulting key experimental detectable signatures include the Dirac, Weyl or nodalline quasiparticles in the bulk, the Fermi arc or drumhead topological surface states on the boundaries, the negative magnetoresistance and nonlocal transport induced by the chiral anomaly^{18, 19}, the surfacetosurface quantum oscillation due to Fermi arcs^{20, 21}, the Kerr and Faraday rotations in optical experiments^{22}, and topological superconductivity and Majorana fermions when superconductivity is induced via doping or proximity effect^{23,24,25,26}. Because all these fascinating properties arise from the band crossings, there has been growing interest in the search for new TMs with new types of band crossings^{27, 28}, including a classification of 3, 6, and 8fold band degeneracies that appear at highsymmetry points in nonsymmorphic crystals^{28}. Such efforts can lead to new protected phenomena in transport and spectroscopic experiments, which can be potentially utilized in device applications.
In this paper, we identify a class of TMs featuring a type of band crossing beyond the Dirac, Weyl and nodalline cases. Specifically, we find that the new TM features a pair of triplydegenerate nodes, which are interconnected by multisegments of lines with twofold degeneracy. The triplydegenerate node realizes emergent fermionic quasiparticles beyond the Dirac and Weyl fermions in quantum field theory. Moreover, the new band crossing evades the classification of TMs based dimensionality because it is neither 0D nor 1D but rather a hybrid. We show that this band crossing gives rise to a distinct Landau level spectrum, suggesting novel magnetotransport responses. Further, we identify the space groups, in which this new TM state can occur and present material candidates for each space group. Our results highlight the exciting possibilities to realize new particles beyond highenergy textbook examples and to search for new topologically protected lowenergy phenomena in transport and spectroscopic experiments beyond the Weyl/Dirac paradigm.
Theory of the new band crossing
We first present a physical picture of the new band crossing without going into mathematical details. We consider an inversion breaking crystal lattice with a threefold rotational symmetry along the \(\hat{z}\) direction (\({\tilde{C}}_{3z}\)) and a mirror symmetry that sends x → −x (\({\tilde{ {\mathcal M} }}_{x}\)). Note that the \({\tilde{C}}_{3z}\) rotational symmetry replicates the \({\tilde{ {\mathcal M} }}_{x}\) twice. In momentum space there are thus in total three mirror planes intersecting along the k _{ z } axis as shown in Fig. 1a,b. We first consider the case without spinorbit coupling (SOC). The \({\tilde{C}}_{3z}\) operator has three eigenvalues, namely, \({e}^{i\frac{2\pi }{3}},{e}^{i\frac{2\pi }{3}}\), and 1, and we denote the corresponding eigenstates as ψ _{1}, ψ _{2}, and ψ _{3}, respectively. Under the mirror reflection \({\tilde{ {\mathcal M} }}_{x}\), ψ _{3} remains unchanged (\({\tilde{ {\mathcal M} }}_{x}{\psi }_{3}={\psi }_{3}\)), whereas ψ _{1} and ψ _{2} will transform into each other \({\tilde{ {\mathcal M} }}_{x}{\psi }_{1}={\psi }_{2}\); \({\tilde{ {\mathcal M} }}_{x}{\psi }_{2}={\psi }_{1}\). Thus \({\tilde{C}}_{3z}\) and \({\tilde{ {\mathcal M} }}_{x}\) do not commute and cannot be simultaneously diagonalized in the space spanned by ψ _{1} and ψ _{2}. Both \({\tilde{C}}_{3z}\) and \({\tilde{ {\mathcal M} }}_{x}\) leave every momentum point along the k _{ z } axis invariant. Thus, at each point along the k _{ z } axis, the Bloch states that form a possibly degenerate eigenspace (band) of the Hamiltonian must be invariant under both \({\tilde{C}}_{3z}\) and \({\tilde{ {\mathcal M} }}_{x}\). Failure of \({\tilde{C}}_{3z}\) and \({\tilde{ {\mathcal M} }}_{x}\) to be simultaneously diagonalizable thus enforces a twofold band degeneracy of bands with the same eigenvalues as ψ _{1} and ψ _{2}. Therefore, in the absence of SOC, along the k _{ z } axis, the three bands with the three different \({\tilde{C}}_{3z}\) eigenvalues always appear as a singlydegenerate band (ψ _{3}) and a doublydegenerate band (ψ _{1} and ψ _{2}). If the single degenerate and the doublydegenerate bands cross each other accidentally, a triplydegenerate node will form because their different \({\tilde{C}}_{3z}\) eigenvalues prohibit hybridization.
When spin is added to the picture, all bands discussed above gain an additional double degeneracy in absence of SOC. However, SOC generically lifts the resulting 6fold degeneracy into two threefold degeneracies in absence of inversion symmetry away from the timereversal symmetric momenta. These threefold degeneracies are protected for very similar reasons as in the spinless case. The three eigenvalues of the spinful C _{3z } operator are \({e}^{i\frac{\pi }{3}}\), \({e}^{i\frac{\pi }{3}}\), and e ^{iπ}. The same symmetry argument combining C _{3z } and the spinful mirror operator \({ {\mathcal M} }_{x}\) will show that the two states with \({e}^{\pm i\frac{\pi }{3}}\) eigenvalues must be degenerate. Considering all these conditions collectively, the six bands appear as two singlydegenerate bands with the C _{3z } eigenvalue of e ^{iπ} and two doublydegenerate bands with the C _{3z } eigenvalues of \({e}^{\pm i\frac{\pi }{3}}\). An accidental crossing between a singlydegenerate and a doublydegenerate band will give rise to a triplydegenerate node along the k _{ z } axis. Away from the k _{ z } axis, all of the three bands can hybridize and the degeneracies will be lifted. Along the k _{ z } axis, a twofold degenerate nodal line emanates from the threefold degeneracy, but this degeneracy occurs between the lowest and the middle band on one side of the threefold degeneracy and between the middle and the highest band on the other side. This structure of degeneracies is reminiscent of but yet distinct from the threefold degeneracy found for space group 220 in ref. 28, where pairs of nodal lines emanate from a threefold degenerate point. The latter is pinned to a highsymmetry point and the symmetries realizing it are quite different from the scenario discussed here.
Now we present the effective Hamiltonian near the triplydegenerate node. In the presence of spinorbit coupling, we denote three eigenstates of C _{3z } with the eigenvalues of \({e}^{i\frac{\pi }{3}},{e}^{i\frac{\pi }{3}}\), and e ^{iπ} as \({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} }\,{\rm{and}}\,{\psi }_{3}^{^{\prime} }\), respectively. Using the basis \(({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} },{\psi }_{3}^{^{\prime} })\), the C _{3z } and \({ {\mathcal M} }_{x}\) operators have the representations
It can be seen that C _{3z } and \({ {\mathcal M} }_{x}\) do not commute with each other, \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) form a twodimensional irreducible representation. Therefore, \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) have to be degenerate at all k points along the k _{ z } axis. We present a k · p model for the bands in the vicinity of one triplydegenerate fermion. We denote the momentum relative to the triplydegenerate node as q = (q _{ x }, q _{ y }, q _{ z }). The k · p Hamiltonian to linear order in q _{ z } and quadratic order q _{ x } and q _{ y } can be written as
where \({q}_{\pm }={q}_{x}\pm i{q}_{y}\), the parameters t, Δ_{ t }, λ and λ_{R} are real, and λ′ is a complex parameter. We now explain how the threefold band crossing can arise through a band inversion process. Consider a material whose lowest valence and conduction bands are the singlydegenerate band (\({\psi }_{3}^{^{\prime} }\)) and the doublydegenerate band (\({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\)), respectively. As shown in Fig. 1a, if we turn off the hopping of electrons between atomic sites (this can be conceptually done by increasing the lattice constants to infinity), then all bands are flat and the system is an insulator. Now as we gradually increase the magnitude of hopping (this can be conceptually done by decreasing the lattice constant from infinity), bands will gain dispersion. When the band width is large enough relative to the energy offset between the bands, the two bands will be inverted in some interval along the k _{ z } axis (Fig. 1a) and cross each other at two points on the opposite sides of the Γ point along the k _{ z } axis. These two crossings are the triplydegenerate nodes. This process shows that the triplydegenerate nodes in our new TM always come in pairs and they can move along the k _{ z } axis as the band dispersion is varied.
We find that the new band crossing can be classified into two classes, namely Class α and Class β, depending on whether the mirror symmetry \({ {\mathcal M} }_{z}\) is present (Class α) or not (Class β). (On the level of the effective Hamiltonian (2), \({ {\mathcal M} }_{z}\) enforces that λ′ is real and λ_{R} = 0). The momentum configurations of band degeneracies in both classes are shown in Fig. 1b,c, respectively. They differ in the line degeneracies that connect the triplydegenerate nodes. In Class α, the energy eigenvalues are
two of which are degenerate on the k _{ z } axis. Specifically, two isolated triplydegenerate nodes are located on the opposite sides of the Γ point, which arise from the degeneracy between all three \(({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} }\,{\rm{and}}\,{\psi }_{3}^{^{\prime} })\) bands. These two triplydegenerate nodes are linked by nonclosed 1D segments with twofold degeneracy, which arise from the degeneracy between the \({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} }\) bands. At any generic k point on the twofold degenerate segments, the inplane (k _{ x } or k _{ y }) dispersion is a quadratic touching of the \({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} }\) bands. The Berry phase along a closed loop around the open segment is 2π, which is trivial. By contrast, in Class β, the twofold degenerate 1D band crossings form four strands at every cut of constant k _{ z } that join at the triplydegenerate nodes. At any generic k point on the twofold degenerate lines, the inplane (k _{ x } or k _{ y }) dispersion is a linear touching of the \({\psi }_{1}^{^{\prime} },{\psi }_{2}^{^{\prime} }\) bands, and the Berry phase around each line is ±π, which is nontrivial. One of the four twofold degenerate segments is pinned to align with the k _{ z } axis. The distinction between Class α and Class β can be understood by an analogy between the Hamiltonian at a generic slice of constant k _{ z } with both realspace nexus in 3HeA^{29, 30} and momentum space nexus of bilayer graphene^{30, 31}. In the bilayer graphene, the addition of skew interlayer hopping turns a quadratic band touching, corresponding to one degeneracy line segment in Class α, into a quadruplet of Dirac points, corresponding four degeneracy line segments in Class β ^{31}. However, an important difference is that the bands described by this effective Hamiltonian in graphene are doubly degenerate due to spin, while the effective model for the triply degenerate fermion already includes spinorbit coupling, and therefore describes bands without residual degeneracies.
We can further classify the triplydegenerate node by its band dispersion into typeI and typeII, in analogy to a recently introduced notion for Weyl semimetals^{11}. In our case, in typeI, the singlydegenerate band and the doublydegenerate band have Fermi velocities of opposite sign, whereas in typeII all Fermi velocities are of the same sign along the k _{ z } axis. The two situations are separated by a Lifschitz transition.
To explicitly reveal the novelty of the new TM, we show that the new band crossing cannot be describe by the established results in topological band theory, which have successfully characterized the topology of the Dirac and Weyl nodes. We first briefly review the established results for the Dirac/Weyl cases. A Weyl node is a point crossing between two singly degenerate bands (bands “1” and “2” in Fig. 2a). We enclose the Weyl node by a sphere in k space as shown in Figs. 2b,c. We notice that the sphere satisfies following two crucial conditions: (1) The sphere is a 2D closed manifold; (2) Bands 1 and 2 are separated by a band gap at all k points on the sphere. These two facts guarantee that one can calculate the Chern number of the filled valence bands on this sphere. Because the Weyl nodes are Berry curvature monopoles, it has been shown^{1} the Chern number (C) of the sphere equals the chiral charge (χ) of the enclosed Weyl node, which serves as the topological invariant of the Weyl node. Specifically, for a single Weyl node, we have χ = ±1 where the sign depends on the chirality of the Weyl node. For a Dirac node, we have a band crossing between two doublydegenerate bands. Same as the Weyl node case, we can enclose the Dirac node by a sphere in k space and it is evident that the sphere will also satisfy the two conditions above, which allow the definition of a Chern number on the sphere. Because a Dirac node can be viewed as two degenerate Weyl nodes of opposite chirality, it can be shown that the chiral charge of a Dirac node is always zero, i.e., χ = 0.
Now we show why this established method cannot characterize the topology of the new band crossing. As shown in Fig. 2g–i, we enclose the triply degenerate point by a sphere. Now, if we want to use the method described above, we need to find two bands that are separated by a full energy gap at all k points on this sphere. However, we see that, between bands 1 and 2, the band gap vanishes at the leftpole of the sphere (Fig. 2h). Similarly, between bands 2 and 3, the band gap is zero at the rightpole of the sphere (Fig. 2i). Therefore, due to the exotic configuration of the new band crossing, it is impossible to enclose it with a 2D closed manifold on which the band structure is fully gapped. This fact demonstrates that the established results in topological band theory for the Dirac/Weyl cases cannot be used for the new band crossing. While the topological invariant for the new band crossing is an open question in theory that deserves further investigation, the fact that it cannot be described by the established results already demonstrates that it represents a breakthrough beyond the Dirac/Weyl paradigm. This fact also shows that the new band crossing cannot be viewed as a simple composition of a 0D point plus a 1D nodal line (Fig. 2f). In that case, the 0D point and the 1D nodal line are isolated with respect to each other, and each of them separately and independently admits its own topological classification. The 0D point can be enclosed by a 2D sphere where as the 1D nodal line can be enclosed by a 1D loop. By contrast, in the new band crossing in our case, the 0D triple point serves as the connection point of the 1D nodal lines, meaning that they cannot be separated. Because of this very fact, it is impossible to enclose the triple point with a 2D closed manifold on which the band structure is fully gapped so that one can define a Chern number, as we have shown above. Therefore, the exotic configuration of the new band crossing excludes a welldefined dimensionality.
Zeeman Coupling
In order to understand how the new TM responds to magnetism or magnetic doping in experiments, we study the Zeeman coupling and contrast it with Dirac semimetals. A topological Dirac semimetal system has timereversal symmetry, spaceinversion symmetry, and a uniaxial rotational symmetry along the k _{ z } direction. The presence of timereversal and spaceinversion symmetries requires all bands to be doublydegenerate because spin up and spin down states have the same energy (Fig. 3a). The crossing between two doublydegenerate bands is realized by a pair of fourfold degenerate points, Dirac nodes, which are protected by the uniaxial rotational symmetry. We consider the effect of a Zeeman field in the z direction, which can be realized by a magnetization or an external magnetic field. Because the Zeeman coupling will lift the spin degeneracy, two doublydegenerate bands become four singlydegenerate bands. However, since the bands can be distinguished by their rotation eigenvalue, protected twofold band crossings remain, as shown in Fig. 3b. This corresponds to splitting each Dirac node into a pair of Weyl nodes with opposite chiral charge. Each blue shaded area shows the separation between the pair of Weyl nodes that arise from the splitting of a Dirac node in energy and momentum space. These areas also define the regions with nonzero Chern number. Specifically, we consider a 2D k _{ x }, k _{ y } slice of the BZ perpendicular to the k _{ z } axis, and we calculate the Chern number of the band structure on such a slice for all bands below some energy E. The Chern number of the slice is only nonzero if the pair (k _{ z }, E) lies within the blue shaded region.
The effect of Zeeman field in z direction, which breaks \({ {\mathcal M} }_{x}\), is quite different for the new TM. As shown in Fig. 3c, the twofold degeneracy between the \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) bands is lifted. As a result the doublydegenerate (blue) band splits into two singlydegenerate bands, each of which crosses with the third band to form a Weyl node. Therefore, each triplydegenerate fermion splits into a pair of Weyl nodes with opposite chiral charge. We point out a number of key distinctions between the Dirac semimetal and the new TM cases. First, in a Dirac semimetal, the immediate pair of Weyl nodes that emerge from the same Dirac node (e.g., \({W}_{1}^{}\) and \({W}_{1}^{+}\) in Fig. 3b) arise from crossings between the same two bands (the yellow and red bands). By contrast, in the new TM, the pair of Weyl nodes that emerge from the same triplydegenerate node (e.g., \({W}_{1}^{}\) and \({W}_{2}^{+}\) in Fig. 3d) arise from the crossings between three different bands. Specifically, \({W}_{1}^{}\) is due to the crossing between the black and the yellow bands whereas \({W}_{2}^{+}\) is due to the crossing between the yellow and the red bands. As a result, the energymomentum region with nonzero Chern number (the blue shaded area) in the new TM is drastically different from that of in a Dirac semimetal and spans across all k _{ z }. Figures 3e,f further show how a triplydegenerate node splits into a pair of Weyl nodes under a Zeeman coupling, for the typeI and II cases, respectively.
Landau level spectrum
In order to understand the magneto transport property of the new TM, we now compare and study the Landau level spectrum arising from triplydegenerate fermions and Weyl fermions. The application of an external magnetic field quantizes the 3D band structure into effective 1D Landau bands that disperse along the kdirection that is parallel to the field. In Fig. 3g the Landau level spectrum along the k _{ z } is shown for a magnetic field applied along the z direction. The Weyl fermion is shown to have a gapless chiral Landau level spectrum. Specifically, besides many parabolic bands away from the Fermi level forming the conduction and valence bands, and we observe a zeroth Landau band (red) extending across the Fermi level. The sign of the velocity of the chiral zeroth Landau level is determined by the chirality of the Weyl fermion.
This is contrasted with Fig. 3h, showing the Landau level spectrum along k _{ z } for typeI and typeII triplydegenerate fermions in the left and right panel, respectively. We first point out the similarities between the Weyl fermion and the triplydegenerate fermion cases. We see that the Landau levels found in the Weyl fermion case, i.e., the gapped Landau levels away from the Fermi level and the gapless chiral zeroth Landau level crossing the Fermi level, are also observed in the triplydegenerate fermion case. We now emphasize the differences. Essentially, we see a number of equally spaced Landau levels parallel to the zeroth (red) one, which are not present in the Weyl case. We can qualitatively understand these results by visualizing the triplydegenerate band crossing as a Weyl cone plus a third band. This can be clearly seen in the cartoon in Fig. 1d,e. For the Landau level structure, the greenblue cone acts like a Weyl cone, while the yellow surface is the third band. They overlap each other on a line that is along the k _{ z } direction. The Landau level spectrum can be explained using this picture. While the Weyl cone will contribute its characteristic Landau level sturcture, additional Landau levels observed can be explained by the third band. Particularly, if the third band were like a completely flat surface, meaning that it has no dispersion along the inplane k _{ x } and k _{ y } directions, then all additional bands would be degenerate with the zeroth chiral Landu level, and the zeroth chiral Landau level would have a huge degeneracy. In real materials, the third band will have finite inplane dispersion. Hence the additional bands become nondegenerate with the zeroth chiral Landau level. This demonstrates that the Landau level spectrum of the triplydegenerate fermion is distinctly different from that of Weyl semimetals. This finding suggests novel magnetotransport responses and further demonstrates the exotic and unique properties of TMs with emergent triplydegenerate fermions.
Material realizations
We have determined the space groups in which the new TM state can occur and identified material candidates for each space group. Importantly, the material candidates that we identified cover both Class α/β and type I/II. The space groups include #187–#190 for Class α and #156–#159 for Class β. A list of the candidate materials is presented in Table 1. Here, we take the example of tungsten carbide, WC, as shown in Fig. 4. WC crystalizes in a hexagonal Bravais lattice, space group P6m2 (#187). The unit cell is shown in Fig. 4a), which obviously breaks spaceinversion symmetry. The crystal has the C _{3z } rotational symmetry and both horizontal (\({ {\mathcal M} }_{z}\)) and vertical (\({ {\mathcal M} }_{x}\)) mirror planes. Hence we expect the new band crossing to be Class α. Figures 4c,d show the firstprinciples calculated band structures without and with SOC. Triplydegenerate band crossings are seen in both cases. We discuss the band crossing in the presence of SOC in detail. Figure 4e, left panel, shows the zoomedin energy dispersion of the band crossing along k _{ z }. It can be seen that the doublydegenerate band (the blue curve) crosses with two singlydegenerate bands (black curves) forming two triplydegenerate nodes. The right panel shows the inplane (k _{ a }) dispersion that goes through one of the triplydegenerate nodes, where we clearly see that three singlydegenerate bands cross each other at one point. Finally, in Fig. 4f, we show that the triplydegenerate nodes in WC indeed split into pairs of Weyl nodes of opposite chirality in the presence of a Zeeman coupling.
We now study the surface states of WC. We choose the (100) surface so that the triplydegenerate nodes are not projected onto the same k point on the surface. The color plot in Fig. 5b shows the surface state band structure along the k _{ z } direction (the \(\tilde{{\rm{\Gamma }}}\tilde{A}\) line). We also superimpose the bulk bands (the white lines) along k _{ z } onto this plot. We clearly see a pair of surface Fermi arcs emerging out of the triplydegenerate nodes with a higher energy (T1). On the other hand, because the lower triplydegenerate nodes (T2) are masked by other irrelevant bulk bands when projected onto the surface, we cannot tell whether they are also connected by the surface Fermi arcs. The constant energy contour map at energy E = E _{ T1} reveal the same phenomenon. That is, each triplydegenerate node is connected by two surface state Fermi arcs. We discuss a few essential aspects in connection to the arc character of the surface states: The Zeeman coupling effect shown in Figs. 3a–f shows that the triplydegenerate node splits into a pair of Weyl nodes of opposite chirality. From this angle, it makes sense that each triplydegenerate node is connected by a pair of Fermi arcs. Now, the question is that whether the arc character of the surface states is protected. Or in other words, whether the surface states are required to go through the triplydegenerate nodes. We know that the arc character is guaranteed in the case for Weyl semimetals because a Weyl node carry a net chiral charge. We also know that it is not guaranteed in Dirac semimetals because a Dirac node does not carry a net chiral charge^{44}. In our case, a definite conclusion is currently not possible because whether there exists a topological invariant for the triplydegenerate node is unknown. This is a highly valuable open question that requires further investigations in theory. On the other hand, one can define 2D topological invariants such as a mirror Chern number or a \({{\mathscr{Z}}}_{2}\) number on a 2D slice of the BZ as discussed above. Our calculation shows that the mirror Chern number \({n}_{ {\mathcal M} }=\,\,1\) for the k _{ z } = π plane, indicating that there should be one surface state connecting the bulk band gap along the \(\bar{Z}\bar{M}\) segment. In Fig. 5d, we see three surface state along \(\bar{Z}\bar{M}\). Two of them that enclose the \(\bar{M}\) point are trivial because they do not connect across the band gap. The third one, which is the Fermi arc that goes all the way from the triplydegenerate point to the \(\bar{Z}\bar{M}\) line is nontrivial. Therefore, the observed surface states are consistent with the \({n}_{ {\mathcal M} }=\,1\) at k _{ z } = π.
As was described in the discussion above, the distinguishing properties of Class α and Class β are understood to be a manifestation of the presence and absence of \({ {\mathcal M} }_{z}\) mirror symmetry, respectively. As shown in Fig. 6a, WC has a \({ {\mathcal M} }_{z}\) mirror symmetry plane at k _{ z } = 0, which, according to its band structure around the triply degenerate band crossing shown in Fig. 6b, is of Class α. At a point away from the triply degenerate node but along the k _{ z }direction, the doubly degenerate band along k _{ y }direction, Fig. 6c, possess a quadratic band dispersion and touching point between the \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) bands. Now, by computing the Berry phase around a closed loop around the open segment, we observe a trivial 2π result, as shown in Fig. 6d. By now proceeding with breaking the \({ {\mathcal M} }_{z}\) mirror symmetry through shifting the W atom along the clattice constant direction, Fig. 6e, we observe that the twofold degenerate 1D band crossings now form four strands at every cut of constant k _{ z }, one of which is pinned to align with the k _{ z } axis. Similar to before, by looking at the inplane dispersion along k _{ y }, the initial quadratic dispersion of the \({\psi }_{1}^{^{\prime} }\) and \({\psi }_{2}^{^{\prime} }\) bands and their touching point results in two linearly dispersing touching points in the absence of \({ {\mathcal M} }_{z}\), as shown in Fig. 6g. By computing the Berry phase around the new touching points, enclosed by a red circle in Fig. 6h, we observe a nontrivial ±π value around each line. In the supplementary information (SI), we include band structure calculations of other Nexus fermion compounds.
In summary, the exploration of TMs has recently experienced a lot of progress and interest. While initially the attraction in TMs was amplified by the realization that the analogues of fermionic particles (e.g. Dirac, Weyl and Majorana fermions) in quantum field theory could be realized in a crystals kspace, we are now reaching a point in our understanding that is allowing the study of quasiparticle excitations arising from protected band crossings that do not have a direct analogy in the Standard Model. A crucial insight into the understanding in TMs is the importance of the band crossing dimensionality. While Weyl and Dirac semimetals have zerodimensional points, the band crossing of nodalline semimetals forms a onedimensional closed loop. In this paper, we reported on a new TM that features a triplydegenerate band crossing thereby realizing quasiparticles that have no analog in quantum field theory. Furthermore, the band crossing is neither 0D or 1D, but a combination of both since the two isolated triplydegenerate nodes are interconnected by multiple segments of lines that are doublydegenerate. We also present a list of crystalline candidate crystals that may realize this new TM. To further elucidate the distinguishing properties of this new threefold degenerate band degeneracy, we performed detailed calculations on the material candidate WC and studied the Landau level spectrum arising from the node, which is distinct from Dirac and Weyl semimetals. Our results are not only pivotal to the development of our understanding of topological phases of quantum matter, but also provide suitable platforms to experimentally elucidate the transport anomalies and spectroscopic responses in these new TM crystals, which have nontrivial band topology that go beyond the Weyl/Dirac paradigm.
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Acknowledgements
Work at Princeton University was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under DEFG0205ER46200. Work at the National University of Singapore was supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its NRF fellowship (NRF award no. NRFNRFF2013 03). T.R.C. and H.T.J. were supported by the National Science Council, Taiwan. H.T.J. also thanks the National Center for HighPerformance Computing, Computer and Information Network Center, National Taiwan. University, and the National Center for Theoretical Sciences, Taiwan, for technical support. Work at Northeastern. University was supported by U.S. DOE/BES grant no. DEFG0207ER46352 and benefited from Northeastern University’s Advanced Scientific Computation Center and the National Energy Research Scientific Computing Center supercomputing center through DOE grant no. DEAC0205CH11231. S.M.H., G.C., and T.R.C. acknowledge their visiting scholar positions at Princeton University, which were funded by the Gordon and Betty Moore Foundation EPiQS Initiative through grant GBMF4547 (Hasan).
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All authors contributed to the intellectual contents of this work. Preliminary material search and initial design of the prediction were done by S.Y.X. and G.C. with help from all authors. The band structure calculations were then performed by G.C., S.M.H., C.H.H., T.R.C., H.T.J., H.L.; Theoretical analysis were done by G.C., S.M.H., T.N., and H.L.; Landau level calculations were performed by Z.M.Y. and S.A.Y.; G.C., S.Y.X., T.N., D.S.S., G.B., I.B., N.A., H.Z., H.L., and M.Z.H. wrote the manuscript; S.Y.X., H.L. and M.Z.H. were responsible for the overall research direction, planning and integration among different research units.
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Chang, G., Xu, SY., Huang, SM. et al. Nexus fermions in topological symmorphic crystalline metals. Sci Rep 7, 1688 (2017). https://doi.org/10.1038/s41598017015238
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