On the topological character of three-dimensional Nexus triple point degeneracies

Band theory of electronic structure occupies a venerable place in quantum condensed matter. Historically, the basic ideas were established quite soon after the development of quantum mechanics. Yet, it is still an active area of research with many surprises. Among the surprises, topological band insulators and superconductors have captured our imagination in a big way. Hasan and Kane (2010) They had their antecedent in the integer quantum Hall effect. Thouless et al. (1982); Klitzing et al. (1980) The electronic structure of these quantum states of matter have interesting and robust phenomenology, e.g. the edge states of topological bands. Wen (1991); Hasan and Kane (2010) Another surprise has been the wealth of physics present in two and three dimensional ($2d$ and $3d$) semimetals. Vafek and Vishwanath (2014); Armitage et al. (2018) The low energy excitations in semimetals also often possess a topological character. This can lead to a certain robustness against back-scattering. Wehling et al. (2014) Already the low density of semimetallic carriers at the Fermi energy makes the effect of interactions less relevant. The combination of these two effects holds promise for technological applications of semimetals. Yan and Felser (2017)

From a theoretical point of view, what gives the semimetallic carriers their topological character is the global structure of their wavefunction geometry in the Brillouin zone. Dirac and Weyl semimetals are the well-known examples in $2d$ and $3d$. These semimetals have two-fold degeneracies (not counting spin) at isolated points in the Brillouin zone often protected by certain symmetries. Vafek and Vishwanath (2014); Das et al. (2020) They can be thought as “topological defects” in the space of the band wavefunctions. The semimetallic character obtains when the Fermi energy is near these degeneracies. Such two-fold point degeneracies are generic only in $3d$, while they are exceptional in $2d$ thus requiring symmetry protection. von Neuman and Wigner (1929) Recently, generalization of Dirac and Weyl fermions have also been found by symmetry-protecting higher-fold point degeneracies. Bradlyn et al. (2016)

While there has been tremendous activity on semimetals with point degeneracies, it has also been realized that band structures with two-fold line degeneracies are another possibility in the universe of possible band structures. Line degeneracies are exceptional in $3d$, and symmetry protection is required to obtain them. Several symmetry protected possibilities have been identified recently. Burkov et al. (2011); Phillips and Aji (2014); Zhu et al. (2016); Chang et al. (2017) Among these, there is a class of band structures where two-fold line degeneracies meet at three-fold or triple point degeneracies. They have been dubbed as Nexus fermions. Chang et al. (2017); Heikkilä and Volovik (2015) There have been material proposals Chang et al. (2017); Zhang et al. (2017); Feng et al. (2018); Weng et al. (2016a, b); Hyart and Heikkilä (2016) and experimental observations Lv et al. (2017); Ma et al. (2018) on this class of fermions. Their spectral structure is intriguing, and their band topology has been analysed previously in terms of the line degeneracies and $\mathbb{Z}_{2}$ topological numbers. Winkler et al. (2019) The goal of this paper is to shed more light on the band topology of Nexus fermions in a different manner which particularly emphasizes Nexus triple points themselves. We want to characterize the topology of these triple point degeneracies when thought of as defects in the space of band wavefunctions.

The topological character of point degeneracies can be understood by studying the band topology in one lower dimension. Schnyder et al. (2008) One generally considers a surface in the momentum space that encloses the $3d$ point degeneracy in question. Since the surface can be chosen to be gapped everywhere, one then computes the Chern number on this surface which serves as a topological charge for the point degeneracy. This discrete topological charge can not be changed by small deformations to the Hamiltonian. This approach will fail to characterize a Nexus triple point degeneracy, because any surface enclosing it will have gapless points where the line degeneracies intersect with the chosen surface. Thus the general principle of calculating a topological charge on an enclosing surface will not work. This is why Ref. Chang et al., 2017 called Nexus fermions as “beyond-Weyl”. If we restrict ourselves to use only gapped lower dimensional spaces, one can characterize the topology of the line degeneracies by considering gapped loops around them. Heikkilä and Volovik (2015); Zhu et al. (2016); Winkler et al. (2019) Also, one can enclose two or more different triple points together by a gapped surface in some cases and calculate a topological invariant on that surface. Zhu et al. (2016)

The question then is how to proceed in order to characterize the band topology of a Nexus system. This includes the basic issue of whether Nexus triple points have a topological character or not. This is a relevant question not just as a conceptual issue, but also because of the following physical point: in Weyl systems, the surface Fermi arcs have a protection in the sense that they have to end at the projection of the bulk Weyl points on to the surface. Armitage et al. (2018) This protection is linked to the fact that the Weyl points in the bulk possess a topological character. Ref. Chang et al., 2017 raised the analogous question on whether the surface Fermi arcs numerically observed in their chosen Nexus systems have a topological protection in the sense of Weyl Fermi arcs. See the discussion on the Nexus Fermi arcs in Ref. Chang et al., 2017 for more on this point. Our paper gives a constructive method to capture the topological character of different Nexus triple points. This method is the main result of this paper. Thus, we give an affirmative answer to the question raised in Ref. Chang et al., 2017, i.e. there will be surface Fermi arcs in Nexus systems that will have to end at the projection of the bulk triple points on to surface. We note here that Ref. Zhu et al., 2016 give an alternate argument for the presence of protected Fermi arcs in these systems based on mirror Chern numbers Teo et al. (2008) without concerning directly with the topological character of the Nexus points.

Our method relies crucially on the analytic properties of the band wavefunctions near the line degeneracies. This builds on the results of Ref. Das and Pujari, 2019 where a toy $2d$ band structure was considered which had a certain likeness to the Nexus band structures. In particular, specific $2d$ cuts of some Nexus band structure resembles the toy band structure considered in Ref. Das and Pujari, 2019. The wavefunctions of this toy model were written down which made the band topology explicit. The $2d$ topology could be captured by a generalization of winding numbers. Park and Marzari (2011); Das and Pujari (2019) This taught us the bigger lesson that near line degeneracies, analytic continuation or movement in the space of wavefunctions is key to exposing the band topology even in $3d$.

Motivated by the above, we will study in detail the analyticity properties of several $3d$ Nexus band structures. We will use Dirac and Weyl systems as scaffolding for the analyticity discussions of Nexus band structures. In the process, we will come to an important notion of the generalized domain when dealing with degeneracies. This will be necessitated by the presence of degenerate points on the surface enclosing the Nexus triple point degeneracy. For point degeneracies like Weyl points, this notion is not necessitated because we can easily find a gapped surface to surround the Weyl point.

Equipped with the generalized domain, we can finally state data on the band topology of a Nexus band structure. This scheme will consist of specifying and counting the distinct analytic loops that can be drawn on the generalized domain around a triple point. Thus we will have the desired scheme to distinguish different triple point degeneracies based on their distinct band topology data. This idea is very similar to the homology classes of 1-cycles used to distinguish the topology of different geometric objects. Nakahara (2003) The familiar example is that of a sphere vs. a torus. The sphere admits no loops that can’t be contracted to a point, whereas a torus admits two distinct classes of loops that can’t be contracted to a point. Our scheme will do a similar characterization of the triple points, with the structure of the homology classes being dictated by the structure of the line degeneracies. In this way, we will be able to describe several Nexus band structures written down in the literature Chang et al. (2017) as well as some obtained as $3d$ extensions of the toy band structure in Ref. Das and Pujari, 2019. This is the culminating result of this paper. Furthermore, this scheme can also potentially reveal the inter-relationships between different kinds of triple points.

We give a brief outline of the paper: Sec. II sets the stage by recapitulating some $2d$ band structures from the point of view of analyticity. We will be paying close attention to what happens near degeneracies, since that is the main roadblock in understanding the band topology of Nexus band structures. Doing this will introduce the notion of the generalized domain. We then go on $3d$ in Sec. III. We start by discussing the familiar Weyl system to give a clear contrast to Nexus band structures in terms of their analyticity properties. We then discuss several Nexus band structures. Sec. IV will finally give the method to state the band topology data in terms of homology classes of analytic loops on a generalized domain around the triple point. This won’t be hindered by a lack of gapped property, because analyticity near the degeneracies constrain the wavefunctions enough to enable stating the topology. This will conclude our exposition on the band topology of Nexus fermions. We end the paper in Sec. V with a summary and outlook. We also discuss here our take on the Fermi arc phenomenology of Nexus systems including a conjecture regarding the charge of these surface states.

In this section, we will start with the analyticity discussion in a $2d$ beyond-Dirac Nexus system. Let’s reconsider the band structure introduced in Ref. Das and Pujari, 2019 to set up the discussion:

The eigensystem of $H(\mathbf{p})$ is

where $\theta_{\mathbf{p}}=\arctan\left(\frac{p_{y}}{p_{x}}\right)\in[0,2\pi)$. $\omega=e^{i\frac{2\pi}{3}}$, $\omega^{2}=e^{-i\frac{2\pi}{3}}$ are the complex cube roots of unity and $\alpha=0,1,2$. This band structure possesses a three-fold degeneracy at $\mathbf{p}=0$ clearly, and has two line degeneracies coming out from the triple point which is a signature feature of Nexus wavefunctions. Because of the line degeneracies, a standard Berry phase description of the wavefunction geometry is not applicable. However, we had used generalized winding numbers Das and Pujari (2019); Park and Marzari (2011) to understand this $2d$ wavefunction geometry (cf. Table I and Sec. II of Ref. Das and Pujari, 2019) and contrasted with other known $2d$ Dirac-like wavefunction geometries. In $3d$, such winding number description is not generally applicable for classification of point degeneracies. Thus, we will take the approach to be described below and in future sections.

Our main point of view will be to understand and write down the key aspects of the analytic behavior of various band structures. This is a different way of communicating invariant data of the wavefunction geometry than winding numbers and Berry phases. For example, we often view the familiar two-fold Dirac system

with the eigensystem as

by calculating Berry phase or chiral winding number Ryu et al. (2010); Schnyder et al. (2008) on gapped region in one lower dimension (e.g. any closed loop around the degeneracy). We rather want to include the degeneracy to be a part of the analysis.

Firstly, on a gapped loop we clearly have the analyticity property

However, we also have the following analyticity property of Dirac wavefunctions

which connects the two bands. In fact, this relation tells us how to consistently arrive at the two-fold degeneracy from all sides without running into analytic ambiguities. Thus, we can interpret this as the way to move analytically across the point degeneracy. This is illustrated in the two figures from the left in Fig. 1.

Eq. 5 and 6 are nothing but an alternate way of describing the wavefunction geometry that is captured by Berry phase and chiral winding numbers, with the additional benefit of allowing to move across the degeneracy in an analytically smooth way. This alternate viewpoint will prove useful for us because Nexus triple points can not be enclosed by a gapped region in one lower dimension. As notation, we refer to analyticity relations with the same band index on left and right hand sides as “index-preserving” (e.g. Eq. 5), while analyticity relations with different band indices on both sides as “index-connecting” (e.g. Eq. 6).

For a quadratic band touching (QBT), the analyticity relation is in fact

We can understand this in terms of two Dirac points (of same winding) sitting on top of each other. Let’s first imagine these two Dirac points are not on top of each other, and we move analytically across both the degeneracies in a single go. In this process, we will return back to the same band that we started from as illustrated in Fig. 2. Now, imagine moving these two Dirac points till they fall on top of each other to obtain a QBT. Analyticity thus forces us that we will stay in the same band when we cross the QBT (bottom panel of Fig. 2), i.e. Eq. 7. This argument also works when the QBT splits into more Dirac points, e.g. in Bernal-stacked honeycomb bilayer lattice in presence of “trigonal” warping terms when it splits into three Dirac points of same winding and a fourth one with opposite winding. Mikitik and Sharlai (2008)

With the above discussion in hand, we can revisit the $2d$ system in Eq. 1 and 2 in terms of its key analyticity information. The index-connecting relation is

which may be contrasted with Eq. 6 in the Dirac case. Similarly, in contrast to Eq. 5, we arrive at the index-preserving relation

This unusual “$+6\pi$” structure is a result of the Nexus lines emanating from the three-fold degeneracy. We emphasize that the above two relations are an alternate way of describing the wavefunction geometry when compared to a generalized winding number description. Das and Pujari (2019) This way of stating the wavefunction geometry via the analyticity will be our approach to tackle the $3d$ Nexus geometry in the next sections.

We end this section with a final conceptual point. Even though Eq. 1 and 3 are multi-band systems as expressed through band-indexed eigenfunctions in Eq. 2 and 4, the analytic structure actually tells us that this band distinction is a matter of convenience or convention and not fundamental when considering the wavefunction geometry. We can imagine a single function defined on a generalized domain that describes the multi-band wavefunctions in analogy with Riemann surfaces. This analogy can be made exact for Eq. 2 by re-writing as $\epsilon_{\alpha}(\mathbf{p})=2p~{}\text{Re}\left[\omega^{2+\alpha}e^{i\frac{\theta_{\mathbf{p}}}{3}}\right]$, $v_{\alpha}(\mathbf{p})=\frac{1}{\sqrt{3}}\left(\omega^{2+\alpha}e^{-i\frac{2\theta_{\mathbf{p}}}{3}}~{}~{}~{}(\omega^{*})^{2+\alpha}e^{i\frac{2\theta_{\mathbf{p}}}{3}}~{}~{}~{}1\right)^{T}$ whereby we can essentially drop the $\alpha$ index to write as $\epsilon(\mathbf{p})=2p~{}\text{Re}\left[e^{i\frac{\theta_{\mathbf{p}}}{3}}\right]$, $v(\mathbf{p})=\frac{1}{\sqrt{3}}\left(e^{-i\frac{2\theta_{\mathbf{p}}}{3}}~{}~{}~{}e^{i\frac{2\theta_{\mathbf{p}}}{3}}~{}~{}~{}1\right)^{T}$ that is defined on a three-fold Riemann surface connected by branch cuts of the complex cube root function. This generalized domain restatement succinctly tells us how to analytically move in the space of wavefunctions, which is of course a key requirement to understand the wavefunction geometry.

A similar generalized domain restatement can be done for the case of Dirac eigensystem Eq. 4. The generalized domain is composed of two copies of the $p_{x}$-$p_{y}$ plane connected at the point-degeneracy. The band-connection relation (Eq. 6) gives us the rule of moving through the “connecting point” in the generalized domain from one copy of the $p_{x}$-$p_{y}$ plane to the other (see the rightmost figure of Fig. 1). In the Dirac case, there is no branch cut structure since the eigensystem (Eq. 4) is perfectly analytic. The generalized domain will be used when we discuss the $3d$ Nexus wavefunctions in the next sections.

In this section, we start with the actual discussion on the analyticity properties of $3d$ Nexus fermions. As mentioned in Sec. I, line degeneracies are exceptional in $3d$ and require symmetry protection, whereas they are fine-tuned in $2d$. Thus the analyticity discussion in the previous section is for a fine-tuned case, but it will help us in the following discussions. Before we go towards Nexus analyticity properties, let us start with the familiar case of Weyl point degeneracies to set the stage.

In the previous sections, we established the rules to move smoothly in our parameter space. Here, parameter space refers to the generalized domain. In this section, we will describe a (topological) characterization scheme for different kinds of Nexus triple points by making use of these rules. Given a Nexus system, the basic idea will be to consider an enclosing surface around the triple point in the generalized domain. As remarked at the end of the previous section, the enclosing surface in the generalized domain consists of three copies of the surface (e.g. spheres) joined at the points where the line degeneracies cross them. On this generalized enclosing surface, we will categorize the various topologically distinct ways in which one may analytically loop back to the start point. This is reminiscent of the concept of homology classes of 1-cycles Munkres (2013) in topological classification of geometric objects. A very familiar example of this are the non-trivial loops that one draws on a torus that can not be shrunk to a point, whereas on a sphere there are no such loops. Importantly, the analyticity relations discussed before allows us to focus only on the enclosing surface to capture the topological data of the wavefunction geometry without the full knowledge of the wavefunctions themselves.

Let’s start with $H^{8}$ in this case. The enclosing surface for this is shown in Fig. 5. Let us imagine drawing topologically distinct loops on this. Clearly there exist (trivial) loops that can be shrunk to a point (not shown in the figures). $H^{8}$ also hosts non-trivial loops which are shown in Fig. 6. We see there are two kinds of loops:

1. 1.

   those that stay on the same sphere. The drawing of such loops relies on the index-preserving kind of analytic relations.

2. 2.

   those that straddle different spheres. The drawing of such loops relies on the index-connecting kind of analytic relations.

Close to the connecting point on the $2d$ enclosing surface, we can imagine a small flat coordinate patch giving us our local coordinate system in which we may use Eq. 6. Therefore, in the drawing of the loop through the connecting point, we have to use the step illustrated in Fig 1. A corollary is that there can not be a non-trivial loop on a single sphere that touches the Dirac-like connecting point.

With these basic steps in hand, we can enumerate all the non-trivial homological classes and they are shown in Fig. 6. There are three categories of non-trivial loops. They are

1. 1.

   loops involving only two connecting points, they can be either on the left-middle sphere pair, or middle-right sphere pair.

2. 2.

   Loops involving all the connecting points, the two connecting points on the left and right spheres have to be joined, while on the middle sphere we have the two choices shown in Fig. 6.

3. 3.

   Loops on the same sphere that enclose the connecting points.

For the case of $H^{3}$, the generalized enclosing surface is shown in second panel of Fig. 7. For this case there is only one possible non-contractible loop in the middle sphere. This captures the band topology of $H^{3}$ and shows its distinction from $H^{8}$ (and other cases). From the above discussions, we can immediately conclude that the $\Lambda^{8}$ triple point and two different $\Lambda^{3}$ and $\tilde{\Lambda}^{3}$ triple points inside the enclosing surface are not topologically different. In our scheme, the distinction between different topological cases are categorized using the non-contractible loops or 1-cycles. The number of distinct loops only depends on the number (and kind) of the connecting points (Dirac-like, or possibly QBT as in the examples to follow) on the enclosing surface. Thus one cannot distinguish between pair of $\Lambda^{3},\tilde{\Lambda}^{3}$ triple points and a single $\Lambda^{8}$ triple point which gives us a thumb rule for composition of these triple point topological defects.

We end with an application of our scheme to recent Nexus triple points discussed in the literature which have possible material realizations. Chang et al. (2017) For the type II nexus system as notated by Chang et al, there are four line degeneracies coming out of the triple point: one along the $z$-axis and the rest three oriented at $\frac{2\pi}{3}$ angular separation about the $z$-axis lying in high symmetry planes. See Eq. 2,3 in Ref. Chang et al., 2017 for the low-energy Hamiltonian, and Fig. 1 for the line degeneracy structure. This happens due to the presence of $\mathcal{C}_{3z}$ crystal symmetry. Chang et al. (2017); Winkler et al. (2019) In this case, the generalized domain for the surface enclosing the triple point degeneracy will have three spheres connected to each other at the points where they intersect the line degeneracies. The topology of this system can thus be similarly understood using the homology classes as discussed above. On this surface the loops are again of three main types (diagram not show due to proliferation of non-contractible loops): 1) loop enclosing one connecting point, 2) loop spanning two spheres, 3) loop spanning through 3 spheres. Even though these three types were also present in the case of $H^{8}$, the count of each type is different which topologically distinguishes the two cases.

The case of type I as notated by Chang et al is worth noting. The generalized enclosing surface in this case looks similar to that of $H^{3}$ (Fig. 7). However, the characterization of non-contractible loops is different than the $H^{3}$ case. This is due to the degeneracies being QBT-like in this case. Chang et al. (2017) Thus while drawing the loops, we have to follow the rule as shown in Fig. 2’s bottom panel. This allows for a new kind of non-contractible loop on the same sphere which goes through the connecting point. We show the various possible loops in Fig. 8. This new kind of loops as in middle and bottom panel of Fig. 8 are not possible when the connecting points are Dirac-like because in that case we are necessarily forced to go to the connected sphere due to analyticity (Fig. 1). We remark here that Ref. Chang et al., 2017’s statement that the line degeneracies are characterized by a $2\pi$ Berry phase does not paint the full picture. Such a characterization strictly can only be applied to the non-contractible loop shown on the top panel of Fig. 8 and not in general. Our scheme helps to make clear which loops have a topological property based on analyticity. Once we have such loops in hand, we may compute familiar topological invariants Heikkilä and Volovik (2015) on the gapped ones among them.

In summary, we laid a general scheme to describe the band topology of the so-called Nexus triple point fermions. This was based on an understanding of the analyticity properties near (line-)degeneracies which are an integral part of the Nexus band structure. This scheme is built on the insight gained in Ref. Das and Pujari, 2019 where we could see the analyticity properties near a line degeneracy explicitly. The discussion started with the known cases of Dirac and QBT bands in $2d$ in Sec. II. We use analyticity to define a generalized domain where we go smoothly across the “connecting” points at the degeneracies (see Fig. 1). We emphasise that in the original domain there is a non-analyticity in the space of wavefunctions at a (non-accidental) degeneracy which is then considered as a topological defect. In the generalized domain, however, this issue is not there. For the $2d$ Nexus case, the generalized domain is a familiar object – Riemann surfaces associated with $z^{1/3}$ – known from the study of complex analysis. However, the general idea is applicable in any situation. So we take this scheme to $3d$ and define generalized domains for 3d Nexus triple points in Sec. III.

In analogy with Weyl points and Chern numbers on associated enclosing surfaces, we characterize the $3d$ Nexus points by enclosing them in the generalized domain (see bottom panel of Fig. 5, 7) in a departure from existing literature. Sec. IV describes the triple point defect topology in terms of non-contractible loops that can be drawn on this enclosing surface. These are the 1-cycle homology classes of the generalized domain. Different Nexus triple points have their unique data of these 1-cycle homology classes. This discrete set of data gives the triple point its topological character, since they will be stable to small deformations of the Hamiltonian. We reiterate here again that this way of describing the topology is actually more general (e.g. we can enclose multiple Nexus points, etc.), however, we have principally concerned ourselves with single Nexus triple points. Line-degeneracies on the other hand are characterizable by using topological invariants defined on the gapped loops around them. Heikkilä and Volovik (2015) Our enclosing scheme is finally applied to examples of Nexus triple point in the literature which has possible material realizations, Zhu et al. (2016); Chang et al. (2017) whereas only the topology of gapped enclosing loops around the line degeneracies and their evolution across the triple point had previously been discussed. Chang et al. (2017); Zhu et al. (2016); Winkler et al. (2019)