Universality and stability of the edge states of chiral-symmetric topological semimetals and surface states of the Luttinger semimetal

TL;DR

This paper demonstrates that chiral symmetry in nodal semimetals guarantees universal edge state properties, which are robust against certain symmetry-breaking perturbations, extending the understanding of topological semimetals.

Contribution

It provides a general analysis of edge states in chiral-symmetric nodal semimetals, deriving boundary conditions, classifying universal edge state behaviors, and establishing their robustness.

Findings

Edge states are determined by the topological winding number N.

There are N+1 universal classes of boundary conditions.

Edge states persist within a finite stability region despite chiral symmetry breaking.

Abstract

We theoretically demonstrate that the chiral structure of the nodes of nodal semimetals is responsible for the existence and universal local properties of the edge states in the vicinity of the nodes. We perform a general analysis of the edge states for an isolated node of a 2D semimetal, protected by {\em chiral symmetry} and characterized by the topological winding number $N$. We derive the asymptotic chiral-symmetric boundary conditions and find that there are $N+1$ universal classes of them. The class determines the numbers of flat-band edge states on either side off the node in the 1D spectrum and the winding number $N$ gives the {\em total} number of edge states. We then show that the edge states of chiral nodal semimetals are {\em robust}: they persist in a finite-size {\em stability region} of parameters of chiral-asymmetric terms. This significantly extends the notion of 2D and 3D topological nodal semimetals. We demonstrate that the Luttinger model with a quadratic node for $j=\frac32$ electrons is a 3D topological semimetal in this new sense and predict that $\alpha$-Sn, HgTe, possibly Pr$_2$Ir$_2$O$_7$, and many other semimetals described by it are topological and exhibit surface states.

Figures

Equations

$$\hat{H}_{N}(p_{x},p_{y})=\left(\begin{array}[]{cc}0&p_{-}^{N}\\ p_{+}^{N}&0\end{array}\right),\mbox{ }p_{\pm}=p_{x}\pm{\text{i}}p_{y},$$

$$\hat{\psi }\to {\tau }_z\hat{\psi },\text{mbox}{\tau }_z=\text{diag}(1,-1),$$

$$c_{an}{\partial }_y^n{\psi }_a+c_{bn}{\partial }_y^n{\psi }_b=0$$

$${\partial }_y^n{\psi }_a=0\text{ and }{\partial }_y^n{\psi }_b=0$$

$${\partial }_y^n{\psi }_a=0\text{ or }{\partial }_y^n{\psi }_b=0.$$

$${\partial }_y^n{\psi }_{\lambda }=0,\text{mbox}(\lambda ,n)\in \Lambda .$$

$$N_a+N_b=N,$$

$$j_y(x,y=0)=0.$$

$$j_y=-\text{i}(j_{+}-j_{+}^{\ast }),\text{mbox}{j}_{+}=\sum _{n=0}^{N-1}({\hat{p}}_{+}^{N-1-n}{\psi }_a{)}^{\ast }{\hat{p}}_{-}^n{\psi }_b.$$

$${\psi }_a,\dots ,{\partial }_y^{N_a-1}{\psi }_a,{\psi }_b,\dots ,{\partial }_y^{N_b-1}{\psi }_b=0.$$

$${\hat{\psi }}_n(p_x>0,y)=(1,0{)}^{\text{T}}{y}^n{\text{e}}^{-p_{x}y},\text{mbox}n=N_a,\dots ,N-1,$$

$${\hat{\psi }}_n(p_x<0,y)=(0,1{)}^{\text{T}}{y}^n{\text{e}}^{+p_{x}y},\text{mbox}n=N_b,\dots ,N-1,$$

$$\hat{H}_{2}^{\beta}(p_{x},p_{y})=\left(\begin{array}[]{cc}(\beta_{0}+\beta_{z})p_{+}p_{-}&\beta_{\perp}p_{-}^{2}\\ \beta_{\perp}p_{+}^{2}&(\beta_{0}-\beta_{z})p_{+}p_{-}\end{array}\right)$$

$${\psi }_a,{\psi }_b=0$$

$${\mathcal{E}}_{\pm }(p_x)=2|{\beta }_{\perp }|\frac{{\beta }_0|{\beta }_{\perp }|\pm {\beta }_z\sqrt{|{\beta }_{\perp }{|}^2+{\beta }_z^2-{\beta }_0^2}}{|{\beta }_{\perp }{|}^2+{\beta }_z^2}{p}_x^2,$$

$${\hat{H}}^{\text{L}}(\mathbf{p})=({\alpha }_0+\frac{5}{2}{\alpha }_z){\mathbf{p}}^2{\hat{1}}_4-2{\alpha }_z(\hat{\mathbf{J}}\cdot \mathbf{p}{)}^2+{\alpha }_{\square }{\hat{M}}_{\square }(\mathbf{p}).$$

$${\hat{\psi }}^{\text{L}}=\hat{0}.$$

Full text

Introduction. Edge states in 2D nodal semimetals have been demonstrated in numerous theoretical calculations Fujita ; Nakada ; Brey ; Akhmerov ; FZhang ; Castro ; Arikawa ; MartinBlanter ; Xu ; LiTao ; Kunstmann ; Bahamon ; Jaskolski ; DeshpandeWinkler ; Chiu ; ChiuRMP , mainly for models describing monolayer and bilayer graphene. Their existence is attributed Chiu ; ChiuRMP to the topological invariants characterizing the nodes, the winding numbers $N$, which are well-defined in the presence of chiral symmetry. Still, up to now, the general structure of the edge states of chiral-symmetric 2D nodal semimetals and its relation to the winding numbers have not yet been explicitly established for arbitrary $N$.

In this Letter, we carry out this task locally in the Brillouin zone (BZ), by performing a general analytical analysis of the edge states in the vicinity of an isolated node. As the main advancement, we derive the most general form of the asymptotic boundary conditions (BCs) that respect chiral symmetry. We find that there are $N+1$ discrete universal classes of them. These classes describe all possible universal structures of the edge states (Fig. 1) and establish their connection to the winding number $N$.

We then address the stability properties of the edge states and show that they are robust under the effects of chiral symmetry breaking. This allows for a significant extension of the notion of a topological nodal semimetal in both 2D and 3D. As an important application of the developed framework, we demonstrate that the Luttinger model L with a quadratic node for $j=\frac{3}{2}$ electrons, describing materials like $\alpha$-Sn BP ; W ; Madelung , HgTe BP ; W ; Madelung , and $\text{Pr}_{2}\text{Ir}_{2}{\text{O}}_{7}$ Pr , exhibits surface states and is a 3D topological semimetal in this new more general sense.

2D chiral-symmetric nodal semimetal. First, we consider an isolated chiral node of a 2D semimetal arising at some point in the BZ from a degeneracy of two electron levels, to be denoted $a$ and $b$. We assume that the Hamiltonian for the two-component wave function $\hat{\psi}=(\psi_{a},\psi_{b})^{\text{T}}$ has the form

[TABLE]

to the leading order in momentum $(p_{x},p_{y})$ in the vicinity of the node; $N$ is a positive integer.

The Hamiltonian has chiral symmetry ChiuRMP : under the transformation

[TABLE]

it changes its sign, $\tau_{z}\hat{H}_{N}(p_{x},p_{y})\tau_{z}^{\dagger}=-\hat{H}_{N}(p_{x},p_{y})$. Due to chiral symmetry, $\hat{H}_{N}$ is characterized by a well-defined topological invariant, the winding number $N$, ChiuRMP related to the Berry phase $\pi N$.

We stress that the winding number $N$ is a local topological characteristic of the node in the BZ. The properties of the edge states that we study are also local, and we do not address the reasons for the existence of the node. We assume that the node is isolated from possible other nodes at different points in the BZ, also in the presence of the edge isolatednode .

Universal asymptotic chiral-symmetric boundary conditions. We first derive the most general form of the BCs for the Hamiltonian $\hat{H}_{N}$ that satisfy chiral symmetry. This derivation is free of any microscopic assumptions and, beside chiral symmetry, invokes only two natural requirements: long-wavelength limit and vanishing of the probability current perpendicular to the boundary.

We assume the sample occupies the half-plane $y>0$. Since the Schrödinger equation $\hat{H}_{N}(\hat{p}_{x},\hat{p}_{y})\hat{\psi}=\epsilon\hat{\psi}$ ($\hat{p}_{x,y}=-{\text{i}}\partial_{x,y}$) is a differential equation of order $N$ in each component $\psi_{a,b}$, BCs at the boundary $y=0$ are a set of $N$ linear homogeneous (meaning that the linear combinations are equated to zero) relations for the derivatives $\partial_{y}^{n}\psi_{a,b}\equiv\partial^{n}_{y}\psi_{a,b}(x,y=0)$, $n=0,\ldots,N-1$ ($\partial^{0}_{y}\psi_{a,b}=\psi_{a,b}$ being the components themselves); we drop the arguments of the functions in the BCs formulas for brevity.

The long-wavelength limit means the following. Any linear relation involving derivatives of different order necessarily contains spatial scales. Consider, for example, a relation $\partial_{y}\psi_{a}+l\partial_{y}^{2}\psi_{a}=0,$ characterized by a spatial scale $l$. In the long-wavelength limit, at spatial scales larger than $l$, the second term $l\partial_{y}^{2}\psi_{a}$ becomes negligible and the relation reduces to $\partial_{y}\psi_{a}=0$. Thus, in the BCs satisfying the requirement of the long-wavelength limit, to be referred to as asymptotic BCs, only the derivatives of the same order can be present in one relation. Therefore, for a given order $n$, there is either no BCs, or one BC

[TABLE]

with dimensionless coefficients $c_{an,bn}$, or two BCs

[TABLE]

with both derivatives vanishing individually.

Demanding chiral symmetry, we find that BC (3) remains invariant under the transformation (2) only if one of the coefficients $c_{an,bn}$ is zero, so that BC (3) reduces to either

[TABLE]

Combined with the possibility (4), we find that under chiral symmetry the most general form of the asymptotic BCs is when some $N$ out of $2N$ derivatives are individually nullified:

[TABLE]

Here, $\lambda=a,b$ and $n=0,\ldots,N-1$, and $\Lambda$ is a subset of size $N$ of $2N$ indices $(\lambda,n)$ labelling the said derivatives.

These $C^{N}_{2N}$ types of BCs can be sorted into $N+1$ groups $(N_{a},N_{b})$ with $N_{a,b}=0,...,N$, such that

[TABLE]

according to the numbers $N_{a,b}$ of constraints imposed on the derivatives $\partial^{n}_{y}\psi_{a,b}$ of a given component.

Finally, the hermiticity of the full Hamiltonian demands that the probability current perpendicular to the boundary must vanish at the boundary

[TABLE]

The expressions for the probability current for $\hat{H}_{N}$ [Eq. (1)] read [see Supplemental Material (SM) SM ]

[TABLE]

The bilinear form (7) must vanish identically for any $\hat{\psi}$. Inspecting Eq. (8) SM , we find that, for a given group $(N_{a},N_{b})$, only one BC among (5) is allowed, the one with the lowest-order derivatives nullified:

[TABLE]

These are all possible asymptotic chiral-symmetric current-conserving BCs for the chiral-symmetric Hamiltonian $\hat{H}_{N}$ [Eq. (1)]. There are $N+1$ classes $(N_{a},N_{b})$ of them. This is the first key result of this work. For $N=1,2$, all BC classes $(N_{a},N_{b})$ are shown in Fig. 1.

Edge states and the winding number $N$. Further, the edge states for the Hamiltonian $\hat{H}_{N}$ [Eq. (1)] and BCs (9) can be found explicitly SM . Taking the plane-wave form $\hat{\psi}(x,y)=\hat{\psi}(p_{x},y){\text{e}}^{{\text{i}}p_{x}x}$ with momentum $p_{x}$ along the edge, we find $N_{b}$ edge-state solutions

[TABLE]

at $p_{x}>0$ and $N_{a}$ edge-state solutions

[TABLE]

at $p_{x}<0$. All solutions have zero energy $\epsilon=0$ and thus represent flat bands.

Thus, we have shown that for a 2D chiral-symmetric nodal semimetal with both the bulk Hamiltonian [Eq. (1)] and BCs [Eq. (9)] obeying chiral symmetry, a set of flat-band edge states always exists asymptotically in the vicinity of an isolated node. The sum of the numbers $N_{a,b}$ of the edge-state bands on both sides $p_{x}\gtrless 0$ off the node in the 1D edge spectrum is equal to the winding number $N$, Eq. (6) and Fig. 1. This is the second key result of this work graphene .

And so, the total number of the edge-state bands is determined solely by the local in the BZ bulk characteristic of the node, the winding number $N$, irrespective of the chiral BC class $(N_{a},N_{b})$, which determines the numbers of the edge-state bands on either side off the node. The specific class $(N_{a},N_{b})$ is in general determined by the bulk Hamiltonian also away from the node, as well as by the orientation and microscopic structure of the edge. Still, $(N_{a},N_{b})$ are also topological numbers, since they cannot be changed by continuously changing the system parameters or surface orientation while preserving the chiral symmetry. Changes in the numbers $(N_{a},N_{b})$ of the edge-state bands can occur by changing the surface orientation only when projections of different nodes onto the surface collapse, which necessarily requires the presence of more than one node in the BZ. The BC class $(N/2,N/2)$, however, possible for even $N$, could be realized for just a single node in the whole BZ; the BC class $(1,1)$ will be relevant below.

Stability of chiral-symmetric edge states under breaking of chiral symmetry. The found edge states of a 2D chiral-symmetric nodal semimetal [Eqs. (1) and (9)] are stable under the effects of chiral-symmetry breaking. Both the bulk Hamiltonian and BCs may contain terms that break chiral symmetry, making them deviate from their chiral-symmetric forms (1) and (9). As chiral-asymmetric terms are introduced, edge states can disappear only by merging with particle or hole continua of bulk states. Since for preserved chiral symmetry the flat edge-state bands are positioned at $\epsilon=0$, it takes finite strength of chiral-asymmetric terms to force edge states merge with bulk states. We thus introduce the notion of the stability region of chiral-symmetric edge states: it is a finite-size region in the parameter space of chiral-asymmetric terms around the point of chiral symmetry, within which edge states persist. This is the third key result of this work.

We illustrate the effect of the chiral asymmetry of the bulk Hamiltonian for $N=2$ by considering the model

[TABLE]

with complex $\beta_{\perp}$ and real $\beta_{0,z}$. The terms due to $\beta_{0,z}$ break chiral symmetry; at $\beta_{0,z}=0$, $\hat{H}_{2}^{\beta}$ reduces to the chiral-symmetric form $\hat{H}_{2}$ [Eq. (1)]. The bulk spectrum of $\hat{H}_{2}^{\beta}$ is $\varepsilon_{\pm}(p_{\perp})=(\beta_{0}\pm\sqrt{|\beta_{\perp}|^{2}+\beta_{z}^{2}})p_{\perp}^{2},\mbox{ }p_{\perp}^{2}=p_{x}^{2}+p_{y}^{2}.$ For $\beta_{0}^{2}<|\beta_{\perp}|^{2}+\beta_{z}^{2}$, the system is in the nodal semimetal regime we will focus on, with particle and hole bands.

We calculate the edge states for chiral-symmetric BCs

[TABLE]

of class $(1,1)$, Fig. 1(b). We obtain SM the edge-state dispersion relations

[TABLE]

at $p_{x}\gtrless 0$, respectively, Fig. 2(a),(b),(c). In the plane $(\beta_{z},\beta_{0})$ of chiral-asymmetry parameters, Fig. 2(a), the semimetal region $|\beta_{0}|<\sqrt{|\beta_{\perp}|^{2}+\beta_{z}^{2}}$ consists of three subregions 2,1,0, labelled according to the numbers of the edge-state bands: stability region 2 (green), which contains the point of chiral symmetry $\beta_{0,z}=0$, and in which both bands $\mathcal{E}_{\pm}(p_{x})$ at $p_{x}\gtrless 0$, originating from the chiral-symmetric edge states of the BC class $(1,1)$ [Fig. 1(b)], persist, Fig. 2(b); stability region 1 (orange), where only one of the bands $\mathcal{E}_{\pm}(p_{x})$ on one side off $p_{x}=0$ exists, Fig. 2(c); and region 0 (magenta), where the edge states are absent. Regions 2 and 1 and regions 1 and 0 are separated by the curves $|\beta_{0}|=\pm\frac{|\beta_{\perp}|^{2}-\beta_{z}^{2}}{\sqrt{|\beta_{\perp}|^{2}+\beta_{z}^{2}}}$, respectively.

We illustrate the effect of chiral asymmetry of the BCs for the linear node ($N=1$) in SM SM .

Extended notion of 2D and 3D topological nodal semimetals. The above findings offer a significant extension of the notion of a 2D topological nodal semimetal: one may regard a 2D nodal semimetal as topological if it belongs to the stability region of some chiral-symmetric 2D nodal semimetal. The edge states of the latter are ensured by a well-defined topological invariant, the winding number $N$. Yet, exact chiral symmetry is not required, and the edge states will persist in the former as long as chiral-symmetric terms are dominant. The above example, Eqs. (10) and (11), Fig. 2, is a 2D topological nodal semimetal in this sense. This definition is then readily extended to 3D: one may regard a 3D nodal semimetal as topological, if its 2D reductions to at least some planes in momentum space passing through the node(s) are 2D topological nodal semimetals in the above sense. In this case, the 3D nodal semimetal will exhibit surface states of topological origin anisotropy .

This viewpoint has wide-reaching implications, since chiral terms are ubiquitous in the Hamiltonians of 2D and 3D nodal semimetals, though exact chiral symmetry is not necessarily present. It allows one to prove the existence and topological origin of the edge or surface states in semimetal systems by relating them to 2D chiral-symmetric models, even in cases when a precise topological invariant may be hard or impossible to define.

Luttinger model for $j=\frac{3}{2}$ electrons as a 3D topological semimetal. One important example of a 3D semimetal with a quadratic node that may be regarded as topological in this new sense is the 4-band Luttinger model L (LM) for electrons with $j=\frac{3}{2}$ angular momentum (Luttinger semimetal):

[TABLE]

Here, $\hat{1}_{4}$ is the unit matrix of order 4, and $\hat{{\bf J}}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$ are the angular-momentum matrices.

It describes the local electron band structure around the $\Gamma$ point of a material with full cubic point group $\mathbb{O}_{h}$ with inversion and time-reversal symmetry. It is the most general form up to quadratic order in momentum ${\bf p}=(p_{x},p_{y},p_{z})$ allowed by these symmetries. All four $j=\frac{3}{2}$ states are degenerate at ${\bf p}={\bf 0}$ due to $\mathbb{O}_{h}$ symmetry. Odd-${\bf p}$ terms are prohibited by inversion. The terms ${\bf p}^{2}\hat{1}_{4}$ and $(\hat{{\bf J}}\cdot{\bf p})^{2}$ are invariants of the full spherical symmetry group ${\text{O}}(3)$ with inversion. Their linear combination $\hat{H}^{\text{L}}({\bf p})|_{\alpha_{\square}=0}$, characterized by two parameters $\alpha_{0,z}$, is the ${\text{O}}(3)$-symmetric LM; its bulk spectrum has two double-degenerate bands $\varepsilon^{{\text{L}},{\text{O}}(3)}_{\pm}(p)=(\alpha_{0}\pm 2|\alpha_{z}|)p^{2}$, $p=|{\bf p}|$; $|\alpha_{0}|<2|\alpha_{z}|$ is the nodal semimetal regime. The additional term $\hat{M}_{\square}({\bf p})=\hat{J}_{x}^{2}p_{x}^{2}+\hat{J}_{y}^{2}p_{y}^{2}+\hat{J}_{z}^{2}p_{z}^{2}-\frac{2}{5}(\hat{{\bf J}}\cdot{\bf p})^{2}-\frac{1}{5}\hat{{\bf J}}^{2}{\bf p}^{2}$ with parameter $\alpha_{\square}$ is a cubic anisotropy term, which arises from lowering the symmetry from spherical to cubic, ${\text{O}}(3)\rightarrow\mathbb{O}_{h}$.

The LM Hamiltonian $\hat{H}^{\text{L}}({\bf p})$ must be supplemented by proper physical BCs. We find SM that the asymptotic BCs for the wave function $\hat{\psi}^{\text{L}}=(\psi^{\text{L}}_{+\frac{3}{2}},\psi^{\text{L}}_{+\frac{1}{2}},\psi^{\text{L}}_{-\frac{1}{2}},\psi^{\text{L}}_{-\frac{3}{2}})^{\text{T}}$ (subscripts indicate $j_{z}$) of the LM following from the 6-band Kane model with hard-wall BCs, describing an interface with vacuum, have the form

[TABLE]

The Kane model describes materials like $\alpha$-Sn and HgTe (see below).

At $p_{z}=0$, $\hat{H}^{\text{L}}(p_{x},p_{y},0)$ is block-diagonal: the pairs $(\psi^{\text{L}}_{+\frac{3}{2}},\psi^{\text{L}}_{-\frac{1}{2}})$ and $(\psi^{\text{L}}_{+\frac{1}{2}},\psi^{\text{L}}_{-\frac{3}{2}})$ of states decouple; the BCs (14) lead to the chiral-symmetric BCs (11) of class $(1,1)$ for each pair. For ${\text{O}}(3)$ symmetry, the respective $2\times 2$ blocks of $\hat{H}^{\text{L}}(p_{x},p_{y},0)|_{\alpha_{\square}=0}$ are of the form $\hat{H}_{2}^{\beta}(p_{x},p_{y})$ [Eq. (10)] of opposite chiralities, with parameters $\beta_{0,z}=\alpha_{0,z}$ and $\beta_{\perp}=-\sqrt{3}\alpha_{z}$. The LM is thus on the line $|\beta_{z}|/|\beta_{\perp}|=1/\sqrt{3}$ in the parameter plane $(\beta_{0},\beta_{z})$ of $\hat{H}_{2}^{\beta}$, Fig. 2(a), and always belongs to the stability regions 2 or 1, as determined by the ratio $|\alpha_{0}|/|\alpha_{z}|$; $|\alpha_{0}|=|\alpha_{z}|$ is the transition point between regions 2 and 1. For $y>0$ sample, the surface-state dispersion relations for $(\psi^{\text{L}}_{+\frac{3}{2}},\psi^{\text{L}}_{-\frac{1}{2}})$ are $\mathcal{E}_{\pm}^{{\text{L}},{\text{O}}(3)}(p_{x},p_{z}=0)=\frac{\sqrt{3}}{2}(\sqrt{3}\alpha_{0}\pm{\rm sgn}\,\alpha_{z}\sqrt{4\alpha_{z}^{2}-\alpha_{0}^{2}})p_{x}^{2}$ at $p_{x}\gtrless 0$, respectively [Eq. (12)], shown in Fig. 2(b) and Fig. 2(c) for $\alpha_{0}=0$ and the parameters $\alpha_{0}/\alpha_{z}=1.7$ of HgTe (see below) belonging to regions 2 and 1, respectively. When combined with the dispersion relations for $(\psi^{\text{L}}_{+\frac{1}{2}},\psi^{\text{L}}_{-\frac{3}{2}})$, $\mathcal{E}_{\mp}^{{\text{L}},{\text{O}}(3)}(p_{x},p_{z}=0)$ at $p_{x}\gtrless 0$, respectively, full surface-state bands of the LM are obtained.

Since for spherical symmetry the same holds for any other orientation of the momentum plane, we conclude that the ${\text{O}}(3)$-symmetric 3D LM exhibits 2D surface states in the whole nodal semimetal regime $|\alpha_{0}|<2|\alpha_{z}|$, two bands for $|\alpha_{0}|<|\alpha_{z}|$ and one band for $|\alpha_{z}|<|\alpha_{0}|<2|\alpha_{z}|$, for any orientation of the surface and any direction of 2D momentum along the surface. According to the above stability arguments, this result also holds upon including the cubic-anisotropy term $\alpha_{\square}\hat{M}_{\square}({\bf p})$, as long as $\alpha_{\square}$ is small enough. We have thus proven that the $\mathbb{O}_{h}$-symmetric LM $\hat{H}^{\text{L}}({\bf p})$ [Eq. (13)] with the BCs (14) is a 3D topological nodal semimetal in the sense of this work. This is the fourth key result of this work.

Our results on the LM are relevant to a multitude of real materials either with exact cubic symmetry $\mathbb{O}_{h}$ or in which deviations from it are small. Among materials with $\mathbb{O}_{h}$ that exhibit a quadratic node is $\alpha$-Sn BP ; W ; Madelung ; a prime example of a material with weakly broken $\mathbb{O}_{h}$ is HgTe BP ; W ; Madelung with a tetrahedral point group $\mathbb{T}_{d}$. The LM parameters for $\alpha$-Sn and HgTe with neglected inversion asymmetry, extracted from Ref. Madelung, , are $(\alpha_{0},\alpha_{z})=(9.31,5.94)/m_{e}$ and $(\alpha_{0},\alpha_{z},\alpha_{\square})=(7.28,4.29,-0.44)/m_{e}$, respectively, where $m_{e}$ is the electron mass. For ${\text{O}}(3)$ symmetry, they both belong to region 1, as indicated in Fig. 2(a), and thus exhibit one band of 2D surface states. Fig. 2(d) shows the surface states for $y>0$ sample at $p_{z}=0$ for the parameters of HgTe including the cubic anisotropy $\alpha_{\square}$. Recently, a quadratic node was predicted and likely observed in $\text{Pr}_{2}\text{Ir}_{2}{\text{O}}_{7}$ Pr ; according to our findings, one or two bands of surface states can be anticipated for this material, although a separate analysis would be desirable. We thus predict that $\alpha$-Sn, HgTe, and many other semimetal materials described by the LM are topological in the sense of this work.

Deviations from cubic symmetry $\mathbb{O}_{h}$ due to breaking of inversion, rotational (strain, confinement), or time-reversal (magnetism) symmetries modify the low-energy band structure, causing the quadratic node of the LM to gap out or split into linear nodes. A variety of resulting topological phases, such as a topological insulator BrueneHgTe3DTI ; BaumHgTe3DTI ; BrueneHgTe3DTIb , Weyl semimetal Ruan , and quantum anomalous Hall insulator Liu , has been predicted or observed. According to our findings, the $\mathbb{O}_{h}$-symmetric quadratic nodal semimetal of the LM [Eqs. (13) and (14)] can be regarded as the parent, highest-symmetry topological phase for these phases, with its own surface states of topological origin.

Future directions. Relations between the topological properties of these phases is an interesting future direction. Among other possible applications and extensions of this work are: relation of the local in the BZ properties of the edge and surface states established here to their global properties, such as those of 3D Weyl semimetals ChiuRMP ; addressing the edge states in graphene and similar 2D systems Fujita ; Nakada ; Brey ; Akhmerov ; FZhang ; Castro ; Arikawa ; MartinBlanter ; Xu ; LiTao ; Kunstmann ; Bahamon ; Jaskolski ; DeshpandeWinkler ; Chiu ; ChiuRMP within this framework graphene ; the role of electron interactions for the edge and surface states in quadratic nodal semimetals in 2D (bilayer graphene) and 3D (LM), where interactions are predicted Sun ; Lemonik ; Cvetkovic ; Murray ; Abrikosov ; Balents ; Herbut ; Goswami to result in interesting physics in the bulk.

Acknowledgements. We appreciate discussions with B. Trauzettel and S. Juergens. We acknowledge financial support from the DFG via SFB 1170 “ToCoTronics” and the ENB Graduate School on Topological Insulators.