Topological transport in Dirac nodal-line semimetals

Topological transport in Dirac nodal-line semimetals

W. B. Rui Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany    Y. X. Zhao Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China    Andreas P. Schnyder Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany
July 26, 2019
Abstract

Nodal-line semimetals are characterized by one-dimensional Dirac nodal rings that are protected by the combined symmetry of inversion and time-reversal . The stability of these Dirac rings is guaranteed by a quantized Berry phase and their low-energy physics is described by a one-parameter family of (2+1)-dimensional quantum field theories exhibiting the parity anomaly. Here we study the Berry-phase supported topological transport of invariant nodal-line semimetals. We find that small inversion breaking allows for an electric-field induced anomalous transverse current, whose universal component originates form the parity anomaly. Due to this Hall-like current, carriers at opposite sides of the Dirac nodal ring flow to opposite surfaces when an electric field is applied. To detect the topological currents, we propose a dumbbell device, which uses surface states to filter charges based on their momenta. Suggestions for experiments and potential device applications are discussed.

pacs:
03.65.Vf, 71.20.-b, 73.20.At,71.90.+q

The last decade witnessed a growing interest in anomalous transport properties of topological semimetals Chiu et al. (2016); Hosur and Qi (2013); Zee (2010); Ryu et al. (2012); Burkov (2015); Parameswaran et al. (2014), such as the axial current in Weyl semimetals Zyuzin and Burkov (2012) and the valley Hall effect in graphene Xiao et al. (2007, 2010). These topological currents have their origin in quantum anomalies of the relativistic field theories describing the low-energy physics of semimetals. Quantum anomalies arise whenever a symmetry of the classical theory is broken by the regularization of the quantum theory. For example, in Weyl semimetals the famous (3+1)-dimensional chiral anomaly Adler (1969); Bell and Jackiw (1969); Nielsen and Ninomiya (1983); Aji (2012); Son and Spivak (2013); Yang et al. (2011); Liu et al. (2013) manifests itself by the non-conservation of the chiral charge, i.e., as an axial current flowing between Weyl points with opposite chiralities. Recent experiments on TaAs Huang et al. (2015); Zhang et al. (2016) and on NaBi Xiong et al. (2015) have revealed signatures of the chiral anomaly in magneto-transport measurements. As has become clear over the last few years, the chiral anomaly of Weyl semimetals is intimately connected to the nontrivial topology of the Berry bundle Hořava (2005); Volovik (2003, 2013); Zhao and Wang (2013), which endows the Weyl points with a nonzero topological charge.

Another example of an anomaly leading to topological currents is the (2+1)-dimensional parity anomaly Niemi and Semenoff (1983); Redlich (1984); Haldane (1988); Dunne (1999), which is realized in graphene Gorbachev et al. (2014); Shimazaki et al. (2015); Sui et al. (2015) and graphene-like systems Mak et al. (2014); Zeng et al. (2012); Mak et al. (2012). The fermionic excitations near the Dirac cones of graphene are described by a (2+1)-dimensional quantum field theory exhibiting the parity anomaly. Any gauge symmetric regularization of this quantum field theory must necessarily break spacetime inversion symmetry, which manifests itself by a parity-breaking Chern-Simons term in the electromagnetic response theory of a single graphene Dirac cone. This Chern-Simons term gives rise to the valley Hall effect, where fermions from different Dirac cones flow to opposite transverse edges, upon applying an electric field. The valley Hall effect has been observed in numerous experiments Gorbachev et al. (2014); Sui et al. (2015); Shimazaki et al. (2015); Mak et al. (2014) and has attracted a lot of attention due to possible applications in valleytronics devices Gorbachev et al. (2014); Rycerz et al. (2007).

Parallel to these developments, recent research has shown that there exist topological semi-metals not just with Fermi points, but also with finite-dimensional Fermi surfaces, such as, e.g., Dirac or Weyl nodal lines Volovik (2003, 2013); Hořava (2005); Zhao and Wang (2013); Burkov et al. (2011); Matsuura et al. (2013); Shiozaki and Sato (2014); Chiu and Schnyder (2014); Zhao et al. (2016); Kim et al. (2015); Chan et al. (2016); Yamakage et al. (2016); Zhao and Schnyder (2016). The topological charges of these finite-dimensional Fermi surfaces are defined in a similar way as for Weyl and Dirac points, namely, by the topology of the Berry bundle on a -dimensional sphere that encloses the Fermi surface from its transverse dimension Volovik (2003, 2013); Hořava (2005); Zhao and Wang (2013). Here, is called the co-dimension of the topological Fermi surface. Since topologically nontrivial Berry bundles are closely connected to quantum anomalies, one may wonder whether the quantum field theories describing nodal-line semimetals exhibit any anomalies and, if so, whether they lead to unusual transport phenomena.

This is the question we want to address in this Letter for the case of (3+1)-dimensional Dirac nodal-line semimetals (DNLSMs) protected by the combined symmetry of time-reversal and inversion with  Kim et al. (2015); Chan et al. (2016); Yamakage et al. (2016). A complete classification of symmetric semi-metals has been recently established by relating the topology of symmetric band structures to orthogonal K theory in algebraic topology Zhao et al. (2016). invariant DNLSMs are experimentally realized in several different materials, e.g., in CaP Xie et al. (2015), in CaAgAs Okamoto et al. (2016); Emmanouilidou et al. (2017), in carbon allotropes Weng et al. (2015); HeikkilÃ¤ and Volovik (2015), and in other systems Zhang et al. (2016); Wu et al. (2016); Yu et al. (2015); Xu et al. (2016); Mikitik and Sharlai (2004); Hirayama et al. (2017). We find that the low-energy fermionic excitations of these DNLSMs are described by a one-parameter family of (2+1)-dimensional quantum field theories with a parity anomaly. We show that in the presence of small inversion breaking, this parity anomaly leads to a Hall-like topological current, which can be controlled using electric fields. To detect this anomalous current, we propose a dumbbell-shaped device, which utilizes the drumhead surface states of DNLSMs to filter electrons based on their momenta.

Topological charge and parity anomaly.— We begin our analysis by discussing the relation between the topological charge of symmetric DNLSMs and the parity anomaly. The Fermi surface of Dirac nodal-line semimetals consists of one-dimensional Dirac rings, which have co-dimension in the three-dimensional Brillouin zone (BZ). Without loss of generality, we assume that the DNLSM exhibits only a single Dirac ring, which is located within the plane [see Fig. 1(a)]. Its low-energy Hamiltonian reads Chan et al. (2016)

 H(k)=1Λ[k20−(k2x+k2y)−b2k2z]σ3+vzkzσ2+mσ1, (1)

where for later use we have introduced a small breaking mass . In a DNLSM material this mass term could be generated, for example, by inversion breaking uniaxial strain or pressure. In the absence of the Hamiltonian is symmetric with the operator . The symmetry protection of the Dirac ring (1) is guaranteed by a quantized topological charge , which is given by the parity of the Berry phase along a loop that interlinks with the Dirac ring [green loop in Fig. 1(a)]. That is, is expressed as

 ν[S1]=1π∫S1dϕ trA(φ)mod2, (2)

where the integration is along the loop , parametrized by , and denotes the Berry connection of the occupied Bloch eigenstates . symmetry ensures that can only take on the quantized values and . Loops that interlink with a Dirac ring have a nontrivial Berry bundle, which results in a nonzero topological charge . In two dimensions, Eq. (2) assures the stability of the Dirac points in graphene. In fact, since graphene is symmetric and its Dirac points have co-dimension , it belongs to the same entry in the classification of topological semimetals as DNLSMs Zhao et al. (2016).

Guided by this observation, we introduce cylindrical coordinates and decompose the (3+1)-dimensional DNLSM into a family of (2+1)-dimensional subsystems parameterized by the angle , as shown in Fig. 1(a). Each subsystem exhibits two Dirac points with opposite Berry phase 111To see this, one may move the green integration loop in Fig. 1 along the Dirac ring from one Dirac point to the other. This demonstrates that the green loop encloses the two Dirac points with opposite orientations.. The low-energy physics of a single Dirac point in a given subsystem is described by a (2+1)-dimensional quantum field theory with the action

 Sϕ=∫d3x¯ψ[iγμ(∂μ+ieAμ)+m]ψ, (3)

where is a two-component Dirac spinor coupled to the electromagnetic gauge field . Here, , , and . The mass term breaks spacetime inversion symmetry, since the spinors transform under as and . In the absence of the mass term , Eq. (3) is symmetric (with ) and can be viewed as a classical action of -dimensional Dirac fields. It is however impossible to quantize this classical action without breaking the spacetime inversion symmetry, i.e., symmetry is broken by the regularization of the quantum theory. To see this, let us consider the Pauli-Villars regularization of the effective action of Eq. (3), which is obtained from the fermion determinant by integrating out the Dirac spinors. The effective action with zero mass needs to be regularized due to ultraviolet divergences, which can be achieved by the standard Pauli-Villars method, i.e., . While this regularization scheme preserves gauge symmetry, it breaks invariance, since the Pauli-Villars mass leads in the limit to the Chern-Simons term Redlich (1984); Dunne (1999)

 SϕCS=ηe24π∫d3xϵμνλAμ∂νAλ, (4)

where is the sign of the Dirac point Berry phase. As discussed in Eq. (2), the Berry phase is related to the topological charge via .

From the modern condensed matter viewpoint, the parity anomaly is attributed to the topological charge of the symmetric Dirac point. That is, because of the topological obstruction from the nontrivial topological charge, there exists no symmetric lattice ultraviolet regularization for a single (2+1)-dimensional Dirac point. In other words, any lattice regularization has to involve an even number of nontrivial Dirac points, since the sum over all topological charges in the BZ torus must be zero. This is consistent with the nature of the parity anomaly, since a doublet of (2+1)-dimensional Dirac points coupled to gauge fields can be quantized without breaking symmetry.

To conclude, in the process of quantizing the classical action (3) we have broken symmetry due to the Chern-Simons term (4). Thus, although the parity anomaly strictly speaking occurs only in (2+1) dimensions, it also appears in (3+1)-dimensional DNLSMs.

Topological transport in DNLSMs.— Next we discuss the anomalous transport phenomena that are associated with the parity anomaly. Varying the Chern-Simons term (4) with respect to the electromagnetic gauge field yields the anomalous transverse current

 jμt,ϕ=ηe24πϵμνλ∂νAλ (5)

for a single Dirac cone in a given (2+1)-dimensional subsystem. Thus, electromagnetic fields projected onto a two-dimensional subsystem induce a topological current, which flows transverse (i.e., perpendicular) to the applied field. Since the energy bands of DNLSMs are, to a first approximation, nondispersive along the direction, one might expect that the electromagnetic response of DNLSMs in the presence of a small breaking term is dominated by this topological current. However, for each two-dimensional subsystem there are two Dirac points that contribute to the transverse current with opposite signs . Since these two contributions cancel out to zero, the topological current can only be measured by a device that filters electrons based on their momenta, as we will explain below.

But before doing so, let us give a second derivation of the transverse topological currents in terms of semiclassical response theory Xiao et al. (2010). In the presence of an electric field, the semiclassical equations of motion for Bloch electrons contain an anomalous velocity proportional to the Berry curvature. This gives rise to a transverse Hall-like current Xiao et al. (2007, 2010), given by , where is the Fermi-Dirac distribution function, denotes the electric field, and represents the Berry curvature of the Bloch eigenstate , which is defined as . From a symmetry analysis it follows that the Berry curvature in a gapped system vanishes identically, unless either time-reversal or inversion symmetry are broken. Indeed, using Eq. (1) with we find that is zero in the entire BZ, except at the Dirac nodal line, where it becomes singular, i.e., . To regularize this divergent Berry curvature, symmetry needs to be broken, for example, by uniaxial strain, pressure, disorder, or circularly polarized light, which leads to a small non-zero mass in Eq. (1) and, consequently, a well-behaved Berry curvature. For the conduction band is given by Sup ()

 Ω(k)=mvzkρ/Λ[(2k0Λqρ)2+v2zk2z+m2]32^eϕ, (6)

where we have neglected terms of higher order in and . Here, is the radial distance from the Dirac ring. As shown in Fig. 2 the Berry curvature is peaked at and points in opposite directions at opposite sides of the Dirac ring. The latter is a consequence of time-reversal symmetry, which requires that .

From Eq. (6) we can now compute the transverse current contributed by states with momentum angle by performing the momentum integral only over the two cylindrical coordinates and . Assuming that the chemical potential lies within the conduction band, just above the gap opened by , we obtain the following -dependent Hall current at zero temperature Sup ()

 jt,ϕ≃e2ℏk08π2(1−mμ)E×^eϕ, (7)

where we have neglected terms of order . Interestingly, when the chemical potential is bigger than the gap energy , the transverse current is dominated by the first term, which originates from the parity anomaly foo (). Indeed, the first term of Eq. (7) is consistent with Eq. (5) as it differs only by the differential element of the cylindrical coordinate system. Figure 2 displays the distribution of the transverse currents (green arrows) along the Dirac ring for a constant electric field applied along the and directions. We observe that carriers on opposing sides of the Dirac ring flow into opposite directions transverse to the electric field. This leads to an accumulations of charge on opposite surfaces of the DNLSM.

Dumbbell filter device.— From the above analysis it is now clear that the parity anomaly in DNLSMs gives rise to transverse topological currents. However, since the currents contributed by modes on opposing sides of the Dirac ring have opposite sign, the total transverse current vanishes (i.e., the anomaly cancels). Nevertheless, it is possible to detect anomalous currents by use of a dumbbell filter device, which is based on a ballistic constriction with (001) surface states [Fig. 3(a)]. To explain this, we consider a lattice version of the effective Hamiltonian (1), which is given by , with a two-component spinor describing electrons in and orbitals, and Chan et al. (2016)

 HL(k)=[μz−2t∥(coskx+cosky)−2t⊥coskz]σ3−2t′⊥sinkzσ2+mσ1. (8)

Here, is an on-site energy, and , , and represent intra- and inter-orbital hopping amplitudes on the cubic lattice. Assuming that and , Eq. (8) describes a single Dirac ring located within the plane. The topologically nontrivial Berry bundle of leads to the appearance of drumhead surface states. This can be seen by deforming the green integration loop in Fig. 1(a) into two lines along the (001) direction, denoted by “L” in Fig. 1(b). It follows from the bulk-boundary correspondence Vanderbilt and King-Smith (1993) that in-gap surface states appear at the (001) face of DNLSMs whenever . This corresponds to regions of the surface BZ that are bounded by the projected Dirac ring, since moving along transverse directions without crossing the Dirac ring preserves . Since the drumhead surface states are of topological origin, their existence does not depend on the surface termination or any other microscopic details of the crystal surface.

The dumbbell filter device that we propose consists of two bulk regions connected by a ballistic constriction with drumhead surface states, as illustrated in Fig. 3(a). Such a device could be manufactured, for example, using focused ion beam micromachining Moll et al. (2016). The electronic states in the constriction are confined in the direction, such that their low-energy spectrum is dominated by the drumhead surface states. We show the dispersion relation of the constriction with dimensions and in Fig. 4(a), which reveals that for this parameter choice all states with energies within the interval are surface states. When a voltage is applied across the device, a current passes through the constriction, whose conductance can be determined using the multi-channel Landauer formula Büttiker et al. (1985); Groth et al. (2014), , where are the transmission coefficients Sup (). Assuming that the chemical potential lies slightly above (or below) the gap energy, transport through the constriction is mediated mainly by the modes of the drumhead surface states. Indeed, as shown in Fig. 4, for the current flows entirely within the surface states, leading to plateaus of quantized conductance with steps in multiples of . For , however, bulk modes start to contribute. Since the right propagating surface modes all have positive , only electrons from the right half of the Dirac ring [red area in Fig. 3(b)] with can pass through the constriction. Electrons from the left half of the Dirac ring [blue area in Fig. 3(b)], on the other hand, are reflected. Therefore, the dumbbell device acts as a filter for modes with . The effectivness of the filter can be estimated by the polarization , where and denote the total conductance and the conductance contributed by the surface modes, respectively. We find that is close to 100% for , while it decreases once bulk modes start to mix in [inset of Fig. 4(b)].

Now, since the electric field is oriented along the direction in the dumbbell device, electrons with give rise to a transverse current that flows upwards along the direction [see Fig. 2(a)]. Thus, a voltage difference develops between the upper and lower surfaces of the right weight plate of Fig. 3(a). This voltage difference can be measured experimentally and is a clear signature of the parity anomaly in DNLSMs.

Conclusion.— We have investigated topological transport phenomena of invariant DNLSMs. Specifically, we have shown that the parity anomaly in DNLSMs gives rise to an anomalous transverse current. To detect this anomaly-induced topological current we have proposed a dumbbell filter device, which utilizes surface states to filter charges based on their momenta. It should be noted that the anomaly-induced current is robust to small perturbations, since it is of topological origin. The same applies to the dumbbell device, as its properties originate from topologically protected surface states. In fact, both the topological current and the dumbbell device are robust to perturbations that are small compared to the mean energy separation between the two bands forming the nodal line. This includes perturbations due to spin-orbit coupling, finite dispersion of the nodal ring, as well as disorder. In the Supplemental Material Sup () we have performed numerical calculations to explicitly demonstrate this. From these calculations we conclude that a finite spin-orbit coupling and a small non-zero dispersion of the nodal ring does not qualitatively affect the transverse Hall current and the dumbbell filter device.

Regarding experimental realizations of our proposal, the hexagonal pnictides CaAgAs and CaAgP Okamoto et al. (2016); Emmanouilidou et al. (2017) are particularly promising candidate materials for observing topological currents. These materials are available in single crystal from Emmanouilidou et al. (2017) and exhibit just a single Dirac ring at the Fermi energy. Their drumhead surface states lie within a gap of the order of  meV, which means that the properties of the drumhead device are robust against perturbations of the order of, say,  meV. Therefore, it should be possible to observe anomaly-induced currents in the hexagonal pnictides, as their spin-orbit coupling is smaller than  meV.

While the observation of the parity anomaly in DNLSMs would be of fundamental interest, the dumbbell device used for this purpose could also lead to new electronic devices, such as a topological current rectifier. We anticipate that similar devices could also be realized in three-dimensional Dirac or Weyl semi-metals, whose Fermi arc surface states could be used as a valley filter. We leave these interesting topics for future research.

Acknowledgements.
We thank Philip Moll and Ali Bangura for useful discussions. A.P.S. is grateful to the KITP at UC Santa Barbara for hospitality during the preparation of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.

References

Supplemental Material

Authors: W. B. Rui, Y. X. Zhao, and A. P. Schnyder

In this Supplemental Material we present the derivation of the Berry curvature (Sec. I), compute the transverse Hall current (Sec. II), and give the details of the numerical simulations (Sec. III). We also show that a finite spin-orbit coupling and a small non-zero dispersion of the band crossing does not qualitatively affect the transverse Hall current and the dumbbell filter device (Secs. IV and V). Furthermore, we demonstrate that the dumbbell filter device is not affected by moderately strong disorder (Sec. VI).

I. Derivation of the Berry curvature of DNLSMs

To compute the Berry curvature of DNLSMs we use the low-energy effective model given by Eq. (1) in the main text, i.e.,

 H(k)=1Λ[k20−(k2x+k2y)−b2k2z]σ3+vzkzσ2+mσ1. (9)

where is a term that breaks symmetry. The Berry curvature is given in terms of the Berry connection by

 Ωα(k)=∇k×Aαα(k) (10)

where are the Bloch eigenstates of Eq. (9). The positive and negative energy eigenstates of are calculated as

 |−,k⟩ = 1√N−(1Λ(k20−k2ρ−b2k2z)−λm+ivzkz), (11a) |+,k⟩ = 1√N+(1Λ(k20−k2ρ−b2k2z)+λm+ivzkz), (11b) with the eigenenergies E±=±λ=±√1Λ2(k20−k2ρ−b2k2z)2+v2zk2z+m2. (11c)

Here we have used the short hand notation and . The Berry connection of the conduction band is given by

 Az++(k) = mvz[1Λ(k20−k2ρ−b2k2z)−λ]2λ(v2zk2z+m2), (12a) while Ax++(k)=Ay++(k)=0. Similarly, for the valence band |−,k⟩ the Berry connection takes the form Az−−(k) = −mvz[1Λ(k20−k2ρ−b2k2z)+λ]2λ(v2zk2z+m2), (12b)

while . Using the Berry connection (12), it is straightforward to compute the topological charge of the Dirac ring of using formula (2) from the main text.

From Eq. (12) we now compute the Berry curvature using Eq. (10)

 Ωx+(k) = −mvzky/Λ[1Λ2(k20−k2ρ−b2k2z)2+v2zk2z+m2]32, (13a) Ωy+(k) = mvzkx/Λ[1Λ2(k20−k2ρ−b2k2z)2+v2zk2z+m2]32, (13b) and Ωz+(k)=0 . Using cylindrical coordinates {kρ,ϕ,kz} and Eq. (11c) the Berry curvature can be written in a more compact form Ω(k)=mvzkρ/Λλ3^eϕ, (13c)

with the unit vector . We observe from the above expressions that the Berry curvature is peaked at , where is the radial distance from the Dirac ring. If we neglect terms of order and , the Berry curvature simplifies to

 Ω(k)=mvzkρ/Λ[(2k0Λqρ)2+v2zk2z+m2]32^eϕ+O[q3ρ,k2z]. (14)

We will use this expression to compute the transverse Hall current.

II. Computation of the transverse Hall current

The transverse Hall current is given by

 jt=e2ℏ∫d3k(2π)3f(k) E×Ω(k), (15)

where is the Fermi-Dirac distribution function and denotes the electric field. To compute the transverse current we assume that the chemical potential lies within the conduction band, slightly above the gap energy , i.e., , see Fig. 5. Hence, at zero temperature the integral in Eq. (15) is over states with energies that lie within the interval . In the vicinity of the Dirac ring the energy dispersion of the conduction band, Eq. (11c), can be approximated by

 E+(qρ,kz)≃√(2k0Λqρ)2+v2zk2z+m2, (16)

where we have neglected terms of order and .

Let us now compute the transverse current contributed by states with momentum angle by performing the integral in Eq. (15) over the two cylindrical coordinates and

 jt,ϕ = e2ℏ∫m

Using Eq. (14) the integral can be expressed as

 I(μ)=∫m

With the substitutions and we obtain

 I(μ) = ∫~q2+~k2<μ2−m2d~qd~k(2π)3m2k0(Λ2k0~q+k0)2[~q2+~k2+m2]32 (19) = k08π2[1−mμ(1−Λ2μ28k40)]+O[m2], (20)

where we have neglected terms of order . The second term in the round brackets of Eq. (20) can be rewritten in terms of the Fermi wave vector , which is related to the chemical potential by [cf. Eq. (16)]

 μ2=4k20Λ2q2Fρ+v2zk2Fz+m2. (21)

We have

 Λ2μ28k40=q2Fρ2k20+Λ2v2zk2Fzk40+Λ2m28k40. (22)

Since and , it follows that the first two terms in Eq. (22) are small and the third term is of order . Hence, we find the following approximate form for the -dependent transverse current

 jt,ϕ≃e2ℏk08π2(1−mμ)E×^eϕ+O[m2]. (23)

III. Details of the numerical simulations

To numerically simulate the band structure and the conductance we discretize the lattice Hamiltonian , Eq. (8), from the main text on a cubic lattice. In real space, Hamiltonian (8) reads

 (24)

where is the nearest-neighbor intra-orbital hopping amplitude in the and directions, is the nearest-neighbor intra-orbital hopping amplitude in the direction, denotes the inter-orbital hopping amplitude, is an onsite energy, and represent the gap energy. For the numerical computations we have used the following parameters and have set the gap energy to . The conductance is computed using the recursive Greens function method with the Kwant code [51].

IV. Effects of a small dispersion of the band crossing

In the main text we have neglected a possible dispersion of the Dirac nodal line, which can be described by the symmetry-allowed term

 c(k)σ0=−[c1(coskx+cosky−1)+c2(coskz−1)]σ0, (25)

that is added to the Hamiltonian , Eq. (8), in the main text. In the presence of the Fermi surface of the DNLSM becomes a thin anisotropic torus. We note that in many DNLSM materials is relatively small, such as, e.g., in CaP Chan et al. (2016); Xie et al. (2015) and CaAgAs Yamakage et al. (2016); Okamoto et al. (2016); Emmanouilidou et al. (2017). Nevertheless, since the term (25) is symmetry allowed, one expects it to be present in general. We therefore study how modifies the Berry curvature and the properties of the dumbbell filter device.

Using the tight-binding model Eq. (8) in the presence of the term (25), with and , we have numerically computed the Berry curvature, see Fig. 6(a). We observe that the Berry curvature shows a similar structure as in the absence of [compare Fig. 2 with Fig. 6(a)]. That is, is peaked at the Dirac ring and is oriented along the torus-like Fermi surface. It follows that the transverse Hall current , which is given by , is qualitatively unchanged by .

Next, we study how the properties of the dumbbell filter device are modified by the term . Comparing Fig. 4(a) with Fig. 6(b), we find that the band structure of the filter device is modified only within a small energy range of around the Fermi energy. In accordance with this, the conductance and the polarization of the filter device is largely unchanged by , see Figs. 6(c) and 6(d).

V. Effects of spin-orbit coupling

There exist DNLSM materials with both weak and strong spin-orbit coupling. In the main text we have focused on the case of very weak spin-orbit coupling, which is relevant for, e.g., CaP. For materials with heavier elements, such as, CaAgAs, spin-orbit coupling cannot be neglected. In order to study the effects of spin-orbit coupling we need to explicitly include the spin degree of freedom in the tight-binding Hamiltonian , Eq. (8). Therefore we consider

 ^HL(k)=(HL(k)Λ(k)Λ†(k)H∗L(−k)), (26)

where represents a spin-orbit coupling term, which breaks spin-rotation symmetry. Time-reversal symmetry acts on Hamiltonia (26) as , with the time-reversal operator , where denotes the second Pauli matrix in spin space. Reflection and inversion symmetry act on as and , with the operators and , respectively. The spin-orbit coupling term is assumed to take the following symmetry-allowed form

 Λ(k)=b1(sinkx+isinky)σ1. (27)

We observe that with Eq. (27) satisfies time-reversal symmetry, reflection symmetry, and parity symmetry, but breaks SU(2) spin-rotation symmetry. In the presence of spin-orbit coupling the nodal line of the DNLSM becomes gapped and the drumhead surface states split. However, the Berry curvature remains finite and is of similar form as without spin-orbit coupling, see Fig. 7(a). Hence, we conclude that a transverse Hall current of the form also exists in DNLSMs with strong spin-orbit coupling.

Spin-orbit coupling splits the spin degeneracy of the drumhead surface states and therefore is expected to modify the properties of the filter device. In order to study this, we have numerically computed the band structure, the conductance, and the polarization of the dumbbell device for the DNLSM (26) with spin-orbit coupling strength , see Figs. 7(b), 7(c), and 7(d). By comparing Fig. 4(a) with Fig. 7(b) we see that spin-orbit coupling spin splits the band structure of the dumbbell device. Within a small energy interval of around the Fermi energy there exist now both left propagating and right propagating surface modes with positive . Hence, for the dumbbell device is no longer a perfect filter for modes with . However, for the filtering property remains intact, see Fig. 7(d). We conclude that the dumbbell device works also for DNLSM with spin-orbit coupling.

VI. Robustness against disorder

Here, we study the effects of disorder on the dumbbell filter device. A simple renormalization group argument shows that short-range correlated disorder is a marginal perturbation to the DNLSM Hamiltonian. Therefore, it is expected that the DNLSM and the dumbbell filter device are robust against weak short-range correlated disorder Burkov et al. (2011); Wang and Nandkishore (2017). To verify this expectation we numerically compute the conductance of the filter device in the presence of short-range impurity scatterers of the form

 V(r)=∑nv(rn)δ(r−rn), (28)

where denotes the scattering potential at the lattice site . For each lattice site the onsite potential is drawn from a Gaussian distribution with width . In Fig. 8 we present the conductance of the filter device as a function of disorder strength . We find that for the conductance steps in multiples of are still clearly visible. This indicates that the filter device is robust against moderately strong disorder.

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