Generalized Triple-Component Fermions: Lattice Model, Fermi arcs and Anomalous Transport

# Generalized Triple-Component Fermions: Lattice Model, Fermi arcs and Anomalous Transport

Snehasish Nandy Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Stra. 38, 01187 Dresden, Germany Department of Physics, Indian Institute of Technology Kharagpur, WB 721302, India    Sourav Manna Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Stra. 38, 01187 Dresden, Germany    Dumitru Călugăru Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Stra. 38, 01187 Dresden, Germany Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom    Bitan Roy Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Stra. 38, 01187 Dresden, Germany
July 27, 2019
###### Abstract

We generalize the construction of time-reversal symmetry-breaking triple-component semimetals, transforming under the pseudo-spin-1 representation, to arbitrary (anti-)monopole charge , with in the crystalline environment. The quasi-particle spectra of such systems are composed of two dispersing bands with pseudo-spin projections and energy dispersions , where , and one completely flat band at zero energy with . We construct simple tight-binding models for such spin-1 excitations in a cubic lattice and address the symmetry protection of the generalized triple-component nodes. In accordance to the bulk-boundary correspondence, triple-component semimetals support branches of topologically protected Fermi arc surface states and also accommodate a large anomalous Hall conductivity (in the plane), given by (in units of ). Furthermore, we compute the longitudinal magneto-, planar Hall and magneto thermal-conductivities in this system, which increase as (due to the non-trivial Berry curvature in the medium) with the external magnetic field (), when it is sufficiently weak. A generalization of our construction to arbitrary integer spin system is also highlighted.

## I Introduction

Energy branches available for electrons to occupy in solid state compounds (also known as bands) can often touch each other at few isolated and specific points in the Brillouin zone herring (); dornhaus (); RyuTeo (); Barnevig_2016 (); kane-prb (); hasan-review (); armitage-review (). In the close proximity to the band-touching points, the system can be described in terms of emergent pseudo-spin degrees of freedom, with the distinct eigenvalues of the pseudo-spin projection representing different bands. Some well known examples of such gapless systems are Dirac and Weyl semimetals. Respectively in these two systems, Kramers degenerate and non-degenerate valence and conduction bands, transforming under half-integer pseudo-spin representations, touch each other. Such special points act as defects or singularities in the reciprocal space. For example, pseudo-spin-1/2 Weyl nodes in three dimensions assume the texture of a hedgehog or anti-hedgehog and stand as sources or sinks of Abelian Berry curvature, respectively. Nonetheless, it is also conceivable to realize band touching points around which the system can be described in terms of arbitrary pseudo-spin- representation, where can be any half-integer or integer. In the simplest incarnation of such higher pseudo-spin system, the energy spectra are described in terms of effective Fermi velocities, when is a half-integer; a phenomena known as multifringence luttinger (); kennett-3 (); Dora (); Lan-1 (); liangfu (); Chen (). By contrast, for integer , energy spectra display effective Fermi velocities and a completely flat band (described by the trivial eigenvalue of the pseudo-spin projections) Barnevig_2016 (); Dai_2016 (); Ding_2017 (); Chen_2017 (); Hasan_2017 (); Zhang_2017 (); Soluyanov_2016 (); Hasan1_2017 (); Chang_arXiv (); Chen_2018 (); fulga_2017 (). The present work is devoted to unveil some quintessential topological features of time-reversal symmetry-breaking semimetals, transforming under the pseudo-spin-1 representation, also known as triple-component semimetals, within the framework of both effective low-energy as well as representative tight-binding models in a cubic lattice.

Irrespective of these details, the entire family of pseudo-spin- Dirac or Weyl fermions can be described by the following effective low-energy Hamiltonian

 Hs(k)=d(k)⋅S,

where momenta is measured from the band-touching points, and are three spin- matrices 111In this work we focus only on three-dimensional systems.. In the simplest realization of pseudo-spin- system , where bears the dimension of the Fermi velocity. The energy spectra are then given by , where , with (when is an half-integer) or (for integer ). We here concentrate on integer pseudo-spin- systems. Even though the following discussion can be generalized to any integer value of , for the sake of concreteness we restrict ourselves to pseudo-spin-1 systems, which can be realized, for example, in -TaN, MoP, RhSi, TiS, ZrSe, HfTe Barnevig_2016 (); Dai_2016 (); Ding_2017 (); Chen_2017 (); Hasan_2017 (); Zhang_2017 (); Soluyanov_2016 (); Hasan1_2017 (); Chang_arXiv (); Chen_2018 (). The electronic excitations in such a setup is also known as triple-component fermions (due to three energy bands), and we here consider such peculiar band touching in time-reversal symmetry-breaking systems. In what follows, such band-touching points are referred as triple-component points or nodes. Now we present a brief synopsis of our main results.

We show that the triple component points act as sources and sinks of Abelian Berry curvature with charges (with , when ), respectively. In this work we generalize the construction of triple component fermions for arbitrary monopole charge (assuming ), with in the crystalline environment (see Sec. II). For arbitrary integer value of , the spectra always accommodate one topologically trivial flat band, while the energy of two dispersive bands scales as and (assuming that the triple-component points are separated along the direction), where . The integer topological invariant of the system is given by . We also present simple lattice realizations of generalized triple-component fermions in a cubic lattice and show that the generalized triple component nodes are protected by discrete rotational symmetries in a tetragonal environment (see Sec. III).

The integer topological invariant of triple-component semimetals () manifest through copies of topologically protected Fermi arc surface states connecting two triple-component points, see Figs. 1 and  2. This observation is in accordance with the bulk-boundary correspondence, discussed in Sec. IV. The time-reversal symmetry-breaking topological triple-component systems also support a large anomalous Hall conductivity in a plane perpendicular to the separation of the two triple-component points (see Sec. V). The anomalous Hall conductivity acquires its largest value, given by (in units of )

 σ3Dxy,max=2n×separation of triple-% component points2π,

when the chemical potential is pinned at the band touching points. Such a large anomalous Hall conductivity solely arises from the underlying Berry curvature of the medium 222The flat band at zero-energy is topologically trivial and possesses net zero Berry curvature. Hence, it does not affect any topological responses of this system.. The Berry curvature can also leave its signature on other transport quantities, when, for example, the system is placed in a weak magnetic field (hence no Landau quantization in the system). Specifically, we here compute the (a) longitudinal magneto-conductivity, (b) planer Hall conductivity and (c) longitudinal magneto-thermal conductivity, within the framework of the semiclassical Boltzmann theory and show that these quantities increases as , in the weak magnetic field regime, see Sec. VI. The enhancement of various components of the magneto-conductivity tensor possibly captures the imprint of the chiral anomaly in triple-component semimetals lepori ().

The rest of the paper is organized as follows. In the next section, we introduce the effective low-energy models for generalized triple-component fermions and compute its topological invariant. Sec. III is devoted to the construction of generalized triple-component fermions from simple tight-binding models in a cubic lattice. The topologically protected Fermi arc surface states and anomalous Hall conductivity are respectively discussed in Sec. IV and Sec. V. Signatures of Berry curvature on magneto transport are discussed in Sec. VI. Concluding remarks and discussions on related issues are presented in Sec. VII. Additional technical details are relegated to the Appendix.

## Ii Generalized triple component fermions: Low-energy model

We begin the discussion with the low-energy models for general triple-component fermions (TCFs). The Hamiltonian operator describing such a system can be written compactly as

 \leavevmode\nobreak HTCFn,τ(k)=dn1(k)Sx+dn2(k)Sy+τd3(k)Sz, (1)

where , and are the spin-1 matrices, given by

 \leavevmode\nobreak Sx = 1√2⎛⎜⎝010101010⎞⎟⎠=1√2(λ1+λ6), Sy = 1√2⎛⎜⎝0−i0i0−i0i0⎞⎟⎠=1√2(λ2+λ7), (2) Sz = ⎛⎜⎝10000000−1⎞⎟⎠=12(λ3+√3λ8),

and s are the standard Gell-Mann matrices Gell-Mann (). Here, represents the two valleys or triple-component points. The energy spectra for TCFs in the vicinity of each valley are given by

 Eσk=σ√[dn1(k)]2+[dn2(k)]2+d23(k),E0k=0, (3)

for , respectively describing the conduction and valence bands. Hence, TCFs accommodate two dispersive bands () and one completely flat band (). Different bands with energy dispersions is characterized by distinct pseudo-spin projection .

Next we discuss the nature of the dispersive bands for various choices of integer . For any value of , , whereas for

 d11(k)=α1kx,d12(k)=α1ky, (4)

where bears the dimension of the Fermi velocity. We name the system linear-triple-component semimetal. For linear-TCFs energy dispersions scale linearly with all three components of momentum. By contrast, for

 d21(k)=α2(k2x−k2y),d22(k)=α2(2kxky), (5)

and bears the dimension of inverse mass. We name the system quadratic-triple-component semimetal. For quadratic-TCFs, dispersion scales linearly only with , but quadratically with the in-plane components of momenta . Finally, for

 d31(k)=α3(k3x−3kxk2y),d22(k)=α3(k3y−3kyk2x), (6)

and we name this system cubic-triple-component semimetal. In this system the energy dispersion relation scales as . This construction can be envisioned as a generalization of the multi-Weyl systems composed of (pseudo-)spin-1/2 excitations Fang-HgCrSe (); bergevig-MWS (); nagaosa (); bera-roy-sau (); ezawa-2 () to the spin-1 systems. In Sec. III we promote simple tight-binding models leading to the lattice realizations of such unconventional quasiparticles.

The topological charge of the triple-component points can be computed from the underlying Berry curvature of the bands. For cimplicity, we now consider only one valley (say ). The Berry curvature of the band for a Bloch Hamiltonian [see Eq. (1)] is given by Xiao:2010 ()

 Ωmθϕ=2i∑m≠m′⟨m|∂H∂θ|m′⟩⟨m′|∂H∂ϕ|m⟩(ϵm−ϵm′)2, (7)

where and are the polar and azimuthal angles in the momentum space, respectively and , are the band indices. For convenience, we here use the spherical polar co-ordinate. The wavefunctions for three bands read as

 ⟨E−k| = (e−2inϕsin2θ2,−e−inϕsinθ√2,cos2θ2), ⟨E+k| = (e−2inϕcos2θ2,e−inϕsinθ√2,sin2θ2), (8) ⟨E0k| = (−e−2inϕsinθ√2,e−inϕcosθ,sinθ√2).

The flat band possesses exactly zero Berry curvature, while it is finite for the two dispersive bands. The monopole charge () of the corresponding triple-component point can now be computed by integrating the Berry curvature over a unit sphere () in the momentum space enclosing this point, yielding

This analysis can be generalized for arbitrary integer () spin systems, for which the monopole charge is . The monopole charge also determines the integer topological invariant of such gapless phase of matter.

## Iii Lattice model and symmetry protection

We now propose simple tight-binding models that allow us to realize various members of the generalized TCF family in a cubic lattice. Such simple lattice construction will also allow us to demonstrate the symmetry protection of triple-component points in the Brillouin zone, and demonstrate the topological features, such as (a) Fermi arc surface states (see Sec. IV) and (b) anomalous Hall effect (see Sec. V). For arbitrary integer value of , the corresponding tight-binding model takes a compact form

 \leavevmode\nobreak H=∑kΨ†k[N(k)⋅S]Ψk, (10)

where is a three-component spinor, is the fermion annihilation operator with momenta and pseudo-spin projection , and [see Eq. (II)]. The momentum dependent form factors , appearing in Eq. (10) for various values of arise from (setting the lattice constant calugaru ()

 Nx(k)=t⎧⎪⎨⎪⎩sin(kx)forn=1,cos(kx)−cos(ky)forn=2,sin(kx)[3cos(ky)−cos(kx)−2]forn=3,
 Ny(k)=t⎧⎪⎨⎪⎩sin(ky)forn=1,sin(kx)sin(ky)forn=2,sin(ky)[3cos(kx)−cos(ky)−2]forn=3,
 \leavevmode\nobreak N1z(k)=tzcos(kz),N2z(k)=m[cos(kx)+cos(ky)−2], (11)

with . In this construction and give rise to the desired form factors for and , respectively, when they are expanded around for , when . On the other hand, produces two triple-component points at , whereas plays the role of a Wilson mass that only vanishes when for . Therefore, with this construction, we end up with a minimal model for a time-reversal symmetry breaking general TCF, for which the triple-component points are located at , see also Ref. roy-slager-juricic (). The continuum models discussed in Sec. II are realized from the above simple tight-binding models in the low-energy limit. We now highlight the symmetry protection of generalized triple-component nodes.

The generalized triple-component nodes are protected by discrete rotational symmetries. A rotation by an angle in the pseudo-spin space about its quantization axis (namely ) is captured by the unitary operator

 RPS(θPS)=exp[iθPSSz]. (12)

On the other hand, a rotation of the momentum axis about the axis by an angle is captured by

 Rk(θk)=⎛⎜⎝1000cosθk−sinθk0sinθkcosθk⎞⎟⎠. (13)

Under a rotation by in the pseudo-spin space

 RPS(π2)(Sx,Sy,Sz)R†PS(π2)=(−Sy,Sx,Sz), (14)

whereas, under a rotation by for

 Rk(π2n)(dn1,dn2,dn3)(k)=(±dn1,±dn2,dn3)(k). (15)

The “” sign appears only for . Therefore, under the combined rotations in the pseudo-spin space [by an angle ] and momentum space [by an angle ] the low-energy Hamiltonian from Eq. (1) remains invariant for separately. Note that for two components of the -vector change sign, and thus under the above mentioned combined rotation in the pseudo-spin and momentum space, the source () and sink () remain invariant. Therefore, the generalized triple-component nodes are symmetry-protected and hence stable. Next we demonstrate the bulk-boundary correspondence and constructing the topological Fermi arc surface states by numerically diagonalizing the above tight-binding models in a cubic lattice for different .

## Iv Fermi arc surface states

The hallmark signature of a topologically non-trivial phase of matter is the existence of surface or edge states, capturing the bulk-boundary correspondence. The structure of such boundary modes, however, crucially depends on the actual nature of the bulk topological phase. For example, a three-dimensional strong topological insulator supports two-dimensional massless helical Dirac fermions on all six surfaces of a cubic system hasan-kane-review (); qi-zhang-review (). The surface modes of a topological semimetal are somewhat different from the ones of a topological insulator. Note that a time-reversal symmetry-breaking triple-component semimetal can be constructed by stacking two-dimensional layers of quantum anomalous Hall insulators of spin-1 fermions in the momentum space along the directions between two triple-component points, located at  333Since the flat band is topologically trivial, we characterize each two-dimensional slice of the system as an “insulator” even though there exists a dispersionless flat band at the middle of the band gap between the dispersive valence and conduction bands.. Each copy of two-dimensional anomalous Hall insulator supports one-dimensional chiral edge modes, accommodating one state precisely at zero energy. The collection or locus of such zero-energy states between the two triple-component points ultimately constitute the topological Fermi arcs, shown in Figs. 1 and  2 for . Such a seemingly hypothetical construction of time-reversal symmetry-breaking topological semimetals, nonetheless, leaves its signature in the anomalous Hall response of these systems, discussed in Sec. V.

In order to construct the Fermi arc surface states, we impose periodic boundaries in the and directions (hence leaving momenta and as good quantum numbers) and implement open boundary in the direction, along which the linear dimension of the system is set to be . For numerical diagonalization of the lattice models we set . In this construction we can observe localized Fermi arcs states (in the mixed Wannier-Bloch representation) on the top and bottom surfaces slager (). The resulting Fermi arcs for , and are shown in Fig. 1(a), (b) and (c), respectively. Note that

 number of Fermi arc=2n≡ monopole charge of triple-component points. (16)

Hence the bulk-boundary correspondence remains operative for spin-1 semimetals, as there exist two, four and six branches of the Fermi arc surface states, respectively for , and . Next we discuss some additional salient features of the arc states.

Note that the surface localization of the Fermi arc states is maximal at its center (). As we approach the two triple-component nodes the surface localization decreases monotonically. At the two triple-component points the arc states become completely delocalized, and the top and bottom surfaces get connected through the bulk triple-component points, as shown in Fig. 2. This observation does not depend on the choice of the integer value of and can be appreciated in the following way. Note that the localization length of the Fermi arc state for each value of is inversely proportional to the size of the gap of the corresponding two-dimensional layer of the quantum anomalous Hall insulator. From Eq. (11), one can appreciate that the bulk gap for the underlying two dimensional Hall insulating phase is largest when (center of the arc) 444Note that for any , one segment of the Fermi arc extends from to , see Fig. 1. The center of such an arc is located at , where it is maximally localized on the surface. Also note that the size of the insulating gap of the underlying two-dimensional anomalous Hall insulator at is bigger than that at . Consequently, the surface localization of the arc states at is stronger than that at , see Fig. 2. . Otherwise, such a gap decreases monotonically as we approach the singular points, located at , from the center of the Fermi arcs, and vanishes at . Consequently, the arc state at these two points become completely delocalized and two opposite surfaces get connected through the bulk triple-component points, which can be seen from Fig. 2. Hence, our numerical analysis of the Fermi arc surface states provides strong evidence in favor of the bulk-boundary correspondence in triple-component semimetals, with arbitrary monopole charge .

## V Anomalous Hall Effect

Yet another hallmark signature of a time-reversal symmetry breaking topological semimetal is the non-trivial anomalous Hall effect in a plane perpendicular to the separation of two band touching points. We discussed in the previous section that both Weyl semimetals and triple-component semimetals can be seen as stacking of two-dimensional quantum anomalous Hall insulators in the momentum space along a specific direction (in our construction ). Since each layer produces quantized (thus large) anomalous Hall conductivity (AHC), the resulting time-reversal symmetry-breaking semimetallic phase also accommodates finite (and large) AHC.

Notice that for any value of the system describes an anomalous Hall insulator in the plane that supports quantized AHC, given by

 \leavevmode\nobreak σ2Dxy(μ=0,kz=0)=e2h×(2n). (17)

The AHC can directly computed from the underlying Berry curvature ()

 \leavevmode\nobreak σ2Dxy(μ,kz=0)=e2h∫d2k⊥(2π)2Ωzf0ms(μ), (18)

where is the equilibrium Fermi-Dirac distribution at finite chemical doping (), given by

 f0ms(μ)=[1+exp(Emsk−μkBT)]−1, (19)

where . Here the Boltmann constant is set to be unity. A direct correspondence between the AHC and the first Chern number of the underlying anomalous Hall insulator, obtained from the continuum models of these systems is presented in Appendix A. First, we compute the AHC for two-dimensional time-reversal symmetry breaking insulator for from the lattice model, shown in Sec. III, upon setting . The results are displayed in Fig. 3(a) for , Fig. 3(b) for and Fig. 3(c) for , as a function of varying chemical doping ().

For completely empty bands the Berry curvature from the conduction and valence bands cancels each other, and the system supports precisely zero AHC. As one increases from the bottom of the valence band, AHC starts to increase monotonically and reaches its quantized value when meets the top of the valence band. Note that when the chemical potential is pinned within the bulk insulating gap the remains constant and quantized, given by Eq. (17). Upon further increasing , the AHC starts to decrease, as the Berry curvature from the fully filled valence band gets partially canceled by that from the partially filled conduction band. Ultimately, when the conduction band becomes fully occupied, the Berry curvature from these two bands completely cancel each other and the AHC once again drops back to zero. Otherwise, this feature is common for . Note that the topologically trivial flat band residing at zero energy does not influence the AHC in two dimensions.

The AHC for a three-dimensional time-reversal symmetry-breaking triple-component semimetal () can be obtained by accumulating the contributions from each constituting two-dimensional layer of anomalous Hall insulator and is given by Balents_2017 ()

 \leavevmode\nobreak σ3Dxy(μ)=∫K02−K02dkz2πσ2Dxy(μ,kz), (20)

where is the separation of two triple-component points. Since for each copy of underlying two-dimensional anomalous Hall insulator yields the largest and quantized AHC, the three-dimensional triple-component semimetal also yields largest AHC, given by

 \leavevmode\nobreak σ3Dxy,max=σ3Dxy(0)=e2h×(2n)×K02π→ne2h, (21)

when , as shown in Fig. 3(a) for , Fig. 3(b) for and Fig. 3(c) for , since in our lattice realization of triple-component semimetals (see Sec. III). With increasing or decreasing chemical doping, the AHC decreases monotonically, as the contribution of Berry curvature from the conduction and valence bands cancels each other. Scaling of the AHC as a function of chemical doping for three-dimensional triple-component semimetals is displayed in Fig. 3 (lower panel). Finally note that the AHC in three-dimensions is insensitive to the presence of the trivial flat band. Next we investigate the influence of the non-trivial Berry curvature in the medium on magnetotransport, such as longitudinal magnetoconductivity.

## Vi Semiclassical Boltzmann transport

In this section, we investigate the imprint of non-trivial Berry curvature of a triple-component semimetal on various transport quantities within the framework of the semiclassical kinetic theory. Specifically, we focus on (a) longitudinal magnetoconductivity [see Sec. VI.1], (b) planer Hall conductivity [see Sec. VI.2], and (c) longitudinal magneto-thermal conductivity [see Sec. VI.3]. In what follows we compute these quantities in the weak magnetic field () limit, such that , where is the cyclotron frequency and is the average time between two successive collisions (should not be confused with the valley index , introduced in Sec. II). In this limit one can neglect the Landau quantization, and treat to be independent of the strength of the external magnetic field. This approximation is justified since the radius of the cyclotron orbit in the weak field limit is sufficiently large, allowing us the treat the path between two successive collisions as a straight line (approximately). Furthermore, we also assume that there exists a single scattering life-time in the medium, determined by the elastic scattering of triple-component fermions from the impurities. For sufficiently strong magnetic field () the Landau levels are sharply formed and one needs to account for the quantum corrections to due to the -field, as the path between two successive collisions can no longer be treated as a straight line, and henceforth .

In the presence of an external electric field () and a temperature gradient (), the charge current () and thermal current () are related to each other via the linear response equations, compactly written as

 (22)

where is the conductivity tensor, is the Seebeck coefficient tensor, and is the thermal conductivity tensor. Also note that and are related to each other by Onsager’s relation . Within the framework of linear response theory the electrical and thermal currents can respectively be written as

 Ja = σabEb+αab(−∇bT)\leavevmode\nobreak , (23) Qa = TαabEb+lab(−∇bT)\leavevmode\nobreak , (24)

where are the spatial indices. We now set up the general formalism for the Boltzmann transport equations to compute these quantities in the presence of an underlying Berry curvature in the medium.

The Boltzmann transport equation reads as Ziman ()

 \leavevmode\nobreak (∂t+˙rm⋅∇rm+˙km⋅∇km)fm,k,r,t=C{fm,k,r,t}, (25)

where is the band index and is the collision integral, which in principle incorporates electron correlations (inelastic scattering) as well as elastic scattering from impurities, and is the electronic distribution function. For the sake of simplicity, we here focus only on the impurity scattering, which is the dominant source of relaxation process in weakly correlated and dirty systems. Within the relaxation time approximation, the collision integral takes the form

 \leavevmode\nobreak C{fm,k,r,t}=f0m−fm,kτm(k), (26)

where is the relaxation time and is the equilibrium Fermi-Dirac distribution function in the absence of any external field. To proceed further with the analysis, we ignore the momentum and band dependence of and assume it to be a constant with (a phenomenological parameter) in the semiclassical limit, the single scattering-time approximation.

Upon incorporating the effects of the Berry curvature, the semiclassical equation of motion takes the following form Niu_1999 (); Niu_2006 ()

 ˙rm = 1ℏ∇ϵkm−(˙km×Ωm,k), (27) ˙km = −eℏE−eℏ(˙rm×B), (28)

where the second term of the Eq. (27) represents the anomalous velocity originating from the non-trivial Berry curvature. The solutions of the coupled equations for and are respectively given by Son_2012 (); Duval:2006 ()

 ˙rm = 1Dm[vm,k+eℏ(E×Ωm,k)+eℏ(vm,k⋅Ωm,k)B], (29) ˙km = 1ℏDm[eE+eℏ(vm,k×B)+e2ℏ(E⋅B)Ωm,k]. (30)

For brevity we use in the above two equations, where

 Dm(B,Ωm,k)=[1+eℏ(B⋅Ωm,k)]

modifies the invariant phase space volume according to and gives rise to a noncommutative mechanical model, since the Poisson brackets of co-ordinates is now nonzero Duval:2006 (). We are now equipped to proceed to the computation of various conductivity tensors introduced in Eqs. (22),  (23) and  (24). The rest of the analysis is presented only for linear-triple component semimetals (with ). The following discussion can be generalized to address similar effects in quadratic and cubic-triple component semimetals.

### vi.1 Longitudinal magnetoconductivity

For the computation of the longitudinal magnetoconductivit (LMC), we assume that and are always parallel to each other, otherwise applied along an arbitrary direction . After solving the Boltzmann equation using Eqs. (29) and  (30), we find Son:2013 (); Kim:2014 (); Lundgren:2014 (); Sharma:2016 ()

 \leavevmode\nobreak σuu(B) = −e2τ∑m∫d3k(2π)3[1+eℏ(B⋅Ωm,k)]−1 (31) × [vm,u+eBℏ(vm,k⋅Ωm,k)]2(∂ϵf0m).

For concreteness, we compute the LMC in the direction () from the linearized model, introduced in Sec. II. Note that only the two dispersive bands contribute to the LMC, as the carriers in flat band are localized. As temperature , the above expression for the LMC simplifies to (after setting )

 \leavevmode\nobreak σzz(B)=e2τ∫d3k(2π)3[vz+eB(vk⋅Ωk)]21+eBΩzδ(μ−ϵk), (32)

where we have used the fact that . For rest of the analysis, we set , so that only the upper band contributes to the LMC. Upon introducing the spherical polar coordinates in which , Eq. (32) can be written as

 σzz(B) =e2τ(2π)2∫π0dθsinθ∫∞0dkk2(cosθ+eBk2)21+eBcosθk2δ(μ−k)=e2τμ2(2π)2[∫π0dθsinθcos2θ(1+eBcosθμ2)−1 +2eBμ2∫π0dθsinθcosθ(1+eBcosθμ2)−1+e2B2μ4∫π0dθsinθ(1+eBcosθμ2)−1] =e2τμ2(2π)2[−2μ4e2B2+2μ6e3B3tanh−1(eBμ2)+2eBμ2{2μ2eB−2μ4e2B2tanh−1(eBμ2)}+2eBμ2tanh−1(eBμ2)]. (33)

While arriving at the last expression we set . Note that semiclassical theory is applicable in the parameter regime where quantum corrections can be neglected. Such condition at is achieved only if , so that the chemical potential provides the infrared cut-off in the system. Therefore, we can expand appearing in the above expression for , yielding

 tanh−1(x)=x−x33+O(x5), (34)

where . Finally, accounting for the contributions from two triple-component nodes, we find

 \leavevmode\nobreak σzz(B)=2×(e2τμ24π2)[23+1615e2B2μ4]. (35)

The first term in the above expression is the standard metallic conductivity arising from the Drude contribution, while the second term shows a enhancement of the LMC. However, in the single scattering time approximation we cannot attribute such an enhancement solely to the chiral anomaly, as both regular and axial charges are relaxed by the same scattering mechanism (characterized by roy-surowka (). Nonetheless, the system still displays a positive LMC and . In order to isolate the contribution from the chiral anomaly we need to introduce two scattering times in the collision integral [see Eq. (26)], and , respectively denoting the inter- and intravalley scattering lifetimes. In particular, when only the contribution from the chiral anomaly survives zyuzin (). Explicit demonstration for this lengthy analysis is left for a future investigation.

### vi.2 Planer Hall Conductivity

The planar Hall effect corresponds to the appearance of an in-plane transverse voltage () in the presence of external, but co-planar electric and magnetic fields; specifically when they are not perfectly aligned to each other. The experimental setup for the measurement of planar Hall conductivity (PHC) is schematically shown in Fig. 4. Notice that in this configuration the conventional Hall effect vanishes. To evaluate the PHC, we conveniently align the electric field () along the axis, while the magnetic field () is directed at a finite angle from the -axis (but in the plane), thus

 \leavevmode\nobreak E=E^x,andB=B(cosγ^x+sinγ^y), (36)

where is the angle between and (see Fig. 4). The PHC is then given by Burkov_2017 (); Nandy_2017 (); Nandy1_2017 ()

 \leavevmode\nobreak σPHCyx(γ)=−e2τ∑m∫d3k(2π)3vm,y+eBsinγℏ(vm,k⋅Ωm,k)1+eℏ(B⋅Ωm,k)[vm,x+eBcosγℏ(vm,k⋅Ωm,k)](∂f0∂ϵ). (37)

As in terms of the spherical polar coordinates introduced earlier the PHC reads as (now setting )

 σyx(γ)=e2τμ2(2π)3∫2π0dϕ∫π0dθ[1+eBsinθcos(ϕ−γ)μ2]−1[sin3θsin2ϕ2+eBμ2sin2θsin(ϕ+γ)+eB2μ4sinθsin2γ]. (38)

We numerically compute the PHC from the low-energy model for a linear triple-component semimetal. The amplitude of the PHC shows a dependence, as shown in Fig. 5(a), for any value of except when and , where PHC vanishes. Also note that the PHC scales as , see Fig. 5(b). We find that the PHC for TCF does not satisfy the antisymmetry property of the regular Hall conductivity since it does not originate from the conventional Lorentz force.

### vi.3 Longitudinal Magneto-thermal Conductivity

Next we compute the longitudinal magneto-thermal conductivity (LMTC) for the linear triple-component semimetal from its low-energy model. To compute the LMTC, we align the external magnetic field and the temperature gradient along an arbitrary direction , such that . After solving the Boltzmann equation using Eqs. (29) and (30), and comparing with Eq. (24), we arrive at the following expression for the LMTC Lundgren:2014 (); Sharma:2016 (); Nandy2_2017 ()

 \leavevmode\nobreak ℓuu = τ∑m∫d3k(2π)3[1+eℏ(B⋅Ωm,k)]−1 × [vm,u+eBℏ(vm,k⋅Ωm,k)]2(ϵm−μ)2T(−∂ϵf0m).

Since the flat band is topologically trivial (possessing zero Chern number), and chemical potential is placed above the triple-component points, only upper band contributes to the LMTC. We compute LMTC separately for each triple-component node and finally add their contributions. For concreteness, we compute the LMTC along the direction. In terms of the spherical polar coordinates and after setting , , we arrive at the final expression for the LMTC at finite-, given by

 \leavevmode\nobreak ℓzz =e2τ(2π)2∫π0sinθdθ∫∞0dk(k−μ)2T2k2 ×(cosθ+eBk2)21+eBcosθk2f0(1−f0), (40)

where we have used the fact that . The scaling of the LMTC, specifically , as a function of the magnetic field strength () is displayed in Fig. 5(c). We find that LMTC also scales as , i.e. for triple-component semimetals.

## Vii Conclusions and Discussions

To summarize, we generalize the notion of time-reversal symmetry-breaking pseudo-spin-1 or triple component semimetals to arbitrary integer (anti-)monopole charge [see Secs. II and III] and address its topological properties, such as Fermi arc surface states [see Secs. IV]. In addition, we also compute the influence of non-trivial Berry curvature in this system on various transport quantities, such as the anomalous Hall conductivity [see Sec. V] within the framework of the Kubo formalism, as well as the longitudinal magnetotransport and the planer Hall conductivity using the semiclassical Boltzmann theory in the single scattering time approximation [see Sec. VI].

In particular, we show that in a simple cubic lattice one can realize triple component nodes with monopole charge , where in a crystalline environment [see Sec. III]. At the triple-component points three bands with pseudo-spin quantum number touch each other. While two bands with pseudo-spin projection are dispersive away from the triple-component points, the one with is completely flat and topologically trivial. In our lattice realization of the spin-1 fermions, the triple-component points are separated along the -direction. For any the energy dispersion (for bands) always scales linearly with , but , where . We also show that such unusual band touching is symmetry protected and can therefore be realized from a simple tight-binding model [see Sec. III].

The topological invariant for triple-component points manifests through the Fermi arc surface states following the bulk-boundary correspondence. We argue that a system with a pair of triple-component points with (anti-)monopole charge accommodates branches of Fermi arc states on the surface, see Sec IV. To establish the bulk-boundary correspondence for spin-1 triple-component fermions, we numerically diagonalize the tight-binding models for these systems [introduced in Sec. III] with periodic boundary in the and directions (hence momenta along these two directions are good quantum numbers) and a open boundary in the -direction. Fig. 1 depicts the Fermi arcs in the plane (the top surface), and we find that there exists exactly branches of the Fermi arc surface states on the top surface connecting two triple-component points. Additional salient features of the arc states can be appreciated from their localization in the -direction, as shown in Fig. 2. We find that while the Fermi arc states are well localized on the top or bottom surfaces away from the triple-component points, at these two points (representing singularities in the momentum space) they are completely delocalized. Specifically, the arcs states from the top surface leak through the bulk triple-component points, and get connect to the ones on the bottom surface. This feature is insensitive to the precise value of and also occurs for spin-1/2 Weyl fermions.

The two dispersive bands possess non-trivial Berry curvature, whereas the flat band is topologically trivial. The signature of non-trivial Berry curvature can, for example, be observed in the anomalous Hall conductivity [see Sec. V]. Note that time-reversal symmetry breaking triple-component semimetal can be envisioned as stacking of two-dimensional anomalous Hall or Chern insulators of spin-1 fermions in the momentum space along direction between two triple-component nodes. As a result for we obtain a quantized anomalous Hall conductivity, given by , for , see Figs. 3(a)-(c), when the chemical potential lies within the bulk band gap. The anomalous Hall conductivity of a triple-component semimetal can then be obtained by accumulating the quantized contribution from each two-dimensional constituting layers in between two nodes, and the results are summarized in Figs. 3(d)-(f). Therefore, the generalization of spin-1 topological semimetals opens up a new route to achieve large anomalous Hall conductivity. A large anomalous Hall conductivity can also be accommodated by spin-1/2 Weyl fermions, which can be germane for PrIrO inside a metallic spin-ice ordered phase goswami-roy-dassarma (). Therefore, possible material realizations of spin-1 Weyl fermions in strongly correlated systems should be an interesting future avenue of research.

The signature of the Berry curvature can also be found in various other transport quantities, such as longitudinal magneto- and magneto-thermal conductivities, planer Hall conductivity [see Sec. VI]. We here compute these quantities using the semiclassical Boltzmann transport theory and for sufficiently weak magnetic field, when the Landau levels are not sharply formed. At least when the strength of the external magnetic field () is sufficiently weak, all of them increase as . Even though it is commonly believed that such a seemingly counter intuitive enhancement of the longitudinal magnetoconductivity, for example, arising from the non-trivial Berry curvature captures the signature of the chiral anomaly, there exists no concrete proof demonstrating this connection. Therefore, it will be interesting to investigate these quantities in the strong magnetic field limit when the inter-particle scattering time () explicitly depends on the magnetic field, and establish the relevance of chiral anomaly in spin-1 system lepori (); argyres-adams (); li-roy-dassarma ().

Our discussion is, however, not limited to spin-1 or triple-component fermions. For example, our lattice construction for spin-1 fermions from Sec. III can immediately be generalized to any integer spin- fermions by replacing the spin-1 matrices by spin- matrices. In that construction, there always exists a topological trivial flat band, and dispersive bands ( number of valence and conduction bands), characterized by distinct Fermi velocities. Therefore, our theoretical analysis should stand as a good starting point to begin the voyage into the world of integer spin topological phases of matter. Besides the topological features of integer-spin Weyl systems, its (in)stability against electronic correlations, which can give birth to exotic superconducting rahul-spin-1-SC () and excitonic phases, is yet another interesting avenue, which we will explore in future.

###### Acknowledgements.
SN acknowledges MHRD, India for research fellowship. B. R. is thankful to Nordita for hospitality during the program “Topological Matter Beyond the Ten-Fold Way”, and Vladimir Juričić, Soumya Bera for valuable discussions.

## Appendix A Ultraviolet regularization and Chern number

The anomalous Hall conductivity (AHC) of two-dimensional constituting layers of the anomalous Hall insulators (AHI) is intimately tied with the first Chern number of the system in the following way

 σ2Dxy(μ=0,kz=0)=e2h×Chern number of% the AHI. (41)

We established this connection by explicitly computing the AHC from tight-binding models of the generalized triple-component semimetals, introduced in Sec. III, after setting (thus yielding AHI), see also Fig. 3(a)-(c). This appendix is devoted to illustrate how the correct Chern number can be extracted from the continuum models of these systems.

We begin the discussion with the linear triple-component semimetal. For , the effective low-energy model of this system is given by

 H=kxSx+kySy+[1−α2(k2x+k2y)]Sz. (42)

Even though in the specific tight binding model [see Eq. (11)], we treat as a free-parameter in this appendix. Note that higher-gradient terms proportional to are irrelevant in comparison to the dominant -linear terms at small momentum. However, as we show here such higher gradient terms play paramount important role in properly capturing the topological invariant or the Chern number of the system. At least for , the Berry curvature of three bands can be computed analytically, yielding

 \leavevmode\nobreak Ω±k=∓4(2+k2x+k2y)2π[4+(k2x+k2y)2]32,Ω0k=0, (43)

from which one can find the Chern number of each band

 Cτ=∫′d2k(2π)2Ωτk. (44)

In the continuum model the momentum integral is restricted up to an ultraviolet cut-off (denoted by the prime symbol in the integral), and we obtain and . These numbers do not depend on . We also numerically compute the Chern number for several other values of the parameter , and find that this number does not depend on , see Fig. 6. Note that in the low-energy Hamiltonian from Eq. (42), the -linear terms dominates at small momentum, whereas the higher gradient terms are more important for large momentum. The term proportional to then plays the role of a “band-inverted” Wilson mass that ensures the topological nature of the insulating phase. In brief, in the entire construction of topological phases of matter the Wilson mass plays a crucial role, which we further investigate below for triple-component semimetals with and .

The effective low-energy models for quadratic and cubic triple-component semimetals for (representing AHIs) can compactly be written as

 \leavevmode\nobreak Hn=dn1(k)Sx+dn2Sy(k)+[1−α2f(k⊥)]Sz, (45)

respectively with and . The functional form of depends on the choice of the Wilson mass and the order up to which we expand it in momentum. For example, if we expand [see Eq. (11)] to quadratic order then