Magnetic topological Kondo semimetal phases of matter

Topologically protected band structure degeneracies have commanded considerable attention in the last decade^1–14 as widespread and relatively practical opportunities to study topology in condensed matter systems, for their great potential for technological applications^15,16 and as platforms for table-top experiments on quasiparticles, both analogous to high-energy particles and beyond this framework.^17–25 A major undertaking to study these systems in the presence of correlations further suggests far richer physics beyond the weakly correlated regime.^26–38 Key challenges in this effort are experimental realization and isolation of this physics for greater understanding and more effective application. These challenges are well-addressed by heavy-fermion systems, which serve as important guides in the study of correlations. Kondo physics has furthermore played a key role in past study of topological matter and magnetic correlations.^39,40 In this work, we therefore introduce magnetic topological Kondo semimetal phases of matter into the literature through first construction of both the magnetic Weyl Kondo semimetal and magnetic Dirac Kondo semimetal phases.

We focus on symmetry-protected Weyl and Dirac cones brought about by ferromagnetic and antiferromagnetic order, respectively. While based on a proposal for weakly correlated Weyl semimetals,² our work goes beyond this proposal by (i) instead deriving topological semimetal phases from topologically non-trivial electronic structures resulting from intrinsically strongly correlated Kondo physics,⁴¹ (ii) including the case of three dimensional magnetic Dirac Kondo semimetals, (iii) including realization of both type-I and type-II Weyl/Dirac Kondo semimetal phases, and (iv) utilizing the strong correlations of hexaborides as the sources of requisite magnetic orders as opposed to magnetic dopants.^42,43 We note that SmB₆ furthermore exhibits evidence of exotic correlated topological surface states,^44–48 which could enrich the physics of topological Kondo semimetals constructed from SmB₆ surface states. (v) Additionally, we perform density functional theory calculations of the heterostructure of EuB₆ (CeB₆) and SmB₆, confirming the presence of Weyl (Dirac) nodes predicted by tight-binding calculations.

We consider the case where interactions strongly renormalize band parameters but low-energy excitations are described by well-defined Fermi-liquid quasiparticles. The Fermi-liquid quasiparticles in such a state are then accurately described by a non-interacting Hamiltonian with renormalized parameters^49–53  $t̃f(j)→tf(j)=z2t̃f(j),Ṽ→V=zṼ, ε̃f→εf=ε̃f+λ$⁠, and $U→0$⁠, while the others remain the same. The additional parameters z and λ are expressed by the self-energy expansion coefficients by $z=(1−b)−1/2$ and $λ=a/(1−b)$⁠, thus depend on the band parameters of the original Hamiltonian containing the quartic interaction terms between f-orbital electrons. The predominant effect of U is to move the f-electron band closer to the Fermi level, thereby enabling topological band inversion.

To model the heterostructure, we endow the onsite energies $εγ,r$ with a spatial dependence to change their values between the two materials comprising the heterostructure. The parameter $εf−εd$ changes the bulk band topology of the translationally invariant model between normal insulator, weak topological insulator, and strong topological insulator.^49,54

The heterostructure is then modeled as $NTI$ layers of the topological Kondo insulator SmB₆, using the parameter set⁴⁹  $tf(1)=−0.1td(1), td(2)=−0.4td(1), tf(2)=0.04td(1), td(3)=tf(3)=0, V=0.5td(1), εd=1.76td(1), εf=−0.24td(1)$⁠, where we set $td(1)=$ 100 meV in order to produce the correct band gaps, and $NBI$ layers of a trivial Kondo insulator with the same parameters except for $εd=4.02td(1), εf=−0.78td(1)$⁠, corresponding to our example AB₆. The bulk band structures of these two parameter sets are shown in Figs. 1(c) and 1(d). The (topological) bandgap of SmB₆ is then 24 meV, a value in good agreement with past work.⁵⁵ For AB₆, we choose a bandgap of 84 meV. Additional numerical results on the non-magnetic heterostructure are also provided in the supplementary material.

Ferromagnetic order could emerge via various mechanisms. In this work, we consider two cases: ferromagnetic order in SmB₆ or in the trivial Kondo insulator (corresponding to, e.g., ferromagnetic EuB₆). We construct phase diagrams showing the number of Weyl cones formed between the two middle bands as a function of the number of AB₆ layers and the magnitude of the magnetization for the former case [see Fig. 2(a)] and latter case [see Fig. 2(b)]. Focusing first on the former case, we find four different phases depending on $NBI$ and m, corresponding to 0, 2, 4, and 6 Weyl cones present in the system, which are located along the high-symmetry lines Γ–Z and X–R. Depending on the parameters, these Weyl cones can either be of type I or type II.^56,57 A representative band structure for magnetization $m=0.25td(1)$ is shown in Fig. 2(c).

We also compute the spectral function of the heterostructure in a slab geometry, using again the magnetization $m=0.25td(1)$⁠. With the heterostructure stacked in the $ẑ$ direction, we open the system in the $x̂$ direction and compute the surface spectral function $Asurf(ω,ky,kz)=Im Tr [G(ω,ky,kz)Psurf]$⁠, where $G(ω,ky,kz)$ is the Green's function, and $Psurf$⁠, the projector on one surface layer. This result is presented in Fig. 2(d), where a Fermi arc, a signature of a Weyl semimetal,⁴ is clearly visible.

We can similarly compute the dispersion, phase diagram, and spectral function for the case of a net magnetization in the trivial Kondo insulator, $A=Eu$⁠. The same three topologically non-trivial phases—characterized by 2, 4, and 6 Weyl cones, respectively—that appear in Fig. 2(a) also occur in this second phase diagram, shown in Fig. 2(b). They occur over regions in phase space that are comparable in size, indicating the phases are largely unaffected by whether magnetization is derived from the trivial or topological Kondo insulator layers. However, topologically non-trivial regions of the phase diagram Fig. 2(b) are slightly smaller than those of Fig. 2(a), which may be due to the fact that the effective magnetization strength seen by the topologically non-trivial surface states of SmB₆ is less than the actual magnetization strength of the trivial Kondo insulator layers.

These results are further supported by a low-energy effective theory of the electronic structure derived in the supplementary material, which illustrate emergence of Weyl nodes in the heterostructure via a mechanism similar to that of Burkov and Balents.² 

We now consider the full heterostructure Hamiltonian with finite magnetization in SmB₆ layers corresponding to antiferromagnetic order oriented in the stacking direction. We physically motivate this case as follows: there is evidence of antiferromagnetic order induced in bulk SmB₆ by compression.^42,43 Also, Eu_xCa_1−xB₆, where x = 0.4 and 0.6, was reported to show intrinsic antiferromagnetism below 3 K.⁵⁸ 

Two examples of Dirac semimetal dispersions of the heterostructure are shown in Fig. 3. Both type-I and type-II¹⁰ Dirac cones can occur in the full model, shown in Figs. 3(b) and 3(d), respectively. Similarly to the ferromagnetic case, the low-energy effective theory of the electronic structure derived in the supplementary material also supports our numerical findings for the antiferromagnetic case. Notably, the heterostructure has a fourfold rotational symmetry about the z-axis, $C4z$⁠, that protects Dirac cones on the $C4z$ invariant lines, such as $kx=ky=0$⁠, as a crossing of two pairs of bands with different $C4z$ eigenvalues.

To further support the above-mentioned tight-binding calculations, we calculate the band structure for a heterostructure of SmB₆ and EuB₆, employing density functional theory (DFT) (additional details of these calculations are provided in the supplementary material). Such band structure calculations are shown in Fig. 4(a) for the 2 × 2 EuB₆/SmB₆ supercell. We confirm the presence of a Weyl node near the Fermi level along the $Γ−Z$ line as expected from the tight-binding calculations, highlighted by a blue box, indicating realization of the magnetic Weyl Kondo semimetal phase in the heterostructure.

We also further support the tight-binding calculations for the magnetic Dirac Kondo semimetal phase by calculating the band structure for a 2 × 2 CeB₆/SmB₆ supercell. We confirm the presence of a Dirac node in the DFT calculations along the M–A line as shown in Fig. 4(b) corresponding to the magnetic topological Dirac semimetal phase, but further from the Fermi level than the Weyl node in (a).

In addition to the Weyl and Dirac nodes predicted by tight-binding calculations and low-energy effective models, however, we also find additional topologically protected degeneracies near the Fermi level in Fig. 4(b). These are Class I threefold-degenerate fermions with zero topological charge.^19,61,62 This illustrates the exceptionally rich phenomenology of magnetic topological Kondo semimetal physics realized through the heterostructure construction and will be explored further in future work.

In conclusion, we introduce magnetic topological Kondo semimetal phases of matter into the literature, showing topological Kondo insulator heterostructures are versatile platforms for the realization of magnetic Weyl and Dirac Kondo semimetal phases. We find ferromagnetism (antiferromagnetism) in the heterostructure generically realizes topologically protected type-I and type-II Weyl (Dirac) cones sufficiently isolated from other states and proximate in energy to the Fermi level to realize type-I and type-II magnetic Weyl (Dirac) Kondo semimetal phases near half filling. Given, in particular, that the magnetic Weyl semimetal phase is the so-called atomic realization of a Weyl semimetal, capable of realizing just two Weyl nodes, our work fills an essential gap in understanding of correlation effects on topological semimetal phases.

We note that thin films of SmB₆ have already been grown via molecular beam epitaxy (MBE)⁶³ and Kondo superlattices of other compounds have been grown via MBE for study of quantum criticality.⁶⁴ Furthermore, evaporation of boron and most rare-earth lanthanides is possible at operating temperatures for effusion cells.⁶⁵ Negative pressure on SmB₆ (lattice constant $a=$ 4.13 Å^66,67) due to interfaces with EuB₆ (⁠ $a=$ 4.19 Å⁶⁸), YbB₆ (⁠ $a=$ 4.18 Å⁶⁹), or CeB₆ (⁠ $a=$ 4.14 Å⁷⁰) may also permit the observation of the desired Kondo physics at much higher temperatures of up to 240 K as well as enhancement of one or both parent compound band gaps, given observed effects of tensile strain in SmB₆.⁷¹ 

Since our work was first made public, other later works have considered magnetic topological Kondo semimetal phases,^72–74 further supporting our work here introducing the first instances of magnetic topological Kondo semimetal phases of matter.