Symmetry-protected ideal Weyl semimetal in HgTe-class materials

Ideal Weyl semimetals with all Weyl nodes exactly at the Fermi level and no coexisting trivial Fermi surfaces in the bulk, similar to graphene, could feature deep physics such as exotic transport phenomena induced by the chiral anomaly. Here, we show that HgTe and half-Heusler compounds, under a broad range of in-plane compressive strain, could be materials in nature realizing ideal Weyl semimetals with four pairs of Weyl nodes and topological surface Fermi arcs. Generically, we find that the HgTe-class materials with nontrivial band inversion and noncentrosymmetry provide a promising arena to realize ideal Weyl semimetals. Such ideal Weyl semimetals could further provide a unique platform to study emergent phenomena such as the interplay between ideal Weyl fermions and superconductivity in the half-Heusler compound LaPtBi.

The line node structure without strain in the full bulk BZ. The blue, red, and green lines represent line nodes lying in the mirror plane but not along any diagonal direction. The black solid lines label line nodes in diagonal directions. The red (blue) dashed lines cross the k y = 0 (k x = 0) plane where the crossing points are indicated by red (blue) points. These dashed line nodes, except those crossing points, split once strain is applied due to the breaking of their mirror symmetries. (c) After applying a small strain, the line nodes in the k x = ±k y mirror planes gradually shrink, as shown by the green lines. The red and blue points represent the Weyl nodes in the k x = 0 and k y = 0 planes, protected by C 2T symmetry. (d) When the compressive strain exceeds a critical value, i.e. δ < δ Weyl c , there are no line nodes any more and the Weyl nodes, indicated by the red and blue points, are type-I such that they are all located exactly at the Fermi level and the system is in the ideal Weyl semimetal phase.   Supplementary Note 1: Systematic evolution of the band structure under strain As mentioned in the main text, the HgTe-class materials which carry T d point group symmetry, do not respect inversion symmetry. The inversion asymmetry in these systems results in qualitatively different band structures compared to those with inversion symmetry, e.g., α-Sn. Instead of a simple quadratic band touching point (at Γ) exactly at Fermi level, two intermediate bands of the four Γ 8 bands in HgTe-class materials touch along a line node in the six mirror planes. These line nodes are protected by mirror symmetry even though they are generically away from the Fermi level. As a consequence, the HgTe-class materials have small electron and hole pockets, even at stoichiometry.
To illustrate these line nodes, we diagonalize Hamiltonian in the mirror plane, e.g., the k x = −k y plane as shown by the shaded plane in Supplementary Figure 1a. The unstrained Hamiltonian is given in the main text, i.e., H unstrained = H Luttinger + H BIA . For simplicity, only the linear term in H BIA is taken into consideration and higherorder terms in H BIA won't affect the results we obtain in this section qualitatively. The line nodes lying in the k x = −k y mirror plane is given by the following equation: where c i is defined in Supplementary Note 3. This bow-shape line node is shown in Supplementary Figure 1a  Upon applying strain in the xy plane, the crystalline symmetry of the material is lowered from T d to D 2d . The mirror symmetries in the k x = ±k z and k y = ±k z planes are broken, while those in k x = ±k y planes survive. As a result, for small strain the line nodes located at k x = ±k y plane survive as they are protected by the unbroken mirror symmetry, even though they contract as the applied strain increases. The evolution of the line nodes is shown in Supplementary Figure 1c where the closed green lines indicate the shrinking line nodes under a small strain. When the applied strain exceeds critical value, these line nodes eventually disappear.
On the other hand, the line nodes in k x = ±k z and k y = ±k z planes are split immediately , upon applying strain except the eight discrete points. These eight discrete points are the intersecting points between the k y = 0 (k x = 0) plane and the line nodes in the k x = ±k z (k y = ±k z ) planes, as shown by red or blue color points in Supplementary  Figure 1b. As mentioned in the main text, these eight discrete points are protected from gapping out by C 2T ≡ C 2 · T symmetries. Applying strain only gradually shift these points in the k y = 0 or k x = 0 planes. Supplementary Figure  1(c) shows these discrete points upon applying a small strain. These eight discrete points are actually Weyl points, which reside in the k x = 0 or k y = 0 planes protected by the C 2T symmetry (see Supplementary Note 3 for details).
When the strain is sufficiently small but finite, these points are type-II Weyl points. These type-II Weyl points in the k y = 0 plane shift to |k z | > |k x | region if compressive strain is applied while they shift to |k z | < |k x | region if tensile strain is applied, and Weyl points in the k x = 0 have the similar behavior related by symmetry. Thanks to this qualitatively different response between tensile and compressive strain, the system evolve into different phases at sufficiently large strain. When the tensile strain increases, these type-II Weyl points first move to the k x or k y axis in the k z = 0 plane, then move within the k z = 0 plane (since there is C 2T symmetry in the k z = 0 plane), and finally they annihilate one another with opposite chiralities in k z = 0, k x = ±k y lines when the tensile strain reaches a critical value. When the strain exceeds the critical value, the system enters into a strong topological insulator phase with nontrivial Z 2 topological-invariant.
On the other hands, when the compressive strain increases from zero, the eight type-II Weyl nodes shift towards a larger |k z | direction. When the compressive strain exceeds a critical value, all trivial Fermi surface vanishes and the eight type-II Weyl nodes evolve to type-I ideal Weyl nodes located exactly at the Fermi level, as shown in Supplementary Figure 1d.

Supplementary Note 2: The Weyl nodes under different strains
In this section, we calculate locations of ideal Weyl nodes for different in-plane compressive strains for both HgTe and the half-Heusler compound LaPtBi, as shown in Supplementary Table 1. The Weyl nodes move slowly towards larger momentum points in both k x and k z directions for increasing strain. As the Weyl nodes can only be pair-annihilated in the k z = π plane, the slow motion of Weyl nodes with increasing strain indicates that the ideal Weyl semimetal phase is stable under a broad range of strain. Indeed, for the large strain of a/a 0 = 0.971 and c/a 0 = 1.06, the Weyl nodes are still far away from the k z = π plane. Moreover, the larger separation between Weyl nodes in momentum space under increasing in-plane compressive strain can make the observation of them by Angle resolved photoemission spectroscopy (ARPES) experiments easier.

Supplementary Note 3: The effective k·p theory at finite strain
In this section, we use following Gamma matrices [1]: to write the Luttinger Hamiltonian: where . Now, we explore the behavior of HgTe under sufficiently large strain (exceeds the critical value g Weyl c ) using the effective k·p theory. From Clebsch-Gordan coefficients, we know where | 3 2 , J z ⟩ labels the wave function of a single electron with J z = ± 3 2 , ± 1 2 . Electrons with J z = ±3/2 have definite orbital angular momentum l z = ±1 while electrons with J z = ±1/2 are superpositions of the wave functions with l z = ±1 and l z = 0. Since compressive strain in the xy plane shortens the lattice distance along the x and y directions, the wave functions with orbital angular momentum l z = ±1 get more overlapped than that of l z = 0, and their energy shift is larger than that of l z = 0. This indicates that the main effect induced by the strain in the xy plane is given by the following perturbation: where g is related to the strength of applied strain δ. Moreover, we obtain g < 0 for compressive strain (δ < 0) and g > 0 for tensile strain (δ > 0). We shall focus on the case that g < g Weyl c here. For this case, as explained in the main text, treating H 0 ≡ H Luttinger +H strain as unperturbed Hamiltonian and the BIA part H BIA as perturbation is a better way to characterize the band features around Weyl nodes. The quadratic touching point in Luttinger Hamiltonian H Luttinger is split into two Dirac points locating at k z axis for compressive strain because H strain has only D 4h symmetry which is lower than the O h symmetry of the Luttinger Hamiltonian. Specifically, the dispersion is given by , leading to two Dirac points at (0, 0, ± √ g/c 2 ). Expanding the Hamiltonian around the touching point (0, 0, √ g/c 2 ), we obtain the Dirac Hamiltonian: where These Dirac points are protected by inversion symmetry, time reversal symmetry, as well as the S 4 (z) symmetry of H 0 . In HgTe, inversion symmetry is actually broken which has important consequences even though the breaking is weak. As a result of BIA, linear, cubic, as well as higher-order terms compatible with the T d symmetry are allowed in the Hamiltonian and we treat them as perturbations. We first consider the effect of the linear term in H BIA [2]: For simplicity, we approximate H linear by its form at the Dirac point (0, 0, √ g/c 2 ) and denote it as H m = mΓ 35 , where m = α √ 3g/c 2 . In the presence of H m , the low-energy dispersion is given by