Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2

Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{1}$$\end{document}1¯, which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.

We can choose our Γ-matrices to commute with the A operator 1 . Requiring this gives that the commutators are all zero, so the effective model can be written using only the 5 Γ-matrices. We can represent our symmetry operations as 1 : Where K is complex conjugation. This gives the 5 Γ-matrices commuting with A and satisfying the Clifford algebra as: Note that Γ 1 equals P. We can then expand the Hamiltonian as: Now we note that: From which it follows that and M are even in k → −k, whereas the {A i } are odd.
Furthermore, S anticommutes with Γ 1 , which which it follows that: To get the surface states from this bulk Hamiltonian, we organize our Hamiltonian into a term H 0 which contains constants and k x dependent terms, and a perturbation H 1 which does not contain any k x dependence. As there is no preferred direction under the symmetry operations, H 0 contains all 5 Γ-matrices, and is therefore not block diagonal. We can then in principle solve H 0 on a finite slab geometry, which would give us four states, two of which are regular on the boundary. These can be interpreted as the edge states. From equation 6, it follows that the opposite parity states will be associated with an opposite sign for matrix S. Explicitly diagonalizing H 0 gives the eigenvectors: Where the states (v 1 , v 2 ) and (v 3 , v 4 ) have pairwise the same eigenvalues, and the coefficients are complicated functions of the free parameters. These are not block diagonal as H 0 is not block diagonal. We could now in principle investigate which of these four states remain regular at the boundary, given some boundary conditions, and then project the perturbation H 1 in this basis. Note, however, that the resulting expression will be rather convoluted, with many free parameters. As we are only interested in the surface dispersion to compute the DOS, we choose to instead build a model for the surface directly.
The surface close to the TRIM Z exhibits time-reversal symmetry. Inversion symmetry is broken at the surface. However, there will be some residual effects of inversion symmetry, as otherwise H 0 should contain more terms, coming from the commutators of the Γ-matrices.
Thus, inversion symmetry is broken down to some effective symmetry on the surface. The exact nature of this symmetry will depend on the orbital contents. Note in particular that if inversion acted purely as an effective C 2 symmetry, taking (k y , The time-reversal symmetry Θ = iσ y K acts in spin space on the surface by constraining: We can write a two-band model for the surface states as: Hermiticity then requires that d(k) should be real. Imposing time-reversal symmetry gives: With energy given by: Expanding to second order then gives: With energy If we want to account for strain, we should expand to third order which gives an effective model: With energy: Note that our energy is symmetric under (k y , k z ) → (−k y , −k z ) even though the Hamiltonian is not. This explains the observed effective C 2 symmetry in the FS.
The free parameters are fit using simulated annealing. This gives the result shown in supplementary table I and II.  Although the triclinic structure of RhBi 2 only has inversion symmetry, k → −k, when projected on the (100) surface, the inversion symmetry in 2D gives an effective twofold rotation symmetry. This can also be understood in the 2D cuts of the 3D FS as shown in A first look into the band structure of the k · p model suggest the existence of two saddle points in the SBZ [see Fig.4 (a)]. Interestingly, as shown in Fig.3, the lower band ε −,k is almost flat along the diagonal direction parametrized by where α ≈ 4.15 and (kZ ,y , kZ ,z ) ≈ (0, 0.5) π a , denotes the position theZ point. Zooming into the region aroundZ clearly shows two local minima along this flat direction at (k y −kZ ,y , k z − kZ ,z ) = ±(−0.042, 0.010) π a ≡ ±k 0 . These points, on the other hand, behave as local maxima of the dispersion along the direction orthogonal to Eq. (16). This behavior is evidenced in where κ = 2e γ /π ≈ 1.13, and γ is Euler's constant. For the second integral in Eq.(18), a simple integration by parts yields Since Λ/T c 1 and log 2 (y) sech 2 (y/2) remains finite as y → ∞, we can approximate where C = 2.669 is a numerical constant. Substituting Eqs. (19) and (21) into Eq.(18), we Upper (lower) band is shown in yellow (blue). The dispersion is centered at the saddle point k 0 = (−0.042, 0.010) π a . Panels (c) and (d) show the norm of the gradient |∇ε k | of the dispersion in a and b, respectively. The momenta k y and k z are given in units of π/a. Accordingly, |∇ε k | has dimensions of energy.
Since typically we have D > Λ 2 , it is convenient to rewrite the logarithms in Eq.(22) in terms of D/T c rather than Λ/T c . This simplifies Eq.(22) significantly, since the linear term in log(κD/T c ) vanishes identically. We thus find The strongest divergence comes from the term log 2 (κD/T c ), so that we can neglect the three last terms in the right-rand-side of Eq.(23), which yields T c as given in Eq.(6) of the main text.