Gapless Topological Peierls-like instabilities in more than one dimension

Abstract

A periodic lattice distortion that reduces the translational symmetry folds electron bands into a reduced Brillouin zone, leading to band mixing and a tendency to gap formation, as in the Peierls transition in one-dimensional systems. However, in higher dimensions, the resulting phase can present topological obstructions preventing a complete gap opening. We discuss two different mechanisms for such obstructions, emergent Weyl nodes and symmetry protected band crossings. Based on density-functional calculations, we show these mechanisms are at play in trigonal PtBi$_2$.

Figures (4)

• Figure 1: Examples of distortions of the triangular lattice that reduce translational symmetry while preserving threefold rotational and reflection symmetries. (a) Atoms shown as darker shaded disks are displaced out of the plane, forming a buckled, decorated honeycomb lattice. (b) The equilateral triangles are distorted into triangles of varying sizes, creating sequences of alternating small and large side-sharing triangles along the mirror planes.
• Figure 2: (a) Fragment of the distorted lattice [see Fig. \ref{['fig:dim']}b)] indicating the hopping amplitudes for short ($t+\delta$) and long ($t-\delta$) bonds. (b) Brillouin zones corresponding to the triangular lattice (dashed blue lines) and to the $\sqrt{3}\times\sqrt{3}$ supercell (black continous lines). The supercell Brillouin zone is rotated by $30^\circ$ and reduced in area by a factor of three relative to the original lattice. (c) Undistorted triangular lattice band structure folded into the supercell Brillouin zone. The reflection symmetry eigenvalues are indicated by $+$ and $-$ signs next to the bands. The linewidth indicates the unfolding weight which is $1$ for the lower band and $0$ for the upper bands in this case. The Fermi level ($\varepsilon_F =0$) corresponds to an occupancy of a single electron per unit cell. (d,e) Band structure for the distorted case with (d) $\delta/t = -0.2$ and (e) $\delta/t=0.2$. (f) Density of states for $\delta/t =0.2$, $0$, and $0.2$. Inset: Total electronic energy relative to the undistorted limit as a function of $\delta/t$.
• Figure 3: Band structure of a triangular AB stacked bilayer (a) without distortion and (b,c) with one of the layers dimerized: (b) $\delta/t=0.1$ and (c) $\delta/t=0.5$. Energies are measured with respect to the 1/3 filling chemical potential. As in Fig. \ref{['fig:monolayer']}, the linewidth indicates the unfolding weight and the reflection symmetry eigenvalues are indicated with $+$ and $-$ signs.
• Figure 4: (a) Crystal structure of PtBi$_2$. (b) Density of states in the absence of SOC for the undistorted (space group P$\bar{3}$2/m1) and distorted (P31m) crystal structures. (c,d,e) Band structure of in the absence of spin-orbit coupling for different crystal structures, the size of the red dots indicates the unfolding weight: c) centrosymmetric limit ($\alpha=0$) where the unfolding weight exctly distinguished folded bands; d) intermediate value of $\alpha=0.39$ where Weyl nodes emerge; e) noncentrosymmetric case ($\alpha=1$).