Interplay between inversion and translation symmetries in trigonal PtBi$_2$

TL;DR

This work shows that in trigonal PtBi$_2$, inversion-symmetry breaking via a crystal distortion introduces a translational-symmetry reduction that dominates the energy landscape, producing large hopping asymmetries and band folding which underlie the metal-to-semimetal transition and the formation of Weyl nodes. Spin-orbit coupling primarily shapes the low-energy nodal structure and Fermi arcs, but the Weyl nodes near the Fermi energy can originate from orbital physics in the absence of SOC due to the reduced translational symmetry. The two Weyl-node sets have distinct origins—one from orbital Weyl points and the other from mirror-protected nodal lines—both ultimately rooted in band folding from the symmetry-lowered structure. The results connect PtBi$_2$ topology to broader phenomena where translational symmetry is spontaneously reduced and motivate phonon- and strain-based tuning of the distorted phase.

Abstract

The trigonal Weyl semimetal PtBi$_2$ presents an intriguing superconducting phase, previously reported to be confined to its topological Fermi arcs within a certain temperature range. This observation highlights the importance of a thorough understanding of its normal phase, particularly the roles that spin-orbit coupling (SOC) and inversion-symmetry breaking play in shaping its band structure. Our density-functional theory calculations reveal that the semimetallic nature of trigonal PtBi$_2$ can be interpreted as stemming from a noncentrosymmetric crystal distortion of a parent structure that drives a metal-to-semimetal transition. This distortion breaks inversion symmetry and, crucially, reduces translational symmetry. Due to its interplay with translational symmetry, inversion-symmetry breaking emerges as the dominant energy scale producing substantial asymmetries ($\sim$ 0.6\,eV) in certain short-range hopping amplitudes, superseding the effects of SOC, whose primary role is to define the characteristics of the low-energy nodal structure and of the topological Fermi arcs. This also applies to the formation of the Weyl nodes closest to the Fermi energy, which are found to exist even in the absence of SOC as a result of the orbital physics associated with the reduced translational symmetry.

Figures (9)

• Figure 1: (a) Crystal structure of trigonal PtBi$_2$ with space group $P\bar{3}m1$. This inversion-symmetric ($I$-symmetric) structure consists of stacked triangular lattices with Pt ions serving as inversion centers. The corresponding first Brillouin zone (FBZ) is outlined in blue, where the green arrows correspond to the primitive lattice vectors $G_i$. (b,c) Band structures calculated for the $I$-structure (b) without spin-orbit coupling (SOC) and (c) with SOC. Red dots indicate the nonfolded bands. (d) Crystal structure of PtBi$_2$ in the inversion-broken structure ($P31m$). The corresponding FBZ is outlined in black. The red dashed segments outside the $I$-broken FBZ are translated by $G_1$ and $G_2$ to $\Gamma$-$K$. e,f) Band structures calculated for the $I$-broken structure (e) without SOC and (f) with SOC included.
• Figure 2: Density of states without (a) and with SOC (b).
• Figure 3: a,b) Top view of the upper Bi plane for the $I$-symmetric and $I$-broken structures, respectively. The vectors $\mathbf{R_1}$ and $\mathbf{R_2}$ have distinct lengths in the noncentrosymmetric case. c) Amplitude of the matrix elements between Bi 6$p_y$ orbitals as a function of the crystalline distortion [$\alpha$ in Eq. (\ref{['eq_interpolation']})] along bonds $\mathbf{R_1}$ and $\mathbf{R_2}$ illustrated in panels a,b. d,e) Side view of the undistorted and distorted crystal structures, highlighting four bonds which become inequivalent in the latter case. f) Amplitude of the hopping along these four bonds between Bi 6$p_y$ and Pt 5$d_{x^2-y^2}$ orbitals as a function of the crystalline distortion parameter $\alpha$.
• Figure 4: (a) Evolution of the Weyl nodes closest to the Fermi energy as a function of the spin-orbit coupling ($\lambda$ in Eq. \ref{['eq_interpolation2']}). Red and blue correspond to different Chern number signs. For $\lambda=0$, two Weyl nodes of identical Chern number overlay at the yellow triangles. Each of these corresponds to a monopole of the orbital Berry curvature. The yellow circles correspond to the fully relativistic limit. The same phenomenology occurs at $C_3$-related planes and, due to time-reversal symmetry, at opposite $\mathbf{k}$. (b) Projection of the nodal points onto the $k_x$-$k_y$ plane. The area covered by the graphic corresponds to the small rectangle shown in panel (a). Dots with lighter shades correspond to weaker spin-orbit coupling. (c) For $\lambda=0$, trajectory of the Weyl nodes as a function of the crystalline distortion ($\alpha$ in Eq. \ref{['eq_interpolation']}). At $\alpha\approx0.39$, a pair of Weyl nodes emerges from the $\Gamma$-$M$ line (gray star). The arrows indicate the sense of increasing $\lambda$ (b) or $\alpha$ (c).
• Figure 5: Band structure of PtBi$_2$ along one of the Weyl nodes without SOC (a) or with SOC (b). In the first case, the point $w=(0.416,-0.041,0.124)2\pi/a$ while in the second $w=(0.324,-0.041,0.153)2\pi/a$. The width of the red lines measures the unfolding weight, which tends to zero for folded bands.