Symmetry-adapted models for multifold fermions with spin-orbit coupling

TL;DR

This work develops symmetry-adapted effective models for multifold fermions with spin–orbit coupling, focusing on threefold and eightfold degeneracies protected by nonsymmorphic space groups. The authors construct both $k \cdot \mathbf{p}$ Hamiltonians and complementary tight-binding models that respect crystal symmetries and include external-field couplings (magnetic field and strain). A key result is the magnetic-field–induced splitting of the threefold node into multiple magnetic monopoles with conserved total charge, and the tight-binding treatment revealing monopole–antimonopole pair annihilation at higher field strengths. Together, these models provide a robust framework for predicting external-field responses and transport phenomena in multifold fermions and guide experimental exploration in topological quantum materials.

Abstract

Multifold fermions, quasiparticles with multiple degeneracy protected by crystalline symmetries, exhibit a variety of intriguing phenomena stemming from their large topological charges and unique band structures. A comprehensive understanding of their response to external stimuli remains challenging, especially for types protected by nonsymmorphic symmetries where various degrees of freedom are intricately coupled. Here, we systematically construct effective models for multifold fermions that incorporate external fields based on crystalline symmetry. Specifically, we develop a $\boldsymbol{k} \cdot \boldsymbol{p}$ model for the threefold fermion protected by space group I2$_1$3 (No.~199) in the presence of spin--orbit coupling, and derive the terms for external fields. By complementing this with a tight-binding model, we investigate the magnetic field response and reveal the pair annihilation of magnetic monopoles. Furthermore, we construct a $\boldsymbol{k} \cdot \boldsymbol{p}$ model for the eightfold fermion in space group P$\bar{4}3n1'$ (No.~218), including its coupling to external fields. This work provides a robust theoretical foundation for advancing the study of external field responses and transport phenomena in multifold fermions, opening new avenues to explore their rich physics.

Figures (3)

• Figure 1: (a) Primitive unit cell which respects $I2_13$ symmetry Momma2011-ve. The green sites are included in the cell, while the gray sites are located in the next cells. The primitive lattice vectors of bcc, $\bm{a}_1, \bm{a}_2$, and $\bm{a}_3$, are indicated. (b) Brillouin zone. (c) Band structure and density of states (DOS). The parameters are taken as $\lambda_1 = 2.0, \, \lambda_2 = 1.0, \, \lambda_3 = -1.5$.
• Figure 2: Splitting of magnetic monopoles in the $\bm k \cdot \bm p$ model when a magnetic field is applied along [(a) and (b)] the [111] axis and [(c) and (d)] the [100] axis. (a) and (c) are shown for both cases with and without the field, while (b) and (d) are shown for the case with the field. The parameters $v_1$, $v_2$, $g_1$, and $g_2$ are obtained from Eqs. \ref{['eq:v1']}--\ref{['eq:g2']} using $\lambda_1=2.0$, $\lambda_2=1.0$, and $\lambda_3=-1.5$.
• Figure 3: Splitting of magnetic monopoles by a magnetic field in the Brillouin zone for a lattice system with space group $\mathrm{I}2_131'$ (No. 199) symmetry. The monopole charge is calculated for the second band from the highest in energy, which is the highest among the threefold fermions. (a) The magnetic monopoles without a magnetic field. The labels, TF, DW, W1, W2, and W3, correspond to those in Fig. \ref{['fig:tb199']}. (b) Splitting of the threefold fermion at the P point when a magnetic field is applied along the [111] axis. As the magnetic field increases, the magnetic monopoles shift as indicated by the arrows, then annihilate in pairs at the points denoted by the red stars. A $C=+1$ magnetic monopole, which splits from the threefold fermion, is on the $\Gamma$-P line (not shown).