Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry (1111.7309v2)

Abstract: We perform a complete classification of two-band $\bk\cdot\mathbf{p}$ theories at band crossing points in 3D semimetals with $n$-fold rotation symmetry and broken time-reversal symmetry. Using this classification, we show the existence of new 3D topological semimetals characterized by $C_{4,6}$-protected double-Weyl nodes with quadratic in-plane (along $k_{x,y}$) dispersion or $C_6$-protected triple-Weyl nodes with cubic in-plane dispersion. We apply this theory to the 3D ferromagnet HgCr$_2$Se$_4$ and confirm it is a double-Weyl metal protected by $C_4$ symmetry. Furthermore, if the direction of the ferromagnetism is shifted away from the [001]- to the [111]-axis, the double-Weyl node splits into four single Weyl nodes, as dictated by the point group $S_6$ of that phase. Finally, we discuss experimentally relevant effects including splitting of multi-Weyl nodes by applying $C_n$ breaking strain and the surface Fermi arcs in these new semimetals.

• The paper derives detailed constraints and effective theories for $C_m$ invariant lines within $C_n$ invariant systems, focusing on band structure and band crossings.
• It introduces the classification of band eigenvalues based on $C_m$ representations and identifies the condition $p eq q$ as crucial for band crossings along invariant lines.
• The research highlights the concept of semi-conservation of total angular momentum $J_z$, conserved up to multiples of $m$, impacting band coupling in systems with discrete rotational symmetry.

Constraints and Effective Theories in $C_m$ Invariant Systems

The research paper articulates a detailed derivation of constraints and effective theories applicable to $C_m$ invariant lines within $C_n$ invariant systems, with $m$ as a divisor of $n$. The discussion centers around the representations of cyclic groups and how these representations influence the band structure in spinless and spinful systems.

Key Findings and Constraints

• Eigenvalue Representations: The paper explicates how the eigenvalues of $C_m$ are organized, positing that a characteristic eigenvalue for a band is formatted as $\alpha_p = \exp(i2\pi p/m + iF\pi/m)$, where $p$ spans from 0 to $m-1$. The classification based on these eigenvalues is central to understanding band crossings and gap openings in these invariant systems.
• Band Crossing Conditions: A critical finding is the condition $p \neq q$ for the presence of a band crossing along a $C_m$-invariant line; otherwise, the occurrence of a constant off-diagonal term can induce a gap. The symmetry involved allows for such terms, impacting the matrix representation of $C_m$ and its transformative effects on effective Hamiltonians.
• Effective Hamiltonians and Constraints: Through mathematical transformations, the paper derives expressions that relate the $C_m$ operations to effective Hamiltonians. This is articulated through formulas that describe the action of $C_m$ on $H_{eff}$ and yield constraints on the functions $f(q_+, q_-)$ and $g(q_+, q_-)$. These constraints are formulated in terms of the eigenvalue relations of conduction to valence bands, specifically involving their ratio $u_c/u_v$.
• Table I Analysis: The manuscript devotes significant attention to constraints presented in Table I, deriving entries via expansions of $f(q_+, q_-)$ and identifying non-zero coefficients subject to relation $n_2-n_1 = p-q\;\text{mod}\;m$. This process clarifies how semi-conservation laws of angular momentum impact term structure in the effective Hamiltonian.

Theoretical Implications

The research emphasizes the concept of 'semi-conservation' of total angular momentum $J_z$ within discrete rotational symmetries. This notion proposes that $J_z$ is conserved up to multiples of $m$ due to the discretization of rotational symmetry, critically affecting band coupling through terms dependent on $k_-$ and $k_+$ powers. This semi-conservation principle extends theoretical understanding of band interactions and symmetry reductions in crystalline solids possessing $C_m$ symmetry.

Practical Implications and Future Directions

Practically, understanding these symmetry constraints and effective Hamiltonian structures allows for better predictions in material science applications, particularly in designing materials with desired electronic properties. The constraints presented might influence future explorations into topological properties of bands in complex crystals.

Future developments could consider expanding these theoretical frameworks to explore non-linear effects, introducing interaction terms or exploring their implications in non-equilibrium systems. The foundational nature of this work suggests various paths towards integrating these constraints with novel computational techniques or experimental methods to further the understanding of band topology in crystalline materials.