Symmetries of Spin-Splitting Induced by Spin-Orbit Coupling in Non-magnetic Crystals

TL;DR

This work develops a symmetry-driven, irrep-projection framework to classify SOC-induced spin-splitting in time-reversal–invariant, non-magnetic crystals. By analyzing $ ext{I}$-odd irreps of the cubic and hexagonal parent groups, the authors show that linear-in-$k$ spin-splitting can be captured by four SOC types—Rashba, Dresselhaus, Weyl, and Ising—and provide reciprocal-space Hamiltonians and minimal TB models for all relevant order parameters, including higher-order Dresselhaus-cubic terms. They connect these SOC terms to electric and electrotoroidal multipoles, revealing how order parameters map to spin textures and how secondary orders shape nodal lines and points near the zone center; external fields can drive topological transitions between nodal configurations. The paper also catalogs material realizations for each SOC type, linking spin-splitting symmetries to possible superconducting and spintronic phenomena, and situates the results within a broader context that includes altermagnetism and chirality-driven effects.

Abstract

Spin-orbit coupling (SOC) leads to splitting of otherwise spin-degenerate bands in noncentrosymmetric materials, even if time-reversal symmetry is present. While this gives rise to well-known phenomena such as the Rashba and Dresselhaus effects, various other terms are allowed based on the point group of the crystal and the electronic Hamiltonian. In this study, we utilize point group representations to illustrate that four different types of SOC terms (Rashba, Dresselhaus, Weyl, and Ising) can emerge in periodic solids. We construct reciprocal space energy expressions for each type of SOC-induced splitting of opposite spin bands, and follow a similar procedure to also obtain minimal tight-binding models that capture all types of spin-splittings for subgroups of the cubic parent group $m\bar{3}m$. Furthermore, we also obtain a complete list of nodal features in the electronic band structure in these systems, distinguishing between crystallographic-symmetry-imposed nodal lines and those imposed by time-reversal-symmetry only. Finally, we conclude by presenting a list of materials that host each type of inversion-breaking SOC effects. Our classification of the spin-splitting symmetries in non-magnetic systems with SOC is the counterpart of the recent classification of spin-splitting symmetries in unconventional magnetic systems without SOC, such as altermagnets and odd-parity magnets. More broadly, our work provides a basis for studying superconductivity and other collective electronic phenomena that are impacted by SOC-induced band splittings in noncentrosymmetric materials.

Figures (10)

• Figure 1: Classification of different $\mathcal{H}_{SOC}(\vb*{k},\vb*{\sigma})$ corresponding to the $\mathcal{I}$-odd irreps of the most symmetric cubic group, $m\bar{3}m$. The actual Hamiltonian is given by this term multipled by a (possibly multi-component) order parameter that transforms as the corresponding irrep. (a) Irreps of $m\bar{3}m$ according to which each class of $\mathcal{H}_{SOC}$ terms transform, and the lowest-order bases listed in pseudo-vector form. Each set of bases are chosen to be orthogonal and to span the space of the particular irrep. Schematics of resultant spin patterns are plotted on $k_x$-$k_y$ plane around $\Gamma$ with $k_z$ slightly above 0 in order to demonstrate the feature of out-of-plane spin components. The direction of the arrows corresponds to the in-plane components, whereas their color denotes the out-of-plane components (red for spin-up; blue for spin-down; gray denotes no out-of-plane component). (b) Lowest-order isomorphic E/ET-multipole equivalence for each class of $\mathcal{H}_{SOC}$ that transforms in the same way in free space as the lowest-order $\mathcal{H}_{SOC}$ bases. For ET-multipoles, the lowest-order E-$n$ pole equivalence in $m\bar{3}m$ is listed in parentheses. Explicit mathematical expressions (normalization factor omitted) and schematics are listed for each E-multipole. (c) Coefficient matrices $M$ of TB models for each class (see Section \ref{['sec:tight binding']} for details) and resultant spin patterns around $\Gamma$ in the lowest spin-polarized TB band. Color scheme represents out-of-spin components; golden circles represent equi-energy contours with a fixed interval. (d) NCS point groups that are subgroups of $m\bar{3}m$ in which each class of spin terms are allowed to emerge.
• Figure 2: (a) NCS subgroups of $m\bar{3}m$ and allowed spin-splitting terms therein. Coordinate axes are chosen in the same way as in Fig. \ref{['fig:cubic']}. For monoclinic and orthorhombic groups (labeled with $\dagger$) that have multiple possible symmetry alignments, D-c is specifically allowed when $m\perp\langle 110\rangle$ and $2\parallel\langle 100\rangle$ in the coordinates of $m\bar{3}m$. (b) NCS subgroups of $6/mmm$ and allowed spin-splitting terms therein. Definitions of spin-splitting terms in the hexagonal case, including the Ising SOC, are given in Fig. \ref{['fig:hex']}. For subgroups of $mm2$ (labeled with $\ddagger$), alignments of axes can be different for R-D or R or D to be allowed.
• Figure 3: Hierarchy tree of all NCS crystallographic point groups as subgroups of $m\bar{3}m$ or $6/mmm$. Each group is connected to its maximal subgroups by arrows, labeled with specific $\mathcal{I}$-odd irreps and their directions that break the corresponding symmetries. All irreps are that of the parent groups $m\bar{3}m$ (black) or $6/mmm$ (red). Irrep directions are not specified for connections between orthorhombic and lower groups, as multiple choices are possible. Common subgroups of $m\bar{3}m$ and $6/mmm$ are colored in blue.
• Figure 4: Classification of the different $\mathcal{H}_{SOC}(\vb*{k},\vb*{\sigma})$ terms in the form of vectors corresponding to the $\mathcal{I}$-odd irreps of the most symmetric hexagonal point group, $6/mmm$. The actual Hamiltonian is given by the vector multiplied by a (possibly multi-component) order parameter that transforms as the corresponding irrep. (a) Irreps of $6/mmm$ according to which each class of $\mathcal{H}_{SOC}$ terms transform. Bases are chosen to be orthogonal and to span the space of the corresponding irrep, with schematics of the resultant spin patterns shown on $k_x-k_y$ plane around $\Gamma$. $k_z$ is set to be slightly above 0 in order to demonstrate the out-of-plane spin features (red for spin-up; blue for spin-down). (b) Lowest-order isomorphic E/ET-multipole equivalence for each class of $\mathcal{H}_{SOC}$ which transforms in the same way in free space as the lowest-order $\mathcal{H}_{SOC}$ bases. For ET-multipoles, the lowest-order E-n pole equivalence in $6/mmm$ is listed in parentheses. (c) NCS point groups that are subgroups of $6/mmm$ in which each class of spin terms are allowed to emerge.
• Figure 5: (a) Schematics of the E-512 multipole (i.e., the $l=9$ electric multipole) corresponding to the Weyl SOC term of a cubic crystal, viewed from different crystal directions. Panel (b) shows the $A_{1u}^+$-projected charge density distribution around the central atom (Ce) in the hypothetical material LaCeBe$_{26}$ (chiral space group $F432$) calculated by DFT. Purple/red lobes represent charge-excessive regions and green/blue ones represent charge-deficient regions. Charge density is processed using the package ProDenCeR Prodencer_1Prodencer_2.