Nonsymmorphic Band Degeneracy

Updated 26 August 2025

Nonsymmorphic symmetry-protected band degeneracy is a phenomenon where combined point-group operations and fractional translations enforce unavoidable band crossings in crystalline solids.
These symmetries induce momentum-dependent phase factors that prevent band hybridization, leading to robust topological phases such as node-surfaces, nodal lines, and point nodes.
The mechanism remains effective under spin–orbit coupling and is evidenced in materials like BaTaS₃, PdSb₂, and ZrSiS, linking band topology with novel quantum transport properties.

Nonsymmorphic symmetry-protected band degeneracy refers to the phenomenon where band crossings in the electronic structure of crystalline solids are enforced by nonsymmorphic space group symmetries—operations that combine a point-group action (such as rotation or mirror reflection) with a fractional lattice translation. Unlike symmorphic symmetries, these operations cannot be reduced to a pure point-group operation followed by a full lattice translation, and they fundamentally dictate band connectivity by making certain degeneracies unavoidable in the Brillouin zone. This mechanism underpins a wide range of topological phases, including nodal points, lines, and surfaces, often exhibiting robust protection even in the presence of spin–orbit coupling and in both bulk and surface band structures.

1. Fundamental Mechanisms of Nonsymmorphic Symmetry Protection

Nonsymmorphic space group symmetries include glide mirrors, screw axes, and off-centered operations (where the translation is not within the invariant subspace of the point-group operation). For a symmetry operator written as $G = \{g | t\}$ , $g$ is a point-group operation and $t$ is a fractional translation. Nonsymmorphic symmetries induce momentum-dependent phase factors in the Bloch states, leading to nontrivial algebraic relations that enforce band degeneracies.

A central result is that these symmetries can force the existence of "sticky bands"—degeneracies along high-symmetry lines or planes—because bands with distinct symmetry eigenvalues cannot hybridize without breaking the nonsymmorphic symmetry. For example, an off-centered twofold rotation $\widetilde{C}_{2z}^{\perp} = \{C_{2z} | \frac{1}{2}\hat{x} + \frac{1}{2}\hat{y}\}$ enforces quantized momentum-independent eigenvalues (e.g., $\pm i$ ), and, together with inversion ( $P$ ) or time reversal, the anticommutation relation $\{P, \widetilde{C}_{2z}^{\perp}\} = 0$ protects doublet pairs and higher-order degeneracies (Yang et al., 2016, Zhang et al., 2018).

Glide symmetries, $\widetilde{M}_{x} = \{M_{x} | \frac{1}{2}\hat{x}\}$ , likewise square to $-1$ (for spinful electrons), so their eigenvalues are $\pm i$ and force twofold (or fourfold, when combined with additional symmetries) degeneracies along mirror-invariant planes (Li et al., 2017, Cuamba et al., 2017).

2. Topological Band Connectivity and Types of Degeneracies

Nonsymmorphic symmetry-protected degeneracies result in an arsenal of topological semimetal phases distinguished by the dimension and structure of the band crossing:

Node-surfaces: Two-dimensional manifolds of band degeneracy, as realized in the $k_z = \pi/c$ plane of BaVS $_3$ , protected by a skew axial symmetry $\mathcal{S}_z = \{ C_{2z} | \, c/2 \}$ in combination with time-reversal and inversion (Liang et al., 2016).
Node-lines: One-dimensional degeneracies, often fourfold in spinful systems, enforced along high-symmetry lines by operations such as mirror or off-centered screw symmetries. In BaTaS $_3$ , strong spin–orbit coupling reduces the node-surface to robust node-lines along $k_x = 0$ and $k_x = \pm \sqrt{3}k_y$ (Liang et al., 2016, Li et al., 2017, Hao et al., 2021).
Point nodes (Dirac or Weyl points): Isolated fourfold or twofold crossings, enforced for example at the R point in nonsymmorphic cubic pyrite-type PdSb $_2$ (sixfold crossing due to nonsymmorphic plus Kramers degeneracy) or as Weyl/Dirac points in hexagonal or trigonal materials (Chapai et al., 2019, Chan et al., 2019, Zhang et al., 2018).

Topological invariants associated with these degeneracies include Berry phases (e.g., $\pi$ for a nontrivial nodal line loop), Chern numbers for Weyl points, and even global $\mathbb{Z}_2$ charges for symmetry-enforced gapless points (Zhao et al., 2016, Lee et al., 2018).

3. Algebraic and Representation-Theoretic Basis

Band connectivity enforced by nonsymmorphic symmetry can be derived both from the algebra of symmetry operators and from compatibility relations between irreducible representations (irreps):

Algebraic relations: For $G^n = \{g^n | nt\} = - p T_a$ , the minus sign accounts for spin– $\frac{1}{2}$ electrons under $2\pi$ rotation. The eigenvalues for $G$ are momentum-dependent, e.g., $G|\psi_m(k)\rangle = e^{i\pi(2m+1)/n}e^{-ip(k\cdot a)/n}|\psi_m(k)\rangle$ .
Representation mismatch: In flat band systems, the symmetry representation of the compact localized state (CLS) may not match that of the constituent orbitals at high-symmetry points, enforcing a "symmetry representation (SR)"-enforced band crossing (Hwang et al., 2021).
Global topological invariants: In two-band systems, the presence of nonsymmorphic symmetry enforces a $\mathbb{Z}_2$ global topology; band crossings cannot be gapped out unless the system is doubled, in sharp contrast to local topological charges, as in Weyl semimetals (Zhao et al., 2016).

The momentum dependence of symmetry eigenvalues leads to "partner switching"—bands with differing symmetry labels at different high-symmetry points must cross, as shown in the comparison of representation irreps between endpoints of symmetry-invariant paths (Zhang et al., 2018, Chan et al., 2019).

4. Robustness under Spin–Orbit Coupling, Breaking and Surface Effects

A fundamental property of nonsymmorphic symmetry-protection is its robustness to spin–orbit coupling (SOC):

In many cases (BaTaS $_3$ , Dirac semimetals with off-centered symmetries, etc.), the doublet pair structure and quantized symmetry eigenvalues remain, preserving fourfold or twofold degeneracies even for strong SOC (Liang et al., 2016, Yang et al., 2016, Li et al., 2017).
However, the inclusion of SOC may "detrimentally" change the algebra, turning an enforced anticommutation (and hence doublet pairing) into a commutation, leading to lifted degeneracies—leaving only crossings at certain high-symmetry points (Wu et al., 2022).

Surface effects are pronounced: when a nonsymmorphic crystal is cleaved, the reduction in symmetry lifts the enforced degeneracies, resulting in surface "floating" two-dimensional bands that are neither Shockley nor conventional topological states (Topp et al., 2017). The alteration of surface mass terms creates new states closely tied to the parent bulk band's degeneracy points.

5. Material Realizations and Experimental Signatures

Empirical studies confirm nonsymmorphic symmetry protection in a range of quantum materials:

BaMX $_3$ family (M = V, Nb, Ta; X = S, Se): Node-surfaces and node-lines observed via first-principles calculations and symmetry analysis (Liang et al., 2016).
Ta $_3$ SiTe $_6$ , Nb $_3$ SiTe $_6$ : Hourglass Dirac loops, nodal lines, and Dirac points protected by glide mirror and screw symmetries, validated by ARPES (Li et al., 2017, Sato et al., 2018).
PtPb $_4$ : Dirac nodal lines along BZ boundaries, their bandwidth drastically reduced (to $\sim$ 0.2 eV) from DFT predictions, indicating strong correlation effects on top of nonsymmorphic symmetry (Wu et al., 2022).
PdSb $_2$ : Sixfold-degenerate fermion crossing at R, observable as nearly massless carriers with Berry phase $\approx \pi$ , leading to unconventional magnetoresistance and a pressure-induced superconducting dome (Chapai et al., 2019).
ZrSiS and analogs: Surface floating bands distinguishable from Shockley or topological insulator surface states, observed via DFT and ARPES and triggered by broken nonsymmorphic symmetry at the surface (Topp et al., 2017).
Molecular crystal OsOF $_5$ : Coexistence of 0D, 1D, and 2D band crossings (Weyl points, nodal lines, nodal surfaces), all symmetry-enforced and characterized by topological invariants (Berry phase, Chern number, $\mathbb{Z}_2$ index) (Lee et al., 2018).

Key experimental observables include Dirac-like dispersions in ARPES; topological drumhead and arc surface states; and anomalous magnetotransport signatures tied to the presence of symmetry-enforced nodal structures.

6. Topological Response and Device Implications

Symmetry-protected band degeneracy leads to several emergent devices-relevant phenomena:

Drumhead surface states: Associated with nodal lines, offering enhanced density of states and, potentially, unconventional superconductivity or correlated electronic phases (Li et al., 2017, Liang et al., 2016, Lee et al., 2018).
Fermi arc states: Appear between symmetry-enforced Weyl points on the boundary (Zhang et al., 2018, Chan et al., 2019).
Anomalous Hall and magnetoelectric response: Rooted in the Berry curvature distribution near enforced band crossings (Chan et al., 2019).
Band tunability via external parameters: Magnetic/Zeman field can split Dirac points into Weyl nodes or gap the system into a topological insulator with chiral edge states; chemical doping can tune the Fermi surface topology near type-II Dirac points (Yang et al., 2016, Cuamba et al., 2017).
Multipole topology: Glide symmetries can quantize Wannier multipole moments, enabling bulk-edge-corner correspondences beyond conventional topological insulator paradigms and multiplexing of topological effects in engineered metamaterials (Zhang et al., 2018).

In several platforms, especially in metacrystals and synthetic gauge field systems, nonsymmorphic symmetries can be engineered to access non-Abelian chiral symmetry algebras, protecting Kramers quartet states and enabling new classes of double Dirac semimetals (Yang et al., 2022).

7. Theoretical Generalizations and Future Research

Nonsymmorphic symmetry-protection extends to:

Global topological charges: Enforcement of band crossings results in $\mathbb{Z}_2$ (rather than $\mathbb{Z}$ ) stability—odd numbers of crossings are robust, even numbers can be gapped (Zhao et al., 2016).
No-go theorems: SR-enforced crossings preclude the possibility of isolated flat bands for CLSs with certain symmetry representations, constraining possible electronic (or bosonic) band structures and flat Chern bands in lattice models (Hwang et al., 2021).
Synthetic systems: Nonsymmorphic symmetry matrices can be constructed in cold atom or photonic systems, resulting in unusual projective representations and band degeneracies not accessible in natural crystals (Zhao et al., 2020, Yang et al., 2022).

Further research directions include exploration of quantum confinement in lower-dimensional systems, the stability of topological response under interactions and disorder, and the use of symmetry design to engineer desired band topology for next-generation quantum materials.

This technical overview assembles the key principles, mathematical formulations, representative material platforms, and topological implications of nonsymmorphic symmetry-protected band degeneracy, as established across recent theoretical and experimental advances.