---
title: Nonsymmorphic Band Degeneracy
url: https://www.emergentmind.com/topics/nonsymmorphic-symmetry-protected-band-degeneracy
type: topic
---

# Nonsymmorphic Band Degeneracy

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 [1604.00843, 1805.05120].

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 [1710.08376, 1708.01202].

## 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 [1601.01440].
- **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 $k_x = \pm \sqrt{3}k_y$ [1601.01440, 1710.08376, 2104.02221].
- **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 [1904.08014, 1908.00901, 1805.05120].

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 [1606.03698, 1808.06346].

## 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 [2105.14919].
- **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 [1606.03698].

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 [1805.05120, 1908.00901].

## 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 [1601.01440, 1604.00843, 1710.08376].
- 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 [2202.06927].

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 [1709.00319]. 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 [1601.01440].
- **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 [1710.08376, 1809.10348].
- **PtPb$_4$:** Dirac nodal lines along BZ boundaries, their bandwidth drastically reduced (to $\sim$0.2 eV) from DFT predictions, indicating strong correlation effects on top of nonsymmorphic symmetry [2202.06927].
- **PdSb$_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 [1904.08014].
- **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 [1709.00319].
- **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) [1808.06346].

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 [1710.08376, 1601.01440, 1808.06346].
- **Fermi arc states**: Appear between symmetry-enforced Weyl points on the boundary [1805.05120, 1908.00901].
- **Anomalous Hall and magnetoelectric response**: Rooted in the Berry curvature distribution near enforced band crossings [1908.00901].
- **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 [1604.00843, 1708.01202].
- **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 [1811.05514].

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 [2209.06697].

## 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 [1606.03698].
- **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 [2105.14919].
- **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 [2007.00575, 2209.06697].

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.

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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.

Source: https://www.emergentmind.com/topics/nonsymmorphic-symmetry-protected-band-degeneracy