Nonsymmorphic Time-Reversal Symmetry

Updated 16 October 2025

Nonsymmorphic time-reversal symmetry is defined as the fusion of conventional TRS with a fractional lattice translation, leading to momentum-dependent symmetry constraints.
It enables a novel Z4 topological classification in 1D superconductors, predicting robust Majorana zero modes and unique hourglass or Möbius band connectivities.
Experimental fingerprints, observable via ARPES and topolectric circuits, include distinct zero-bias peaks and symmetry-enforced band degeneracies confirming theoretical predictions.

Nonsymmorphic time-reversal symmetry refers to the situation where the conventional time-reversal operator is "twisted" or combined with a non-primitive (fractional) lattice translation, resulting in a symmetry operation that has profound consequences for the topological classification, physical phenomena, and band structure degeneracies of quantum systems. This symmetry arises naturally in crystals with nonsymmorphic space group symmetries—those featuring symmetry operations that combine point group elements (like rotation or reflection) with fractional lattice translations—and deeply modifies the standard picture of time-reversal-invariant phases, particularly in one-dimensional and higher-dimensional systems exhibiting superconductivity or correlated topological phenomena.

1. Mathematical Structure of Nonsymmorphic Time-Reversal Symmetry

A nonsymmorphic time-reversal symmetry (NTRS) acts as the product of ordinary time-reversal symmetry $\mathcal{T}$ (an antiunitary operation) and a fractional translation $\tau$ (often by half a unit cell), i.e.,

$\widetilde{\mathcal{T}} = \mathcal{T} \cdot t_{a/2}$

where $t_{a/2}$ acts as a translation by $a/2$ .

In momentum space, this leads to a $k$ -dependent symmetry action on the Bloch Hamiltonian. The symmetry constraint becomes

$U_{\mathcal{T}}(k, a/2) H^*(k) U_{\mathcal{T}}^\dagger (k, a/2) = H(-k)$

with $U_{\mathcal{T}}(k, a/2)$ the matrix representation of the combined operation. This extra $k$ -dependence is what distinguishes NTRS from symmorphic cases, where the time-reversal operator is $k$ -independent.

The matrix $U_{\mathcal{T}}(k, a/2)$ can be constructed by dressing the usual unitary part of time reversal with the translation operator in the projective representation:

$U_{\mathcal{T}}(k, a/2) = U_k^\dagger(k, a/2)\, U_{\mathcal{T}}(0)\, U_k(k, a/2)$

where $U_k$ generates the translation in Bloch momentum. This gives rise to momentum-dependent symmetry constraints that can alter allowed topological invariants.

2. Topological Classification: $\mathbb{Z}_4$ and Beyond

The introduction of nonsymmorphic time-reversal symmetry allows new topological phases beyond the $\mathbb{Z}$ and $\mathbb{Z}_2$ indices of the tenfold way classification. In particular, the interplay of NTRS and particle-hole symmetry in 1D Bogoliubov–de Gennes (BdG) Hamiltonians enables a $\mathbb{Z}_4$ classification for 1D superconductors in class D with NTRS (Shiozaki et al., 2015, Tymczyszyn et al., 13 Oct 2025).

The topological index can be computed using generalized winding numbers and Pfaffian formulas. For a four-band BdG model, the $\mathbb{Z}_4$ invariant is

$N_{\mathrm{D}^{\mathrm{NS}}} = -\frac{2}{\pi} \arg\left\{ \operatorname{Pf}\left[ \sigma_x Z(\pi/a) \right] \right\} + \frac{1}{\pi} \int_{0}^{\pi/a} dk\, \frac{\partial}{\partial k} \arg\left\{ \det\left[ \sigma_x Z(k) \right] \right\} \mod 4$

where $Z(k)$ arises from bringing $H(k)$ to an off-diagonal "Q-matrix" form in the Majorana basis, and $\sigma_x$ enforces the appropriate antiunitary symmetry structure (Tymczyszyn et al., 13 Oct 2025).

This classification predicts four distinct phases: two supporting robust Majorana zero modes (MZM) and two that are topologically trivial regarding edge MZMs. The $4\pi$ periodicity inherent to the nonsymmorphic modification means the phase traced by $\prod_j \lambda_j(k)$ (the product of eigenvalues of $q(k)$ , the off-diagonal block of $Q(k)$ ) does not return after a $2\pi$ sweep, but after $4\pi$ , matching the fourfold structure of the invariant.

Comparison of various symmetry settings:

Symmetry Type	Index Type	Possible Phases
Symmorphic TRS (class DIII)	$\mathbb{Z}_2$	Trivial / Nontrivial
Nonsymmorphic TRS (class D)	$\mathbb{Z}_4$	0, 1, 2, 3
Nonsymmorphic Unitary Only	$\mathbb{Z}$ , $\mathbb{Z}_2$	(no new types)

(Adapted from (Shiozaki et al., 2015, Tymczyszyn et al., 13 Oct 2025))

3. Band Structure and Symmetry-Enforced Degeneracies

Nonsymmorphic time-reversal symmetry generically enforces extra degeneracies and modifies band connectivity. When combined with other nonsymmorphic operations such as glides or screw rotations, band crossings—either symmetry-enforced or protected—can occur at high-symmetry points or along high-symmetry lines. The hallmark is that the momentum-dependent eigenvalues of the symmetry operators feature extra phase windings, resulting in Möbius or hourglass-type connectivities in the band structures (Shiozaki et al., 2015, Zhang et al., 2022, Shiozaki et al., 2015).

For example, in 2D systems:

A glide and time-reversal symmetry can enforce Kramers-like degeneracies even at time-reversal-invariant momenta that otherwise would not have them under symmorphic TRS.
The combination leads to surface states with unique features: Möbius twists and hourglass dispersions where the eigenstates must "switch partners" as momentum traverses the Brillouin zone (Ezawa, 2016, Zhang et al., 2022, Gao et al., 2022).
These features result directly from the algebra [e.g., $\{ \hat{g}_x, \hat{m}_z \} = 0$ ] between nonsymmorphic symmetry and other crystalline symmetries in the presence of TRS.

4. Experimental Fingerprints and Majorana Zero Modes

The experimental consequences of nonsymmorphic time-reversal symmetry include:

Edge Majorana zero modes in 1D topological superconductors, with solitonic domain walls supporting zero-bias peaks corresponding to nontrivial values of the $\mathbb{Z}_4$ index (Tymczyszyn et al., 13 Oct 2025).
Möbius or hourglass fermion surface states, observable by ARPES as band branches that invert their symmetry sector across the surface Brillouin zone (Shiozaki et al., 2015, Ezawa, 2016).
In photonics, symmetry-adapted finite element modeling enables the systematic identification of nonsymmorphic-symmetry-protected modes and degeneracies—where time-reversal symmetry enters the boundary constraints and thus band classification (Liu et al., 26 May 2025).
In non-Hermitian systems, nonsymmorphic symmetry in combination with TRS enforces non-Hermitian skin effects and topologically robust point-gap closings at zero energy, with winding numbers predicting bulk-boundary correspondence in open boundary conditions (Tanaka et al., 2023).

5. Disorder, Local Symmetries, and Experimental Realization

Disorder generally weakens nonsymmorphic protection since translational invariance is only approximate in realistic systems. However, specific local symmetries at domain walls or soliton locations—such as additional symmorphic symmetry preserved precisely at the defect—can stabilize Majorana modes against moderate disorder (Tymczyszyn et al., 13 Oct 2025).

Topolectric circuit networks are a powerful platform to realize such systems in experiment. The nodal structure of the circuit Laplacian can be tuned to emulate the lattice Hamiltonian, and two-point impedance measurements probe the presence of boundary MZM or hourglass connectivity.

6. Relation to Other Symmetries and Broader Implications

Nonsymmorphic time-reversal symmetry is conceptually distinct from standard time-reversal positivity, which is a property of operators invariant under time reversal and ensures sign-free quantum Monte Carlo simulations by guaranteeing positive-definite traces (Wei, 27 Sep 2025). NTRS, on the other hand, involves spatial structure—namely fractional translations—combined with TRS, yielding new classes of robust, quantized topological invariants and corresponding physical boundary phenomena.

Broadly, the paper of NTRS:

Expands the topological periodic table to include $\mathbb{Z}_4$ phases (Shiozaki et al., 2015, Tymczyszyn et al., 13 Oct 2025).
Enables the existence of protected Majorana zero modes in settings not possible with ordinary TRS or symmorphic symmetries.
Underpins the stability of hourglass/ Möbius fermions and symmetry-enforced boundary states in both Hermitian and non-Hermitian condensed matter and photonic systems (Zhang et al., 2022, Tanaka et al., 2023, Liu et al., 26 May 2025).

7. Key Formulas and Computational Approaches

The modified time-reversal operator for a translation by $a/2$ :

$U_{\mathcal{T}}(k, a/2) = U_{k}^\dagger(k, a/2) U_{\mathcal{T}}(0) U_{k}(k, a/2)$

$\mathbb{Z}_4$ invariant for 1D nonsymmorphic BdG chains (Tymczyszyn et al., 13 Oct 2025):

The appearance of Majorana modes at domain walls signaled by abrupt changes in this topological index.

Summary

Nonsymmorphic time-reversal symmetry combines time reversal with non-primitive lattice translations, leading to $k$ -dependent symmetry constraints, expanded topological classifications (notably $\mathbb{Z}_4$ indices for 1D superconductors), and new types of robust boundary states including Majorana zero modes and hourglass-type fermion surface bands. This symmetry is fundamentally distinct from and richer than ordinary TRS, and its experimental signatures are accessible in both quantum materials and engineered metamaterials such as topolectric circuits and symmetry-adapted photonic crystals. Its mathematical framework is built on generalized winding numbers, Pfaffian invariants, and group cohomology, and it is central to ongoing developments in topological quantum matter and advanced computational modeling in condensed matter and photonics (Shiozaki et al., 2015, Tymczyszyn et al., 13 Oct 2025, Liu et al., 26 May 2025, Shiozaki et al., 2015, Tanaka et al., 2023, Zhang et al., 2022, Ezawa, 2016).