Topological gyromorphs

Isidora Araya Day

Phys. Rev. B 114, L080201 (2026)

21 September 2026

Higher-order topology

First-order topology

Chiral Majorana edge modes

Class D in 2D

Second-order topology

Majorana corner modes

Particle–hole and C₄ symmetry in 2D

All clean higher-order phases have been classified,

but how do we identify the ones that are not?

Topology beyond crystals

Ammann–Beenker tiling with eight Majorana corner modes — Varjas et al., Phys. Rev. Lett. 123, 196401 (2019)

Local density of in-gap edge states in an amorphous Weaire–Thorpe model — Marsal et al., Proc. Natl. Acad. Sci. U.S.A. 117, 30260–30265 (2020)

Disorder can drive topological phase transitions

Topological Anderson insulator

Topological Anderson phase diagram with disorder-driven transitions — Groth et al., Phys. Rev. Lett. 103, 196805 (2009)
Li et al., Phys. Rev. Lett. 102, 136806 (2009)

Intrinsic statistical topological phases

Topological-crystal construction of an inversion-protected axion insulator, Chen et al. Fig. 1a — Chen et al., Phys. Rev. Lett. 134, 226601 (2025)

Intrinsic statistical topological-insulator region separated from the trivial insulator by a metal, Chen et al. Fig. 2d — Chen et al., Phys. Rev. Lett. 134, 226601 (2025)

Gyromorphs: rotational order without a lattice

Annotated panels: real-space gyromorph point pattern and reciprocal-space sixtyfold structure factor

Average local rotational symmetry

Local environments fluctuate. Rotating a neighborhood need not map its sites onto one another.

Angular correlations recur around many sites within a single sample.

Diffraction reveals the order: sharp angular peaks on a ring, without a periodic lattice.

Casiulis et al., Phys. Rev. Lett. 135, 196101 (2025)

Quasicrystals and gyromorphs: a duality

Quasicrystals

Aperiodic, with long-range quasiperiodic order and sharp diffraction peaks.

Gyromorphs

Liquid-like translational disorder, with sharp rotational order on a ring in diffraction.

S_{\mathrm{gyro}}(\mathbf k)\;\longleftrightarrow\;g_{\mathrm{quasi}}(\mathbf r)

Fourteenfold gyromorph structure factor on the left and quasicrystal pair correlations on the right, with matching ring features

Gyromorph diffraction ↔︎ quasicrystal correlations

g_{\mathrm{gyro}}(\mathbf r)\;\longleftrightarrow\;S_{\mathrm{quasi}}(\mathbf k)

Gyromorph pair correlations on the left and fourteenfold quasicrystal structure factor on the right, with similar patterns

Gyromorph correlations ↔︎ quasicrystal diffraction

14-fold examples.
Casiulis et al., Phys. Rev. Lett. 135, 196101 (2025)

Model choice

Two coupled superconducting copies

\mathcal H=\sum_j\mathbf c_j^\dagger\epsilon_j\mathbf c_j +\sum_{j,k}f(r_{jk})\mathbf c_j^\dagger T_{jk}\mathbf c_k

\epsilon_j=\mu\,\sigma_z\otimes\tau_z

\begin{aligned} T_{jk}&=\frac{t}{2}\sigma_z\otimes\bigl[\tau_z-i(\tau_x\cos\theta_{jk}+\tau_y\sin\theta_{jk})\bigr]+\frac V2\cos(4\theta_{jk})\,\sigma_y\otimes\tau_0,\\ f(r)&=e^{1-r/R}\Theta(R-r). \end{aligned}

Symmetries: particle–hole and average local C_8

Figure 1c: spectrum and corner-state LDOS

Are these corner modes protected?

Gómez Paz et al., Phys. Rev. B 114, L080201 (2026)

Methods

Effective Hamiltonian
Scattering invariants
Symmetry-reduced spectral localizer

1. Effective Hamiltonian

Project the real-space Green’s function onto plane waves.

G_{\mathrm{eff}}(\mathbf k)_{ab}=\langle\mathbf k,a|(\mathcal H+i\eta)^{-1}|\mathbf k,b\rangle

H_{\mathrm{eff}}=\frac12\left[G_{\mathrm{eff}}^{-1}+(G_{\mathrm{eff}}^{-1})^\dagger\right]

Find momenta where the rotation becomes a good label:

\|[H_{\mathrm{eff}}(\mathbf k),C_8]\|\approx0.

At Γ and A, symmetry eigenvalues invert.

Figure 2 panels a and b: rotation commutator and effective gap

Figure 2 panel c: symmetry-labeled band inversions at Gamma and A

Caveat: C_8 is only approximate and H_{\mathrm{eff}} gets less accurate as k increases.

Gómez Paz et al., Phys. Rev. B 114, L080201 (2026)

2. Scattering theory of higher-order phases

Symmetric sample with threaded flux and spatially symmetric leads

(H-E)\left(\Psi^{\mathrm{in}}\alpha^{\mathrm{in}}+\Psi^{\mathrm{out}}S(E)\alpha^{\mathrm{in}}+\Psi^{\mathrm{loc}}\alpha^{\mathrm{in}}\right)=0

S=\begin{pmatrix}r_{LL}&0\\0&r_{RR}\end{pmatrix},\qquad rr^\dagger=1

Recipe

Construct a symmetric sample.
Thread flux \phi.
Attach spatially symmetric leads.
Compute r(k,\ldots,\phi).
Apply symmetry constraints.
Compute the invariant of r.

Fulga et al., Phys. Rev. B 85, 165409 (2012) · Zijderveld, Araya Day & Akhmerov, SciPost Phys. 19, 058 (2025)

2. Scattering invariant of the gyromorph

Construct an exactly C_8-symmetric annulus from a ⅛ gyromorph sector.

C_8:\quad r(\phi)=Q_{C_8}\,r(\phi)\,V_{C_8}^{\dagger}

\mathcal P:\quad r(\phi)=r^*(-\phi)

At \phi=\pi, the +1 rotation block r' is real in \mathcal{P}-basis

\nu=\det r'(\pi) \in \{\pm1\}.

Figure 3d: gyromorph scattering determinant versus staggered mass, with transport geometry inset

Caveat: repeating one ⅛ sector imposes exact C_8 symmetry.

Gómez Paz et al., Phys. Rev. B 114, L080201 (2026)

3. Symmetry-reduced spectral localizer

Combine the Hamiltonian, position and chiral symmetry:

\widetilde L_{\mathbf x}= \left[i\mathcal H+\kappa\,\hat{\vphantom{X}e}_d\cdot(\mathbf X-\mathbf x)\right]\mathcal C.

Its half-signature counts local chiral charge:

\zeta_{\mathbf x}=\frac12\operatorname{sig}\widetilde L_{\mathbf x} =\frac12(N_+-N_-).

Figure 3c: half-signature of the symmetry-reduced localizer versus staggered mass

Figure S2b: local gap of the symmetry-reduced localizer versus staggered mass

Caveat: Spectral localizers did not exist for spatially symmetric systems.

Gómez Paz et al., Phys. Rev. B 114, L080201 (2026) · Cerjan, Loring & Schulz-Baldes, Phys. Rev. Lett. 132, 073803 (2024)

A consistent topological phase diagram

Conclusions:

Gyromorphs combine an amorphous-like absence of translational order with quasi-long-range rotational order.
They host higher-order topology protected by average local rotational symmetry.
Photonic lattices are the natural platform to look for it.

All four results together: real-space spectrum, effective-Hamiltonian symmetry inversions, localizer and scattering invariant, Figure 3

Symmetric approximant formalism for statistical topological matter

arXiv:2601.00784

Simple invariants for band topology

arXiv:2609.07775

Gómez Paz et al., Phys. Rev. B 114, L080201 (2026)