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arXiv:2604.07313 · Physical Review Research

Z 2 q parafermionic hinge states in a three-dimensional array of coupled nanowires

Sarthak Girdhar, Viktoriia Pinchenkova, Even Thingstad, Jelena Klinovaja

Published Aug 8, 2026• 0 citations
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Abstract

We construct a model of a three-dimensional helical second-order topological superconductor formed by an array of weakly coupled Rashba nanowires. We identify the parameter regime in which there are energy gaps in both the bulk and surface energy spectra, while a pair of helical Z 2 q parafermionic modes (with q being an odd integer) remains gapless along a closed path of one-dimensional hinges. The precise trajectory of these hinge modes is dictated by the hierarchy of interwire couplings and the boundary termination of the sample. In the noninteracting limit q = 1 , the system hosts gapless Majorana hinge modes.

DOI
10.1103/4jdx-k37q

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We construct a model of a three-dimensional helical second-order topological superconductor formed by an array of weakly coupled Rashba nanowires. We identify the parameter regime in which there are energy gaps in both the bulk and surface energy spectra, while a pair of helical Z 2 q parafermionic modes (with q being an odd integer) remains gapless along a closed path of one-dimensional hinges. The precise trajectory of these hinge modes is dictated by the hierarchy of interwire couplings and the boundary termination of the sample. In the noninteracting limit q = 1 , the system hosts gapless Majorana hinge modes.
In this work, we present a collection of three-dimensional higher-order symmetry-protected topological phases (HOSPTs) with gapless hinge modes that exist only in strongly interacting systems subject to subsystem symmetry constraints. We use a coupled wire construction to generate three families of microscopic lattice models: insulators with helical hinge modes, superconductors with chiral Majorana hinge modes, and fractionalized insulators with helical hinge modes that carry fractional charge. In particular, these HOSPTs do not require spatial symmetry protection, but are instead protected by subsystem symmetries, and support ``fractonic'' quasiparticle excitations that move within only a low-dimensional submanifold of the system. We analyze the anomaly structure for the boundary theory and the entanglement Hamiltonian, and show that the side surfaces of these HOSPTs, despite being partially gapped, exhibit symmetry anomalies, and can only be realized as the boundary of three-dimensional HOSPT phases.
We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting C_{2n}T symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time-reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a C_{2n}T preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between C_{2n}T-conjugate domains. However, using a combination of K-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence "anomalous." Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.
Topological nodal superconductors possess gapless low energy excitations that are characterized by point or line nodal Fermi surfaces. In this work, using a coupled wire construction, we study topological nodal superconductors that have protected Dirac nodal points. In this construction, the low-energy electronic degrees of freedom are confined in a three-dimensional array of wires, which emerge as pairing vortices of a microscopic superconducting system. The vortex array harbors an antiferromagnetic time-reversal and a mirror glide symmetry that protect the massless Dirac fermion in the single-body noninteracting limit. Within this model, we demonstrate exact-solvable many-body interactions that preserve the underlying symmetries and introduce a finite excitation energy gap. These gapping interactions support fractionalization and generically lead to nontrivial topological order. We also construct a special case of N=16N=16N=16 Dirac fermions where the corresponding gapping interaction leads to a trivial E8{E}_{8}E8​ topological order that is closely related to the cancellation of the large gravitational anomaly.
Classification of symmetry-protected topological phases such as the topological superconductor (TSC) is well known in the free fermion picture. Recently, it was observed that including many-body interactions that preserves symmetry gives rise to novel surface phases and breaks down the classification scheme. A complete understanding of the microscopic theory for such phases is missing. This paper is an attempt to understand the interaction that gives rise to these phases in the case of the three-dimensional TSC. The authors build a coupled wire model Hamiltonian for the gapless surface and show that the addition of a time-reversal symmetric two-body interaction, constructed in this paper, opens an energy gap in the spectrum. They also show this gapped surface phase is topologically nontrivial and is classified differently compared to the noninteracting case.
Finite-length one-dimensional topological superconductor wires host localized Majorana zero modes at their ends. In realistic models, these appear only after a topological quantum critical point is crossed by external tuning of parameters. Thus, there is a universal finite-size scaling, governed by the critical point, that dictates the evolution of the energy of the Majorana modes near the transition. We first describe this scaling, then apply it in detail to an explicit synthetic topological superconductor model. Our work not only connects Ising quantum criticality with realistic nanowires in the presence of spin-orbit coupling, Zeeman splitting, and superconductivity, but also provides a viable experimental route for discerning the existence of the topological quantum critical point.
We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.