Symplectic Hopf Insulator: Delicate Topology in Bosonic Bogoliubov-de Gennes Systems
Abstract
Recent advances in topological phases have highlighted the role of symplectic (Krein-space) topology in the classification of bosonic Bogoliubov-de Gennes (BBdG) systems. In this work, we construct a BBdG realization of Hopf topology, which we dub the symplectic Hopf insulator, starting from a microscopic Bose-Hubbard generalization of the Moore-Ran-Wen model with weak on-site interactions treated within a Bogoliubov approximation. The resulting BBdG system admits a symplectic Hopf invariant, which we show to be integer-quantized for isolated bands. We establish that this topology is intrinsically delicate, requiring exactly two bosonic modes per unit cell, while remaining robust against weak interactions over a range of mass parameters. Upon terminating the three-dimensional insulator at a boundary, we find topologically protected in-gap surface states at finite excitation energy, whose protection is itself delicate. Our results establish the symplectic Hopf insulator as a robust yet delicate topological phase in weakly interacting bosonic systems lying beyond the tenfold-way classification.
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BibTeX
@misc{tesfaye2026symplectic,
title = {{Symplectic Hopf Insulator: Delicate Topology in Bosonic Bogoliubov-de Gennes Systems}},
author = {Isaac Tesfaye and Giandomenico Palumbo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10541},
eprint = {2609.10541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10541},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symplectic Hopf Insulator: Delicate Topology in Bosonic Bogoliubov-de Gennes Systems
AU - Isaac Tesfaye
AU - Giandomenico Palumbo
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.10541
UR - https://arxiv.org/abs/2609.10541
AB - Recent advances in topological phases have highlighted the role of symplectic (Krein-space) topology in the classification of bosonic Bogoliubov-de Gennes (BBdG) systems. In this work, we construct a BBdG realization of Hopf topology, which we dub the symplectic Hopf insulator, starting from a microscopic Bose-Hubbard generalization of the Moore-Ran-Wen model with weak on-site interactions treated within a Bogoliubov approximation. The resulting BBdG system admits a symplectic Hopf invariant, which we show to be integer-quantized for isolated bands. We establish that this topology is intrinsically delicate, requiring exactly two bosonic modes per unit cell, while remaining robust against weak interactions over a range of mass parameters. Upon terminating the three-dimensional insulator at a boundary, we find topologically protected in-gap surface states at finite excitation energy, whose protection is itself delicate. Our results establish the symplectic Hopf insulator as a robust yet delicate topological phase in weakly interacting bosonic systems lying beyond the tenfold-way classification.
ER -