SC♯: superconductivity intertwined with topological order
Abstract
Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge 2ne. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC∗, and finally to SC♯. Our main finding is the family SC♯, in which the elementary vortex carrying h/(2ne) flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least √6 and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC♯ phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.
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BibTeX
@misc{gao2026superconductivity,
title = {{SC$^\sharp$: superconductivity intertwined with topological order}},
author = {Zhi-Qiang Gao and Yan-Qi Wang and Hui Yang and Zhaoyu Han and Ashvin Vishwanath},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10679},
eprint = {2610.10679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10679},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - SC^♯: superconductivity intertwined with topological order
AU - Zhi-Qiang Gao
AU - Yan-Qi Wang
AU - Hui Yang
AU - Zhaoyu Han
AU - Ashvin Vishwanath
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10679
UR - https://arxiv.org/abs/2610.10679
AB - Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge 2ne. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC^∗, and finally to SC^♯. Our main finding is the family SC^♯, in which the elementary vortex carrying h/(2ne) flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least √6 and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC^♯ phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.
ER -