Remarks on invertible phases with non-onsite symmetry
Abstract
We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic ℤ4F symmetry in (2+1) dimensions. For an onsite ℤ4F symmetry, symmetric invertible phases have integer chiral central charge c-∈ℤ, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local ℤ4F symmetry of a p+ip superconductor with c-=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c-∈ ℤ invertible states. The allowed invertible phases are consequently shifted to c-∈ℤ+1/2. By gauging the fermion parity of the p+ip superconductor, one obtains an Ising topological order enriched by a ℤ4 symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ip state admits an enlarged non-onsite U(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)f symmetry again forbids Chern insulators carrying c-∈ℤ, while being compatible with invertible states with c-∈ℤ+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
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BibTeX
@misc{kobayashi2026remarks,
title = {{Remarks on invertible phases with non-onsite symmetry}},
author = {Ryohei Kobayashi and Kansei Inamura and Ken Shiozaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12042},
eprint = {2609.12042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12042},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on invertible phases with non-onsite symmetry
AU - Ryohei Kobayashi
AU - Kansei Inamura
AU - Ken Shiozaki
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12042
UR - https://arxiv.org/abs/2609.12042
AB - We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic ℤ_4^F symmetry in (2+1) dimensions. For an onsite ℤ_4^F symmetry, symmetric invertible phases have integer chiral central charge c_-∈ℤ, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local ℤ_4^F symmetry of a p+ip superconductor with c_-=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c_-∈ ℤ invertible states. The allowed invertible phases are consequently shifted to c_-∈ℤ+1/2. By gauging the fermion parity of the p+ip superconductor, one obtains an Ising topological order enriched by a ℤ_4 symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ip state admits an enlarged non-onsite U(1)^f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)^f symmetry again forbids Chern insulators carrying c_-∈ℤ, while being compatible with invertible states with c_-∈ℤ+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
ER -