Invertible subalgebras for chiral semion topological order
Abstract
Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the ℤ4 toric code, we condense the boson e2m2 and identify the semion and anti-semion algebras. We then decompose each ℤ4 qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the z direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a U(1)4 ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions d=4k−2, with one qubit or ℤ4 qudit on each (2k−1)-cell for the semion or U(1)4 ISA, respectively.
Related
- Monstrous parafermionic defects and other non-invertible symmetries in chiral CFTs
- Non-Clifford quantum cellular automata from invertible topological quantum field theories
- A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology
- Logarithmic transport at topological phase transitions of one-dimensional chiral quantum systems
- Chiral Spin(7) Sigma Models and Topological Modular Forms
BibTeX
@misc{chen2026invertible,
title = {{Invertible subalgebras for chiral semion topological order}},
author = {Yu-An Chen and Meng Sun and Nathanan Tantivasadakarn and Zongyuan Wang and Bowen Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18953},
eprint = {2610.18953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18953},
}
RIS
TY - UNPB
TI - Invertible subalgebras for chiral semion topological order
AU - Yu-An Chen
AU - Meng Sun
AU - Nathanan Tantivasadakarn
AU - Zongyuan Wang
AU - Bowen Yang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18953
UR - https://arxiv.org/abs/2610.18953
AB - Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the ℤ_4 toric code, we condense the boson e^2m^2 and identify the semion and anti-semion algebras. We then decompose each ℤ_4 qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the z direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a U(1)_4 ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions d=4k−2, with one qubit or ℤ_4 qudit on each (2k−1)-cell for the semion or U(1)_4 ISA, respectively.
ER -