Absence of topological order in the $U(1)$ checkerboard toric code, by Maximilian Vieweg, Viktor Kott, Lea Lenke, Andreas Schellenberger, Kai Phillip Schmidt

SciPost Physics Home Authoring Refereeing Submit a manuscript About Absence of topological order in the $U(1)$ checkerboard toric code Maximilian Vieweg, Viktor Kott, Lea Lenke, Andreas Schellenberger, Kai Phillip Schmidt SciPost Phys. 20, 056 (2026) · published 24 February 2026 doi: 10.21468/SciPostPhys.20.2.056 pdf BiBTeX RIS Submissions/Reports Abstract We investigate the $U(1)$ checkerboard toric code which corresponds to the $U(1)$-symmetry enriched toric code with two distinct star sublattices. One can therefore tune from the limit of isolated stars to the uniform system. The uniform system has been conjectured to possess topological order based on quantum Monte Carlo simulations suggesting a non-trivial ground-state degeneracy depending on the compactification of the finite clusters. Here we show that these non-trivial properties can be naturally explained in the perturbative limit of isolated stars. Indeed, the compactification dependence of the ground-state degeneracy can be traced back to geometric constraints stemming from the plaquette operators. Further, the ground-state degeneracy is fully lifted in fourth-order degenerate perturbation theory giving rise to a non-topological phase with confined fracton excitations. These fractons are confined for small perturbations so that they cannot exist as single low-energy excitation in the thermodynamic limit but only as topologically trivially composite particles. However, the confinement scale is shown to be surprisingly large so that gaps are extremely small on finite clusters up to the uniform limit which is calculated explicitly by high-order series expansions. Our findings suggest that these gaps were not distinguished from finite-size effects by the recent quantum Monte Carlo simulation in the uniform limit. All our results therefore point towards the absence of topological order in the $U(1)$ checkerboard toric code along the whole parameter axis. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.20.2.056TI - Absence of topological order in the $U(1)$ checkerboard toric codePY - 2026/02/24UR - https://scipost.org/SciPostPhys.20.2.056JF - SciPost PhysicsJA - SciPost Phys.VL - 20IS - 2SP - 056A1 - Vieweg, MaximilianAU - Kott, ViktorAU - Lenke, LeaAU - Schellenberger, AndreasAU - Schmidt, Kai PhillipAB - We investigate the $U(1)$ checkerboard toric code which corresponds to the $U(1)$-symmetry enriched toric code with two distinct star sublattices. One can therefore tune from the limit of isolated stars to the uniform system. The uniform system has been conjectured to possess topological order based on quantum Monte Carlo simulations suggesting a non-trivial ground-state degeneracy depending on the compactification of the finite clusters. Here we show that these non-trivial properties can be naturally explained in the perturbative limit of isolated stars. Indeed, the compactification dependence of the ground-state degeneracy can be traced back to geometric constraints stemming from the plaquette operators. Further, the ground-state degeneracy is fully lifted in fourth-order degenerate perturbation theory giving rise to a non-topological phase with confined fracton excitations. These fractons are confined for small perturbations so that they cannot exist as single low-energy excitation in the thermodynamic limit but only as topologically trivially composite particles. However, the confinement scale is shown to be surprisingly large so that gaps are extremely small on finite clusters up to the uniform limit which is calculated explicitly by high-order series expansions. Our findings suggest that these gaps were not distinguished from finite-size effects by the recent quantum Monte Carlo simulation in the uniform limit. All our results therefore point towards the absence of topological order in the $U(1)$ checkerboard toric code along the whole parameter axis.ER - × @Article{10.21468/SciPostPhys.20.2.056, title={{Absence of topological order in the $U(1)$ checkerboard toric code}}, author={Maximilian Vieweg and Viktor Kott and Lea Lenke and Andreas Schellenberger and Kai Phillip Schmidt}, journal={SciPost Phys.}, volume={20}, pages={056}, year={2026}, publisher={SciPost}, doi={10.21468/SciPostPhys.20.2.056}, url={https://scipost.org/10.21468/SciPostPhys.20.2.056},} Ontology / Topics See full Ontology or Topics database. Quantum many-body systems Topological codes Authors / Affiliation: mappings to Contributors and Organizations See all Organizations. 1 Maximilian Vieweg, 1 Viktor Kott, 1 Lea Lenke, 1 Andreas Schellenberger, 1 Kai Phillip Schmidt 1 Friedrich-Alexander-Universität Erlangen-Nürnberg / University of Erlangen-Nuremberg [FAU] Funder for the research work leading to this publication Deutsche Forschungsgemeinschaft / German Research FoundationDeutsche Forschungsgemeinschaft [DFG]