Isotropic 3D topological phases with broken time reversal symmetry, by Helene Spring, Anton R. Akhmerov, Daniel Varjas

SciPost Physics Home Authoring Refereeing Submit a manuscript About Isotropic 3D topological phases with broken time reversal symmetry Helene Spring, Anton R. Akhmerov, Daniel Varjas SciPost Phys. 20, 051 (2026) · published 20 February 2026 doi: 10.21468/SciPostPhys.20.2.051 pdf BiBTeX RIS Submissions/Reports Abstract Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a three-dimensional medium with average isotropy and inversion symmetry where time-reversal symmetry is systematically broken. We devise a model of an amorphous material with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the bonds. The presence of only average spatial symmetries—continuous rotation and inversion—is sufficient to protect a topological phase, yielding a statistical topological insulator. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant for the effective continuum model, which guarantees gapless surface spectrum on any surface with an odd number of Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.20.2.051TI - Isotropic 3D topological phases with broken time reversal symmetryPY - 2026/02/20UR - https://scipost.org/SciPostPhys.20.2.051JF - SciPost PhysicsJA - SciPost Phys.VL - 20IS - 2SP - 051A1 - Spring, HeleneAU - Akhmerov, Anton R.AU - Varjas, DanielAB - Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a three-dimensional medium with average isotropy and inversion symmetry where time-reversal symmetry is systematically broken. We devise a model of an amorphous material with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the bonds. The presence of only average spatial symmetries—continuous rotation and inversion—is sufficient to protect a topological phase, yielding a statistical topological insulator. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant for the effective continuum model, which guarantees gapless surface spectrum on any surface with an odd number of Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.ER - × @Article{10.21468/SciPostPhys.20.2.051, title={{Isotropic 3D topological phases with broken time reversal symmetry}}, author={Helene Spring and Anton R. Akhmerov and Daniel Varjas}, journal={SciPost Phys.}, volume={20}, pages={051}, year={2026}, publisher={SciPost}, doi={10.21468/SciPostPhys.20.2.051}, url={https://scipost.org/10.21468/SciPostPhys.20.2.051},} Supplementary Information External links to supplemental resources; opens in a new tab. Code repository Ontology / Topics See full Ontology or Topics database. Amorphous solids Magnetic phases Topological insulators Topological materials Authors / Affiliations: mappings to Contributors and Organizations See all Organizations. 1 Helene Spring, 1 Anton R. Akhmerov, 2 3 4 5 Daniel Varjas 1 Technische Universiteit Delft / Delft University of Technology [TU Delft] 2 Max-Planck-Institut für Physik komplexer Systeme / Max Planck Institute for the Physics of Complex Systems 3 Budapesti Műszaki és Gazdaságtudományi Egyetem / Budapest University of Technology and Economics [BUTE] 4 Stockholm University [Univ Stockholm] 5 Leibniz-Institut für Festkörper- und Werkstoffforschung Dresden / Leibniz Institute for Solid State and Materials Research [IFW] Funders for the research work leading to this publication Deutsche Forschungsgemeinschaft / German Research FoundationDeutsche Forschungsgemeinschaft [DFG] Knut och Alice Wallenbergs Stiftelse / Knut and Alice Wallenberg Foundation Magyar Tudományos Akadémia / Hungarian Academy of Sciences [MTA] Ministerie van Onderwijs, Cultuur en Wetenschap / Ministry of Education Culture and Science [OCW] Nederlandse Organisatie voor Wetenschappelijk Onderzoek / Netherlands Organisation for Scientific Research [NWO] Nemzeti Kutatási, Fejlesztési és Innovációs Hivatal / National Research, Development and Innovation Office [NKFIH] Vetenskapsrådet / Swedish Research Council