nLab unstable classification of topological phases -- references

Unstable classification of topological phases

Arguments for unstable topological phases of matter, saying that that some effects in topological phases of matter are “unstable” (“fragile” or “delicate”) in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an un-stable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:

Influential precursor discussion:

• Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig: Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) [doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786]

More explicit highlighting of the role of the unstable case and coinage of the term “fragile topologucal phase”:

• Hoi Chun Po, Haruki Watanabe, Ashvin Vishwanath: Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121 (2018) 126402 [doi:10.1103/PhysRevLett.121.126402]

• Hoi Chun Po, Liujun Zou, T. Senthil, Ashvin Vishwanath: Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99 (2019) 195455 [doi:10.1103/PhysRevB.99.195455, arXiv:1808.02482]

• Adrien Bouhon, Tomáš Bzdušek, Robert-Jan Slager: Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102 (2020) 115135 [doi:10.1103/PhysRevB.102.115135, arXiv:2005.02044]

Coinage of the term “delicate topological phase”:

• Aleksandra Nelson, Titus Neupert, Tomáš Bzdušek, Aris Alexandradinata: Multicellularity of delicate topological insulators, Phys. Rev. Lett. 126 (2021) 216404 [doi:10.1103/PhysRevLett.126.216404, arXiv:2009.01863]

• Aleksandra Nelson, Titus Neupert, Aris Alexandradinata, Tomáš Bzdušek: Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond, Phys. Rev. B 106 (2022) 075124 [doi:10.1103/PhysRevB.106.075124, arXiv:2111.09365]

• Aleksandra Nelson: Delicate topological insulators: a new world of phases between trivial and fragile, PhD thesis, Zürich (2022) $[$webpage, pdf]

With focus on Bloch Hamiltonian classifying spaces with non-abelian fundamental groups:

• QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek: Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 [doi:10.1126/science.aau8740, arXiv:1808.07469]

Applications:

• Adrien Bouhon, Robert-Jan Slager, around equation (3) in: Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]

• Zory Davoyan, Wojciech J. Jankowski, Adrien Bouhon, Robert-Jan Slager, section II.A in: Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index [doi:10.1103/PhysRevB.109.165125 arXiv:2308.15555]

Expositions with an eye towards non-abelian braiding of band nodes in momentum space:

• Gunnar F. Lange: Multi-gap topology & non-abelian braiding in $\mathbf{k}$-space, talk at University of Oslo (Feb 2023) $[$pdf]

• Adrien Bouhon: Non-Abelian and Euler multi-gap topologies in crystalline materials, talk at: Quantum Information and Quantum Matter, CQTS @ NYU Abu Dhabi (May 2023) $[$pdf]

Further discussion:

• Barry Bradlyn, Zhijun Wang, Jennifer Cano, B. Andrei Bernevig: Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice, Phys. Rev. B 99 (2019) 045140 [doi:10.1103/PhysRevB.99.045140, arXiv:1807.09729]

• Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle, Phys. Rev. X 9 (2019) 021013 [doi:10.1103/PhysRevX.9.021013, arXiv:1808.05375]

• Yoonseok Hwang, Junyeong Ahn, Bohm-Jung Yang: Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop, Phys. Rev. B 100 (2019) 205126 [doi:10.1103/PhysRevB.100.205126, arXiv:1905.08128]

• Zhi-Da Song, Luis Elcoro, B. Andrei Bernevig: Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology, Science 367 6479 (2020) 794-797 [doi:10.1126/science.aaz7650, arXiv:1910.06869]

• Zhida Song, L. Elcoro, Nicolas Regnault, B. Andrei Bernevig: Fragile Phases As Affine Monoids: Classification and Material Examples, Phys. Rev. X 10 031001 (2020) [doi:10.1103/PhysRevX.10.031001, arXiv:1905.03262]

• Xiaoming Wang, Tao Zhou: Fragile topology in nodal-line semimetal superconductors, New Journal of Physics 24 (2022) [doi:10.1088/1367-2630/ac8306, arXiv:2106.06928]

• Piet W. Brouwer, Vatsal Dwivedi: Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology, Phys. Rev. B 108 (2023) 155137 [doi:10.1103/PhysRevB.108.155137, arXiv:2306.13713]

• Simon Becker, Zhongkai Tao, Mengxuan Yang: Fragile topology on solid grounds: a mathematical perspective [arXiv:2502.03442]

• Hisham Sati, Urs Schreiber: Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems [arXiv:2507.00138]