The symmetries of a crystal form the guiding principle to understand the topology of its band structure. They dictate the location and degrees of stable band crossings which lead to significant sources of Berry curvature. Here we show how non-crystalline quasi-symmetries stabilize near-degeneracies of bands over extended regions in energy and in the Brillouin zone. Specifically, a quasi-symmetry is an exact symmetry of a $kcdot p$ Hamiltonian to lower-order that is broken by higher-order terms. Hence quasi-symmetric points are gapped, yet the gap is parametrically small and therefore does not influence the physical properties of the system. We demonstrate that in the eV-bandwidth semi-metal CoSi an internal quasi-symmetry stabilizes gaps in the 1-2 meV range over a large near-degenerate plane. This quasi-symmetry is key to explaining the surprising simplicity of the experimentally observed quantum oscillations of four interpenetrating Fermi surfaces around the R-point. Untethered from limitations of crystalline symmetry, quasi-symmetries can source large Berry curvature over wide ranges of energy and on low symmetry points - thereby impacting quasiparticle dynamics in unexpected places. Quasi-symmetries also lead to new types of Wigner-Von Neumann classifications.
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