Hierarchy of quasisymmetries and degeneracies in the CoSi family of chiral crystal materials

In materials, certain approximated symmetry operations can exist in a lower-order approximation of the effective model but are good enough to influence the physical responses of the system, and these approximated symmetries were recently dubbed "quasisymmetries" [Nat. Phys. 18, 813 (2022)]. In this paper, we reveal a hierarchy structure of the quasisymmetries and the corresponding nodal structures that they enforce via two different approaches of the perturbation expansions for the effective model in the chiral crystal material CoSi. In the first approach, we treat the spin-independent linear momentum (k) term as the zero-order Hamiltonian. Its energy bands are fourfold degenerate due to an SU(2) x SU(2) quasisymmetry. We next consider both the k-independent spin-orbit coupling (SOC) and full quadratic k terms as the perturbation terms and find that the first-order perturbation leads to a model described by a self-commuting "stabilizer code" Hamiltonian with a U(1) quasisymmetry that can protect nodal planes. In the second approach, we treat the SOC-free linear k term and k-independent SOC term as the zero order. They exhibit an SU(2) quasisymmetry, which can be reduced to U(1) quasisymmetry by a choice of quadratic terms. Correspondingly, a twofold degeneracy for all the bands due to the SU(2) quasisymmetry is reduced to twofold nodal planes that are protected by the U(1) quasisymmetry. For both approaches, including higher-order perturbation will break the U(1) quasisymmetry and induce a small gap similar to 1 meV for the nodal planes. These quasisymmetry protected near degeneracies play an essential role in understanding recent quantum oscillation experiments in CoSi.