Emergent quasi-symmetries in crystals: a physical picture

Recently, the concept of quasi-symmetry has been introduced to describe emergent symmetries that appear only in the first-order term H₁(r) of the Löwdin expansion within the kp method. This phenomenon manifests as small anti-crossings, influencing electronic properties such as non-trivial topological phases. A physical picture for these extra symmetries remains elusive, and recent works have identified quasi-symmetry operators without proper physical interpretation.

In this presentation, we discuss physical pictures for the quasi-symmetry (QS) operators as emergent elements of a space group and provide physical interpretations. In all cases, we find that within certain subspaces the eigenstates are (approximately) invariant under QS, while the lattice itself is not invariant. Using density functional theory (DFT) and the DFT2kp code, we identify the subspaces that enable quasi-symmetries in various materials. For the Sn/SiC and MoS₂ systems, we find that the QS corresponds to an emergent mirror symmetry, while for wurtzite structures, it is an emergent spatial inversion. In both cases, we propose a metric to quantify the degree of QS based on the matrix elements of the QS operator within its relevant subspace. For some materials, e.g., AgLa, we find that the QS picture is useful but not entirely necessary to explain the small value of band anti-crossings. All our findings combine into a proper physical picture, unifying the concepts of quasi-symmetries and pseudo-symmetries.