I am dealing with a Chiral-symmetric Hamiltonian such that
$$ ๐๐ป๐^{โ1}=โ๐ป. $$
Two of its eigenstates have zero eigenvalue and fulfill $๐โฃ๐_{\alpha}โฉ=๐^{๐๐_{\alpha}}โฃ๐_{\alpha}โฉ$, while the rest have finite eigenvalues and are pairwise symmetric. When I diagonalize my Hamiltonian numerically however, the zero energy states get mixed and the resulting eigenstates are therefore not symmetric.
When I have a symmetry that commutes with my Hamiltonian, $[๐ป,๐]=0$, and two degenerate eigenstates, to obtain the symmetric ones I can just add a small perturbation to the Hamiltonian proportional to $๐$ which breaks the degeneracy and does not modify the eigenstates. Is there a similar trick one can do to obtain symmetric degenerate eigenstates for symmetries that anticommute with the Hamiltonian?