General procedure for component-dependent BCD in multiband systems with symmetry-protected crossings

Develop a general algorithmic procedure for constructing a k-dependent complex deformation h(k) in the Brillouin Complex Deformation method that either detects block-diagonal structure or deforms different Hamiltonian components in distinct k-space directions, thereby ensuring that all bands near a target energy are shifted downward in the complex plane even in the presence of symmetry-protected band crossings.

Background

The BCD method can fail at non–van Hove energies when symmetry-protected crossings mix gradients from unrelated bands, potentially pushing some eigenvalues upward. The authors suggest that a more sophisticated approach should detect block structure or apply component-wise deformations tailored to different parts of the Hamiltonian.

They explicitly state that they have not found a general procedure to implement such an approach, identifying a concrete methodological gap needed to make BCD robust in multiband systems with complex symmetry constraints.