Diagnosis of explicit symmetry breaking in the tight-binding constructions for symmetry-protected topological systems

It has been well established that for symmetry-protected topological systems, the nontrivial topology prevents a real space representation using exponentially localized Wannier wave functions (WFs) in all directions, unless the protecting symmetry is sacrificed as an on-site transformation. This makes it challenging to determine the symmetry of various physical observables represented using such WFs. In this work, we propose a practical method for overcoming such challenges using the Kane-Mele model as a concrete example. We present a systematic procedure for diagnosing the symmetry of any observables, as well as a method for constructing symmetric operators up to arbitrary truncation accuracy.