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It has been well established that for symmetry protected topological systems,the non-trivial topology prevents a real space representation usingexponentially localized Wannier wavefunctions (WFs) in all directions, unlessthe protecting symmetry is sacrificed as an on-site transformation. This makesit challenging to determine the symmetry of various physical observablesrepresented using such WFs. In this work, we propose a practical method forovercoming such challenges using the Kane-Mele model as a concrete example. Wepresent a systematic procedure for diagnosing the symmetry of any observables,as well as a method for constructing symmetric operators up to arbitrarytruncation accuracy.

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