Z2Invariant

Z2Invariant is a Z2-valued topological invariant used to classify the electronic band structure of noninteracting fermionic systems that respect time-reversal symmetry. It can take the values 0 or 1, with 0 corresponding to a conventional (trivial) insulating phase and 1 indicating a nontrivial topological insulator. A nontrivial Z2 invariant implies robust boundary modes that persist as long as the bulk gap and the protecting symmetry remain intact.

In two dimensions, the Z2 invariant distinguishes quantum spin Hall insulators from ordinary insulators. A 2D system with Z2 = 1 exhibits helical edge states protected by time-reversal symmetry, while Z2 = 0 corresponds to a trivial edge structure. In three dimensions, time-reversal symmetry yields one strong Z2 invariant and three weak Z2 invariants. A nonzero strong invariant signals gapless surface states on most surfaces, whereas the weak invariants reflect layered, quasi-two-dimensional character and do not guarantee robust surface states on all surfaces.

Computationally, several approaches exist. For centrosymmetric crystals, the Fu–Kane–Mele parity criterion expresses the 2D or 3D Z2 invariant in terms of parity eigenvalues at time-reversal-invariant momenta. More generally, the invariant can be obtained from the winding of the non-Abelian Berry connection (Wilson loop) or from Pfaffians constructed from time-reversal symmetry. These methods track how occupied bands transform under time-reversal.

Physically, a nontrivial Z2 invariant implies protected edge or surface states tied to topology and symmetry, with potential applications in spintronics and quantum computing. In mathematics, Z2 invariants also appear as obstructions captured by mod 2 characteristic classes, linking to Stiefel-Whitney theory.