We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to $-\mathbb{1}$. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a $\mathbb{Z}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four $\mathbb{Z}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.

1 aFiorenza, Domenico1 aMonaco, Domenico1 aPanati, Gianluca uhttps://doi.org/10.1007/s00220-015-2552-000480nas a2200109 4500008004100000245009100041210006900132490001100201100001900212700002000231856011900251 2014 eng d00aZeros of Large Degree Vorob'ev-Yablonski Polynomials via a Hankel Determinant Identity0 aZeros of Large Degree VorobevYablonski Polynomials via a Hankel 0 vrnu2391 aBertola, Marco1 aBothner, Thomas uhttps://www.math.sissa.it/publication/zeros-large-degree-vorobev-yablonski-polynomials-hankel-determinant-identity