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Register a new user$mathbb{Z}_2$ invariants of topological insulators as geometric obstructions
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We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to -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.
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Solid state systems with time reversal symmetry and/or particle-hole symmetry often only have $mathbb{Z}_2$-valued strong invariants for which no general local formula is known. For physically relevant values of the parameters, there may exist approximate symmetries or almost conserved observables, such as the spin in a quantum spin Hall system with small Rashba coupling. It is shown in a general setting how this allows to define robust integer-valued strong invariants stemming from the complex theory, such as the spin Chern numbers, which modulo $2$ are equal to the $mathbb{Z}_2$-invariants. Moreover, these integer invariants can be computed using twist
For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^*$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.
We investigate spin transport in 2-dimensional insulators, with the long-term goal of establishing whether any of the transport coefficients corresponds to the Fu-Kane-Mele index which characterizes 2d time-reversal-symmetric topological insulators. Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator, we define the Kubo-like spin conductance $G_K^{s_z}$ and spin conductivity $sigma_K^{s_z}$. We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically well-defined and the equality $G_K^{s_z} = sigma_K^{s_z}$ holds true. Moreover, we argue that the physically relevant condition to obtain the equality above is the vanishing of the mesoscopic average of the spin-torque response, which holds true under our hypotheses on the Hamiltonian operator. This vanishing condition might be relevant in view of further extensions of the result, e.g. to ergodic random discrete Hamiltonians or to Schrodinger operators on the continuum. A central role in the proof is played by the trace per unit volume and by two generalizations of the trace, the principal value trace and it directional version.
We proposed a group-theory method to calculate topological invariant in bi-isotropic photonic crystals invariant under crystallographic point group symmetries. Spin Chern number has been evaluated by the eigenvalues of rotation operators at high symmetry k-points after the pseudo-spin polarized fields are retrieved. Topological characters of photonic edge states and photonic band gaps can be well predicted by total spin Chern number. Nontrivial phase transition is found in large magnetoelectric coupling due to the jump of total spin Chern number. Light transport is also issued at the {epsilon}/{mu} mismatching boundary between air and the bi-isotropic photonic crystal. This finding presents the relationship between group symmetry and photonic topological systems, which enables the design of photonic nontrivial states in a rational manner.
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are $64$ such real index pairings of real $K$-theory with real $K$-homology. For $16$ of them, the Noether index of the pairing vanishes, but there is a secondary $mathbb{Z}_2$-valued invariant. The first set of results provides index formulas expressing each of these $16$ $mathbb{Z}_2$-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from a Fredholm module and shows that the $mathbb{Z}_2$-invariant can be computed as the sign of its Pfaffian and in $8$ of the cases as the sign of the determinant of its off-diagonal entry. This is of relevance for the numerical computation of invariants of topological insulators.
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