Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess–Zumino, and Fu–Kane–Mele

Abstract

We establish a connection between two recently-proposed approaches to the understanding of the geometric origin of the Fu-Kane-Mele invariant FKM ∈ Z2, arising in the context of 2-dimensional time-reversal symmetric topological insulators. On the one hand, the Z2 invariant can be formulated in terms of the Berry connection and the Berry curvature of the Bloch bundle of occupied states over the Brillouin torus. On the other, using techniques from the theory of bundle gerbes it is possible to provide an expression for FKM containing the square root of the Wess-Zumino amplitude for a certain U (N )-valued field over the Brillouin torus.We link the two formulas by showing directly the equality between the above mentioned Wess-Zumino amplitude and the Berry phase, as well as between their square roots. An essential tool of independent interest is an equivariant version of the adjoint Polyakov-Wiegmann formula for fields T 2 → U (N ), of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

“…Finally, the 1-dimensional localisation formulas KM E − ,τ 3 = exp π i X 1 ρ 1 = hol RK(A, T ), X 1 (7.46) are analogous to the geometric formulas for the weak Kane-Mele invariants in terms of Berry phases [28], though again we have not found any canonical choices equating the 1-forms ρ 1 and ν with the trace of the Berry connection on E − . The equivalence between the Berry phase formula of [28] and the bundle gerbe holonomy of [31] is demonstrated explicitly by [59].…”

“…Our representation of the Kane-Mele invariant as a bundle gerbe holonomy again expresses it in terms of 2-dimensional weak Kane-Mele invariants over X 2 [28,33]. The intermediate expression of the Kane-Mele invariant (2.39) as the holonomy of an equivariant line bundle on X 2 is equivalent to the formulation of the discrete Pfaffian formula as a geometric obstruction which expresses it as a kind of Berry phase [28] (see also [46]); the relation between this Berry phase formula and the continuous representation as a Wess-Zumino-Witten amplitude is derived in [59]. However, these representations are only derived for the Kane-Mele invariant in two dimensions, whereas our computation of the Kane-Mele invariant stems from purely 3-dimensional considerations.…”

“…Kane-Mele invariants for semimetals have also been discussed in [71]. A variety of mathematical approaches to the Kane-Mele invariant are known: they range from algebraic topology perspectives [27,59] over index theory [19,54] and C * -algebras [56,57] to geometric approaches [31][32][33]. An overview relating various different treatments of the Kane-Mele invariant is provided by [46].…”

Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators

“…Remark. Proposition 6 implies, in particular, that the right hand side of (9.5) does not dependent on the choice of the submanifold with boundary F ⊂ R forming the closure of a fundamental domain for the involution ρ of R. Although the right hand side of (9.5) may be often defined using a homotopic non-local formula (7.2) for the square root of gerbe holonomy, the local approach based on gerbe theory is useful to establish such a result that, in application to topological insulators, is a powerful source of equalities between different forms of invariants [12,20].…”

Eigenvalue crossings in Floquet topological systems

“…Notice that a similar filtration of Hermitian matrices is used in [Ar]. Moreover, originally developed in the context of conformal field theory, gerbes have been recently used already in the context of topological insulators, with no driving and with time-reversal symmetry [CDFGT,Ga,Ga 2 ,MT]. The present paper is another application of this geometrical concept.…”