A reader informed us of an error in Theorem 3.3, which characterizes magnetic symmetries; more precisely, items (ii) and (iii) are only true in d = 1 and generally false in d > 1. Therefore, Theorem 1.5 is false in dimension d > 1—under the stated assumptions in Ref. 2, the existence of a magnetic time-reversal symmetry is not automatic. Indeed, the Haldane model is an explicit counterexample in dimension 2.3 

Consequently, we need to add this as an extra hypothesis to our assumptions in our main results of Ref. 2, namely, Theorems 1.6–1.8 only hold once we impose

Assumption
(Presence of time-reversal symmetry). Suppose that in addition to Assumptions 1.1 on the fields and potentials and the Gap Condition 2.1, there exists an antiunitary operator T with
THAk=HAkT.
(1)

The error lies in the Proof of item (ii), where we incorrectly claim that

ixj,λ1=Âjλ1.

This is false. Since item (iii) hinges on (ii), this mistake also invalidates (iii).

The main body of our paper consists of two parts: the first part, Sec. III, is functional analytic and introduces the notion of magnetic symmetries. The second part, comprised of Secs. IV and V, is algebraic in nature and covers the Bloch vector bundle and its topological classification.

The first part is false since it depends on the pivotal Theorem 3.3. Nevertheless, the second part is unaffected once we add the assumption that a time-reversal symmetry satisfying (1) is present. Importantly, Theorem 4.6 (i) remains true in this setting and states that the Bloch vector bundle ξ satisfies the relation

f*(ξ)ξ*,
(2)

where f* is the map induced by the involution f:BdBd, k ↦ −k on the level of vector bundles.

Our subsequent discussion from Sec. V B onwards makes no reference to magnetic fields and only hinges on the validity of (2). Indeed, it is this relation between the Bloch bundle ξ and its conjugate bundle ξ* that ensures the vanishing of all odd Chern classes. Therefore, it implies the triviality of the Bloch bundle in dimensions d ⩽ 3. Our discussion for higher dimensions also holds true verbatim: when d = 4, we need to assume in addition that the second Chern class vanishes, and when d ⩾ 5, we need to be aware of the unstable regime (d ≰ 2m, where m is the rank of the vector bundle and m > 1).

Even under these stronger hypotheses, our work contributes to the state of the art at the time of publication, for it extends the classification beyond d ⩽ 4—including explicit counter examples where the classification of vector bundles over the torus by Chern classes fails. Such a classification is of immediate relevance to physics since, recently, topologically non-trivial systems with artificial dimensions up to d = 6 have been studied theoretically4,6,7 and the case d = 4 has been realized experimentally.8 

Moreover, we expect that our work extends to magnetic systems in various limiting regimes, e.g., in the case of a constant but very small magnetic field by adapting the ideas of, e.g., Refs. 1 and 5 (Theorem 5.2).

The authors thank the attentive anonymous reader for pointing out the mistake in our paper.

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