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I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is known that, in this case, the first Chern number describes the topological invariant for the entire Brillouin zone. However, what happens if we take a closed adiabatic loop in the Brillouin torus associated with Haldane's model? Then we have a manifold with a boundary.

I might be wrong, but to my understanding, the linked reference suggests, in equations 8.2 and 8.5 that the Atiyah-Singer index theorem implies that the second Chern class will be involved in this case and that we will still have a quantized topological invariant.

However, I am having trouble wrapping my head around this because Haldane's model is a 2D system, and other resources (like the top-most paragraph in page 1630003-19 of this paper, for example) imply that it is not useful to look at the second Chern number in 2D. What then, does this topological invariant mean? Won't it be useless if it is not applicable?

Clearly, there is a gap in my understanding, so I would appreciate it if someone could help me see what I missed. Thanks!

Reference 1: Gravitation, Gauge Theories and Differential Geometry, by Eguchi, Gilkey & Hanson

Reference 2: Notes on topological insulators, by R. Kaufmann & D. Li

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  • $\begingroup$ Which manifold has boundary? Space? $\endgroup$ – Ryan Thorngren Sep 23 '18 at 6:16
  • $\begingroup$ @RyanThorngren by manifold, I’m thinking of the local area inside the loop in momentum space (Brillouin torus). The loop itself will be the boundary. $\endgroup$ – TribalChief Sep 23 '18 at 6:25
  • $\begingroup$ What is the physical meaning of such a thing though? $\endgroup$ – Ryan Thorngren Sep 23 '18 at 16:59
  • $\begingroup$ @RyanThorngren I’m trying to figure that out. :’-) $\endgroup$ – TribalChief Sep 23 '18 at 18:10
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Indeed, the second Chern class cannot be a topological invariant in two dimensions. This class is represented by a rank $4$ form on smooth manifolds, thus its associated topological quantum numbers can be obtained only by integration over $4$ dimensional cycles which do not exist in this case. The index theorem mentioned in the question from Eguchi-Gilkey-Hanson's work is explicitly given for $4$ dimensional manifolds.

However, the second Chern class is indirectly connected to the classification of vector bundles over $2$ dimensional systems as follows:

Just to remind, vector bundles over two dimensional manifolds are classified by means of the first Chern Class. But when the bundle is time reversal invariant (in this case the first Chern class vanishes) there exists a binary invariant: the Fu–Kane–Mele $Z_2$ invariant which discriminates between cases where the Berry phase is always $2\pi$ (the trivial case), or it can assume the value of $\pi$ (the nontrivial case).

One way to understand the $Z_2$ invariant is to look at higher dimensional systems from which our two-dimensional system can be obtained by dimensional reduction. The dimensional reduction is to be understood according to the Kaluza-Klein paradigm in which the extra dimensions are compactified and in the infrared only the low energy modes which are constants along these directions reside.

Qi, Hughes and Zhang showed in a seminal work that a $2+1$ dimensional system with nontrivial $Z_2$ invariant can be obtained from dimensional reduction of a $4+1$ dimensional free fermion system only if the latter's second Chern class is odd.

This phenomenon is related to the dimensional ladder of chiral anomalies, please see Nakahara section 13.6.2.

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  • $\begingroup$ Thank you very much for your detailed response! Do you know whether the idea of a second Chern class in the 2D Haldane model would make sense in the context of inter-band quantum systems? We know that the first Chern number is for intra-band systems. I know that this sounds absurd but I am still trying to think of other (even unrealistic) scenarios that the second Chern class will make sense in the 2D Haldane model. $\endgroup$ – TribalChief Sep 27 '18 at 4:55
  • $\begingroup$ Does this mean the Kane-Mele invariant is the Z/2 in degree 2 in KO theory? Or did I misunderstand? I usually think of this as the Arf invariant. $\endgroup$ – Ryan Thorngren Sep 27 '18 at 5:38
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    $\begingroup$ TribalChief: The second Chern class density has the form $\mathrm{tr}F\wedge F$. The only way to obtain a quantum number from this density is to integrate it over a $4$ dimensional space. In the example that you brought, the Brillouin zone is two dimensional, thus there is no way to integrate the second Chern density to obtain a topological quantum number. $\endgroup$ – David Bar Moshe Sep 30 '18 at 7:32
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    $\begingroup$ Cont. Nakahara's book was written prior to the discovery of the $Z_2$ invariant. Thus, there is no specific mention of it in the book. I gave the Nakahara reference because it has a detailed description of the anomaly dimensional Ladder. Qi, Hughes and Zhang mentioned that their work relates to the anomaly dimensional Ladder in the middle of the second column of page 40. $\endgroup$ – David Bar Moshe Sep 30 '18 at 7:32
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    $\begingroup$ Cont. The connection of the anomaly ladder to topological insulators and the $Z_2$ invariant was also mentioned in the following work by Li Kaufmann and Wehefritz-Kaufmann: de.arxiv.org/abs/1501.02874v1 (last paragraph of section 3.4). It was elaborated more explicitly in the context of the quantum Hall effects by Hasebe arxiv.org/abs/1612.05853v2. $\endgroup$ – David Bar Moshe Sep 30 '18 at 7:32

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