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In the field of topological insulators

  1. What topological space do they talk of?
  2. Looking for some resources that sheds light on the topology part of topological insulator
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  1. Put "bluntly", in the IQHE for example, one refers to the space of classes of vector bundles over the two-dimensional torus (the Brillouin zone), which apparently is isomorphic to $\mathbb{Z}$, the isomorphism given by the Hall conductivity.

  2. It really depends on what perspective you want, and how realistic you want to be from the physics side. Very generally you need to know algebraic topology, vector bundle theory, K-theory (of vector bundles or C-star algebras). There are plenty of resources about this already. I propose for you to start with the book of Prodan and Schulz-Baldes: https://www.springer.com/gp/book/9783319293509

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  • $\begingroup$ How is the Brillouin zone a torus? What rank are these (I guess smooth?) vector bundles? If your claim about the set of isomorphism classes of vector bundles on the torus, then the rank is at least greater than $1$: The group of isomorphism classes of (holomorphic) vector bundles on the torus is isomorphic to $\mathbb{C}/\Lambda \oplus \mathbb{Z}$, where $\Lambda \subset \mathbb{C}$ is a lattice. $\endgroup$
    – AmorFati
    Commented Sep 17, 2020 at 10:30
  • $\begingroup$ Also, what is the $K$-theory of $C$-star algebras? There is topological $K$-theory and algebraic $K$-theory, neither of which relate to $C$-star algebras (at least to my knowledge). $\endgroup$
    – AmorFati
    Commented Sep 17, 2020 at 10:32
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    $\begingroup$ The Brillouin zone is a torus because momentum space is periodic, so at dimension $d$ Hamiltonians are smooth maps $\mathbb{T}^d\to \mathrm{Gr}_n(\mathbb{C}^\infty)$ the latter being the Grassmannian manifold. Hence they define a rank $n$ vector bundle above the $d$-torus. The group of isomorphism classes of $\mathbb{T}^2\to \mathrm{Gr}_1(\mathbb{C}^\infty)$ is $\mathbb{Z}$. The K-theory of C-star algebras can be studied in the book by Rordam web.math.ku.dk/~rordam/K-theory.html $\endgroup$
    – PPR
    Commented Sep 17, 2020 at 14:10

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