When can we take the Brillouin zone to be a sphere?

Question

When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen the usage of spherical BZ in these answers(1,2) by SE user Heidar. I can think of two possibilities:

(1)Some physical system has a spherical BZ. This is hard to imagine, since it seems to me that all lattice systems with translational symmetries will have a torodal BZ, by the periodicity of Bloch wavefunctions. The closest scenario I can imagine is a continuous system having $\mathbf{R}^n$ as BZ, and somehow(in a way I cannot think of) acquires an one-point compactification.

(2)A trick that makes certain questions easier to deal with, while the true BZ is still a torus.

Can someone elaborate the idea behind a spherical BZ for me?

Update: I recently came across these notes(pdf) by J.Moore. In the beginning of page 9 he mentioned

We need to use one somewhat deep fact: under some assumptions, if $π_1(M) = 0$ for some target space $M$, then maps from the torus $T^ 2\to M$ are contractible to maps from the sphere $S^2 \to M$

I think this is a special case of the general math theorem I want to know, but unfortunately Moore did not give any reference so I'm not sure where to look.

EDIT: The above math theorem is intuitively acceptable to me although I'm not able to prove it. I can take this theorem as a working hypothesis for now, what I'm more interested in is, granted such theorem, what makes a $\pi_1(M)=0$ physical system candidate for strong topological insulators(robust under local perturbations), and why in $\pi_1(M)\neq 0$ case we can only have weak topological insulators.

Crossposted: When can we take the Brillouin zone to be a sphere?

Answer 1

The answer to your question is covered in these two papers: Homotopy and Quantization in Condensed Matter Physics

J. E. Avron, R. Seiler, and B. Simon http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51

and

Homotopy Theory of Strong and Weak Topological Insulators

Ricardo Kennedy, Charles Guggenheim

http://arxiv.org/abs/1409.2529

The first one explains how to use maps from the sphere in the case of the quantum Hall effect, whereas the second one discusses a more general context of all symmetry classes and dimensions.

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Let $X$ be some topological space and $\mathbb{T}^2$ be the $2$-torus. Consider the task of classifying the set $[\mathbb{T}^2\to X]$, the set of homotopy classes the maps $\mathbb{T}^2\to X$.

Then as Avron, Seiler and Simon argue the following:

As you should know, $\pi_1(\mathbb{T}^2)\cong\mathbb{Z}\times \mathbb{Z}$, that is, there are two basic loops $S^1\to \mathbb{T}^2$ which are not homotopic to each other. Call them $\gamma_1$ (blue) and $\gamma_2$ (red).

Then any map $f:\mathbb{T}^2\to X$ induces two maps $S^1\to X$ via $$f\circ\gamma_i:S^1\to X$$

Then one can prove that for two maps $f_1:\mathbb{T}^2\to X$ and $f_2:\mathbb{T}^2\to X$, if the either $f_1\circ\gamma_1$ is not homotopic to $f_2\circ\gamma_1$ OR $f_1\circ\gamma_2$ is not homotopic to $f_2\circ\gamma_2$, then $f_1$ is not homotopic to $f_2$.

However, "even if the two induced maps $S^1\to X$ are pair-wise homotopic, there could still be a left over piece, which is a map $S^2\to X$.". In this way, you could classify maps $\mathbb{T}^2\to X$ by two pieces of $\pi_1(X)$, one piece of $\pi_2(X)$, etc...

For our case, $X$ is the set of gapped Hamiltonians, which topologically is the Grassmannian ($G_m(\mathbb{C}^n)$ for an $n$ level system with the Fermi energy above $m$ levels).

For instance, for a two-level system with 1 occupied level, we have $$G_1(\mathbb{C}^2)\cong S^2$$. But the the $2$-sphere is simply connected (that is, $\pi_1(S^2)=0$), so the two aforementioned loops $\gamma_1$ and $\gamma_2$ will not contribute to the classification, and we would only have the "left over" piece $$S^2\to S^2$$ And it is well known that $$[S^2\to S^2]\equiv\pi_2(S^2)\cong\mathbb{Z}$$ The integer being the degree of the map $\mathbb{T}^2\to S^2$ which would be the same degree as the induced map $S^2\to S^2$ which in a more general setting (of a complex line bundles over $\mathbb{T^2}$) would be the Chern number of that line bundle. Indeed, isomorphism classes of line bundles over $\mathbb{T^2}$ are isomorphic to homotopy classes of maps $\mathbb{T^2}\to Gr_1\left(\mathbb{C}^2\right)\cong S^2$.