Line bundles in Bloch band theory

In this post, I would like to present an alternative viewpoint of Bloch band theory, in light of bundle theory, which perhaps some people are not aware of. 

Before doing so, I want to give you an idea of how line bundles over tori, i.e., manifolds of the form $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$, arise in terms of $\mathbb{Z}^n$-group actions.  Let $p:\mathbb{R}^n\to\mathbb{T}^n$ be the quotient map. Let $L\to \mathbb{T}^n$ be a smooth complex line bundle. Then, the pullback bundle $p^*L$, whose fiber over $x\in\mathbb{R}^n$ is the fiber of $L$ over $p(x)$, is a line bundle over a contractible space. It follows that $p^*L$ admits a global section $\sigma: \mathbb{R}^n\to p^*L$. Now observe that if $x$ and $x'=x+\gamma$ differ by an element $\gamma\in \mathbb{Z}^n$, i.e., they satisfy $p(x)=p(x')$, then $\sigma(x)$ and $\sigma(x')$ lie in the same vector space, namely $L_{p(x)}$. As a consequence, there exists an invertible complex number $e_{\gamma}(x)$ such that

$$\sigma(x+\gamma)=\sigma(x)e_{\gamma}(x)^{-1}.$$

Using associativity of the sum, it follows that

$$e_{\gamma_1+\gamma_2}(x)= e_{\gamma_2}(x+\gamma_1)e_{\gamma_1}(x).$$

The above condition is known as the cocycle condition and the collection of *smooth* functions $(e_{\gamma})_{\gamma\in\mathbb{Z}^n}$ is know as a system of multipliers for $L$. More on that below.

For any $v\in \mathbb{C}$, we have 

$$\sigma(x)v= \sigma(x+\gamma)e_{\gamma}(x)v,$$

hence, the pair $(x,v)$ which determines $\sigma(x)v$ in the fiber $(p^*L)_{x}=L_{\pi(x)}$ defines the same element as $(x+\gamma,e_{\gamma}(x)v)$ over the fiber $(p^*L)_{x+\gamma}=L_{\pi(x+\gamma)}=L_{\pi(x)}$. The system of multipliers $(e_{\gamma})_{\gamma\in\mathbb{Z}^n}$ defines an action of $\mathbb{Z}^n$ on the trivial line bundle $\mathbb{R}^n\times \mathbb{C}\to \mathbb{R}^n$ by

$$\gamma\cdot (x,v)= (x+\gamma, e_{\gamma}(x)v).$$

The cocycle condition ensures that this is indeed an action. The quotient $\left(\mathbb{R}^n\times \mathbb{C}\right)/\mathbb{Z}^n$ is naturally a bundle over $\mathbb{T}^n$ which is identified with $L$. By the way, we can also use this to give a description of the space of sections of $L$, denoted by $\Gamma(L)$, in terms of a certain space of functions in $\mathbb{R}^n$ satisfying twisted boundary conditions. Namely, a section of $L$, $s: \mathbb{T}^n\to L$, will be expressible in terms of $\sigma$. In particular, if we take a representative $x$ for $p(x)$, we a unique element $f(x)\in\mathbb{C}$ such that

$$s(x)=\sigma(x)f(x).$$

Clearly, due to the relations satisfied by $\sigma$, $f(x+\gamma)=e_{\gamma}(x)f(x)$. Conversely, given a function $f: \mathbb{R}^n\to \mathbb{C}$ satisfying these boundary conditions will determine a section by the above formula.

Now a particular kind of system of multipliers arises if we have a $1$-dimensional irreducible representation (irrep) of $\mathbb{Z}^n$. In particular, if we take $k\in\mathbb{R}^n$, the assignment 

$$\gamma\mapsto \chi(\gamma)=e^{2\pi i k\cdot \gamma}$$

defines an irrep of $\mathbb{Z}^n$. Observe that $k$ and $k+ G$, with $G\in \mathbb{Z}^d$, determines the same character $\chi$. We learn that the set of unitarry irreps of $\mathbb{Z}^d$ is another torus $\mathbb{T}^n$, which is nothing but what condensed matter physicists (like me) call the Brillouin zone! In the condensed matter physics context $k$ is known as the quasimomentum. 

Setting $e_{\gamma}(x)=\chi(\gamma)$, it trivially follows that we have a system of multipliers. Namely, we have a line bundle $L_{\chi}=\mathbb{R}^n\times_{\chi} \mathbb{C}$ consisting of equivalence classes of pairs $(x,v)\in\mathbb{R}^n\times \mathbb{C}$ under the equivalence relation

$$(x,v)\sim (x+\gamma, e^{2\pi ik\cdot \gamma}v).$$

Now, by the above discussion, it is clear that sections of $L_{\chi}$ are in one-to-one correspondence with functions $f:\mathbb{R}^n\to\mathbb{C}$ satisfying

$$f(x+\gamma)=e^{2\pi ik\cdot \gamma}f(x).$$

Bloch band theory arises, for example, in the setting of the Schroedinger equation in $\mathbb{R}^d$ in the presence of a lattice-periodic potential, say $\mathbb{Z}^d$. In such case, the lattice acts unitarily on the Hilbert space via translations and this action commutes with the Hamiltonian. Because of this we can find a basis of common eigenvectors. The eigenvectors of the translations---known as Bloch wavefunctions---are precisely functions satisfying

$$f(x+\gamma)=\chi(\gamma)f(x)=e^{2\pi ik\cdot \gamma}f(x),$$

for some quasimomentum $k$. We then learn that the set of Bloch wavefunctions consistent with quasimomentum $k$ are sections of $L_{\chi}$ with $\chi=e^{2\pi i k\bullet}$.