For my master's thesis, I decided to dive into the world of topological insulators, this led me to the applications of Non-commutative Geometry into homogeneous materials a.k.a. disordered crystals. We followed Prodan et.al. on how K-theory and Cyclic cohomology provides topological invariants for Hamiltonians describing disordered crystals. The results from Prodan et.al. also provide a proof for the quantization of such invariants under weak disorder. For more information refer to my thesis.
We will work with effective models over lattices, these are called tight-binding models. The Fourier analysis is a powerful tool to analyze tight-binding models over periodic lattices a.k.a. periodic crystals, because it allows to translate problems from $L^2(\mathbb{Z}^d)\otimes \mathbb{C}^n$ into $L^2(\mathbb{T}^d)\otimes \mathbb{C}^n$ , that is, we go from a Hamiltonian over a lattice to an equivalent operator over the $n$ dimensional Brillouin zone $L^2(\mathbb{T}^d) \otimes \mathbb{C}^n \simeq (L^2(\mathbb{T}^d))^n$. We are interested in translation invariant Hamiltonians, those take the form
$$ H = \sum_{y \in \mathbb{Z}^d} W_y S^y, \; \text{a finte sum,} $$
with $S^y \in B(L^2(\mathbb{Z}^d)), \; S^y \ket{x} = \ket{x+y}$ . A Hamiltonian over $L^2(\mathbb{Z}^d) \otimes \mathbb{C}$ describes a system without boundaries, which is also know as a bulk model. These types of systems describe perfect crystals, thus, are not well suited for dealing with elements that contain random imperfections. Besides the random imperfections in the crystals we would like to take into account the effect of a magnetic field, henceforth, we need to modify the form of the Hamiltonians.
Schematics of hopping matrices in a 1d model, on the horizontal axis are depicted the lattices sites and on the vertical axis are depicted the energy levels per site, notice that each site has the same energy levels because we are working with translation invariant systems. We have that $W_y \in M_n(\mathbb{C})$, such that, we are depicting the components $(l,l-1), (l, l), (l, l+1)$ (from top to bottom) of the following hopping matrices: in blue we have depicted $W_0$ (inter hopping terms), in red we have depicted $W_{-1}$ (nearest neighbor on the left) and in green we have $W_1$ (nearest neighbor on the right). Taken from Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport
We will work with crystals containing random imperfections, and we care about the macroscopic features that stay invariant under those random imperfections, therefore, we will pass from working with one Hamiltonian to work with a covariant family of Hamiltonians. A covariant family of Hamiltonians looks like
$$ H = \{ H_{\omega} \}{\omega \in \Omega}, \; \text{such that, } V^y H{\omega} (V^y)^* = H_{\alpha(y)(\omega)}, \; \forall y \in \mathbb{Z}^d $$
with $\alpha: \mathbb{Z}^d \to \text{Aut}(\Omega)$ a group action an a compact, contractible and second countable Hausdorff space. In the previous setting $V^y$ are direct magnetic translation, and take the form
$$ V^y := e^{i \bra{X} B_+ \ket{y}} S^y, \text{ that is, } \; V^y \ket{x} = e^{i \bra{x+y} B_+ \ket{y}} \ket{x + y}, $$
with $B_+$ a the lower triangular part of a real anti-symmetric matrix having entries on $[-\pi, \pi)$ , in this setting $B_+$ contains the effect of a constant magnetic field over the disordered crystal. The prototypical form of $H$ is $\{ \sum_{y \in \mathcal{R}} W_y(\omega) \otimes U^y\}_{\omega \in \Omega}$ with $W_y : \Omega \to M_n(\mathbb{C})$ continuous functions, and $U^y$ the dual magnetic translations i.e.
$$ U^y := S^y e^{i \bra{y} B_+ \ket{X}}, \text{ that is, } U^y \ket{x} = e^{i \bra{y} B_+ \ket{x}} \ket{x + y}. $$
The action $\alpha$ induces and action $\tilde{\alpha}: \mathbb{Z}^d \to \text{Aut}(C(\Omega))$ with $\tilde{\alpha}(y)(f)(\omega):= f(\alpha(-y)(\omega)),\; y \in \mathbb{Z}^d$, such that, if $H = \{ H_{\omega}\}{\omega \in \Omega}$ is a covariant family of Hamiltonians then $H \in M_n(C(\Omega) \rtimes{\tilde{\alpha},\zeta}\mathbb{Z}^d)$. In this scenario $C(\Omega) \rtimes_{\tilde{\alpha},\zeta}\mathbb{Z}^d$ is a C* algebra whose elements are families of operators, each member of such family is an operator over $L^2(\mathbb{Z}^d)$ and their relation stores the information about the disorder of the system (encoded in $\Omega$) and an external magnetic field (encoded in $\zeta$). If you want more information about this family of operators refer to the section Math.
In the literature the most usually from that $\Omega$ takes is
$$ \Omega = \prod_{s \in \mathbb{Z}^d} \Omega_0, \; \alpha(y): \Omega \to \Omega, \; \alpha(y)\left((l \mapsto \omega_l){l \in \mathbb{Z}^d}\right) = (l \mapsto \omega{l - y})_{l \in \mathbb{Z}^d}, $$
with $\Omega_0$ a topological space that parameterize the disorder associated to the interactions with a given set of nearest neighbors on the tight-binding model. This $\Omega$ comes motivated from the disorder introduced in a pure crystal due to thermal fluctuations. For example, if we want to work the closest neighbors inside the cube of side $2L$, we can take $\Omega_0 = \prod_{y \in V_L} \Omega_0^y, \; V_L = \{ (y_1, \cdots, y_d) \in \mathbb{Z}^d | |y_i| \leq L \}$ with $\Omega_0^y$ a compact, metrizable and convex space for all $y \in V_L$.
Let $H = \{ H_{\omega} \}{\omega \in \Omega} \in C(\Omega)\rtimes{\alpha,\zeta} \mathbb{Z}^d \otimes M_n(\mathbb{C})$ be a covariant family of Hamiltonians with a gap $g$ in its spectrum, if the Fermi levels lies in that gap ($\mu \in g$) then, in the limit of zero temperature and infinite relaxation time, we have that