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.
For our intends and purposes the Non-Commutative Brillouin torus is a non-commutative smooth manifold, in which will reside the Hamiltonians of the disordered crystals. In the context of Non-Commutative Geometry, we translate topological/geometrical concepts into algebraic/analytic ones, thus, while the Brillouin torus is the torus $\mathbb{T}^n$ with its smooth and topological features, then Non-Commutative Brillouin torus corresponds to a pair of topological algebras, one with the topological information (a C* algebra), and the other with the smooth structure (a smooth sub-algebra). The C* algebra takes the form of a twisted crossed product, which we denote by $A \rtimes_{\alpha,\zeta}\mathbb{Z}^d$, and the smooth sub algebra, which we denote by $\mathcal{A}{\alpha, \zeta}$, takes the form of a dense sub *algebra of $A \rtimes{\alpha,\zeta}\mathbb{Z}^d$ that is invariant under the holomorphic functional calculus of $A \rtimes_{\alpha,\zeta}\mathbb{Z}^d$ and has a Frechet topology that is stronger than the topology of $A \rtimes_{\alpha,\zeta}\mathbb{Z}^d$.
The twisted crossed product is a C* algebra that is constructed from a twisted dynamical system, whose ingredients are:
We have the following comparative between then Brillouin torus and the Non-Commutative Brillouin torus,
| Brillouin Torus | Non-Commutative Brillouin torus |
|---|---|
| $C(\mathbb{T}^d)$ is the C* algebra generated by elements of the form $u_j$ with $1 \leq j \leq d$ which follow the commutation relations $u_j u_i = u_i u_j$ | $C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d$ is the C* algebra generated by elements of the form $a u_j$ with $a \in C(\Omega), \; 1 \leq j \leq d$ which follow the commutation relations $a_0 u_i a_1 u_j = a_0 \alpha(e_i)(a_1) u_i u_j$ , $a_0 \alpha(e_i)(a_1) u_i u_j = \zeta(i,j) a_0 \alpha(e_i)(a_1) u_j u_i.$ |
| Every $f \in C(\mathbb{T}^d)$ is uniquely determined by a set $\{ \mathcal{F}(f)(s) \}{s \in \mathbb{Z}^d}, \; \mathcal{F}(f)(s) \in \mathbb{C},$ where $\mathcal{F}(f)(s) = \int{\mathbb{T}^d} \gamma_{-s}(\lambda) f(\lambda) d \mu(\lambda)$. In this expression, $\gamma_{-s}(\lambda)$ is the character over $\mathbb{T}^d$ given by $\lambda^{-s}$ | Every $p \in C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d$ is uniquely determined by a set $\{ \Phi_s (p) \}{s \in \mathbb{Z}^d}, \; \Phi_s(p) \in C(\Omega),$ where $\Phi_s(p) u^s = \int{\mathbb{T}^d} \gamma_{-s}(\lambda) \tau(\lambda)(p) d \mu(\lambda).$ In this expression, $\tau$ is a strongly continuous action of $\mathbb{Z}^d$ over $C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d$ and $\gamma_{-s}(\lambda)$ is the character over $\mathbb{T}^d$ given by $\lambda^{-s}$. |
| $C(\mathbb{T}^d) \simeq \underbrace{C(\mathbb{T}) \otimes \cdots \otimes C(\mathbb{T})}_{d \text{ times }},$ and for any $1 \leq j \leq d-1$, $C(\mathbb{T}^j)$ is a sub C* algebra of $C(\mathbb{T}^d)$. | $C(\Omega) \rtimes_{\alpha,\zeta} \mathbb{Z}^d$ can be written as an iterated crossed product with $\mathbb{Z}$, that is, there are the d canonical C* algebras $\{ A_i \}{0 \leq i \leq d-1}$ and group actions $\alpha_i : \mathbb{Z} \to \text{Aut}(A{i-1})$ such that $A_{i-1} \simeq A_i \rtimes_{\alpha_i} \mathbb{Z}$ and $C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d \simeq A_{d-1} \rtimes_{\alpha_d} \mathbb{Z}$ with $C(\Omega) = A_0$. For $1 \leq j \leq d-1$, $A_{j}$ is the C* algebra generated by the elements of the form $a u_j, \; i \leq j \leq i, \; a \in C(\Omega)$ following the commutation relations $a_0 u_k a_i u_l = a_0 \alpha(e_k)(a_1) u_k u_l$, $a_0 \alpha(e_k)(a_1) u_k u_l = \zeta(k,l) a_0 \alpha(e_k)(a_1) u_l u_k.$ Additionally, for $0 \leq j\leq d-1$, $A_{j}$ is a sub C* algebra of $C(\Omega) \rtimes_{\alpha,\zeta} \mathbb{Z}^d$. |
| $C^{\infty}(\mathbb{T}^d)$ is Fréchet m-convex algebra with $C^{\infty}(\mathbb{T}^d) \subset C(\mathbb{T}^d),$ such that, $f \in C^{\infty}(\mathbb{T}^d)$ iff for every $n \in \mathbb{N}^d$ there is $K_n < \infty$ with $ | s |
| The derivations $\partial_j : C^{\infty} (\mathbb{T}^d) \to C^{\infty} (\mathbb{T}^d),$ given by, $\partial_j f := \frac{\partial f}{ \partial \lambda_j} ,$ are continuous and commute. | The maps given by $\partial_j : \mathcal{C}(\Omega){\alpha, \zeta} \to \mathcal{C}(\Omega){\alpha, \zeta},$ with, $\Phi_s(\partial_j a) = i s_j \Phi_s(a),$ are continuous commuting derivations over $\mathcal{C}(\Omega)_{\alpha, \zeta}$. |
| The map $f \mapsto \mathcal{F}(f)(0)$ with $\mathcal{F}(f)(0) = \int_{\mathbb{T}^d} f(\lambda) \frac{d \mu(\lambda)}{ (2 \pi)^{d} }$ is a faithful continuous trace over $C(\mathbb{T}^d)$. Also, $f \mapsto \mathcal{F}(f)(0)$ is a continuous trace over $C^{\infty}(\mathbb{T}^d).$ | We have that $\Phi_0( (C (\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d){\text{pos}} ) = C(\Omega){\text{pos}},$ also, if $\mu$ is a Radon measure with full support over $\Omega$ such that $\mu \circ \alpha(s) = \mu$ for all $s \in \mathbb{Z}^d$, then $\int : C(\Omega) \to \mathbb{C}, \; \int(f) = \int_{\Omega} f(\omega) d \mu(\omega)$ is a faithful continuous trace over $C(\Omega)$ and $\int \circ \Phi_0$ is a continuous trace over $C(\Omega)\rtimes_{\alpha,\zeta}\mathbb{Z}^d$. Also, $\int \circ \Phi_0$ is a continuous trace over $\mathcal{C}(\Omega)_{\alpha,\zeta}.$ |
| We have that $C(\mathbb{T}^d)\otimes C_0(\Omega) \simeq C_0(\Omega, C(\mathbb{T}^d)),$ and $C_0(\Omega, C(\mathbb{T}^d)) \subseteq C_b(\Omega, C(\mathbb{T}^d)),$ such that, for $f \in C_0(\Omega, \mathbb{T}^d)$, $|f| = \sup_{\omega \in \Omega} | f(\omega)|$ | Take $\Omega$ a second countable compact Hausdorff space, then $\Omega$ is a metric space, so, assume that $\alpha$ preserves the metric i.e. $d(x,y) = d(\alpha(s)(x), \alpha(s)(x)), \; s \in \mathbb{Z}^d,$ then, $C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d \subseteq C(\Omega, B(L^2(\mathbb{Z}^d)))$ and $a \in C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d$ can be seen as $a = \{ a_{\omega}\}{\omega \in \Omega} \subset C(\Omega, B(L^2(\mathbb{Z}^d)))$, with $|a |{C(\Omega) \rtimes_{\alpha, \zeta} \mathbb{Z}^d} = \sup_{\omega \in \Omega} | a_{\omega} |_{B(L^2(\mathbb{Z}^d))}$. |
Notice that the Non-Commutative Brillouin torus falls back into the Brillouin torus when $\alpha$ is trivial and $\zeta$ is trivial i.e. when we take away the non-commutativity on the generators of the Non-Commutative Brillouin torus.
Let $A$ be a C* algebra and $\mathcal{A}$ a smooth sub algebra of $A$, then, the groups $K_0(A), \; K_0(\mathcal{A})$ are constructed from homotopy classes of projections in matrix algebras of $A$ and $\mathcal{A}$ respectively. The groups $K_1(A), \; K_1(\mathcal{A})$ are constructed from homotopy classes of unitaries in matrix algebras of $A$ and $\mathcal{A}$ respectively. Turns out that the homotopy classes of projections/unitaires of smooth sub algebras are in one to one correspondence with the homotopy classes of their C* algebras, thus, there are isomorphisms
$$ K_0(A) \simeq K_0(\mathcal{A}), \; K_1(A) \simeq K_1(\mathcal{A}). $$
Using the Toeplitz-extension associated to a crossed product with $\mathbb{Z}$ is possible to compute the the K groups $C(\Omega) \rtimes_{\alpha,\zeta} \mathbb{Z}^d$, which happen to take the following form