Quantization of Edge Currents Along Magnetic Interfaces: A $K$-Theory Approach

Abstract

The purpose of this paper is to investigate the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material. We consider tight-binding magnetic models associated to generic magnetic multi-interfaces and describe the $K$-theoretical setting in which a bulk-interface duality can be derived. Then, the (trivial) case of a localized magnetic field and the (non trivial) case of the Iwatsuka magnetic field are considered in full detail. This is a pedagogical preparatory work that aims to anticipate the study of more complicated multi-interface magnetic systems.

Appendices

Appendix A: Crossed Product Structure

In Sect. 2.3 we provided an explicit construction of the magnetic $C^{*}$-algebra $\mathcal{A}_{A_{B}}$ associated to a vector potential $A_{B}$ for the magnetic field $B:\mathbb{Z}^{2}\to \mathbb{R}$. In this section we will show that the magnetic $C^{*}$-algebra $\mathcal{A}_{A_{B}}$ is the ‘‘concrete’’ realization of an abstract twisted crossed product $C^{*}$-algebra over $\mathbb{Z}^{2}$. For the general theory of the crossed product $C^{*}$-algebras we will refer to classic monographs [47, 64]. The special case of discrete crossed product $C^{*}$-algebras, which is the most related to our construction, is discussed in detail in [15, Chapter VIII].

Let $(\mathcal{C}(\Omega _{B}),\tau ,\mathbb{Z}^{2})$ be the $C^{*}$-dynamical system associated with the dynamical system $(\Omega _{B}, \tau ^{*},\mathbb{Z}^{2})$ described in Sect. 2.6. Consider a pair of (abstract) unitary elements $u_{1}$, $u_{2}$ and for every $\gamma :=(\gamma _{1},\gamma _{2})\in \mathbb{Z}^{2}$ set $u_{\gamma }:=u_{1}^{\gamma _{1}}u_{2}^{\gamma _{2}}$. Let $\mathcal{C}(\Omega _{B})[\mathbb{Z}^{2}]$ be the set of finite sums

$$ G\;:=\;\sum _{\gamma \in \Lambda }g_{\gamma }\; u_{\gamma }$$

with $g_{\gamma }\in \mathcal{C}(\Omega _{B})$ for all $\gamma \in \Lambda$ and $\Lambda \in \mathcal{P}_{0}(\mathbb{Z}^{2})$ a finite subset of $\mathbb{Z}^{2}$. The product in $\mathcal{C}(\Omega _{B})[\mathbb{Z}^{2}]$ is defined by the rules

$$ u_{2}\;u_{1}\;=\;\overline{f_{B}}\;u_{1}\;u_{2}\;,\qquad u_{\gamma }\;g \;u_{-\gamma }\;=\; \tau _{\gamma }(g) $$

for every $\gamma \in \mathbb{Z}^{2}$ and $g\in \mathcal{C}(\Omega _{B})$, where $f_{B}:=\,\mathrm{e}^{\,\mathrm{i}\,B}\,\in \mathcal{C}(\Omega _{B})$ is the magnetic phase associated to $B$. The involution is provided by

$$ (g\; u_{\gamma })^{*}\;:=\;u_{-\gamma }\;\overline{g}\;=\;\tau _{-\gamma } \big(\overline{g}\big)\;u_{-\gamma }\;. $$

Endowed with these operations $\mathcal{C}(\Omega _{B})[\mathbb{Z}^{2}]$ acquires the structure of a unital ∗-algebra. Moreover it can be completed to a Banach ∗-algebra with respect to the norm

$$ \|G\|_{1}\;:=\;\sum _{\gamma \in \Lambda }\|g_{\gamma }\|_{\infty }\;. $$

The (universal) enveloping $C^{*}$-algebra [17, Sect. 2.7] obtained from this Banach ∗-algebra is called the $B$-twisted crossed product of $\mathcal{C}(\Omega _{B})$ and is denoted with $\mathcal{C}(\Omega _{B})\rtimes _{\tau ,B}\mathbb{Z}^{2}$.

In order to better understand the twisted structure of the crossed product $\mathcal{C}(\Omega _{B})\rtimes _{\tau ,B}\mathbb{Z}^{2}$ one can observe that the mapping $\gamma \mapsto u_{\gamma }$ provides a projective (abstract) unitary representation of $\mathbb{Z}^{2}$ defined by

$$ u_{\gamma }\;u_{\xi }\;=\;\Theta _{B}(\gamma ,\xi )\;u_{\gamma +\xi }\;, \qquad \forall \; \gamma ,\xi \in \mathbb{Z}^{2}\;, $$

with phase given by

$$ \Theta _{B}(\gamma ,\xi )\;:=\;\prod _{\nu \in \Lambda (\gamma ,\xi )} \overline{\tau _{\nu }({f}_{B})} $$

and the product is extended on the finite cell

$$ \Lambda (\gamma ,\xi )\;:=\;\big([\gamma _{1},\gamma _{1}+\xi _{1}-1] \times [0,\gamma _{2}-1]\big)\;\cap \;\mathbb{Z}^{2}\;. $$

The map $\Theta _{B}: \mathbb{Z}^{2}\times \mathbb{Z}^{2}\to \mathcal{U}( \mathcal{C}(\Omega _{B}))$ takes value on the unitary elements of $\mathcal{C}(\Omega _{B})$ and a direct check shows that it satisfies the cocycle condition

$$ \Theta _{B}(\gamma +\xi ,\zeta )\; \Theta _{B}(\gamma ,\xi )\;=\; \tau _{\gamma }\big(\Theta _{B}(\xi ,\zeta )\big)\;\Theta _{B}(\gamma , \xi +\zeta ) $$

for all $\gamma ,\xi ,\zeta \in \mathbb{Z}^{2}$.

Given a vector potential $A_{B}$ for the magnetic field $B$ one can consider the representation $\pi _{A_{B}}:\mathcal{C}(\Omega _{B})\rtimes _{\tau ,B}\mathbb{Z}^{2} \to \mathcal{B}(\ell ^{2}(\mathbb{Z}^{2}))$ defined by

$$ \begin{aligned} \pi _{A_{B}}(u_{\gamma })\;&:=\;(\mathfrak{s}_{A_{B},1})^{\gamma _{1}} \;(\mathfrak{s}_{A_{B},2})^{\gamma _{2}} \\ \pi _{A_{B}}(g)\;&:=\;\iota ^{-1}(g) \end{aligned} \;,\qquad \forall \gamma \in \mathbb{Z}^{2}\;,\quad \forall g\in \mathcal{C}(\Omega _{B}) $$

where $\mathfrak{s}_{A_{B},1}$, $\mathfrak{s}_{A_{B},2}$ are the magnetic translations defined in Sect. 2.2 and $\iota ^{-1}(g)\in \mathcal{F}_{B}$ is given by the isomorphism defined in Lemma 2.26. The map $\pi _{A_{B}}$ coincides with the tensor product of the isomorphism $\iota ^{-1}$ with the $B$-twisted (left) regular representation of $\mathbb{Z}^{2}$. This representation turns out to be faithful (cf. [15, p. 218]) and as a consequence one gets

$$ \mathcal{A}_{A_{B}}\;=\;\pi _{A_{B}}\left (\mathcal{C}(\Omega _{B}) \rtimes _{\tau ,B}\mathbb{Z}^{2}\right )\;. $$

It is also interesting to observe that $\mathcal{A}_{A_{B}}$ can be represented as an iterated crossed product algebra as discussed in [49, Sect. 3.1.1]. Indeed one can check that

$$ \mathcal{C}(\Omega _{B})\rtimes _{\tau ,B}\mathbb{Z}^{2}\;\simeq \; \mathcal{Y}_{B,j}\rtimes _{\alpha _{k}}\mathbb{Z}\;,\qquad \mathcal{Y}_{B,j} \;:=\;\mathcal{F}_{B}\rtimes _{\alpha _{j}}\mathbb{Z}, $$

where $\{j,k\}=\{1,2\}$, the crossed product algebra $\mathcal{Y}_{B,j}$ is generated by $\mathcal{F}_{B}$ and $u_{j}$ along with the relation

$$ \alpha _{j}(g):=u_{j} gu_{j}^{*}\;:=\;\left \{ \begin{aligned} &\tau _{(1,0)}(g)&\text{if }j=1 \\ &\tau _{(0,1)}(g)&\text{if }j=2 \end{aligned} \right \} \;=:\tau _{j}(g) $$

for all $g\in \mathcal{C}(\Omega _{B})$ and the crossed product algebra $\mathcal{Y}_{B,j}\rtimes _{\alpha _{k}}\mathbb{Z}$ is generated by $\mathcal{Y}_{B,j}$ and $u_{k}$ along with the relation $\alpha _{k}(gu_{j}):=u_{k}(gu_{j})u_{k}^{*}=\tau _{k}(g)f_{B}u_{j}$ for all $g\in \mathcal{F}_{B}$.

In the special case the magnetic field $B$ is constant in the $n_{2}$-direction (as in the case of the Iwatsuka magnetic field) it follows that the $\alpha _{2}$-action is trivial, meaning that it reduces to the identity $\alpha _{2}(g)=g$ for all $g\in \mathcal{C}^{*}(\Omega _{B})$. In this situation the crossed product algebra $\mathcal{Y}_{B,2}$ acquires the following very simple structure

$$ \mathcal{Y}_{B,2}\;\simeq \;\mathcal{C}^{*}(\Omega _{B})\otimes \;C^{*}_{r}( \mathbb{Z})\;\simeq \;\mathcal{C}^{*}(\Omega _{B})\otimes \mathcal{C}( \mathbb{S}^{1})\;\simeq \;\mathcal{C}(\Omega _{B}\times \mathbb{S}^{1}) $$

The first isomorphism involves the (reduced) group $C^{*}$-algebra $C^{*}_{r}(\mathbb{Z})$ and is proved in [64, Lemma 2.73] (along with the nuclearity of the various $C^{*}$-algebras). The isomorphism $C^{*}_{r}(\mathbb{Z})\simeq \mathcal{C}(\mathbb{S}^{1})$ is a consequence of the Pontryagin duality [15, Proposition VII.1.1].

Appendix B: The $K$-Theory of the Iwatsuka Magnetic Hull

Let $\Omega _{I}=\mathbb{Z}\cup \{-\infty \}\cup \{+\infty \}$ be the Iwatsuka magnetic hull described in Example 2.24 and consider the short exact sequence

$$ 0\;\stackrel{}{\longrightarrow }\;\mathcal{C}_{0}(\mathbb{Z})\; \stackrel{\imath }{\longrightarrow }\;\mathcal{C}(\Omega _{I})\; \stackrel{\mathrm{ev}}{\longrightarrow }\mathbb{C}\oplus \mathbb{C}\; \stackrel{}{\longrightarrow }0 $$

(B.1)

where $\mathcal{C}_{0}(\mathbb{Z})$ is the $C^{*}$-algebra of sequences vanishing at infinity, $\imath$ is the inclusion homomorphism and the evaluation homomorphism $\mathrm{ev}$ compute the left and right limits of elements in $\mathcal{C}(\Omega _{I})$. The sequence (B.1) is split exact in view of the homomorphism

$$ \mathbb{C}\oplus \mathbb{C}\;\ni \;(\ell _{-},\ell _{+})\; \stackrel{\jmath }{\longrightarrow }\;c_{(\ell _{-},\ell _{+})}\;\in \; \mathcal{C}(\Omega _{I}) $$

where the element $c_{(\ell _{-},\ell _{+})}$ is specified by

$$ c_{(\ell _{-},\ell _{+})}(n)\;:=\; \left \{ \begin{aligned} &\ell _{-}&\quad &\text{if}\quad n< 0 \\ &\ell _{+}&\quad &\text{if}\quad n\geqslant 0\;. \end{aligned} \right . $$

Then, it follows that [63, Corollary 8.2.2]

$$ K_{j}\big(\mathcal{C}(\Omega _{I})\big)\;\simeq \;K_{j}\big(\mathcal{C}_{0}( \mathbb{Z})\big)\;\oplus \;K_{j}\big(\mathbb{C}\oplus \mathbb{C} \big)\;,\qquad j=1,2\;. $$

The $K$-theory of $\mathbb{C}\oplus \mathbb{C}$ is easily calculated as $K_{0}(\mathbb{C}\oplus \mathbb{C})=\mathbb{Z}\oplus \mathbb{Z}$ and $K_{1}(\mathbb{C}\oplus \mathbb{C})=0$. The $K$-theory of $\mathcal{C}_{0}(\mathbb{Z})$ is given by $K_{0}(\mathcal{C}_{0}(\mathbb{Z}))=\mathbb{Z}^{\oplus \mathbb{Z}}$ and $K_{1}(\mathcal{C}_{0}(\mathbb{Z}))=0$. The latter fact follows from the isomorphism $K_{j}(\mathcal{C}_{0}(\mathbb{Z}))\simeq K^{j}_{\mathrm{top}}(\mathbb{Z}) \simeq K^{j}_{\mathrm{top}}(\ast )^{\oplus \mathbb{Z}}$ between the algebraic and the topological $K$-theory [12, Theorem 5]. Another way of achieving the same result is to consider the Pontryagin duality $\mathbb{S}^{1}=\widehat{\mathbb{Z}}$ and the isomorphism $\mathcal{C}_{0}(\mathbb{Z})\simeq C^{*}_{r}(\mathbb{S}^{1})$ where $C^{*}_{r}(\mathbb{S}^{1})$ is the (reduced) group algebra of the circle [15, Proposition VII.1.1]. Therefore, one has that $K_{0}(C^{*}_{r}(\mathbb{S}^{1}))\simeq {\mathrm{Rep}}(\mathbb{S}^{1}) \simeq \mathbb{Z}^{\oplus \mathbb{Z}}$ and $K_{0}(C^{*}_{r}(\mathbb{S}^{1}))\simeq 0$ where $\mathrm{Rep}(\mathbb{S}^{1})$ denotes the complex representation ring of $\mathbb{S}^{1}$ [12, Sect. 7]. The generators of $K_{j}(\mathcal{C}_{0}(\mathbb{Z}))$ are the classes $[\pi _{i}]$, $i\in \mathbb{Z}$, of the projections $\pi _{i}(n):=\delta _{i,n}$. After putting all the information together, we can describe the $K$-theory of the Iwatsuka magnetic hull as

$$ K_{0}\big(\mathcal{C}(\Omega _{I})\big)\;\simeq \;\bigoplus _{i\in \mathbb{Z}}\mathbb{Z}[\pi _{i}]\;\oplus \;\mathbb{Z}[\pi _{-}]\; \oplus \;\mathbb{Z}[\pi _{+}]\;,\qquad K_{1}\big(\mathcal{C}(\Omega _{I}) \big)\;=\;0\;, $$

where $\pi _{-}:=\jmath ((1,0))$ and $\pi _{+}:=\jmath ((0,1))-\pi _{0}$ are the projections at infinity.

Appendix C: The Pimsner-Voiculescu Exact Sequence

In this section we will provide a brief overview on the Pimsner-Voiculescu six-term exact sequence which is the main tool to compute the $K$-theory for crossed product $C^{*}$-algebras by ℤ. For the interested reader we refer to the original work [51] and the monograph [9, Chapter V].

Let $\mathcal{Y}$ be a $C^{*}$-algebra, $\alpha \in {\mathrm{Aut}}(\mathcal{Y})$ and automorphism and $\mathcal{Y}\rtimes _{\alpha }\mathbb{Z}$ the crossed product generated by $\mathcal{Y}$ and the unitary $u$ with the relation

$$ \alpha (a)\;=\:uau^{*}\;,\qquad \forall \; a\in \mathcal{Y}\;. $$

The first step of the construction is to define an appropriate short exact sequence of $C^{*}$-algebras. This is done by considering the tensor product $\mathcal{Y}\otimes \mathcal{K}$, where $\mathcal{K}$ denotes the $C^{*}$-algebra of compact operators, and the $C^{*}$-algebra $\mathcal{T}_{\alpha }$ generated in $\mathcal{Y}\otimes {C^{*}}(v)$ by $\mathcal{Y}\otimes {\mathbf{1}}$ and $V=u\otimes v$, with $v$ a non-unitary (abstract) isometry. It is useful to think at elements of $\mathcal{K}$ as infinite matrices acting on $\ell ^{2}(\mathbb{N}_{0})$ with respect its canonical basis. Let us consider the map $\varphi :\mathcal{Y}\otimes \mathcal{K}\to \mathcal{T}_{\alpha }$ defined by

$$ \varphi (a\otimes e_{j,k})\;:=\;V^{j}(a\otimes (\mathbf{1}-vv^{*}))(V^{*})^{k} \;=\;(\alpha ^{j}(a)\otimes {\mathbf{1}})V^{j}P(V^{*})^{k} $$

where $P$ is the (non-trivial) self-adjoint projection given by $P:=\mathbf{1}-VV^{*}=\mathbf{1}\otimes (\mathbf{1}-vv^{*})$ and $e_{j,k}$ are the rank one operators which generates $\mathcal{K}$. Then, there exists a short exact sequence of $C^{*}$-algebras

$$ 0\;\longrightarrow \; \mathcal{Y}\otimes \mathcal{K} \; \stackrel{\varphi }{\longrightarrow }\; \mathcal{T}_{\alpha } \; \stackrel{\psi }{\longrightarrow }\; \mathcal{Y}\rtimes _{\alpha } \mathbb{Z}\;\longrightarrow \; 0, $$

(C.1)

where the map $\psi :\mathcal{T}_{\alpha }\to \mathcal{Y}\rtimes _{\alpha }\mathbb{Z}$ defined by

$$ \psi (a\otimes {\mathbf{1}})\;:=\;a\;,\quad \psi (V)\;:=\;u\;. $$

For this reason $\mathcal{T}_{\alpha }$ is called the Toeplitz extension of the stabilized algebra $\mathcal{Y}\otimes \mathcal{K}$ by the crossed product $\mathcal{Y}\rtimes _{\alpha }\mathbb{Z}$.

The Pimsner-Voiculescu (six-term) exact sequence is a cyclic sequence which connects the $K$-theory of $\mathcal{Y}$ and $\mathcal{Y}\rtimes _{\alpha }\mathbb{Z}$, and is given by

$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} K_{0}(\mathcal{Y})&\stackrel{\beta _{\ast }}{\longrightarrow } &K_{0}( \mathcal{Y}) & \stackrel{{\imath }_{*}}{\longrightarrow } &K_{0}( \mathcal{Y}\rtimes _{\alpha }\mathbb{Z}) \\ &&&&\\ {\partial _{1}}\Big\uparrow &&&&\Big\downarrow {\partial _{0}} \\ &&&&\\ K_{1}(\mathcal{Y}\rtimes _{\alpha }\mathbb{Z})& \underset{{\imath }_{*}}{\longleftarrow } &K_{1}(\mathcal{Y}) & \underset{\beta _{\ast }}{\longleftarrow } &K_{1}(\mathcal{Y}) \\ \end{array} $$

(C.2)

and it is worth pointing out that this is not exactly the standard six-term exact sequence associated with the short exact sequence (C.1), although it is closely related. The maps ${\imath }_{*}$ are induced by the canonical inclusion $\imath :\mathcal{Y}\hookrightarrow \mathcal{Y}\rtimes _{\alpha } \mathbb{Z}$ and the maps $\beta _{\ast }$ are induced by the map $\beta :\mathcal{Y}\to \mathcal{Y}$ defined as $\beta :=\mathrm{Id}-\alpha ^{-1}$. The vertical maps are related with the index and the exponential maps for the standard six-term exact sequence in $K$-theory emerging from the short exact sequence (C.1) (cf. [63, Theorem 9.3.2]). More precisely one has that $\partial _{0}:=\kappa _{0}^{-1}\circ {\mathrm{ind}}$ and $\partial _{1}:=\kappa _{0}^{-1}\circ {\mathrm{exp}}$ where $\mathrm{ind}:K_{1}(\mathcal{Y}\rtimes _{\alpha }\mathbb{Z})\to K_{0}( \mathcal{Y}\otimes \mathcal{K})$ and $\mathrm{exp}:K_{0}(\mathcal{Y}\rtimes _{\alpha }\mathbb{Z})\to K_{1}( \mathcal{Y}\otimes \mathcal{K})$ are the usual index and the exponential maps related to the short exact sequence (C.1) and $\kappa _{0}: K_{j}(\mathcal{Y})\to K_{j}(\mathcal{Y}\otimes \mathcal{K})$, with $j=0,1$, is the stabilization isomorphism induced by $a\mapsto a\otimes e_{0,0}$ for every $a\in \mathcal{Y}$.

Appendix D: $K$-Theory for a Constant Magnetic Field

In this section the $K$-theory of the magnetic $C^{*}$-algebra $\mathcal{A}_{b}$ associated with a constant field of strength $b$ will be described. The key observation is that $\mathcal{A}_{b}$ is a faithful representation of the noncommutative torus $\mathbb{A}_{\theta _{b}}$ provided that $\theta _{b}:=b(2\pi )^{-1}$. Let us observe that $b$ enters in the definition of $\mathbb{A}_{\theta _{b}}$ only modulo $2\pi$. For this reason, without loss of generality, we can assume $0< \theta _{b} <1$ as the condition for a non trivial magnetic field. The $K$-theory of the noncommutative torus $\mathbb{A}_{\theta _{b}}$ has been investigated in [50–52] and is described in several textbooks like [63, Sect. 12.3] or [22, Chap. 12]. As a consequence of the isomorphism $\mathcal{A}_{b}\simeq \mathbb{A}_{\theta _{b}}$ we get

$$ \begin{aligned} K_{0}(\mathcal{A}_{b})\;&=\;\mathbb{Z}[{\mathbf{1}}]\;\oplus \;\mathbb{Z}[ \mathfrak{p}_{\theta _{b}}]\;\simeq \;\mathbb{Z}^{2} \;, \\ K_{1}(\mathcal{A}_{b})\;&=\;\mathbb{Z}[\mathfrak{s}_{b,1}]\;\oplus \;\mathbb{Z}[\mathfrak{s}_{b,2}]\;\simeq \;\mathbb{Z}^{2}\;. \end{aligned} $$

(D.1)

The generators of the $K$-theory of $\mathcal{A}_{b}$ are quite explicit except for the projection $\mathfrak{p}_{\theta _{b}}\in \mathcal{A}_{b}$ which is known as Powers-Rieffel projection. Our next task is to provide a presentation of $\mathfrak{p}_{\theta _{b}}$ optimized for the aims of this work. We will set

$$ \mathfrak{p}_{\theta _{b}}\;:=\mathfrak{s}_{b,1}^{*}\;\mathfrak{d}_{1} \;+\;\mathfrak{d}_{0}\;+\;\mathfrak{d}_{1}\;\mathfrak{s}_{b,1} $$

where $\mathfrak{d}_{1}:=g(\mathfrak{s}_{2})$ and $\mathfrak{d}_{0}:=f(\mathfrak{s}_{2})$ are suitable self-adjoint elements of $C^{*}(\mathfrak{s}_{2})\subset \mathcal{A}_{b}$. Here we are using the coincidence $\mathfrak{s}_{2}=\mathfrak{s}_{b,2}$ between the ordinary shift and magnetic translation in view of the election of the Landau gauge for the constant magnetic field. The requirement for $\mathfrak{p}_{\theta _{b}}$ of being a projection is automatically satisfied it the following relations hold true:

$$ \begin{aligned} (\mathfrak{s}_{b,1}^{*}\mathfrak{d}_{1}\mathfrak{s}_{b,1}) \mathfrak{d}_{1}\;&=\;0\;, \\ \mathfrak{d}_{1}(\mathfrak{d}_{0}+\mathfrak{s}_{b,1}\mathfrak{d}_{0} \mathfrak{s}_{b,1}^{*})\;&=\;\mathfrak{d}_{1}\;, \\ \mathfrak{d}_{0}^{2}+\mathfrak{d}_{1}^{2}+(\mathfrak{s}_{b,1}^{*} \mathfrak{d}_{1}\mathfrak{s}_{b,1})^{2}\;&=\;\mathfrak{d}_{0}\;. \end{aligned} $$

(D.2)

The way of implementing these relation is by the isomorphism (induced by the Fourier transform) $C^{*}(\mathfrak{s}_{2})\simeq C_{\mathrm{per}}([0,1])$ where on the right-hand side one has the $C^{*}$-algebra of continuous function on $[0,1]$ with periodic boundary conditions, i.e. $f(0)=f(1)$. Under this isomorphism $\mathfrak{s}_{2}\mapsto e$ where $e(k):=\,\mathrm{e}^{\,\mathrm{i}\,2\pi k}$ and $\mathfrak{s}_{b,1}\mathfrak{s}_{2}\mathfrak{s}_{b,1}^{*}=\, \mathrm{e}^{\,\mathrm{i}\,b}\, \mathfrak{s}_{2}\mapsto e(\cdot + \theta _{b})$ Consider a $0<\delta <\theta _{b}$ such that $\theta _{b}+\delta <1$ and the function $f$ such that

$$ f(e(k))\;:=\;\frac{k}{\delta }\chi _{[0,\delta ]}(k)+\chi _{(\delta , \theta _{b})}(k)+\left (1+\frac{\theta _{b}-k}{\delta }\right )\chi _{[ \theta _{b},\theta _{b}+\delta ]}(k)\;. $$

Define

$$ g(e(k))\;:=\;\sqrt{f(e(k))(1-f(e(k)))}\;\chi _{[0,\delta ]}(k)\;=\; \sqrt{\frac{k}{\delta }\left (1-\frac{k}{\delta }\right )}\;\chi _{[0, \delta ]}(k)\;. $$

One can check that by using $f$ and $g$ above to define $\mathfrak{d}_{0}$ and $\mathfrak{d}_{1}$ respectively, then the conditions (D.2) are automatically verified. The crucial identities are $\mathfrak{s}_{b,1}^{*}\mathfrak{d}_{1}\mathfrak{s}_{b,1}\mapsto g(e( \cdot -\theta _{b}))$ (which is supported in $[\theta _{b},\theta _{b}+\delta ]$) and $\mathfrak{s}_{b,1}\mathfrak{d}_{0}\mathfrak{s}_{b,1}^{*}\mapsto f(e( \cdot +\theta _{b}))$ (which must be defined periodically on $[0,1]$). Let us point out that with a standard ‘‘smoothing argument’’ it is possible to replace the continuous function $f$ and $g$ with smooth functions. This implies that it is possible to define the Powers-Rieffel projection inside the smooth algebra $\mathcal{A}_{b}^{\infty }$.

The relations (D.2) provide other useful identities. Let $\mathfrak{L}:=\mathfrak{L}(\mathfrak{d}_{1})$ be the support projection of $\mathfrak{d}_{1}$ (in the von Neumann algebra generated by $\mathcal{A}_{b}$). This is by definition the smallest projection such that $\mathfrak{L}\mathfrak{d}_{1}=\mathfrak{d}_{1}=\mathfrak{d}_{1} \mathfrak{L}$. It is immediate to conclude that $\mathfrak{L}$ is mapped into the characteristic function on the support of $g\circ e$ under the isomorphism used above, i.e. $\mathfrak{L}\mapsto \chi _{[0, \delta ]}$. Combining $\mathfrak{L}$ with the first relation in (D.2) one gets $(\mathfrak{s}_{b,1}^{*}\mathfrak{d}_{1}\mathfrak{s}_{b,1}) \mathfrak{L}=0$. This relation combined with the third equation in (D.2) provides

$$ \mathfrak{d}_{1}^{2}\;=\;\mathfrak{L}(\mathfrak{d}_{0}- \mathfrak{d}_{0}^{2})\;=\;(\mathfrak{d}_{0}-\mathfrak{d}_{0}^{2}) \mathfrak{L}\;. $$

(D.3)

For the next result we need to recall that two unitary operators $\mathfrak{u}_{0},\mathfrak{u}_{1}\in \mathcal{A}_{b}$ are said to be homotopic equivalent, denoted $\mathfrak{u}_{0}\sim \mathfrak{u}_{1}$, if there is a continuous map $[0,1]\ni t\mapsto \mathfrak{u}(t)\in \mathcal{A}_{b}$ such that $\mathfrak{u}(0)=\mathfrak{u}_{0}$, $\mathfrak{u}(1)=\mathfrak{u}_{1}$ and $\mathfrak{u}(t)$ is unitary for every $t\in [0,1]$.

Lemma D.1

The unitary operators $\mathrm{e}^{-\,\mathrm{i}\,2\pi \mathfrak{d}_{0}\mathfrak{L}}$ and $\mathfrak{s}_{b,2}^{*}$ are homotopic equivalent in $\mathcal{A}_{b}$, i.e. $\mathrm{e}^{-\,\mathrm{i}\,2 \pi \mathfrak{d}_{0}\mathfrak{L}}\,\sim \mathfrak{s}_{b,2}^{*}$.

Proof

In view of the isomorphism $C^{*}(\mathfrak{s}_{2})\simeq C_{\mathrm{per}}([0,1])$ it is enough to find an homotopy between the functions $t(k):=\,\mathrm{e}^{-\,\mathrm{i}\,2\pi \frac{k}{\delta }\chi _{[0, \delta ]}(k)}\ \mbox{and}\ e(k)^{-1}:=\,\mathrm{e}^{-\,\mathrm{i}\,2\pi k}$. Such an homotopy is explicitly given by

$$ [0,1]\ni t\;\longmapsto \;{u}_{t}(k)\;:=\quad \mathrm{e}^{-\, \mathrm{i}\,2\pi \frac{k}{\delta +t(1-\delta )}\chi _{[0,\delta +t(1- \delta )]}(k)} $$

and this completes the proof. □

Let $\mathfrak{L}'$ be the support projection of the shifted operator $\mathfrak{s}_{b,1}^{*}\mathfrak{d}_{1}\mathfrak{s}_{b,1}$. It turns out that $\mathfrak{L}'$ is isomorphically mapped into the characteristic function $\chi _{[\theta _{b},\theta _{b}+\delta ]}$.

Lemma D.2

The unitary operators $\mathrm{e}^{-\,\mathrm{i}\,2\pi \mathfrak{d}_{0}\mathfrak{L}'}$ and $\mathfrak{s}_{b,2}$ are homotopic equivalent in $\mathcal{A}_{b}$, i.e. $\mathrm{e}^{-\,\mathrm{i}\,2 \pi \mathfrak{d}_{0}\mathfrak{L}'}\,\sim \mathfrak{s}_{b,2}$.

Proof

As above it is enough to find an homotopy between the functions $t'(k):= \mathrm{e}^{-\,\mathrm{i}\,2\pi \left (1+ \frac{\theta _{b}-k}{\delta }\right )\chi _{[\theta _{b},\theta _{b}+ \delta ]}(k)}$ and $e(k):=\,\mathrm{e}^{\,\mathrm{i}\,2\pi k}$. Such an homotopy is explicitly given by

$$ [0,1]\ni t\;\longmapsto \;{u}'_{t}(k)\;:=\quad \mathrm{e}^{-\, \mathrm{i}\,2\pi \left ( \frac{(1-t)(\theta _{b}-k+\delta -1)-t\delta k}{\delta } \right ) \chi _{[(1-t)\theta _{b},(1-t)(\theta _{b}+\delta -1)+1]}(k)} $$

and this completes the proof. □

Let us end this appendix with a more precise description of the $K_{0}$-group of $\mathcal{A}_{b}$. Let $[\mathfrak{p}]\in K_{0}(\mathcal{A}_{b})$. Then, from the first equation of (D.1) one infers the existence of $M,N\in \mathbb{Z}$ such that

$$ [\mathfrak{p}]\;\simeq \;M\;[{\mathbf{1}}]\;+\;N\;[\mathfrak{p}_{\theta _{b}}] \;. $$

The number $N$ can be deduced by using the pairing (3.15) along with $\prec [{\mathbf{1}}],[\xi _{b}]\succ =0$ and $\prec [\mathfrak{p}_{\theta _{b}}],[\xi _{b}]\succ =1$. This implies that $N=\mathrm{Ch}_{b}(\mathfrak{p})$. The number $M$ can be deduced from the pairing $\tau :K_{0}(\mathcal{A}_{b})\to \mathbb{Z}+\theta _{b} \mathbb{Z}$ induced by the trace, i.e. $\tau ([\mathfrak{p}]):=\mathscr{T}( \mathfrak{p})$. Since $\tau ([{\mathbf{1}}])=1$ and $\tau ([\mathfrak{p}_{\theta _{b}}])=\theta _{b}$ one gets that $M=\mathscr{T}(\mathfrak{p})-N\theta _{b}$.