Fredholm Homotopies for Strongly-Disordered 2D Insulators

Abstract

We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (Commun Math Phys 259(1):185–221, 2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We furthermore provide a proof of the stability of the $\mathbb {Z}_2$ index in the mobility gap regime. These two-dimensional results serve as a model for the study of higher dimensional $\mathbb {Z}_2$ indices.

Appendices

Appendices

A A Schatten Class Lemma

A fundamental tool for our arguments is the observation that if $P\in \textrm{WLOC}{}$ and f(X) is a multiplication operator in the position basis such that $|f(x)-f(y)|$ decays suitably as $x,y\rightarrow \infty$, then [P, f(X)] is compact. In fact we will show that this operator is Schatten-3, i.e. $\left\| [P,f(X)]\right\| _3^3 = {\text {tr}}|[P,f(X)]|^3<\infty$, under natural conditions on f. The proof of this fact follows similar arguments to those presented in [ASS94a, AG98].

Lemma A.1

Let $P\in \textrm{WLOC}$ be such that $\left\| P\right\| \le 1$ and $f\in \ell ^\infty (\mathbb {Z}^2)$ be such that

$$\begin{aligned} |f(x)-f(y)| \ \le \ D\frac{\left\| x-y\right\| }{1+\left\| x\right\| } \end{aligned}$$

(A.1)

with $D<\infty$. Then [P, f(X)] is Schatten-3.

Proof

We have $[P,f(X)]_{xy} = P_{xy}(f(x)-f(y))$ and

$$\begin{aligned} \left\| [P,f(X)]\right\| _3 \ \le \ \sum _{b\in \mathbb {Z}^2}\left( \sum _{x\in \mathbb {Z}^2}\left\| P_{x+b,x}\right\| ^3|f(x)-f(x+b)|^3\right) ^{1/3}\,. \end{aligned}$$

Let $B\subseteq \mathbb {Z}^2$ be a finite set, to be specified below. Applying the estimate (2.2) for $x\in B$ (since $P\in \textrm{WLOC}$) and, for $x\in B^c$, noting that $\left\| P_{x,x+b}\right\| \le 1$, we conclude that there is $\nu \in \mathbb {N}$ such that for any $\mu \in \mathbb {N}$ there is $C_\mu <\infty$ with which

$$\begin{aligned}{} & {} \sum _{x\in \mathbb {Z}^2}\left\| P_{x+b,x}\right\| ^3|f(x)-f(x+b)|^3 \ \le \ 2\Vert f\Vert _\infty ^3 \sum _{x\in B} C_\mu ^3 \frac{(1+\left\| x\right\| )^{3\nu }}{ (1+\left\| b\right\| )^{3\mu }} \nonumber \\{} & {} \quad + \sum _{x\in B^c} D^3 \frac{\left\| b\right\| ^3}{(1+\left\| x\right\| )^3}\,. \end{aligned}$$

(A.2)

Now pick B to be the set of x such that $1+\left\| x\right\| \le (1+\left\| b\right\| )^\alpha$, with $\alpha$ still to be determined.Then we find that the second term on the right hand side of (A.2) is bounded above by

$$\begin{aligned} D^3\left\| b\right\| ^3(1+\left\| b\right\| )^{-\alpha /2}\sum _{x}(1+\left\| x\right\| )^{-2.5} \ \le \ {\tilde{D}}^3\left\| b\right\| ^3(1+\left\| b\right\| )^{-\alpha /2}\, , \end{aligned}$$

while the first term on the right hand side is bounded above by

$$\begin{aligned} 8 C_\mu ^3(1+\left\| b\right\| )^{-3\mu }\sum _{x\in B}(1+\left\| x\right\| )^{3\nu } \le 8 C_\mu ^3(1+\left\| b\right\| )^{-3\mu +3\nu \alpha }|B|\,. \end{aligned}$$

Since B is a ball about the origin of radius $(1+\left\| b\right\| )^\alpha -1$, B is bounded above by ${\tilde{C}} (1+\left\| b\right\| )^{2\alpha }$ for some universal ${\tilde{C}}<\infty$. Hence the first term on the right hand side of (A.2) is bounded above by

$$\begin{aligned} 8 C_\mu ^3 \tilde{{\tilde{C}}}(1+\left\| b\right\| )^{-3\mu +3\nu \alpha +2\alpha }\,. \end{aligned}$$

To make the sum $\sum _{b\in \mathbb {Z}^2}$ finite we choose $\alpha /2-3>6$ and $3\mu -(3\nu +2)\alpha >6$ (6 since we are taking the 1/3 power and we need at least power, say, 2 to make this summable on $\mathbb {Z}^2$). Both of these may be arranged since $\alpha$ was arbitrary and $\mu$ may be taken arbitrarily large. $\square$

B The SULE Basis

In this section let $\mathcal {H}=\ell ^2(\mathbb {Z}^d)\otimes \mathbb {C}^N$ and $V\subset \mathcal {H}$ a closed subspace.

Definition B.1

(SULE basis) A Semi-Uniformly Localized basis for V is an orthonormal basis $\left\{ \psi _n\right\} _n$ such that there are a sequence of “localization centers” $\left\{ x_n\right\} _n\subseteq \mathbb {Z}^d$ and $\nu \in \mathbb {N}$ so that for any $\mu >0$ it holds that

$$\begin{aligned} \left\| \psi _n(x)\right\| \le C_\mu (1+\left\| x-x_n\right\| )^{-\mu }(1+\left\| x_n\right\| )^{\nu }\qquad (x\in \mathbb {Z}^d)\, \end{aligned}$$

(B.1)

with $C_\mu <\infty$.

Remark

When a semi-uniformly localized basis $\left\{ \psi _n\right\} _n$ consists of eigenfunctions for a self-adjoint operator, it is called a Semi-Uniformly Localized Eigenfunction (SULE) basis. This notion was originally defined in [Rio+96].

It is shown in [Rio+96, Corollary 7.3] that the localization centers $\left\{ x_n\right\} _n$ obey

$$\begin{aligned} \sum _{n}\frac{1}{(1+\left\| x_n\right\| )^{d+\varepsilon }}\ < \ \infty \qquad (\varepsilon >0) \ , \end{aligned}$$

(B.2)

a fact we shall use below. The estimate (B.1) of course implies that the operator on $\ell ^2(\mathbb {Z}^d)$ with matrix elements $\left\| \psi _n(x)\right\| \left\| \psi _n(y)\right\|$ is $\textrm{WLOC}$.

The following proof appeared in [EGS05, Section 3.6]. We include it here for completeness.

Lemma B.2

For an interval $\Delta \subseteq \mathbb {R}$, if H is a $\Delta$-insulator on $\mathcal {H}$ in the sense of Definition 2.5 then there exists a SULE basis for the vector subspace ${\text {im}}(\chi _\Delta (H))$ consisting of eigenfunctions for H.

Proof

We let $P:=\chi _\Delta (H)$ and $P_\lambda := \chi _{\left\{ \lambda \right\} }(H)$ for each eigenvalue $\lambda \in \Delta$ of H. Since $\chi _{\left\{ \lambda \right\} }$ is a bounded Borel function on $\Delta$, we conclude from Definition 2.5 that there is $\nu$ such that for any $\mu$ we have

$$\begin{aligned} \left\| (P_{\lambda })_{x,x_0}\right\| \ \le \ C_\mu (1+\left\| x-x_0\right\| )^{-\mu }(1+\left\| x_0\right\| )^{\nu }\, \end{aligned}$$

(B.3)

for every eigenvalue $\lambda \in \Delta$, with $C_\mu <\infty$.

Since all eigenfunctions are of finite multiplicity (see Definition 2.5), we have ${\text {tr}}(P_\lambda )<\infty$, so $a_x=\left\| (P_\lambda )_{xx}\right\| \le {\text {tr}}(P_\lambda )_{xx}$ is a summable sequence. Let $x_0\in \mathbb {Z}^d$ be a point at which $a_x$ attains its maximum value and let $\textbf{v}_0\in \mathbb {C}^N$ with $\left\| \textbf{v}_0\right\| =1$ and $\textbf{v}_0^\dagger (P_\lambda )_{xx}\textbf{v}_0=a_{x_0}$. Now define

$$\begin{aligned} \psi (x) \ :=\ \frac{1}{\sqrt{a_{x_0}}} (P_{\lambda })_{x,x_0}\textbf{v}\,. \end{aligned}$$

One verifies that $\psi$ is an eigenvector for H with eigenvalue $\lambda$, and it is normalized so that $\left\| \psi \right\| ^2=1$. We have the bound

$$\begin{aligned}{} & {} \left\| (P_{\lambda })_{x,x_0}\mathbf {v_0}\right\| \ = \ \max _{\left\| \textbf{v}\right\| =1} |\langle \delta _x\otimes \textbf{v}, P_\lambda \delta _{x_0}\otimes \textbf{v}_0 \rangle | \ = \ \max _{\left\| \textbf{v}\right\| =1} |\langle P_\lambda \delta _x\otimes \textbf{v}, P_\lambda \delta _{x_0}\otimes \textbf{v}_0 \rangle | \nonumber \\{} & {} \quad \le \ \sqrt{a_x}\sqrt{a_{x_0}} \ \le \ a_{x_0} \end{aligned}$$

(B.4)

where in the penultimate step the Cauchy-Schwarz inequality was used and in the last step the fact that a achieves its maximum at $x_0$. Combining (B.4), (B.3) and noting that $a_{x_0}\le 1$, we find that

$$\begin{aligned} \left\| \psi (x)\right\| \ \le \ \sqrt{C_\mu } (1+\left\| x-x_0\right\| )^{-\mu /2}(1+\left\| x_0\right\| )^{\nu /2}\,. \end{aligned}$$

Applying the same process again now to $P_\lambda -\psi \otimes \psi ^*$, whose rank is smaller by 1 compared to $P_\lambda$, we obtain the result by induction. $\square$

One further consequence of our definition of an insulator (Definition 2.5) is that matrix elements of the resolvent decay, as expressed in the following.

Lemma B.3

If H is a $\Delta$-insulator as in Definition 2.5 then there is a (fixed, H-dependent) subset $S\subseteq \Delta$ of full Lebesgue measure such that, for any $E\in S$ one has

$$\begin{aligned} \left( H-\left( E+{\text {i}}\varepsilon \right) \mathbb {1}\right) ^{-1} \in \textrm{WLOC}\end{aligned}$$

(B.5)

with $\textrm{WLOC}$ estimates uniform in $\varepsilon \in [-1,1]\setminus \left\{ 0\right\}$.

Remark B.4

For random operators, the decay manifested in (B.5) is an almost-sure consequence of the various methods used to prove localization, and so, in principle could have been included in the definition of a deterministic insulator. We preferred to keep that definition however as is and provide the deterministic proof below.

Proof

Let $R(z)=(H-z)^{-1}$ for ${\mathbb {I}\mathbb {m}}\{z\} >0$. It suffices to show, for any closed interval $\Delta '\subset \textrm{Int}\Delta$ (the interior of $\Delta$), that there is a full measure set $S'\subset \Delta '$ such that (B.5) holds for $E\in S'$. Fix $\Delta '$ and let $\phi :\mathbb {R}\rightarrow [0,1]$ be a smooth function with $\phi (x)=1$ on $\Delta '$ and $\phi (x)=0$ on $\Delta ^c$. Then

$$\begin{aligned} R(z) \ = \ \phi (H) (H-z)^{-1} \ + \ (1-\phi (H)) (H-z)^{-1} \ =: \ R_\phi (z) + R_{1-\phi }(z) \ . \end{aligned}$$

We will bound $R_\phi \in \textrm{WLOC}$ and $R_{1-\phi }\in \textrm{WLOC}$ separately. The result for $R_{1-\phi }$ is immediate, since $\sigma (H)\ni \lambda \mapsto (1-\phi (\lambda ))(\lambda -E-{\text {i}}\varepsilon )^{-1}$ is a smooth function with derivatives bounded uniformly in $\varepsilon \in [0,1]$, so for $E\in \textrm{Int}\Delta '$ we have $R_{1-\phi }(E+{\text {i}}\varepsilon ) \in \textrm{LOC}_{}\subseteq \textrm{WLOC}$, uniformly in $\varepsilon \in [0,1]$. To bound $R_\phi$ we will use the fact that $\Delta$ is a mobility gap. We have

$$\begin{aligned} R_\phi (z) \ = \ \sum _{\lambda \in \mathcal {E}(\Delta )} \phi (\lambda ) (\lambda -z)^{-1} P_\lambda \ , \end{aligned}$$

where $\mathcal {E}(\Delta )$ denotes the (countable) set of eigenvalues of H in $\Delta$.

For $j=1,\ldots ,N$ let $\textbf{e}_j$ denote the elements of the standard basis for $\mathbb {C}^N$. Fix $x,y\in \mathbb {Z}^2$ and $i,j\in \{1,\ldots ,N\}$ and let

$$\begin{aligned}{} & {} f_{x,y}^{i,j}(z)\ := \ \langle \textbf{e}_i, R_\phi (z)_{x,y}\mathbf {e_j} \rangle \ = \ \sum _{\lambda \in \mathcal {E}(\Delta )} \phi (\lambda ) (\lambda -z)^{-1} \langle \textbf{e}_i, (P_\lambda )_{x,y}\textbf{e}_j \rangle \\{} & {} \quad = \ \int _{\Delta } (\lambda -z)^{-1} \phi (t) {\text {d}}{m_{x,y}^{i,j}}(t) \end{aligned}$$

with

$$\begin{aligned} {\text {d}}{m_{x,y}^{i,j}}(t) \ := \ \sum _{\lambda \in \mathcal {E}(\Delta )} \langle \textbf{e}_i, (P_\lambda )_{x,y}\textbf{e}_j \rangle \, \delta (t-\lambda ){\text {d}}{\lambda } \ . \end{aligned}$$

Because $f_{x,y}^{i,j}$ is the Borel transform of the finite Borel measure $m_{x,y}^{i,j}$, it is well known that the limit $\lim _{\varepsilon \rightarrow 0} f_{x,y}^{i,j}(E+{\text {i}}\varepsilon )$ exists for almost every E and satisfies

$$\begin{aligned} \left| \{E \in \mathbb {R}\ : \ |f_{x,y}^{i,j}(E+{\text {i}}0)| > \alpha \} \right| \ \le \ \frac{C}{\alpha }\int _{\mathbb {R}} \phi (t){\text {d}}{|m_{x,y}^{i,j}|(t)} \ , \end{aligned}$$

(B.6)

where $| \cdot |$ denotes Lebesgue measure, ${\text {d}}{|m_{x,y}^{i,j}|}(t) = \sum _{\lambda } |\langle \textbf{e}_i, (P_\lambda )_{x,y}\textbf{e}_j \rangle | \, \delta (t-\lambda ) {\text {d}}{\lambda }$ is the total variation measure for ${\text {d}}{m_{x,y}^{i,j}}$, and C is a universal constant. To see this, recall that $\lim _{\varepsilon \rightarrow 0} \int {\mathbb {I}\mathbb {m}}\{\frac{1}{t-E-{\text {i}}\varepsilon }\}{\text {d}}{m_{x,y}^{i,j}}(t)=0$ a.e., since $m_{x,y}^{i,j}$ is purely singular, while $\lim _{\varepsilon \rightarrow 0} \int {\mathbb {R}\mathbb {e}}\{\frac{1}{t-E-{\text {i}}\varepsilon }\}{\text {d}}{m_{x,y}^{i,j}}(t)=0 = Hm_{x,y}^{i,j}(E)$, the Hilbert transform of $m_{x,y}^{i,j}$, a.e.. Thus (B.6) follows from Loomis’s weak $L^1$ bound on the Hilbert transform of a measure [Loo46].

The integral on the right hand side of (B.6) may be bounded as follows

$$\begin{aligned}{} & {} \int _{\mathbb {R}} \phi {\text {d}}{|m_{x,y}^{i,j}|} \ = \ \sum _{\lambda \in \mathcal {E}(\Delta )} \phi (t) |\langle \textbf{e}_i, (P_\lambda )_{x,y}\textbf{e}_j \rangle | \ \le \ \left| \sup _{|g|\le 1} \sum _{\lambda \in \mathcal {E}(\Delta )} g(\lambda ) \langle \textbf{e}_i, (P_\lambda )_{x,y}\textbf{e}_j \rangle \right| \nonumber \\{} & {} \quad = \ \sup _{|g|\le 1} \langle \delta _x\otimes \textbf{e}_i, P_\Delta (H) g(H) \delta _y \otimes \textbf{e}_j \rangle \ \le \ \sup _{|g|\le 1} \left\| (P_\Delta (H) g(H))_{x,y}\right\| \ . \end{aligned}$$

(B.7)

Using Definition 2.5 to bound the right hand side, we see that we have shown the following

There are $\nu \in \mathbb {N}$ such that for every $\mu \in \mathbb {N}$ and $\alpha >0$ we have

$$\begin{aligned} \left| \{E \in \mathbb {R}\ : \ |f_{x,y}^{i,j}(E+{\text {i}}0)| > \alpha \} \right| \ \le \ C_\mu (1+\left\| x\right\| )^{\nu } (1+ \left\| x-y\right\| )^{-\mu } \frac{1}{\alpha } \ .\nonumber \\ \end{aligned}$$

(B.8)

To prove (B.5) we need to extend (B.8) off the real axis. For the moment let x, y, i, j be fixed and write $f\equiv f_{x,y}^{i,j}$, $m\equiv m_{x,y}^{i,j}$ to simplify notation. For this purpose, let $0<s<1$ and note that the function $|f(z)|^s$ is sub-harmonic in the upper half plane. Let ${\tilde{\Delta }}= \{t : \textrm{dist}(t,\Delta )\le 1\}$ and define

$$\begin{aligned} g(E) = |f(E+{\text {i}}0)|^s \chi _{{\tilde{\Delta }}}(E) \quad \text {and} \quad h(E) = |f(E+{\text {i}}0)|^s (1-\chi _{{\tilde{\Delta }}}(E)) \ . \end{aligned}$$

By the subharmonicity of $|f(z)|^s$, we have

$$\begin{aligned} |f(z)|^s \ \le \ \mathcal {P}g(z) + \mathcal {P}h(z) \end{aligned}$$

for all z in the upper half plane, $\mathcal {P}g(z)=\frac{1}{\pi }\int g(t){\mathbb {I}\mathbb {m}}\{\frac{1}{t-z}\}{\text {d}}{t}$ denotes the Poisson integral. Because the Poisson kernel is a radially decreasing function,

$$\begin{aligned} |f(E+{\text {i}}\varepsilon )|^s \ \le \ Mg(E) + Mh(E) \ , \end{aligned}$$

with Mg, Mh the Hardy-Littlewood maximal functions of g, h, respectively. Since g is compactly supported, it follows from (B.6) that $g\in L^p(\mathbb {R})$ for, say, $p=\frac{2}{1+s}>1$, with

$$\begin{aligned}{} & {} \left\| g\right\| _{L_p}^p \ = \ \frac{1}{p} \int _0^\infty t^{p-1} |\{ |g|>t\}| {\text {d}}{t} \ \le \ C \int _0^\infty t^{p-1} \min \left( |{\tilde{\Delta }}|, t^{-1/s}\int _{\mathbb {R}}\phi {\text {d}}{|m|} \right) {\text {d}}{t} \\{} & {} \quad \le \ C |{\tilde{\Delta }}|^{1-sp} \left( \int _{\mathbb {R}} \phi {\text {d}}{|m|} \right) ^{sp} \ . \end{aligned}$$

Because $|f(z)|^s\le \frac{1}{\textrm{dist}(z,\Delta )^s} \left( \int \phi {\text {d}}{|m|} \right) ^s$, we have $h\in L^q(\mathbb {R})$, for, say, $q=\frac{2}{s}>1$, with

$$\begin{aligned} \left\| h\right\| _{L_q}^q \ \le \ \int _{{\tilde{\Delta }}^c} \frac{1}{\textrm{dist}(t,\Delta )^{qs}} \left( \int \phi {\text {d}}{|m|} \right) ^{qs} {\text {d}}{t} \ \le \ C \left( \int \phi {\text {d}}{|m|} \right) ^{qs} \ . \end{aligned}$$

Thus

$$\begin{aligned} \left| \{ E \ : \ \sup _{\varepsilon \in (0,1]} |f(E+{\text {i}}\varepsilon )|^s > \alpha \} \right| \ \le&\ \left| \{ E \ : \ Mg(E)\ge \tfrac{\alpha }{2} \} \right| \ + \ \left| \{ E \ : \ Mh(E) \ge \tfrac{\alpha }{2}\} \right| \\ \le&C \left( \frac{1}{\alpha ^p} \left( \int _{\mathbb {R}} \phi {\text {d}}{|m|} \right) ^{sp} + \frac{1}{\alpha ^q} \left( \int \phi {\text {d}}{|m|} \right) ^{qs} \right) \ , \end{aligned}$$

by the Hardy-Littlewood maximal inequality. Using (B.7) and Definition 2.5, we find that we have shown:

There are $\nu \in \mathbb {N}$, $s<1$, and $p,q>1$ such that for $\mu \in \mathbb {N}$ and $\alpha >0$ we have

$$\begin{aligned}{} & {} \left| \{ E \ : \ \sup _{\varepsilon \in (0,1]} |f^{i,j}_{x,y}(E+{\text {i}}\varepsilon )|^s > \alpha \} \right| \nonumber \\{} & {} \quad \le \ C_\mu \left( \frac{1}{\alpha ^p} (1+\left\| x\right\| )^{sp\nu } (1+ \left\| x-y\right\| )^{-sp\mu } + \frac{1}{\alpha ^q} (1+\left\| x\right\| )^{sq\nu } (1+ \left\| x-y\right\| )^{-sq\mu }\right) \ ,\nonumber \\ \end{aligned}$$

(B.9)

for every $x,y\in \mathbb {Z}^2$ and $i,j\in \{1,\ldots ,N\}$.

To prove (B.5) we now apply a Borel-Cantelli argument. Fix $\nu$, $s<1$ and $p,q>1$ as above and let $\nu '\in \mathbb {N}$ be such that $\nu '=\nu + 3/\min (sp,sq)$. Let $\mu '>0$ and apply (B.9) with $\alpha =(1+\left\| x\right\| )^{\nu '} (1+ \left\| x-y\right\| )^{-\mu '}$ and $\mu > \mu ' +3/\min (sp,sq)$ to conclude that

$$\begin{aligned}{} & {} \sum _{x,y,i,j} \left| \left\{ E \ : \ \sup _{\varepsilon \in (0,1]} |f^{i,j}_{x,y}(E+{\text {i}}\varepsilon )|^s > (1+\left\| x\right\| )^{\nu '} (1+ \left\| x-y\right\| )^{-\mu '} \right\} \right| \\{} & {} \quad \le \ C_{\mu }\sum _{x,y,i,j} (1+ \left\| x\right\| )^{-3} (1+\left\| x-y\right\| )^{-3} \ < \ \infty \ . \end{aligned}$$

We conclude from the Borel-Cantelli lemma that there is a full measure set of energies on which

$$\begin{aligned} \sup _{\varepsilon \in (0,1]} |f^{i,j}_{x,y}(E+{\text {i}}\varepsilon )|^s \ \le \ (1+\left\| x\right\| )^{\nu '} (1+ \left\| x-y\right\| )^{-\mu '} \end{aligned}$$

for all but finitely many x, y, i, j. Since for each i, j, x, y we have also have $\sup _{\varepsilon \in (0,1]} |f^{i,j}_{x,y}(E+{\text {i}}\varepsilon )|^s < \infty$ on a full measure set of E, we conclude that there is a full measure set of E on which

$$\begin{aligned} \sup _{\varepsilon \in (0,1]} |f^{i,j}_{x,y}(E+{\text {i}}\varepsilon )|^s \ \le \ C_{\mu '} (1+\left\| x\right\| )^{\nu '} (1+ \left\| x-y\right\| )^{-\mu '} \end{aligned}$$

Repeating this for each $\mu '\in \mathbb {N}$ (a countable set) we find that (B.5) holds for E in a set of full measure. $\square$

C Proof of Proposition 4.3

In this section we prove that ${\text {index}}\mathbb {Q}U = 0\,$ for Q a projection onto a subset of the mobility gap (this is Proposition 4.3). Since Q projects onto localized states of H, we know it is spanned by a SULE basis $\left\{ \psi _n\right\} _n$ as in Definition B.1.

Let us define an operator V on ${\text {im}}Q$, diagonal in the SULE basis, via

$$\begin{aligned} V \psi _n := \exp ({\text {i}}\arg (x_n\cdot e_1+{\text {i}}x_n\cdot e_2)) \psi _n\,. \end{aligned}$$

We extend V to $\mathcal {H}$ by defining $V\psi =\psi$ for $\psi \in {\text {im}}Q^\perp$. Clearly, V is unitary and commutes with Q. Thus ${\text {index}}\mathbb {Q} V = 0$, so it suffices to prove $(U-V)Q=:B$ is compact. We shall show it is Schatten. For this, it suffices to show

$$\begin{aligned} \sum _y\left( \sum _x|B_{x,x+y}|^p\right) ^{1/p}<\infty \,. \end{aligned}$$

The proof is similar to that of Lemma A.1, but here we have the added complication of having to control the infinite collection $\left\{ \psi _n\right\} _n$.

Note that

$$\begin{aligned} B_{x,y} = \sum _{n=1}^\infty ({\text {e}}^{{\text {i}}\arg (x)}-{\text {e}}^{{\text {i}}\arg (x_n)}))\psi _n(x)\overline{\psi _n(y)}\,. \end{aligned}$$

Now defining $f(x) := \exp ({\text {i}}\arg (x))$ we have

$$\begin{aligned}{} & {} |B_{x,y}|^p \ \le \ \left( \sum _{n=1}^\infty |f(x)-f(x_n)||\psi _n(x)||\psi _n(y)|\right) ^p \\{} & {} \quad \le \ \sum _{n}|f(x)-f(x_n)|^p|\psi _n(x)||\psi _n(y)| \ , \end{aligned}$$

where we have used Hölder’s inequality in the form

$$\begin{aligned} (\sum _j a_j b_j c_j)^p \ \le \ (\sum _j a_j^p b_j c_j) \, (\sum _j b_j^2)^{\frac{p-1}{2}}\, (\sum _j c_j^2)^{\frac{p-1}{2}} \end{aligned}$$

as well as the fact that $\sum _n |\psi _n(x)|^2 \le 1$, i.e., $\sum _n \psi _n\otimes \psi _n^*= Q \le \mathbb {1}$.

Now we observe that f obeys the estimate in (A.1), so that using (B.1) we find

$$\begin{aligned} |B_{x,x+y}|^p&\le \ D^pC_\mu ^2\sum _{n} \frac{\left\| x-x_n\right\| ^p}{(1+\left\| x_n\right\| )^{p/2}(1+\left\| x\right\| )^{p/2}}(1+\left\| x-x_n\right\| )^{-\mu }\\&\quad (1+\left\| x+y-x_n\right\| )^{-\mu +p}(1+\left\| x_n\right\| )^{2\nu }\\&\le \ D^pC_\mu ^2\sum _{n} (1+\left\| x-x_n\right\| )^{-\mu +p}(1+\left\| x+y-x_n\right\| )^{-\mu +p}(1+\left\| x_n\right\| )^{2\nu -p/2}\\&\quad (1+\left\| x\right\| )^{-p/2}\\&\le \ D^pC_\mu ^2(1+\left\| y\right\| )^{-\mu /2+p/2}(1+\left\| x\right\| )^{-p/2}\sum _{n} (1+\left\| x_n\right\| )^{2\nu -p} \end{aligned}$$

where in the last step we have used the triangle inequality in the form $(1+\left\| a\right\| )(1+\left\| a+b\right\| ) \ge 1+\left\| b\right\|$ as well as $(1+\left\| x+y-x_n\right\| )^{-\mu /2+p/2}\le 1$, $(1+\left\| x-x_n\right\| )^{-\mu /2+p/2}\le 1$.

The result now follows thanks to (B.2) and the fact $p,\mu$ may be chosen arbitrarily large.