Magic angle (in)stability and mobility edges in disordered Chern insulators

Simon Becker [email protected] ETH Zurich, Institute for Mathematical Research, Raemistrasse 101, 8092 Zurich, Switzerland , Izak Oltman [email protected] University of California, Department of Mathematics, Berkeley, CA 94720, USA and Martin Vogel [email protected] Université de Strasbourg, Institut de Recherche Mathématique Avancée, 67084 Strasbourg Cedex, France

Abstract.

Why do experiments only exhibit one magic angle if the chiral limit of the Bistritzer-MacDonald Hamiltonian suggest a plethora of them? - In this article, we investigate the remarkable stability of the first magic angle in contrast to higher (smaller) magic angles. More precisely, we examine the influence of disorder on magic angles and the Bistritzer-MacDonald Hamiltonian. We establish the existence of a mobility edge near the energy of the flat band for small disorder. We also show that the mobility edges persist even when all global Chern numbers become zero, leveraging the $C_{2z}T$ symmetry of the system to demonstrate non-trivial sublattice transport. This effect is robust even beyond the chiral limit and in the vicinity of perfect magic angles, as is expected from experiments.

1. Introduction

Twisted bilayer graphene is a highly tunable material that exhibits approximately flat bands at special twisting angles, the so-called magic angles [TKV19, BiMa11].

In this article, we study the question of why the largest magic angle, as predicted by the chiral model, is more robust than smaller magic angles within the chiral limit. We analyze why the chiral model’s largest magic angle exhibits greater resilience compared to smaller magic angles within the chiral limit. This finding potentially elucidates why, up until now, only the first magic angle has been experimentally observed. We also study the impact of disorder in the chiral limit and its interplay with the flat bands at magic angles. In quantum systems, disorder-induced dynamical localization is a well-known phenomenon, wherein spatially localized wavepackets do not significantly diffuse under time evolution. While the underlying mechanisms behind this phenomenon are relatively well understood, the opposite behavior—diffusive behavior in disordered systems—is only understood in specific cases [AS19, AW13, BH22, GKS07, JSS03].

It is widely believed that large classes of two-dimensional quantum systems, even under minor disorder, exclusively exhibit localization, as conjectured by Problem 2 on Simon’s list of open problems for Schrödinger operators [Si00]. As we will show below, the Hamiltonian describing twisted bilayer graphene at a magic angle or other related materials is an exception. These new classes of materials, so-called Chern insulators, exhibit non-zero Chern numbers in the absence of external magnetic fields [Li21]. In particular, in twisted bilayer graphene, for wavepackets localized sufficiently close to the perturbed flat band at zero energy, the time-evolution is, in a suitable sense, ballistic. Our argument here is an adaptation of an argument by Germinet, Klein, and Schenker showing a form of delocalization for the Landau Hamiltonian [GKS07]. The physical intuition behind this delocalization argument is straightforward: the Landau Hamiltonian exhibits non-zero Hall conductivity at each Landau level. Moreover, as the Hall conductivity, a topological quantity, remains invariant under minor disorder, the existence of substantial spectral gaps between the Landau levels prevents strong localization across the spectrum.

The properties of the flat bands for twisted bilayer graphene are analogous to Landau levels with the crucial difference that no magnetic field is required. At the first magic angle, the two flat bands exhibit only a Chern number zero. However, the two flat bands individually carry non-zero Chern numbers $\pm 1$ , allowing for an anomalous quantum Hall effect when the TBG substrate is e.g. aligned with hexagonal boron nitride [Li21]. Mathematically, the effect of aligning the substrate with hBN is modeled by adding an effective mass term to the Hamiltonian thereby splitting the two flat bands each carrying a non-zero Chern number. In addition, the flat bands are gapped from the rest of the spectrum.

We also establish a localized regime that rests on the multi-scale analysis framework of Germinet-Klein [GK01, GK03]. Here, the only difficulty is to allow for a sufficiently large class of random perturbations which requires us to extend the estimate on the number of eigenvalues (NE) and thus the Wegner estimate (W).

The chiral limit of the massive continuum model for twisted bilayer graphene, which can also be thought of as a model Hamiltonian for twisted transition metal dichalcogenides (TMDs) [CRQ23], is the Hamiltonian $H(m,\alpha)$ acting on $L^{2}({\mathbb{C}};{\mathbb{C}}^{4})$ with domain given by the Sobolev space $H^{1}({\mathbb{C}};{\mathbb{C}}^{4})$

H(m,\alpha)=\begin{pmatrix}m&D(\alpha)^{*}\\ D(\alpha)&-m\end{pmatrix}\text{ with }D(\alpha)=\begin{pmatrix}2D_{\bar{z}}&\alpha U(z)\\ \alpha U(-z)&2D_{\bar{z}}\end{pmatrix},

(1.1)

where $D_{\bar{z}}=-i\partial_{\bar{z}}$ , $\alpha\in\mathbb{C}\setminus\{0\}$ is an effective parameter that is inversely proportional to the twisting angle and $m\geq 0$ a mass parameter. Let $\Gamma:=4\pi i\omega({\mathbb{Z}}\oplus\omega{\mathbb{Z}})$ be a triangular lattice with $\omega=e^{2\pi i/3}$ . The tunnelling potentials $U$ are $\Gamma$ -periodic functions satisfying for $\mathbf{a}=4\pi ia_{1}\omega/3+4\pi ia_{2}\omega^{2}/3$ with $a_{i}\in{\mathbb{Z}}$ , i.e. $\mathbf{a}\in\Gamma_{3}:=\Gamma/3$

U(z+\mathbf{a})=\bar{\omega}^{a_{1}+a_{2}}U(z),\quad U(\omega z)=\omega U(z),\quad\overline{U(z)}=U(\bar{z}).

(1.2)

The central object in the one-particle picture of twisted bilayer graphene are the so-called magic angles. We say that $\alpha\in\mathbb{C}\setminus\{0\}$ is magic if and only if the Bloch-Floquet transformed Hamiltonian, see [Be*22, (2.11)], with mass parameter $m\geq 0$ exhibits a flat band at energy $\pm m$ , i.e.

\pm m\in\bigcap_{k\in{\mathbb{C}}}\operatorname{Spec}_{L^{2}({\mathbb{C}}/\Gamma)}(H_{k}(m,\alpha))\text{ with }H_{k}(m,\alpha)=\begin{pmatrix}m&D(\alpha)^{*}+\bar{k}\\ D(\alpha)+k&-m\end{pmatrix},

(1.3)

with $\operatorname{Spec}_{X}(S)$ denoting the spectrum of the linear operator $S$ on the Hilbert space $X$ on a suitable dense domain, where $H_{k}(m,\alpha):H^{1}({\mathbb{C}}/\Gamma;{\mathbb{C}}^{2})\to L^{2}({\mathbb{C}}/\Gamma;{\mathbb{C}}^{2}).$ The set of $\alpha$ under which there exists a flat band at energy $\pm m$ is independent of $m$ ¹¹1Observe that $\operatorname{Spec}H_{k}(m,\alpha)=\pm\sqrt{\operatorname{Spec}H_{k}(0,\alpha)^{2}+m^{2}}$ [T92, (5.66)].. In the sequel, we shall suppress the mass parameter $m\geq 0$ in the notation.

For the study of magic angles we also introduce a translation operator

\mathscr{L}_{\mathbf{a}}w(z):=\begin{pmatrix}\omega^{a_{1}+a_{2}}&0\\ 0&1\end{pmatrix}w(z+\mathbf{a}),\ \ \ \mathbf{a}\in\Gamma_{3},

(1.4)

and a rotation operator $\mathscr{C}u(z)=u(\omega z).$ We can then define subspaces

L^{2}_{k,p}:=\{u\in L^{2}({\mathbb{C}}/\Gamma;{\mathbb{C}}^{2});\mathscr{L}_{\mathbf{a}}u(z)=\omega^{k}u(z)\text{ and }\mathscr{C}u(z)=\bar{\omega}^{p}u(z)\}

(1.5)

and similarly $L^{2}_{k}:=\bigoplus_{p\in{\mathbb{Z}}_{3}}L^{2}_{k,p}$ .

The set of such magic parameters $\alpha$ , that we shall denote by $\mathcal{A},$ is characterized by [Be*22, Theo. $2$ ]

\alpha^{-1}\in\operatorname{Spec}_{L^{2}_{0}}(T_{k})\text{ with }T_{k}=(2D_{\bar{z}}-k)^{-1}\begin{pmatrix}0&U(z)\\ U(-z)&0\end{pmatrix}\text{ for some }k\notin\Gamma^{*}

(1.6)

with $\Gamma^{*}$ the dual lattice.

We then define the set of generic magic angles. The terminology generic is motivated by [BHZ23, Theo. $3$ ] which shows that for a generic choice of tunnelling potentials all magic angles are of the following form.

Definition 1.1 (Generic magic angles).

We say that $\alpha\in\mathcal{A}$ is a simple or two-fold degenerate magic angle if $1/\alpha\in\operatorname{Spec}_{L^{2}_{0}}(T_{k})$ and $\dim\ker_{L^{2}_{0}}(T_{k}-1/\alpha)=\nu$ with $\nu=1,2$ , respectively. In the following sections, we will denote the combination of these magic angles as the collection of generic magic angles.

1.1. Magic angle (in)stability

The first aim of this article is to study perturbations of the operator $T_{k}$ , i.e. for $\delta>0$

T_{k,\delta}=(2D_{\bar{z}}-k)^{-1}\begin{pmatrix}0&U(z)+\delta V_{+}\\ U(-z)+\delta V_{-}&0\end{pmatrix}

with bounded linear perturbations $V_{\pm}$ . We then obtain in Theorem 6 a bound on the spread of the magic angles under such perturbations. This is a non-trivial result as the operator $T_{k}$ , whose eigenvalues are the magic angles, is a non-normal operator. In particular, small perturbations in norm can lead to substantial perturbations of the spectrum, see [TE05]. On the other hand, we show that even simple rank $1$ -perturbations of exponentially small size in $1/|\mu|$ suffice to generate eigenvalues $\mu$ in the spectrum of $T_{k}$ .

Theorem 1 (Instability).

Let $\mu\in\mathbb{C}$ and $k\notin\Gamma^{*}$ , then there exists a rank- $1$ operator $R$ with $\|R\|=\mathcal{O}(e^{-c/|\mu|})$ and $c(k)>0$ independent of $\mu$ such that $\mu\in\operatorname{Spec}(T_{k}+R)$ . Here, $T_{k}$ characterizes the set of magic parameters as explained in (1.6).

1.2. Anderson model and IDS

One consequence of having a flat band is the occurrence of jump discontinuities in the integrated density of states (IDS). The integrated density of states is defined as follows, see [Sj89] and others:

Definition 1.2.

The integrated density of states (IDS) for energies $E_{2}>E_{1}$ and $I=[E_{1},E_{2}]$ is defined by

N(I):=\lim_{L\to\infty}\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{I}(H_{\Lambda_{L}}(\alpha)))}{L^{2}}

with $\Lambda_{L}={\mathbb{C}}/(L\Gamma)$ and $H_{\Lambda_{L}}$ has periodic boundary conditions, i.e. $H_{\Lambda_{L}}:H^{1}(\Lambda_{L})\subset L^{2}(H_{\Lambda_{L}})\to L^{2}(H_{\Lambda_{L}}).$

For ergodic random operators, the almost sure existence of this limit is shown using the subadditive ergodic theorem, see for instance [K89, Sec. 7.3]. Alternatively, one may define for $f\in C_{c}^{\infty}({\mathbb{R}})$ the regularized trace

\widetilde{\operatorname{tr}}(f(H(\alpha)))=\lim_{L\to\infty}\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{{\Lambda_{L}}}f(H(\alpha)))}{|\Lambda_{L}|}.

(1.7)

By Riesz’s theorem on the representation of positive functionals, one has that

\widetilde{\operatorname{tr}}(f(H(\alpha)))=\int_{{\mathbb{R}}}f(\lambda)\ d\rho(\lambda),

where $\rho$ is the density of states (DOS) measure of $H(\alpha).$ This way, $N(I)=\int_{I}\ d\rho(\lambda).$

Remark 1.

For Schrödinger operators it is common to consider Dirichlet approximations of the finite-size truncation. It is known that Dirac operators do in general not have any self-adjoint Dirichlet realizations. However, self-adjoint Neumann-type boundary conditions are possible, see [BM87] and for instance the introduction of [SV19] for a mathematical discussion. The independence of the definition of the IDS of the boundary conditions can then be shown using spectral shift function techniques if the operator contains a gap in the spectrum, see for instance the work by Nakamura [N01] on Schrödinger operators.

A periodic Hamiltonian that exhibits a flat band at energy $E$ possesses a jump discontinuity in the IDS at $E$ . In particular, the Lebesgue decomposition of $\rho$ has a pure point contribution at $E.$ As a consequence, if we define the associated cumulative distribution function $N_{E_{0}}:(E_{0},\infty)\to\mathbb{R}$ by $N_{E_{0}}(E):=N([E_{0},E])$ , then this function is monotonically increasing and right-continuous (càdlàg). At a magic angle, the function $N_{E_{0}}$ for $E_{0}<\pm m$ exhibits a jump discontinuity at $E=\pm m.$ Indeed, this can easily be seen from the following formula which shows that for a periodic Hamiltonian one just has [Sj89, (1.29)]

N(I)=\int_{\mathbb{C}/\Gamma^{*}}\Bigg{(}k\mapsto\sum_{\lambda\in\operatorname{Spec}_{L^{2}({\mathbb{C}}/\Gamma)}(H_{k}(\alpha))}\operatorname{1\hskip-2.75ptl}_{I}(\lambda)\Bigg{)}\ \frac{dk}{4\pi^{2}}.

Let $\alpha\in\mathcal{A}$ be a generic magic angle, as in Def. 1.1, then we define the energy gap between the flat bands and the rest of the spectrum

E_{\operatorname{gap}}:=\inf_{\lambda\in\operatorname{Spec}(H(\alpha)^{2})\setminus\{0\}}\sqrt{\lambda}>0.

(1.8)

That this quantity is non-zero by [BHZ22, Theo. $2$ ] for simple and by [BHZ23, Theo. $4$ ] for two-fold degenerate magic angles and is illustrated in Figure 1. In particular, the following union of intervals is in the resolvent set of the Hamiltonians

\Big{(}-\sqrt{E_{\operatorname{gap}}^{2}+m^{2}},-m\Big{)}\cup\big{(}-m,m\big{)}\cup\Big{(}m,\sqrt{E_{\operatorname{gap}}^{2}+m^{2}}\Big{)}\subset\mathbb{R}\setminus\operatorname{Spec}(H).

(1.9)

Let $P_{X}$ be the orthogonal projection onto a closed subspace $X$ . For $\alpha\in\mathcal{A}$ generic it has been shown in [BHZ22b, Theo. $4$ ] and [BHZ23, Theo. $5$ ] that the Chern number of the flat band at energy $\pm m$ is $\mp 1$ or more generally (including $m=0$ ) for the Hamiltonian in (1.1)

\operatorname{Cher}(P_{\ker(D(\alpha))})=-1\text{ and }\operatorname{Cher}(P_{\ker(D(\alpha)^{*})})=1.

(1.10)

The Chern number can be computed from the Hall conductivity $\Omega(P)$ , see (4.4), by using that

\operatorname{Cher}(P)=-2\pi i\Omega(P).

In particular, the net Chern number of the flat bands for $m=0$ is zero

\operatorname{Cher}(P_{\ker(D(\alpha))}\oplus P_{\ker(D(\alpha)^{*})})=\operatorname{Cher}(P_{\ker(H(\alpha))})=0.

Refer to caption — Figure 1. Band structure of non-disordered twisted bilayer graphene (1.1) at the first real positive magic angle $\alpha\approx 0.58566$ with zero effective mass (top) and non-zero effective mass (bottom).

Assumption 1 (Anderson model).

We introduce the Anderson-type Hamiltonian with alloy-type potentials and (possible) lattice relaxation effects for $\lambda>0$ and $u\in C^{\infty}_{c}({\mathbb{C}};{\mathbb{C}}^{4}).$

H_{\lambda}=H+\lambda V_{\omega}\text{ where }V_{\omega}=\sum_{\gamma\in\Gamma}\omega_{\gamma}u(\bullet-\gamma-\xi_{\gamma}),

(1.11)

where $(\omega_{\gamma})_{\gamma}$ and $(\xi_{\gamma})_{\gamma}$ are families of i.i.d. random variables with absolutely continuous bounded densities. For $(\omega_{\gamma})$ the density is a function $g$ with $\operatorname{supp}(g)\subset[-1,1]$ and in case of $(\xi_{\gamma})_{\gamma}$ a density $h$ supported within a compact domain $D\subset{\mathbb{C}}$ where we allow $\xi\equiv\operatorname{const}$ . Random variables $\xi_{\gamma}$ model small inhomogeneities of the moiré lattice due to relaxation effects. We assume that either

(1)

Case $1$ : The disorder $u$ in (1.11) is of the form

$u(z)=\begin{pmatrix}Y(z)&Z(z)^{*}\\ Z(z)&-Y(z)\end{pmatrix}\in C^{\infty}_{c}({\mathbb{C}};{\mathbb{C}}^{4})$ (1.12)

where $\inf_{\xi\in D^{\Gamma}}\inf_{z\in{\mathbb{C}}}\sum_{\gamma\in\Gamma}Y(z-\gamma-\xi_{\gamma})>0.$
(2)

Case $2$ : The disorder $u\in C^{\infty}_{c}({\mathbb{C}};{\mathbb{C}}^{4})$ with $u\geq 0$ and $\inf_{z\in B_{\varepsilon}(z_{0}),\gamma\in\Gamma,\xi\in D}u(z-\gamma-\xi)>0$ for some $B_{\varepsilon}(z_{0})$ .

For normalization purposes, we assume that $\sup_{\xi\in D^{\Gamma}}\|\sum_{\gamma\in\Gamma}u(\bullet-\gamma-\xi_{\gamma})\|_{\infty}\leq 1$ and $\operatorname{supp}u\subset\Lambda_{R}(0)$ for some fixed $R>0$ where $\Lambda_{L}:={\mathbb{C}}/(L\Gamma)$ and $\Lambda_{L}(z)=\Lambda_{L}+z.$

We emphasize that under assumption (1), the matrix $u$ is neither positive nor negative definite. This usually poses an obstruction to proving Wegner estimates as the eigenvalues are not monotone in the noise parameter. We can overcome this obstacle here by using the off-diagonal structure of the Hamiltonian. The probability space is the Polish space $\Omega=(\operatorname{supp}(g))^{\Gamma}\times(\operatorname{supp}(h))^{\Gamma}$ with the product measure. Then $(H_{\lambda})$ is an ergodic (with respect to lattice translations) family of self-adjoint operators with continuous dependence $\Omega\ni(\omega,\xi)\mapsto(H_{\lambda}+i)^{-1}.$ Thus, there is $\Sigma\subset\mathbb{R}$ closed such that

\operatorname{Spec}_{L^{2}({\mathbb{C}})}(H_{\lambda})=\Sigma\text{ a.s.},

(1.13)

see [CFKS87, Pa80]. In addition, using ergodicity arguments, see e.g. [W95], the density of states measure for the random operator, $\rho^{H_{\lambda}}$ exists almost surely and is almost surely non-random, i.e. is almost surely equal to a non-random measure $\rho^{H_{\lambda}}=\rho$ a.s. with $\rho$ non-random. An extension of our work to unbounded disorder is possible. In the context of Schrödinger operators this extension has been shown for magnetic Landau Hamiltonians [GKS09, GKM09]. Related proofs of localization for Dirac operators under a spectral gap assumption have also been obtained in [BCZ19].

For $\lambda\neq 0$ , the infinitely-degenerate point spectrum of $H$ at energy zero, i.e. the flat band, gets non-trivially perturbed and expands in energy. To capture this, we then introduce constants $K_{\pm}:=\sqrt{E_{\operatorname{gap}}^{2}+m^{2}}\pm\lambda\sup_{\omega\in\Omega}\|V_{\omega}\|_{\infty}$ and $k_{\pm}:=m\pm|\lambda|\sup_{\omega\in\Omega}\|V_{\omega}\|_{\infty}.$ One thus finds analogously to (1.9) for the disordered Hamiltonian

(-K_{-},-k_{+})\cup(-k_{-},k_{-})\cup(k_{+},K_{-})\subset\mathbb{R}\setminus\Sigma,

(1.14)

where all three intervals are non-trivial for $\lambda>0$ sufficiently small and $m>0$ . We then also define

J_{-}:=[-k_{+},-k_{-}]\text{ and }J_{+}:=[k_{-},k_{+}].

(1.15)

When perturbing away from perfect magic angles, we may do so by either using a random potential or perturbing $\alpha$ slightly. In both cases, for sufficiently small perturbations, this leaves the spectral gap to the remaining bands open.

Given a finite domain $\Lambda_{L}:={\mathbb{C}}/(L\Gamma)\subset{\mathbb{C}}$ , we introduce the Hamiltonian

H_{\lambda,\Lambda_{L}}=H_{\Lambda_{L}}+\lambda V_{\omega,\Lambda_{L}},

with periodic boundary conditions where $\sum_{\gamma\in\tilde{\Lambda}_{L}}\omega_{\gamma}(u\operatorname{1\hskip-2.75ptl}_{\Lambda_{L}})(\bullet-\gamma-\xi_{\gamma})\text{ with }\widetilde{\Lambda}_{L}:=\Lambda_{L}\cap\Gamma$ . In general we shall denote by $S_{\Lambda_{L}}$ the restriction of an operator $S$ to the domain $\Lambda_{L}$ with periodic boundary conditions in case that $S$ is a differential operator.

While the occurrence of a flat band for the unperturbed Hamiltonian (1.1) leads to a jump discontinuity in the IDS, one has that the random Hamiltonian (1.11) has a Lipschitz continuous IDS for all $\lambda\neq 0$ . Since the randomly perturbed Hamiltonian is no longer periodic, it is customary to measure the destruction of the flat band by studying the regularity of the IDS.

Theorem 2 (Continuous IDS).

Consider the Anderson Hamiltonian as in Assumption (1) with $m\geq 0$ and coupling constant $\lambda\in(-\varepsilon(m),\varepsilon(m))\setminus\{0\}$ with $\varepsilon(m)>0$ sufficiently small, then the integrated density of states (IDS) is a.s. Hölder continuous in Hausdorff distance $d_{H}$ for all $\beta\in(0,1)$ , i.e.

|N(I)-N(I^{\prime})|\lesssim_{\beta,I,I^{\prime}}d_{\text{H}}(I,I^{\prime})^{\beta},

•

Case 1 disorder: for intervals $I,I^{\prime}\subset[-k_{+},k_{+}]$ with $m>0$ and
•

Case 2 disorder: arbitrary bounded intervals $I,I^{\prime}\Subset{\mathbb{R}}$ , $\lambda\in{\mathbb{R}}\setminus\{0\}$ and $m\geq 0.$ If we assume in addition that $u$ is globally positive, i.e.

$\inf_{\xi\in D^{\Gamma}}\inf_{z\in{\mathbb{C}}}\sum_{\gamma\in\Gamma}u(z-\gamma-\xi_{\gamma})>0,$ (1.16)

then the IDS is a.s. Lipschitz continuous

$|N(I)-N(I^{\prime})|\lesssim_{I,I^{\prime}}d_{\text{H}}(I,I^{\prime}).$

In particular, the IDS is a.s. differentiable and its Radon-Nikodym derivative, the density of states (DOS), exists a.s. and is a.s. bounded.

The above results follow directly from the following estimate on the number of eigenvalues (NE) that directly lead to Wegner estimates (4.2).

Proposition 1.3 (NE).

Under the assumptions of Theorem 2, we have that there is $\beta\in(0,1)$

\mathbf{E}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda,\Lambda_{L}}))\lesssim_{\beta}|I|^{\beta}|\Lambda_{L}|.

If in Case 2 we assume in addition that (1.16) holds, then we may take $\beta=1$

\mathbf{E}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda,\Lambda_{L}}))\lesssim|I||\Lambda_{L}|.

1.3. Mobility edges

In the works of Germinet–Klein [GK01, GK03, GK04] dynamical measures of transport have been introduced. The dynamical localization implies a strong form of decaying eigenfunctions, see Def. 4.1. To measure dynamical localization/delocalization one introduces the following Hilbert-Schmidt norm

M_{\lambda}(p,\chi,t)=\left\lVert\langle\bullet\rangle^{p/2}e^{-itH_{\lambda}}\chi(H_{\lambda})\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2},

(1.17)

where $\Gamma_{3}:=\Gamma/3$ , for some non-negative $\chi\in C_{c}^{\infty}$ with time average

\mathcal{M}_{\lambda}(p,\chi,T)=\frac{1}{T}\int_{0}^{\infty}\mathbf{E}\Big{(}M_{\lambda}(p,\chi,t)\Big{)}e^{-t/T}\ dt.

Recall that $\frac{1}{T}\int_{0}^{\infty}t^{n}e^{-t/T}\ dt=T^{n}\Gamma(n+1),$ to see that $\mathcal{M}_{\lambda}(p,\chi,T)$ indicates a time-averaged power scaling of $M_{\lambda}(p,\chi,t).$ Here, $M_{\lambda}(p,\chi,t)$ measures the spread of mass in a spectral energy window of the Hamiltonian from the origin under the free Schrödinger evolution.

We shall then show that the random Hamiltonian (1.11) exhibits diffusive behavior in the vicinity of magic angles.

Theorem 3 (Dynamical delocalization).

Let $\alpha_{*}$ be a generic magic angles as in Definition 1.1. We consider a coupling constant $\lambda\in(-\varepsilon(m,\alpha_{*}),\varepsilon(m,\alpha_{*}))$ , $\alpha\in(\alpha_{*}-\delta(m,\alpha_{*}),\alpha_{*}+\delta(m,\alpha_{*}))$ , mass $m\geq 0$ and sufficiently small $\varepsilon(m,\alpha_{*}),\delta(m,\alpha_{*})>0$ . The random Hamiltonian $H_{\lambda}$ exhibits diffusive behavior for $m>0$ at at least two energies $E_{\pm}(\lambda)$ located close to energies $\pm m$ , respectively, and at at least one energy $E(\lambda)$ for $m=0$ . In particular, for every $\chi\in C_{c}^{\infty}$ that equals to one on an open interval $J$ containing at least one of $E_{\pm}(\lambda)$ and $p>0$ we have for all $T>0$

\mathcal{M}_{\lambda}(p,\chi,T)\gtrsim_{p,J}T^{\frac{p}{4}-6}.

We do not have a very precise understanding how close $E_{\pm}(\lambda)$ are to $\pm m$ . By choosing suitable disorder (of fixed support but rescaled probability), one can show that $E_{\pm}$ and $\pm m$ can get arbitrarily close, see Theorem 7, at least when $\alpha\in\mathcal{A}$ is a magic angle, as the bands of the unperturbed Hamiltonian are perfectly flat.

Remark 2.

Transport behavior can also be characterized by the $p$ -dependence of the estimate in the previous theorem and using a local transport exponent

\beta_{\lambda}(E)=\sup_{p>0}\inf_{\begin{subarray}{c}I\ni E\\ I\text{ open }\end{subarray}}\sup_{\chi\in C_{c}^{\infty}(I;[0,\infty))}\liminf_{T\to\infty}\frac{\log_{+}\mathcal{M}_{\lambda}(p,\chi,T)}{p\log(T)}.

The region of dynamical localization is then defined as the open set

\Sigma^{\operatorname{DL}}:=\{E\in{\mathbb{R}};\beta_{\lambda}(E)=0\}

(1.18)

whereas the region of dynamical delocalization $\Sigma^{\operatorname{DD}}$ is defined as its complement. A mobility edge is an energy $E\in\Sigma^{\operatorname{DD}}\cap\overline{\Sigma^{\operatorname{DL}}\cap\Sigma}.$ It follows from [GK04, Theo. $2.10$ , $2.11$ ] that Theorem 3 implies $\beta_{\lambda}(E)>1/4.$ Theorem 7 then proves the existence of mobility edges for the disordered Hamiltonian.

While Theorem 3 describes the dynamical features of the Hamiltonian, it is equally valid to ask for a spectral theoretic interpretation of transport and localization. The interpretation of the nature of the spectrum in the dynamically localized phase is captured by the concept of SUDEC, see Def. 4.1.

The existence of a dynamical delocalization, in the above sense, does not imply the existence of a.c. or s.c. spectrum. Given that at magic angles the Hamiltonian $H_{0}(\alpha)$ only exhibits (infinitely degenerate) point spectrum at energies $\pm m$ , it is unknown if such phases can occur for our disordered Hamiltonian in a neighborhood of the flat bands. We conjecture that this is not the case.

As we will explain below, see Remark 3, the point spectrum of the Hamiltonian within an energy window, cannot be too localized.

This can be made precise within the framework of generalized Wannier functions [CMM19, MMP]:

Definition 1.4 (Wannier basis).

Let $P$ be an orthogonal projection onto $L^{2}({\mathbb{C}})$ . We say an orthonormal basis $(\psi_{\beta})_{\beta\in I}\in L^{2}({\mathbb{C}})$ for an index set $I\subset\mathbb{N}$ is an $s$ -localized generalized Wannier basis for $P$ for some $s>0$ if:

•

$\overline{\operatorname{span}}(\psi_{\beta})=\operatorname{ran}(P).$

•

There exists $M<\infty$ and a collection of localization centers $(\mu_{\beta})\subset{\mathbb{C}}$ such that for all $\beta\in I$

\int_{{\mathbb{C}}}\langle z-\mu_{\beta}\rangle^{2s}|\psi_{\beta}(z)|^{2}d\lambda(z)\leq M,\text{ with }\lambda\text{ Lebesgue measure.}

One then has for the random Hamiltonian $H_{\lambda}:$

Theorem 4 (Slow decay; $m>0$ ).

Under the assumptions of Theorem 3, we define the orthogonal projection $P_{\lambda}:=\operatorname{1\hskip-2.75ptl}_{J_{\pm}}(H_{\lambda})$ on $L^{2}({\mathbb{C}})$ with $J_{\pm}$ as in (1.15) for $m>0.$ For any $\delta>0$ and for any $\lambda\in(-\varepsilon(m),\varepsilon(m))$ with $\varepsilon(m)>0$ sufficiently small and independent of $\delta>0$ , $P_{\lambda}$ does not admit a $1+\delta$ -localized generalized Wannier basis.

However, the projection admits a $1-\delta$ -localized generalized Wannier basis for small disorder.

In this article, we have not considered disorder that only perturbs the off-diagonal entries of the Hamiltonian (1.1), since no techniques to show Wegner estimates for such disorder are known on which the multi-scale analysis rests.

Wegner estimates are however not needed to study the decay of Wannier functions and thus we shall consider such perturbations now, by looking at the Hamiltonian

H_{\lambda}=\begin{pmatrix}m&(D(\alpha)+\lambda W)^{*}\\ D(\alpha)+\lambda W&-m\end{pmatrix}

(1.19)

where $W\in L^{\infty}({\mathbb{C}};{\mathbb{C}}^{2\times 2})$ is a (possibly random) potential which we assume without loss of generality to satisfy $\|W\|_{\infty}\leq 1.$ The result of Theorem 4 cannot be directly extended to $m=0$ , since the net Chern number of the Hamiltonian is zero. However, the square of the Hamiltonian (1.19) exhibits a diagonal form

H_{\lambda}^{2}=\operatorname{diag}((D(\alpha)+\lambda W)^{*}(D(\alpha)+\lambda W)+m^{2},(D(\alpha)+\lambda W)(D(\alpha)+\lambda W)^{*}+m^{2}).

(1.20)

Thus, to capture the low-lying spectrum, we may study the projections

\begin{split}P_{+,\lambda}&:=\operatorname{1\hskip-2.75ptl}_{[0,\mu]}((D(\alpha)+\lambda W)^{*}(D(\alpha)+\lambda W))\text{ and }\\ P_{-,\lambda}&:=\operatorname{1\hskip-2.75ptl}_{[0,\mu]}((D(\alpha)+\lambda W)(D(\alpha)+\lambda W)^{*}),\end{split}

(1.21)

separately, where we dropped the $m\geq 0$ , dependence as it does not affect the spectrum apart from a constant shift. We then have

Theorem 5 (Slow decay; $m\geq 0$ ).

Let $\mu<E_{\text{gap}}^{2}/2$ with $E_{\text{gap}}$ as in (1.8) and $P_{\pm,\lambda}$ be as in (1.21). For any $\delta>0$ and for any $\lambda\in(-\varepsilon,\varepsilon)$ with $\varepsilon>0$ sufficiently small and independent of $\delta>0$ , the projection $P_{\pm,\lambda}$ does not admit a $1+\delta$ -localized generalized Wannier basis. However, the projections admit a $1-\delta$ -localized generalized Wannier basis for small disorder.

We make a few observations related to Theorem 4 and the notion of Wannier bases. First, these theorems imply a lower bound on the uniform decay of eigenfunctions for the random Hamiltonian. In particular, it implies that if the random Hamiltonian exhibits pure point spectrum, then the decay is not too fast in a uniform sense which should be compared with the notion of SUDEC, see Def. 4.1 which one obtains by applying the multiscale analysis. In particular, one has

Remark 3 (Lower bound on uniform eigenfunction decay).

If the Hamiltonian only exhibits point spectrum in the interval $I$ , for which the associated spectral projections does not admit a $1+\delta$ generalized Wannier basis, then we can choose an orthonormal basis of eigenfunctions $(\psi_{\beta})$ such that $\overline{\operatorname{span}}(\psi_{\beta})=\operatorname{ran}(P)$ and any sequence of localization centers $\mu_{\beta}$

\sup_{\beta}\int_{{\mathbb{C}}}\langle z-\mu_{\beta}\rangle^{2+\delta}|\psi_{\beta}(z)|^{2}\ dz=\infty.

In this sense, Theorem 4 gives a lower-bound on the decay of eigenfunctions in case that the random Hamiltonian exhibits only pure point spectrum.

Outline of article.

•

In Section 2, we study the stability of the first magic angle under small perturbations.
•

In Section 3, we study the regularity of the integrated density of states by stating a estimate on the number of eigenvalues (NE) under Assumption 1.
•

In Section 4, we derive the existence of a mobility edge in a neighborhood of the perturbed flat bands.
•

In Section 5, we prove Theorem 4.

2. (In)stability of magic angles

In this section, we obtain stability bounds on magic angles with respect to perturbations.

We recall the definition of the compact Birman-Schwinger operator $T_{k}$ , with $k=(\omega^{2}k_{1}-\omega k_{2})/\sqrt{3}$ , where $(k_{1},k_{2})\in\mathbb{R}^{2}\setminus(3{\mathbb{Z}}^{2}+\{(0,0),(-1,-1)\})$ . This operator is defined by

T_{k}:=(2D_{\bar{z}}-k)^{-1}\begin{pmatrix}0&U(z)\\ U(-z)&0\end{pmatrix}:L^{2}_{0}(\mathbb{C}/\Gamma;\mathbb{C}^{2})\to(H^{1}\cap L^{2}_{0})(\mathbb{C}/\Gamma;\mathbb{C}^{2}),

where

L^{2}_{p}(\mathbb{C}/\Gamma;\mathbb{C}^{2}):=\Big{\{}u\in L^{2}(\mathbb{C}/\Gamma,{\mathbb{C}}^{2}):\mathscr{L}_{\mathbf{a}}u(z)=e^{2\pi i(a_{1}p+a_{2}p)}u(z+\mathbf{a}),\ a_{j}\in\tfrac{1}{3}\mathbb{Z}\Big{\}},

for $\mathbf{a}=4\pi i(\omega a_{1}+\omega^{2}a_{2}).$ For scalar functions, we also define spaces $L^{2}_{p}({\mathbb{C}}/\Gamma;{\mathbb{C}}^{2})$ where we replace the translation operator by its first component (1.4). As described in (1.6), $\alpha\neq 0$ is magic if and only if $1/\alpha\in\operatorname{Spec}_{L^{2}_{0}}(T_{k})\setminus\{0\}.$ One can then show that $1/\alpha\in\operatorname{Spec}_{L^{2}_{0}}(T_{k})\setminus\{0\}$ if and only if $1/\alpha\in\operatorname{Spec}_{L^{2}_{1}}(T_{k})\setminus\{0\},$ see [BHZ22b]. By squaring the operator, we define new compact operators $A_{k},B_{k}$

\begin{split}&T_{k}^{2}=:3\operatorname{diag}(A_{k},B_{k})\text{ with }A_{k}:=(2D_{\bar{z}}-k)^{-1}U(z)(2D_{\bar{z}}-k)^{-1}U(-z)\text{ and }\\ &B_{k}:=(2D_{\bar{z}}-k)^{-1}U(-z)(2D_{\bar{z}}-k)^{-1}U(z).\end{split}

(2.1)

Since the operator $(2D_{\bar{z}}-k)^{-1}$ is compact, it follows that

\operatorname{Spec}_{L^{2}_{1}}(T_{k}^{2})\setminus\{0\}=3\operatorname{Spec}_{L^{2}_{1}}(A_{k})\setminus\{0\}=3\operatorname{Spec}_{L^{2}_{2}}(B_{k})\setminus\{0\}.

This implies that $\operatorname{tr}((T_{k}^{2})^{n})=2\cdot 3^{n}\operatorname{tr}(A_{k}^{n})$ for $n>1$ which is well-defined as $A_{k}$ is a Hilbert-Schmidt operator. In particular, the spectrum of $A_{k}$ is independent of $k$ , see [Be*22, Prop.3.1.] and we have $\operatorname{Spec}_{L^{2}_{1}}(A_{0})=\operatorname{Spec}_{L^{2}_{1}}(A_{k})$ for any $k\notin\Gamma^{*}.$ The traces of powers of $A_{k}$ are illustrated in Table 1. We thus have that

\mathbb{C}\setminus\{0\}\ni\alpha\text{ is magic }\Leftrightarrow 1/\alpha\in\operatorname{Spec}_{L^{2}_{1}}(T_{k})\Leftrightarrow 1/(3\alpha^{2})\in\operatorname{Spec}_{L^{2}_{1}}(A_{k}),

where the right-hand side depends only on $\alpha$ and the unperturbed operator $A.$

$p$	$\sigma_{p}\frac{\sqrt{3}}{\pi}$
1	$2/9$
2	$4/9$
3	$32/63$
4	$40/81$

$p$	$\sigma_{p}\frac{\sqrt{3}}{\pi}$
5	${9560}/{20007}$
6	${245120}/{527877}$
7	${1957475168}/{4337177481}$
8	${13316086960}/{30360242367}$

Table 1. Traces of

A_{k}

\sigma_{p}=\operatorname{tr}(A_{k}^{p})

, where

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma_{1}}

is not absolutely summable as

A_{k}

is not of trace-class.

We then consider a perturbation of the potentials which gives us a new operator

T_{k,\delta}=(2D_{\bar{z}}-k)^{-1}\begin{pmatrix}0&U(z)+\delta V_{+}\\ U(-z)+\delta V_{-}&0\end{pmatrix}:L^{2}_{1}\to L^{2}_{1},

with bounded potentials $V_{\pm}\in C^{\infty}({\mathbb{C}}/\Gamma)$ and $\delta>0$ where $V_{\pm}$ satisfies the same symmetries as $U(\pm\bullet)$ , respectively, cf. (1.2). By squaring the operator, similar to (LABEL:eq:Tk), we define

T_{k,\delta}^{2}=:3\operatorname{diag}(A_{k,\delta},B_{k,\delta})

(2.2)

such that $\operatorname{Spec}_{L^{2}_{1}}(T_{k,\delta}^{2})\setminus\{0\}=3\operatorname{Spec}_{L^{2}_{1}}(A_{k,\delta})\setminus\{0\}=3\operatorname{Spec}_{L^{2}_{2}}(B_{k,\delta})\setminus\{0\}.$

To describe the spectral (in)-stability of non-normal operators one resorts to the pseudospectrum, see also the book [ET05].

Definition 2.1.

Let $P$ be a bounded linear operator. We denote the $\varepsilon$ -pseudospectrum of $P$ , for every $\varepsilon>0$ , by

\operatorname{Spec}_{\varepsilon}(P):=\bigcup_{K\in L(H);\|K\|\leq\varepsilon}\operatorname{Spec}(P+K),

(2.3)

with $L(H)$ the space of bounded linear operators. It is equivalently characterized by

\operatorname{Spec}_{\varepsilon}(P):=\operatorname{Spec}(P)\cup\{z\notin\operatorname{Spec}(P);\|(z-P)^{-1}\|>1/\varepsilon\}.

2.1. Stability of magic angles

In order to study the stability of small magic angles, characterized by the eigenvalues of $A_{k}$ ( $\alpha$ is magic if and only if $(3\alpha^{2})^{-1}\in\operatorname{Spec}_{L^{2}_{1}}(A_{k})$ ), we start with a resolvent bound and recall the definition of the regularized determinant for a Hilbert-Schmidt operator $T$ [Si77]

\det_{2}(1+T):=\prod_{\lambda\in\operatorname{Spec}(T)}(1+\lambda)e^{-\lambda}.

The following estimate is non-trivial, as the operator $A_{k}$ is non-normal:

Lemma 2.2.

Let $A=A_{0}$ be as above, then for $\alpha\in{\mathbb{C}}$ such that $1\notin\operatorname{Spec}_{L^{2}_{1}}(3\alpha^{2}A)$

\|(1-3\alpha^{2}A)^{-1}\|\leq 1+\frac{e^{(6|\alpha|^{2}+e)}}{|\det_{2}(1-3\alpha^{2}A)|}.

Before stating the proof of this lemma we state a perturbation estimate that limits by how much the eigenvalues of $A_{\delta}$ can spread. This bound is illustrated in Fig. 2.

Theorem 6.

Let $A:=A_{0}$ , as in (LABEL:eq:Tk), with $\sqrt{\operatorname{Spec}(A)}\setminus\{0\}$ the magic angles, and define $A_{\delta}:=A_{0,\delta}$ as in (2.2). The perturbed operator $A_{\delta}$ does not have any eigenvalues $1/(3\alpha^{2})$ with $\alpha\in{\mathbb{C}}\setminus\{0\}$ as long as the size of the perturbation satisfies

\|A_{\delta}-A\|\leq\frac{|\det_{2}(1-3\alpha^{2}A)|}{3|\alpha^{2}|\Big{(}|\det_{2}(1-3\alpha^{2}A)|+e^{(6|\alpha|^{2}+e)}\Big{)}}.

(2.4)

Before stating the proof of this result, we shall briefly discuss the interpretation of (2.4). The right hand side of (2.4) is small for large $|\alpha|$ , i.e. small twisting angles as well as for $1/(3\alpha^{2})$ close to $\operatorname{Spec}_{L^{2}_{1}}(A)$ , i.e. for $\alpha$ that are almost magic. This means that for such $\alpha$ even small perturbations of the potential may generate eigenvalues of the form $1/(3\alpha^{2})$ of the perturbed operator $A_{\delta}$ . This shows that such $\alpha$ are inherently unstable, as small perturbations can generate and destroy them. Conversely, for large twisting angles, i.e. small $\alpha,$ it is in general impossible to generate spectrum $1/(3\alpha^{2})$ of the perturbed operator. In particular, this bound implies a spectral stability for small $\alpha$ , i.e. large magic angles, since they cannot move by much. The regularized determinant in (2.4) can be controlled (from above and below) by Lemma 2.3.

Proof of Theo. 6.

On the one hand by (2.3), we find

\operatorname{Spec}_{1}(3\alpha^{2}A_{\delta})\subset\operatorname{Spec}_{1,3\|\alpha^{2}(A_{\delta}-A)\|}(3\alpha^{2}A).

This implies that if $1\in\operatorname{Spec}_{1}(3\alpha^{2}A_{\delta}),$ then by the characterization of the pseudo-spectrum and Lemma 2.2

\frac{1}{3\|\alpha^{2}(A_{\delta}-A)\|}\leq\|(1-3\alpha^{2}A)^{-1}\|\leq 1+\frac{e^{(6|\alpha|^{2}+e)}}{|\operatorname{det}_{2}(1-3\alpha^{2}A)|}.

Rearranging this estimate implies the result. ∎

We now give the proof of the auxiliary Lemma 2.2.

Proof of Lemma 2.2.

We recall from [Si77, Theo. $5.1$ ] that together with $\|S^{*}S\|_{1}\leq\|S\|_{2}^{2}$ for $S$ a Hilbert-Schmidt operator and $K$ a trace-class operator with $\nu=1+e^{1/2}$

|\det_{2}(1+S+K)|\leq e^{\frac{\|S\|_{2}^{2}}{2}+\nu\|K\|_{1}}.

(2.5)

Assuming $1+S$ is invertible and $S$ a finite rank operator, we have for the usual determinant

\begin{split}\det(1+S+\mu K)&=\det(1+S)\det(1+\mu(1+S)^{-1}K)\\ &=\det(1+S)(1+\mu\operatorname{tr}((1+S)^{-1}K))+\mathcal{O}(\mu^{2}).\end{split}

This shows that

\partial_{\mu}|_{\mu=0}\det(1+S+\mu K)=\det(1+S)\operatorname{tr}((1+S)^{-1}K))

which shows

\partial_{\mu}|_{\mu=0}\log\det(1+S+\mu K)=\operatorname{tr}((1+S)^{-1}K)).

Using that

\log\det_{2}(1+S+\mu K)=\log\det(1+S+\mu K)-\operatorname{tr}(S+\mu K),

(2.6)

we find the log-derivative of the regularized 2-determinant

\partial_{\mu}|_{\mu=0}\log(\det_{2}(1+S+\mu K))=\operatorname{tr}((1+S)^{-1}K)-\operatorname{tr}(K).

By using a density argument it follows that this formula also holds for $S$ Hilbert-Schmidt, i.e. we can drop the assumption that $S$ is of finite rank. Thus, one finds from (2.6) by specializing to $K=\langle\phi,\bullet\rangle\psi$ , with $\|\phi\|=\|\psi\|=1$ and multiplying by $\det_{2}(1+S)$

\det_{2}(1+S)\langle\phi,(1+S)^{-1}\psi\rangle=\partial_{\mu}\Big{|}_{\mu=0}\det_{2}(1+S+\mu K)-\det_{2}(1+S)\langle\phi,\psi\rangle.

Hence, using a Cauchy estimate $|\partial_{\mu}|_{\mu=0}f(\mu)|\leq\sup_{|\mu|=1}|f(\mu)|$ for $f(\mu):=\det_{2}(1+S+\mu K)$ , we find

\begin{split}\|\det_{2}(1+S)(1+S)^{-1}\|&\leq\sup_{|\mu|=1}|\det_{2}(1+S+\mu K)|+|\det_{2}(1+S)|\end{split}

it thus follows together with (2.5) that

\begin{split}\|(1+S)^{-1}\|&\leq 1+\sup_{|\mu|=1}\frac{|\det_{2}(1+S+\mu K)|}{|\det_{2}(1+S)|}\leq 1+\frac{e^{\|S\|_{2}^{2}/2+\nu}}{|\det_{2}(1+S)|}\end{split}.

Specializing the estimate to $S=-3\alpha^{2}A$ , we find by using that $\|A_{0}\|_{2}\leq 2$ , see [BHZ22, Lemma $4.1$ ] and $\nu<e$ , see [Si77],

\|(1-3\alpha^{2}A)^{-1}\|\leq 1+\frac{e^{(6|\alpha|^{2}+e)}}{|\det_{2}(1-3\alpha^{2}A)|}

which was to be shown. ∎

Consequently, if $\alpha_{*}$ is a magic angle, we can estimate $\det_{2}(1-3\alpha^{2}A_{0})$ in (2.4) by using [BHZ22b, Lemma $5.1$ ], which in a reduced version states that

Lemma 2.3.

The entire function ${\mathbb{C}}\ni\alpha\mapsto\operatorname{det}_{2}(1-3\alpha^{2}A_{k})$ satisfies for any $n\geq 0$

\begin{split}\left\lvert\operatorname{det}_{2}(1-3\alpha^{2}A_{k})-\sum_{k=0}^{n}\mu_{k}\frac{(-3)^{k}\alpha^{2k}}{k!}\right\rvert&\leq\sum_{j=n+1}^{\infty}\Bigg{(}\frac{\sqrt{e}\|A_{0}\|_{2}3|\alpha|^{2}}{\sqrt{j}}\Bigg{)}^{j}\end{split}

with $\|A_{0}\|_{2}\leq 2$ , where

{\mu}_{j}=\operatorname{det}\begin{pmatrix}0&j-1&0&\cdots&0\\ \sigma_{2}&0&j-2&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ \sigma_{j-1}&\sigma_{j-2}&\cdots&0&1\\ \sigma_{j}&\sigma_{j-1}&\sigma_{j-2}&\cdots&0\end{pmatrix},\text{ with }\sigma_{j}=\operatorname{tr}A_{k}^{j}.

(2.7)

The first few traces $\sigma_{j}$ are summarized in Table 1.

2.2. Instability of magic angles

We shall now give the proof of Theorem 1. Arbitrary low-lying eigenvalues of $T_{k}$ , which correspond to large magic angles in the unperturbed case, can be produced by rank $1$ perturbations of $T_{k}$ that are exponentially small in the spectral parameter. Let $\mu$ be one such low-lying eigenvalue of $T_{k}$ . On the Hamiltonian side, this indicates that zero modes with quasi-momentum $k$ and $\alpha=1/\mu$ can be generated by rank one perturbations of the Bloch-Floquet Hamiltonian, $H_{k}(\alpha)$

Proof of Theo. 1.

We recall that by [Be*22, Theo $4$ ] there exists for each $k\in\mathbb{C}$ an $L^{2}$ -normalized $u_{\mu}\in C_{c}^{\infty}({\mathbb{C}};{\mathbb{C}}^{2})$ such that the operator

P(\mu)=\begin{pmatrix}2\mu D_{\bar{z}}&U(z)\\ U(-z)&2\mu D_{\bar{z}}\end{pmatrix}

satisfies $\|(P(\mu)-\mu k)u_{\mu}\|=\mathcal{O}(e^{-c/|\mu|})$ with $\|u_{\mu}\|_{L^{2}}=1$ and $c>0.$ This implies that there is a constant $K>0$ , which we allow to change throughout this proof, such that $\|(P(\mu)-\mu k)^{-1}\|\geq Ke^{c/|\mu|}.$ Hence, we define the normalized $v_{\mu}:=\frac{(P(\mu)-\mu k)u_{\mu}}{\|(P(\mu)-\mu k)u_{\mu}\|},$ then $\|(P(\mu)-\mu k)^{-1}v_{\mu}\|>Ke^{c/|\mu|}.$ We recall that

(P(\mu)-\mu k)^{-1}=-(T_{k}-\mu)^{-1}(2D_{\bar{z}}-k)^{-1}.

This implies that, since $\|(2D_{\bar{z}}-k)^{-1}\|=1/d(k,\Gamma^{*})$ , where $d$ denotes the Hausdorff distance

\left\lVert(T_{k}-\mu)^{-1}\right\rVert\geq Ke^{c/|\mu|}.

Hence, for the normalized $s_{\mu}:=\frac{(2D_{\bar{z}}-k)^{-1}v_{\mu}}{\|(2D_{\bar{z}}-k)^{-1}v_{\mu}\|}$ , we have

(T_{k}-\mu)^{-1}s_{\mu}=t_{\mu}\text{ with }\|t_{\mu}\|\geq Ke^{c/|\mu|}.

Thus, we can define $R\varphi:=\frac{\langle\varphi,t_{\mu}\rangle}{\|t_{\mu}\|^{2}}s_{\mu}$ with norm $\|R\|=\mathcal{O}(e^{-c/|\mu|})$ such that

\mu\in\operatorname{Spec}(T_{k}-R).

∎

3. Integrated DOS and Wegner estimate

In this section we prove Theorem 2 by stating a proof of Prop. 1.3, i.e. study the regularity of the integrated density of states and prove a corresponding estimate on the number of eigenvalues of the disordered Hamiltonian. This then also implies a Wegner estimate by (4.2). We start by giving the proof of Hölder continuity, which uses the spectral shift function, see [CHK07, CHK03, CHN01], and then subsequently explain the modifications to obtain Lipschitz continuity, which uses spectral averaging. In the following, we will write $\chi_{x,L}:=\operatorname{1\hskip-2.75ptl}_{\Lambda_{L}}(x)$ with $\chi_{x}:=\chi_{x,1}$ , $\Lambda_{L}:={\mathbb{C}}/(L\Gamma)$ , and $\Lambda_{L}(z):=z+\Lambda_{L}$ and often drop subscripts to simplify the notation. Let $A$ be a compact operator then $\|A\|_{k}$ denotes the $k$ -th Schatten class norm.

3.1. Proof of Prop. 1.3

In this subsection we shall give the proof of Prop. 1.3, up to two crucial estimates that are provided in different subsections, namely the Hölder estimate (3.13) in Subsection 3.2 and the Lipschitz estimate (3.15) in Subsection 3.3.

Proof of Prop. 1.3.

In the proof, we shall focus on Case 1 disorder as Case 2 disorder follows along the same lines but more care is needed since the potential $u$ is not positive in Case 1. We shall emphasize the differences of the two cases in our proof. Since the spectrum in Case 1 exhibits a spectral gap, see (1.14), we may focus without loss of generality on the spectrum around $m$ . The argument around $-m$ is analogous. In Case 2, we do not have to restrict ourselves to those neighborhoods. Let $E_{0}\in\Delta\subset\tilde{\Delta}\subset(k_{-},k_{+})$ for two closed bounded intervals $\Delta,\tilde{\Delta}$ , with $\Delta$ of non-empty interior, centered at $E_{0}$ and $d_{0}:=d(E_{0},{\mathbb{R}}\setminus\tilde{\Delta})$ . We decompose

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))=\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}))+\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})).

(3.1)

We then write for the second term in (3.1)

\begin{split}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))&=\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})(H_{\lambda,\Lambda_{L}}-E_{0})(H_{0,\Lambda_{L}}-E_{0})^{-1}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))\\ &-\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\lambda V_{\omega,\Lambda_{L}}(H_{0,\Lambda_{L}}-E_{0})^{-1}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})).\end{split}

(3.2)

The first term in (3.2) satisfies by Hölder’s inequality and the definition of $d_{0}$

|\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})(H_{\lambda,\Lambda_{L}}-E_{0})(H_{0,\Lambda_{L}}-E_{0})^{-1}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))|\leq\frac{|\Delta|}{2d_{0}}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})).

We then use the inequality

\begin{split}&\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\lambda V_{\omega,\Lambda_{L}}(H_{0,\Lambda_{L}}-E_{0})^{-1}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))\\ &\leq\frac{|\lambda|}{d_{0}}\|\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})V_{\omega,\Lambda_{L}}\|_{2}\|\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\|_{2}\\ &\leq\frac{\zeta\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{2d_{0}}+\frac{\lambda^{2}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})V_{\omega,\Lambda_{L}}^{2})}{2\zeta d_{0}},\end{split}

with $\zeta>0$ . We can then bound (3.2), in terms of

\tilde{V}_{\omega,\Lambda_{L}}:=\sum_{\gamma\in\tilde{\Lambda}_{L}}u(\bullet-\gamma-\xi_{\gamma}),

by choosing $\zeta>0$ sufficiently small

\begin{split}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))&\leq\frac{|\Delta|}{d_{0}}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))+\frac{\lambda^{2}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})V_{\omega,\Lambda_{L}}^{2})}{\zeta d_{0}}\\ &\lesssim\frac{|\Delta|}{d_{0}}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))+\frac{\lambda^{2}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}})}{\zeta d_{0}}.\end{split}

(3.3)

Notice that while we do not have that $V_{\omega,\Lambda_{L}}^{2}\lesssim\tilde{V}_{\omega,\Lambda_{L}}$ , at least in case (1), since $\tilde{V}_{\omega,\Lambda_{L}}$ is not positive, we still have that

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})V_{\omega,\Lambda_{L}}^{2})\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}).

(3.4)

To see this, observe that

\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{0,\Lambda_{L}})=P_{\ker(D(\alpha)_{\Lambda_{L}})}\oplus 0.

(3.5)

Indeed, since $\Lambda_{L}={\mathbb{C}}/(L\Gamma)$ , we have by periodicity of the Hamiltonian $H_{0}$ that $\operatorname{Spec}(H_{0,\Lambda_{L}})\subset\operatorname{Spec}(H_{0}).$ Since the spectrum of $H_{\lambda,\Lambda_{L}}$ is uniformly gapped for $\lambda$ small, it follows that the spectral projection $\lambda\mapsto\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})$ is norm-continuous. We conclude from (3.5) that for $\varphi=(\varphi_{1},\varphi_{2})$

\varphi=\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\varphi\Rightarrow\|\varphi_{2}\|\leq\varepsilon(\lambda)\|\varphi_{1}\|.

(3.6)

with $\varepsilon(0)=0$ and $\lambda\mapsto\varepsilon(\lambda)\geq 0$ continuous. Indeed, applying norms to (3.6), we find by substituting

\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})=\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{0,\Lambda_{L}})+(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})-\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{0,\Lambda_{L}}))

in (3.6) that, since $\|\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})-\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{0,\Lambda_{L}})\|=\mathcal{O}(|\lambda|)$ , there is $C>0$ such that

\sqrt{\|\varphi_{1}\|^{2}+\|\varphi_{2}\|^{2}}\leq\|P_{\ker(D(\alpha)_{\Lambda_{L}})}\varphi_{1}\|+C\lambda\sqrt{\|\varphi_{1}\|^{2}+\|\varphi_{2}\|^{2}}.

Rearranging this, we find

(1-C\lambda)\sqrt{\|\varphi_{1}\|^{2}+\|\varphi_{2}\|^{2}}\lesssim\|P_{\ker(D(\alpha)_{\Lambda_{L}})}\varphi_{1}\|\leq\|\varphi_{1}\|

which implies that, since

(1-C\lambda)\sqrt{1+\|\varphi_{2}\|^{2}/\|\varphi_{1}\|^{2}}\leq 1\text{ that }\|\varphi_{2}\|\lesssim\lambda/(1-C\lambda)\|\varphi_{1}\|

showing (3.6). This then directly implies (3.4), since in the notation of (1.12)

\begin{split}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}})&=\sum_{\begin{subarray}{c}\varphi\text{ ONB of }\\ \operatorname{ran}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\end{subarray}}\Big{(}\langle\varphi_{1},Y\varphi_{1}\rangle-\langle\varphi_{2},Y\varphi_{2}\rangle+2\operatorname{Re}(\langle\varphi_{1},Z\varphi_{2}\rangle)\Big{)}\\ &\geq\sum_{\begin{subarray}{c}\varphi\text{ ONB of }\\ \operatorname{ran}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\end{subarray}}\Big{(}\|\varphi_{1}\|^{2}\inf Y-\|\varphi_{2}\|^{2}\sup Y-\|\varphi_{1}\|\|\varphi_{2}\|\sup Z\Big{)}\\ &\gtrsim\sum_{\begin{subarray}{c}\varphi\text{ ONB of }\\ \operatorname{ran}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\end{subarray}}\|\varphi_{1}\|^{2}(\inf Y-\lambda^{2}\sup Y-\lambda\sup Z)\\ &\gtrsim\sum_{\begin{subarray}{c}\varphi\text{ ONB of }\\ \operatorname{ran}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\end{subarray}}\|\varphi_{1}\|^{2}\inf Y\text{ for }\lambda\text{ small enough}.\end{split}

We can easily obtain, along the same lines, an upper bound on the left-hand side of (3.4)

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})V_{\omega,\Lambda_{L}}^{2})\lesssim\sum_{\begin{subarray}{c}\varphi\text{ ONB of }\\ \operatorname{ran}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\end{subarray}}\|\varphi_{1}\|^{2}\sup(V_{\omega,\Lambda_{L}}^{2})_{11}

to see that (3.4) holds.

Finally, we have for the first term in (3.1)

\begin{split}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}&(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}))\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\\ &\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}(\operatorname{1\hskip-2.75ptl}-\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})))\\ &\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})(\operatorname{1\hskip-2.75ptl}-\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))\tilde{V}_{\omega,\Lambda_{L}}-\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))\\ &\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})(\tilde{V}_{\omega,\Lambda_{L}}-\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\\ &\qquad-\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})-\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))).\end{split}

(3.7)

Here, we used in the first line of (3.7) the following identity that we shall verify below

\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\lesssim\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}).

(3.8)

Since $H_{0,\Lambda_{L}}$ is the unperturbed Hamiltonian, the eigenvectors associated to spectrum in $\tilde{\Delta}$ are supported in the first two entries of the wavefunction, cf. (3.5). Let $\pi_{1}:=\operatorname{diag}(1,1,0,0)$ be the projection onto the first two entries.

We can then define another auxiliary potential $\hat{V}_{\Lambda_{L}}(z):=\inf_{\xi\in D^{\Gamma}}\sum_{\gamma\in\tilde{\Lambda}_{L}}\pi_{1}u(z-\gamma-\xi)\pi_{1}.$ Thus, one has that $0\leq\hat{V}_{\Lambda_{L}}\leq\pi_{1}\tilde{V}_{\omega,\Lambda_{L}}\pi_{1}$ . The projection onto the first two components is redundant for case $2$ disorder since $u\geq 0$ in that case.

Finally to show (3.8) it suffices to show that

\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\lesssim\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\hat{V}_{\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}).

Since $H_{0}$ is a periodic Hamiltonian with respect to any lattice $L\Gamma$ it suffices by Bloch-Floquet theory to prove the estimate for the Bloch functions of the full Hamiltonian $H_{0}.$ Indeed, let $(v_{i}(k))_{i\in I(k)}$ be the Bloch functions associated with the spectral projection $\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0}),$ where $I(k)$ is the set of Bloch eigenvalues inside $\tilde{\Delta}$ with quasimomentum $k$ , where $H_{0,\Lambda_{L}}$ has a finite subset (in $k$ ) of those as eigenvectors. It then suffices to show that $M(k):=(\langle v_{i}(k),\hat{V}_{\Lambda_{L}}v_{j}(k)\rangle_{L^{2}({\mathbb{C}})})_{i,j}$ is strictly positive definite for all $k$ . If not, then there is $k_{0}\in{\mathbb{C}}$ and $w(k_{0}):=\sum_{j}\beta_{j}v_{j}$ with $\beta_{j}$ not all zero, such that $M(k_{0})w(k_{0})=0$ and by strict positivity of $\hat{V}_{\Lambda_{L}}$ , see (1.12), we find $w(k_{0})|_{B_{\varepsilon}(z_{0})}\equiv 0,$ but this implies that $w\equiv 0$ by real-analyticity of $w(k_{0}),$ since $H_{0}$ is elliptic with real-analytic coefficients, which is a contradiction. Thus $M_{k}$ is a strictly positive matrix and using continuity in $k$ and compactness of ${\mathbb{C}}/\Gamma^{*}$ , we also see that $M_{k}>c_{0}>0$ for all $k$ .

For the second term in (3.7) observe that by the boundedness of the potential

|\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))|\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))

where the last term can be estimated using (3.3).

We shall now estimate the third and fourth term at the end of (3.7) for $\delta>0$ , using Young’s, the Cauchy-Schwarz inequality, and that $\|A\|_{2}=\|A^{*}\|_{2}$

\begin{split}&|\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}))|\\ &\leq\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{2\delta}+\frac{\delta}{2}\|\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\|_{2}^{2}\\ &\lesssim\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{2\delta}+\frac{\delta}{2}\|\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\|_{2}^{2}\end{split}

and similarly

\begin{split}&|\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))|\\ &\lesssim\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{2\delta}+\frac{\delta}{2}\|\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}})\|_{2}^{2}.\end{split}

Inserting the last two estimates into (3.7) and choosing $\delta>0$ small enough

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{\tilde{\Delta}}(H_{0,\Lambda_{L}}))\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}})+\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{\delta}.

Inserting this estimate into (3.1) yields

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))+\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}})+\frac{\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\operatorname{1\hskip-2.75ptl}_{{\mathbb{R}}\setminus\tilde{\Delta}}(H_{0,\Lambda_{L}}))}{\delta}.

Thus, by choosing $|\Delta|$ sufficiently small in (3.3)

\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\lesssim\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}}).

Applying expectation values and using (3.13), which we show below, we find for $q\in(0,1)$

\mathbf{E}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\lesssim\mathbf{E}\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\tilde{V}_{\omega,\Lambda_{L}})\lesssim_{q}|\Delta|^{q}|\Lambda_{L}|

(3.9)

which shows the result by using a partition of small intervals $\Delta$ covering $I$ . ∎

3.2. Spectral shift function and Hölder continuity

To obtain the Hölder estimate, used to show (3.9), we recall the definition of the spectral shift function, first. Let $H_{0}$ and $H_{1}$ be two self-adjoint operator such that $H_{1}-H_{0}$ is trace-class, then the spectral shift function is defined as, see [Y92, Ch. $8$ , Sec. $2$ , Theo. $1$ ]

\xi(\lambda,H_{1},H_{0}):=\frac{1}{\pi}\lim_{\varepsilon\downarrow 0}\operatorname{arg}\det(\operatorname{id}+(H_{1}-H_{0})(H_{0}-\lambda-i\varepsilon)^{-1}).

In particular for any $p\geq 1$ one has the $L^{p}$ bound [CHN01, Theorem $2.1$ ]

\|\xi(\bullet,H_{1},H_{0})\|_{L^{p}}\leq\|H_{1}-H_{0}\|_{1/p}^{1/p}

(3.10)

where the right-hand side is defined as the generalized Schatten norm

\|T\|_{q}=\Big{(}\sum_{\lambda\in\operatorname{Spec}(T^{*}T)}\lambda^{q/2}\Big{)}^{1/q}.

We then start by setting $\varphi(x):=\arctan(x^{n})$ , with $n\in 2\mathbb{N}_{0}+1$ sufficiently large, such that $h_{1}-h_{0}$ is trace-class, with $h_{0}:=\varphi(H_{0})$ and $h_{1}:=\varphi(H_{1})$ . Then, we have the Birman-Krein formula, see [Y92, Ch.8, Sec. $11$ , Lemma $3$ ] stating that for absolutely continuous $f$

\operatorname{tr}(f(H_{1})-f(H_{0}))=\int_{{\mathbb{R}}}\xi(\varphi(\lambda),h_{1},h_{0})\ df(\lambda).

Let $\Delta=[a,b]$ then we start by defining

s(x):=\begin{cases}0&x\leq 0\\ 3x^{2}-2x^{3}&0\leq x\leq 1\\ 1&1\leq x\\ \end{cases}

and

f_{\Delta}(t):=1-s\Bigg{(}\tfrac{t-a+\tfrac{1}{2}|\Delta|}{2|\Delta|}\Bigg{)}.

(3.11)

We observe that this function satisfies $\inf_{t\in\Delta}(-f_{\Delta}^{\prime}(t))=9/8$ .

Thus, we have for $C>0$

\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\leq-C|\Delta|f_{\Delta}^{\prime}(H_{\lambda,\Lambda_{L}})

which implies

\begin{split}\operatorname{tr}(\lambda\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))&\leq-C|\Delta|\operatorname{tr}(\lambda\tilde{V}_{\omega,\Lambda_{L}}f_{\Delta}^{\prime}(H_{\lambda,\Lambda_{L}}))\\ &=-C|\Delta|\sum_{\gamma\in\tilde{\Lambda}_{L}}\partial_{\omega_{\gamma}}\operatorname{tr}(f_{\Delta}(H_{\lambda,\Lambda_{L}})).\end{split}

Applying the expectation value to this inequality, we find by positivity of $g$ , the density of $\omega_{\gamma},$ that for $\mathbf{E}_{\gamma}$ the expectation value with respect to all random variables $(\xi_{\gamma^{\prime}})$ and all $\omega_{\gamma^{\prime}}$ apart from $\gamma^{\prime}=\gamma$

\begin{split}\mathbf{E}\operatorname{tr}(\lambda&\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\leq-\sum_{\gamma\in\tilde{\Lambda}_{L}}\mathbf{E}_{\gamma}C|\Delta|\int_{0}^{1}g(\omega_{\gamma})\partial_{\omega_{\gamma}}\mathbf{\operatorname{tr}}(f_{\Delta}(H_{\lambda,\Lambda_{L}}))\ d\omega_{\gamma}\\ &\leq-\sum_{\gamma\in\tilde{\Lambda}_{L}}\mathbf{E}_{\gamma}C|\Delta|\|g\|_{\infty}\int_{0}^{1}\partial_{\omega_{\gamma}}\mathbf{\operatorname{tr}}(f_{\Delta}(H_{\lambda,\Lambda_{L}}))\ d\omega_{\gamma}\\ &\leq-C|\Delta|\|g\|_{\infty}\sum_{\gamma\in\tilde{\Lambda}_{L}}\mathbf{E}_{\gamma}\operatorname{tr}(f_{\Delta}(H_{\lambda,\Lambda_{L}}(\omega_{\gamma}=1))-f_{\Delta}(H_{\lambda,\Lambda_{L}}(\omega_{\gamma}=0)))\\ &=C|\Delta|\|g\|_{\infty}\sum_{\gamma\in\tilde{\Lambda}_{L}}\int_{\operatorname{supp}(f_{\Delta})}f_{\Delta}^{\prime}(t)\mathbf{E}_{\gamma}\xi(\varphi(t),\varphi(H_{\lambda,\Lambda_{L}}(\omega_{\gamma}=1)),\varphi(H_{\lambda,\Lambda_{L}}(\omega_{\gamma}=0)))\ dt,\end{split}

(3.12)

where $H_{\lambda,\Lambda_{L}}(\omega_{\gamma}=\zeta)$ is the Hamiltonian $H_{\lambda,\Lambda_{L}}$ with $\omega_{\gamma}$ replaced by the constant $\zeta$ and $|\operatorname{supp}(f_{\Delta})|=\mathcal{O}(|\Delta|).$ Thus, using Hölder’s inequality, we find for any $\beta\in(0,1)$ with (3.10) and $n$ in the $\arctan$ regularization $\varphi$ sufficiently large²²2using $\varphi(t)-\varphi(t_{0})=\int_{t_{0}}^{t}\frac{ns^{n-1}}{1+s^{2n}}\ ds$ we can create, by choosing $n$ sufficiently large, arbitrarily large powers of the resolvent. This yields the desired trace-class condition.

\mathbf{E}\operatorname{tr}(\lambda\tilde{V}_{\omega,\Lambda_{L}}\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}}))\lesssim|\Delta|^{\beta}|\Lambda_{L}|

(3.13)

which is the identity used to obtain (3.9).

3.3. Spectral averaging and Lipschitz continuity

We now complete the proof of Lipschitz continuity and follow an argument developed initially by Combes and Hislop [CH94, Corr. $4.2$ ] for Schrödinger operators.

Proof of Theorem 2 (Lipschitz continuity).

Let $E=\max\{|E_{1}|,|E_{2}|\}$ where $\Delta=[E_{1},E_{2}],$ then

\begin{split}\mathbf{E}(\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})))&\leq e^{E^{2}}\mathbf{E}(\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})e^{-H^{2}_{\lambda,\Lambda_{L}}}))\\ &\leq e^{E^{2}}\sum_{j\in\tilde{\Lambda}_{L}}\Bigg{(}\|\mathbf{E}(\chi_{j}\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\chi_{j})\|\sup_{\omega\in\Omega}\operatorname{tr}\Big{(}\chi_{j}e^{-H^{2}_{\lambda,\Lambda_{L}}}\Big{)}\Bigg{)}\\ &\lesssim e^{E^{2}}\sum_{j\in\tilde{\Lambda}_{L}}\|\mathbf{E}(\chi_{j}\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})\chi_{j})\|,\end{split}

(3.14)

where we used that $\sup_{\omega\in\Omega}\operatorname{tr}\Big{(}\chi_{j}e^{-H^{2}_{\lambda,\Lambda_{L}}}\Big{)}$ is uniformly bounded in all parameters. Under the assumptions of Case $2$ , we know that $u_{j}$ are strictly positive on $\operatorname{supp}(\chi_{j})$ thus also $0\leq\chi_{j}^{2}\lesssim u_{j}$ which is the necessary condition [CH94, (4.2)] to apply spectral averaging which readily implies together with (3.14) that

\mathbf{E}(\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{\Delta}(H_{\lambda,\Lambda_{L}})))\lesssim|\Delta||\Lambda_{L}|

(3.15)

which is the identity (3.9) with $\beta=1$ for Case $2$ disorder. ∎

4. Mobility edge

To prove Theorem 3 we recall Germinet and Klein’s notion of summable uniform decay of correlations (SUDEC), see [GK06].

Definition 4.1 (SUDEC).

The Hamiltonian $H_{\lambda}$ is said to exhibit a.e. SUDEC in an interval $J$ if its spectrum is pure point and for every closed $I\subset J$ , for $\{\varphi_{n}\}$ an orthonormal set of eigenfunctions of $H_{\lambda}$ with eigenvalues $E_{n}\in I$ , we define $\beta_{n}:=\|\langle z\rangle^{-2}\varphi_{n}\|^{2}.$ Then for $\zeta\in(0,1)$ there is $C_{I,\zeta}<\infty$ such that

\|\chi_{z}(\varphi_{n}\otimes\varphi_{n})\chi_{w}\|\leq C_{I,\zeta}\beta_{n}\langle z\rangle^{2}\langle w\rangle^{2}e^{-|z-w|^{\zeta}}\text{ for }w,z\in{\mathbb{C}}

and in addition one has $\mathbf{P}$ -almost surely

\sum_{n\in\mathbb{N}}\beta_{n}<\infty.

(4.1)

The strategy to establish delocalization is to show that if the Hamiltonian would exhibit only SUDEC-type localization (SUDEC), then this would contradict the non-vanishing Chern numbers of the flat bands.

4.1. The ingredients to the multi-scale analysis

For the applicability of the multi-scale analysis à la Germinet-Klein we require six ingredients of our Hamiltonian often referred to in their works by the acronyms, as introduced in [GK01],

•

Strong generalized eigenfunction expansion SGEE (Lemma 4.2),
•

Simon-Lieb inequality SLI and exponential decay inequality EDI (both Lemma 4.3),
•

Number of eigenvalues estimate NE and Wegner estimate W (both (4.2) and Prop. 1.3), and
•

Independence at a distance IAD.

Here, independence at a distance (IAD) just follows from the choice of Anderson-type randomness and means that the disorder of the potentials at a certain distance are independent of each other.

We then start with the strong generalized eigenfunction expansion (SGEE). Therefore, we introduce Hilbert spaces

\mathcal{H}_{\pm}:=L^{2}({\mathbb{C}},{\mathbb{C}}^{4};\langle z\rangle^{\pm 4\nu}\ dz).

Lemma 4.2 (SGEE).

Let $\nu\geq 1/2$ . The set $D^{\omega}_{+}:=\{\phi\in D(H_{\lambda})\cap\mathcal{H}_{+};H_{\lambda}\phi\in\mathcal{H}_{+}\}$ is dense in $\mathcal{H}_{+}$ and a core of $H_{\lambda}.$ Moreover, for $\mu\in\mathbb{R}\setminus\{0\}$ we have

\mathbf{E}\Bigg{(}\operatorname{tr}\Big{(}\langle z\rangle^{-2\nu}(H_{\lambda}-i\mu)^{-4}\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda})\langle z\rangle^{-2\nu}\Big{)}^{2}\Bigg{)}<\infty.

Proof.

The statement about the core is immediate, as $C_{c}^{\infty}({\mathbb{C}};{\mathbb{C}}^{4})$ is a core, see for instance Theorem 8. The second statement follows as $\langle z\rangle^{-2\nu}(H_{\lambda}-i\mu)^{-2}$ is a uniformly bounded (in $\omega$ ) Hilbert-Schmidt operator. This is easily seen by ∎

The Simon-Lieb inequality (SLI), relating resolvents at different scales, and the eigenfunction decay inequality (EDI), relating decay of finite-volume resolvents to the decay of generalized eigenfunctions and thus Anderson localization, are discussed in the next Lemma. We thus define $\Xi_{\Lambda_{L}(z)}$ to be the characteristic function of the belt

\Upsilon_{L}(z):=\Lambda_{L-1}(z)\setminus\Lambda_{L-3}(z).

For $z\in\Gamma$ and $l>4$ , we define smooth cut-off functions $\tilde{\chi}_{\Lambda_{l}(z)}\in C_{c}^{\infty}({\mathbb{C}};[0,1])$ that are equal to one on $\Lambda_{l-3}(z)$ and $0$ on ${\mathbb{C}}\setminus\Lambda_{l-5/2}(z).$

Lemma 4.3 (SLI & EDI).

Let $J$ be a compact interval. For $L,l^{\prime},l^{\prime\prime}\in 2\mathbb{N}$ and $x,y^{\prime},y^{\prime\prime}\in\Gamma$ with $\Lambda_{l^{\prime\prime}}(y)\subsetneq\Lambda_{l^{\prime}}(y^{\prime})\subsetneq\Lambda_{L}(x)$ , then $\mathbf{P}$ -a.s.: If $E\in J\cap(\operatorname{Spec}(H_{\lambda,\Lambda_{L}(x)})\cap\operatorname{Spec}(H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}))^{c}$ then the Simon-Lieb inequality holds

\begin{split}\|\Xi_{\Lambda_{L}(x)}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}\chi_{\Lambda_{l^{\prime\prime}}(y)}\|&\lesssim_{J}\|\Xi_{\Lambda_{l^{\prime}}(y^{\prime})}(H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}-E)^{-1}\chi_{\Lambda_{l^{\prime\prime}}(y)}\|\\ &\times\|\Xi_{\Lambda_{L}(x)}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}\Xi_{\Lambda_{l^{\prime}}(y^{\prime})}\|.\end{split}

In addition, we have $\mathbf{P}$ -a.s. that any $z\in\Gamma$ , and any generalized eigenfunction $\psi$ , i.e. $\psi$ solving $(H_{\lambda}-E)\psi=0$ and growing at most polynomially, with $E\in J\cap\operatorname{Spec}(H_{\lambda,\Lambda_{L}(x)})^{c}$ one has the eigenfunction decay inequality

\begin{split}\|\chi_{z}\psi\|&\lesssim_{J}\|\Xi_{\Lambda_{L}(x)}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}\chi_{z}\|\|\Xi_{\Lambda_{L}(x)}\psi\|.\end{split}

Proof.

(1)

The proof of the SLI can be streamlined for linear differential operators with disorder of Anderson-type. We start from the following resolvent identity

\begin{split}(H_{\lambda}-E)\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}&=[(H_{\lambda}-E),\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}](H_{\lambda,\Lambda_{L}(x)}-E)^{-1}\\ &\ +\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}(H_{\lambda}-E)(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}.\end{split}

Using that by assumption $\Lambda_{l^{\prime}}(y^{\prime})\subset\Lambda_{L}(x)$ we have $\chi_{\Lambda_{l^{\prime}}(y^{\prime})}H_{\lambda}=\chi_{\Lambda_{l^{\prime}}(y^{\prime})}H_{\lambda,\Lambda_{L}(x)}$ and find by substituting $\chi_{\Lambda_{l^{\prime}}(y^{\prime})}H_{\lambda}$ in the last line above

(H_{\lambda}-E)\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}=[H_{\lambda},\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}](H_{\lambda,\Lambda_{L}(x)}-E)^{-1}+\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}.

Since $H_{\lambda}\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}=H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}$ we find by multiplying the previous line by $(H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}-E)^{-1}$ that

\begin{split}\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}(H_{\lambda,\Lambda_{L}(x)}-E)^{-1}&=(H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}-E)^{-1}[H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime}),},\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}](H_{\lambda,\Lambda_{L}(x)}-E)^{-1}\\ &\quad+(H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime})}-E)^{-1}\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}.\end{split}

Multiplying this equation from the left by $\chi_{\Lambda_{l^{\prime\prime}}(y)}$ and from the right by $\Xi_{\Lambda_{L}(x)}$ , the SLI ready follows from the boundedness of $[H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime}),},\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}]$ and submultiplicativity of the operator norm, as $\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}\Xi_{\Lambda_{L}(x)}=0$ implies that the second term on the right vanishes and

[H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime}),},\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}]=\Xi_{\Lambda_{l^{\prime}}(y^{\prime})}[H_{\lambda,\Lambda_{l^{\prime}}(y^{\prime}),},\tilde{\chi}_{\Lambda_{l^{\prime}}(y^{\prime})}]\Xi_{\Lambda_{l^{\prime}}(y^{\prime})}.

(4.1)

(2)

For the proof of the EDI, it suffices to choose $\psi$ as in the Lemma and observe the resolvent identity $(H_{\lambda,x,L}-E)^{-1}[H_{\lambda},\tilde{\chi}_{\Lambda_{L}(x)}]\psi=\tilde{\chi}_{\Lambda_{L}(x)}\psi$ which is easily verified by using that $(V_{\omega}-V_{\omega,\Lambda_{L}(x)})\tilde{\chi}_{\Lambda_{L}(x)}=0$ as well as $H_{\lambda}\psi=E\psi.$ Using then an analogue of (4.1), $[H_{\lambda},\tilde{\chi}_{\Lambda_{L}(x)}]=\Xi_{\Lambda_{L}(x)}[H_{\lambda},\tilde{\chi}_{\Lambda_{L}(x)}]\Xi_{\Lambda_{L}(x)},$ together with the boundedness of the commutator shows the claim.

∎

We complete our preparations by discussing the estimate on the number of eigenvalues (NE) and the Wegner estimate (W). The estimate on the number of eigenvalues (NE) is the estimate stated in Proposition 1.3. The Wegner estimate is then obtained by applying the estimate in Proposition 1.3 to the last expression in this set of inequalities

\begin{split}\mathbf{P}(d(\operatorname{Spec}(H_{\lambda,\Lambda_{L}}),E)<\eta)&=\mathbf{P}(\operatorname{rank}\operatorname{1\hskip-2.75ptl}_{(E-\eta,E+\eta)}(H_{\lambda,\Lambda_{L}})\geq 1)\\ &\leq\mathbf{E}(\operatorname{tr}(\operatorname{1\hskip-2.75ptl}_{(E-\eta,E+\eta)}(H_{\lambda,\Lambda_{L}}))).\end{split}

(4.2)

4.2. Dynamical delocalization

In this subsection we prove Theorem 3. To imitate the proof of delocalization in [GKS07], we shall study the third power of the random Hamiltonian (1.11), since $H^{3}(M)\hookrightarrow L^{2}(M)$ , for $M$ a two-dimensional compact manifold, is a trace-class embedding³³3Recall that $\lambda_{k}\sim_{M}k$ is the Weyl asymptotics of the negative Laplacian in dimension 2; thus $\sum_{k}k^{-3/2}<\infty$ and $x\mapsto x^{3}$ is bijective, by defining

S_{\lambda}:=H_{\lambda}^{3}.

Let $\mathcal{C}_{\pm}:=\partial B_{|\lambda|\sup_{\omega\in\Omega}\|V_{\omega}\|_{\infty}}(\pm m)$ such that $\mathcal{C}_{\pm}$ encircles the spectrum of the random perturbation of a single flat band, but nothing else (if $m=0$ , then $\mathcal{C}_{\pm}$ both coincide, we shall explain the modifications of this case at the end of this section). This is possible for sufficiently small noise $\lambda>0$ as the flat band at energies $\pm m$ are strictly gapped (1.8) from all higher bands, in the absence of disorder. We then define the $L^{2}({\mathbb{C}};\mathbb{C}^{4})$ -valued spectral projection

P_{\lambda,\pm}=-\frac{1}{2\pi i}\int_{\mathcal{C}_{\pm}^{3}}(S_{\lambda}-z)^{-1}\ dz,

(4.3)

where by $\mathcal{C}_{\pm}^{3}$ we just mean the set of elements in $\mathcal{C}_{\pm}$ raised to the third power. The delocalization argument rests on the following two pillars:

•

If the random Hamiltonian exhibits only dynamical localization close to $\pm m$ , then this implies that the partial Chern numbers of $P_{\lambda,\pm}$ , defined in section B, have to vanish, see Prop. B.2.
•

The partial Chern numbers of $P_{\lambda,\pm}$ are invariant under disorder as well as small perturbations in $\alpha$ away from perfect magic angles.

As a consequence, the Hamiltonian exhibits dynamical delocalization at energies close to $\pm m$ . To simplify the notation, we drop the $\pm$ and just focus on $+m$ , since $-m$ can be treated analogously.

The central object in this discussion is the Hall conductance. Assuming $\|P[[P,\Theta_{1}],[P,\Theta_{2}]]\|_{1}<\infty$ for a spectral projection $P$ and multiplication operators $\Theta_{1}(z):=\operatorname{1\hskip-2.75ptl}_{[1/2,\infty)}(\operatorname{Re}z)$ and $\Theta_{2}(z):=\operatorname{1\hskip-2.75ptl}_{[1/2,\infty)}(\operatorname{Im}z)$ the Hall conductance is defined by

\Omega(P):=\operatorname{tr}(P[[P,\Theta_{1}],[P,\Theta_{2}]])=\operatorname{tr}([P\Theta_{1}P,P\Theta_{2}P]).

(4.4)

Here, $\kappa=-i[P\Theta_{1}P,P\Theta_{2}P]$ is also called the adiabatic curvature with Hall charge transport $Q=-2\pi\operatorname{tr}(\kappa).$ That $Q$ is an integer is shown for example in [ASS94, Theorem $8.2$ ] or [BES94, (49),(58)] where it is related to Chern characters and Fredholm indices, respectively, and then [BES94, Theorem $1$ ] where this quantity is discussed for periodic and quasi-periodic operators.

Proof of Theo. 3.

Since $H^{3}(M)\hookrightarrow L^{2}(M)\xrightarrow[]{\text{extension}}L^{2}({\mathbb{C}})$ is a trace-class embedding, for bounded open sets $M$ , it follows that there is a universal constant $K_{1}>0$ such that for sufficiently small disorder $\lambda$ and $\mu\in\mathcal{C}^{3}$ with $\mathcal{C}^{3}$ as above in trace norm

\|(S_{\lambda}-\mu)^{-1}\chi_{z}\|_{1}\leq K_{1}\text{ for all }z\in\Gamma.

(4.5)

Next, we are going to construct an analogue of the Combes-Thomas estimate (CTE) for the operator $S_{\lambda}$ :

By conjugating the operator $S_{\lambda}$ with $e^{f}$ where $f$ is some smooth function, we find

e^{f}S_{\lambda}e^{-f}=S_{\lambda}+R_{f},

where

\|R_{f}\|_{L(H^{3},L^{2})}\lesssim\varepsilon\text{ if }\|\partial^{\beta}f\|_{\infty}\leq\varepsilon\ll 1\text{ for all }1\leq|\beta|\leq 3.

This implies that for $z\notin\operatorname{Spec}(S_{\lambda})$

e^{f}(S_{\lambda}-z)e^{-f}=(\operatorname{id}+R_{f}(S_{\lambda}-z)^{-1})(S_{\lambda}-z).

Thus, for $z\notin\operatorname{Spec}(S_{\lambda})$ and $\varepsilon>0$ sufficiently small such that $\|R_{f}(S_{\lambda}-z)^{-1}\|<1$ ,

\|e^{-f}(S_{\lambda}-z)^{-1}e^{f}\|_{L(L^{2},H^{3})}=\mathcal{O}(\langle d(\operatorname{Spec}(S_{\lambda}),z)^{-1}\rangle).

We conclude that for $f(z):=\varepsilon\langle z-w_{0}\rangle$ with $w_{0}\in{\mathbb{C}}$ fixed, we have for all $w\in{\mathbb{C}}$

\begin{split}\|\chi_{w_{0}}(S_{\lambda}-z)^{-1}\chi_{w}\|&=\|\chi_{w_{0}}e^{f}(e^{-f}(S_{\lambda}-z)^{-1}e^{f})e^{-f}\chi_{w}\|=\mathcal{O}\Big{(}\tfrac{e^{-\varepsilon\langle w-w_{0}\rangle}}{d(\operatorname{Spec}(S_{\lambda}),z)}\Big{)},\end{split}

(CTE)

as well as

\begin{split}\|\chi_{w_{0}}(S_{\lambda}-S_{0})(S_{\lambda}-z)^{-1}\chi_{w}\|&=\|\chi_{w_{0}}e^{f}\|\|e^{-f}(S_{\lambda}-S_{0})e^{f}\|_{L(H^{3},L^{2})}\\ &\quad\times\|e^{-f}(S_{\lambda}-z)^{-1}e^{f}\|_{L(L^{2},H^{3})}\|e^{-f}\chi_{w}\|\\ &=\mathcal{O}(\tfrac{\|e^{-f}\chi_{w}\|}{d(\operatorname{Spec}(S_{\lambda}),z)})=\mathcal{O}\Big{(}\tfrac{e^{-\varepsilon\langle w-w_{0}\rangle}}{d(\operatorname{Spec}(S_{\lambda}),z)}\Big{)}.\end{split}

(4.6)

From the Combes-Thomas estimate (CTE) and (4.3) we find the exponential estimate

\|\chi_{w_{0}}P_{\lambda}\chi_{w}\|\lesssim e^{-\varepsilon|w-w_{0}|}.

(4.7)

By [GKS07, Lemma $3.1$ ], this implies that

\|P_{\lambda}[[P_{\lambda},\Theta_{1}],[P_{\lambda},\Theta_{2}]]\|_{1}<\infty,

which implies that the Hall conductance is well-defined. In fact, using (4.5) we have

\begin{split}\|\chi_{w}P_{\lambda}\chi_{w_{0}}\|_{1}=\mathcal{O}(1)\text{ and }\|\chi_{w}P_{\lambda}\chi_{w_{0}}\|^{2}_{2}\leq\|\chi_{w}P_{\lambda}\chi_{w_{0}}\|_{1}\|\chi_{w}P_{\lambda}\chi_{w_{0}}\|=\mathcal{O}(e^{-\varepsilon|w-w_{0}|}).\end{split}

(4.8)

To obtain the invariance of the Chern number under small disorder, we now define

Q_{\lambda,\zeta}:=P_{\zeta}-P_{\lambda}=\frac{\zeta-\lambda}{2\pi i}\int_{\mathcal{C}^{3}}(S_{\lambda}-z)^{-1}\frac{(S_{\zeta}-S_{\lambda})}{(\zeta-\lambda)}(S_{\zeta}-z)^{-1}\ dz.

(4.9)

then by (4.8) we find

\|\chi_{w}Q_{\lambda,\zeta}\chi_{w_{0}}\|_{2}^{2}=\mathcal{O}(e^{-\varepsilon|w-w_{0}|}).

(4.10)

If the random potential has compact support, i.e. $H_{\lambda}$ in (1.11) is replaced by

H_{\lambda}(L)=H+\lambda V_{\omega}\text{ where }V_{\omega}=\sum_{\gamma\in\tilde{\Lambda}_{L}}\omega_{\gamma}u(\bullet-\gamma-\xi_{\gamma}),

(4.11)

for some $L>0$ , then by using a partition of unity and (4.5), we find $\|Q_{\lambda,\zeta}\|_{1}<\infty$ and consequently the traces of all commutators vanish

\begin{split}\Omega(P_{\zeta})-\Omega(P_{\lambda})=&\operatorname{tr}([Q_{\lambda,\zeta}\Theta_{1}P_{\zeta},P_{\zeta}\Theta_{2}P_{\zeta}]+[P_{\lambda}\Theta_{1}Q_{\lambda,\zeta},P_{\zeta}\Theta_{2}P_{\zeta}]\\ &+[P_{\lambda}\Theta_{1}P_{\lambda},Q_{\lambda,\zeta}\Theta_{2}P_{\zeta}]+[P_{\lambda}\Theta_{1}P_{\lambda},P_{\lambda}\Theta_{2}Q_{\lambda,\zeta}])=0.\end{split}

(4.12)

So the integer-valued map $\lambda\mapsto\Omega(P_{\lambda})$ is constant for $\lambda$ small around zero, under the assumption of a compactly supported random potential in (4.11).

It remains now to drop the compact support constraint on the random potential in (4.11). Let $S_{\lambda}(L)=H_{\lambda}(L)^{3}$ , then we define

Q_{\lambda,>L}:=P_{\lambda}-P_{\lambda}(L)=\frac{\lambda}{2\pi i}\int_{\mathcal{C}^{3}}(S_{\lambda}-z)^{-1}(S_{\lambda}-S_{\lambda}(L))(S_{\lambda}(L)-z)^{-1}\ dz,

(4.13)

where $P_{\lambda}(L)$ is the corresponding spectral projection associated with $S_{\lambda}(L).$ By the Combes-Thomas estimates (CTE) and the resolvent identity (4.13), we find

\|\chi_{w}Q_{\lambda,>L}\chi_{w_{0}}\|\lesssim e^{-c\varepsilon((L-R-|w|)_{+}+(L-R-|w_{0}|)_{+}+|w-w_{0}|)}

for some $c>0$ , where we used that $S_{\lambda}-S_{\lambda}(L)$ is zero on $\Lambda_{L-R}(0).$ Thus, writing the difference of Hall conductivities yields the desired limit

\begin{split}\Omega(P_{\lambda})-\Omega(P_{\lambda}(L))=&\operatorname{tr}(Q_{\lambda,>L}[[P_{\lambda},\Theta_{1}],[P_{\zeta},\Theta_{2}]]+P_{\lambda,L}[[Q_{\lambda,>L},\Theta_{1}],[P_{\lambda},\Theta_{2}]]\\ &+P_{\lambda,L}[[P_{\lambda},\Theta_{1}],[Q_{\lambda,>L},\Theta_{2}]])\to 0\text{ as }L\to\infty.\end{split}

(4.14)

Here, one uses the strong limit $s-\lim_{L\to\infty}Q_{\lambda,>L}=0$ to show the non-vanishing of the first term on the right-hand side in (4.14) and that

|\operatorname{tr}(P_{\lambda,L}[[Q_{\lambda,>L},\Theta_{1}],[P_{\lambda},\Theta_{2}]])|\leq 2\sum_{\gamma,\gamma^{\prime}\in\Gamma}\|\chi_{\gamma}[Q_{\lambda,>L},\Theta_{1}]\chi_{\gamma^{\prime}}\|_{2}\|\chi_{\gamma^{\prime}}[P_{\lambda},\Theta_{2}]\chi_{\gamma}\|_{2}

with a similar estimate for the last term in (4.14). The last bound converges to zero for $L\to\infty$ by using (4.8) and (4.10), see [GKS07, Lemma 3.1 (i)] for details. Thus, the conductivity derived from $P_{\lambda}$ is locally constant in $\lambda$ and $\alpha$ , see (4.12), which shows using (1.10) that Chern numbers stay $\pm 1$ , for $m>0$ , respectively.

For $m=0$ we repeat the previous computation with our modified $\Omega_{i}$ (B.3) to arrive at the same conclusion. Thus, if, in the notation of (1.14), $\Sigma\cap(-K_{-},K_{-})\subset\Sigma^{\operatorname{DL}}$ then this would contradict the non-vanishing of the (partial) Chern number, see (B.6), in regions of full localization as shown in Prop. B.2.

The bound in the statement of Theorem 3 follows then from [GK04, Theo $2.10$ ].

∎

4.3. Dynamical localization

Working under assumptions (1), we shall now study the localized phase of the Anderson model of the form

H_{\lambda}=H+V_{\omega}\text{ where }V_{\omega}=\sum_{\gamma\in\Gamma}\omega_{\gamma}u(\bullet-\gamma-\xi_{\gamma}).

(4.15)

Here we got rid of the $\lambda$ parameter but instead consider random variables $\omega_{\gamma}$ that are distributed according to a bounded density $g_{\lambda}$ of compact support in $[-\delta,\delta]$ with $\delta<\min(m,E_{\operatorname{gap}})$ for $m>0$ and $\delta<E_{\operatorname{gap}}$ for $m=0$ . Here $g_{\lambda}$ is a rescaled distribution $g_{\lambda}(u)=c_{\lambda}g(u/\lambda)/\lambda\operatorname{1\hskip-2.75ptl}_{[-\delta,\delta]}$ , with $g>0$ , such that as $\lambda\downarrow 0$ the mass gets concentrated near zero and $c_{\lambda}\leq C$ , uniformly in $\lambda$ , is the normalizing constant.

By (1.13), with probability $1$ , the spectrum $\Sigma$ is independent of $\lambda.$ Our next theorem shows that the mobility edges can be shown to be located arbitrarily close to the original flat bands, by choosing $\lambda$ small, while keeping the support of the disorder fixed, within the interval $[-\delta,\delta]$ which is the motivation for our modifications of the Hamiltonian.

Theorem 7 (Mobility edge).

Let $\langle\bullet\rangle^{\gamma}g$ be bounded for some $\gamma>3$ and let $\tau\in(0,\tfrac{\gamma-3}{\gamma+1})$ . Let $H_{\lambda}$ be as in Assumption 1 with the modification that $\lambda$ is incorporated in the rescaled density, as described in (4.15) and $D\subset\mathbb{C}$ small enough. Then for any $m>0$ there exist at least two distinct dynamical mobility edges, denoted by $\mathscr{E}_{+}(\lambda)>\mathscr{E}_{-}(\lambda)$ such that

\left\lvert\mathscr{E}_{+}(\lambda)-m\right\rvert+\left\lvert\mathscr{E}_{-}(\lambda)+m\right\rvert\lesssim\lambda^{1-\frac{4}{\gamma+1}-\tau}\xrightarrow[\lambda\downarrow 0]{}0.

In particular,

\left\{E\in\Big{(}-\sqrt{E_{\operatorname{gap}}^{2}/2+m^{2}},\sqrt{E_{\operatorname{gap}}^{2}/2+m^{2}}\Big{)};|E\pm m|\gtrsim\lambda^{1-\frac{4}{\gamma+1}-\tau}\right\}\subset\Sigma^{\operatorname{DL}},

where the region of dynamical localization $\Sigma^{\operatorname{DL}}$ has been defined in (1.18). In the case of $m=0$ the same result holds but with only at least one guaranteed mobility edge.

Proof.

We start by observing that using the $L^{\infty}$ bound on $\langle\bullet\rangle^{\gamma}g$ , we have for any $\varepsilon>0$ and $X\sim g_{\lambda}$

\begin{split}\mathbf{P}(|X|\geq\varepsilon)&=\int_{\delta\geq|x|\geq\varepsilon}g_{\lambda}(x)\ dx\lesssim\int_{\delta/\lambda\geq|x|\geq\varepsilon/\lambda}g(x)\ dx\lesssim\langle\lambda/\varepsilon\rangle^{\gamma-1}.\end{split}

(4.16)

Thus, for the probability of the low-lying spectrum to be contained in a small interval $[-\varepsilon,\varepsilon]$ , we find for $L_{0}\gg 1$ fixed

\begin{split}\mathbf{P}&\Big{(}\operatorname{Spec}(H_{\lambda,\Lambda_{L_{0}}})\cap\Big{(}-\sqrt{E_{\operatorname{gap}}^{2}/2+m^{2}},\sqrt{E_{\operatorname{gap}}^{2}/2+m^{2}}\Big{)}\subset\pm m+[-\varepsilon,\varepsilon]\Big{)}\\ \overset{\text{union bound}}{\geq}&\mathbf{P}(|\omega_{\gamma}|\leq\varepsilon/2;\gamma\in\tilde{\Lambda}_{L_{0}+R}(x))\\ \overset{\eqref{eq:flambda}}{\geq}&(1-C(\lambda/\varepsilon)^{\gamma-1})^{(L_{0}+R)^{2}}\\ \overset{\text{Bernoulli}}{\geq}&1-C(\lambda/\varepsilon)^{\gamma-1}L_{0}^{2}\text{ for small }\lambda/\varepsilon,\end{split}

for small enough $\lambda/\varepsilon$ . We recall that $R>0$ is such that $\operatorname{supp}u\subset\Lambda_{R}(0).$ This probability is large, if we choose

\varepsilon=C\lambda L_{0}^{\frac{2}{\gamma-1}}

(4.17)

for $C\gg 1.$ To prove localization, one chooses $L_{0}\gg 1$ large enough, as specified in [GK03, (2.16)] and $\lambda\ll 1$ . We now fix an energy $\sqrt{E_{\operatorname{gap}}^{2}/2+m^{2}}\geq|E|$ such that $|E\pm m|\geq 2\varepsilon$ with $E\in\Sigma$ . Then $E$ is, with high probability, a distance $\varepsilon>0$ away from the spectrum of the finite-size Hamiltonian $H_{\lambda,\Lambda_{L_{0}}}$ .

In order to show localization, we shall satisfy the finite-size criterion of [GK03, Theorem $2.4$ ]. This will give us another condition aside from (4.17). Indeed, in our setting the finite-size criterion stated in [GK03, Theorem $2.4$ ] takes the following form

\frac{C_{1}L_{0}^{25/3}}{\lambda\varepsilon}e^{-C_{2}\varepsilon L_{0}}<1

(4.18)

for two constants $C_{1},C_{2}>0.$ The term $L_{0}^{25/3}$ is obtained from [GK03, Theorem $2.4$ ] by choosing (in the notation of [GK03]) $b=1,d=2$ , and performing a union bound over a partition of $\Gamma_{0}$ and $\chi_{0,L_{0}/3}$ which accounts for another $L_{0}^{3}.$ The $\lambda$ in the denominator is due to the scaling of the constant in the Wegner estimate which for us is proportional to the supremum norm of the density, which for us is $\|g_{\lambda}\|_{\infty}=\mathcal{O}(1/\lambda)$ .

By [GK03, Theorem $2.4$ ] one concludes localization if both (4.17) and (4.18) hold.

Setting then $\varepsilon:=C_{3}\lambda L_{0}^{\frac{2}{\gamma-1}}$ with $C_{3}>0$ sufficiently large to satisfy (4.17), we find that (4.18) becomes

\frac{C_{1}L_{0}^{25/3}}{C_{3}\lambda^{2}L_{0}^{\frac{2}{\gamma-1}}}e^{-C_{2}C_{3}\lambda^{2}L_{0}^{\frac{\gamma+1}{\gamma-1}}}<1.

We now also set $L_{0}^{\frac{2}{\gamma-1}}=\lambda^{-\frac{4}{\gamma+1}-\tau}$ with $\tau(\gamma)>0$ small such that $-\frac{4}{\gamma+1}-\tau>-1.$ This means that $L_{0}^{\frac{\gamma+1}{\gamma-1}}=\lambda^{-2-\tau\frac{\gamma+1}{2}}$ which implies that for $\lambda$ small enough, (4.18) holds as well.

The characterization of the localized regime then follows from [GK03, Theorem $2.4$ ], the existence of a mobility edge follows together with Theorem 3. ∎

5. Decay of point spectrum and Wannier bases

We now give the proof of Theo. 4 and 5. We shall mainly focus on the first case and only explain the modifications for the second result at the very end.

Proof of Theo.4 & 5.

We first reduce the analysis to $\lambda=0.$ By $\lambda$ -continuity of the random perturbation, the spectral projections $P_{0}=\operatorname{1\hskip-2.75ptl}_{J_{\pm}}(H_{0})$ and $P_{\lambda}=\operatorname{1\hskip-2.75ptl}_{J_{\pm}}(H_{\lambda})$ with $J_{\pm}$ as in (1.15)

\|P_{0}-P_{\lambda}\|=\mathcal{O}(|\lambda|)

by using e.g. the resolvent identity and holomorphic functional calculus and the spectral gap of the Hamiltonian. Thus, for $\lambda$ small enough there is an isometry [BES94, Lemma $10$ ] $U$ such that $UU^{*}=P_{0}$ and $U^{*}U=P_{\lambda}.$ In particular $P_{0}U=UP_{\lambda}.$ It then follows that $U$ has a Schwartz kernel $K$ that is exponentially close to the identity, cf. [CMM19, Lemma $8.5$ ]. By this we mean that there is $\gamma>0$ such that

|K(z,z^{\prime})-1|=\mathcal{O}(e^{-\gamma|z-z^{\prime}|}).

The Schur test for integral operators implies that $\tilde{U}:=\langle\bullet-z_{0}\rangle U\langle\bullet-z_{0}\rangle^{-1}$ is a family of operators uniformly bounded in $z_{0}\in{\mathbb{C}}$ . This implies that for any $\varphi\in L^{2}({\mathbb{C}};{\mathbb{C}}^{4})$

\langle\bullet-z_{0}\rangle^{1+\delta}P_{0}U\varphi=\langle\bullet-z_{0}\rangle^{1+\delta}U\langle\bullet-z_{0}\rangle^{-1-\delta}\langle\bullet-z_{0}\rangle^{1+\delta}P_{\lambda}\varphi.

Taking norms, we find, using that $\|\langle\bullet-z_{0}\rangle^{1+\delta}P_{\lambda}\varphi\|<\infty$ by assumption, that

\|\langle\bullet-z_{0}\rangle^{1+\delta}P_{0}U\varphi\|<\infty.

This implies, by choosing for $U\varphi$ an orthonormal basis of $\overline{\operatorname{ran}}(P_{0})$ , i.e. $(\psi_{n})$ is an orthonormal basis of $\overline{\operatorname{ran}}(P_{0})$ , then $\varphi_{n}:=U^{*}\psi_{n}$ , that $P_{0}$ exhibits a $(1+\delta)$ -localized generalized Wannier basis. Since $P_{0}$ is precisely the projection onto $\ker(D(\alpha))$ , we deduce that $P_{0}$ exhibits a non-zero Chern number, see (1.10), and therefore do not possess a $(1+\delta)$ -localized Wannier basis, see [LS21] which gives a contradiction.

Conversely, let $P_{k}(\alpha)=(2\pi i)^{-1}\oint_{\gamma}(z-H_{k}(0,\alpha))^{-1}\ dz$ , where $\gamma$ is a sufficiently small circle around zero encircling only the flat band eigenvalue but nothing else in the spectrum of $H_{k}(0,\alpha)$ . Then $P_{k}(\alpha)$ is the spectral projection onto the flat band eigenfunction of $H_{k}$ . Since $k\mapsto H_{k}$ is real-analytic, this implies that $k\mapsto P_{k}$ is real-analytic. Moreover, since $H_{k-\gamma^{*}}(\alpha)=\tau(\gamma^{*})H_{k}(\alpha)\tau(\gamma^{*})^{-1}$ with $\tau_{\gamma}(z):=e^{i\operatorname{Re}(z\overline{\gamma^{*}})}$ with $\gamma^{*}\in\Gamma_{3}^{*},$ the spectral projection satisfies the covariance relation

P_{k-\gamma}(\alpha)=\tau(\gamma^{*})P_{k}(\alpha)\tau(\gamma^{*})^{-1}.

It then follows from [MPPT18, Theo. $2.4$ ] that there exists an associated Wannier basis satisfying $\|\langle\bullet\rangle^{p/2}w_{\gamma}\|^{2}_{L^{2}({\mathbb{C}})}\leq C<\infty$ for $p<1$ and all $\gamma\in\Gamma$ for the unperturbed periodic problem. Reversing the argument provided in the first part of the proof, it follows that the randomly perturbed problem also exhibits such a Wannier basis.

To show Theorem 5 one proceeds analogously and notices that $P_{\pm,\lambda=0}$ correspond to the projections onto $\ker(D(\alpha))$ and $\ker(D(\alpha)^{*})$ , each one exhibiting a non-zero Chern number. ∎

With this result at hand, we are able to evaluate the quantity (1.17) for the unperturbed Hamiltonian providing a link between the dynamical and spectral theoretic notion of (de)-localization.

Proposition 5.1.

Let $\alpha$ be a simple magic angle, as in Def. 1.1, then for all $p\geq 1$

\left\lVert\langle\bullet\rangle^{p/2}e^{-itH(\alpha)}P_{\ker(H(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}=\infty,

while the left-hand side is finite for $p<1.$

Proof.

We start by observing that for an orthonormal basis $(f_{n})$ of $L^{2}({\mathbb{C}}/\Gamma_{3})$ and $(e_{i})$ the standard basis of ${\mathbb{C}}^{4}$

\begin{split}\left\lVert\langle\bullet\rangle^{p/2}e^{-itH(\alpha)}P_{\ker(H(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}&=\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(H(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}\\ &=\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))\oplus\ker(D(\alpha)^{*})}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}\\ &=\sum_{i=1}^{4}\sum_{n\in\mathbb{N}}\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))\oplus\ker(D(\alpha)^{*})}f_{n}\otimes e_{i}\right\rVert^{2}\\ &=\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}+\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha)^{*})}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}.\end{split}

Without loss of generality, we shall focus on the first summand. Consider the unitary Bloch-Floquet transform $\mathcal{B}u(z,k):=\sum_{\gamma\in\Gamma_{3}}e^{i\langle z+\gamma,k\rangle}\mathscr{L}_{\gamma}u(z)$ , where $L_{\gamma}$ has been defined in (1.4), with the convention that $\langle z,z_{0}\rangle:=\operatorname{Re}(z\bar{z}_{0}),$ and its inverse/adjoint $\mathcal{C}v(z):=\int_{{\mathbb{C}}/\Gamma_{3}^{*}}v(z,k)e^{-i\langle z,k\rangle}\ \frac{dm(k)}{|{\mathbb{C}}/\Gamma_{3}^{*}|}$ . We then find that

\mathscr{L}_{\gamma}\mathcal{C}v(z):=\int_{{\mathbb{C}}/\Gamma_{3}^{*}}v(z,k)e^{-i\langle z+\gamma,k\rangle}\ \frac{dm(k)}{|{\mathbb{C}}/\Gamma_{3}^{*}|}=\mathcal{C}(e^{-i\langle\gamma,k\rangle}v(z,k)).

(5.1)

Since by assumption $\ker(D(\alpha)+k)=\operatorname{span}\{\varphi(\bullet,k)\}$ , we see that

(e^{-i\langle\gamma,k\rangle}\varphi(z,k))_{\gamma\in\Gamma}\text{, for }\varphi(\bullet,k)\in L^{2}({\mathbb{C}}/\Gamma_{3})\text{ normalized,}

(5.2)

forms a basis of the space $\int^{\oplus}_{{\mathbb{C}}/\Gamma_{3}^{*}}\ker(D(\alpha)+k)dk$ . Indeed, orthonormality just follows from

\begin{split}\langle e^{-i\langle\gamma,k\rangle}\varphi(z,k),e^{i\langle\gamma^{\prime},k\rangle}\varphi(z,k)\rangle&=\int_{{\mathbb{C}}/\Gamma_{3}^{*}}\int_{{\mathbb{C}}/\Gamma_{3}}|\varphi(z,k)|^{2}e^{-i\langle\gamma-\gamma^{\prime},k\rangle}\frac{dz\ dk}{|{\mathbb{C}}/\Gamma_{3}^{*}|}\\ &=\int_{{\mathbb{C}}/\Gamma_{3}^{*}}e^{-i\langle\gamma-\gamma^{\prime},k\rangle}\frac{dk}{|{\mathbb{C}}/\Gamma_{3}^{*}|}=\delta_{\gamma,\gamma^{\prime}}\end{split}

(5.3)

and completeness from the completeness of the regular Fourier expansion, i.e. a general element in this subspace is of the form

\sum_{\gamma\in\Gamma_{3}^{*}}f(\gamma)e^{-i\langle\gamma,k\rangle}v(z,k)\text{ for }f\in\ell^{2}(\Gamma_{3}^{*}).

We then have that $\mathcal{B}D(\alpha)\mathcal{C}\varphi(x,k)=(D(\alpha)+k)\varphi(x,k).$ Recall the trivial decomposition of $L^{2}$ given by $L^{2}({\mathbb{C}})=L^{2}({\mathbb{C}}/\Gamma_{3})\oplus L^{2}({\mathbb{C}}\setminus({\mathbb{C}}/\Gamma_{3})).$

We then find for the Hilbert-Schmidt norm using an orthonormal basis $(e_{n})$ of $L^{2}({\mathbb{C}}/\Gamma_{3})$

\begin{split}&\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}=\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}\\ &=\sum_{n\in{\mathbb{Z}}}\|\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))}e_{n}\|^{2}_{L^{2}({\mathbb{C}})}=\sum_{n\in{\mathbb{Z}}}\|\langle\bullet\rangle^{p/2}\mathcal{C}P_{\ker(D(\alpha)+k)}\mathcal{B}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}e_{n}\|^{2}_{L^{2}({\mathbb{C}})}.\end{split}

Since by assumption $P_{\ker(D(\alpha)+k)}=\varphi(\bullet,k)\otimes\varphi(\bullet,k)$ is a rank $1$ projection, we have

\begin{split}\|\langle\bullet\rangle^{p/2}\mathcal{C}P_{\ker(D(\alpha)+k)}\mathcal{B}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}e_{n}\|^{2}_{L^{2}({\mathbb{C}})}&=\|\langle\bullet\rangle^{p/2}\mathcal{C}\varphi\|^{2}_{L^{2}({\mathbb{C}})}|\langle\varphi,\mathcal{B}(\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}e_{n})\rangle_{L^{2}({\mathbb{C}}/\Gamma_{3}\times{\mathbb{C}}/\Gamma_{3}^{*})}|^{2}\\ &=\|\langle\bullet\rangle^{p/2}\mathcal{C}\varphi\|^{2}_{L^{2}({\mathbb{C}})}|\langle\mathcal{C}\varphi,\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}e_{n}\rangle_{L^{2}({\mathbb{C}})}|^{2}.\end{split}

This implies that

\begin{split}\left\lVert\langle\bullet\rangle^{p/2}P_{\ker(D(\alpha))}\operatorname{1\hskip-2.75ptl}_{{\mathbb{C}}/\Gamma_{3}}\right\rVert_{2}^{2}&=\|\langle\bullet\rangle^{p/2}\mathcal{C}\varphi\|^{2}_{L^{2}({\mathbb{C}})}\sum_{n\in\mathbb{Z}}|\langle\mathcal{C}\varphi,e_{n}\rangle_{L^{2}({\mathbb{C}}/\Gamma_{3})}|^{2}\\ &=\|\langle\bullet\rangle^{p/2}\mathcal{C}\varphi\|^{2}_{L^{2}({\mathbb{C}})}\|\mathcal{C}\varphi\|_{L^{2}({\mathbb{C}}/\Gamma_{3})}.\end{split}

However, a Wannier basis is obtained from $\mathcal{C}\varphi$ by defining $w_{\gamma}:=\mathscr{L}_{\gamma}\mathcal{C}\varphi$ . Indeed, using (5.3) functions $w_{\gamma}$ are an orthonormal basis of $\ker(D(\alpha))$ as

\langle w_{\gamma},w_{\gamma^{\prime}}\rangle_{L^{2}({\mathbb{C}})}=\langle\mathscr{L}_{\gamma}\mathcal{C}\varphi,\mathscr{L}_{\gamma^{\prime}}\mathcal{C}\varphi\rangle_{L^{2}({\mathbb{C}})}=\delta_{\gamma,\gamma^{\prime}}

and span $\ker(D(\alpha))$ due to (5.1) and (5.2). Thus, we obtain since $\mathscr{L}_{\gamma}$ is an isometry that

\|\langle\bullet\rangle^{p/2}w_{0}\|^{2}_{L^{2}({\mathbb{C}})}=\|\mathscr{L}_{\gamma}\langle\bullet\rangle^{p/2}w_{0}\|^{2}_{L^{2}({\mathbb{C}})}=\|\langle\bullet+\gamma\rangle^{p/2}w_{\gamma}\|_{L^{2}({\mathbb{C}})}.

From the non-existence of a $1$ -localized Wannier basis and the existence of a $(1-\delta)$ Wannier basis, for any $\delta>0$ , see for instance [MPPT18], we find that $\|\langle\bullet\rangle^{p/2}w_{0}\|^{2}_{L^{2}({\mathbb{C}})}=\infty$ for $p\geq 1$ and is finite for $p<1.$

∎

Appendix A Essential self-adjointness

In this appendix, we recall the essential self-adjointness of our Hamiltonian with even possibly unbounded disorder on $C_{c}^{\infty}({\mathbb{C}})$ .

Theorem 8.

The Hamiltonian $H_{\lambda}(\alpha)$ (1.11) is, under the more general assumptions, with $L^{\infty}({\mathbb{R}})$ -bounded density $g$ for random variables $(\omega_{\gamma})$ and arbitrary density $h$ is almost surely essentially self-adjoint on $C_{c}^{\infty}({\mathbb{C}}).$

Proof.

To see that $H_{\lambda}(\alpha)$ is essentially self-adjoint, we first observe that it is symmetric on $C_{c}^{\infty}({\mathbb{C}}).$ It thus suffices to show that for any $L^{2}$ -normalized $\psi$

(H_{\lambda}(\alpha)\pm i)\psi=0\text{ implies }\psi\equiv 0,

i.e. the deficiency indices are zero. Elliptic regularity and the assumption that $u\in L^{\infty}$ implies that $\psi\in C^{\infty}({\mathbb{C}})$ . We then pick a cut-off function $\eta_{n}(z):=\eta(z/n)$ with $\eta\in C_{c}^{\infty}({\mathbb{C}})$ and $\eta|_{B_{1}(0)}\equiv 1$ and find

(H_{\lambda}(\alpha)\pm i)\eta_{n}\psi=\begin{pmatrix}0&2D_{z}\eta_{n}\cdot\operatorname{id}_{\mathbb{C}^{2}}\\ 2D_{\bar{z}}\eta_{n}\cdot\operatorname{id}_{\mathbb{C}^{2}}&0\end{pmatrix}\psi.

We conclude that

\|\eta_{n}\psi\|_{2}^{2}+\|H_{\lambda}(\alpha)\eta_{n}\psi\|_{2}^{2}=\|(H_{\lambda}(\alpha)\pm i)\eta_{n}\psi\|_{2}^{2}\lesssim\|\nabla\eta_{n}\|_{\infty}^{2}=\mathcal{O}(1/n^{2})\xrightarrow[n\to\infty]{}0.

Since $\eta_{n}\psi\to\psi$ by dominated convergence, we conclude that $\psi\equiv 0.$ ∎

Appendix B Partial Chern numbers

Let $P$ be an orthogonal projection on $L^{2}({\mathbb{C}};{\mathbb{C}}^{2n})$ such that for some $\xi\in(0,1)$ , $\kappa>0$ , and $K_{P}<\infty$ we have

\|\chi_{z_{0}}P\chi_{z_{1}}\|_{2}\leq K_{P}\langle z_{0}\rangle^{\kappa}\langle z_{1}\rangle^{\kappa}e^{-|z_{0}-z_{1}|^{\xi}}\text{ for all }z_{0},z_{1}\in\Gamma.

(B.1)

This condition is satisfied for spectral projections of Hamiltonians under the assumption of (SUDEC), cf. Def. 4.1. Let $\pi_{1}:=\operatorname{diag}(\operatorname{id}_{{\mathbb{C}}^{n}},0)$ and $\pi_{2}:=\operatorname{diag}(0,\operatorname{id}_{{\mathbb{C}}^{n}})$ . By the definition of the Hilbert-Schmidt norm one finds for all $i,j$

\|\chi_{z_{0}}\pi_{i}P\pi_{j}\chi_{z_{1}}\|_{2}\leq\|\chi_{z_{0}}P\chi_{z_{1}}\|_{2}\leq K_{P}\langle z_{0}\rangle^{\kappa}\langle z_{1}\rangle^{\kappa}e^{-|z_{0}-z_{1}|^{\xi}}\text{ for all }z_{0},z_{1}\in\Gamma.

(B.2)

We define the new $\hat{\Theta}_{j,i}:=\pi_{i}\Theta_{j}=\Theta_{j}\pi_{i}$ and replace (4.4) by

\Omega_{i}(P):=\operatorname{tr}(P[[P,\hat{\Theta}_{1}(i)],[P,\hat{\Theta}_{2}(i)]])

(B.3)

which is well-defined for

|\Omega_{i}(P)|:=\|P[[P,\hat{\Theta}_{1}(i)],[P,\hat{\Theta}_{2}(i)]]\|_{1}<\infty.

(B.4)

Remark 4.

It is convenient to modify $\hat{\Theta}_{i}$ rather than $P$ in the definition of $\Omega$ , since $\pi_{i}P\pi_{j}$ is in general no longer a projection, even for $i=j.$

Since we still have that $[\hat{\Theta}_{i},\hat{\Theta}_{j}]=0$ we find the equivalent formulation of (B.3)

\Omega_{i}(P)=\operatorname{tr}[P\hat{\Theta}_{1}(i)P,P\hat{\Theta}_{2}(i)P].

(B.5)

In particular, if $P$ is a finite-rank projection then, we always find $\Omega_{i}(P)=0$ , as (B.5) is a commutator of trace-class operators.

To provide further motivation for the above definition (B.3), we shall consider the unperturbed Hamiltonian $H_{0}=\begin{pmatrix}m&D^{*}\\ D&-m\end{pmatrix}$ then $H_{0}^{2}=\operatorname{diag}(D^{*}D+m^{2},DD^{*}+m^{2})$ and consequently any spectral projection of $H_{0}^{2}$ is also diagonal and thus of the form $P_{0}=\operatorname{diag}(P_{0}(1),P_{0}(2)).$ Thus, we have

\Omega_{i}(P_{0})=\operatorname{tr}([P_{0}\hat{\Theta}_{1}(i)P_{0},P_{0}\hat{\Theta}_{2}(i)P_{0}])=\Omega(P_{0}(i)),

where we recall from (1.10) that for a generic magic angle and $P_{0}=\operatorname{1\hskip-2.75ptl}_{[0,\mu]}(H_{0}^{2})$ with $\mu\in(0,E_{\text{gap}}^{2})$

\Omega_{1}(P_{0})=\frac{i}{2\pi}\text{ and }\Omega_{2}(P_{0})=-\frac{i}{2\pi}.

(B.6)

Thus, while $\Omega(P_{0})=0$ for $m=0$ , we have $\Omega_{1}(P_{0}),\Omega_{2}(P_{0})\neq 0.$ The definition of $\Omega_{i}$ captures the non-trivial sublattice Chern numbers of twisted bilayer graphene while the total Chern number vanishes.

One also readily verifies the usual properties of Chern characters, see for instance [GKS07, Lemma 3.1], [BES94]:

Proposition B.1.

Let $P$ be an orthogonal projection satisfying (B.1), then

(1)

$|\Omega_{i}(P)|\lesssim_{\kappa,\xi}K_{P}^{2}.$

(2)

Let $s\in\mathbb{R}$ and define $\hat{\Theta}^{(s)}_{j,i}(t):=\pi_{i}\Theta_{j}(t-s)$ , then

\Omega_{i}^{r,s}(P):=\operatorname{tr}(P[[P,\hat{\Theta}_{1,i}^{(s)}],[P,\hat{\Theta}_{2,i}^{(r)}]])\text{ for }r,s\in{\mathbb{R}}.

In particular,

\Omega_{i}^{r,s}=\Omega_{i}.

(B.7)

(3)

Let $P,Q$ be two orthogonal projections, each satisfying (B.1), such that $PQ=QP=0$ , then

$\Omega_{i}(P+Q)=\Omega_{i}(P)+\Omega_{i}(Q).$

Proof.

The first property follows readily from combining (B.2) with the proof [GKS07, Lemma 3.1 (i)].

The second property follows from a direct computation, see [GKS07, Lemma 3.1 (ii)].

The last property follows from $P[Q,\hat{\Theta}_{i}]=-P\hat{\Theta}_{i}Q$ and evaluating (B.3) since one finds for the cross-terms

\begin{split}\operatorname{tr}\Big{(}-P\hat{\Theta}_{1}Q\hat{\Theta}_{2}+P\hat{\Theta}_{1}Q\hat{\Theta}_{2}P-Q\hat{\Theta}_{1}P\hat{\Theta}_{2}+Q\hat{\Theta}_{1}P\hat{\Theta}_{2}Q\Big{)}=0.\end{split}

∎

We also want to mention reference [ASS94, Sec.6] showing full details on how to obtain the second point and [BES94, Lemma $8$ ] for the third point.

The independence of switch functions $\hat{\Theta}^{(s)}_{j,i}$ in Prop. B.1 implies that $\Omega_{i}$ is an almost surely constant quantity

\Omega_{i}(P)=\mathbf{E}\Omega_{i}(P)\text{ for }\mathbf{P}-a.s.

The purpose of the first and last point in Prop. B.1 is to conclude that in regions of $\operatorname{SUDEC}$ , cf. Definition 4.1, all $\Omega_{i}$ vanish.

Proposition B.2.

Let $H_{\lambda}$ exhibit $\operatorname{SUDEC}$ in an interval $J$ , then for all closed $I\subset J$ we have

\Omega_{i}(\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda}))=0\text{ for }\mathbf{P}-a.s.

Proof.

Let $M\subset\mathbb{N}$ be a (finite or infinite) enumeration (counting multiplicities) of all point spectrum of $H_{\lambda}$ . We can then write the spectral projection as

\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda})=\sum_{m\in M}P_{m}

where $P_{m}$ are rank one projections. In addition, we have $K_{P}:=\sum_{m\in M}\alpha_{m}$ where $\alpha_{m}$ are defined in (4.1). Using the third item in Prop. B.1 we then have for any $\{1,...,N\}\subset M$

\Omega_{i}(\operatorname{1\hskip-2.75ptl}_{I}(H_{\lambda}))=\sum_{m=1}^{N}\underbrace{\Omega_{i}(P_{m})}_{=0}+\Omega_{i}\Bigg{(}\sum_{m\in M\setminus\{1,..,N\}}P_{m}\Bigg{)}=\Omega_{i}\Bigg{(}\sum_{m\in M\setminus\{1,..,N\}}P_{m}\Bigg{)}.

Invoking then the first item in Prop. B.1, we find that as we make $N$ arbitrarily large or equal to $|M|$ , if $M$ is finite, we obtain $\Omega_{i}(\sum_{m\in M\setminus\{1,..,N\}}P_{m})\to 0.$ ∎

Acknowledgements:. We thank Jie Wang for suggesting the relevance of different disorder types for TBG and Maciej Zworski for initial discussions on the project. M. Vogel was partially funded by the Agence Nationale de la Recherche, through the project ADYCT (ANR-20-CE40-0017). I. Oltman was jointly funded by the National Science Foundation Graduate Research Fellowship under grant DGE-1650114 and by grant DMS-1901462.

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Magic angle (in)stability and mobility edges in disordered Chern insulators

Abstract.

1. Introduction

Definition 1.1 (Generic magic angles).

1.1. Magic angle (in)stability

Theorem 1 (Instability).

1.2. Anderson model and IDS

Definition 1.2.

Remark 1.

Assumption 1 (Anderson model).

Theorem 2 (Continuous IDS).

Proposition 1.3 (NE).

1.3. Mobility edges

Theorem 3 (Dynamical delocalization).

Remark 2.

Definition 1.4 (Wannier basis).

Theorem 4 (Slow decay; m>0m>0).

Theorem 5 (Slow decay; m≥0m\geq 0).

Remark 3 (Lower bound on uniform eigenfunction decay).

2. (In)stability of magic angles

Definition 2.1.

2.1. Stability of magic angles

Lemma 2.2.

Theorem 6.

Proof of Theo. 6.

Proof of Lemma 2.2.

Lemma 2.3.

2.2. Instability of magic angles

Proof of Theo. 1.

3. Integrated DOS and Wegner estimate

3.1. Proof of Prop. 1.3

Proof of Prop. 1.3.

3.2. Spectral shift function and Hölder continuity

3.3. Spectral averaging and Lipschitz continuity

Proof of Theorem 2 (Lipschitz continuity).

4. Mobility edge

Definition 4.1 (SUDEC).

4.1. The ingredients to the multi-scale analysis

Lemma 4.2 (SGEE).

Proof.

Lemma 4.3 (SLI & EDI).

Proof.

4.2. Dynamical delocalization

Proof of Theo. 3.

4.3. Dynamical localization

Theorem 7 (Mobility edge).

Proof.

5. Decay of point spectrum and Wannier bases

Proof of Theo.4 & 5.

Proposition 5.1.

Proof.

Appendix A Essential self-adjointness

Theorem 8.

Proof.

Appendix B Partial Chern numbers

Remark 4.

Proposition B.1.

Proof.

Proposition B.2.

Proof.

References

Theorem 4 (Slow decay; $m>0$ ).

Theorem 5 (Slow decay; $m\geq 0$ ).