Inventiones mathematicae

, Volume 218, Issue 3, pp 979–1041 | Cite as

Cantor spectrum of graphene in magnetic fields

  • Simon BeckerEmail author
  • Rui Han
  • Svetlana Jitomirskaya
Open Access


We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux \(\Phi / 2\pi \) through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

1 Introduction

Graphene is a two-dimensional material that consists of carbon atoms at the vertices of a hexagonal lattice. Its experimental discovery, unusual properties, and applications led to a lot of attention in physics, see e.g. [44]. Electronic properties of graphene have been extensively studied rigorously in the absence of magnetic fields [20, 21, 22, 39].

Magnetic properties of graphene have also attracted strong interest in physics (e.g. [27, 55]). The purpose of this paper is to provide for the first time an analysis of the spectrum of honeycomb structures in magnetic fields with constant flux.

The fact that magnetic electron spectra have fractal structures was first predicted by Azbel [3] and then numerically observed by Hofstadter [31], for the Harper’s model. The scattering plot of the electron spectrum as a function of the magnetic flux is nowadays known as Hofstadter’s butterfly. Verifying such results experimentally has been restricted for a long time due to the extraordinarily strong magnetic fields required. Only recently, self-similar structures in the electron spectrum in graphene have been observed [15, 17, 23, 25].

With this work, we provide a rigorous foundation for self-similarity by showing that for irrational fluxes, the electron spectrum of a model of graphene is a Cantor set. We say A is a Cantor set if it is closed, nowhere dense and has no isolated points (so compactness not required). The Schrödinger operator \(H^B\) we study, see (3.7), is defined on a metric honeycomb graph1 and is a direct sum, over all edges \(\vec e\) of the graph, of Schrödinger operators
$$\begin{aligned}H^B_{\vec e} = (-\,i\partial _x-A_{\vec e})^2 + V_{\vec e}\end{aligned}$$
with magnetic potential \(A_{\vec e}\), describing a constant magnetic field, and potential \(V_{\vec e}\in L^2(\vec {e}).\) We write \(\sigma ^{\Phi }\),\(\sigma _{cont}^\Phi ,\sigma _{ess}^\Phi \) for the (continuous, essential) spectra of \(H^B\) and set \(H^D\) to be the Dirichlet operator (no magnetic field) defined in (2.14) (2.11), and denote by \(\sigma (H^D)\) its spectrum. Let \(\sigma _p^\Phi \) be the collection of eigenvalues of \(H^B.\) Then we have the following description of the topological structure and point/continuous decomposition of the spectrum

Theorem 1

For any symmetric Kato-Rellich potential \(V_{\vec {e}}\in L^2(\vec {e})\) we have
  1. (1)

    \(\sigma ^\Phi = \sigma _{ess}^\Phi ,\)

  2. (2)

    \(\sigma _p^\Phi = \sigma (H^D),\)

  3. (3)
    \(\sigma _{cont}^\Phi \) is
    • a Cantor set of measure zero for \(\Phi \notin 2\pi {\mathbb {Q}},\)

    • a countable union of disjoint intervals for \(\Phi \in 2\pi {\mathbb {Q}},\)

  4. (4)

    \(\sigma _p^\Phi \cap \sigma _{cont}^\Phi =\emptyset \) for \(\Phi \notin 2\pi {\mathbb {Z}}\),

  5. (5)

    the Hausdorff dimension \(\dim _H (\sigma ^{\Phi })\le 1/2\) for generic2\(\Phi \).


Thus for irrational flux, the spectrum is a zero measure Cantor set plus a countable collection of flux-independent isolated eigenvalues, each of infinite multiplicity, while for rational flux the Cantor set is replaced by a countable union of intervals.

Furthermore, we can also describe the spectral decomposition of \(H^B\).

Theorem 2

For any symmetric Kato-Rellich potential \(V_{\vec {e}}\in L^2(\vec {e})\) we have
  1. (1)

    For \(\Phi \notin 2\pi {\mathbb {Q}}\), the spectrum on \(\sigma _{cont}^\Phi \) is purely singular continuous.

  2. (2)

    For \(\Phi \in 2\pi {\mathbb {Q}}\), the spectrum on \(\sigma _{cont}^\Phi \) is absolutely continuous.


Of course our results only describe the quantum graph model of graphene in a magnetic field, which is both single-electron and high contrast. In particular, we believe that the isolated eigenvalues are unphysical, being an artifact of the graph model which does not allow something similar to actual Coulomb potentials close to the carbon atoms or dissolving of eigenstates supported on edges in the bulk. However, there are reasons to expect that continuous spectrum of the quantum graph operator (thus the Cantor set described in this paper) does adequately capture the experimental properties of graphene in the magnetic field [14]. In particular, certain properties of the density of states of our model (which starts from actual differential operator and is exact in every step) better correspond to the experimental observations [24] than those of the commonly used tight-binding model [4]. We refer the reader to [13, 14] for detail. Finally, our analysis provides full description of the spectrum of the tight-binding Hamiltonian as well. Moreover, the applicability of our model is certainly not limited to graphene.

Earlier work showing Cantor spectrum on quantum graphs with magnetic fields, e.g. for the square lattice [11] and magnetic chains studied in [19], has been mostly limited to applications of the Cantor spectrum of the almost Mathieu operator [5, 45]. On the honeycomb graph, we can no longer resort to this operator. The discrete operator is then matrix-valued and can be further reduced to a one-dimensional discrete quasiperiodic operator using supersymmetry. The resulting discrete operator is a singular Jacobi matrix3 Cantor spectrum (in fact, a stronger, dry ten martini type statement) for Jacobi matrices of this type has been studied in the framework of the extended Harper’s model [29]. However, the method of [29] that goes back to that of [6] relies on (almost) reducibility, and thus in particular is not applicable in absence of (dual) absolutely continuous spectrum which is prevented by singularity. Similarly, the method of [5] breaks down in presence of singularity in the Jacobi matrix as well. Instead, we present a novel way that exploits singularity rather than circumvents it by showing that the singularity leads to vanishing of the measure of the spectrum, and thus Cantor structure and singular continuity, once (4) of Theorem 1 is established.4 Our method applies also to proving zero measure Cantor spectrum of the extended Harper’s model whenever the corresponding Jacobi matrix is singular and either the Lyapunov exponent is zero on the spectrum or one can estimate the measure of the spectrum for the rational frequency. The latter is also useful for estimating the Hausdorff dimension and was only available previously for the almost Mathieu operator [9, 42] with, in particular, the method of [9] extendable only to situations when measure of the spectrum is not zero, and the method of [42] very almost Mathieu specific. Here we develop a novel method, that applies to general singular Jacobi matrices (see e.g. Lemma 6.8) for which one can establish a Chambers-type formula.

As mentioned, our first step is a reduction to a matrix-valued tight-binding hexagonal model. This leads to an operator \(Q_\Lambda \) defined in (4.1). This operator has been studied before for the case of rational magnetic flux (see [28] and references therein). Our analysis gives complete spectral description for this operator as well.

Theorem 3

The spectrum of \(Q_\Lambda (\Phi )\) is
  • a finite union of intervals and purely absolutely continuous for \({\Phi }/{2\pi }=p/q\), which is a reduced rational number, with the following measure estimate
    $$\begin{aligned} |\sigma (Q_{\Lambda }(\Phi ))|\le \frac{C}{\sqrt{q}}, \end{aligned}$$
    where \(C>0\) is an absolute constant.
  • singular continuous and a zero measure Cantor set for \(\Phi \notin 2\pi {\mathbb {Q}}\),

  • a set of Hausdorff dimension \(\dim _H(\sigma (Q_{\Lambda }(\Phi ))) \le 1/2\) for generic5\(\Phi \).

Remark 1

We will show that the constant C in the first item can be bounded by \(\frac{8\sqrt{6 \pi }}{9}\).

The theory of magnetic Schrödinger operators on graphs can be found in [41]. The effective one-particle graph model for graphene without magnetic fields was introduced in [39]. After incorporating a magnetic field according to [41] in the model of [39], the reduction of differential operators on the graph to a discrete tight-binding operator can be done using Krein’s extension theory for general self-adjoint operators on Hilbert spaces. This technique has been introduced in [46] for magnetic quantum graphs on the square lattice. The quantum graph nature of the differential operators causes, besides the contribution of the tight-binding operator to the continuous spectrum, a contribution to the point spectrum that consists of Dirichlet eigenfunctions vanishing at every vertex.

In this paper we develop the corresponding reduction for the hexagonal structure and derive spectral conclusions in a way that allows easy generalization to other planar graphs spanned by two basis vectors. In particular, our techniques should be applicable to study quantum graphs on the triangular lattice, which will be pursued elsewhere.

One of the striking properties of graphene is the presence of a linear dispersion relation which leads to the formation of conical structures of the dispersion surfaces in the Brillouin zone, see Fig. 5. The points where the cones match are called Dirac points to account for the special dispersion relation. We use a spectral equivalence between the magnetic Schrödinger operators on the graph and tight-binding operators that is based on Krein’s theory in a version introduced in [47, 48]. In particular, the bands of the graph model always touch at the Dirac points and are shown to have open gaps at the band edges of the associated Hill operator if the magnetic flux is non-trivial. We obtain the preceding results by first proving a bound on the operator norm of the tight-binding operator and analytic perturbation theory.

In [39] it was shown that the Dirichlet contribution to the spectrum in the non-magnetic case is generated by compactly supported eigenfunctions and that this is the only contribution to the point spectrum of the Schrödinger operator on the graph. We extend this result to magnetic Schrödinger operators on hexagonal graphs. Let \(H_{pp}\) be the pure point subspace accociated with \(H^B.\) Then

Theorem 4

For any \(\Phi \), \(H_{pp}\) is spanned by compactly supported eigenfunctions (in fact, by double hexagonal states).

While for the rational \(\Phi \) the proof is based on ideas similar to those of [39], for the irrational \(\Phi \) we no longer have an underlying periodicity thus cannot use the arguments of [36]. After showing that there are double hexagonal state eigenfunctions for each Dirichlet eigenvalue, it remains to show their completeness. While there are various ways to show that all \(\ell ^1\) (in a suitable sense) eigenfunctions are in the closure of the span of double hexagonal states, the \(\ell ^2\) condition is more elusive. Bridging the gap between \(\ell ^1\) and \(\ell ^2\) has been a known difficult problem in several other scenarios [1, 7, 10, 32]. Here we achieve this by constructing, for each \(\Phi \), an operator that would have all slowly decaying \(\ell ^2\) eigenfunctions in its kernel and showing its invertibility. This is done using constructive arguments and properties of holomorphic families of operators. We note that, to the best of our knowledge, Theorem 4 is the first result of this sort in absence of periodicity, and our way of bridging the gap between \(\ell ^1\) and \(\ell ^2\) is also a novel argument.

1.1 Outline

Section 2 serves as background, in particular it reviews results on the honeycomb quantum graph model without magnetic fields. In Sect. 3, we introduce the magnetic Schrödinger operator \(H^B\) show that this one is unitarily equivalent to a non-magnetic Schrödinger operator \(\Lambda ^B\) with magnetic contributions moved into the boundary conditions. In Sect. 4, we present several key ingredients of the proofs of the main theorems: Lemmas 4.14.4. Lemma 4.1 involves a further reduction from \(\Lambda ^B\) to a two-dimensional tight-binding Hamiltonian \(Q_{\Lambda }(\Phi )\), and Lemmas 4.24.4 reveal the topological structure of \(\sigma (Q_{\Lambda }(\Phi ))\) (thus proving the topological part of Theorem 3). The proofs of Lemmas 4.14.4 are given is Sects. 5, 6 and 7. Section 8 is devoted to a complete spectral analysis of \(H^B\), thus proving Theorem 1, with the analysis of Dirichlet spectrum in Sect. 8.2, where, in particular, we prove Theorem 4; absolutely continuous spectrum for rational flux in Sect. 8.3, singular continuous spectrum for irrational flux in Sect. 8.4 (thus proving Theorem 2). Since most of the proofs for different parts of Theorems 14 are distributed throughout the paper, we give an index to them, for the reader’s convenience in Sect. 8.5.

2 Preliminaries

2.1 Notation

Given a graph G, we denote the set of edges of G by \({\mathcal {E}}(G),\) the set of vertices by \({\mathcal {V}}(G)\), and the set of edges adjacent to a vertex \({{\varvec{v}}} \in {\mathcal {V}}(G)\) by \({\mathcal {E}}_{{{\varvec{v}}}}(G).\)

For an operator H, let \(\sigma (H)\) be its spectrum and \(\rho (H)\) be the resolvent set.

The space \(c_{00}\) is the space of all infinite sequences with only finitely many non-zero terms (finitely supported sequences). We denote by \(\Omega ^i({\mathbb {R}}^2)\) the vector space of all i-covectors or differential forms of degree i on \({\mathbb {R}}^2.\)

For a set \(U\subseteq {\mathbb {R}}\), let |U| be its Lebesgue measure. We define \({\mathbb {T}}_2^*:={\mathbb {R}}^2/(2\pi {\mathbb {Z}})^2\) and \({\mathbb {T}}:={\mathbb {T}}_1:={\mathbb {R}}/{\mathbb {Z}}\).

List of main symbols used in this article.
  • \({{\varvec{r}}}_0\) and \({{\varvec{r}}}_1\) are the vertices of the fundamental cell (2.1).

  • \(\vec {f},\vec {g},\vec {h}\) are the vectors of the fundamental cell (2.2).

  • \(W_{\Lambda }:=\left\{ \vec {f},\vec {g},\vec {h},{{\varvec{r}}}_0,{{\varvec{r}}}_1 \right\} \) is the fundamental cell.

  • \(\vec {b}_1,\vec {b}_2\) are the basis vectors of the lattice (2.3).

  • \(\Lambda \) is the metric honeycomb graph (2.4).

  • \([{{\varvec{v}}}],[\vec {e}]\) denotes the translate of a vertex \({{\varvec{v}}}\) or edge \(\vec {e}\) into the fundamental cell (2.5).

  • \(v=(\gamma _1, \gamma _2, [{{\varvec{v}}}]),\vec {e}=(\gamma _1, \gamma _2, [\vec {e}])\) are defined in the paragraph below (2.5).

  • it map edges to their respective initial and terminal vertex (2.6).

  • \(\kappa _{\vec {e}}\) is the chart defined in (2.9).

  • \({\mathcal {H}}^n\) are the Sobolev spaces (2.10).

  • \(H_{\vec {e}}\) is the maximal Schrödinger operator on an edge \(\vec {e}\) (2.11).

  • V is the potential as defined in (2.12).

  • \(H^D\) is the Schrödinger operator with Dirichlet boundary conditions (2.14).

  • H is the Schrödinger operator without magnetic field (2.16).

  • \(T^{\text {st}}_{{{\varvec{\gamma }}}}\) are lattice translations (2.17).

  • \(H({{\varvec{k}}})\) are non-magnetic Schrödinger operators satisfying Floquet boundary conditions (2.21).

  • \(\psi _{\lambda ,1},\psi _{\lambda ,2}\) are solutions to the boundary value problem stated in (2.23).

  • \(\eta (\lambda )\) is introduced below (2.25).

  • \(c_{\lambda ,\vec {e}}\) and \(s_{\lambda ,\vec {e}}\) are defined in (2.27).

  • Hill potential \(V_{\text {Hill}}\) and Hill operator \(H_{\text {Hill}}\) are defined in (2.33) and (2.34).

  • \(\Delta (\lambda )\) is the Floquet discriminant defined in (2.37).

  • \(H_{\Phi , \theta }\in {\mathcal {L}}(l^2({\mathbb {Z}}))\) is the Jacobi operator defined in (2.41), with spectrum \(\Sigma _{\Phi ,\theta }\) and \(\Sigma _{\Phi }:=\bigcup _{\theta \in {\mathbb {T}}_1} \Sigma _{\Phi ,\theta }.\)

  • \(\Theta \) is the set of zeros of \(c(\theta )\) as defined in Sect. 2.3.1.

  • \(A^{\lambda },A_n^{\lambda },{\widetilde{A}}^{\lambda }\) and are the transfer, n-step transfer, and normalized transfer matrix defined in (2.42), (2.43), and (2.46).

  • \(D^{\lambda }\) and \(D_n^{\lambda }\) are derived from transfer matrices in (2.49).

  • \(L(\lambda ,\Phi )\) is the Lyapunov exponent defined in (2.44).

  • Vector potential A, integrated vector potential \(\beta \), and flux \(\Phi \) are defined in (3.1) and (3.2).

  • \(H^B\) is the Schrödinger operator introduced in (3.7)

  • \(\Lambda ^B\) is the Schrödinger operator introduced in (3.14).

  • \(Q_{\Lambda }(\Phi )\) is the tight-binding operator stated in (4.1).

  • \(\tau _0\) and \(\tau _1\) are discrete magnetic translation operators defined in (4.2).

  • \(\gamma (\lambda )\) is defined in (7.8), \(M(\lambda ,\Phi )\) is given in (7.9), \(K_{\lambda }\) is defined in (7.10).

  • \(T^{B}_{{{\varvec{\gamma }}}}\) are magnetic translation defined in (8.1).

2.2 Hexagonal quantum graphs

This subsection is devoted to reviewing hexagonal quantum graphs without magnetic fields. The readers could refer to [39] for details. We include some material here that serves as a preparation for the study of quantum graphs with magnetic fields in Sect. 3.

A model for effective one electron behavior in graphene is given by a hexagonal graph with Schrödinger operators defined on each edge [39]. The hexagonal graph \(\Lambda \) is obtained by translating its fundamental cell \(W_{\Lambda }\), the red colored part of Fig. 1, consisting of vertices
$$\begin{aligned} {{\varvec{r}}}_0:=(0,0) \text { and } {{\varvec{r}}}_1:=\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right) \end{aligned}$$
and edges
$$\begin{aligned} \begin{aligned} \vec {f}&:={\text {conv}}\left( \{{{\varvec{r}}}_0,{{\varvec{r}}}_1 \}\right) \ \backslash \ \{{{\varvec{r}}}_0,{{\varvec{r}}}_1 \}, \\ \vec {g}&:={\text {conv}}\left( \{{{\varvec{r}}}_0,\left( -\,1,0\right) \}\right) \ \backslash \{{{\varvec{r}}}_0,\left( -\,1,0\right) \} , \text { and } \\ \vec {h}&:={\text {conv}}\left( \left\{ {{\varvec{r}}}_0, \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \right\} \right) \ \backslash \ \left\{ {{\varvec{r}}}_0, \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \right\} , \end{aligned} \end{aligned}$$
along the basis vectors of the lattice. The basis vectors are
$$\begin{aligned} \vec {b}_1:= \left( \frac{3}{2}, \frac{\sqrt{3}}{2} \right) \text { and } \vec {b}_2:= \left( 0,\sqrt{3}\right) \end{aligned}$$
and so the hexagonal graph \(\Lambda \subset {\mathbb {R}}^2\) is given by the range of a \({\mathbb {Z}}^2\)-action on the fundamental domain \(W_{\Lambda }\)
$$\begin{aligned} \Lambda := \left\{ {{\varvec{x}}} \in {\mathbb {R}}^2:\ x = \gamma _1 \vec {b}_1+\gamma _2 \vec {b}_2+{{\varvec{y}}} \text { for } {{\varvec{\gamma }}} \in {\mathbb {Z}}^2 \text { and } {{\varvec{y}}} \in W_{\Lambda } \right\} . \end{aligned}$$
The fundamental domain of the dual lattice can be identified with the dual 2-torus \({\mathbb {T}}_2^*\).
For any vertex \({{ {{\varvec{v}}}}}\in {\mathcal {V}}(\Lambda )\), we denote by \([{{\varvec{v}}}]\in {\mathcal {V}}(W_{\Lambda })\) the unique vertex, \({{\varvec{r}}}_0\) or \({{\varvec{r}}}_1\), for which there is \({{\varvec{\gamma }}}\in {\mathbb {Z}}^2\) such that
$$\begin{aligned} {{\varvec{v}}}=\gamma _1 \vec {b}_1+\gamma _2 \vec {b}_2+[{{\varvec{v}}}]. \end{aligned}$$
We will occasionally denote \({{\varvec{v}}}\) by \((\gamma _1, \gamma _2, [{{\varvec{v}}}])\) to emphasize the location of \({{\varvec{v}}}\). We also introduce a similar notation for edges. For an edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\), we will sometimes denote it by \((\gamma _1, \gamma _2, [\vec {e}])\). Finally, for any \({{\varvec{x}}}\in \Lambda \), we will also denote its unique preimage in \(W_{\Lambda }\) by \([{{\varvec{x}}}]\).6
Fig. 1

The fundamental cell \(W_{\Lambda }\), colored in red and including points \(r_0,r_1\), and lattice basis vectors of \(\Lambda \) (color figure online)

We can orient the edges in terms of initial and terminal maps
$$\begin{aligned} i:{\mathcal {E}}(\Lambda ) \rightarrow {\mathcal {V}}(\Lambda ) \text { and } t:{\mathcal {E}}(\Lambda ) \rightarrow {\mathcal {V}}(\Lambda ) \end{aligned}$$
where i and t map edges to their initial and terminal ends respectively. It suffices to specify the orientation on the edges of the fundamental domain \(W_{\Lambda }\) to obtain an oriented graph \(\Lambda \)
$$\begin{aligned} \begin{aligned} i(\vec {f})&=i(\vec {g})=i(\vec {h})={{\varvec{r}}}_0, \\ t(\vec {f})&={{\varvec{r}}}_1, \ t(\vec {g})={{\varvec{r}}}_1-\vec {b}_1,\text { and } t(\vec {h})={{\varvec{r}}}_1-\vec {b}_2. \end{aligned} \end{aligned}$$
For arbitrary \(\vec {e} \in {\mathcal {E}}(\Lambda )\), we then just extend those maps by
$$\begin{aligned} i(\vec {e}):=\gamma _1 \vec {b}_1+\gamma _2 \vec {b}_2+i([\vec {e}]) \text { and } t(\vec {e}):=\gamma _1 \vec {b}_1+\gamma _2 \vec {b}_2+t([\vec {e}]). \end{aligned}$$
Let \(i(\Lambda )=\{{{\varvec{v}}}\in {\mathcal {V}}(\Lambda ): {{\varvec{v}}}=i(\vec {e}) \text { for some }\vec {e}\in {\mathcal {E}}(\Lambda )\}\) be the collection of initial vertices, and \(t(\Lambda )=\{{{\varvec{v}}}\in {\mathcal {V}}(\Lambda ): {{\varvec{v}}}=t(\vec {e}) \text { for some }\vec {e}\in {\mathcal {E}}(\Lambda )\}\) be the collection of terminal ones. It should be noted that based on our orientation, \({\mathcal {V}}(\Lambda )\) is a disjoint union of \(i(\Lambda )\) and \(t(\Lambda )\).
Every edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\) is of length one and thus has a canonical chart
$$\begin{aligned} \kappa _{\vec {e}}:\ \,&\vec {e} \rightarrow (0,1), \nonumber \\&(i(\vec {e})x+t(\vec {e})(1-x))\mapsto x \end{aligned}$$
that allows us to define function spaces and operators on \(\vec {e}\) and finally on the entire graph. For \(n \in {\mathbb {N}}_0\), the Sobolev space \({\mathcal {H}}^n({\mathcal {E}}\left( \Lambda \right) )\) on \(\Lambda \) is the Hilbert space direct sum
$$\begin{aligned} {\mathcal {H}}^n({\mathcal {E}}\left( \Lambda \right) ):= \bigoplus _{\vec {e} \in {\mathcal {E}}(\Lambda )} {\mathcal {H}}^n(\vec {e}). \end{aligned}$$
Fig. 2

The potential is the same on all edges and symmetric with respect to the centre of the edge

On every edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\) we define the maximal Schrödinger operator
$$\begin{aligned} \begin{aligned} H_{\vec {e}}&: {\mathcal {H}}^2(\vec {e}) \subset L^2(\vec {e}) \rightarrow L^2(\vec {e}) \\ H_{\vec {e}}&\psi _{\vec {e}} := -\,\psi ''_{\vec {e}}+V_{\vec {e}} \psi _{\vec {e}} \end{aligned} \end{aligned}$$
with Kato-Rellich potential \(V_{\vec {e}} \in L^2(\vec {e})\) that is the same on every edge and even with respect to the center of the edge, see Fig. 2. Let
$$\begin{aligned} V(t)=V_{\vec {e}}((\kappa _{\vec {e}})^{-1}(t)). \end{aligned}$$
$$\begin{aligned} V(t)=V(1-t). \end{aligned}$$
One self-adjoint restriction of (2.11) is the Dirichlet operator
$$\begin{aligned} \begin{aligned}&H^D:=\bigoplus _{\vec {e} \in {\mathcal {E}}(\Lambda )}\left( {\mathcal {H}}_0^1(\vec {e})\cap {\mathcal {H}}^2(\vec {e}) \right) \subset L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) ) \\&(H^D\psi )_{\vec {e}}:=H_{\vec {e}}\psi _{\vec {e}}, \end{aligned} \end{aligned}$$
where \({\mathcal {H}}_0^1(\vec {e})\) is the closure of compactly supported smooth functions in \({\mathcal {H}}^1(\vec {e})\). The Hamiltonian we will use to model the graphene without magnetic fields is the self-adjoint [36] operator H on \(\Lambda \) with Neuman type boundary conditions
$$\begin{aligned} \begin{aligned} D(H):= \biggl \{ \psi =(\psi _{\vec {e}}) \in {\mathcal {H}}^2({\mathcal {E}}(\Lambda )):&\text { for all }{{\varvec{v}}} \in {\mathcal {V}}(\Lambda ), \psi _{\vec {e}_1}({{\varvec{v}}})\\&=\psi _{\vec {e}_2}({{\varvec{v}}}) \text { if }\vec {e}_1,\vec {e}_2 \in {\mathcal {E}}_{{{\varvec{v}}}}(\Lambda ) \\&\text { and } \sum _{\vec {e} \in {\mathcal {E}}_{{{\varvec{v}}}}(\Lambda )} \psi _{\vec {e}}^\prime ({{\varvec{v}}})=0\biggr \} \end{aligned} \end{aligned}$$
and defined by
$$\begin{aligned} \begin{aligned}&\ H: D(H) \subset L^2({\mathcal {E}}(\Lambda )) \rightarrow L^2({\mathcal {E}}(\Lambda )) \\&(H\psi )_{\vec {e}}:=H_{\vec {e}}\psi _{\vec {e}}. \end{aligned} \end{aligned}$$

Remark 2

The self-adjointness of H will also follow from the self-adjointness of the more general family of magnetic Schrödinger operators that is obtained in Sec. 7.

Remark 3

The orientation is chosen so that all edges at any vertex are either all incoming or outgoing. Thus, there is no need to distinguish those situations in terms of a directional derivative in the boundary conditions (2.15).

2.2.1 Floquet–Bloch decomposition

Operator H commutes with the standard lattice translations
$$\begin{aligned} \begin{aligned} T^{\text {st}}_{\varvec{\gamma }} :&L^2({\mathcal {E}}(\Lambda )) \rightarrow L^2({\mathcal {E}}(\Lambda ))\\&f \mapsto f(\cdot -\gamma _1 \vec {b}_1-\gamma _2 \vec {b}_2) \end{aligned} \end{aligned}$$
for any \({{\varvec{\gamma }}} \in {\mathbb {Z}}^2.\) In terms of those, we define the Floquet–Bloch transform for \({{\varvec{x}}} \in {\mathcal {E}}(W_{\Lambda })\) and \({{\varvec{k}}} \in {\mathbb {T}}_2^*\) first on function \(f \in C_c({\mathcal {E}}(\Lambda ))\)
$$\begin{aligned} (Uf)({{\varvec{k}}},{{\varvec{x}}}):= \sum _{{{\varvec{\gamma }}} \in {\mathbb {Z}}^2} (T^{\text {st}}_{\varvec{\gamma }}f)({{\varvec{x}}})e^{i\langle {{\varvec{k}}},\varvec{\gamma }\rangle } \end{aligned}$$
and then extend it to a unitary map \(U \in {\mathcal {L}}(L^2({\mathcal {E}}(\Lambda )),L^2({\mathbb {T}}_2^* \times {\mathcal {E}}(W_{\Lambda })))\) with inverse
$$\begin{aligned} (U^{-1}\varphi )({{\varvec{x}}})= \int _{{\mathbb {T}}_2^*} \varphi ({{\varvec{k}}},[{{\varvec{x}}}]) e^{-i \langle \varvec{\gamma },{{\varvec{k}}} \rangle } \frac{d{{\varvec{k}}}}{(2 \pi )^2}, \end{aligned}$$
where \([{{\varvec{x}}}] \in {\mathcal {E}}\left( W_{\Lambda }\right) \) is the unique pre-image of \({{\varvec{x}}}\) in \(W_{\Lambda }\), and \(\varvec{\gamma }\in {\mathbb {Z}}^2\) is defined by \({{\varvec{x}}}=\gamma _1 \vec {b}_1+\gamma _2 \vec {b}_2+[{{\varvec{x}}}]\).
Then standard Floquet–Bloch theory implies that there is a direct integral representation of H
$$\begin{aligned} UHU^{-1}= \int _{{\mathbb {T}}_2^{*}}^{\oplus } H({{\varvec{k}}}) \frac{d{{\varvec{k}}}}{(2\pi )^2} \end{aligned}$$
in terms of self-adjoint operators \(H({{\varvec{k}}})\)
$$\begin{aligned} \begin{aligned}&\ H({{\varvec{k}}}):D(H({{\varvec{k}}})) \subset L^2({\mathcal {E}}\left( W_{\Lambda }\right) ) \rightarrow L^2({\mathcal {E}}\left( W_{\Lambda }\right) ) \\&(H({{\varvec{k}}})\psi )_{\vec {e}}:=(H_{\vec {e}}\psi _{\vec {e}}) \end{aligned} \end{aligned}$$
on the fundamental domain \(W_{\Lambda }\) with Floquet boundary conditions
$$\begin{aligned} \begin{aligned} D(H({{\varvec{k}}})):=\biggl \{&\psi \in {\mathcal {H}}^2({\mathcal {E}}\left( W_{\Lambda }\right) ): \psi _{\vec {f}}({{\varvec{r}}}_0)=\psi _{\vec {g}}({{\varvec{r}}}_0)=\psi _{\vec {h}}({{\varvec{r}}}_0) \text { and } \\&\sum _{\vec {e} \in {\mathcal {E}}_{{{\varvec{r}}}_0}(\Lambda )} \psi _{\vec {e}}^\prime ({{\varvec{r}}}_0)=0, \\&\text { as well as }\psi _{\vec {f}}({{\varvec{r}}}_1)=e^{ik_1}\psi _{\vec {g}}({{\varvec{r}}}_1-\vec {b}_1)=e^{ik_2}\psi _{\vec {h}}({{\varvec{r}}}_1-\vec {b}_2)\\&\ \text {and }\psi _{\vec {f}}^\prime ({{\varvec{r}}}_1)+e^{ik_1}\psi _{\vec {g}}^\prime ({{\varvec{r}}}_1-\vec {b}_1)+e^{ik_2}\psi _{\vec {h}}^\prime ({{\varvec{r}}}_1-\vec {b}_2)=0 \biggr \}. \end{aligned} \end{aligned}$$
Fix an edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\) and \(\lambda \notin \sigma (H^D)\). There are linearly independent \({\mathcal {H}}^2(\vec {e})\)-solutions \(\psi _{\lambda ,1,\vec {e}}\) and \(\psi _{\lambda ,2,\vec {e}}\) to the equation \(H_{\vec {e}}\psi _{\vec {e}}=\lambda \psi _{\vec {e}}\) with the following boundary condition (Fig. 3)
$$\begin{aligned}&\psi _{\lambda ,1,\vec {e}}(i(\vec {e}))=1, \quad \psi _{\lambda ,1,\vec {e}}(t(\vec {e}))=0, \quad \nonumber \\&\psi _{\lambda ,2,\vec {e}}(i(\vec {e}))=0,\quad \text{ and } \psi _{\lambda ,2,\vec {e}}(t(\vec {e}))=1. \end{aligned}$$
Fig. 3

For zero potential, we illustrate functions (2.23) and (2.27) for \(\lambda =16\)

Any eigenfunction to operators \(H({{\varvec{k}}})\), with eigenvalues away from \(\sigma (H^D)\), can therefore be written in terms of those functions for constants \(a,b \in {\mathbb {C}}\)
$$\begin{aligned} \psi := {\left\{ \begin{array}{ll} a \ \psi _{\lambda ,1,\vec {f}}+ b \ \psi _{\lambda ,2,\vec {f}} &{}\text{ along } \text{ edge } \vec {f} \\ a \ \psi _{\lambda ,1,\vec {g}} + e^{-ik_1}b \ \psi _{\lambda ,2,\vec {g}} &{} \text{ along } \text{ edge } \vec {g} \\ a \ \psi _{\lambda ,1,\vec {h}} + e^{-ik_2}b \ \psi _{\lambda ,2,\vec {h}} &{} \text{ along } \text{ edge } \vec {h} \end{array}\right. } \end{aligned}$$
with the continuity conditions of (2.22) being already incorporated in the representation of \(\psi .\) Imposing the conditions stated on the derivatives in (2.22) shows that \(\psi \) is non-trivial (ab not both equal to zero) and therefore an eigenfunction with eigenvalue \(\lambda \in {\mathbb {R}}\) to \(H({{\varvec{k}}})\) iff
$$\begin{aligned} \eta (\lambda )^2 = \frac{\left| 1+ e^{ik_1}+ e^{ik_2} \right| ^2}{9} \end{aligned}$$
with \(\eta (\lambda ):=\frac{\psi _{\lambda ,2,\vec {e}}'(t(\vec {e}))}{\psi _{\lambda ,2,\vec {e}}'(i(\vec {e}))}\) well-defined away from the Dirichlet spectrum.

By noticing that the range of the function on the right-hand side of (2.25) is [0, 1],  the following spectral characterization is obtained [39, Theorem 3.6].

Theorem 5

As a set, the spectrum of H away from the Dirichlet spectrum is given by
$$\begin{aligned} \sigma (H)\backslash \sigma (H^D) = \left\{ \lambda \in {\mathbb {R}}:\ \left|\eta (\lambda ) \right|\le 1 \right\} \backslash \sigma (H^D). \end{aligned}$$

2.2.2 Dirichlet-to-Neuman map

Fix an edge \(\vec {e}\in {\mathcal {E}}(\Lambda )\). Let \(c_{\lambda , \vec {e}}, s_{\lambda , \vec {e}}\), which for \(V_{\vec {e}}=0\) reduce to just \(c_{\lambda , \vec {e}}=\cos (\sqrt{\lambda }\bullet )\) and \(s_{\lambda , \vec {e}}=\sin (\sqrt{\lambda }\bullet )/\sqrt{\lambda }\), be solutions to \(H_{\vec {e}}\psi _{\vec {e}}=\lambda \psi _{\vec {e}}\) with the following boundary condition
$$\begin{aligned} \left( \begin{matrix} c_{\lambda , \vec {e}}(i(\vec {e}))\, &{}s_{\lambda , \vec {e}}(i(\vec {e}))\\ c'_{\lambda , \vec {e}}(i(\vec {e}))\, &{}s'_{\lambda , \vec {e}}(i(\vec {e})) \end{matrix} \right) = \left( \begin{matrix} 1\, \, &{}\quad 0\\ 0\, \, &{}\quad 1 \end{matrix} \right) . \end{aligned}$$
We point out that \(c_{\lambda }(t):=c_{\lambda , \vec {e}}(\kappa _{\vec {e}}^{-1}(t))\) and \(s_{\lambda }(t):=s_{\lambda , \vec {e}}(\kappa _{\vec {e}}^{-1}(t))\) are independent of \(\vec {e}\). They are clearly solutions to \(-\,\psi ^{\prime \prime }+V\psi =\lambda \psi \) on (0, 1), with \(c_{\lambda }(0)=1, c^{\prime }_{\lambda }(0)=0, s_{\lambda }(0)=0, s^{\prime }_{\lambda }(0)=1\), where V is defined in (2.12).
Then for \(\lambda \notin \sigma (H^D)\), namely when \(s_{\lambda }(1)\ne 0\), any \({\mathcal {H}}^2(\vec {e})\)-solution \(\psi _{\lambda , \vec {e}}\) can be written as a linear combination of \(c_{\lambda , \vec {e}}, s_{\lambda , \vec {e}}\)
$$\begin{aligned} \psi _{\lambda , \vec {e}} ({{\varvec{x}}}) = \frac{\psi _{\lambda , \vec {e}}(t(\vec {e}))-\psi _{\lambda , \vec {e}}(i(\vec {e}))c_{\lambda }(1)}{s_{\lambda }(1)}s_{\lambda , \vec {e}}({{\varvec{x}}})+\psi _{\lambda , \vec {e}}(i(\vec {e})) c_{\lambda , \vec {e}}({{\varvec{x}}}). \end{aligned}$$
The Dirichlet-to-Neuman map is defined by
$$\begin{aligned} m(\lambda ) := \frac{1}{s_{\lambda }(1)} \left( \begin{matrix} -\,c_{\lambda }(1) &{}\quad 1 \\ 1 &{}\quad -\,s'_{\lambda }(1) \end{matrix} \right) \end{aligned}$$
with the property that for \(\psi _{\lambda , \vec {e}}\) as in (2.28), one has
$$\begin{aligned} \left( \begin{matrix} \psi ^{\prime }_{\lambda , \vec {e}}(i(\vec {e})) \\ -\,\psi ^{\prime }_{\lambda , \vec {e}}(t(\vec {e})) \end{matrix} \right) = m(\lambda ) \left( \begin{matrix} \psi _{\lambda , \vec {e}}(i(\vec {e})) \\ \psi _{\lambda , \vec {e}}(t(\vec {e})) \end{matrix} \right) . \end{aligned}$$
For the second component, the constancy of the Wronskian is used. Since V(t) is assumed to be even, the intuitive relation
$$\begin{aligned} c_{\lambda }(1)=s'_{\lambda }(1) \end{aligned}$$
remains also true for non-zero potentials.
For \(\lambda \notin \sigma (H^D)\), by expressing \(c_{\lambda }(1)\) in terms of \(\psi _{\lambda ,1, \vec {e}}\) and \(\psi _{\lambda ,2, \vec {e}}\), it follows immediately that
$$\begin{aligned} \eta (\lambda )=s^\prime _{\lambda }(1). \end{aligned}$$

2.2.3 Relation to Hill operators

Using the potential V(t) (2.12), we define the \({\mathbb {Z}}\)-periodic Hill potential \(V_{\text {Hill}} \in L^2_{\text {loc}}({\mathbb {R}}).\)
$$\begin{aligned} V_{\text {Hill}}(t):=V(t\ ({\text {mod}}1)), \end{aligned}$$
for \(t \in {\mathbb {R}}.\) The associated self-adjoint Hill operator on the real line is given by
$$\begin{aligned} \begin{aligned} H_{\text {Hill}}&: {\mathcal {H}}^2({\mathbb {R}}) \subset L^2({\mathbb {R}}) \rightarrow L^2({\mathbb {R}}) \\ H_{\text {Hill}}&\psi :=-\,\psi ''+ V_{\text {Hill}} \psi . \end{aligned} \end{aligned}$$
Then \(c_{\lambda },s_{\lambda }\in {\mathcal {H}}^2(0,1)\), extending naturally to \({\mathcal {H}}^2_{\text {loc}}({\mathbb {R}})\), become solutions to
$$\begin{aligned} H_{\text {Hill}}\psi = \lambda \psi . \end{aligned}$$
The monodromy matrix associated with \(H_{\text {Hill}}\) is the matrix valued function
$$\begin{aligned} Q(\lambda ):= \left( \begin{matrix} c_{\lambda }(1) &{}\quad s_{\lambda }(1) \\ c_{\lambda }'(1) &{}\quad s_{\lambda }'(1) \end{matrix}\right) \end{aligned}$$
and depends by standard ODE theory holomorphically on \(\lambda .\) Its normalized trace
$$\begin{aligned} \Delta ({\lambda }):= \frac{{\text {tr}}(Q(\lambda ))}{2}=s_{\lambda }'(1) \end{aligned}$$
is called the Hill (aka Floquet) discriminant. In the simplest case when \(V_{\text {Hill}}=0,\) the Floquet discriminant is just \(\Delta (\lambda )=\cos \left( \sqrt{\lambda }\right) \) for \(\lambda \ge 0.\)
By the well-known spectral decomposition of periodic differential operators on the line [49], the spectrum of the Hill operator is purely absolutely continuous and satisfies
$$\begin{aligned} \sigma (H_{\text {Hill}})= \left\{ \lambda \in {\mathbb {R}}: \left|\Delta (\lambda ) \right|\le 1 \right\} =\bigcup _{n=1}^{\infty } [\alpha _n,\beta _n] \end{aligned}$$
where \(B_n:=[\alpha _n,\beta _n]\) denotes the n-th Hill band with \(\beta _n \le \alpha _{n+1}\). We have \(\Delta \vert _{{\text {int}}(B_n)}'(\lambda ) \ne 0.\)
Putting (2.32) and (2.37) together, we get the following relation
$$\begin{aligned} \Delta (\lambda )=\eta (\lambda ), \text { for }\lambda \notin \sigma (H^D), \end{aligned}$$
that connects the Hill spectrum with the spectrum of the graphene Hamiltonian.
Fig. 4

The Floquet discriminant for a Mathieu potential \(V(t)=4\cos (6t)\). Energies in shaded regions are inside the band spectrum. Dirichlet eigenvalues are located at the band edges

Also, if \(\lambda \in \sigma (H^D)\), then by the symmetry of the potential, the Dirichlet eigenfunction are either even or odd with respect to \(\frac{1}{2}\). Thus, Dirichlet eigenvalues can only be located at the edges of the Hill bands, see Fig. 4. Namely,
$$\begin{aligned} \Delta (\lambda )=\pm \, 1, \text { for }\lambda \in \sigma (H^D). \end{aligned}$$

2.2.4 Spectral decomposition

The singular continuous spectrum of H is empty by the direct integral decomposition (2.20) [26]. Due to Thomas [51] there is the characterization, stated also in [37, Corollary 6.11], of the pure point spectrum of fibered operators: \(\lambda \) is in the pure point spectrum iff the set \(\{ {{\varvec{k}}} \in {\mathbb {T}}^{*}_2; \lambda _j({{\varvec{k}}})=\lambda \}\) has positive measure where \(\lambda _j({{\varvec{k}}})\) is the j-th eigenvalue of \(H({{\varvec{k}}})\). Away from the Dirichlet spectrum, the condition \({\mathbb {R}} \ni \lambda =\lambda _j({{\varvec{k}}})\) is by (2.25) equivalent to \(\Delta (\lambda )^2 = \frac{\left|1+e^{ik_1}+e^{ik_2} \right|^2}{9}.\) Yet, the level-sets of this function are of measure zero. The spectrum of H away from the Dirichlet spectrum is therefore purely absolutely continuous. The Dirichlet spectrum coincides with the point spectrum of H and is spanned by so-called loop states that consist of six Dirichlet eigenfunctions wrapped around each hexagon of the lattice [39, Theorem 3.6(v)]. Hence, the spectral decomposition in the case without magnetic field is given by

Theorem 6

The spectra of \(\sigma (H)\) and \(\sigma (H_{\text {Hill}})\) coincide as sets. Aside from the Dirichlet contribution to the spectrum, H has absolutely continuous spectrum as in Fig. 5 with conical cusps at the points (Dirac points) where two bands on each Hill band meet. The Dirichlet spectrum is contained in the spectrum of H, is spanned by loop states supported on single hexagons, and is thus infinitely degenerated.

Fig. 5

The first two bands of the Schrödinger operator on the graph with Mathieu potential \(V(t)=20\cos (2\pi t)\) and no magnetic field showing the characteristic conical Dirac points where the two differently colored bands touch. The two bands are differently colored

2.3 One-dimensional quasi-periodic Jacobi matrices

The proof of the main Theorems will involve the study of a one-dimensional quasi-periodic Jacobi matrix. We include several general facts that will be useful.

Let \(H_{\Phi , \theta }\in {\mathcal {L}}(l^2({\mathbb {Z}}))\) be a quasi-periodic Jacobi matrix, that is given by
$$\begin{aligned} (H_{\Phi , \theta }u)_m= & {} c\left( \theta +m\frac{\Phi }{2\pi } \right) u_{m+1}+\overline{c\left( \theta +(m-1)\frac{\Phi }{2\pi }\right) }u_{m-1}\nonumber \\&+\,v\left( \theta +m \frac{\Phi }{2\pi }\right) u_m. \end{aligned}$$
Let \(\Sigma _{\Phi , \theta }:=\sigma (H_{\Phi , \theta })\) be the spectrum of \(H_{\Phi ,\theta }\) and \(\Sigma _{\Phi }=\bigcup _{\theta \in {\mathbb {T}}_1}\Sigma _{\Phi , \theta }\). It is a well known result that for irrational \(\frac{\Phi }{2\pi }\), the set \(\Sigma _{\Phi , \theta }\) is independent of \(\theta \), thus \(\Sigma _{\Phi , \theta }={{\Sigma _{\Phi }}}\). It is also well known that, for any \(\Phi \), \(\Sigma _{\Phi }\) has no isolated points.7

2.3.1 Transfer matrix and Lyapunov exponent

We assume that \(c(\theta )\) has finitely many zeros (counting multiplicity), and label them as \(\theta _1, \theta _2,\ldots , \theta _m\).8 Let \(\Theta :=\cup _{j=1}^m \cup _{k\in {\mathbb {Z}}}\left\{ \theta _j+k\frac{\Phi }{2\pi }\right\} \), in particular if \(\frac{\Phi }{2\pi }\in {\mathbb {Q}}\), then \(\Theta \) is a finite set in \({\mathbb {T}}\).

For \(\theta \notin \Theta \), the eigenvalue equation \(H_{\Phi , \theta }u=\lambda u\) has the following dynamical reformulation:
$$\begin{aligned} \left( \begin{matrix} u_{n+1}\\ u_n \end{matrix} \right) = A^{\lambda }\left( \theta +n\frac{\Phi }{2\pi } \right) \left( \begin{matrix} u_n\\ u_{n-1} \end{matrix} \right) , \end{aligned}$$
$$\begin{aligned} \text {GL}(2, {\mathbb {C}})\ni A^{\lambda }(\theta )= \frac{1}{c(\theta )} \left( \begin{matrix} \lambda -v(\theta )\, &{}-\overline{c(\theta -\frac{\Phi }{2\pi })}\\ c(\theta )\, &{}0 \end{matrix} \right) \end{aligned}$$
is called the transfer matrix. Let
$$\begin{aligned} A^{\lambda }_{n}(\theta )=A^{\lambda }(\theta +(n-1)\frac{\Phi }{2\pi })\cdots A^{\lambda }(\theta +\frac{\Phi }{2\pi })A^{\lambda }(\theta ) \end{aligned}$$
be the n-step transfer matrix.
We define the Lyapunov exponent of \(H_{\Phi , \theta }\) at energy \(\lambda \) as
$$\begin{aligned} L(\lambda , \Phi ):=\lim _{n\rightarrow \infty }\frac{1}{n}\int _{{\mathbb {T}}_1}\log {\Vert A^{\lambda }_n(\theta )\Vert }\ \text {d}\theta . \end{aligned}$$
By a trivial bound \(\Vert A\Vert ^2\ge |\det {A}|\), which comes from the fact A is a \(2\times 2\) matrix, we get
$$\begin{aligned} L(\lambda , \Phi )\ge \lim _{n\rightarrow \infty }\frac{1}{2n}\int _{{\mathbb {T}}_1}\log \left( \frac{|c(\theta -\frac{\Phi }{2\pi })|}{|c(\theta +(n-1)\frac{\Phi }{2\pi }|}\right) \ \text {d}\theta =0. \end{aligned}$$

2.3.2 Normalized transfer matrix

Let \(|c(\theta )|=\sqrt{c(\theta )\overline{c(\theta )}}\). We introduce the normalized transfer matrix:
$$\begin{aligned} \text {SL}(2, {\mathbb {R}})\ni {\tilde{A}}^{\lambda }(\theta )= \frac{1}{\sqrt{|c(\theta )||c(\theta -\frac{\Phi }{2\pi })|}} \left( \begin{matrix} \lambda -v(\theta )\, &{}-|c(\theta -\frac{\Phi }{2\pi })|\\ |c(\theta )|\, &{}0 \end{matrix} \right) \end{aligned}$$
and the n-step normalized transfer matrix\({\tilde{A}}^{\lambda }_{n}(\theta )\).
The following connection between \(A^{\lambda }\) and \({\tilde{A}}^{\lambda }\) is clear:
$$\begin{aligned} {\tilde{A}}^{\lambda }(\theta )= \frac{c(\theta )}{\sqrt{|c(\theta )||c(\theta -\frac{\Phi }{2\pi })|}} \left( \begin{matrix} 1\, &{}0\\ 0\, &{}{{\frac{\overline{{c}(\theta )}}{|c(\theta )|}}} \end{matrix} \right) A^{\lambda }(\theta ) \left( \begin{matrix} 1\, &{}0\\ 0\, &{}{{\frac{\overline{{c}(\theta -\frac{\Phi }{2\pi })}}{|c(\theta -\frac{\Phi }{2\pi })|}}} \end{matrix} \right) ^{-1}. \end{aligned}$$
When \(\frac{\Phi }{2\pi }=\frac{p}{q}\) is rational, (2.47) yields
$$\begin{aligned} {{\,\mathrm{tr}\,}}({\tilde{A}}^{\lambda }_q(\theta ))=\frac{\prod _{j=0}^{q-1}c(\theta +j\frac{p}{q})}{\prod _{j=0}^{q-1}|c(\theta +j\frac{p}{q})|}{{\,\mathrm{tr}\,}}(A^{\lambda }_q(\theta )). \end{aligned}$$
$$\begin{aligned} D^{\lambda }(\theta )=c(\theta )A^{\lambda }(\theta )=\left( \begin{matrix} \lambda -v(\theta )\, &{}-\overline{c(\theta -\frac{\Phi }{2\pi })}\\ c(\theta )\, &{}0 \end{matrix} \right) \end{aligned}$$
and \(D_n^\lambda (\theta )=D^\lambda (\theta +(n-1)\frac{\Phi }{2\pi })\cdots D^\lambda (\theta +\frac{\Phi }{2\pi })D^\lambda (\theta )\). Then when \(\frac{\Phi }{2\pi }=\frac{p}{q}\) is rational, (2.48) becomes
$$\begin{aligned} {{\,\mathrm{tr}\,}}({\tilde{A}}^{\lambda }_q(\theta ))=\frac{{{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta ))}{\prod _{j=0}^{q-1}|c(\theta +j\frac{p}{q})|}. \end{aligned}$$
Note that although \(A^{\lambda }_n(\theta )\) is not well-defined for \(\theta \in \Theta \), \(D^{\lambda }_n(\theta )\) is always well-defined.

3 Magnetic Hamiltonians on quantum graphs

3.1 Magnetic potential

Given a vector potential \(A({{\varvec{x}}})=A_1(x_1, x_2)\ dx_1+A_2(x_1, x_2)\ dx_2 \in \Omega ^1({\mathbb {R}}^2)\), the scalar potential \(A_{\vec {e}} \in C^{\infty }(\vec {e})\) along edges \(\vec {e} \in {\mathcal {E}}(\Lambda )\) is obtained by evaluating the form A on the graph along the vector field generated by edges \([\vec {e}] \in {\mathcal {E}}(W_{\Lambda })\)
$$\begin{aligned} A_{\vec {e}}(x) := A(x)\left( [\vec {e}]_1 \partial _1 + [\vec {e}]_2 \partial _2 \right) . \end{aligned}$$
The integrated vector potentials are defined as \(\beta _{\vec {e}}:=\int _{\vec {e}} A_{\vec {e}}({{\varvec{x}}}) d{{\varvec{x}}}\) for \(\vec {e} \in {\mathcal {E}}(\Lambda ).\)

Assumption 1

The magnetic flux \(\Phi \) through each hexagon Open image in new window of the latticeis assumed to be constant.
Let us mention that the assumption above is equivalent to the following equation, in terms of the integrated vector potentials
$$\begin{aligned}&\beta _{\gamma _1, \gamma _2, \vec {f}}-\beta _{\gamma _1, \gamma _2+1, \vec {h}}+\beta _{\gamma _1, \gamma _2+1, \vec {g}}-\beta _{\gamma _1-1, \gamma _2+1,\vec {f}}+\beta _{\gamma _1-1, \gamma _2+1,\vec {h}}-\beta _{\gamma _1, \gamma _2, \vec {g}}\nonumber \\&\quad =\Phi , \end{aligned}$$
for any \(\gamma _1, \gamma _2\in {\mathbb {Z}}\).

Example 1

The vector potential \(A \in \Omega ^1({\mathbb {R}}^2)\) of a homogeneous magnetic field \(B \in \Omega ^2({\mathbb {R}}^2)\)
$$\begin{aligned} B(x)=B_0 \ dx_1 \wedge dx_2 \end{aligned}$$
can be chosen as
$$\begin{aligned} A(x):=B_0 x_1 \ dx_2. \end{aligned}$$
This scalar potential is invariant under \(\vec {b}_2\)-translations. The integrated vector potentials \(\beta _{\vec {e}}\) are given by
$$\begin{aligned} \beta _{\gamma _1, \gamma _2, \vec {f}}= \frac{\Phi }{2}\left( \gamma _1+\frac{1}{6} \right) ,\ \beta _{\gamma _1, \gamma _2, \vec {g}}=0, \text { and } \beta _{\gamma _1, \gamma _2, \vec {h}}=-\beta _{\gamma _1, \gamma _2, \vec {f}}, \end{aligned}$$
where, in this case, the magnetic flux through each hexagon is \(\Phi =\frac{3\sqrt{3}}{2}B_0\).

3.2 Magnetic differential operator and modified Peierls’ substitution

In terms of the magnetic differential operator \((D^B \psi )_{\vec {e}}:= -\, i \psi _{\vec {e}}' - A_{\vec {e}}\psi _{\vec {e}} \), the Schrödinger operator modeling graphene in a magnetic field reads
$$\begin{aligned} \begin{aligned} H^B&:D(H^B) \subset L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) )\\ (H^B&\psi )_{\vec {e}}:=(D^B D^B \psi )_{\vec {e}}+V_{\vec {e}}\psi _{\vec {e}}, \end{aligned} \end{aligned}$$
and is defined on
$$\begin{aligned} \begin{aligned} D(H^B):= \Bigg \{&\psi \in {\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ): \psi _{\vec {e}_1}({{\varvec{v}}})=\psi _{\vec {e}_2}({{\varvec{v}}}) \text { for any } \vec {e}_1,\vec {e}_2 \in {\mathcal {E}}_{{{\varvec{v}}}}(\Lambda ) \\&\text { and } \sum _{\vec {e} \in {\mathcal {E}}_{{{\varvec{v}}}}(\Lambda )} \left( D^B\psi \right) _{\vec {e}}({{\varvec{v}}}) = 0 \Bigg \}. \end{aligned} \end{aligned}$$
Let us first introduce a unitary operator U on \(L^2({\mathcal {E}}(\Lambda ))\), defined as
$$\begin{aligned} U\psi _{\gamma _1, \gamma _2, \vec {e}}=\zeta _{\gamma _1, \gamma _2}\psi _{\gamma _1, \gamma _2, \vec {e}}\ \text { for } \vec {e}=\vec {f},\vec {g},\vec {h}, \end{aligned}$$
the factors \(\zeta _{\gamma _1, \gamma _2}\) are defined as follows. First, choose a path \(p(\cdot ): {\mathbb {N}}\rightarrow {\mathbb {Z}}^2\) connecting (0, 0) to \((\gamma _1, \gamma _2)\) with
$$\begin{aligned} p(0)=(0,0)\ \text { and }\ p(|\gamma _1|+|\gamma _2|)=(\gamma _1, \gamma _2). \end{aligned}$$
Note that (3.10) implies that both components of \(p(\cdot )\) are monotonic functions. Then we define \(\zeta _{\gamma _1, \gamma _2}\) recursively through the following relations along \(p(\cdot )\):
$$\begin{aligned} \begin{aligned} \zeta _{0, 0}&=1, \\ \zeta _{\gamma _1+1, \gamma _2}&=e^{i\beta _{\gamma _1, \gamma _2, \vec {f}}\ -i\beta _{\gamma _1+1, \gamma _2, \vec {g}}} \zeta _{\gamma _1, \gamma _2}, \\ \zeta _{\gamma _1, \gamma _2+1}&=e^{i\beta _{\gamma _1, \gamma _2, \vec {f}}\ -i\beta _{\gamma _1, \gamma _2+1, \vec {h}}\ -i\Phi \gamma _1}\zeta _{\gamma _1, \gamma _2}. \end{aligned} \end{aligned}$$
Due to (3.3), it is easily seen that the definition of \(\zeta _{\gamma _1, \gamma _2}\) is independent of the choice of \(p(\cdot )\), hence is well-defined.
The unitary Peierls’ substitution9 is the multiplication operator
$$\begin{aligned} \begin{aligned} P:&L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) ) \\&(\psi _{\vec {e}}) \mapsto \left( \left( \vec {e}\ni {{\varvec{x}}}\mapsto e^{i \int _{i(\vec {e}) \rightarrow {{\varvec{x}}}} A_{\vec {e}}(\varvec{s})d\varvec{s}}\right) \psi _{\vec {e}} \right) , \end{aligned} \end{aligned}$$
where \(i(\vec {e})\rightarrow {{\varvec{x}}}\) denotes the straight line connecting \(i(\vec {e})\) with \({{\varvec{x}}} \in \vec {e}\). It reduces the magnetic Schrödinger operator to non-magnetic ones with magnetic contribution moved into boundary condition, with multiplicative factors at terminal edges given by \(e^{i\beta _{\vec {e}}}\).
We will define a modified Peierls’ substitution that allows us to reduce the number of non-trivial multiplicative factors to one, by taking
$$\begin{aligned} \begin{aligned} {\tilde{P}}=PU. \end{aligned} \end{aligned}$$
It transforms \(H^B\) into
$$\begin{aligned} \Lambda ^B:=\left( -\, \frac{d^2}{dt_{\vec {e}}^2}+V_{\vec {e}} \right) _{\vec {e} \in {\mathcal {E}}(\Lambda )}={{\tilde{P}}}^{-1} H^B {\tilde{P}}. \end{aligned}$$
The domain of \(\Lambda ^B\) is
$$\begin{aligned} \begin{aligned} D({\Lambda }^B)=\Bigg \{&\psi \in {\mathcal {H}}^2({\mathcal {E}}(\Lambda )):\ \text {any } \vec {e}_1, \vec {e}_2\in {\mathcal {E}}(\Lambda ) \text { with } i(\vec {e}_1)=i(\vec {e}_2)={{\varvec{v}}} \text { satisfy} \\&\ \psi _{\vec {e}_1}({{\varvec{v}}})=\psi _{\vec {e}_2}({{\varvec{v}}}) \text { and } \sum _{i(\vec {e})={{\varvec{v}}}} \psi ^\prime _{\vec {e}}({{\varvec{v}}})=0; \text { whilst at edges for which }\\&\, t(\vec {e}_1)=t(\vec {e}_2)={{\varvec{v}}}, e^{i\tilde{\beta }_{\vec {e}_1}}\psi _{\vec {e}_1}({{\varvec{v}}})=e^{i\tilde{\beta }_{\vec {e}_2}}\psi _{\vec {e}_2}({{\varvec{v}}}) \text { and }\\&\sum _{t(\vec {e})={{\varvec{v}}}}e^{i\tilde{\beta }_{\vec {e}}}\psi ^\prime _{\vec {e}}({{\varvec{v}}})=0 \Bigg \}, \end{aligned} \end{aligned}$$
$$\begin{aligned} \tilde{\beta }_{\gamma _1, \gamma _2, \vec {g}}\equiv \tilde{\beta }_{\gamma _1, \gamma _2, \vec {f}}\equiv 0\ \text { and }\ \tilde{\beta }_{\gamma _1, \gamma _2, \vec {h}}=-\,\Phi \gamma _1. \end{aligned}$$
Thus, the problem reduces to the study of non-magnetic Schrödinger operators with magnetic contributions moved into the boundary conditions.
Observe that the magnetic Dirichlet operator
$$\begin{aligned} \begin{aligned} H^{D,B}&:\bigoplus _{\vec {e} \in {\mathcal {E}}(\Lambda )}\left( {\mathcal {H}}_0^1(\vec {e})\cap {\mathcal {H}}^2(\vec {e}) \right) \subset L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) ) \\ (H^{D,B}&\psi )_{\vec {e}}:=(D^B D^B \psi )_{\vec {e}}+V_{\vec {e}}\psi _{\vec {e}} \end{aligned} \end{aligned}$$
is by the (modified) Peierls’ substitution unitary equivalent to the Dirichlet operator without magnetic field
$$\begin{aligned} H^D={\tilde{P}}^{-1}H^{D,B}{\tilde{P}}=P^{-1}H^{D,B}P. \end{aligned}$$
Consequently, the spectrum of the Dirichlet operator \(H^D\) is invariant under perturbations by the magnetic field.

4 Main lemmas

First, let us introduce the following two-dimensional tight-binding Hamiltonian
$$\begin{aligned} Q_{\Lambda }(\Phi ) := \frac{1}{3} \left( \begin{matrix} 0 &{}&{}\quad 1+\tau _0+\tau _1 \\ \left( 1 +\tau _0+\tau _1 \right) ^* &{}&{}\quad 0 \end{matrix} \right) \end{aligned}$$
with translation operators \(\tau _0, \tau _1 \in {\mathcal {L}}(l^2({\mathbb {Z}}^2; {\mathbb {C}}))\) which for \({{\varvec{\gamma }}} \in {\mathbb {Z}}^2\) and \(u \in l^2({\mathbb {Z}}^2;{\mathbb {C}})\) are defined as
$$\begin{aligned} (\tau _0(u))_{\gamma _1, \gamma _2}:= u_{\gamma _1-1,\gamma _2} \text { and } (\tau _1(u))_{\gamma _1, \gamma _2}:= e^{-i \Phi \gamma _1}u_{\gamma _1,\gamma _2-1}. \end{aligned}$$
The following lemma connects the spectrum of \(H^B\) with \(\sigma (Q_{\Lambda })\). We have

Lemma 4.1

Let \(\Delta (\lambda )\) be the Hill discriminant defined in (2.37). A number \(\lambda \in \rho (H^{D})\) lies in \(\sigma (H^B)\) iff \(\Delta (\lambda )\in \sigma (Q_{\Lambda }(\Phi ))\). Such \(\lambda \) is in the point spectrum of \(H^B\) iff \(\Delta (\lambda )\in \sigma _p(Q_{\Lambda }(\Phi ))\).

Remark 4

We will show in Lemma 5.2 that \(\sigma _p(Q_{\Lambda }(\Phi ))\) is empty, thus \(H^B\) has no point spectrum away from \(\sigma (H^D)\).

Lemma 4.2 below shows \(\sigma (Q_{\Lambda }(\Phi ))\) is a zero-measure Cantor set for irrational flux \(\frac{\Phi }{2\pi }\), Lemma 4.3 gives a measure estimate for rational flux, and Lemma 4.4 provides an upper bound on the Hausdorff dimension of the spectrum of \(Q_{\Lambda }(\Phi )\). These three lemmas prove the topological structure part of Theorem 3.

Lemma 4.2

For \(\frac{\Phi }{2\pi } \in {\mathbb {R}}{\setminus } {\mathbb {Q}}\), \(\sigma (Q_{\Lambda }(\Phi ))\) is a zero-measure Cantor set.

Lemma 4.3

If \(\frac{\Phi }{2\pi }=\frac{p}{q}\) is a reduced rational number, then \(\sigma (Q_{\Lambda }(\Phi ))\) is a finite union of intervals, with measure estimate
$$\begin{aligned} |\sigma (Q_{\Lambda }(\Phi ))|\le \frac{8\sqrt{6 \pi }}{9\sqrt{q}}. \end{aligned}$$

Lemma 4.4

For generic \(\Phi \), the Hausdorff dimension of \(\sigma (Q_{\Lambda }(\Phi ))\) is \(\le \frac{1}{2}\).

5 Reduction of \(Q_{\Lambda }(\Phi )\) to a one-dimensional Jacobi matrix

5.1 Symmetric property of \(Q_{\Lambda }\)

Lemma 5.1

The spectrum of \(Q_{\Lambda }\) has the following properties:
  1. (1)

    \(\sigma (Q_{\Lambda }(\Phi ))\) is symmetric with respect to 0.

  2. (2)

    \(0\in \sigma (Q_{\Lambda }(\Phi ))\).



(1). Conjugating \(Q_{\Lambda }\) in (4.1) by
$$\begin{aligned} \Omega = \left( \begin{matrix} -{\text {id}} &{}\quad 0\\ 0 &{}\quad {\text {id}} \\ \end{matrix}\right) \end{aligned}$$
shows that \(\sigma (Q_{\Lambda }(\Phi ))\) is symmetric with respect to 0 [40, Prop. 3.5].
(2). If we view \(Q_{\Lambda }(\Phi )\) as an operator-valued function of the flux \(\Phi ,\) then
$$\begin{aligned} \Phi \mapsto \langle Q_{\Lambda }(\Phi )x,y \rangle , \end{aligned}$$
for \(x,y \in c_{00}\) arbitrary, is analytic and \(Q_{\Lambda }\) therefore is a bounded analytic map. If there was \({\Phi _0}/{2\pi } \in {\mathbb {R}} \backslash {\mathbb {Q}}\) where \(Q_{\Lambda }(\Phi _0)\) was invertible, then \(Q_{\Lambda }(\Phi )\) would also be invertible in a sufficiently small neighborhood of \(\Phi _0\) (e.g. [38, Ch.7.1]). Yet, in [28, Prop.4.1] it has been shown that for rational \(\Phi /2\pi \), \(0 \in \sigma (Q_{\Lambda }(\Phi )).\) Thus, by density \(0 \in \sigma (Q_{\Lambda }(\Phi ))\), independent of \(\Phi \in {\mathbb {R}}.\)\(\square \)

5.2 Reduction to the one-dimensional Hamiltonian

Relating the spectrum of \(Q_{\Lambda }\) to that of \(Q_{\Lambda }^2\), we obtain the following characterization of \(\sigma (Q_{\Lambda })\).

Lemma 5.2

  1. (1)
    The spectrum of the operator \(Q_{\Lambda }(\Phi )\) as a set is given by
    $$\begin{aligned} \sigma (Q_{\Lambda }(\Phi ))= \pm \, \sqrt{\frac{\bigcup _{\theta \in {\mathbb {T}}_1}\sigma (H_{\Phi , \theta })}{9}+\frac{1}{3}} \bigcup \left\{ 0\right\} . \end{aligned}$$
    where \(H_{\Phi , \theta } \in {\mathcal {L}}( l^2({\mathbb {Z}}) )\) is the one-dimensional quasi-periodic Jacobi matrix defined as in (2.41) with
    $$\begin{aligned} c(\theta )=1+e^{-2\pi i \theta },\ \ \text {and}\ \ \ v(\theta )=2\cos {2\pi \theta }. \end{aligned}$$
  2. (2)

    \(Q_{\Lambda }(\Phi )\) has no point spectrum.



(1). Let \(A:=\frac{1}{3}\left( 1+\tau _0+\tau _1\right) \). Then squaring the operator \(Q_{\Lambda }(\Phi )\) yields
$$\begin{aligned} Q_{\Lambda }^2(\Phi )=\left( \begin{matrix} AA^* &{} 0 \\ 0 &{} A^*A \end{matrix} \right) . \end{aligned}$$
The spectral mapping theorem implies that \(\sigma (Q_{\Lambda }^2(\Phi ))=\sigma (Q_{\Lambda }(\Phi ))^2\) and from Lemma 5.2 we conclude that \(\sigma (Q_{\Lambda }(\Phi ))= \pm \, \sqrt{\sigma (Q_{\Lambda }^2(\Phi ))}.\) Clearly, the operator \(AA^* \vert _{{\text {ker}}(A^*)^{\perp }}\) and \(A^*A \vert _{{\text {ker}}(A)^{\perp }}\) are unitarily equivalent. Thus, the spectrum can be expressed by
$$\begin{aligned} \sigma (Q_{\Lambda }(\Phi ))=\pm \, \sqrt{\sigma (AA^*)} \cup \left\{ 0\right\} \end{aligned}$$
where we are able to use either of the two (\(AA^*\) or \(A^*A\)) since \(0 \in \sigma (Q_{\Lambda }(\Phi ))\) due to Lemma 5.1.
Then, it follows that
$$\begin{aligned} AA^*&= \frac{{\text {id}}}{3}+\frac{{\text {id}}}{9}\underbrace{\left( (\tau _0+\tau _0^*)+(\tau _1+\tau _1^*)+\tau _0\tau _1^* + \tau _1\tau _0^*\right) }_{=:H_{\Phi }}. \end{aligned}$$
Observe that
$$\begin{aligned} H_{\Phi }\psi _{m,n}&= \psi _{m-1,n}+\psi _{m+1,n}+e^{- i \Phi m } \psi _{m,n-1}+e^{i \Phi m} \psi _{m,n+1} \nonumber \\&\quad +\,e^{ i \Phi (m-1)} \psi _{m-1,n+1}+e^{-i \Phi m}\psi _{m+1,n-1}. \end{aligned}$$
Since \(H_{\Phi }\) is invariant under discrete translations in n, the operator is unitarily equivalent to the direct integral operator \(\int _{{\mathbb {T}}_1}^{\oplus } H_{\Phi , \theta }\ d \theta \), which gives the claim.

(2). It follows from a standard argument that the two dimensional operator \(H_{\Phi }\) has no point spectrum. Indeed, assume \(H_{\Phi }\) has point spectrum at energy E, then \(H_{\Phi , \theta }\) would have the same point spectrum E for a.e. \(\theta \in {\mathbb {T}}_1\). This implies the integrated density of states of \(H_{\Phi , \theta }\) has a jump discontinuity at E, which is impossible. Therefore the point spectrum of \(H_{\Phi }\) is empty, hence the same holds for \(Q_{\Lambda }(\Phi )\). \(\square \)

6 Proof of Lemmas 4.24.3 and 4.4

For a set U, let \(\text {dim}_H(U)\) be its Hausdorff dimension.

Lemma 4.2 follows as a direct consequence of (5.3) and the following Theorem 7. Let \(\Sigma _{\Phi }\) be defined as in Sect. 2.3.

Theorem 7

For \(\frac{\Phi }{2\pi } \in {\mathbb {R}}\backslash {\mathbb {Q}}\), \(\Sigma _{\Phi }\) is a zero-measure Cantor set.

We will postpone the proof of Theorem 7 till the end of this section. We will first present the proofs of Lemmas 4.3 and 4.4, which are based on the following three lemmas. First, we have

Lemma 6.1

Let \(\frac{\Phi }{2\pi }=\frac{p}{q}\) be a reduced rational number, then \(\Sigma _{2\pi p/q}\) is a union of q (possibly touching) bands with \(|\Sigma _{2\pi p/q}|<\frac{16\pi }{3q}\).

Lemma 6.1 will be proved in Sects. 6.4 and 6.5 after some further preparation. The following lemma addresses the continuity of the spectrum \(\Sigma _{\Phi }\) in \(\Phi \), extending a result of [9] (see Proposition 7.1 therein) from quasiperiodic Schrödinger operators to Jacobi matrices.

Lemma 6.2

There exist absolute constants \(C_1, C_2>0\) such that if \(\lambda \in \Sigma _{\Phi }\) and \(|\Phi -\Phi ^\prime |<C_1\), then there exists \(\lambda ^\prime \in \Sigma _{\Phi ^\prime }\) such that
$$\begin{aligned} |\lambda -\lambda ^\prime | \le C_2|\Phi -\Phi '|^{\frac{1}{2}}. \end{aligned}$$

We will prove Lemma 6.2 in “Appendix C”.

The next lemma provides an upper bound on the Hausdorff dimension of a set.

Lemma 6.3

(Lemma 5.1 of [42]) Let \(S\subset {\mathbb {R}}\), and suppose that S has a sequence of covers: \(\{S_n\}_{n=1}^{\infty }\), \(S\subset S_n\), such that each \(S_n\) is a union of \(q_n\) intervals, \(q_n\rightarrow \infty \) as \(n\rightarrow \infty \), and for each n,
$$\begin{aligned} |S_n|<\frac{C}{q_n^{\beta }}, \end{aligned}$$
where \(\beta \) and C are positive constants, then
$$\begin{aligned} \text {dim}_H(S)\le \frac{1}{1+\beta }. \end{aligned}$$

6.1 Proof of Lemma 4.3

The fact that \(\sigma (Q_{\Lambda }(\Phi ))\) is a finite union of intervals follows from (5.3) and Lemma 6.1.

It suffices to prove the measure estimate. It is clear that for any \(\varepsilon >0\), we have
$$\begin{aligned} \sqrt{\Sigma _{2 \pi p/q}+3}\subseteq [\ 0, \sqrt{\varepsilon }\ ]\bigcup \sqrt{\left( \Sigma _{2 \pi p/q}+3\right) \bigcap \ (\varepsilon , \infty )}. \end{aligned}$$
Hence by Lemma 6.1, we have
$$\begin{aligned} |\sqrt{\Sigma _{2 \pi p/q}+3}|\le \sqrt{\varepsilon }+\frac{|\Sigma _{2\pi p/q}|}{2\sqrt{\varepsilon }}\le \sqrt{\varepsilon }+\frac{8\pi }{3\sqrt{\varepsilon } q}. \end{aligned}$$
Optimizing in \(\varepsilon \) leads to
$$\begin{aligned} |\sqrt{\Sigma _{2 \pi p/q}+3}|\le \frac{4\sqrt{6\pi }}{3\sqrt{q}}. \end{aligned}$$
Then (5.3) implies
$$\begin{aligned} |\sigma (Q_{\Lambda }(2\pi p/q))| \le \frac{8\sqrt{6 \pi }}{9\sqrt{q}}. \end{aligned}$$
\(\square \)

6.2 Proof of Lemma 4.4

We will show that if \(\frac{\Phi }{2\pi }\) is an irrational obeying
$$\begin{aligned} q_n^4 \left| \frac{\Phi }{2\pi }-\frac{p_n}{q_n}\right| <C, \end{aligned}$$
for some constant C, and a sequence of reduced rationals \(\{p_n/q_n\}\) with \(q_n\rightarrow \infty \), then \(\dim _H(\sigma (Q_{\Lambda }(\Phi )))\le 1/2\). It is easy to see that the \(\Phi \)’s satisfying (6.2) form a dense \(G_{\delta }\) set of \({\mathbb {R}}\), hence is generic.

Without loss of generality, we may assume \(\frac{\Phi }{2\pi }\in (0,1)\).

First, by (5.3), we have that
$$\begin{aligned} \text {dim}_H(\sigma (Q_{\Lambda }(\Phi )))=\sup _{k\ge 2}\ \text {dim}_H\left( \pm \, \sqrt{\left( \frac{\Sigma _{\Phi }}{9}+\frac{1}{3}\right) \cap [\frac{1}{k}, 1]}\right) , \end{aligned}$$
where we used a trivial bound \(\Vert H_{\Phi , \theta }\Vert \le 6.\) Hence it suffices to show that for each \(k\ge 2\),
$$\begin{aligned} \text {dim}_H\left( \sqrt{\left( \frac{\Sigma _{\Phi }}{9}+\frac{1}{3}\right) \cap [\frac{1}{k}, 1]}\right) \le \frac{1}{2}. \end{aligned}$$
The rest of the argument is similar to that of [42]. By Lemma 6.2, taking any \(\lambda \in \Sigma _{\Phi }\), for \(n\ge n_0\), there exists \(\lambda ^\prime \in \Sigma _{2\pi p_n/q_n}\) such that \(|\lambda -\lambda ^\prime |\le {{C_2}} |\frac{\Phi }{2\pi }-\frac{p_n}{q_n}|^{\frac{1}{2}}\). This means \(\Sigma _{\Phi }\) is contained in the \({{C_2}} |\frac{\Phi }{2\pi }-\frac{p_n}{q_n}|^{\frac{1}{2}}\) neighbourhood of \(\Sigma _{2\pi p_n/q_n}\). By Lemma 6.1, \(\Sigma _{2\pi p_n/q_n}\) has \(q_n\) (possibly touching) bands with total measure \(|\Sigma _{2\pi p_n/q_n}|\le \frac{16\pi }{3q_n}\). Hence \(\Sigma _{\Phi }\) has cover \(S_n\) such that \(S_n\) is a union of (at most) \(q_n\) intervals with total measure
$$\begin{aligned} |S_n|\le \frac{16\pi }{3q_n}+ 2{{C_2}} q_n \left| \frac{\Phi }{2\pi }-\frac{p_n}{q_n}\right| ^{\frac{1}{2}}. \end{aligned}$$
Since \(q_n^4 \left| \frac{\Phi }{2\pi }-\frac{p_n}{q_n}\right| \le C\), we have, by (6.4),
$$\begin{aligned} |S_n|\le \frac{16\pi }{3q_n}+\frac{2C_2\sqrt{C}}{q_n}=:\frac{{\tilde{C}}}{q_n}. \end{aligned}$$
This implies \(\left( \frac{\Sigma _{\Phi }}{9}+\frac{1}{3} \right) \cap [\frac{1}{k}, 1]\) has cover \({\tilde{S}}_n\) such that \({\tilde{S}}_n\) is a union of (at most) \(q_n\) intervals with total measure
$$\begin{aligned} |{\tilde{S}}_n|\le \frac{\sqrt{k}{\tilde{C}}}{2q_n}. \end{aligned}$$
Then Lemma 6.3 yields (6.3). \(\square \)

6.3 Proof of Theorem 7

Note that Lemmas 6.1 and 6.2 already imply zero measure (and thus Cantor nature) of the spectrum for fluxes \(\Phi /2\pi \) with unbounded coefficients in the continued fraction expansion, thus for a.e. \(\Phi \), by an argument similar to that used in the proof of Lemma 4.4. However extending the result to the remaining measure zero set this way would require a slightly stronger continuity in Lemma 6.2, which is not available. We circumvent this by the following strategy:

(1). Use quantization of acceleration techniques to prove the Lyapunov exponent of operator \(H_{\Phi ,\theta }\) identically vanishes on the spectrum, see Proposition 6.4;

(2). employ the singularity of the Jacobi matrix to show the absolutely continuous spectrum of \(H_{\Phi ,\theta }\) is empty, see Proposition 6.5;

(3). apply Kotani theory for Jacobi matrices, see Theorem 8.

Let \(\Sigma _{ac}(H_{\Phi , \theta })\) be the absolutely continuous spectrum of \(H_{\Phi , \theta }\). Let \(L(\lambda , \Phi )\) be the Lyapunov exponent of \(H_{\Phi , \theta }\) at energy \(\lambda \), as defined in (2.44). For a set \(U\subseteq {\mathbb {R}}\), let \({\overline{U}}^{ess}\) be its essential closure.

First, we are able to give a characterization of the Lyapunov exponent on the spectrum.

Proposition 6.4

For \(\frac{\Phi }{2\pi }\in {\mathbb {R}}{\setminus } {\mathbb {Q}}\), \(L(\lambda , \Phi )=0\) if and only if \(\lambda \in \Sigma _{\Phi }\).

The proof of this is similar to that for the almost Mathieu operator as given in [2] and the extended Harper’s model [34]. The general idea is to complexify \(\theta \) to \(\theta +i\varepsilon \), and obtain asymptotic behavior of the Lyapunov exponent when \(|\varepsilon |\rightarrow \infty \). Convexity and quantization of the acceleration (see Theorem 5 of [2]) then bring us back to the \(\varepsilon =0\) case. We will give the proof in “Appendix A”.

Exploiting the fact that \(c(\theta )=1+e^{-2\pi i \theta }\) has a real zero \(\theta _1=\frac{1}{2}\), we have

Proposition 6.5

([18], see also Proposition 7.1 of [34]) For \(\frac{\Phi }{2\pi }\in {\mathbb {R}}{\setminus } {\mathbb {Q}}\), and a.e. \(\theta \in {\mathbb {T}}_1\), \(\Sigma _{ac}(H_{\Phi , \theta })\) is empty.

Hence our operator \(H_{\Phi , \theta }\) has zero Lyapunov exponent on the spectrum and empty absolutely continuous spectrum. Celebrated Kotani theory identifies the essential closure of the set of zero Lyapunov exponents with the absolutely continuous spectrum, for general ergodic Schrödinger operators. This has been extended to the case of non-singular (that is \(|c(\cdot )|\) uniformly bounded away from zero) Jacobi matrices in Theorem 5.17 of [52]. In our case \(|c(\cdot )|\) is not bounded away from zero, however a careful inspection of the proof of Theorem 5.17 of [52] shows that it holds under a weaker requirement: \(\log {(|c(\cdot )|)} \in L^1\). Namely, let \(H_{c,v}(\theta )\) acting on \(\ell ^2({\mathbb {Z}})\) be an ergodic Jacobi matrix,
$$\begin{aligned} (H_{c,v}(\theta )u)_m=c(T^m\theta )u_{m+1}+\overline{c(T^{m-1}\theta )}u_{m-1}+v(T^m\theta )u_m \end{aligned}$$
where \(c:M\rightarrow {\mathbb {C}}\), \(v:M\rightarrow {\mathbb {R}}\), are bounded measurable functions, and \(T:M\rightarrow M\) is an ergodic map. Let \(L_{c,v}(\lambda )\) be the corresponding Lyapunov exponent. We have

Theorem 8

(Kotani theory) Assume \(\log {(|c(\cdot )|)} \in L^1(M)\). Then for a.e. \(\theta \in M\), \(\Sigma _{ac}(H_{c,v}(\theta ))=\overline{\{\lambda :\ L_{c,v}(\lambda )=0\}}^{ess}\).


The proof of Theorem 5.17 of [52] works verbatim. \(\square \)

Proof of Theorem 7

In our concrete model, \(\log {(|c(\theta )|)}=\log {(2|\cos {\pi \theta }|)}\in L^1({\mathbb {T}}_1)\), thus Theorem 8 applies, and combining with Propositions 6.46.5, it follows that \(\Sigma _{\Phi }\) must be a zero measure set. \(\square \)

The rest of this section will be devoted to proving Lemma 6.1.

6.4 Quick observations about \(H_{2\pi p/q, \theta }\)

Let \(A^{\lambda }(\cdot ), {\tilde{A}}^{\lambda } (\cdot ), D^{\lambda }(\cdot ), \Theta \) be defined as in Sect. 2.3.1. We start with several quick observations about \(H_{2\pi p/q, \theta }\).

Obervation 1

The sampling function \(c(\theta )=0\) yields a unique solution \(\theta =\frac{1}{2}\) (mod 1), hence \(\Theta =\frac{1}{2}+\frac{1}{q}{\mathbb {Z}}\). Then,
  • for \(\theta \notin \Theta \), we have \(c(\theta +n\frac{p}{q})\ne 0\) for any \(n\in {\mathbb {Z}}\)

  • for \(\theta \in \Theta \), there exists \(k_0\in \{0,1,\ldots ,q-1\}\) such that \(c(\theta +n\frac{p}{q})=0\) if and only if \(n\equiv k_0\) (mod q).

Note that \(|c(\theta )|=2|\cos {\pi \theta }|\), so a simple computation yields that \(\prod _{j=0}^{q-1}|c(\theta +j\frac{p}{q})|=2|\sin {\pi q(\theta +\frac{1}{2})}|\). Thus (2.50) becomes
$$\begin{aligned} {{\,\mathrm{tr}\,}}({\tilde{A}}^{\lambda }_q(\theta ))=\frac{{{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta ))}{2|\sin {\pi q(\theta +\frac{1}{2})}|}. \end{aligned}$$
We have the following characterization of \(\Sigma _{2\pi p/q, \theta }\).

6.4.1 Case 1.

If \(\theta \in \Theta \), we have the following

Obervation 2

For \(\theta \in \Theta \), the infinite matrix \(H_{2\pi p/q, \theta }\) is decoupled into copies of the following block matrix \(M_q\) of size q:
$$\begin{aligned} \left( \begin{array}{cccccccc} &{}v(\frac{1}{2}+\frac{p}{q})\ \ &{}c(\frac{1}{2}+\frac{p}{q})\ \ &{}\ \ &{}\ \ &{} \ \ &{}\ \ &{}\\ &{}\overline{c(\frac{1}{2}+\frac{p}{q})}\ \ &{}v(\frac{1}{2}+2\frac{p}{q})\ \ &{}\ddots \ \ &{}\ \ &{} \ \ &{}\ \ &{}\\ &{}\ddots &{} &{}\ddots &{} &{} &{} &{}\\ &{} &{} &{} &{} &{} &{} &{}\\ \\ \\ &{} &{} &{} &{} &{} &{}v(\frac{1}{2}+(q-1)\frac{p}{q}) &{}c(\frac{1}{2}+(q-1)\frac{p}{q})\\ &{} &{} &{} &{} &{} &{}\overline{c(\frac{1}{2}+(q-1)\frac{p}{q})} &{}v(\frac{1}{2}) \end{array} \right) . \end{aligned}$$
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }=\{\text {eigenvalues of }M_q\},\ \ \ \text {for } \theta \in \Theta . \end{aligned}$$

6.4.2 Case 2.

If \(\theta \notin \Theta \), by Floquet theory, we have
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }=\{\lambda :\ |{{\,\mathrm{tr}\,}}{{{\tilde{A}}^{\lambda }_{q}(\theta )}}| \le 2 \}. \end{aligned}$$
Furthermore, the set \(\{\lambda : {{\,\mathrm{tr}\,}}{{{\tilde{A}}^{\lambda }_{q}(\theta )}}=2\cos {2\pi \nu }\}\) contains q individual points (counting multiplicities), which are eigenvalues of the following \(q\times q\) matrix \(M_{q, \nu }\):
$$\begin{aligned} M_{q, \nu }(\theta )= \left( \begin{matrix} v(\theta +\frac{p}{q}) &{}|c(\theta +\frac{p}{q})| &{} &{} &{} &{} &{}e^{2\pi i \nu } |c(\theta )|\\ |c(\theta +\frac{p}{q})| &{}v(\theta +2\frac{p}{q}) &{}\ddots &{} &{} &{} &{}\\ &{}\ddots &{}\ddots &{} &{}\, &{} &{} &{}\, \\ \\ &{} &{} &{} &{} &{}v(\theta +(q-1)\frac{p}{q}) &{}|c(\theta +(q-1)\frac{p}{q})|\\ e^{-2\pi i \nu } |c(\theta )| &{} &{} &{} &{} &{}|c(\theta +(q-1)\frac{p}{q})| &{}v(\theta ) \end{matrix} \right) \end{aligned}$$
Combining (6.10) with (6.7), we arrive at an alternative representation
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }=\left\{ \lambda : |{{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta )) | \le 4|\sin {\pi q (\theta +\frac{1}{2})}|\right\} . \end{aligned}$$

6.5 Key lemmas

$$\begin{aligned} d_q(\theta )={{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta )). \end{aligned}$$
We have

Lemma 6.6

(Chambers’ type formula) For all \(\theta \in {\mathbb {T}}_1\), we have
$$\begin{aligned} d_q(\theta )=-\,2\cos {2\pi q\theta }+G_q (\lambda ), \end{aligned}$$
where \(G_q(\lambda )\) (defined by (6.14)) is independent of \(\theta \).

Remark 5

Chambers’ formula is well-known for the celebrated almost Mathieu operator. It was also recently developed for various models including the tight-binding model \(Q_{\Lambda }(\Phi )\) in [28]. Here we do not use the Chambers’ formula for \(Q_{\Lambda }(\Phi )\), rather we develop one for one-dimensional Hamiltonian \(H_{\Phi ,\theta }\).


It is easily seen that \(d_q(\cdot )\) is a 1 / q-periodic function, thus
$$\begin{aligned} d_q(\theta )=G_q(\lambda )+a_q e^{2\pi i q\theta }+a_{-q} e^{-2\pi i q\theta }, \end{aligned}$$
in which the \(G_q(\lambda )\) part is independent of \(\theta \). One can easily compute the coefficients \(a_q, a_{-q}\), and get \(a_q=a_{-q}={{-\,1}}\). \(\square \)

Lemma 6.7

For \(\theta \in \Theta \),
$$\begin{aligned} \det {(\lambda \cdot \text {Id} -M_q(\theta ))}={{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta )). \end{aligned}$$

The proof of this lemma is stated in “Appendix B”.

Combining (6.12), (6.9) and Lemma 6.7 with the fact that \(|\sin {\pi q(\theta +\frac{1}{2})}|=0\) for \(\theta \in \Theta \), we arrive at
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }=\{\lambda :\ |{{\,\mathrm{tr}\,}}(D^{\lambda }_q(\theta ))|\le 4|\sin {\pi q(\theta +\frac{1}{2})}| \} \end{aligned}$$
holds uniformly for \(\theta \in {\mathbb {T}}_1\).
By (6.14), we get the following alternative characterization of \(\Sigma _{2\pi p/q, \theta }\).
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }&=\left\{ \lambda : -\,4|\sin {\pi q(\theta +\frac{1}{2})}|+2\cos {2\pi q\theta }\le G_q (\lambda )\right. \nonumber \\&\left. \le 4|\sin {\pi q(\theta +\frac{1}{2})}|+2\cos {2\pi q\theta } \right\} . \end{aligned}$$
Let us denote \(L_q(\theta ):=4|\sin {\pi q(\theta +\frac{1}{2})}|+2\cos {2\pi q\theta }\), and \(l_q(\theta ):=-\,4|\sin {\pi q(\theta +\frac{1}{2})}|+2\cos {2\pi q\theta }\). Then (6.17) translates into
$$\begin{aligned} \Sigma _{2\pi p/q, \theta }=\{ \lambda : l_q(\theta )\le G_q(\lambda )\le L_q(\theta )\}. \end{aligned}$$
This clearly implies
$$\begin{aligned} \Sigma _{2\pi p/q}=\{ \lambda : \min _{{\mathbb {T}}_1} l_q(\theta ) \le G_q (\lambda ) \le \max _{{\mathbb {T}}_1} L_q(\theta ) \}. \end{aligned}$$
Note that \(G_q(\lambda )\) is a polynomial of \(\lambda \) of degree q with leading coefficient 1, \(\Sigma _{2\pi p/q}\) consists of q non-overlapping (but possibly touching) bands and \(G'(\lambda )\ne 0\) in the interior of each band, see e.g. Section 6 of [28].

The following lemma provides estimates of \(|\Sigma _{2\pi p/q, \theta }|\) and holds for any Jacobi matrix (2.41).

Lemma 6.8

We have
$$\begin{aligned} |\Sigma _{2\pi p/q, \theta }|\le 4|c(\theta )|. \end{aligned}$$


For \(\theta \in \Theta \), by (6.9), \(|\Sigma _{2\pi p/q, \theta }|=0\). It then suffices to consider \(\theta \notin \Theta \).

By (6.10), we have
$$\begin{aligned} \Sigma _{\frac{2\pi p}{q}, \theta }=\{\lambda :\ -\,2\le {{\,\mathrm{tr}\,}}\,({\tilde{A}}_q^{\lambda }(\theta ))\le 2\}. \end{aligned}$$
Note that \({{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))\) is a polynomial of degree q in \(\lambda \) with leading coefficient 1. By standard Floquet theory, see e.g. [52, 53], we have
$$\begin{aligned} \frac{d}{d\lambda } {{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))\ne 0, \end{aligned}$$
holds for any \(\lambda \) such that \({{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))\in (-\,2,2)\). Hence \(\Sigma _{2\pi p/q, \theta }\) is completely determined by the \(\lambda \)’s such that \({{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))=\pm \, 2\). By (6.10) and the explanation below it,
$$\begin{aligned} {\left\{ \begin{array}{ll} \{\lambda :\ {{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))=2\}=\{\text {eigenvalues of }M_{q, 0}(\theta )\} \\ \{\lambda :\ {{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))=-\,2\}\ =\{\text {eigenvalues of }M_{q, \frac{1}{2}}(\theta )\}. \end{array}\right. } \end{aligned}$$
Let \(\{\lambda _i(\theta )\}_{i=1}^{q}\) be eigenvalues of \(M_{q, 0}(\theta )\), labelled in the increasing order. Let \(\{\tilde{\lambda }_i{(\theta )}\}_{i=1}^q\) be eigenvalues of \(M_{q, \frac{1}{2}}(\theta )\), labelled also in the increasing order. Then we have
$$\begin{aligned} |\Sigma _{\frac{2\pi p}{q}, \theta }|&=\sum _{k=1}^q (-1)^{q-k} \left( \lambda _k (\theta )-\tilde{\lambda }_k{(\theta )} \right) \nonumber \\&=\sum _{k=1}^{[\frac{q+1}{2}]} \left( \lambda _{q-2k+2}{(\theta )}-\tilde{\lambda }_{q-2k+2}{(\theta )}\right) \nonumber \\&\quad - \sum _{k=1}^{[\frac{q-1}{2}]} \left( \lambda _{q-2k+1}{(\theta )}- \tilde{\lambda }_{q-2k+1}{(\theta )} \right) . \end{aligned}$$
Note the coefficient of \((\lambda _q(\theta )-\tilde{\lambda }_q(\theta ))\) is 1 rather than \(-\,1\). This is due to the fact that the leading coefficient of \({{\,\mathrm{tr}\,}}({\tilde{A}}_q^{\lambda }(\theta ))\) is positive.
Consider the difference matrix
$$\begin{aligned} M_{q, 0}(\theta )-M_{q, \frac{1}{2}}(\theta ) = \left( \begin{array}{ccccccc} &{} &{} &{} &{} &{} &{}2|c(\theta )|\\ &{} &{} &{} &{} &{} &{}\\ \\ &{} &{} &{} &{} &{} &{}\\ 2|c(\theta )| &{} &{} &{} &{} &{} &{} \end{array} \right) \end{aligned}$$
whose eigenvalues we denote by \(\{E_i(\theta )\}_{i=1}^{q}\), namely,
$$\begin{aligned} E_{1}(\theta )=-\,2|c(\theta )|< 0=E_{2}(\theta )=\cdots =E_{q-1}(\theta )=0< 2|c(\theta )|=E_{q}{(\theta )}. \end{aligned}$$
Recall the following Lidskii inequality and dual Lidskii inequality: Let \(M_j\), \(j=1,2\) be \(n\times n\) self-adjoint matrices, let \(E_1(M_j)\le E_2(M_j)\le \cdots \le E_n(M_j)\) be the eigenvalues of \(M_j\). Then, for the eigenvalues of the sum of the two matrices, we have
$$\begin{aligned} {\left\{ \begin{array}{ll} \sum _{l=1}^k E_{i_l}(M_1+M_2)\le \sum _{r=1}^k E_{i_r}(M_1)+\sum _{s=n-k+1}^n E_s(M_2)\\ \sum _{l=1}^k E_{i_l}(M_1+M_2)\ge \sum _{r=1}^k E_{i_r}(M_1)+\sum _{s=1}^k E_s(M_2) \end{array}\right. } \end{aligned}$$
for any \(1\le i_1<\cdots <i_k\le n\).
By Lidskii inequalities (6.22), we have
$$\begin{aligned} \sum _{k=1}^{[\frac{q+1}{2}]} \lambda _{q-2k+2}{(\theta )}&\le \sum _{k=1}^{[\frac{q+1}{2}]} \tilde{\lambda }_{q-2k+2}{(\theta )} +\sum _{k=1}^{[\frac{q+1}{2}]} E_{q-k+1}{(\theta )}\nonumber \\&=\sum _{k=1}^{[\frac{q+1}{2}]} \tilde{\lambda }_{q-2k+2}{(\theta )}+2|c(\theta )|, \end{aligned}$$
$$\begin{aligned} \sum _{k=1}^{[\frac{q-1}{2}]}\lambda _{q-2k+1}{(\theta )}&\ge \sum _{k=1}^{[\frac{q-1}{2}]}\tilde{\lambda }_{q-2k+1}{(\theta )} +\sum _{k=1}^{[\frac{q-1}{2}]} E_{k}{(\theta )}\nonumber \\&= \sum _{k=1}^{[\frac{q-1}{2}]}\tilde{\lambda }_{q-2k+1}{(\theta )} -2|c(\theta )|. \end{aligned}$$
Hence combining (6.21) with (6.23) (6.24), we get,
$$\begin{aligned} |\Sigma _{\frac{2\pi p}{q}, \theta }|\le 4|c(\theta )|. \end{aligned}$$
\(\square \)

6.6 Proof of Lemma 6.1 for even q

For sets/functions that depend on \(\theta \), we will sometimes substitute \(\theta \) in the notation with \(A\subseteq {\mathbb {T}}_1\), if corresponding sets/functions are constant on A.

Since q is even, a simple computation shows
$$\begin{aligned} {\left\{ \begin{array}{ll} \max _{{\mathbb {T}}_1} L_q(\theta )=L_q(\frac{6{\mathbb {Z}}+1}{6q})=L_q(\frac{6{\mathbb {Z}}+5}{6q})=3,\\ \\ \min _{{\mathbb {T}}_1} l_q(\theta )=l_q(\frac{2{\mathbb {Z}}+1}{2q})=-\,6. \end{array}\right. } \end{aligned}$$
A simple computation also shows \(l_q(\frac{6{\mathbb {Z}}+1}{6q})=-1\) and \(L_q(\frac{2{\mathbb {Z}}+1}{2q})=2\). Thus we have, by (6.19),
$$\begin{aligned} \Sigma _{2\pi p/q}&=\{\lambda : -\,6\le G_q(\lambda )\le 3\}\\&=\{\lambda : -\,6\le G_q(\lambda )\le 2\} \bigcup \{\lambda : -\,1\le G_q(\lambda )\le 3\}\\&=\Sigma _{\frac{2\pi p}{q}, \frac{2{\mathbb {Z}}+1}{2q}}\bigcup \Sigma _{\frac{2\pi p}{q}, \frac{6{\mathbb {Z}}+1}{6q}}. \end{aligned}$$
This implies
$$\begin{aligned} |\Sigma _{2\pi p/q}| \le |\Sigma _{\frac{2\pi p}{q}, \frac{2{\mathbb {Z}}+1}{2q}}|+|\Sigma _{\frac{2\pi p}{q}, \frac{6{\mathbb {Z}}+1}{6q}}|. \end{aligned}$$
Now it remains to estimate \(|\Sigma _{\frac{2\pi p}{q}, \frac{2{\mathbb {Z}}+1}{2q}}|\) and \(|\Sigma _{\frac{2\pi p}{q}, \frac{6{\mathbb {Z}}+1}{6q}}|\). Since q is even, let us consider \(\Sigma _{\frac{2\pi p}{q}, \frac{q+1}{2q}}\) and \(\Sigma _{\frac{2\pi p}{q}, {{\frac{3q+1}{6q}}}}\).
By Lemma 6.8, we have
$$\begin{aligned} {\left\{ \begin{array}{ll} |\Sigma _{\frac{2\pi p}{q}, \frac{q+1}{2q}}|\le 4|c(\frac{q+1}{2q})|<\frac{4\pi }{q},\\ |\Sigma _{\frac{2\pi p}{q}, {{\frac{3q+1}{6q}}}}|\le 4|c({{\frac{3q+1}{6q}}})|< \frac{4\pi }{3q}. \end{array}\right. } \end{aligned}$$
Hence putting (6.26), (6.27) together, we have
$$\begin{aligned} |\Sigma _{\frac{2\pi p}{q}}|< \frac{16\pi }{3q}. \end{aligned}$$

6.7 Proof of Lemma 6.1 for odd q

Since the proof for odd q is very similar to that for even q, we only sketch the steps here.

For odd q, similar to (6.26), we have
$$\begin{aligned} |\Sigma _{\frac{2\pi p}{q}}|\le |\Sigma _{\frac{2\pi p}{q}, \frac{3{\mathbb {Z}}+1}{3q}}|+|\Sigma _{\frac{2\pi p}{q}, \frac{{\mathbb {Z}}}{q}}|. \end{aligned}$$
By Lemma 6.8, we have
$$\begin{aligned} {\left\{ \begin{array}{ll} |\Sigma _{\frac{2\pi p}{q}, \frac{3q-1}{6q}}| \le 4|c(\frac{3q-1}{6q})|< \frac{4\pi }{3q},\\ |\Sigma _{\frac{2\pi p}{q}, \frac{q-1}{2q}}|\le 4|c(\frac{q-1}{2q})| < \frac{4\pi }{q}. \end{array}\right. } \end{aligned}$$
Hence putting (6.29), (6.30) together, we have
$$\begin{aligned} |\Sigma _{\frac{2\pi p}{q}}|< \frac{16\pi }{3q}. \end{aligned}$$
This proves the claimed result. \(\square \)

7 Proof of Lemma 4.1

Lemma 4.1 is the reduction from \(\Lambda ^B\) to the tight-binding model \(Q_{\Lambda }\). We now present its proof below.

Using ideas from [11, 46], we can express the resolvent of the operator \(\Lambda ^B\) (3.14) by Krein’s resolvent formula in terms of the resolvent of the Dirichlet Hamiltonian and the resolvent of \(Q_{\Lambda }\).

For this we need to introduce a few concepts first. The \(l^2\)-space on the vertices \(l^2({\mathcal {V}}(\Lambda ))\) carries the inner product
$$\begin{aligned} \langle f,g \rangle :=\sum _{{{\varvec{v}}} \in {\mathcal {V}}(\Lambda )} 3 f({{\varvec{v}}}) \overline{g({{\varvec{v}}})} \end{aligned}$$
where the factor three accounts for the number of incoming or outgoing edges at each vertex.

A convenient method from classical extension theory required to state Krein’s resolvent formula, and thus to link the magnetic Schrödinger operator \(H^B\) with an effective Hamiltonian, is the concept of boundary triples.

Definition 7.1

Let \({\mathcal {T}}: D({\mathcal {T}}) \subset {\mathscr {H}} \rightarrow {\mathscr {H}}\) be a closed linear operator on the Hilbert space \({\mathscr {H}}\), then the triple \((\pi ,\pi ',{\mathscr {H}}'),\) with \({\mathscr {H}}'\) being another Hilbert space and \(\pi , \pi ': D({\mathcal {T}})\rightarrow {\mathscr {H}}'\), is a boundary triple for \({\mathcal {T}}\), if
  • Green’s identity holds on \(D({\mathcal {T}}),\) i.e. for all \(\psi ,\varphi \in D({\mathcal {T}})\)
    $$\begin{aligned} \langle \psi ,{\mathcal {T}}\varphi \rangle _{{\mathscr {H}}} - \langle {\mathcal {T}} \psi , \varphi \rangle _{{\mathscr {H}}} = \langle \pi \psi ,\pi '\varphi \rangle _{{\mathscr {H}}'}-\langle \pi ' \psi ,\pi \varphi \rangle _{{\mathscr {H}}'}. \end{aligned}$$
  • \({\text {ker}}(\pi ,\pi ')\) is dense in \({\mathscr {H}}.\)

  • \((\pi ,\pi '):D({\mathcal {T}}) \rightarrow {\mathscr {H}}' \oplus {\mathscr {H}}'\) is a linear surjection.

The following lemma applies this concept to our setting.

Lemma 7.2

The operator \({\mathcal {T}}^B:D({\mathcal {T}}^B) \subset L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) )\) acting as the maximal Schrödinger operator (2.11) on every edge with domain
$$\begin{aligned} D({\mathcal {T}}^B):= \Bigg \{\psi \in {\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ):&\text { any }\vec {e}_1, \vec {e}_2 \in {\mathcal {E}}_{{{\varvec{v}}}}(\Lambda ) \text { such that } i(\vec {e}_1)=i(\vec {e}_2)={{\varvec{v}}} \text { satisfy } \nonumber \\&\ \psi _{\vec {e}_1}({{\varvec{v}}})=\psi _{\vec {e}_2}({{\varvec{v}}}) \text { and if }t(\vec {e}_1)=t(\vec {e}_2)={{\varvec{v}}}, \nonumber \\&\text { then } e^{ i\tilde{\beta }_{\vec {e}_1}}\psi _{\vec {e}_1}({{\varvec{v}}})=e^{ i\tilde{\beta }_{\vec {e}_2} }\psi _{\vec {e}_2}({{\varvec{v}}})\Bigg \} \end{aligned}$$
is closed. The maps \(\pi ,\pi '\) on \(D({\mathcal {T}}^B)\) defined by
$$\begin{aligned} \pi (\psi )({{\varvec{v}}})&:=\frac{1}{3} \Bigg (\sum _{i(\vec {e})={{\varvec{v}}}} \psi _{\vec {e}}({{\varvec{v}}})+\sum _{t(\vec {e})={{\varvec{v}}}} e^{ i\tilde{\beta }_{\vec {e}}} \psi _{\vec {e}}({{\varvec{v}}})\Bigg ) \nonumber \\ \pi '(\psi )({{\varvec{v}}})&:= \frac{1}{3} \Bigg (\sum _{i(\vec {e})={{\varvec{v}}}} \psi _{\vec {e}}'({{\varvec{v}}})-\sum _{t(\vec {e})={{\varvec{v}}}} e^{ i\tilde{\beta }_{\vec {e}}} \psi _{\vec {e}}'({{\varvec{v}}}) \Bigg ) \end{aligned}$$
form together with \({\mathscr {H}}':=l^2({\mathcal {V}}(\Lambda ))\) a boundary triple associated to \({\mathcal {T}}^B\).


The proof follows the same strategy as in [46]. The operator \({\mathcal {T}}^B\) is closed iff its domain is a closed subspace (with respect to the graph norm) of the domain of some closed extension of \({\mathcal {T}}^B\). Such a closed extension is given by \(\bigoplus _{\vec {e} \in {\mathcal {E}}(\Lambda )}{{{\mathcal {H}}^2_{\vec {e}}}}\) on \({\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ).\) To see that \(D({\mathcal {T}}^B)\) is a closed subspace of \({\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ),\) observe that in terms of continuous functionals
$$\begin{aligned} l_{\vec {e}_i, \vec {e}_j}&: {\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow {\mathbb {C}},\ \ l_{\vec {e}_i, \vec {e}_j}(\psi )= \psi _{\vec {e}_i}(i(\vec {e}_i))-\psi _{\vec {e}_j}(i(\vec {e}_j)) \nonumber \\ k_{\vec {e}_i, \vec {e}_j}&: {\mathcal {H}}^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow {\mathbb {C}},\ \ k_{\vec {e}_i, \vec {e}_j}(\psi )= e^{ i\tilde{\beta }_{\vec {e}_i}}\psi _{\vec {e}_i}(t(\vec {e}_i))-e^{ i\tilde{\beta }_{\vec {e}_j}}\psi _{\vec {e}_j}(t(\vec {e}_j)) \end{aligned}$$
we obtain
$$\begin{aligned} D({\mathcal {T}}^B)=\bigcap _{\vec {e}_i,\vec {e}_j \in {\mathcal {E}}(\Lambda ) \text { with }i(\vec {e}_i)=i(\vec {e}_j)} {\text {ker}}\left( l_{\vec {e}_i,\vec {e}_j}\right) \cap \bigcap _{\vec {e}_i,\vec {e}_j \in {\mathcal {E}}(\Lambda ) \text { with }t(\vec {e}_i)=t(\vec {e}_j)} {\text {ker}}\left( k_{\vec {e}_i,\vec {e}_j}\right) \end{aligned}$$
which proves closedness of \({\mathcal {T}}^B\). Green’s identity follows directly from integration by parts on the level of edges. The denseness of \({\text {ker}}(\pi ,\pi ')\) is obvious since this space contains \(\bigoplus _{\vec {e} \in {\mathcal {E}}(\Lambda )} C_c^{\infty }(\vec {e})\). To show surjectivity, it suffices to consider a single edge. On those however, the property can be established by explicit constructions as in Lemma 2 in [46]. \(\square \)
Any boundary triple for \({\mathcal {T}}\) as in Definition 7.1 and any self-adjoint relation \(A \subseteq {\mathscr {H}}'\bigoplus {\mathscr {H}}'\) gives rise [50] to a self-adjoint restriction \({\mathcal {T}}_A\) of \({\mathcal {T}}\) with domain
$$\begin{aligned} D({\mathcal {T}}_A)=\left\{ \psi \in D({\mathcal {T}}): (\pi (\psi ), \pi '(\psi ))\in A \right\} . \end{aligned}$$
The restriction of \({\mathcal {T}}^B\) satisfying Dirichlet type boundary conditions on every edge is obtained by selecting \(A_1:=\left\{ 0\right\} \oplus l^2({\mathcal {V}}(\Lambda ))\) and coincides with \(H^D\) (2.14). The operator \(\Lambda ^B\) (3.14) is recovered from \({\mathcal {T}}^B\) by picking the relation \(A_2:=l^2({\mathcal {V}}(\Lambda )) \oplus \left\{ 0\right\} .\)

Definition 7.3

Given the boundary triple for \({\mathcal {T}}^B\) as above, the gamma-field \(\gamma : \rho (H^D) \rightarrow {\mathcal {L}}(l^2({\mathcal {V}}(\Lambda )),L^2({\mathcal {E}}\left( \Lambda \right) ))\) is given by \(\gamma (\lambda ):= \left( \pi \vert _{{\text {ker}}({\mathcal {T}}^B - \lambda )} \right) ^{-1}\) and the Weyl function \(M(\cdot , \Phi ): \rho (H^D) \rightarrow {\mathcal {L}}(l^2({\mathcal {V}}(\Lambda )))\) is defined as \( M(\lambda , \Phi ):=\pi ' \gamma (\lambda ).\)

A computation shows that those maps are well-defined.

Lemma 7.4

For the operator \({\mathcal {T}}^B\), the gamma-field \(\gamma \) and Weyl function M can be explicitly written in terms of the solutions \(s_{\lambda },c_{\lambda }\) (2.35) on an arbitrary edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\) for \(\lambda \in \rho (H^D)\) and \(z \in l^2({\mathcal {V}}(\Lambda ))\) by
$$\begin{aligned} (\gamma (\lambda )z)_{\vec {e}}({{\varvec{x}}}) = \frac{\left( s_{\lambda }(1)c_{\lambda , \vec {e}}({{\varvec{x}}})-s_{\lambda , \vec {e}}({{\varvec{x}}})c_{\lambda }(1)\right) z(i(\vec {e})) +e^{-i\tilde{\beta }_{\vec {e}}}s_{\lambda , \vec {e}}({{\varvec{x}}}) z(t(\vec {e}))}{s_{\lambda }(1)} \end{aligned}$$
$$\begin{aligned} M(\lambda , \Phi ) = \frac{K_{\Lambda }(\Phi )-\Delta (\lambda )}{s_{\lambda }(1)} \end{aligned}$$
$$\begin{aligned} (K_\Lambda (\Phi ) z)({{\varvec{v}}}):=\frac{1}{3} \left( \sum _{\vec {e}:\ i(\vec {e})={{\varvec{v}}} } e^{-i \tilde{\beta }_{\vec {e}}} z(t(\vec {e})) + \sum _{\vec {e}:\ t(\vec {e})={{\varvec{v}}} } e^{i \tilde{\beta }_{\vec {e}}} z(i(\vec {e})) \right) \end{aligned}$$
defines an operator in \({\mathcal {L}}(l^2({\mathcal {V}}(\Lambda )))\) with \(\Vert K_{\Lambda }(\Phi ) \Vert \le 1.\)


For \(\lambda \in \rho (H^D)\) and \(z \in l^2({\mathcal {V}}(\Lambda ))\) we define for \(\vec {e} \in {\mathcal {E}}(\Lambda )\) arbitrary
$$\begin{aligned} \psi _{\vec {e}}:=(\gamma (\lambda )z)_{\vec {e}} = ((\pi \vert _{{\text {ker}}({\mathcal {T}}^B-\lambda )})^{-1}z)_{\vec {e}} \end{aligned}$$
with \(\psi :=(\psi _{\vec {e}}).\) In particular, \(\psi _{\vec {e}}\) is the solution to \(-\psi ''_{\vec {e}} + V_{\vec {e}}\psi _{\vec {e}} = \lambda \psi _{\vec {e}}\) with the following boundary condition: \(\psi _{\vec {e}}(i(\vec {e}))=z(i(\vec {e}))\) and \(\psi _{\vec {e}}(t(\vec {e}))=e^{-i\tilde{\beta }_{\vec {e}} }z(t(\vec {e}))\). The representation (7.8) is then an immediate consequence of (2.28).
The expression for the Weyl function on the other hand, follows from the Dirichlet-to-Neuman map (2.29).
$$\begin{aligned} (M(\lambda , \Phi )z)({{\varvec{v}}})&= (\pi ' \gamma (\lambda ) z)({{\varvec{v}}}) \nonumber \\&= \frac{1}{3} \left( \sum _{\vec {e}:\ i(\vec {e})={{\varvec{v}}} } \psi _{\vec {e}}'({{\varvec{v}}}) - \sum _{\vec {e}:\ t(\vec {e})={{\varvec{v}}}} e^{i \tilde{\beta }_{\vec {e}}} \psi _{\vec {e}}'({{\varvec{v}}}) \right) \nonumber \\&= \frac{(K_\Lambda (\Phi ) z)({{\varvec{v}}})}{s_{\lambda }(1)}- \left( \underbrace{\frac{c_{\lambda }(1)}{s_{\lambda }(1)} \delta _{{{\varvec{v}}} \in i({\mathcal {V}}(\Lambda ))}z({{\varvec{v}}}) + \frac{s_{\lambda }'(1)}{s_{\lambda }(1)}\delta _{{{\varvec{v}}} \in t({\mathcal {V}}(\Lambda ))}z({{\varvec{v}}})}_{= \frac{s_{\lambda }'(1)}{s_{\lambda }(1)} z({{\varvec{v}}})}\right) \nonumber \\&= \frac{\left( K_\Lambda (\Phi ) z-s_{\lambda }'(1)z \right) ({{\varvec{v}}}) }{s_{\lambda }(1)}, \end{aligned}$$
here we used (2.31). The formula (7.9) then follows from (7.12) and (2.37). Since \(i(\Lambda )\cap t(\Lambda )=\emptyset \), we have \(\Vert K_{\Lambda }(\Phi )\Vert \le 1\). \(\square \)

The resolvents of \(H^D={\mathcal {T}}_{A_1}^B\) and \(\Lambda ^B={\mathcal {T}}_{A_2}^B\) are then related by Krein’s resolvent formula [50, Theorem 14.18] and a unitary equivalence between \(\Lambda ^B\) and \(K_{\Lambda }(\Phi )\) away from the Dirichlet spectrum holds [47, 48]

Theorem 9

Let \((l^2({\mathcal {V}}(\Lambda )),\pi ,\pi ')\) be the boundary triple for \({\mathcal {T}}^B\) and \(\gamma ,M\) as above, then for \(\lambda \in \rho (H^D)\cap \rho (\Lambda ^B)\) there is also a bounded inverse of \(M(\lambda , \Phi )\) and
$$\begin{aligned} (\Lambda ^B-\lambda )^{-1}-(H^D-\lambda )^{-1} = -\gamma (\lambda )M(\lambda , \Phi )^{-1} \gamma ({\overline{\lambda }})^*. \end{aligned}$$
In particular, \(\sigma (\Lambda ^B)\backslash \sigma (H^D) = \{\lambda \in {\mathbb {R}} \cap \rho (H^D):\ 0 \in \sigma (M(\lambda , \Phi ))\}\) and for intervals \(J \subset {\mathbb {R}} \backslash \sigma (H^D)\)
$$\begin{aligned} \Delta \left( \Lambda ^B \mathbb {1}_{J}(\Lambda ^B)\right) = U\left[ K_{\Lambda }(\Phi )\mathbb {1}_{\Delta (J)}(K_{\Lambda }(\Phi )) \right] U^* \end{aligned}$$
with unitary operator \(U: {\text {ran}}\left( K_{\Lambda }(\Phi )\mathbb {1}_{\Delta (J)}(K_{\Lambda }(\Phi )) \right) \rightarrow {\text {ran}} \left( \Lambda ^B \mathbb {1}_{J}(\Lambda ^B)\right) \) given by
$$\begin{aligned} U = \int _J \sqrt{ \tfrac{\partial _{\lambda } s_{\lambda }(1)}{\Delta '(\lambda )}} \gamma (\lambda ) \ d{\mathcal {E}}_{K_{\Lambda }(\Phi )}(\Delta (\lambda )) \end{aligned}$$
and \({\mathcal {E}}_{K_{\Lambda }(\Phi )}\) is the spectral measure of the self-adjoint operator \(K_{\Lambda }(\Phi ).\)

Since all vertices are integer translates of either of the two vertices \({{\varvec{r}}}_0,{{\varvec{r}}}_1 \in W_{\Lambda }\) by basis vectors \(\vec {b}_1, \vec {b}_2\), we conclude that \(l^2({\mathcal {V}}(\Lambda )) \simeq l^2({\mathbb {Z}}^2;{\mathbb {C}}^2)\). Our next Lemma shows \(K_{\Lambda }(\Phi )\) and \(Q_{\Lambda }(\Phi )\) are unitary equivalent under this identification.

Lemma 7.5

\(K_{\Lambda }(\Phi )\) is unitary equivalent to operator \(Q_{\Lambda }(\Phi )\).


By (3.16), (7.10),
$$\begin{aligned} {\left\{ \begin{array}{ll} (K_{\Lambda }(\Phi )z)(\gamma _1, \gamma _2, {{\varvec{r}}}_0)= \frac{1}{3} \left( z(\gamma _1, \gamma _2, {{\varvec{r}}}_1) +z(\gamma _1-1, \gamma _2, {{\varvec{r}}}_1) + e^{-i \Phi \gamma _1}z(\gamma _1, \gamma _2-1, {{\varvec{r}}}_1)\right) ,\\ (K_{\Lambda }(\Phi )z)(\gamma _1, \gamma _2, {{\varvec{r}}}_1)= \frac{1}{3} \left( z(\gamma _1, \gamma _2, {{\varvec{r}}}_0)+z(\gamma _1+1, \gamma _2, {{\varvec{r}}}_0) +e^{i\Phi \gamma _1}z(\gamma _1, \gamma _2+1, {{\varvec{r}}}_0)\right) . \end{array}\right. } \end{aligned}$$
In order to transform \(K_{\Lambda }\) to \(Q_{\Lambda }\) we use the unitary identification \( W:l^2({\mathcal {V}}(\Lambda )) \rightarrow l^2({\mathbb {Z}}^2, {\mathbb {C}}^2)\)
$$\begin{aligned} (Wz)_{\gamma _1, \gamma _2}:= \left( \begin{matrix} z(\gamma _1, \gamma _2, {{\varvec{r}}}_0) \ ,z(\gamma _1, \gamma _2, {{\varvec{r}}}_1 )\end{matrix} \right) ^T. \end{aligned}$$
This way, \(Q_{\Lambda }(\Phi ) = WK_{\Lambda }(\Phi ) W^*.\)\(\square \)

Remark 6

In terms of \(a \in l^2({\mathbb {Z}}^2,{\mathbb {C}}^2)\) defined as
$$\begin{aligned} a_{(0,0)}&:=\frac{1}{3} \left( \begin{matrix} 0 &{}{}&{}{}\quad 1 \\ 1 &{}{}&{}{}\quad 0 \end{matrix}\right) , \ a_{(0,1)}:=\frac{1}{3} \left( \begin{matrix} 0 &{}{}&{}{}\quad 1 \\ 0 &{}{}&{}{}\quad 0 \end{matrix}\right) \nonumber \\ a_{(1,0)}&:=\frac{1}{3} \left( \begin{matrix} 0 &{}{}&{}{}\quad 1 \\ 0 &{}{}&{}{}\quad 0 \end{matrix}\right) , \ a_{(0,-1)}:=\frac{1}{3} \left( \begin{matrix} 0 &{}{}&{}{}\quad 0 \\ 1 &{}{}&{}{}\quad 0 \end{matrix}\right) \nonumber \\ a_{(-1,0)}&:=\frac{1}{3} \left( \begin{matrix} 0 &{}{}&{}{}\quad 0 \\ 1 &{}{}&{}{}\quad 0 \end{matrix}\right) , \text{ and } a_{{{\varvec{\gamma }}}}:=0 \text{ for } \text{ other } {{\varvec{\gamma }}} \in {\mathbb {Z}}^2, \end{aligned}$$
we can express (4.1) in the compact form
$$\begin{aligned} Q_{\Lambda }(\Phi )=\sum _{{{\varvec{\gamma }}} \in {\mathbb {Z}}^2; |{{\varvec{\gamma }}} |\le 1}a_{{{\varvec{\gamma }}}}(\tau _0)^{\gamma _1} (\tau _1)^{\gamma _2}, \end{aligned}$$
where \(|\varvec{\gamma }|:=|\gamma _1|+|\gamma _2|\). This operator has already been studied, in different contexts, for rational flux quanta in [4, 28, 40].

Finally, we point out that Lemma 4.1 follows from a combination of Theorem 9 and Lemmas 7.4 and 7.5. \({} \Box \)

8 Spectral analysis

This section is devoted to complete spectral analysis of \(H^B\).

In view of Lemmas 4.1 and 5.1, an important technical fact is:

Lemma 8.1

The operator norm of \(Q_{\Lambda }(\Phi )\) for non-trivial flux quanta \(\Phi \notin 2 \pi {\mathbb {Z}}\) is strictly less than 1.

Indeed, then, away from the Dirichlet spectrum \(\sigma (H^D)\), which are located on the edges of the Hill bands (2.40), we have the following characterization of \(\sigma (H^B)\). Let \(B_n\) and \(\Delta \) be defined as in Sect. 2.2.3.

Lemma 8.2

For the magnetic Schrödinger operator \(H^B\), the following properties hold.
  1. (1)

    The level of the Dirac points \(\Delta \vert _{{\text {int}}(B_n)}^{-1}(0)\) always belongs to the spectrum of \(H^B\), i.e. \(0 \in \Delta \vert _{{\text {int}}(B_n)}(\sigma (H^B))\).

  2. (2)

    \(\lambda \in \Delta \vert _{{\text {int}}(B_n)}(\sigma (H^B))\) iff \(-\lambda \in \Delta \vert _{{\text {int}}(B_n)}(\sigma (H^B)).\) Consequently, the property \(\Delta '({{\Delta |_{\text {int}(B_n)}^{-1}(0)}})\ne 0\) implies that locally with respect to the Dirac points, the spectrum of \(H^B\) is symmetric.

  3. (3)

    \(H^B\) has no point spectrum away from \(\sigma (H^D)\).

  4. (4)

    For non-trivial flux \(\Phi \notin 2\pi {\mathbb {Z}}\), \(H^B\) has purely continuous spectrum bounded away from \(\sigma (H^D)\).


In this paper, we only show the energy \(\Delta \vert _{{\text {int}}(B_n)}^{-1}(0)\) belongs to the spectrum of \(H^B\). In [12] the first two authors show that not only this energy belongs to the spectrum, but also that Dirac cones actually form around this energy for any \(\Phi \in 2\pi {\mathbb {Q}}.\)

Combining Lemma 8.2 with Lemma 4.4, we get

Lemma 8.3

For generic \(\Phi \), \(\dim _H(\sigma ^{\Phi })\le \frac{1}{2}\).

8.1 Proof of Lemma 8.3

Lemmas 4.4 and 8.1, which implies that \(\Delta ^{-1}\mid _{B_n}\) is Lipschitz on \(\sigma (Q_{\Lambda }(\Phi ))\) for \(\Phi \notin 2\pi {\mathbb {Z}}\), show that for generic \(\Phi \),
$$\begin{aligned} \dim _H\left( \Delta \vert _{{\text {int}}(B_n)}^{-1}(\sigma (Q_{\Lambda }(\Phi ))\right) \le \frac{1}{2}. \end{aligned}$$
Hence since
$$\begin{aligned} \sigma ^{\Phi }=\sigma (H^D)\bigcup \left( \cup _{n\in {\mathbb {N}}}\ \Delta \vert _{{\text {int}}(B_n)}^{-1}(\sigma (Q_{\Lambda }(\Phi ))\right) , \end{aligned}$$
we have
$$\begin{aligned} \dim _H(\sigma ^{\Phi })\le \sup \left\{ \dim _H(\sigma (H^D)),\ \sup _{n\in {\mathbb {N}}}\ \dim _H\left( \Delta \vert _{{\text {int}}(B_n)}^{-1}(\sigma (Q_{\Lambda }(\Phi ))\right) \right\} \le \frac{1}{2}. \end{aligned}$$
This proves Lemma 8.3. \(\square \)

8.2 Proof of Lemma 8.2

(1), (2) follow from a quick combination of Lemmas 4.1 and 5.1, and (3) follows from Part (2) of Lemma 5.2. (4) is a corollary of Lemma 8.1 and (3). \({} \Box \)

8.3 Proof of Lemma 8.1

Without loss of generality \(\Phi \in (0, 2\pi )\). By (5.3), it suffices to show \(\Vert H_{\Phi , \theta }\Vert<c_{\Phi }<6\) for some constant \(c_{\Phi }\) independent of \(\theta \in {\mathbb {T}}_1\). Let us take \(\varphi \in \ell ^2({\mathbb {Z}})\) with \(\Vert \varphi \Vert _{\ell ^2({\mathbb {Z}})}=1\). Consider
$$\begin{aligned} (H_{\Phi , \theta }\varphi )_n&=\underbrace{c\left( \theta +n\frac{\Phi }{2\pi }\right) \varphi _{n+1}}_{=:(h_1\varphi )_n}+\underbrace{\overline{c\left( \theta +(n-1)\frac{\Phi }{2\pi }\right) }\varphi _{n-1}}_{=:(h_2\varphi )_n}\\&\quad +\underbrace{v\left( \theta +n\frac{\Phi }{2\pi }\right) \varphi _n}_{=:(h_3\varphi )_n}, \end{aligned}$$
in which \(h_1, h_2, h_3\in {\mathcal {L}}(\ell ^2({\mathbb {Z}}))\). Hence
$$\begin{aligned} \Vert H_{\Phi , \theta }\varphi \Vert _{\ell ^2({\mathbb {Z}})}^2&\le 3\left( \Vert h_1\varphi \Vert _{\ell ^2({\mathbb {Z}})}^2+\Vert h_2\varphi \Vert _{\ell ^2({\mathbb {Z}})}^2+\Vert h_3\varphi \Vert _{\ell ^2({\mathbb {Z}})}^2\right) \\&\le 3\ \sup _{n\in {\mathbb {Z}}}\left( \left| c\left( \theta +(n-1)\frac{\Phi }{2\pi }\right) \right| ^2+\left| \overline{c\left( \theta +n\frac{\Phi }{2\pi }\right) }\right| ^2+\left| v\left( \theta +n\frac{\Phi }{2\pi }\right) \right| ^2\right) \\&\le 12\sup _{\theta \in {\mathbb {T}}_1}\left( {{\cos ^2}} \pi \left( \theta -\frac{\Phi }{2\pi }\right) +{{\cos ^2}} (\pi \theta )+\cos ^2 (2\pi \theta )\right) \\&=:c_{\Phi }^2<36. \end{aligned}$$
\(\square \)

In order to investigate further the Dirichlet spectrum and spectral decomposition of the continuous spectrum into absolutely and singular continuous parts, we start with constructing magnetic translations.

8.4 Magnetic translations

Below, let \(\varvec{\gamma }=(\gamma _1,\gamma _2)\) be in \({\mathbb {Z}}^2\) and \(\vec {e}=(\tilde{\gamma }_1,\tilde{\gamma }_2,[\vec {e}])\) an arbitrary edge.

In general, \(\Lambda ^B\) does not commute with lattice translations \(T^{st}_{\varvec{\gamma }}\). Yet, there is a set of modified translations, introduced by [54], that do still commute with \(\Lambda ^B,\) although they in general no longer commute with each other. We define those magnetic translations \(T^{B}_{{{\varvec{\gamma }}}} : L^2({\mathcal {E}}\left( \Lambda \right) ) \rightarrow L^2({\mathcal {E}}\left( \Lambda \right) )\) as unitary operators given by
$$\begin{aligned} (T^{B}_{{{\varvec{\gamma }}}} \psi )_{\vec {e}} := u^B_{{{\varvec{\gamma }}}}(\vec {e}) (T^{\text {st}}_{\varvec{\gamma }}\psi )_{\vec {e}} \end{aligned}$$
for any \(\psi :=(\psi _{\vec {e}})_{\vec {e} \in {\mathcal {E}}(\Lambda )} \in L^2({\mathcal {E}}\left( \Lambda \right) )\) and \({{\varvec{\gamma }}} \in {\mathbb {Z}}^2.\) The lattice translation \(T^{st}_{{{\varvec{\gamma }}}}\) is defined by \((T^{st}_{{{\varvec{\gamma }}}}\psi )_{\vec {e}}(x)=\psi _{\vec {e}-\gamma _1\vec {b}_1-\gamma _2\vec {b}_2}(x-\gamma _1 \vec {b}_1-\gamma _2\vec {b}_2)\) as before. The function \(u^B_{{{\varvec{\gamma }}}}\) is constant on each copy of the fundamental domain, and defined as follows
$$\begin{aligned} u^B_{{{\varvec{\gamma }}}}(\tilde{\gamma }_1, \tilde{\gamma }_2, {{[\vec {e}])}}= e^{i\Phi \gamma _1 \tilde{\gamma }_2},\ \ \ \text {for } [\vec {e}]=\vec {f}, \vec {g}\text { or }\vec {h}, \ \ \varvec{\gamma }=(\gamma _1, \gamma _2)\in {\mathbb {Z}}^2. \end{aligned}$$
By the definition (8.1), (8.2), it is clear that for any \(\psi \in L^2({\mathcal {E}}(\Lambda ))\),
$$\begin{aligned} \frac{d}{dt} T^B_{\varvec{\gamma }}\psi =T^B_{\varvec{\gamma }}\frac{d}{dt}\psi \text { and } VT^B_{\varvec{\gamma }}\psi =T^B_{\varvec{\gamma }}V\psi . \end{aligned}$$
In order to make sure \(D(\Lambda ^BT^B_{\varvec{\gamma }})=D(T^B_{\varvec{\gamma }}\Lambda ^B)\), it suffices to check \(T^B_{\varvec{\gamma }}(D(\Lambda ^B))=D(\Lambda ^B)\), which translates into
$$\begin{aligned} \left\{ \begin{matrix} u_{\varvec{\gamma }}(\vec {e}_1)=u_{\varvec{\gamma }}(\vec {e}_2)\ \text { whenever } i(\vec {e}_1)=i(\vec {e}_2)\\ \\ \frac{e^{i\tilde{\beta }_{\vec {e}_2}}\, u_{\varvec{\gamma }}(\vec {e}_2)}{e^{i\tilde{\beta }_{\vec {e}_1}}\, u_{\varvec{\gamma }}(\vec {e}_1)}=\frac{e^{i\tilde{\beta }_{\vec {e}_2-\gamma _1\vec {b}_1-\gamma _2 \vec {b}_2}}}{e^{i\tilde{\beta }_{\vec {e}_1-\gamma _1 \vec {b}_1-\gamma _2 \vec {b}_2}}}\ \text { whenever } t(\vec {e}_1)=t(\vec {e}_2). \end{matrix} \right. \end{aligned}$$
This, by (3.16) is in turn equivalent to the following: for any \(\tilde{\gamma }_1, \tilde{\gamma }_2\in {\mathbb {Z}}\):
$$\begin{aligned} \left\{ \begin{matrix} u_{\varvec{\gamma }}(\tilde{\gamma }_1, \tilde{\gamma }_2, \vec {f})=u_{\varvec{\gamma }}(\tilde{\gamma }_1, \tilde{\gamma }_2, \vec {g}) =u_{\varvec{\gamma }}(\tilde{\gamma }_1, \tilde{\gamma }_2, \vec {h})\\ \\ u_{\varvec{\gamma }}(\tilde{\gamma }_1, \tilde{\gamma }_2, \vec {f})=u_{\varvec{\gamma }}(\tilde{\gamma }_1+1, \tilde{\gamma }_2, \vec {g})=e^{-i\Phi \gamma _1}u_{\varvec{\gamma }}(\tilde{\gamma }_1, \tilde{\gamma }_2+1, \vec {h}) \end{matrix} \right. \end{aligned}$$
The definition of \(u^B_{\varvec{\gamma }}\) (8.2) clearly satisfies this requirement.
Therefore, although magnetic translations do not necessarily commute with one another, they commute with \(\Lambda ^B\)
$$\begin{aligned} T_{\varvec{\gamma }}^B \Lambda ^B = \Lambda ^B T_{\varvec{\gamma }}^B. \end{aligned}$$

8.5 Dirichlet spectrum

In this subsection, we will study the energies belonging to the Dirichlet spectrum \(\sigma (H^D)\). Lemma 8.4 below shows that \(\sigma (H^D)\) is contained in the point spectrum of \(H^B\), hence the only point spectrum of \(H^B\), due to Part (3) of Lemma 8.2.

Consider a compactly supported simply closed loop, which is a path with vertices of degree 2 enclosing q hexagons, see e.g. Fig. 7. Then this loop passes (proceeding in positive direction from the center of an edge \(\vec {e}_1\) such that the first vertex we reach is \(t(\vec {e}_1)\)) n edges \(\vec {e}_1,\ldots , \vec {e}_{n}\) in \({\mathcal {E}}(\Lambda )\), where n is an even number. For a solution vanishing outside this loop, the boundary conditions imposed by (3.15) on the derivatives can be represented in a matrix equation
$$\begin{aligned} T_{\Phi }(n)\psi ^\prime (n)=0, \end{aligned}$$
$$\begin{aligned} T_{\Phi }(n)&:= \left( \begin{matrix} e^{i\tilde{\beta }_{\vec {e}_1}} &{}\quad e^{i\tilde{\beta }_{\vec {e}_2}} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad e^{i\tilde{\beta }_{\vec {e}_3}} &{}\quad e^{i \tilde{\beta }_{\vec {e}_4}} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad 0 &{}\quad 0 &{}\quad 0 &{}\quad e^{i \tilde{\beta }_{\vec {e}_{n-1}}} &{}\quad e^{i \tilde{\beta }_{\vec {e}_n}} \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{matrix}\right) \ \text {and}\ \nonumber \\ \psi ^\prime (n)&:= \left( \begin{matrix} \psi ^\prime _{\vec {e}_1}(1)\\ \psi ^\prime _{\vec {e}_2}(1)\\ \psi ^\prime _{\vec {e}_3}(1)\\ \vdots \\ \psi ^\prime _{\vec {e}_{n-1}}(1)\\ \psi ^\prime _{\vec {e}_n}(1) \end{matrix} \right) . \end{aligned}$$

Remark 7

We observe that \(T_{\Phi }(n)\) can be row-reduced to an upper triangular matrix with diagonal
$$\begin{aligned}&(e^{i\tilde{\beta }_{\vec {e}_1}}, 1, e^{i\tilde{\beta }_{\vec {e}_3}}, 1, e^{i\tilde{\beta }_{\vec {e}_5}},\ldots ,1, e^{i\tilde{\beta }_{\vec {e}_{n-1}}}, 1-e^{i\sum _{j=1}^n (-1)^j \tilde{\beta }_{\vec {e}_j}})\\&=(e^{i\tilde{\beta }_{\vec {e}_1}}, 1, e^{i\tilde{\beta }_{\vec {e}_3}}, 1, e^{i\tilde{\beta }_{\vec {e}_5}},\ldots ,1, e^{i\tilde{\beta }_{\vec {e}_{n-1}}}, 1-e^{{{\pm \, i q\Phi }}}), \end{aligned}$$
where q is the number of enclosed hexagons. Hence \({\text {rank}}(T_{\Phi }(n))=n\) iff \( q \Phi \notin 2\pi {\mathbb {Z}}\) and \({\text {rank}}(T_{\Phi }(n))=n-1\) otherwise.

Lemma 8.4

The Dirichlet eigenvalues \(\lambda \in \sigma (H^D)\) are contained in the point spectrum of \(H^B\).


For \(\Phi \in 2\pi {\mathbb {Z}}\) the statement is known [39, Theroem 3.6], thus we focus on \(\Phi \notin 2\pi {\mathbb {Z}}.\) By unitary equivalence, it suffices to construct an eigenfunction to \(\Lambda ^B.\) We will construct an eigenfunction on two adjacent hexagons \(\Gamma \) as in Fig. 6. Thus, \(q=2\), the total number of edges is \(m=11,\) of which \(n=10\) are on the outer loop. Let us denote the slicing edge by \(\vec {e}\) and the edges on the outer loop by \(\vec {e}_1, \vec {e}_2,\ldots , \vec {e}_{10}\) (see Fig. 6). Recall that \(s_{\lambda , \vec {e}}\) is the Dirichlet eigenfunction on \(\vec {e}\).

By Remark 7, for \(2 \Phi \in 2\pi {\mathbb {Z}}\), operator \(T_{\Phi }(10)\) has a non-trivial nullspace. We could take
$$\begin{aligned} a=(a_j) \in {\text {ker}}\left( T_{\Phi }(10)\right) \backslash \{0\}, \end{aligned}$$
and an eigenfunction \(\psi \) on \(\Gamma \) such that \(\psi _{\vec {e}}=0\) and \(\psi _{\vec {e}_j}=a_j s_{\lambda , \vec {e}_j}\).
If \(2\Phi \notin 2\pi {\mathbb {Z}}\), we take a vector \(y \in {\mathbb {C}}^{10}\) such that \(y_2=-1\), \(y_7=-e^{i\tilde{\beta }_{\vec {e}}}\) and \(y_j=0\) otherwise. Since in this case \(T_{\Phi }(10)\) is invertible, there exists a unique solution \(a=(a_j)\) to the following equation:
$$\begin{aligned} T_{\Phi }(10) a = y. \end{aligned}$$
Let us take \(\psi \) on \(\Gamma \) such that \(\psi _{\vec {e}_j}=a_j s_{\lambda , \vec {e}_j}\) and \(\psi _{\vec {e}}=s_{\lambda , \vec {e}}\), then one can easily check \(\psi \) is indeed an eigenfunction on \(\Gamma \). \(\square \)
Fig. 6

Black arrows describe the double hexagon with slicing edge \(\vec {e}\) indicated by the dashed arrow

As a corollary of Lemmas 8.1 and 8.4, we have the following:

Corollary 8.5

The spectrum of \(H^B\) must always have open gaps for \(\Phi \notin 2 \pi {\mathbb {Z}}\) at the edges of the Hill bands.

Remark 8

If the magnetic flux is trivial, i.e. \(\Phi \in 2 \pi {\mathbb {Z}}\), then there do not have to be gaps. In particular, for zero potential in the non-magnetic case discussed in Theorem 6 all gaps of the absolutely continuous spectrum are closed and \(\sigma _{\text {ac}}(H^B)=[0,\infty )\).

The next lemma concerns the general feature of eigenspace of \(H^B\). Before proceeding, let us introduce the degree of a vertex in order to distinguish different types of eigenfunctions.

Definition 8.6

An eigenfunction is said to have a vertex of degree d if there is a vertex with exactly d adjacent edges on which the eigenfunction does not vanish.

Lemma 8.7

For the point spectrum of \(H^B\) it follows that
  1. (1)

    Every eigenspace of \(H^B\) is infinitely degenerated.

  2. (2)

    Eigenfunctions of \(H^B\) vanish at every vertex and are thus eigenfunctions of \(H^D\) as well.

  3. (3)

    Eigenfunctions of \(H^B\) with compact support cannot have vertices of degree 1. In particular, they must contain loops and the boundary edges of their support form loops as well.



(1). This follows immediately using magnetic translations (8.1) and studying \(\Lambda ^B\), instead. Assume there was a finite-dimensional eigenspace of \(\Lambda ^B.\) Because magnetic translations commute with \(\Lambda ^B\), they leave the eigenspaces of \(\Lambda ^B\) invariant. Magnetic translations are unitary, thus there is for any magnetic translation a normalized eigenfunction \(\psi \) with eigenvalue \(\lambda \) on the unit circle in \({\mathbb {C}}\). For \(\psi \), there is a sufficiently large ball B(0, R) such that
$$\begin{aligned} \Vert \psi \Vert _{L^2({\mathcal {E}}(\Lambda ) \cap B(0,R))}>1-\varepsilon . \end{aligned}$$
Upon n-fold application of the magnetic translation, the point 0 gets translated to some point \(x_n\) whereas the eigenfunction \(\psi \) acquires only a complex phase \(\lambda ^n\). Thus, (8.10) still holds and we must also have that
$$\begin{aligned} \left||\psi \right||_{L^2({\mathcal {E}}(\Lambda ) \cap B(x_n,R))}>1-\varepsilon . \end{aligned}$$
Yet, there exists n such that \(B(0,R)\cap B(x_n,R)=\emptyset .\) Therefore, (8.10) and (8.11) cannot hold at the same time for arbitrarily large n. This contradicts the existence of an eigenfunction to magnetic translations and thus the existence of a finite-dimensional eigenspace.
(2). If there is an eigenfunction to \(H^B\) with eigenvalue \(\lambda \) that does not vanish at a vertex, by (modified) Peierls’ substitution (3.13), there is one to \(\Lambda ^B,\) denoted as \(\varphi ,\) as well. We may expand the function in local coordinates on every edge \(\vec {e} \in {\mathcal {E}}(\Lambda )\) as \(\varphi _{\vec {e}}=a_{\vec {e}} c_{\lambda , \vec {e}} + b_{\vec {e}} s_{\lambda , \vec {e}}\) according to (2.35). Recall also that the Dirichlet eigenfunction \(s_{\lambda }\) is either even or odd. Thus, using (2.31) we conclude that \(\left|c_{\lambda }(0) \right|= \left|c_{\lambda }(1) \right|\) and thus \(\varphi \) cannot be compactly supported. In particular, \(\varphi \) has the same absolute value at any vertex by boundary conditions (3.15). Due to
$$\begin{aligned} \sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} \left|\varphi _{\vec {e}}(i(\vec {e})) \right|^2 \le \left||\varphi \right||^2_{{\mathcal {H}}^2}< \infty \end{aligned}$$
\(\varphi \) has to vanish at every vertex. Thus \(\varphi \) is also an eigenfunction to \(H^{D}.\)

(3) clearly follows from (2) and (3.15). \(\square \)

8.5.1 Dirichlet spectrum for rational flux quanta

In this section, the flux quanta are assumed to be reduced fractions \(\frac{\Phi }{2\pi } =\frac{p}{q}.\)

If magnetic fields are absent, the point spectrum is spanned by hexagonal simply closed loop states, i.e. states supported on a single hexagon [39]. We will see in the following that similar statements remain true in the case of rational flux quanta and derive such a basis as well. The natural extension of loop states supported on a single hexagon, in the case of magnetic fields, are simply closed loops enclosing an area \(\frac{q\Phi }{B_0}\) rather than just \(\frac{\Phi }{B_0}\), see Fig. 7.
Fig. 7

Simply closed loop state supported on black arrows encloses area \(\frac{4\Phi }{B_0}\)

Lemma 8.8

Any simply closed loop enclosing an area of \(\frac{q \Phi }{B_0}\) has a unique (up to normalization) eigenfunction of \(H^B\) supported on it.


The existence of eigenfunctions on simply closed loops enclosing this flux follows directly from the non-trivial kernel of (8.7), see Remark 7. Due to \({\text {dim}}({\text {ker}}(T_{\Phi }))=1\), such eigenfunctions are also unique (up to normalization). \(\square \)

Lemma 8.9

The nullspaces \({\text {ker}}(H^B-\lambda )\) where \(\lambda \in \sigma (H^D)\) are generated by compactly supported eigenfunctions.


Unitary equivalence allows us to work with \(\Lambda ^B\) rather than \(H^B\). Without loss of generality, we assume that the Dirichlet eigenfunction to \(\lambda \) is even. Due to Lemma 8.7, eigenfunctions of \(\Lambda ^B\) to Dirichlet eigenvalues vanish at every vertex. Thus, on every edge \(\vec {e}\in {\mathcal {E}}(V)\), they are of the form \(\varphi _{\vec {e}}=a_{\vec {e}}s_{\lambda , \vec {e}}\) for some \(a_{\vec {e}}\).

Let \(\varphi \) be such a function. We define the sequence \((u({{\varvec{v}}}))_{{{\varvec{v}}} \in {\mathcal {V}}(\Lambda )}\) as follows
$$\begin{aligned} \left\{ \begin{matrix} u(\gamma _1, \gamma _2, {{\varvec{r}}}_0):= \varphi _{\gamma _1, \gamma _2, \vec {g}}'(\gamma _1, \gamma _2, {{\varvec{r}}}_0) \\ u(\gamma _1, \gamma _2, {{\varvec{r}}}_1):= \varphi _{\gamma _1, \gamma _2, \vec {f}}'(\gamma _1, \gamma _2, {{\varvec{r}}}_1). \end{matrix} \right. \end{aligned}$$
Observe that the sequence \((u({{\varvec{v}}}))\) determines the eigenfunction on every edge. Indeed, \(a_{\gamma _1, \gamma _2, \vec {g}}=u(\gamma _1, \gamma _2, {{\varvec{r}}}_0)\) and \(a_{\gamma _1, \gamma _2, \vec {f}}=u(\gamma _1, \gamma _2, {{\varvec{r}}}_1)\), since \(s^\prime _{\lambda }(1)=s^\prime _{\lambda }(0)\). At the same time, \(a_{\gamma _1, \gamma _2, \vec {h}}\) can be determined in two different ways, one for each endpoint, from the boundary condition (3.15). Let us now introduce an operator \(A \in {\mathcal {L}}( l^2({\mathcal {V}}(\Lambda )))\) that has precisely the sequences \((u({{\varvec{v}}}))\) with matching boundary conditions for \(a_{\gamma _1, \gamma _2, \vec {h}}\) in its kernel. Then,
$$\begin{aligned} (Au)(\gamma _1, \gamma _2, {{\varvec{r}}}_0)&:= u(\gamma _1, \gamma _2, {{\varvec{r}}}_0)+u(\gamma _1, \gamma _2, {{\varvec{r}}}_1) \nonumber \\&\ \ \ \, -e^{2\pi i \frac{p\gamma _1}{q}} \Big (u(\gamma _1+1, \gamma _2-1, {{\varvec{r}}}_0)+u(\gamma _1, \gamma _2-1, {{\varvec{r}}}_1) \Big ) \text { and } \nonumber \\ (Au)(\gamma _1, \gamma _2, {{\varvec{r}}}_1)&:=0. \end{aligned}$$
The operator A is then a \({\mathbb {Z}}^2\)-periodic finite-order difference operator. Any eigenfunction \(\varphi \) satisfying \((\Lambda ^B - \lambda )\varphi =0\) leads by standard arguments to a square-summable sequence \((u({{\varvec{v}}}))\) as defined above in the nullspace of A. Conversely, any such element in the nullspace of A uniquely defines an eigenfunction \(\varphi =a_{\vec {e}} s_{\lambda , \vec {e}}\) to \(\Lambda ^B.\) Theorem 8 in [36] implies then that the nullspace of A is generated by sequences in \(c_{00}({\mathcal {V}}(\Lambda ))\). It suffices now to observe that those compactly supported sequences also give rise to compactly supported eigenfunctions to conclude the claim. \(\square \)

Lemma 8.10

Let \(\Phi \notin 2 \pi {\mathbb {Z}}.\) The eigenspaces are spanned by the set of double hexagonal states, see Fig. 6.


By Lemma 8.7, all eigenfunctions vanish at every vertex. Compactly supported eigenfunctions are dense in the eigenspace by the previous Lemma 8.9. Thus, it suffices, as in the non-magnetic [39] case, to show that any compactly supported eigenfunction is a linear combination of double hexagonal states. Let \(\varphi \) be a compactly supported eigenfunction of \(\Lambda ^B\) to some Dirichlet eigenvalue \(\lambda \). Consider an edge \(\vec {d} \in {\mathcal {E}}(\Lambda )\) on the boundary loop of the support of \(\varphi \). It exists due to (3) of Lemma 8.7. The boundary loop, which cannot be just a loop around a single hexagon, as this one does not support such eigenfunctions, necessarily encloses a double hexagon \(\Gamma \), as in Fig. 6, which contains the chosen edge \(\vec {d}.\) Then, there is by the proof of Lemma 8.4 a state \(\psi \) on \(\Gamma \) so that the wavefunction \(\psi _{\vec {d}}\) on \(\vec {d}\) coincides with \(\varphi _{\vec {d}}.\) Subtracting \(\psi \) from \(\varphi \) leaves us with an eigenfunction to \(\Lambda ^B\) that encloses at least one single hexagon less than \(\psi .\) Thus, iterating this procedure shows that compactly supported eigenfunctions are spanned by double hexagonal states which implies the claim. \(\square \)

8.5.2 Dirichlet spectrum for irrational flux quanta

After proving Theorem 4 for rational flux quanta, we now prove the analogous result for irrational magnetic fluxes. We start by introducing the following definition.

Definition 8.11

The Hilbert space \(l^2({\mathcal {E}}(\Lambda ))\) is defined as
$$\begin{aligned} l^2({\mathcal {E}}(\Lambda )):=\left\{ z: {\mathcal {E}}(\Lambda ) \rightarrow {\mathbb {C}},\ \Vert z \Vert _{l^2({\mathcal {E}}(\Lambda ))}^2:=\sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} \left| z(\vec {e}) \right| ^2 < \infty \right\} . \end{aligned}$$

Theorem 10

The double hexagonal states generate the eigenspaces of Dirichlet spectrum of \(H^B\) for irrational flux quanta.

We will give a proof of this theorem after a couple of auxiliary observations. For this entire discussion to follow we consider a fixed \(\lambda \in \sigma (H^D).\)

Definition 8.12

We denote the closed \(L^2({\mathcal {E}}(\Lambda ))\) subspace generated by linear combinations of all double hexagonal states on the entire graph \(\Lambda \) by \(DH_{{\mathcal {E}}(\Lambda )}(\Phi ).\)

There is a countable orthonormal system of states \(V(\Phi )\subset DH_{{\mathcal {E}}(\Lambda )}(\Phi )\) such that
$$\begin{aligned} \overline{{\text {span}}(V(\Phi ))}= DH_{{\mathcal {E}}(\Lambda )}(\Phi ). \end{aligned}$$
We may label elements of \(V(\Phi )\) by \(\varphi _{{{\varvec{\gamma }}}}(\Phi )\) with \({{\varvec{\gamma }}} \in {\mathbb {Z}}^2\). Without loss of generality, \(\varphi _{\varvec{\gamma }}(\Phi )\) can be chosen to depend analytically on \(\Phi \in (0,1)\). Every element \(\varphi _{\varvec{\gamma }}(\Phi ) \in V(\Phi )\) is due to Lemma 8.7 of the form
$$\begin{aligned} \varphi _{\varvec{\gamma }}(\Phi )=\sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} \varphi _{\varvec{\gamma },\vec {e}}(\Phi ) s_{\lambda ,\vec {e}} \end{aligned}$$
because it is an element of \({\text {ker}}(H^B-\lambda ).\)

Now assume that the statement of Theorem 10 does not hold, this is equivalent to saying that \(Z(\Phi ):= {\text {ker}}(H^B-\lambda ) \cap DH_{{\mathcal {E}}(\Lambda )}(\Phi )^{\perp }\) is not the zero space, i.e. there are eigenfunctions not spanned by double hexagonal states. Our goal is to characterize \(Z(\Phi )\) as the nullspace of a suitable operator we define next.

Definition 8.13

Let \(A(\Phi ) \in {\mathcal {L}}(l^2({\mathcal {E}}(\Lambda )))\) be defined as
$$\begin{aligned} \begin{aligned} (A(\Phi )u)(\varvec{\gamma },\vec {f})&:=u(\varvec{\gamma },\vec {f})+u(\varvec{\gamma },\vec {g})+u(\varvec{\gamma },\vec {h}) \\ (A(\Phi )u)(\varvec{\gamma },\vec {g})&:=u(\gamma _1, \gamma _2-1 ,\vec {f})+u(\gamma _1+1, \gamma _2-1,\vec {g})\\&\quad +e^{-i \Phi \gamma _1 } u(\gamma _1, \gamma _2,\vec {h}) \\ (A(\Phi )u)(\varvec{\gamma },\vec {h})&:=\left\langle u, (\varphi _{\varvec{\gamma },\vec {e}}(\Phi )) \right\rangle _{l^2({\mathcal {E}}(\Lambda ))}, \end{aligned} \end{aligned}$$
for any \(u\in l^2({\mathcal {E}}(\Lambda ))\).

Remark 9

The first two lines of this definition resemble the boundary conditions for the derivatives at outgoing/incoming vertices (3.17) and with the third line we monitor the orthogonality of \(\sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} u_{\vec {e}} s_{\lambda ,\vec {e}}\) to \(DH_{{\mathcal {E}}(\Lambda )}(\Phi ).\)

In particular, there is an isometric isomorphism \(\eta \in {\mathcal {L}}( {\text {ker}}(A(\Phi )), Z(\Phi ))\) with
$$\begin{aligned} \eta (u):=\sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} \frac{u_{\vec {e}}}{\left||s_{\lambda ,\vec {e}}\right||_{L^2(\vec {e})}} s_{\lambda ,\vec {e}}. \end{aligned}$$
We observe that by Lemma 8.9 and the isomorphism (8.19) the operator \(A(\Phi )\) is injective for \(\frac{\Phi }{2\pi } \in {\mathbb {Q}} \cap (0,1)\). To prove Theorem 10 we only need the following Lemma:

Lemma 8.14

The operator \(A(\Phi )\) is surjective for \(\frac{\Phi }{2\pi } \in (0,1)\). In particular, for any \((a(\vec {e}))\in l^2({\mathcal {E}}(\Lambda ))\), there exists \((u(\vec {e}))\in l^2({\mathcal {E}}(\Lambda ))\) such that \(A(\Phi )u=a\) and
$$\begin{aligned} \Vert u\Vert _{l^2({\mathcal {E}}(\Lambda ))}\le \frac{C}{|1-e^{-i\Phi }|} \Vert a\Vert _{l^2({\mathcal {E}}(\Lambda ))} \end{aligned}$$
holds for a universal constant C.
Combining Lemma 8.14 with the already established injectivity result, we have \(A(\Phi )\) is continuously invertible for \(\frac{\Phi }{2\pi }\in {\mathbb {Q}}\cap (0,1)\) with the following control of its norm
$$\begin{aligned} \Vert A(\Phi )^{-1}\Vert \le \frac{C}{|1-e^{-i\Phi }|}. \end{aligned}$$
Now let us give the proof of Theorem 10, assuming the result of Lemma 8.14.

8.6 Proof of Theorem 10

Since \(\Vert A(\Phi )\Vert \) is uniformly bounded by a constant and \(\Phi \mapsto \langle x,A(\Phi )y \rangle \) is analytic for \(x,y \in c_{00}({\mathcal {E}}(\Lambda ))\), \(A(\Phi )\) is an analytic operator in \(\Phi \). Thus for any \(\frac{\tilde{\Phi }}{2\pi } \in (0,1)\), there exists \(\varepsilon _1(\tilde{\Phi })\) and \(C(\tilde{\Phi })\) such that
$$\begin{aligned} \Vert A(\Phi )-A(\tilde{\Phi })\Vert \le C(\tilde{\Phi })|\Phi -\tilde{\Phi }|,\ \ \ \ \ \text {for}\ |\Phi -\tilde{\Phi }|<\varepsilon _1(\tilde{\Phi }). \end{aligned}$$
Also by (8.21), for any irrational \(\frac{\tilde{\Phi }}{2\pi }\in (0,1)\) and rational \(\frac{\Phi }{2\pi }\) with \(|\Phi -\tilde{\Phi }|<\varepsilon _2(\tilde{\Phi })\), we have
$$\begin{aligned} \Vert A(\Phi )^{-1}\Vert \le \frac{2C}{|1-e^{-i\tilde{\Phi }}|}. \end{aligned}$$
Hence, taking \(\frac{\Phi }{2\pi }\in {\mathbb {Q}}\cap (0,1)\) that is close to \(\frac{\tilde{\Phi }}{2\pi }\) such that \(|\Phi -\tilde{\Phi }|<\min (\varepsilon _1(\tilde{\Phi }), \varepsilon _2(\tilde{\Phi }), \frac{|1-e^{-i\tilde{\Phi }}|}{2C(\tilde{\Phi })C})\), we would get
$$\begin{aligned} \Vert A(\Phi )^{-1}(A(\tilde{\Phi })-A(\Phi ))\Vert <1. \end{aligned}$$
This implies that
$$\begin{aligned} A(\tilde{\Phi })=A(\Phi )\left( \text {Id}+A(\Phi )^{-1}(A(\tilde{\Phi })-A(\Phi ))\right) \end{aligned}$$
is invertible. Thus, we conclude that also for irrational fluxes \({\text {ker}}(A(\Phi ))=\left\{ 0\right\} \) and by (8.19) therefore \(Z(\Phi )=\left\{ 0\right\} \) which shows the claim. \(\square \)

8.7 Proof of Lemma 8.14

We prove this Lemma by showing that there is a sufficiently sparse set of elements in \(l^2({\mathcal {E}}(\Lambda ))\) that gets mapped under \(A(\Phi )\) on the standard basis of \(l^2({\mathcal {E}}(\Lambda )).\)

Let \(\alpha _{\vec {e},(\varvec{\gamma },\vec {h})}:= \varphi _{\varvec{\gamma },\vec {e}} \left||s_{\lambda ,\vec {e}} \right||^2_{L^2(\vec {e})}.\) Since functions \(\varphi _{\varvec{\gamma }}\) satisfy the continuity conditions (3.17) and form an \(L^2\) orthonormal system, we obtain the standard basis vectors \(\delta _{\bullet ,(\varvec{\gamma },\vec {h})} \in l^2({\mathcal {E}}(\Lambda ))\) under \(A(\Phi )\)
$$\begin{aligned} \begin{aligned} (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {h})})(\varvec{\gamma }',\vec {f})&:=0, \\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {h})})(\varvec{\gamma }',\vec {g})&:=0, \text { and }\\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {h})})(\varvec{\gamma }',\vec {h})&:=\delta _{\varvec{\gamma },\varvec{\gamma }'}. \end{aligned} \end{aligned}$$
Fig. 8

Labelling of hexagon \(\Gamma _{\varvec{\gamma }}\)

To obtain also the remaining basis vectors, let us define \(L^2\) functions \(\tilde{\psi }_{(\varvec{\gamma },\vec {f})}\) and \(\tilde{\psi }_{(\varvec{\gamma },\vec {g})}\) supported on a single hexagon \(\Gamma _{\varvec{\gamma }}\) as shown in Fig. 8. The indices of \(\tilde{\psi }_{(\varvec{\gamma },[\vec {e}])}\) are chosen to indicate the standard basis vectors \(\delta _{\bullet ,(\varvec{\gamma },[\vec {e}])} \in l^2({\mathcal {E}}(\Lambda ))\) in the range of \(A(\Phi )\) that we will construct from those functions. To define \(\tilde{\psi }_{(\varvec{\gamma },\vec {f})}\) and \(\tilde{\psi }_{(\varvec{\gamma },\vec {g})}\), we introduce coefficients \(\zeta _{\bullet ,{(\varvec{\gamma },\vec {f})}}\) and \(\zeta _{\bullet ,{(\varvec{\gamma },\vec {g})}}\) such that \(\tilde{\psi }_{(\varvec{\gamma },\vec {f})}:=\sum _{\vec {e} \in {\mathcal {E}}(\Gamma _{\varvec{\gamma }})} \zeta _{\vec {e},(\varvec{\gamma },\vec {f})} s_{\lambda ,\vec {e}}\) and \(\tilde{\psi }_{(\varvec{\gamma },\vec {g})}:=\sum _{\vec {e} \in {\mathcal {E}}(\Gamma _{\varvec{\gamma }})} \zeta _{\vec {e},(\varvec{\gamma },\vec {g})} s_{\lambda ,\vec {e}},\) respectively.

We do this in such a way that all continuity conditions for \(\tilde{\psi }_{(\varvec{\gamma },\vec {f})}\) at the vertices of \(\Gamma _{\varvec{\gamma }}\) are satisfied up to a single one at the (initial) vertex \({{\varvec{v}}}_1:=i((\varvec{\gamma },\vec {g}))=i((\varvec{\gamma },\vec {h}))\), see Fig. 8. We define for fixed \(\vec {e}=(\varvec{\gamma },\vec {f})\)
$$\begin{aligned} \begin{aligned}&\zeta _{(\varvec{\gamma },\vec {h}),\vec {e}}:=\frac{1}{1-e^{-i\Phi }}, \ \zeta _{(\varvec{\gamma }-(0,1),\vec {f}),\vec {e}}:=\frac{-e^{-i\Phi \gamma _1}}{1-e^{-i\Phi }}, \ \zeta _{(\varvec{\gamma }-(0,1),\vec {g}),\vec {e}}:=\frac{e^{-i\Phi \gamma _1}}{1-e^{-i\Phi }},\\&\zeta _{(\varvec{\gamma }-(1,0),\vec {h}),\vec {e}}:=\frac{-\,e^{-i\Phi }}{1-e^{-i\Phi }}, \, \zeta _{(\varvec{\gamma }-(1,0),\vec {f}),\vec {e}}:=\frac{e^{-i\Phi }}{1-e^{-i\Phi }},\ \zeta _{(\varvec{\gamma },\vec {g}),\vec {e}}:=\frac{-\,e^{-i\Phi }}{1-e^{-i\Phi }} \end{aligned} \end{aligned}$$
and all other \(\zeta _{\bullet ,\vec {e}}\) are taken to be zero. Since for \(\tilde{\psi }_{(\varvec{\gamma },\vec {f})}\) all but one continuity conditions are satisfied, we obtain for the first two components of (8.18)
$$\begin{aligned} (A(\Phi )\zeta _{\bullet ,(\varvec{\gamma },\vec {f})})(\varvec{\gamma }',\vec {f}):=\delta _{\varvec{\gamma },\varvec{\gamma }'} \text { and } (A(\Phi )\zeta _{\bullet ,(\varvec{\gamma },\vec {f})})(\varvec{\gamma }',\vec {g}):=0. \end{aligned}$$
To ensure that we also get constant zero in the third component of (8.18), we project onto the orthogonal complement of the double hexagonal states \(\psi _{(\varvec{\gamma },\vec {f})}:={\widetilde{\psi }}_{(\varvec{\gamma },\vec {f})}-P_{DH_{{\mathcal {E}}(\Lambda )}(\Phi )}{\widetilde{\psi }}_{(\varvec{\gamma },\vec {f})}\) where \(P_{DH_{{\mathcal {E}}(\Lambda )}(\Phi )}\) is the orthogonal projection onto \(DH_{{\mathcal {E}}(\Lambda )}(\Phi ).\) Let now \(\alpha _{\vec {e},(\varvec{\gamma },\vec {f})}\) be such that
$$\begin{aligned} \psi _{(\varvec{\gamma },\vec {f})}=\sum _{\vec {e} \in {\mathcal {E}}(\Gamma _{\varvec{\gamma }})} \alpha _{\vec {e},(\varvec{\gamma },\vec {f})} s_{\lambda ,\vec {e}}, \end{aligned}$$
then it follows that
$$\begin{aligned} (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {f})})(\varvec{\gamma }',\vec {f})&:=\delta _{\varvec{\gamma },\varvec{\gamma }'}, \nonumber \\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {f})})(\varvec{\gamma }',\vec {g})&:=0, \text { and } \nonumber \\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {f})})(\varvec{\gamma }',\vec {h})&:=0. \end{aligned}$$
Similarly, we choose coefficients \(\zeta _{\bullet ,\vec {e}}\) with \(\vec {e}=(\varvec{\gamma },\vec {g})\), such that the boundary conditions are satisfied up to the one at the (terminal) vertex \({{\varvec{v}}}_2:=t((\varvec{\gamma },\vec {h}))=t((\varvec{\gamma }-(0,1),\vec {f})),\) see Fig. 8.
$$\begin{aligned}&\zeta _{(\varvec{\gamma }-(0,1),\vec {f}),\vec {e}}:=\frac{1}{1-e^{-i\Phi }}, \ \zeta _{(\varvec{\gamma }-(0,1),\vec {g}),\vec {e}}:=\frac{-1}{1-e^{-i\Phi }}, \ \zeta _{(\varvec{\gamma }-(1,0),\vec {h}),\vec {e}}\nonumber \\&\quad :=\frac{e^{i\Phi (\gamma _1-1)}}{1-e^{-i\Phi }} \nonumber \\&\zeta _{(\varvec{\gamma }-(1,0),\vec {f}),\vec {e}}:=\frac{-e^{i\Phi (\gamma _1-1)}}{1-e^{-i\Phi }},\ \zeta _{(\varvec{\gamma },\vec {g}),\vec {e}}:=\frac{e^{i\Phi (\gamma _1-1)}}{1-e^{-i\Phi }}, \ \zeta _{(\varvec{\gamma },\vec {h}),\vec {e}}:=\frac{-e^{i\Phi (\gamma _1-1)}}{1-e^{-i\Phi }} \end{aligned}$$
and all other coefficients \(\zeta _{\bullet ,\vec {e}}\) equal to zero. Thus, we get for the first two components of (8.18)
$$\begin{aligned} (A(\Phi )\zeta _{\bullet ,(\varvec{\gamma },\vec {g})})(\varvec{\gamma }',\vec {f})= 0 \text { and } (A(\Phi )\zeta _{\bullet ,(\varvec{\gamma },\vec {g})})(\varvec{\gamma }',\vec {g})=\delta _{\varvec{\gamma },\varvec{\gamma }'}. \end{aligned}$$
To ensure that we also get constant zero in the third component of (8.18), we project again on the orthogonal complement of the double hexagonal states \(\psi _{(\varvec{\gamma },\vec {g})}:={\widetilde{\psi }}_{(\varvec{\gamma },\vec {g})}-P_{DH_{{\mathcal {E}}(\Lambda )}(\Phi )}{\widetilde{\psi }}_{(\varvec{\gamma },\vec {g})}\). Let now \(\psi _{(\varvec{\gamma },\vec {g})}=\sum _{\vec {e} \in {\mathcal {E}}(\Gamma _{\varvec{\gamma }})} \alpha _{\vec {e},(\varvec{\gamma },\vec {g})} s_{\lambda ,\vec {e}}\), then
$$\begin{aligned} \begin{aligned} (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {g})})(\varvec{\gamma }',\vec {f})&:=0 \\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {g})})(\varvec{\gamma }',\vec {g})&:=\delta _{\varvec{\gamma },\varvec{\gamma }'}, \text { and} \\ (A(\Phi )\alpha _{\bullet ,(\varvec{\gamma },\vec {g})})(\varvec{\gamma }',\vec {h})&:=0. \end{aligned} \end{aligned}$$
Hence, we obtained in (8.24), (8.28), and (8.31) sequences
$$\begin{aligned} \left\{ \alpha _{\bullet ,(\varvec{\gamma },\vec {f})}, \alpha _{\bullet ,(\varvec{\gamma },\vec {g})}, \text { and } \alpha _{\bullet ,(\varvec{\gamma },\vec {h})} ; {{\varvec{\gamma }}} \in {\mathbb {Z}}^2 \right\} \end{aligned}$$
in \(l^2({\mathcal {E}}(\Lambda ))\) that get mapped under \(A(\Phi )\) onto the standard unit basis of \(l^2({\mathcal {E}}(\Lambda )).\)
To conclude surjectivity of \(A(\Phi )\) from this, it suffices to show that for all \((a({\vec {e}})) \in l^2({\mathcal {E}}(\Lambda ))\) we can bound \(u(\vec {e}):=\sum _{\vec {d} \in {\mathcal {E}}(\Lambda )} a({\vec {d}}) \ \alpha _{\vec {e},\vec {d}}\) as follows
$$\begin{aligned} \left||u \right||^2_{l^2({\mathcal {E}}(\Lambda ))} \le \frac{C^2}{\left|1-e^{-i\Phi } \right|^2} \sum _{\vec {e} \in {\mathcal {E}}(\Lambda )} \left|a({\vec {e}}) \right|^2. \end{aligned}$$
We then define
$$\begin{aligned} \begin{aligned} \sigma _{\vec {e}}=\sum _{\vec {d} \in {\mathcal {E}}(\Lambda ); [\vec {d}] \ne \vec {h}} a({\vec {d}}) \ \alpha _{\vec {e},\vec {d}} \text { and } \nu _{\vec {e}}= \sum _{\vec {d} \in {\mathcal {E}}(\Lambda ); [\vec {d}] \ne \vec {h}} a({\vec {d}}) \ \zeta _{\vec {e},\vec {d}}. \end{aligned} \end{aligned}$$
Since \(\psi _{(\varvec{\gamma },\vec {f})},\psi _{(\varvec{\gamma },\vec {g})}\in DH_{{\mathcal {E}}(\Lambda )}(\Phi )^{\perp }\) and \((\varphi _{\varvec{\gamma }})\) forms an orthonormal system in \(DH_{{\mathcal {E}}(\Lambda )}(\Phi )\), to prove (8.33) it suffices to show
$$\begin{aligned} \Vert \sigma \Vert ^2_{l^2({\mathcal {E}}(\Lambda ))} \le \frac{C^2}{\left| 1-e^{-i\Phi } \right| ^2} \sum _{\vec {e} \in {\mathcal {E}}(\Lambda );[\vec {e}] \ne \vec {h} } \left| a({\vec {e}}) \right| ^2. \end{aligned}$$
Due to \(\left||\sigma \right||_{l^2({\mathcal {E}}(\Lambda ))} \le \left||\nu \right||_{l^2({\mathcal {E}}(\Lambda ))}+ \left||\sigma -\nu \right||_{l^2({\mathcal {E}}(\Lambda ))}\) we may establish estimate (8.33) for each term on the right-hand side of the triangle inequality, individually.
For two edges \( \vec {d}, \vec {e} \in {\mathcal {E}}(\Lambda )\) we define a function \(M(\vec {d},\vec {e}):=1\) if there are \(\varvec{\gamma },\varvec{\gamma }' \in {\mathbb {Z}}^2\) and two hexagons \(\Gamma _{\varvec{\gamma }}, \Gamma _{\varvec{\gamma }'}\) satisfying \(\Gamma _{\varvec{\gamma }} \cap \Gamma _{\varvec{\gamma }'}\ne \emptyset \) such that \(\vec {d} \in \Gamma _{\varvec{\gamma }}\) and \(\vec {e} \in \Gamma _{\varvec{\gamma }'},\) and \(M(\vec {d},\vec {e}):=0\) otherwise. Choosing \(\tau _1\) such that \(\sum _{\vec {d} \in {\mathcal {E}}(\Lambda ); [\vec {d}] \ne \vec {h}} M(\vec {d},\vec {e})\le \tau _1\) for any \(\vec {e} \in {\mathcal {E}}(\Lambda ),\) then
$$\begin{aligned} \begin{aligned} \left||\nu \right||^2_{l^2({\mathcal {E}}(\Lambda ))}&\le \sum _{\vec {d},\vec {e} \in {\mathcal {E}}(\Lambda ); [\vec {d}],[\vec {e}] \ne \vec {h}} \left|a({\vec {d}}) \right|\left|a({\vec {e}}) \right|\underbrace{\left||\zeta _{\bullet ,\vec {d}} \right||_{l^2({\mathcal {E}}(\Lambda ))} \left||\zeta _{\bullet ,\vec {e}} \right||_{l^2({\mathcal {E}}(\Lambda ))} }_{\le \frac{7}{\left|1-e^{-i\Phi } \right|^2}} M(\vec {d},\vec {e}) \\&\le \frac{{{6}} \tau _1 }{\left|1-e^{-i\Phi } \right|^2} \sum _{\vec {e} \in {\mathcal {E}}(\Lambda );[\vec {e}] \ne \vec {h}} \left|a({\vec {e}}) \right|^2. \end{aligned} \end{aligned}$$
For the second term \(\left||\sigma -\nu \right||_{l^2({\mathcal {E}}(\Lambda ))}\), we use that functions \(\tilde{\psi }_{(\varvec{\gamma },[{{\vec {e}}}])}\) with \([{{\vec {e}}}] \ne [{{\vec {h}}}]\) are supported on hexagons \(\Gamma \) and can therefore only overlap with finitely many linearly independent double hexagonal states. Thus, we define a function N with \(N(\vec {d}, \vec {e}):=1\) if \(\vec {d},\vec {e}\) belong to two hexagons \(\Gamma _{\varvec{\gamma }}, \Gamma _{\varvec{\gamma }'}\) for which there are two double hexagons \(\Gamma _{1},\Gamma _{2}\) with the property that all intersections \(\Gamma _{\varvec{\gamma }} \cap \Gamma _{1}\), \(\Gamma _{1} \cap \Gamma _{2}\), \(\Gamma _{2}\cap \Gamma _{\varvec{\gamma }'}\) are not empty. Otherwise, we set \(N(\vec {d}, \vec {e}):=0.\) Choosing \(\tau _2\) such that \(\sum _{\vec {d} \in {\mathcal {E}}(\Lambda ); [\vec {d}] \ne \vec {h}} N(\vec {d}, \vec {e})\le \tau _2\) for any \(\vec {e} \in {\mathcal {E}}(\Lambda ),\) then
$$\begin{aligned} \begin{aligned} \Vert \sigma -\nu \Vert _{l^2({\mathcal {E}}(\Lambda ))}^2&=\sum _{\vec {d},\vec {e} \in {\mathcal {E}}(\Lambda );[\vec {d}],[\vec {e}] \ne \vec {h}} \frac{N( \vec {d},\vec {e}) a({\vec {d}})\ \overline{a({\vec {e}})}}{\Vert s_{\lambda } \Vert _{L^2((0,1))}^2} \\&\quad \quad \left\langle P_{DH_{{\mathcal {E}}(\Lambda )}(\Phi )}\tilde{\psi }_{\vec {d}}, P_{DH_{{\mathcal {E}}(\Lambda )}(\Phi )}\tilde{\psi }_{\vec {e}} \right\rangle _{L^2({\mathcal {E}}(\Lambda ))} \\&\le \sum _{\vec {d},\vec {e} \in {\mathcal {E}}(\Lambda );[\vec {d}],[\vec {e}] \ne \vec {h}} \left| a({\vec {d} }) \right| \left| a({\vec {e}}) \right| \Vert \zeta _{\bullet ,\vec {d}} \Vert _{l^2({\mathcal {E}}(\Lambda ))}\\&\quad \quad \Vert \zeta _{\bullet ,\vec {e}} \Vert _{l^2({\mathcal {E}}(\Lambda ))} N(\vec {d},\vec {e}) \\ {}&\le \frac{{{6}}\tau _2}{\left| 1-e^{-i\Phi } \right| ^2} \sum _{\vec {e} \in {\mathcal {E}}(\Lambda );[\vec {e}] \ne \vec {h}} \left| a({\vec {e}}) \right| ^2. \end{aligned} \end{aligned}$$
\(\square \)
Fig. 9

Touching bands for \(\frac{\Phi }{2\pi }=\frac{1}{2}\) on the first Hill band of a Schrödinger operator with Mathieu potential \(V(t)=20\cos (2\pi t)\). Different bands are differently colored

8.8 Absolutely continuous spectrum for rational flux quanta

Lemma 8.15

For \(\frac{\Phi }{2\pi }=\frac{p}{q} \in {\mathbb {Q}},\) the spectrum of \(H^B\) away from the Dirichlet spectrum is absolutely continuous and has possibly touching, but non-overlapping band structure. An interval \(I \subset [-1,1]\) is a band of \(Q_{\Lambda }(\Phi )\) if and only if its pre-image under \(\Delta \), on each fixed band of the Hill operator, is a band of \(H^B.\)


That the bands of \(Q_{\Lambda }(\Phi )\) do not overlap is shown in Section 6 of [28]. Thus, the unique correspondence between bands of \(Q_{\Lambda }(\Phi )\) and \(H^B\), following from the unitary equivalence (7.14), shows that the non-overlapping of bands holds true for \(H^B\) as well. \(\square \)

Remark 10

For \(\frac{\Phi }{2\pi }= \frac{1}{2}\) the spectral bands of \(Q_{\Lambda }(\Phi )\) are touching and given by [28]
$$\begin{aligned} \left[ -\sqrt{\tfrac{2}{3}},-\sqrt{\tfrac{1}{3}}\right] , \ \left[ -\sqrt{\tfrac{1}{3}},0\right] ,\ \left[ 0,\sqrt{\tfrac{1}{3}}\right] , \text { and } \left[ \sqrt{\tfrac{1}{3}}, \sqrt{\tfrac{2}{3}}\right] . \end{aligned}$$
Thus, by Lemma 8.15 the bands of \(H^B\) on each Hill band are touching as well, see Fig. 9. Bands belonging to different Hill bands do, as a rule for \(\Phi \in (0,2\pi )\), not touch by Lemma 8.1.
Fig. 10

Only the third and fourth band touch at the Dirac points for \(\frac{\Phi }{2\pi }=\frac{1}{3}\) on the first Hill band of a Schrödinger operator with Mathieu potential \(V(t)=20\cos (2\pi t)\). Different bands are differently colored

In the case of \(\frac{\Phi }{2\pi }= \frac{1}{3}\) however, only the bands at the Dirac points touch, see also Fig. 10. The touching at the Dirac points is always satisfied by Lemma 8.2.

8.9 Singular continuous Cantor spectrum for irrational flux quanta


By Lemma 4.2, the spectrum of \(Q_{\Lambda }(\Phi )\) for irrational \(\frac{\Phi }{2\pi }\) is a Cantor set of measure zero. Thus, the pullback of \(\sigma (Q_{\Lambda })\) by \(\Delta \vert _{{\text {int}}(B_n)}\) is still a Cantor set of zero measure that coincides with \(\sigma (H^B)\backslash \sigma (H^D).\) Therefore, the absolutely continuous spectrum of \(H^B\) has to be empty. The Cantor spectrum part of (3) of Theorem 1, and (1) of Theorem 2 then follows from (4) of Lemma 8.2. \(\square \)

Fig. 11

The Hofstadter butterfly for \(H^B\) with \(V=0\) on the first five Hill bands \(B_k=[\pi ^2(k-1)^2,\pi ^2 k^2]\) for \(k \in \left\{ 1,..,5\right\} \) and magnetic flux quanta \(\frac{\Phi }{2\pi }=\frac{p}{q} \in [0,1]\) with \(q\le 50\)

8.10 Proofs of Theorems 14

This section serves as an index to the proofs of our main theorems that are distributed in different sections throughout the paper.

8.10.1 Proof of Theorem 1

  1. (1).

    Follows from (1) of Lemma 8.7.

  2. (2).

    Combine (3) of Lemma 8.2 with Lemma 8.4.

  3. (3).

    This is proved in Sects. 8.3 and  8.4 (Fig. 11).

  4. (4).

    Follows from (4) of Lemma 8.2.

  5. (5).

    This is Lemma 8.3. \(\square \)


8.10.2 Proof of Theorem 2

This is proved in Sects. 8.3 and 8.4. \(\square \)

8.10.3 Proof of Theorem 3

This is proved in Lemmas 4.24.3 and 4.4. \(\square \)

8.10.4 Proof of Theorem 4

This is proved in Lemma 8.10 and Theorem 10. \(\square \)


  1. 1.

    Schrödinger operators defined on metric graphs are also called quantum graphs.

  2. 2.

    In this paper, “generic” refers to a dense \(G_{\delta }\) set. Recently, a stronger continuity of spectra statement was proved in [33], which combined with Lemma 4.3 allowed the authors to extend the Hausdorff dimension statement to all irrational \(\Phi \).

  3. 3.

    A Jacobi matrix is called singular if its off-diagonal entries are not bounded away from zero.

  4. 4.

    We note that singular continuity of the spectrum of critical extended Harper’s model (including for parameters leading to singularity in the corresponding Jacobi matrix) has been proved recently in [7, 30] without establishing the Cantor nature.

  5. 5.

    See Footnote 2.

  6. 6.

    So that \({{\varvec{y}}}\) in (2.4)=\([{{\varvec{x}}}]\).

  7. 7.

    For rational \(\frac{\Phi }{2\pi }\) and singular \(H_{\Phi , \theta },\)\(\Sigma _{\Phi , \theta }\) may consist of infinitely degenerate isolated eigenvalues, if c vanishes somewhere on the orbit of rotation of \(\theta \) by \(\frac{\Phi }{2\pi }\).

  8. 8.

    In our concrete model, \(c(\theta )=1+e^{-2\pi i\theta }\), see (5.4), hence has a single zero \(\theta _1=1/2\).

  9. 9.

    This transform is also known as minimal coupling.



This research was partially supported by the NSF DMS-1401204 and DMS-1901462. Support by the EPSRC grant EP/L016516/1 for the University of Cambridge CDT, the CCA is gratefully acknowledged (S.B.). R.H. would like to thank the Institute for Advanced Study, Princeton, for its hospitality during the 2017–2018 academic year. R.H. was supported in part by NSF DMS-1638352 and NSF-DMS-1800689.


  1. 1.
    Avila, A.: On point spectrum with critical coupling (preprint).
  2. 2.
    Avila, A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215, 1–54 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Azbel, M.: Energy spectrum of a conduction electron in a magnetic field. Sov. Phys. JETP 19(3), 634–645 (1964)Google Scholar
  4. 4.
    Agazzi, A., Eckmann, J.-P., Graf, G.M.: The colored Hofstadter butterfly for the Honeycomb lattice. J. Stat. Phys. 156(3), 417–426 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Avila, A., Jitomirskaya, S.: The Ten Martini problem. Ann. Math. 170(1), 303–342 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12(1), 93–131 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Avila, A., Jitomirskaya, S., Marx, C.: Spectral theory of extended Harper’s model and a question by Erdős and Szekeres. Invent. Math. 210(1), 283–339 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Avila, A., Jitomirskaya, S., Sadel, C.: Complex one-frequency cocycles. J. Eur. Math. Soc. 16(9), 1915–1935 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Avron, J., v Mouche, P., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132(1), 103–118 (1990)Google Scholar
  10. 10.
    Aizenman, M., Warzel, S.: Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. 15, 1167–1222 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brüning, J., Geyler, V., Pankrashkin, K.: Cantor and band spectra for periodic quantum graphs with magnetic fields. Commun. Math. Phys. 269(1), 87–105 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Becker, S., Han, R.: In preparationGoogle Scholar
  13. 13.
    Becker, S., Zworski, M.: Magnetic Oscillations in a Model of Graphene. Commun. Math. Phys. 367(3), 941–989 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Becker, S., Han, R., Jitomirskaya, S., Zworski, M.: In preparationGoogle Scholar
  15. 15.
    Chen, X., Wallbank, A., Patel, A., Mucha-Kruczynski, M., McCann, E., Fal’ko, V.: Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices. Phys. Rev. B 89(7), 075401 (2014)CrossRefGoogle Scholar
  16. 16.
    Duren, P.: Theory of Hp Spaces, Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)Google Scholar
  17. 17.
    Dean, C.R., Wang, L., Maher, P., Forsythe, C., Ghahari, F., Gao, Y., Katoch, J., Ishigami, M., Moon, P., Koshino, M., Taniguchi, T., Watanabe, K., Shepard, K.L., Hone, J., Kim, P.: Hofstadter’s butterfly in moire superlattices: a fractal quantum Hall effect. Nature 497, 598–602 (2013)CrossRefGoogle Scholar
  18. 18.
    Dombrowsky, J.: Quasitriangular matrices. Proc. Am. Math. Soc. 69, 95–96 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Exner, P., Vaata, D.: Cantor spectra of magnetic chain graphs. J. Phys. A Math. Theor. 50(16), 165201 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fefferman, C., Weinstein, M.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25(4), 1169–1220 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fefferman, C., Weinstein, M.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326, 251–286 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fefferman, C., Lee-Thorp, M.J.P., Weinstein, M.: Honeycomb Schroedinger operators in the strong binding regime. Commun. Pure Appl. Math. 71, 6 (2018)zbMATHCrossRefGoogle Scholar
  23. 23.
    Garcia-C., H., Gaggero-S., L., Díaz-G., D.S., Sotolongo-C., O., Rodríguez-V., I.: Self-similar conductance patterns in graphene Cantor-like structures. Sci. Rep. 7(1), 617 (2017)Google Scholar
  24. 24.
    Gomes, K., Mar, W., Ko, W., Guinea, F., Manoharan, H.: Designer Dirac fermions and topological phases in molecular graphene. Nature 483(7389), 306–310 (2012)CrossRefGoogle Scholar
  25. 25.
    Ponomarenko, L., Gorbachev, R., Yu, G., Elias, D., Jalil, R., Patel, A., Mishchenko, A., Mayorov, A., Woods, C., Wallbank, J., Mucha-Kruczynski, M., Piot, B., Potemski, M., Grigorieva, I., Novoselov, K., Guinea, F., Fal’ko, V., Geim, A.: Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013)CrossRefGoogle Scholar
  26. 26.
    Gérard, C., Nier, F.: The Mourre theory for analytically fibered operators. J. Funct. Anal. 152(1), 202–219 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Guinea, F., et al.: Strain-induced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles. Science 329(5991), 544–547 (2010)CrossRefGoogle Scholar
  28. 28.
    Helffer, B., Kerdelhué, P., Royo-Letelier, J.: Chambers’s formula for the graphene and the Hou model. Ann. Henri Poincaré 17(4), 795–818 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Han, R.: Dry Ten Martini problem for the non-self-dual extended Harper’s model. Trans. Amer. Math. Soc. 370, 197–217 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Han, R.: Absence of point spectrum for the self-dual extended Harper’s model. Int. Math. Res. Not. 2018(9), 801–2809 (2018)MathSciNetGoogle Scholar
  31. 31.
    Hofstadter, D.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14(6), 2239–2249 (1976)CrossRefGoogle Scholar
  32. 32.
    Jaksic, V., Last, Y.: Surface states and spectra. Commun. Math. Phys. 218, 459–477 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Jitomirskaya, S., Krasovsky, I.: Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum (preprint) (2019)Google Scholar
  34. 34.
    Jitomirskaya, S., Marx, C.: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Commun. Math. Phys. 316(1), 237–267 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Jitomirskaya, S., Marx, C.: Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Commun. Math. Phys. 317, 269–271 (2013)CrossRefGoogle Scholar
  36. 36.
    Kuchment, P.: Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A Math. Gen. 38(22), 4887 (2005)Google Scholar
  37. 37.
    Kuchment, P.: An overview of periodic elliptic operators. Bull. Am. Math. Soc. 53(3), 343–414 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)zbMATHCrossRefGoogle Scholar
  39. 39.
    Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275(3), 805–82 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Kerdelhué, P., Royo-Letelier, J.: On the low lying spectrum of the magnetic Schrödinger operator with kagome periodicity. Rev. Math. Phys. 26(10), 1450020 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kostrykin, V., Schrader, R.: Quantum wires with magnetic fluxes. Commun. Math. Phys. 237(1), 161–179 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421–432 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Marx, C.: Dominated splittings and the spectrum of almost periodic Jacobi operators. Nonlinearity 27, 3059–3072 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Novoselov, K.: Nobel lecture: graphene: materials in the flatland. Rev. Mod. Phys. 83, 837–849 (2011)CrossRefGoogle Scholar
  45. 45.
    Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244(2), 297–309 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Pankrashkin, K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77(2), 139–154 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Pankrashkin, K.: An example of unitary equivalence between self-adjoint extensions and their parameters. J. Funct. Anal. 265(2013), 2910-2936, 640-655 (2013)Google Scholar
  48. 48.
    Pankrashkin, K.: Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures. J. Math. Anal. Appl. 396, 640–655 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Reed, M., Simon, B.: Analysis of Operators, Methods of Modern Mathematical Physics, vol. IV. Elsevier, Amsterdam (1978)zbMATHGoogle Scholar
  50. 50.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics. Springer, Berlin (2012)zbMATHCrossRefGoogle Scholar
  51. 51.
    Thomas, L.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33(4), 335–343 (1973)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, vol. 72. Amer. Math. Soc, Providence (2000)zbMATHGoogle Scholar
  53. 53.
    Toda, M.: Theory of Nonlinear Lattices. Springer, Berlin (1981)zbMATHCrossRefGoogle Scholar
  54. 54.
    Zak, J.: Magnetic translation group. Phys. Rev. 134(6A), A1602 (1964)zbMATHCrossRefGoogle Scholar
  55. 55.
    Zhang, Y., Tan, Y.-W., Stormer, H.-L., Kim, P.: Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438(7065), 201–204 (2005)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.DAMTPUniversity of CambridgeCambridgeUK
  2. 2.School of MathGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA

Personalised recommendations