Cantor spectrum of graphene in magnetic fields

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 Φ/2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi / 2\pi $$\end{document} through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

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 graph 1 and is a direct sum, over all edges e of the graph, of Schrödinger operators with magnetic potential A e , describing a constant magnetic field, and potential V e ∈ L 2 ( e). We write σ ,σ cont , σ ess 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 σ (H D ) its spectrum. Let σ p 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 e ∈ L 2 ( e) we have σ cont is • a Cantor set of measure zero for / ∈ 2π Q, • a countable union of disjoint intervals for ∈ 2π Q, (4) σ p ∩ σ cont = ∅ for / ∈ 2π Z, (5) the Hausdorff dimension dim H (σ ) ≤ 1/2 for generic 2 .
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 e ∈ L 2 ( e) we have (1) For / ∈ 2π Q, the spectrum on σ cont is purely singular continuous.
(2) For ∈ 2π Q, the spectrum on σ cont 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 matrix 3 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 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 ( ) is
• a finite union of intervals and purely absolutely continuous for /2π = p/q, which is a reduced rational number, with the following measure estimate where C > 0 is an absolute constant. • singular continuous and a zero measure Cantor set for / ∈ 2π Q, • a set of Hausdorff dimension dim H (σ (Q ( ))) ≤ 1/2 for generic 5 .
Remark 1 We will show that the constant C in the first item can be bounded by 8 √ 6π 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 , H pp is spanned by compactly supported eigenfunctions (in fact, by double hexagonal states).
While for the rational the proof is based on ideas similar to those of [39], for the irrational 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 1 (in a suitable sense) eigenfunctions are in the closure of the span of double hexagonal states, the 2 condition is more elusive. Bridging the gap between 1 and 2 has been a known difficult problem in several other scenarios [1,7,10,32]. Here we achieve this by constructing, for each , an operator that would have all slowly decaying 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 1 and 2 is also a novel argument.

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 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 4.1-4.4. Lemma 4.1 involves a further reduction from B to a two-dimensional tight-binding Hamiltonian Q ( ), and Lemmas 4.2-4.4 reveal the topological structure of σ (Q ( )) (thus proving the topological part of Theorem 3). The proofs of Lemmas 4.1-4.4 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 1-4 are distributed throughout the paper, we give an index to them, for the reader's convenience in Sect. 8.5.

Notation
Given a graph G, we denote the set of edges of G by E(G), the set of vertices by V(G), and the set of edges adjacent to a vertex v ∈ V(G) by E v (G).
For an operator H , let σ (H ) be its spectrum and ρ(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 i (R 2 ) the vector space of all i-covectors or differential forms of degree i on R 2 .
For a set U ⊆ R, let |U | be its Lebesgue measure. We define T * 2 := R 2 /(2π Z) 2 and T := T 1 := R/Z. List of main symbols used in this article.
• r 0 and r 1 are the vertices of the fundamental cell (2.1).
• f , g, h are the vectors of the fundamental cell (2.2).
• b 1 , b 2 are the basis vectors of the lattice (2.3).

• [v], [ e]
denotes the translate of a vertex v or edge e into the fundamental cell (2.5).
) are defined in the paragraph below (2.5). • i, t map edges to their respective initial and terminal vertex (2.6).
• κ e is the chart defined in (2.9).
• H e is the maximal Schrödinger operator on an edge 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 • B is the Schrödinger operator introduced in (3.14).
• T B γ are magnetic translation defined in (8.1).

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 is obtained by translating its fundamental cell W , the red colored part of Fig. 1, consisting of vertices r 0 := (0, 0) and r 1 := 1 2 , and edges along the basis vectors of the lattice. The basis vectors are and so the hexagonal graph ⊂ R 2 is given by the range of a Z 2 -action on the fundamental domain W The fundamental domain of the dual lattice can be identified with the dual 2-torus T * 2 . For any vertex v ∈ V( ), we denote by [v] ∈ V(W ) the unique vertex, r 0 or r 1 , for which there is γ ∈ Z 2 such that (2.5) We will occasionally denote v by (γ 1 , γ 2 , [v]) to emphasize the location of v. We also introduce a similar notation for edges. For an edge e ∈ E( ), we will sometimes denote it by (γ 1 , γ 2 , [ e]). Finally, for any x ∈ , we will also denote its unique preimage in W by [x]. 6 We can orient the edges in terms of initial and terminal maps 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 to obtain an oriented graph For arbitrary e ∈ E( ), we then just extend those maps by (2.8) for some e ∈ E( )} be the collection of initial vertices, and t ( ) = {v ∈ V( ) : v = t ( e) for some e ∈ E( )} be the collection of terminal ones. It should be noted that based on our orientation, V( ) is a disjoint union of i( ) and t ( ).
Every edge e ∈ E( ) is of length one and thus has a canonical chart that allows us to define function spaces and operators on e and finally on the entire graph. For n ∈ N 0 , the Sobolev space H n (E ( )) on is the Hilbert space direct sum Then (2.13) One self-adjoint restriction of (2.11) is the Dirichlet operator (2.14) where H 1 0 ( e) is the closure of compactly supported smooth functions in H 1 ( e). The Hamiltonian we will use to model the graphene without magnetic fields is the self-adjoint [36] operator H on with Neuman type boundary conditions 15) and defined by (2.16)

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).

Theorem 5 As a set, the spectrum of H away from the Dirichlet spectrum is given by
(2.26)

Dirichlet-to-Neuman map
Fix an edge e ∈ E( ). Let c λ, e , s λ, e , which for V e = 0 reduce to just c λ, e = cos( √ λ•) and s λ, e = sin( √ λ•)/ √ λ, be solutions to H e ψ e = λψ e with the following boundary condition The Dirichlet-to-Neuman map is defined by with the property that for ψ λ, e as in (2.28), one has (2.30) For the second component, the constancy of the Wronskian is used. Since V (t) is assumed to be even, the intuitive relation remains also true for non-zero potentials.

Relation to Hill operators
Using the potential V (t) (2.12), we define the Z-periodic Hill potential V Hill ∈ L 2 loc (R).
for t ∈ R. The associated self-adjoint Hill operator on the real line is given by Then c λ , s λ ∈ H 2 (0, 1), extending naturally to H 2 loc (R), become solutions to The monodromy matrix associated with H Hill is the matrix valued function and depends by standard ODE theory holomorphically on λ. Its normalized trace is called the Hill (aka Floquet) discriminant. In the simplest case when V Hill = 0, the Floquet discriminant is just (λ) = cos √ λ for λ ≥ 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 (2. 38) where B n := [α n , β n ] denotes the n-th Hill band with β n ≤ α n+1 . We have | int(B n ) (λ) = 0. Putting (2.32) and (2.37) together, we get the following relation that connects the Hill spectrum with the spectrum of the graphene Hamiltonian. Also, if λ ∈ σ (H D ), then by the symmetry of the potential, the Dirichlet eigenfunction are either even or odd with respect to 1 2 . Thus, Dirichlet eigenvalues can only be located at the edges of the Hill bands, see Fig. 4. Namely, (2.40)

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: . 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 The spectra of σ (H ) and σ (H 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.

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 ,θ ∈ L(l 2 (Z)) be a quasi-periodic Jacobi matrix, that is given by It is a well known result that for irrational 2π , the set ,θ is independent of θ , thus ,θ = . It is also well known that, for any , has no isolated points. 7
We define the Lyapunov exponent of H ,θ at energy λ as By a trivial bound A 2 ≥ | det A|, which comes from the fact A is a 2 × 2 matrix, we get

Normalized transfer matrix
We introduce the normalized transfer matrix: (2.46) and the n-step normalized transfer matrixÃ λ n (θ ). The following connection between A λ andÃ λ is clear: (2.50)

Magnetic potential
, the scalar potential A e ∈ C ∞ ( e) along edges e ∈ E( ) is obtained by evaluating the form A on the graph along the vector field generated by edges The integrated vector potentials are defined as β e := e A e (x)d x for e ∈ E( ).

Assumption 1
The magnetic flux through each hexagon of the lattice is assumed to be constant.
Let us mention that the assumption above is equivalent to the following equation, in terms of the integrated vector potentials for any γ 1 , γ 2 ∈ Z.

Example 1 The vector potential
can be chosen as This scalar potential is invariant under b 2 -translations. The integrated vector potentials β e are given by where, in this case, the magnetic flux through each hexagon is = 3 √ 3

Magnetic differential operator and modified Peierls' substitution
In terms of the magnetic differential operator (D B ψ) e := − iψ e − A e ψ e , the Schrödinger operator modeling graphene in a magnetic field reads and is defined on Note that (3.10) implies that both components of p(·) are monotonic functions. Then we define ζ γ 1 ,γ 2 recursively through the following relations along p(·): Due to (3.3), it is easily seen that the definition of ζ γ 1 ,γ 2 is independent of the choice of p(·), hence is well-defined. The unitary Peierls' substitution 9 is the multiplication operator where i( e) → x denotes the straight line connecting i( e) with x ∈ 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 iβ e . We will define a modified Peierls' substitution that allows us to reduce the number of non-trivial multiplicative factors to one, by taking 14) The domain of B is 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 is by the (modified) Peierls' substitution unitary equivalent to the Dirichlet operator without magnetic field Consequently, the spectrum of the Dirichlet operator H D is invariant under perturbations by the magnetic field.

Main lemmas
First, let us introduce the following two-dimensional tight-binding Hamiltonian with translation operators τ 0 , τ 1 ∈ L(l 2 (Z 2 ; C)) which for γ ∈ Z 2 and u ∈ l 2 (Z 2 ; C) are defined as The following lemma connects the spectrum of H B with σ (Q ). We have Remark 4 We will show in Lemma 5.2 that σ p (Q ( )) is empty, thus H B has no point spectrum away from σ (H D ).    (1) σ (Q ( )) is symmetric with respect to 0.

Reduction to the one-dimensional Hamiltonian
Relating the spectrum of Q to that of Q 2 , we obtain the following characterization of σ (Q ).

Lemma 5.2 (1)
The spectrum of the operator Q ( ) as a set is given by (2) Q ( ) has no point spectrum.
Then squaring the operator Q ( ) yields The spectral mapping theorem implies that σ (Q 2 ( )) = σ (Q ( )) 2 and from Lemma 5.2 we conclude that σ (Q ( )) = ± σ (Q 2 ( )). Clearly, the operator A A * | ker(A * ) ⊥ and A * A| ker(A) ⊥ are unitarily equivalent. Thus, the spectrum can be expressed by where we are able to use either of the two (A A * or A * A) since 0 ∈ σ (Q ( )) due to Lemma 5.1. Then, it follows that Observe that Since H is invariant under discrete translations in n, the operator is unitarily equivalent to the direct integral operator ⊕ T 1 H ,θ dθ , which gives the claim. (2). It follows from a standard argument that the two dimensional operator H has no point spectrum. Indeed, assume H has point spectrum at energy E, then H ,θ would have the same point spectrum E for a.e. θ ∈ T 1 . This implies the integrated density of states of H ,θ has a jump discontinuity at E, which is impossible. Therefore the point spectrum of H is empty, hence the same holds for Q ( ).

Proof of Lemmas 4.2, 4.3 and 4.4
For a set U , let dim H (U ) be its Hausdorff dimension.
Lemma 4.2 follows as a direct consequence of (5.3) and the following Theorem 7. Let be defined as in Sect. 2.3.

Theorem 7
For 2π ∈ R\Q, 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 2π = p q be a reduced rational number, then 2π p/q is a union of q (possibly touching) bands with | 2π p/q | < 16π 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 in , 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 λ ∈ and | − | < C 1 , then there exists λ ∈ such that 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 ⊂ R, and suppose that S has a sequence of covers: {S n } ∞ n=1 , S ⊂ S n , such that each S n is a union of q n intervals, q n → ∞ as n → ∞, and for each n, where β and C are positive constants, then

Proof of Lemma 4.3
The fact that σ (Q ( )) 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 ε > 0, we have Hence by Lemma 6.1, we have Optimizing in ε leads to

Proof of Lemma 4.4
We will show that if 2π is an irrational obeying for some constant C, and a sequence of reduced rationals { p n /q n } with q n → ∞, then dim H (σ (Q ( ))) ≤ 1/2. It is easy to see that the 's satisfying (6.2) form a dense G δ set of R, hence is generic. Without loss of generality, we may assume 2π ∈ (0, 1). First, by (5.3), we have that where we used a trivial bound H ,θ ≤ 6. Hence it suffices to show that for each k ≥ 2, 3) The rest of the argument is similar to that of [42]. By Lemma 6.2, taking any λ ∈ , for n ≥ n 0 , there exists λ ∈ 2π p n /q n such that |λ − λ | ≤ C 2 | 2π − p n q n | 1 2 . This means is contained in the C 2 | 2π − p n q n | 1 2 neighbourhood of 2π p n /q n . By Lemma 6.1, 2π p n /q n has q n (possibly touching) bands with total measure | 2π p n /q n | ≤ 16π 3q n . Hence has cover S n such that S n is a union of (at most) q n intervals with total measure Since q 4 n 2π − p n q n ≤ C, we have, by (6.4), This implies 9 + 1 3 ∩ [ 1 k , 1] has coverS n such thatS n is a union of (at most) q n intervals with total measure Then Lemma 6.3 yields (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 /2π with unbounded coefficients in the continued fraction expansion, thus for a.e. , 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 ,θ 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 ,θ is empty, see Proposition 6.5; (3). apply Kotani theory for Jacobi matrices, see Theorem 8. Let ac (H ,θ ) be the absolutely continuous spectrum of H ,θ . Let L(λ, ) be the Lyapunov exponent of H ,θ at energy λ, as defined in (2.44). For a set U ⊆ R, let U ess be its essential closure.
First, we are able to give a characterization of the Lyapunov exponent on the spectrum. 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 θ to θ + iε, and obtain asymptotic behavior of the Lyapunov exponent when |ε| → ∞. Convexity and quantization of the acceleration (see Theorem 5 of [2]) then bring us back to the ε = 0 case. We will give the proof in "Appendix A". Exploiting the fact that c(θ ) = 1 + e −2πiθ has a real zero θ 1 = 1 2 , we have Proposition 6.5 ([18], see also Proposition 7.1 of [34]) For 2π ∈ R\Q, and a.e. θ ∈ T 1 , ac (H ,θ ) is empty.
Hence our operator H ,θ 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(·)| uniformly bounded away from zero) Jacobi matrices in Theorem 5.17 of [52]. In our case |c(·)| 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(·)|) ∈ L 1 . Namely, let H c,v (θ ) acting on 2 (Z) be an ergodic Jacobi matrix, Proof of Theorem 7 In our concrete model, log (|c(θ )|) = log (2| cos πθ|) ∈ L 1 (T 1 ), thus Theorem 8 applies, and combining with Propositions 6.4, 6.5, it follows that must be a zero measure set.
The rest of this section will be devoted to proving Lemma 6.1.

Case 1.
If θ ∈ , we have the following Obervation 2 For θ ∈ , the infinite matrix H 2π p/q,θ is decoupled into copies of the following block matrix M q of size q:
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 ( ) in [28]. Here we do not use the Chambers' formula for Q ( ), rather we develop one for one-dimensional Hamiltonian H ,θ .
Proof It is easily seen that d q (·) is a 1/q-periodic function, thus in which the G q (λ) part is independent of θ . One can easily compute the coefficients a q , a −q , and get a q = a −q = − 1.
The following lemma provides estimates of | 2π p/q,θ | and holds for any Jacobi matrix (2.41).

Proof of Lemma 6.1 for even q
For sets/functions that depend on θ , we will sometimes substitute θ in the notation with A ⊆ T 1 , if corresponding sets/functions are constant on A.
Since q is even, a simple computation shows A simple computation also shows l q ( 6Z+1 6q ) = −1 and L q ( 2Z+1 2q ) = 2. Thus we have, by (6.19), This implies  This proves the claimed result.

Proof of Lemma 4.1
Lemma 4.1 is the reduction from B to the tight-binding model Q . We now present its proof below. Using ideas from [11,46], we can express the resolvent of the operator B (3.14) by Krein's resolvent formula in terms of the resolvent of the Dirichlet Hamiltonian and the resolvent of Q .
For this we need to introduce a few concepts first. The l 2 -space on the vertices l 2 (V( )) carries the inner product 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.
The following lemma applies this concept to our setting. is closed. The maps π, π on D(T B ) defined by e iβ e ψ e (v) (7.4) form together with H := l 2 (V( )) a boundary triple associated to T B .
Proof The proof follows the same strategy as in [46]. The operator 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 T B . Such a closed extension is given by e∈E( ) H 2 e on H 2 (E ( )). To see that D(T B ) is a closed subspace of H 2 (E ( )), observe that in terms of continuous functionals ker k e i , e j (7.6) which proves closedness of T B . Green's identity follows directly from integration by parts on the level of edges. The denseness of ker(π, π ) is obvious since this space contains e∈E( ) C ∞ c ( 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].  defines an operator in L(l 2 (V( ))) with K ( ) ≤ 1.
The resolvents of H D = T B A 1 and B = T B A 2 are then related by Krein's resolvent formula [50,Theorem 14.18] and a unitary equivalence between B and K ( ) away from the Dirichlet spectrum holds [47,48] Theorem 9 Let (l 2 (V( )), π, π ) be the boundary triple for T B and γ, M as above, then for λ ∈ ρ(H D ) ∩ ρ( B ) there is also a bounded inverse of M(λ, ) and with unitary operator U : ran K ( )1 (J ) and E K ( ) is the spectral measure of the self-adjoint operator K ( ).
Since all vertices are integer translates of either of the two vertices r 0 , r 1 ∈ W by basis vectors b 1 , b 2 , we conclude that l 2 (V( )) l 2 (Z 2 ; C 2 ). Our next Lemma shows K ( ) and Q ( ) are unitary equivalent under this identification.
Finally, we point out that Lemma 4.1 follows from a combination of Theorem 9 and Lemmas 7.4 and 7.5.

Spectral analysis
This section is devoted to complete spectral analysis of H B .

(3) H B has no point spectrum away from σ (H D ).
(4) For non-trivial flux / ∈ 2π Z, H B has purely continuous spectrum bounded away from σ (H D ).
In this paper, we only show the energy | −1 int(B n ) (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 ∈ 2π Q.
Combining Lemma 8.2 with Lemma 4.4, we get

Proof of Lemma 8.3
Lemmas 4.4 and 8.1, which implies that −1 | B n is Lipschitz on σ (Q ( )) for / ∈ 2π Z, show that for generic , Hence since This proves Lemma 8.3.
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.
In general, B does not commute with lattice translations T st γ . Yet, there is a set of modified translations, introduced by [54], that do still commute with B , although they in general no longer commute with each other. We define those magnetic translations T B γ : L 2 (E ( )) → L 2 (E ( )) as unitary operators given by for any ψ := (ψ e ) e∈E( ) ∈ L 2 (E ( )) and γ ∈ Z 2 . The lattice translation T st as before. The function u B γ is constant on each copy of the fundamental domain, and defined as follows By the definition (8.1), (8.2), it is clear that for any ψ ∈ L 2 (E( )), In order to make sure . (8.4) This, by (3.16) is in turn equivalent to the following: for anyγ 1 ,γ 2 ∈ Z: The definition of u B γ (8.2) clearly satisfies this requirement. Therefore, although magnetic translations do not necessarily commute with one another, they commute with B

Dirichlet spectrum
In this subsection, we will study the energies belonging to the Dirichlet spec- Remark 7 We observe that T (n) can be row-reduced to an upper triangular matrix with diagonal where q is the number of enclosed hexagons. Hence rank(T (n)) = n iff q / ∈ 2π Z and rank(T (n)) = n − 1 otherwise.

Lemma 8.4 The Dirichlet eigenvalues λ ∈ σ (H D ) are contained in the point spectrum of H B .
Proof For ∈ 2π Z the statement is known [39, Theroem 3.6], thus we focus on / ∈ 2π Z. By unitary equivalence, it suffices to construct an eigenfunction to B . We will construct an eigenfunction on two adjacent hexagons 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 e and the edges on the outer loop by e 1 , e 2 , . . . , e 10 (see Fig. 6). Recall that s λ, e is the Dirichlet eigenfunction on e. By Remark 7, for 2 ∈ 2π Z, operator T (10) has a non-trivial nullspace. We could take a = (a j ) ∈ ker (T (10)) \{0}, (8.8) and an eigenfunction ψ on such that ψ e = 0 and ψ e j = a j s λ, e j .
If 2 / ∈ 2π Z, we take a vector y ∈ C 10 such that y 2 = −1, y 7 = −e iβ e and y j = 0 otherwise. Since in this case T (10) is invertible, there exists a unique Upon n-fold application of the magnetic translation, the point 0 gets translated to some point x n whereas the eigenfunction ψ acquires only a complex phase λ n . Thus, (8.10) still holds and we must also have that Yet, there exists n such that B(0, R) ∩ B(x n , R) = ∅. 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 λ that does not vanish at a vertex, by (modified) Peierls' substitution (3.13), there is one to B , denoted as ϕ, as well. We may expand the function in local coordinates on every edge e ∈ E( ) as ϕ e = a e c λ, e + b e s λ, e according to (2. 35). Recall also that the Dirichlet eigenfunction s λ is either even or odd. Thus, using (2.31) we conclude that |c λ (0)| = |c λ (1)| and thus ϕ cannot be compactly supported. In particular, ϕ has the same absolute value at any vertex by boundary conditions (3.15). Due to ϕ has to vanish at every vertex. Thus ϕ is also an eigenfunction to H D .

Dirichlet spectrum for rational flux quanta
In this section, the flux quanta are assumed to be reduced fractions 2π = 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 Proof 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 dim(ker(T )) = 1, such eigenfunctions are also unique (up to normalization).
Proof Unitary equivalence allows us to work with B rather than H B . Without loss of generality, we assume that the Dirichlet eigenfunction to λ is even. Due to Lemma 8.7, eigenfunctions of B to Dirichlet eigenvalues vanish at every vertex. Thus, on every edge e ∈ E(V ), they are of the form ϕ e = a e s λ, e for some a e .
Conversely, any such element in the nullspace of A uniquely defines an eigenfunction ϕ = a e s λ, e to B . Theorem 8 in [36] implies then that the nullspace of A is generated by sequences in c 00 (V( )). It suffices now to observe that those compactly supported sequences also give rise to compactly supported eigenfunctions to conclude the claim. Proof 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 ϕ be a compactly supported eigenfunction of B to some Dirichlet eigenvalue λ. Consider an edge d ∈ E( ) on the boundary loop of the support of ϕ. 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 , as in Fig. 6, which contains the chosen edge d. Then, there is by the proof of Lemma 8.4 a state ψ on so that the wavefunction ψ d on d coincides with ϕ d . Subtracting ψ from ϕ leaves us with an eigenfunction to B that encloses at least one single hexagon less than ψ. Thus, iterating this procedure shows that compactly supported eigenfunctions are spanned by double hexagonal states which implies the claim.

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 (E( )) is defined as

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 λ ∈ σ (H D ).
Definition 8. 12 We denote the closed L 2 (E( )) subspace generated by linear combinations of all double hexagonal states on the entire graph by There is a countable orthonormal system of states We may label elements of V ( ) by ϕ γ ( ) with γ ∈ Z 2 . Without loss of generality, ϕ γ ( ) can be chosen to depend analytically on ∈ (0, 1). Every element ϕ γ ( ) ∈ V ( ) is due to Lemma 8.7 of the form because it is an element of ker(H B − λ). Now assume that the statement of Theorem 10 does not hold, this is equivalent to saying that Z ( ) := ker(H B − λ) ∩ D H E( ) ( ) ⊥ is not the zero space, i.e. there are eigenfunctions not spanned by double hexagonal states. Our goal is to characterize Z ( ) as the nullspace of a suitable operator we define next.

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 e∈E( ) u e s λ, e to D H E( ) ( ).
Combining Lemma 8.14 with the already established injectivity result, we have A( ) is continuously invertible for 2π ∈ Q ∩ (0, 1) with the following control of its norm Now let us give the proof of Theorem 10, assuming the result of Lemma 8.14.

Proof of Lemma 8.14
We prove this Lemma by showing that there is a sufficiently sparse set of elements in l 2 (E( )) that gets mapped under A( ) on the standard basis of l 2 (E( )). To obtain also the remaining basis vectors, let us define L 2 functionsψ (γ , f ) andψ (γ , g) supported on a single hexagon γ as shown in Fig. 8. The indices of ψ (γ , [ e]) are chosen to indicate the standard basis vectors δ •,(γ ,[ e]) ∈ l 2 (E( )) in the range of A( ) that we will construct from those functions. To defineψ (γ , f ) andψ (γ , g) , we introduce coefficients ζ •,(γ , f ) and ζ •,(γ , g) such thatψ (γ , f ) := e∈E( γ ) ζ e,(γ , f ) s λ, e andψ (γ , g) := e∈E( γ ) ζ e,(γ , g) s λ, e , respectively. We do this in such a way that all continuity conditions forψ (γ , f ) at the vertices of γ are satisfied up to a single one at the (initial) vertex v 1 := i((γ , g)) = i ((γ , h)), see Fig. 8. We define for fixed e = (γ , f ) (8.25) and all other ζ •, e are taken to be zero. Since forψ (γ , f ) all but one continuity conditions are satisfied, we obtain for the first two components of (8.18) 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  and all other coefficients ζ •, e equal to zero. Thus, we get for the first two components of (8.18) 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 To conclude surjectivity of A( ) from this, it suffices to show that for all (a( e)) ∈ l 2 (E( )) we can bound u( e) := d∈E( ) a( d) α e, d as follows Due to σ l 2 (E( )) ≤ ν l 2 (E( )) + σ − ν l 2 (E( )) we may establish estimate (8.33) for each term on the right-hand side of the triangle inequality, individually.
For two edges d, e ∈ E( ) we define a function M( d, e) := 1 if there are γ , γ ∈ Z 2 and two hexagons γ , γ satisfying γ ∩ γ = ∅ such that d ∈ γ and e ∈ γ , and M( d, e) := 0 otherwise. Choosing τ 1 such that  Proof That the bands of Q ( ) do not overlap is shown in Section 6 of [28]. Thus, the unique correspondence between bands of Q ( ) and H B , following from the unitary equivalence (7.14), shows that the non-overlapping of bands holds true for H B as well.

Singular continuous Cantor spectrum for irrational flux quanta
Proof By Lemma 4.2, the spectrum of Q ( ) for irrational 2π is a Cantor set of measure zero. Thus, the pullback of σ (Q ) by | int(B n ) is still a Cantor set of zero measure that coincides with σ (H B )\σ (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.

Proofs of Theorems 1-4
This section serves as an index to the proofs of our main theorems that are distributed in different sections throughout the paper.

Proof of Theorem 2
This is proved in Sects. 8.3 and 8.4.

Proof of Theorem 3
This is proved in Lemmas 4.2, 4.3 and 4.4.

Proof of Theorem 4
This is proved in Lemma 8.10 and Theorem 10.