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 $\Phi/2\pi$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.


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. [N11]. Electronic properties of graphene have been extensively studied rigorously in the absence of magnetic fields [FW12,KP07].
Magnetic properties of graphene have also become a major research direction in physics that has been kindled recently by the observation of the quantum Hall effect [Zh05] and strain-induced pseudo-magnetic fields [Gu10] in graphene. The purpose of this paper is to provide for the first time an analysis of the spectrum of graphene in magnetic fields with constant flux.
The fact that magnetic electron spectra have fractal structures was first predicted by Azbel [A64] and then numerically observed by Hofstadter [Ho76] 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 [Ch14], [De13], [G17], and [Gor13].
With this work, we provide a rigorous foundation for self-similarity by showing that for irrational flux quanta, the electron spectrum 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). Let σ Φ , σ Φ cont , σ Φ ess be the (continuous, essential) spectra of H B , the Hamiltonian of the quantum graph graphene in a magnetic field with constant flux Φ, as defined in (3.7) (3.8) (3.1) (3.2), with some Kato-Rellich potential V e ∈ L 2 ( e). Let H D be the Dirichlet operator (no magnetic field) defined in (2.15) (2.12), and σ(H D ) its spectrum. Let σ Φ 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 • a Cantor set of measure zero for Φ 2π ∈ R \ Q, • a countable union of disjoint intervals for Φ 2π ∈ Q, 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 the 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 Φ 2π ∈ R \ Q the spectrum on σ Φ cont is purely singular continuous.
Since molecular bonds in graphene are equivalent and delocalized, we use an effective one-particle electron model [KP07] on a hexagonal graph with magnetic field [KS03]. While closely related to the commonly used tight-binding model [AEG14], we note that unlike the latter, our model starts from actual differential operator and is exact in every step, so does not involve any approximation. Moreover, while the density of states of the tight-binding model is symmetric around the Dirac point [C09,Fig. 5] and [HKL16, Fig. 3], it is not what is found in experiments [Go12], while the density of states for the quantum graph model [BZ18, Fig. 7 (A)] has a significant similarity with the one observed experimentally [BHJZ]. We note though that isolated eigenvalues are likely 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. Thus while the isolated eigenvalues are probably unphysical, there are reasons to expect that continuous spectrum of the quantum graph operator described in this paper does adequately capture the experimental properties of graphene in the magnetic field. 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. Many atoms and even particles confined to the same lattice structure show similar physical properties [Go12] that are described well by this model (compare [BZ18, Fig. 7 (A)] with [Go12, Fig. 2c]).
Earlier work showing Cantor spectrum on quantum graphs with magnetic fields, e.g. for the square lattice [BGP07] and magnetic chains studied in [EV17], has been mostly limited to applications of the Cantor spectrum of the almost Mathieu operator [AJ09,P04]. In the case of graphene, 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. 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 [H1]. However, the method of [H1] that goes back to that of [AJ10] 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 [AJ09] breaks down in presence of singularity in the Jacobi matrix as well. Instead, we present a new way that exploits singularity rather than circumvent it by showing that the singularity leads to vanishing of the measure of the spectrum, thus Cantor structure and singular continuity, once 4 of Theorem 1 is established. 1 Our method applies to also prove zero measure Cantor spectrum of the extended Harper's model whenever the corresponding Jacobi matrix is singular.
As mentioned, our first step is a reduction to a matrix-valued tight-binding hexagonal model. This leads to an operator Q Λ defined in (4.1). This operator has been studied before for the case of rational magnetic flux (see [HKL16] and references therein). Our analysis gives complete spectral description for this operator as well.
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 [KS03]. The effective one-particle graph model for graphene without magnetic fields was introduced in [KP07]. After incorporating a magnetic field according to [KS03] in the model of [KP07], 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 1 We note that singular continuity of the spectrum of critical extended Harper's model (including for parameters leading to singularity) has been proved recently in [AJM17,H2] without establishing the Cantor nature.
on Hilbert spaces. This technique has been introduced in [Pa06] 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 as well. 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 in the Brillouin zone. The points where the cones match are called Dirac points to account for the special dispersion relation. Using magnetic translations introduced by [Z64] we establish a one-to-one correspondence between bands of the magnetic Schrödinger operators on the graph and of the tight-binding operators for rational flux quanta that only relies on Krein's theory. 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. This way, the conical Dirac points are preserved in rational magnetic fields. We obtain the preceding results by first proving a bound on the operator norm of the tight-binding operator and analytic perturbation theory.
In [KP07] it was shown that the Dirichlet contribution to the spectrum in the nonmagnetic 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 While for rational Φ the proof is based on ideas similar to those of [KP07], for irrational Φ we no longer have an underlying periodicity thus cannot use the arguments of [K05]. 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 [A,AJM17,AW13,JL01]. Here we achieve this by constructing, for each Φ, 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. This paper is organized as follows. Section 2 serves as background. In Section 3, we introduce the magnetic Schrödinger operator H B and (modified) Peierls' substitution, which enables us to reduce H B to a non-magnetic Schrödinger operator Λ B with magnetic contributions moved into the boundary conditions. In Section 4, we present several key ingredients of the proofs of the main theorems: Lemmas 4.1 and 4.2 -4.4. Lemma 4.1 involves a further reduction from Λ B to a two-dimensional tightbinding Hamiltonian Q Λ (Φ), and Lemmas 4.2 -4.4 reveal the topological structure of σ(Q Λ (Φ)) (thus proving the topological part of Theorem 3). The proofs of Lemmas 4.1, 4.2, 4.3 and 4.4 will be given is Sections 7, 5 and 6 respectively. Section 8 is devoted to a complete spectral analysis of H B , thus proving Theorem 1, with the analysis of Dirichlet spectrum in Section 8.2, where in particular we prove Theorem 4; absolutely continuous spectrum for rational flux in Section 8.3, singular continuous spectrum for irrational flux in Section 8.4 (thus proving Theorem 2).

Preliminaries
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 Ω 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.
2.1. Hexagonal quantum graphs. This subsection is devoted to reviewing hexagonal quantum graphs without magnetic fields. The readers could refer to [KP07] for details. We include some material here that serves as a preparation for the study of quantum graphs with magnetic fields in Section 3.
The effective one electron behavior in graphene can be described by a hexagonal graph with Schrödinger operators defined on each edge [KP07]. The hexagonal graph Λ is obtained by translating its fundamental cell W Λ shown in Figure 1 along the basis vectors of the lattice. The basis vectors are and so the hexagonal graph Λ ⊂ R 2 is given by the range of a Z 2 -action on the fundamental domain W Λ The fundamental domain of the dual lattice can be identified with the dual 2-torus where the dual tori are defined as (2.6) We will sometimes 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(Λ), we will sometimes denote it by γ 1 , γ 2 , [e]. Finally, for any x ∈ Λ, we will also denote its unique preimage in W Λ by [x] 2 .
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 Λ to obtain an oriented graph Λ (2.8) For arbitrary e ∈ E(Λ), we then just extend those maps by for some e ∈ E(Λ)} be the collection of initial vertices, and t(Λ) = {v ∈ V(Λ) : v = t( e) for some e ∈ E(Λ)} be the collection of terminal ones. It should be noted that based on our orientation, V(Λ) is a disjoint union of i(Λ) and t(Λ).
Every edge e ∈ E(Λ) is of length one and thus has a canonical chart κ e : e → (0, 1), (i( e)x + t( e)(1 − x)) → x (2.10) 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 (Λ)) on Λ is the Hilbert space direct sum (2.11) On every edge e ∈ E(Λ) we define the maximal Schrödinger operator H e ψ e := −ψ e + V e ψ e (2.12) with Kato-Rellich potential V e ∈ L 2 ( e) that is the same on every edge and even with respect to the center of the edge. Let (2.14) One self-adjoint restriction of (2.12) is the Dirichlet operator 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 selfadjoint [K05] operator H on Λ with Neumann type boundary conditions (2.17) 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 (2.16).
(2.24) Any eigenfunction to operators H k , with eigenvalues away from σ(H D ), can therefore be written in terms of those functions for constants a, b ∈ C ψ := with the continuity conditions of (2.23) being already incorporated in the representation of ψ. Imposing the conditions stated on the derivatives in (2.23) shows that ψ is non-trivial (a, b not both equal to zero) and therefore an eigenfunction with eigenvalue λ ∈ R to H k iff η(λ) 2 = 1 + e ik 1 + e ik 2 2 9 (2.26) with η(λ) := ψ λ,2, e (t( e)) ψ λ,2, e (i( e)) well-defined away from the Dirichlet spectrum. By noticing that the range of the function on the right-hand side of (2.26) is [0, 1], the following spectral characterization is obtained [KP07, Theorem 3.6].
(2.33) 2.1.3. Relation to Hill operators. Using the potential V (t) (2.13), we define the Zperiodic Hill potential V Hill ∈ L 2 loc (R). V Hill (t) := V (t (mod 1)), (2.34) 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 H Hill ψ = λψ. (2.36) 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 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 [RS78], the spectrum of the Hill operator is purely absolutely continuous and satisfies where B n := [α n , β n ] denotes the n-th Hill band with β n ≤ α n+1 . We have ∆| int(Bn) (λ) = 0.
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. Namely, (2.41) 2.1.4. Spectral decomposition. The singular continuous spectrum of H is empty by the direct integral decomposition (2.21) [GN98]. Due to Thomas [T73] there is the characterization, stated also in [K16, Corollary 6.11], of the pure point spectrum of fibered operators: λ is in the pure point spectrum iff the set {k ∈ T * 2 ; λ j (k) = λ} has positive measure where λ j (k) is the j-th eigenvalue of H k . Away from the Dirichlet spectrum, the condition R λ = λ j (k) is by (2.26) equivalent to ∆(λ) 2 = |1+e ik 1 +e ik 2 | 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 [KP07, Theorem 3.6(v)]. Hence, the spectral decomposition in the case without magnetic field is given by Theorem 6. 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.2 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.
The proof of the main Theorems will involve the study of a one-dimensional quasiperiodic Jacobi matrix. We include several general facts that will be useful.

Transfer matrix and Lyapunov exponent.
We assume that c(θ) has finitely many zeros (counting multiplicity), and label them as For θ / ∈ Θ, the eigenvalue equation H Φ,θ u = λu has the following dynamical reformulation: be the n-step transfer matrix.
We define the Lyapunov exponent of H Φ,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 We introduce the normalized transfer matrix: and the n-step normalized transfer matrixÃ λ n (θ).
2.3. Continued fraction expansion. Let α ∈ R \ Q, then α has the following continued fraction expansion with a 0 being the integer part of α and a n ∈ N + for n ≥ 1.
For α ∈ (0, 1), let the reduced rational number be the continued fraction approximants of α.
The following property of continued fraction expansions is well-known: .
(2.50) 2.4. Lidskii inequalities. Let A be an n × n self-adjoint matrices, let E 1 (A) ≥ E 2 (A) ≥ · · · ≥ E n (A) be its eigenvalues. Then, for the eigenvalues of the sum of two self-adjoint matrices, we have for any 1 ≤ i 1 < · · · < i k ≤ n. These two inequalities are called Lidskii inequality and dual Lidskii inequality, respectively.

Magnetic Hamiltonians on quantum graphs
3.1. Magnetic potential.

Given a vector potential
, the scalar vector potential A e ∈ C ∞ ( e) along edges e ∈ E(Λ) is obtained by evaluating the form A on the graph along the vector field generated by edges (3.1) The integrated vector potentials are defined as β e := e A e (x)dx for e ∈ E(Λ).
Assumption 1. The magnetic flux Φ through each hexagon Q 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.
can be chosen as This scalar vector 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 Φ = 3 √ 3 3.2. 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 (3.8) Let us first introduce a unitary operator U on L 2 (E(Λ)), defined as the factors ζ γ 1 ,γ 2 are defined as follows. First, choose a path p(·) : N → Z 2 connecting (0, 0) to (γ 1 , γ 2 ) with 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(·): (3.11) 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 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 P = P U. (3.13) (3.14) The domain of Λ 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.
Lemma 4.2 below shows σ(Q Λ (Φ)) is a zero-measure Cantor set for irrational flux Φ 2π , 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 Λ (Φ). These three lemmas prove the topological structure part of Theorem 3.

Reduction to the one-dimensional Hamiltonian.
Relating the spectrum of Q Λ to that of Q 2 Λ , we obtain the following characterization of σ(Q Λ ).
Then, it follows that Observe that Since H Φ is invariant under discrete translations in n, the operator is unitarily equivalent to the direct integral operator which gives the claim.
(2). It follows from a standard argument that the two dimensional operator H Φ has no point spectrum. Indeed, assume H Φ has point spectrum at energy E, then H Φ,θ would have the same point spectrum E for a.e. θ ∈ T 1 . This implies the integrated density of states of H Φ,θ has a jump discontinuity at E, which is impossible. Therefore the point spectrum of H Φ is empty, hence the same holds for Q Λ (Φ).
Lemma 4.2 follows as a direct consequence of (5.3) and the following Theorem 7. Let Σ Φ be defined as in Section 2.2.
We will prove Theorem 7 in the next section.
6. Proof of Lemmas 4.3, 4.4, and Theorem 7 For a set U , let dim H (U ) be its Hausdorff dimension.
We will need the following three lemmas.
First, we have Lemma 6.1. Let Φ 2π = p q be a reduced rational number, then Σ 2πp/q is a union of q (possibly touching) bands with |Σ 2πp/q | < 16π 3q .
Lemma 6.1 will be proved in subsections 6.4 and 6.5 after some further preparation. The following lemma addresses the continuity of the spectrum Σ Φ in Φ, extending a result of [AMS78] (see Proposition 7.1 therein) from quasiperiodic Schrödinger operators to Jacobi matrices.
Lemma 6.2. There exists absolute constants We will prove Lemma 6.2 in the appendix.
The next Lemma gives a way to bound the Hausdorff dimension from above. Lemma 6.3. (Lemma 5.1 of [L94]) 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. This is a quick consequence of Lemma 6.1. It is clear that for any ε > 0, we have Hence by Lemma 6.1, we have Optimizing in ε leads to for some constant C, and a sequence of reduced rationals {p n /q n } with q n → ∞, then dim H (σ(Q Λ (Φ))) ≤ 1/2.
6.1. 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 α with unbounded coefficients in the continued fraction expansion, thus for a.e. α, 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 combining quantization of localization-type arguments, singularity-induced absence of absolutely continuous spectrum, and Kotani theory for Jacobi matrices.
Let Σ ac (H Φ,θ ) be the absolutely continuous spectrum of H Φ,θ . Let L(λ, Φ) be the Lyapunov exponent of H Φ,θ at energy λ, as defined in (2.43). 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 [A15] and the extended Harper's model [JM12]. 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 [A15]) then bring us back to the ε = 0 case. We will leave the proof to the appendix.
Hence our operator H Φ,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 nonsingular (that is |c(·)| uniformly bounded away from zero) Jacobi matrices in Theorem 5.17 of [Te00]. In our case |c(·)| is not bounded away from zero, however a careful inspection of the proof of Theorem 5.17 of [Te00] 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. The proof of Theorem 5.17 of [Te00] works verbatim. 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 Σ Φ must be a zero measure set.
The rest of this section will be devoted to proving Lemma 6.1.
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 proof of this lemma will be included in the appendix.
By Lemma 6.14, we get the following alternative characterization of Σ 2πp/q,θ .
Proof. Let us point out that, due to (6.10) and the explanation below it, be eigenvalues of M q,0 (θ), labelled in the increasing order. Let {λ i (θ)} q i=1 be eigenvalues of M q, 1 2 (θ), labelled also in the decreasing order. Then by (6.18) and (6.20),
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 the proof for odd q is very similar to that for even q, we only sketch the steps here.
For odd q, similar to (6.25), we have (6.28) By Lemma 6.8, we have (6.29) Hence putting (6.28), (6.29) together, we have Using ideas from [Pa06] and [BGP07], we can express the resolvent of the operator Λ B (3.14) by Krein's resolvent formula in terms of the resolvent of the Dirichlet Hamiltonian and the resolvent of Q Λ .
For this we need to introduce a few concepts first. The l 2 -space on the vertices l 2 (V(Λ)) 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. • ker(π, π ) is dense in H .
The following lemma applies this concept to our setting. Proof. The proof follows the same strategy as in [Pa06]. 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(Λ) H e on H 2 (E (Λ)). To see that D(T B ) is a closed subspace of H 2 (E (Λ)), observe that in terms of continuous functionals l e i , e j : H 2 (E (Λ)) → C, l e i , e j (ψ) = ψ e i (i( e i )) − ψ e j (i( e j )) k e i , e j : H 2 (E (Λ)) → C, k e i , e j (ψ) = e iβ e i ψ e i (t( e i )) − e iβ e j ψ e j (t( e j )) (7.5) we obtain 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(Λ) 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 [Pa06].
The expression for the Weyl function on the other hand, follows from the Dirichletto-Neumann map (2.30).
here we used (2.32). The formula (7.10) then follows from (7.13) and (2.38). Since Since all vertices are integer translates of either of the two vertices r 0 , r 1 ∈ W Λ by basis vectors b 1 , b 2 , we conclude that l 2 (V(Λ)) l 2 (Z 2 ; C 2 ). Our next Lemma shows K Λ (Φ) and Q Λ (Φ) are unitary equivalent under this identification.
Finally, we point out that Lemma 4.1 follows from a combination of Theorem 9, Lemma 7.4 and Lemma 7.5.

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 Λ (Φ) for non-trivial flux quanta Φ / ∈ 2πZ is strictly less than 1.
Indeed, then, away from the Dirichlet spectrum σ(H D ), which are located on the edges of the Hill bands (2.41), we have the following characterization of σ(H B ). Let B n and ∆ be defined as in Section 2.1.3. (1) The level of the Dirac points ∆| −1 int(Bn) (0) always belongs to the spectrum of H B , i.e. 0 ∈ ∆| int(Bn) (σ(H B )).
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.

Magnetic translations.
In general, Λ B does not commute with lattice translations T st γ . Yet, there is a set of modified translations that do still commute with Λ B , although they in general no longer commute with each other. We define those magnetic translations T B γ : L 2 (E (Λ)) → L 2 (E (Λ)) as unitary operators given by for any ψ := (ψ e ) e∈E(Λ) ∈ L 2 (E (Λ)) and γ ∈ Z 2 . The lattice translation T st γ is defined by (T st γ ψ) e (x) = ψ e−γ 1 b 1 −γ 2 b 2 (x − γ 1 b 1 − γ 2 b 2 ) as before. The function u B γ is constant on each copy of the fundamental domain, and defined as follows where ω(µ, γ) := µ 1 γ 2 − µ 2 γ 1 is the standard symplectic form on R 2 . It also follows hence T B γ is unitary. By the definition (8.1), (8.2), it is clear that for any ψ ∈ L 2 (E(Λ)), 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 (8.4), they commute with Λ B One can check that T B γ (D(T B )) = D(T B ) is also equivalent to (8.7), hence by (8.6), Note that we also have which is due to (8.6) and T B γ (H 1 0 ( e)) = H 1 0 ( e − γ 1 b 1 − γ 2 b 2 ) for any e ∈ E(Λ). We will now study the structure of eigenfunctions and the nature of the continuous spectrum of H B .

Dirichlet spectrum.
In this subsection, we will study the energies belonging to the Dirichlet spectrum Consider a compactly supported simply closed loop, which is a path with vertices of degree 2 enclosing q hexagons, see e.g. Fig. 4. Then this loop passes (proceeding in positive direction from the center of an edge e 1 such that the first vertex we reach is t( e 1 )) n := 2 + 4q edges e 1 , ..., e n in E(Λ). where q is the number of enclosed hexagons. Hence rank(T Φ (n)) = n iff qΦ / ∈ 2πZ and rank(T Φ (n)) = n − 1 otherwise. Proof. For Φ ∈ 2πZ the statement is known [KP07, Theroem 3.6], thus we focus on Φ / ∈ 2πZ. By unitary equivalence, it suffices to construct an eigenfunction to Λ B . We will construct an eigenfunction on two adjacent hexagons Γ as in Fig. 3. 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.  3). Recall that s λ, e is the Dirichlet eigenfunction on e.
If 2Φ / ∈ 2πZ, 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 Φ (10) is invertible, there exists a unique solution a = (a j ) to the following equation: (8.14) Let us take ψ on Γ such that ψ e j = a j s λ, e j and ψ e = s λ, e , then one can easily check ψ is indeed an eigenfunction on Γ.
As a corollary of Lemma 8.1 and Lemma 8.4, we have the following: Corollary 8.5. The spectrum of H B must always have open gaps for Φ / ∈ 2πZ at the edges of the Hill bands.
Remark 7. If the magnetic flux is trivial, i.e. Φ ∈ 2πZ, 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 σ ac (H B ) = [0, ∞).
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. 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, (8.15) still holds and we must also have that Yet, there exists n such that B(0, R) ∩ B(x n , R) = ∅. Therefore, (8.15) and (8.16) 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 Λ B , denoted as ϕ, as well. We may expand the function in local coordinates on every edge e ∈ E(Λ) as ϕ e = a e c λ, e + b e s λ, e according to (2.36). Recall also that the Dirichlet eigenfunction s λ is either even or odd. Thus, using (2.32) 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 .
In this section, the flux quanta are assumed to be reduced fractions Φ 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 [KP07]. 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 qΦ B 0 rather than just Φ B 0 , see Fig. 4.
Lemma 8.8. Any simply closed loop enclosing an area of qΦ B 0 has a unique (up to normalization) eigenfunction of H B supported on it.
Proof. The existence of eigenfunctions on simply closed loops enclosing this flux follows directly from the non-trivial kernel of (8.12), see Remark 6. Due to dim(ker(T Φ )) = 1, such eigenfunctions are also unique (up to normalization).
Proof. Unitary equivalence allows us to work with Λ 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 Λ 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 .
Lemma 8.10. Let Φ / ∈ 2πZ. The eigenspaces are spanned by the set of double hexagonal states Fig. 3.
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 [KP07] case, to show that any compactly supported eigenfunction is a linear combination of double hexagonal states. Let ϕ be a compactly supported eigenfunction of Λ B to some Dirichlet eigenvalue λ. Consider an edge d ∈ E(Λ) 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 Γ, as in Fig. 3, which contains the chosen edge d. Then, there is by the proof of Lemma 8.4 a state ψ on Γ so that the wavefunction ψ d on d coincides with ϕ d . Subtracting ψ from ϕ leaves us with an eigenfunction to Λ 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. 8.2.2. 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 (E(Λ)) 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(Λ)) subspace generated by linear combinations of all double hexagonal states on the entire graph Λ by DH E(Λ) (Φ).
Now assume that the statement of Theorem 10 does not hold, this is equivalent to saying that Z(Φ) := ker(H B − λ) ∩ DH E(Λ) (Φ) ⊥ is not the zero space, i.e. there are eigenfunctions not spanned by double hexagonal states. Our goal is to characterize Z(Φ) as the nullspace of a suitable operator we define next.
Combining Lemma 8.14 with the already established injectivity result, we have A(Φ) is continuously invertible for Φ 2π ∈ Q ∩ (0, 1) with the following control of its norm Now let us give the proof of Theorem 10, assuming the result of Lemmas 8.14.

This implies that
is invertible. Thus, we conclude that also for irrational fluxes ker(A(Φ)) = {0} and by (8.23) therefore Z(Φ) = {0} which shows the claim. Proof of Lemma 8.14. We prove this Lemma by showing that there is a sufficiently sparse set of elements in l 2 (E(Λ)) that gets mapped under A(Φ) on the standard basis of l 2 (E(Λ)).
We do this in such a way that all continuity conditions forψ (γ, f ) at the vertices of Γ γ are satisfied up to a single one at the (initial) vertex i((γ, g)) = i((γ, h)). We define for fixed e = (γ, f ) 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 (8.22) To ensure that we also get constant zero in the third component of (8.22), we project onto the orthogonal complement of the double hexagonal states ψ (γ, . Let now α e,(γ, f ) be such that To conclude surjectivity of A(Φ) from this, it suffices to show that for all (a( e)) ∈ l 2 (E(Λ)) we can bound u( e) := d∈E(Λ) a( d) α e, d as follows Since ψ (γ, f ) , ψ (γ, g) ∈ DH E(Λ) (Φ) ⊥ and (ϕ γ ) forms an orthonormal system in DH E(Λ) (Φ), to prove (8.37) it suffices to show (8.39) Due to σ l 2 (E(Λ)) ≤ ν l 2 (E(Λ)) + σ − ν l 2 (E(Λ)) we may establish estimate (8.37) for each term on the right-hand side of the triangle inequality, individually.
For two edges d, e ∈ E(Λ) we define a function M ( d, e) := 1 if there are γ, γ ∈ Z 2 and two hexagons Γ γ , Γ γ satisfying Γ γ ∩ Γ γ = ∅ such that d ∈ Γ γ and e ∈ Γ γ , and M ( d, e) := 0 otherwise. Choosing τ 1 such that d∈E(Λ);[ d] = h M ( d, e) ≤ τ 1 for any e ∈ E(Λ), then  Proof. Note that σ Φ p \σ(H D ) = ∅ is due to (3) of Lemma 8.2. The absence of singular continuous spectra is clear because H B is invariant under magnetic translations (8.1), some of which commute with one another in the case of rational flux quanta. In particular, T B 0,q and T B q,0 always generate a group of commutative magnetic translations due to (8.4). Thus, standard arguments from Floquet-Bloch theory show that H B has no singular continuous spectrum [GN98]. Now we will show the non-overlapping band structure. We recall that Λ B , T B , H D all commute with magnetic translations T B µ (8.8) (8.9) (8.10) where we assume that µ ∈ qZ. Consequently, T B µ leaves all eigenspaces of those operators invariant. For λ ∈ ρ(H D ) we define discrete magnetic translations τ B µ : l 2 (V) → l 2 (V), Since Λ B , H D both commute with T B µ , Krein's formula (7.8) implies that for λ ∈ ρ(H D ) ∩ ρ(Λ B ), Multiplying with π(λ) and π(λ) * from both sides respectively, it follows that To see that ∆ restricts on every Hill band B n (because ∆| Bn is one-to-one) to an isomorphism from each band of Q Λ (Φ) to a unique band of Λ B , it suffices to note that the preceding calculation shows that Krein's formula (7.8) holds true for the Floquet-Bloch transformed operators. Let U cont be the Gelfand transform generated by translations T B µ and U discrete the Gelfand transform generated by discrete translations τ B µ , i.e. for k in the Brioullin zone Using (8.42) and γ(λ)π(λ) = id V(Λ) we obtain from where we used that T B µ preserves ker(T B − λ) and γ(λ) * | ker(T B −λ) ⊥ = 0, on some fun- where γ(λ)(k) ∈ L(l 2 (W Φ Λ ), L 2 (W Φ Λ )) and M (λ, Φ)(k) ∈ L(l 2 (W Φ Λ )) are the restrictions of γ(λ) and M (λ, Φ) on l 2 (W Φ Λ ) satisfying Floquet boundary conditions. Both Λ B and H D commute with translations T B µ , and thus fiber upon conjugation by U cont , so that for λ ∈ ρ(Λ B (k)) ∩ ρ(H D ) Krein's formula remains true for each k In particular, for λ ∈ ρ(H D ) γ(λ)(k) ker(M (λ, Φ)(k)) = ker(Λ B (k) − λ).
(8.49) Therefore bands of Q Λ (Φ) are in one-to-one correspondence with bands of Λ B , and thus also with bands of H B . That the bands of Q Λ (Φ) do not overlap is shown in Section 6 of [HKL16]. Thus, the unique correspondence among bands of Q Λ (Φ) and H B shows that the non-overlapping of bands holds true for H B as well.
Remark 11. Similar to the Hofstadter butterflies for discrete tight binding operators, the explicit spectrum for rational flux quanta allows us to plot the spectrum of H B for different rational flux quanta in Fig. 8. 8.4. Singular continuous Cantor spectrum for irrational flux quanta.
Proof. By Lemma 4.2, the spectrum of Q Λ (Φ) for irrational Φ 2π is a Cantor set of measure zero. Thus, the pullback of σ(Q Λ ) by ∆| int(Bn) 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. (2) of Theorem 1 then follows from (4) of Lemma 8.2.
Appendix A. Proof of Proposition 6.4 The proof of this result is very similar to that for the almost Mathieu operator and the extended Harper's model. We will present it briefly here for completeness. Readers could refer to Theorem 3.2 (together with its proof in Appendix 2) of [AJM17] for a more detailed discussion.  By Theorem 1 of [JM13], since det(D λ (θ + iε)) = 0 for ε = 0, we have ω(λ, Φ; ε) ∈ Z, for ε = 0. (A.5) This is usually referred to as quantization of acceleration.
Lemma 6.2 follows from Lemma C.1 by taking Φ = 2πα and Φ = 2πα . Lemma C.1 is in turn the argument of [AMS78] adapted to the Jacobi setting.