Cantor Spectrum of Graphene in Magnetic Fields

CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS

SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

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. [No]. Electronic properties of graphene have been extensively studied rigorously in the absence of magnetic fields [FW,KP]. 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 [Zh] and strain-induced pseudo-magnetic fields [Gu] in graphene. The purpose of this paper is to provide for the first time a full 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 [Az] and then numerically observed by Hofstadter [Ho] 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 [Ch], [De], [Ga], and [Gor]. 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 Φ Φ compactness not required). Let σΦ, σcont, σess be the (continuous, essential) spectra of HB, 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 2 D V~e ∈ L (~e). Let H be the Dirichlet operator (no magnetic field) defined in (2.15) D Φ B (2.12), and σ(H ) its spectrum. Let σp be the collection of eigenvalues of H . Then we have the following description of the topological structure and point/continuous decomposition of the spectrum

2 Theorem 1. For any symmetric Kato-Rellich potential V~e ∈ L (~e) we have

Φ Φ (1) σ = σess, Φ D (2) σp = σ(H ), 1 2 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Φ (3) σcont is Φ • a Cantor set of measure zero for 2π ∈ R \ Q, Φ • a countable union of disjoint intervals for 2π ∈ Q, Φ Φ Φ (4) σp ∩ σcont = ∅ for 2π ∈/ Z, Φ (5) the Hausdorﬀ dimension dimH (σ ) ≤ 1/2 for generic Φ.

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 HB.

2 Theorem 2. For any symmetric Kato-Rellich potential V~e ∈ L (~e) we have

Φ Φ (1) For 2π ∈ R \ Q the spectrum on σcont is purely singular continuous. Φ Φ (2) For 2π ∈ Q, the spectrum on σcont is absolutely continuous.

Since molecular bonds in graphene are equivalent and delocalized, we use an effective one-particle electron model [KP] on a hexagonal graph with magnetic field [KS]. While closely related to the commonly used tight-binding model [AEG], 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 tight-binding spectrum is symmetric around 0, it is not what is found in graphene experiments [Zh], while the integrated density of states for the quantum graph model 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 un- physical, 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 [Go] that are described well by this model (compare Figure2 and Figure 2c in [Go]). Earlier work showing Cantor spectrum on quantum graphs with magnetic fields, e.g. for the square lattice [BGP] and magnetic chains studied in [EV], has been mostly limited to applications of the Cantor spectrum of the almost Mathieu operator [AJ, Pu]. 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 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 3

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 [AJ2] 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 [AJ] 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 Theorem1 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 hexag-

onal model. This leads to an operator QΛ deﬁned in (4.1). This operator has been studied before for the case of rational magnetic ﬂux (see [HKL] and references therein). Our analysis gives complete spectral description for this operator as well.

Theorem 3. The spectrum of QΛ(Φ) is Φ p • a finite union of intervals and purely absolutely continuous for 2π = q ∈ Q, with the following measure estimate C |σ(Q (Φ))| ≤ √ , Λ q Φ • singular continuous and a zero measure Cantor set for 2π ∈ R \ Q, • a set of Hausdorff dimension dimH (σ(QΛ(Φ))) ≤ 1/2 for generic Φ. √ 8 6π Remark 1. We will show that the constant C in the first item can be bounded by 9 . The theory of magnetic Schrödingeroperators on graphs can be found in [KS]. The effective one-particle graph model for graphene without magnetic fields was introduced in [KP]. After incorporating a magnetic field according to [KS] in the model of [KP], 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 [P] 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

1We note that singular continuity of the spectrum of critical extended Harper’s model (including for parameters leading to singularity) has been proved recently in [AJM, H2] without establishing the Cantor nature. 4 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA 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 Brioullin zone. The points where the cones match are called Dirac points to account for the special dispersion relation. Using magnetic translations introduced by [Zak] we establish a one-to-one correspondence between bands of the magnetic Schrödingeroperators 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 [KP] 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 Schrödingeroperator on the graph. We extend this result to magnetic Schrödingeroperators on hexagonal graphs. Let Hpp be the pure point subspace accociated with HB. Then

Theorem 4. For any Φ, Hpp is spanned by compactly supported eigenfunctions (in fact, by double hexagonal states).

While for rational Φ the proof is based on ideas similar to those of [KP], for irrational Φ we no longer have an underlying periodicity thus cannot use the arguments of [K2]. 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 [A1, AJM, AW,1]. 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 by constructive arguments plus orthogonality and open mapping type considerations. This paper is organized as follows. Section2 serves as background. In Section3, we introduce the magnetic Schrödingeroperator HB and (modified) Peierls’ substitution, which enables us to reduce HB to a non-magnetic Schrödingeroperator ΛB with magnetic contributions moved into the boundary conditions. In Section4, 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 tight- binding Hamiltonian QΛ(Φ), and Lemmas 4.2- 4.4 reveal the topological structure of CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 5

σ(QΛ(Φ)) (thus proving the topological part of Theorem3). The proofs of Lemmas 4.1, 4.2, 4.3 and 4.4 will be given is Sections7,5 and6 respectively. Section8 is devoted to a complete spectral analysis of HB, thus proving Theorem1, with the analysis of Dirichlet spectrum in Section 8.2, where in particular we prove Theorem4; absolutely continuous spectrum for rational ﬂux in Section 8.3, singular continuous spectrum for irrational ﬂux in Section 8.4 (thus proving Theorem2).

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 Ev(G). The set of points on the graph that are located on edges, and thus not on vertices, is denoted by PE(G). For an operator H, let σ(H) be its spectrum and ρ(H) be the resolvent set. 1 2 2 2 Deﬁne c00, Γ(Λ (R )) and Γ(Λ (R )). 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 [KP] for details. We include some material here that serves as a preparation for the study of quantum graphs with magnetic fields in Section3. The effective one electron behavior in graphene can be described by a hexagonal graph with Schrödingeroperators defined on each edge [KP]. The hexagonal graph

Λ is obtained by translating its fundamental cell WΛ shown in Figure1, consisting of vertices √ ! 1 3 r := (0, 0) and r := , (2.1) 0 1 2 2

and edges ~ f := conv ({r0, r1}) \{r0, r1},

~g := conv ({r0, (−1, 0)}) \{r0, (−1, 0)}, and ( √ !)! ( √ !) 1 3 1 3 ~h := conv r , , − \ r , , − , (2.2) 0 2 2 0 2 2 along the basis vectors of the lattice. The basis vectors are √ ! 3 3 √ ~b := , and ~b := 0, 3 (2.3) 1 2 2 2 6 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

r1  f

g r0  h

b2 b1

Figure 1. The fundamental cell and lattice basis vectors of Λ . and so the hexagonal graph Λ ⊂ R2 is given by the range of a Z2-action on the fundamental domain WΛ n 2 ~ ~ 2 o Λ := x ∈ R : x = γ1b1 + γ2b2 + y for γ ∈ Z and y ∈ WΛ . (2.4)

The fundamental domain of the dual lattice can be identiﬁed with the dual 2-torus where the dual tori are deﬁned as

∗ k k Tk := R /(2πZ) . (2.5)

For any vertex v ∈ V(Λ), we denote by [v] ∈ V(WΛ) the unique vertex, r0 or r1, for which there is γ ∈ Z2 such that ~ ~ v = γ1b1 + γ2b2 + [v]. (2.6)

We will sometimes denote v by (γ1, γ2, [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 γ1, γ2, [e]. Finally, for any x ∈ Λ, we will also denote its unique preimage in WΛ by [x] 2.

We can orient the edges in terms of initial and terminal maps i : E(Λ) → V(Λ) and t : E(Λ) → V(Λ) (2.7) 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 WΛ to obtain an

2so that y in (2.4)=[x] CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 7 oriented graph Λ ~ ~ i(f) = i(~g) = i(h) = r0, ~ ~ ~ ~ t(f) = r1, t(~g) = r1 − b1, and t(h) = r1 − b2. (2.8)

For arbitrary ~e ∈ E(Λ), we then just extend those maps by

~ ~ ~ ~ ~ ~ i(~e) := γ1b1 + γ2b2 + i([e]) and t(~e) := γ1b1 + γ2b2 + t([e]). (2.9)

Let i(Λ) = {v ∈ V(Λ) : v = i(~e) 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)) 7→ x (2.10) that allows us to deﬁne function spaces and operators on ~e and ﬁnally on the entire n graph. For n ∈ N0, the Sobolev space H (PE (Λ)) on Λ is the Hilbert space direct sum M Hn(PE (Λ)) := Hn(~e). (2.11) ~e∈E(Λ)

On every edge ~e ∈ E(Λ) we deﬁne the maximal Schr¨odingeroperator

2 2 2 H~e : H (~e) ⊂ L (~e) → L (~e) 00 H~eψ~e := −ψ~e + V~eψ~e (2.12)

2 with Kato-Rellich potential V~e ∈ L (~e) that is the same on every edge and even with respect to the center of the edge. Let

−1 V (t) = V~e((κ~e) (t)). (2.13)

Then

V (t) = V (1 − t). (2.14)

One self-adjoint restriction of (2.12) is the Dirichlet operator

D M 1 2 2 2 H := H0(~e) ∩ H (~e) ⊂ L (PE (Λ)) → L (PE (Λ)) ~e∈E(Λ) D (H ψ)~e := H~eψ~e, (2.15) 8 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

1 1 where H0(~e) is the closure of compactly supported smooth functions in H (~e). The Hamiltonian we will use to model the graphene without magnetic ﬁelds is the self- adjoint [K2] operator H on Λ with Neumann type boundary conditions 2 D(H) := ψ = (ψ~e) ∈ H (PE(Λ)) : for all v ∈ V(Λ), ψ~e1 (v) = ψ~e2 (v) if ~e1,~e2 ∈ Ev(Λ) X 0 and ψ~e(v) = 0 (2.16)

~e∈Ev(Λ) and deﬁned by

H : D(H) ⊂ L2(PE(Λ)) → L2(PE(Λ))

(Hψ)~e := (H~eψ~e). (2.17) Remark 2. The self-adjointness of H will also follow from the self-adjointness of the more general family of magnetic Schr¨odingeroperators 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).

2.1.1. Floquet-Bloch decomposition. Operator H commutes with the standard lattice translations

st 2 2 Tγ :L (PE(Λ)) → L (PE(Λ)) ~ ~ f 7→ f(· − γ1b1 − γ2b2) (2.18) for any γ ∈ Z2. In terms of those, we deﬁne the Floquet-Bloch transform for x ∈ ∗ PE(WΛ) and k ∈ T2 ﬁrst on function f ∈ Cc(PE(Λ))

X st ihk,γi (Uf)(k, x) := (Tγ f)(x)e (2.19) γ∈Z2 2 2 ∗ and then extend it to a unitary map U ∈ L(L (PE(Λ)),L (T2 ×PE(WΛ)) with inverse Z −1 −ihγ,ki dk (U ϕ)(x) = ϕ(k, [x])e 2 , (2.20) ∗ (2π) T2 2 where [x] ∈ PE (WΛ) is the unique pre-image of x in WΛ, and γ ∈ Z is deﬁned by ~ ~ x = γ1b1 + γ2b2 + [x]. Then standard Floquet-Bloch theory implies that there is a direct integral representation of H Z ⊕ −1 k dk UHU = H 2 (2.21) ∗ (2π) T2 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 9

in terms of self-adjoint operators Hk

k k 2 2 H : D(H ) ⊂ L (PE (WΛ)) → L (PE (WΛ)) k (H ψ)~e := (H~eψ~e) (2.22) on the fundamental domain WΛ with Floquet boundary conditions k 2 X 0 D(H ) := ψ ∈ H (PE (WΛ)) : ψf~(r0) = ψ~g(r0) = ψ~h(r0) and ψ~e(r0) = 0,

~e∈Er0 (Λ)

ik1 ~ ik2 ~ as well as ψf~(r1) = e ψ~g(r1 − b1) = e ψ~h(r1 − b2) and ψ0 (r ) + eik1 ψ0 (r −~b ) + eik2 ψ0 (r −~b ) = 0 . (2.23) f~ 1 ~g 1 1 ~h 1 2

Fix an edge ~e ∈ E(Λ) and λ∈ / σ(HD). There are linearly independent H2(~e)-

solutions ψλ,1,~e and ψλ,2,~e to the equation H~eψ~e = λψ~e with the following boundary condition

ψλ,1,~e(i(~e)) = 1, ψλ,1,~e(t(~e)) = 0, ψλ,2,~e(i(~e)) = 0, and ψλ,2,~e(t(~e)) = 1. (2.24)

Any eigenfunction to operators Hk, with eigenvalues away from σ(HD), can therefore be written in terms of those functions for constants a, b ∈ C  ~ a ψ ~ + b ψ ~ along edge f  λ,1,f λ,2,f −ik1 ψ := a ψλ,1,~g + e b ψλ,2,~g along edge ~g (2.25)  −ik2 ~ a ψλ,1,~h + e b ψλ,2,~h along edge h

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 Hk iﬀ 2 1 + eik1 + eik2 η(λ)2 = (2.26) 9 0 ψλ,2,~e(t(~e)) with η(λ) := 0 well-deﬁned away from the Dirichlet spectrum. ψλ,2,~e(i(~e)) 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 [KP].

Theorem 5. As a set, the spectrum of H away from the Dirichlet spectrum is given by D D σ(H)\σ(H ) = {λ ∈ R : |η(λ)| ≤ 1}\σ(H ). (2.27) 10 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

2.1.2. Dirichlet-to-Neumann map. Fix an edge ~e ∈ E(Λ). Let cλ,~e, sλ,~e be solutions to H~eψ~e = λψ~e with the following boundary condition cλ,~e(i(~e)) sλ,~e(i(~e)) 1 0 0 0 = . (2.28) cλ,~e(i(~e)) sλ,~e(i(~e)) 0 1

−1 −1 We point out that cλ(t) := cλ,~e(κ~e (t)) and sλ(t) := sλ,~e(κ~e (t)) are independent of 00 0 ~e. They are clearly solutions to −ψ + V (t)ψ = λψ on (0, 1), with cλ(0) = 1, cλ(0) = 0 0, sλ(0) = 0, sλ(0) = 1, where V (t) is deﬁned in (2.13). D 2 Then for λ∈ / σ(H ), namely when sλ(1) 6= 0, any H (~e)-solution ψλ,~e can be written as a linear combination of cλ,~e, sλ,~e

ψλ,~e(t(~e)) − ψλ,~e(i(~e))cλ(1) ψλ,~e(x) = sλ,~e(x) + ψλ,~e(i(~e))cλ,~e(x). (2.29) sλ(1) The Dirichlet-to-Neumann map is deﬁned by 1 −cλ(1) 1 m(λ) := 0 (2.30) sλ(1) 1 −sλ(1) with the property that for ψλ,~e as in (2.29) 0 ψλ,~e(i(~e)) ψλ,~e(i(~e)) 0 = m(λ) . (2.31) −ψλ,~e(t(~e)) ψλ,~e(t(~e)) For the second component, the constancy of the Wronskian is used. Since V (t) is assumed to be even, the intuitive relation

0 cλ(1) = sλ(1) (2.32) remains also true for non-zero potentials. D For λ∈ / σ(H ), by expressing cλ(1) in terms of ψλ,1,~e and ψλ,2,~e, it follows immedi- ately that 0 η(λ) = sλ(1). (2.33)

2.1.3. Relation to Hill operators. Using the potential V (t)(2.13), we deﬁne the Z- 2 periodic Hill potential VHill ∈ Lloc(R).

VHill(t) := V (t (mod 1)), (2.34) for t ∈ R. The associated self-adjoint Hill operator on the real line is given by 2 2 2 HHill : H (R) ⊂ L (R) → L (R) 00 HHillψ := −ψ + VHillψ. (2.35)

2 2 Then cλ, sλ ∈ H (0, 1), extending naturally to Hloc(R), become solutions to

HHillψ = λψ. (2.36) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 11

1.4

1.2

0.8

0.6 Density of states

0.4

0.2

0 0 1 2 3 4 5 6 7 8 9 10 Energy

Figure 2. The density of states for the free Schrödingeroperator without magnetic fields on the first Hill band [0, π2].

The monodromy matrix associated with HHill is the matrix valued function cλ(1) sλ(1) Q(λ) := 0 0 (2.37) cλ(1) sλ(1) and depends by standard ODE theory holomorphically on λ. Its normalized trace tr(Q(λ)) ∆(λ) := = s0 (1) (2.38) 2 λ is called the Floquet discriminant. By the well-known spectral decomposition of periodic diﬀerential operators on the line [RS4], the spectrum of the Hill operator is purely absolutely continuous and sat- isﬁes ∞ [ σ(HHill) = {λ ∈ R : |∆(λ)| ≤ 1} = [αn, βn] (2.39) n=1 where B := [α , β ] denotes the n-th Hill band with β ≤ α . We have ∆|0 (λ) 6= n n n n n+1 int(Bn) 0. Putting (2.33) and (2.38) together, we get the following relation ∆(λ) = η(λ), for λ∈ / σ(HD), (2.40) that connects the Hill spectrum with the spectrum of the graphene Hamiltonian. 12 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Also, if λ ∈ σ(HD), then by the symmetry of the potential, the Dirichlet eigenfunc- 1 tion are either even or odd with respect to 2 . Thus, Dirichlet eigenvalues can only be located at the edges of the Hill bands. Namely, ∆(λ) = ±1, for λ ∈ σ(HD). (2.41)

2.1.4. Spectral decomposition. The singular continuous spectrum of H is empty by the direct integral decomposition (2.21)[GeNi]. Due to Thomas [T] there is the characterization, stated also in [K] as Corollary 6.11, of the pure point spectrum of fibered ∗ operators: λ is in the pure point spectrum iff the set {k ∈ T2; λj(k) = λ} has positive k measure where λj(k) is the j-th eigenvalue of H . Away from the Dirichlet spectrum, ik ik 2 2 |1+e 1 +e 2 | the condition R 3 λ = λj(k) is by (2.26) equivalent to ∆(λ) = 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 [KP]. Hence, the spectral decomposition in the case without magnetic field is given by

Theorem 6. The spectra of σ(H) and σ(HHill) coincide as sets. Aside from the Dirich- let contribution to the spectrum, H has absolutely continuous spectrum as in Fig.3 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.

2.2. 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. 2 Let HΦ,θ ∈ L(l (Z)) be a quasi-periodic Jacobi matrix, that is given by Φ Φ Φ (H u) = c θ + m u +c θ + (m − 1) u +v θ + m u . (2.42) Φ,θ m 2π m+1 2π m−1 2π m Let Σ be the spectrum of H , and Σ = S Σ . It is a well known result Φ,θ Φ,θ Φ θ∈T1 Φ,θ Φ that for irrational 2π , the set ΣΦ,θ is independent of θ, thus ΣΦ,θ = σΦ. It is also well known that, for any Φ, ΣΦ has no isolated points.

2.2.1. Transfer matrix and Lyapunov exponent. We assume that c(θ) has ﬁnitely many zeros (counting multiplicity), and label them 3 m Φ Φ as θ1, θ2, ..., θm. Let Θ := ∪j=1 ∪k∈Z θj + k 2π , in particular if 2π ∈ Q, then Θ is a ﬁnite set.

3 −2πiθ In our concrete model, c(θ) = 1 + e , see (5.4), hence has a single zero θ1 = 1/2. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 13

-1.5

-2

-2.5

-3 Dirac point Dirac point Energy value -3.5

-4

-4.5 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3

Figure 3. The ﬁrst two bands of the free Schr¨odingeroperator without magnetic perturbation showing the characteristic conical Dirac points at π2 energy level 4 where the two bands touch.

For θ∈ / Θ, the eigenvalue equation HΦ,θu = λu has the following dynamical refor- mulation: u Φ u n+1 = Aλ θ + n n , un 2π un−1 where ! 1 λ − v(θ) −c(θ − Φ ) GL(2, ) 3 Aλ(θ) = 2π C c(θ) c(θ) 0 is called the transfer matrix. Let Φ Φ Aλ(θ) = Aλ(θ + (n − 1) ) ··· Aλ(θ + )Aλ(θ) n 2π 2π be the n-step transfer matrix.

We deﬁne the Lyapunov exponent of HΦ,θ at energy λ as Z 1 λ L(λ, Φ) := lim log kAn(θ)k dθ. (2.43) n→∞ n T1 14 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

q 2.2.2. Normalized transfer matrix. Let |c(θ)| = c(θ)c(θ). We introduce the normalized transfer matrix:

Φ λ 1 λ − v(θ) −|c(θ − )| SL(2, R) 3 A˜ (θ) = 2π q Φ |c(θ)| 0 |c(θ)||c(θ − 2π )|

˜λ and the n-step normalized transfer matrix An(θ). The following connection between Aλ and A˜λ is clear:  −1 1 0 ! 1 0 ˜λ c(θ) λ r A (θ) = q c(θ) A (θ)  c(θ− Φ )  . (2.45) q Φ 2π |c(θ)||c(θ − )| 0 c(θ) 0 Φ 2π c(θ− 2π )

Φ p When 2π = q is rational, (2.45) yields Qq−1 c(θ + j p ) tr(A˜λ(θ)) = j=0 q tr(Aλ(θ)). (2.46) q Qq−1 p q j=0 |c(θ + j q )| Let ! λ − v(θ) −c(θ − Φ ) Dλ(θ) = c(θ)Aλ(θ) = 2π (2.47) c(θ) 0

λ λ Φ λ Φ λ Φ p and Dn(θ) = D (θ + (n − 1) 2π ) ··· D (θ + 2π )D (θ). Then when 2π = q is rational, (2.46) becomes tr(Dλ(θ)) tr(A˜λ(θ)) = q . (2.48) q Qq−1 p j=0 |c(θ + k q )|

λ λ Note that although An(θ) is not well-deﬁned for θ ∈ Θ, Dn(θ) is always well-deﬁned.

2.3. Continued fraction expansion. Let α ∈ R \ Q, then α has the following continued fraction expansion 1 α = a0 + 1 , a1 + 1 a2+ a3+··· + with a0 being the integer part of α and an ∈ N for n ≥ 1. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 15

For α ∈ (0, 1), let the reduced rational number

pn 1 = 1 (2.49) qn a1 + 1 a2+ ···+ 1 an be the continued fraction approximants of α. The following property of continued fraction expansions is well-known: p 1 |α − n | ≤ . (2.50) qn qnqn+1

2.4. Lidskii inequalities. Let A be an n × n self-adjoint matrices, let E1(A) ≥ E2(A) ≥ · · · ≥ En(A) be its eigenvalues. Then, for the eigenvalues of the sum of two self-adjoint matrices, we have ( Ei1 (A + B) + ··· + Eik (A + B) ≤ Ei1 (A) + ··· + Eik (A) + E1(B) + ··· + Ek(B)

Ei1 (A + B) + ··· + Eik (A + B) ≥ Ei1 (A) + ··· + Eik (A) + En−k+1(B) + ··· + En(B) (2.51) for any 1 ≤ i1 < ··· < ik ≤ n. These two inequalities are called Lidskii inequality and dual Lidskii inequality, respectively.

3. Magnetic Hamiltonians on quantum graphs

3.1. Magnetic potential. 1 2 Given a vector potential A(x) = A1(x1, x2) dx1 + A2(x1, x2) dx2 ∈ Γ(Λ (R )), 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 [e] ∈ E(WΛ) ~ ~ A~e(x) := A(x) [e]1∂1 + [e]2∂2 . (3.1) R 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 Z Φ := dA (3.2) Q 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 β − β + β − β + β − β = Φ, (3.3) γ1,γ2,f~ γ1,γ2+1,~h γ1,γ2+1,~g γ1−1,γ2+1,f~ γ1−1,γ2+1,~h γ1,γ2,~g

for any γ1, γ2 ∈ Z. 16 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Example 1. The vector potential A ∈ Γ(Λ1(R2)) of a homogeneous magnetic ﬁeld B ∈ Γ(Λ2(R2)) B(x) = B0 dx1 ∧ dx2 (3.4) can be chosen as

A(x) := B0x1 dx2. (3.5) ~ This scalar vector potential is invariant under b2-translations. The integrated vector potentials β~e are given by Φ 1 β ~ = γ1 + , βγ ,γ ,~g = 0, and β ~ = −β ~, (3.6) γ1,γ2,f 2 6 1 2 γ1,γ2,h γ1,γ2,f √ 3 3 where, in this case, the magnetic ﬂux through each hexagon is Φ = 2 B0.

3.2. Magnetic differential operator and modified Peierls’ substitution. B 0 In terms of the magnetic differential operator (D ψ)~e := −iψ~e−A~eψ~e, the Schrödinger operator modeling graphene in a magnetic field reads HB : D(HB) ⊂ L2(PE (Λ)) → L2(PE (Λ)) B B B (H ψ)~e := (D D ψ)~e + V~eψ~e, (3.7) and is defined on ( B 2 D(H ) := ψ ∈ H (PE (Λ)) : ψ~e1 (v) = ψ~e2 (v) for any ~e1,~e2 ∈ Ev(Λ) ) X B and D ψ ~e (v) = 0 . ~e∈Ev(Λ) (3.8)

Let us first introduce a unitary operator U on L2(PE(Λ)), defined as ~ ~ Uψγ1,γ2,~e = ζγ1,γ2 ψγ1,γ2,~e for ~e = f,~g, h, (3.9) 2 the factors ζγ1,γ2 are defined as follows. First, choose a path p(·): N → Z connecting (0, 0) to (γ1, γ2) with

p(0) = (0, 0) and p(|γ1| + |γ2|) = (γ1, γ2). (3.10) Note that (3.10) implies that both components of p(·) are monotonic functions. Then we deﬁne ζγ1,γ2 recursively through the following relations along p(·):

ζ0,0 = 1,

iβ ~ −iβγ +1,γ ,~g γ1,γ2,f 1 2 ζγ1+1,γ2 = e ζγ1,γ2 ,

iβ ~ −iβ ~ −iΦγ1 γ1,γ2,f γ1,γ2+1,h ζγ1,γ2+1 = e ζγ1,γ2 . (3.11) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 17

Due to (3.3), it is easily seen that the deﬁnition of ζγ1,γ2 is independent of the choice of p(·), hence is well-deﬁned. The unitary Peierls’ substitution is the multiplication operator P :L2(PE (Λ)) → L2(PE (Λ)) R i A~e(s)ds (ψ~e) 7→ ~e 3 x 7→ e i(~e)→x ψ~e , (3.12)

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 eiβ~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)

It transforms HB into 2 B d ˜−1 B ˜ Λ := − 2 + V~e = P H P. (3.14) dt~e ~e∈E(Λ) The domain of ΛB is ( B 2 D(Λ ) = ψ ∈ H (PE(Λ)) : any ~e1,~e2 ∈ E(Λ) with i(~e1) = i(~e2) = v satisfy

X 0 ψ~e1 (v) = ψ~e2 (v) and ψ~e(v) = 0; whilst at edges for which i(~e)=v ) ˜ ˜ X ˜ iβ~e1 iβ~e2 iβ~e 0 t(~e1) = t(~e2) = v, e ψ~e1 (v) = e ψ~e2 (v) and e ψ~e(v) = 0. , t(~e)=v (3.15) where β˜ ≡ β˜ ≡ 0 and β˜ = −Φγ . (3.16) γ1,γ2,~g γ1,γ2,f~ γ1,γ2,~h 1 Thus, the problem reduces to the study of non-magnetic Schr¨odingeroperators with magnetic contributions moved into the boundary conditions. Observe that the magnetic Dirichlet operator

D,B M 1 2 2 2 H : H0(~e) ∩ H (~e) ⊂ L (PE (Λ)) → L (PE (Λ)) ~e∈E(Λ) D,B B B (H ψ)~e := (D D ψ)~e + V~eψ~e (3.17) 18 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

is by the (modiﬁed) Peierls’ substitution unitary equivalent to the Dirichlet operator without magnetic ﬁeld

HD = P˜−1HD,BP˜ = P −1HD,BP. (3.18)

Consequently, the spectrum of the Dirichlet operator HD is invariant under perturba- tions by the magnetic ﬁeld.

4. Main lemmas

First, let us introduce the following two-dimensional tight-binding Hamiltonian 1 0 1 + τ0 + τ1 QΛ(Φ) := ∗ (4.1) 3 (1 + τ0 + τ1) 0

2 2 2 2 2 with translation operators τ0, τ1 ∈ L(l (Z ; C)) which for γ ∈ Z and u ∈ l (Z ; C) are deﬁned as

−iΦγ1 (τ0(u))γ1,γ2 := uγ1−1,γ2 and (τ1(u))γ1,γ2 := e uγ1,γ2−1. (4.2)

B The following lemma connects the spectrum of H with σ(QΛ). With ∆(λ) deﬁned in (2.38), we have

D B Lemma 4.1. A number λ ∈ ρ(H ) lies in σ(H ) iﬀ ∆(λ) ∈ σ(QΛ(Φ)). Such λ is in B the point spectrum of H iﬀ ∆(λ) ∈ σp(QΛ(Φ)).

B Remark 4. We will show in Lemma 5.2 that σp(QΛ(Φ)) is empty, thus H has no point spectrum away from σ(HD).

Lemma 4.2 below shows σ(QΛ(Φ)) is a zero-measure Cantor set for irrational flux Φ 2π , Lemma 4.3 gives measure estimate for rational flux, and Lemma 4.4 shows upper bound on the Hausdorff dimension of the spectrum of QΛ(Φ). These three lemmas prove the topological structure part of Theorem3.

Φ Lemma 4.2. For 2π ∈ R \ Q, σ(QΛ(Φ)) is a zero-measure Cantor set.

Φ p Lemma 4.3. If 2π = q is a reduced rational number, then we have √ 8 6π |σ(Q (Φ))| ≤ √ . Λ 9 q

1 Lemma 4.4. For generic Φ, the Hausdorﬀ dimension of σ(QΛ(Φ)) is ≤ 2 . CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 19

5. Proof of Lemma 4.2

5.1. Symmetric property of QΛ.

Lemma 5.1. The spectrum of QΛ has the following properties:

(1) σ(QΛ(Φ)) is symmetric with respect to 0. (2)0 ∈ σ(QΛ(Φ)).

Proof. (1). Conjugating QΛ in (4.1) by − id 0 Ω = (5.1) 0 id shows that σ(QΛ(Φ)) is symmetric with respect to 0 [KL].

(2). If we view QΛ(Φ) as an operator-valued function of the ﬂux Φ, then

Φ 7→ hQΛ(Φ)x, yi, (5.2) for x, y ∈ c00 arbitrary, is analytic and QΛ therefore is a bounded analytic map. If there was Φ0/2π ∈ R\Q where QΛ(Φ0) was invertible, then QΛ(Φ) would also be invertible in a suﬃciently small neighborhood of Φ0 (e.g. [Ka]). Yet, in [HKL] it has been shown that for rational Φ/2π, the Dirac points are in the spectrum, i.e. 0 ∈ σ(QΛ(Φ)). Thus, by density 0 ∈ σ(QΛ(Φ)), independent of Φ ∈ R.

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

Lemma 5.2. (1) The spectrum of the operator QΛ(Φ) as a set is given by rS θ∈ σ(HΦ,θ) 1 [ σ(Q (Φ)) = ± T1 + {0} . (5.3) Λ 9 3 2 where HΦ,θ ∈ L(l (Z)) is the one-dimensional quasi-periodic Jacobi matrix de- ﬁned as in (2.42) with c(θ) = 1 + e−2πiθ, and v(θ) = 2 cos 2πθ. (5.4)

(2) QΛ(Φ) has no point spectrum.

1 Proof. (1). Let A := 3 (1 + τ0 + τ1). Then squaring the operator QΛ(Φ) yields AA∗ 0 Q2 (Φ) = . (5.5) Λ 0 A∗A 20 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

2 2 The spectral mapping theorem implies that σ(QΛ(Φ)) = σ(QΛ(Φ)) and from Lemma p 2 ∗ 5.2 we conclude that σ(QΛ(Φ)) = ± σ(QΛ(Φ)). Clearly, the operator AA |ker(A∗)⊥ ∗ and A A|ker(A)⊥ are unitarily equivalent. Thus, the spectrum can be expressed by

p ∗ σ(QΛ(Φ)) = ± σ(AA ) ∪ {0} (5.6) ∗ ∗ where we are able to use either of the two (AA or A A) since 0 ∈ σ(QΛ(Φ)) due to Lemma 5.1. Then,

∗ id id ∗ ∗ ∗ ∗ AA = + ((τ0 + τ0 ) + (τ1 + τ1 ) + τ0τ1 + τ1τ0 ) (5.7) 3 9 | {z } =:HΦ Observe that −iΦm iΦm HΦψm,n =ψm−1,n + ψm+1,n + e ψm,n−1 + e ψm,n+1 iΦ(m−1) −iΦm + e ψm−1,n+1 + e ψm+1,n−1. (5.8)

Since HΦ is invariant under discrete translations in n, it is unitarily equivalent to the direct integral operator Z ⊕ HΦ,θ dθ, (5.9) T1 which gives the claim.

(2). It follows from a standard argument that the two dimensional operator HΦ has no point spectrum. Indeed, assume HΦ has point spectrum at energy E, then HΦ,θ would have the same point spectrum E for a.e. θ ∈ T1. This implies the integrated density of states of HΦ,θ has a jump discontinuity at E, which is impossible. Therefore the point spectrum of HΦ is empty, hence the same holds for QΛ(Φ). Lemma 4.2 follows as a direct consequence of (5.3) and the following Theorem7.

Let ΣΦ be deﬁned as in Section 2.2. Φ Theorem 7. For 2π ∈ R\Q, ΣΦ is a zero-measure Cantor set. We will prove Theorem7 in the next section.

6. Proof of Lemmas 4.3, 4.4, and Theorem7

For a set U, let dimH (U) be its Hausdorﬀ dimension. We will need the following three lemmas. First, we have

Φ p Lemma 6.1. Let 2π = q be a reduced rational number, then Σ2πp/q is a union of q 16π (possibly touching) bands with |Σ2πp/q| < 3q . CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 21

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 ΣΦ in Φ, extending a result of [AMS] (see Proposition 7.1 therein) from quasiperiodic Schr¨odingeroperators to Jacobi matrices.

Lemma 6.2. There exists absolute constants C1,C2 > 0 such that if λ ∈ ΣΦ and 0 0 |Φ − Φ | < C1, then there exists λ ∈ ΣΦ0 such that

0 0 1 |λ − λ | < C2|Φ − Φ | 2 .

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 [L]) Let S ⊂ R, and suppose that S has a sequence of ∞ covers: {Sn}n=1, S ⊂ Sn, such that each Sn is a union of qn intervals, qn → ∞ as n → ∞, and for each n, C |Sn| < β , qn where β and C are positive constants, then 1 dim (S) ≤ . H 1 + β

Proof of Lemma 4.3. This is a quick consequence of Lemma 6.1. It is clear that for any ε > 0, we have

p √ [ q \ Σ2πp/q + 3 ⊆ [ 0, ε ] Σ2πp/q + 3 (ε, ∞).

Hence by Lemma 6.1, we have

√ |Σ | √ 8π |pΣ + 3| ≤ ε + 2√πp/q ≤ ε + √ . 2πp/q 2 ε 3 εq

Optimizing in ε leads to √ 4 6π |pΣ + 3| ≤ √ . 2πp/q 3 q

Then (5.3) implies √ 8 6π |σ(Q (2πp/q))| ≤ √ . (6.1) Λ 9 q 22 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Φ Proof of Lemma 4.4. We will show that if 2π is an irrational obeying

4 Φ pn qn − < C, (6.2) 2π qn

for some constant C, and a sequence of reduced rationals {pn/qn} with qn → ∞, then dimH (σ(QΛ(Φ))) ≤ 1/2. Φ Without loss of generality, we may assume 2π ∈ (0, 1). First, by (5.3), we have that s ! ΣΦ 1 1 dimH (σ(QΛ(Φ))) = sup dimH ± + ∩ [ , 1] , k≥2 9 3 k

where we used a trivial bound kHΦ,θk ≤ 6. Hence it suﬃces to show that for each k ≥ 2, s ! Σ 1 1 1 dim Φ + ∩ [ , 1] ≤ . (6.3) H 9 3 k 2

The rest of the argument is similar to that of [L]. By Lemma 6.2, taking any λ ∈ ΣΦ, 1 0 0 Φ pn for n ≥ n , there exists λ ∈ Σ such that |λ − λ | < C˜ | − | 2 . This means Σ 0 2πpn/qn 2 2π qn Φ Φ p 1 is contained in the C˜ | − n | 2 neighbourhood of Σ . By Lemma 6.1,Σ 2 2π qn 2πpn/qn 2πpn/qn has q (possibly touching) bands with total measure |Σ | ≤ 16π . Hence Σ has n 2πpn/qn 3qn Φ cover Sn such that Sn is a union of (at most) qn intervals with total measure 1 2 16π ˜ Φ pn |Sn| ≤ + 2C2qn − . (6.4) 3qn 2π qn

Since q4 Φ − pn ≤ C, we have, by (6.4), n 2π qn √ ˜ 16π 2C2 C C |Sn| ≤ + =: . (6.5) 3qn qn qn

ΣΦ 1 1 ˜ ˜ This implies 9 + 3 ∩ [ k , 1] has cover Sn such that Sn is a union of (at most) qn intervals with total measure √ ˜ ˜ kC |Sn| ≤ . (6.6) 2qn Then Lemma 6.3 yields (6.3). 6.1. Proof of Theorem7. 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 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 23

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 Σac(HΦ,θ) be the absolutely continuous spectrum of HΦ,θ. Let L(λ, Φ) be the ess Lyapunov exponent of HΦ,θ at energy λ, as deﬁned in (2.43). For a set U ⊆ R, let U be its essential closure. First, we are able to give a characterization of the Lyapunov exponent on the spectrum.

Φ Proposition 6.4. For 2π ∈ R \ Q, L(λ, Φ) = 0 if and only if λ ∈ ΣΦ. The proof of this is similar to that for the almost Mathieu operator as given in [A] and the extended Harper’s model [JM1]. 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 [A]) then bring us back to the ε = 0 case. We will leave the proof to the appendix. −2πiθ 1 Exploiting the fact that c(θ) = 1 + e has a real zero θ1 = 2 , we have Φ Proposition 6.5. ([D], see also Proposition 7.1 of [JM1]) For 2π ∈ R \ Q, and a.e. θ ∈ T1, Σac(HΦ,θ) is empty.

Hence our operator HΦ,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 Schr¨odinger 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 [T]. In our case |c(·)| is not bounded away from zero, however a careful inspec- tion of the proof of Theorem 5.17 of [T] shows that it holds under a weaker requirement: 1 2 log (|c(·)|) ∈ L . Namely, let Hc,v(θ) acting on ` (Z) be an ergodic Jacobi matrix,

m m−1 m (Hc,v(θ)u)m = c(T θ)um+1 + c(T θ)um−1 + v(T θ)um where c : M → C, v : M → R, are bounded measurable functions, and T : M → M is an ergodic map. Let Lc,v(λ) be the corresponding Lyapunov exponent. We have Theorem 8. (Kotani theory) Assume log (|c(·)|) ∈ L1(M). Then for a.e. θ ∈ M, ess Σac(Hc,v(θ)) = {λ : Lc,v(λ) = 0} .

Proof. The proof of Theorem 5.17 of [T] works verbatim. 1 In our concrete model, log (|c(θ)|) = log (2| cos πθ|) ∈ L (T1), thus Theorem8 applies, and combining with Propositions 6.4, 6.5, it follows that ΣΦ must be a zero measure set. 24 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

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

6.2. Quick Observations about H2πp/q,θ. Let Aλ(·), A˜λ(·),Dλ(·), Θ be deﬁned as in Section 2.2.1. We start with several quick observations about H2πp/q,θ.

1 Observation 1. The sampling function c(θ) = 0 yields a unique solution θ = 2 (mod 1 1 1), hence Θ = 2 + q Z. Then, p • for θ∈ / Θ, we have c(θ + n q ) 6= 0 for any n ∈ Z p • for θ ∈ Θ, there exists k0 ∈ {0, 1, ..., q − 1} such that c(θ + n q ) = 0 if and only if n ≡ k0 (mod q).

tr(Dλ(θ)) ˜λ q tr(Aq (θ)) = 1 . (6.7) 2| sin πq(θ + 2 )|

We have the following characterization of Σ2πp/q,θ.

6.2.1. Case 1. If θ ∈ Θ, we have the following

Observation 2. For θ ∈ Θ, the inﬁnite matrix H2πp/q,θ is decoupled into copies of the following block matrix Mq of size q:

 1 1 p  v( 2 ) c( 2 − q )  1 p 1 p .   c( − ) v( − ) ..   2 q 2 q   .. ..   . .        (6.8)          v( 1 − (q − 2) p ) c( 1 − (q − 1) p )  2 q 2 q  1 p 1 p c( 2 − (q − 1) q ) v( 2 − (q − 1) q )

Thus

Σ2πp/q,θ = {eigenvalues of Mq}, for θ ∈ Θ. (6.9) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 25

6.2.2. Case 2. If θ∈ / Θ, by Floquet theory, we have ˜λ Σ2πp/q,θ = {λ : | tr(A p (θ))| ≤ 2}. (6.10) q ,q

λ Furthermore, the set {λ : tr A˜ p (θ) = 2 cos 2πν} contains q individual points (counting q ,q multiplicities), which are eigenvalues of the following q × q matrix Mq,ν:  p 2πiν  v(θ) |c(θ − q )| e |c(θ)| .  p p ..   |c(θ − q )| v(θ − q )     ......  Mq,ν(θ) =        p p   v(θ − (q − 2) q ) |c(θ − (q − 1) q )|  −2πiν p p e |c(θ)| |c(θ − (q − 1) q )| v(θ − (q − 1) q ) (6.11)

Combining (6.10) with (6.7), we arrive at an alternative representation 1 Σ = λ : | tr(Dλ(θ))| ≤ 4| sin πq(θ + )| . (6.12) 2πp/q,θ q 2

6.3. Key lemmas. Let

λ dq(θ) = tr(Dq (θ)). (6.13) We have

Lemma 6.6. (a Chambers’ type formula) For all θ ∈ T1, we have

dq(θ) = −2 cos 2πqθ + Gq(λ), (6.14)

where Gq(λ) (deﬁned by (6.14)) is independent of θ.

Proof. It is easily seen that dq(·) is a 1/q-periodic function, thus

2πiqθ −2πiqθ dq(θ) = Gq(λ) + aqe + a−qe , in which the Gq(λ) part is independent of θ. One can easily compute the coeﬃcients aq, a−q, and get aq = a−q = 1. Lemma 6.7. For θ ∈ Θ,

λ det (λ · Id − Mq(θ)) = tr(Dq (θ)). (6.15)

The proof of this lemma will be included in the appendix. 26 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

1 Combining (6.12), (6.9) and Lemma 6.7 with the fact that | sin πq(θ + 2 )| = 0 for θ ∈ Θ, we arrive at 1 Σ = {λ : | tr(Dλ(θ))| ≤ 4| sin πq(θ + )|} (6.16) 2πp/q,θ q 2 holds uniformly for θ ∈ T1.

By Lemma 6.14, we get the following alternative characterization of Σ2πp/q,θ. 1 1 Σ = λ : −4| sin πq(θ + )| + 2 cos 2πqθ ≤ G (λ) ≤ 4| sin πq(θ + )| + 2 cos 2πqθ . 2πp/q,θ 2 q 2 (6.17)

1 1 Let us denote Lq(θ) := 4| sin πq(θ + 2 )| + 2 cos 2πqθ, and lq(θ) := −4| sin πq(θ + 2 )| + 2 cos 2πqθ. Then (6.17) translates into

Σ2πp/q,θ = {λ : lq(θ) ≤ Gq(λ) ≤ Lq(θ)}. (6.18) This clearly implies

Σ2πp/q = {λ : min lq(θ) ≤ Gq(λ) ≤ max Lq(θ)}. (6.19) T1 T1

Since Gq(λ) is a polynomial of λ of degree q,(6.19) implies Σ2πp/q has q (possibly touching) bands.

The following lemma provides estimates of |Σ2πp/q,θ|.

Lemma 6.8. We have

|Σ2πp/q,θ| ≤ 4|c(θ)|.

Proof. Let us point out that, due to (6.10) and the explanation below it, ( {λ : G (λ) = L (θ)} = {eigenvalues of M (θ)} q q q,0 (6.20) {λ : Gq(λ) = lq(θ)} = {eigenvalues of M 1 (θ)} q, 2

q Let {λi(θ)}i=1 be eigenvalues of Mq,0(θ), labelled in the increasing order. Let ˜ q {λi(θ)} be eigenvalues of M 1 (θ), labelled also in the decreasing order. Then by i=1 q, 2 (6.18) and (6.20),

q X q−k ˜ |Σ 2πp | = (−1) λk(θ) − λk(θ) (6.21) q ,θ k=1 q+1 q−1 [ 2 ] [ 2 ] X ˜ X ˜ = λq−2k+2(θ) − λq−2k+2(θ) − λq−2k+1(θ) − λq−2k+1(θ) . k=1 k=1 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 27

Consider the diﬀerence matrix  2|c(θ)|      Mq,0(θ) − Mq, 1 (θ) =   2     2|c(θ)| q whose eigenvalues we denote by {Ei(θ)}i=1, namely,

E1(θ) = −2|c(θ)| < 0 = E2(θ) = ··· = Eq−1(θ) = 0 < 2|c(θ)| = Eq(θ).

By Lidskii inequalities (2.51), we have

q+1 q+1 q+1 q+1 [ 2 ] [ 2 ] [ 2 ] [ 2 ] X X ˜ X X ˜ λq−2k+2(θ) ≤ λq−2k+2(θ) + Eq−k+1(θ) = λq−2k+2(θ) + 2|c(θ)|, (6.22) k=1 k=1 k=1 k=1 and

q−1 q−1 q−1 q−1 [ 2 ] [ 2 ] [ 2 ] [ 2 ] X X ˜ X X ˜ λq−2k+1(θ) ≥ λq−2k+1(θ) + Ek(θ) = λq−2k+1(θ) − 2|c(θ)|. (6.23) k=1 k=1 k=1 k=1 Hence combining (6.21) with (6.22)(6.23), we get,

|Σ 2πp | ≤ 4|c(θ)|. (6.24) q ,θ

6.4. 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 ⊆ T1, if corresponding sets/functions are constant on A. Since q is even, a simple computation shows  6Z+1 6Z+5 max 1 Lq(θ) = Lq( ) = Lq( ) = 3,  T 6q 6q

 2Z+1 minT1 lq(θ) = lq( 2q ) = −6.

6Z+1 2Z+1 A simple computation also shows lq( 6q ) = −1 and Lq( 2q ) = 2. Thus we have, by (6.19),

Σ2πp/q ={λ : −6 ≤ Gq(λ) ≤ 3} [ ={λ : −6 ≤ Gq(λ) ≤ 2} {λ : −1 ≤ Gq(λ) ≤ 3} [ =Σ 2πp 2Z+1 Σ 2πp 6Z+1 . q , 2q q , 6q 28 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

This implies

Now it remains to estimate |Σ 2πp 2Z+1 | and |Σ 2πp 6Z+1 |. Since q is even, let us consider q , 2q q , 6q Σ 2πp q+1 and Σ 2πp 1 . q , 2q q , 6q By Lemma 6.8, we have

 q+1 4π |Σ 2πp , q+1 | ≤ 4|c( 2q )| < q , q 2q (6.26) 1 4π |Σ 2πp 1 | ≤ 4|c( )| < .  q , 6q 6q 3q Hence putting (6.25), (6.26) together, we have 16π |Σ 2πp | < . (6.27) q 3q

6.5. 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 (6.25), we have

|Σ 2πp | ≤ |Σ 2πp 3Z+1 | + |Σ 2πp Z |. (6.28) q q , 3q q , q By Lemma 6.8, we have

 3q−1 4π |Σ 2πp , 3q−1 | ≤ 4|c( 6q )| < 3q , q 6q (6.29) q−1 4π |Σ 2πp q−1 | ≤ 4|c( )| < .  q , 2q 2q q Hence putting (6.28), (6.29) together, we have 16π |Σ 2πp | < . (6.30) q 3q

7. Proof of Lemma 4.1

Using ideas from [P] and [BGP], we can reduce the operator ΛB (3.14) by Krein’s

resolvent formula to a term containing the Dirichlet spectrum and QΛ. For this we need to introduce a few concepts first. The l2-space on the vertices l2(V(Λ)) carries the inner product X hf, gi := 3f(v)g(v) (7.1) v∈V(Λ) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 29 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 Schrödingeroperator HB with an effective Hamiltonian, is the concept of boundary triples.

Deﬁnition 7.1. Let T : D(T ) ⊂ H → H be a closed linear operator on the Hilbert space H , then the triple (π, π0, H 0), with H 0 being another Hilbert space and π, π0 : D(T ) → H 0, is a boundary triple for T , if • Green’s identity holds on D(T ), i.e. for all ψ, ϕ ∈ D(T )

0 0 hψ, T ϕiH − hT ψ, ϕiH = hπψ, π ϕiH 0 − hπ ψ, πϕiH 0 . (7.2)

• ker(π, π0) is dense in H . • (π, π0): D(T ) → H 0 ⊕ H 0 is a linear surjection.

The following lemma applies this concept to our setting.

Lemma 7.2. The operator T B : D(T B) ⊂ L2(PE (Λ)) → L2(PE (Λ)) acting as the maximal Schr¨odingeroperator (2.12) on every edge with domain ( B 2 D(T ) := ψ ∈ H (PE (Λ)) : any ~e1,~e2 ∈ Ev(Λ) such that i(~e1) = i(~e2) = v satisfy

ψ~e1 (v) = ψ~e2 (v) and if t(~e1) = t(~e2) = v, ) ˜ ˜ iβ~e1 iβ~e2 then e ψ~e1 (v) = e ψ~e2 (v) (7.3) is closed. The maps π, π0 on D(T B) deﬁned by ! 1 X X ˜ π(ψ)(v) := ψ (v) + eiβ~e ψ (v) 3 ~e ~e i(~e)=v t(~e)=v ! 1 X X ˜ π0(ψ)(v) := ψ0 (v) − eiβ~e ψ0 (v) (7.4) 3 ~e ~e i(~e)=v t(~e)=v form together with H 0 := l2(V(Λ)) a boundary triple associated to T B.

Proof. The proof follows the same strategy as in [P]. The operator T B is closed iﬀ its domain is a closed subspace (with respect to the graph norm) of the domain of B L some closed extension of T . Such a closed extension is given by ~e∈E(Λ) H~e on H2(PE (Λ)). To see that D(T B) is a closed subspace of H2(PE (Λ)), observe that in 30 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

terms of continuous functionals

2 l~ei,~ej : H (PE (Λ)) → C, l~ei,~ej (ψ) = ψ~ei (i(~ei)) − ψ~ej (i(~ej)) 2 iβ˜ iβ˜ ~ei ~ej k~ei,~ej : H (PE (Λ)) → C, k~ei,~ej (ψ) = e ψ~ei (t(~ei)) − e ψ~ej (t(~ej)) (7.5) we obtain

B \ \ D(T ) = ker l~ei,~ej ∩ ker k~ei,~ej (7.6)

~ei,~ej ∈E(Λ) with i(~ei)=i(~ej ) ~ei,~ej ∈E(Λ) with t(~ei)=t(~ej ) 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, π0) is obvious since this space L ∞ contains ~e∈E(Λ) Cc (~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 [P].

Any boundary triple for T as in Def. 7.1 and any self-adjoint relation A ⊆ H 0 L H 0 gives rise [Sch] to a self-adjoint restriction TA of T with domain

0 D(TA) = {ψ ∈ D(T ):(π(ψ), π (ψ)) ∈ A} . (7.7) The restriction of T B satisfying Dirichlet type boundary conditions on every edge is 2 D obtained by selecting A1 := {0}⊕l (V(Λ)) and coincides with H (2.15). The operator B B 2 Λ (3.14) is recovered from T by picking the relation A2 := l (V(Λ)) ⊕ {0} . Definition 7.3. Given the boundary triple for T B as above, the gamma-field γ : D 2 2 −1 ρ(H ) → L(l (V(Λ)),L (PE (Λ))) is given by γ(λ) := π|ker(T B −λ) and the Weyl function M(·, Φ) : ρ(HD) → L(l2(V(Λ))) is defined as M(λ, Φ) := π0γ(λ).

D B A computation shows that those maps are well-deﬁned. The resolvents of H = TA1 B B and Λ = TA2 are then related by Krein’s resolvent formula [Sch]. Theorem 9. Let (l2(V(Λ)), π, π0) be the boundary triple for T B and γ, M as above, then for λ ∈ ρ(HD) ∩ ρ(ΛB) there is also a bounded inverse of M(λ, Φ) and (ΛB − λ)−1 − (HD − λ)−1 = −γ(λ)M(λ, Φ)−1γ(λ)∗. (7.8)

In particular, σ(ΛB)\σ(HD) = {λ ∈ R ∩ ρ(HD) : 0 ∈ σ(M(λ, Φ))}. For λ ∈ ρ(HD) we have γ(λ) ker(M(λ, Φ)) = ker(ΛB − λ). This implies that both null-spaces are of equal dimension.

Remark 5. To Simon: shall we add a short comment/proof that in general we have (ΛB − λ)−1 − (HD − λ)−1 = −γ(λ)(T − M(λ, Φ))−1γ(λ)∗

where D(ΛB) = {ψ : π0ψ = T πψ}, hence here T = 0? CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 31

Lemma 7.4. For the operator T B, the gamma-ﬁeld γ and Weyl function M can be

explicitly written in terms of the solutions sλ, cλ (2.36) on an arbitrary edge ~e ∈ E(Λ) for λ ∈ ρ(HD) and z ∈ l2(V(Λ)) by

−iβ˜ (sλ(1)cλ,~e(x) − sλ,~e(x)cλ(1)) z(i(~e)) + e ~e sλ,~e(x)z(t(~e)) (γ(λ)z)~e(x) = (7.9) sλ(1) and K (Φ) − ∆(λ) M(λ, Φ) = Λ (7.10) sλ(1) where   1 X ˜ X ˜ (K (Φ)z)(v) := e−iβ~e z(t(~e)) + eiβ~e z(i(~e)) (7.11) Λ 3   ~e: i(~e)=v ~e: t(~e)=v

2 deﬁnes an operator in L(l (V(Λ))) with kKΛ(Φ)k ≤ 1.

Proof. For λ ∈ ρ(HD) and z ∈ l2(V(Λ)) we deﬁne for ~e ∈ E(Λ) arbitrary

−1 ψ~e := (γ(λ)z)~e = ((π|ker(T B −λ)) z)~e (7.12)

00 with ψ := (ψ~e). In particular, ψ~e is the solution to −ψ~e +V~eψ~e = λψ~e with the following −iβ˜ boundary condition: ψ~e(i(~e)) = z(i(~e)) and ψ~e(t(~e)) = e ~e z(t(~e)). The representation (7.9) is then an immediate consequence of (2.29). The expression for the Weyl function on the other hand, follows from the Dirichlet- to-Neumann map (2.30).

(M(λ, Φ)z)(v) = (π0γ(λ)z)(v)   1 X X ˜ = ψ0 (v) − eiβ~e ψ0 (v) 3  ~e ~e  ~e: i(~e)=v ~e: t(~e)=v     (K (Φ)z)(v) c (1) s0 (1)  Λ  λ λ  = −  δv∈i(V(Λ))z(v) + δv∈t(V(Λ))z(v) sλ(1) sλ(1) sλ(1)  | {z } s0 (1) = λ z(v) sλ(1) (K (Φ)z − s0 (1)z)(v) = Λ λ , (7.13) sλ(1) here we used (2.32). The formula (7.10) then follows from (7.13) and (2.38). Since i(Λ) ∩ t(Λ) = ∅, we have kKΛ(Φ)k ≤ 1. 32 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Since all vertices are integer translates of either of the two vertices r0, r1 ∈ WΛ by ~ ~ 2 2 2 2 basis vectors b1, b2, we conclude that l (V(Λ)) ' l (Z ; C ). Our next Lemma shows KΛ(Φ) and QΛ(Φ) are unitary equivalent under this identiﬁcation.

Lemma 7.5. KΛ(Φ) is unitary equivalent to operator QΛ(Φ).

Proof. By (3.16), (7.11),

( 1 −iΦγ1 (KΛ(Φ)z)(γ1, γ2, r0) = 3 z(γ1, γ2, r1) + z(γ1 − 1, γ2, r1) + e z(γ1, γ2 − 1, r1) , 1 iΦγ1 (KΛ(Φ)z)(γ1, γ2, r1) = 3 z(γ1, γ2, r0) + z(γ1 + 1, γ2, r0) + e z(γ1, γ2 + 1, r0) . 2 In order to transform KΛ to QΛ we use the unitary identiﬁcation W : l (V(Λ)) → l2(Z2, C2) T (W z)γ1,γ2 := z(γ1, γ2, r0) , z(γ1, γ2, r1) . (7.14)

∗ This way, QΛ(Φ) = WKΛ(Φ)W . Remark 6. In terms of a ∈ l2(Z2, C2) deﬁned as 1 0 1 1 0 1 a := , a := (0,0) 3 1 0 (0,1) 3 0 0 1 0 1 1 0 0 a := , a := (1,0) 3 0 0 (0,−1) 3 1 0 1 0 0 a := , and a := 0 for other γ ∈ 2, (7.15) (−1,0) 3 1 0 γ Z we can express (4.1) in the compact form

X γ1 γ2 QΛ(Φ) = aγ(τ0) (τ1) . (7.16) γ∈Z2;|γ|≤1 This operator has already been studied, in diﬀerent contexts, for rational ﬂux quanta in [KL], [HKL], and [AEG].

Finally, we point out that Lemma 4.1 follows from a combination of Theorem9, Lemma 7.4 and Lemma 7.5.

8. Spectral analysis

This section is devoted to complete spectral analysis of HB. In view of Lemmas 4.1 and 5.1, an important technical fact is:

Lemma 8.1. The operator norm of QΛ(Φ) for non-trivial ﬂux quanta Φ ∈/ 2πZ is strictly less than 1. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 33

Indeed, then, away from the Dirichlet spectrum σ(HD), which are located on the edges of the Hill bands (2.41), we have the following characterization of σ(HB). Let

Bn and ∆ be deﬁned as in Section 2.1.3.

Lemma 8.2. For the magnetic Schrödingeroperator HB, the following properties hold. (1) The level of the Dirac points ∆|−1 (0) always belongs to the spectrum of HB, int(Bn) B i.e. 0 ∈ ∆|int(Bn)(σ(H )). B B (2) λ ∈ ∆|int(Bn)(σ(H )) iff −λ ∈ ∆|int(Bn)(σ(H )). Consequently, the property ∆0(0) 6= 0 implies that locally with respect to the Dirac points, the spectrum of HB is symmetric. (3) HB has no point spectrum away from σ(HD). (4) For non-trivial flux Φ ∈/ 2πZ, HB has purely continuous spectrum bounded away from σ(HD).

Combining Lemma 8.2 with Lemma 4.4, we get

Φ 1 Lemma 8.3. For generic Φ, dimH (σ ) ≤ 2 .

Proof of Lemma 8.3. Lemma 4.4 and (2) of Lemma 8.2 together imply that for generic Φ,

−1 1 dimH ∆| (σ(QΛ(Φ)) ≤ . int(Bn) 2 Hence since [ σΦ = σ(HD) ∪ ∆|−1 (σ(Q (Φ)) , n∈N int(Bn) Λ we have 1 dim (σΦ) ≤ sup dim (σ(HD)), sup dim ∆|−1 (σ(Q (Φ)) ≤ . H H H int(Bn) Λ n∈N 2 Remark 7. To Simon: Shall we insert a picture here, to illustrate the ∆ function, the symmetry of ∆(σ(HB)) as well as the Dirac points?

Proof of Lemma 8.2. (1), (2) follow from a quick combination of Lemma 4.1 and Lemma 5.1, and (3) follows from Part (2) of Lemma 5.2. (4) is a corollary of Lemma 8.1 and (3).

Alternate proof of Lemma 8.1. Without loss of generality Φ ∈ (0, 2π). By (5.3), it suﬃces to show kHΦ,θk < cΦ < 6 for some constant cΦ independent of θ ∈ T1. Let us 34 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

2 take ϕ ∈ ` (Z) with kϕk`2(Z) = 1. Consider Φ Φ Φ (H ϕ) = c θ + n ϕ + c θ + (n − 1) ϕ + v θ + n ϕ , Φ,θ n 2π n+1 2π n−1 2π n | {z } | {z } | {z } =:(h1ϕ)n =:(h2ϕ)n =:(h3ϕ)n

2 in which h1, h2, h3 ∈ L(` (Z)). Hence

2 2 2 2 kH ϕk 2 ≤3 kh ϕk 2 + kh ϕk 2 + kh ϕk 2 Φ,θ ` (Z) 1 ` (Z) 2 ` (Z) 3 ` (Z)  2  Φ 2 Φ Φ 2

≤3 sup  c θ + (n − 1) + c θ + n + v θ + n  n∈Z 2π 2π 2π Φ ≤12 sup sin2 π θ − + sin2(πθ) + cos2(2πθ) θ∈T1 2π 2 =:cΦ < 36.

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.

8.1. Magnetic translations. B st In general, Λ does not commute with lattice translations Tγ . Yet, there is a set of modiﬁed translations that do still commute with ΛB, although they in general no longer B 2 commute with each other. We deﬁne those magnetic translations Tγ : L (PE (Λ)) → L2(PE (Λ)) as unitary operators given by

B B st (Tγ ψ)~e := uγ (~e)(Tγ ψ)~e (8.1) 2 2 st for any ψ := (ψ~e)~e∈E(Λ) ∈ L (PE (Λ)) and γ ∈ Z . The lattice translation Tγ is deﬁned by (T stψ) (x) = ψ (x − γ ~b − γ ~b ) as before. The function uB is constant γ ~e ~e−γ1~b1−γ2~b2 1 1 2 2 γ on each copy of the fundamental domain, and deﬁned as follows

B ~ iΦγ1γ˜2 ~ ~ ~ 2 uγ (˜γ1, γ˜2, [e]) = e , for [e] = f,~g or h, γ = (γ1, γ2) ∈ Z . (8.2) ~ ~ ~ For ~e =γ ˜1b1 +γ ˜2b2 + [e] we have (T BT Bψ) =u (~e)u (~e − µ ~b − µ ~b )ψ µ γ ~e µ γ 1 1 2 2 ~e−µ1~b1−µ2~b2−γ1~b1−γ2~b2 =eiΦµ1γ˜2+iΦγ1(˜γ2−µ2)ψ , ~e−µ1~b1−µ2~b2−γ1~b1−γ2~b2 (T BT Bψ) =u (~e)u (~e − γ ~b − γ ~b )ψ γ µ ~e γ µ 1 1 2 2 ~e−µ1~b1−µ2~b2−γ1~b1−γ2~b2 =eiΦγ1γ˜2+iΦµ1(˜γ2−γ2)ψ . (8.3) ~e−µ1~b1−µ2~b2−γ1~b1−γ2~b2 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 35

Hence

B B B B iΦω(µ,γ) Tµ Tγ = Tγ Tµ e , (8.4)

2 where ω(µ, γ) := µ1γ2 − µ2γ1 is the standard symplectic form on R . It also follows B −1 −iΦγ1γ2 B from (8.3) that (Tγ ) = e T−γ. One can also check that

B ∗ −iΦγ1γ2 B B −1 (Tγ ) = e T−γ = (Tγ ) , (8.5)

B hence Tγ is unitary. By the deﬁnition (8.1), (8.2), it is clear that for any ψ ∈ L2(PE(Λ)), d d T Bψ = T B ψ and VT Bψ = T BV ψ. (8.6) dt γ γ dt γ γ

B B B B B B In order to make sure D(Λ Tγ ) = D(Tγ Λ ), it suﬃces to check Tγ (D(Λ )) = D(ΛB), which translates into   uγ(~e1) = uγ(~e2) whenever i(~e1) = i(~e2)  ˜ ˜ (8.7) iβ~e iβ ~ ~  e 2 uγ (~e2) e ~e2−γ1b1−γ2b2  ˜ = ˜ whenever t(~e1) = t(~e2).  iβ~e iβ ~ ~ e 1 uγ (~e1) e ~e1−γ1b1−γ2b2

This, by (3.16) is in turn equivalent to the following: for anyγ ˜1, γ˜2 ∈ Z:  u (˜γ , γ˜ , f~) = u (˜γ , γ˜ ,~g) = u (˜γ , γ˜ ,~h)  γ 1 2 γ 1 2 γ 1 2

 ~ −iΦγ1 ~ uγ(˜γ1, γ˜2, f) = uγ(˜γ1 + 1, γ˜2,~g) = e uγ(˜γ1, γ˜2 + 1, h)

B The deﬁnition of uγ (8.2) clearly satisﬁes this requirement. Therefore, although magnetic translations do not necessarily commute with one another (8.4), they commute with ΛB

B B B B Tγ Λ = Λ Tγ . (8.8)

B B B One can check that Tγ (D(T )) = D(T ) is also equivalent to (8.7), hence by (8.6),

B B B B Tγ T = T Tγ . (8.9) Note that we also have

B D D B Tγ H = H Tγ , (8.10)

B 1 1 ~ ~ which is due to (8.6) and Tγ (H0(~e)) = H0(~e − γ1b1 − γ2b2) for any ~e ∈ E(Λ). We will now study the structure of eigenfunctions and the nature of the continuous spectrum of HB. 36 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

8.2. Dirichlet spectrum. In this subsection, we will study the energies belonging to the Dirichlet spectrum σ(HD). Lemma 8.4 below shows that σ(HD) is contained in the point spectrum of HB, hence the only point spectrum of HB, 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.5. Then this loop passes (proceeding in

positive direction from the center of an edge ~e1 such that the ﬁrst vertex we reach is t(~e1)) n := 2 + 4q edges ~e1, ...,~en in E(Λ). For a solution vanishing outside this loop, the boundary conditions imposed by (3.15) on the derivatives can be represented in a matrix equation 0 TΦ(n)ψ (n) = 0, (8.11) where  iβ˜ iβ˜  e ~e1 e ~e2 0 0 0 ··· 0  ψ0 (1)  ~e1  0 1 1 0 0 ··· 0  ψ0 (1)    ~e2  iβ˜ iβ˜  0   0 0 e ~e3 e ~e4 0 ··· 0  ψ (1)   0  ~e3  TΦ(n) :=  . . . .  and ψ (n) :=  .  .  . . 0 .. .. 0 0   .      iβ˜ ˜  0   ~en−1 iβ~en  ψ (1)  0 0 0 0 0 e e   ~en−1  ψ0 (1) 1 0 0 0 0 0 1 ~en (8.12)

Remark 8. We observe that TΦ(n) can be row-reduced to an upper triangular matrix with diagonal

iβ˜ iβ˜ iβ˜ iβ˜ i Pn (−1)j β˜ (e ~e1 , 1, e ~e3 , 1, e ~e5 , ..., 1, e ~en−1 , 1 − e j=1 ~ej )

iβ˜ iβ˜ iβ˜ iβ˜ iqΦ =(e ~e1 , 1, e ~e3 , 1, e ~e5 , ..., 1, e ~en−1 , 1 − e ), where q is the number of enclosed hexagons. Hence rank(TΦ(n)) = n iﬀ qΦ ∈/ 2πZ and rank(TΦ(n)) = n − 1 otherwise. Lemma 8.4. The Dirichlet eigenvalues λ ∈ σ(HD) are contained in the point spectrum of HB.

Proof. For Φ ∈ 2πZ the statement is known [KP], thus we focus on Φ ∈/ 2πZ. By unitary equivalence, it suﬃces to construct an eigenfunction to ΛB. We will construct an eigenfunction on two adjacent hexagons X as in Fig.4. 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 ~e1,~e2, ...,~e10 (see Fig.4). Recall that sλ,~e is the Dirichlet eigenfunction on ~e.

By Remark8, for 2Φ ∈ 2πZ, operator TΦ(10) has a non-trivial nullspace. We could take

a = (aj) ∈ ker (TΦ(10)) \{0}, (8.13) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 37

e6 e4

e7 e3

e

e8 e2

e9 e1

e10

Figure 4. Black arrows describe the double hexagon with slicing edge indicated by the dashed arrow.

and an eigenfunction ψ on X such that ψ~e = 0 and ψ~ej = ajsλ,~ej . 10 iβ˜ If 2Φ ∈/ 2πZ, we take a vector y ∈ C such that y2 = −1, y7 = −e ~e and yj = 0 otherwise. Since in this case TΦ(10) is invertible, there exists a unique solution a = (aj) to the following equation:

TΦ(10)a = y. (8.14)

Let us take ψ on X such that ψ~ej = ajsλ,~ej and ψ~e = sλ,~e, then one can easily check ψ is indeed an eigenfunction on X.

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

Corollary 8.5. The spectrum of HB must always have open gaps for Φ ∈/ 2πZ at the edges of the Hill bands.

Remark 9. 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 B 6 all gaps of the absolutely continuous spectrum are closed and σac(H ) = [0, ∞).

The next lemma concerns the general feature of eigenspace of HB. 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. We have 38 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

(1) Every eigenspace of HB is inﬁnitely degenerated. (2) Eigenfunctions of HB vanish at every vertex and are thus eigenfunctions of HD as well. (3) Eigenfunctions of HB 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.

Proof. (1). This is a standard consequence of the existence of magnetic translations (8.1), here we only give a sketch of the proof. One could consult e.g. the proof of Lemma 3.5 in [KP] for details. Assume there was a ﬁnite dimensional eigenspace of HB, then the magnetic translations (8.1) would form a family of unitary translation operators on this space. By normality, there was for any unitary translation operator an eigenfunction with some eigenvalue located on the unit circle in C. This situation however is clearly impossible for eigenfunctions in L2(PE(Λ)). (To Simon: Let us make this argument a little bit clearer.)

(2). If there is an eigenfunction to HB with eigenvalue λ that does not vanish at a vertex, by (modiﬁed) Peierls’ substitution (3.13), there is one to ΛB, denoted as ϕ, as well. We may expand the function in local coordinates on every edge ~e ∈ E(Λ) as

ϕ~e = a~ecλ,~e + b~esλ,~e according to (2.36). Recall also that the Dirichlet eigenfunction sλ is either even or odd. Thus, using (2.32) we conclude that |cλ(0)| = |cλ(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

X 2 2 |ϕ~e(i(~e))| ≤ kϕkH2 < ∞ (8.15) ~e∈E(Λ)

ϕ has to vanish at every vertex. Thus ϕ is also an eigenfunction to HD. (3) clearly follows from (2) and (3.15).

8.2.1. Dirichlet spectrum for rational flux quanta. Φ p In this section, the flux quanta are assumed to be reduced fractions 2π = 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 [KP]. 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 B0 rather than just Φ , see Fig.5 B0 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 39

Figure 5. Simply closed loop state supported on black arrows encloses area 4Φ . B0

Lemma 8.8. Any simply closed loop enclosing an area of qΦ has a unique (up to B0 normalization) eigenfunction of HB 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 Remark8. Due to dim(ker( TΦ)) = 1, such eigenfunctions are also unique (up to normalization). Lemma 8.9. The nullspaces ker(HB −λ) where λ ∈ σ(HD) are generated by compactly supported eigenfunctions.

Proof. Unitary equivalence allows us to work with ΛB rather than HB. Without loss of generality, we assume that the Dirichlet eigenfunction to λ is even. Due to Lemma 8.7, eigenfunctions of ΛB to Dirichlet eigenvalues vanish at every vertex. Thus, on every edge ~e ∈ E(V ), they are of the form ϕ~e = a~esλ,~e for some a~e.

Let ϕ be such a function. We deﬁne the sequence (u(v))v∈V(Λ) as follows ( u(γ , γ , r ) := ϕ0 (γ , γ , r ) 1 2 0 γ1,γ2,~g 1 2 0 0 u(γ1, γ2, r1) := ϕ (γ1, γ2, r1). γ1,γ2,f~ Observe that the sequence (u(v)) determines the eigenfunction on every edge. Indeed, a = u(γ , γ , r ) and a = u(γ , γ , r ), since s0 (1) = s0 (0). At the same γ1,γ2,~g 1 2 0 γ1,γ2,f~ 1 2 1 λ λ time, a can be determined in two diﬀerent ways, one for each endpoint, from the γ1,γ2,~h boundary condition (3.15). Let us now introduce an operator A ∈ L(l2(V(Λ))) that has precisely the sequences (u(v)) with matching vertex conditions for a at its γ1,γ2,~h 40 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

endpoints in its kernel. Then,

(Au)(γ1, γ2, r0) := u(γ1, γ2, r0) + u(γ1, γ2, r1)

2πi pγ1 − e q u(γ1 + 1, γ2 − 1, r0) + u(γ1, γ2 − 1, r1) and

(Au)(γ1, γ2, r1) := 0. (8.16)

The operator A is then a Z2-periodic finite-order difference operator. Any eigenfunction ϕ satisfying (ΛB −λ)ϕ = 0 leads by standard arguments to a square-summable sequence (u(v)) as defined above in the nullspace of A. Conversely, any such element in the B nullspace of A uniquely defines an eigenfunction ϕ = a~esλ,~e to Λ . Theorem 8 in [K2] implies then that the nullspace of A is generated by sequences in c00(V(Λ)). It suffices now to observe that those compactly supported sequences also give rise to compactly supported eigenfunctions to conclude the claim.

Lemma 8.10. Let Φ ∈/ 2πZ. The eigenspaces are spanned by the set of double hexagonal states Fig.4.

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 [KP] case, to show that any compactly supported eigenfunction is a linear combination of double hexagonal states. Let ϕ be a compactly supported eigenfunction of ΛB to some Dirichlet eigenvalue λ. Consider an edge ~e ∈ 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 on a single hexagon, necessarily encloses a double hexagon X, as in Fig.4, which contains the chosen edge ~e. Then, there is by the proof

of Lemma 8.4 a state ψ on X so that the wavefunction ψ~e on ~e coincides with ϕ~e. Subtracting ψ from ϕ leaves us with an eigenfunction 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 Theorem4 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 l2(E(Λ)) is defined as   2  2 X 2  l (E(Λ)) := z : E(Λ) → C, kzkl2(E(Λ)) := |z(~e)| < ∞ . (8.17)  ~e∈E(Λ)  Theorem 10. The double hexagonal states generate the eigenspaces of Dirichlet spectrum of HB for irrational flux quanta. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 41

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(HD). Deﬁnition 8.12. We denote the closed L2(PE(Λ)) subspace generated by linear com-

binations of all double hexagonal states on the entire graph Λ by DHE(Λ)(Φ).

There is, as for any separable Hilbert space, a countable orthonormal system of

states V (Φ) ⊂ DHE(Λ)(Φ) such that

span(V (Φ)) = DHE(Λ)(Φ). (8.18) Because the set V (Φ) is countably inﬁnite, there is a bijection ζ : Z2 → V (Φ). This 2 way, we can address the elements of V (Φ) by ϕγ(Φ) with γ ∈ Z . One can make sure that for each γ, ϕγ(Φ) is analytic in Φ ∈ (0, 1). Every element ϕγ(Φ) ∈ V (Φ) is of the form X ϕγ(Φ) = ϕγ,~e(Φ)sλ,~e (8.19) ~e∈E(Λ) because it is an element of ker(HB − λ). Now assume that the statement of Theorem 10 does not hold, this is equivalent to B ⊥ saying that Z(Φ) := ker(H − λ) ∩ DHE(Λ)(Φ) is not the zero space, i.e. there are eigenfunctions not spanned by double hexagonal states. Our goal is to characterize this space as the nullspace of a suitable operator we deﬁne next.

Deﬁnition 8.13. Let A(Φ) ∈ L(l2(E(Λ))) on u ∈ l2(E(Λ)) for γ ∈ Z2 be given by (A(Φ)u)(γ, f~) := u(γ, f~) + u(γ,~g) + u(γ,~h)

~ −iΦγ1 ~ (A(Φ)u)(γ,~g) := u(γ1, γ2 − 1, f) + u(γ1 + 1, γ2 − 1,~g) + e u(γ1, γ2, h) ~ X (A(Φ)u)(γ, h) := u(~e)ϕγ,~e(Φ). (8.20) ~e∈E(Λ) Remark 10. The motivation behind this deﬁnition is that we can check with the ﬁrst two lines in (8.20) the boundary condition at the vertices and with the third line in ⊥ (8.20) we monitor the orthogonality on DHE(Λ)(Φ) .

This way, ker(A(Φ)) and Z(Φ) are isometrically isomorphic where the isomorphism is given by X u~e η ∈ L(ker(A(Φ)),Z(Φ)) and η(u) := sλ,~e. (8.21) ksλ,~ek 2 ~e∈E(Λ) L (~e) Φ Recall that we already know that for 2π ∈ Q ∩ (0, 1) the operator A(Φ) is injective. This is so, since double hexagonal states span the entire eigenspace ker(HB − λ), hence Z(Φ) = {0} , and thus by the isomorphism (8.21) injectivity of ker(A(Φ)) = {0} follows. To prove Theorem 10 we only need the following Lemmas: 42 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Φ Lemma 8.14. The operator A(Φ) is surjective for 2π ∈ (0, 1). In particular, for any (a(~e)) ∈ l2(E(Λ)), there exists (ψ(~e)) ∈ l2(E(Λ)) such that A(Φ)ψ = a and C kψk 2 ≤ kak 2 , (8.22) l (E(Λ)) |1 − e−iΦ| l (E(Λ)) holds for a universal constant C.

Combining Lemma 8.14 with the already established injectivity result, we have A(Φ) Φ is continuously invertible for 2π ∈ Q ∩ (0, 1) with the following control of its norm C kA(Φ)−1k ≤ . (8.23) |1 − e−iΦ|

Now let us give the proof of Theorem 10, assuming the result of Lemmas 8.14.

Proof of Theorem 10. Since kA(Φ)k is uniformly bounded by a constant and Φ 7→ hx, A(Φ)yi is analytic for x, y ∈ c00(E(Λ)), A(Φ) is an analytic operator in Φ. Thus Φ˜ ˜ ˜ for any 2π ∈ (0, 1), there exists ε1(Φ) and C(Φ) such that ˜ ˜ ˜ ˜ ˜ kA(Φ) − A(Φ)k ≤ C(Φ)|Φ − Φ|, for |Φ − Φ| < ε1(Φ). (8.24)

Φ˜ Φ ˜ ˜ Also by (8.23), for any irrational 2π ∈ (0, 1) and rational 2π with |Φ − Φ| < ε2(Φ), we have C kA(Φ)−1k ≤ . (8.25) 2|1 − e−iΦ˜ |

−iΦ˜ Hence, taking Φ ∈ ∩ (0, 1) with |Φ − Φ˜| < min(ε (Φ)˜ , ε (Φ)˜ , 2|1−e | ), we would get 2π Q 1 2 C(Φ)˜ C kA(Φ)−1(A(Φ)˜ − A(Φ))k < 1. This implies that A(Φ)˜ = A(Φ) Id + A(Φ)−1(A(Φ)˜ − A(Φ))

is invertible. Thus, we conclude that also for irrational ﬂuxes ker(A(Φ)) = {0} and by (8.21) therefore Z(Φ) = {0} which shows the claim.

Proof of Lemma 8.14. We do this by showing that there is a suﬃciently sparse set of elements in l2(E(Λ)) that gets mapped under A(Φ) on the standard basis of l2(E(Λ)).

The functions ϕγ deﬁned in (8.19) satisfy the continuity condition at vertices, from ! ϕγ,~e which we conclude that for A(Φ) 2 the ﬁrst two lines of (8.20) are zero. sλ,e k kL2((0,1)) Similarly, the orthonormality of ϕγ ensures that also the third line of (8.20) vanishes CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 43

( ,g) γ 

( -e , f ) ( ,h) 1   γ γ

( -e ,h) ( -e , f ) 1  2  γ γ

( -e2,g) γ 

Figure 6. Γγ

for γ0 6= γ and is equal to one if γ0 = γ. Thus, for γ0 ∈ Z2, we obtain the following standard basis vectors of l2(E(Λ)) 0 ~ (A(Φ)(α~e,(γ,~h)))(γ , f) := 0 0 (A(Φ)(α~e,(γ,~h)))(γ ,~g) := 0 0 ~ (A(Φ)(α~e,(γ,~h)))(γ , h) := δγ,γ0 . (8.26) 2 ˜f~ ˜~g To obtain also the remaining basis vectors, let us define L functions ψγ and ψγ ˜~e on a single hexagon Γγ. The indices of ψγ are chosen to indicate the standard basis ˜f~ vectors (δ•,(γ,~e)) we shall eventually obtain from those functions. Starting with ψγ , we ~ ~ introduce coefficients (˜cγ,f ) such that ψ˜f~ := P c˜γ,f s . ~e γ ~e∈E(Γγ ) ~e λ,~e ˜f~ We do this in such a way that all continuity conditions for ψγ at the vertices of Γγ are satisfied up to a single one at the (initial) vertex i((γ,~g)) = i((γ,~h)). We define

−iΦγ1 −iΦγ1 γ,f~ 1 γ,f~ −e γ,f~ e c˜ ~ := , c˜ ~ := , c˜(γ−e ,~g) := , (γ,h) 1 − e−iΦ (γ−e2,f) 1 − e−iΦ 2 1 − e−iΦ −iΦ −iΦ −iΦ γ,f~ −e γ,f~ e γ,f~ −e c˜ ~ := , c˜ ~ := , c˜(γ,~g) := (8.27) (γ−e1,h) 1 − e−iΦ (γ−e1,f) 1 − e−iΦ 1 − e−iΦ γ,f~ ˜f~ and all otherc ˜~e are taken to be zero. Since for ψγ all but one continuity conditions are satisﬁed, we obtain for the ﬁrst two components of (8.20)

γ,f~ 0 ~ (A(Φ)(˜c~e ))(γ , f) := δγ,γ0 γ,f~ 0 (A(Φ)(˜c~e ))(γ ,~g) := 0. (8.28) To ensure that we also get constant zero in the third component of (8.20), we project f~ f~ f~ onto the orthogonal complement of the double hexagonal states ψγ := ψeγ −PDHE(Λ)(Φ)ψeγ 44 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

where PDHE(Λ)(Φ) is the orthogonal projection onto DHE(Λ)(Φ). Let now

f~ X ψγ := α~e,(γ,f~)sλ,~e, (8.29) ~e∈E(Γγ ) then, since subtracting linear combinations of double hexagonal states does not aﬀect the continuity conditions, 0 ~ (A(Φ)(α~e,(γ,f~)))(γ , f) := δγ,γ0 , 0 (A(Φ)(α~e,(γ,f~)))(γ ,~g) := 0, and 0 ~ (A(Φ)(α~e,(γ,f~)))(γ , h) := 0. (8.30) Similarly, taking coeﬃcients introducing a discontinuity at the (terminal) vertex ~ ~ t((γ, h)) = t((γ − e2, f)) only

iΦ(γ1−1) γ,~g 1 γ,~g −1 γ,~g e c˜ ~ := , c˜(γ−e ,~g) := , c˜ ~ := (γ−e2,f) 1 − e−iΦ 2 1 − e−iΦ (γ−e1,h) 1 − e−iΦ iΦ(γ1−1) iΦ(γ1−1) iΦ(γ1−1) γ,~g −e γ,~g e γ,~g −e c˜ ~ := , c˜(γ,~g) := , c˜ ~ := (8.31) (γ−e1,f) 1 − e−iΦ 1 − e−iΦ (γ,h) 1 − e−iΦ and all other coeﬃcients being zero, we obtain for the ﬁrst two components of (8.20) γ,~g 0 ~ (A(Φ)(˜c~e )(γ , f) := 0 γ,~g 0 (A(Φ)(˜c~e )(γ ,~g) := δγ,γ0 . (8.32) To ensure that we also get constant zero in the third component of (8.20), we project ~g ~g again on the orthogonal complement of the double hexagonal states ψγ := ψeγ − ~g ~g P PDHE(Λ)(Φ)ψeγ. Let now ψγ = ~e∈E(Γ) α~e,(γ,~g)sλ,~e, then 0 ~ (A(Φ)(α~e,(γ,~g)))(γ , f) := 0 0 (A(Φ)(α~e,(γ,~g)))(γ ,~g) := δγ,γ0 0 ~ (A(Φ)(α~e,(γ,~g)))(γ , h) := 0. (8.33) Hence, what we accomplished in (8.30),(8.33) and (8.26) so far is that we exhibited a set of sequences

n 2o (α~e,(γ,f~)), (α~e,(γ,~g)) and (α~e,(γ,~h)); γ ∈ Z (8.34) in l2(E(Λ)) that gets mapped under A(Φ) onto the standard unit basis of l2(E(Λ)). To conclude surjectivity of A(Φ) from this, it suﬃces to show that (8.34) satisﬁes 2 for all (a~e) ∈ l (E(Λ)) 2

X X 2 a~e α~•,(γ,~g) |a~e| . (8.35) . ~e=(γ,[~e])∈E(Λ) ~e∈E(Λ) l2(E(Λ)) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 45

Φ By the open mapping theorem, equation (8.35) implies then that for 2π ∈ Q ∩ (0, 1) the set (8.34) is a Riesz basis for l2(E(Λ)). f~ ~g ⊥ First, we observe that instead of showing (8.35) it suﬃces, since ψγ , ψγ ∈ DHE(Λ)(Φ) and (ϕγ) forms an orthonormal system of the orthogonal complement DHE(Λ)(Φ), to prove only that

X X 2 a~e α~•,(γ,~g) |a~e| . (8.36) . ~ ~ ~e=(γ,[~e])∈E(Λ);[~e]6=h l2(E(Λ)) ~e∈E(Λ);[~e]6=h

The left hand side of (8.36) reads

X X a a h(α ), (α )i 2 (γ1,δ~1) (γ2,δ~2) ~e,(γ1,δ~1) ~e,(γ2,δ~2) l (E(Λ)) ~ ~ ~ γ ,γ ∈ 2 δ1,δ2∈{f,~g} 1 2 Z

X X γ1,δ~1 γ2,δ~2 = − a a h(˜c ), (˜c )i 2 (γ1,δ~1) (γ2,δ~2) ~e ~e l (E(Λ)) ~ ~ ~ γ ,γ ∈ 2 δ1,δ2∈{f,~g} 1 2 Z | {z } =:τ1

X X γ1,δ~1 γ2,δ~2 + a a h(α − c˜ ), (α − c˜ )i 2 (γ1,δ~1) (γ2,δ~2) ~e,(γ1,δ~1) ~e ~e,(γ2,δ~2) ~e l (E(Λ)) ~ ~ ~ γ ,γ ∈ 2 δ1,δ2∈{f,~g} 1 2 Z | {z } =:τ2

X X γ1,δ~1 γ2,δ~2 + 2 Re a a h(α − c˜ ), c˜ i 2 (8.37) (γ1,δ~1) (γ2,δ~2) ~e,(γ1,δ~1) ~e ~e l (E(Λ)) ~ ~ ~ γ ,γ ∈ 2 δ1,δ2∈{f,~g} 1 2 Z | {z } =:τ3

We will now bound τ1,τ2, and τ3 separately. Let us denote by

N(γ) := {γ, γ + e2, γ + e1, γ + e1 − e2, γ − e2, γ − e2 − e1, γ + e2 − e1} (8.38) 46 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

0 the set of all γ such that Γγ ∩Γγ0 6= ∅. Starting with τ1 and using that the inner-product in τ1 is zero unless γ2 ∈ N(γ1) by the disjoint support

X X γ1,δ~1 γ2,δ~2 |τ1| ≤ a ~ a ~ (˜c ) (˜c ) 1{γ ∈N(γ )} (γ1,δ1) (γ2,δ2) ~e ~e 2 1 2 δ~1,δ~2∈ f,~g~ γ1,γ2∈Z { } | {z7 } ≤ 2 |1−e−iΦ| 7 X X X ≤ a ~ a ~ −iΦ 2 (γ1,δ) (γ2,δ) |1 − e | 2 δ~1,δ~2∈ f,~g~ γ1∈Z γ2∈N(γ1) { } | {z } 2 ≤max~ ~ 0 a 0 ~ δ∈{f,~g},γ ∈N(γ1)| (γ ,δ)| 7 · 72 · 22 X X 2 ≤ a ~ |1 − e−iΦ|2 (γ,δ) ~δ∈{f,~g~ } γ∈Z2

1372 X 2 = |a~e| . (8.39) |1 − e−iΦ|2 ~e∈E(Λ);[e]6=~h

For τ2 we ﬁrst rewrite

a(γ ,δ~ )a(γ ,δ~ ) ~ ~ X X 1 1 2 2 ˜δ1 ˜δ2 τ2 = hPDH (Φ)ψ ,PDH (Φ)ψ i. (8.40) ks k2 E(Λ) γ1 E(Λ) γ2 ~ ~ ~ γ ,γ ∈ 2 λ L2((0,1)) δ1,δ2∈{f,~g} 1 2 Z

We deﬁne DHΓ (Φ) ⊂ DHE(Λ)(Φ) to be the set of (normalized) double hexagonal γ states whose support intersects Γγ. It is clear that DHΓγ (Φ) < ∞ because there are only ﬁnitely many (normalized) double hexagonal states whose support intersects Γγ. ˜~δ However, only their linear span contributes to the projection PDHE(Λ)(Φ)ψγ such that

P ψ˜~δ = P ψ˜~δ . (8.41) DHE(Λ)(Φ) γ span(DHΓγ (Φ)) γ From this, we conclude that the cardinality of

2 DH(γ, Φ) := γ ∈ Z ; DHΓγ (Φ) ∩ DHΓγ (Φ) 6= ∅ (8.42) can also be bounded by a ﬁnite number N. Observe that this bound is uniform in

γ because the vicinity of double hexagonal states is for every γ the same. If γ2 ∈/ DH(γ1, Φ) then this implies by disjointness of supports that

~ ~ ~ ~ ˜δ1 ˜δ2 ˜δ1 ˜δ2 hPDH (Φ)ψ ,PDH (Φ)ψ i = hPspan(DH (Φ))ψ ,Pspan(DH (Φ))ψ i = 0. (8.43) E(Λ) γ1 E(Λ) γ2 Γγ1 γ1 Γγ2 γ2

Thus, the inner-product in (8.40) is non-zero only if γ2 ∈ DH(γ1, Φ). So, we obtain for (8.40) that

a(γ ,δ~ )a(γ ,δ~ ) ~ ~ X X X 1 1 2 2 ˜δ1 ˜δ2 τ2 = hPDH (Φ)ψ ,PDH (Φ)ψ i. (8.44) ks k2 E(Λ) γ1 E(Λ) γ2 ~ ~ ~ γ ∈ 2 γ ∈DH(γ ,Φ) λ L2((0,1)) δ1,δ2∈{f,~g} 1 Z 2 1 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 47

This expression can be estimated in a similar way as |τ1| 2 sup ψ˜~δ γ∈Z2,~δ∈{f,~g~ } γ X X X |τ2| ≤ a ~ a ~ 2 (γ1,δ1) (γ2,δ2) ksλkL2((0,1)) 2 δ~1,δ~2∈ f,~g~ γ1∈Z γ2∈DH(γ1,Φ) { } | {z } 2 ≤max ~ ~ a ~ γ∈DH(γ1,Φ),δ∈{f,~g}| (γ,δ)| 2 7 X X 2 2 ≤ 2 N a ~ |1 − e−iΦ|2 (γ,δ) ~δ∈{f,~g~ } γ∈Z2 2 28N X 2 = |a~e| . (8.45) |1 − e−iΦ|2 ~e∈E(Λ);[e]6=~h

Analogously, a bound for τ3 is found by 2 sup ψ˜~δ γ∈Z2,~δ∈{f,~g~ } γ X X X |τ3| ≤ 2 a ~ a ~ 2 (γ1,δ1) (γ2,δ2) ksλkL2((0,1)) 2 δ~1,δ~2∈ f,~g~ γ1∈Z γ2∈DH(γ1,Φ) { } | {z } 2 ≤max ~ ~ a ~ γ∈DH(γ1,Φ),δ∈{f,~g}| (γ,δ)| 2 14 X X 2 2 ≤ 2 · N a ~ |1 − e−iΦ|2 (γ,δ) ~δ∈{f,~g~ } γ∈Z2 2 56N X 2 ≤ |a~e| . (8.46) |1 − e−iΦ|2 ~e∈E(Λ);[e]6=~h

8.3. Absolutely continuous spectrum for rational ﬂux quanta.

Φ p B Lemma 8.15. For 2π = q ∈ Q, the spectrum of H away from the Dirichlet spectrum is absolutely continuous and has possibly touching, but non-overlapping band structure.

An interval I ⊂ [−1, 1] is a band of QΛ(Φ) if and only if its pre-image under ∆, on each ﬁxed band of the Hill operator, is a band of HB.

Φ D Proof. Note that σp \σ(H ) = ∅ is due to (3) of Lemma 8.2. The absence of singular continuous spectra is clear because HB is invariant under magnetic translations (8.1), some of which commute with one another in the case of rational ﬂux quanta. In B B particular, T0,q and Tq,0 always generate a group of commutative magnetic translations due to (8.4). Thus, standard arguments from Floquet-Bloch theory show that HB has no singular continuous spectrum [GeNi]. Now we will show the non-overlapping band structure. We recall that ΛB, T B, HD B all commute with magnetic translations Tµ (8.8)(8.9)(8.10) where we assume that B µ ∈ qZ. Consequently, Tµ leaves all eigenspaces of those operators invariant. 48 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

D B 2 2 For λ ∈ ρ(H ) we deﬁne discrete magnetic translations τµ : l (V) → l (V),

B B τµ := π(λ)Tµ γ(λ), (8.47)

−1 with γ as in Def. 7.3. Recall now that γ(λ) = (π|ker(T B −λ)) maps given vertex values B B B onto a function in ker(T − λ) with T as in Lemma 7.2. The operator Tµ translates this function and multiplies it by a function taking values on the unit circle of C. Hereupon, π(λ) recovers the translated and phase shifted vertex values. Thus, the deﬁnition of those discrete translation operators is in fact independent of λ ∈ ρ(HD) B∗ B and they form a family of unitary operators with τµ = τ−µ, due to (8.5). B D B Since Λ , H both commute with Tµ , Krein’s formula (7.8) implies that for λ ∈ ρ(HD) ∩ ρ(ΛB),

B −1 ∗ −1 ∗ B Tµ γ(λ)M(λ, Φ) γ(λ) = γ(λ)M(λ, Φ) γ(λ) Tµ . (8.48)

Multiplying with π(λ) and π(λ)∗ from both sides respectively, it follows that

B −1 −1 ∗ B ∗ τµ M(λ, Φ) = M(λ, Φ) γ(λ) Tµ π(λ) −1 B ∗ = M(λ, Φ) π(λ)T−µγ(λ) −1 B = M(λ, Φ) τµ , (8.49) thus M(λ, Φ)−1 commutes with discrete magnetic translations.

To see that ∆ restricts on every Hill band Bn (because ∆|Bn is one-to-one) to an B isomorphism from each band of QΛ(Φ) to a unique band of Λ , 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 B discrete translations Tµ and U the Gelfand transform generated by discrete translations B τµ , i.e. for k in the Brioullin zone

cont X B ihk,µi (U f)(k, x) := Tµ f (x)e and µ∈qZ2 discrete X B ihk,µi (U ξ)(k, δ) := τµ ξ (δ)e . (8.50) µ∈qZ2

Using (8.47) and γ(λ)π(λ) = idV(Λ) we obtain from

B ∗ B ∗ ∗ ∗ B ∗ ∗ ∗ B τµ γ(λ) = τ−µ γ(λ) = γ(λ) Tµ π(λ) γ(λ) = γ(λ) Tµ , (8.51) CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 49

B B ∗ where we used that Tµ preserves ker(T − λ) and γ(λ) |ker(T B−λ)⊥ = 0, on some fun- Φ damental domain WΛ − U contγ(λ)M(λ, Φ)−1γ(λ)∗f (k, x) = −γ(λ)(k) U discreteM(λ, Φ)−1γ(λ)∗f (k, x) = −γ(λ)(k)M(λ, Φ)(k)−1 U discreteγ(λ)∗f (k, x) = −γ(λ)(k)M(λ, Φ)(k)−1γ(λ)(k)∗(U contf)(k, x). (8.52)

2 Φ 2 Φ 2 Φ where γ(λ)(k) ∈ L(l (WΛ ),L (WΛ )) and M(λ, Φ)(k) ∈ L(l (WΛ )) are the restrictions 2 Φ B of γ(λ) and M(λ, Φ) on l (WΛ ) satisfying Floquet boundary conditions. Both Λ and D B cont H commute with translations Tµ , and thus fiber upon conjugation by U , so that for λ ∈ ρ(ΛB(k)) ∩ ρ(HD) Krein’s formula remains true for each k B −1 D −1 −1 ∗ (Λ (k) − λ) − (H | 2 Φ − λ) = −γ(λ)(k)M(λ, Φ)(k) γ(λ)(k) . (8.53) L (WΛ ) In particular, for λ ∈ ρ(HD) γ(λ)(k) ker(M(λ, Φ)(k)) = ker(ΛB(k) − λ). (8.54) B Therefore bands of QΛ(Φ) are in one-to-one correspondence with bands of Λ , and B thus also with bands of H . That the bands of QΛ(Φ) do not overlap is shown in B Section 6 of [HKL]. Thus, the unique correspondence among bands of QΛ(Φ) and H B shows that the non-overlapping of bands holds true for H as well. Φ 1 Remark 11. For 2π = 2 the spectral bands of QΛ(Φ) are touching and given by [HKL] " # " # " # " # r2 r1 r1 r1 r1 r2 − , − , − , 0 , 0, , and , . (8.55) 3 3 3 3 3 3 Thus, by Lemma 8.15 the bands of HB on each Hill band are touching as well, see Fig. 7. Bands belonging to different Hill bands do, as a rule for Φ ∈ (0, 2π), not touch by Lemma 8.1. Φ 1 In the case of 2π = 3 however, only the bands at the Dirac points touch, see also Fig. 8. The touching at the Dirac points is always satisfied by Lemma 8.2. Φ p Remark 12. The spectrum of QΛ(Φ) for rational 2π = q are precisely the eigenvalues [HKL] of ik ik 0 id q +e 1 J + e 2 K C p,q q (8.56) −ik1 ∗ −ik2 ∗ idCq +e Jp,q + e Kq 0 p ∗ 2πi(m−1) q for k ∈ T2 where Jp,q := δm,ne and (Kq)mn := 1 if n = (m + 1) mod q mn and 0 otherwise. Remark 13. Similar to the Hofstadter butterflies for discrete tight binding operators, the explicit spectrum for rational flux quanta allows us to plot the spectrum of HB for different rational flux quanta in Fig. ??. 50 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

Φ 1 2 Figure 7. Touching bands for 2π = 2 on the ﬁrst Hill band B1 = [0, π ] with V = 0.

Figure 8. Only the third and fourth band touch at the Dirac points Φ 1 2 for 2π = 3 on the ﬁrst Hill band B1 = [0, π ] with V = 0. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 51

8.4. Singular continuous Cantor spectrum for irrational ﬂux quanta.

Φ Proof. By Lemma 4.2, the spectrum of QΛ(Φ) for irrational 2π is a Cantor set of mea-

sure zero. Thus, the pullback of σ(QΛ) by ∆|int(Bn) is still a Cantor set of zero measure that coincides with σ(HB)\σ(HD). Therefore, the absolutely continuous spectrum of B H has to be empty. (2) of Theorem1 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 [AJM] for a more detailed discussion. Let Dλ be deﬁned as in (2.47), in which v(θ) = 2 cos 2πθ and c(θ) = 1+e−2πiθ, hence

2πiθ −2πiθ 2πi(θ− Φ ) λ − e − e −1 − e 2π Dλ(θ) = . (A.1) 1 + e−2πiθ 0

λ Let us complexify θ and define Dε for ε ∈ R as follows λ λ Dε (θ) := D (θ + iε). (A.2) Let Z 0 λ 1 Y λ Φ L(Dε , Φ) := lim log k Dε (θ + j )k dθ, (A.3) n→∞ n 2π T1 j=n−1 be the complexified Lyapunov exponent. By Hardy’s convexity theorem, see e.g. The- λ orem 1.6 in [Du], L(Dε , Φ) is convex in ε. Let 1 L(Dλ , Φ) − L(Dλ, Φ) ω(λ, Φ; ε) := lim ε+h ε (A.4) 2π h→0+ h be the right-derivative of the complexified Lyapunov exponent, which has been dubbed acceleration in [A]. By Theorem 1 of [JM2], since det(Dλ(θ + iε)) 6= 0 for ε 6= 0, we have ω(λ, Φ; ε) ∈ Z, for ε 6= 0. (A.5) This is usually referred to as quantization of acceleration. One can also easily compute the following asymptotic behaviour −e2πε 0 Dλ(θ) = + O(1), ε → ∞ ε e2πε 0 (A.6) −e−2πε − e−iΦe−2πε Dλ(θ) = + O(1), ε → −∞, ε 0 0 52 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

hence by (A.5), ( L(Dλ, Φ) = ε, ε > ε > 0, ε 0 (A.7) λ L(Dε , Φ) = −ε, ε < −ε0.

λ Hence convexity of L(Dε , Φ) and quantization of acceleration force either λ • L(D0 , Φ) = 0 or λ • L(D0 , Φ) > 0 with ω(0, Φ; ε) = 0. Φ λ By Theorem 1.2 of [AJS], the second case is equivalent to ( 2π ,D0 ) inducing a dominated splitting. This is equivalent to λ∈ / ΣΦ, by [Ma]. Finally note that we always have Z λ −2πiθ λ L(λ, Φ) = L(D0 , Φ) − log |1 + e | dθ = L(D0 , Φ). (A.8) T1 Hence L(λ, Φ) = 0 if and only if λ ∈ ΣΦ.

Appendix B. Proof of Lemma 6.7

1 p Assume θ = 2 − k0 q . Let (H2πp/q,θ)|[0,k−1] be the restriction of H2πp/q,θ onto interval [0, k − 1] with Dirichlet boundary condition. Let Pk(θ) = det (λ − (H2πp/q,θ)|[0,k−1]) be the determinant of this k × k matrix. One can prove by induction (in k) that the following holds

p p ! λ Pk(θ) c(θ − q )Pk−1(θ + q ) Dk (θ) = p p p p . (B.1) c(θ + (k − 1) q )Pk−1(θ) −c(θ − q )c(θ + (k − 1) q )Pk−2(θ + q ) Thus p p tr(Dλ(θ)) = P (θ) + |c(θ − )|2P (θ + ). (B.2) q q q q−2 q It then suﬃces to note that  tr(Dλ(θ − (k − 1) p )) = tr(Dλ(θ)),  q 0 q q  p c(θ − k0 q ) = 0, (B.3)  (H 2πp p )|[0,q−1] = Mq. q ,θ−(k0−1) q )

Appendix C. 1/2-Holder¨ continuity of spectra of Jacobi matrices

Proof of Lemma 6.2. We will prove the following general result for quasi-periodic 2 Jacobi matrices. Let Hα,θ ∈ L(l (Z)) be deﬁned as

(Hα,θu)n = c(θ + nα)un+1 + c(θ + (n − 1)α)un−1 + v(θ + nα)un. (C.1)

Let σα := ∪θ∈T1 σ(Hα,θ). CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 53

1 ˜ Lemma C.1. Let c(·), v(·) ∈ C (T1, C). There exist constants C(c, v),C(c, v) > 0 0 0 ˜ 0 such that if λ ∈ σα and α ∈ T1 is such that |α − α | < C(c, v), then there is a λ ∈ σα0 such that 0 0 1 |λ − λ | < C(c, v)|α − α | 2 .

Lemma 6.2 follows from Lemma C.1 by taking Φ = 2πα and Φ0 = 2πα0. Lemma C.1 is in turn the argument of [AMS] adapted to the Jacobi setting.

2 Proof of Lemma C.1. Let L ≥ 1 be given. There exists φL ∈ l (Z) and θ such that 1 k(H − λ)φ k ≤ kφ k. (C.2) α,θ L L L Let ηj,L be the test function centered at j, ( (1 − |n − j|/L), |n − j| ≤ L, ηj,L(n) = . 0, |n − j| ≥ L Then for large L, X (L − 1)(2L − 1) (η (n))2 = 1 + ≡ a . (C.3) j,L 3L L j is independent of n. Clearly,

X aL 1 X kη (H − λ)φ k2 = a k(H − λ)φ k2 ≤ kφ k2 = kη φ k2. (C.4) j,L α,θ L L α,θ L L2 L L2 j,L L j j Since ku + vk2 ≤ 2kvk2 + 2kuk2, by (C.4), we get

X 2 X 2 X 2 k(Hα,θ − λ)ηj,LφLk ≤2 kηj,L(Hα,θ − λ)φLk + 2 k[ηj,L,Hα,θ]φLk j j j 2 X X ≤ kη φ k2 + 2 k[η ,H ]φ k2, (C.5) L2 j,L L j,L α,θ L j j where [ηj,L,Hα,θ] = ηj,LHα,θ − Hα,θηj,L is the commutator. Note that

([ηj,l,Hα,θ]φ)n = c(θ + nα)(ηj,L(n)−ηj,L(n + 1))φn+1

+c(θ + (n − 1)α)(ηj,L(n) − ηj,L(n − 1))φn−1, which implies X 8kck2 8kck2 X k[η ,H ]φ k2 ≤ ∞ kφ k2 ≤ ∞ kη φ k2. j,L α,θ L L L La j,L L j L j 2 Combining this with (C.5) and taking into account that aL ∼ 3 L, we get X 2 + 25kck2 X k(H − λ)η φ k2 ≤ ∞ kη φ k2, α,θ j,L L L2 j,L L j j 54 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

for L > L0. Hence for certain j, ηj,LφL 6= 0 and 2 1 (2 + 25kck ) 2 k(H − λ)η φ k ≤ ∞ kη φ k2. (C.6) α,θ j,L L L j,L L Given α0 near α, choose θ0 such that θ + jα = θ0 + jα0.

Then on supp(ηj,Lφε), 0 0 0 0 |f(θ + nα) − f(θ + nα )| ≤ Lkf k∞|α − α |, (C.7) holds for f = c, v. Thus, by (C.6) and (C.7),

k(Hα0,θ0 − λ)ηj,LφLk ≤ C1(c, v)kηj,LφLk, where 2 1 (2 + 25kck∞) 2 0 2 0 2 1 0 C (c, v) = + (6kc k + 3kv k ) 2 L|α − α |. 1 L ∞ ∞ Finally, taking 0 − 1 L = C2(c, v)|α − α | 2 > L0, we get 0 1 k(Hα0,θ0 − λ)ηj,LφLk ≤ C(c, v)|α − α | 2 kηj,LφLk.

References [AEG] Andrea Agazzi, J-P. Eckmann, and Gian Michele Graf, The colored Hofstadter butterfly for the Honeycomb lattice , Journal of Statistical Physics 156.3, 2014. [AW] Michael Aizenman and Simone Warzel, Resonant delocalization for random Schrdinger operators on tree graphs. J. Eur. Math. Soc. 15 , 1167-1222 (2013). [A] Artur Avila, Global theory of one-frequency Schrödingeroperators, Acta Math. 215, 1–54 (2015). [A1] Artur Avila, On point spectrum with critical coupling Preprint available on www.impa.br/ avila/. [BGP] Jochen Brüning,Vladimir Geyler, and Konstantin Pankrashkin, Cantor and band spectra for periodic quantum graphs with magnetic fields, Communications in mathematical physics, 269(1), 87-105, 2007. [GeNi] Christian Gérardand Francis Nier, The Mourre Theory for Analytically Fibered Operators, Journal of Functional Analysis, 1998. [HKL] Bernard Helffer, Philippe Kerdelhué,and Jimena Royo-Letelier Chambers’s formula for the graphene and the Hou model, Annales Henri Poincaré,Volume 17, Issue 4, pp 795818, 2016. with kagome periodicity and applications. Ann. Henri Poincaré17: 795. [HS1] Bernard Helffer and Johannes Sjöstrand. Equation de Schrödingeravec champ magnétiqueet équationde Harper, Schrödingeroperators. Springer Berlin Heidelberg, 118-197, 1989. [K] Peter Kuchment, An overview of periodic elliptic operators, Bulletin of the American Mathemat- ical Society, 53(3), pp.343-414, 2016. [Ka] Tosio Kato, Perturbation Theory for Linear Operators, Springer, 1995. CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 55

[K1] Peter Kuchment, Quantum graphs: I. Some basic structures, Waves in Random media, 14(1), pp.107-128, 2004. [K2] Peter Kuchment, Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs, Journal of Physics A: Mathematical and General 38.22, 2005. [KP] Peter Kuchment and Olaf Post, On the spectra of carbon nano-structures, Communications in Mathematical Physics, 275(3), pp.805-82, 2007. [KL] Philippe Kerdelhué, and Jimena Royo-Letelier. On the low lying spectrum of the magnetic Schrödingeroperator with kagome periodicity, Reviews in Mathematical Physics 26(10). 2014. [L] Yoram Last. Zero measure spectrum for the almost Mathieu operator. Comm. Math. Phys. 164, 421– 432 (1994). [P] Konstantin Pankrashkin Spectra of Schrödinger operators on equilateral quantum graphs, Letters in Mathematical Physics, 77(2), 139-154, 2006. [Pu] Joaquim Puig, Cantor spectrum for the almost Mathieu operator. Comm. Math. Phys. 244 (2004), no. 2, 297–309. [RS4] Michael Reed and Barry Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathe- matical Physics, Elsevier, 1978. [Sch] Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, Springer, 2012 [T] Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Com- munications in Mathematical Physics, 33(4), pp.335-343, 1973. [AS] Joseph Avron and Barry Simon. The asymptotics of the gap in the Mathieu equation. Annals of Physics 134.1. 76-84. 1981. [Az] Mark Azbel. Energy spectrum of a conduction electron in a magnetic field. Sov. Phys. JETP 19.3. 634-645. 1964. [BGP] Jochen Brüning,Vladimir Geyler, and Konstantin Pankrashkin, Cantor and band spectra for periodic quantum graphs with magnetic fields, Communications in mathematical physics, 269(1), 87-105, 2007. [Ch] Xi Chen et al. Dirac edges of fractal magnetic minibands in graphene with hexagonal moir superlattices. Physical Review B 89.7: 075401. 2014. [De] Cory R. Dean et al. Hofstadter’s butterfly in moire superlattices: A fractal quantum Hall effect. Nature 497.745: 598. 2013. [EV] Pavel Exner and Daniel Vaata. ”Cantor spectra of magnetic chain graphs.” Journal of Physics A: Mathematical and Theoretical 50.16: 165201. 2017. [FW] Charles Fefferman and Michael Weinstein. Honeycomb lattice potentials and Dirac points. Jour- nal of the American Mathematical Society 25.4. 1169-1220. 2012. [Ga] Luis Gaggero et al. Self-similar conductance patterns in graphene Cantor-like structures. Scien- tific Reports 7.1. 617. 2017. [Go] Kenjiro K. Gomes et al. Designer Dirac fermions and topological phases in molecular graphene. Nature 483.7389. 2012. [Gor] Roman Gorbachev et al. Cloning of Dirac fermions in graphene superlattices. Nature. 2013. [BHJZ] Simon Becker, Rui Han, Svetlana Jitomirskaya, Maciej Zworsky, In preparation. [Gu] Francisco Guinea et al. Strain-induced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles. Science 329.5991. 544-547. 2010. [HKL] Bernard Helffer, Philippe Kerdelhué,and Jimena Royo-Letelier Chambers’s formula for the graphene and the Hou model, Annales Henri Poincaré,Volume 17, Issue 4, pp 795?818, 2016. 56 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA

[Ho] Douglas R. Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14.6. 1976. [1] Vojkan Jaksic and Yoram Last: Surface states and spectra. Comm. Math. Phys. 218, 459–477 (2001). [KP] Peter Kuchment and Olaf Post, On the spectra of carbon nano-structures. Communications in Mathematical Physics, 275(3), pp.805-82, 2007. [KS] Vadim Kostrykin and Robert Schrader. Quantum wires with magnetic fluxes. Communications in mathematical physics 237.1. 161-179. 2003. [N] Konstantin Novoselov, Nobel lecture: Graphene: Materials in the flatland, Reviews of Modern Physics, 837849 (2011). [P] Konstantin Pankrashkin Spectra of Schrödinger operators on equilateral quantum graphs, Letters in Mathematical Physics, 77(2), 139-154, 2006. [T] Gerald Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices Mathematical Sur- veys and Monographs, Volume 72, Amer. Math. Soc., Providence 2000. [Zak] Paul Zak Magnetic translation group. Physical Review 134.6A (1964): A1602. [Zh] Yuanbo Zhang et al. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438.7065. 201-204. 2005. [AJ] Artur Avila and Svetlana Jitomirskaya The Ten Martini Problem. Annals of Mathematics, 170(1) ,303-342, 2009. [AJ2] Artur Avila and Svetlana Jitomirskaya Almost localization and almost reducibility. Journal of the European Mathematical Society, 12(1), 93-131, 2010. [AJM] Artur Avila, Svetlana Jitomirskaya and Christoph Marx Spectral theory of extended Harper’s model and a question by Erd˝osand Szekeres. Inventiones mathematicae. 210 (2017), no. 1, 283339 [H1] Rui Han Dry Ten Martini problem for the non-self-dual extended Harper’s model. Transactions of the American Mathematical Society, in press. [H2] Rui Han Absence of point spectrum for the self-dual extended Harper’s model International Mathematics Research Notices, in press. [No] Konstantin Novoselov, 2011. Nobel lecture: Graphene: Materials in the flatland, Reviews of Modern Physics, pp. 837-849. [AMS] Joseph Avron, Pierre v Mouche and Barry Simon On the measure of the spectrum for the almost Mathieu operator. Comm. Math. Phys. 132 (1990), no. 1, 103-118. [D] J. Dombrowsky, Quasitriangular matrices, Proc. Amer. Math. Soc. 69, 95 – 96 (1978). [JM1] Svetlana Jitomirskaya and Christoph Marx Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Comm. Math. Phys. 316 (2012), no. 1, 237-267. [JM2] Svetlana Jitomirskaya and Christoph Marx Erratum to: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Comm. Math. Phys. 317, (2013), 269-271. [Du] Peter L. Duren, Theory of Hp spaces. Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970. [AJS] Artur Avila, Svetlana Jitomirskaya and Christian Sadel Complex one-frequency cocycles. Jour- nal of the European Mathematical Society, (2014), 16(9), 1915-1935. [Ma] Christoph Marx Dominated splittings and the spectrum of almost periodic Jacobi operators. Nonlinearity, 27, (2014), 3059-3072.

E-mail address: [email protected] CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 57

Department of Mathematics, University of California, Berkeley, CA 94720, USA

E-mail address: [email protected]

Department of Mathematics, University of California, Irvine

E-mail address: [email protected]

Department of Mathematics, University of California, Irvine