CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS

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1 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 compactness not required). Let σ Φ, σcont, Φ σess Φ be the (continuous, essential) spectra of H B, the Hamiltonian of the quantum graph graphene in a magnetic field with constant flux Φ, as defined in (3.7) (3.8) (3.1) (3.2), with some Kato-Rellich potential V e L 2 ( e). Let H D be the Dirichlet operator (no magnetic field) defined in (2.15) (2.12), and σ(h D ) its spectrum. Let σp Φ be the collection of eigenvalues of H B. Then we have the following description of the topological structure and point/continuous decomposition of the spectrum Theorem 1. For any symmetric Kato-Rellich potential V e L 2 ( e) we have (1) σ Φ = σ Φ ess, (2) σ Φ p = σ(h D ), 1

2 2 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA (3) σcont Φ is a Cantor set of measure zero for Φ R \ Q, 2π a countable union of disjoint intervals for Φ Q, 2π (4) σp Φ σcont Φ = for Φ / Z, 2π (5) the Hausdorff dimension dim H (σ Φ ) 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 H B. Theorem 2. For any symmetric Kato-Rellich potential V e L 2 ( e) we have (1) For Φ 2π R \ Q the spectrum on σφ cont is purely singular continuous. (2) For Φ 2π Q, the spectrum on σφ cont 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 unphysical, there are reasons to expect that continuous spectrum of the quantum graph operator described in this paper does adequately capture the experimental properties of graphene in the magnetic field. Finally, our analysis provides full description of the spectrum of the tight-binding Hamiltonian as well. Moreover, the applicability of our model is certainly not limited to graphene. Many atoms and even particles confined to the same lattice structure show similar physical properties [Go] that are described well by this model (compare Figure 2 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

3 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 Theorem 1 is established. 1 Our method applies to also prove zero measure Cantor spectrum of the extended Harper s model whenever the corresponding Jacobi matrix is singular. As mentioned, our first step is a reduction to a matrix-valued tight-binding hexagonal model. This leads to an operator Q Λ defined in (4.1). This operator has been studied before for the case of rational magnetic flux (see [HKL] and references therein). Our analysis gives complete spectral description for this operator as well. Theorem 3. The spectrum of Q Λ (Φ) is a finite union of intervals and purely absolutely continuous for Φ 2π = p q Q, with the following measure estimate σ(q Λ (Φ)) C q, singular continuous and a zero measure Cantor set for Φ 2π R \ Q, a set of Hausdorff dimension dim H (σ(q Λ (Φ))) 1/2 for generic Φ. Remark 1. We will show that the constant C in the first item can be bounded by 8 6π 9. The theory of magnetic Schrödinger operators on graphs can be found in [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 1 We note that singular continuity of the spectrum of critical extended Harper s model (including for parameters leading to singularity) has been proved recently in [AJM, H2] without establishing the Cantor nature.

4 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 Schrödinger operators on the graph and of the tight-binding operators for rational flux quanta that only relies on Krein s theory. In particular, the bands of the graph model always touch at the Dirac points and are shown to have open gaps at the band edges of the associated Hill operator if the magnetic flux is non-trivial. This way, the conical Dirac points are preserved in rational magnetic fields. We obtain the preceding results by first proving a bound on the operator norm of the tight-binding operator and analytic perturbation theory. In [KP] it was shown that the Dirichlet contribution to the spectrum in the nonmagnetic case is generated by compactly supported eigenfunctions and that this is the only contribution to the point spectrum of the Schrödinger operator on the graph. We extend this result to magnetic Schrödinger operators on hexagonal graphs. Let H pp be the pure point subspace accociated with H B. Then Theorem 4. For any Φ, H pp 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 l 1 (in a suitable sense) eigenfunctions are in the closure of the span of double hexagonal states, the l 2 condition is more elusive. Bridging the gap between l 1 and l 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 l 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. Section 2 serves as background. In Section 3, we introduce the magnetic Schrödinger operator H B and (modified) Peierls substitution, which enables us to reduce H B to a non-magnetic Schrödinger operator Λ B with magnetic contributions moved into the boundary conditions. In Section 4, we present several key ingredients of the proofs of the main theorems: Lemmas 4.1 and Lemma 4.1 involves a further reduction from Λ B to a two-dimensional tightbinding Hamiltonian Q Λ (Φ), and Lemmas reveal the topological structure of

5 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 5 σ(q Λ (Φ)) (thus proving the topological part of Theorem 3). The proofs of Lemmas 4.1, 4.2, 4.3 and 4.4 will be given is Sections 7, 5 and 6 respectively. Section 8 is devoted to a complete spectral analysis of H B, thus proving Theorem 1, with the analysis of Dirichlet spectrum in Section 8.2, where in particular we prove Theorem 4; absolutely continuous spectrum for rational flux in Section 8.3, singular continuous spectrum for irrational flux in Section 8.4 (thus proving Theorem 2). 2. Preliminaries Given a graph G, we denote the set of edges of G by E(G), the set of vertices by V(G), and the set of edges adjacent to a vertex v V(G) by E v (G). 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. Define c 00, Γ(Λ 1 (R 2 )) and Γ(Λ 2 (R 2 )). For a set U R, let U be its Lebesgue measure 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 Section 3. The effective one electron behavior in graphene can be described by a hexagonal graph with Schrödinger operators defined on each edge [KP]. The hexagonal graph Λ is obtained by translating its fundamental cell W Λ shown in Figure 1, consisting of vertices ( ) 1 3 r 0 := (0, 0) and r 1 := 2, (2.1) 2 and edges f := conv ({r 0, r 1 }) \ {r 0, r 1 }, g := conv ({r 0, ( 1, 0)}) \{r 0, ( 1, 0)}, and ( )}) { ( )} 1 3 h := conv ({r 0, 2, 1 3 \ r 0, 2 2,, (2.2) 2 along the basis vectors of the lattice. The basis vectors are ( ) 3 3 b1 := 2, and 2 ( b 2 := 0, ) 3 (2.3)

6 6 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA f r 1 g r 0 h b 2 b 1 Figure 1. The fundamental cell and lattice basis vectors of Λ. and so the hexagonal graph Λ R 2 is given by the range of a Z 2 -action on the fundamental domain W Λ } Λ := {x R 2 : x = γ 1 b1 + γ 2 b2 + y for γ Z 2 and y W Λ. (2.4) The fundamental domain of the dual lattice can be identified with the dual 2-torus where the dual tori are defined as T k := R k /(2πZ) k. (2.5) For any vertex v V(Λ), we denote by [v] V(W Λ ) the unique vertex, r 0 or r 1, for which there is γ Z 2 such that v = γ 1 b1 + γ 2 b2 + [v]. (2.6) We will sometimes denote v by (γ 1, γ 2, [v]) to emphasize the location of v. We also introduce a similar notation for edges. For an edge e E(Λ), we will sometimes denote it by γ 1, γ 2, [e]. Finally, for any x Λ, we will also denote its unique preimage in W Λ by [x] 2. We can orient the edges in terms of initial and terminal maps 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 W Λ to obtain an 2 so that y in (2.4)=[x]

7 oriented graph Λ CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 7 i( f) = i( g) = i( h) = r 0, t( f) = r 1, t( g) = r 1 b 1, and t( h) = r 1 b 2. (2.8) For arbitrary e E(Λ), we then just extend those maps by i( e) := γ 1 b1 + γ 2 b2 + i( [e]) and t( e) := γ 1 b1 + γ 2 b2 + 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)) x (2.10) that allows us to define function spaces and operators on e and finally on the entire graph. For n N 0, the Sobolev space H n (PE (Λ)) on Λ is the Hilbert space direct sum H n (PE (Λ)) := H n ( e). (2.11) e E(Λ) On every edge e E(Λ) we define the maximal Schrödinger operator H e : H 2 ( e) L 2 ( e) L 2 ( e) H e ψ e := ψ e + V e ψ e (2.12) with Kato-Rellich potential V e L 2 ( e) that is the same on every edge and even with respect to the center of the edge. Let Then V (t) = V e ((κ e ) 1 (t)). (2.13) V (t) = V (1 t). (2.14) One self-adjoint restriction of (2.12) is the Dirichlet operator H D := ( H 1 0 ( e) H 2 ( e) ) L 2 (PE (Λ)) L 2 (PE (Λ)) e E(Λ) (H D ψ) e := H e ψ e, (2.15)

8 8 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA where H0( e) 1 is the closure of compactly supported smooth functions in H 1 ( e). The Hamiltonian we will use to model the graphene without magnetic fields is the selfadjoint [K2] operator H on Λ with Neumann type boundary conditions { D(H) := ψ = (ψ e ) H 2 (PE(Λ)) : for all v V(Λ), ψ e1 (v) = ψ e2 (v) if e 1, e 2 E v (Λ) and } ψ e(v) = 0 (2.16) and defined by e E v(λ) H : D(H) L 2 (PE(Λ)) L 2 (PE(Λ)) (Hψ) e := (H e ψ e ). (2.17) Remark 2. The self-adjointness of H will also follow from the self-adjointness of the more general family of magnetic Schrödinger operators that is obtained in Sec. 7. Remark 3. The orientation is chosen so that all edges at any vertex are either all incoming or outgoing. Thus, there is no need to distinguish those situations in terms of a directional derivative in the boundary conditions (2.16) Floquet-Bloch decomposition. Operator H commutes with the standard lattice translations T st γ :L 2 (PE(Λ)) L 2 (PE(Λ)) f f( γ 1 b1 γ 2 b2 ) (2.18) for any γ Z 2. In terms of those, we define the Floquet-Bloch transform for x PE(W Λ ) and k T 2 first on function f C c (PE(Λ)) (Uf)(k, x) := γ Z 2 (T st γ f)(x)e i k,γ (2.19) and then extend it to a unitary map U L(L 2 (PE(Λ)), L 2 (T 2 PE(W Λ )) with inverse (U 1 ϕ)(x) = ϕ(k, [x])e i γ,k dk (2π), (2.20) 2 T 2 where [x] PE (W Λ ) is the unique pre-image of x in W Λ, and γ Z 2 is defined by x = γ 1 b1 + γ 2 b2 + [x]. Then standard Floquet-Bloch theory implies that there is a direct integral representation of H UHU 1 = H k dk (2.21) (2π) 2 T 2

9 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 9 in terms of self-adjoint operators H k H k : D(H k ) L 2 (PE (W Λ )) L 2 (PE (W Λ )) (H k ψ) e := (H e ψ e ) (2.22) on the fundamental domain W Λ with Floquet boundary conditions { D(H k ) := ψ H 2 (PE (W Λ )) : ψ f (r 0 ) = ψ g (r 0 ) = ψ h (r 0 ) and e E r0 (Λ) ψ e(r 0 ) = 0, as well as ψ f (r 1 ) = e ik 1 ψ g (r 1 b 1 ) = e ik 2 ψ h (r 1 b 2 ) } and ψ f (r 1 ) + e ik 1 ψ g(r 1 b 1 ) + e ik 2 ψ h (r 1 b 2 ) = 0. (2.23) Fix an edge e E(Λ) and λ / σ(h D ). There are linearly independent H 2 ( e)- solutions ψ λ,1, e and ψ λ,2, e to the equation H e ψ e = λψ e with the following boundary condition ψ λ,1, e (i( e)) = 1, ψ λ,1, e (t( e)) = 0, ψ λ,2, e (i( e)) = 0, and ψ λ,2, e (t( e)) = 1. (2.24) Any eigenfunction to operators H k, with eigenvalues away from σ(h D ), can therefore be written in terms of those functions for constants a, b C a ψ λ,1, f + b ψ λ,2, f along edge f ψ := a ψ λ,1, g + e ik 1 b ψ λ,2, g along edge g (2.25) a ψ λ,1, h + e ik 2 b ψ λ,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 H k iff 1 + e ik 1 η(λ) 2 + e ik 2 2 = (2.26) 9 with η(λ) := ψ λ,2, e (t( e)) well-defined away from the Dirichlet spectrum. ψ λ,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 σ(h)\σ(h D ) = {λ R : η(λ) 1} \σ(h D ). (2.27)

10 10 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Dirichlet-to-Neumann map. Fix an edge e E(Λ). Let c λ, e, s λ, e be solutions to H e ψ e = λψ e with the following boundary condition ( ) ( ) cλ, e (i( e)) s λ, e (i( e)) 1 0 c λ, e (i( e)) s λ, e (i( e)) =. (2.28) 0 1 We point out that c λ (t) := c λ, e (κ 1 e (t)) and s λ(t) := s λ, e (κ 1 e (t)) are independent of e. They are clearly solutions to ψ + V (t)ψ = λψ on (0, 1), with c λ (0) = 1, c λ (0) = 0, s λ (0) = 0, s λ (0) = 1, where V (t) is defined in (2.13). Then for λ / σ(h D ), namely when s λ (1) 0, any H 2 ( e)-solution ψ λ, e can be written as a linear combination of c λ, e, s λ, e ψ λ, e (x) = ψ λ, e(t( e)) ψ λ, e (i( e))c λ (1) s λ, e (x) + ψ λ, e (i( e))c λ, e (x). (2.29) s λ (1) The Dirichlet-to-Neumann map is defined by m(λ) := 1 s λ (1) ( cλ (1) 1 1 s λ (1) ) (2.30) with the property that for ψ λ, e as in (2.29) ( ) ( ) ψ λ, e (i( e)) ψλ, e (i( e)) ψ λ, e (t( e)) = m(λ). (2.31) ψ λ, e (t( e)) For the second component, the constancy of the Wronskian is used. assumed to be even, the intuitive relation remains also true for non-zero potentials. Since V (t) is c λ (1) = s λ(1) (2.32) For λ / σ(h D ), by expressing c λ (1) in terms of ψ λ,1, e and ψ λ,2, e, it follows immediately that η(λ) = s λ(1). (2.33) Relation to Hill operators. Using the potential V (t) (2.13), we define the Z- periodic Hill potential V Hill L 2 loc (R). V Hill (t) := V (t (mod 1)), (2.34) for t R. The associated self-adjoint Hill operator on the real line is given by H Hill : H 2 (R) L 2 (R) L 2 (R) H Hill ψ := ψ + V Hill ψ. (2.35) Then c λ, s λ H 2 (0, 1), extending naturally to Hloc 2 (R), become solutions to H Hill ψ = λψ. (2.36)

11 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS Density of states Energy Figure 2. The density of states for the free Schrödinger operator without magnetic fields on the first Hill band [0, π 2 ]. The monodromy matrix associated with H Hill is the matrix valued function ( ) cλ (1) s Q(λ) := λ (1) c λ (1) s λ (1) and depends by standard ODE theory holomorphically on λ. Its normalized trace (2.37) (λ) := tr(q(λ)) = s 2 λ(1) (2.38) 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 satisfies σ(h Hill ) = {λ R : (λ) 1} = [α n, β n ] (2.39) where B n := [α n, β n ] denotes the n-th Hill band with β n α n+1. We have int(b n) (λ) 0. Putting (2.33) and (2.38) together, we get the following relation n=1 (λ) = η(λ), for λ / σ(h D ), (2.40) that connects the Hill spectrum with the spectrum of the graphene Hamiltonian.

12 12 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Also, if λ σ(h D ), then by the symmetry of the potential, the Dirichlet eigenfunction are either even or odd with respect to 1. Thus, Dirichlet eigenvalues can only be 2 located at the edges of the Hill bands. Namely, (λ) = ±1, for λ σ(h D ). (2.41) 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 T 2; λ j (k) = λ} has positive measure where λ j (k) is the j-th eigenvalue of H k. Away from the Dirichlet spectrum, the condition R λ = λ j (k) is by (2.26) equivalent to (λ) 2 = 1+eik 1 +e ik 2 2. Yet, 9 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(h Hill ) coincide as sets. Aside from the Dirichlet 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 One-dimensional quasi-periodic Jacobi matrices. The proof of the main Theorems will involve the study of a one-dimensional quasiperiodic Jacobi matrix. We include several general facts that will be useful. Let H Φ,θ L(l 2 (Z)) be a quasi-periodic Jacobi matrix, that is given by ( (H Φ,θ u) m = c θ + m Φ ) ( u m+1 +c θ + (m 1) Φ ) ( u m 1 +v θ + m Φ ) u m. (2.42) 2π 2π 2π Let Σ Φ,θ be the spectrum of H Φ,θ, and Σ Φ = θ T 1 Σ Φ,θ. It is a well known result Φ that for irrational, the set Σ 2π Φ,θ is independent of θ, thus Σ Φ,θ = σ Φ. It is also well known that, for any Φ, Σ Φ has no isolated points Transfer matrix and Lyapunov exponent. We assume that c(θ) has finitely many { zeros (counting multiplicity), and label them as θ 1, θ 2,..., θ m. 3 Let Θ := m j=1 k Z θj + k 2π} Φ, in particular if Φ Q, then Θ is a 2π finite set. 3 In our concrete model, c(θ) = 1 + e 2πiθ, see (5.4), hence has a single zero θ 1 = 1/2.

13 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS Energy value Dirac point -3.5 Dirac point Figure 3. The first two bands of the free Schrödinger operator without magnetic perturbation showing the characteristic conical Dirac points at energy level π2 where the two bands touch. 4 For θ / Θ, the eigenvalue equation H Φ,θ u = λu has the following dynamical reformulation: ( ) ( un+1 = A λ θ + n Φ ) ( ) un, u n 2π u n 1 where ( ) GL(2, C) A λ (θ) = 1 λ v(θ) c(θ Φ ) 2π c(θ) c(θ) 0 is called the transfer matrix. Let be the n-step transfer matrix. A λ n(θ) = A λ (θ + (n 1) Φ 2π ) Aλ (θ + Φ 2π )Aλ (θ) We define the Lyapunov exponent of H Φ,θ at energy λ as 1 L(λ, Φ) := lim log A λ n n n(θ) dθ. (2.43) T 1

14 14 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA By a trivial bound A det A, we get 1 L(λ, Φ) lim log n n T 1 ( c(θ Φ 2π ) c(θ + (n 1) Φ 2π ) dθ = 0. (2.44) Normalized transfer matrix. Let c(θ) = c(θ)c(θ). We introduce the normalized transfer matrix: ( 1 λ v(θ) c(θ Φ SL(2, R) Ãλ (θ) = ) ) 2π c(θ) c(θ Φ ) c(θ) 0 2π and the n-step normalized transfer matrix Ãλ n(θ). The following connection between A λ and Ãλ is clear: ( ) Ã λ c(θ) 1 (θ) = c(θ) c(θ Φ ) c(θ) A λ (θ) 0 c(θ) 0 2π c(θ Φ 2π ) c(θ Φ 2π ) 1. (2.45) Let When Φ 2π = p q is rational, (2.45) yields tr(ãλ q (θ)) = q 1 j=0 c(θ + j p q ) q 1 j=0 c(θ + j p q ) tr(aλ q (θ)). (2.46) ( ) D λ (θ) = c(θ)a λ λ v(θ) c(θ (θ) = Φ ) 2π c(θ) 0 (2.47) and Dn(θ) λ = D λ (θ + (n 1) Φ ) 2π Dλ (θ + Φ 2π )Dλ (θ). Then when Φ = p is rational, 2π q (2.46) becomes tr(ãλ q (θ)) = tr(dq λ (θ)) q 1 j=0 c(θ + k (2.48) p ). q Note that although A λ n(θ) is not well-defined for θ Θ, D λ n(θ) is always well-defined Continued fraction expansion. Let α R \ Q, then α has the following continued fraction expansion α = a , a 1 + a a 3 + with a 0 being the integer part of α and a n N + for n 1.

15 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 15 For α (0, 1), let the reduced rational number p n q n = be the continued fraction approximants of α. 1 a a an The following property of continued fraction expansions is well-known: (2.49) α p n q n 1 q n q n+1. (2.50) 2.4. Lidskii inequalities. Let A be an n n self-adjoint matrices, let E 1 (A) E 2 (A) E n (A) be its eigenvalues. Then, for the eigenvalues of the sum of two self-adjoint matrices, we have { Ei1 (A + B) + + E ik (A + B) E i1 (A) + + E ik (A) + E 1 (B) + + E k (B) E i1 (A + B) + + E ik (A + B) E i1 (A) + + E ik (A) + E n k+1 (B) + + E n (B) (2.51) for any 1 i 1 < < i k n. These two inequalities are called Lidskii inequality and dual Lidskii inequality, respectively Magnetic potential. 3. Magnetic Hamiltonians on quantum graphs Given a vector potential A(x) = A 1 (x 1, x 2 ) dx 1 + A 2 (x 1, x 2 ) dx 2 Γ(Λ 1 (R 2 )), 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(W Λ ) ( A e (x) := A(x) [e]1 1 + [e] ) 2 2. (3.1) The integrated vector potentials are defined as β e := e A e(x)dx for e E(Λ). Assumption 1. The magnetic flux Φ through each hexagon Q of the lattice Φ := da (3.2) 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 Q β γ1,γ 2, f β γ 1,γ 2 +1, h + β γ 1,γ 2 +1, g β γ1 1,γ 2 +1, f + β γ 1 1,γ 2 +1, h β γ 1,γ 2, g = Φ, (3.3) for any γ 1, γ 2 Z.

16 16 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Example 1. The vector potential A Γ(Λ 1 (R 2 )) of a homogeneous magnetic field B Γ(Λ 2 (R 2 )) B(x) = B 0 dx 1 dx 2 (3.4) can be chosen as A(x) := B 0 x 1 dx 2. (3.5) This scalar vector potential is invariant under b 2 -translations. The integrated vector potentials β e are given by β γ1,γ 2, f = Φ ( γ ), β γ1,γ 2 6 2, g = 0, and β γ1,γ 2, h = β γ 1,γ 2, f, (3.6) where, in this case, the magnetic flux through each hexagon is Φ = B Magnetic differential operator and modified Peierls substitution. In terms of the magnetic differential operator (D B ψ) e := iψ e A eψ e, the Schrödinger operator modeling graphene in a magnetic field reads and is defined on { D(H B ) := H B : D(H B ) L 2 (PE (Λ)) L 2 (PE (Λ)) (H B ψ) e := (D B D B ψ) e + V e ψ e, (3.7) ψ H 2 (PE (Λ)) : ψ e1 (v) = ψ e2 (v) for any e 1, e 2 E v (Λ) and e E v(λ) Let us first introduce a unitary operator U on L 2 (PE(Λ)), defined as ( D B ψ ) e (v) = 0 }. (3.8) Uψ γ1,γ 2, e = ζ γ1,γ 2 ψ γ1,γ 2, e for e = f, g, h, (3.9) the factors ζ γ1,γ 2 are defined as follows. First, choose a path p( ) : N Z 2 connecting (0, 0) to (γ 1, γ 2 ) with p(0) = (0, 0) and p( γ 1 + γ 2 ) = (γ 1, γ 2 ). (3.10) Note that (3.10) implies that both components of p( ) are monotonic functions. Then we define ζ γ1,γ 2 recursively through the following relations along p( ): ζ 0,0 = 1, ζ γ1 +1,γ 2 = e iβ γ 1,γ 2, f iβ γ 1 +1,γ 2, g ζ γ1,γ 2, ζ γ1,γ 2 +1 = e iβ γ 1,γ 2, f iβ γ 1,γ 2 +1, h iφγ 1 ζ γ1,γ 2. (3.11)

17 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 17 Due to (3.3), it is easily seen that the definition of ζ γ1,γ 2 is independent of the choice of p( ), hence is well-defined. The unitary Peierls substitution is the multiplication operator P :L 2 (PE (Λ)) L 2 (PE (Λ)) (ψ e ) (( e x e i ) ) i( e) x A e(s)ds 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 e iβ eẇe will define a modified Peierls substitution that allows us to reduce the number of non-trivial multiplicative factors to one, by taking It transforms H B into The domain of Λ B is { D(Λ B ) = where Λ B := ( d2 dt 2 e P = P U. (3.13) + V e ) e E(Λ) = P 1 H B P. (3.14) ψ H 2 (PE(Λ)) : any e 1, e 2 E(Λ) with i( e 1 ) = i( e 2 ) = v satisfy ψ e1 (v) = ψ e2 (v) and i( e)=v ψ e(v) = 0; whilst at edges for which t( e 1 ) = t( e 2 ) = v, e i β e1 ψ e1 (v) = e i β e2 ψ e2 (v) and t( e)=v e i β e ψ e(v) = 0. }, (3.15) β γ1,γ 2, g β γ1,γ 2, f 0 and β γ1,γ 2, h = Φγ 1. (3.16) Thus, the problem reduces to the study of non-magnetic Schrödinger operators with magnetic contributions moved into the boundary conditions. Observe that the magnetic Dirichlet operator ( H D,B : H 1 0 ( e) H 2 ( e) ) L 2 (PE (Λ)) L 2 (PE (Λ)) e E(Λ) (H D,B ψ) e := (D B D B ψ) e + V e ψ e (3.17)

18 18 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA is by the (modified) Peierls substitution unitary equivalent to the Dirichlet operator without magnetic field H D = P 1 H D,B P = P 1 H D,B P. (3.18) Consequently, the spectrum of the Dirichlet operator H D is invariant under perturbations by the magnetic field. 4. Main lemmas First, let us introduce the following two-dimensional tight-binding Hamiltonian Q Λ (Φ) := 1 ( ) τ 0 + τ 1 3 (1 + τ 0 + τ 1 ) (4.1) 0 with translation operators τ 0, τ 1 L(l 2 (Z 2 ; C)) which for γ Z 2 and u l 2 (Z 2 ; C) are defined as (τ 0 (u)) γ1,γ 2 := u γ1 1,γ 2 and (τ 1 (u)) γ1,γ 2 := e iφγ 1 u γ1,γ 2 1. (4.2) The following lemma connects the spectrum of H B with σ(q Λ ). With (λ) defined in (2.38), we have Lemma 4.1. A number λ ρ(h D ) lies in σ(h B ) iff (λ) σ(q Λ (Φ)). Such λ is in the point spectrum of H B iff (λ) σ p (Q Λ (Φ)). Remark 4. We will show in Lemma 5.2 that σ p (Q Λ (Φ)) is empty, thus H B has no point spectrum away from σ(h D ). Lemma 4.2 below shows σ(q Λ (Φ)) is a zero-measure Cantor set for irrational flux Φ, Lemma 4.3 gives measure estimate for rational flux, and Lemma 4.4 shows upper 2π bound on the Hausdorff dimension of the spectrum of Q Λ (Φ). These three lemmas prove the topological structure part of Theorem 3. Lemma 4.2. For Φ 2π R \ Q, σ(q Λ(Φ)) is a zero-measure Cantor set. Lemma 4.3. If Φ 2π = p q is a reduced rational number, then we have σ(q Λ (Φ)) 8 6π 9 q. Lemma 4.4. For generic Φ, the Hausdorff dimension of σ(q Λ (Φ)) is 1 2.

19 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS Symmetric property of Q Λ. 5. Proof of Lemma 4.2 Lemma 5.1. The spectrum of Q Λ has the following properties: (1) σ(q Λ (Φ)) is symmetric with respect to 0. (2) 0 σ(q Λ (Φ)). Proof. (1). Conjugating Q Λ in (4.1) by Ω = ( ) id 0 0 id (5.1) shows that σ(q Λ (Φ)) is symmetric with respect to 0 [KL]. (2). If we view Q Λ (Φ) as an operator-valued function of the flux Φ, then Φ Q Λ (Φ)x, y, (5.2) for x, y c 00 arbitrary, is analytic and Q Λ therefore is a bounded analytic map. If there was Φ 0 /2π R\Q where Q Λ (Φ 0 ) was invertible, then Q Λ (Φ) would also be invertible in a sufficiently small neighborhood of Φ 0 (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(q Λ (Φ)). Thus, by density 0 σ(q Λ (Φ)), independent of Φ R Reduction to the one-dimensional Hamiltonian. Relating the spectrum of Q Λ to that of Q 2 Λ, we obtain the following characterization of σ(q Λ ). Lemma 5.2. (1) The spectrum of the operator Q Λ (Φ) as a set is given by θ T σ(q Λ (Φ)) = ± 1 σ(h Φ,θ ) + 1 {0}. (5.3) 9 3 where H Φ,θ L(l 2 (Z)) is the one-dimensional quasi-periodic Jacobi matrix defined as in (2.42) with (2) Q Λ (Φ) has no point spectrum. c(θ) = 1 + e 2πiθ, and v(θ) = 2 cos 2πθ. (5.4) Proof. (1). Let A := 1 (1 + τ τ 1 ). Then squaring the operator Q Λ (Φ) yields ( ) AA Q 2 0 Λ(Φ) = 0 A. (5.5) A

20 20 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA The spectral mapping theorem implies that σ(q 2 Λ (Φ)) = σ(q Λ(Φ)) 2 and from Lemma 5.2 we conclude that σ(q Λ (Φ)) = ± σ(q 2 Λ (Φ)). Clearly, the operator AA ker(a ) and A A ker(a) are unitarily equivalent. Thus, the spectrum can be expressed by σ(q Λ (Φ)) = ± σ(aa ) {0} (5.6) where we are able to use either of the two (AA or A A) since 0 σ(q Λ (Φ)) due to Lemma 5.1. Then, Observe that AA = id 3 + id 9 ((τ 0 + τ0 ) + (τ 1 + τ1 ) + τ 0 τ1 + τ 1 τ0 ) (5.7) }{{} =:H Φ H Φ ψ m,n =ψ m 1,n + ψ m+1,n + e iφm ψ m,n 1 + e iφm ψ m,n+1 + e iφ(m 1) ψ m 1,n+1 + e iφm ψ m+1,n 1. (5.8) Since H Φ is invariant under discrete translations in n, it is unitarily equivalent to the direct integral operator T 1 H Φ,θ dθ, (5.9) which gives the claim. (2). It follows from a standard argument that the two dimensional operator H Φ has no point spectrum. Indeed, assume H Φ has point spectrum at energy E, then H Φ,θ would have the same point spectrum E for a.e. θ T 1. This implies the integrated density of states of H Φ,θ has a jump discontinuity at E, which is impossible. Therefore the point spectrum of H Φ is empty, hence the same holds for Q Λ (Φ). Lemma 4.2 follows as a direct consequence of (5.3) and the following Theorem 7. Let Σ Φ be defined as in Section 2.2. Theorem 7. For Φ 2π R\Q, Σ Φ is a zero-measure Cantor set. We will prove Theorem 7 in the next section. 6. Proof of Lemmas 4.3, 4.4, and Theorem 7 For a set U, let dim H (U) be its Hausdorff dimension. We will need the following three lemmas. First, we have Φ Lemma 6.1. Let = p be a reduced rational number, then Σ 2π q 2πp/q is a union of q (possibly touching) bands with Σ 2πp/q < 16π 3q.

21 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 Σ Φ in Φ, extending a result of [AMS] (see Proposition 7.1 therein) from quasiperiodic Schrödinger operators to Jacobi matrices. Lemma 6.2. There exists absolute constants C 1, C 2 > 0 such that if λ Σ Φ and Φ Φ < C 1, then there exists λ Σ Φ such that λ λ < C 2 Φ Φ 1 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: {S n } n=1, S S n, such that each S n is a union of q n intervals, q n as n, and for each n, S n < C, qn β where β and C are positive constants, then dim H (S) 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 Σ2πp/q + 3 [ 0, ε ] ( Σ2πp/q + 3 ) (ε, ). Hence by Lemma 6.1, we have Σ 2πp/q + 3 ε + Σ 2πp/q 2 ε ε + 8π 3 εq. Optimizing in ε leads to Σ 2πp/q π 3 q. Then (5.3) implies σ(q Λ (2πp/q)) 8 6π 9 q. (6.1)

22 22 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Proof of Lemma 4.4. We will show that if Φ is an irrational obeying 2π qn 4 Φ 2π p n < C, (6.2) for some constant C, and a sequence of reduced rationals {p n /q n } with q n, then dim H (σ(q Λ (Φ))) 1/2. Without loss of generality, we may assume Φ (0, 1). 2π First, by (5.3), we have that (ΣΦ dim H (σ(q Λ (Φ))) = sup k 2 q n dim H (± ) [ 1k, 1] ) where we used a trivial bound H Φ,θ 6. Hence it suffices to show that for each k 2, ( (ΣΦ dim H ) ) [ 1k 3, 1] 1 2. (6.3) The rest of the argument is similar to that of [L]. By Lemma 6.2, taking any λ Σ Φ, for n n 0, there exists λ Σ 2πpn/qn such that λ λ < C 2 Φ pn 2π q n 1 2. This means Σ Φ is contained in the C 2 Φ pn 2π q n 1 2 neighbourhood of Σ 2πpn/qn. By Lemma 6.1, Σ 2πpn/qn has q n (possibly touching) bands with total measure Σ 2πpn/qn 16π 3q n. Hence Σ Φ has cover S n such that S n is a union of (at most) q n intervals with total measure S n 16π + 2 3q C 2 q n Φ n 2π p 1 n 2 q n. (6.4) Since qn 4 Φ pn 2π q n C, we have, by (6.4), S n 16π + 2C 2 C C =:. (6.5) 3q n q n q n This implies ( ) Σ Φ [ 1, 1] has cover S k n such that S n is a union of (at most) q n intervals with total measure S k C n. (6.6) 2q n Then Lemma 6.3 yields (6.3) Proof of Theorem 7. Note that Lemmas 6.1 and 6.2 already imply zero measure (and thus Cantor nature) of the spectrum for fluxes 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,

23 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 Σ ac (H Φ,θ ) be the absolutely continuous spectrum of H Φ,θ. Let L(λ, Φ) be the Lyapunov exponent of H Φ,θ at energy λ, as defined in (2.43). For a set U R, let U ess be its essential closure. First, we are able to give a characterization of the Lyapunov exponent on the spectrum. Proposition 6.4. For Φ 2π R \ Q, L(λ, Φ) = 0 if and only if λ Σ Φ. 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. Exploiting the fact that c(θ) = 1 + e 2πiθ has a real zero θ 1 = 1, we have 2 Proposition 6.5. ([D], see also Proposition 7.1 of [JM1]) For Φ 2π θ T 1, Σ ac (H Φ,θ ) is empty. R \ Q, and a.e. Hence our operator H Φ,θ has zero Lyapunov exponent on the spectrum and empty absolutely continuous spectrum. Celebrated Kotani theory identifies the essential closure of the set of zero Lyapunov exponents with the absolutely continuous spectrum, for general ergodic Schrödinger operators. This has been extended to the case of nonsingular (that is c( ) uniformly bounded away from zero) Jacobi matrices in Theorem 5.17 of [T]. In our case c( ) is not bounded away from zero, however a careful inspection of the proof of Theorem 5.17 of [T] shows that it holds under a weaker requirement: log ( c( ) ) L 1. Namely, let H c,v (θ) acting on l 2 (Z) be an ergodic Jacobi matrix, (H c,v (θ)u) m = c(t m θ)u m+1 + c(t m 1 θ)u m 1 + v(t m θ)u m where c : M C, v : M R, are bounded measurable functions, and T : M M is an ergodic map. Let L c,v (λ) be the corresponding Lyapunov exponent. We have Theorem 8. (Kotani theory) Assume log ( c( ) ) L 1 (M). Then for a.e. θ M, Σ ac (H c,v (θ)) = {λ : L c,v (λ) = 0} ess. Proof. The proof of Theorem 5.17 of [T] works verbatim. In our concrete model, log ( c(θ) ) = log (2 cos πθ ) L 1 (T 1 ), thus Theorem 8 applies, and combining with Propositions 6.4, 6.5, it follows that Σ Φ must be a zero measure set.

24 24 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA The rest of this section will be devoted to proving Lemma Quick Observations about H 2πp/q,θ. Let A λ ( ), Ãλ ( ), D λ ( ), Θ be defined as in Section We start with several quick observations about H 2πp/q,θ. Observation 1. The sampling function c(θ) = 0 yields a unique solution θ = 1 (mod 2 1), hence Θ = Z. Then, 2 q for θ / Θ, we have c(θ + n p ) 0 for any n Z q for θ Θ, there exists k 0 {0, 1,..., q 1} such that c(θ + n p ) = 0 if and only q if n k 0 (mod q). Note that c(θ) = 2 cos πθ, so a simple computation yields that q 1 j=0 c(θ + j p ) = q 2 sin πq(θ + 1 ). Thus (2.48) becomes 2 tr(ãλ q (θ)) = tr(dq λ (θ)) 2 sin πq(θ + 1 (6.7) ). 2 We have the following characterization of Σ 2πp/q,θ Case 1. If θ Θ, we have the following Observation 2. For θ Θ, the infinite matrix H 2πp/q,θ is decoupled into copies of the following block matrix M q of size q: Thus v( 1 2 ) c( 1 2 p q ) c( 1 2 p q ) v( 1 2 p q ) v( 1 (q 2) p ) c( 1 (q 1) p ) 2 q 2 q c( 1 (q 1) p ) v( 1 (q 1) p ) 2 q 2 q (6.8) Σ 2πp/q,θ = {eigenvalues of M q }, for θ Θ. (6.9)

25 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS Case 2. If θ / Θ, by Floquet theory, we have Σ 2πp/q,θ = {λ : tr(ãλ p,q(θ)) 2}. (6.10) q Furthermore, the set {λ : tr Ãλ p,q(θ) = 2 cos 2πν} contains q individual points (counting q multiplicities), which are eigenvalues of the following q q matrix M q,ν : v(θ) c(θ p ) q e2πiν c(θ) c(θ p ) v(θ p )... q q M q,ν (θ) = v(θ (q 2) p ) c(θ (q 1) p ) q q e 2πiν c(θ) c(θ (q 1) p ) v(θ (q 1) p ) q q (6.11) Combining (6.10) with (6.7), we arrive at an alternative representation Σ 2πp/q,θ = {λ : tr(d λq (θ)) 4 sin πq(θ + 12 } ). (6.12) 6.3. Key lemmas. Let We have d q (θ) = tr(d λ q (θ)). (6.13) Lemma 6.6. (a Chambers type formula) For all θ T 1, we have where G q (λ) (defined by (6.14)) is independent of θ. d q (θ) = 2 cos 2πqθ + G q (λ), (6.14) Proof. It is easily seen that d q ( ) is a 1/q-periodic function, thus d q (θ) = G q (λ) + a q e 2πiqθ + a q e 2πiqθ, in which the G q (λ) part is independent of θ. One can easily compute the coefficients a q, a q, and get a q = a q = 1. Lemma 6.7. For θ Θ, det (λ Id M q (θ)) = tr(d λ q (θ)). (6.15) The proof of this lemma will be included in the appendix.

26 26 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Combining (6.12), (6.9) and Lemma 6.7 with the fact that sin πq(θ + 1 ) = 0 for 2 θ Θ, we arrive at holds uniformly for θ T 1. Σ 2πp/q,θ = {λ : tr(dq λ (θ)) 4 sin πq(θ + 1 ) } (6.16) 2 By Lemma 6.14, we get the following alternative characterization of Σ 2πp/q,θ. Σ 2πp/q,θ = {λ : 4 sin πq(θ + 12 ) + 2 cos 2πqθ G q(λ) 4 sin πq(θ + 12 } ) + 2 cos 2πqθ. (6.17) Let us denote L q (θ) := 4 sin πq(θ ) + 2 cos 2πqθ, and l q(θ) := 4 sin πq(θ ) + 2 cos 2πqθ. Then (6.17) translates into This clearly implies Σ 2πp/q,θ = {λ : l q (θ) G q (λ) L q (θ)}. (6.18) Σ 2πp/q = {λ : min l q (θ) G q (λ) max L q (θ)}. (6.19) T 1 T 1 Since G q (λ) is a polynomial of λ of degree q, (6.19) implies Σ 2πp/q has q (possibly touching) bands. The following lemma provides estimates of Σ 2πp/q,θ. Lemma 6.8. We have Σ 2πp/q,θ 4 c(θ). Proof. Let us point out that, due to (6.10) and the explanation below it, { {λ : Gq (λ) = L q (θ)} = {eigenvalues of M q,0 (θ)} {λ : G q (λ) = l q (θ)} = {eigenvalues of M q, 1 (θ)} 2 (6.20) Let {λ i (θ)} q i=1 be eigenvalues of M q,0(θ), labelled in the increasing order. Let { λ i (θ)} q i=1 be eigenvalues of M q, 1 2 (θ), labelled also in the decreasing order. Then by (6.18) and (6.20), Σ 2πp q q,θ = ( 1) (λ q k k (θ) λ ) k (θ) k=1 [ q+1 2 ] = k=1 ( λ q 2k+2 (θ) λ ) q 2k+2 (θ) [ q 1 2 ] k=1 ( λ q 2k+1 (θ) λ ) q 2k+1 (θ). (6.21)

27 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 27 Consider the difference matrix M q,0 (θ) M q, 1 (θ) = 2 2 c(θ) whose eigenvalues we denote by {E i (θ)} q i=1, namely, 2 c(θ) E 1 (θ) = 2 c(θ) < 0 = E 2 (θ) = = E q 1 (θ) = 0 < 2 c(θ) = E q (θ). By Lidskii inequalities (2.51), we have [ q+1 2 ] k=1 [ q+1 2 ] k=1 [ q+1 2 ] k=1 [ q+1 2 ] λ q 2k+2 (θ) λ q 2k+2 (θ) + E q k+1 (θ) = λ q 2k+2 (θ) + 2 c(θ), (6.22) and [ q 1 2 ] k=1 [ q 1 2 ] k=1 [ q 1 2 ] k=1 k=1 [ q 1 2 ] λ q 2k+1 (θ) λ q 2k+1 (θ) + E k (θ) = λ q 2k+1 (θ) 2 c(θ). (6.23) Hence combining (6.21) with (6.22) (6.23), we get, 6.4. Proof of Lemma 6.1 for even q. k=1 Σ 2πp,θ 4 c(θ). (6.24) q For sets/functions that depend on θ, we will sometimes substitute θ in the notation with A T 1, if corresponding sets/functions are constant on A. Since q is even, a simple computation shows max T1 L q (θ) = L q ( 6Z+1) = L 6q q( 6Z+5) = 3, 6q min T1 l q (θ) = l q ( 2Z+1) = 6. 2q A simple computation also shows l q ( 6Z+1) = 1 and L 6q q( 2Z+1 ) = 2. Thus we have, by 2q (6.19), Σ 2πp/q ={λ : 6 G q (λ) 3} ={λ : 6 G q (λ) 2} {λ : 1 G q (λ) 3} =Σ 2πp Σ q, 2Z+1 2πp 2q q,. 6Z+1 6q

28 28 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA This implies Now it remains to estimate Σ 2πp Σ 2πp q, q+1 2q and Σ 2πp q,. 1 6q By Lemma 6.8, we have Σ 2πp/q Σ 2πp q, 2Z+1 2q Σ 2πp q, q+1 2q Σ 2πp q, 1 6q q, 2Z+1 2q and Σ 2πp + Σ 2πp q, 6Z+1 6q q, 6Z+1 6q 4 c( q+1 2q ) < 4π q, 4 c( 1 6q Hence putting (6.25), (6.26) together, we have 6.5. Proof of Lemma 6.1 for odd q.. (6.25). Since q is even, let us consider ) < 4π 3q. (6.26) Σ 2πp < 16π q 3q. (6.27) 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 By Lemma 6.8, we have Σ 2πp Σ 2πp q Σ 2πp q Σ 2πp q, 3q 1 6q, q 1 2q q, 3Z+1 3q 4 c( 3q 1 6q Hence putting (6.28), (6.29) together, we have + Σ 2πp q,. (6.28) Z q ) < 4π 3q, 4 c( q 1 2q ) < 4π q. (6.29) Σ 2πp < 16π q 3q. (6.30) 7. Proof of Lemma 4.1 Using ideas from [P] and [BGP], we can reduce the operator Λ B (3.14) by Krein s resolvent formula to a term containing the Dirichlet spectrum and Q Λ. For this we need to introduce a few concepts first. The l 2 -space on the vertices l 2 (V(Λ)) carries the inner product f, g := 3f(v)g(v) (7.1) v V(Λ)

29 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 Schrödinger operator H B 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, π, H ), with H being another Hilbert space and π, π : D(T ) H, is a boundary triple for T, if Green s identity holds on D(T ), i.e. for all ψ, ϕ D(T ) ψ, T ϕ H T ψ, ϕ H = πψ, π ϕ H π ψ, πϕ H. (7.2) ker(π, π ) is dense in H. (π, π ) : D(T ) H H is a linear surjection. The following lemma applies this concept to our setting. Lemma 7.2. The operator T B : D(T B ) L 2 (PE (Λ)) L 2 (PE (Λ)) acting as the maximal Schrödinger operator (2.12) on every edge with domain { D(T B ) := ψ H 2 (PE (Λ)) : any e 1, e 2 E v (Λ) such that i( e 1 ) = i( e 2 ) = v satisfy ψ e1 (v) = ψ e2 (v) and if t( e 1 ) = t( e 2 ) = v, } then e i β e1 ψ e1 (v) = e i β e2 ψ e2 (v) (7.3) is closed. The maps π, π on D(T B ) defined by ( π(ψ)(v) := 1 ψ e (v) + ) e i β e ψ e (v) 3 i( e)=v t( e)=v ( π (ψ)(v) := 1 ψ 3 e(v) ) e i β e ψ e(v) i( e)=v t( e)=v (7.4) form together with H := l 2 (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 some closed extension of T B. Such a closed extension is given by e E(Λ) H e on H 2 (PE (Λ)). To see that D(T B ) is a closed subspace of H 2 (PE (Λ)), observe that in

30 30 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA terms of continuous functionals l ei, e j : H 2 (PE (Λ)) C, l ei, e j (ψ) = ψ ei (i( e i )) ψ ej (i( e j )) k ei, e j : H 2 (PE (Λ)) C, k ei, e j (ψ) = e i β ei ψ ei (t( e i )) e i β ej ψ ej (t( e j )) (7.5) we obtain D(T B ) = ker ( ) l ei, e j ker ( ) k ei, e j e i, e j E(Λ) with i( e i )=i( e j ) e i, e j E(Λ) with t( e i )=t( e j ) (7.6) which proves closedness of T B. Green s identity follows directly from integration by parts on the level of edges. The denseness of ker(π, π ) is obvious since this space contains e E(Λ) C c ( e). To show surjectivity, it suffices to consider a single edge. On those however, the property can be established by explicit constructions as in Lemma 2 in [P]. Any boundary triple for T as in Def. 7.1 and any self-adjoint relation A H H gives rise [Sch] to a self-adjoint restriction T A of T with domain D(T A ) = {ψ D(T ) : (π(ψ), π (ψ)) A}. (7.7) The restriction of T B satisfying Dirichlet type boundary conditions on every edge is obtained by selecting A 1 := {0} l 2 (V(Λ)) and coincides with H D (2.15). The operator Λ B (3.14) is recovered from T B by picking the relation A 2 := l 2 (V(Λ)) {0}. Definition 7.3. Given the boundary triple for T B as above, the gamma-field γ : ρ(h D ) L(l 2 (V(Λ)), L 2 (PE (Λ))) is given by γ(λ) := ( π ker(t B λ)) 1 and the Weyl function M(, Φ) : ρ(h D ) L(l 2 (V(Λ))) is defined as M(λ, Φ) := π γ(λ). A computation shows that those maps are well-defined. The resolvents of H D = TA B 1 and Λ B = TA B 2 are then related by Krein s resolvent formula [Sch]. Theorem 9. Let (l 2 (V(Λ)), π, π ) be the boundary triple for T B and γ, M as above, then for λ ρ(h D ) ρ(λ B ) there is also a bounded inverse of M(λ, Φ) and (Λ B λ) 1 (H D λ) 1 = γ(λ)m(λ, Φ) 1 γ(λ). (7.8) In particular, σ(λ B )\σ(h D ) = {λ R ρ(h D ) : 0 σ(m(λ, Φ))}. For λ ρ(h D ) we have γ(λ) ker(m(λ, Φ)) = ker(λ B 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 (Λ B λ) 1 (H D λ) 1 = γ(λ)(t M(λ, Φ)) 1 γ(λ) where D(Λ B ) = {ψ : π ψ = T πψ}, hence here T = 0?

31 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 s λ, c λ (2.36) on an arbitrary edge e E(Λ) for λ ρ(h D ) and z l 2 (V(Λ)) by and where (γ(λ)z) e (x) = (s λ(1)c λ, e (x) s λ, e (x)c λ (1)) z(i( e)) + e i β esλ, e (x)z(t( e)) s λ (1) (K Λ (Φ)z)(v) := 1 3 M(λ, Φ) = K Λ(Φ) (λ) s λ (1) e: i( e)=v e i β e z(t( e)) + defines an operator in L(l 2 (V(Λ))) with K Λ (Φ) 1. e: t( e)=v Proof. For λ ρ(h D ) and z l 2 (V(Λ)) we define for e E(Λ) arbitrary (7.9) (7.10) e i β e z(i( e)) (7.11) ψ e := (γ(λ)z) e = ((π ker(t B λ)) 1 z) e (7.12) with ψ := (ψ e ). In particular, ψ e is the solution to ψ e +V eψ e = λψ e with the following boundary condition: ψ e (i( e)) = z(i( e)) and ψ e (t( e)) = e i β ez(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 Dirichletto-Neumann map (2.30). (M(λ, Φ)z)(v) = (π γ(λ)z)(v) = 1 ψ 3 e(v) e: i( e)=v = (K Λ(Φ)z)(v) s λ (1) e: t( e)=v e i β e ψ e(v) c λ (1) s λ (1) δ v i(v(λ))z(v) + s λ (1) s λ (1) δ v t(v(λ))z(v) }{{} = s λ (1) s λ (1) z(v) = (K Λ(Φ)z s λ (1)z) (v), (7.13) s λ (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 K Λ (Φ) 1.

32 32 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Since all vertices are integer translates of either of the two vertices r 0, r 1 W Λ by basis vectors b 1, b 2, we conclude that l 2 (V(Λ)) l 2 (Z 2 ; C 2 ). Our next Lemma shows K Λ (Φ) and Q Λ (Φ) are unitary equivalent under this identification. Lemma 7.5. K Λ (Φ) is unitary equivalent to operator Q Λ (Φ). Proof. By (3.16), (7.11), { ( (KΛ (Φ)z)(γ 1, γ 2, r 0 ) = 1 z(γ1, γ 3 2, r 1 ) + z(γ 1 1, γ 2, r 1 ) + e iφγ 1 z(γ 1, γ 2 1, r 1 ) ), ( (K Λ (Φ)z)(γ 1, γ 2, r 1 ) = 1 z(γ1, γ 3 2, r 0 ) + z(γ 1 + 1, γ 2, r 0 ) + e iφγ 1 z(γ 1, γ 2 + 1, r 0 ) ). In order to transform K Λ to Q Λ we use the unitary identification W : l 2 (V(Λ)) l 2 (Z 2, C 2 ) (W z) γ1,γ 2 := ( z(γ 1, γ 2, r 0 ), z(γ 1, γ 2, r 1 ) ) T. (7.14) This way, Q Λ (Φ) = W K Λ (Φ)W. Remark 6. In terms of a l 2 (Z 2, C 2 ) defined as ( ) 0 1 ( ) 0 1 a (0,0) := 1, a (0,1) := a (1,0) := 1 ( ) 0 1, a (0, 1) := 1 ( ) a ( 1,0) := 1 ( ) 0 0, and a γ := 0 for other γ Z 2, (7.15) we can express (4.1) in the compact form Q Λ (Φ) = γ Z 2 ; γ 1 a γ (τ 0 ) γ 1 (τ 1 ) γ 2. (7.16) 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 Theorem 9, Lemma 7.4 and Lemma Spectral analysis This section is devoted to complete spectral analysis of H B. In view of Lemmas 4.1 and 5.1, an important technical fact is: Lemma 8.1. The operator norm of Q Λ (Φ) for non-trivial flux quanta Φ / 2πZ is strictly less than 1.

33 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 33 Indeed, then, away from the Dirichlet spectrum σ(h D ), which are located on the edges of the Hill bands (2.41), we have the following characterization of σ(h B ). Let B n and be defined as in Section Lemma 8.2. For the magnetic Schrödinger operator H B, the following properties hold. (1) The level of the Dirac points 1 int(b n) (0) always belongs to the spectrum of HB, i.e. 0 int(bn)(σ(h B )). (2) λ int(bn)(σ(h B )) iff λ int(bn)(σ(h B )). Consequently, the property (0) 0 implies that locally with respect to the Dirac points, the spectrum of H B is symmetric. (3) H B has no point spectrum away from σ(h D ). (4) For non-trivial flux Φ / 2πZ, H B has purely continuous spectrum bounded away from σ(h D ). Combining Lemma 8.2 with Lemma 4.4, we get Lemma 8.3. For generic Φ, dim H (σ Φ ) 1 2. Proof of Lemma 8.3. Lemma 4.4 and (2) of Lemma 8.2 together imply that for generic Φ, ( ) dim H 1 int(b (σ(q n) Λ(Φ)) 1 2. Hence since σ Φ = σ(h D ) ( ) n N 1 int(b (σ(q n) Λ(Φ)), we have dim H (σ Φ ) sup { dim H (σ(h D )), sup n N dim H ( 1 int(b n) (σ(q Λ(Φ))) } 1 2. Remark 7. To Simon: Shall we insert a picture here, to illustrate the function, the symmetry of (σ(h B )) 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 H Φ,θ < c Φ < 6 for some constant c Φ independent of θ T 1. Let us

34 34 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA take ϕ l 2 (Z) with ϕ l 2 (Z) = 1. Consider ( (H Φ,θ ϕ) n = c θ + n Φ ) ( ϕ n+1 + c θ + (n 1) Φ ) ϕ n 1 2π 2π }{{}}{{} =:(h 1 ϕ) n =:(h 2 ϕ) n in which h 1, h 2, h 3 L(l 2 (Z)). Hence H Φ,θ ϕ 2 l 2 (Z) ( h 3 1 ϕ 2 l 2 (Z) + h 2ϕ 2 l 2 (Z) + h 3ϕ 2 l 2 (Z) 3 sup (θ c + (n 1) Φ ) 2 + n Z 2π c 12 sup θ T 1 =:c 2 Φ < 36. ) ( + v ( θ + n Φ ) 2 + 2π ) ( ( sin 2 π θ Φ ) + sin 2 (πθ) + cos 2 (2πθ) 2π ) θ + n Φ ϕ n 2π }{{} =:(h 3 ϕ) n (θ v + n Φ 2π, ) 2 In order to investigate further the Dirichlet spectrum and spectral decomposition of the continuous spectrum into absolutely and singular continuous parts, we start with constructing magnetic translations Magnetic translations. In general, Λ B does not commute with lattice translations Tγ st. Yet, there is a set of modified translations that do still commute with Λ B, although they in general no longer commute with each other. We define those magnetic translations Tγ B : L 2 (PE (Λ)) L 2 (PE (Λ)) as unitary operators given by (T B γ ψ) e := u B γ ( e)(t st γ ψ) e (8.1) for any ψ := (ψ e ) e E(Λ) L 2 (PE (Λ)) and γ Z 2. The lattice translation Tγ st is defined by (Tγ st ψ) e (x) = ψ e γ1 b1 γ 2 b2 (x γ 1 b1 γ 2 b2 ) as before. The function u B γ is constant on each copy of the fundamental domain, and defined as follows u B γ ( γ 1, γ 2, [e]) = e iφγ 1 γ 2, For e = γ 1 b1 + γ 2 b2 + [e] we have for [e] = f, g or h, γ = (γ 1, γ 2 ) Z 2. (8.2) (Tµ B Tγ B ψ) e =u µ ( e)u γ ( e µ 1 b1 µ 2 b2 )ψ e µ1 b1 µ 2 b2 γ 1 b1 γ 2 b2 =e iφµ 1 γ 2 +iφγ 1 ( γ 2 µ 2 ) ψ e µ1 b1 µ 2 b2 γ 1 b1 γ 2 b2, (Tγ B Tµ B ψ) e =u γ ( e)u µ ( e γ 1 b1 γ 2 b2 )ψ e µ1 b1 µ 2 b2 γ 1 b1 γ 2 b2 =e iφγ 1 γ 2 +iφµ 1 ( γ 2 γ 2 ) ψ e µ1 b1 µ 2 b2 γ 1 b1 γ 2 b2. (8.3)

35 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 35 Hence T B µ T B γ = T B γ T B µ e iφω(µ,γ), (8.4) where ω(µ, γ) := µ 1 γ 2 µ 2 γ 1 is the standard symplectic form on R 2. It also follows from (8.3) that (T B γ ) 1 = e iφγ 1γ 2 T B γ. One can also check that (T B γ ) = e iφγ 1γ 2 T B γ = (T B γ ) 1, (8.5) hence Tγ B is unitary. By the definition (8.1), (8.2), it is clear that for any ψ L 2 (PE(Λ)), d dt T B γ ψ = T B γ d dt ψ and V T B γ ψ = T B γ V ψ. (8.6) In order to make sure D(Λ B Tγ B ) = D(Tγ B Λ B ), it suffices to check Tγ B (D(Λ B )) = D(Λ B ), which translates into u γ ( e 1 ) = u γ ( e 2 ) whenever i( e 1 ) = i( e 2 ) (8.7) e i β e2 β e2 uγ( e 2 ) e i β e1 = ei γ 1b1 γ 2b2 whenever t( e uγ( e 1 ) e i β e1 γ 1b1 γ 1 ) = t( e 2 ). 2b2 This, by (3.16) is in turn equivalent to the following: for any γ 1, γ 2 Z: u γ ( γ 1, γ 2, f) = u γ ( γ 1, γ 2, g) = u γ ( γ 1, γ 2, h) u γ ( γ 1, γ 2, f) = u γ ( γ 1 + 1, γ 2, g) = e iφγ 1 u γ ( γ 1, γ 2 + 1, h) The definition of u B γ (8.2) clearly satisfies this requirement. Therefore, although magnetic translations do not necessarily commute with one another (8.4), they commute with Λ B T B γ Λ B = Λ B T B γ. (8.8) One can check that T B γ (D(T B )) = D(T B ) is also equivalent to (8.7), hence by (8.6), Note that we also have T B γ T B = T B T B γ. (8.9) T B γ H D = H D T B γ, (8.10) which is due to (8.6) and T B γ (H 1 0( e)) = H 1 0( e γ 1 b1 γ 2 b2 ) for any e E(Λ). We will now study the structure of eigenfunctions and the nature of the continuous spectrum of H B.

36 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(h D ). Lemma 8.4 below shows that σ(h D ) is contained in the point spectrum of H B, hence the only point spectrum of H B, due to Part (3) of Lemma 8.2. Consider a compactly supported simply closed loop, which is a path with vertices of degree 2 enclosing q hexagons, see e.g. Fig. 5. Then this loop passes (proceeding in positive direction from the center of an edge e 1 such that the first vertex we reach is t( e 1 )) n := 2 + 4q edges e 1,..., e n in E(Λ). For a solution vanishing outside this loop, the boundary conditions imposed by (3.15) on the derivatives can be represented in a matrix equation T Φ (n)ψ (n) = 0, (8.11) where e i β e1 e i β e ψ e (1) 0 0 e T Φ (n) := e3 e i β e4 ψ e 2 (1) and ψ ψ e (n) := 3 (1) e i β en 1 e i β. en ψ e n 1 (1) ψ e n (1) (8.12) Remark 8. We observe that T Φ (n) can be row-reduced to an upper triangular matrix with diagonal (e i β e1, 1, e i β e3, 1, e i β e5,..., 1, e i β en 1, 1 e i n j=1 ( 1)j β ej ) =(e i β e1, 1, e i β e3, 1, e i β e5,..., 1, e i β en 1, 1 e iqφ ), where q is the number of enclosed hexagons. Hence rank(t Φ (n)) = n iff qφ / 2πZ and rank(t Φ (n)) = n 1 otherwise. Lemma 8.4. The Dirichlet eigenvalues λ σ(h D ) are contained in the point spectrum of H B. 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 Λ B. 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 e 1, e 2,..., e 10 (see Fig. 4). Recall that s λ, e is the Dirichlet eigenfunction on e. By Remark 8, for 2Φ 2πZ, operator T Φ (10) has a non-trivial nullspace. We could take a = (a j ) ker (T Φ (10)) \{0}, (8.13)

37 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 37 e 5 e 6 e 4 e 7 e 3 e e 8 e 2 e 9 e 1 e 10 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 = a j s λ, ej. If 2Φ / 2πZ, we take a vector y C 10 such that y 2 = 1, y 7 = e i β e and yj = 0 otherwise. Since in this case T Φ (10) is invertible, there exists a unique solution a = (a j ) to the following equation: T Φ (10)a = y. (8.14) Let us take ψ on X such that ψ ej = a j s λ, ej and ψ e = s λ, 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 H B 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 6 all gaps of the absolutely continuous spectrum are closed and σ ac (H B ) = [0, ). The next lemma concerns the general feature of eigenspace of H B. Before proceeding, let us introduce the degree of a vertex in order to distinguish different types of eigenfunctions. Definition 8.6. An eigenfunction is said to have a vertex of degree d if there is a vertex with exactly d adjacent edges on which the eigenfunction does not vanish. Lemma 8.7. We have

38 38 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA (1) Every eigenspace of H B is infinitely degenerated. (2) Eigenfunctions of H B vanish at every vertex and are thus eigenfunctions of H D as well. (3) Eigenfunctions of H B with compact support cannot have vertices of degree 1. In particular, they must contain loops and the boundary edges of their support form loops as well. 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 H B, 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 L 2 (PE(Λ)). (To Simon: Let us make this argument a little bit clearer.) (2). If there is an eigenfunction to H B with eigenvalue λ that does not vanish at a vertex, by (modified) Peierls substitution (3.13), there is one to Λ B, denoted as ϕ, as well. We may expand the function in local coordinates on every edge e E(Λ) as ϕ e = a e c λ, e + b e s λ, e according to (2.36). Recall also that the Dirichlet eigenfunction s λ is either even or odd. Thus, using (2.32) we conclude that c λ (0) = c λ (1) and thus ϕ cannot be compactly supported. In particular, ϕ has the same absolute value at any vertex by boundary conditions (3.15). Due to ϕ e (i( e)) 2 ϕ 2 H < (8.15) 2 e E(Λ) ϕ has to vanish at every vertex. Thus ϕ is also an eigenfunction to H D. (3) clearly follows from (2) and (3.15) Dirichlet spectrum for rational flux quanta. In this section, the flux quanta are assumed to be reduced fractions Φ 2π = p q. If magnetic fields are absent, the point spectrum is spanned by hexagonal simply closed loop states, i.e. states supported on a single hexagon [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 B 0 rather than just Φ B 0, see Fig. 5

39 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 39 Figure 5. Simply closed loop state supported on black arrows encloses area 4Φ B 0. Lemma 8.8. Any simply closed loop enclosing an area of qφ B 0 normalization) eigenfunction of H B supported on it. has a unique (up to Proof. The existence of eigenfunctions on simply closed loops enclosing this flux follows directly from the non-trivial kernel of (8.12), see Remark 8. Due to dim(ker(t Φ )) = 1, such eigenfunctions are also unique (up to normalization). Lemma 8.9. The nullspaces ker(h B λ) where λ σ(h D ) are generated by compactly supported eigenfunctions. Proof. Unitary equivalence allows us to work with Λ B rather than H B. Without loss of generality, we assume that the Dirichlet eigenfunction to λ is even. Due to Lemma 8.7, eigenfunctions of Λ B to Dirichlet eigenvalues vanish at every vertex. Thus, on every edge e E(V ), they are of the form ϕ e = a e s λ, e for some a e. Let ϕ be such a function. We define the sequence (u(v)) v V(Λ) as follows { u(γ1, γ 2, r 0 ) := ϕ γ1,γ2, g (γ 1, γ 2, r 0 ) u(γ 1, γ 2, r 1 ) := ϕ γ 1,γ 2, f (γ 1, γ 2, r 1 ). Observe that the sequence (u(v)) determines the eigenfunction on every edge. Indeed, a γ1,γ 2, g = u(γ 1, γ 2, r 0 ) and a γ1,γ 2, f = u(γ 1, γ 2, r 1 ), since s λ (1) = s λ (0). At the same time, a γ1,γ 2, h can be determined in two different ways, one for each endpoint, from the boundary condition (3.15). Let us now introduce an operator A L(l 2 (V(Λ))) that has precisely the sequences (u(v)) with matching vertex conditions for a γ1,γ 2, h at its

40 40 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA endpoints in its kernel. Then, (Au)(γ 1, γ 2, r 0 ) := u(γ 1, γ 2, r 0 ) + u(γ 1, γ 2, r 1 ) ) e 2πi pγ 1 q (u(γ 1 + 1, γ 2 1, r 0 ) + u(γ 1, γ 2 1, r 1 ) (Au)(γ 1, γ 2, r 1 ) := 0. (8.16) The operator A is then a Z 2 -periodic finite-order difference operator. Any eigenfunction ϕ satisfying (Λ B 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 nullspace of A uniquely defines an eigenfunction ϕ = a e s λ, e to Λ B. Theorem 8 in [K2] implies then that the nullspace of A is generated by sequences in c 00 (V(Λ)). It suffices now to observe that those compactly supported sequences also give rise to compactly supported eigenfunctions to conclude the claim. Lemma 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 Λ B 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 Dirichlet spectrum for irrational flux quanta. After proving Theorem 4 for rational flux quanta, we now prove the analogous result for irrational magnetic fluxes. We start by introducing the following definition. Definition The Hilbert space l 2 (E(Λ)) is defined as l 2 (E(Λ)) := z : E(Λ) C, z 2 l 2 (E(Λ)) := z( e) 2 <. (8.17) e E(Λ) Theorem 10. The double hexagonal states generate the eigenspaces of Dirichlet spectrum of H B for irrational flux quanta. and

41 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(h D ). Definition We denote the closed L 2 (PE(Λ)) subspace generated by linear combinations of all double hexagonal states on the entire graph Λ by DH E(Λ) (Φ). There is, as for any separable Hilbert space, a countable orthonormal system of states V (Φ) DH E(Λ) (Φ) such that span(v (Φ)) = DH E(Λ) (Φ). (8.18) Because the set V (Φ) is countably infinite, there is a bijection ζ : Z 2 V (Φ). This way, we can address the elements of V (Φ) by ϕ γ (Φ) with γ Z 2. One can make sure that for each γ, ϕ γ (Φ) is analytic in Φ (0, 1). Every element ϕ γ (Φ) V (Φ) is of the form ϕ γ (Φ) = ϕ γ, e (Φ)s λ, e (8.19) because it is an element of ker(h B λ). e E(Λ) Now assume that the statement of Theorem 10 does not hold, this is equivalent to saying that Z(Φ) := ker(h B λ) DH E(Λ) (Φ) is not the zero space, i.e. there are eigenfunctions not spanned by double hexagonal states. Our goal is to characterize this space as the nullspace of a suitable operator we define next. Definition Let A(Φ) L(l 2 (E(Λ))) on u l 2 (E(Λ)) for γ Z 2 be given by (A(Φ)u)(γ, f) := u(γ, f) + u(γ, g) + u(γ, h) (A(Φ)u)(γ, g) := u(γ 1, γ 2 1, f) + u(γ 1 + 1, γ 2 1, g) + e iφγ 1 u(γ 1, γ 2, h) (A(Φ)u)(γ, h) := u( e)ϕ γ, 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 DH E(Λ) (Φ). This way, ker(a(φ)) and Z(Φ) are isometrically isomorphic where the isomorphism is given by η L(ker(A(Φ)), Z(Φ)) and η(u) := u e s λ, e. (8.21) s λ, e L 2 ( e) e E(Λ) Recall that we already know that for Φ Q (0, 1) the operator A(Φ) is injective. 2π This is so, since double hexagonal states span the entire eigenspace ker(h B 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 42 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Lemma The operator A(Φ) is surjective for Φ (0, 1). In particular, for any 2π (a( e)) l 2 (E(Λ)), there exists (ψ( e)) l 2 (E(Λ)) such that A(Φ)ψ = a and ψ l 2 (E(Λ)) holds for a universal constant C. C 1 e iφ a l 2 (E(Λ)), (8.22) Combining Lemma 8.14 with the already established injectivity result, we have A(Φ) is continuously invertible for Φ Q (0, 1) with the following control of its norm 2π A(Φ) 1 C 1 e iφ. (8.23) Now let us give the proof of Theorem 10, assuming the result of Lemmas Proof of Theorem 10. Since A(Φ) is uniformly bounded by a constant and Φ x, A(Φ)y is analytic for x, y c 00 (E(Λ)), A(Φ) is an analytic operator in Φ. Thus for any Φ 2π (0, 1), there exists ε 1( Φ) and C( Φ) such that A(Φ) A( Φ) C( Φ) Φ Φ, for Φ Φ < ε 1 ( Φ). (8.24) Also by (8.23), for any irrational have Φ 2π (0, 1) and rational Φ 2π with Φ Φ < ε 2 ( Φ), we A(Φ) 1 Hence, taking Φ 2π Q (0, 1) with Φ Φ < min(ε 1 ( Φ), ε 2 ( Φ), A(Φ) 1 (A( Φ) A(Φ)) < 1. C. (8.25) 2 1 e i Φ 2 1 e i Φ ), we would get C( Φ)C 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 We do this by showing that there is a sufficiently sparse set of elements in l 2 (E(Λ)) that gets mapped under A(Φ) on the standard basis of l 2 (E(Λ)). The functions ϕ γ defined in (8.19) ( satisfy the ) continuity condition at vertices, from which we conclude that for A(Φ) ϕ γ, e s λ,e 2 L 2 ((0,1)) the first two lines of (8.20) are zero. Similarly, the orthonormality of ϕ γ ensures that also the third line of (8.20) vanishes

43 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 43 (γ,g ) (γ-e 1,f ) (γ,h ) (γ-e 1,h ) (γ-e 2,f ) (γ-e 2,g ) Figure 6. Γ γ for γ γ and is equal to one if γ = γ. Thus, for γ Z 2, we obtain the following standard basis vectors of l 2 (E(Λ)) (A(Φ)(α e,(γ, h) ))(γ, f) := 0 (A(Φ)(α e,(γ, h) ))(γ, g) := 0 (A(Φ)(α e,(γ, h) ))(γ, h) := δ γ,γ. (8.26) To obtain also the remaining basis vectors, let us define L 2 functions ψ f γ and ψ g γ on a single hexagon Γ γ. The indices of ψ e γ are chosen to indicate the standard basis vectors (δ,(γ, e) ) we shall eventually obtain from those functions. Starting with ψ f γ, we introduce coefficients ( c γ, f e ) such that ψ f γ := e E(Γ γ) cγ, f e s λ, e. We do this in such a way that all continuity conditions for ψ f γ at the vertices of Γ γ are satisfied up to a single one at the (initial) vertex i((γ, g)) = i((γ, h)). We define c γ, f (γ, := 1 h) 1 e, f iφ cγ, e iφγ1 (γ e 2, f) := 1 e, f iφ cγ, (γ e 2, g) := e iφγ 1 1 e, iφ c γ, f (γ e 1, h) := e iφ 1 e iφ, cγ, f (γ e 1, f) := e iφ 1 e iφ, cγ, f (γ, g) := e iφ 1 e iφ (8.27) and all other c γ, f e are taken to be zero. Since for ψ f γ all but one continuity conditions are satisfied, we obtain for the first two components of (8.20) (A(Φ)( c γ, f e ))(γ, f) := δ γ,γ (A(Φ)( c γ, f e ))(γ, g) := 0. (8.28) To ensure that we also get constant zero in the third component of (8.20), we project onto the orthogonal complement of the double hexagonal states ψ f γ := ψ f γ P DHE(Λ) (Φ) ψ f γ

44 44 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA where P DHE(Λ) (Φ) is the orthogonal projection onto DH E(Λ) (Φ). Let now f ψ γ := α e,(γ, f) s λ, e, (8.29) e E(Γ γ) then, since subtracting linear combinations of double hexagonal states does not affect the continuity conditions, (A(Φ)(α e,(γ, f) ))(γ, f) := δ γ,γ, (A(Φ)(α e,(γ, f) ))(γ, g) := 0, and (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 e 2, f)) only c γ, g (γ e 2, f) := 1 1 e, cγ, g iφ (γ e 2, g) := (γ, g) := eiφ(γ 1 1) 1 eiφ(γ1 1), cγ, g 1 e iφ (γ e 1, := h) 1 e iφ c γ, g eiφ(γ1 1) eiφ(γ1 1) (γ e 1, f) :=, cγ, g, cγ, g 1 e iφ 1 e iφ (γ, := (8.31) h) 1 e iφ and all other coefficients being zero, we obtain for the first two components of (8.20) (A(Φ)( c γ, g e )(γ, f) := 0 (A(Φ)( c γ, g e )(γ, g) := δ γ,γ. (8.32) To ensure that we also get constant zero in the third component of (8.20), we project again on the orthogonal complement of the double hexagonal states ψγ g := ψ γ g P ψ DHE(Λ) (Φ) γ. g Let now ψγ g = e E(Γ) α e,(γ, g)s λ, e, then (A(Φ)(α e,(γ, g) ))(γ, f) := 0 (A(Φ)(α e,(γ, g) ))(γ, g) := δ γ,γ (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 {(α e,(γ, f) ), (α e,(γ, g) ) and (α e,(γ, h) ); γ Z 2 } (8.34) in l 2 (E(Λ)) that gets mapped under A(Φ) onto the standard unit basis of l 2 (E(Λ)). To conclude surjectivity of A(Φ) from this, it suffices to show that (8.34) satisfies for all (a e ) l 2 (E(Λ)) 2 a e α,(γ, g) a e 2. (8.35) e=(γ,[ e]) E(Λ) l 2 (E(Λ)) e E(Λ)

45 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 45 By the open mapping theorem, equation (8.35) implies then that for Φ Q (0, 1) 2π the set (8.34) is a Riesz basis for l 2 (E(Λ)). First, we observe that instead of showing (8.35) it suffices, since ψ f γ, ψ g γ DH E(Λ) (Φ) and (ϕ γ ) forms an orthonormal system of the orthogonal complement DH E(Λ) (Φ), to prove only that e=(γ,[ e]) E(Λ);[ e] h a e α,(γ, g) 2 l 2 (E(Λ)) a e 2. (8.36) e E(Λ);[ e] h The left hand side of (8.36) reads a (γ1, δ 1 ) a (γ 2, δ 2 ) (α e,(γ 1, δ 1 ) ), (α e,(γ 2, δ 2 ) ) l 2 (E(Λ)) δ 1, δ 2 { f, g} γ 1,γ 2 Z 2 = a (γ1, δ 1 ) a (γ 2, δ 2 ) ( cγ1, δ1 e ), ( c γ 2, δ 2 e ) l 2 (E(Λ)) δ 1, δ 2 { f, g} γ 1,γ 2 Z 2 }{{} =:τ 1 + a (γ1, δ 1 ) a (γ 2, δ 2 ) (α e,(γ 1, δ 1 ) cγ1, δ1 e ), (α e,(γ2, δ 2 ) cγ 2, δ 2 e ) l 2 (E(Λ)) δ 1, δ 2 { f, g} γ 1,γ 2 Z 2 }{{} + 2 Re δ 1, δ 2 { f, g} γ 1,γ 2 Z 2 =:τ 2 ( a (γ1, δ 1 ) a (γ 2, δ 2 ) (α e,(γ 1, δ 1 ) cγ 1, δ 1 e ), c γ 2, δ 2 e l 2 (E(Λ)) } {{ } =:τ 3 ) (8.37) We will now bound τ 1,τ 2, and τ 3 separately. Let us denote by N(γ) := {γ, γ + e 2, γ + e 1, γ + e 1 e 2, γ e 2, γ e 2 e 1, γ + e 2 e 1 } (8.38)

46 46 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA the set of all γ such that Γ γ Γ γ. Starting with τ 1 and using that the inner-product in τ 1 is zero unless γ 2 N(γ 1 ) by the disjoint support τ 1 a (γ1, δ 1 ) a (γ2, δ 2 ) ( c γ 1, δ 1 e ) ( c γ 2, δ 2 e ) 1 {γ2 N(γ 1 )} }{{} δ 1, δ 2 { f, g} γ 1,γ 2 Z e iφ e iφ 2 = For τ 2 we first rewrite τ 2 = δ 1, δ 2 { f, g} γ 1 Z 2 γ 2 N(γ 1 ) δ { f, g} γ Z e iφ 2 δ 1, δ 2 { f, g} γ 1,γ 2 Z 2 e E(Λ);[e] h a (γ, δ) 2 a (γ1, δ 1 ) a (γ 2, δ 2 ) s λ 2 L 2 ((0,1)) 7 1 e iφ 2 a (γ1, δ) a (γ2, δ) }{{} max δ { f, g},γ N(γ1 ) a (γ, δ) 2 a e 2. (8.39) P DHE(Λ) (Φ) ψ δ 1 γ 1, P DHE(Λ) (Φ) ψ δ 2 γ 2. (8.40) We define DH Γγ (Φ) DH E(Λ) (Φ) to be the set of (normalized) double hexagonal states whose support intersects Γ γ. It is clear that DHΓγ (Φ) < because there are only finitely many (normalized) double hexagonal states whose support intersects Γ γ. However, only their linear span contributes to the projection P DHE(Λ) (Φ) ψ δ γ such that From this, we conclude that the cardinality of P DHE(Λ) (Φ) ψ δ γ = P span(dhγγ (Φ)) ψ δ γ. (8.41) DH(γ, Φ) := { γ Z 2 ; DH Γγ (Φ) DH Γγ (Φ) } (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 γ 2 / DH(γ 1, Φ) then this implies by disjointness of supports that P DHE(Λ) (Φ) ψ δ 1 γ 1, P DHE(Λ) (Φ) ψ δ 2 γ 2 = P span(dhγγ1 (Φ)) ψ δ 1 γ 1, P span(dhγγ2 (Φ)) ψ δ 2 γ 2 = 0. (8.43) Thus, the inner-product in (8.40) is non-zero only if γ 2 DH(γ 1, Φ). So, we obtain for (8.40) that τ 2 = δ 1, δ 2 { f, g} γ 1 Z 2 γ 2 DH(γ 1,Φ) a (γ1, δ 1 ) a (γ 2, δ 2 ) s λ 2 L 2 ((0,1)) P DHE(Λ) (Φ) ψ δ 1 γ 1, P DHE(Λ) (Φ) ψ δ 2 γ 2. (8.44)

47 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 47 This expression can be estimated in a similar way as τ 1 sup γ Z τ 2 2, δ { f, g} ψ δ 2 γ s λ 2 L 2 ((0,1)) 7 1 e iφ 2 δ { f, g} = 28N 2 1 e iφ 2 e E(Λ);[e] h δ 1, δ 2 { f, g} γ 1 Z 2 γ 2 DH(γ 1,Φ) 2 2 N 2 2 a (γ, δ) γ Z 2 Analogously, a bound for τ 3 is found by sup γ Z τ 3 2 2, δ { f, g} ψ δ 2 γ s λ 2 L 2 ((0,1)) 14 1 e iφ 2 δ { f, g} 56N 2 1 e iφ 2 e E(Λ);[e] h a (γ1, δ 1 ) a (γ2, δ 2 ) }{{} max γ DH(γ1,Φ), δ { f, g} a (γ, δ) 2 a e 2. (8.45) δ 1, δ 2 { f, g} γ 1 Z 2 γ 2 DH(γ 1,Φ) 2 2 N 2 2 a (γ, δ) γ Z 2 a (γ1, δ 1 ) a (γ2, δ 2 ) }{{} max γ DH(γ1,Φ), δ { f, g} a (γ, δ) 2 a e 2. (8.46) 8.3. Absolutely continuous spectrum for rational flux quanta. Lemma For Φ 2π = p q Q, the spectrum of HB 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 Q Λ (Φ) if and only if its pre-image under, on each fixed band of the Hill operator, is a band of H B. Proof. Note that σp Φ \σ(h D ) = is due to (3) of Lemma 8.2. The absence of singular continuous spectra is clear because H B is invariant under magnetic translations (8.1), some of which commute with one another in the case of rational flux quanta. In particular, T0,q B and Tq,0 B always generate a group of commutative magnetic translations due to (8.4). Thus, standard arguments from Floquet-Bloch theory show that H B has no singular continuous spectrum [GeNi]. Now we will show the non-overlapping band structure. We recall that Λ B, T B, H D all commute with magnetic translations Tµ B (8.8) (8.9) (8.10) where we assume that µ qz. Consequently, Tµ B leaves all eigenspaces of those operators invariant.

48 48 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA For λ ρ(h D ) we define discrete magnetic translations τ B µ : l 2 (V) l 2 (V), τ B µ := π(λ)t B µ γ(λ), (8.47) with γ as in Def Recall now that γ(λ) = (π ker(t B λ)) 1 maps given vertex values onto a function in ker(t B λ) with T B as in Lemma 7.2. The operator Tµ B 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(h D ) and they form a family of unitary operators with ( ) τµ B = τ B µ, due to (8.5). Since Λ B, H D both commute with Tµ B, Krein s formula (7.8) implies that for λ ρ(h D ) ρ(λ B ), T B µ ( γ(λ)m(λ, Φ) 1 γ(λ) ) = ( γ(λ)m(λ, Φ) 1 γ(λ) ) T B µ. (8.48) Multiplying with π(λ) and π(λ) from both sides respectively, it follows that τµ B M(λ, Φ) 1 = M(λ, Φ) ( 1 γ(λ) Tµ B π(λ) ) = M(λ, Φ) ( 1 π(λ)t µγ(λ) ) B = M(λ, Φ) 1 τµ B, (8.49) thus M(λ, Φ) 1 commutes with discrete magnetic translations. To see that restricts on every Hill band B n (because Bn is one-to-one) to an isomorphism from each band of Q Λ (Φ) to a unique band of Λ B, it suffices to note that the preceding calculation shows that Krein s formula (7.8) holds true for the Floquet-Bloch transformed operators. Let U cont be the Gelfand transform generated by translations Tµ B and U discrete the Gelfand transform generated by discrete translations τµ B, i.e. for k in the Brioullin zone (U cont f)(k, x) := µ qz 2 (U discrete ξ)(k, δ) := µ qz 2 Using (8.47) and γ(λ)π(λ) = id V(Λ) we obtain from ( T B µ f ) (x)e i k,µ and ( τ B µ ξ ) (δ)e i k,µ. (8.50) τ B µ γ(λ) = ( τ B µ) γ(λ) = γ(λ) T B µ π(λ) γ(λ) = γ(λ) T B µ, (8.51)

49 CANTOR SPECTRUM OF GRAPHENE IN MAGNETIC FIELDS 49 where we used that T B µ preserves ker(t B λ) and γ(λ) ker(t B λ) = 0, on some fundamental domain W Φ Λ ( U cont γ(λ)m(λ, Φ) 1 γ(λ) f ) (k, x) = γ(λ)(k) ( U discrete M(λ, Φ) 1 γ(λ) f ) (k, x) = γ(λ)(k)m(λ, Φ)(k) 1 ( U discrete γ(λ) f ) (k, x) = γ(λ)(k)m(λ, Φ)(k) 1 γ(λ)(k) (U cont f)(k, x). (8.52) where γ(λ)(k) L(l 2 (WΛ Φ), L2 (WΛ Φ)) and M(λ, Φ)(k) L(l2 (WΛ Φ )) are the restrictions of γ(λ) and M(λ, Φ) on l 2 (WΛ Φ) satisfying Floquet boundary conditions. Both ΛB and H D commute with translations Tµ B, and thus fiber upon conjugation by U cont, so that for λ ρ(λ B (k)) ρ(h D ) Krein s formula remains true for each k (Λ B (k) λ) 1 (H D L 2 (W Φ Λ ) λ) 1 = γ(λ)(k)m(λ, Φ)(k) 1 γ(λ)(k). (8.53) In particular, for λ ρ(h D ) γ(λ)(k) ker(m(λ, Φ)(k)) = ker(λ B (k) λ). (8.54) Therefore bands of Q Λ (Φ) are in one-to-one correspondence with bands of Λ B, and thus also with bands of H B. That the bands of Q Λ (Φ) do not overlap is shown in Section 6 of [HKL]. Thus, the unique correspondence among bands of Q Λ (Φ) and H B shows that the non-overlapping of bands holds true for H B as well. Remark 11. For Φ = 1 the spectral bands of Q 2π 2 Λ(Φ) are touching and given by [HKL] [ ] [ ] [ ] [ ] ,, 3 3, 0, 0,, and 3 3,. (8.55) 3 Thus, by Lemma 8.15 the bands of H B 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. In the case of Φ = 1 however, only the bands at the Dirac points touch, see also Fig. 2π 3 8. The touching at the Dirac points is always satisfied by Lemma 8.2. Φ Remark 12. The spectrum of Q Λ (Φ) for rational = p are precisely the eigenvalues 2π q [HKL] of ( ) 0 id C q +e ik 1 J p,q + e ik 2 K q id C q +e ik 1 Jp,q + e ik 2 Kq (8.56) 0 ( ) for k T 2 where J p,q := δ m,n e 2πi(m 1) p q and (K q ) mn := 1 if n = (m + 1) mod q and 0 otherwise. mn 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 H B for different rational flux quanta in Fig.??.

50 50 SIMON BECKER, RUI HAN, AND SVETLANA JITOMIRSKAYA Figure 7. Touching bands for Φ 2π = 1 2 on the first Hill band B 1 = [0, π 2 ] with V = 0. Figure 8. Only the third and fourth band touch at the Dirac points for Φ 2π = 1 3 on the first Hill band B 1 = [0, π 2 ] with V = 0.