Honeycomb structures in magnetic fields

We consider reduced-dimensionality models of honeycomb lattices in magnetic fields and report results about the spectrum, the density of states, self-similarity, and metal/insulator transitions under disorder. We perform a spectral analysis by which we discover a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, prove the existence of zero energy Dirac cones for each rational flux, obtain an explicit expansion of the density of states near the conical points, and show the existence of mobility edges under Anderson-type disorder. Our results give a precise description of de Haas-van Alphen and Quantum Hall effects, and provide quantitative estimates on transport properties. In particular, our findings explain experimentally observed asymmetry phenomena by going beyond the perfect cone approximation.

Reduced-dimensionality models are of central importance in condensed matter physics as they are often analytically solvable and allow for a qualitative description of material properties.
A phenomenologically rich class is composed of tightbinding and infinite-contrast models on periodic lattices with constant magnetic fields. Following a thorough study of these models over the past fourty years, rigorous results on the fractal spectrum on the Z 2 -lattice (Harper's model) [1]- [13], the location of the low-lying spectrum [14][15][16][17], and the disordered model (Anderson model) [18][19][20] have been obtained.
The purpose of this letter is to report rigorous results on tight-binding and infinite-contrast models for honeycomb structures in constant magnetic fields and emphasize new theoretical approaches. We consider the Hamiltonian, H B , with constant magnetic field B, defined on edges e of a honeycomb graph Λ by H B e := (−i∂ x − A e ) 2 + V e , e (0, 1).
We assume Kirchhoff boundary conditions [21] at the vertices and that V e is symmetric with respect to the centre of the edge e. Such Hamiltonians are called quantum graphs or wave guides [21,22] and represent the infinitecontrast limits [23] of continuous Schrödinger operators on R 2 with honeycomb lattice potential [24][25][26][27]. Apart from the interest in such models as limits (see [28,29] for related work on Harper's model), quantum graphs are natural models for molecular graphene [30,31] and wave guides [32], see Fig.3. In this letter, we explain three key physical phenomena: • Spectral theoretic: We provide a full spectral analysis. In particular, we show that for irrational magnetic flux through a single honeycomb, the spectrum is a Cantor set of measure zero and Hausdorff dimension at most 1/2 [33,34].
• Semiclassical : We derive a semiclassical expansion for the density of states (DOS) supported on geometric Landau levels and relate it to the Shubnikovde Haas, de Haas-van Alphen, and Quantum Hall effects [35,37]. We note a remarkable agreement of the asymptotic result with exact spectral calculations, see Fig.5. This analysis holds near each conical singularity, which we show to exist for each rational flux, thus providing a foundation for selfsimilarity appearing in Fig.6.
• Dynamical : For Anderson-type potentials in weak magnetic fields under weak disorder, we identify insulating regions away from the Landau levels in which Anderson localization occurs, and regions of metallic transport close to the Landau levels [37].
For vector potentials A(x) = A 1 (x)dx 1 + A 2 (x)dx 2 the scalar potential A e in (1) along edges e is A e (x) := A(x) (e 1 ∂ 1 + e 2 ∂ 2 ) [22]. The magnetic flux h := dA through each honeycomb of the lattice is taken to be constant. The Hamiltonian H B on the graph can be identified with the standard tight-binding operator t h on the honeycomb lattice: After a simple change of geometry, the tight-binding operator t h is with the magnetic translations τ 0 (r)(γ) := r(γ 1 − 1, γ 2 ), τ 1 (r)(γ) := e ihγ1 r(γ 1 , γ 2 − 1) for γ ∈ Z 2 , and r ∈ 2 (Z 2 ; C). Solving −y λ (x) + V e y λ (x) = λy λ (x), y λ (0) = 1, y λ (0) = 0, we put ∆(λ) := y λ (1). Then λ ∈ Spec(H B ) \ Spec(H D ) (H D is the operator (1) on a single edge with Dirichlet boundary conditions) if and only if ∆(λ) ∈ Spec(t h ) -see [29,33,[38][39][40]. Since t h < 1 [29,33] for non-trivial magnetic flux h / ∈ 2πZ the spectrum of H B decomposes into the disjoint union of continuous spectrum ∆ −1 (Spec(t h )) and infinitely degenerate eigenvalues λ ∈ Spec(H D ), see [33]. Cantor spectrum:. The fractal structure of magnetic electron spectra was first predicted by Azbel [41] and then numerically confirmed by Hofstadter [42] for Harper's model, see Fig.2. Verifying this experimentally is difficult as the smallness of the cell requires extraordinarily strong magnetic fields to obtain observable magnetic flux. Only recently, self-similar structures in the electron spectrum of graphene have been observed [43][44][45][46]. Earlier experiments involved modeling of periodic structures by microwaves [47]. Here, we first assume that the normalized magnetic flux h 2π = p q is rational, as then the Floquet-Bloch theory implies that the spectrum of (2) has band structure, see Fig.1. We can then express the spectrum of (2) using a 1D-Jacobi operator with quasi-momentum k ∈ T * 1 = [0, 2π] (Ju) m = 1 + e i(k+mh) u m+1 + 2 cos (k + mh) u m and from the study of such singular Jacobi operators [33,Lemma 4.3] we estimate the Lebesgue measure The spectrum of (2) However, the spectral nature for irrational fluxes changes dramatically, see [33,Thm. 3]. If h 2π is irrational, the spectrum of (2), and the continuous spectrum of (1), is a fully disconnected and nowhere dense set without isolated points of measure zero with Hausdorff dimension at most (4) and the continuity estimate (5). Since the spectrum is always closed and, as can be shown, it has no isolated points, this implies Cantor-type spectrum. Using Kotani's theory, the Cantor structure of the spectrum can also be shown to hold for all irrational fluxes h 2π . The bound on the Hausdorff dimensions follows from an almost Lipschitz continuity estimate on the spectrum of singular quasiperiodic Jacobi operators obtained in [34]. Semiclassical analysis of the DOS:. The density of states is a generalized function ρ H B defined in terms of the regularized trace where B r (0) is the ball of radius r, see Figs. 3 and 4. By spectral equivalence of (1) and (2), for energies close to the Dirac point energy it suffices to analyze the DOS of t h . The magnetic translations in (2) satisfy the Weyl commutation relations τ 1 τ 0 = e ih τ 1 τ 2 and the same commutation relation is obtained for D x := −i ∂ ∂x by e ihDx e ix = e ih e ix e ihDx where e ihDx = Op w h (e iξ ) is the Weyl quantization of the symbol e iξ [48]. This different representation reduces the analysis of the DOS of (2) to the study of the DOS of the operator Through a symplectic change of variables, y = a(x + ξ), ) one finds that at the Dirac points we have 1 + e ix + e iξ = c(η ∓ iy) + O(y 2 + η 2 ), Classical-quantum correspondence implies that by the symplectic change of variables (classical ), κ(nh, h) are the solutions to the Bohr-Sommerfeld con- In particular, we show that F 1 (s) = 0 for all such 2 × 2 operators with only off-diagonal contributions. Writing g(x) = F 0 (∆(x) 2 ), we obtain a leading order approximation of Landau levels In (8), |f | C α := sup x |f (x)| + sup x =y |f (x)−f (y)| |x−y| α : it is essential to allow non-smooth test functions f in view of applications to magnetic oscillations. (See for instance [36] for a physics perspective on semiclassical approximation in this setting.)  To formulate it we introduce the grand-canonical potential with inverse temperature β and chemical potential µ: for smooth η localizing to energy intervals contained in ∆ −1 (−δ, δ). (Note that |f β | α is uniformly bounded for α ≤ 1 but not for α > 1.) The magnetization is defined by [49] M β (µ, h) := − 3 and at zero temperature, we can derive from this a sawtooth approximation, with σ(y) := y − [y] − 1 2 , as the O(h 1 2 ) approximation of (12) given as This provides a refinement of results found in [50][51][52]. The remarkable agreement of the different expressions for the magnetization is illustrated in Fig.5: the characteristic sawtooth pattern (13) is compared with the magnetization computed from (12), using either the operator spectrum or the semiclassical limit (8).
QHE and self-similarity:. One of the striking properties of graphene is the presence of Dirac points, which has remarkable physical and technological implications [53]. It turns out Dirac points are present at E = 0 for any magnetic flux h = 2πp/q, [37,Thm. 2]. To study transport properties on honeycomb structure (see [54][55][56][57]) we consider operator (2) with additive disorder where (V 1,2 ω(z) ) z∈Z 2 are i.i.d. random variables with compactly supported probability distribution and small κ > 0. For discrete operators A with C 2 -valued kernel (A(x, y)), we define a regularized trace trA := lim r→∞ 1 |Br(0)| γ∈Λ∩Br(0) Here, t B0 j (k) is the j-th Floquet eigenvalue to t B0 with quasimomentumk where B 0 is the magnetic field associated to the flux h 0 = 2πp q . This study is inherently connected with self-similarity in the Hofstadter butterfly, see Fig.6, and the occurrence of magnetic mini-bands [44]. Since t B is an element of the rotation algebra, so is its Fermi projection P = 1l [z D ,µ) (t B ) for Fermi energies µ inside a spectral gap of t B . By [58][59][60], there is γ ∈ Z 2 , such that where by (16) we see that γ = (0, 2qn) and n is the number of Landau levels between z D and µ. Combining (16) with (4) implies the existence of spectral gaps between a finite number of disjoint intervals B n (h) z n (h) up to some small disorder strength κ 0 > 0. The Hall conductivity, which by universality (see [61,62]) is invariant under weak disorder, is given by Středa's formula [63] as c H (µ) := ∂ ∂h tr(1l [z D ,µ) (t h κ,ω )) = γ2 2π with Fermi energies µ in the interval I n between B n (h) and B n+1 (h) [ (16) and (17) we then find , µ ∈ I n , n ≥ 0 (2n−1)q 2π , µ ∈ I n−1 , n ≤ 0.
This expression is only valid for Fermi energies close to the conical point. The Hall conductivity for arbitrary Fermi energies is far more intricate, see Fig.6, [54,64]. Metal/insulator transition:. The Hall conductivity allows us also to analyze transport properties of t h κ,ω . Transport in disordered media at energy E is measured by transport coefficients β h κ (E) [20,[65][66][67][68]. This quantity allows us to define two complementary energy regions, the insulator region Σ h,DL κ = E ∈ R; β h κ (E) = 0 and the metallic transport region Σ h,DD κ = E ∈ R; β h κ (E) > 0 . Energies E ∈ Σ h,DD κ at which the transport coefficient β h κ jumps from zero to a non-zero value are called mobility edges, while energies E ∈ Σ h,DL κ (H h κ,ω ), that also belong to the spectrum of (14), are eigenvalues of finite multiplicity with exponentially decaying eigenfunctions (Anderson localization). From the jumps of the Hall conductivity, we conclude [37, Thm. 1] that there exist mobility edges E close to each Landau level with non-trivial transport β h λ (E) ≥ 1/4. In contrast to this, we show by verifying the starting criteria of the multi-scale analysis [19,37,66] that the spectral gaps between the Landau levels can only be filled with spectrum belonging to the insulating region [37,Prop.5.5] in which the operator (14) therefore exhibits Anderson localization.