Honeycomb Lattice Potentials and Dirac Points

We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

1. Introduction and Outline. In this article we study the spectral properties of the Schrödinger operator H V "´∆`V pxq, x P R 2 , where the potential, V , is periodic and has honeycomb structure symmetry. For general periodic potentials the spectrum of H V , considered as an operator on L 2 pR 2 q, is the union of closed intervals of continuous spectrum called the spectral bands. Associated with each spectral band are a band dispersion function, µpkq, and Floquet-Bloch states, upx; kq " ppx; kqe ik¨x , where Hupx; kq " µpkqupx; kq and ppx; kq is periodic with the periodicity of V pxq. The quasi-momentum, k, varies over B, the first Brillouin zone [8]. Therefore, the time-dependent Schrödinger equation has solutions of the form e ipk¨x´µpkqtq ppx; kq. Furthermore, any finite energy solution of the initial value problem for the timedependent Schrödinger equation is a continuum weighted superposition, an integral dk, over such states. Thus, the time-dynamics are strongly influenced by the character of µpkq on the spectral support of the initial data.
We investigate the properties of µpkq in the case where V " V h is a honeycomb lattice potential, i.e. V h is periodic with respect to a particular lattice, Λ h , and has honeycomb structure symmetry; see Definition 2.1. There has been intense interest within the fundamental and applied physics communities in such structures; see, for example, the survey articles [12,14]. Graphene, a single atomic layer of carbon atoms, is a two-dimensional structure with carbon atoms located at the sites of a honeycomb structure. Most remarkable is that the associated dispersion surfaces are observed to have conical singularities at the vertices of B h , which in this case is a regular hexagon. That is, locally about any such quasi-momentum vertex, k « K ‹ , one has is the Schrödinger equation with inverse effective mass related to the local curvature of the band dispersion relation at the band edge. The presence of Dirac points has many physical implications with great potential for technological applications [20]. Refractive index profiles with honeycomb lattice symmetry and their applications are also considered in the context of electro-magnetics [5,19]. Also, linear and nonlinear propagation of light in a two-dimensional refractive index profile with honeycomb lattice symmetry, generated via the interference pattern of plane waves incident on a photorefractive crystal, has been investigated in [15,3] . In such structures, wave-packets of light with spectral components concentrated near Dirac points, evolve diffractively (rather than dispersively) with increasing propagation distance into the crystal.
Previous mathematical analyses of such honeycomb lattice structures are based upon extreme limit models: 1. the tight-binding / infinite contrast limit (see, for example, [18,12,11]) in which the potential is taken to be concentrated at lattice points or edges of a graph; in this limit, the dispersion relation has an explicit analytical expression, or 2. the weak-potential limit, where formal perturbation theory is used [5,1].
The goal of the present paper is to provide a rigorous construction of conical singularities (Dirac points) for essentially any potential with a honeycomb structure. No assumptions on smallness or largeness of the potential are made. More precisely, consider the Schrödinger operator H pεq "´∆`εV h pε realq (1.2) where V h pxq denotes a honeycomb lattice potential. These potentials are real-valued, smooth, Λ h -periodic and, with respect to some origin of coordinates, inversion symmetric px Ñ´xq and invariant under a 2π{3-rotation (R-invariance); see Def. 2.1. We also make a simple, explicit genericity assumption on V h pxq; see equation (5.2).
Our main results are: 1. Theorem 5.1, which states that for fixed honeycomb lattice potential V h , the dispersion surface of H pεq has conical singularities at each vertex of the hexagonal Brillouin zone, except possibly for ε in a countable and closed set,C. We do not know whether exceptional non-zero ε can occur, i.e. whether the above countable closed set can be taken to be t0u. However our proof excludes exceptional ε from p´ε 0 , ε 0 qzt0u, for some ε 0 ą 0. Moreover, for small ε these conical singularities occur either as intersections between the first and second band dispersion surfaces or between the second and third dispersion surfaces. As ε increases, there continue to be such conical intersections of dispersion surfaces, but we do not control which dispersion surfaces intersect.
2. Theorem 9.1, which states that the conical singularities of the dispersion surface of H pεq for ε RC, are robust in the following sense: Let W be Λ h -periodic and inversion-symmetric, but not necessarily R-invariant. Then, for all sufficiently small real η, the operator Hpηq " H pεq`η W has a dispersion surface with a conical-type singularity. Furthermore, these conical singularities will typically not occur at the vertices of the Brillouin zone, B h ; see also the numerical results in [3]. In Remark 9.2 we show instability of Dirac points to certain perturbations, e.g. perturbations W which are Λ h -periodic but not inversion-symmetric. The dispersion surface is smooth in this case.
The paper is structured as follows. In section 2 we briefly outline the spectral theory of general periodic potentials. We then introduce Λ h , the particular lattice (Bravais lattice) used to generate a honeycomb structure or "honeycomb lattice", the union of two interpenetrating triangular lattices. Section 2 concludes with implications for Fourier analysis in this setting. Section 3 contains a discussion of the spectrum of the Laplacian on L 2 k , the subspace of L 2 satisfying pseudo-periodic boundary conditions with quasi-momentum k P B h , the Brillouin zone. We observe that degenerate eigenvalues of multiplicity three occur at the vertices of B h . In section 4 we state and prove Theorem 4.1 which reduces the construction of conical singularities of the dispersion surface at the vertices of B h to establishing the existence of two-dimensional R´invariant eigenspaces of H pεq for quasi-momenta at the vertices of B h . In section 5 we give a precise statement of our main result, Theorem 5.1, on conical singularities of dispersion surfaces at the vertices of B h . In section 6 we prove for all ε sufficiently small and non-zero, by a Lyapunov-Schmidt reduction, that the degenerate, multiplicity three eigenvalue of the Laplacian splits into a multiplicity two eigenvalue and a multiplicity one eigenvalue, with associated R-invariant eigenspaces. In order to continue this result to ε large we introduce, in section 7, a globally-defined analytic function, Epµ, εq, whose zeros, counting multiplicity, are the eigenvalues of H pεq . Eigenvalues occur where an operator I`Cpµ, εq, Cpµ, εq compact, is singular. Since Cpµ, εq is not trace-class but is Hilbert-Schmidt, we work with Epµ, εq " det 2 pI`Cpµ, εqq, a renormalized determinant. In section 8, Epµ, εq and λ ε 7 (see (1.1)) are studied using techniques of complex function theory to establish the existence of Dirac points for arbitrary real values of ε, except possibly for a countable closed subset of R. In section 9 we prove Theorem 9.1, which gives conditions for the local persistence of the conical singularities. Remark 9.2 discusses perturbations which break the conical singularity and for which the dispersion surface is smooth. Appendix A contains a counterexample, illustrating the topological obstruction discussed in section 8.3.
Acknowledgments: CLF was supported by US-NSF Grant DMS-09-01040. MIW was supported in part by US-NSF Grant DMS-10-08855. The authors would like to thank Z.H. Musslimani for stimulating discussions early in this work. We are also grateful to M.J. Ablowitz, B. Altshuler, J. Conway, P. Kuchment, A. Millis and C. Marianetti.

Notation.
1. z P C ùñ z denotes the complex conjugate of z.
2.1. Floquet-Bloch Theory. Let tv 1 , v 2 u be a linearly independent set in R 2 . Consider the lattice The fundamental period cell is denoted Denote by L 2 per,Λ " L 2 pR 2 {Λq, the space of L 2 loc functions which are periodic with the respect to the lattice Λ, or equivalently functions in L 2 on the torus R 2 {Λ " T 2 : More generally, we consider functions satisfying a pseudo-periodic boundary condition: We shall suppress the dependence on the period-lattice, Λ, and write L 2 k , if the choice of lattice is clear from context. For f and g in L 2 k,Λ , f g is locally integrable and Λperiodic and we define their inner product by: In a standard way, one can introduce the Sobolev spaces H s k,Λ . The dual lattice, Λ˚, is defined to be Λ˚" tm 1 k 1`m2 k 2 : m 1 , m 2 P Zu " Zk 1 ' Zk 2 , (2.5) where k 1 and k 2 are dual lattice vectors, satisfying the relations: If f P L 2 per,Λ then f can be expanded in a Fourier series with Fourier coefficientŝ f " tf m u mPZ 2 : Let V pxq denote a real-valued potential which is periodic with respect to Λ, i.e.
V px`vq " V pxq, for x P R 2 , v P Λ Throughout this paper we shall also assume the potential, V pxq, under consideration is C 8 . Thus, We expect that this smoothness assumption can be relaxed considerably without much extra work.
For each k P R 2 we consider the Floquet-Bloch eigenvalue problem where An L 2 k -solution of (2.9)-(2.10) is called a Floquet-Bloch state. Since the eigenvalue problem (2.9)-(2.10) is invariant under the change k Þ Ñ k`k, wherek P Λ˚, the dual period lattice, the eigenvalues and eigenfunctions of (2.9)-(2.10) can be regarded as Λ˚´periodic functions of k, or functions on T 2 k " R 2 k {Λ˚. Therefore, it suffices to restrict our attention to k varying over any primitive cell. It is standard to work with the first Brillouin zone, B, the closure of the set of points k P R 2 , which are closer to the origin than to any other lattice point.
An alternative formulation is obtained as follows. For every k P B we set φpx; kq " e ik¨x ppx; kq (2.12) Then ppx; kq satisfies the periodic elliptic boundary value problem: where The eigenvalue problem (2.9)-(2.10), or equivalently (2.13)-(2.14), has a discrete spectrum: with eigenpairs p b px; kq, µ b pkq : b " 1, 2, 3, . . . . The set tp b px; kqu bě1 can be taken to be a complete orthonormal set in L 2 per pR 2 {Λq. The functions µ b pkq are called band dispersion functions. Some general results on their regularity appear in [2]. As k varies over B, µ b pkq sweeps out a closed real interval. The spectrum of´∆`V pxq in L 2 pR 2 q is the union of these closed intervals: where the sum converges in the L 2 norm.
, a ą 0. (2.18) The dual lattice Λh " Zk 1 ' Zk 2 is spanned by the dual basis vectors: The Brillouin zone, B h , is a hexagon in R 2 ; see figure 2.1. Denote by K and K 1 the vertices of B h given by: All six vertices of B h can be generated by application of the rotation matrix, R, which rotates a vector in R 2 clockwise by 2π{3. R is given by and the vertices of B h fall into to groups, generated by action of R on K and K 1 : Remark 2.1 (Symmetry Reduction). Let p φpx, ; kq, µpkq q denote a Floquet-Bloch eigenpair for the eigenvalue problem (2.9)-(2.10) with quasi-momentum k. Since V is real, pφpx; kq " φpx; kq, µpkq q is a Floquet-Bloch eigenpair for the eigenvalue problem with quasi-momentum´k. Recall the relations (2.25) and the Λh-periodicity of: k Þ Ñ µpkq, are k Þ Ñ φpx; kq. It follows that the local character of the dispersion surfaces in a neighborhood of any vertex of B h is determined by its character about any other vertex of B h .
In our computations using Fourier series, we shall frequently make use of the following relations: Moreover, R˚maps the period lattice Λ h to itself and, in particular,

Honeycomb lattice potentials.
For any function f , defined on R 2 , introduce Rrf spxq " f pR˚xq, (2.30) where R is the 2ˆ2 rotation matrix displayed in (2.24).

Definition 2.1. [Honeycomb lattice potentials]
Let V be real-valued and V P C 8 pR 2 q. V is a honeycomb lattice potential if there exists x 0 P R 2 such thatṼ pxq " V px´x 0 q has the following properties: 1.Ṽ is Λ h´p eriodic, i.e.Ṽ px`vq "Ṽ pxq for all x P R 2 and v P Λ h . 2.Ṽ is even or inversion-symmetric, i.e.Ṽ p´xq "Ṽ pxq.
RrṼ spxq "Ṽ pR˚xq "Ṽ pxq, where, R˚is the counter-clockwise rotation matrix by 2π{3, i.e. R˚" R´1, where R is given by (2.24). Thus, a honeycomb lattice potential is smooth, Λ h -periodic and, with respect to some origin of coordinates, both inversion symmetric and R-invariant. Remark 2.2. As the spectral properties are independent of translation of the potential we shall assume in the proofs, without any loss of generality, that x 0 " 0. Remark 2.3. A consequence of a honeycomb lattice potential being real-valued and even is that if pφpx; kq, µq is an eigenpair with quasimomentum k of the Floquet-Bloch eigenvalue problem, then´φp´x; kq, µ¯is also an eigenpair with quasimomentum k.
Remark 2.4. We present two constructions of honeycomb lattice potentials. Example 1: "Atomic" honeycomb lattice potentials: Start with the two points A " p0, 0q, and B " aˆ1 ? 3 , 0˙, (2.31) which lie within the unit period cell of Λ h ; see (2.18). Define the triangular lattices of A-type and B-type points: We define the honeycomb structure, H, to be the union of these two triangular lattices: (2.33) see Figure 2.1. Let V 0 be a smooth, radial and rapidly decreasing function, which we think of as an "atomic potential". Then, is a potential associated with "atoms" at each site of the honeycomb structure H. Moreover, V pxq is a honeycomb lattice potential in the sense of Definition 2.1 with x 0 " B. Example 2: Optical honeycomb lattice potentials: The electric field envelope of a nearly monochromatic beam of light propagating through a dielectric medium with two-dimensional refractive index profile satisfies a linear Schrödinger equation iB z ψ "´∆ x,y ψ`V px, yqψ " 0. Here, z denotes the direction of propagation of the beam and px, yq the transverse directions. Honeycomb lattice potentials have been generated taking advantage of nonlinear optical phenomena. It was demonstrated in [15] that a honeycomb lattice potential (a honeycomb "photonic lattice"), V px, yq, can generated through an optical induction technique based on the interference of three plane wave beams of light within a photorefractive crystal, exhibiting the defocusing (nonlinear) optical Kerr effect. The refractive-index variations are governed by a potential of the approximate form: It is straightforward to check, in view of (2.26), that a potential of this type is a honeycomb lattice potential in the sense of Definition 2.1 with x 0 " 0. In fact, in Proposition 2.3 below we assert that with respect to some origin of coordinates, any honeycomb lattice potential can be expressed as a Fourier series of terms of this type. . The following proposition plays a key role. It states that at distinguished points in k´space, namely the K and K 1 type points, H V with quasi-momentum dependent boundary conditions (2.10) or equivalently, H V pkq, with Λ h periodic boundary conditions, has an extra rotational invariance property.
Proposition 2.2. Assume V is a honeycomb lattice potential, as in Definition 2.1. Assume K ‹ is a point of K or K 1 type; see (2.25). Then, H and R map a dense subspace of L 2 K‹ to itself. Furthermore, restricted to this dense subspace of L 2 K‹ , the commutator rH, Rs " H R´R H vanishes. In particular, if φpx, kq is a solution of the Floquet-Bloch eigenvalue problem (2.9)-(2.10) with k " K ‹ , then Rrφp¨, kqspxq is also a solution of (2.9)-(2.10) with k " K ‹ .
Proof. Take as a dense subspace C 8 K‹ , the space of C 8 functions satisfying f pxv q " e iK‹¨v f pxq for all x P R 2 and v P Λ h . Clearly, H maps C 8 K‹ to itself. Define φ R pxq " Rrφp¨, K ‹ qspxq " φpR˚x, K ‹ q. Without loss of generality, assume K ‹ " K.
Thus, we have R maps C 8 K‹ to itself. Next note that by invariance of the Laplacian under rotations,´∆ x φ R pxq " ∆ y φpy, K ‹ q| y"R˚x . Furthermore, by R´invariance of V pxq, that V pxqφ R pxq " V pR˚xqφpR˚x, K ‹ q " V pyqφpy, K ‹ q| y"R˚x . Therefore, rH, Rs vanishes on on C 8 K‹ . In particular, we have that This completes the proof of the proposition.
We conclude this section with a discussion of the Fourier representation of honeycomb lattice potentials in the sense of Definition 2.1. Let V pxq be such a potential with Fourier series: Since V pxq " RrV spxq, we have Therefore, V m1,m2 " V´m 2 ,m1´m2 . Similarly, V pxq " R 2 rV spxq implies that V m1,m2 " V m2´m1,´m1 . Introduce the mappingR : Z 2 Ñ Z 2 acting on the indices of the Fourier coefficients of V : Rpm 1 , m 2 q " p´m 2 , m 1´m2 q and thereforẽ R 2 pm 1 , m 2 q " pm 2´m1 ,´m 1 q, andR 3 pm 1 , m 2 q " pm 1 , m 2 q . (2.35) Note thatR0 " 0 and that 0 is the unique element of the kernel ofR. Furthermore, any m ‰ 0 lies on anR´orbit of length exactly three. Indeed, m "Rm Ø pm 1 , m 2 q " p´m 2 , m 1´m2 q ùñ m 1 " m 2 " 0 and m "R 2 m Ø pm 1 , m 2 q " p´m 1`m2 ,´m 1 q ùñ m 1 " m 2 " 0 .
Suppose m and n are non-zero. We say that m " n if m and n lie on the same 3ć ycle. The relation " is an equivalence relation, which partitions Zzt0u into equivalence classes, pZzt0uq { ". LetS denote a set consisting of exactly one representative from each equivalence class. We now have the following characterization of Fourier series of honeycomb lattice potentials: Proposition 2.3. Let V pxq denote a honeycomb lattice potential. Then,

2.38)
The relation V pxq " pV pxq`V p´xqq{2 and (2.38) imply V pxq "V p0q`ÿ mPS V m´c ospmk¨xq`cos´pRmqk¨x¯`cos´pR 2 mqk¨x¯( 2.39) Moreover, since V is real and even, V m is real for m P Z 2 . This completes the proof.

Fourier analysis in L 2
K‹ . We characterize the Fourier series of functions φ P L 2 K‹ , i.e. functions φpx; K ‹ q, satisfying the quasiperiodic boundary condition: The discussion is analogous to that preceding Proposition 2.3.
We are interested in the general Fourier expansion of functions in each of the eigenspaces of R: Since R is unitary these subspaces are pairwise orthogonal.
Fix, without loss of generality, K ‹ " K. We first consider the action of R on general φ P L 2 K . Applying R to φ, given by (2.42), we obtain: Thus, c Rφ p´m 2 , m 1´m2`1 q " c φ pm 1 , m 2 q, or equivalently c Rφ pm 1 , m 2 q " c φ pm 2´m1´1 ,´m 1 q . (2.47) Similarly, by a second application of R, and using the relation we have c R 2 φ pm 2´m1´1 ,´m 1 q " c φ pm 1 , m 2 q, or equivalently c R 2 φ pm 1 , m 2 q " c φ p´m 2 , m 1´m2`1 q (2.49) Finally, since R 3 " I, c R 3 φ pm 1 , m 2 q " c φ pm 1 , m 2 q. R acting in L 2 K induces a decomposition of Z 2 into orbits of length three: For convenience we shall abuse notation and write Rm " Rpm 1 , m 2 q " p´m 2 , m 1´m2`1 q R 2 m " R 2 pm 1 , m 2 q " pm 2´m1´1 ,´m 1 q, R 3 m " Id pm 1 , m 2 q " pm 1 , m 2 q (2.51) Using the notation (2.51), relations (2.47), (2.49) and (2.51) can be expressed as: c Rφ pmq " c φ pR 2 mq " c φ pm 2´m1´1 ,´m 1 q c R 2 φ pmq " c φ pRmq " c φ p´m 2 , m 1´m2`1 q (2.52) Furthermore, by (2.46) and (2.48) Each point in Z 2 lies on an orbit of R of precisely length 3, a 3-cycle. To see this, note that by (2.51) R 3 m " m for all m P Z 2 . So we need only check that there are no solutions to either Rm " m or to R 2 m " m. First, suppose Rm " m. Then, R 2 m " m as well. So, on the one hand the centroid of m, Rm and R 2 m is equal to m P Z 2 . On the other hand, by (2.51) their centroid is p´1{3, 1{3q R Z 2 , a contradiction. Therefore, there are no Z 2 solutions of Rm " m. Now if R 2 m " m, then applying R to this relation yields m " R 3 m " Rm, and we're back to the previous case.
We shall say that two points in Z 2 , m and n are equivalent, m " n, if they lie on the same 3-cycle of R. We identify all equivalent points by introducing the set of equivalence classes, Z 2 { " . Definition 2.4. We denote by S a set consisting of exactly one representative of each equivalence class in Z 2 { " . For example, tp0, 0q, p0, 1q, p´1, 0qu P Z 2 { ", from which we choose p0, 1q as its representative in S.
Using the relations (2.51), we can express the Fourier series of an arbitrary φ P L 2 K as a sum over 3-cycles of R: where R j m, j " 1, 2 is given in (2.51).
We now turn to the Fourier representation of elements of the subspaces L 2 K,1 , L 2 Moreover, where Rm and R 2 m are defined in (2.51).
Proposition 2.5 can now be used to find a representation of the eigenspaces of R. We state the result for an arbitrary point, K ‹ , of K or K 1 type. (2.62) 3. If φ 1 P L 2 K‹,τ is given by (2.65) We summarize the preceding in a result which facilitates the study of H on L 2 K‹ in terms of the action of R on invariant subspaces of H.
Proposition 2.7. Let K ‹ denote a point of K or K 1 type, R denote the 2π{3 clockwise rotation matrix (see (2.24)) and Rrf spxq " f pR˚xq. Then R, acting on L 2 K‹ has eigenvalues 1, τ andτ " τ 2 inducing a corresponding orthogonal sum decomposition of L 2 K‹ into eigenspaces: The elements of each summand are represented as in Proposition 2.6. Remark 2.5. Since, by Proposition 2.2, H and R commute on L 2 K‹ , the spectral theory of H in L 2 K‹ can be reduced to its independent study in each of the eigenspaces in the orthogonal sum (2.66).

Spectral properties of H p0q in L 2
K‹ -Degeneracy at K and K 1 points . Our starting point for the study of H pεq on L 2 K‹ is the study of H p0q "´∆. Consider the eigenvalue problem The eigenvalue problem (3.1), (3.2) has solutions of the form: with associated eigenvalues 2. Restricted to each of the R´invariant subspaces of H p0q has an eigenvalue µ p0q " |K ‹ | 2 of multiplicity one with eigenspaces: 3. µ p0q is the lowest eigenvalue of H p0q in L 2 K‹ . Proof. Without loss of generality, let K ‹ " K. Since R is orthogonal, |K| " |RK| " |R 2 K|. Therefore,´∆Ψ " |K| 2 Ψ for Ψ " e iK¨x , e iRK¨x and e iR 2 K¨x . It follows that µ p0q " |K| 2 is an eigenvalue of multiplicity at least three. To show that the multiplicity is exactly three, we seek all m " pm 1 , m 2 q for which |K m | 2 " |K| 2 .
Recall that for each k P B h , the L 2 k eigenvalues of H p0q are ordered (2.16): We shall see in section 6 that for small ε, the spectrum L 2 K perturbs to either In either case, the multiplicity three eigenvalue splits into a multiplicity two eigenvalue and a simple eigenvalue. The connection between the double eigenvalue and conical singularities of the dispersion surface is explained in the next section; see Theorem 4.1. We shall see from Theorem 5.1, or rather its proof (in section 6) that for all small ε, conical singularities occur at all vertices K ‹ of B h , and that these occur at the intersection point of the first and second band dispersion surfaces in the case of (3.7), and at the intersection of the second and third bands in the case of (3.8). As ε increases, we continue to have such conical intersections of dispersion surfaces, but we do not control which dispersion surfaces intersect.
4. Multiplicity two L 2 K eigenvalues of H pεq and conical singularities. Let K ‹ a point of K or K 1 type. In this section we show that if H acting in L 2 K‹ has a dimension two eigenspace E τ ' Eτ , where E τ and Eτ are dimension one subspaces of L 2 K‹,τ and L 2 K‹,τ , respectively, then the dispersion surface is conical in a neighborhood of K ‹ . Recall that we assume V P C 8 pR 2 {Λ h q.
Then H acting on L 2 has a dispersion surface which, in a neighborhood of k " K ‹ , is conical. That is, for k´K ‹ near 0, there are two distinct branches of eigenvalues of the Floquet-Bloch eigenvalue problem with quasi-momentum, k: where E˘pκq " Op|κ|q as |κ| Ñ 0 and E˘are Lipschitz continuous functions in a neighborhood of 0. Remark 4.1.
1. Elliptic regularity implies that the eigenfunctions Φ j , j " 1, 2 are in H 2 pR 2 q. Therefore, ř mPS p1`|m| 2 q 2 |cpmq| 2 ă 8. We conclude that the sum defining λ 7 converges. 2. In section 6 we study the case of "weak" or small potentials, i.e. V " εV h with ε small. For all ε such that 0 ă |ε| ă ε 0 , where ε 0 is a sufficiently small positive number, we will: (i) verify the double eigenvalue hypothesis (h1) of Proof of Theorem 4.1: By Symmetry Remark 2.1, we may without loss of generality consider the specific B h vertex: K ‹ " K. The local character of all others is identical. We consider a perturbation of K, K`κ, with |κ| small. We express Φ P L 2 k as Φpx; kq " e ik¨x ψpx; kq, where ψpx; kq is Λ-periodic. The eigenvalue problem for k " K`κ takes the form:´p Let µ 0 " µ p0q " µpKq be the double eigenvalue and let ψ p0q be in the corresponding two-dimensional eigenspace. Express µpK`κq and ψpx; K`κq as: where ψ p1q is to be chosen orthogonal to the nullspace of HpKq´µ p0q I, and µ p1q are corrections to be determined. Substituting (4.6) into the eigenvalue problem (4.4)-(4.5) we obtain: Since ψ p0q is in the L 2 per,Λ -nullspace of HpKq´µ 0 I, we write it as Here φ 1 and φ 2 are normalized eigenstates with Fourier expansions as in part 3 of Proposition 2.6 and α, β are constants to be determined.
We now turn to the construction of ψ p1q . Introduce the orthogonal projections: Q , onto the two-dimensional kernel of HpKq´µ 0 I, and Q K " I´Q . Note that (4.10) We next seek a solution to (4.7) by solving the following system for ψ p1q and µ p1q : Equation (4.12) is a system of two equations obtained by setting the projections of F p1q onto φ 1 and φ 2 equal to zero. Our strategy is to solve (4.11) for ψ p1q as a continuous functional of α, β, κ, µ p1q with appropriate estimates, then substitute the result into (4.12) to obtain a closed bifurcation equation. This is a linear homogeneous system of the form Mpµ p1q , κqpα, βq t " 0. The function κ Þ Ñ µ p1q pκq is then determined by the condition that det Mpµ p1q , κq " 0.
Written out in detail, the system (4.11)-(4.12) becomes: Introduce the resolvent operator: for ψ p1q can be rewritten as: In several equations above we have used (4.10). By elliptic regularity, the mapping is a bounded operator on H s pR 2 {Λ h q, for any s. Furthermore, for |κ|`|µ p1q | sufficiently small, the operator norm of A is less than one, pI`Aq´1 exists, and hence (4.15) is uniquely solvable in Q K H 2 pR 2 {Λ h q: Since ψ p0q is given by (4.8), ψ p1q is clearly linear in α and β and we write: where pκ, µ p1q q Þ Ñ c pjq rκ, µ p1q s is a smooth mapping from a neighborhood of p0, 0q P R 2ˆC into H 2 pR 2 {Λ h q satisfying the bound: Note that Q c pjq " 0, j " 1, 2. We next substitute (4.16) into (4.14) to obtain a system of two homogeneous linear equations for α and β. Using the relations: we have: where Mpµ p1q , κq is the 2ˆ2 matrix given by: Thus, µpK`κq " µ p0q`µp1q is an eigenvalue for the spectral problem Equation (4.19) is an equation for µ p1q , which characterizes the splitting of the double eigenvalue at κ ‰ 0. We now proceed to show that if the nondegeneracy condition (4.1) holds, then the solution set of (4.19) is locally conic.
We anticipate that a solution µ p1q " Op|κ|q and hence C pjq " Op|κ|q. This motivates expanding M as: Now we claim that, in a neighborhood of κ " 0, the solutions of (4.19) are well approximated by those of the truncated equation: We shall first solve (4.23) (Proposition 4.2) and then show that the solutions of (4. 19) are small corrections to these (Proposition 4.3).
We prove Proposition 4.2 just below. A consequence is that M 0 simplifies to and therefore Therefore, the truncation of M to M 0 , yields det M 0 pν, κq " 0 and a locally conical dispersion relation, provided λ 7 ‰ 0.
To complete the proof of Theorem 4.1 we next show that the local character of solutions to (4.19) is, for |κ| small, essentially that derived in Proposition 4.2 for the solutions of (4.23). Proposition 4.3. Suppose λ 7 , defined in (4.33), is non-zero. Then, in a neighborhood of any K or K 1 point, the dispersion surface is conic. Specifically, the eigenvalue equation det Mpµ p1q , κq " 0 (see (4.19)) defines, in a neighborhood U Ă R 2 of κ " 0, two functions: where E˘pκq Ñ 0 as κ Ñ 0 and E˘pκq is Lipschitz continuous in κ.

Main
Theorem: Conical singularity in dispersion surfaces. Assume that V is a honeycomb lattice potential in the sense of Definition 2.1. Since V P C 8 pR 2 {Λ h q, its Fourier coefficients satisfŷ Theorem 5.1. Conical singularities and the dispersion surfaces of H pεq Let V pxq honeycomb lattice potential. Assume further that the Fourier coefficient of V , V 1,1 , is non-vanishing, i.e.
There exists a countable and closed setC Ă R such that for any vertex K ‹ of B h and all ε RC the following holds: 1. There exists a Floquet-Bloch eigenpair Φ ε px; K ‹ q, µ ε pK ‹ q such that µ ε pK ‹ q is an L 2 K,τ -eigenvalue of H pεq of multiplicity one, with corresponding eigenfunction, Φ ε px; K ‹ q.
µ ε pK ‹ q is an L 2 K,τ -eigenvalue of H pεq of multiplicity one, with corresponding eigenfunction, Φ ε p´x; K ‹ q.
Remark 5.2. Part 3 of Theorem 5.1 gives conditions for intersections of the first and second band dispersion surfaces or interesections of the second and third. As the magnitude of ε is increased it is possible that there are crossings among the L 2 K,σeigenvalues of H pεq , so in general the theorem does not specify which band dispersion surfaces intersect.

5.1.
Outline of the proof of Theorem 5.1. By Symmetry Remark 2.1, it suffices to prove Theorem 5.1 for K ‹ " K. We have seen that the central point is to verify for all ε, except possibly those in a closed countable exceptional set, that hypotheses (h1-h3) of Theorem 4.1 hold. These hypotheses state that H pεq has simple L 2 K‹,τ and L 2 K‹,τ eigenvalues which are related by symmetry, which are not L 2 K‹,1eigenvalues, and moreover that λ ε 7 ‰ 0. We proceed as follows. In section 6 we show that there is a positive number, ε 0 , such that for all ε P p´ε 0 , ε 0 qzt0u (h1-h3) of Theorem 4.1 hold. That is, the conclusions of Theorem 5.1 hold for all sufficiently small, non-zero ε. In section 7.8 we introduce the key tool, a renormalized determinant, to detect and track the L 2 K,σ eigenvalues of H pεq for σ " 1, τ,τ . A continuation argument is then implemented using tools from complex function theory in section 8, to pass to large ε. We now embark on the detailed proofs.
6. Proof of Main Theorem 5.1 for small ε. We begin the proof of Theorem 5.1 by first establishing it for some interval´ε 0 ă ε ă ε 0 , where ε 0 is positive but possibly small. We shall consider the eigenvalue problem for H pεq on the three eigenspaces of R: L 2 K‹,τ , L 2 K‹,τ and L 2 K‹,1 : An eigenstate Φpx; K ‹ q in L 2 K‹,σ is, by Proposition 2.6, of the form: The summation is over the set, S, introduced in Definition 2.4. Note that by Proposition 2.6 and Remark 2.3, solutions to the eigenvalue problem on L 2 K‹,τ can be obtained from those in L 2 K‹,τ via the symmetry: Φpxq Þ Ñ Φp´xq. Recall that c Φ pmq or cpm; Φq denote the L 2 K‹,σ -Fourier coefficients of Φ. Our next task is to reformulate the eigenvalue problem (6.1) as an equivalent algebraic problem for the Fourier coefficients tcpm; Φp¨; K ‹ qqu mPS . First, applying´∆´µ to Φ, given by (6.2), and using that R is orthogonal, we have that Therefore, by Proposition 2.6, V Φp¨; K ‹ q has the expansion Furthermore, with the notation qk¨x " pq 1 k 1`q2 k 2 q¨x, where (recall (2.51)) Summarizing, we have Proposition 6.1. Let σ P t1, τ,τ u. Then, the spectral problem (6.1) on L 2 K‹,σ is equivalent to algebraic eigenvalue problem for cpmq " cpm; Φq and µ: where tcpmqu mPS P l 2 pSq.
To fix ideas, let K ‹ " K; starting with K 1 , we would proceed similarly. For ε " 0, we have the algebraic eigenvalue problem: Equation (6.8), viewed as an eigenvalue problem for ptcpmqu mPZ 2 , µq is equivalent to the eigenvalue problem for´∆ on L 2 K treated in Proposition 3.1. Restated in terms of Fourier coefficients, Proposition 3.1 states that µ p0q " |K| 2 is an eigenvalue of multiplicity three with corresponding eigenvectors: Recall from Definition 2.4 that the equivalence class of indices tp0, 0q, p0, 1q, p´1, 0qu has as its representative in S the point p0, 1q.
The eigenvalue problem (6.8) has a one dimensional L 2 K,σ -eigenspace with eigenpair: e seek a solution of (6.7) for ε varying in a small open interval about ε " 0. We proceed via a Lyapunov-Schmidt reduction argument. First, decompose the system (6.7) into coupled equations for: c " cp0, 1q P C, and tc K pmqu mPS K P l 2 pS K q , (6.9) where S K " Sztp0, 1qu . (6.10) and rewrite (6.7) as a coupled system for c and c K : We next seek a solution of (6.11)-(6.12), for ε small, in a neighborhood of the solution to the ε " 0 problem: c 0 " 1, µ p0q " |K| 2 , c K prq " 0, r P S K .
We begin by solving the second equation in (6.12) for c K as a function of the scalar parameter c . For ε small, the operator to be inverted is diagonally dominant with diagonal elements: |K m | 2´µ , which we bound from below for m P S K . By the relations (2.22) we have 3 .

(6.30)
This completes the proof of our main theorem, Theorem 5.1, for the case where ε is taken to be sufficiently small. We now turn to extending Theorem 5.1 to large ε.

Characterization of eigenvalues of H pεq for large ε.
To extend the assertions of Theorem 5.1 to large values of ε, we introduce a characterization of the L 2 K,σ -eigenvalues of the eigenvalue problem (6.1) as zeros of an analytic function of ε.
Since we can add an arbitrary constant to the potential, by redefinition of the eigenvalue parameter, µ, we may assume without loss of generality that 0 ď V pxq ď V max .
Assume first that ε P C and ℜε ą 0. Then, H pεq´µ I "´∆`εV´µI " p´∆`εV`Iq´pµ`1qI. The eigenvalue problem (6.1) may be rewritten as p´∆`εV`Iq Φ´pµ`1qΦ " 0, u P L 2 K,σ . (7.1) Now for any real ε ą 0 we have´∆`εV`I ě I. Hence we introduce ; T pεq " pI´∆`εV q´1, which exists as a bounded operator from L 2 K,σ to H 2 K,σ and obtain the following Lippmann -Schwinger equation, equivalent to the eigenvalue problem (6.1): We now show that if ℜε ă 0, we also obtain an equation of the same type as in (7.3). In this case, we observe that ε pV´V max q ě 0. Therefore,´∆`ε pV´V max qÌ ě I and we rewrite (6.1) as p´∆`ε pV´V max q`Iq Φ´pµ`1´εV max q Φ " 0, Φ P L 2 K,σ . If for ε ă 0 we defineT pεq " I´∆`ε pV´V max q, then (6.1) is equivalent to For the remainder of this section we shall assume ℜε ą 0 and work with the form of the eigenvalue problem given in (7.3). The analysis below applies with only trivial modifications to the case ε ă 0 and the form of the eigenvalue problem given in (7.5).
For each ε ą 0, we would like to characterize L 2 K,σ -eigenvalues, µpεq, as points where the determinant of the operator I´pµ`1q T pεq vanishes. To define the determinant of I´zT , one requires that T be trace class. Although T pεq is compact on L 2 K,σ , it is not trace class. Indeed, in spatial dimension two, λ j , the j th eigenvalue of ∆ K`W acting in L 2 per,Λ satisfies the asymptotics λ j " |j| (Weyl). Therefore The divergence of the determinant can be removed if we work with the regularized or renormalized determinant; see [7,13]. Note that T pεq is Hilbert Schmidt, i.e.
For a Hilbert-Schmidt operator, A, i.e. trpA 2 q ă 8, define R 2 pAq " rI`A s e´A´I. Note that I`A is singular if and only if I`R 2 pAq " pI`Aqe´A is singular.
3. For ε real, µ is an L 2 K,σ -eigenvalue of the eigenvalue problem (6.1) if and only if E σ pµ, εq " 0 . (7.10) 4. For ε real, µ is an L 2 K,σ eigenvalue of (6.1) of geometric multiplicity m if and only if µ is a root of (7.10) of multiplicity m.
We next study the persistence of properties I.-IV. for ε of arbitrary size. Our continuation strategy is based on the following general Lemma 8.1. Let A Ă pε 0 , 8q with ε 0 ą 0. Then one of the following assertions holds: (1) A is contained in a closed countable set.
(2) There exists ε c P p0, 8q for which the set A X r0, ε c q is contained in a closed countable set, but for any ε 1 ą ε c , the set A X r0, ε 1 q is not contained in a closed countable set.
The main work of this section is to prove (8.2) for A " A by precluding option (2) of Lemma 8.1. This suggests introducing the notion of a critical value of ε: Definition 8.2 (Critical ε c ). Call a real and positive number ε c critical if there is an increasing sequence tε ν u tending to ε c and a corresponding sequence of geometric multiplicity-two L 2 K -eigenvalues, tµ ν u, such that (a) properties I.-IV. above, with ε replaced by ε ν and µ ε replaced by µ ν , hold for all ν " 1, 2, . . . , and (b) for ε " ε c and µ c " µ εc " lim νÑ8 µ ν ă 8 at least one of the properties I.-IV. does not hold.
First let's use Lemma 8.3 to prove Lemma 8.1. We then give the proof of Lemma 8.3.
Proof of Lemma 8.1: Let ε c " suptε P p0, 8q : A X r0, εq is contained in a closed countable set.u. Clearly 0 ă ε 0 ď ε c ď 8. If ε c " 8, then option (1) holds, thanks to Lemma 8.3. And if ε c ă 8, then by definition, A X r0, ε 1 q is not contained in a closed countable set for any ε 1 ą ε c . Again applying Lemma 8.3 shows that A X r0, ε c q is contained in a countable closed set. In this case, (2) holds and the proof of Lemma 8.1 is complete.
Remark 8.1. Our proof will in fact show that defined on R`is a function ε Þ Ñ βpεq, analytic on the connected components of the complement of a countable closed set C and such that: (a) βpεq is a multiplicity two L 2 K -eigenvalue of H pεq and (b) λ ε 7 " λ 7 pβpεq, εq ‰ 0, outside C.
In section 8.2 we develop general complex function theoretic tools concerning the variation of zeros, λ, of a real-analytic function, P pλ, zq as a parameter, z, is varied. This result is later applied to P " E σ pµ, εq for σ " 1 and σ " τ , the renormalized determinant whose zeros are L 2 K,σ eigenvalues of H pεq ; see section 7. In section 8.3 we study the general problem of constructing non-trivial null-vectors of NˆN matrices of rank N´1, which depend analytically on the matrix entries. The results of sections 8.2 and 8.3 are applied to the construction of an eigenfunction of H pεq and a corresponding simple L 2 K,τ -eigenvalue, βpεq, which vary analytically with ε. It follows via (8.1) that λ ε 7 " λ 7 pβpεq, εq, varies analytically with ε. These results play a central role in the continuation strategy outlined above.

Picking a branch. Let
where ε 1 and ε 2 are given positive numbers. Suppose we are given an analytic function P : U Ñ C and an analytic mapping F : U Ñ C m . We make the following Assumptions: (A1) If pλ, zq P U, P pλ, zq " 0 and z P R, then λ P R.
Remark 8.2. With the above setup, we have centered the analysis about pz, λq " p0, 0q. We shall apply the results of this section to an appropriate analytic function of pµ, εq centered about pµ c , ε c q.
Under assumptions (A1) and (A2) we will prove the following Lemma 8.4. There exist δ ą 0 and a real-analytic function βpzq, defined for z P p0, δq, such that for all but at most countably many z P p0, δq we have: Moreover, lim zÑ0`β pzq " 0.
Moreover, g j pzq are analytic in |z| ă ε 4 . Note that D ě 1, since Assumption (A2) implies P p0, 0q " 0. For k ě 1, define The right hand side of (8.7) is a symmetric polynomial in α 1 pzq, . . . , α D pzq and is therefore a polynomial in the coefficients g j pzq of P pλ, zq [6], which are analytic in z. Consequently, each Q k pzq is an analytic function of z. Moreover, when z is real, the α ν pzq are also real, and therefore, for z P R, Q k pzq ‰ 0 if and only if λ Þ Ñ P pλ, zq has at least k distinct zeros. In particular, for k ě D`1, Q k pzq " 0 for all real z, since λ Þ Ñ P pλ, zq has only D zeros; see (8.5). Hence, there existsk with 1 ďk ď D such that Qkpzq is not identically zero, but Q k pzq " 0 for all k ąk.
Unfortunately, the above j may depend on z. However, we may simply fix some x 0 P p0, δq, and pick j 0 such that m j0 " 1 and F pβ j0 px 0 q, x 0 q ‰ 0. The function z Þ Ñ β j0 pzq is a real-analytic function of z P p0, δq. Moreover, we know that the real analytic function z Þ Ñ F pβ j0 pzq, zq is not identically zero on p0, δq, since it is nonzero for z " x 0 . So, it can vanish only on a set of discrete points which accumulates at 0 or at δ. The proof of Lemma 8.4 is complete.
Remark 8.3. We have proven more than asserted in Lemma 8.4. In fact, P pβpzq, zq " 0 and B λ P pβpzq, zq ‰ 0 for all z P p0, δq; and F pβpzq, zq ‰ 0 for all z P p0, δq except perhaps for countably many z tending to 0. Note that we can arrange for this countable sequence not to accumulate at δ by simply taking δ to be slightly smaller.

Linear Algebra.
Given an NˆN (complex) matrix A of rank N´1, we would like to produce a nonzero vector in the nullspace of A, depending analytically on the entries of A. In general there is a topological obstruction to this; see Appendix A. However, the following result will be enough for our purposes.
Fix N ě 1. Let MatpN q be the space of all complex NˆN matrices. We denote an NˆN matrix by A P MatpN q. We say that a map Γ : MatpN q Ñ C N is a polynomial map if the components of ΓpAq are polynomials in the entries of A. Polynomial maps are therefore analytic in the entries of A.
In this section we prove Lemma 8.5. There exist polynomial maps Γ jk : MatpN q Ñ C N , where j, k " 1, . . . , N , with the following property: Let A P MatpN q have rank N´1. Then all the vectors Γ jk pAq belong to the nullspace of A, and at least one of these vectors is non-zero.
Proof of Lemma 8.5: We begin by setting up some notation. Given A P MatpN q, we write A pj,kq to denote the matrix obtained from A by deleting row j and column k. We write ColpA, kq to denote the k th column of A. If v " pv 1 , . . . , v N q t P C N is a column vector, then we write v j to denote the j th coordinate of v, and writev pkq to denote the column vector obtained from v by deleting the k th coordinate. Thus, v pkq P C N´1 .
From linear algebra, we recall that For any A P MatpN q of rank N´1, we have det´A pj,kq¯‰ 0 for some pj, kq . If rankpAq " N´1 and det A pj,kq ‰ 0, then the space of solutions of (8.13), and the nullspace of A are one-dimensional; hence, in this case the nullspace of A consists precisely of the solutions of (8.13).
We now define Γ jk pAq P C N to be the element, v, in the nullspace of A, whose N components are constructed as follows: det A pj,kq ı 2 (8.14) and obtain the other N´1 entries comprising the vectorv pkq by solving If det A pj,kq ‰ 0, we can solve (8.15) uniquely forv pkq by Cramer's rule and together with (8.14) construct v.
Note that each component of the vector Γ jk pAq has the form det A pj,kqˆP olynomial in the entries of A .
If rankpAq " N´1, then some Γ jk pAq is non-zero, thanks to (8.12). Moreover, each Γ jk pAq always belongs to the nullspace of A. Indeed, fix j, k; if det A pj,kq " 0 then Γ jk pAq " 0 P NullspacepAq. If instead det A pj,kq ‰ 0, then NullspacepAq consists of the solutions of (8.13); and we defined Γ jk to solve (8.13). Thus, in all cases we have Γ jk pAq P NullspacepAq if rankpAq " N´1. The proof of Lemma 8.5 is complete. Let ε c and µ c be as in Definition 8.2.
Without loss of generality we can assume K ‹ " K. We work in the Hilbert spaces We will apply the results of section 8.2 and section 8.3, with the analysis centered at pµ c , ε c q rather than at p0, 0q; see Remark 8.2. We shall use that Let M be a positive integer, chosen below to be sufficiently large. We regard L 2 K,τ as the direct sum L 2 lo ' L 2 hi , where L 2 lo consists of Fourier series as in (8.16) such that the cpmq " 0 whenever |m| ą M , and L 2 hi consists of Fourier series as in (8.16) such that the cpmq " 0 whenever |m| ď M . Similarly, we regard H s K,τ as the direct sum H s lo ' H s hi using (8.17). We set N " dimpL 2 lo q. Let Π lo and Π hi be the projections that map a Fourier series as in (8.16) or (8.17) to the truncated Fourier series obtained by setting all the cpmq with |m| ą M , or with |m| ď M , respectively, equal to zero. We may view H pεq as the mappinĝ By choosing the frequency cutoff, M , to be sufficiently large, we have Therefore, for all pµ, εq in some fixed small neighborhood of pµ c , ε c q we have The eigenvalue problem is equivalent to the system That is, where we regard A pεq´µ I as an operator from H 2 hi to L 2 hi . Note also that B pεq ψ lo P L 2 since ψ lo P H 2 K,τ ; hence`A pεq´µ I˘´1 B pεq ψ lo P H 2 hi , thanks to (8.19), which holds under our assumption that pµ, εq is near pµ c , ε c q. It follows that Let us write out the Fourier expansions of the ψ jk pµ, εq. We have The coefficients c jk pm, µ, εq depend analytically on pµ, εq P U , where U is a small neighborhood of pµ c , ε c q, which is independent of m. Moreover, since ψ jk pµ, εq is an analytic H 2 K,τ -valued function, it follows that ÿ mPS`1`| m| 2˘2ˇcjk pm, µ, εqˇˇ2 is bounded as pµ, εq varies over U .
(8.28) (Perhaps we must shrink U to achieve (8.28).) With a view toward continuation of the Properties I.-IV. (enumerated at the start of section 8) as ε traverses any critical value (Definition 8.2), ε c we state the following Lemma 8.6. Suppose there exists a sequence of eigenvalues pµ ν , ε ν q Ñ pµ c , ε c q with 0 ă ε ν ă ε c , such that for each ν the following properties (A1)-(A4) hold: (A1) µ ν is a simple eigenvalue of H pεν q on L 2 K,τ , with eigenfunction (A2) µ ν is a simple eigenvalue of H pεν q on L 2 K,τ , with eigenfunction The following non-degeneracy condition (λ ε 7 ‰ 0) holds: where twpmqu mPS are fixed weights, such that Our choice of weights (see (4.1)) is: Then, there exist a (non-empty) open interval I " pε c , ε c`δ q, a real-valued realanalytic function βpεq defined on I, a function ϕ ε P L 2 K,τ depending on the parameter ε P I, and a countable subset C Ă I, such that the following hold: (i) p´∆`εV h q ϕ pεq " βpεqϕ pεq for each ε P I.
(iii) C has no accumulation points in I, although ε c may be an accumulation point of C. (iv) For each ε in IzC (a) βpεq is a simple eigenvalue of´∆`εV h on L 2 K,τ , (b) βpεq is not an eigenvalue of´∆`εV h on L 2 K,1 , and (c) the quantity λ ε 7 , arising from the eigenfunction ϕ pεq via formula (4.1) (with Φ 1 replaced by ϕ pεq q is non-zero. Proof of Lemma 8.6: Recall that the zeros, µ, of the renormalized determinant, E 1 pµ, εq, defined in section 7.8, are precisely the set of L 2 K,1 eigenvalues of H pεq . Thus, tracking the set of pµ, εq such that Assumptions (A1)-(A4) and in particular (A3) suggests that we introduce, for pµ, εq P U , the matrix-valued function: F jk pµ, εq is an analytic function on U . We define F pµ, εq "`F jk pµ, εq˘j ,k"1,...,N (8.32) Thus, F : U Ñ C N 2 is an analytic map. Now for each ν, (8.26) applies to pµ ν , ε ν q , since µ ν is a simple eigenvalue. Thus, for some jk, the function ψ jk pµ ν , ε ν q is a non-zero null-vector of H pεν q´µ ν I, i.e. an eigenfunction of H pεν q . Since by hypothesis ψ ν is an eigenfunction of H pεν q satisfying (8.29) with eigenvalue µ ν and since µ ν is a simple eigenvalue of H pεν q , the corresponding eigenfunction Ψ ν satisfies: (8.31) and (8.29).
We complete the proof of Lemma 8.6 by application of Lemmata 8.4 and 8.5 for appropriate choices of P pµ, εq and F pµ, εq. Let E τ pµ, εq, denote the renormalized determinant (7.9). Let P pµ, εq " E τ pµ, εq and F pµ, εq be given by (8.31), (8.32). We now check the hypotheses of Lemma 8.4. First note P pµ, εq " 0 if and only if µ is an L 2 K,τ eigenvalue of H pεq , and the multiplicity of µ as a zero of P pµ, εq is equal to its multiplicity as an eigenvalue of H pεq . (8.34) Because H pεq is self-adjoint for real ε, we see from (8.34) that if pµ, εq P U, ε is real, and P pµ, εq " 0, then µ P R.
(8.38) Moreover, for all but countably many ε P pε c , ε c`δ q, with their only possible accumulation point at ε c , we have F pβpεq, εq ‰ 0. Now unfortunately the pair pj, kq in (8.43) may depend on ε. However, (8.43) implies that for some fixed pj, kq " pj 0 , k 0 q, the function defined for ε P pε c , ε c`δ q is not identically zero. Since this function is analytic in ε, it is equal to zero at most at countably many ε. Moreover, the zeros of the function (8.44) in pε c , ε c`δ q can accumulate only at ε c and at ε c`δ . By taking δ smaller, we may assume that the zeros of the function (8.44) can only accumulate at ε c . We now set, for ε P pε c , ε c`δ q: Ψ pεq " Ψ j0,k0 pβpεq, εq P H 2 K,τ and Ψ pεq pxq " ÿ mPS c j0,k0 pm; βpεq, εq " e iK m¨x`τ e iRK m¨x`τ e iR 2 K m¨x ı .
We now have that the function µ " βpεq satisfies Properties I.-IV. for all ε P pε c , ε c`δ q except possibly along a sequence of "bad" ε's which tends to ε c . Properties I.-III., that βpεq is a eigenvalue in each of the subspaces L 2 K,τ and L 2 K,τ , and not an L 2 K,1´e igenvalue, hold for all ε P pε c , ε c`δ q, except possibly along the above sequence of bad ε's. This completes the proof of Lemma 8.6.
To complete the proof of Theorem 5.1 for ε of arbitrary size we require Lemma 8.7. For all ε P p0, 8q outside a countable closed set there exists a Floquet-Bloch eigenpair µ P R, ϕ P L 2 K,τ for´∆`εV h , with the following properties: a. |µ| ď C 0 ε`C 1 , where C 0 and C 1 depend only on V h . b. µ is a multiplicity one eigenvalue of´∆`εV h on L 2 K,τ . c. µ is not an eigenvalue of´∆`εV h on L 2 K,1 .
Proof of Lemma 8.7: Set C 0 " max |V h |. By our the analysis of section 6, there exists ε 0 ą 0, a sufficiently large constant C 1 , such that for ε P p0, ε 0 q there exist µ, ϕ satisfying (a.)-(d.) . Now suppose that Lemma 8.7 fails. Then, by Lemma 8.1 there exists ε c P p0, 8q such that for all ε P p0, ε c q outside a countable closed set, there exist µ, ϕ satisfying (a.)-(d.) but for all ε 1 c ą ε c , assertions (a.-d.) fail on a subset of p0, ε 1 c q that is not contained in any countable closed set. (8.45) We will deduce a contradiction, from which we conclude Lemma 8.7.
Sinceε was taken to be an arbitrary point of IzC, and since IzC is dense in I by (iii) of Lemma 8.6, we haveˇˇˇˇd βpεq dεˇˇˇˇď C 0 , for all ε P I. Recall that I " pε c , ε c`δ q. Our desired estimate (8.47) now follows at once from (8.46), (8.52) and (ii) of Lemma 8.6. The proof of Lemma 8.7 and therefore of Theorem 5.1 is now complete.
9. Deformed honeycomb structures and robust conical singularities. In the previous sections we established the existence of conical singularities, Dirac points, in the dispersion surface for honeycomb lattice potentials. These Dirac points are at the vertices of the Brillouin zone, B h . In this section we explore the structural stability question of whether such Dirac points persist under small, even and Λ hperiodic perturbations of a base honeycomb lattice potential. We prove the following Theorem 9.1. Let V pxq denote a honeycomb lattice potential in the sense of Definition 2.1. Let W pxq denote a smooth, even and Λ h -periodic function, which does not necessarily have honeycomb structure symmetry, i.e. W pxq is not necessarily R-invariant. Consider the operator Hpηq "´∆`V pxq`ηW pxq , (9.1) Fork " 1, we have βpεq ă λ 2 pεq.
Remark 9.2 below shows that Dirac points are unstable to typical perturbations, W P C 8 pR 2 {Λ h q, which are not even.
Remark 9.1. For η " 0, Theorem 9.1 reduces to Theorem 5.1, which covers the case of the undeformed honeycomb lattice potential. In particular, if the perturbation W is itself a honeycomb lattice potential, then K pηq ‹ " K ‹ .
(9.32) 2. By self-adjointness, for η P R and K 1 P R 2 , if µ p1q is a solution of (9.32) then µ p1q is real. 3. µ pηq is a geometric multiplicity two L 2 K pηq -eigenvalue of Hpηq if and only if the triple`µ p1q , K 1 , η˘is such that the 2ˆ2 Hermitian matrix, M`µ p1q , K 1 , ηh as zero as a double eigenvalue, i.e. M`µ p1q , K 1 , η˘is the zero matrix.
Now, up to this point we have not used the hypothesis that W pxq is an even function (inversion symmetry). We now impose this condition on W . For the case where W is not even, see Remark 9.2 at this end of this section. Claim 1: W pxq " W p´xq ùñ xΦ 1 , W Φ 1 y " xΦ 2 , W Φ 2 y and a 11 " a 22 . By (9.30), it follows that A 11 " A 22 .
Furthermore, one checks easily that a 11 " a 22 .
To prove part (2) of Theorem 9.1, we need to display a conical singularity in the dispersion surface about the point pK pηq , µ pηq q, (9.2). For this we make strong use of the calculations in the proof of Theorem 4.1. In particular,´∆`V , µ p0q , K and φ j , j " 1, 2 of the proof of Theorem 4.1 are replaced by Hpηq, µ pηq , K pηq from the proof of part (1) and tφ pηq 1 , φ pηq 2 u, now denotes an orthonormal spanning set for the L 2 pR 2 {Λ h q nullspace of Hpη; K pηq q´µ pηq I. Then, tΦ pηq 1 , Φ pηq 2 u " te iK¨x φ pηq 1 , e iK¨x φ pηq 2 u is an orthonormal spanning set for the L 2 K pηq nullspace of Hpηq´µ pηq I. Note also that Φ pη"0q j " Φ j , the Floquet-Bloch states associated with the unperturbed honeycomb lattice potential, V .
Remark 9.2 (Instability of the Dirac Point and smooth dispersion surfaces). We here note a class of perturbing potentials, W , such that although´∆`V has Dirac (conical) points, the operator´∆`V`ηW has a locally smooth dispersion surface near the vertices of B h . Assume that V is a honeycomb lattice potential, which is inversion-symmetric with respect to x " 0, i.e. x 0 " 0 in Definition 2.1, i.e. V p´xq " V pxq. Let W P C 8 pRq, Λ h -periodic, but without the requirement that W pxq " W p´xq for all x. Then, typically xΦ 1 , W Φ 1 y ‰ xΦ 2 , W Φ 2 y. In this case, A 11 pµ p1q , K 1 , ηq ‰ A 22 pµ p1q , K 1 , ηq; see (9.30) . For µ pηq " µ p0q`η µ 1,η to be an L 2 K pηq eigenvalue, we found that it is necessary and sufficient that: det M´µ p1,ηq , K 1 , η¯" 0. When, A 11´A22 ‰ 0, each sign in (9.56) gives rise to an equation to which we may apply the implicit function theorem to obtain a smooth function pK 1 , ηq Þ Ñ µ p1,ηq pK 1 , ηq.
Appendix A. Topological obstruction.
In section 8 there arises the situation of an NˆN complex matrix A varying within the space of rank N´1 matrices. It was of interest to know whether one can construct a non-zero nullvector which is an analytic function of the entries of A. In this section we provide a 2ˆ2 matrix counterexample that exhibits a topological obstruction.
Let M Ă Matp2q denote the space of 2ˆ2 complex matrices of rank 1. We prove the following Proposition A.1. There is no continuous map φ : M Ñ C 2 zt0u such that φpAq P Nullspace(A) for each A P M.
Proof. Let φ denote such a map. We proceed to derive a contradiction. For vectors v "ˆv 1 v 2˙P C 2 zt0u, define the 2ˆ2 complex rank 1 matrix: where J is skew symmetric and non-singular. Note: v Þ Ñ Apvq is a continuous map from C 2 zt0u to M. By skew-symmetry of J, Apvqv " 0 and therefore NullspacepApvqq " Cˆv for each v P C 2 zt0u .
Hence, for each v P C 2 zt0u there is one and only one non-zero complex number λpvq such that φpApvqq " λpvqv .
Taking t " 0, we have ζpθ; 0q " λ pê 2 q , for all θ P S 1 ; and taking t " 1, we have ζpθ; 1q " λ`e iθê 1˘" e´i θ λ pê 1 q , for all θ P S 1 , by (A.3). Thus by varying t between 0 and 1 we obtain a continuous deformation of the unit circle to a point, remaining in Czt0u. This is impossible.