Symmetry and Dirac points in graphene spectrum

# Symmetry and Dirac points in graphene spectrum

Gregory Berkolaiko Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA  and  Andrew Comech
###### Abstract.

Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by and inversion, rotation by and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by and vertical reflection.

All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by . Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.

## 1. Introduction

Many interesting physical properties of graphene [32, 9, 24, 16], are consequences of presence of special conical points in the dispersion relation, where its different sheets touch to form a conical singularity. These points are often referred to as Dirac points or as diabilical points.

Most mathematical analyses of the dispersion relation of graphene are performed in physics literature in the tight-binding approximation, starting from the work of Wallace [40] and Slonczewski and Weiss [37]. This is equivalent to modeling the material as a discrete graph with vertices at the carbon molecules’ locations and with edges indicating chemical bonds. A richer mathematical model for graphene was considered by Kuchment and Post in [27], who studied honeycomb quantum graphs with even potential on edges.

The Schrödinger operator in with the real-valued potential that has honeycomb symmetry was considered by Grushin [20]. A condition for a multiple eigenvalue to be a conical point was established and checked in the perturbative regime of a weak potential (small ). The multiplicity two of the eigenvalue was proved from the symmetry point of view, an approach that we fully develop here.

The case of potential of arbitrary strength was studied by Fefferman and Weinstein [15] (see also [14] for further results). The results of [15] can be schematically broken into three parts: (a) establish that the dispersion relation has a double degeneracy at certain known values of quasi-momenta; (b) establish that for almost all the dispersion relation is conical in the vicinity of the degeneracy; (c) prove that the conical singularities survive under weak perturbation which destroys some of the symmetries of the potential (namely, the rotational symmetry). These results are contained in [15, Thms 5.1(1), 4.1 and 9.1] with proofs which are rather technical.

The purpose of this article is to make explicit the role of symmetry in existence and stability of Dirac points and to give proofs that are at the same time simpler and more general. Our methods apply to many different settings: graphs (discrete or quantum) and Schrödinger and Dirac operators on . We use Schrödinger operator as our primary focus, and give numerical examples based on discrete graphs. We also consider the effect of different symmetries, substituting inversion symmetry, usually assumed in the literature, with horizontal reflection symmetry (the results are analogous or stronger, as explained below).

We will now briefly review our results and the methods employed. The Schrödinger operator is assumed to be shift-invariant with respect to the hexagonal lattice. We also consider the following symmetries (see Fig. 1 for an illustration): rotation by (henceforth, “rotation”), inversion (reflection with respect to the point ), horizontal reflection and, to a lesser extent, vertical reflection. We remark that horizontal and vertical reflections are substantially different because the hexagonal lattice is not invariant with respect to rotation by . We study the question of existence and stability of Dirac points when the operator has various subsets of the above symmetries.

We show that existence of the degeneracy is a direct consequence of symmetries of the operator. The natural tool for studying this is, of course, the representation theory. It is well known that existence of a two- (or higher-) dimensional irreducible representation suggests that some eigenvalues will be degenerate. However, rotation combined with inversion — the most usual choice of symmetries [20, 15] — is an abelian group, whose irreps are all one-dimensional. The resolution of this question lies in the fact that the relevant symmetry is the inversion combined with complex conjugation and one should look at representations combining unitary and antiunitary operators, the so-called corepresentations introduced and fully classified by Wigner [43, Chap. 26].

To prove the existence of the degeneracy in the spectrum (Lemma 4.3) we identify the 2-dimensional (co)representation responsible for it and describe the subspace of the Hilbert space that carries this representation. We also relate our results to the proofs of isospectrality, in particular the isospectrality condition of Band–Parzanchevski–Ben-Shach [5, 33].

The conical nature of the dispersion relation is known to be generic (see, for example, [2, Appendix 10]); to prove this in a particular case one uses perturbation theory, as done in [20] and, implicitly, in [15]. Again, we seek to make the effect of symmetry most explicit here. This is done on two levels. First, in Lemma 2.1 and Lemma 3.1 we show that the dispersion relation also has rotational symmetry and thus, by Hilbert-Weyl theory of invariant functions, is restricted to be a circular cone (which could be degenerate) plus higher order terms. Then, in Lemma 5.2, we show that the symmetries also enforce certain relations on the first order terms of the perturbative expansion of the operator, which restricts the possible form of the terms. In spirit, this conclusion parallels the Hilbert-Weyl theory, but is more powerful: for example, it allows us to conclude that at quasi-momentum , where we discover persistent degeneracies with only the rotational symmetry, the dispersion relation is locally flat.

Part (c) of the above classification, the survival of the Dirac points when a weak perturbation breaks the rotational symmetry, can be established by perturbation theory and implicit function theorem, as done in [15]. However, such resilience of singularities indicates that there are topological obstacles to their disappearance [30, 29, 31]. The method familiar to physicists is to use the Berry phase [6, 36], which works when the operator has inversion symmetry (Section 7.1). Interestingly, when instead of inversion symmetry we have horizontal reflection symmetry, Berry phase is not restricted to the integer multiples of and the topological obstacle has a different nature. The survival of the Dirac cone is shown to be a consequence of the structure of representation of the reflection symmetry (Section 7.2), which combines eigenfunctions of different parities at the degeneracy point. As a consequence of our proof we conclude that the perturbed cone, although shifted from the corner of the Brillouin zone, remains on a certain explicitly defined line. In particular, this restricts the location of points in the Brillouin zone where Dirac cones can be destroyed by merging with their symmetric counterparts. Naturally, this effect is also present when there is horizontal reflection symmetry in addition to the inversion symmetry. We remark that experimentally created potentials usually possess the reflection symmetry, [4, 38].

In connection with the survival of the Dirac points, we would like to mention the complementary result of by Colin de Verdière in [10], who considered the Schrödinger operator with periodic, real and inversion-symmetric, but not -rotation invariant. In this case, for small , there are also conical singularities of the dispersion in the vicinity of the same special quasi-momenta. The proof uses the transversality condition of von Neumann–Wigner [39] and Arnold [1]. The method of [10] or, on a more basic level, the implicit function theorem, could also be used to prove our results, but we feel that the Berry phase technique is both beautiful and relatively unknown in the mathematics literature and thus deserves an appearance.

To summarize, in addition to providing simpler and shorter symmetry-based proofs to existing results, we discover some previously unknown consequences. In particular, we consider the case of rotational symmetry coupled with horizontal reflection symmetry; in this case, when the rotational symmetry is weakly destroyed, the conical points travel on a special line. We observe degeneracies at quasi-momentum in presence of rotational symmetry only; the dispersion relation at this point is shown to be locally flat. Finally, we explain why the coupling of rotation and vertical reflection does not, in general, lead to the appearance of Dirac points. The tools developed in this article would be easily extensible to other lattice structures [11] and graphene superlattices [44, 34].

### 1.1. Symmetries

The periodicity lattice of the operators that we consider is the 2-dimensional hexagonal lattice with the basis vectors

 (1) →a1=(√3/21/2),→a2=(√3/2−1/2),

see Fig. 1(a). The operator considered will always be assumed to be invariant with respect to the shifts by this lattice.

In addition to the shifts, the lattice has several other symmetries. We now describe some of them as operators acting on functions on (or on a graph embedded into ).

• Rotation by in the positive (counter-clockwise) direction:

 R:ψ(x1,x2)↦ψ(−12x1+√32x2,−√32x1−12x2)=ψ(MR→x).
• Inversion :

 V:ψ(x1,x2)↦ψ(−x1,−x2).
• Horizontal reflection :

 F:ψ(x1,x2)↦ψ(−x1,x2).

Note that and together form the abelian group of rotations by multiples of .

We denote by the antiunitary operation of taking complex conjugation (or “time-reversal” in physics terminology),

 (2) C:ψ(x1,x2)↦¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ψ(x1,x2).

In what follows, we will assume our operator has symmetries generated by a subset of the following: complex conjugation , rotation , reflection , conjugate inversion .

As the base operator (i.e. before we apply Floquet-Bloch analysis) we will always take an operator with real coefficients, thus it will be symmetric with respect to complex conjugation. As it turns out, an important role is played by the product of inversion and complex conjugation, known as the (parity-time) transformation:

Finally, we will also consider the vertical reflection symmetry:

• Vertical reflection :

 FV:ψ(x1,x2)↦ψ(x1,−x2).

Its effect is not the same as that of the horizontal reflection because the two symmetries are aligned differently with respect to the lattice . In fact, in contrast to , the presence of (in addition to symmetry ) does not generally lead to the appearance of conical points in the dispersion relation. This negative result is also important to understand; we explain it in section 4.1.4.

In Fig. 1(b-d) we show the fundamental domain of the lattice with defects that have symmetry in addition to , or , correspondingly.

### 1.2. Operators

As our primary motivational example we use the two-dimensional Schrödinger operator

 (3) H=−Δ+q(→x),→x∈R2,

with the real-valued potential assumed to be bounded and periodic with respect to the lattice . For general properties of the dispersion relation of such operators we refer the reader to [3, 25, 26].

To generate simple numerical examples we use discrete Schrödinger operators with potentials crafted to break or retain some of the symmetries listed above. More precisely, denote by an infinite graph embedded in , with vertex set and edge set . The embedding is realized by the mapping which gives the location in of the given vertex. A transformation preserves the graph structure if implies existence of such that and are connected by an edge if and only if are connected.

The graph is -periodic if the graph structure is preserved by the shifts defining the lattice. A graph with space symmetry is defined analogously.

The Schrödinger operator is defined on the functions from by

 (4) (Hf)v=∑(v,u)∈Emv,u(fv−fu)+qvfv,

where the sum is over all vertices adjacent to , are weights associated to edges (often, they are taken inversely proportional to edge length) and is the discrete potential. In our examples, the graph structure will be compatible with all symmetries of the lattice , while and will be breaking some of the point symmetries (however, they will always be periodic). The simplest -periodic graph is shown in Fig. 8(a). This is the graph arising as the tight-binding approximation of graphene.

Note that the discrete Schrödinger operator on graphs with more than two atoms per unit cell is not a mere mathematical curiosity since it arises in studying the twisted graphene and graphene in a periodic potential (superlattice); see [28, 44, 41] and references therein.

###### Remark 1.1.

We will always assume that is time-reversal (TR) invariant, or, in other words, satisfies

 CHC=H,

with the complex conjugation from (2). In particular, in the case (3), this means that is real-valued, while in the case (4) this means that both and are real-valued.

### 1.3. Floquet-Bloch reduction

Floquet theory can be thought of as a version of Fourier expansion, mapping the spectral problem on a non-compact manifold into a continuous sum of spectral problems on a compact manifold. The compact spectral problems are parametrized by the representations of the abelian group of periods (shifts).

Denote by , the space of Bloch functions, i.e. locally functions satisfying

 (5) ψ(→x+n1→a1+n2→a2)=ei(n1k1+n2k2)ψ(x),n1,n2∈Z.

For functions which also belong to the domain of it can be immediately seen that

 (Hψ)(x+n1→a1+n2→a2)=ei(n1k1+n2k2)Hψ(x),

i.e. the space is invariant under . By we will denote the restriction of the operator to the space . Its domain is , the dense subspace of consisting of functions that locally belong to together with their derivatives up to the second order.

Choosing a fundamental domain111a domain having the property that each trajectory has exactly one representative in it of the action of the group of periods, we can reduce the problem to the fundamental domain with quasi-periodic boundary conditions. The result of the Floquet-Bloch reduction is shown in Fig. 2. In Fig. 1(a), the lattice generating vectors and were shown together with a convenient choice of the fundamental region (shaded) and its four translations, by , , and . We will denote this choice of the fundamental domain by . The values of a Bloch function in surrounding regions, according to equation (5), are indicated in Fig. 2(a); we use the notation

 (6) ωj=eikj,j=1,2.

The continuity of the function and its derivative across the boundaries of copies of the fundamental region impose boundary conditions shown schematically in Fig. 2(b). They should be understood as follows: taking the bottom and top boundaries as an example, and parametrizing them left to right, the conditions read

where the normal derivative is taken in the outward direction (this causes the minus sign to appear). We stress that in Fig. 2(c) we use letters , and as placeholder labels, connecting the values of the function and its derivative on similarly labeled sides.

To represent the exponent of the Bloch phase as a scalar product, we introduce the vectors

 (7) →b1=(1√3,1)T,→b2=(1√3,−1)T,

see Fig. 3(a). Then

 (8) →biT⋅→aj=δi,j.

The vectors , define a lattice which is known as the dual lattice. For a hexagonal lattice, the dual lattice is also hexagonal. The lattice spanned by the vectors , will be denoted .

Due to (8), one can write as the dot product

Let us comment on using coordinates which are the coordinates with respect to the basis versus the corresponding Cartesian coordinates given by

 (9) →ϰ=(1√31√31−1)→k=:B→kor, equivalently,→k=B−1→ϰ=⎛⎜⎝√3212√32−12⎞⎟⎠→ϰ.

In Fig. 3(b) we show two choices of the Brillouin zone222By “Brillouin zone” we understand any choice of the fundamental domain of the dual lattice. What is known as the “first Brillouin zone” is the hexagonal domain in blue in Fig. 3(c) drawn in terms of coordinates and coordinates . One arrives at the first picture if one uses and as parameters for the dispersion relation (which is natural) ranging from to (black square) and then plots the result using and as Cartesian coordinates. The resulting plot of the dispersion relation will be skewed similarly to the blue hexagon in Fig. 3(b) (cf. Figures 5 and 6 of [27]). A more correct way of plotting is over a domain in Fig. 3(c), as it will highlight the symmetries of the result (see Figs. 4 and 5 and the explanations in the following section).

## 2. Formulation of results

For each value of the quasi-momentum , the operator has discrete spectrum. Its eigenvalues as functions of form what is known as the dispersion relation. Our results are concerned with the structure of the dispersion relation for the operators we described in Section 1.2. A typical example is shown in Fig. 4; it was computed for a discrete Laplacian described in detail in Example 4.5.

In the figure, one can see two conical points where the lowest two sheets of the dispersion relation touch. In terms of coordinates, they touch at , where

 →k∗=(2π3,−2π3),or, % correspondingly,→ϰ∗=(0,4π3).

The middle and the top sheets also touch, at the point ; at the point of contact both surfaces are locally flat. We will show that these features are typical: conical singularities at the point and flat contact at the point .

We start with formulating the following well-known result, summarizing the effects the different symmetries of have on the structure of the dispersion relation.

###### Lemma 2.1.
1. If the operator is -periodic (i.e. invariant with respect to the shifts by the lattice ), then the dispersion relation is -periodic, i.e. invariant with respect to the shifts

 (10) →ϰ↦→ϰ+2πb1and→ϰ↦→ϰ+2πb2.
2. If the operator is invariant with respect to complex conjugation or inversion , then the dispersion relation is invariant with respect to the inversion .

3. If the operator is invariant with respect to horizontal reflection , then the dispersion relation is invariant with respect to the reflection .

4. If the operator is invariant with respect to rotation , then the dispersion relation is invariant with respect to rotation by around the point .

1. If, in addition to symmetry , the operator is -periodic, then the dispersion relation is also invariant with respect to rotation by around the points .

2. If, in addition to symmetry , the operator has symmetry or , the dispersion relation is invariant with respect to rotation by around the point .

For completeness, we provide the proof in Section 3.

###### Remark 2.2.

When is invariant with respect to complex conjugation, inversion symmetry of the operator does not result in any additional symmetries of the dispersion relation.

###### Example 2.3.

Figure 4 was produced for a -periodic graph operator which has symmetries , and (but not ). Its dispersion relation therefore has symmetry groups around the point , and around the points ( and are the groups of symmetries of equilateral triangle and hexagon). This can be seen clearly if we plot the level curves of the dispersion surfaces, Fig. 5.

###### Theorem 2.4.

Let the self-adjoint -periodic operator be invariant under rotation . Let be one of the points , or . The space splits into the orthogonal sum

 (11) X(→k0)=X0(→k0)⊕X⊥(→k0),

where . This splitting is -invariant. Additionally,

1. if is also invariant with respect to at least one of the following: reflection or the conjugated inversion , then all eigenvalues of the operator restricted to have even multiplicity. Hence, if , has some eigenvalues with multiplicity at least 2. If, moreover, the multiplicity of an eigenvalue is exactly 2, the dispersion relation in coordinates is, to the leading order, a circular cone:

 (12) (λ−λ0)2=|α|2|→ϰ−→ϰ0|2+O(|→ϰ−→ϰ0|3),α∈C.
2. If is also invariant under the complex conjugation , then all eigenvalues of the operator restricted to have even multiplicity. Hence, if , has some eigenvalues with multiplicity at least 2. If, moreover, the multiplicity of an eigenvalue is exactly 2, then the dispersion relation at this point is flat:

 (13) (λ−λ0)2=O(|→ϰ−→ϰ0|3).

Theorem 2.4 will follow from Lemma 3.1, Lemma 4.3 (for the points ) and Lemma 6.1 (for the point ). In addition, in Lemma 5.3 we will give a convenient expression for of (12). We will also discuss a further splitting of the spaces and will give an explicit description of the restriction of to the constituent subspaces.

By Theorem 2.4, we are guaranteed to have conical points (i.e. points where the dispersion relation is of the form (12)) whenever two conditions are satisfied: an eigenvalue of on has minimal multiplicity (two) and is not in the spectrum of on , and the parameter . Intuitively, it is clear that both conditions are “generic”: if either of them is broken, any typical small perturbation of the potential should restore it.

To make this intuition precise, we consider the operator , where we are able to say more about the parameter and the exact multiplicity of eigenvalues.

###### Theorem 2.5.

Let with bounded measurable real potential which is invariant under the shifts by lattice , rotation , and at least one of the following: reflection or inversion . Further, assume that the condition

 (14) ∫ΩHe4πi√3x1q(→x)d→x≠0

is satisfied. Then the following conditions hold for all except possibly on a discrete set:

1. there is an eigenvalue of on of multiplicity exactly two and it is the smallest eigenvalue of on for small ,

2. the eigenvalue is not an eigenvalue of on ,

3. the corresponding value of in equation (12) is non-zero.

Theorem 2.5 will be proved in Section 5.4. We mention that condition (14) above is equivalent to condition (5.2) of [15] when one takes into account symmetries (such as (2.36) of [15]).

We now consider the fate of a conical point when the rotational symmetry is broken by a small perturbation. The following theorem is proved in Section 7.

###### Theorem 2.6.

Let be an operator satisfying the conditions of Theorem 2.4, part 1. Assume that its dispersion relation has a nondegenerate conical point at the point . Consider the perturbed operator , where the relatively bounded perturbation has the same symmetries as (namely, -invariance and either - or -invariance) except the -invariance.

Then, for small , the dispersion relation of has a nondegenerate conical point in the neighborhood of . Furthermore, if is invariant with respect to reflection , the conical point remains on the line modulo .

We remark that a complementary result in the case when is the pure Laplacian () and is a and -invariant (but not necessarily -invariant) potential satisfying a Fourier condition akin to (14) was obtained by Colin de Verdière in [10]. This highlights the fact that conical singularities are very typical in 2-dimensional problems.

## 3. Symmetries in the dual space; proof of Lemma 2.1

We recall that the operator is the restriction of the operator to the space . Equivalently, it can be considered as an operator on the compact domain of Fig. 2(c) with the specified boundary conditions333if the operator is specified on discrete graphs, the “boundary conditions” require special interpretation, see Section 4.4 for some examples. It is immediate from the definition of that the dispersion relation is invariant with respect to shifts by ,

 (15) →k↦→k+(2π,0)and→k↦→k+(0,2π).

In other words, the dispersion relation is periodic with respect to the lattice . We will now study other symmetries of the dispersion relation.

For given values of (or, equivalently, , where ), the operator may no longer have all the symmetries of the original operator : while the differential expression defining the operator is still invariant, the domain of definition has been restricted and may not be invariant anymore.

We start with the rotation operator . We first need to understand the effect of on the space . This can be understood by rotating the picture in Fig. 2(b) by and finding the “new , ”:

 ω′1=¯¯¯¯¯¯ω1ω2,ω′2=¯¯¯¯¯¯ω1,¯¯¯¯¯¯ω′2ω′1=ω2.

The last equation clearly follows from the first two. For the exponents , , defined as in (6), we have

 (16) (k′1k′2)=(−11−10)(k1k2)=:^R(k1k2).

With respect to the dual basis , the matrix is unitary: in terms of coordinates the action of is given by

 B^RB−1=M∗R=(−1/2−√3/2√3/2−1/2).

Therefore, the action of is the rotation of coordinates by , see Fig. 3(a), and acts as a unitary operator from to .

More formally, denote by the operator of the shift , with

 (17) A:=(→a1,→a2)=(√3/2√3/21/2−1/2).

Then, for a function satisfying

 ψ(→x+A→n)=ei→k⋅→nψ(→x),

we have

 S→nRψ =ψ(MR(→x+A→n))=ψ(MR→x+A(A−1MRA)→n) =ei→k⋅(A−1MRA)→nψ(MR→x)=ei((A−1MRA)∗→k)⋅→nRψ,

and therefore maps functions from to with .

Since the operator is the restriction of the operator , which is invariant under the rotation , to the space , we get

 (18) H(→k)=R∗H(^R→k)R,

i.e.  is unitarily equivalent to . As a consequence, the dispersion relation is invariant under the mapping

 (19) →k↦^R→kmod2πZ2,

which maps a Brillouin zone to itself (here we assumed that is -periodic). The fixed points of this mapping are the points

 (20) →k∗:=(2π/3,−2π/3),−→k∗:=(−2π/3,2π/3),→0:=(0,0),

and their shifts by . In coordinates , the fixed points are

 (21) →ϰ∗:=(0,4π/3),−→ϰ∗:=(0,−4π/3),→0:=(0,0).

Analogous considerations for the horizontal reflection result in

 ω′1=¯¯¯¯¯¯ω2,ω′2=¯¯¯¯¯¯ω1,

and, eventually, in

 (22)

The matrix is a reflection with respect to the line and it leaves the points of this line invariant. In coordinates the mapping acts as .

Both complex conjugation and inversion result in

 ω′1=¯¯¯¯¯¯ω1,ω′2=¯¯¯¯¯¯ω2,

and possess a unique fixed point . However, their composition preserves the space for all values of . To be more precise, using the antiunitary operation of taking complex conjugation , we have

 (23) CH(→k)C−1=H(−→k)=VH(→k)V∗.

Equations (18), (22) and (23) show that the symmetries of the operator result in the symmetries of the dispersion relation. These symmetries have been summarized in Lemma 2.1 above.

An important consequence of symmetry is a restriction on the possible local form of the dispersion relation. In particular, the dispersion relation must be a circular cone (which could be degenerate) around a symmetry point of multiplicity two.

###### Lemma 3.1.

Let be one of the symmetry points, or .

1. If is a simple eigenvalue, the dispersion relation is given locally by

 (24) λ−λ0=a|→ϰ−→ϰ0|2+O(|→ϰ−→ϰ0|3),a∈R.
2. If is a double eigenvalue, the dispersion relation is given locally by

 (25) λ−λ0=±|α||→ϰ−→ϰ0|+O(|→ϰ−→ϰ0|2),α∈C.

Note that may be equal to zero.

We note that using perturbation theory together with symmetry in Sections 5.3 and 6 below, it will be possible to make further conclusion about appearing in equation (25)..

###### Proof.

We start by remarking that by standard perturbation theory the number of eigenvalues close to in the vicinity of the point remains equal to the multiplicity of at .

We know from general theory of analytic Fredholm operators [45] that the dispersion relation is an analytic variety, i.e. given by an equation

 (26) F(λ,→ϰ)=0,

where is a real-analytic function. Without loss of generality, consider the point . It is an easy special case of Hilbert-Weyl theorem on invariant functions [42] (see also [17, XII.4]), that if a real-analytic function is symmetric with respect to rotations by around the origin, it can be represented as , with some real-analytic . Therefore, (26) takes the form

 (27) G(λ,ϰ21+ϰ22,ϰ31−3ϰ1ϰ22,ϰ32−3ϰ2ϰ21)=0,

with real-analytic in all the variables.

If is a simple root,

 ∂λG|(λ0,0,0,0)=∂λF|(λ0,→0)≠0;

by the implicit function theorem, (27) defines , with analytic in all three variables, and (24) follows.

If is a double root, we have

 ∂λG|(λ0,0,0,0)=∂λF|(λ0,→0)=0,∂2λG|(λ0,0,0,0)=∂2λF|(λ0,→0)≠0.

Without loss of generality, we assume that . Then we have

 F(λ,→ϰ)=G(λ,ϰ21+ϰ22,ϰ31−3ϰ1ϰ22,ϰ32−3ϰ2ϰ21) (28) =(λ−λ0)2+a|→ϰ|2+O((λ−λ0)3)+O((λ−λ0)|→ϰ|2)+O(|→ϰ|3).

Note that the coefficient at satisfies or else would be strictly positive for close to ; thus, there would be no eigenvalues for arbitrarily close to .

If , there is small enough and large enough so that for the function changes sign for between and also for between . Thus, the eigenvalue satisfies (25).

If , then we need higher order terms in the expansion of (3):

 G(λ,ϰ21+ϰ22,ϰ31−3ϰ1ϰ22,ϰ32−3ϰ2ϰ21) =(λ−λ0)2+c0(λ−λ0)|→ϰ|2+c1(ϰ31−3ϰ1ϰ22)+c2(ϰ32−3ϰ2ϰ21) +O((λ−λ0)2|→ϰ|2)+O((λ−λ0)3)+O(|→ϰ|4)

where . We claim that . For example, if we had , then there would be such that is positive-definite for , , and ; thus, there would be no eigenvalues for particular arbitrarily close to , leading to a contradiction. Once , the relation allows us to conclude that , which results in (25) with . ∎

## 4. Degeneracies in the spectrum at the point ±→k∗

We have seen in Section 3 that the points are special in that the operator has a large symmetry group. In the next subsection we give a review of the mechanism due to which symmetries give rise to degeneracies in the spectrum.

### 4.1. A review of representation theory background

Let be a self-adjoint operator (“Hamiltonian”) acting on a separable Hilbert space . Let be a finite group of unitary operators on (the “symmetries” of ) which commute with .

###### Remark 4.1.

It is assumed implicitly that the domain of is invariant under the action of operators . Such technical details will be omitted unless they have some importance to the task at hand.

It is well-known (see, e.g. [43, 18]) that in the circumstances described above, there is an isotypic decomposition of into a finite orthogonal sum of subspaces each carrying copies of an irreducible representation of . More precisely,

 X=⨁ρXρ,

where for any two vectors , there is an isomorphism between the spaces

 [Sv1]=span{Sv1:S∈S}and[Sv2]=span{Sv2:S∈S},

which preserves the group action on the spaces (i.e. commutes with all ). The dimension of is coincides with the dimension of the representation .

###### Example 4.2.

Let and be the cyclic group of order 2 generated by the reflection or, more precisely,

 S:f(x)↦f(−x).

Then , where

 Xeven={f∈X:f(−x)=f(x)},Xodd={f∈X:f(−x)=−f(x)}.

Then carries infinitely many copies of the trivial representation of :

 Id↦(1),S↦(1),

while carries infinitely many copies of the alternating representation of :

 Id↦(1),S↦(−1).

Both representations are one-dimensional. Note that the decomposition of a into irreducible copies is not unique.

Each isotypic component is invariant with respect to : either or provides an isomorphism between subspaces and .

If has discrete spectrum then the restriction of to has eigenvalues with multiplicities divisible by the dimension of . Indeed, by commuting and we see that if is an eigenvector of , then the entire subspace is an eigenspace of with the same eigenvalue.

It is sometimes stated in the physics literature that if the group of symmetries of an operator has an irreducible representation , the operator will have eigenspaces carrying this irreducible representation; in particular, the corresponding eigenvalue will have multiplicity equal to the dimension of . This implicitly assumes that the isotypic component corresponding to this representation is present in the domain of operator (for examples to the contrary, see e.g. [5, Sec. 7.2] or Example 6.5 below). Thus the fundamental question in describing spectral degeneracies is finding the isotypic decomposition of the domain of the operator.

#### 4.1.1. R and F symmetry

Suppose the operator on the whole space has and symmetry. The symmetries satisfy the relations and and the symmetries group is thus isomorphic to the symmetric group . The representations are

 (29) R↦(1), F↦(1) trivial'', (30) R↦(1), F↦(−1) alternating'',

and

 (31) R↦(τ00¯¯¯τ),F↦(0110)"standard",

where is the third root of unity,

 (32) τ:=e2πi/3.

We thus expect that the two-dimensional representation will give rise to eigenvalues of of multiplicity at least 2.

#### 4.1.2. R and V symmetry

On the face of it, the group generated by and is the group of rotations by , which is abelian and therefore has one-dimensional representations only. This would normally suggest there are no persistent degeneracies in the spectrum. However, the symmetry relevant to us, as explained in section 3, is combined with complex conjugation. The representation must be an antiunitary operator, i.e. an operator satisfying

 (33) A(αv)=¯¯¯¯α(Av),⟨Av,Au⟩=⟨u,v⟩,

which is a complex conjugation followed by the multiplication by a unitary matrix. Representations combining unitary and antiunitary operators have been fully classified by Wigner [43, Chap. 26] (see also [7] for a summary of the method), who called them “corepresentations”. In short, one looks at the representation of the maximal unitary subgroup (in our case, the cyclic group of rotations ) and, from them, follows a simple prescription to construct all corepresentations. This prescription is essentially constructing the induced representation à la Frobenius, although in the case when the induced representation decomposes into two copies of an irrep, one takes only one copy.

The group has two corepresentations, given by

 (34) R:z↦z, ¯¯¯¯V:z↦¯¯¯z, (35) R:(z1z2)↦(τz1¯¯¯τz2), ¯¯¯¯V:(z1z2)↦(¯¯¯¯¯z2¯¯¯¯¯z1).

To see how they arise, we start with the representation of the subgroup , acting on a 1-dimensional space spanned by . We denote and calculate

 (36) R→v1=τ→v1, ¯¯¯¯V→v1=→v2 (37) R→v2=R¯¯¯¯V→v1=¯¯¯¯VR→v1=¯¯¯¯Vτ→v1=¯¯¯τ¯¯¯¯V→v1=¯¯¯τ→v2, ¯¯¯¯V→v2=¯¯¯¯V2→v1=→v1.

This is the representation (35) shown above.

The induced representation of is the same, after the change of basis .

The induced representation of the trivial representation of turns out to be

 (38) R→v1=→v1, ¯¯¯¯V→v1=→v2 (39) R→v2=R¯¯¯¯V→v1=¯¯¯¯VR→v1=¯¯¯¯V→v1=→v2, ¯¯¯¯V→v2=¯¯¯¯V2→v1=→v1.

After the change of basis