Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other.


Introduction
This paper is about discrete Schrödinger operators on Archimedean tilings, a class of periodic two-dimensional lattices that were already investigated by Johannes Kepler in 1619 [Kep19].They are natural candidates for the geometry of two-dimensional nanomaterials and due to advances in this field, most prominently represented by graphene, they have become increasingly a focus of attention.
Much work has been devoted to understanding physical properties of such (new) materials [SYY22,TFGK22,dLFM19].Most importantly, it can be expected that the underlying geometry, that is the particular lattice, is a key feature determining physical properties of the system.In fact, in particular in the mathematical physics literature, investigations of the connection between the geometry (or topology) of a system and the spectral properties of the associated Hamiltonian have become ubiquitous.Classical examples in this context are so-called quantum waveguides [EK15,Exn20,Exn22] as well as quantum graphs [BK13,BE22]; see also [KP07] for a relatively recent reference relevant in our context.
A closely related research direction is superconductivity: the existence of a boundary leads to boundary states in a superconductor with a higher critical temperature than the one of the bulk [SB20, SB21,HRS].In this spirit, it seems very promising to also study the interplay of geometry and many-particle phenomena on Archimedean tilings.Yet another related investigation can be found in [JBT21,SYY22] where another important quantum phenomenon, namely Bose-Einstein condensation, is examined.It turns out that so-called flat bands, that are infinitely degenerate eigenvalues of the Hamiltonian, play an important role in understanding such many-particle effects, and for other physical phenomena [KFSH19].One of the central motivations for this paper is to study robustness of flat bands under certain natural perturbations.
Two Archimedean tilings, the (3.6) 2 Kagome lattice and the (3.12 2 ) tiling 1 , which we shall dub Super-Kagome lattice for reasons that will become clear over the course of the article, stand out: they are the only Archimedean lattices on which the discrete, unweighted Laplacian has flat bands.In particular the Kagome lattice is a prominent model in physics that has recently enjoyed increasing interest [BM18,MDY22,Dia21].From a mathematical point of view, our paper is motivated by [PT21] where flat bands for the discrete, unweighted Laplacian on Archimedean tilings have been studied in great detail, in combination with an explicit calculation of the integrated density of states.
A priory, the flat-band phenomena on the Kagome and Super-Kagome lattice seem very sensitive to perturbations: if one replaces the adjacency matrix or the Laplacian by a variant with periodically chosen edge weights, one will generically destroy flat bands.However, the results of this paper suggest that, if one looks at proper, meaningful variants of the discrete Laplacian which respect certain, natural symmetries of the tiling (we call them monomeric Laplacians in Definition 3), then flat bands will persist.Since monomericity is a physically justifiable assumption, this makes a strong case that flat bands are a robust phenomenon, caused by the geometry of the lattice alone and specific to these two lattices, see Theorems 6, and 10.
Other questions of interest on periodic graphs concern existence, persistence and estimates on the width of spectral bands and the gaps between them [KS19, KS19, MW89].We will completely identify the spectra as a function of the perturbation in these cases, see Theorems 8, and 11 as well as Figures 3, and 5.This provides an exhaustive description of all nanomaterials based on Archimedean tilings on which discrete Laplacians can exhibit flat bands.
Our paper is organized as follows: Sections 2, and 3 are of introductory nature, introducing the notion of and arguing for the relevance of Archimedean tilings, and defining a proper notion of a discrete Laplace operator with non-uniform edge weights.Section 3 also introduces the notion of flat bands and argues why it suffices to restrict our attention to the (3.6) 2 Kagome and the (3.12 2 ) Super-Kagome lattice.Sections 4, and 5 contain our main results on the Kagome and Super Kagome lattice, respectively.The contributions of this paper are: (i) We identify the Kagome and Super-Kagome lattice as the only Archimedean lattices on which a natural class of periodic, weighted Laplacians can have flat bands (Proposition 5).(ii) We describe all periodic edge weights which lead to the maximal possible number of bands on the Kagome and Super-Kagome lattice, and prove that this is equivalent to so-called monomericity of the edge weights (Theorems 6 and 10).(iii) We completely describe the spectrum in the monomeric Kagome and Super-Kagome lattice (Theorems 8 and 11).In particular, the monomeric Super-Kagome lattice has a surprisingly rich spectrum-perturbation phase diagram (Figure 5) which might bear relevance for various applications.(iv) In the Super-Kagome lattice, under a weaker condition than monomericity, namely constant vertex weight, we explicitely describe all remaining "spurious" edge weights which have only one flat band.We describe the topology of this set within the parameter space and show in particular that it is disconnected from the monomeric two-band set (Theorem 12).

Archimedean tilings
Archimedean, Keplerian or regular tilings are edge-to-edge tesselations of the Euclidean plane by regular convex polygons such that every vertex is surrounded by the same pattern of adjacent polygons.We will adopt the notation of [GS89] and use the (counterclockwise) order of polygons arranged around a vertex as a symbol for a tiling (this is unique up to cyclic permutations), see Figure 1 for the (3.6)2Kagome lattice and the (3.12 2 ) Super-Kagome lattice which will be investigated in this paper.
(3.6) 2 Kagome lattice (3.12 2 ) Super-Kagome lattice The first systematic investigation from 1619 is due to Kepler who identified all 11 such tilings [Kep19] 2 .Most importantly, Archimedean tilings provide natural candidates for geometries of two-dimensional nanomaterials since they form natural, symmetric arrangements of a single buiding block, positioned at every vertex.And indeed, these lattices can be observed in many naturally occurring materials [FK58, FK59, KHZ + 20].
From a physical point of view, two-dimensional materials such as graphene are interesting since they feature so-called Dirac points which are related to a specific behaviour of the electronic band structure of the material [FW12,LWL13,HC15].
Also note that there are deep connections between Laplacians on these lattices, percolation, and self-avoiding walks which have also been studied extensively [SE64, Kes80, Nie82, SZ99, Ves04, Par07, Jac14, JSG16].An important quantity in this context is the so-called connective constant, which is known only in few cases, for example on the hexagonal lattice [DCS12].

Defining a suitable Hamiltonian
Every Archimedean tiling can be regarded as an infinite discrete graph G = (V, E) with (countable) vertex set V and (countable) edge set E. We write v ∼ w if the vertices v and w are joined by an edge and denote by |v| := #{w ∈ V : v ∼ w} the vertex degree of v (which in the case of Archimedean lattice graphs is v-independent).Archimedean lattices are Z 2 -periodic, and there exists a cofinite Z 2 -action that is a group of graph isomorphisms (intuitively understood as a group of shifts) isomorphic to the group Z 2 .Let Q ⊂ V be a minimal (in particular finite) fundamental domain of this action, i.e. the quotient of V under the equivalence relation generated by the group of isomorphisms (T β ) β∈Z 2 .
In the unweighted case, a natural, normalized choice for the Hamiltonian is the discrete Laplacian as used for instance in [PT21].It can be written as ∆f = Id − 1 |v| Π where Π is the adjacency matrix, that is Π(v, w) = 1 if v ∼ w and 0 else.The following is standard: Lemma 1.The unweighted, normalized Laplacian (1) with a uniformly bounded vertex degree boasts the following properties: (i) All restrictions of ∆ to finitely many vertices are real-symmetric M-matrices, that is, their off-diagonal elements are non-positive¸and all their eigenvalues are nonnegative.(ii) The infimum of the spectrum of ∆ is 0. (iii) All rows and columns of ∆ sum to zero.Furthermore, the spectrum is always contained in the interval [0, 2].
Introducing non-trivial edge weights, we would like to keep a form of the Laplacian that preserves properties (i) to (iii).A natural candidate, similar to formula (2.11) in [KS], is where the edge weights γ vw = γ wv > 0 and vertex weights µ(v) satisfy the relation As long as the vertex weights µ(v) (and thus also the γ vw ) are uniformly bounded, this will lead to an operator with properties (i) to (iii) and spectrum contained in [0, 2].
Remark 2. In the literature, one often finds the definition as a normalized, discrete Laplacian.Note that, whenever µ(v) = µ(w) for some v ∼ w, then this will not lead to a self-adjoint operator, but it can be made self-adjoint on a suitably weighted ℓ 2 (V )-space, cf.[KLW21].If all µ(v) are the same, then this definition coincides with (2), and can be simplified to Now, one can prescribe various degrees of the symmetry of the underlying Archimedean lattice to be respected by the Laplacian: Definition 3. Consider an Archimedean tiling (V, E) with periodic edge weights γ vw = γ wv > 0, that is γ vw = γ T β vT β w for all v, w ∈ V and β ∈ Z 2 , and corresponding vertex weights µ(v) = w∼v γ vw .Define the Laplacian ∆ γ as in (2).Then, we say that the Archimedean tiling with Laplacian ∆ γ (1) has constant vertex weight, if there is µ > 0 such that µ(v) = µ for all v ∈ V .
(2) is monomeric if for all vertices v ∈ V the list of edge weights, arranged cyclically around v, coincides (up to cyclic permutations).
Clearly, (2) is stronger than (1).However, in either case, the Laplacian reduces to (4).The term "monomeric" is inspired by the fact that the associated operators can be interpreted as describing properties of nanomaterials formed from one type of monomeric building block, positioned at every vertex of an Archimedean tiling.Clearly, monomeric Laplacians on Archimedean lattices have constant vertex weights, but the converse is not true in general.However, we will see in Theorems 6 and 10 that on the Kagome and Super-Kagome lattice, the validity of the converse implication is equivalent to existence (or persistence) of all flat bands.Also, monomericity seems a physically reasonable assumption for nanomaterials, which suggests that the emergence of flat bands, while a priori very sensitive to perturbations of coefficients in the operator, might nevertheless be robust within the class of physically relevant operators.
Next, let T 2 = R 2 /Z 2 be the flat torus and define for every Given the Laplacian (4) on ℓ 2 (V ) with properties described in Definition 3, we define on ℓ 2 (V ) θ the operator Clearly, (5) can be represented as a |Q|-dimensional Hermitian matrix.Due to Floquet theory, we have and the following statement holds.
Proposition 4 (See [PT21] and references therein).Let E ∈ R.Then, the following are equivalent: for a positive measure subset of θ ∈ T 2 .(iii) There is an infinite orthonormal family eigenfunctions of ∆ γ to the eigenvalue E.
Each of them can be chosen to be supported on a finite number of vertices.
If any of (i) to (iii) is satisfied, we say that ∆ γ has a flat band (at energy E).
Note that, in the ℓ ∞ (V ) setting instead of the ℓ 2 (V ) setting, such infinitely degenerate eigenvalues are also referred to as "black hole eigenvalues" in [BL09].Also, the existence of flat bands can be interpreted as a breakdown of the unique continuation principle [PTV17].
Proposition 5 is proved by a series of straightforward but somewhat lengthy calculations in which one calculates the associated characteristic polynomials, and shows that there are no θ-independent roots, employing Proposition 4 (this should be compared to the proofs of Theorems 6 and 10 below).We omit them here for the sake of conciseness.In any case, Proposition 5 justifies to restrict our attention to the (perturbed) Kagome and Super-Kagome lattices from now on.

The perturbed Kagome lattice
In this section we discuss the Kagome lattice with non-uniform (periodic) edge weights.The elementary cell of the Kagome lattice contains three vertices and six edges (one can think of the edges as arranged around a hexagon).A priori, periodicity allows for six . Fundamental domain of the Kagome lattice with edge weights.In the monomeric case, all edge weights around downwards pointing triangles are γ 2 = γ 4 = γ 6 =: α and all edge weights on upwards pointing triangles are γ 1 = γ 3 = γ 5 =: β, where 2α + 2β = µ.edge weights γ 1 , ..., γ 6 > 0, and the Floquet Laplacian ∆ θ γ can be written as the Hermitian matrix where w := e iθ 1 and z := e iθ 2 .We denote the three real eigenvalues of ∆ θ γ by λ Note that the six degrees of freedom are to be further reduced, depending on the following symmetry conditions: • If we merely assume a constant vertex weight µ > 0, then identity (3) will impose the three additional linearly independent conditions and we end up with three degrees of freedom.
• If we also assume monomericity, then it is easy to see that the only choice is the breathing Kagome lattice, cf.[HKdP + 22], with an edge weight α > 0 on all edges belonging to upwards pointing triangles and edge weight β > 0 on all edges belonging to downwards pointing triangles, where 2(α + β) = µ.After fixing the vertex weight µ, this amounts to only one degree of freedom.
(ii) The vertex weights are monomeric.More explicitly, there are α, β > 0 with 2(α + The rest of this subsection is devoted to the proof of Theorem 6.We start with identifying flat bands using the weighted adjacency matrix which is spectrally equivalent to ∆ θ γ up to scaling and shifting via the relation In order to find flat bands, we will identify conditions for θ-independent eigenvalues of Π θ γ and therefore calculate where A := γ 3 + wγ 6 , B := wγ 4 + zγ 1 and C := γ 2 + zγ 5 .Rearranging the terms yields ) .The prefactors w + w = 2 cos θ 1 , z + z = 2 cos θ 2 , and wz + zw = 2 cos(θ 1 − θ 2 ) , are linearly independent as measurable functions of θ on T 2 .Consequently, since all γ i are positive, θ-independent eigenvalues exist if and only if the w and z-independent terms in every line are zero.This is only possible for negative λ, which (possibly after scaling the γ i and µ for the moment) can be assumed to equal −1.Therefore, we obtain the conditions and Lemma 7. The only positive solutions (meaning all γ i are non-zero) of (7), ( 9), (10) are with x, y ∈ (0, 1) and x + y = 1.
We are now in the position to prove Theorem 6.
Proof of Theorem 6. Comparing Π θ γ with ∆ θ γ we conclude that ∆ θ γ has a flat band with edge weights γ 1 , ..., γ 6 if and only if there exists δ > 0 such that Π θ γ has a flat band for edge weights δγ 1 , ..., δγ 6 .From this observation the statement follows directly taking Lemma 7 into account.4.2.The spectrum and band gaps in the monomeric Kagome lattice.In the case where the perturbed Kagome lattice has a flat band, we further study the structure of the rest of the spectrum.We reiterate that, due to Theorem 6, the existence of a flat band is equivalent to the weights being monomeric.
As shown for instance in [PT21], in the case where all edge weights are equal, the two other spectral bands, generated by the two other θ-dependent eigenvalues of ∆ θ γ , touch at E = 3/4, and the derivative of the integrated density of states at E = 3/4 vanishes -an indication that the spectral density at 3/4 is sufficiently thin for a gap to form under perturbation.And indeed, this is the statement of the next theorem, which also characterises the width of the gap.
Theorem 8 (Band gaps in the perturbed Kagome lattice).Consider the perturbed Kagome lattice with fixed vertex weight µ > 0, and monomeric edge weights α, β > 0, satisfying 2(α + β) = µ as characterized in Theorem 6.Then, the spectrum is given by Furthermore, there is always a flat band at 3 2 .Remark 9. Theorem 8 states that, as soon as α = β, or alternatively, α = µ 4 , a spectral gap of width will form around 3 4 , see also Figure 3.The flat band at 3 2 will always be connected to the energy band below it which means that the "touching" of the flat band at 3 2 is protected in the class of monomeric perturbations.

The perturbed Super-Kagome lattice
In this section, we investigate the Archimedean tiling (3.12 2 ) which we call Super-Kagome lattice.Its minimal elementary cell contains six vertices and nine edges: three edges on upwards pointing triangles, three edges on downwards pointing triangles, and three edges bordering two dodecagons, see Figure 4. Given a constant vertex weight µ > 0, the Floquet Laplacian (5) is a 6 × 6-matrix given by where w := e iθ 1 , z := e iθ 2 .
• If we fix a constant vertex weight µ > 0, the condition w∼v γ vw = µ for all v ∈ V leads to This can be seen to be a linear system of 6 linearly independent equations with 9 unknowns, so the solution space is 3-dimensional.More precisely, by appropriate additions, we infer the three identities which imply γ 1 = γ 4 .The identities γ 2 = γ 5 , and γ 3 = γ 6 follow by completely analogous calculations.This leaves us with 6 independent variables γ 1 , γ 2 , γ 3 , and γ 7 , γ 8 , γ 9 which are however still subject to the three conditions 14).Therefore, we are left with three degrees of freedom.• If we additionally prescribe monomericity, it is easy to see that there is only one degree of freedom: All edges around triangles carry the weight α > 0, and all remaining edges (separating two dodecagons) carry the weight β > 0 under the condition 2α + β = µ.
(ii) The Super-Kagome lattice is monomeric.More explicitly, there are α, β > 0 such that 2α + β = µ together with Proof.Recall that in the constant vertex weight case, we have γ 1 = γ 4 , γ 2 = γ 5 , and γ 3 = γ 6 , and consider the weighted adjacency matrix which is a shifted and scaled version of ∆ θ γ .We calculate Since w + w = 2 cos(θ 1 ), z + z = 2 cos(θ 2 ), and wz + wz = 2 cos(θ 1 − θ 2 ) are linearly on T 2 , λ is a θ-independent eigenvalue if and only if the conditions as well as hold.3 Conditions (17) imply that any θ-independent eigenvalue of the matrix Π θ γ must satisfy Since all γ i are positive, the only way for these three equations to have the same set of solutions, that is for two flat bands to exist, is therefore This implies that the matrix Π θ γ can only have two θ-independent eigenvalues if there are α, β > 0 with α = γ 7 = γ 8 = γ 9 and β = γ 1 = γ 2 = γ 3 , that is the monomeric case, and the only candidates for these eigenvalues are −β ± α.To see that they are indeed eigenvalues, one verifies by an explicit calculation that condition (18) is also fulfilled.This shows the stated equivalence.
Next, we further describe the spectrum of the monomeric Super-Kagome lattice.
Theorem 11 (Band gaps in the perturbed Super-Kagome lattice).Consider the perturbed Super-Kagome lattice with Laplacian (4) with fixed vertex weight µ > 0 and monomeric edge weights α, β > 0, satisfying 2α + β = µ as characterized in Theorem 10.Then, the spectrum is given by with flat bands at 3α µ and 2 − α µ .The spectrum and the position of the flat bands have been plotted in Figure 5.The spectrum generically consists of two distinct intervals (bands) except for the case 3α = 2β, that is α = 2µ 7 , in which the two bands touch and the spectrum consists of one interval with an embedded flat band in the middle as well as a flat band at its maximum.This case α = 2µ 7 connects two regimes with different spectral pictures: • If α > 2µ 7 the spectrum consists of two intervals the upper one of which has two flat bands at its endpoints.In the special case of uniform edge weights (that is α = µ 3 , this has already been observed, for instance in [PT21].
7 , the spectrum will again consist of two intervals each of which will have a flat band at its maximum.Somewhat surprisingly, the lower flat band has now attach itself to the lower interval I 2 upon passing the critical parameter α = 2µ 7 .Another noteworthy observation is that no gap opens within the intervals I 1 and I 2 , despite them being generated by two distinct Floquet eigenvalues and the density of states measure vanishing at a point in the interior of the bands, see again [PT21] for plots of the integrated density of states in the case of constant edge weights.In particular, this distinguishes the monomeric Super-Kagome lattice from the monomeric Kagome lattice where such a gap indeed opens within the spectrum at points of zero spectral density.

Constant edge weights
Figure 5. Spectrum of the monomeric (3.12 2 ) "Super-Kagome" lattice with vertex weight µ > 0 as a function of the parameter α ∈ (0, µ 2 ), describing the edge weights on edges adjacent to triangles.
The solution space is invariant under those permutations of the γ i which correspond to rotations of the lattice by 2π 3 , and 4π 3 .Modulo these permutations, the two connected components can be described as follows • A one-dimensional submanifold, isomorphic to an interval, and explicitely descibed in equation (26), • Two one-dimensional submanifolds each isomorphic to an interval, explicitely described in (28), and (30), which intersect in a single point.
Proof of Theorem 12. Recall that due to the reductions made at the beginning of the section, after fixing the constant vertex weight µ > 0, the space of edge weights is a 3-dimensional manifold in the 6-dimensional parameter space {γ 1 , γ 2 , γ 3 , γ 7 , γ 8 , γ 9 > 0}, subject to the conditions Furthermore, from the proof of Theorem 10 we infer that ∆ γ has a flat band at λ if and only if the weighted adjacency matrix Π θ γ has the θ-independent eigenvalue λ := µ(1 − λ).This requires in particular that λ holds with a certain combination of plus and minus signs.Now, if equality in (22) holds with all three signs positive or all three signs negative, respectively, then the argument in the proof of Theorem 10 shows that this already implies that the edge weights are monomeric, the identities also hold with the opposite sign, the additional condition (18) is fulfilled, and there are two flat bands.As a consequence, the only chance for the existence of exactly one flat band is (22) to hold with different signs in front of γ 7 , γ 8 , γ 9 .Also, it is immediately clear that (22) with different signs does not allow for a monomeric and non-zero solution and hence the solution space consists of at most six mutually disjoint components which have no intersection with the two-flat-band manifold, identified in Theorem 10.By symmetry, it suffices to investigate two out of these six cases: and Case(+ --): To solve Case(-+ +), combine the second identities in in ( 21) and ( 23), to deduce which, recalling γ i > 0, is only possible if γ 1 = γ 3 .But then, by (23), γ 7 = γ 8 .Calling α ′ := γ 2 , and β ′ := γ 9 , we can use (21), to further express Next, we eliminate β ′ by resolving the yet unused first identity in (23), which yields This only has real solutions if α ′ > 8 17 µ > 1 3 µ, thus only can be a positive solution.Furthermore, we need β ′ ∈ (0, µ), which is the case if and only if We therefore find the one-parameter solution set Case (-+ +) Finally, an explicit calculation shows that with these parameters, (18) is indeed fulfilled.As for Case(+ --), we combine the second identity in (21) with the second identity in (24) to deduce Identity ( 27) has two types of solutions: Case(+ --)(a): γ 1 = γ 3 .

Case(+ --)(b):
The other solution of ( 27) is We set α ′′ := γ 1 , β ′′ := γ 3 , whence γ 2 = α ′′ β ′′ α ′′ + β ′′ , and use (21) to infer Plugging (29) into the yet unused first identity in (24), we arrive at We observe that only the solution with a plus has a chance to be positive and it is easy to see that this solution takes values in (0, µ) for all α ′′ ∈ (0, µ).We obtain the one-parameter solution set Case (+ --) (b) Again, an explicit calculation verifies that with these choices, (18) is fullfilled.Finally, to conclude the claimed topological properties of the manifolds, we need to verify that the solution space (28) in Case(+ --)(a) intersects the solution space (30) in Case(+ --)(b) if and only if Remark 13.Theorems 10 and 12 imply that the six one-flat-band components and the two-flat-band component are mutually disjoint.However, a closer analysis of the extremal cases in Formulas (26), (28), and (30), as well as of the monomeric case, implies that when sending the parameters to their extremal values, the three one-dimensional manifolds corresponding to Case(+ --) (a), and the two-flat-band-manifold of solutions converge to the two points , 0, 0, 0 , which themselves do no longer belong to the space of admissible parameters.Likewise, the limit of solutions of Case(+ --) in (26) corresponding to α ′ = µ 2 corresponds to the the point X 2 , see also Figure 6.

Figure 1 .
Figure 1.The two Archimedean tilings primarily investigated in this article.

γ 1 =Figure 6 .
Figure6.Schematic overview of the topology of the six "spurious" oneflat-band solution sets, and the monomeric two-flat-band manifold within the constant-vertex weight parameter space.Case(-+ +) solutions asymptotically meet the limit points of the two-flat-band manifold at one end of the parameter range, whereas Case(+ --) (a) solutions asymptotically meet it at both ends of the parameter range.