1 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 [21]. 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 increasingly become a focus of attention.

Much work has been devoted to understanding physical properties of such (new) materials [7, 42, 44]. 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 [8,9,10] as well as quantum graphs [1, 3]; see also [26] 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 [17, 38, 40]. 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 [19, 42] 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 [23]. Flat bands have recently become a topic of increasing attention [2, 39]. 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 \((3.12^2)\) tilingFootnote 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 [4, 6, 32]. From a mathematical point of view, our paper is motivated by [36] 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 2), 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 5, and 9.

Other questions of interest on periodic graphs concern existence, persistence and estimates on the width of spectral bands and the gaps between them [28, 28, 33]. We will completely identify the spectra as a function of the perturbation in these cases, see Theorems 7, and 10 as well as Figs. 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:

  1. (i)

    We identify the Kagome and Super-Kagome lattice as the only Archimed-ean lattices on which a natural class of periodic, weighted Laplacians can have flat bands (Theorem 4).

  2. (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 5 and 9).

  3. (iii)

    We completely describe the spectrum in the monomeric Kagome and Super-Kagome lattice (Theorems 7 and 10). In particular, the monomeric Super-Kagome lattice has a surprisingly rich spectrum-perturbation phase diagram (Fig. 5), which might bear relevance for various applications.

  4. (iv)

    In the Super-Kagome lattice, under a weaker condition than monomericity, namely constant vertex weight, we explicitly 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 11).

2 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 [14] 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 Fig. 1 for the \((3.6)^2\) Kagome lattice and the \((3.12^2)\) Super-Kagome lattice, which will be investigated in this paper.

Fig. 1
figure 1

The two Archimedean tilings primarily investigated in this article

Full size image

The first systematic investigation from 1619 is due to Kepler who identified all 11 such tilings [21]Footnote 2. Most importantly, Archimedean tilings provide natural candidates for geometries of two-dimensional nanomaterials since they form natural, symmetric arrangements of a single building block, positioned at every vertex. And indeed, these lattices can be observed in many naturally occurring materials [11, 12, 24].

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 [13, 15, 31].

Also note that there are deep connections between Laplacians on these lattices, percolation, and self-avoiding walks, which have also been studied extensively [18, 20, 22, 34, 35, 41, 43, 46]. 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 [5].

3 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 \(v \sim w\) if the vertices v and w are joined by an edge and denote by

$$\begin{aligned} |v |:= \# \{ w \in V :v \sim w \} \end{aligned}$$

the vertex degree of v (which in the case of Archimedean lattice graphs is v-independent). Archimedean lattices are \(\mathbb {Z}^2\)-periodic, and there exists a cofinite \(\mathbb {Z}^2\)-action

$$\begin{aligned} \mathbb {Z}^2 \ni \beta \mapsto T_\beta :V \rightarrow V\ , \end{aligned}$$

that is a group of graph isomorphisms isomorphic to the group \(\mathbb {Z}^2\). It can be intuitively understood as a group of shifts, generated by two linearly independent vectors \(\omega _1\in \mathbb {R}^2\) and \(\omega _2 \in \mathbb {R}^2\), see Figs. 2 and 4 for illustrations. Let \(Q \subset V\) 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 \((T_\beta )_{\beta \in \mathbb {Z}^2}\).

In the unweighted case, a natural, normalized choice for the Hamiltonian is the discrete Laplacian

$$\begin{aligned} (\Delta f)(v) := \frac{1}{|v |} \sum _{w \sim v} \left( f(v) - f(w) \right) = f(v) - \frac{1}{|v |} \sum _{w \sim v} f(w)\ , \end{aligned}$$
(1)

as used for instance in [36]. It can be written as \(\Delta f = {\text {Id}}- \frac{1}{|v |} \Pi \) where \(\Pi \) is the adjacency matrix, that is \(\Pi (v,w) = 1\) if \(v \sim w\) and 0 else.

Introducing non-trivial edge weights, a natural candidate for a (normalized) Laplacian—similar to formula (2.11) in [27]—is:

$$\begin{aligned} (\Delta _{\gamma }f)(v):=\frac{1}{\sqrt{\mu (v)}}\sum _{w\sim v}\gamma _{vw}\left( \frac{f(v)}{\sqrt{\mu (v)}}-\frac{f(w)}{\sqrt{\mu (w)}}\right) \end{aligned}$$
(2)

where the edge weights \(\gamma _{vw}=\gamma _{wv} > 0\) and vertex weights \(\mu (v)\) satisfy the relation

$$\begin{aligned} \sum _{w \sim v} \gamma _{vw} = \mu (v) \quad \text {for every}\,\, v \in V. \end{aligned}$$
(3)

As for (1), the spectrum of (2) is contained in [0, 2].

Remark 1

In the literature, one often finds the definition

$$\begin{aligned} (\Delta _\gamma f) (v) = \frac{1}{\mu (v)} \sum _{w \sim v} \gamma _{vw} \left( f(v) - f(w) \right) \end{aligned}$$

as a normalized, discrete Laplacian. Note that, whenever \(\mu (v) \ne \mu (w)\) for some \(v \sim w\), then this will not lead to a self-adjoint operator, but it can be made self-adjoint on a suitably weighted \(\ell ^2(V)\)-space, cf. [25]. If all \(\mu (v)\) are the same, then this definition coincides with (2) and can be simplified to

$$\begin{aligned} (\Delta _{\gamma }f)(v)=f(v)-\frac{1}{\mu }\sum _{w \sim v} \gamma _{wv}f(w)\ . \end{aligned}$$
(4)

Now, one can prescribe various degrees of the symmetry of the underlying Archimedean lattice to be respected by the Laplacian:

Definition 2

Consider an Archimedean tiling (V, E) with periodic edge weights \(\gamma _{vw} = \gamma _{wv} > 0\), that is \(\gamma _{vw} = \gamma _{T_\beta v T_{\beta } w}\) for all \(v,w \in V\) and \(\beta \in \mathbb {Z}^2\), and corresponding vertex weights \(\mu (v) = \sum _{w \sim v} \gamma _{vw}\). Define the Laplacian \(\Delta _\gamma \) as in (2). Then, we say that the Archimedean tiling with Laplacian \(\Delta _\gamma \)

  1. (1)

    has constant vertex weight, if there is \(\mu > 0\) such that \(\mu (v) = \mu \) for all \(v \in V\).

  2. (2)

    is monomeric if for all vertices \(v \in V\) 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 5 and 9 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. Let us note that in the literature, one also finds investigations of other perturbations to periodic operators such as by potentials [4, 39] or magnetic fields [4].

Next, let \(\mathbb {T}^2 = \mathbb {R}^2 / \mathbb {Z}^2\) be the flat torus and define for every \(\theta \in \mathbb {T}^2\) the |Q|-dimensional Hilbert space

$$\begin{aligned} \ell ^2(V)_{\theta }:=\{f:V \rightarrow \mathbb {C}\ |\ f(T_{\beta }v)=e^{i\langle \theta ,\beta \rangle }f(v) \}\ \end{aligned}$$

with inner product

$$\begin{aligned} \langle f,g\rangle _{\theta }:=\sum _{v \in Q}f(v)\overline{g(v)}\ . \end{aligned}$$

Note that this inner product is independent of the choice of the fundamental cell. Given the Laplacian (4) on \(\ell ^2(V)\) with properties described in Definition 2, we define on \(\ell ^2(V)_{\theta }\) the operator

$$\begin{aligned} (\Delta ^{\theta }_{\gamma }f)(v):=f(v)-\frac{1}{\mu }\sum _{w \sim v} \gamma _{wv}f(w)\ . \end{aligned}$$
(5)

Clearly, (5) can be represented as a \(|Q |\)-dimensional Hermitian matrix. Due to Floquet theory, we have

$$\begin{aligned} \sigma (\Delta _\gamma ) = \bigcup _{\theta \in \mathbb {T}^2} \sigma (\Delta _\gamma ^\theta )\ , \end{aligned}$$

and the following statement holds.

Proposition 3

Let \(E \in \mathbb {R}\). Then, the following are equivalent:

  1. (i)

    \(E \in \sigma ( \Delta _\gamma ^\theta )\) for all \(\theta \in \mathbb {T}^2\).

  2. (ii)

    \(E \in \sigma ( \Delta _\gamma ^\theta )\) for a positive measure subset of \(\theta \in \mathbb {T}^2\).

  3. (iii)

    There is an infinite orthonormal family eigenfunctions of \(\Delta _\gamma \) 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 \(\Delta _\gamma \) has a flat band (at energy E).

The proof of Proposition 3 can be found in [29, 30]. Note that, in the \(\ell ^\infty (V)\) setting instead of the \(\ell ^2(V)\) setting, such infinitely degenerate eigenvalues are also referred to as “black hole eigenvalues” in [45]. Also, the existence of flat bands can be interpreted as a breakdown of the unique continuation principle [37].

In the Hilbert space setting working on \(\ell ^2(V)\), it is known that for constant edge weights, the discrete Laplacian has flat bands only on two of the 11 Archimedean lattices, namely the \((3.6)^2\) Kagome lattice and the \((3.12^2)\) Super-Kagome lattice [36]. Before turning to perturbed versions of those two lattices, one should verify that there won’t be any surprises on the other lattices:

Theorem 4

On the Archimedean lattices \((4^4)\), \((3^6)\), \((6^3)\), \((3^3.4^2)\), \((4.8^2)\), \((3^2.4.3.4)\), (3.4.6.4), (4.6.12), \((3^4.6)\), there is no choice of periodic (with respect to the fundamental cell on the lattice) edge weights \(\gamma _{vw} = \gamma _{wv} > 0\) which will make the weighted adjacency matrix

$$\begin{aligned} \Pi _\gamma (v,w) = {\left\{ \begin{array}{ll} \gamma _{vw} &{} \quad \textrm{if }\,\,v \sim w,\\ 0 &{} \quad \textrm{else} \end{array}\right. } \end{aligned}$$

have a flat band. Consequently, also the Laplacian with constant or monomeric edge weights has no flat bands on these lattices.

Theorem 4 is proved in Appendix A by a series of straightforward but somewhat lengthy calculations in which one identifies certain terms in the associated characteristic polynomials and shows that there are no \(\theta \)-independent roots, employing Proposition 3 (this should be compared to the proofs of Theorems 5 and 9). Theorem 4 justifies to restrict our attention to the (perturbed) Kagome and Super-Kagome lattices from now on.

4 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).

Fig. 2
figure 2

Fundamental domain of the Kagome lattice with edge weights and generating vectors \(\omega _1\), \(\omega _2\). In the monomeric case, all edge weights on downwards pointing triangles are \(\gamma _2 = \gamma _4 = \gamma _6 =: \alpha \) and all edge weights on upwards pointing triangles are \(\gamma _1 = \gamma _3 = \gamma _5 =: \beta \), where \(2 \alpha + 2 \beta = \mu \)

Full size image

A priori, periodicity allows for six edge weights \(\gamma _1,...,\gamma _6 > 0\), and the Floquet Laplacian \(\Delta _\gamma ^\theta \) can be written as the Hermitian matrix

$$\begin{aligned} \Delta ^{\theta }_{\gamma } = {\text {Id}}-\frac{1}{\mu }\begin{pmatrix} 0 &{} \quad \gamma _3+w\gamma _6 &{} \quad w\gamma _4+z\gamma _1 \\ \gamma _3+\overline{w}\gamma _6 &{} \quad 0 &{} \quad \gamma _2+z\gamma _5 \\ \overline{w}\gamma _4+\overline{z}\gamma _1 &{} \quad \gamma _2+\overline{z}\gamma _5 &{} \quad 0 \end{pmatrix}\ , \end{aligned}$$
(6)

where \(w:=e^{i\theta _1}\) and \(z:=e^{i\theta _2}\). We denote the three real eigenvalues of \(\Delta ^{\theta }_{\gamma }\) by \(\lambda _1(\theta ,\gamma ) \le \lambda _2(\theta ,\gamma ) \le \lambda _3(\theta ,\gamma )\).

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 \(\mu > 0\), then identity (3) will impose the three additional linearly independent conditions

    $$\begin{aligned} \begin{aligned} \gamma _1+\gamma _4&=\gamma _2+\gamma _5\ , \\ \gamma _3+\gamma _6&=\gamma _2+\gamma _5\ ,\\ \gamma _1+\gamma _3+\gamma _4+\gamma _6&= \mu \ , \end{aligned} \end{aligned}$$
    (7)

    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. [16], with an edge weight \(\alpha > 0\) on all edges belonging to upwards pointing triangles and edge weight \(\beta > 0\) on all edges belonging to downwards pointing triangles, where \(2 (\alpha + \beta ) = \mu \). After fixing the vertex weight \(\mu \), this amounts to only one degree of freedom.

4.1 Flat Bands in the Perturbed Kagome Lattice

Theorem 5

Consider the perturbed Kagome lattice with Laplacian (4), fixed vertex weight \(\mu > 0\) and periodic edge weights \(\gamma _1,...,\gamma _6 > 0\), satisfying the condition (3) on vertex and edge weights. Then, the following are equivalent:

  1. (i)

    There exists a flat band.

  2. (ii)

    The vertex weights are monomeric. More explicitly, there are \(\alpha , \beta > 0\) with \(2 (\alpha + \beta ) = \mu \) such that

    $$\begin{aligned}\begin{aligned} \gamma _{2}&=\gamma _{4}=\gamma _{6}:=\alpha , \\ \gamma _{1}&=\gamma _{3}=\gamma _{5}:=\beta . \end{aligned} \end{aligned}$$

The rest of this subsection is devoted to the proof of Theorem 5. We start by identifying flat bands using the weighted adjacency matrix

$$\begin{aligned} \Pi ^{\theta }_{\gamma }:= \begin{pmatrix} 0 &{} \quad \gamma _3+w\gamma _6 &{} \quad w\gamma _4+z\gamma _1 \\ \gamma _3+\overline{w}\gamma _6 &{} \quad 0 &{} \quad \gamma _2+z\gamma _5 \\ \overline{w}\gamma _4+\overline{z}\gamma _1 &{} \quad \gamma _2+\overline{z}\gamma _5 &{} \quad 0 \end{pmatrix} \end{aligned}$$
(8)

which is spectrally equivalent to \(\Delta _\gamma ^\theta \) up to scaling and shifting via the relation

$$\begin{aligned} \Delta _\gamma ^\theta = {\text {Id}}- \frac{1}{\mu } \Pi _\gamma ^\theta . \end{aligned}$$

In order to find flat bands, we will identify conditions for \(\theta \)-independent eigenvalues of \(\Pi _\gamma ^\theta \) and therefore calculate

$$\begin{aligned}\begin{aligned} \det (\lambda {\text {Id}}-\Pi ^{\theta }_{\gamma })&=-\lambda ^3+\lambda (|A|^2+|B|^2+|C|^2)+2 \Re (A\overline{B}C) \end{aligned} \end{aligned}$$

where \(A:=\gamma _3+w\gamma _6\), \(B:=w\gamma _4+z\gamma _1\) and \(C:=\gamma _2+z\gamma _5\). Rearranging the terms yields

$$\begin{aligned}\begin{aligned} \det (\lambda {\text {Id}}-\Pi ^{\theta }_{\gamma })&= (w + {\overline{w}}) (\lambda \gamma _6 \gamma _3+\gamma _3\gamma _2\gamma _4+\gamma _6\gamma _5\gamma _1) \\&\quad + (z + {\overline{z}}) (\lambda \gamma _5 \gamma _2+\gamma _6\gamma _5\gamma _4+\gamma _1\gamma _3\gamma _2)\\&\quad + (w {\overline{z}} + z\overline{w}) (\lambda \gamma _1\gamma _4+\gamma _3 \gamma _5\gamma _4+\gamma _6\gamma _2\gamma _1) \\&\quad + (-\lambda ^3+\lambda (\gamma ^2_1+...+\gamma ^2_6)+2(\gamma _4\gamma _6\gamma _2+\gamma _3\gamma _5\gamma _1))\ . \end{aligned} \end{aligned}$$

The prefactors

$$\begin{aligned} w + \overline{w} = 2 \cos \theta _1\ , \quad z + \overline{z} = 2 \cos \theta _2\ , \quad \text {and} \quad w \overline{z} + z \overline{w} = 2 \cos (\theta _1 - \theta _2)\ , \end{aligned}$$

are linearly independent as measurable functions of \(\theta \) on \(\mathbb {T}^2\). Consequently, since all \(\gamma _i\) are positive, \(\theta \)-independent eigenvalues exist if and only if the w and z-independent terms in every line are zero. This is only possible for negative \(\lambda \), which (possibly after scaling the \(\gamma _i\) and \(\mu \) for the moment) can be assumed to equal \(-1\). Therefore, we obtain the conditions

$$\begin{aligned} \begin{aligned} \gamma _{3}\gamma _{6}&=\gamma _{2}\gamma _{3}\gamma _{4}+\gamma _{1}\gamma _{5}\gamma _{6}\ , \\ \gamma _{2}\gamma _{5}&=\gamma _{4}\gamma _{5}\gamma _{6}+\gamma _{1}\gamma _{2}\gamma _{3}\ , \\ \gamma _{1}\gamma _{4}&=\gamma _{3}\gamma _{4}\gamma _{5}+\gamma _{1}\gamma _{2}\gamma _{6}\ , \end{aligned} \end{aligned}$$
(9)

and

$$\begin{aligned} \begin{aligned} 1-(\gamma ^2_{1}+\dots +\gamma ^2_{6})+2\left( \gamma _{2}\gamma _{4}\gamma _{6}+\gamma _{1}\gamma _{3}\gamma _{5}\right) =0\ . \end{aligned} \end{aligned}$$
(10)

Lemma 6

The only positive solutions (meaning all \(\gamma _i\) are nonzero) of (7), (9), (10) are

$$\begin{aligned} \begin{aligned} \gamma _{2}&=\gamma _{4}=\gamma _{6} = x \\ \gamma _{1}&=\gamma _{3}=\gamma _{5} = y \end{aligned} \end{aligned}$$
(11)

with \(x,y \in (0,1)\) and \(x+y=1\).

Proof

By a direct calculation (11) solves (7), (9), (10).

Conversely, assume that there are positive solutions \(\gamma _1,...,\gamma _6 > 0\). From (9) we obtain

$$\begin{aligned} \begin{aligned} \gamma _3=\frac{\gamma _{1}\gamma _{5}\gamma _{6}}{\gamma _6-\gamma _2\gamma _4}\ , \quad \gamma _1=\frac{\gamma _{3}\gamma _{4}\gamma _{5}}{\gamma _4-\gamma _2\gamma _6}\ , \end{aligned} \end{aligned}$$

and this implies \(\gamma _6 > \gamma _2 \gamma _4\) and \(\gamma _4 > \gamma _2\gamma _6\). Hence, combining both equations yields \(\gamma ^2_2\gamma _6 < \gamma _6\) which shows that \(\gamma _2 < 1\). In the same way, one proves \(\gamma _i < 1\) for every other i.

Next, let \(\gamma _2+\gamma _5:=\Lambda \). By (7) one immediately concludes \(\gamma _1+\gamma _4=\gamma _3+\gamma _6=\Lambda \). Now, we add (9) and (10) and rearrange the equations to obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2}\left( \gamma ^2_{1}+\dots +\gamma ^2_{6}-1\right) +\gamma _{3}\gamma _{6}+\gamma _{2}\gamma _{5}+\gamma _{1}\gamma _{4}&=\gamma _{2}\gamma _{4}\gamma _{6}+\gamma _{1}\gamma _{3}\gamma _{5}\\&\quad +\gamma _{2}\gamma _{3}\gamma _{4}+\gamma _{1}\gamma _{5}\gamma _{6}+\gamma _{4}\gamma _{5}\gamma _{6}\\&\quad +\gamma _{1}\gamma _{2}\gamma _{3}+\gamma _{3}\gamma _{4}\gamma _{5}+\gamma _{1}\gamma _{2}\gamma _{6}\ . \end{aligned} \end{aligned}$$

By repeated factorization, the right-hand side simplifies to

$$\begin{aligned} \gamma _{1}\gamma _{3}(\gamma _2+\gamma _5) + \gamma _3\gamma _4(\gamma _2+\gamma _5) + \gamma _1\gamma _6(\gamma _2+\gamma _5) + \gamma _4\gamma _6(\gamma _2+\gamma _5) = \Lambda ^3,\qquad \quad \end{aligned}$$
(12)

and since for the left-hand side one has

$$\begin{aligned} \frac{1}{2}\left( \gamma ^2_{1}+\dots +\gamma ^2_{6}-1\right) +\gamma _{3}\gamma _{6}+\gamma _{2}\gamma _{5}+\gamma _{1}\gamma _{4}=\frac{3\Lambda ^2-1}{2}\ , \end{aligned}$$

we arrive at the polynomial \(\Lambda ^3-\frac{3\Lambda ^2}{2}+\frac{1}{2}=0\) the only positive solution of which is \(\Lambda =1\). Finally, adding the first the two equations of (9) yields

$$\begin{aligned} \begin{aligned} \gamma _3 \gamma _6+\gamma _2 \gamma _5= (\gamma _6\gamma _5+\gamma _2\gamma _3)(\gamma _1+\gamma _4)=\gamma _6\gamma _5+\gamma _2\gamma _3 \end{aligned} \end{aligned}$$

and this implies \(\gamma _5=\gamma _3\). Furthermore, adding the last two equations gives

$$\begin{aligned} \begin{aligned} \gamma _2 \gamma _5+\gamma _1 \gamma _4= (\gamma _4\gamma _5+\gamma _1\gamma _2)(\gamma _3+\gamma _6) =\gamma _4\gamma _5+\gamma _1\gamma _2 \end{aligned} \end{aligned}$$

giving \(\gamma _4=\gamma _2\). Conditions (7) hence give \(\gamma _1=\gamma _5\) and \(\gamma _6=\gamma _2\). This proves the statement. \(\square \)

We are now in the position to prove Theorem 5.

Proof of Theorem 5

Comparing \(\Pi ^{\theta }_\gamma \) with \(\Delta ^{\theta }_{\gamma }\) we conclude that \(\Delta ^{\theta }_\gamma \) has a flat band with edge weights \(\gamma _1,...,\gamma _6\) if and only if there exists \(\delta > 0\) such that \(\Pi ^{\theta }_{\gamma }\) has a flat band for edge weights \(\delta \gamma _1,...,\delta \gamma _6\). From this observation the statement follows directly taking Lemma 6 into account. \(\square \)

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 5, the existence of a flat band is equivalent to the weights being monomeric.

As shown, for instance, in [36], in the case where all edge weights are equal, the two other spectral bands, generated by the two other \(\theta \)-dependent eigenvalues of \(\Delta ^{\theta }_{\gamma }\), 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 characterizes the width of the gap.

Theorem 7

(Band gaps in the perturbed Kagome lattice). Consider the perturbed Kagome lattice with fixed vertex weight \(\mu > 0\), and monomeric edge weights \(\alpha , \beta > 0\), satisfying \(2 (\alpha + \beta ) = \mu \) as characterized in Theorem 5. Then, the spectrum is given by

$$\begin{aligned} I_1 \cup I_2 := \left[ 0, \frac{3}{4} - \left| \frac{3\alpha }{\mu }-\frac{3}{4}\right| \right] \bigcup \left[ \frac{3}{4} + \left| \frac{3\alpha }{\mu }-\frac{3}{4}\right| , \frac{3}{2} \right] . \end{aligned}$$

Furthermore, there is always a flat band at \(\frac{3}{2}\).

Remark 8

Theorem 7 states that, as soon as \(\alpha \ne \beta \), or alternatively, \(\alpha \ne \frac{\mu }{4}\), a spectral gap of width

$$\begin{aligned} \left| \frac{6\alpha }{\mu }-\frac{3}{2}\right| = \frac{3}{\mu } |\alpha - \beta |\end{aligned}$$

will form around \(\frac{3}{4}\), see also Fig. 3. The flat band at \(\frac{3}{2}\) will always be connected to the energy band below it which means that the “touching” of the flat band at \(\frac{3}{2}\) is protected in the class of monomeric perturbations.

Fig. 3
figure 3

Spectrum of the monomeric \((3.6)^2\) Kagome lattice with vertex weight \(\mu > 0\) as a function of the parameter \(\alpha \in (0, \frac{\mu }{2})\), describing the edge weights on edges adjacent to downwards pointing triangles

Full size image

Proof

A calculation shows that the eigenvalues of \(\Delta _\gamma ^\theta \) with the choice \(2( \alpha + \beta )=\mu \) as in Theorem 5 are given by

$$\begin{aligned} \lambda _{1,2}(\theta ,\gamma )=\frac{3}{4}\pm \frac{1}{4}\sqrt{1+8\left( 1+(F(\theta )-3)\left( \frac{2\alpha }{\mu }-\frac{4\alpha ^2}{\mu ^2}\right) \right) } \end{aligned}$$

and

$$\begin{aligned} \lambda _{3}(\theta ,\gamma )=\frac{3}{2} \end{aligned}$$

where \(F(\theta ):=\cos (\theta _1) + \cos (\theta _2) + \cos (\theta _1 - \theta _2)\). The function \(\mathbb {T}^2 \ni \theta \mapsto F(\theta )\) takes all values in \([-3/2, 3]\), see Lemma 3.1 in [36], whence \(\lambda _1(\theta , \gamma )\) and \(\lambda _2(\theta , \gamma )\) take all values in the intervals

$$\begin{aligned} \left[ 0, \frac{3}{4} - \left| \frac{3\alpha }{\mu }-\frac{3}{4}\right| \right] , \quad \text {and} \quad \left[ \frac{3}{4} + \left| \frac{3\alpha }{\mu }-\frac{3}{4}\right| , \frac{3}{2} \right] , \quad \text {respectively}. \end{aligned}$$

\(\square \)

5 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 Fig. 4.

Fig. 4
figure 4

Fundamental domain of the \((3.12^2)\) Super-Kagome tiling with edge weights and generating vectors \(\omega _1\), \(\omega _2\). In the monomeric case, all edge weights around triangles are \(\gamma _1 = \dots = \gamma _6 =: \alpha \) and the remaining weights are \(\gamma _7 = \gamma _8 = \gamma _9 =: \beta \)

Full size image

Given a constant vertex weight \(\mu > 0\), the Floquet Laplacian (5) is a \(6\times 6\)-matrix given by

$$\begin{aligned} \Delta ^{\theta }_{\gamma }={\text {Id}}- \frac{1}{\mu } \begin{pmatrix} 0&{} \quad \gamma _4&{} \quad \gamma _6&{} \quad 0&{} \quad z\gamma _9&{} \quad 0 \\ \gamma _4&{} \quad 0&{} \quad \gamma _5&{} \quad 0&{} \quad 0&{} \quad w\gamma _8 \\ \gamma _6&{} \quad \gamma _5&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad \gamma _3&{} \quad \gamma _2\\ \overline{z}\gamma _9&{} \quad 0&{} \quad 0&{} \quad \gamma _3&{} \quad 0&{} \quad \gamma _1\\ 0&{} \quad \overline{w}\gamma _8&{} \quad 0&{} \quad \gamma _2&{} \quad \gamma _1&{} \quad 0 \end{pmatrix}, \end{aligned}$$
(13)

where \(w:=e^{i\theta _1}\), \(z:=e^{i\theta _2}\).

  • If we fix a constant vertex weight \(\mu > 0\), the condition \(\sum _{w \sim v} \gamma _{vw} = \mu \) for all \(v \in V\) leads to

    $$\begin{aligned} \begin{aligned} \mu = \gamma _2+\gamma _3+\gamma _7 = \gamma _5+\gamma _6+\gamma _7&= \gamma _1+\gamma _2+\gamma _8 = \gamma _4+\gamma _5+\gamma _8 \\&= \gamma _1+\gamma _3+\gamma _9 = \gamma _4+\gamma _6+\gamma _9. \end{aligned} \end{aligned}$$
    (14)

    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

    $$\begin{aligned} \begin{aligned} 2 \gamma _1 + \gamma _8 + \gamma _9&= 2 \gamma _7 + \gamma _2 + \gamma _3, \\ 2 \gamma _4 + \gamma _8 + \gamma _9&= 2 \gamma _7 + \gamma _5 + \gamma _6, \\ \gamma _2 + \gamma _3&= \gamma _5 + \gamma _6 \end{aligned} \end{aligned}$$
    (15)

    which imply \(\gamma _1 = \gamma _4\). The identities \(\gamma _2 = \gamma _5\), and \(\gamma _3 = \gamma _6\) follow by completely analogous calculations. This leaves us with 6 independent variables \(\gamma _1, \gamma _2, \gamma _3\), and \(\gamma _7, \gamma _8, \gamma _9\) which are, however, still subject to the three conditions

    $$\begin{aligned} \gamma _2 + \gamma _3 + \gamma _7 = \gamma _1 + \gamma _2 + \gamma _8 = \gamma _1 + \gamma _3 + \gamma _9 = \mu \end{aligned}$$

    from (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 \(\alpha > 0\), and all remaining edges (separating two dodecagons) carry the weight \(\beta > 0\) under the condition \(2 \alpha + \beta = \mu \).

5.1 Flat Bands in the Perturbed Super-Kagome Lattice

Theorem 9

Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed vertex weight \(\mu > 0\), and periodic edge weights \(\gamma _1, \dots , \gamma _9 > 0\) satisfying the condition (3) on vertex and edge weights. Then, the following are equivalent:

  1. (i)

    There exist exactly two flat bands.

  2. (ii)

    The Super-Kagome lattice is monomeric. More explicitly, there are \(\alpha ,\beta > 0\) such that \( 2\alpha + \beta = \mu \) together with

    $$\begin{aligned} \begin{aligned} \gamma _1=\gamma _2=\gamma _3=\gamma _4=\gamma _5=\gamma _6&=\alpha \ , \\ \gamma _7=\gamma _8=\gamma _9&=\beta \ . \end{aligned} \end{aligned}$$

Proof

Recall that in the constant vertex weight case, we have

$$\begin{aligned} \gamma _1=\gamma _4\ , \quad \gamma _2=\gamma _5\ , \quad \text {and} \quad \gamma _3=\gamma _6\ , \end{aligned}$$

and consider the weighted adjacency matrix

$$\begin{aligned} \Pi ^{\theta }_{\gamma }:=\begin{pmatrix} 0&{} \quad \gamma _4&{} \quad \gamma _6&{} \quad 0&{} \quad z\gamma _9&{} \quad 0 \\ \gamma _4&{} \quad 0&{} \quad \gamma _5&{} \quad 0&{} \quad 0&{} \quad w\gamma _8 \\ \gamma _6&{} \quad \gamma _5&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad \gamma _3&{} \quad \gamma _2\\ \overline{z}\gamma _9&{} \quad 0&{} \quad 0&{} \quad \gamma _3&{} \quad 0&{} \quad \gamma _1\\ 0&{} \quad \overline{w}\gamma _8&{} \quad 0&{} \quad \gamma _2&{} \quad \gamma _1&{} \quad 0\end{pmatrix} =\begin{pmatrix} 0&{} \quad \gamma _1&{} \quad \gamma _3&{} \quad 0&{} \quad z\gamma _9&{} \quad 0 \\ \gamma _1&{} \quad 0&{} \quad \gamma _2&{} \quad 0&{} \quad 0&{} \quad w\gamma _8 \\ \gamma _3&{} \quad \gamma _2&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad \gamma _7&{} \quad 0&{} \quad \gamma _3&{} \quad \gamma _2\\ \overline{z}\gamma _9&{} \quad 0&{} \quad 0&{} \quad \gamma _3&{} \quad 0&{} \quad \gamma _1\\ 0&{} \quad \overline{w}\gamma _8&{} \quad 0&{} \quad \gamma _2&{} \quad \gamma _1&{} \quad 0\end{pmatrix}\nonumber \\ \end{aligned}$$
(16)

which is a shifted and scaled version of \(\Delta ^{\theta }_{\gamma }\). We calculate

$$\begin{aligned} \det (\lambda {\text {Id}}-\Pi ^{\theta }_{\gamma })&=\lambda ^6 - \lambda ^4 \left( 2 \gamma _1^2 + 2 \gamma _2^2 + 2 \gamma _3^2 + \gamma _7^2 + \gamma _8^2 + \gamma _9^2 \right) - 4 \lambda ^3 \gamma _1 \gamma _2 \gamma _3 \\&\quad +\lambda ^2 \big ( \gamma _1^4 + \gamma _2^4 + \gamma _3^4 + 2 \gamma _1^2 \gamma _2^2 + 2 \gamma _2^2 \gamma _3^2 + 2 \gamma _3^2 \gamma _1^2 + 2 \gamma _1^2 \gamma _7^2 + 2 \gamma _2^2 \gamma _9^2 + 2 \gamma _3^2 \gamma _8^2 + \\&\quad + \gamma _7^2 \gamma _8^2 + \gamma _8^2 \gamma _9^2 + \gamma _9^2 \gamma _7^2 \big ) \\&\quad + 4 \lambda \gamma _1 \gamma _2 \gamma _3 \left( \gamma _1^2 + \gamma _2^2 + \gamma _3^2 \right) \\&\quad - \gamma _1^4 \gamma _7^2 - \gamma _2^4 \gamma _9^2 - \gamma _3^4 \gamma _8^2 - \gamma _7^2 \gamma _8^2 \gamma _9^2 + 4 \gamma _1^2 \gamma _2^2 \gamma _3^2 \\&\quad - \left( w + \overline{w} \right) \left( \lambda ^2 \gamma _2^2 \gamma _7 \gamma _8 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 \gamma _7 \gamma _8 + \gamma _1^2 \gamma _3^2 \gamma _7 \gamma _8 - \gamma _2^2 \gamma _7 \gamma _8 \gamma _9^2 \right) \\&\quad - \left( z + \overline{z} \right) \left( \lambda ^2 \gamma _3^2 \gamma _7 \gamma _9 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 \gamma _7 \gamma _9 + \gamma _1^2 \gamma _2^2 \gamma _7 \gamma _9 - \gamma _3^2 \gamma _7 \gamma _8^2 \gamma _9 \right) \\&\quad - \left( w \overline{z} + \overline{w} z \right) \left( \lambda ^2 \gamma _1^2 \gamma _8 \gamma _9 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 \gamma _8 \gamma _9 + \gamma _2^2 \gamma _3^2 \gamma _8 \gamma _9 - \gamma _1^2 \gamma _7^2 \gamma _8 \gamma _9 \right) . \end{aligned}$$

Since \(w + {\overline{w}} = 2 \cos (\theta _1)\), \(z + {\overline{z}} = 2 \cos (\theta _2)\), and \(w {\overline{z}} + {\overline{w}} z = 2 \cos (\theta _1 - \theta _2)\) are linearly independent on \(\mathbb {T}^2\), \(\lambda \) is a \(\theta \)-independent eigenvalue if and only if the conditions

$$\begin{aligned} \begin{aligned} \lambda ^2 \gamma _2^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _1^2 \gamma _3^2 - \gamma _2^2 \gamma _9^2&= 0, \\ \lambda ^2 \gamma _3^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _1^2 \gamma _2^2 - \gamma _3^2 \gamma _8^2&= 0, \\ \lambda ^2 \gamma _1^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _2^2 \gamma _3^2 - \gamma _1^2 \gamma _7^2&= 0 , \end{aligned} \end{aligned}$$
(17)

as well as

$$\begin{aligned} \begin{aligned}&\lambda ^6 - \lambda ^4 \left( 2 \gamma _1^2 + 2 \gamma _2^2 + 2 \gamma _3^2 + \gamma _7^2 + \gamma _8^2 + \gamma _9^2 \right) - 4 \lambda ^3 \gamma _1 \gamma _2 \gamma _3 \\&+\lambda ^2 \big ( \gamma _1^4 + \gamma _2^4 + \gamma _3^4 + 2 \gamma _1^2 \gamma _2^2 + 2 \gamma _2^2 \gamma _3^2 + 2 \gamma _3^2 \gamma _1^2 + 2 \gamma _1^2 \gamma _7^2 + 2 \gamma _2^2 \gamma _9^2 + 2 \gamma _3^2 \gamma _8^2 \\&+ \gamma _7^2 \gamma _8^2 + \gamma _8^2 \gamma _9^2 + \gamma _9^2 \gamma _7^2 \big )+ 4 \lambda \gamma _1 \gamma _2 \gamma _3 \left( \gamma _1^2 + \gamma _2^2 + \gamma _3^2 \right) \\&- \gamma _1^4 \gamma _7^2 - \gamma _2^4 \gamma _9^2 - \gamma _3^4 \gamma _8^2 - \gamma _7^2 \gamma _8^2 \gamma _9^2 + 4 \gamma _1^2 \gamma _2^2 \gamma _3^2=0 \end{aligned} \end{aligned}$$
(18)

hold.Footnote 3 Conditions (17) imply that any \(\theta \)-independent eigenvalue of the matrix \(\Pi ^{\theta }_{\gamma }\) must satisfy

$$\begin{aligned} \lambda = - \frac{\gamma _1 \gamma _3}{\gamma _2} \pm \gamma _9, \quad \lambda = - \frac{\gamma _1 \gamma _2}{\gamma _3} \pm \gamma _8, \quad \text {and} \quad \lambda = - \frac{\gamma _2 \gamma _3}{\gamma _1} \pm \gamma _7. \end{aligned}$$

Since all \(\gamma _i\) 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

$$\begin{aligned} -\frac{\gamma _1\gamma _3}{\gamma _2}+\gamma _9=-\frac{\gamma _1\gamma _2}{\gamma _3}+\gamma _8=-\frac{\gamma _2\gamma _3}{\gamma _1}+\gamma _7 \end{aligned}$$
(19)

together with

$$\begin{aligned} -\frac{\gamma _1\gamma _3}{\gamma _2}-\gamma _9=-\frac{\gamma _1\gamma _2}{\gamma _3}-\gamma _8=-\frac{\gamma _2\gamma _3}{\gamma _1}-\gamma _7. \end{aligned}$$
(20)

This implies that the matrix \(\Pi _\gamma ^\theta \) can only have two \(\theta \)-independent eigenvalues if there are \(\alpha , \beta > 0\) with

$$\begin{aligned} \alpha = \gamma _7=\gamma _8=\gamma _9 \qquad \text{ and } \qquad \beta = \gamma _1=\gamma _2=\gamma _3 , \end{aligned}$$

that is the monomeric case, and the only candidates for these eigenvalues are \(-\beta \pm \alpha \). To see that they are indeed eigenvalues, one verifies by an explicit calculation that condition (18) is also fulfilled. This shows the stated equivalence.\(\square \)

Next, we further describe the spectrum of the monomeric Super-Kagome lattice.

Theorem 10

(Band gaps in the perturbed Super-Kagome lattice). Consider the perturbed Super-Kagome lattice with Laplacian (4) with fixed vertex weight \(\mu > 0\) and monomeric edge weights \(\alpha , \beta > 0\), satisfying \(2 \alpha + \beta = \mu \) as characterized in Theorem 9. Then, the spectrum is given by:

$$\begin{aligned} I_1 \cup I_2 := \left[ 0, \left( 1 - \frac{\alpha }{2 \mu } \right) - \frac{|3 \alpha - 2 \beta |}{2 \mu } \right] \bigcup \left[ \left( 1 - \frac{\alpha }{2 \mu } \right) + \frac{|3 \alpha - 2 \beta |}{2 \mu } , 2 - \frac{\alpha }{\mu } \right] \end{aligned}$$

with flat bands at \(\frac{3 \alpha }{\mu }\) and \(2 - \frac{\alpha }{\mu }\).

The spectrum and the position of the flat bands are plotted in Fig. 5. The spectrum generically consists of two distinct intervals (bands) except for the case \(3 \alpha = 2 \beta \), that is \(\alpha = \frac{2 \mu }{7}\), 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 \(\alpha = \frac{2 \mu }{7}\) connects two regimes with different spectral pictures:

  • If \(\alpha > \frac{2 \mu }{7}\), 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 \(\alpha = \frac{\mu }{3})\), this has already been observed, for instance in [36].

  • If \(\alpha < \frac{2 \mu }{7}\), 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 attached itself to the lower interval \(I_2\) upon passing the critical parameter \(\alpha = \frac{2 \mu }{7}\).

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 [36] 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.

Fig. 5
figure 5

Spectrum of the monomeric \((3.12^2)\) “Super-Kagome” lattice with vertex weight \(\mu > 0\) as a function of the parameter \(\alpha \in (0, \frac{\mu }{2})\), describing the edge weights on edges adjacent to triangles

Full size image

Proof of Theorem 10

In the monomeric case, the characteristic polynomial \( \det (\lambda {\text {Id}}- \Pi _\gamma ^\theta )\) of the matrix \(\Pi _\gamma ^\theta \) simplifies to

$$\begin{aligned} \begin{aligned} ( (&\alpha + \lambda )^2 - \beta ^2 )\cdot \\&( \lambda ^4 - 2 \alpha \lambda ^3 - (3 \alpha ^2 + 2 \beta ^2) \lambda ^2 + (4 \alpha ^3 + 2 \alpha \beta ^2) \lambda + 4 \alpha ^4+ \alpha ^2 \beta ^2 + \beta ^4 - 2 \alpha ^2 \beta ^2 F(\theta _1,\theta _2))\ , \end{aligned} \end{aligned}$$

where \(F(\theta _1, \theta _2) = \cos (\theta _1) + \cos (\theta _2) + \cos (\theta _1 + \theta _2)\). Its six roots are

$$\begin{aligned} \left\{ -\alpha \pm \beta , \frac{1}{2} \left( \alpha \pm \sqrt{9 \alpha ^2 + 4 \beta ^2 \pm 4 \alpha \beta \sqrt{3 + 2 F(\theta _1,\theta _2)} } \right) \right\} \ , \end{aligned}$$

whence the eigenvalues of \(\Delta _\gamma ^\theta \) are given by

$$\begin{aligned} \lambda _1(\theta , \gamma )&= 1 - \frac{1}{2 \mu } \left( \alpha + \sqrt{9 \alpha ^2 + 4 \beta ^2 + 4 \alpha \beta \sqrt{3 + 2 F(\theta _1,\theta _2)} } \right) \ , \\ \lambda _2(\theta , \gamma )&= 1 - \frac{1}{2 \mu } \left( \alpha + \sqrt{9 \alpha ^2 + 4 \beta ^2 - 4 \alpha \beta \sqrt{3 + 2 F(\theta _1,\theta _2)} } \right) \ , \\ \lambda _3(\theta , \gamma )&= 1 + \frac{\alpha - \beta }{\mu } = \frac{3 \alpha }{\mu } = {\left\{ \begin{array}{ll} 1 - \frac{\alpha - |3 \alpha - 2 \beta |}{2 \mu } &{}\text {if}\,\, 3 \alpha \ge 2 \beta \ , \\ 1 - \frac{\alpha - |3 \alpha - 2 \beta |}{2 \mu } &{} \text {if}\,\, 3 \alpha < 2 \beta \ , \end{array}\right. } \\ \lambda _4(\theta , \gamma )&= 1 - \frac{1}{2 \mu } \left( \alpha - \sqrt{9 \alpha ^2 + 4 \beta ^2 - 4 \alpha \beta \sqrt{3 + 2 F(\theta _1,\theta _2)} } \right) \ , \\ \lambda _5(\theta , \gamma )&= 1 - \frac{1}{2 \mu } \left( \beta - \sqrt{9 \alpha ^2 + 4 \beta ^2 + 4 \alpha \beta \sqrt{3 + 2 F(\theta _1,\theta _2)} } \right) \ , \\ \lambda _6(\theta , \gamma )&= 1 + \frac{ \alpha + \beta }{\mu } = 2 - \frac{\alpha }{\mu }\ . \end{aligned}$$

Using that the map \(\mathbb {T}^2 \ni (\theta _1, \theta _2) \mapsto F(\theta _1, \theta _2)\) takes all values in the interval \((- 3/2, 3)\), we conclude that the bands, generated by \(\lambda _1(\theta , \gamma )\) and \(\lambda _2(\theta , \gamma )\), as well as the bands generated by \(\lambda _4(\theta , \gamma )\) and \(\lambda _5(\theta , \gamma )\) always touch, and the spectrum consists of the two intervals

$$\begin{aligned}&\left[ \min _{\theta \in \mathbb {T}^2} \lambda _1(\theta , \gamma ) , \max _{\theta \in \mathbb {T}^2} \lambda _2(\theta , \gamma ) \right] \bigcup \left[ \min _{\theta \in \mathbb {T}^2} \lambda _4(\theta , \gamma ) , \max _{\theta \in \mathbb {T}^2} \lambda _5(\theta , \gamma ) \right] \\&\qquad = \left[ 0, 1 - \frac{\alpha + |3 \alpha - 2 \beta |}{2 \mu } \right] \bigcup \left[ 1 - \frac{\alpha - |3 \alpha - 2 \beta |}{2 \mu } , 2 - \frac{\alpha }{2 \mu } \right] \\&\qquad = \left[ 0, \left( 1 - \frac{\alpha }{2 \mu } \right) - \frac{|3 \alpha - 2 \beta |}{2 \mu } \right] \bigcup \left[ \left( 1 - \frac{\alpha }{2 \mu } \right) + \frac{|3 \alpha - 2 \beta |}{2 \mu } , 2 - \frac{\alpha }{\mu } \right] . \end{aligned}$$

\(\square \)

One might now wonder under which conditions only one flat band exists. The next theorem completely identifies all parameters for which one flat band exists:

Theorem 11

Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed vertex weight \(\mu > 0\), and periodic edge weights \(\gamma _1, \dots , \gamma _9 > 0\) satisfying the condition (3) on vertex and edge weights. The set of \((\gamma _i)\) such that exactly one flat band exists consists of six connected components which have no mutual intersections and have no intersection with the two-flat-band parameter set, identified in Theorem 9.

The solution space is invariant under those permutations of the \(\gamma _i\), which correspond to rotations of the lattice by \(\frac{2\pi }{3}\), and \(\frac{4\pi }{3}\). 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 11

Recall that due to the reductions made at the beginning of the section, after fixing the constant vertex weight \(\mu > 0\), the space of edge weights is a 3-dimensional manifold in the 6-dimensional parameter space \(\{ \gamma _1, \gamma _2, \gamma _3, \gamma _7, \gamma _8, \gamma _9 > 0 \}\), subject to the conditions

$$\begin{aligned} \gamma _1+\gamma _3+\gamma _9= \gamma _1+\gamma _2+\gamma _8=\gamma _2+\gamma _3+\gamma _7=\mu . \end{aligned}$$
(21)

Furthermore, from the proof of Theorem 9 we infer that \(\Delta _\gamma \) has a flat band at \(\lambda \) if and only if the weighted adjacency matrix \(\Pi _\gamma ^\theta \) has the \(\theta \)-independent eigenvalue \(\tilde{\lambda }:= \mu (1-\lambda )\). This requires in particular that

$$\begin{aligned} \tilde{\lambda }= -\frac{\gamma _1\gamma _3}{\gamma _2}\pm \gamma _9=-\frac{\gamma _1\gamma _2}{\gamma _3}\pm \gamma _8=-\frac{\gamma _2\gamma _3}{\gamma _1}\pm \gamma _7 \end{aligned}$$
(22)

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 9 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 \( \gamma _7, \gamma _8, \gamma _9 \). Also, it is immediately clear that (22) with different signs does not allow for a monomeric and nonzero 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 9.

By symmetry, it suffices to investigate two out of these six cases:

$$\begin{aligned} {{\textbf {Case(- + +):}}} \qquad -\frac{\gamma _1\gamma _3}{\gamma _2} - \gamma _9=-\frac{\gamma _1\gamma _2}{\gamma _3}+\gamma _8=-\frac{\gamma _2\gamma _3}{\gamma _1}+\gamma _7 = \tilde{\lambda }\ , \end{aligned}$$
(23)

and

$$\begin{aligned} {{{\textbf {Case(+ - -):}}}} \qquad -\frac{\gamma _1\gamma _3}{\gamma _2} + \gamma _9=-\frac{\gamma _1\gamma _2}{\gamma _3}-\gamma _8=-\frac{\gamma _2\gamma _3}{\gamma _1}-\gamma _7 = \tilde{\lambda }\ . \end{aligned}$$
(24)

To solve Case(- + +), combine the second identities in  (21) and (23), to deduce

$$\begin{aligned} \gamma _3 - \gamma _1 = \frac{\gamma _2}{\gamma _1 \gamma _3} (\gamma _1^2 - \gamma _3^2) \end{aligned}$$

which, recalling \(\gamma _i > 0\), is only possible if \(\gamma _1 = \gamma _3\). But then, by (23), \(\gamma _7 = \gamma _8\). Calling \(\alpha ' := \gamma _2\), and \(\beta ' := \gamma _9\), we can use (21), to further express

$$\begin{aligned} \gamma _1 = \gamma _3 = \frac{\mu - \beta '}{2}, \quad \text {and} \quad \gamma _7 = \gamma _8 = \frac{\mu + \beta '}{2} - \alpha '. \end{aligned}$$
(25)

Next, we eliminate \(\beta '\) by resolving the yet unused first identity in (23), which yields

$$\begin{aligned}&- \frac{( \mu - \beta ')^2}{4 \alpha '} - \beta ' = - \alpha ' + \frac{\mu + \beta '}{2} - \alpha ' \\ \Leftrightarrow \quad&\beta ' = \mu - 3 \alpha ' \pm \sqrt{17 \alpha '^2 - 8 \alpha ' \mu }. \end{aligned}$$

This only has real solutions if \(\alpha '> \frac{8}{17} \mu > \frac{1}{3} \mu \), thus only

$$\begin{aligned} \beta ' = \mu - 3 \alpha ' + \sqrt{17 \alpha '^2 - 8 \alpha ' \mu }. \end{aligned}$$

can be a positive solution. Furthermore, we need \(\beta ' \in (0, \mu )\), which is the case if and only if

$$\begin{aligned} \gamma _2 = \alpha ' \in \left( \frac{\mu }{2}, \mu \right) . \end{aligned}$$

We therefore find the one-parameter solution set

$$\begin{aligned} {{\textbf {Case (-\ +\ +)}}} \quad {\left\{ \begin{array}{ll} \gamma _1 = \gamma _3 &{}= \frac{\mu - \beta '}{2},\\ \gamma _2 = \alpha ' &{}\in \left( \frac{\mu }{2}, \mu \right) ,\\ \gamma _7 = \gamma _8 &{}= \frac{\mu + \beta '}{2} - \alpha ',\\ \gamma _9 = \beta ' &{}:= \mu - 3 \alpha ' + \sqrt{17 \alpha '^2 - 8 \alpha \mu } \end{array}\right. } \end{aligned}$$
(26)

with energy

$$\begin{aligned} \tilde{\lambda }= - \gamma _2 + \gamma _7 = -2 \alpha ' + \frac{\mu + \beta '}{2} = -2 \alpha ' + \frac{2 \mu - 3 \alpha ' + \sqrt{17 \alpha '^2-8 \alpha ' \mu }}{2}. \end{aligned}$$

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

$$\begin{aligned} \gamma _3 - \gamma _1 = \frac{\gamma _2}{\gamma _1 \gamma _3} (\gamma _3^2 - \gamma _1^2)\ . \end{aligned}$$
(27)

Identity (27) has two types of solutions:

Case(+ - -)(a): \(\gamma _1 = \gamma _3\).

As before we find \(\gamma _7 = \gamma _8\). Let \(\alpha ' := \gamma _2\), \(\beta ' := \gamma _9\), and combine the remaining first identity in (24) with (25) to solve for \(\beta '\), finding

$$\begin{aligned}&- \frac{(\mu - \beta ')^2}{4 \alpha '} + \beta ' = - \frac{\mu + \beta '}{2} \\ \Leftrightarrow \quad&\beta ' = \mu + 3 \alpha ' \pm \sqrt{9 \alpha '^2 + 8 \alpha ' \mu }. \end{aligned}$$

Only the solution

$$\begin{aligned} \beta ' = \mu + 3 \alpha ' - \sqrt{9 \alpha '^2 + 8 \alpha ' \mu } \end{aligned}$$

has a chance to be in \((0, \mu )\), and, indeed, this is the case if and only if

$$\begin{aligned} \gamma _2 = \alpha ' \in \left( 0, \frac{\mu }{2} \right) . \end{aligned}$$

We obtain the one-parameter solution set

$$\begin{aligned} {{\textbf {Case(+ - -)(a)}}} \quad {\left\{ \begin{array}{ll} \gamma _1 = \gamma _3 &{}= \frac{\mu - \beta '}{2},\\ \gamma _2 = \alpha ' &{}\in \left( 0, \frac{\mu }{2} \right) ,\\ \gamma _7 = \gamma _8 &{}= \frac{\mu + \beta '}{2} - \alpha ',\\ \gamma _9 = \beta ' &{}:= \mu + 3 \alpha ' - \sqrt{9 \alpha '^2 + 8 \alpha ' \mu } \end{array}\right. } \end{aligned}$$
(28)

with energy

$$\begin{aligned} \tilde{\lambda } = - \gamma _2 - \gamma _7 = - \frac{\mu + \beta '}{2} = - \frac{2 \mu + 3 \alpha ' - \sqrt{9 \alpha '^2+8 \alpha ' \mu }}{2}. \end{aligned}$$

Again, an explicit calculation shows that (18) is fullfilled.

Case(+ - -)(b): The other solution of (27) is

$$\begin{aligned} \gamma _1 \gamma _3 = \gamma _2 (\gamma _1 + \gamma _3). \end{aligned}$$

We set \(\alpha '' := \gamma _1\), \(\beta '' := \gamma _3\), whence

$$\begin{aligned} \gamma _2 = \frac{\alpha '' \beta ''}{\alpha '' + \beta ''}, \end{aligned}$$

and use (21) to infer

$$\begin{aligned} \gamma _7 = \mu - \frac{2 \alpha '' \beta '' + \beta ''^2}{\alpha '' + \beta ''} , \quad \gamma _8 = \mu - \frac{\alpha ''^2 + 2 \alpha '' \beta ''}{\alpha '' + \beta ''}, \quad \gamma _9 = \mu - \alpha '' - \beta ''.\qquad \end{aligned}$$
(29)

Plugging (29) into the yet unused first identity in (24), we arrive at

$$\begin{aligned}&- (\alpha '' + \beta '') + \mu - \alpha '' - \beta '' = - \frac{\alpha ''^2}{\alpha '' + \beta ''} - \mu + \frac{\alpha ''^2 + 2 \alpha '' \beta ''}{\alpha '' + \beta ''} \\ \Leftrightarrow \quad&\beta '' = \frac{\mu - 3 \alpha '' \pm \sqrt{(\mu - 3 \alpha '')^2 + 4 \alpha ''(\mu - \alpha '')}}{2} = \frac{\mu - 3 \alpha '' \pm \sqrt{\mu ^2 - 2 \alpha '' \mu + 5 \alpha ''^2}}{2} \end{aligned}$$

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 \((0, \mu )\) for all \(\alpha '' \in (0, \mu )\). We obtain the one-parameter solution set

$$\begin{aligned} \text {{\textbf {Case (+\ -\ -) (b)}}} \quad {\left\{ \begin{array}{ll} \gamma _1 = \alpha '' &{}\in \left( 0 , \mu \right) ,\\ \gamma _2 &{}= \frac{\alpha '' \beta ''}{\alpha '' + \beta ''},\\ \gamma _3 = \beta '' &{}:= \frac{\mu - 3 \alpha '' + \sqrt{\mu ^2 - 2 \alpha '' \mu + 5 \alpha ''^2}}{2},\\ \gamma _7 &{}= \mu - \frac{2 \alpha '' \beta '' + \beta ''^2}{\alpha '' + \beta ''},\\ \gamma _8 &{}= \mu - \frac{\alpha ''^2 + 2 \alpha '' \beta ''}{\alpha '' + \beta ''},\\ \gamma _9 &{}= \mu - \alpha '' - \beta '' \end{array}\right. } \end{aligned}$$
(30)

at energy

$$\begin{aligned} \tilde{\lambda } = - \frac{\gamma _1 \gamma _3}{\gamma _2} + \gamma _9 = \mu - 2 \alpha '' - 2 \beta '' = \alpha - \sqrt{\mu ^2 - 2 \alpha '' \mu + 5 \alpha ''^2}. \end{aligned}$$

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

$$\begin{aligned} \gamma _1 = \gamma _3 = \gamma _7 = \gamma _8 = \frac{2 \mu }{5}, \quad \gamma _2 = \gamma _9 = \frac{\mu }{5}. \end{aligned}$$

\(\square \)

Fig. 6
figure 6

Schematic overview of the topology of the six “spurious” one-flat-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

Full size image

Remark 12

Theorems 9 and 11 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

$$\begin{aligned} X_1 := \left( 0,0,0,\frac{\mu }{2},\frac{\mu }{2},\frac{\mu }{2} \right) \quad \text {and} \quad X_2 := \left( \frac{\mu }{2},\frac{\mu }{2},\frac{\mu }{2},0,0,0 \right) , \end{aligned}$$

which themselves do no longer belong to the space of admissible parameters. Likewise, the limit of solutions of Case(+ - -) in (26) corresponding to \(\alpha ' = \frac{\mu }{2}\) corresponds to the point \(X_2\), see also Fig. 6.