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

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 \(\gamma _7 = \gamma _8 = \gamma _9 =: \beta \)

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

$$\begin \Delta ^_=}- \frac \begin 0& \quad \gamma _4& \quad \gamma _6& \quad 0& \quad z\gamma _9& \quad 0 \\ \gamma _4& \quad 0& \quad \gamma _5& \quad 0& \quad 0& \quad w\gamma _8 \\ \gamma _6& \quad \gamma _5& \quad 0& \quad \gamma _7& \quad 0& \quad 0 \\ 0& \quad 0& \quad \gamma _7& \quad 0& \quad \gamma _3& \quad \gamma _2\\ \overline\gamma _9& \quad 0& \quad 0& \quad \gamma _3& \quad 0& \quad \gamma _1\\ 0& \quad \overline\gamma _8& \quad 0& \quad \gamma _2& \quad \gamma _1& \quad 0 \end, \end$$

(13)

where \(w:=e^\), \(z:=e^\).

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

$$\begin \begin \mu = \gamma _2+\gamma _3+\gamma _7 = \gamma _5+\gamma _6+\gamma _7&= \gamma _1+\gamma _2+\gamma _8 = \gamma _4+\gamma _5+\gamma _8 \\&= \gamma _1+\gamma _3+\gamma _9 = \gamma _4+\gamma _6+\gamma _9. \end \end$$

(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

$$\begin \begin 2 \gamma _1 + \gamma _8 + \gamma _9&= 2 \gamma _7 + \gamma _2 + \gamma _3, \\ 2 \gamma _4 + \gamma _8 + \gamma _9&= 2 \gamma _7 + \gamma _5 + \gamma _6, \\ \gamma _2 + \gamma _3&= \gamma _5 + \gamma _6 \end \end$$

(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

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

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:

(i)

There exist exactly two flat bands.

(ii)

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

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

Proof

Recall that in the constant vertex weight case, we have

$$\begin \gamma _1=\gamma _4\ , \quad \gamma _2=\gamma _5\ , \quad \text \quad \gamma _3=\gamma _6\ , \end$$

and consider the weighted adjacency matrix

$$\begin \Pi ^_:=\begin 0& \quad \gamma _4& \quad \gamma _6& \quad 0& \quad z\gamma _9& \quad 0 \\ \gamma _4& \quad 0& \quad \gamma _5& \quad 0& \quad 0& \quad w\gamma _8 \\ \gamma _6& \quad \gamma _5& \quad 0& \quad \gamma _7& \quad 0& \quad 0 \\ 0& \quad 0& \quad \gamma _7& \quad 0& \quad \gamma _3& \quad \gamma _2\\ \overline\gamma _9& \quad 0& \quad 0& \quad \gamma _3& \quad 0& \quad \gamma _1\\ 0& \quad \overline\gamma _8& \quad 0& \quad \gamma _2& \quad \gamma _1& \quad 0\end =\begin 0& \quad \gamma _1& \quad \gamma _3& \quad 0& \quad z\gamma _9& \quad 0 \\ \gamma _1& \quad 0& \quad \gamma _2& \quad 0& \quad 0& \quad w\gamma _8 \\ \gamma _3& \quad \gamma _2& \quad 0& \quad \gamma _7& \quad 0& \quad 0 \\ 0& \quad 0& \quad \gamma _7& \quad 0& \quad \gamma _3& \quad \gamma _2\\ \overline\gamma _9& \quad 0& \quad 0& \quad \gamma _3& \quad 0& \quad \gamma _1\\ 0& \quad \overline\gamma _8& \quad 0& \quad \gamma _2& \quad \gamma _1& \quad 0\end\nonumber \\ \end$$

(16)

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

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

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

$$\begin \begin \lambda ^2 \gamma _2^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _1^2 \gamma _3^2 - \gamma _2^2 \gamma _9^2&= 0, \\ \lambda ^2 \gamma _3^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _1^2 \gamma _2^2 - \gamma _3^2 \gamma _8^2&= 0, \\ \lambda ^2 \gamma _1^2 + 2 \lambda \gamma _1 \gamma _2 \gamma _3 + \gamma _2^2 \gamma _3^2 - \gamma _1^2 \gamma _7^2&= 0 , \end \end$$

(17)

as well as

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

(18)

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

$$\begin \lambda = - \frac \pm \gamma _9, \quad \lambda = - \frac \pm \gamma _8, \quad \text \quad \lambda = - \frac \pm \gamma _7. \end$$

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

$$\begin -\frac+\gamma _9=-\frac+\gamma _8=-\frac+\gamma _7 \end$$

(19)

together with

$$\begin -\frac-\gamma _9=-\frac-\gamma _8=-\frac-\gamma _7. \end$$

(20)

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

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

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:

$$\begin I_1 \cup I_2 := \left[ 0, \left( 1 - \frac \right) - \frac \right] \bigcup \left[ \left( 1 - \frac \right) + \frac , 2 - \frac \right] \end$$

with flat bands at \(\frac\) and \(2 - \frac\).

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 \(\alpha = \frac\), 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 \(\alpha = \frac\) connects two regimes with different spectral pictures:

If \(\alpha > \frac\), 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 \(\alpha = \frac)\), this has already been observed, for instance in [36].

If \(\alpha < \frac\), 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 \(\alpha = \frac\).

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

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

Proof of Theorem 10

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

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

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

$$\begin \left\ \left( \alpha \pm \sqrt } \right) \right\} \ , \end$$

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

$$\begin \lambda _1(\theta , \gamma )&= 1 - \frac \left( \alpha + \sqrt } \right) \ , \\ \lambda _2(\theta , \gamma )&= 1 - \frac \left( \alpha + \sqrt } \right) \ , \\ \lambda _3(\theta , \gamma )&= 1 + \frac = \frac = 1 - \frac &\text \,\, 3 \alpha \ge 2 \beta \ , \\ 1 - \frac & \text \,\, 3 \alpha < 2 \beta \ , \end\right. } \\ \lambda _4(\theta , \gamma )&= 1 - \frac \left( \alpha - \sqrt } \right) \ , \\ \lambda _5(\theta , \gamma )&= 1 - \frac \left( \beta - \sqrt } \right) \ , \\ \lambda _6(\theta , \gamma )&= 1 + \frac = 2 - \frac\ . \end$$

Using that the map \(\mathbb ^2 \ni (\theta _1, \theta _2) \mapsto F(\theta _1, \theta _2)\) 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

$$\begin&\left[ \min _^2} \lambda _1(\theta , \gamma ) , \max _^2} \lambda _2(\theta , \gamma ) \right] \bigcup \left[ \min _^2} \lambda _4(\theta , \gamma ) , \max _^2} \lambda _5(\theta , \gamma ) \right] \\&\qquad = \left[ 0, 1 - \frac \right] \bigcup \left[ 1 - \frac , 2 - \frac \right] \\&\qquad = \left[ 0, \left( 1 - \frac \right) - \frac \right] \bigcup \left[ \left( 1 - \frac \right) + \frac , 2 - \frac \right] . \end$$

\(\square \)

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 \(\frac\), and \(\frac\). 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 \(\\), subject to the conditions

$$\begin \gamma _1+\gamma _3+\gamma _9= \gamma _1+\gamma _2+\gamma _8=\gamma _2+\gamma _3+\gamma _7=\mu . \end$$

(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 \(\tilde:= \mu (1-\lambda )\). This requires in particular that

$$\begin \tilde= -\frac\pm \gamma _9=-\frac\pm \gamma _8=-\frac\pm \gamma _7 \end$$

(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:

$$\begin }} \qquad -\frac - \gamma _9=-\frac+\gamma _8=-\frac+\gamma _7 = \tilde\ , \end$$

(23)

and

$$\begin }}} \qquad -\frac + \gamma _9=-\frac-\gamma _8=-\frac-\gamma _7 = \tilde\ . \end$$

(24)

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

$$\begin \gamma _3 - \gamma _1 = \frac (\gamma _1^2 - \gamma _3^2) \end$$

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

$$\begin \gamma _1 = \gamma _3 = \frac, \quad \text \quad \gamma _7 = \gamma _8 = \frac - \alpha '. \end$$

(25)

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

$$\begin&- \frac - \beta ' = - \alpha ' + \frac - \alpha ' \\ \Leftrightarrow \quad&\beta ' = \mu - 3 \alpha ' \pm \sqrt. \end$$

This only has real solutions if \(\alpha '> \frac \mu > \frac \mu \), thus only

$$\begin \beta ' = \mu - 3 \alpha ' + \sqrt. \end$$

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

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

We therefore find the one-parameter solution set

$$\begin }} \quad \gamma _1 = \gamma _3 &= \frac,\\ \gamma _2 = \alpha ' &\in \left( \frac, \mu \right) ,\\ \gamma _7 = \gamma _8 &= \frac - \alpha ',\\ \gamma _9 = \beta ' &:= \mu - 3 \alpha ' + \sqrt \end\right. } \end$$

(26)

with energy

$$\begin \tilde= - \gamma _2 + \gamma _7 = -2 \alpha ' + \frac = -2 \alpha ' + \frac}. \end$$

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

$$\begin \gamma _3 - \gamma _1 = \frac (\gamma _3^2 - \gamma _1^2)\ . \end$$

(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

$$\begin&- \frac + \beta ' = - \frac \\ \Leftrightarrow \quad&\beta ' = \mu + 3 \alpha ' \pm \sqrt. \end$$

Only the solution

$$\begin \beta ' = \mu + 3 \alpha ' - \sqrt \end$$

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

$$\begin \gamma _2 = \alpha ' \in \left( 0, \frac \right) . \end$$

We obtain the one-parameter solution set

$$\begin }} \quad \gamma _1 = \gamma _3 &= \frac,\\ \gamma _2 = \alpha ' &\in \left( 0, \frac \right) ,\\ \gamma _7 = \gamma _8 &= \frac - \alpha ',\\ \gamma _9 = \beta ' &:= \mu + 3 \alpha ' - \sqrt \end\right. } \end$$

(28)

with energy

$$\begin \tilde = - \gamma _2 - \gamma _7 = - \frac = - \frac}. \end$$

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

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

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

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

$$\begin \gamma _2 = \frac, \end$$

and use (21) to infer

$$\begin \gamma _7 = \mu - \frac , \quad \gamma _8 = \mu - \frac, \quad \gamma _9 = \mu - \alpha '' - \beta ''.\qquad \end$$

(29)

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

$$\begin&- (\alpha '' + \beta '') + \mu - \alpha '' - \beta '' = - \frac - \mu + \frac \\ \Leftrightarrow \quad&\beta '' = \frac} = \frac} \end$$

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

$$\begin \text }} \quad \gamma _1 = \alpha '' &\in \left( 0 , \mu \right) ,\\ \gamma _2 &= \frac,\\ \gamma _3 = \beta '' &:= \frac},\\ \gamma _7 &= \mu - \frac,\\ \gamma _8 &= \mu - \frac,\\ \gamma _9 &= \mu - \alpha '' - \beta '' \end\right. } \end$$

(30)

at energy

$$\begin \tilde = - \frac + \gamma _9 = \mu - 2 \alpha '' - 2 \beta '' = \alpha - \sqrt. \end$$

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

$$\begin \gamma _1 = \gamma _3 = \gamma _7 = \gamma _8 = \frac, \quad \gamma _2 = \gamma _9 = \frac. \end$$

\(\square \)

Fig. 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

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

$$\begin X_1 := \left( 0,0,0,\frac,\frac,\frac \right) \quad \text \quad X_2 := \left( \frac,\frac,\frac,0,0,0 \right) , \end$$

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