# Kagome network with vertex coupling
of a preferred orientation

Marzieh Baradaran1,2 and Pavel Exner3,4

(1) Department of Physics, Faculty of Science, University of Hradec Králové,
Rokitanského 62, 500 03 Hradec Králové, Czechia
2) Department of Mathematics, Faculty of Nuclear Sciences and Physical
Engineering, Czech Technical University, Břehová 7, 11519 Prague, Czechia
3) Doppler Institute for Mathematical Physics and Applied Mathematics, Czech
Technical University, Břehová 7, 11519 Prague, Czechia
4) Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy
of Sciences, 25068 Řež near Prague, Czechia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate spectral properties of periodic quantum graphs in the form of a
kagome or a triangular lattice in the situation when the condition matching
the wave functions at the lattice vertices is chosen of a particular form
violating the time-reversal invariance. The positive spectrum consists of
infinite number of bands, some of which may be flat; the negative one has at
most three and two bands, respectively. The kagome lattice example shows that
even in graphs with such an uncommon vertex coupling spectral universality may
hold: if its edges are incommensurate, the probability that a randomly chosen
positive number is contained in the spectrum is $\approx 0.639$.

##  Introduction

Quantum graphs proved to be a useful tool to model quantum transport in
periodically structured environments, both natural and artificially prepared;
the latter are gaining importance in connection with the progress in
metamaterial physics. The fact that the way in which the wave functions are
coupled in the nodes of the network is vital for the band structure is quite
old; one can trace it back the celebrated paper of Kronig and Penney [10]. As
long as the only requirement one imposes is the conservation of the
probability current, mathematically expressed as the self-adjointness of the
corresponding Hamiltonian, there is a number of ways how to do that [4, 8, 9]:
for a vertex $v$ in which $N$ edges meet, the self-adjointness is ensured
provided the the boundary-value vectors of the wave functions and their
derivatives are matched through the condition

$(U-I)\psi(v)+i\ell(U+I)\psi^{\prime}(v)=0,$ (1.1)

where $\ell>0$ is the parameter fixing the length scale and $U$ is an $N\times
N$ unitary matrix.

Given such a multitude, one naturally asks about the meaning of the couplings
parametrized by different matrices $U$. The simplest class, the so-called
$\delta$ coupling with the wave functions continuous at the vertex and the
condition (1.1) being reduced to
$\sum_{j=1}^{N}\psi_{j}^{\prime}(v)=\alpha\psi(v)$ with a real parameter
$\alpha$, can be understood easily as modeling a potential sharply localized
around the vertex [6]. The general coupling (1.1) can also be interpreted in
terms of properly scaled potentials, however, the approximation is
considerably more complicated [5] and the result has mostly the existence
meaning.

A pragmatic approach is to choose the coupling that suits the model in
question. Recently a class of couplings unnoticed so far attracted attention,
with the motivation coming from an attempt to model the anomalous Hall effect
using a lattice graph [11]. The said model used the $\delta$ coupling at the
lattice nodes which forced the authors to impose by hand a preferential
direction. That was a flaw since such an assumption cannot justified on the
lattice edges, but it inspired the observation that the family specified by
the condition (1.1) includes vertex couplings that may not be invariant with
respect to the time reversal [7]. The simplest among them corresponds to the
matrix $U$ of the circulant type, with the entries equal to one at the first
side diagonal and in the opposite corner, and zero otherwise. A rotational
motion associated with such a matrix becomes obvious if we realize that $U$ is
nothing but the on-shell scattering matrix of vertex at the momentum
$k=\ell^{-1}$.

It appeared that such a coupling has a remarkable topological property, namely
that the transport properties of the vertex at _high energies_ depend on the
vertex _parity_ ; in this asymptotic regime the vertex remain transparent if
the parity is even, while for the odd one we get an effective decoupling of
the edges. This effect was illustrated in [7] through comparison of band
spectra two lattices, the square and the hexagonal one. The spectrum appeared
to be dominated by the bands and gaps, respectively, in the sense that the
probability that a randomly chosen positive energy belong to the spectrum, as
defined by Band and Berkolaiko [1], equals one and zero, respectively.

However, things may not be that simple. In [2, 3], we investigated another
graph with the described vertex coupling, a periodic chain of rings connected
either tightly, or loosely through connecting links. The loosely connected
chain with vertices of degree three appeared to be effectively decoupled at
high energies, but for the tightly connected one the probability of being in
the spectrum might or might not equal to one; this happened if the chain had
(vertically) the mirror symmetry, otherwise the said probability equaled one
half. We also found that despite these differences the spectrum of the loose
chain converges to that of the tight one as the lengths of the connecting
links shrink to zero, but the convergence was rather non-uniform.

The aim of the present paper is to analyze another class of quantum graphs
with the indicated vertex coupling violating the time-reversal invariance,
namely lattices of _kagome_ and _triangular type_ , where the latter can be
regarded as the degenerate case of the former. As in other periodic graphs we
find that under appropriate rationality conditions such a system exhibit flat
bands. Apart from them, there is no effective decoupling here since the vertex
parities are always even in the present situation. The probability that a
positive energy belongs to a spectral band is equal to two thirds if the
kagome lattice is equilateral or degenerate to the triangular one. If the
lattice is asymmetric, the probability is different, however, if the edge
lengths are _incommensurate_ , it takes a fixed value $\approx 0.639$ showing
that the universality result derived in [1] for Kirchhoff graphs may be valid
for a much wider class of vertex couplings.

Let us mention briefly the contents of the paper. In the next section we
collect the needed information about the vertex coupling. The spectral problem
for the kagome and triangular lattices are solved respectively in Secs. 3 and
4. We derive the appropriate spectral conditions and solve then separately for
the positive and negative part of the spectrum.

##  Preliminaries

Let us describe now the basic setting in more technical terms. We suppose that
the motion on the graph edges is free away from the vertices, so that the
Hamiltonian acts there as $-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$. Consider
a vertex $v$ of degree $N$. Writing the coupling condition (1.1) with the
circulant matrix $U$ described above in components, we get

$(\psi_{j+1}-\psi_{j})+i\ell(\psi_{j+1}^{\prime}+\psi_{j}^{\prime})=0,$ (2.1)

where $\psi_{j},\,j=1,\dots,N,$ are the components of $\psi(v)$ and similarly
for $\psi^{\prime}(v)$. The corresponding on-shell scattering matrix,
$S(k)=\frac{k\ell-1+(k\ell+1)U}{k\ell+1+(k\ell-1)U}$, can be also expressed in
components [7] being

$S_{ij}(k)=\frac{1-\eta^{2}}{1-\eta^{N}}\bigg{\\{}-\eta\,\frac{1-\eta^{N-2}}{1-\eta^{2}}\,\delta_{ij}+(1-\delta_{ij})\,\eta^{(j-i-1)(\textstyle{\rm
mod}\;N)}\bigg{\\}},$ (2.2)

where $\eta\mathrel{\mathop{\ordinarycolon}}=\tfrac{1-k\ell}{1+k\ell}$.
Inspecting the behavior of this expression in the high-energy limit, $\eta\to
1-$, we find that $\lim_{k\to\infty}S(k)=I$ if $N$ is odd, while for $N$ even
the limit is different from the unit matrix describing the full separation of
the edges. The root of this difference is the fact that $-1$ is an eigenvalue
of $U$ if and only if $N$ is even.

The peculiar feature of quantum graphs is that they have a single propagating
mode. As a consequence, the motion is free away from the vertex and the
scattering matrix makes sense irrespective of the edge lengths. If the edges
are semi-infinite, the Hamiltonian of such a star graph has a nonempty
discrete spectrum for any $N\geq 3$ and the eigenvalues are

$E=-\tan^{2}\frac{m\pi}{N},$ (2.3)

with $m$ running through $1,\cdots,[\tfrac{N}{2}]$ for odd $N$ and
$1,\cdots,[\tfrac{N-1}{2}]$ for even $N$.

##  Kagome lattice

###  3.1 The spectral condition

Our main topic in this paper is spectral properties of periodic quantum graphs
of _kagome_ type sketched in Fig. 1. We suppose that the edge lengths $b$ and
$c$ are both positive, postponing the degenerate case of a triangular lattice
to the next section, and assume that their sum is fixed, $b+c=d$.

Figure 1:  An elementary cell of the kagome network.

Since the system is periodic, its spectral analysis relies on the Floquet-
Bloch decomposition [4, Chap. 4] which makes it possible to reduce the task to
investigation of an elementary cell of the graph which contains three vertices
of degree four. Choosing the coordinates on the edges to increase from the
left to right, we employ the following Ansatz for the wave function components

$\displaystyle\psi_{j}(x)=B_{j}^{+}\mathrm{e}^{ikx}+B_{j}^{-}\mathrm{e}^{-ikx},\qquad
x\in[0,\textstyle{\frac{1}{2}}c],\;\;\qquad j=1,2,$
$\displaystyle\psi_{j}(x)=B_{j}^{+}\mathrm{e}^{ikx}+B_{j}^{-}\mathrm{e}^{-ikx},\qquad
x\in[0,\textstyle{\frac{1}{2}}b],\;\;\qquad j=3,4,$
$\displaystyle\varphi_{j}(x)=C_{j}^{+}\mathrm{e}^{ikx}+C_{j}^{-}\mathrm{e}^{-ikx},\qquad
x\in[-\textstyle{\frac{1}{2}}b,0],\qquad j=2,3,$
$\displaystyle\varphi_{j}(x)=C_{j}^{+}\mathrm{e}^{ikx}+C_{j}^{-}\mathrm{e}^{-ikx},\qquad
x\in[-\textstyle{\frac{1}{2}}c,0],\qquad j=1,4,$
$\displaystyle\chi_{1}(x)=D_{1}^{+}\mathrm{e}^{ikx}+D_{1}^{-}\mathrm{e}^{-ikx},\qquad
x\in[0,\textstyle{\frac{1}{2}}c],$ (3.1)
$\displaystyle\chi_{2}(x)=D_{2}^{+}\mathrm{e}^{ikx}+D_{2}^{-}\mathrm{e}^{-ikx},\qquad
x\in[-\textstyle{\frac{1}{2}}b,0],$
$\displaystyle\chi_{3}(x)=D_{3}^{+}\mathrm{e}^{ikx}+D_{3}^{-}\mathrm{e}^{-ikx},\qquad
x\in[0,\textstyle{\frac{1}{2}}b],$
$\displaystyle\chi_{4}(x)=D_{4}^{+}\mathrm{e}^{ikx}+D_{4}^{-}\mathrm{e}^{-ikx},\qquad
x\in[-\textstyle{\frac{1}{2}}c,0].$

The network is periodic in two independent directions, for the sake of
definiteness we assume that they are associated with the unit vectors $(1,0)$
and $\frac{1}{2}(1,\sqrt{3})$; the periodicity in the remaining direction is a
superposition of those two. Consequently, the Floquet conditions at the free
ends of the graph cell are

$\displaystyle\chi_{1}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i\theta_{1}}\varphi_{4}(-\textstyle{\frac{1}{2}}c),\;$
$\displaystyle\chi_{1}^{\prime}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i\theta_{1}}\varphi_{4}^{\prime}(-\textstyle{\frac{1}{2}}c),$
$\displaystyle\psi_{1}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i\theta_{2}}\varphi_{1}(-\textstyle{\frac{1}{2}}c),\;$
$\displaystyle\psi_{1}^{\prime}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i\theta_{2}}\varphi_{1}^{\prime}(-\textstyle{\frac{1}{2}}c),$
$\displaystyle\psi_{2}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i(\theta_{2}-\theta_{1})}\chi_{4}(-\textstyle{\frac{1}{2}}c),\;$
$\displaystyle\psi_{2}^{\prime}(\textstyle{\frac{1}{2}}c)$
$\displaystyle=\mathrm{e}^{i(\theta_{2}-\theta_{1})}\chi_{4}^{\prime}(-\textstyle{\frac{1}{2}}c),$
(3.2)

referring to the parameters $\theta_{1},\theta_{2}\in[-\pi,\pi)$; for
simplicity we will speak of them as of quasimomentum components, even if the
true quasimomentum is $\frac{1}{d}(\theta_{1},\theta_{2})$. In addition, the
functions have to be matched smoothly at the segment midpoints, that is,

$\displaystyle\chi_{2}(0)$ $\displaystyle=\psi_{4}(0),\;$
$\displaystyle\chi_{2}^{\prime}(0)$ $\displaystyle=\psi_{4}^{\prime}(0),$
$\displaystyle\varphi_{3}(0)$ $\displaystyle=\psi_{3}(0),\;$
$\displaystyle\varphi_{3}^{\prime}(0)$ $\displaystyle=\psi_{3}^{\prime}(0),$
$\displaystyle\varphi_{2}(0)$ $\displaystyle=\chi_{3}(0),\;$
$\displaystyle\varphi_{2}^{\prime}(0)$ $\displaystyle=\chi_{3}^{\prime}(0).$
(3.3)

Most important, we have to impose the matching conditions (2.1) at the
vertices of graph cell. Remembering that the derivatives have to be taken in
the outward direction, we get

$\displaystyle\psi_{2}(0)-\psi_{1}(0)+i\ell\left(\psi_{2}^{\prime}(0)+\psi_{1}^{\prime}(0)\right)=0,$
$\displaystyle\psi_{3}(\textstyle{\frac{1}{2}}b)-\psi_{2}(0)+i\ell\left(-\psi_{3}^{\prime}(\textstyle{\frac{1}{2}}b)+\psi_{2}^{\prime}(0)\right)=0,$
$\displaystyle\psi_{4}(\textstyle{\frac{1}{2}}b)-\psi_{3}(\textstyle{\frac{1}{2}}b)-i\ell\left(\psi_{4}^{\prime}(\textstyle{\frac{1}{2}}b)+\psi_{3}^{\prime}(\textstyle{\frac{1}{2}}b)\right)=0,$
$\displaystyle\psi_{1}(0)-\psi_{4}(\textstyle{\frac{1}{2}}b)+i\ell\left(\psi_{1}^{\prime}(0)-\psi_{4}^{\prime}(\textstyle{\frac{1}{2}}b)\right)=0,$
$\displaystyle\varphi_{2}(-\textstyle{\frac{1}{2}}b)-\varphi_{1}(0)+i\ell\left(\varphi_{2}^{\prime}(-\textstyle{\frac{1}{2}}b)-\varphi_{1}^{\prime}(0)\right)=0,$
$\displaystyle\varphi_{3}(-\textstyle{\frac{1}{2}}b)-\varphi_{2}(-\textstyle{\frac{1}{2}}b)+i\ell\left(\varphi_{3}^{\prime}(-\textstyle{\frac{1}{2}}b)+\varphi_{2}^{\prime}(-\textstyle{\frac{1}{2}}b)\right)=0,$
$\displaystyle\varphi_{4}(0)-\varphi_{3}(-\textstyle{\frac{1}{2}}b)+i\ell\left(-\varphi_{4}^{\prime}(0)+\varphi_{3}^{\prime}(-\textstyle{\frac{1}{2}}b)\right)=0,$
(3.4)
$\displaystyle\varphi_{1}(0)-\varphi_{4}(0)-i\ell\left(\varphi_{1}^{\prime}(0)+\varphi_{4}^{\prime}(0)\right)=0,$
$\displaystyle\chi_{2}(-\textstyle{\frac{1}{2}}b)-\chi_{1}(0)+i\ell\left(\chi_{2}^{\prime}(-\textstyle{\frac{1}{2}}b)+\chi_{1}^{\prime}(0)\right)=0,$
$\displaystyle\chi_{3}(\textstyle{\frac{1}{2}}b)-\chi_{2}(-\textstyle{\frac{1}{2}}b)+i\ell\left(-\chi_{3}^{\prime}(\textstyle{\frac{1}{2}}b)+\chi_{2}^{\prime}(-\textstyle{\frac{1}{2}}b)\right)=0,$
$\displaystyle\chi_{4}(0)-\chi_{3}(\textstyle{\frac{1}{2}}b)-i\ell\left(\chi_{4}^{\prime}(0)+\chi_{3}^{\prime}(\textstyle{\frac{1}{2}}b)\right)=0,$
$\displaystyle\chi_{1}(0)-\chi_{4}(0)+i\ell\left(\chi_{1}^{\prime}(0)-\chi_{4}^{\prime}(0)\right)=0.$

Substituting now from (3.1) into (3.1), and using (3.1) and (3.1), we get a
system of twelve linear equations for the coefficients
$B_{3}^{\pm},B_{4}^{\pm},C_{1}^{\pm},C_{4}^{\pm},D_{3}^{\pm},D_{4}^{\pm}$;
computing the corresponding determinant, taking into account that $b=d-c$, and
neglecting the inessential multiplicative factor $65536\,i$, we arrive at the
spectral condition

$\mathrm{e}^{2i\theta_{2}}k^{9}\ell^{3}\;\sin\dfrac{kc}{2}\sin\dfrac{kd}{2}\sin\dfrac{k(d-c)}{2}\;\big{(}\lambda_{1}(k)-\lambda_{2}(k)\,f_{\theta}-\lambda_{3}(k)\,g_{\theta}\big{)}=0,$
(3.5)

where

$\displaystyle\lambda_{1}(k)\mathrel{\mathop{\ordinarycolon}}=2(k^{2}\ell^{2}+1)\big{(}4(k^{2}\ell^{2}+1)^{2}\big{(}\cos
k(c+d)+\cos k(c-2d)+2\cos kd+\cos 2kd\big{)}$
$\displaystyle\qquad\qquad+\left(k^{4}\ell^{4}+14k^{2}\ell^{2}+1\right)(2\cos
kd+1)\cos k(2c-d)+\left(3k^{4}\ell^{4}+18k^{2}\ell^{2}+3\right)$
$\displaystyle\qquad\qquad+\left(5k^{4}\ell^{4}+22k^{2}\ell^{2}+5\right)\big{(}\cos
k(d-c)+\cos kc\big{)}\big{)},$
$\displaystyle\lambda_{2}(k)\mathrel{\mathop{\ordinarycolon}}=8\left(k^{2}\ell^{2}+1\right)\left(k^{2}\ell^{2}-1\right)^{2}\cos\dfrac{k(d-c)}{2}\cos\dfrac{kc}{2}\Big{(}\cos\dfrac{k(2c-d)}{2}+2\cos\dfrac{kd}{2}\Big{)},$
$\displaystyle\lambda_{3}(k)\mathrel{\mathop{\ordinarycolon}}=16k\ell\left(k^{2}\ell^{2}-1\right)^{2}\sin\dfrac{k(d-c)}{2}\sin\dfrac{kc}{2}\sin\dfrac{k(d-2c)}{2},$

and the quasimomentum-dependent quantities in (3.5),

$\displaystyle
f_{\theta}\mathrel{\mathop{\ordinarycolon}}=\cos\theta_{1}+\cos(\theta_{1}-\theta_{2})+\cos\theta_{2},$
$\displaystyle
g_{\theta}\mathrel{\mathop{\ordinarycolon}}=\sin\theta_{2}+\sin(\theta_{1}-\theta_{2})-\sin\theta_{1},$

range through $[-\textstyle{\frac{3}{2}},3]$ and
$[-\textstyle{\frac{3\sqrt{3}}{2}},\textstyle{\frac{3\sqrt{3}}{2}}]$,
respectively. In particular, for $d=2c$, in which case the graph is
equilateral exhibiting a repeated David-star pattern, the coefficient
$\lambda_{3}(k)$ vanishes and the spectral condition (3.5) reduces to

$\displaystyle 4\big{(}k^{2}\ell^{2}+1\big{)}\big{(}2\cos kc+1\big{)}\sin
kc\;\sin^{2}\frac{kc}{2}$
$\displaystyle\times\left(\left(k^{4}\ell^{4}+14k^{2}\ell^{2}+1\right)\cos
kc+\left(k^{2}\ell^{2}+1\right)^{2}\big{(}2\cos 2kc+2\cos
3kc+1\big{)}-\big{(}\cos
kc+1\big{)}\left(k^{2}\ell^{2}-1\right)^{2}f_{\theta}\right)=0.$

###  3.2 Positive spectrum

According to the spectral conditions (3.5) and (3.1), the positive spectrum
consists of two parts:

####  _i_. Infinitely degenerate eigenvalues

  * •

In the general case, the number $k^{2}$ belongs to the spectrum for
$k=\frac{2n\pi}{l}$ with $l=\\{d-c,c,d\\}$ and $n\in\mathbb{N}$, may or may
not be embedded in the continuous spectrum. In particular, in the equilateral
case, they merge into $k^{2}=(\tfrac{n\pi}{c})^{2}$ which may not be embedded
in the continuous spectrum; inspecting the large bracket in (3.1) for
$k=\tfrac{n\pi}{c}$, we get $-12\pi^{2}c^{-2}n^{2}\ell^{2}<0$ and
$3\left(c^{2}+\pi^{2}n^{2}\ell^{2}\right)^{2}-f_{\theta}\left(c^{2}-\pi^{2}n^{2}\ell^{2}\right)^{2}+6\pi^{2}c^{2}n^{2}\ell^{2}>0$
for odd and even $n$, respectively, the latter can be easily checked in view
of the inequality
$\left(c^{2}+\pi^{2}n^{2}\ell^{2}\right)^{2}>\left(c^{2}-\pi^{2}n^{2}\ell^{2}\right)^{2}$.

  * •

In the equilateral case, the number $k^{2}$ belongs to the spectrum for
$k=\left((-1)^{n+1}+(6n-3)\right)\frac{\pi}{6c}$ with $n\in\mathbb{N}$, may or
may not be embedded in the continuous spectrum.

  * •

It is possible that some positive bands degenerate to the points; in the
general case, this happens at $k=\ell^{-1}$ for
$\\{d-c,c,d\\}=\ell\big{(}(-1)^{n+1}+(6n-3)\big{)}\frac{\pi}{6}$ with
$n\in\mathbb{N}$; accordingly, in the equilateral case, this happens at
$k=\ell^{-1}$ for $c=\ell\big{(}(-1)^{n+1}+(6n-3)\big{)}\frac{\pi}{12}$.

####  _ii_. Continuous bands

Away from the flat bands mentioned above, the rest of the spectrum is
continuous having a band-and-gap structure determined by vanishing of the
bracket in (3.5), that is

$\lambda_{1}(k)=\lambda_{2}(k)\,f_{\theta}+\lambda_{3}(k)\,g_{\theta}.$ (3.7)

In order to describe the bands and gaps in a more explicit way, one can
inspect the right-hand side of this equation as a function of two variables
$(\theta_{1},\theta_{2})$; using the Hessian method of determining extrema of
multivariate functions as well as checking the boundaries of our rectangular
domain, we find that the global extrema of the function may happen at one of
the points $(0,0)$ and $\left(\pm\tfrac{2\pi}{3},\mp\tfrac{2\pi}{3}\right)$.
Hence, the positive spectrum is determined by the intersection of the function
$k\mapsto\lambda_{1}(k)$ with the region bordered from below and above by the
curves $k\mapsto\lambda^{0}(k)$ and $k\mapsto\lambda^{\pm}(k)$, where

$\displaystyle\lambda^{0}(k)\mathrel{\mathop{\ordinarycolon}}=3\,\lambda_{2}(k),$
$\displaystyle\lambda^{\pm}(k)\mathrel{\mathop{\ordinarycolon}}=-\frac{3}{2}\left(\lambda_{2}(k)\pm\sqrt{3}\,\lambda_{3}(k)\right).$

In other words, depending on the signs of the functions $\lambda_{2}(k)$ and
$\lambda_{3}(k)$, a number $k^{2}$ belongs to a spectral band if and only if

$k\in\left\\{k\mathrel{\mathop{\ordinarycolon}}\;\lambda^{\pm}(k)\leq\lambda_{1}(k)\leq\lambda^{0}(k)\quad\cup\quad\lambda^{0}(k)\leq\lambda_{1}(k)\leq\lambda^{\pm}(k)\right\\}.$
(3.8)

The band-and-gap pattern of the general model in dependence on $d$ and $c$ is
illustrated in Figs. 3 and 4, respectively. Moreover, two other examples for
specific values of $c$ and $d$ are shown in Fig. 5. The spectrum of the
equilateral model in dependence on, and for specific values of $c$ is
illustrated in Figs. 6 and 7, respectively. We see that: For $d\geq
2\sqrt{3}\,\ell$, the positive spectrum starts at zero, and in contrast, for
$d<2\sqrt{3}\,\ell$, the first positive band remains separated from zero. To
see that, let us inspect the behavior of the band condition (3.8) for
$k\rightarrow 0+$; considering the first inequality condition, and using the
Taylor expansion to the second order, we arrive at

$0\;\leq\;108-9k^{2}\left(5c^{2}-5cd+7d^{2}-36\ell^{2}\right)\;\leq\;108-9k^{2}\left(5c^{2}-5cd+3d^{2}+12\ell^{2}\right),$

where we have added $36-3k^{2}\left(5c^{2}-5cd+3d^{2}+12\ell^{2}\right)$ to
the inequality, all the terms with a relative error $\mathcal{O}(k^{4})$.
Simplifying the last two parts of the inequality, we get $d^{2}\geq
12\ell^{2}$, hence, small values of $k$ with $d<2\sqrt{3}\,\ell$ cannot belong
to the spectral bands. Needless to say, it is obvious that the second
inequality in (3.8) does not hold for small values of $k$. Consequently, the
first positive band of the equilateral model starts at zero if
$c\geq\sqrt{3}\,\ell$, otherwise, it remains separated from zero. In the high-
energy regime, we have two types of asymptotic behavior. To take a closer look
at their structure, we rewrite the spectral condition (3.7) in the form

$\alpha(k)\cdot\,k^{6}+\mathcal{O}(k^{5})=0,$ (3.9)

where

$\displaystyle\alpha(k)=4\left(\cos\frac{k(2c-d)}{2}+2\cos\frac{kd}{2}\right)\;\times$
(3.10) $\displaystyle\qquad\left(\left(2\cos k(c-d)+4\cos
kd-1\right)\cos\frac{kd}{2}+\cos\frac{k(2c+d)}{2}-2\,f_{\theta}\cos\frac{kc}{2}\cos\frac{k(c-d)}{2}\right).$

Hence, as $k\rightarrow\infty$, the function $\alpha(k)$ should be close to
zero which results in two types of spectral bands. We begin the discussion
with the general model and discuss the equilateral model as a special case:
Pairs of narrow bands in the vicinity of the roots of
$\cos\textstyle{\frac{k(2c-d)}{2}}+2\cos\textstyle{\frac{kd}{2}}$
The bands appear in pairs centered around the points $k$ which solve the
equation $\cos\textstyle{\frac{k(2c-d)}{2}}+2\cos\textstyle{\frac{kd}{2}}=0$.
Note that these bands do not appear in the high energy regime of the
equilateral model. Indeed, the left-hand side of the mentioned equation with
$d=2c$ appears as a multiplicative factor in (3.9) corresponding to the flat
bands of the second bullet point in Sec. 3.2. Wide bands
The bands and the gaps between them grow asymptotically but not at the same
rate; they correspond to those values of $k$ for which the function in the
second bracket in (3.10) is close to zero; without loss of generality,
dividing the corresponding equation by $\cos\frac{kc}{2}\cos\frac{k(c-d)}{2}$
and after simple manipulations, the sufficient condition of belonging to the
spectral bands for large $k$ is obtained as

$0\;\leq\;\frac{5}{4}+\frac{\cos
kd\;\cos\frac{k(2c-d)}{2}+\cos\frac{3kd}{2}}{\cos\frac{k(2c-d)}{2}+\cos\frac{kd}{2}}\;\leq\;\frac{9}{4}.$
(3.11)

As mentioned in the introduction, we are interested in the probability that a
randomly chosen energy lies in the spectrum, introduced in [1] as

$P_{\sigma}(H)\mathrel{\mathop{\ordinarycolon}}=\lim_{K\to\infty}\frac{1}{K}\left|\sigma(H)\cap[0,K]\right|.$
(3.12)

In the general case, we are unable to find it in a closed form and as Fig. 10
shows it may take different values, however, for $c$ and $d$ incommensurate,
the value is the same being $\approx 0.639$. There are two ways to see that.
First of all, it is clear from Fig. 10 that for rational
$\frac{c}{d}=\frac{p}{q}$ with large coprime $p$ and $q$, which we may regard
as rational approximation to a given irrational number, the probability is
near to the indicated value. Secondly, keeping the leading order in the band
condition (3.8) we get asymptotically

$\Big{(}2\cos\frac{k(c-2d)}{2}+\cos\frac{kc}{2}\Big{)}\Big{(}\cos\frac{k(c-d)}{2}+2\cos\frac{k(c+d)}{2}\Big{)}\Big{(}\cos\frac{k(2c-d)}{2}+2\cos\frac{kd}{2}\Big{)}\geq
0$ (3.13)

We can rewrite the left-hand side of (3.13) as a function of
$\sin\frac{kb}{2}$ and $\sin\frac{kc}{2}$ only. The resulting expression is
quite complicated and gives no hope to solve the inequality, however, if $b$
and $c$ are incommensurate, one can regard the two sines as a pair of
independent identically distributed random variables and compute numerically
the probability that such a quantity will be non-negative; this yields again
the value mentioned above. Indeed, if we calculate the area of the gray parts
in Fig. 11, i.e. the region where the left-hand side of (3.13) is non-
negative, and divide it by $4\pi^{2}$, we get $\approx 0.639081$. Hence,
despite the vertex coupling is in the present case substantially different, we
find again the universality demonstrated in [1] for periodic graphs with
Kirchhoff vertices.

Let us turn to the _equilateral case_ , $b=c$. Requiring the second bracket in
(3.10) to vanish, we get

$\cos^{2}\frac{kc}{2}\big{(}4\cos kc-4\cos
2kc+f_{\theta}-3\big{)}=\mathcal{O}(k^{-1}),$ (3.14)

indicating that, again, we have two types of spectral bands: [–] Pairs of
narrow bands in the vicinity of $k=(2n-1)\frac{\pi}{c}$, $n\in\mathbb{N}$
The two bands around the points $k=(2n-1)\frac{\pi}{c}$ and the gap between
them have asymptotically constant width as $n\rightarrow\infty$. To see that,
we rewrite the expression in the large brackets in (3.1) in the asymptotic
form

$\beta_{1}(k)+\frac{\beta_{2}(k)}{k^{2}}=\mathcal{O}(k^{-4}),$ (3.15)

with

$\displaystyle\beta_{1}(k)=-2\ell^{4}\cos^{2}\frac{kc}{2}\left(4\cos kc-4\cos
2kc+f_{\theta}-3\right),$ $\displaystyle\beta_{2}(k)=2\ell^{2}\left(\left(\cos
kc+1\right)f_{\theta}+7\cos kc+2\cos 2kc+2\cos 3kc+1\right).$

Then, setting $k=(2n-1)\frac{\pi}{c}+\delta$, we get
$k^{-2}=\frac{c^{2}}{4n^{2}\pi^{2}}+\mathcal{O}(n^{-3})$ as $n\to\infty$.
Substituting these into (3.15) and solving the resulting equation for
$\delta$, we obtain

$\delta=\frac{\sqrt{6}}{\pi\ell\sqrt{11-f_{\theta}}}\,\frac{1}{n}+\mathcal{O}(n^{-3}).$

Since the band edges correspond to $f_{\theta}=-\tfrac{3}{2}$ and $3$, the
width of the bands and the gap between them are (on the energy scale)
respectively determined as $\frac{2}{5c\ell}\sqrt{3}+\mathcal{O}(n^{-1})$ and
$\frac{16}{5c\ell}\sqrt{3}+\mathcal{O}(n^{-1})$ as $n\rightarrow\infty$. [–]
Wide bands
Both bands and gaps grow asymptotically, again, not at the same rate;
manipulating the expression in the bracket in (3.14), or equivalently,
substituting $d=2c$ in (3.11), we find that large values of $k$ belong to the
spectral bands if and only if

$0\leq\xi(k)\leq\frac{9}{8}\;;\quad\xi(k)\mathrel{\mathop{\ordinarycolon}}=\cos
kc-\cos 2kc,$ (3.16)

with a relative error $\mathcal{O}(k^{-1})$. The function $\xi(k)$ is periodic
with the period $T=\tfrac{2\pi}{c}$ and one can easily check that the maximum
value of this function is $\tfrac{9}{8}$ which happens at
$k=\tfrac{1}{c}|2m\pi\pm\operatorname{arcsec}4|$ with $m\in\mathbb{Z}$.

It remains to calculate the probability that $\xi(k)$ is positive for a
randomly chosen value of $k$. The roots of $\xi(k)$ in the period are
$\tfrac{2\pi}{3c}$ and $\tfrac{4\pi}{3c}$. On the other hand, we have
$\xi(\tfrac{\pi}{c})=-2$, hence, $\xi(k)$ is negative over the domain
$\left(\tfrac{2\pi}{3c},\tfrac{4\pi}{3c}\right)$ and thus, the probability
(3.12) is for any $c$ equal to

$P_{\sigma}(H)=1-\frac{1}{T}\left(\frac{4\pi}{3c}-\frac{2\pi}{3c}\right)=\frac{2}{3}.$
(3.17)

This differs from the universal value obtained above which is not surprising:
the equilateral character of the graph, as a particular case of
commensurability, means that the ergodicity of the flow which was crucial for
the universality [1] is lost.

The fact that the lattice exhibits a nonvanishing transport in the sense of
the probability (3.12) follows from the fact that the high-energy limit of the
vertex scattering matrix is nontrivial. In fact, for vertices of degree four
we get from (2.2) that

$\underset{k\to\infty}{\text{lim}}S(k)={\scriptsize\frac{1}{2}\left(\begin{array}[]{cccc}\phantom{-}1&\phantom{-}1&-1&\phantom{-}1\\\
\phantom{-}1&\phantom{-}1&\phantom{-}1&-1\\\
-1&\phantom{-}1&\phantom{-}1&\phantom{-}1\\\
\phantom{-}1&-1&\phantom{-}1&\phantom{-}1\\\ \end{array}\right)},$

which means that the probabilities of leaving the vertex in any of the four
direction are asymptotically the same. A comparison with other lattices and
chains with vertices of degree four [3, 7] shows, however, that the quantity
(3.12) is different in different situations depending on the topology of the
structure. While generically the gaps are open, it may happen that some of
them close for some particular values of the parameters. This happens when the
boundaries of neighboring bands touch as illustrated in Fig. 3; note that such
crossing points may occur in sequences with the same energy. In general, it is
not easy to find their coordinates in a closed form, however, introducing the
symbol
$\Delta\mathrel{\mathop{\ordinarycolon}}=\big{(}\lambda_{1}(k)-\lambda_{2}(k)\,f_{\theta}-\lambda_{3}(k)\,g_{\theta}\big{)}$
for the bracket in (3.5) we can identify such situations using the sufficient
condition,

$\frac{\partial\Delta}{\partial\theta_{1}}=\frac{\partial\Delta}{\partial\theta_{2}}=\frac{\partial\Delta}{\partial
k}=\frac{\partial\Delta}{\partial d}=0,$ (3.18)

supposing that $c$ and $\ell$ are fixed. For the first two derivatives we get
the expressions

$\displaystyle\frac{\partial\Delta}{\partial\theta_{1}}=\sin\Big{(}\theta_{1}-\frac{\theta_{2}}{2}\Big{)}\left(\lambda_{2}(k)\cos\frac{\theta_{2}}{2}-\lambda_{3}(k)\sin\frac{\theta_{2}}{2}\right),$
$\displaystyle\frac{\partial\Delta}{\partial\theta_{2}}=\sin\Big{(}\theta_{2}-\frac{\theta_{1}}{2}\Big{)}\left(\lambda_{2}(k)\cos\frac{\theta_{1}}{2}+\lambda_{3}(k)\sin\frac{\theta_{1}}{2}\right),$

which vanish at $(\theta_{1},\theta_{2})=(0,0)$ and
$(\pm\tfrac{2\pi}{3},\mp\tfrac{2\pi}{3})$ corresponding to the band edges.
Inspecting then the last two derivatives in (3.18) at these values, one
obtains a system of two equations in variables $k$ and $d$ that may always be
fulfilled for particular values of parameters. The spectral bands are
symmetric with respect to the exchange of $c$ to $d-c$, as seen in Fig. 4. The
band condition (3.8) depends on the three functions $\lambda_{1}(k)$,
$\lambda_{2}(k)$ and $\lambda_{3}(k)$; the invariance of the first two
functions under $c\leftrightarrow d-c$ are easily checked since $\cos$ is an
even function; in the case of $\lambda_{3}(k)$, although it is an odd
function, the functions $\lambda^{\pm}(k)$ in the band condition (3.8) contain
it with both the positive and negative signs. If the size of the lattice cell
is large, that is, in the asymptotic regime $d\rightarrow\infty$, the the
number of bands in a fixed energy interval increases, roughly linearly with
$d$, however, the probability to belong to the spectrum remains asymptotically
the same.

###  3.3 Negative Spectrum

Replacing the momentum variable $k$ in (3.5) by $i\kappa$ with $\kappa>0$, we
arrive at the spectral condition

$\mathrm{e}^{2i\theta_{2}}\kappa^{9}\ell^{3}\;\sinh\frac{\kappa
c}{2}\sinh\frac{\kappa
d}{2}\sinh\frac{\kappa(d-c)}{2}\;\Big{(}\tilde{\lambda}_{1}(\kappa)-\tilde{\lambda}_{2}(\kappa)\,f_{\theta}-\tilde{\lambda}_{3}(\kappa)\,g_{\theta}\Big{)}=0,$
(3.19)

where

$\displaystyle\tilde{\lambda}_{1}(\kappa)\mathrel{\mathop{\ordinarycolon}}=2(1-\kappa^{2}\ell^{2})\bigg{(}4(\kappa^{2}\ell^{2}-1)^{2}\Big{(}\cosh\kappa(c+d)+\cosh\kappa(c-2d)+2\cosh\kappa
d+\cosh 2\kappa d\Big{)}$
$\displaystyle\qquad\qquad+\left(\kappa^{4}\ell^{4}-14\kappa^{2}\ell^{2}+1\right)(2\cosh\kappa
d+1)\cosh\kappa(2c-d)+\left(3\kappa^{4}\ell^{4}-18\kappa^{2}\ell^{2}+3\right)$
$\displaystyle\qquad\qquad+\left(5\kappa^{4}\ell^{4}-22\kappa^{2}\ell^{2}+5\right)\big{(}\cosh\kappa(d-c)+\cosh\kappa
c\big{)}\bigg{)},$
$\displaystyle\tilde{\lambda}_{2}(\kappa)\mathrel{\mathop{\ordinarycolon}}=8\left(1-\kappa^{2}\ell^{2}\right)\left(\kappa^{2}\ell^{2}+1\right)^{2}\left(\cosh\frac{\kappa(2c-d)}{2}+2\cosh\frac{\kappa
d}{2}\right)\cosh\frac{\kappa(d-c)}{2}\cosh\frac{\kappa c}{2},$
$\displaystyle\tilde{\lambda}_{3}(\kappa)\mathrel{\mathop{\ordinarycolon}}=16\kappa\ell\left(\kappa^{2}\ell^{2}+1\right)^{2}\sinh\frac{\kappa(d-c)}{2}\sinh\frac{\kappa
c}{2}\sinh\frac{\kappa(d-2c)}{2}.$

Except for the equilateral case, flat bands in the negative part of the
spectrum are obviously absent. Mimicking the argument of Sec. 3.2, we infer
that a number $-\kappa^{2}$ belongs to a spectral band if and only if

$\kappa\in\left\\{\kappa\mathrel{\mathop{\ordinarycolon}}\;\tilde{\lambda}^{\pm}(\kappa)\leq\tilde{\lambda}_{1}(\kappa)\leq\tilde{\lambda}^{0}(\kappa)\quad\cup\quad\tilde{\lambda}^{0}(\kappa)\leq\tilde{\lambda}_{1}(\kappa)\leq\tilde{\lambda}^{\pm}(\kappa)\right\\},$
(3.20)

where $\tilde{\lambda}^{0,\pm}(\kappa)$ are defined as

$\displaystyle\tilde{\lambda}^{0}(\kappa)\mathrel{\mathop{\ordinarycolon}}=3\,\tilde{\lambda}_{2}(\kappa),$
$\displaystyle\tilde{\lambda}^{\pm}(\kappa)\mathrel{\mathop{\ordinarycolon}}=-\frac{3}{2}\left(\tilde{\lambda}_{2}(\kappa)\pm\sqrt{3}\,\tilde{\lambda}_{3}(\kappa)\right).$

In the equilateral case, $\tilde{\lambda}_{3}(\kappa)$ in (3.19) vanishes and
the spectral condition reduces to

$\displaystyle 4\big{(}\kappa^{2}\ell^{2}-1\big{)}(2\cosh\kappa
c+1)\sinh\frac{\kappa c}{2}\sinh\kappa c\;\times$
$\displaystyle\left(\left(\kappa^{2}\ell^{2}-1\right)^{2}(2\cosh 2\kappa
c+2\cosh 3\kappa
c+1)+\left(\kappa^{4}\ell^{4}-14\kappa^{2}\ell^{2}+1\right)\cosh\kappa
c-\left(\kappa^{2}\ell^{2}+1\right)^{2}(\cosh\kappa c+1)f_{\theta}\right)=0.$

The negative spectrum of the general and the equilateral lattices in
dependence on $d$ and $c$ are shown in Figs. 3, 4 and 6, respectively.
Concerning the number of negative bands, since the elementary cell of the
kagome lattice contains three vertices of degree four, the corresponding
matrix $U$ in each of them has one eigenvalue in the upper complex halfplane–
cf. eq. (2.3) – thus the negative spectrum cannot have more than three bands
in accordance with Theorem 2.6 of [3]. It should be noted that in the general
case some gaps may close at specific values of $\kappa$. The crossing points
are given by a relation analogous to (3.18); it again does not allow for
solution in a closed form but one can check that such crossings indeed happen
at the quasimomentum component values
$\theta_{1}=-\theta_{2}=\pm\tfrac{2\pi}{3}$ corresponding to the band edges.
Here, we see that: As indicated, a flat band band occurs only in the
equilateral case, corresponding to the energy $-\ell^{-2}$; it may not be
embedded in the continuous spectrum as we will see below. The number
$-\ell^{-2}$ always belongs to the spectrum. Inspecting (3.19) for
$\kappa=\ell^{-1}$, we get

$-64\;g_{\theta}\,\sinh\frac{c}{2\ell}\sinh\frac{d-2c}{2\ell}\sinh\frac{d-c}{2\ell}=0,$

which holds for $g_{\theta}=0$ independently of other parameters. In
particular, as mentioned above, for $d=2c$, the band containing this number
shrinks to the point $-\ell^{-2}$ corresponding to the flat band of the
equilateral model. For $d\leq 2\sqrt{3}\,\ell$, the first negative band
reaches zero, while for $d>2\sqrt{3}\,\ell$ the negative spectrum remains
separated from zero. Following an argument similar to that of Sec. 3.2, using
the Taylor expansion around $\kappa\rightarrow 0+$ in (3.20), one can easily
check that small values of $\kappa$ with $d>2\sqrt{3}\,\ell$ do not correspond
to the spectral points. In particular, in the equilateral case, the first
negative band reaches zero if $c\leq\sqrt{3}\,\ell$, otherwise it remains
separated from zero. The negative bands are symmetric with respect to the
interchange of $c$ and $d-c$, cf. Fig. 4. This property can be justified by an
argument similar to that used in Sec. 3.2 for the mirror symmetry of the
positive bands. It is also interesting to inspect the situation where one of
the edges, as well as the scale parameter $\ell$, is kept fixed while the
other becomes large assuming, say, that $d\gg 2c$. Since attraction
responsible for the negative spectrum comes from the vertex coupling only and
the transport requires tunneling over the edges which are now classically
forbidden zones, one expects that negative bands shrink to points as
$d\to\infty$. It is indeed the case; the see that we rewrite the spectral
condition (3.19) in the form

$\left(1-\kappa^{2}\ell^{2}\right)f(\ell,c;\kappa)\,\mathrm{e}^{2\kappa
d}+g(\ell,c;\kappa,f_{\theta},g_{\theta})\,\mathrm{e}^{\kappa
d}+h(\ell,c;\kappa)+\mathcal{O}(\mathrm{e}^{-\kappa d})=0,$ (3.22)

with

$\displaystyle
f(\ell,c;\kappa)\mathrel{\mathop{\ordinarycolon}}=4\left(\mathrm{e}^{-\kappa
c}+1\right)\left(\kappa^{2}\ell^{2}-1\right)^{2}+\mathrm{e}^{-2\kappa
c}\left(\kappa^{4}\ell^{4}-14\kappa^{2}\ell^{2}+1\right),$ $\displaystyle
g(\ell,c;\kappa,\theta_{1},\theta_{2})\mathrel{\mathop{\ordinarycolon}}=\left(\mathrm{e}^{-2\kappa
c}+3\mathrm{e}^{-\kappa
c}+2\right)\left(\kappa^{2}\ell^{2}-1\right)\left(\kappa^{2}\ell^{2}+1\right)^{2}\,f_{\theta}-2\kappa\ell\mathrm{e}^{-2\kappa
c}\left(\mathrm{e}^{\kappa
c}-1\right)\left(\kappa^{2}\ell^{2}+1\right)^{2}\,g_{\theta}$
$\displaystyle\quad-\left(\kappa^{2}\ell^{2}-1\right)\left(\mathrm{e}^{-2\kappa
c}\left(\kappa^{4}\ell^{4}-14\kappa^{2}\ell^{2}+1\right)+\mathrm{e}^{-\kappa
c}\left(5\kappa^{4}\ell^{4}-22\kappa^{2}\ell^{2}+5\right)+4\left(\mathrm{e}^{\kappa
c}+2\right)\left(\kappa^{2}\ell^{2}-1\right)^{2}\right),$ $\displaystyle
h(\ell,c;\kappa)\mathrel{\mathop{\ordinarycolon}}=\left(1-\kappa^{2}\ell^{2}\right)\left(\mathrm{e}^{-\kappa
c}\left(\mathrm{e}^{2\kappa
c}+1\right)\left(5\kappa^{4}\ell^{4}-22\kappa^{2}\ell^{2}+5\right)+6\left(\kappa^{4}\ell^{4}-6\kappa^{2}\ell^{2}+1\right)\right).$

For large $d$ the bands are thus in the vicinity of zeros of the first,
$\theta$-independent term, being exponentially narrow with respect to $d$. One
of those limit points is $-\ell^{-2}$, and one of the others in each of the
the intervals $(0,\ell^{-1})$ and $(\ell^{-1},\infty)$ being determined by the
condition $f(\ell,c;\kappa)=0$. To see that, note that $f(\ell,c;\cdot)$ is
continuous assuming the values $9$ and $-12\mathrm{e}^{-\frac{2c}{\ell}}$ at
$\kappa=0,\,\ell^{-1}$, respectively, and
$\lim_{\kappa\to\infty}f(\ell,c;\kappa)=+\infty$; recall that by general
principles the kagome lattice cannot have more than three negative bands.
Consider finally the equilateral case in the asymptotic regime
$c\rightarrow\infty$. We know that $-\ell^{-2}$ is a flat band, the other two
bands now shrink to the same value. To see that, we rewrite the spectral
condition (3.3) in the form

$f_{\theta}=F(\kappa)\mathrel{\mathop{\ordinarycolon}}=\frac{\left(\kappa^{2}\ell^{2}-1\right)^{2}(2\cosh
2\kappa c+2\cosh 3\kappa
c+1)+\left(\kappa^{4}\ell^{4}-14\kappa^{2}\ell^{2}+1\right)\cosh\kappa
c}{\left(\kappa^{2}\ell^{2}+1\right)^{2}(\cosh\kappa c+1)}.$ (3.23)

We note that
$F(\ell^{-1})=-\tfrac{3}{2}\big{(}\tanh^{2}\tfrac{c}{2\ell}+1\big{)}<-\tfrac{3}{2}$
since $\tanh x>0$ holds for $x>0$, hence the flat band is not embedded in the
continuous spectrum. On the other hand, we have $F(0)=3$ and
$\lim_{\kappa\to\infty}F(\kappa)=+\infty$, hence there is one negative band
below and above the energy $-\ell^{-2}$. For large values of $c$, the spectral
condition (3.23) reads
$\left(\kappa^{2}\ell^{2}-1\right)^{2}\mathrm{e}^{3\kappa
c}+\mathcal{O}(\mathrm{e}^{2\kappa c})=0$ implying again that the bands shrink
exponentially fast. To estimate the band widths, we put
$\kappa^{2}=\ell^{-2}+\varepsilon$ in (3.3) obtaining

$\varepsilon=\pm\sqrt{2f_{\theta}+6}\,\ell^{-2}\mathrm{e}^{-\frac{c}{\ell}}+\mathcal{O}(\mathrm{e}^{-\frac{2c}{\ell}}).$

This yields the following asymptotic expression for the width of the two
bands,

$\Delta
E=\sqrt{3}\ell^{-2}\mathrm{e}^{-\frac{c}{\ell}}+\mathcal{O}(\mathrm{e}^{-2\frac{c}{\ell}});$

this behavior is seen in Fig. 6.

##  Triangular lattice

These lattices can be regarded as a degenerate case of a kagome lattice when
one of the edge lengths, $b$ or $c$, shrinks to zero. The elementary cell now
contains a single vertex of degree six, cf. Fig. 2.

Figure 2: The kagome lattice degenerates to the triangular one when one of the
edge lengths shrinks to zero assuming that $d=b+c$ is fixed.

Let us recall that the condition (3.5) is symmetric with respect to the
interchange of $b=0$ and $c=0$. To find the spectral condition of the
triangular lattice, we can either use the natural Ansätze and match them as we
did when deriving (3.5), or to take the limit $c\rightarrow d$ in the latter;
this yields

$\displaystyle\left(k^{2}\ell^{2}+1\right)\;\sin^{2}\textstyle{\frac{kd}{2}}\;\times$
(4.1)
$\displaystyle\quad\Big{(}3\big{(}k^{4}\ell^{4}+6k^{2}\ell^{2}+1\big{)}+\big{(}3k^{4}\ell^{4}+10k^{2}\ell^{2}+3\big{)}(2\cos
kd+\cos
2kd)-4\big{(}k^{2}\ell^{2}-1\big{)}^{2}\cos^{2}\textstyle{\frac{kd}{2}}\,f_{\theta}\Big{)}=0.$

These two methods are almost equivalent, with a small exception which we will
mention in Sec. 4.2 below. Let us look how the corresponding spectrum looks
like considering again the positive and negative part separately.

###  4.1 Positive Spectrum

The positive spectrum of the triangular lattice consists again of two parts:

####  _i_. Infinitely degenerate eigenvalues

  * •

For $k=\frac{2n\pi}{d}$ with $n\in\mathbb{N}$, the number $k^{2}$ belongs to
the spectrum. Inspecting the expression in the ‘large’ bracket in (4.1) for
$k=\frac{2n\pi}{d}$, we get

$\frac{4}{d^{4}}\Big{(}\left(d^{4}+16\pi^{4}n^{4}\ell^{4}\right)(3-f_{\theta})+8\pi^{2}d^{2}n^{2}\ell^{2}\left(f_{\theta}+6\right)\Big{)}>0,$

which implies that the flat bands may not be embedded in the continuous
spectrum.

  * •

As in the kagome lattice case, it may happen that some positive bands
degenerate to a point; here this happens at $k=\ell^{-1}$ for
$d=\ell\big{(}(-1)^{n+1}+(6n-3)\big{)}\frac{\pi}{6}$ with $n\in\mathbb{N}$.

####  _ii_. Continuous bands

The rest of the spectrum has a band-gap structure determined by vanishing of
the ‘large’ bracket in (4.1). Note that the latter reduces to
$\frac{8}{d^{2}}(2n-1)^{2}\ell^{2}\pi^{2}>0$ for $k=(2n-1)\frac{\pi}{d}$,
$n\in\mathbb{N}$; hence dividing the corresponding equation by
$\left(k^{2}\ell^{2}-1\right)^{2}\cos^{2}\textstyle{\frac{kd}{2}}$, it can be
brought into a more convenient form

$f_{\theta}=G(k)\mathrel{\mathop{\ordinarycolon}}=\frac{2k^{2}\ell^{2}\sec^{2}\frac{kd}{2}+\left(3k^{4}\ell^{4}+10k^{2}\ell^{2}+3\right)\cos
kd}{\left(k^{2}\ell^{2}-1\right)^{2}}.$ (4.2)

Consequently, a number $k^{2}$ belongs to the spectral bands if and only if
$G(k)$ lies in the interval $[-\tfrac{3}{2},3]$. The band-gap pattern in
dependence on $d$ is illustrated in Fig. 8, and moreover, two examples for
$d=1$ and $5$ are shown in Fig. 9. We find that: The first positive band
starts at zero if $d\geq 2\sqrt{3}\,\ell$, otherwise it remains separated from
zero since $G(k)=3+\frac{3}{2}(12\ell^{2}-d^{2})k^{2}+\mathcal{O}(k^{4})$ is
greater than three for $d<2\sqrt{3}\,\ell$ and small values of momentum $k$.
In the high-energy regime, we have again two types of asymptotic behavior. To
take a closer look at their structure, we rewrite the spectral condition (4.1)
in the form

$\gamma_{1}(k)+\frac{\gamma_{2}(k)}{k^{2}}=\mathcal{O}(k^{-4}),$ (4.3)

where

$\displaystyle\gamma_{1}(k)=4\ell^{4}\,\cos^{2}\frac{kd}{2}\,\big{(}3\cos kd-
f_{\theta}\big{)},$ $\displaystyle\gamma_{2}(k)=2\ell^{2}\big{(}10\cos
kd+5\cos 2kd+9+2(\cos kd+1)f_{\theta}\big{)}.$

To satisfy the condition (4.3) for large values of $k$, the the leading term
$\gamma_{1}(k)$ should be close to zero; this results in two types of spectral
bands: Pairs of narrow bands in the vicinity of $k=(2n-1)\frac{\pi}{d}$,
$n\in\mathbb{N}$
The two bands around the points $k=(2n-1)\frac{\pi}{d}$ and the gap between
them are of asymptotically constant width. To see that, we set
$k=(2n-1)\frac{\pi}{d}+\delta$. Then we have
$k^{-2}=\frac{d^{2}}{4n^{2}\pi^{2}}+\mathcal{O}(n^{-3})$ as $n\to\infty$;
substituting these values into (4.3) and solving the resulting equation for
$\delta$, we obtain

$\delta=\frac{\sqrt{2}}{\pi\ell\sqrt{f_{\theta}+3}}\,\frac{1}{n}+\mathcal{O}(n^{-3}).$

Since the band edges correspond to $f_{\theta}=-\tfrac{3}{2}$ and $3$, the
width of the bands and the gap between them are respectively equal to
$\frac{4}{d\ell\sqrt{3}}+\mathcal{O}(n^{-1})$ and
$\frac{8}{d\ell\sqrt{3}}+\mathcal{O}(n^{-1})$ as $n\rightarrow\infty$. Pairs
of wide bands in the vicinity of $k=2n\frac{\pi}{d}$, $n\in\mathbb{N}$
These bands and the gaps between them grow asymptotically but not at the same
rate; large values of $k$ belong to the spectrum if and only if

$-\frac{1}{2}\leq\cos kd\leq 1,$ (4.4)

with a relative error $\mathcal{O}(k^{-2})$. The function $\cos kd$ is
periodic with the period $T=\tfrac{2\pi}{d}$, and solutions of the equation
$\cos kd=-\tfrac{1}{2}$ in the period are $\tfrac{2\pi}{3d}$ and
$\tfrac{4\pi}{3d}$. On the other hand, $\cos kd$ for $k=\tfrac{\pi}{d}$ is
$-1$; and $\cos kd$ is less than $-\tfrac{1}{2}$ over the domain
$\left(\tfrac{2\pi}{3d},\tfrac{4\pi}{3d}\right)$. Consequently, the
probability of belonging to the spectrum is equal to

$P_{\sigma}(H)=1-\frac{1}{T}\left(\frac{4\pi}{3d}-\frac{2\pi}{3d}\right)=\frac{2}{3}.$

The triangular lattice is equilateral, hence it makes no sense to speak about
the universality. It is nevertheless interesting that the above value
coincides with (3.17). We note also that these bands appear in pairs centered
around the points $k=2n\frac{\pi}{d}$ marking the flat bands; as mentioned
earlier, they may not be embedded in the continuous spectrum; from (4.4) we
see that these gaps are situated in the vicinity of the Brillouin zone center.
As in the kagome case, the on-shell scattering matrix is nontrivial at high
energies. Now the vertex degree is six and from (2.2) we get

$\underset{k\to\infty}{\text{lim}}S(k)=\frac{2}{3}\,I_{6}\,+\frac{1}{3}{\scriptsize\left(\begin{array}[]{cccccc}\phantom{-}0&\phantom{-}1&-1&\phantom{-}1&-1&\phantom{-}1\\\
\phantom{-}1&\phantom{-}0&\phantom{-}1&-1&\phantom{-}1&-1\\\
-1&\phantom{-}1&\phantom{-}0&\phantom{-}1&-1&\phantom{-}1\\\
\phantom{-}1&-1&\phantom{-}1&\phantom{-}0&\phantom{-}1&-1\\\
-1&\phantom{-}1&-1&\phantom{-}1&\phantom{-}0&\phantom{-}1\\\
\phantom{-}1&-1&\phantom{-}1&-1&\phantom{-}1&\phantom{-}0\\\
\end{array}\right)}.$

As $d\rightarrow\infty$, the number of bands in a fixed interval increases,
roughly linearly with $d$, while the probability to be in the spectrum is
asymptotically constant.

###  4.2 Negative Spectrum

To find the negative spectrum, we may replace $k$ by $i\kappa$ in (4.1), which
leads to the condition

$\displaystyle\left(\kappa^{2}\ell^{2}-1\right)\;\sinh^{2}\frac{\kappa
d}{2}\;\times$ (4.5)
$\displaystyle\left(3\big{(}\kappa^{4}\ell^{4}-6\kappa^{2}\ell^{2}+1\big{)}+\big{(}3\kappa^{4}\ell^{4}-10\kappa^{2}\ell^{2}+3\big{)}\big{(}2\cosh\kappa
d+\cosh 2\kappa
d\big{)}-4\big{(}\kappa^{2}\ell^{2}+1\big{)}^{2}\cosh^{2}\frac{\kappa
d}{2}\;f_{\theta}\right)=0.$

Let us mention first the exception mentioned in the opening of the section. It
may seem that the energy $-\ell^{-2}$ belongs to the spectrum but in reality
it is a spurious solution. To see that, we note that for the triangular
lattice the functions
$\chi_{2},\,\chi_{3},\,\varphi_{2},\,\varphi_{3},\,\psi_{3}$, and $\psi_{4}$
in (3.1) are absent; rewriting then the matching condition (3.1) accordingly,
and computing the determinant of the corresponding system at $k=i\,\ell^{-1}$,
we arrive at the expression

$1024\,i\,\mathrm{e}^{2i\theta_{2}}\,\ell^{-3}\,\sinh^{2}\frac{d}{2\ell}\,\left(3+2\,f_{\theta}+2\left(f_{\theta}+1\right)\cosh\frac{d}{\ell}+\cosh\frac{2d}{\ell}\right),$

which is nonzero taking into account the range of $f_{\theta}$ and the fact
that $\cosh 2x>\cosh x$ holds for $x>0$, and consequently, the point
$-\ell^{-2}$ cannot belong to the spectrum. The true spectral condition comes
from vanishing of the ‘large’ bracket in (4.5); we can rewrite it in the form

$f_{\theta}=\tilde{G}(\kappa)\mathrel{\mathop{\ordinarycolon}}=\frac{\left(3\kappa^{4}\ell^{4}-10\kappa^{2}\ell^{2}+3\right)\cosh\kappa
d-2\kappa^{2}\ell^{2}\text{sech}^{2}\frac{\kappa
d}{2}}{\left(\kappa^{2}\ell^{2}+1\right)^{2}}.$ (4.6)

In other words, a number $-\kappa^{2}$ belongs to the spectrum if
$\tilde{G}(\kappa)$ lies in the interval $[-\tfrac{3}{2},3]$. The negative
spectrum in dependence on the length $d$ is illustrated in Fig. 8. We see
that:

  * •

The spectrum consists of two bands. We first note that
$\tilde{G}(\ell^{-1})=-\cosh\frac{d}{\ell}-(\cosh\frac{d}{\ell}+1)^{-1}<-\tfrac{3}{2}\,$
which can be easily checked by computing the derivative of the right-hand side
with respect to $d$, equal to
$\tfrac{1}{\ell}\sinh\tfrac{d}{\ell}\left((\cosh\tfrac{d}{\ell}+1)^{-2}-1\right)<0$.
Hence $d\mapsto\tilde{G}(\ell^{-1})$ is decreasing reaching the value
$-\frac{3}{2}$ at $d=0$. On the other hand, we have $\tilde{G}(0)=3$ and
$\lim_{\kappa\to\infty}\tilde{G}(\kappa)=+\infty$. Consequently, there is at
least one negative band in each of the domains $(0,\ell^{-1})$ and
$(\ell^{-1},\infty)$, however, according to Theorem 2.6 in [3] the lattice
cannot have more than two negative bands, since the matrix $U$ describing the
coupling (1.1) in a vertex of degree six has by (2.3) exactly two eigenvalues
in the upper complex halfplane.

  * •

For $d\leq 2\sqrt{3}\,\ell$, the first negative reaches zero, while for
$d>2\sqrt{3}\,\ell$, the spectrum remains separated from zero since
$\tilde{G}(\kappa)=3+\textstyle{\frac{3}{2}}(d^{2}-12\ell^{2})\kappa^{2}+\mathcal{O}(\kappa^{4})$
is greater than three for $d>2\sqrt{3}\,\ell$ and small values of $\kappa$.

  * •

For large values of $d$, the negative bends become exponentially narrow and
approach the eigenvalues of a star graph of degree six. Since we have
$\cosh\kappa d\approx\frac{1}{2}\mathrm{e}^{\kappa d}$ as
$d\rightarrow\infty$, one can write the spectral condition (4.5) as

$\tilde{f}(\ell;\kappa)\,\mathrm{e}^{2\kappa
d}+\tilde{g}(\ell;\kappa,f_{\theta})\,\mathrm{e}^{\kappa
d}+\tilde{h}(\ell;\kappa,f_{\theta})+\mathcal{O}(\mathrm{e}^{-\kappa d})=0,$

with

$\displaystyle\tilde{f}(\ell;\kappa)\mathrel{\mathop{\ordinarycolon}}=\textstyle{\frac{1}{2}}\big{(}3\kappa^{4}\ell^{4}-10\kappa^{2}\ell^{2}+3\big{)},$
$\displaystyle\tilde{g}(\ell;\kappa,f_{\theta})\mathrel{\mathop{\ordinarycolon}}=\big{(}3\kappa^{4}\ell^{4}-10\kappa^{2}\ell^{2}+3\big{)}-\left(\kappa^{2}\ell^{2}+1\right)^{2}\,f_{\theta},$
$\displaystyle\tilde{h}(\ell;\kappa,f_{\theta})\mathrel{\mathop{\ordinarycolon}}=3\left(\kappa^{4}\ell^{4}-6\kappa^{2}\ell^{2}+1\right)-2\left(\kappa^{2}\ell^{2}+1\right)^{2}\,f_{\theta}.$

In the limit $d\rightarrow\infty$, the bands therefore shrink to the points
determined by the condition $\tilde{f}(\ell;\kappa)=0$, or explicitly to the
energies $-3\ell^{-2}$ and $-\frac{1}{3}\ell^{-2}$ which fits with (2.3). To
estimate the widths of the shrinking bands, we set
$\kappa_{1}=\sqrt{3}\ell^{-1}+\delta_{1}$ and
$\kappa_{2}=(\ell\sqrt{3})^{-1}+\delta_{2}$; solving then the resulting
equations for $\delta_{1}$ and $\delta_{2}$, we obtain the following
asymptotic expressions for the energy and width of the bands

$\displaystyle E_{1}=-\kappa_{1}^{2}(f_{\theta})$
$\displaystyle=-3\ell^{-2}-4\ell^{-2}\,\mathrm{e}^{-\frac{\sqrt{3}d}{\ell}}\,f_{\theta}+\mathcal{O}(\mathrm{e}^{-\frac{2\sqrt{3}d}{\ell}}),$
$\displaystyle\Delta E_{1}$
$\displaystyle=18\ell^{-2}\mathrm{e}^{-\frac{\sqrt{3}d}{\ell}}+\mathcal{O}(\mathrm{e}^{-\frac{2\sqrt{3}d}{\ell}}),$
$\displaystyle E_{2}=-\kappa_{2}^{2}(f_{\theta})$
$\displaystyle=-\textstyle{\frac{1}{3}}\ell^{-2}+\textstyle{\frac{4}{9}}\ell^{-2}\,\mathrm{e}^{-\frac{d}{\sqrt{3}\ell}}\,f_{\theta}+\mathcal{O}(\mathrm{e}^{-\frac{2d}{\sqrt{3}\ell}}),$
$\displaystyle\Delta E_{2}$
$\displaystyle=2\ell^{-2}\mathrm{e}^{-\frac{d}{\sqrt{3}\ell}}+\mathcal{O}(\mathrm{e}^{-\frac{2d}{\sqrt{3}\ell}}).$

Figure 3:  Spectrum of the general kagome lattice in dependence on $d$ for
$c=\ell=1$. The band edge crossings is indicated by black dots. The red curves
correspond to the flat bands $\frac{4n^{2}\pi^{2}}{d^{2}}$ and
$\frac{4n^{2}\pi^{2}}{(d-1)^{2}}$ (solid and dashed, respectively) with
$n=1,2$; the flat bands $4n^{2}\pi^{2}$ lay outside the picture area. On the
other hand, the values
$d=\left\\{\frac{2\pi}{3},\,\frac{2\pi}{3}+1,\,\frac{4\pi}{3},\,\frac{4\pi}{3}+1\right\\}$
with energy $k^{2}=1$ correspond to the degenerate eigenvalues of the third
bullet point in Sec. 3.2. Figure 4:  Spectrum of the general kagome lattice in
dependence on $c$ for $d=6$ and $\ell=1$. Since $d>2\sqrt{3}$, the positive
spectrum starts at zero while the negative spectrum remains separated from
zero. The red solid lines correspond to the flat bands
$\frac{4n^{2}\pi^{2}}{d^{2}}$ with $n=1,2$. The green dashed curves correspond
to the flat bands $\frac{4n^{2}\pi^{2}}{(d-c)^{2}}$ with $n=1,2$. The blue
dot-dashed curves correspond to the flat bands $\frac{4n^{2}\pi^{2}}{c^{2}}$
with $n=1,2$. On the other hand, the values
$c=\left\\{6-\frac{4\pi}{3},\,\frac{2\pi}{3},\,6-\frac{2\pi}{3},\frac{4\pi}{3}\right\\}$
with energy $k^{2}=1$ correspond to the degenerate eigenvalues of the third
bullet point in Sec. 3.2; this is not well seen in the picture but magnifying
it one can check that the bands indeed shrink to points there.

(a) Spectral bands for $d=3$; the first band remains separated from zero.

(b) Spectral bands for $d=4$; the first band starts at zero.

Figure 5: Spectral condition (3.8) for $d=3,4$ with $c=\ell=1$.  Figure 6:
Spectrum of the equilateral kagome lattice in dependence on $c$ for $\ell=1$.
The red solid curves in the positive spectrum correspond to the flat bands
$\frac{n^{2}\pi^{2}}{c^{2}}$ with $n=1,2,3$ respectively. The blue dashed
curves correspond to the flat bands $k^{2}$ for
$k=\left((6n-3)+(-1)^{n+1}\right)\tfrac{\pi}{6c}$ with $n=1,\ldots,8$
respectively. On the other hand, the values
$c=\left\\{\frac{\pi}{3},\,\frac{2\pi}{3},\,\frac{4\pi}{3}\right\\}$ with
energy $k^{2}=1$ correspond to the degenerate eigenvalues of the third bullet
point in Sec. 3.2. The red line in the negative spectrum corresponds to the
flat band $-\ell^{-2}$.

(a) Spectral bands for $c=1$; the first band remains separated from zero.

(b) Spectral bands for $c=3$; the first band starts at zero.

Figure 7: Spectral condition (3.8) in the equilateral case for $c=1$ and $3$
with $\ell=1$.  Figure 8:  Spectrum of the triangular lattice in dependence on
$d$ for $\ell=1$. The red curves correspond to the flat bands
$\frac{4n^{2}\pi^{2}}{d^{2}}$ for $n=1,2$. On the other hand, the points
$(2\frac{\pi}{3},1)$ and $(4\frac{\pi}{3},1)$ correspond to the flat bands of
the second bullet point in Sec. 4.1.

(a) Spectral bands for $d=1$; the first band remains separated from zero.

(b) Spectral bands for $d=5$; the first band starts at zero.

Figure 9: Spectral condition (4.2) for $d=1$ and $5$ with $\ell=1$.  Figure
10: The probability (3.12) as a function of the edge length ratio. Figure 11:
The probability (3.12) calculated as the area of the indicated region.

### Acknowledgments

The research was supported by the Czech Science Foundation project 21-07129S
and by the EU project CZ.02.1.01/0.0/0.0/16_019/0000778. MB’s work was also
supported by the Internal Postdoc Project UHK for years 2021-2022.

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