Periodic Quantum Graphs Research Articles

Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract One interesting phenomenon of graphene is the presence of the conical singularity
or Dirac points. Using the quantum graph model, we show that there exist three
classes of possible Dirac points for all of the periodic quantum graphs associated with
Archimedean tilings, when the potentials are identical and even. They occur at the
periodic eigenvalues, anti-periodic eigenvalues and other double eigenvalues of the dispersion
relations respectively. We also characterize their associated potentials. Moreover,
we show that there are no other possible Dirac points. Finally, we solve an
inverse spectral problem for the potential, given the knowledge of the pure point and
absolutely continuous spectra.

Vertex coupling interpolation in quantum chain graphs

We analyze the band spectrum of the periodic quantum graph in the form of a chain of rings connected by line segments with the vertex coupling which violates the time reversal invariance, interpolating between the δ coupling and the one determined by a simple circulant matrix. We find that flat bands are generically absent and that the negative spectrum is nonempty even for interpolation with a non-attractive δ coupling; we also determine the high-energy asymptotic behavior of the bands.

Cairo lattice with time-reversal non-invariant vertex couplings

We analyze the spectrum of a periodic quantum graph of the Cairo lattice form. The used vertex coupling violates the time reversal invariance and its high-energy behavior depends on the vertex degree parity; in the considered example both odd and even parities are involved. The presence of the former implies that the spectrum is dominated by gaps. In addition, we discuss two modifications of the model in which this is not the case, the zero limit of the length parameter in the coupling, and the sign switch of the coupling matrix at the vertices of degree three; while different they both yield the same probability that a randomly chosen positive energy lies in the spectrum.

On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use.

A quantum graph approach to metamaterial design

Since the turn of the century, metamaterials have gained a large amount of attention due to their potential for possessing highly nontrivial and exotic properties—such as cloaking or perfect lensing. There has been a great push to create reliable mathematical models that accurately describe the required material composition. Here, we consider a quantum graph approach to metamaterial design. An infinite square periodic quantum graph, constructed from vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport occurs along the edges with vertices acting as scatterers modelling sub-wavelength resonant elements. These resonant elements are constructed with the help of finite quantum graphs attached to each vertex of the lattice with customisable properties controlled by a unitary scattering matrix. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at an interface between two different metamaterials. We extend this treatment to N layered metamaterials using the Transfer Matrix Method. We demonstrate both positive and negative refraction and beam steering. Our proposed quantum graph modelling technique is very flexible and can be easily adjusted making it an ideal design tool for creating metamaterials with exotic band diagram properties or testing promising multi-layer set ups and wave steering effects.

Magnetic ring chains with vertex coupling of a preferred orientation

We discuss spectral properties of a periodic quantum graph consisting of an array of rings coupled either tightly or loosely through connecting links, assuming that the vertex coupling is manifestly non-invariant with respect to the time reversal and a homogeneous magnetic field perpendicular to the graph plane is present. It is shown that the vertex parity determines the spectral behavior at high energies and the Band–Berkolaiko universality holds whenever the edges are incommensurate. The magnetic field influences the probability that an energy belongs to the spectrum in the tight-chain case, and also it can turn some spectral bands into infinitely degenerate eigenvalues.

Kagome network with vertex coupling of a preferred orientation

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 lattice vertices is chosen of a particular form violating the time-reversal invariance. The positive spectrum consists of an 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 ≈0.639.

Dirac particles on periodic quantum graphs.

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasiperiodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

Reducible Fermi Surface for Multi-layer Quantum Graphs Including Stacked Graphene

We construct two types of multi-layer quantum graphs (Schr\"odinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph operator. When the layers are graphene, AA-, AB-, and ABC-stacking are allowed within the same multi-layer structure. Conical singularities (Dirac cones) characteristic of single-layer graphene break when multiple layers are coupled, except for special AA-stacking. One of the tools we introduce is a surgery-type calculus for obtaining the dispersion function for a periodic quantum graph by gluing two graphs together.

Periodic quantum graphs with predefined spectral gaps

Let Γ be an arbitrary -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian on Γ with the action −ɛ −1d2/dx 2 on its edges; here ɛ > 0 is a small parameter. Let . We show that under a proper choice of vertex conditions the spectrum of has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of as ɛ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.

Degenerate band edges in periodic quantum graphs

Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet--Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of $\mathbb{Z}^3$-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.

Topological bulk-edge effects in quantum graph transport

We examine quantum transport in periodic quantum graphs with a vertex coupling non-invariant with respect to time reversal. It is shown that the graph topology may play a decisive role in the conductivity properties illustrating this claim with two examples. In the first, the transport is possible at high energies in the bulk only being suppressed at the sample edges, while in the second one the situation is opposite, the transport is possible at the edge only.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (II)

We continue the work of a previous paper (Luo et al 2019 J. Phys. A. Math. Theor. 52 165201) to derive the dispersion relations of the periodic quantum graphs associated with the remaining 5 of the 11 Archimedean tilings, namely the truncated hexagonal tiling (3,122), rhombi-trihexagonal tiling , snub square tiling (32,4,3,6), snub trihexagonal tiling (34,6), and truncated trihexagonal tiling . The computation is done with the help of the symbolic software Mathematica. With these explicit dispersion relations, we perform more analysis on the spectra.

Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators

This work constructs a class of non-symmetric periodic Schrödinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is an algebraic set that describes all complex wave vectors admissible by the periodic operator at the given energy. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class. The notion of asymmetry class is defined in this article through the introduction of an entire spectral A-function a(\lambda) associated with a potential – two potentials belong to the same asymmetry class if their A-functions are identical. Symmetric potentials correspond to a(\lambda) \equiv 0 . If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. An exception occurs when two copies of certain bipartite graphs are coupled; the Floquet surface in this case is always reducible. This includes AA-stacked bilayer graphene.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (I)

There are 11 kinds of Archimedean tilings for the plane in total. Applying the Floquet–Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tilings, namely the triangular tiling (36), the elongated triangular tiling , the trihexagonal tiling and the truncated square tiling (4,82). The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed the structure of absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in Kuchment and Post (2007 Commun. Math. Phys. 275 805–26) and Korotyaev and Lobanov (2007 Ann. Henri Poincaré 8 1151–76).

Gaps in the Spectrum of a Cuboidal Periodic Lattice Graph

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As the main result, we find a connection between the arrangement of the gaps and the coefficients in a continued fraction associated with the ratio of edge lengths of the lattice. This knowledge enables a straightforward construction of a periodic quantum graph with any required number of spectral gaps and---to some degree---to control their positions; i.e., to partially solve the inverse spectral problem.

Periodic quantum graphs from the Bethe–Sommerfeld perspective

The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To date, its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the ‘Bethe–Sommerfeld property’, that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a δ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows one to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to date.

Quantum graphs with the Bethe-Sommerfeld property

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned $\delta$-coupling at the vertices.

Validity of the NLS approximation for periodic quantum graphs

We consider a nonlinear Schrodinger (NLS) equation on a spatially extended periodic quantum graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.

Spectra of Semi-Infinite Quantum Graph Tubes

The spectrum of a semi-infinite quantum graph tube with square period cells is analyzed. The structure is obtained by rolling up a doubly periodic quantum graph into a tube along a period vector and then retaining only a semi-infinite half of the tube. The eigenfunctions associated to the spectrum of the half-tube involve all Floquet modes of the full tube. This requires solving the complex dispersion relation $D(\lambda,k_1,k_2)=0$ with $(k_1,k_2)\in(\mathbb{C}/2\pi\mathbb{Z})^2$ subject to the constraint $\alpha k_1 + \beta k_2 \equiv 0$ (mod $2\pi$), where $\alpha$ and $\beta$ are integers. The number of Floquet modes for a given $\lambda\in\mathbb{R}$ is $2\max\left\{ \alpha, \beta \right\}$. Rightward and leftward modes are determined according to an indefinite energy flux form. The spectrum may contain eigenvalues that depend on the boundary conditions, and some eigenvalues may be embedded in the continuous spectrum.