Quantum graphs with vertices of a preferred orientation

Abstract

Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.Comment: 11 pages, 3 figure

Ring chains with vertex coupling of a preferred orientation

“…This is not surprising, recall that that the three-edge star graph with the vertex coupling (1.2) has the eigenvalue −3. A similar squeezing of a band pair around this value occurs in hexagonal lattice graphs with long edges [7]. Here, however, the two bands do not converge to a single point but a pair of separate ones, the reason being that the lengths of two of the three edges meeting in a vertex are fixed and only one is getting longer.…”

Topological bulk-edge effects in quantum graph transport

“…(c) Note also that the scattering matrices referring to vertices at the strip edges play no role in the proofs. For vertices of degree four with our coupling all the transmition and reflection probabilities are asymptotically the same [3], hence one expects the quantity |a 1 | 2 −|b 1 | 2 |a 1 | 2 +|b 1 | 2 (and similarly for c 1 , d 1 ) describing the (relative) probability current along the left strip edge to range approximately between 1 and -1 as a function of θ for k large enough, cf. the end of Section 5.2 below.…”

“…The most remarkable property of this coupling is the dependence of its high-energy transport properties on the topology, specifically on the vertex degree parity as we have recalled above. As in [3,5] we are here concerned with the transport properties of infinite periodic graphs, this time in the form of a strip cut from an infinite two-dimensional lattice. Our main observation is that the vertex degrees at the boundary of such a strip typically differ from those of the original lattice which in combination with the above mentioned asymptotic behavior may make the transport different in the 'bulk' of such a sample and at its 'edge'.…”

“…Figure 6 shows in the left panel ||b j | 2 − |a j | 2 | and the analogous expressions for the other edges and in the right panel |a j | 2 + |b j | 2 and its analogues. The x-coordinate of the points from the left to right corresponds to the values of these expressions on the edges g 1 , h 1 , f 1 , e 1 , g 2 , h 2 , f 2 , e 2 , g 3 , h 3 . We see that the indicated expressions with indices 2 are three to four orders of magnitude smaller than the largest expressions (which are in the right panel).…”

Spectral asymptotics of the Laplacian on Platonic solids graphs

“…On the other hand, the vertex coupling can be non-invariant with respect to time reversal. A simple example was proposed in [ET18] and its investigation revealed an interesting topological property of such a vertex coupling, namely that the transport properties of the vertex at high energies depend substantially on the vertex parity; this effect was illustrated in [ET18] through comparison of the band spectra of square and hexagonal lattices.…”