Two-exciton bound state quantum self-trapping in an extended star graph.

In the past years, various network architectures for parallel computers have been proposed, for example, hyper cubes or star graphs. These classes of networks are known as Cayley graphs. In recent years, there have been some proposals of new families of interconnection networks, namely, constant degree networks. In this paper, a new interconnection network named extended star graphs is proposed, and it is proved that the extended star graphs have hypercube structure. We also provide a routing algorithm for node-to-node communication on extended star graphs. Based on the algorithm, we obtain an upper bound 2n-1 on the diameter for the n-th order extended star graph.

In the past years, various network architectures for parallel computers have been proposed, for instance, hyper cubes or star graphs. These classes of networks are known as Cayley graphs. In recent years, there have been some proposals of new families of interconnection net-works, namely, constant degree networks. In this paper, we propose a new interconnection network named extended star graphs, and we prove the extended star graphs have hypercube’s structure. We also propose routing algorithms for node-to-node communication on extended star graphs. Based on the algorithms, we obtain an upper bound 2n − 1 on the diameter for the n-th order extended star graph.KeywordsCayley graphsstar graphsdegree four Cayley graphshypercubesrouting algorithmdiameter

The extended star graph with cross-connections ESC (n, k) is a relatively new interconnection network topology, that has a hierarchical and recursive network that combines the versatility and robustness of star graph architectures. In this paper we have discussed some combinatorial results to find the number of nodes and edges of ESC (n, k).

A tight-binding model is introduced for describing the dynamics of an exciton on an extended star graph whose central node is occupied by a trap. On this graph, the exciton dynamics is governed by two kinds of eigenstates: many eigenstates are associated with degenerate real eigenvalues insensitive to the trap, whereas three decaying eigenstates characterized by complex energies contribute to the trapping process. It is shown that the excitonic population absorbed by the trap depends on the size of the graph, only. By contrast, both the size parameters and the absorption rate control the dynamics of the trapping. When these parameters are judiciously chosen, the efficiency of the transfer is optimized resulting in the minimization of the absorption time. Analysis of the eigenstates reveals that such a feature arises around the superradiance transition. Moreover, depending on the size of the network, two situations are highlighted where the transport efficiency is either superoptimized or suboptimized.

When two identical two-dimensional (2D) periodic lattices are stacked in parallel after rotating one layer by a certain angle relative to the other layer, the resulting bilayer system can lose lattice periodicity completely and become a 2D quasicrystal. Twisted bilayer graphene with 30-degree rotation is a representative example. We show that such quasicrystalline bilayer systems generally develop macroscopically degenerate localized zero-energy states (ZESs) in strong coupling limit where the interlayer couplings are overwhelmingly larger than the intralayer couplings. The emergent chiral symmetry in strong coupling limit and aperiodicity of bilayer quasicrystals guarantee the existence of the ZESs. The macroscopically degenerate ZESs are analogous to the flat bands of periodic systems, in that both are composed of localized eigenstates, which give divergent density of states. For monolayers, we consider the triangular, square, and honeycomb lattices, comprised of homogenous tiling of three possible planar regular polygons: the equilateral triangle, square, and regular hexagon. We construct a compact theoretical framework, which we call the quasiband model, that describes the low energy properties of bilayer quasicrystals and counts the number of ZESs using a subset of Bloch states of monolayers. We also propose a simple geometric scheme in real space which can show the spatial localization of ZESs and count their number. Our work clearly demonstrates that bilayer quasicrystals in strong coupling limit are an ideal playground to study the intriguing interplay of flat band physics and the aperiodicity of quasicrystals.

Preface: phys. stat. sol. (b) 241/9

The attractive Bose-Hubbard model is applied for describing the two-exciton dynamics in a nonlinear quantum star graph. When the excitons are created on the core of the star, it is shown that the interplay between the complex architecture of the network and the nonlinearity favors the occurrence of a real quantum self-trapping. Quite weak in the small nonlinearity limit, this self-localization is enhanced as the nonlinearity increases. This feature originates in the restructuring of the two-exciton eigenstates whose localized nature intensifies with the nonlinearity. Nevertheless, the quantum self-trapping is never complete since it is impossible to localize the entire exciton density, even in the strong nonlinearity limit.

A particle in a box with the δ-function potential Vλ(x)=λδ(x−x0) has been recently explored. We show that this example is solvable in the weak (λ→0±) and the strong (1∕λ→0±) coupling limits. In either limit the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when 1∕λ→0−. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1∕λ→0±.

It is shown that the Green function operator for the electronic excited states of an arbitrary aggregate of identical monomers can be expanded in a series of terms in which the first represents the strong coupling (exciton) limit. Higher terms in this series successively improve the exciton limit by bringing in the effects of vibrational structure of the monomers. By evaluating the contribution to optical spectra of three terms in this series explicit expressions are found for the intensities, positions, and widths of individual exciton peaks near the strong coupling limit. It is shown for example that each exciton peak is narrowed with respect to the monomer band width at most by a factor of M−1/2 where M is the number of monomers in the aggregate. An approximate expression for optical absorption by molecular aggregates (DGS model) derived earlier is shown to satisfy the relations derived here.

Selftrapping has been traditionally studied on the assumption that quasiparticles interact with harmonic phonons and that this interaction is linear in the displacement of the phonon. To complement recent semiclassical studies of anharmonicity and nonlinearity in this context, we present below a fully quantum mechanical analysis of a two-site system, where the oscillator is described by a tunably anharmonic potential, with a square well with infinite walls and the harmonic potential as its extreme limits, and wherein the interaction is nonlinear in the oscillator displacement. We find that even highly anharmonic polarons behave similar to their harmonic counterparts in that selftrapping is preserved for long times in the limit of strong coupling, and that the polaronic tunneling time scale depends exponentially on the polaron binding energy. Further, in agreement, with earlier results related to harmonic polarons, the semiclassical approximation agrees with the full quantum result in the massive oscillator limit of small oscillator frequency and strong quasiparticle-oscillator coupling.

The extended Hubbard model for high temperature superconductors is studied for infinite systems in the strong coupling limit. The attractive interaction between added holes that has been noted for nearly degenerate copper and oxygen energy levels arises because the ground state is near an instability. Adding holes nucleates small regions of this unstable, infinitely compressible phase. Three distinct bound states of two holes occur in the strong coupling limit when double occupation is forbidden. In this limit, the attractive interaction results in phase separation rather than pairing when many holes are present. Perturbative calculations suggest that a pairing state is stable in a narrow parameter range as t increases.

We study the quantum dynamics of a photoexcitation uniformly distributed at the periphery of an extended star network (with N_{B} branches of length L_{B}). More specifically, we address here the question of the energy absorption at the core of the network and how this process can be improved (or not) by the inclusion of peripheral defects with a tunable energy amplitude Δ. Our numerical simulations reveal the existence of optimal value of energy defect Δ^{*} which depends on the network architecture. Around this value, the absorption process presents a strong speedup (i.e., reduction of the absorption time) provided that L_{B}≤L_{B}^{*} with L_{B}^{*}≈12.5/ln(N_{B}). Analytical and numerical developments are then conducted to interpret this feature. We show that the origin of this speedup takes place in the hybridization of two upper-band excitonic eigenstates. This hybridization is important when L_{B}≤L_{B}^{*} and vanishes almost totally when L_{B}>L_{B}^{*}. These structural rules we draw here could represent a potential guide for the practical design of molecular nanonetwork dedicated to the realization of efficient photoexcitation absorption.

Using a tight binding model, we investigate the dynamics of an exciton on a disordered extended star graph whose central site acts as an energy trap. When compared with what happens in an ordered network, our results reveal that the disorder drastically improves the excitonic absorption that becomes complete. Moreover, we show the occurrence of an optimal disorder for which the absorption time is strongly minimized, a surprising effect that originates in a disorder-induced restructuring process of the exciton eigenstates. Finally, we also show the existence of an optimal value of the absorption rate that reduces even more the absorption time. The resulting superoptimized trapping process is interpreted as a positive interplay between both the disorder and the so-called superradiance transition.

An extended star graph: a proposal of a new network topology and its fundamental properties

This paper concerns with some variants of the inverse obnoxious median location problem on tree networks, where the aim is either to augment or to reduce the edge lengths at the minimum total cost such that a prespecified subset of vertices becomes an obnoxious multi-facility median location with respect to the perturbed edge lengths. For both augmentation and reduction models, under the rectilinear norm and the sum-type Hamming distance, we develop novel combinatorial algorithms with polynomial time complexities. Particularly, if the underlying tree is an extended star graph, then the problems can be solved in linear time.