Quantum walk transport on carbon nanotube structures

We study source-to-sink excitation transport on carbon nanotubes using the concept of quantum walks. In particular, we focus on transport properties of Grover coined quantum walks on ideal and percolation perturbed nanotubes with zig-zag and armchair chiralities. Using analytic and numerical methods we identify how geometric properties of nanotubes and different types of a sink altogether control the structure of trapped states and, as a result, the overall source-to-sink transport efficiency. It is shown that chirality of nanotubes splits behavior of the transport efficiency into a few typically well separated quantitative branches. Based on that we uncover interesting quantum transport phenomena, e.g. increasing the length of the tube can enhance the transport and the highest transport efficiency is achieved for the thinnest tube. We also demonstrate, that the transport efficiency of the quantum walk on ideal nanotubes may exhibit even oscillatory behavior dependent on length and chirality.     

n-Particle Quantum Statistics on Graphs

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs.     

Numerical simulation of electron scattering by nanotube junctions

We demonstrate the possibility of computing the intensity of electronic transport through various junctions of three-dimensional metallic nanotubes. In particular, we observe that the magnetic field can be used to control the switch of electron in Y-type junctions. Keeping in mind the asymptotic modeling of reliable nanostructures by quantum graphs, we conjecture that the scattering matrix of the graph should be the same as the scattering matrix of its nanosize-prototype. The numerical computation of the latter gives a method for determining the “gluing” conditions at a graph. Exploring this conjecture, we show that the Kirchhoff conditions (which are commonly used on graphs) cannot be applied to model reliable junctions. This work is a natural extension of the paper [1], but it is written in a self-consistent manner. In memoriam Vladimir A. Geyler (1943–2007)     

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this Correspondence we show that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.     

Transport in graphene nanostructures

Graphene nanostructures are promising candidates for future nanoelectronics and solid-state quantum information technology. In this review we provide an overview of a number of electron transport experiments on etched graphene nanostructures. We briefly revisit the electronic properties and the transport characteristics of bulk, i.e., two-dimensional graphene. The fabrication techniques for making graphene nanostructures such as nanoribbons, single electron transistors and quantum dots, mainly based on a dry etching ??paper-cutting?? technique are discussed in detail. The limitations of the current fabrication technology are discussed when we outline the quantum transport properties of the nanostructured devices. In particular we focus here on transport through graphene nanoribbons and constrictions, single electron transistors as well as on graphene quantum dots including double quantum dots. These quasi-one-dimensional (nanoribbons) and quasi-zero-dimensional (quantum dots) graphene nanostructures show a clear route of how to overcome the gapless nature of graphene allowing the confinement of individual carriers and their control by lateral graphene gates and charge detectors. In particular, we emphasize that graphene quantum dots and double quantum dots are very promising systems for spin-based solid state quantum computation, since they are believed to have exceptionally long spin coherence times due to weak spin-orbit coupling and weak hyperfine interaction in graphene.     

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.     

Measurement-Based Quantum Computation and Undecidable Logic

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs—describing the quantum correlations of the states—are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Gödel’s incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven.     

Quantum Transport of Particles and Entropy

A unified view on macroscopic thermodynamics and quantum transport is presented. Thermodynamic processes with an exchange of energy between two systems necessarily involve the flow of other balancable quantities. These flows are first analyzed using a simple drift-diffusion model, which includes the thermoelectric effects, and connects the various transport coefficients to certain thermodynamic susceptibilities and a diffusion coefficient. In the second part of the paper, the connection between macroscopic thermodynamics and quantum statistics is discussed. It is proposed to employ not particles, but elementary Fermi- or Bose-systems as the elementary building blocks of ideal quantum gases. In this way, the transport not only of particles but also of entropy can be derived in a concise way, and is illustrated both for ballistic quantum wires, and for diffusive conductors. In particular, the quantum interference of entropy flow is in close correspondence to that of electric current.     

The role of nonlinearity in computing graph-theoretical properties of resting-state functional magnetic resonance imaging brain networks

In recent years, there has been an increasing interest in the study of large-scale brain activity interaction structure from the perspective of complex networks, based on functional magnetic resonance imaging (fMRI) measurements. To assess the strength of interaction (functional connectivity, FC) between two brain regions, the linear (Pearson) correlation coefficient of the respective time series is most commonly used. Since a potential use of nonlinear FC measures has recently been discussed in this and other fields, the question arises whether particular nonlinear FC measures would be more informative for the graph analysis than linear ones. We present a comparison of network analysis results obtained from the brain connectivity graphs capturing either full (both linear and nonlinear) or only linear connectivity using 24 sessions of human resting-state fMRI. For each session, a matrix of full connectivity between 90 anatomical parcel time series is computed using mutual information. For comparison, connectivity matrices obtained for multivariate linear Gaussian surrogate data that preserve the correlations, but remove any nonlinearity are generated. Binarizing these matrices using multiple thresholds, we generate graphs corresponding to linear and full nonlinear interaction structures. The effect of neglecting nonlinearity is then assessed by comparing the values of a range of graph-theoretical measures evaluated for both types of graphs. Statistical comparisons suggest a potential effect of nonlinearity on the local measures-clustering coefficient and betweenness centrality. Nevertheless, subsequent quantitative comparison shows that the nonlinearity effect is practically negligible when compared to the intersubject variability of the graph measures. Further, on the group-average graph level, the nonlinearity effect is unnoticeable.     

Percolation on Infinite Graphs and Isoperimetric Inequalities

We consider the Bernoulli bond percolation process (with parameter p) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for p arbitrarily close to one.     

Trapping of continuous-time quantum walks on Erdös–Rényi graphs

We consider the coherent exciton transport, modeled by continuous-time quantum walks, on Erdös–Rény graphs in the presence of a random distribution of traps. The role of trap concentration and of the substrate dilution is deepened showing that, at long times and for intermediate degree of dilution, the survival probability typically decays exponentially with a (average) decay rate which depends non-monotonically on the graph connectivity; when the degree of dilution is either very low or very high, stationary states, not affected by traps, get more likely giving rise to a survival probability decaying to a finite value. Both these features constitute a qualitative difference with respect to the behavior found for classical walks.     

Continuous-time quantum walks on star graphs

In this paper, we investigate continuous-time quantum walk on star graphs. It is shown that quantum central limit theorem for a continuous-time quantum walk on star graphs for N-fold star power graph, which are invariant under the quantum component of adjacency matrix, converges to continuous-time quantum walk on K₂ graphs (complete graph with two vertices) and the probability of observing walk tends to the uniform distribution.     

The number of spanning trees of an infinite family of outerplanar,small-world and self-similar graphs

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.     

Towards the map of quantum gravity

In this paper we point out some possible links between different approaches to quantum gravity and theories of the Planck scale physics. In particular, connections between loop quantum gravity, causal dynamical triangulations, Ho?ava–Lifshitz gravity, asymptotic safety scenario, Quantum Graphity, deformations of relativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantum deformations of the Hypersurface Deformations Algebra and Poincaré algebra, nonlinear structure of phase space, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present an attempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantum gravity. The analysis is performed in the spirit of a mind map, which represents the architectural approach to the studied theory, being a natural way to describe the properties of a complex system. We hope that the constructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity as a particular complex system and serve as a useful guide for the researchers.     

低维纳米材料量子热输运与自旋热电性质 ——非平衡格林函数方法的应用

低维材料不断涌现的新奇性质吸引着科学研究者的目光. 除了电子的量子输运行为之外, 人们也陆续发现和确认了热输运中显著的量子行为, 如 热导低温量子化、声子子带、尺寸效应、瓶颈效应等. 这些小尺度体系的热输运性质可以很好地用非平衡格林函数来描述. 本文首先介绍了量子热输运的特性、声子非平衡格林函数方法及其在低维纳米材料中的研究进展; 其次回顾了近年来在 一系列低维材料中发现的热-自旋输运现象. 这些自旋热学现象展现了全新的热电转换机制, 有助于设计新型的热电转换器件, 同时也给出了用热产生自旋流的新途径; 最后介绍了线性响应理论以及在此理论框架下结合声子、电子非平衡格林函数方法进行的一些有益的探索. 量子热输运的研究对热效应基础研究以及声子学器件、能量转换器件的发展有着不可替代的重要作用.     

Phase Transitions on Markovian Bipartite Graphs—an Application of the Zero-range Process

We analyze the existence and the size of the giant component in the stationary state of a Markovian model for bipartite multigraphs, in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero-range process, the degrees being static on the other set. The analysis is based on approximations by independent variables and on the results of Molloy and Reed for graphs with prescribed degree sequences. The possible types of phase diagrams are identified by studying the behavior below the zero-range condensation point. As a specific example, we consider the so-called Evans interaction. In particular, we examine the values of a critical exponent, describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold. Rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero-range condensation. These results, supplemented with conjectures supported by Monte Carlo simulations, suggest that the phenomenological Landau theory for percolation on graphs is not broken by the fluctuations.     

Coloring random graphs

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable. Depending on q, we find the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.     

Stability and dynamical properties of material flow systems on random networks

The theory of complex networks and of disordered systems is used to study the stability and dynamical properties of a simple model of material flow networks defined on random graphs. In particular we address instabilities that are characteristic of flow networks in economic, ecological and biological systems. Based on results from random matrix theory, we work out the phase diagram of such systems defined on extensively connected random graphs, and study in detail how the choice of control policies and the network structure affects stability. We also present results for more complex topologies of the underlying graph, focussing on finitely connected Erdös-Réyni graphs, Small-World Networks and Barabási-Albert scale-free networks. Results indicate that variability of input-output matrix elements, and random structures of the underlying graph tend to make the system less stable, while fast price dynamics or strong responsiveness to stock accumulation promote stability.     

Transparent quantum graphs

We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrödinger equation on metric graphs. This allows to derive simple constraints, which make usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straightforward.     

Quantum transport in one-dimensional systems via a master equation approach: Numerics and an exact solution

We discuss recent findings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to find a stationary solution of Lindblad master equation. Furthermore, for a specific model an exact solution is presented.