Naomichi Hatano

Vortex pinning and non-Hermitian quantum mechanics

A delocalization phenomenon is studied in a class of non-Hermitian random quantum-mechanical problems. Delocalization arises in response to a sufficiently large constant imaginary vector potential. The transition is related to depinning of flux lines from extended defects in type-II superconductors subject to a tilted external magnetic field. The physical meaning of the complex eigenvalues and currents of the non-Hermitian system is elucidated in terms of properties of tilted vortex lines. The singular behavior of the penetration length describing stretched exponential screening of a perpendicular magnetic field (transverse Meissner effect), the surface transverse magnetization, and the trapping length is determined near the flux-line depinning point. {copyright} {ital 1997} {ital The American Physical Society}

DISPERSIVE TRANSPORT OF IONS IN COLUMN EXPERIMENTS : AN EXPLANATION OF LONG-TAILED PROFILES

We present a novel microscopic model of sorption and convection of ions in heterogeneous media. Our model is based on an analogy to electron transport in a semiconductor. A new feature of our model is a power law random distribution of the adsorption time of ions. Diverging standard deviation of the distribution function yields anomalous ion transport. We show that this anomalous transport explains a concentration profile with a long tail that has been observed in column experiments. We successfully fit recent experimental data. Finally, we propose new experiments by which we can check the validity of our model.

The physics of communicability in complex networks

Abstract A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantum-mechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks.

Communicability betweenness in complex networks

Betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks. They allow us to study the extent to which a node takes part when information is passed around the network. Nodes with high betweenness may be regarded as key players that have a highly active role. At one extreme, betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes. At the other extreme, an alternative type of betweenness has been defined by considering all possible walks of any length. In this work, we propose a betweenness measure that lies between these two opposing viewpoints. We allow information to pass through all possible routes, but introduce a scaling so that longer walks carry less importance. This new definition shares a similar philosophy to that of communicability for pairs of nodes in a network, which was introduced by Estrada and Hatano [E. Estrada, N. Hatano, Phys. Rev. E 77 (2008) 036111]. Having defined this new communicability betweenness measure, we show that it can be characterized neatly in terms of the exponential of the adjacency matrix. We also show that this measure is closely related to a Frechet derivative of the matrix exponential. This allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations. Using illustrative synthetic and real life networks, we show that the new betweenness measure behaves differently to existing versions, and in particular we show that it recovers meaningful biological information from a protein–protein interaction network.

Non-Abelian gauge field theory of the spin-orbit interaction and a perfect spin filter

We point out that the Rashba and Dresselhaus spin-orbit interactions in two dimensions can be regarded as a Yang-Mills non-Abelian gauge field. The physical field generated by the gauge field gives the electron wave function a spin-dependent phase which is frequently called the Aharonov-Casher phase. Applying on an AB ring this non-Abelian field together with the usual vector potential, we can make the interference condition completely destructive for one component of the spin while completely constructive for the other component of the spin over the entire energy range. This enables us to construct a perfect spin filter.

Finding Exponential Product Formulas of Higher Orders

This article is based on a talk presented at a conference “Quantum Annealing and Other Optimization Methods” held at Kolkata, India on March 2–5, 2005. It will be published in the proceedings “Quantum Annealing and Other Optimization Methods” (Springer, Heidelberg) pp. 39–70. In the present article, we review the progress in the last two decades of the work on the Suzuki-Trotter decomposition, or the exponential product formula. The simplest Suzuki-Trotter decomposition, or the well-known Trotter decomposition [1–4] is given by e x(A+B) = e xA e xB + O(x 2 ), (1) where x is a parameter and A and B are arbitrary operators with some commutation relation [A, B] 6 0. Here the product of the exponential operators on the right-hand side is regarded as an approximate decomposition of the exponential operator on the left-hand side with correction terms of the second order of x. Mathematicians put Eq. (1) in the form e xA e xB = e x(A+B)+O(x 2 ) (2)

Non-Hermitian Delocalization and Eigenfunctions

Recent literature on delocalization in non-Hermitian systems has stressed criteria based on sensitivity of eigenvalues to boundary conditions and the existence of a nonzero current. We emphasize here that delocalization {ital also} shows up clearly in eigenfunctions, provided one studies the product of left and right eigenfunctions, as required on physical grounds, and {ital not} simply the squared modulii of the eigenfunctions themselves. We also discuss the right and left eigenfunctions of the ground state in the delocalized regime and suggest that the behavior of these functions, when considered separately, may be viewed as {open_quotes}intermediate{close_quotes} between localized and delocalized. {copyright} {ital 1998} {ital The American Physical Society}

Some Properties of the Resonant State in Quantum Mechanics and Its Computation

The resonant state of open quantum systems is studied from the viewpoint of the eigenfunction with an outgoing momentum flux. We show that the number of particles is conserved for a resonant state if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles in a fixed volume of integration would decay exponentially. Moreover, we introduce new numerical methods of treating the resonant state with the use of an ef fective potential. We fi rst present an umerical method for finding a resonance pole in the complex energy plane. This method seeks an energy eigenvalue iteratively. We found that it leads to super-convergence, i.e., convergence whose rate is exponential with respect to the iteration step. Also, it is independent of the commonly used complex scaling. We also present a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Because the wave function of the resonant state is diverging away from the scattering potential, it is difficult to follow its time evolution numerically in a finite area using previous methods.

Isotropic spin-1 chains with bond alternation: analytic and numerical studies

Isotropic spin-1 chains with bond alternation are studied. Analytic (both variational and perturbative) results are presented. A phase transition is found to separate the S=1 Haldane phase and the dimer phase. The critical point is numerically determined using the Binder parameter. The universality class is predicted to be given by the level-1 SU(2) Wess-Zumin-Witten model, and this prediction is supported by the present exact diagonalization study. The dimer phase is found to be connected to the S=2 Haldane phase in a special limit. The change of the excitation spectra and the string order parameter with bond alternation is discussed.

Topological atomic displacements, Kirchhoff and Wiener indices of molecules

We provide a physical interpretation of the Kirchhoff index of any molecules as well as of the Wiener index of acyclic ones. For the purpose, we use a local vertex invariant that is obtained from first principles and describes the atomic displacements due to small vibrations/oscillations of atoms from their equilibrium positions. In addition, we show that the topological atomic displacements correlate with the temperature factors (B-factors) of atoms obtained by X-ray crystallography for both organic molecules and biological macromolecules.