Naomichi Hatano: Photo Album

The following are some of my current interests. 

Localization, Resonance, and non-Hermitian quantum mechanics

Although Hamiltonians in quantum mechanics are Hermitian, mathematical models for a certain kind of physical systems are given by non-Hermitian Hamiltonian; for example, magnetic flux lines in type-II superconductors, Dirac Fermions in random magnetic fields, QCD with broken chiral symmetry, and population biology. Unfortunately, however, generalization of quantum mechanics with non-Hermitian Hamiltonians have been hardly studied. Recently, I have proposed a large class of non-Hermitian models and explored physical relevance of its complex eigenvalues and complex currents. I have found that the models are relevant to the studies of the Anderson localization as well as the resonant conduction in mesoscopic systems. The topics include the following.

• A non-Hermitian analysis of the Anderson localization

  A metal, upon increasing impurities, becomes an insulator at a critical concentration of impurities. This is because electron waves cannot transmit, heavily scattered by impurities, just like waves on the water surface cannot go forward when scattered by stakes in the water. This phenomenon is called the Anderson localization. Whether the electron wave is spatially localized or not depends on the impurity concentration and the electron energy. Even when it is localized, how strongly localized again depends on the impurity concentration and the electron energy.

  To know whether the wave is localized and how strongly it is, we propose a new model, sometimes called the Hatano-Nelson model. In this model, we introduced to the metal with impurities an imaginary vector potential, which is physically impossible in the real world. As a result, the model becomes non-Hermitian and can have complex eigenvalues. By inspecting the eigenvalue distribution of the intentionally non-Hermitized model, we can know the behabior of the localized states of the original Hermitian model.

  References
  N. Hatano and D.R. Nelson, Phys. Rev. B 56 (1997) 8651--8673.
  N. Hatano and D.R. Nelson, Phys. Rev. B 56 (1997) 8651--8673.
  N. Hatano and D.R. Nelson, Phys. Rev. B 58 (1998) 8384--8390.
  N. Hatano, Physica A 254 (1998) 317--331.

• Definition and compuation of quantum-mechanical resonance

  Many textbooks treat quantum-mechanical resonance only phenomenologically, stating that it is convenient to assume that the resonance possesses a complex eigenvalue. It is acutally possible to define mathematically the resonant state as an eigenstate of the Schroedinger equation of an open quantum system. Its wave function decays in time (due to the imaginary part of the eigenenergy) but diverges far away from the potential (due to the imaginary part of the eigenwave-number). This seemingly strange wave function, in fact, conserves the particle number and is numerically tractable.

  The physical mathematical understanding of the resonant state is still at an elementary level. We plan to develop it highly with application to resonance in quantum dots.

• Eigenvalue distribution of non-Hermitian matrices

  It is necessary to know the eigenvalue distribution of non-Hermitian matrices for analysis of the Anderson localization, the resonant state and non-equilibrium models. In order to know statistical mechanical properties, we have to compute numerically the eigenvalues of huge non-Hermitian matrices such as of the size 106*106. However, there are almost no algorithms that compute the eigenvalue distribution of huge non-Hermitian matrices efficiently. We are developing such an algorithm as a tool for studying phsyical models.

  Specifically, we compute the norm of the Green's function of non-Hermitian matrices on a complex energy plain by combining the conjugate gradient method and the Lanczos method. We can know the structure of the eigenvalue distribution as a three-dimensional plot.

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

The spin-orbit interaction in the two-dimensional electron gas is of great interest because we may manupulate electron spins with electric fields and may generate electric currents with magnetic fields. We are interested in the spin-orbit interactionin quantum wires.

In particular, we recently pointed out that the spin-orbit interaction is treated with the standard non-Abelian gauge field theory (the Yang-Mills theory). On the basis of the understanding, we succeeded to find the structure of a perfect spin filter. The perfect spin filter is an interference circuit of quantum wires such that upon injecting electrons with mixed spins, only downward spins come out. Our spin filter is perfect over the entire energy range of injected electrons.

References
N. Hatano, R. Shirasaki and H. Nakamura, Phys. Rev. A 75 (2007) 032107.
N. Hatano, R. Shirasaki and H. Nakamura, Solid State Commun 141 (2007) 79--83.

Mathematical models for environmental science with fractal structures

There are many fractal phenomena in the environment around us. In many studies in the field of environmental sciencesl, however, fractal effects are rarely taken into account. Since many previous studies carry out numerical simulations with only simple linear processes, the results fail to predict long-term environmental changes.

We have developed a mathematical model of abnormal wind diffusion of environemental pollutants, taking account of fractal fluctuations of the wind velocity. An analitic solution of the model fit remarkably well data of radioactive dusts around Chernobyl over ten years and data of ozone concentration in the Arctic. We also developed a model of abnormal diffution of soil pollutants, taking into account the fractality of adsorption of rocks. The solution reproduced experimental data well.

These studies suggest that the fractality of the natural environment is essential in predicting long-term and large-scale changes. I would like to develop fundamental mathematical models with such effects and thereby contribute to predicting pollution and green-house effect.

References
Y. Hatano and N. Hatano, Water Resources Research 34 (1998) 1027--1033.
Y. Hatano, N. Hatano, H. Amano, T. Ueno, A.K. Sukhoruchkin, and S.V. Kazakov, Atmospheric Environment 32 (1998) 2587--2594.
Y. Hatano and N. Hatano, Z. Geomorpho. N.F., Suupl.-Bd. 116 (1999) 45--58.