Jun-ichi Hori

Degree of localization in three dimensional disordered systems

It is remarked that any matrix element of a Green operator can be expanded in terms of principal minors of the secular determinant of the system under consideration. The proof is given in a way similar to that of Feenherg (1948). For diagonal elements the expansion turns out to be the same as Andersons renormalized perturbation expansion of the self energy. Upon its basis, the interrelation between self energies, principal minors of secular determinants, and the state ratios, which play an important role in the phase-theoretical treatment of the localized of eigenfunctions in disordered systems, is studied. It turns out that in the case of a diagonally disordered one dimensional system with nearest neighbour interactions only, the state ratio and the self energy are essentially the same quantities, and the degree of localization, defined in the phase theory in terms of the state ratio, can be defined also in terms of the self energy. Secondly it is shown, on the basis of the expansion of offdiagonal elements of the Green operator, that in the one dimensional case the degree of localization thus defined gives the exponent of the average exponential decay of the Green function. It is thus concluded that it is natural to define the degree of localization in three dimensional disordered systems by the exponent of average exponential decay of the Green function.

Spectral gaps in vibrational systems II. Generalized special frequencies in the spectra of isotopically disordered two- and three-dimensional lattices

It is shown that the evaporation method is effective for discussing the generalized special frequencies of multi-dimensional lattices and, by combining it with Deans method of obtaining the limits of impurity bands, we can estimate the critical mass ratio of each generalized special frequency. As examples the critical mass ratios of some isotopically disordered two- and three-dimensional lattices with simple structures are actually estimated.

Long‐Time Behavior of a Random Lattice

The time‐dependent behavior of a linear isotopic random lattice is studied. The asymptotic average value of the solution of the equation of motion for the amplitude zk of a normal mode of the virtual regular lattice, composed of atoms with mass equal to the harmonic average of those of the isotopes, is calculated in the limit N → ∞, t → ∞, γn → 0, and γnt=a finite constant (n=1, 2,⋯). Here N is the number of normal modes, and γn the nth cumulant of the distribution of inverse mass. The initial conditions are zj(0)=δkj and (dzj/dt)t=0=0(j=−N/2+1, −N/2+2,⋯, N/2−1, N/2). The result comes out to be 〈zk〉=exp(−iαωk+ ∑ n=1∞γ2nQknqn−1/n!)t, where αωk≡2f12Hωk (f: force constant, H2: average inverse mass) is the eigenfrequency of the kth normal mode, Qk≡8iωkf2/α3, and q=t/N, assumed to be a constant. If q → 0, there remains only the first term, and the result coincides with that for the case of a Gaussian distribution.

Spectral gaps in vibrational systems III. Applications of the evaporation method to non-isotopically disordered lattices

The method of evaporation presented previously by the author is applied to some lattices which are non-isotopically disordered. For the mixed crystal of alkali halide type, the Saxon-Hutner-type statements are again obtained for the common gaps of appropriate constituent systems. For the disordered lattice which contains only the simplest clusters of non-isotopic impurities, some definite conclusions concerning the critical mass ratios of the generalized special frequencies are obtained.

Localization of eigermodes in disordered one-dimensional systems I. General theory and isotopically disordered diatomic chain

The arguments hitherto presented on the localization of eigen-modes of the one-dimensional disordered system(1)–(3) are to a large extent intuitive and qualitative. Borland(4) presented a quantitative theory but it depends on the assumption of ergodicity of the phase distribution and on the restrictive condition of Frechet, which is not fulfilled e. g. for an isotopically disordered diatomic chain. In this paper we present a general theoretical framework which is convenient to discuss the phenomenon of localization without being restricted by any assumption or condition, and upon this basis demonstrate numerically that the eigenmodes of the isotopically disordered chain must always be localized. At the same time it is shown that there exists a unique ensemble phase-density function in spite of the fact that the condition of Frechet is violated, and that the phase distribution has the ergodic property.

WITHDRAWN: Structure of the Spectra of Disordered Systems. I: —– Fundamental Theorems —–

Two theorems are presented, which play a fundamental role in discussing the spectra of disordered systems. The first theorem is valid for any mixed lattice which is described by a set of a number of 2x2 non-singular transfer-matrices with real traces, arranged regularly or randomly. It states that, if a value of frequency or energy lies in one of the spectral gaps for every constituent regular lattice, and if the fixed points of the transformations induced by these matrices are so arranged that there exists a trapping region, the density of frequency- or energy-spectrum of the mixed lattice vanishes at this frequency or energy-value. For some special forms of the transfer-matrices, which often appear in physical problems, the latter condition may be stated in terms of a simple notion “phase”, so that we get the second theorem. Various theorems of Saxon-Hutner-type are to be deduced from these theorems, according to the nature, or the manner of description, of individual systems.