Fumihiko Takano

New Approximate Method for Ising System

A new approximate treatment of the Ising spins is presented, in which the fluctuation of the effective field is taken into consideration. It is shown that the simplest approximation gives the satisfactory results. The approximation is applied to the random spin system and the critical concentration is calculated. The results are again satisfactory compared with those of other method. This idea is applied to the dynamical problem based upon the Galuber model, and it is shown that the approximation equivalent to Bethes one can be obtained in the case of the thermal equilibrium by the improvement of the simplest approximation. Using the improved approximation, the dynamical susceptibility and the pair correlation function in the thermal equilibrium are calculated.

Statistical Theory of Ferro- and Antiferromagnetism

The method of the constant coupling approximation is applied to investigate the effect of the second nearest neighbor interactions. Ferromagnetic, antiferromagnetic, and screw spin arrangements are discussed as functions of a ratio of the magnitude of the nearest neighbor and the second nearest neighbor interactions. The critical value where the spin arrangement changes is obtained for the antiferromagnetic body centered cubic lattice. The condition for screw arrangement of spins to occur in rutile type structure is same as that in the simple molecular field approximation but the transition temperature is different due to the short range order effect. We further improve the constant coupling approximation by taking account of three spin or four spin system. One of the important results is two-dimensional triangular lattice cannot be ferromagnetic, on the contrary to the results obtained by the constant coupling approximation.

Spin Wave Theory of the Spin 1/2 XY Model

The spin wave theory of Kondo and Yamaji is applied to the spin 1/2 XY model in one to three dimensions. In one dimension, the value of the gap Δ which appears in the spectrum of x -component spin remains finite up to 0 K, and no phase transition occurs. The value of the nearest neighbour correlation function is in good agreement with the exact values. In three dimensions, Δ vanishes at a finite temperature, and the second order phase transition occurs. In two dimensions, as T →0, Δ becomes very small for square lattice and tends to zero as exp (- T 0 / T ) for triangular lattice, which means the non-existence of the phase transition. The susceptibility, however, becomes anomalously large for low temperatures, and the extrapolation of Curie-Weiss law gives a “fictitious” transition temperature whose value agrees with those given by the high temperature expansion and the real space renormalization group theory.

Statistics of Electrical Resistance in One-Dimensional Disordered System

The motion of electrons in the one-dimensional disordered system including N δ = function potentials of random locations and strengths is studied by using the random transfer matrix method. The statistical properties of the electrical resistance ρ N are discussed in the limit N → ∞. By studing the cumulant expansion of f N = ln (1 + ρ N ) in the cases when only the locations of potentials are random and the uniform strength is extremely small or large, it is found that the probability distribution of f N is not normal in the large strength limit in contrast to the argument by Anderson et al. and by Abrikosov. In the general case when both locations and strengths are random, the inverse localization length is numerically calculated under the assumption of the normal distribution of f N .

Quantum Theory of Stimulated Raman Effect

A photon theory of the stimulated Raman effect is developed. A system of coupled equations for the amplitudes of scattered photons and molecular vibrations are derived. The molecular vibration coordinates are then eliminated using the random phase approximation. The equations which result assume the form of coupled equations for the first Stokes and anti-Stokes lights. The secular equation which characterizes the solution of these coupled equations is shown to be equivalent to that of Shen and Bloembergen. Using this secular equation, the condition for temporal growth of the scattered lights is obtained and is compared with that for the spatial growth, which was obtained previously by Shen and Bloembergen. It is shown that both types of growth (temporal and spatial) proceed with comparable rates if the damping of the molecular vibration is large. If its damping is small, however, the temporal growth prevails over the spatial growth. Typical line-shape of the anti-Stokes line is obtained in the case of sma...

On the Non-Orthogonality Difficulty in Ferromagnetism

In a many-electron system, one is faced to some difficulties when one-electron wave functions are not orthogonal to each other. For example, the inner product (\(\varPhi\), \(\varPhi\)) (\(\varPhi\) is a Slater determinant made from these non-orthogonal wave functions) loses its meaning as the number of electrons becomes large. We avoid this difficulty by using the expansion coefficients instead of the inner products, in order to define the matrix elements of operators. Applying this method to a magnetic problem, in which there are some doubts about the validity of the Heisenberg model \(\displaystyle \mathcal{H} = -2 \sum_{i>j} J_{i,j}S_{i} \cdot S_{j} \), we can derive the energy of the ground state and the energy spectrum of spin waves. The results are written as

Theory of Excitations in Liquid Helium 4

A modified “ phonon-roton ” model for liquid Helium II is proposed and is shown to be connected smoothly with the theory of Bose condensation. The method used is the direct extension of the method of “ auxiliary variables “ used by Bogolyubov and Zubarev, and is equivalent to that used by Bohm and Pines in the theory of the electron gas in metals. The phonon excitation is described in terms of the density operator, and the roton part by the so-called individual coordinates. The elimination of the kinematical interaction leads to the following model: The excitations for long wave length are described by phonons, and for short wave length by rotons, and the total number of excitations cannot exceed the total number of the particles. The thermal properties of this model is seen to agree with the experimental fact qualitatively.

Phase Transition of the Sherrington-Kirkpatrick Model

The free energy of the Sherrington-Kirkpatrick (SK) model for the spin glass is calculated exactly as a function of local magnetic moments in the form of power expansion. The expression is shown to coincide with that of Thouless, Anderson and Palmer and with that of Oguchi and Takano up to the fourth order term, suggesting that the Bethe or the pair approximation is exact for this model. The transition temperature obtained from the second order term gives the SK transition line. It is shown that the linear susceptibility diverges at the ferromagnetic transition temperature and remains finite at the spin glass one, and that the nonlinear susceptibility diverges negatively both at the spin glass and the ferromagnetic transition temperatures. The nature of the phase transition is seen to be rather peculiar and the more detailed analysis of higher order terms is necessary to obtain the definite conclusion.

Magnetism of d-Electron System with Orbital Degeneracy –Constant Coupling Approximation for Inagaki's Model–

The model expressed by the Hamiltonian \begin{aligned} H{=}-\sum_{{\langle}ij{\rangle}}\{2J_{S}(\textbf{\itshape S}_{i}{\cdot}\textbf{\itshape S}_{j})+2J_{T}(\textbf{\itshape T}_{i}{\cdot}\textbf{\itshape T}_{j})+8J_{0}(\textbf{\itshape S}_{i}{\cdot}\textbf{\itshape S}_{j})(\textbf{\itshape T}_{i}{\cdot}\textbf{\itshape T}_{j})\}, \end{aligned} which is introduced by Inagaki for the discussion of the d-electron system with doubly degenerate orbitals, is investigated by the constant coupling approximation. The result is quite different from that of the molecular field approximation, showing that the effect of the short range order is very important in this system. Especially in the case when all exchange integrals are negative and their magnitudes are slightly different with | J T |>| J 0 |>| J S |, it is shown that the ferromagnetic ordering in S occurs first in contrast to the result of Inagaki that the antiferromagnetic ordering in T occurs first.

Notes on Fermi and Bose Statistics

By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as e n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 A and 14.2 A. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi...