Showing papers in "Progress of Theoretical Physics in 1977"

Phenomenological Theory of Spin-Glasses and Some Rigorous Results

Abstract: A general phenomenological theory of spin-glasses is presented, which predicts the weak singularities (like cusps) of susceptibilities and specific heat near the transition point TsG and also predicts generally the divergence of the second derivative of the nonlinear suscepti­ bility with respect to a magnetic field in zero field at Tsa. A similar second derivative of specific heat is also shown to diverge at Tsa except for the molecular field theory. Scaling equations of state for magnetization m and spin-glass order-parameter q are derived, which yield new scaling relations of critical exponents for m(T), q(T), X,(T), Co(T) and the corresponding nonlinear quantities. These general results explain qualitatively very well the magnetic field dependence of magnetization and specific heat observed experimentally. A non-uniform phenomenological Hamiltonian is also proposed for the rcnormalization group approach, which yields the critical dimensionality d,=6, as usuaL Intermediate random statistics and real replica method are p;oposed. There is also presented an exactly soluble model showing a spin-glass like phase transition.

Monte Carlo Simulation of Quantum Spin Systems. I

Abstract: A general explicit formulation of Monte Carlo simulation for quantum systems is given in this paper on the basis of the previous fundamental proposal by Suzuki. This paper also demonstrates explicitly the possibility of it and gives new interesting physical results on the two-dimensional XY-model. That is, the present preliminary simulation seems to indicate a phase transition with a divergent susceptibility, and a very weak singularity of specific heat if it exists, and without long-range order.

Structure of the Excited States in 12C. II

Abstract: Structure of both positive- and nCc:gative-parily slates in 12C is studied with a microscopic :l u-parlicle model. The c.m. motions of 3 a-clusters are treated by the generator coordinate method. All the levels with T=O below 15 MeV (except the 1:0.7 MeV r+ level) are suc­ cessfully reproclucecl, including the famous o,+ level and the next positive-parity level which is regarded as :o,+. Another 2', :l1+ and two 4+ stales which have a structure quite similar to that of the O, ,_ and :0, + stales are preclictecl with large K-rnixings. Furthermore the pre­ sent investigation predicts an existence of O, + state. The structure of all the above excited positive-parity slates is quite diiferenl from that expected in the shell model, hut rather should he considered that· o£ a finite a-boson ga,. The negative-parity leveis are quite well de":ribecl '" memhcrs of the K"=:l- and I bands in the present model. § 1. Introduetion The nucleus "C has been offering a testing field to vanous nuclear models. It is a stable and tight binding system though it is one of the lightest nuclei. On the assumption of a stable average nuclear Jield, shell model \Vas applied in se\·eral \·ersions to explain low energy properties of 12C.'' Unfortunately, all the efforts have resulted in obtaining only partial success. Above all the second 0 t level at 7.7 MeV and the next positive-parity level at 10.3 MeV have been difficult to be reproduced at such low excitation energies. They are also known to have anomalously large a-decay widths.'' Morinaga suggested that they form an excited rotational band with a linear chain structure of 3 a-particles. 11 According to Ikeda's cliagram5l' 61 one can expect appearance of nuclear states with some cluster structure in the neighbourhood o£ decay-threshold energies of a-particles. In 12C, the 0,' le\·el and the next positive-parity level noted aboYe are expected to be such states, that is, to ha\·e distinct cluster structure relevant to the 'Be -1--CY channel al 7.4 MeV. From the \'iewpoint of a-particle mudel this nucleus has been investigated since the early ages of nuclear study. The classical a-particle model was applied to it on the analogy o£ homonuclear tri-atomic mole­ cule." The modeL howeYer, predicts a :3 state at too low an excitation energy compared with experiment. This dr;cm·back is known to disi!ppear in the micro­ scopic model.RJ Recently sc1·eral dynamical calculations"' of :3 a-particle system haYe been made by the use of 1·arious a-ct interactions. Fuji\\'ara and Tamagaki 9b 1 11 Preliminary results were

Static and Dynamic Finite-Size Scaling Theory Based on the Renormalization Group Approach

Abstract: Fisher's static finite-size scaling law is derived on the basis of the renormalization group theory and it is extended to dynamic critical phenomena in a finite system. This dynamic finite-size scaling law yields a cross-over effect with respect to the size and time-region. This effect is useful in analyzing computer simulations and also in studying the scaling property of the Kondo effect near the absolute zero temperature.

Nonlinear Evolution Equations Generated from the Bäcklund Transformation for the Boussinesq Equation

Abstract: A Backlund transformation for the Boussinesq equation is given in the bilinear form. It is shown that the Backlund transformation generates an important class of nonlinear evolution equations exhibiting N-soliton solutions. They are a modified Boussinesq equation, a higher order water wave equation introduced by Kaup and a coupled equation whose N-soliton solution reduces to that of the nonlinear Schrodinger equation with normal dis­ persion. The relation between the Backlund transformation and the inverse scattering method is also discussed.

Grain Formation through Nucleation Process in Astrophysical Environment

Abstract: General picture of grain formation is preoented based on the nucleation theory. Grain formation process is described by a growth equation of grain radius and an equation of monomer consumption due to the growth of grains. These equations are characterized by two parameters. One depends on the physical conditions of the system and the other reflects the nature of grain materials. An overall feature of the grain formation process is illustrated by the use of an analytic expression of the solutions. After the vapor cools down to the saturated state, a waiting time is necessary until the grain formation begins effectively. Size distribution is relatively sharp in general. The representative size is closely related to the parameter which depends on the physical conditions. Growth by coalescence is not effective until the monomer sticking process is almost completed. The results are applied to the condensation in the primordial solar nebula. It is shown how the picture based on the chemical equilibrium calculations should be modified.

The Effects of Impurities on Superconductors with Kondo Effect

Abstract: The Green's functions of s- and d-electrons in superconductors are obtained on the basis of the interpolation theory'>,'> which includes as impurity effects the pair breaking and the effective repulsive interaction between s-electrons. By the use of these Green's functions, the order parameter and the critical magnetic field at zero temperature in the presence of impurities and its initial decrease are given. The localized excited state in the gap 1s found and shown to be doublet, differently from that of MUller-Hartmann and Zittartz.

A New Fermion Many-Body Theory Based on the SO(2N+1) Lie Algebra of the Fermion Operators

Abstract: A new many-body theory for fermions is proposed which is based on the S0(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A new cannonical transformation, which is the extension of the Bogoliubov transformation to the SO (2N + 1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the S0(2N+ 1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the S0(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the S0(2N+1) canonical transformation. The physical fermion space is shown to be the irredu­ cible spinor representation of the SO (2N + 1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N + 1 dimensional rotator. The conventional standard approach to fermion many-body problems starts with the independent particle approximation (IPA), either the Hartree-Fock (HF) or the Hartree-Bogoliubov (HB) approximation, for the ground state. Excited states are then treated with the random phase approximation (RP A). The RPA treatment of excited states, however, meets a serious difficulty when an instability in the IPA ground state takes place. The lowest excitation energy evaluated by the RP A becomes zero at the instability boundary of the IP A ground state due to the equivalence of the instabilities of the IPA ground state and the RPA excited states.n In the region near the phase transition of the IP A ground state, the amplitudes of collective excitations become large and couplings between collective modes, which are neglected in the RPA, become of essential importance. Since the RPA is the approximation based on an approximate bose quantization for the fermion pair operators,"> attempts were made to take into account the effect of mode couplings with the use of the boson expansion satisfying exactly the commutation relation of the pair operators. 3> However, the boson expansion theory treats mode couplings in a perturbational way and the convergence of the expansion becomes bad when the amplitudes of collective excitations become large. 4>

Theory of Anharmonic Lattice Vibration in Metallic Fine Particles. I

Abstract: \Ve apply the self-consistent Einstein moclel to the theory of anharmonic lattice vibration in metallic fine particles, and discuss the size dependence of the melting and superconducting transition temperatures in the connection to the softening of the surface lattice vibrations. Assuming a simple interatomic potential and certain distribution of the particle size, we express both transition temperatures as functions of the average radius of fine particles and show that the numerical results calculatecl from the theoretical expressions arc in fairly good accord with the experiments. § I. Introduction In a previous paper, one of the present authors has discussed, on the basis of a self-consistent Einstein model, that the atoms on the surface of metal crystal perform oscillations with much larger amplitudes compared with the interior at­ oms.]) The large surface lattice vibration is related to the softening of the fre­ quencies of the surface atoms and also closely related to the so-called surface relaxation of the lattice constant. Since the surface to volume ratio of metallic fine particles is increased with decreasing size, the surface softening should give appreciable effects in the various thermal properties of metallic fine particles vvith sufficiently small radii. Up to the present time, there are many accumulated ex­ perimental facts to sho1v that this is indeed the case. It has long been known for instance that the metallic fine particles have melting points much lower than those of the corresponding bulk metals. The first observation of this depression of the melting point was made by Takagi2l on Pb, Sn and I3i thin films by the use of the electron diffraction method. Since then similar observations have been reported, for example, on Sn by vVronski, 31 on Pb and In by Coombes, 41 on Au by Buffat and BorePJ and so on. This phenomena are undoubtedly due to the surface effects, and phenomenological theories based on the thermodynamic arguments have been presented. 61 A simple microscopic theory 1vas first proposed by one of the present authorsn with a pretty success in explaining the main experimental facts. This paper contains a sophistication of the previous theory. Another example of the phenomena 1vhich seem to be related with the surface

Effect of the Magnetic Field on the des Cloizeaux-Pearson Spin-Wave Spectrum

Abstract: The des Cloizeaux and Pearson's exact solution for the spin-wave spectrum of the S=1/2 antiferromagnetic Heisenberg chain is extended to the case of finite external field. It reproduces naturally the des Cloizeaux-Pearson spectrum in the zero-field limit as well as the spin-wave spectrum in the ferromagnetic state for fields larger than the critical field. The results are discussed by comparing them with predictions of other approximate theories. on CuC12 • 2NC,D,, which is a typical 1D antiferromagnetic Heisenberg spin system with S = 1/2. They showed by neutron scattering that its spin-wave spectrum agrees quite well with the celebrated exact solution of des Cloizeaux and Pearson ( dC-P) ;l and is in disagreement with the Anderson (molecular-field) theory. It clearly demonstrates the importance of exact theoretical studies of 1D systems. Although this experiment was made in the absence of external magnetic field, the magnetic-field dependence of elementary excitations in antiferromagnetic linear chains would be very interesting. The purpose of this paper is to study exactly the magnetic-field dependence of the dC-P spin-wave spectrum. We hope that this work would stimulate further experimental studies on dynamics of 1D Heisenberg antiferromagnets in the presence of external field. The field dependence of spin waves of lD Heisenberg antifer­ romagnets has previously been calculated by Pytte,"l who applied the Bulaevskii (Hartree-Fock) approximation') based on the Fermion representation 5l.Bl of the lD Heisenberg model. We later compare it with our result. We also show that the magnetic-field dependence of the dC-P spin wave is qualitatively different from that of the classical (Anderson) spin wave. The details of the formulation and calculations are presented in § 2. Com-

Multi-Cluster Allowed States and Spectroscopic Amplitude of Cluster Transfer

Abstract: A general construction procedure of the allowed states of the multi-cluster system is presented with the use of the concept of the "coefficient of fractional parentage". We explain in detail in the case of three-cluster system that the introduction of the Elliott SU3 group greatly simplifies the construction process, thereby making the practical calculation significantly rapid. As an important application of the allowed states, we discuss the calculation of the spectroscopic amplitude of cluster transfer and investigate its characteristic features on the basis of the properties of the allowed states.

Entrainment of a Limit Cycle by a Periodic External Excitation

Abstract: Entrainment of a limit cycle by a periodic external excitation 1s investigated with the Prigogine-Lefever·Nicolis model for chemical reaction. In the neighbourhood of the marginal point a quasi-harmonic theory is developed and the theoretical prediction is compared with the numerical computation. The stability of the entrained oscillation is examined in particular. The instabilities are classified into two types, i.e., hard- and soft-mode instabilities, by the use of the Floquet exponents which are, calculated by a non-perturbational method. The hard·mode instability corresponds to the limit of entrainment and the amplitude is subject to a modulation beyond the threshold. The frequency of modulation is estimated from the Floquet exponent, which is compared with the results of numerical computation. The soft. mode instability corresponds to a jump phenomenon in the amplitude. The smaller amplitude branch is liable to a modulation due to a superimposed hard-mode instability. As a whole, a reasonable agreement is obtained between numerical and theoretical results.