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The temperature dependence of the electronic diffusion coefficient in a disordered symplectic system with electron–phonon interactions is numerically calculated. Electron–phonon interactions are in...

The static and dynamical behaviors of an acoustic polaron in a quasi-one-dimensional branching-off system are studied numerically by employing Su–Schrieffer–Heeger's model extended to involve the branching-off structure. At the branching-off site, an electron is found to feel an attractive potential with the strength equal to the transfer integral between one-dimensional neighboring sites. When an acoustic polaron prepared at a site sufficiently apart from the branching-off site is set in motion by an external force towards the branching site, it is totally reflected, totally trapped or partially reflected and partially trapped at the branching site, depending on the extent of the polaron just before it reaches the branching site. It is argued that this behavior is essentially the same as that seen in a one-dimensional system with a single-site-type attractive potential. When the effective attractive potential at the branching site is cancelled out by a real repulsive site-potential, the polaron moving to...

The periodic lattice distortion BOW owing to the electron–lattice interaction and the spin density wave SDW owing to the electron–electron interaction in the two-dimensional Peierls–Hubbard model having a half-filled electronic band are studied numerically within the Hartree–Fock approximation. It is shown that the phase transition from the BOW to the SDW behaves as the first order phase transition when the on-site electron–electron interaction U is increased continuously. There is no region of U where the BOW and the SDW coexist as a uniform phase.

The phonon dispersion relations in the two-dimensional Peierls phase (Peierls insulator) are studied numerically, focusing on a two-dimensional SSH (Su–Shrieffer–Heeger) model having a half-filled ...

Electron transport in a two-dimensional symplectic system with time-dependent perturbations is investigated numerically by the equation of motion method. The effect of time-dependent perturbations on the conductivity is examined in the critical regime as well as in the metallic regime. The universal correction to the conductivity, which is consistent with the weak localization theory, is indeed observed in the metallic regime. In the critical regime, it is found that the dependence of the conductivity on the frequency of perturbations can be described by the one-parameter scaling.

The collision processes between two acoustic polarons, two acoustic bipolarons, and a polaron and a bipolaron in a one-dimensional electron-lattice system are analyzed by numerical simulations based on Su, Schrieffer and Heeger's model extended to include short-ranged Coulombic electron-electron interactions. The collisions between two polarons with the same spin and between two bipolarons are essentially elastic because of a rather strong repulsion due to the Pauli principle. On the other hand, those between two polarons with opposite spins and between a polaron and a bipolaron show some inelastic behaviors depending on the relative velocity of the two objects before the collision. This knowledge will serve to understand the dynamics of the system when many polarons and/or bipolarons are created in the system e.g. by photoexcitation.

The static and dynamical properties of an acoustic polaron in a two-dimensional electron-lattice system is studied numerically on the basis of Su, Schrieffer and Heeger's Hamiltonian (SSH Hamiltonian) extended to two dimensions. As for the static structure of the polaron, the coupling constant dependence of the width is analyzed. It is confirmed that, in contrast to the one-dimensional case, the two-dimensional acoustic polaron is stable only when the dimensionless electron-lattice coupling constant exceeds a certain critical value. As the coupling constant is reduced from a larger value, the polaron width increases, but the polaron becomes unstable before the width diverges. The dynamics of the polaron is studied by applying an external electric field which accelerate the polaron in a natural way. Similarly to the one-dimensional system the polaron velocity shows a saturation, though the saturation velocity is about one quarter of that in the one-dimensional case which is known to be nearly equal to the ...

In order to estimate the mobility of an acoustic bipolaron in a typical non-degenerated polymer such as polydiacetylene (PDA), which can be regarded as a one-dimensional electron-lattice system, the dynamical behavior of an acoustic bipolaron in an external electric field is numerically investigated in the presence of a phenomenological damping of the lattice motion. In these simulations, we use the Su, Schrieffer and Heeger (SSH) Hamiltonian extended to include on-site and nearest-neighbor Coulomb repulsion terms, and the damping is introduced by adding a phenomenological friction term in the lattice equation of motion. The repulsive interactions are treated within the unrestricted time-dependent Hartree–Fock approximation. It is confirmed that the mobility of a bipolaron increases with the increase of the repulsion strength and that the relevant quantity determining the mobility of a bipolaron is its width as far as the phenomenological friction is kept fixed. The mobility of a bipolaron is compared wit...

The properties of an acoustic polaron are numerically studied for an electron-lattice system with a ladder-type structure where an electron transfer between two chains is allowed. Su-Schrieffer-Heeger's (SSH) model extended to the ladder system is used on the assumption that the periodic boundary condition in the chain direction is satisfied. When the interchain electronic transfer integral t x 0 is comparable with the intrachain one t y 0 , the lattice deformation related to the polaron is small compared to the case of one-dimensional systems because the electronic density profile is distributed equally in the two chains, and as a result the effective mass becomes lighter than the single chain case. On the other hand, when t x 0 is much smaller than t y 0 , the polaron is localized in one of the two chains instead of being distributed with a homogeneous weight to the two chains. This transition from the homogeneous to an inhomogeneous state is found to occur suddenly at a certain value of t x 0 as we dec...

Dynamical process of the formation of nonlinear excitations such as charged or neutral solitons from a photogenerated electron-hole pair in polyacetylene is studied numerically, based on Su-Schrieffer-Heeger's model extended to involve on-site as well as nearest-neighbor electron-electron Coulomb repulsion terms. The electron-electron interactions are treated within the time-dependent unrestricted Hartree-Fock approximation. Lattice fluctuations and a weak local disorder are introduced in order to trigger the formation and to represent many different initial conditions. Starting from an initial configuration with an electron at the bottom of the conduction band and a hole at the top of the valence band, separated by the Peierls gap, the time dependent Schrodinger equation for the electron wave functions and the equation of motion for the lattice displacements are solved numerically. It is not systematic but rather stochastic whether nonlinear excitations are created or not. We have obtained branching rati...

The dynamical properties of an acoustic bipolaron in non-degenerate conjugated polymers such as polydiacetylene (PDA), which can be regarded as one-dimensional electron-lattice systems, are studied through numerical simulations. In these simulations, use has been made of Su-Schrieffer-Heeger's model extended to involve on-site as well as nearest-neighbor electron-electron Coulomb repulsion terms. The electron-electron interactions are treated within the unrestricted Hartree-Fock approximation. It is confirmed that the velocity of the acoustic bipolaron accelerated by a constant electric field saturates at the sound velocity of the system, and that the width of the bipolaron shrinks to zero as its velocity approaches the saturation velocity (= sound velocity). The effective mass of a bipolaron is estimated to be several hundred times as heavy as the free electron mass, in the case of PDA. These properties are compared with those of a polaron.

Ballistic transport properties in a two dimensional electron gas are studied numerically, where magnetic fields are perpendicular to the plane of two dimensional electron systemsand periodically modulated both in $x$ and $y$ directions. We show that there are three types of trajectories of classical electron motions in this system; chaotic, pinned and runaway trajectories. It is found that the runaway trajectories can explain the peaks of magnetoresistance as a function of external magnetic fields, which is believed to be related to the commensurability effect between the classical cyclotron diameter and the period of magnetic modulation. The similarity with and difference from the results in the antidot lattice are discussed., Comment: 4 pages, 7 figures, to appear in J. Phys. Soc. Jpn., vol. 67 (1998) November

The magnetization due to the orbital motion of electrons in confined two dimensional disordered systems subject to perpendicular strong magnetic fields is investigated numerically. The effects of the edge states and randomness are discussed from the view point of microscopic current distribution. The magnetization as a function of the filling factor is modified from that for the impurity free case by the fluctuations in the bulk region, which are of the order of the effective Bohr magneton. On the other hand, the contributions of the edge states to the magnetization and to the current distribution are stable against the randomness.

The effects of a site-type impurity potential and a bond-type disorder on the dynamical behaviors of charged and neutral solitons in polyacetylene are studied numerically. The charged soliton feels the site-type potential in the same way as an electron does, except for the broadening of the potential range due to the finite width of the soliton. While the neutral soliton goes through the site-type impurity potential as if there is nothing, reflecting the fact that the excess electron density vanishes everywhere in this case. On the other hand, there is no essential difference between the effects of the bond-type disorder on both solitons. The effective potential in this case looks like a step function, which can be explained in terms of matching or mismatching of the disorder with the dimerization pattern in the region outside of the soliton.

The dynamical behavior of an acoustic polaron in typical non-degenerate conjugated polymer, polydiacetylene, is numerically studied by using Su-Schrieffer-Heeger's model for the one dimensional electron-lattice system. It is confirmed that the velocity of a polaron accelerated by a constant electric field shows a saturation to a velocity close to the sound velocity of the system, and that the width of a moving polaron decreases as a monotonic function of the velocity tending to zero at the saturation velocity. The effective mass of a polaron is estimated to be about one hundred times as heavy as the bare electron mass. Furthermore the linear mode analysis in the presence of a polaron is carried out, leading to the conclusion that there is only one localized mode, i.e. the translational mode. This is confirmed also from the phase shift of extended modes. There is no localized mode corresponding to the amplitude mode in the case of the soliton in polyacetylene. Nevertheless the width of a moving polaron shows small oscillations in time. This is found to be related to the lowest odd symmetry extended mode and to be due to the finite size effect., Comment: 12 pages, latex, 9 figures (postscript figures abailble on request to ono@polaron.ph.sci.toho-u.ac.jp) to be published in J. Phys. Soc. Jpn. vol.65 (1996) No. 5

The activation energy of a charged soliton pinned by a dopant ion is numerically studied as a function of the dopant concentration. To investigate this problem, use is made of Su-Schrieffer-Heeger's model modified to involve the long-range potential due to the dopants. The dopants are assumed to form a regular lattice even in the low concentration region. The activation energy is defined by the energy difference between a state where the soliton is pinned by a dopant and another state where the soliton locates at a mid point between neighboring dopants. It is shown that the activation energy decreases rapidly with increase of the dopant concentration. It is also found through introducing the effect of the intrachain electron-electron interactions that the soliton width is one of the essential factors in determining the activation energy. The threshold electric field for the depinning of the charged soliton from a dopant is also numerically investigated by dynamical simulations. As expected from the result...

In order to estimate the mobility of a charged soliton in polyacetylene, the dynamics of a soliton in an electric field is numerically investigated in the presence of a phenomenological damping of the lattice motion. Except for the damping term in the equation of motion for the lattice displacements, the dynamics is assumed to be determined by Su, Schrieffer and Heeger's model. It is shown that a Drude-type treatment of the soliton motion is applicable. The mobility is found to be µ=88.9 cm 2 /Volt·sec, and the conductivity is estimated to be σ=7.54×10 3 Ω -1 cm -1 on the assumption that the concentration of the solitons is one per 100 carbon atoms. The present result is compatible with experimental data in the non-magnetic regime.

The localization problem in a three dimensional tight binding system in the presence of a spatially random vector potential is investigated. The density of states exhibits tails of localized states that are induced by large regions in space where the vector potential is accidentally constant. In the center of the band the states are shown to be extended by investigating the system size dependence of the inverse participation number. The existence of a metal-insulator transition at a critical energy with non-vanishing density of states is demonstrated. The system size and energy dependent exponential decay length of the modulus of the Green's function satisfies a one-parameter scaling law. The value of the critical exponent is estimated, ν≈1, which is different from those found previously for the disordered tight binding (Anderson) model with and without homogeneous magnetic field.

The conductance of a two-dimensional system subject to a random magnetic field is calculated within the Landauer-Buttiker formalism. Although there is no potential disorder the random fluctuating magnetic field introduces a random vector potential. Electrons in bulk states are found to be strongly scattered by the fluctuating field. In contrast, edge states are found to be quite stable even if the fluctuation of the field is of the order of a uniform background field. In a fully random field, the U(1) model, clear universal conductance fluctuations (UCF) are observed. The magnitude of the fluctuations is compared with that in the time reversal symmetric Z 2 model.

The spacing distributions of unfolded energy levels in the lowest Landau band formed in disordered two-dimensional electron systems are numerically studied for different energy regions. By fitting the distribution function to an appropriate interpolation formula, the level repulsion parameter β is obtained as a function of the unfolded energy x . The functional forms of β( x ) for different system sizes are found to exhibit a certain scaling behavior. From this scaling property, it is possible to estimate the exponent of the localization length as a function of energy, which is found to be 2.4±0.2, consistent with previously reported values. Possible applications to other Anderson transitions are also discussed.

The Anderson transition in three dimensional layered systems is investigated in the presence of a quantizing magnetic field perpendicular to the layers. The density of states is calculated. By computing numerically the modulus of the one-body Green's function, the system size dependence of the exponential decay length of the transmission probability is calculated. It satisfies a one-parameter scaling law. The existence of a metal insulator transition is demonstrated. The mobility edge trajectory, and the critical parameters are obtained. The value of the critical exponent, ν=1.35 ±0.15, is found to be close to that reported previously for the Anderson model without magnetic field.

We study velocity damping of a moving soliton in polyacetylene by using numerical simulations within Su, Schrieffer and Heeger's model. The system is assumed to be at absolute zero of temperature and involves no impurity. In order to get an initial velocity the soliton is accelerated by applying an external electric field. The relaxation time of the soliton velocity has a minimum value when the initial velocity is around the sound velocity of this system. This is a result of the interaction between the moving soliton and the acoustic phonons emitted by the soliton itself. We find that the damping of the soliton velocity is caused by the energy transfer from the moving soliton to the acoustic phonons. The transfer is enhanced when the soliton is surrounded by the acoustic phonons.

The quantum Hall effect is studied in terms of the random matrix model, which has been recently proposed and proved to be quite powerful in investigating the electronic structures in strong magneti...

In numerical simulations on a moving charged soliton in polyacetylene, the behavior of the total, electric, lattice potential and lattice Kinetic energies are analyzed. In order to induce physically natural motion, the soliton is accelerated by a uniform electronic field as in the previous papers of this series. During the acceleration the electronic energy shows an almost monotone decrease while the lattice potential energy increases in an almost monotone way. These two energies have also an oscillating component which is related to the previously reported oscillation of the soliton width. On the contrary the total and the lattice kinetic energies show no oscillation. The relation between the total energy e tot and the soliton velocity υ is well fitted to a functional form e tot (υ)=e tot (0)-( M s υ m 2 /2) ln (1-υ 2 /υ m 2 ) where υ m is the saturation velocity of about three times the sound velocity and M s the soliton effective mass of four to five times the bare electron mass.

By numerical simulations treating the motion of a charged soliton in polyacetylene, we analyze the width of a moving soliton. The motion of the soliton is induced by applying an electric field as in the previous work. The soliton width is estimated by two methods; one is from the excess charge density profile and the other from the distortion in the lattice displacement. Both results indicate that the width decreases as the soliton velocity increases. Quantitatively, however, the amount of decrease is different in the two cases; the maximum change of the width is about 10% in the former case whereas it is 20% in the latter. Furthermore, the width obtained from the lattice distortion is found to oscillate as a function of time, consistently with the excitation of the amplitude mode around a soliton.

Electronic states in a two dimensional random system with a perpendicular strong magnetic field are analyzed by using a model Hamiltonian matrix which has random elements reflecting the characteristics of the random potential such as the correlation length. The random matrix model is suitable for numerical calculations and has advantages compared to preparing the random potential by distributing impurities particularly when discussing correlated potential. As an example, the fractal dimensionality of extended states in the lowest Landau band is considered. It is found that the extended states have multifractal properties. The results are compared with those obtained through the random potential model.

The dynamics of a charged soliton pinned by an impurity in the presence of an electric field is numerically studied by using Su-Schrieffer-Heeger's model for trans-polyacetylene. The field is introduced in terms of a time-dependent vector potential which is included in the Hamiltonian in the form of a phase of the transfer integral. The time-dependent Schrodinger equation for the electronic wave functions and the equation of motion for the lattice displacements are numerically integrated in a self-consistent way. When the field is lower than a certain threshold value, the soliton remains trapped and oscillates periodically around the impurity position. As the field is increased, the time period of the oscillation becomes longer. When the field exceeds the threshold value, the soliton is depinned and begins to move in one direction. The threshold field is estimated to be 10 5 -10 6 V/cm, which is in rough agreement with the value obtained in experiments on polyacetylene lightly doped with AsF 5 .

Real time dynamics of a moving charged soliton in polyacetylene is studied numerically by using Su-Schrieffer-Heeger's model. An electric field is introduced to the system through a time dependent vector potential which can be included into the Hamiltonian through the Peierls substitution of the phase factor to the transfer integral. Several interesting properties of the moving soliton are obtained, e.g. , the Brownian-like motion of the soliton due to the soliton-phonon interaction, the saturation of the soliton velocity, the saturation velocity being independent of the applied electric field strength, while the time needed to attain the saturation velocity is, roughly speaking, linearly dependent on the logarithm of the applied field.

The anisotropic fractal dimensionality of the extended electronic states is studied for a two dimensional disordered system subject to a strong perpendicular magnetic field, through the system size dependence of the inverse participation number (IPN) of the eigenstates. The system is assumed to satisfy the periodic boundary condition in the y -direction, and the Dirichlet boundary condition in the x -direction. The case with short ranged impurity potentials is considered. Two kinds of effective dimensionalities d x * and d y * are obtained from the width ( L x ) and the length ( L y ) dependences of IPN. For the bulk extended states, d x * and d y * are estimated as 0.76±0.05 and 0.77±0.09, respectively. Through the analysis of the energy dependence of d x * and d y * , we find the crossover between very anisotropic extended states (edge states) and almost isotropic extended states (bulk extended states).

Acoustic component of lattice displacement field is investigated for a soliton and a polaron in Su, Schrieffer and Heeger's Model for t -( CH ) x , using a perturbation method. Since this component has a small magnitude of the first order with respect to the ratio between the lattice constant and the soliton and/or polaron width, we take account of the higher order corrections to Takayama, Lin-Liu, and Maki's continuum model. The perturbation calculations determine corrections to electronic wave functions as well as the lattice displacement. Phase shifts of the wave functions are determined to give the correction of the soliton creation energy which turns out to be small. It is not straightforward to proceed to the second order calculations. The difficulty is associated with a new localized state below the valence band. Obtained results are in agreement with numerical simulation works.

The equilibrium and non-equilibrium current distributions are investigated in the quantum Hall regime by using the finite cylinder model with a gate barrier potential at the middle of the system. The energy for the edge states of the n th Landau subband, Δ E ν ≡e ν (θ)-( n +1/2)\hbarω_c, e_ν(θ) b e i n g t h e e i g e n e n e r g y o f t h e e d g e s t a t e s u n d e r t h e g e n e r a l i z e d p e r i o d i c b o u n d a r y c o n d i t i o n f o r t h e w a v e f u n c t i o n Ψ(x, y+L_y)=e^-iθΨ(x, y), i s i n t r o d u c e d t o c h a r a c t e r i z e t h e b e h a v i o r o f e d g e s t a t e s . W h e n ΔE_ν i s m u c h s m a l l e r t h a n t h e g a t e b a r r i e r p o t e n t i a l U_g, t h e e d g e s t a t e s a r e c o m p l e t e l y r e f l e c t e d , w h i l e t h e y a r e t r a n s p a r e n t w h e n ΔE_ν i s l a r g e r t h a n U_g. W h e n ΔE_ν\lesssimU_g, w e h a v e s t a t e s w h i c h h a v e s i g n i f i c a n t a m p l i t u d e b o t h i n t h e b u l k r e g i o n o f...

We reconsider the problem of “binding” of a pair of fractionally charged solitons in “1100” charge order backgrounds, using the variational Monte Carlo method based on the matrix-product state. In the “1100” charge order ground state of the one-dimensional extended Peierls–Hubbard model, the 2kF bond order wave (BOW) coexists with the 4kF BOW. We have found that the fractionally charged soliton with −e/2, −e being an electronic charge, is accompanied with not only a kink in the 2kF BOW but also a kink in the 4kF BOW, which was not allowed to exist in the dimer model used in the previous study. Calculating the total energy of a pair of fractionally charged solitons with the same charge as a function of the soliton–soliton distance, we have found that the soliton–soliton interaction is repulsive. Therefore the “binding” of the solitons does not occur and the elementary charge excitation from the “1100” charge order ground state is a fractionally charged soliton.

The dynamical properties of a phase soliton with a fractional charge \(\pm e/2\) in the bond-charge-density-wave (BCDW) background of one-dimensional quarter-filled electron–lattice systems are studied by numerical and semi-phenomenological methods using the Su–Schrieffer–Heeger (SSH) model. A special focus is on the time evolutions of the velocity and energy of the charged soliton subject to an electric field. Several interesting properties are obtained: the saturation of the soliton velocity, the divergence of the soliton energy and the propagation of condensed acoustic phonons. The saturation velocity, which is independent of the applied field strength, is less than the phason velocity. These properties are different from those of the sine-Gordon model which was believed to describe the commensurate BCDW systems. The velocity–energy relation is also different from that of the sine-Gordon soliton but is described by the same form as that of the acoustic polaron in the SSH model. These results suggest th...

The effect of time-dependent perturbations on transport phenomena in disordered systems has been studied in various context, such as the destruction of localization, the suppression of the interference effect, and the effect of temperature. In general, timedependent perturbations destroy the phase coherence. In fact, a numerical study of the dephasing effect by a time-dependent perturbation in two-dimensional disordered systems has been performed by Nakanishi et al. They have investigated the effect of the timedependent on-site potential oscillating with a frequency ω and demonstrated that the ω-dependent correction to the conductivity takes an universal form in the metallic regime. In the present work, we study a different type of time-dependent perturbation, namely the timedependent transfer-integrals. It is expected that this type of perturbation is more representative for the effect of temperature. We consider a two-dimensional disordered system with spin-orbit interaction, which belongs to the symplectic universality class. The system exhibits the Anderson transition even in two dimensions. We numerically evaluate the conductivity as a function of the frequency of the perturbation using the equation of motion method. In the metallic regime, it is examined whether the universal ω-dependent correction to the conductivity can be observed also for the present type of perturbation. For the critical regime, we find that the frequency dependence of the conductivity can be described by the one-parameter scaling. The model we adopt is the Ando model extend to the case where transfer-integrals oscillate with a frequency ω. The Hamiltonian is given by

In order to study the stability of the multimode Peierls (MMP) state in the θ-type lattice system, the stable state is studied using a modified square-lattice Su–Schrieffer–Heeger (SSH) model. Since the MMP state has been predicted for the 2D square-lattice system with the SSH type electron–lattice (e–l) interaction and with a half-filled electronic band, we study the lowest energy state in the system, mimicking the θ-type lattice structure, where the transfer integral in one of the diagonal directions is added. The MMP state has the lattice distortions not only with the nesting vector Q but also with many others parallel to Q , and there are many degenerate states in the case of the 2D square-lattice electronic system. It is found that the energy of the MMP state decreases by increasing the transfer integral in the diagonal direction, and the amount of the decrease depends on the pattern of the lattice distortions. Consequently, the degeneracy of the MMP state is removed. This phenomenon is peculiar to t...

It is shown that Berry's phase associated with the adiabatic change of local variables in the Hamiltonian can be used to characterize the multimode Peierls state, which has been proposed as a new type of the ground state of the two-dimensional(2D) systems with the electron-lattice interaction., Comment: 2 pages, 2 figures

The effect of anisotropy on phonon softening at finite temperatures in a two-dimensional (2D) electron–lattice system is studied numerically. On the basis of the Su–Schrieffer–Heeger model generalized to a 2D system with a half-filled electronic band on the square lattice, we introduce an anisotropy parameter γ in the electronic transfer integrals and the electron–lattice coupling constants in the x and y directions, t x and t y , and α x and α y , respectively, as t x / t y =α x /α y =(1+γ/2)/(1-γ/2). In the isotropic system, it has been shown that the phonon frequencies of transverse modes with the nesting vector Q =(π/ a ,π/ a ) ( a is the lattice constant) and with wave vectors parallel to Q , and that of the longitudinal mode with Q are all softened at the same Peierls transition temperature. In the presence of anisotropy, it is found that the mode with Q having its amplitude only in the strong-coupling direction is softened first as the temperature is lowered in the metallic phase while other modes ...

The effects of electron–electron (e–e) interactions on the two-dimensional (2D) Peierls state are discussed using the 2D Peierls–Hubbard model (2D PHM) in the limit of U ≫ t 0 , where U is the on-site e–e interaction parameter between up and down spin electrons and t 0 is the electron transfer integral between nearest neighbor sites. In this limit, the 2D PHM can be mapped onto the 2D spin-Peierls model (2D SPM) in which the effect of the lattice distortion is taken into account in the spin exchange interaction of the 2D Heisenberg model. As was demonstrated in a previous paper [J. Phys. Soc. Jpn. 69 (2000) 1769], the ground state of the 2D electron–lattice (e–l) system (the 2D 1/2-filled SSH model) is the so-called multimode Peierls state where the lattice distortion involves Fourier components with the nesting wave vector Q =(π/ a ,π/ a ), a being the lattice constant, and with those paralell to it. Using the exact diagonalization technique it is shown that the spin-Peierls transition with a mutlimode l...

The first order Raman scattering processes due to phonons are considered in a continuum model for t -(CH) x when there is a nonlinear excitation as a soliton or a polaron. The formulation by Ono and Ito is generalized to incorporate the amplitude mode formalism, proposed by Horovitz and later developed further by Mele and Hicks. The transition amplitudes are calculated for the two states, one with a soliton and the other with a polaron. Difference Raman spectra between these states and the perfectly dimerized state have three prominent dips due to bleaching of extended optical phonons. There is one small peak at the lower shift frequency side of each dip for the soliton. The intensity of the peak is about one tenth of the dip for the system with one soliton over 100 lattice sites. The difference spectrum between the states with and without a polaron has two or three small peaks at the lower shift frequency side. If these small peaks are observable, we should be able to determine whether the photoconductio...

Numerical analyses of the various states with nonlinear excitations in one-dimensional nearly one-third filled electron-phonon systems are performed by using Su-Schrieffer-Heeger's model originally proposed for polyacetylene. The states are determined by solving numerically the self-consistent equations satisfied by the lattice displacement and the electronic wave functions. Different solutions are obtained for different values of the excess electron number δ n = N e -2 N /3 with N the total number of the lattice sites and N e the total electron number; the cases with δ n =±1/3, ±2/3, ±1, ±4/3, ±5/3 and ±2 are considered. Through comparison of the energies for δ n =0 and δ n ≠0, the creation energies of various nonlinear excitations are calculated and shown to depend not only on the absolute value but also on the sign of the charge of the excitations.

Various fractionally charged states in nearly quarter-filled one-dimensional electron-phonon systems are numerically investigated for Su-Schrieffer-Heeger's model. Three types of solitons are obtained depending on the excess electron number δ n = N e - N /2 and spin s with N and N e the total numbers of lattice sites and electrons; they may be called phase solitons, charged amplitude solitons and a neutral amplitude soliton for (δ n , s ) = (±1/2, 0), (±1/2, 1/2) and (0, 1/2), respectively. The states for the other δ n 's are found to consist of proper combinations of the above solitons well separated from one another. Electronic levels corresponding to localized states associated with the solitons exist not only within the gap between the conduction and valence bands but also below the valence band. Creation energies and widths of the solitons are calculated for various coupling constants.

The dynamical conductivity due to phonons around a charged soliton is calculated, using Horovitz's three phonon band model for trans-polyacetylene. We find three weak i.r. modes in addition to the three intense ones which were known. The weak modes have their origins in the third localized mode which was found in the Takayama-Lin-Liu-Maki model. The vibration frequencies and relative optical intensities are calculated to give a good agreement with a recent photo-induced absorption experiment. This strongly indicates that there actually is the third localized mode.

The ground states of nearly quarter-filled one-dimensional electron-phonon systems are numerically investigated by using Su-Schrieffer-Heeger's model and a Frohlich-type model. When the electron filling factor is very near to but different from 1/4, the ground state is the soliton lattice. The structure factors of the CDW's in the soliton lattice states are calculated and their square moduli are found to have several peaks, the main one corresponding to the 2 k F mode and others being explained as its higher harmonics. The behaviour of the higher harmonics as functions of the filling factor and the electron-phonon coupling constant is studied and the model dependence is discussed. The distribution of the field gradient at off-chain sites due to the CDW is considered in connection with the quadrupole shift of the recent NMR measurement in Rb 0.3 MoO 3 .

Using the linear response theory, the Hall current distributions in the quantum Hall regime are calculated numerically for two dimensional systems of non- interacting electrons, confined to an area of a finite width. At the low filling factor region, the local current density is enhanced around the localized states. Such current density is much stronger than that in the impurity-free case, although the Hall conductivity, which is obtained by integrating the current density over the system, is much smaller than that in the impurity-free case. With the increase of the filling factor, the local enhancement spreads over the whole system. The current, however, flows uniformly almost in the same way as in the impurity-free case, when the Fermi level is in the bulk band gap, i.e., in the edge state region. It shows that the strong spatial fluctuations of the Hall current are locally compensated.

Effects of short ranged bond-type impurities in polyacetylene are investigated numerically, using a generalized Su-Schrieffer-Heeger's model. Soliton formation and pinning by impurities are studied. Infrared activities are analyzed with relevant phonons. Similarities and differences from site-type impurities are pointed out. It is striking to find that there may be two types of exotic “solitons”. One is a kink in the bond alternation configuration which carries neither charge nor spin. The other is a soliton with a fractional charge and a fractional spin, which appears in a situation where the polaron solution is stable in the absence of impurities. The stabilization of the exotic solitons is associated with the fact that the particle-hole symmetry still holds in the presence of the bond-type impurities.

The two-dimensional electronic states and the quantum Hall conductivity on the cylinder surface with finite width are analyzed numerically. A parameter θ characterizing the generalized boundary condition in the y -direction is introduced. For each θ the sum of the x -component of the center of mass x νν (θ) over the states ν involved in the lowest Landau subband is shown to satisfy a sum rule and to be stable against randomness. Through different interpretations of θ, the numerical data for x νν (θ) can be used to discuss the Hall conductivity in two different cases. One is an infinitely long strip system with finite periodicity L y in the y -direction; θ is the generalized Bloch momentum in this case. The other is a real cylinder with the Aharonov-Bohm flux penetrating the cylinder along the axis, where θ represents the flux. In the latter case we can discuss the fluctuation of the Hall conductivity with respect to the flux.

A self-consistent treatment of the Anderson localization by Vollhardt and Wolfle [Phys. Rev. B22 (1980) 4666] is extended to systems without time reversal symmetry by use of a diagrammatical method. In this paper magnetoresistance in two dimension is calculated for the case where the field effect can be treated quasiclassically. It is found that the magnetic field enhances the conductivity, though the system remains insulator in the absence of inelastic scatterings. The present investigation indicates that the one-parameter scaling theory cannot describe systems without the time reversal symmetry.