Lee, Rice and Anderson, in their monumental paper, have proved the existence of a collective mode describing the coupled motion of electron density and phonons in one-dimensional incommensurate charge density wave (CDW) in the Peierls state. This mod
e, which represents the coherent sliding motion of electrons and lattice distortions and affects low energy transport properties, is described by the phase of the complex order parameter of the Peierls condensate, leading to Frohlich superconductivity in pure systems. Once spatial disorder is present, however, phason is pinned and system is transformed into an insulating ground state: a dramatic change. Since phason can be considered as an ultimate of phonon drag effect, it is of interest to see its effects on thermoelectricity, which has been studied in the present paper based linear response theory of Kubo and Luttinger. The result indicates that a large absolute value of Seebeck coefficient proportional to the square root of resistivity is expected at low temperatures k_B T/Delta <<1 (Delta: Peierls gap) with opposite sign to the electronic contributions in the absence of Peierls gap.
Analytical solutions for the surface state (SS) of an extended Wolff Hamiltonian, which is a common Hamiltonian for strongly spin-orbit coupled systems, are obtained both for semi-infinite and finite-thickness boundary conditions. For the semi-infini
te system, there are three types of SS solutions: (I-a) linearly crossing SSs in the direct bulk band gap, (I-b) SSs with linear dispersions entering the bulk conduction or valence bands away from the band edge, and (II) SSs with nearly flat dispersions entering the bulk state at the band edge. For the finite-thickness system, a gap opens in the SS of solution I-a. Numerical solutions for the SS are also obtained based on the tight-binding model of Liu and Allen [Phys. Rev. B, 52, 1566 (1995)] for Bi$_{1-x}$Sb$_x$ ($0le x le 0.1$). A perfect correspondence between the analytic and numerical solutions is obtained around the $bar{M}$ point including their thickness dependence. This is the first time that the character of the SS numerically obtained is identified with the help of analytical solutions. The size of the gap for I-a SS can be larger than that of bulk band gap even for a thick films ($lesssim 200$ bilayers $simeq 80$ nm) of pure bismuth. Consequently, in such a film of Bi$_{1-x}$Sb$_x$, there is no apparent change in the SSs through the band inversion at $xsimeq 0.04$, even though the nature of the SS is changed from solution I-a to I-b. Based on our theoretical results, the experimental results on the SS of Bi$_{1-x}$Sb$_x$ ($0le x lesssim 0.1$) are discussed.
We derive an exact formula of orbital susceptibility expressed in terms of Bloch wave functions, starting from the exact one-line formula by Fukuyama in terms of Greens functions. The obtained formula contains four contributions: (1) Landau-Peierls s
usceptibility, (2) interband contribution, (3) Fermi surface contribution, and (4) contribution from occupied states. Except for the Landau-Peierls susceptibility, the other three contributions involve the crystal-momentum derivatives of Bloch wave functions. Physical meaning of each term is clarified. The present formula is simplified compared with those obtained previously by Hebborn et al. Based on the formula, it is seen first of all that diamagnetism from core electrons and Van Vleck susceptibility are the only contributions in the atomic limit. The band effects are then studied in terms of linear combination of atomic orbital treating overlap integrals between atomic orbitals as a perturbation and the itinerant feature of Bloch electrons in solids are clarified systematically for the first time.
Based on the Phase Hamiltonian, two types of solitons are found to exist in the crossover region between band insulator and Mott insulator in one-dimension. Both of these solitons have fractional charges but with different spins, zero and 1/2, respec
tively. The results are in accord with the experimental results by Kanoda et al. for TTF-Chloranil under pressure.
Bismuth crystal is known for its remarkable properties resulting from particular electronic states, e. g., the Shubnikov-de Haas effect and the de Haas-van Alphen effect. Above all, the large diamagnetism of bismuth had been a long-standing puzzle so
on after the establishment of quantum mechanics, which had been resolved eventually in 1970 based on the effective Hamiltonian derived by Wolff as due to the interband effects of a magnetic field in the presence of a large spin-orbit interaction. This Hamiltonian is essentially the same as the Dirac Hamiltonian, but with spatial anisotropy and an effective velocity much smaller than the light velocity. This paper reviews recent progress in the theoretical understanding of transport and optical properties, such as the weak-field Hall effect together with the spin Hall effect, and ac conductivity, of a system described by the Wolff Hamiltonian and its isotropic version with a special interest of exploring possible relationship with orbital magnetism. It is shown that there exist a fundamental relationship between spin Hall conductivity and orbital susceptibility in the insulating state on one hand, and the possibility of fully spin-polarized electric current in magneto-optics. Experimental tests of these interesting features have been proposed.
Spin-Hall conductivity $sigma_{{rm s}xy}$ and orbital susceptibility $chi$ are investigated for the anisotropic Wolff Hamiltonian, which is an effective Hamiltonian common to Dirac electrons in solids. It is found that, both for $sigma_{{rm s}xy}$ an
d $chi$, the effect of anisotropy appears only in the prefactors, which is given as the Gaussian curvature of the energy dispersion, and their functional forms are equivalent to those of the isotropic Wolff Hamiltonian. As a result, it is revealed that the relationship between the spin Hall conductivity and the orbital susceptibility in the insulating state, $sigma_{{rm s}xy}=(3mc^2/hbar e)chi$, which was firstly derived for the isotropic Wolff Hamiltonian, is also valid for the anisotropic Wolff Hamiltonian. Based on this theoretical finding, the magnitude of spin-Hall conductivity is estimated for bismuth and its alloys with antimony by that of orbital susceptibility, which has good correspondence between theory and experiments. The magnitude of spin-Hall conductivity turns out to be as large as $esigma_{{rm s}xy} sim 10^4 {Omega}^{-1}{rm cm}^{-1}$, which is about 100 times larger than that of Pt.
Spin-Hall conductivity (SHC) of fully relativistic (4x4 matrix) Dirac electrons is studied based on the Kubo formula aiming at possible application to bismuth and bismuth-antimony alloys. It is found that there are two distinct contributions to SHC,
one only from the states near the Fermi energy and the other from all the occupied states. The latter remains even in the insulating state, i.e., when the chemical potential lies in the band-gap, and turns to have the same dependences on the chemical potential as the orbital susceptibility (diamagnetism), a surprising fact. These results are applied to bismuth-antimony alloys and the doping dependence of the SHC is proposed.
It is shown that the energy $(varepsilon)$ and momentum $(k)$ dependences of the electron self-energy function $ Sigma (k, varepsilon + i0) equiv Sigma^{R}(k, varepsilon) $ are, $ {rm Im} Sigma^{R} (k, varepsilon) = -avarepsilon^{2}|varepsilon - xi_{
k}|^{- gamma (k)} $ where $a$ is some constant, $xi_{k} = varepsilon(k)-mu, varepsilon(k)$ being the band energy, and the critical exponent $ gamma(k) $, which depends on the curvature of the Fermi surface at $ k $, satisfies, $ 0 leq gamma(k) leq 1 $. This leads to a new type of electron liquid, which is the Fermi liquid in the limit of $ varepsilon, xi_{k} rightarrow 0 $ but for $ xi_{k} eq 0 $ has a split one-particle spectra as in the Tomonaga-Luttinger liquid.
