Kazuki Kanki

Hall effect and resistivity in high- T c superconductors: The conserving approximation

The Hall coefficient (R_H) of high-Tc cuprates in the normal state shows the striking non-Fermi liquid behavior: R_H follows a Curie-Weiss type temperature dependence, and |R_H|>>1/|ne| at low temperatures in the under-doped compounds. Moreover, R_H is positive for hole-doped compounds and is negative for electron-doped ones, although each of them has a similar hole-like Fermi surface. In this paper, we give the explanation of this long-standing problem from the standpoint of the nearly antiferromagnetic (AF) Fermi liquid. We consider seriously the vertex corrections for the current which are indispensable to satisfy the conservation laws, which are violated within the conventional Boltzmann transport approximation. The obtained total current J_k takes an enhanced value and is no more perpendicular to the Fermi surface due to the strong AF fluctuations. By virtue of this mechanism, the anomalous behavior of R_H in high-Tc cuprates is neutrally explained. We find that both the temperature and the (electron, or hole) doping dependences of R_H in high-T_c cuprates are reproduced well by numerical calculations based on the fluctuation-exchange (FLEX) approximation, applied to the single-band Hubbard model. We also discuss the temperature dependence of R_H in other nearly AF metals, e.g., V_2O_3, kappa-BEDT-TTF organic superconductors, and heavy fermion systems close to the AF phase boundary.

Theory of Hall Effect and Electrical Transport in High-Tc Cuprates: Effects of Antiferromagnetic Spin Fluctuations

This is a reply to the comment by O. Narikiyo (cond-mat/0012505) on our paper J. Phys. Soc. Jpn. {\bf 68}, (1999) 1614. We point out mistakes about his arguments, and we show that our analysis is compatible with the established Fermi liquid theory, so the obtained result is justified.

Non-divergent representation of a non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas

We propose a non-singular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from a divergence in the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also find that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmanns collision operator in a one-dimensional quantum Lorentz gas in the weak coupling approximation.

Microcanonical Simulation on the First Order Phase Transition of the XY Antiferromagnet on the Stacked Triangular Lattice

The nature of the phase transition of antiferromagnetic vector spin models on the stacked triangular lattice (STA) has been an issue of debate. In order to account for the coplanar non-collinear order in the ground state one needs two vector fields in a Ginzburg–Landau–Wilson Hamiltonian. The field theoretical renormalization group analysis shows that there is a threshold value Nc for the number of spin component N, and for N > Nc the transition is of second order belonging to a new ‘‘chiral’’ universality class, while for N < Nc the chiral fixed points ceases to exist in the real parameter space and the transition is of first order. According to a recent estimate based on a Monte Carlo renormalization group study, 3 < Nc < 8. 2) Therefore a first order transition is expected for the physically relevant XY and Heisenberg spin systems. The phase transition of the XY or Heisenberg STA is of very weak first order and is said to be of ‘‘almost second order’’ with large (but finite) correlation lengths due to a slow velocity region in the renormalization group flow. Monte Carlo simulations on systems with moderate size appear to show second order transitions with pseudo-critical behavior. Strong first order transitions were found for variants of the model on which a constraint of local rigidity was imposed which still belong to the same universality class as the original STA. Recently the first order nature of the phase transition in XY antiferromagnet on the stacked triangular lattice (XY-STA model) was demonstrated directly by canonical Monte Carlo simulations on large systems, in which the energy probability distribution shows a double peak structure near the transition temperature. The purpose of the present paper is to show the first order transition of the XY-STA model by applying a novel technique: a microcanonical method. To distinguish a weak first order transition from a second order one can be difficult, especially if the energy jump, the characteristic of a first order transition, is tiny as is the case for frustrated spin systems. Limited to accessible system sizes one may analyze the different power law behavior of the maximum of specific heat or susceptibility, with an exponent equal to three for a first order transition and less than three for a second order transition. The obvious solution to increase the size of the system runs into difficulties for the canonical simulations (at constant temperature) because of the growing autocorrelation times. Also the histogram method is then restricted to smaller intervals of temperature shrinking as the inverse power of the system size. In this short note we will show that the microcanonical simulations (at constant energy) can help to solve the problems encountered in canonical simulations. The method has been tested for strong first order transitions and also for second order ones. Before presenting the method and the results we have to describe the model we study. It is given by the usual Hamiltonian

Microscopic description of quantum Lorentz gas and extension of the Boltzmann equation to entire space-time scale

Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed.

Band structure and accumulation point in the spectrum of quantum collision operator in a one-dimensional molecular chain

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.

Accumulation Point in the Spectrum of the Collision Operator for a One-Dimensional Polaron System

The spectrum of Hamiltonians has been well investigated in quantum mechanics, since it plays an essential role in analyzing dynamical properties of conservative systems. For example, we know there are Hamiltonians that have a discrete spectrum, or a continuous spectrum, or a band structure, and so on, each of which leads to a characteristic behavior of the system. However, our knowledge on the spectrum of the collision operators in kinetic equations in dissipative systems still remains in a poor stage in spite of the fact that their spectral properties play a fundamental role in non-equilibrium statistical physics. In this paper we give an example of quantum systems that has a rich structure of the spectrum, such as an accumulation point and band structures, in the collision operator for a momentum relaxation process. We consider a one-dimensional (1D) polaron system in which a quantum particle is weakly interacting with a thermal reservoir consisting of an acoustic phonon field. Similar systems have been studied in different contexts, see e.g. Refs. 1)–5). As a consequence of a constraint in 1D due to the resonance condition, the momenta of the particle related successively through the collision operator form a subset separated from other momenta. Momentum relaxation occurs among those momenta in such a subset independently of other momenta. In this case the collision operator is represented by a tridiagonal matrix for each such subset of momenta. Taking advantage of the tridiagonal nature of the matrix, we found a solution of the eigenvalue problem of the collision operator in terms of continued fractions. This paper is organized as follows. In §2 we introduce a model of a quantum particle weakly coupled with a phonon field, and present a kinetic equation for the

Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

We have theoretically investigated the time-symmetry breaking phase transition process for two discrete states coupled with a one-dimensional continuum by solving the nonlinear eigenvalue prob- lem for the effective Hamiltonian associated with the discrete spectrum. We obtain the effective Hamiltonian with use of the Feshbach-Brillouin-Wigner projection method. Strong energy depen- dence of the self-energy appearing in the effective Hamiltonian plays a key role in the time-symmetry breaking phase transition: as a result of competition in the decay process between the Van Hove singularity and the Fano resonance, the phase transition becomes a higher-order transition when both the two discrete states are located near the continuum threshold.

Kinetic equations for classical and quantum Brownian particles and eigenfunction expansions as generalized functions

We study momentum relaxation processes of a classical and a quantum Brownian particle by considering the eigenvalue problem of the collision operators in the kinetic equations. The collision operators are anti-Hermitian with an appropriate inner product defined by an integral with a weight factor given by the inverse of the equilibrium distribution function. Owing to the weight factor, the norm of a momentum distribution function is infinite, if the distribution is characterized by a temperature higher than a threshold temperature determined by the environmental temperature. Although the eigenfunction expansion of a given distribution function with an infinite norm does not converge to a function in the Hilbert space, it has a legitimate meaning as a generalized function and defines a linear functional. We introduce an H-function through the norm which directly reflect the spectral properties of the collision operators. When the norm of the momentum distribution function diverges, the H-function reduces t...

Hofstadter's Butterfly Type of Singular Spectrum of a Collision Operator for a Model of Molecular Chains

We study the eigenvalue problem of a quantum collision operator in an irreversible kinetic equation for momentum relaxation processes in a molecular chain where a vibron is weakly coupled to a thermal phonon bath. As a consequence of a constraint due to the resonance condition represented by the conservation of the unperturbed energy, the momenta of the vibron associated through the collision operator successively with a representative momentum form a subset separated from other momenta. The number of possible values of momenta in each subset and the spectrum of the collision operator crucially depend on the rationality or irrationality of the parameter R ≡ π −1 sin −1 B ,w hereB denotes the ratio of the phonon bandwidth to the vibron bandwidth. The plot of the spectrum as a function of R shows features reminiscent of Hofstadter’s butterfly.