Analytic extension of the coherent potential approximation to clusters

We put forward the homomorphic cluster coherent potential approximation (HCPA) for substitutionally disordered systems as one possible way of avoiding the breakdown of analyticity in average Green's functions even when effects of clusters are properly taken into account. We assert that a cluster CPA yields an analytic physical solution if a one-particle total Hamiltonian is partitioned into the sum of homomorphic single-cluster Hamiltonians and the CPA condition is applied to one of these homomorphic sub-Hamiltonians.     

Analyticity of the coherent potential approximation

A general proof is presented that the coherent potential approximation does not violate analyticity requirements.     

On the homomorphic cluster CPA

Among the various extensions of the CPA, the homomorphic cluster CPA (HCPA) was presented recently as the only approximation (i) having the correct analytical properties, (ii) taking multiple scattering into account and (iii) preserving the symmetry of the original lattice. Though very appealing mathematically, we show in this paper that it does not give the right position for the impurity states in the dilute limit with non zero diagonal disorder. Consequently, even if HCPA satisfies points (i) to (iii) and even if it gives the correct dilute limit for purely off-diagonal disorder, it is to be rejected as a method to study systems with non negligible diagonal disorder.     

Dissipative Bohmian mechanics within the Caldirola–Kanai framework: A trajectory analysis of wave-packet dynamics in viscid media

Classical viscid media are quite common in our everyday life. However, we are not used to find such media in quantum mechanics, and much less to analyze their effects on the dynamics of quantum systems. In this regard, the Caldirola–Kanai time-dependent Hamiltonian constitutes an appealing model, accounting for friction without including environmental fluctuations (as it happens, for example, with quantum Brownian motion). Here, a Bohmian analysis of the associated friction dynamics is provided in order to understand how a hypothetical, purely quantum viscid medium would act on a wave packet from a (quantum) hydrodynamic viewpoint. To this purpose, a series of paradigmatic contexts have been chosen, such as the free particle, the motion under the action of a linear potential, the harmonic oscillator, or the superposition of two coherent wave packets. Apart from their analyticity, these examples illustrate interesting emerging behaviors, such as localization by “quantum freezing” or a particular type of quantum–classical correspondence. The reliability of the results analytically determined has been checked by means of numerical simulations, which has served to investigate other problems lacking of such analyticity (e.g., the coherent superpositions).     

Electronic density of states for the linear random binary alloy in the three-site coherent potential approximation

Morita and Chen's scheme of generalizing the coherent potential approximation in which the diagonal and off-diagonal randomness are included, is applied to the calculation of the electronic density of states for the linear random binary alloy. The result obtained in the three-site coherent potential approximation is reported.     

Electronic density of states for the linear random binary alloy in the two-site coherent potential approximation

Morita and Chen's scheme of generalizing the coherent potential approximation, in which the diagonal and off-diagonal randomness are included, is applied to the calculation of the electronic density of states for the linear random binary alloy. The result obtained in the two-site coherent potential approximation is reported.     

色散关系的简单证明

在此短文中,我们给色散关系一个简单的但不严格的证明。证明的方法为将因果振幅对中间态展开,分别研究能量分母及相应的分子的解析性。能量分母的解析性是较显然的。至于分子的解析性的证明,我们先研究μ²换为—p²时的相应量(μ为介子质量,p为核子在Breit坐标系中的动量,介子核子散射为我们所考虑的具体对象),研究其解析性,通过一个变数的变换而达到我们证明的目的。在此方法中,p²可以大至M²—μ²,M为核子质量。我们也考虑了在位场散射中相位移η(k)的解析开拓问题,证明了如果位能在r→∞处形如e^-αr(a>0),则η(k)可以开拓至|Imk|<1/2α的区域。     

Analytic properties of the Ishida-Yonezawa approximation for liquid and amorphous metals

We generally examine the analytic properties of the Ishida-Yonezawa approximation for the tight binding single s-band model. A sufficient condition is derived on which a solution satisfies the physical conditions; (1) sum rule, (2) reality, (3) definiteness, (4) analyticity, (5) boundary condition and (6) uniqueness.     

微扰展开的解析性

本文指出了Nambu对于微扰展开逐项的解析性的工作有可以简化处,利用简化后的公式讨论了四点方形费曼图的解析性在增加内线后有否改变,当内线都与方形一边平行而成一梯形时,解析性没有基本的改变。当我们引入竖直的与水平的内线时,解析性有了改变,但依然与Mandelstamm在双重色散关系中所引用的假定一致。此外,本文证明了在某一个非物理区中,一个圆形的解析性不因为更多内点内线的引入而有所改变。     

Vibrational properties of bond-diluted triangular lattices

Vibrational properties of two-dimensional (2D) triangular elastic bond percolation systems with scalar, central, and bond-bending forces are studied by using the HCPA, the recursion method and the replica trick. The densities of states calculated by means of the HCPA agree qualtatively well with corresponding results of the recursion method. The low-frequency behaviour of the density of states can be interpreted in terms of a phonon-fracton crossover. The dynamic structure factor calculated with the aid of the HCPA for the scalar model exhibits for any wave vector only a single peak, contrary to threedimensional systems.     

Berechnung der Zustandsdichte ungeordneter Legierungen mit starker Elektronenkorrelation

Calculation of the Density of States of Disordered Alloys with Strong Electron Correlation A method is derived, which is a generalization of the coherent potential approximation. Herein the electron correlation is considered using a simple approximation (alloy analogy). The equation for the coherent potential can be reduced to a real equation of 5-th order. The main advantage of the method is the exclusively simple possibility for numerical analysis.     

An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.     

Excitonic polaron in the molecular crystal model: Nonlocal dynamic coherent-potential approximation

A nonlocal dynamic coherent-potential approximation is formulated as a further development of the dynamic coherent-potential method. The nonlocal dynamic coherent-potential approximation is an efficient method of determining the one-exciton Green’s function in a model with the Hamiltonian in the strong-coupling approximation, where a spectrum of optical phonons is assumed, and the exciton-phonon interaction operator is linear or quadratic in the phonon operators. A system of recursion equations is derived, from which the coherent potential is found as a function of the energy E and the wave vector k. An analytical expression is derived for the one-exciton Green’s function in the case of narrow (in comparison with the phonon energy) exciton bands and exciton-phonon interaction linear in the phonon operators. For broader exciton bands and more complex exciton-phonon interaction the system of equations determining the coherent potential represents a recursion algorithm, which can be effectively implemented by numerical means. Fiz. Tverd. Tela (St. Petersburg) 39, 1560–1563 (September 1997)     

ELECTRON STATES IN SIMPLE METALLIC COMPOSITIONALLY MODULATED STRUCTURE

We present a generalization of the coherent potential approximation (CPA) in order to make it applicable to simple metallic compositionally umdulated structure. The medium Green's function and the coherent potential in the CPA are calculated by using the infinite order perturbation theory and the CPA self-consistent equation.     

Relativistic analysis of four-body final states

The constraints of unitarity and analyticity on four-body final states are studied. It is shown that unitarity alone forces the amplitudes to be coherent and have singular behaviour. The implementation of unitarity with total energy analyticity yields a set of relativistic linear integral equations for the four-body amplitude. This is the minimal set consistent with quantum mechanics and also is the full dynamical set of equations with two-body separable interactions. These equations will provide important ingredients for the phenomenological analysis of four-body final states using the isobar model.     

Methods for analysing transport in disordered systems with long-range traps

The continuous time random walk (CTRW) methods and the homomorphic cluster coherent potential approximation (HCCPA) are known to be useful approximate methods for the analysis of transport between localised states in ordinary disordered systems. In order to examine their applicability to disordered systems in the presence of long-range traps, calculations of the trapping probability were performed by these methods for ordered systems containing such traps, and the results compared with those obtained by the usual coherent potential approximation. It is found that the HCCPA method is totally unreliable for such systems, and the reasons for this are discussed. The CTRW methods are found to be unreliable for quantitative results, but the results that they give have the correct qualitative behaviour.     

Diagrammatic approach to the off-diagonal randomness in the coherent potential approximation

The off-diagonal randomness in the coherent potential approximation (CPA) is self-consistently treated by extending Nickel and Krumhansl's diagram-resummation technique CPA(2). New diagrams are introduced to represent the off-diagonal potentials and multiple occupancy correction is made in a completely self-consistent way.     

Chemical Potential Dependence of Vertices

Based on the rainbow-ladder approximation of the Dyson-Schwinger equations and the assumption of the analyticity of the fermion-boson vertex in the neighborhood of zero chemical potential (μ=0) and neglecting the μ-dependence of the dressed gluon propagator, we apply the method in [Phys. Rev. C 71 (2005) 015205] of studying the dressed quark propagator at finite chemical potential to prove that the general fermion-boson vertex at finite μ can also be obtained from the one at μ=0 by a simple shift of variables. Using this result we extend the results of [Phys. Lett. B 420 (1998) 267] to the situation of finite chemical potential and show that under the approximations we have taken, the Gell-Mann-Oakes-Renner relation also holds at finite chemical potential.     

s-wave bound and scattering state wave functions for a velocity-dependent Kisslinger potential

A relation linking the normalized s-wave scattering and the corresponding bound state wave functions at bound state poles is derived. This is done in the case of a non-local, velocity-dependent Kisslinger potential. Using formal scattering theory, we present two analytical proofs of the validity of the theorem. The first tackles the non-local potential directly, while the other transforms the potential to an equivalent local but energy-dependent one. The theorem is tested both analytically and numerically by solving the Schr?dinger equation exactly for the scattering and bound state wave functions when the Kisslinger potential has the form of a square well. A first order approximation to the deviation from the theorem away from bound state poles is obtained analytically. Furthermore, a proof of the analyticity of the Jost solutions in the presence of a non-local potential term is also given. Received: 3 March 2001 / Accepted: 9 June 2001     

Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface

We present a rigorous discussion of the analyticity properties of the beta function and of the effective potential for the theory of the ground state of a one dimensional system of many spinless fermions. We show that their analyticity domain as a function of the running couplings is a polydisk with positive radius bounded below, uniformly in, all the cut offs (infrared and ultraviolet) necessary to give a meaning to the formal Schwinger functions. We also prove the vanishing of the scale independent part of the beta function showing that this implies the analyticity of the effective potential and of the Schwinger functions in terms of the bare coupling. Finally we show that the pair Schwinger function has an anomalous long distance behaviour.