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Journal of the Physical Society of Japan

Homomorphic Cluster Coherent Potential Approximation for Systems with Site-Diagonal and/or Off-Diagonal Randomness

Fumiko Yonezawa, Takashi Odagaki

Published Aug 15, 1979• 15 citations
Publisher

Abstract

We apply the homomorphic cluster coherent potential approximation (HCPA) to systems with site-diagonal and/or off-diagonal randomness. We show numerically that the average Green's function obtained on the basis of the HCPA is analytic off the real axis on the complex energy plane. We also show that the HCPA reproduces the ordinary single-site CPA and the molecular CPA when systems include only diagonal disorder. The HCPA is advantageous in that it can deal with both site-diagonal and off-diagonal disorder in a unified manner. We also present a numerical result for a disordered chain by taking both kinds of disorder into account. The density of states calculated in the HCPA is in good agreement with the result of a computer simulation.

DOI
10.1143/jpsj.47.388

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We apply the homomorphic cluster coherent potential approximation (HCPA) to systems with site-diagonal and/or off-diagonal randomness. We show numerically that the average Green's function obtained on the basis of the HCPA is analytic off the real axis on the complex energy plane. We also show that the HCPA reproduces the ordinary single-site CPA and the molecular CPA when systems include only diagonal disorder. The HCPA is advantageous in that it can deal with both site-diagonal and off-diagonal disorder in a unified manner. We also present a numerical result for a disordered chain by taking both kinds of disorder into account. The density of states calculated in the HCPA is in good agreement with the result of a computer simulation.
An extension of the usual diagrammatic coherent-potential approximation is considered to account for the presence of disorder both in off-diagonal and inhomogeneous terms. The method is applied to the quenched site-disordered Ising model (S= 1 / 2 ) above the transition, where the fundamental equation for the correlation function is a generalized random-phase approximation. The results for the initial slope of the critical temperature and the critical concentration are presented for the simple cubic lattice with nearest-neighbour interactions, and they are in reasonable agreement with the values derived by other methods. The differences are studied and a discussion is presented to show that, near the critical concentration, terms not considered in the coherent-potential approximation are as important as the terms included. The application of this method to other models is also briefly discussed and, in the particular case of the Heisenberg model, it is shown that the method reproduces the basic results derived by Harris et al. (1974) using an effective-medium approach.
The coherent potential approximation (CPA) has recently been applied to disordered s-wave, p-wave and d-wave superconductors. Here we show that for systems with hypercubic lattices and generalized local s-wave, d-wave, extended s-wave or p-wave superconductivity the exact average Green function is equal to the CPA Green function in the limit where the number of dimensions goes to infinity. As in the case of normal systems, the $\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}$ limit is regularized by demanding that the second moment of the quasiparticle density of states be finite. Surprisingly, this condition leads to the scalings tij=t∗/d{t}_{\mathrm{ij}}{=t}^{*}/\sqrt{d}tij​=t∗/d​ and ${\ensuremath{\Delta}}_{\mathrm{ij}}={\ensuremath{\Delta}}^{*}/\sqrt{d}$ for the hopping integral and the nonlocal pairing potential, respectively, but does not require scaling of the effective electron-electron interaction constant, Uij{U}_{\mathrm{ij}}Uij​ even if i and j refer to different sites.
We investigate the spectral function of Bloch states in a one-dimensional tight-binding non-interacting chain with two different models of static correlated disorder, at zero temperature. We report numerical calculations of the single-particle spectral function based on the Kernel polynomial method, which has an [Formula: see text] computational complexity. These results are then confirmed by analytical calculations, where precise conditions were obtained for the appearance of a classical limit in a single-band lattice system. Spatial correlations in the disordered potential give rise to non-perturbative spectral functions shaped as the probability distribution of the random on-site energies, even at low disorder strengths. In the case of disordered potentials with an algebraic power-spectrum, [Formula: see text] [Formula: see text], we show that the spectral function is not self-averaging for [Formula: see text].
This work reports a theoretical framework that combines the auxiliary coherent potential approximation (ACPA-DMFT) with dynamical mean-field theory to study strongly correlated and disordered electronic systems with both diagonal and off-diagonal disorders. In this method, by introducing an auxiliary coupling space with extended local degree of freedom,the diagonal and off-diagonal disorders are treated in a unified and self-consistent framework of coherent potential approximation, within which the dynamical mean-field theory is naturally combined to handle the strongly correlated Anderson-Hubbard model. By using this approach, we compute matsubara Green's functions for a simple cubic lattice at finite temperatures and derive impurity spectral functions through the maximum entropy method. Our results reveal the critical influence of off-diagonal disorder on Mott-type metal-insulator transitions. Specifically, a reentrant phenomenon is identified, where the system transitions between insulating and metallic states under varying interaction strengths. The ACPA-DMFT method provides an efficient and robust computational method for exploring the intricate interplay of disorder and strong correlations.
The dynamics of a two-component Ising model in a transverse field (IMTF) with randomness during both in the transverse field and the Ising coupling is treated within the single-site coherent potential approximation (CPA). The authors deal primarily with the polarized phase; they replace the fluctuating longitudinal component of the molecular field by two effective parameters, one for each species, which are determined self-consistently. The CPA spectra of the collective spin excitations are found analytically in the dilute case using a Hubbard model density for the Ising interaction, and are also calculated numerically for the general two-component IMTF. It is shown that the static susceptibility cannot diverge within the single site CPA. Characterizing the phase transition by the softening of the lowest mode in the spectrum. The stability conditions for the polarized phase at T=O are then determined.
The previous chapter is devoted to the formulation of DMFT and applications that are exact in the limit of infinite dimensions; however, in finite dimensions this is only a single-site approximation. The present chapter is devoted to clusters and other methods to treat correlation between sites. The equations are derived in a unified way, applicable to a single site or a cluster. For a cluster with more than one site there are various ways to choose the boundary conditions and the embedding procedure, and special care must be taken to satisfy causality and translation invariance. A few selected applications illustrate different techniques and results for Hubbard-type models. The essential features of DMFT are brought out in the previous chapter in the simplest form: the single-site approximation in which correlation between sites can be neglected. In this case the self-energy due to interactions can be calculated using an auxiliary system of a single site embedded in an effective medium that represents the surrounding crystal. The resulting many-body problem is equivalent to an Anderson impurity model that must be solved self-consistently. The results are exact for infinite dimensions d → ∞; however, in any finite dimension, i.e., real systems, this is just a mean-field approximation analogous to the Weiss mean field in magnetic systems or the CPA for disordered alloys. There are, however, many important properties and phenomena that require us to go beyond mean-field approximations. For example, basic questions concerning the nature of a metal–insulator transition can be established only by quantitative assessment of the correlations between different sites. Is a single site sufficient (the central idea in the arguments by Mott discussed in Sec. 3.2) or is correlation between sites (for example, an ordered antiferromagnetic state) essential for interactions to lead to an insulating state? A complete theory must establish the range of correlation needed to understand the mechanisms and to make quantitative calculations. A striking example that requires a k -dependent self-energy is the opening of a “pseudogap” in only part of the Brillouin zone in a high-temperature superconductor, as illustrated in Fig. 2.16. This chapter is devoted to methods that go beyond the single-site approximation to take into account interactions and correlations between electrons on sites.