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arXiv:cond-mat/9912116

Density of States near the Anderson Transition in a Four-dimensional Space. Renormalizable Models

Игорь Михайлович Суслов

Published Dec 7, 1999• 0 citations
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Abstract

Asymptotically exact results are obtained for the average Green function and density of states of a disordered system for a renormalizable class of models (as opposed to the lattice models examined previously [Zh. Eksp. Teor. Fiz. 106 (1994) 560-584]). For N\sim 1 (where N is an order of the perturbation theory), only the parquet terms corresponding to the highest powers of large logarithms are retained. For large N, this approximation is inadequate because of the fast growth with N of the coefficients for the lower powers of the logarithms. The latter coefficients are calculated in the leading order in N from the Callan-Symanzik equation with results of the Lipatov method using as boundary conditions. For calculating the self-energy at finite momentum, a modification of the parquet approximation is used, that allows the calculations to be done in an arbitrary finite logarithmic approximation but in the leading order in N. It is shown that the phase transition point shifts in the complex plane, thereby insuring regularity of the density of states for all energies. The "spurious" pole is avoided in such a way that effective interaction remains logarithmically weak.

DOI
10.48550/arxiv.cond-mat/9912116

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Asymptotically exact results are obtained for the average Green function and density of states of a disordered system for a renormalizable class of models (as opposed to the lattice models examined previously [Zh. Eksp. Teor. Fiz. 106 (1994) 560-584]). For N\sim 1 (where N is an order of the perturbation theory), only the parquet terms corresponding to the highest powers of large logarithms are retained. For large N, this approximation is inadequate because of the fast growth with N of the coefficients for the lower powers of the logarithms. The latter coefficients are calculated in the leading order in N from the Callan-Symanzik equation with results of the Lipatov method using as boundary conditions. For calculating the self-energy at finite momentum, a modification of the parquet approximation is used, that allows the calculations to be done in an arbitrary finite logarithmic approximation but in the leading order in N. It is shown that the phase transition point shifts in the complex plane, thereby insuring regularity of the density of states for all energies. The "spurious" pole is avoided in such a way that effective interaction remains logarithmically weak.
Asymptotically accurate results are obtained for the average Green function and density of states of a disordered system for a renormalizable class of models (as opposed to the lattice model examined previously [I. M. Suslov, Zh. Éksp. Teor. Fiz. 106 , 560 (1994)]. For N ∼1 (where N is the order of perturbation theory), only the parquet terms corresponding to the higher powers of large logarithms are taken into account. For large N , this approximation is inadequate because of the higher rate of increase with respect to N of the coefficients for the lower powers of the logarithms. The latter coefficients are determined from the renormalization condition for the theory expressed in the form of a Callan-Symanzik equation using the Lipatov asymptote as boundary conditions. For calculating the self-energy at finite momentum, a modification of the parquet approximation, is used that allows the calculations to be done in an arbitrary finite logarithmic approximation, including the principal asymptote in N of the expansion coefficients. It is shown that the phase transition point moves in the complex plane, thereby ensuring regularity of the density of states for all energies and avoiding the “false” pole in such a way that the effective interaction remains logarithmically weak.
We present self-consistent Green-function calculations for the properties of the symmetric Anderson impurity model at zero temperature. A stable method for calculating the spectral density directly on the real frequency axis in the spin-fluctuation limit is described. Results using an approximation to the self-energy, previously termed the fluctuation-exchange approximation, which includes the effects of spin and density fluctuations (within the random-phase approximation) and multiple electron-electron scattering are compared with second-order perturbation-theory calculations. The static spin and charge susceptibilities, calculated in a manner consistent with conservation laws, are compared with the exact results from the Bethe ansatz. We find that the fluctuation-exchange approximation gives a good description of the charge susceptibility over a wide range of interaction strengths and is a great improvement over the self-consistent second-order theory. However, the T-linear coefficient of the specific heat is greatly overestimated by the fluctuation-exchange approximation and this leads to the underestimation of the spin susceptibility. The self-consistent second-order perturbation theory, which underestimates the specific-heat coefficient, predicts a magnetic ground state for sufficiently high interaction strengths.
A variety of self-consistent-field techniques have recently been used to calculate correlation functions of the two-dimensional Hubbard model. Such techniques extend Hartree–Fock theory by including a large class of fluctuation processes. The computational requirements of the self-consistent-field approach decrease in lower dimensional systems. For this reason, a thorough test of techniques may be performed by examining the Anderson impurity model, the zero-dimensional limit of the Hubbard model. Using the fluctuation exchange and pseudopotential parquet approximations applied previously in two dimensions, we calculate static and dynamic correlation functions for the Anderson model and compare with quantum Monte Carlo results.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r.1/r.1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.
Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.
Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.