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Journal of Experimental and Theoretical Physics

Density of states near an Anderson transition in a four-dimensional space. Renormalizable models

Игорь Михайлович Суслов

Published Jan 1, 1997• 6 citations
Publisher

Abstract

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.

DOI
10.1134/1.558140

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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.
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.
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.
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.
We investigate how free probability allows us to approximate the density of states in tight-binding models of disordered electronic systems. Extending our previous studies of the Anderson model in one dimension with nearest-neighbor interactions [Chen et al., Phys. Rev. Lett. 109, 036403 (2012)], we find that free probability continues to provide accurate approximations for systems with constant interactions on two- and three-dimensional lattices or with next-nearest-neighbor interactions, with the results being visually indistinguishable from the numerically exact solution. For systems with disordered interactions, we observe a small but visible degradation of the approximation. To explain this behavior of the free approximation, we develop and apply an asymptotic error analysis scheme to show that the approximation is accurate to the eighth moment in the density of states for systems with constant interactions, but is only accurate to sixth order for systems with disordered interactions. The error analysis also allows us to calculate asymptotic corrections to the density of states, allowing for systematically improvable approximations as well as insight into the sources of error without requiring a direct comparison to an exact solution.
The methods used to determine the reduced density matrix (RDM) of the ground and excited states, the finite-temperature systems, and the large systems without using the wave function by solving the density equation were discussed. We examined the foundations to reconstruct the higher-order RDMs of the ground and excited states and the finite-temperature systems in terms of the lower-order RDMs. We presented the equation to determine the RDMs of the finite-temperature systems directly and showed that only the exact RDMs satisfy the equation. Our previous approximation for third- and fourth-order RDMs of the ground state [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] was reformulated, and the accuracy of this approximation for the excited states was examined. The structure of the nthn\mathrm{th}nth order energy density matrix (n(n(n-EDM) was analyzed, and the calculation method which sums up the Parquet diagram of the 2-EDM without explicitly constructing the third- and fourth-order RDMs was reported. This approximation is more accurate than the previous second-order approximation and also includes the infinite series of bubble and ladder Green's function diagrams. Such a method is necessary to apply the density-equation method to large systems, such as polymers, metals, and semiconductors. The new approximation together with the density equation was applied to the ground states of some molecules including CO, C2H2,{\mathrm{C}}_{2}{\mathrm{H}}_{2},C2​H2​, C3H8{\mathrm{C}}_{3}{\mathrm{H}}_{8}C3​H8​, and C4H10,{\mathrm{C}}_{4}{\mathrm{H}}_{10},C4​H10​, and the excited states of the Be atom and Li2{\mathrm{Li}}_{2}Li2​ molecule. The calculated energies were as accurate as the exact or coupled-cluster single and double excitations with triples included noniteratively, and the energy errors of the second-order approximation were significantly reduced. The calculated 2-RDMs almost satisfied important representability conditions while the 1-RDMs were exactly ensemble representable. These results demonstrate that the density equation offers a new quantitative method for treating electron correlations. The relationship between the iterative procedure and the finite-temperature density-equation method was discussed.
For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=dc{\mathit{d}}_{\mathit{c}}dc​-d, where dc{\mathit{d}}_{\mathit{c}}dc​ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy kB{\mathit{k}}_{\mathit{B}}kB​T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.