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Journal of the Physical Society of Japan

Quantitative Aspects of the Dynamical Coherent Potential Approximation in Harmonic Approximation

Toshihito Tamashiro, Shota Nohara, Keisuke Miyagi, Y. Kakehashi

Published Jun 3, 2011• 4 citations
Publisher

Abstract

Magnetic and electronic properties of the Hubbard model on the Bethe and fcc lattices in infinite dimensions have been investigated numerically on the basis of the dynamical coherent potential approximation (CPA) theory combined with the harmonic approximation (HA) in order to clarify the quantitative aspects of the theory. It is shown that the dynamical CPA+HA reproduces well the sublattice magnetizations, the magnetizations, susceptibilities, and the Néel temperatures ( T N ) as well as the Curie temperatures calculated by the Quantum Monte-Carlo (QMC) method. The critical Coulomb interactions ( U c ) for the metal–insulator (MI) transition are also shown to agree with the QMC results above T N . Below T N , U c deviate from the QMC values by about 30% at low temperature regime. These results indicate that the dynamical CPA+HA is applicable to the quantitative description of the magnetic properties in high dimensional systems, but one needs to take into account higher-order dynamical corrections in order to describe the MI transition quantitatively at low temperatures.

DOI
10.1143/jpsj.80.064702

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•Jan 1, 2003•0 citations
Magnetic and electronic properties of the Hubbard model on the Bethe and fcc lattices in infinite dimensions have been investigated numerically on the basis of the dynamical coherent potential approximation (CPA) theory combined with the harmonic approximation (HA) in order to clarify the quantitative aspects of the theory. It is shown that the dynamical CPA+HA reproduces well the sublattice magnetizations, the magnetizations, susceptibilities, and the Néel temperatures ( T N ) as well as the Curie temperatures calculated by the Quantum Monte-Carlo (QMC) method. The critical Coulomb interactions ( U c ) for the metal–insulator (MI) transition are also shown to agree with the QMC results above T N . Below T N , U c deviate from the QMC values by about 30% at low temperature regime. These results indicate that the dynamical CPA+HA is applicable to the quantitative description of the magnetic properties in high dimensional systems, but one needs to take into account higher-order dynamical corrections in order to describe the MI transition quantitatively at low temperatures.
Magnetic and electronic properties of the Hubbard model on the Bethe and fcc lattices in infinite dimensions have been investigated numerically on the basis of the dynamical coherent potential approximation (CPA) theory combined with the harmonic approximation (HA) in order to clarify the quantitative aspects of the theory. It is shown that the dynamical CPA+HA reproduces well the sublattice magnetization, the magnetizations, susceptibilities, and the Néel temperatures (TNT_{\rm N}TN​) as well as the Curie temperatures calculated by the Quantum Monte-Carlo (QMC) method. The critical Coulomb interactions (UcU_{\rm c}Uc​) for the metal-insulator (MI) transition are also shown to agree with the QMC results above TNT_{\rm N}TN​. Below TNT_{\rm N}TN​, UcU_{\rm c}Uc​ deviate from the QMC values by about 30% at low temperature regime. These results indicate that the dynamical CPA+HA is applicable to the quantitative description of the magnetic properties in high dimensional systems, but one needs to take into account higher-order dynamical corrections in order to describe the MI transition quantitatively at low temperatures.
A theory of the single-particle excitation spectrum is presented on the basis of the projection operator method combined with the many-body coherent-potential approximation (CPA). The theory describes the dynamics of the excitations by means of an energy-dependent Liouville operator accompanied by a coherent potential which is determined by the self-consistent CPA condition. It is shown that the present theory is essentially equivalent to the dynamical CPA and the dynamical mean-field theory. The Hubbard III approximation and the modified perturbation theory are rederived from the theory. A renormalized perturbation scheme for the Green function is developed on the basis of a general formula for the memory function. It interpolates between the weak- and strong-Coulomb interaction limits, and yields the metal-insulator transition for half-filled bands. Numerical calculations have been performed for the Gutzwiller-Hubbard model on a hypercubic lattice in infinite dimensions. The results show that the theory describes quantitatively the quasiparticle weight vs Coulomb interaction curve, yielding a reasonable critical Coulomb interaction for the metal-insulator transition. It produces the overall features of the excitation spectra and the momentum distributions for various Coulomb interaction strengths.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
We investigate the Hubbard model on the triangular lattice at half filling using the dynamical cluster approximation (DCA) and dual fermion (DF) methods in combination with continuous-time quantum Monte Carlo (CT QMC) and semiclassical approximation (SCA) methods. We study the one-particle properties and nearest-neighbor spin correlations using the DCA method. We calculate the spectral functions using the CT QMC and SCA methods. The spectral function in the SCA and obtained by analytic continuation of the Pad\'e approximation in CT QMC are in good agreement. We determine the metal-insulator transition and the hysteresis associated with a first-order transition in the double-occupancy and nearest-neighbor spin-correlation functions as functions of temperature. As a further check, we employ the DF method and discuss the advantages and limitation of the dynamical mean-field theory, DCA, and recently developed DF methods by comparing the Green's functions. We find an enhancement of antiferromagnetic correlations and provide evidence for magnetically ordered phases by calculating the spin susceptibility.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
[[abstract]]The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0 < q < pi) into the state with strong antiferromagnetic correlations (q = pi) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.