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Physical review. B, Condensed matter

Cluster method for the Hubbard model: Local moments and short-range correlations

A. P. Kampf

Published Aug 1, 1991• 7 citations
Publisher

Abstract

We use a functional-integral representation for the Hubbard model to describe the formation of local moments and their magnetic correlations. In the absence of long-range magnetic order, the presence of well-established local moments alone is sufficient to produce a gap in the electronic spectrum. The local moments are treated within a coherent-potential approximation, and we apply an embedded-cluster method to incorporate their interactions.

DOI
10.1103/physrevb.44.2637

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We use a functional-integral representation for the Hubbard model to describe the formation of local moments and their magnetic correlations. In the absence of long-range magnetic order, the presence of well-established local moments alone is sufficient to produce a gap in the electronic spectrum. The local moments are treated within a coherent-potential approximation, and we apply an embedded-cluster method to incorporate their interactions.
We study the two-dimensional Hubbard model by means of an extension of the dynamical mean-field approximation (DMFA). The extension is based on the projection technique and the DMFA, in which nonlocal contributions to the dynamics of the system are taken into account through static short-ranged correlations, while dynamical local contributions are kept by the local self-energy. The short-ranged correlations and the local self-energy are determined by mapping the lattice problem onto a self-consistently embedded cluster. We solve the cluster problem by using a noncrossing approximation. The spectral function obtained satisfies the sum rules of its first four moments. The spectra display pronounced nonlocal features including a pseudogap near the Fermi surface due to a competition between the Kondo-singlet formation and short-range spin correlations.
The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.
We present a generalization of the recently proposed variational cluster perturbation theory to extended Hubbard models at half-filling with repulsive nearest neighbor interaction. The method takes into account short-range correlations correctly by the exact diagonalization of clusters of finite size, whereas long-range order beyond the size of the clusters is treated on a mean-field level. For one dimension, we show that quantum Monte Carlo and density-matrix renormalization-group results can be reproduced with very good accuracy. Moreover we apply the method to the two-dimensional extended Hubbard model on a square lattice. In contrast to the one-dimensional case, a first order phase transition between spin density wave phase and charge density wave phase is found as function of the nearest-neighbor interaction at onsite interactions $U\ensuremath{\geqslant}3t$. The single-particle spectral function is calculated for both the one-dimensional and the two-dimensional system.
The local approach to electron correlation in the Hubbard model has been extended to include charge and spin correlations between nearest neighbours. Both these effects are particularly important near the half-filled band and further stabilise local moments. Polarisation induced by local moments is shown to be antiferromagnetic independently of the band filling. A considerable gain in energy connected with the formation of the polarisation cloud indicates that short-range order should exist also at relatively high temperatures.
In order to understand various local phenomena related to electronic phase separation discovered recently in strongly correlated materials, quantum cluster methods are applied to a theoretical model, i.e., Hubbard model, which has successfully predicted basic properties of strongly correlated systems. Our calculations are performed at two levels: exact calculations based on the isolated small clusters via exact diagonalization and approximate calculations applied to large lattices using the variational cluster approximation (VCA).\nWe first study phase separation instabilities with the formation of a pairing gap in the two-dimensional Hubbard model for 8-site Betts clusters. The exact diagonalization method is applied to extract exact ground and excited states of the Hubbard model. The results show that the electronic states with one hole off half filling are unstable and they energetically prefer creating the spatial phase separation. The effect of next nearest hopping is also discussed in this part.\nWe studied the variation of the pairing when the out-of-plane correlation is present by adding an interacting apical site above the two-dimensional cluster plane. The calculations indicate that the out-of-plane correlation can be detrimental to the in-plane pairing effect. When it is not too strong to destroy electron pairs on the plane, the modulation of the apical site can drive the pairing gap to show a sinusoidal variation which is consistent with recent experimental discoveries.\nVariational cluster approximation is introduced to explore local properties of larger lattices. The local nematic state is studied in the square lattice using this method, which indicates that it is possible that the electronic states can locally break the C4 symmetry (90o$ rotational symmetry) to form a pattern where the electronic properties, such as charge and spin correlations, are different in x and y directions.\nThe spatial phase separation in the square and honeycomb lattices is also investigated by the variational cluster approximation. The study shows that Coulomb interaction and lattice geometry are both crucial for electronic phase separation. We propose that electronic phase separation is the result of an instability of the Fermi surface at the boundary of the first Brillouin zone of antiferromagnetic states.
The two-dimensional (2D) Hubbard model is solved in a new way. The zero-frequency components of the Hubbard interaction are separated out and solved to infinite order to avoid the traditional difficulties associated with perturbation theory in the strong-coupling limit. Specifically the zero-frequency spin interaction is decoupled by the introduction of Stratonovich-Hubbard fields, while the zero-frequency charge interaction is treated by "slave" Hartree-Fock. This approach leads to the formation of large fluctuating spin moments, and as a consequence the one-electron spectral density exhibits a pseudogap. A Metropolis Monte Carlo method is used to perform the configuration averaging over Stratonovich-Hubbard fields. Within single Monte Carlo configurations the non-zero frequency components of the Hubbard interaction are treated by a high-order perturbation method, which incorporates the exchange of dynamical particle-hole fluctuations. Since a single configuration has neither translational symmetry nor spin rotational symmetry, these fluctuations do not have a well-defined crystal momentum or total spin. When the dynamical particle-hole fluctuations are included in the calculation, localized electron pairing results at elevated temperatures. Superconductivity presumably onsets when these localized pairs develop coherence, but this is not studied in the present work. The nature of the pairing interaction and the resulting pair eigenfunctions is studied.