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International Journal of Modern Physics B

LOCAL APPROACH TO VARIATIONAL METHOD FOR ELECTRON CORRELATION IN NARROW BAND SYSTEMS

Guozhu Wei, Hui‐Quan Nie, Kaiyi Zhang, K. A. Chao

Published Oct 1, 1987• 1 citation
Publisher

Abstract

Using the local approach, we have performed a third order perturbation calculation to investigate the electron correlation effect in extended Hubbard model with both positive and negative intraatomic and interatomic correlation energies. We found the significant third order correction over the second order terms. Consequently, all published results, especially the phase diagrams which were derived with the second order perturbation calculation are not reliable.

DOI
10.1142/s0217979287001730

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Using the local approach, we have performed a third order perturbation calculation to investigate the electron correlation effect in extended Hubbard model with both positive and negative intraatomic and interatomic correlation energies. We found the significant third order correction over the second order terms. Consequently, all published results, especially the phase diagrams which were derived with the second order perturbation calculation are not reliable.
Using the local approach, we have performed a third·order perturbation calculation to investigate the effects of intra-atomic electron correlation and electron and spin correlation between nearest neighbour sites in the extended Hubbard model. We found that significant correction of the third order over the second order results and, in comparison with the results of the third-order perturbation where only the intra-atomic electron correlation is included, the influence of the electron and spin correlation between nearest neighbour sites on the correlation energy is non-negligible.
Based on the intraatomic electron correlation and the interatomic electron and spin correlation, the local approach has been used to the Hubbard model. The term〈OOH〉which was neglected in the general calculation is retained in our calculation. Camparing our results with those of the general calculation, we find that when U is large, the effects of the term〈OOH〉in the paramagnetic phase can not be heglected for calculating the correlation energy, the ground state energy, the local moments and the antiferromagnetic polarization around the local moments. To estimate more accurately the electron correlation effects, one has to calculate to higher orders and check he convergency of the series.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.
The half-filled extended Hubbard model (containing nearest-neighbor interaction) is considered at strong coupling in fourth-order perturbation theory. The ground-state energies in the spin- and charge-density-wave phases and the resulting phase diagram are calculated in various dimensions: here d=1, d=\ensuremath{\infty}, high dimensions (d\ensuremath{\gg}1), and d=2,3 are discussed separately. In d=1 it is shown that, for U\ensuremath{\gtrsim}6, fourth-order perturbation theory leads to excellent agreement with existing Monte Carlo data. Second-order perturbation theory alone is valid only at unrealistically large values of U and V. In d=\ensuremath{\infty}, one needs fourth-order perturbation theory to obtain any nontrivial contribution to the phase diagram at all. As a consequence the fourth-order corrections to the ground-state energy and to the phase diagram are large also in d=2,3. The formalism used to obtain the perturbative expansion is discussed in an appendix.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc{U}_{c}Uc​ and exhibits strong indications of a lower critical occupation number nc{n}_{c}nc​. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.