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Journal of Physics C Solid State Physics

A new functional integral formalism for strongly correlated Fermi systems

S K Sarker

Published Jun 30, 1988• 15 citations
Publisher

Abstract

A new functional integral mapping of a strongly correlated Fermi system to another Fermi system is presented. The strong correlations appear in the new system through momentum- and frequency-dependent coupling constants characterising correlated multi-particle motions, which can be treated perturbatively. The method is well suited to treat Hubbard and Anderson models. For the Hubbard model, the lowest-order term reproduces Hubbard's two-band Green function. Second-order processes leading to spin fluctuations are identified.

DOI
10.1088/0022-3719/21/18/002

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A new functional integral mapping of a strongly correlated Fermi system to another Fermi system is presented. The strong correlations appear in the new system through momentum- and frequency-dependent coupling constants characterising correlated multi-particle motions, which can be treated perturbatively. The method is well suited to treat Hubbard and Anderson models. For the Hubbard model, the lowest-order term reproduces Hubbard's two-band Green function. Second-order processes leading to spin fluctuations are identified.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
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A nonperturbative method to obtain on- and off-site one-particle Green's function is introduced and applied to noninteracting Hubbard model with next nearest neighbor hopping and interacting Hubbard model in large dimensions, for example. The former gives some lessons on the method and shows the advantage of the method compared with continued fraction formalism. The latter is treated by selecting important dynamic processes contributing to the Green's function when correlation is strong. We consider the model in the Bethe lattice with large connectivity. The dynamic processes describing on-site spin fluctuation clearly shows metal-insulator transition in the paramagnetic ground state at half filling.
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A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal–insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when U ~ W, where U is the Coulomb repulsion parameter and W is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter U/W or W/U, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, d = ∞, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters U and W, and of the electron density n.