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Acta Physica Polonica A

Theory of Strongly Correlated Systems in the Limit of Large On-Site Coulomb Repulsion

Yu. A. Izyumov

Published Jan 1, 1994• 1 citation
Publisher

Abstract

A strongly correlated electronic system is studied within the framework of the t-J model with the help of diagram technique for the Hub- bard X-operators.The summing of some series of diagrams is suggested, corresponding to the following three types of approximations: the generalized random phase approximation, the mean field approximation, and the low density approximation.In generalized random phase approximation the dynamic magnetic susceptibility of paramagnetic phase is calculated, and boundaries of stability of this phase are found on the plane of Hamiltonian parameters and electron concentration n.It is shown that at some critical concentration nc the system undergoes a crossover from the itinerant magnetism (n < n c ) to the magnetism with localized magnetic moments (n > n c ). Simultaneously, the system transfers from the Fermi-liquid beh a v i o u r t o t h e r e g i m e o f s t r o n g e l e c t r o n c o r r e l a t i o n s w i t h n o n q u a s i p a r t i c l e states.In mean field approximation equations for the order parameters were derived and phase transitions temperatures in a ferromagnetic and antiferromagnetic phases were obtained.In low density approximation a state of the saturated ferromagnetism was investigated and a critical concentration n $ was calculated when the nonsaturated ferromagnetism follows the saturated ferromagnetism.

DOI
10.12693/aphyspola.85.103

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Generalized random-phase approximation in the theory of strongly correlated systems

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A strongly correlated electronic system is studied within the framework of the t-J model with the help of diagram technique for the Hub- bard X-operators.The summing of some series of diagrams is suggested, corresponding to the following three types of approximations: the generalized random phase approximation, the mean field approximation, and the low density approximation.In generalized random phase approximation the dynamic magnetic susceptibility of paramagnetic phase is calculated, and boundaries of stability of this phase are found on the plane of Hamiltonian parameters and electron concentration n.It is shown that at some critical concentration nc the system undergoes a crossover from the itinerant magnetism (n < n c ) to the magnetism with localized magnetic moments (n > n c ). Simultaneously, the system transfers from the Fermi-liquid beh a v i o u r t o t h e r e g i m e o f s t r o n g e l e c t r o n c o r r e l a t i o n s w i t h n o n q u a s i p a r t i c l e states.In mean field approximation equations for the order parameters were derived and phase transitions temperatures in a ferromagnetic and antiferromagnetic phases were obtained.In low density approximation a state of the saturated ferromagnetism was investigated and a critical concentration n $ was calculated when the nonsaturated ferromagnetism follows the saturated ferromagnetism.
The Hubbard model in the limit of strong Coulomb interaction (the (t-J) model) is studied by the diagram technique for Hubbard operators. The generalized random-phase approximation (GRPA) is formulated as an approximation taking into account electron loop-type diagrams. Within the framework of this approximation, both the dynamic magnetic and dielectric susceptibilities are calculated for a wide interval of electron concentrations, 0<n<1. The magnetic susceptibility shows a crossover in the magnetic behaviour of the system from pure itinerant magnetism to magnetism with localized magnetic moments. The crossover occurs at a critical point, nc, when the chemical potential for electrons of the lower Hubbard band changes sign. Magnetic phase diagrams are constructed on the (t/U, n) plane at T=0 for different types of crystal lattices. The behaviour of magnetic phases with temperature is also studied. The GRPA leads to a too drastic crossover, especially at low temperatures, because of insufficient allowance for the charge and spin fluctuations on a site. Summation of special diagram series shows that the Gaussian fluctuations of the effective electric and magnetic fields lead to a smoother picture of the crossover.
On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.
The authors construct a diagram technique for the (t-J) model Hamiltonian expressed in terms of Hubbard operators. This technique combines features of the diagram technique for normal Fermi systems and those for the Heisenberg model with spin operators. The general graphic structure is identified for one-particle Green functions composed of Hubbard operators. The (t-J) model has logical consistency, is reasonably simple and can serve as a working tool in exploring various properties of this model. To an approximation that resides in summing ladder-type diagrams with antiparallel electron lines, the authors calculate the system's magnetic susceptibility. A formula is derived which reflects the dual nature of the magnetic states in the Hubbard model; this duality manifests itself in the presence of a localised and an itinerant contribution in the bare susceptibility. The authors trace how the relative value of the localised contribution varies as the electron concentration is increased from 0 to 1. With n>2/3, the paramagnetic phase of the system becomes unstable with respect to the occurrence of ferromagnetic or antiferromagnetic ordering. A magnetic phase diagram for zero temperature is constructed on the (t/U, n) plane.
Strongly correlated electron systems show a rich variety of interesting physical phenomena. However, their theoretical description poses a highly non-trivial task, and even the simplest models cannot be solved exactly. In this thesis the intricate behavior of the single impurity Anderson model (SIAM) and the Hubbard model is investigated. The main emphasis is laid on two-particle Green functions of these systems. To this end, existing approximation schemes are extended to be capable of treating susceptibilities as well. The quality and range of applicability of these approximations is discussed in detail. The physical pictures behind the phenomena encountered in these systems are introduced. A prominent many-body effect is the emergence of temperature dependent low energy quasiparticles in both models. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments, a characteristic feature of the Kondo effect. It is argued, that within the presented approximations, the low temperature phase of these models can usually be described as a Fermi liquid. As a counterexample, the Hubbard model is shown to exhibit pronounced non-Fermi liquid behavior in situations, where a van Hove singularity in the non-interacting density of states is found in the vicinity of the Fermi level. The magnetic properties of the Hubbard model are discussed in view of the two archetypical pictures of magnetism: local moment magnetism and itinerant Stoner-magnetism. Critical temperatures for antiferromagnetic, ferromagnetic and incommensurate phase transitions are calculated, and the influence of the lattice structure and frustration is examined. In strongly correlated metallic situations, the complicated competition between Kondo-screening and local moment magnetism can lead to a reentrant behavior of the N\'eel temperature. A direct signature of the Kondo effect in the Hubbard model is found in the dynamic magnetic susceptibility, where a localized collective mode emerges at very low temperatures. It is connected to the breakup of a singlet formed by the local moments with their Kondo-screening cloud, and consequently the excitation energy is of the order of the low energy scale T*.
We include an additional diagram in the bare electron-electron vertex within the generalised random phase approximation (GRPA) for the t-J model. The equation for the effective electron-electron vertex becomes more complicated but is solved exactly for the simple cubic lattice. The dynamical magnetic susceptibility obtained is the same as in GRPA but with all quantities renormalised. Instability of the paramagnetic phase with respect to the formation of ferromagnetic or antiferromagnetic ordering is studied. Formulae for Curie and Neel temperatures are obtained. The additional diagram leads to an extension of the antiferromagnetic region and a reduction in the ferromagnetic region.
Ground-state properties of the repulsive Hubbard model on a cubic lattice are investigated by means of the auxiliary-field quantum Monte Carlo method. We focus on low-density systems with varying on-site interaction U/tU/tU/t, as a model relevant to recent experiments on itinerant ferromagnetism in a dilute Fermi gas with contact interaction. Twist-average boundary conditions are used to eliminate open-shell effects and large lattice sizes are studied to reduce finite-size effects. The sign problem is controlled by a generalized constrained path approximation. We find no ferromagnetic phase transition in this model. The ground-state correlations are consistent with those of a paramagnetic Fermi liquid.