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Journal of Experimental and Theoretical Physics Letters

Fermi-liquid effects in the Hubbard model

R. O. Zaǐtsev

Published Sep 1, 2007• 2 citations
Publisher

Abstract

Properties of an electronic system with a very strong electron-electron interaction are studied. The equation of state is obtained, and the temperature and concentration dependences of the hydrodynamic sound and magnetic susceptibility are calculated. A low-temperature domain of instability corresponding to phase segregation is found. This domain is located inside the wide region of Cooper instability. The equation for zero-sound oscillations is obtained with regard to the substantial anisotropy of the energy spectrum.

DOI
10.1134/s0021364007130115

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Properties of an electronic system with a very strong electron-electron interaction are studied. The equation of state is obtained, and the temperature and concentration dependences of the hydrodynamic sound and magnetic susceptibility are calculated. A low-temperature domain of instability corresponding to phase segregation is found. This domain is located inside the wide region of Cooper instability. The equation for zero-sound oscillations is obtained with regard to the substantial anisotropy of the energy spectrum.
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The order parameters are constructed for a Fermi liquid with conformational degrees of freedom. Based on them, additional thermodynamic parameters were introduced: the spin unit vector dα (determining the anisotropy in the spin subspace), the unit spatial vectors mi and ni (determining the anisotropy in space), and also three scalar parameters determining the shape of the Cooper pair u, v, q ( first two items are half-axes of ellipsoid of Cooper pair and last item is mutual orientation in space of these half axes). The symmetry properties of the order parameter operator are considered. The equations of ideal hydrodynamics of a Fermi liquid are derived taking into account the influence of conformational degrees of freedom. By conformational degrees of freedom should be understood the parameters associated with the shape and size of the Cooper pair. Expressions are obtained for the flows of thermodynamic quantities of such a Fermi liquid in terms of the density of the energy functional. The energy functional depends both on the additive integrals of motion (classical fluid parameters) and on conformational parameters. The dispersion equation of such a liquid is obtained for a model representation of the energy functional (the work was performed as part of the Fermi-liquid approach). The dispersion equation includes spin modes, first, second, and third sounds. The dispersion equation for the spatial subsystem, including the first, second, and third sounds characteristic of superfluid systems, is analyzed. Particular solutions of the dispersion equation are simulated using the Maple software package (several 3D figures are given for the angular dependence of the speeds of 1 and 2 sounds in a spherical coordinate system). All of the above allows us to conclude that such a Fermi liquid can be considered as a superfluid liquid crystal of a nematic type. The presence of conformational parameters distinguishes the considered phase from the F phase of a superfluid Fermi liquid.
The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby kFk_FkF​.
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.