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# Diagrammatic analysis of the Hubbard model: Stationary property of the thermodynamic potential 

V. A. Moskalenko 1 , $^{2}$, ∗ L. A. Dohotaru $^{3}$, and I. D. Cebotari1 

$^{1}$Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova 

$^{2}$BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia and 

$^{3}$Technical University, Chisinau 2004, Moldova 

(Dated: November 20, 2018) 

Diagrammatic approach proposed many years ago for strong correlated Hubbard model is devel- oped for analyzing of the thermodynamic potential properties. The new exact relation between such renormalized quantities as thermodynamic potential, one-particle propagator and correlation func- tion is established. This relation contains additional integration of the one-particle propagator by the auxiliary constant. The vacuum skeleton diagrams constructed from irreducible Green’s functions and tunneling propagator lines are determined and special functional is introduced. The properties of such functional are investigated and its relation to the thermodynamic potential is established. The stationary properties of this functional with respect to first order changing of the correlation function is demonstrated and as a consequence the stationary properties of the thermodynamic potential is proved. 

The Hubbard model is one of the most important mod- els for the electron of solids which describes quantum mechanical hopping of electron between lattice sites and their short ranged repulsive Coulomb interaction. [1] 

This model was discussed by Hubbard in order to describe a narrow-band system of transition metals and has been revised to investigate the properties of highly correlated electron systems such as the copper oxide su- perconductors and others. 

Hubbard model exhibits various phenomena includ- ing metal-insulator transition, antiferromagnetism, ferro- magnetism and superconductivity. This model assumes that each atom of the crystal lattice has only one electron orbit and the corresponding orbital state is non degener- ate. 

The Hamiltonian of Hubbard model is a sum of the two terms 

where H 0 is the atomic contribution, which contains the Coulomb interaction term U and local electron energy ϵ on the atom 

and hopping Hamiltonian 

PACS numbers: 71.27.+a, 71.10.Fd 

## I. INTRODUCTION 

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Here C + − → i $_{σ}$( C − → i $_{σ}$) are the creation (annihilation) electron operators with local site − → i and spin σ . Because in the thermodynamic perturbation theory we shall use thermal averages in a grand canonical ensemble we have added to the Hamiltonian (1) the term − µ ˆ N$_{e}$ 

where µ is the chemical potential and ˆ N$_{e}$ electron number operator. The quantities U and ˆ N$_{e}$ are the fundamental parameters of the model and because of large value of the Coulomb repulsion it is taken into account in zero approximation of our theory. The operator H $^{′}$, which describes hopping of the electrons between sites of the crystal lattice is regarded as a perturbation. 

For investigating this model new physical and mathe- matical concepts and techniques have been elaborated. We can only enumerate some of them. There are many analytical approximations as Hubbard approxi- mation, noncrossing approximation (NCA), slave-boson method, Dynamical Mean Field Theory (DMFT), com- posite and others projection operator methods and ap- proaches. Also numerical simulations of thermodynamic quantities and density of states have been performed by various methods. Some exact results for one- and two-dimensional Hubbard model are known. Each ap- proximate method has its advantages and disadvan- tages. A short and comprehensive reviews of the meth- ods can be found in papers and books [2 − $^{7]}$. Besides these approaches we must enumerate also the special di- agram techniques elaborated for strongly correlated elec- tron systems. Stasyuk $^{[8]}$, Zaitsev $^{[9]}$, Izyumov [10] and coworkers have developed a diagram technique for Hub- bard model, based on the disentangling of nondiagonal Hubbard operator χmn out of a time-ordered products of other such operators. Due to a more complicate al- 

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