Diagrammatic theory for Periodic Anderson Model. Stationary property of the thermodynamic potential V. A. Moskalenko 1 , $^{2}$, ∗ L. A. Dohotaru $^{3}$, and R. Citro4 $^{1}$Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova $^{2}$BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia $^{3}$Technical University, Chisinau 2004, Moldova and $^{4}$Dipartimento di Fisica E. R. Caianiello, Universit´ a degli Studi di Salerno and CNISM, Unit´ a di ricerca di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy (Dated: November 16, 2018) Diagrammatic theory for Periodic Anderson Model has been developed, supposing the Coulomb repulsion of f − localized electrons as a main parameter of the theory. f − electrons are strongly correlated and c − conduction electrons are uncorrelated. Correlation function for f − and mass operator for c − electrons are determined. The Dyson equation for c − and Dyson-type equation for f − electrons are formulated for their propagators. The skeleton diagrams are defined for correlation function and thermodynamic functional. The stationary property of renormalized thermodynamic potential about the variation of the mass operator is established. The result is appropriate as for normal and as for superconducting state of the system. PACS numbers: 71.27.+a, 71.10.Fd the most essential stages in the development of this model because exists a number of consistent reviews [2 − 7] and books [8 , 9] on this field and we shall use the references to previous our papers. We consider the simplest form of PAM with a spin de- generation of the level of localized f − electrons, a simple energy band of conducting c − electrons, Coulomb one- site repulsion U of correlated f − electrons with opposite spins and one-site hybridization between both group of electrons of this system. The hamiltonian of the system reads: where Here V is the hybridization amplitude assumed con- stant. We have indicated with C + $_{iσ}$( f + $_{iσ}$) the creation op- erator for an uncorrelated (correlated) electron with spin σ and i lattice site, n$_{iσ}$ is the number operator for f − electrons, ϵ ( k ) is the band energy with momentum k of There are already an enormous number of approximate methods and approaches devoted to PAM, as perturba- tion expansions, static and dynamic mean field theories, variational and numerical approaches, large N expan- sion , slave boson methods, non crossing approximations (NCA), Bogoliubov inequality method and others. Also some exact results are known, obtained in special with the Bethe ansatz, renormalization group methods and Bogoliubov inequality method. We will not enlarge upon The study of the systems with strongly correlated electrons has become in the last time one of the cen- tral problem of condensed matter physics. One of the most important models of strongly correlated electrons is periodic Anderson model (PAM) $^{[1]}$. This model is used to describe the physics of mixed valence systems, heavy fermion compounds, high-temperature supercon- ductivity as well as other phenomena in which the strong Coulomb repulsion of the localized electrons is present. This model describes the intermetallic compounds which contain magnetic moments of rare earth or actinide ions included in the host metal. This ions have a partially filled f − shell and can be considered as scattering cen- ters for conduction electrons of the host metal. Because of the strong Coulomb repulsion of the electrons with op- posite spins located at the same site of lattice the mag- netic ion electrons are strongly correlated. There is also the hybridization of states between the uncorrelated con- duction electrons and localized correlated ones when both of them are present on the same lattice site. Magnetic properties of the impurities ions affect in a different man- ner the properties of the host matrix and of the system as a whole. For different regime of physical parameters, determined by Coulomb local interaction, hybridization of the wave functions and exchange interaction, it is pos- sible to obtain different classes of the system phases. II. MODEL HAMILTONIAN I. INTRODUCTION H = H 0 c + H 0 f + H$_{int}$, H 0 c = ∑ k σ ϵ ( k ) C + k $_{σ}$C$_{k}$$_{σ}$ , H 0 f = ϵ$_{f}$ ∑ iσ n$_{iσ}$ + U ∑ i n$_{i}$$_{↑}$n$_{i}$$_{↓}$, (1) H$_{int}$ = V ∑ iσ ( C + $_{iσ}$f$_{iσ}$ + f + $_{iσ}$C$_{iσ}$ ) , ϵ ( k ) = ϵ ( k ) − µ, ϵ$_{f}$ = ϵ$_{f}$ − µ, n$_{iσ}$ = f + $_{iσ}$f$_{iσ}$ , (2) C$_{iσ}$ = 1 √$_{N}$ ∑ k exp ( − i kR $_{i}$) C$_{k}$$_{σ}$. arXiv:1001.1644v1 [cond-mat.str-el] 11 Jan 2010