INTRODUCTION |

Chapter I. | THE DIAGRAMMATIC TECHNIQUE FOR THE HUBBARD OPERATORS |

| § 1. | The atomic representation |

| § 2. | Commutation relations |

| § 3. | The Wick theorem |

| § 4. | The diagrammatic technique |

| § 5. | Conclusions |

Chapter II. | TRANSITION TO THE ATOMIC REPRESENTATION |

| § 1. | Three-level system |

| § 2. | Shubin-Wonsowsky model -- a four-level system |

| § 3. | The atomic representation of p- and d-electrons |

| § 4. | Conclusions |

Chapter III. | MAGNETIC PROPERTIES IN THE HUBBARD MODEL |

| § 1. | The one-loop approximation |

| § 2. | The half-filled band |

| § 3. | Paramagnetic properties of the metallic phase |

| § 4. | Conclusions |

Chapter IV. | SUPERCONDUCTIVITY IN THE HUBBARD MODEL |

| § 1. | Peculiarities of high-Tc superconductors |

| § 2. | Calculation of the scattering amplitude at infinite Hubbard energy |

| § 3. | Superconducting transition temperature |

| § 4. | Taking account of relaxation processes |

| § 5. | Finite Hubbard energies |

| § 6. | Conclusions |

Chapter V. | SUPERCONDUCTIVITY IN A SYSTEM OF p-d-ELECTRONS |

| § 1. | Hamiltonian and formulation of the problem |

| § 2. | Equation of state |

| § 3. | Superconductivity criterion |

| § 4. | Emery -- Hirsch model |

| § 5. | The phase diagram calculated in the generalized p-d-model |

| § 6. | Conclusions |

Chapter VI. | FERROMAGNETISM IN TRANSITION METALS |

| § 1. | General relations |

| § 2. | Ferromagnetism in nickel |

| § 3. | Ferromagnetism in cobalt |

| § 4. | Ferromagnetism in a-iron |

| § 5. | The region 3 < h_{d} < 4; the one-loop approximation |

| § 6. | The region 4 < h_{d} < 5; the one-loop approximation |

| § 7. | Conclusions |

REFERENCES |

The consistent study of strongly interacting electron systems
started with the works of J.Hubbard. He suggested that the Coulomb
electron-electron interaction (the largest part of the interaction) be
regarded as the zeroth approximation, while the kinetic energy of
electron hopping into a neighboring cell should be treated as the
perturbation.

Using such an approach, in 1964 Hubbard managed to solve one
of the most important problems of solid-state physics, namely to
determine the conditions under which the transition of dielectrics
into metals occurs.

Another, even more important, problem of high temperature
superconductivity was solved experimentally when superconductors
with transition temperatures ranging from 40 К to 153 К were discovered.
Immediately after that an attempt was made to describe
the newly discovered superconductors theoretically by introducing
an extremely large BCS coupling constant. But it turned out that
even the electronic structure of the normal phase of these compounds
cannot be explained without taking into account the strong
electron-electron interaction. A non-phonon mechanism of superconductivity
is exhibited in Cu compounds in which the Hubbard
energy exceeds the Fermi energy. Such compounds change from
metals to semiconductors at low dopant concentrations.

The third problem of transition metal physics is to explain why,
among all elements of the periodic system possessing metal conductivity,
only Ni, Co and alpha-Fe display ferromagnetic properties
within a wide range of temperature, while only Cr, Mn and gamma-Fe
display antiferromagnetic properties. It seems impossible to determine
the criterion for ferromagnetism in these elements without
taking account of strong Hubbard repulsion at short distances.

To solve these problems a special theory of perturbations was
worked out which treated single atom states as the zeroth approximation,
and a Hamiltonian connected with the overlapping of the
wave functions belonging to neighboring atoms was considered as
the energy of interaction.

In Chapter I general theorems are proved and the rules establishing
connections between each term of the series of the perturbation
theory and the corresponding diagram are formulated. To
determine essential differences from the standard Matsubara diagrammatic
method [1], we consider 3- and 4-level atomic systems.
The classification of transitions is made by means of separation into
Fermi- and Bose-transitions with further expansion of X-operators
in terms of the Cartan--Weyl canonical basis.

In Chapter II we consider the atomic representation of specific
simple systems which will be studied in detail in the following
chapters. In this chapter the atomic representation of s-, p- and
d-electrons in a crystal of cubic symmetry is given.

Chapter III studies the classical Hubbard model for s-electrons.
Electron spectra are obtained, the semiconductor gap is calculated,
and the dependence of the magnetic susceptibility on temperature
and electron concentration is also obtained.

Chapter IV is devoted to peculiarities of high temperature superconductors.
The residual interaction, which is present in an electron
system with the strong electron-electron repulsion, essentially
depends on energy. The scattering amplitude of two excitations
with opposite spins decreases with increasing energy of relative motion,
and it could change its sign on the entire Fermi surface. Thus,
taking into account correctly the electron-electron interaction provides
an opportunity to explain the strong dependence of the superconducting
transition temperature on the location of the Fermi-level
within an incomplete relatively narrow electron sub-band. It has
been shown that in the classical Hubbard model there is a range
of concentrations for which superconductivity with sufficiently high
transition temperature is observed. Spin fluctuations, which decrease
the superconducting transition temperature, are also taken
into account.

Chapter V considers a system of cations and anions with overlapping
incomplete d- and p-shells. It has been shown that taking
into account the strong intra-atomic Coulomb repulsion leads to the
occurrence of specific ranges of concentration of d-and p-electrons
for which non-phonon superconductivity is observed. The phase diagram
of the superconducting state can be obtained in qualitative
agreement with the experimental data, related to doping both by
2-valent and 4-valent cations.

In Chapter VI we study ferromagnetism of elements of the 3d
transition group within the framework of the Hubbard model for
electrons in high spin states. The dependence of ferromagnetic ordering
on the 3d-shell occupation numbers in a-Fe, Co, and Ni is
studied. The reasons for the absence of the ferromagnetic instability
in Pd, Pt and also in gamma-Fe, Cr, and Ni are determined. It appears
possible to prove that as far as Ni, Pd and Pt are concerned,
ferromagnetism can be observed only within a rather narrow interval
of hole concentrations. At the same time, it turns out that in
this case the number of 4d-holes in Pd is too small and the number
of 5d-holes in Pt is too large to ensure their presence within
a ferromagnetic concentration interval, where the number of 3d-hole
states of Ni can be observed. The experimental values of saturated
magnetic moments for Ni, Co and alpha-Fe correspond to theoretical
values of ferromagnetic intervals of concentration. As far as gamma-Fe
is concerned, it turns out that the number of 3d-holes exceeds the
maximum possible value of the theoretically predicted range for the
existence of ferromagnetism.

**Rogdai Olegovich Zaitsev** was born in Moscow in 1938. He graduated from
the physical faculty of the Moscow State Uni-versity; since 1965 he has been
working in the Kurchatov Institute of Nuclear Energy and in the Moscow
Institute of Physical Technology.

He is a professor of theoretical physics and his scientific interests are
devoted to theories of solid state, superconductivity, ferromagnetism, and
the non-linear theory of noise. The book "Diagrammatic methods in the theory
of superconductivity and ferromagnetism" discusses some of these problems.

Rogdai Olegovich Zaitsev was born in Moscow in 1938. He graduated from the physical faculty of the Moscow State University in 1961; worked in the Kurchatov Institute of Nuclear Energy from 1965 to 2008. Now he works in the Moscow Institute of Physics and Technology as a professor of theoretical physics.

His scientific interests are devoted to theories of solid state, superconductivity, ferromagnetism, and the non-linear theory of noise.