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Diagrammatic method in the theory superconductivity and ferromagnetism

URSS. 168 pp. (English). ISBN 978-5-484-01012-7.
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Diagrammatic methods are applied to strongly interacting electron systems. Several problems are studied in the atomic representation. It is assumed that electron-electron interaction in an atom is stronger that the interaction between electrons belonging to different atoms. The theory of high-temperature superconductivity is considered in the Hubbard model. Then the theory of ferromagnetism in transition metals is also presented. (More)

 § 1.The atomic representation
 § 2.Commutation relations
 § 3.The Wick theorem
 § 4.The diagrammatic technique
 § 5.Conclusions
 § 1.Three-level system
 § 2.Shubin-Wonsowsky model – a four-level system
 § 3.The atomic representation of p- and d-electrons
 § 4.Conclusions
 § 1.The one-loop approximation
 § 2.The half-filled band
 § 3.Paramagnetic properties of the metallic phase
 § 4.Conclusions
 § 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
 § 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
 § 1.General relations
 § 2.Ferromagnetism in nickel
 § 3.Ferromagnetism in cobalt
 § 4.Ferromagnetism in a-iron
 § 5.The region 3 < hd < 4; the one-loop approximation
 § 6.The region 4 < hd < 5; the one-loop approximation
 § 7.Conclusions


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.

About the author
photoZaitsev R.O.
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.