Ideas by S.V. Vonsovsky and Modern Model Treatment of Magnetism

A review of fundamental works by Shubin and Vonsovsky on the formulation of the polar and s-d(f) exchange models is given. Their ideas are compared with subsequent developments in the theory of magnetism in d- and f-metals and their compounds. Modern approaches including different slave-boson and slave-fermion representations, formation of exotic quasiparticles etc. are discussed. Internal connections between various many-electron models (the Heisenberg, Hubbard, t-J, Anderson Hamiltonians) are presented. Description of anomalous rare-earth and actinide compounds (Kondo lattices, systems with heavy fermions and non-Fermi-liquid behavior) within the framework of the s-d(f) exchange model and related models is considered.


Introduction
The papers by S.P. Shubin and S.V. Vonsovsky on the polar model [1] put forward a program of building up a systematic theory of solids with account of electron correlations. Such a theory should explain simultaneously electric and magnetic properties of metals and combine localized and itinerant features of d-electrons. This program was extensively developed by many investigators, but is not fully completed up to now.
The s-d(f) exchange model [2] provided a basis to describe transport properties of transition 3dmetals and magnetism of 4f-metals. Later this model was applied to explain electronic properties of anomalous rare-earth and actinides compounds, including Kondo lattices and heavy-fermion systems.
The present paper is devoted to evolution of the ideas of many-electron models developed in the works by S.V. Vonsovsky and his colleagues.

Polar model
The many-electron polar model of a crystal [1] was proposed as a synthesis of the homeopolar Heisenberg model describing a localized-moment system and the Slater determinant approach treating many-electron system of a metal. The initial formulation of the model included the electron hopping and all the types of electron-electron interaction. The corresponding Hamiltonian in the second quantization representation was written down by Bogolyubov: This function mixes the Fermi-and Bose-type excitations and thereby does not satisfy the Pauli principle. Nevertheless, the quasiclassical approximation provides a rough description of metalinsulator transition in spirit of the Gutzwiller approximation (see [7]). The interest in the Hubbard model has been greatly revived after the discovery of hightemperature superconductivity. In particular, the electron states in CuO 2 -planes of copper-oxide perovskites may be described by the so called Emery Hamiltonian † † † † † ( ) where ε and Δ are positions of pand d-levels for Cu and O ions, and the k-dependence of matrix elements of p-d hybridization for the square lattice is given by At |V pd | ≪ ε -Δ the Emery model is reduced by a canonical transformation to the Hubbard model with strong Coulomb repulsion and the effective Cu-Cu transfer integrals In connection with the high-temperature superconductor theory Anderson [8] put forward the idea of separating spin and charge degrees of freedom in two-dimensional systems by using the representation of slave Bose and Fermi operators † † The physical sense of such excitations may be explained as follows. Consider the lattice with one electron per site with strong Hubbard repulsion, so that each site is neutral. In the ground resonance valence bond (RVB) state each site takes part in one bond. When a bond becomes broken, two uncoupled sites occur which possess spins of 1/2. The corresponding excitations (spinons) are uncharged. On the other hand, the empty site (hole) in the system carries the charge, but not spin. On the other hand, the empty site (hole) in the system carries the charge, but not spin.
In the half-filled case only spinon excitations with the kinetic energy of order of Heisemberg exchange | are present. At doping the system by holes, there occur the current carriers which are described by the holon operators . In the simplest gapless version, the Hamiltonian of the system for a square lattice may be presented as with ζ being the chemical potential, Δ the RVB order parameter determined by anomalous averages of the spinon operators, the hole concentration. Thus a spin-liquid state can arise (even in purely spin systems without conduction electrons) with long-range magnetic order suppressed, a small energy scale J, and a large linear term in specific heat, which is owing to existence of the spinon Fermi surface, † e e δ = 〈 〉 Later, more complicated versions of the RVB theory were developed which use topological consideration and analogies with the fractional quantum Hall effect. These ideas led to rather unusual and beautiful results. For example, it was shown that spinons may obey fractional statistics, i.e. the wavefunction of the system acquires a complex factor at permutation of two quasiparticles.
Taking into account a concrete physical problem, various representation of Hubbard's operators can be used. In the paper [9] a representation of four bosons i p σ , , was proposed which project onto the states Γ = σ, 2 and 0. Then the Hubbard Hamiltonian takes the form This representation enables one to reproduce old results on the metal-insulator transition yielding a Gutzwiller-type picture. Wang [10] proposed the representation with two kinds of Fermi operators and corresponding to holes and doubles: , The physical spin operators are connected with the pseudospin operators i s α by the relation Besides that, supersymmetric representations for the Hubbard operators were developed [11]. At theoretical consideration of highly-correlated compounds, including copper-oxide highsuperconductors, the t-J model (the Hubbard model with U and Heisenberg exchange included) is widely exploited. Its Hamiltonian in many-electron representation reads At derivation of the t-J model from the large-U Hubbard model, is the antiferromagnetic kinetic exchange integral. Using the above Fermi-type holon representation, in the case of hole conductivity ( ) we get

s s s s
This representation was applied to the magnetic polaron problem in an antiferromagnet.
In the ME representation one can demonstrate that the t-J model is a particular case of the narrow-band s-d exchange model, corresponding to , being replaced by (the factor of 2 occurs because of equivalence of both electrons in the Hubbard model). Being first proposed to describe transport properties of transition d-metals [2,12], this model turned out to be very successful to treat the properties of various dand f-systems. Recently, the narrow-band s-d exchange model with large |I| has been applied to colossal magnetoresistance manganites (the double-exchange problem).
The s-d(f) exchange model describes also correlation effects in the half-metallic ferromagnets [13]. These materials have an unusual electronic structure: the states with only one spin projection are present at the Fermi level E F . Thus an important role belongs to the so-called non-quasiparticle (NQP, incoherent) states which arise in the minority-(majority-) spin gap above (below) the Fermi level owing to the electron-magnon interaction. With increasing temperature, the capacity of the spin-polaron tail rapidly increases. The NQP states make considerable contributions to the electronic properties and can be probed, in particular, by spin-polarized scanning tunneling microscopy (STM). They also lead to observable effects in core-hole spectroscopy, nuclear magnetic relaxation and temperature dependence of impurity resistivity, etc. [13].
Similar (and even more strong) NQP effects occur in a Hubbard ferromagnet with large U and small concentration of doubles c. The corresponding Green's function of spin-up electrons has a non-pole form be obtained directly in the s-d(f) exchange model. A special mean-field approximation for the ground state of magnetic Kondo lattices [15] exploits the Abrikosov pseudofermion representation for spin operators and reduces the s-f exchange model to an effective hybridization model. The corresponding energy spectrum contains narrow density-of-states peaks owing to the pseudofermion contribution. Thus f-pseudofermions become itinerant in the situation under consideration. This fact, although being not obviously understandable, is confirmed by observation of large electron mass in de Haas -van Alphen experiments.

1/ 2 S =
Starting from the middle of 1980s [16,17], anomalous rare-earth and actinide compounds are extensively studied. They include Kondo lattices (with moderately enhanced electronic specific heat) and heavy-fermion systems demonstrating a huge linear specific heat. Main role in the physics of the Kondo lattices belongs to the interplay of the on-site Kondo screening and intersite exchange interactions. Following to the Doniach criterion [18], it was believed in early works that the total suppression of either magnetic moments or the Kondo anomalies takes place. However, more recent experimental data and theoretical investigations made clear that the Kondo lattices as a rule demonstrate magnetic ordering or are close to this. This concept was consistently formulated and justified in the papers [19] treating the mutual renormalization of two characteristic energy scales: the Kondo temperature and spin-fluctuation frequency K T ω . A simple scaling consideration of this renormalization process in the s-f exchange model [19] yields, depending on the values of bare parameters, both the usual states (a non-magnetic Kondo lattice or a magnet with weak Kondo corrections) and the peculiar magnetic Kondo-lattice state. In the latter state, small variations of parameters result in strong changes of the ground-state moment. Thereby high sensitivity of the ground-state moment to external factors like pressure and doping by a small amount of impurities (a characteristic feature of heavy fermion magnets) is naturally explained. During 1990s, a number of anomalous f-systems (U x Y 1-x Pd 3 , UPt 3-x Pd x , CeCu 6-x Au x , Ce 7 Ni 3 etc.) demonstrating the non-Fermi-liquid (NFL) behavior have become a subject of great interest (see the review [20]). These systems possess unusual logarithmic or power-law temperature dependences of electron and magnetic properties. The NFL behavior is typical for Kondo systems lying on the boundary of magnetic ordering and demonstrating strong spin fluctuations.

Summary
Being formulated already in the first half of XX century, the polar and s-d(f) exchange models still work successfully in the solid state physics. They provide a basis for new theoretical concepts describing physical phenomena discovered by experimentators. The model approaches which include effects of strong electron correlations in dand f-compounds turn out to be very useful from the point of view of the qualitative microscopic description.
The spectrum of highly-correlated systems is often described in terms of auxiliary (slave) Fermi and Bose operators, which correspond to quasiparticles with exotic properties (neutral fermions, charged bosons etc.). Last time such ideas have been extensively applied in connection with the unusual spectra of high-T c superconductors and heavy-fermion systems. Investigation of these problems leads to complicated mathematics, which uses the whole variety of modern quantum field theory methods, and very beautiful physics. For example, description of the Fermi-liquid state in terms of Bose excitations becomes possible. These concepts change essentially classical notions of the solid state theory. Modern many-particle physics is intimately connected to other fields of science: nuclear and elementary-particle physics, cosmology, quantum technologies, biology etc.
The work was supported in part by the Program "Quantum Physics of Condensed Matter" from Presidium of Russian Academy of Sciences.