Sd-model with strong exchange coupling and a metal-insulator phase transition

Sd-exchange model (Kondo lattice model) is formulated for strong sd-coupling within the framework of the Xoperators technique and the generating functional approach. The X-operators are constructed based on the exact eigen functions of a single-site sd-exchange Hamiltonian. Such representation enables us to develop a perturbation theory near the atomic level. A locator-type representation was derived for the electron Green’s function. The electron self-energy includes interaction of electrons and spin fluctuations. An integral equation for the self-energy was obtained in the limit of infinite localized spins. A solution of this equation in the static approximation for spin fluctuations leads to a structure of electron Green’s function showing a metal-insulator phase transition. This transition is similar to that in the Hubbard model at half filling.


Introduction
The sd-exchange model is one of the fundamental models in the theory of strongly correlated systems.This model is frequently referred to as the Kondo lattice model in the West.Its Hamiltonian takes into account the hopping of conduction electrons over a lattice and the exchange interaction of them with the localized atomic moments (value spin S) placed at the lattice sites.The Hamiltonian is written as Here c iσ (c † iσ ) is a Fermi-operator of annihilation (creation) of an electron on a site i with spin projection σ.
For the case of strong sd-exchange coupling S|J| W (W is width of the electron band) the sd-model was applied for description of a metal-insulator phase transition [1,2] and physics of magnetic semiconductors [3].Exact solution of eigenvalue problem for a single site exchange Hamiltonian was applied [1][2][3].Two energy levels E + and E − for the states with the total spin on a site j + = S + 1/2 and j + = S − 1/2 are spread into two bands by hopping term, that determines the physics of the model in the strong coupling limit.
In recent works [4,5] we applied the generating functional approach (GFA) to this problem which, in a most general form, permits to construct a perturbation theory "near the atomic limit".Equations for electron Green's functions (GF) were derived in terms of functional derivatives over fluctuating fields introduced in GFA, and the simplest approximations for their solution were tested.In the present paper we return to the problem on a most general basis.Particularly, we introduced four-component electron operator instead of two-component one used in works [4,5].Such generalization of mathematical structure makes it possible to construct more correctly the perturbation theory near the atomic limit and to consider the states with more complicated order parameters such as magnetic ordering or superconductivity.
In the case of strong sd-exchange coupling one has to use the exact solution of the problem on one site.Wave functions of the Hamiltonian J/2 Sσ are known [1,2]: where A wave function |M0 describes the state of a site without an electron when the atom is in |M state with projection of spin M; |M2 is a state with two electrons on a site, and |M α α with only one electron with projection of total spin j α and (1.6) In |M α α state the wave function is a superposition with both projections of an electron spin with mixing coefficients being Klebs-Gordan coefficients.Selfenergy of one site exchange Hamiltonian is as follows: Because the self-energy is known it is possible to develop a perturbation theory over small parameter W/JS.The best way to do it is GFA combined with the formalism of X-operators.

Formalism of X-operators
Any one-site operator A i can be expanded over the system of X-operators determined on the base of functions |p .This explanation is written as The calculation of a matrix element of an annihilation operator c iσ leads to its representation in terms of X-operators [2,[4][5][6][7]: as well as a line Ψ † iσ (I) including the conjugated X-operators.Operator Ψ 1σ (I) describes the creation of an electron on site i at time τ (1 = iτ ) with spin σ when the atom has an angular momentum projection M, with two possible combinations of these momenta labeled by indexes α = +, −.Components of spinor are numbered by index ν = 1, 2, 3, 4. Its combined index I = (Mν).
In such representation the Hamiltonian of sd-model is written as where ) and T is a 4 × 4 matrix with all elements being equal to 1.
Let us introduce a supermatrix GF each of its elements is a matrix 4 × 4.
Here T • • • V is a statistical average of some chronological product of operators of the system in fluctuating fields depending on both sites and times where 2 )α 2 1 . (2.12) Herein below a sum over the repeated primed indexes is implied.The rest of the definition corresponds to the standard technique of the temperature GFs [8].
The introduction of the fluctuating fields makes it possible to derive an equation of motion for electron GF in terms of functional derivatives with respect to these fields.This is a principle of GFA.
We have to write an equation of motion for each matrix element of GF (2.9).For example for an element "11" a general form of the equation is: where we introduced a short notation T . . .V = Tr e −βH T . . .e −V . (2.14) Having matrix elements for GF constructed on X-operators we are able to write down the electron GF determined on Fermi operators.To this end, keeping in mind formula (2.3) we must sum up over indexes M 1 and M 2 giving the states of atoms.It is reasonable to do this summation directly in equation (3.1).Then one can obtain the equation for "averaged" GF if we define it by relation: where with diagonal matrix ϑ σ (M) (2.8).Multiplicative character of matrix (2.7), for the electron hopping on the lattice makes it possible to derive an equation for averaged GF L. We can write this equation as follows: Here all overlined quantities are determined by the relations of type (3.7).For example operator A 1 and effective hopping T 12 are given by matrices: (matrix T was defined following the equation (2.8)).In equation (3.10) quantity F is defined by relation: and so on.An explicit form of matrix elements F, Q and their conjugated ones can be easily written if necessary.Notice only that for 4 × 4 matrix and in matrix Q only elements do not vanish (spin indexes σ 1 and σ 2 are dropped).Zero order GF L 0V in the fluctuating fields is given by a relation

Solution of equation for the electron Green's function
Equation (3.9) has a standard form of equation for basic models of strongly correlated systems.As usual we look for its solution in the multiplicative form: We see that the first order terms in λ i (i = 1, 2, 3, 4) are expressed by averages of X-operators and the second order terms by T -products of two X-operators, being bose-type GFs.
In equations (4.9) and (4.10) some combinations of electron GFs were introduced, namely We shall look for a solution of Dyson equation for propagator G σ (k), when the self-energy Σ is determined by equation 4.7) and equation (4.8).Due to a specific matrix (4.8), the solution can be written as where z n = iω n + µ.Explicit expressions for matrices G and G are complicated and we do not write them down.
One can see that in expressions for λ i determining the electron GF, some linear combinations of X-operators may appear which determine the components of the total spin on a site as well as the components for the pseudospin [5] Along with these combinations in expressions for λ i there are terms containing operators of the type which describe the transfers on a site with change of the total spin S + α 2 on S − α 2 .At large values of sd-exchange integral such transfers give a small contribution to statistical averages.We shall neglect such terms.Therefore, all off diagonal matrix elements F ij can be omitted.In the rest of the expressions for λ i , the following combinations of matrix elements appear: where k = {k, iω n }.G(k) denotes a Fourier transform of the quantity tG (1) − tG (2) involved in expression (5.5).
Using the spectral representation for Fermi-like GFs g(k) and G(k) and Bose-like GF D(k) we present the expression (5.7) for ξ(k) in a form where f [ω ] and n[Ω ] are functions of Fermi and Bose, respectively.After analytical continuation iω n → ω + iδ we obtain an equation for ξ k (ω), which is an integral one over frequency and momentum.It should be solved together with the equation for chemical potential (5.12) Now we do an estimation of quantity ξ(k) under the following approximation: consider limit of static fluctuations ImD(q, Ω) ≈ π a q δ(Ω) (5.13) and neglect q-dependence of the spectral density a q ≈ a. Then for temperature T = 0 from (5.11) it follows where we use a rectangular form of the bare band and mean field approximation for electron GF g(k), equation (5.9) with ξ(k) = 0. with (5.16) determine two branches of quasiparticle spectrum.When electron density n 1 chemical potential µ lies in the lower band, and ε means energy ε q corresponding to fixed µ; ε obeys equaion E 1 (ε) = µ.One can see that ξ(ω) has logarithmic singularity at ω = 0 for account of electrons near Fermi-level.
In conclusion, we notice that, contrary to paper [4], here we developed an approximation for the sd-model with strong sd-exchange coupling beyond the mean-field theory to take into account fluctuations in the system.Electron GF (5.9) contains contribution of magnetic fluctuations via ξ(k)-quantity.Further analysis of the problem reduces now to the calculation ξ(k) as a function of momentum and frequency.
Preliminary analysis of equations (5.9)-(5.12)shows that at half filling a gap between two branches of quasiparticle spectrum vanishes at large enough exchange parameter I, and an insulatormetal phase transition occurs.A detailed numerical analysis of equations (5.9)-(5.12)at half filling and beyond it will be done elsewhere.
This work was supported by Russian Foundation for Support of Scientific School, grant NS-4640.2006.2.