-Model in Terms of Equations With Variational Derivatives

For a tJ -model in the X -operators representation a generating functional of the field describing fluctuations of matrix elements of electron hopping on a lattice is presented. The first order functional derivative with respect to this field determines the electron Green function, while the second order derivatives determine the boson Green functions of collective excitations in the system. Thus, the Kadanoff-Baym approach in the theory of fermi system with a weak Coulomb interaction is generalized on the opposite limit of systems with strong correlations. A chain of equations for different order variational derivatives were obtained, and a method was suggested based on iterations over the parameters of a tJ -model: the hopping matrix element and the exchange integral. This approach corresponds to a self-consistent Born approximation, not for the effective but for the original Hamiltonian. A scheme of calculation of the dynamical spin susceptibility is analyzed with self-consistent corrections of the first and second order. Connection of this approach with the diagram technique for X -operators is discussed.


Introduction
A tJ-model is the basic working model in the theory of strongly correlated electron systems.It is convenient for the study of an interaction of charge and spin degrees of freedom, because the model describes the correlated motion of electrons on the lattice.The model is given by the Hamiltonian (see 1]): (1 ?n i )C y i C j (1 ?n j ) + X ij J ij S i S j ? 1   4 n i n j : (1.1) Here C i (C y i ) is an operator of annihilation (creation) of an electron on site i with spin , S i | operator of spin, and n i | operator of the electron number on a c Yu.A.Izyumov, N.I.Chashchin site.Hopping matrix elements t ij and exchange integrals J ij are usually taken in the nearest neighbours approximations, so the model contains only two energy parameters: t and J.
At half-lling, when the number of electrons on a site is n = 1, the model (1.1) reduces to the Heisenberg model and has a dielectric antiferromagnetic ground state.When a deviation from half-lling takes place then some concentration of holes = 1 ?n appears.They strongly interact with the magnetic order and locally deform it.As a result, a compound quasiparticle (magnetic polaron) appears, being a carrier of an electron charge in the system.A mathematical description of this situation is achieved by the transformation of Hamiltonian (1.1) into an e ective Hamiltonian of a hole-magnon interaction.The self-consistent Born approximation (SCBA) is an approximation leading to a magnetic polaron picture inside the antiferromagnetic phase.The last one exists when < c , where critical concentration c 1 1].It is well known that in high-T c -compounds the superconducting state appears outside the antiferromagnetic phase when > c ; therefore, the question about applicability of the magnetic polaron picture for this concentration region is left open.For this reason a lot of new attempts are made to study the properties of the model (1.1) at > c 1].
We nd rather perspective here an approach on the basis of equations with variational derivatives in the spirit of the Kadano -Baym scheme 2].At the beginning it was applied to the description of usual Fermi systems with a weak Coulomb interaction.Our aim is to develop a similar approach for the opposite case of systems with a strong Coulomb interaction, for example, for Hamiltonian (1.1) describing the correlated motion of electrons.The most convenient way to do that is to use the Hamiltonian of the tJ-model in terms of the Hubbard X-operators.Then, by the analogy with usual Fermi operators, one can introduce uctuating elds, corresponding to the hopping term in the Hamiltonian, in contrast to Kadano -Baym who introduced uctuating elds of the potential interaction.As the result of such an approach, equations for generating functional Z and variational derivatives of Z over these elds are derived.These equations are convenient for iterations with respect to parameters t and J, in contrast to Kadano -Baym equations convenient for iterations with respect to Coulomb interaction U.
The obtained equations allow one to develop a scheme of the self-consistent Born approximation (SCBA), not for the e ective Hamiltonian, adopted for the description of the antiferromagnetic phase, but for the original Hamiltonian (1.1) and for the paramagnetic state as a ground state of the system.
The rst attempt of such an approach was given by us in 3].

Generating potential in terms of X -operators
Because of the projection factors (1 ?n i ) in the hopping term of the Hamiltonian, it is convenient to rewrite it in terms of X-operators: Here the rst X-operator is Fermi-like (f-type), and the others are Bose-like (btype).One can get then 4] H We included external magnetic eld h, that is why term " = ?h=2 appeared.
We also introduced in a formal way hopping matrix element t ij by spin index .We shall denote by gures complex indices including site i and imaginary time , so that 1 = (i ), etc. Equations of motion for f-and b-operators are written in the form: As usual, the summation (integrating) is implied over repeated indices because we introduced the quantities t 11 0 = ( ? 0)t ii 0 ; J 11 0 = ( ? 0)J ii 0 : Here and further we use a notation for matrix products, for example, (tX 0 )(1) t 11 0 X 0 1 0 ; : : : Finally, by n i and m i we denote the number of electrons on a site and local magnetization n 1 = X 1 ; m 1 = n 1 ?n 1 : Instead of quantities n 1 it is convenient to use F 1 = 1 ?n 1 , being an anticommutator of two f-operators X 0 1 ; X 0 1 ] + = F 1 : (2.5) Now we introduce the generating potential where 1 0 2 0 is an auxiliary external eld depending on thermodynamic time and spin.Here = 1=T and T is a time-ordering operator.Since F ] coincides in the form with the hopping term in the Hamiltonian, the quantity t 1 0 2 0 is to be considered as a uctuating eld of the electron hopping on the lattice.
The quantity Z t ] is useful for determining the Green functions of the system, as its variational derivatives over t 12 G (12) = ?It is clear from these de nitions that G is the Green function of itinerant electrons, and two others are the Green functions of collective Bose-like excitations: magnons and plasmons.The Bose-like Green functions are actually two-particle functions because b-operator may be presented as a product of f-operators: X 1 = X 0 1 X 0 1 ; n 1 = X 0 1 X 0 1 : For the generating functional one can derive the equation of motion with variational derivatives: or in the conjugated form Z t 12 0 K (2 0 2) = ?12 hF 1 i Z + 2 Z t 12 0 t 3 0 3 0 e (3 0 2 0 ; 2) + 2 Z t 13 0 t 3 0 2 0 e (3 0 2 0 ; 2): (2.12) Here K is a di erential operator, and e are matrix elements of the Hamiltonian.
K (12) = ?@ @ 1 + " !(1 ?2) ?t 12 ; (2.13) (1; 23) = t 12 13 + 12 J 13 e (32; 1) = 31 t 21 + J 31 21 9 = ; : (2.14) One must add the obvious relation to equations (2.11) and (2.12): Equation (2.11) is the basic equation corresponding to the Kadano -Baym method applied to a strongly correlated system.In contrast to the theory of a usual Fermi-system here variational derivatives are taken over the uctuating eld of the hopping term but not over the potential one.For this reason equation (2.11) is convenient for iterations with respect to parameters t and J, while in the usual Fermi-system variational derivatives are taken over the parameters of an electron interaction.
The basic equation (2.11) connects the rst derivative of Z with the second one.
To nd an equation for the second derivative it is necessary to take a variational derivative of equation (2.11) over t 34 .Then one obtains an equation connecting the second derivative with the third one: K (11 0 ) We write the following equation of the in nite chain obtained by the di erentiation of the previous equation with respect to t 56 : K (11 0 ) (2.17) After di erentiation in these equations one has to put variable t 12 equal to matrix element t 12 standing in the hopping term of the Hamiltonian.
In gures 1 and 2 a graph representation of the basic equation is given.A four-

Calculation of variational derivatives
In accordance with the graph representation for the second derivative shown in gure 1, we shall look for the form: 1 Z 2 Z t 12 t 34 = t 11 0 t 33 0 ?2 ( 1 0 2 0 ; 3 0 4 0 ) t 2 0 2 t 4 0 4 : (3.1) The quantity = t is an electron Green function (Z = e i ), while ?2 is to be considered a vertex part.In the theory of the usual Fermi system ?2 should be a real vertex part of an electron-electron interaction.In the case of strongly correlated electrons the situation is di erent and in representation (3.1) ?2 should be an operator.It acts on the Green function standing by the left-and the righthand side and can transform them into other Green functions.
The representation (3.1) is, however, convenient because it opens a way to calculate the second and higher derivatives.For this purpose equation (3.1) has to be multiplied by the left-and the right-hand side operator quantities (2.13).Then it is written: In the right-hand side of this relation we may use equations (2.11) and (2.12), and in the left-hand side { equation (2.16) (and a conjugated one).It allows one to nd ?2 as a series in powers of and .After rather cumbersome calculations we nd quantities ?2 for two second variation derivatives with equal and opposite spin indices.In the zero approximation ?0 2 ( 12; 34) = K 0 (12)K 0 (34) ?K 0 (14)K 0 (32) hF 1 i hF 2 i hF 3 i hF 4 i d 0 (13) + hF 1 i hF We see that expressions (3.3) and (3.4) contain operators K 0 which act in relation (3.1) on electron Green functions.In this way the expression for the second derivatives of Z splits into a number of terms containing not four electron Green functions but a smaller number of them, three or two.We have used the fact that at the zero approximation, according to equation (2.11), K 0 ( = t ) is just hF i.As a result, the second derivative of Z can be written in a graph form ( gures 3 and 4).When writing analytical expressions, one has to keep in mind the following rule: a complex vertex creates a numerical factor corresponding to each outgoing and ingoing electron line.For example, to the rst connected graphs in gure 3 there corresponds factor hF 1 i ?1 hF 4 i ?1 .Coe cients A and B for the triple graphs in gure 4 are equal to A 1 = hF 1 i + hF 1 i ? 1  2 hm 1 i ; B 2 = hF 2 i ?hF 2 i + 1 2 hm 1 i : Besides, according to the mentioned rule, to these two graphs additional factors hF i ?2 hF i ?1 or hF i ?1 hF i ?2 should be prescribed.The third order variational derivative is calculated in the same way.It is necessary to start from the relation of type (3.1) not with four but six factors = t.The complete expression is too complicated to write it down.It contains nonconnected diagrams corresponding to the product of the second derivative and the rst one.Among the connected diagrams there are graphs constructed only from electron lines ( ve lines) and graphs with one boson line and four electron lines converging to one point (as in the pure electron graphs in gure 4).Finally, there are peculiar graphs containing a structure corresponding to a six-tailed one with pairs of joint electron lines (a ower with three clovers).Such type of graphs is shown in gure 5.One can check that a complete set of graphs for the third derivative provides the necessary symmetry arising from the possibility to change the order of di erentiation.Besides, the symmetry is achieved according to the permutation of pair indices (3,5) and (4,6).
In principle, in the same way it would be possible to obtain a graphical representation for the fourth derivative (eight-tailed) of the zero approximation.

The magnon Green function
The magnon Green function can be found if the second derivative 2 Z= t 12 t 34 is calculated in some approximation.According to de nition (2.9), it is enough to equate indices 4 = 1 and 3 = 2.We calculate rst D 1 (12) at the zero approximation.In the left-and the right-hand sides of the graph relation in gure 4 we make these indices equal to each other.Then, an equation for D 1 (12) appears, which we write in the analytical form: One can check here the rules of writing analytical expressions.
For the paramagnetic phase we have from here: We neglect the last term in expression (4.2) which changes only a numerical coe cient.If in this expression sites 1 and 2 are put to be the nearest neighbours and also time 1 and 2 to be equal, we obtain a relation between the spin and electron correlators coinciding with the one obtained in 4].
To include corrections of the rst order in the magnon Green function we must calculate the second derivative with the accuracy up to the rst order with respect to t and J.Then, making the indices equal in the same way as in the zero approximation, we come to the equation, presented in gure 6.Its solution gives the magnon Green function of the form: D 1 (q) = ?(q) 1 ?n ?n 2 Q(q) + (q)] ?J(q) (q) ; (4.3) where f (q); Q(q); (q)g = X k f1; "(k); "(k ?q)gG(k ?q)G(k) : (4.4) corresponds to an electron loop and the loops with an inserted wavy line.Such expressions appeared earlier in the diagram technique with X-operators 7].Result (4.3) is consistent with the expression obtained by us 3], with a di erence in numerical coe cients.At q = 0 the last term in the denominator can be neglected, thus, the ferromagnetic instability of the system is determined by the hopping term in the Hamiltonian.In contrast, at q = Q = ( ; ; : : :)=a Q(q)+ (q) = 0 the antiferromagnetic instability is determined by the exchange term.In 3] we showed that in the Hubbard-1 approximation for electrons (Q; 0) (1 ?n) and antiferromagnetic instability occurs only when J t, which is not consistent with the known idea 1] that critical value J=t for the appearance of antiferromagnetic order should be small, as half-lling (1 ?n) is approached.It means that near n = 1 for the analysis of antiferromagnetic instability it is probably necessary to go beyond the mean eld approximation for electrons.The other possibility is to take into account the second order corrections for t and J.In the next chapter we will give a preliminary analysis of these corrections without explicit calculations.

On the second order corrections in the magnon Green function
It would be di cult to look for these corrections by the method of 3] because one should know self-energy of electrons up to the third order.In the present approach it is su cient to have corrections up to only the second order for the second derivative.We need for this the forth order derivative (eight-tailed) in the zero approximation.As we saw earlier, the number of terms increases very fast with the order of the derivative, and a selection of actual diagrams is necessary.In our case (nearly half-lling) this selection might be done on the basis of parameter 1 ?n 1.
Let us look for the magnon Green function (4.2) of the zero approximation.
Factor 1 ?n in the denominator does not yet mean a singularity because it may be cancelled by the numerator.However, it is necessary to pay attention to all the terms containing this factor in the denominator.Notice that this factor appeared due to the coe cient in the left-hand side of equation (4.1).We try to search for such factors in expressions for the third and fourth derivatives, however, we shall use now another method based on the Wick theorem for X-operators 5,7].
We start from the second order derivative written in the form: 1 where (1) X 0 1 is a Fermi-like operator.When averaging in (4.4) with the Hamiltonian of the zero approximation, the average of T-product would be reduced to the pair-averages by the procedure based on the Wick theorem.We should take all the possible systems of pairing.First, consider the systems with the pairing of only and y operators, for example: Here G (12) is the fermion Green function of the zero approximation.Our approximation now is the one, in which we take only -operators pairing and in expressions of type (5.2) the Green functions are replaced by the exact ones.They are related to the electron Green functions by the identity G (12) = G (12) hF 2 i : ( In this approximate relation we put indices 4=1 and 3=2, then we have an equation for the magnon Green function: " 1 ?hn 1 i hF 1 i hn 2 i hF 2 i # D 1 (12) = ?G (12)G (21) 1 + d 0 (21) hF 1 i hF 2 i : (5.5) Here we use the value for the electron Green function with coincided arguments: G (11) = hX 1 i = hn 1 i .
For the paramagnetic phase the coe cient in the left-hand side is proportional to 1 ?n, and we come to the result consistent with (4.2).Thus, the decoupling of the Wick theorem type leads for the second derivative to the same result as on the basis of representation (3.1).Consider now the third order derivative 1 Z 3 Z t 12 t 34 t 56 = D T (1) y (2) (3) y (4) ( 5) y (6) E : (5.6) Let us consider a system of pairing, when only and y operators are paired, for example, h T (1) y (2) (3) y ( 4) -( 5) y ( 6 (5.9) This third order derivative with equal arguments corresponds to a six-tailed diagram with joint lines.Comparing relation (5.8) with the graphs in gure 5 shows that decoupling by the Wick theorem corresponds to the two graphs explicitly shown in gure 5.One of them is a ower with three clovers.As we see from equation (5.9), the ower does not have a singular factor 1 ?n.
A di erent situation occurs if the forth order derivative (eigt-tailed) is studied by the same method.When calculating T-product of eight -operators, we take into account only two systems of pairing: D T (1) y (2) (3) y (4) (5) y (6) (7)    (5.11)We see from here that the derivative has a singular factor 1?n in the denominator.The di erence in the third and fourth derivatives is caused by di erent signs in diagrams of the ower type in six-tailed and eight-tailed diagrams.It is easy to see that the singular factor should always appear in the ower type diagrams with an even number of clovers.
If in expression (5.10) for the fourth derivative we take only the last term, we obtain a second order correction in the magnon Green function presented in gure 8.For this graph an analytical contribution into the denominator of expression J(q 1 )D 1 (q 1 )G (k ?q 1 ) ! : (5.12)Of course, a symmetrical graph in gure 8 should be added which gives expression (5.12) with the change q !?q.
Due to singular factor 1 ?n, the second order corrections are as important, when n ! 1, as the rst order ones.Zero in the denominator of the magnon Green function (4.3) at ! = 0 and q=Q determines a boundary of the paramagnetic phase stability with respect to the appearance of an antiferromagnetic order.As one can see from (4.3) and (5,12), this boundary goes along the line of type 1 ?n J=t, as it should be.Detailed analysis of the magnetic phase diagram will be given elsewhere.

Conclusions
We have generalized the Kadano -Baym approach to systems with strong electron correlations and applied it to a tJ-model.The X-operator representation allows one to derive equations for variational derivatives of the generating functional in a form convenient for iteration with respect to t and J. Within the framework of the general approach we suggested an approximation of the SCBA type, where electron and boson Green functions are considered as exact, and vertex parts are expanded over t and J.In such an approach the problem of calculation of the Green functions reduces to the calculation of the variational derivatives in the "zero" approximation with exact Green functions.It opens a possibility to obtain self-consistent equations for Green functions.
We have suggested a method for the calculation of variational derivatives in the zero approximation, which gives a correct symmetry arising from the possibility to change the order of di erentiation.The results for such calculations are too cumbersome and we presented them only for the second derivatives.It is remarkable that they are consistent with the diagram technique for X-operators based on the generalized Wick theorem 7].
We used this formalism for the calculation of the magnon Green function in the paramagnetic phase.To avoid cumbersome calculations of higher order derivatives based on the representation of the (3.1)-type we calculated the third order derivative by using the Wick theorem, because in this case the exact symmetry is not important.We have found graphs giving a growing contribution when the system tends to half-lling.In n ! 1 limit only graphs of the second order containing singular factor (1 ?n) ?1 should be taken into account.They produce instability of the paramagnetic phase with respect to antiferromagnetic ordering.Singular contribution (5.12) in the denominator of the magnon Green function comes from the graph in gure 8.One can see that it involves a magnon in the intermediate state.The corresponding physical process is the following: a magnon creates an electron-hole pair with the same spins and still it exists itself.Then, one of particles of the pair absorbs this magnon.As a nal result, an electron-hole pair with the opposite spins appears.Notice that the possibility of such a virtual process is connected with the existence of two types of vertices for the electron-magnon interaction in the diagram technique for the tJ-model: elastic and inelastic ones.Both vertices are involved in the second order graph.
We consider the results of the calculation of the magnon Green function as preliminary ones.It is necessary to look for singular terms among high order diagrams.It is also necessary to calculate the plasmon Green functions (2.10), particularly for the longitudinal spin deviations, and also to calculate the second order correction to the electron Green function.In this way a self-consistent system ефективного, а для вихідного гамільтоніану.Аналізується схема розрахунку динамічної спінової сприйнятливості з самоузгодженими поправками першого і другого порядку.Обговорюється зв'язок цього підходу з діаграмною технікою для X -операторів.

Figure 1 .
Figure 1.Graphic elements.taildiagram denoting a second order variational derivative is not a vertex part but a two-particle Green function.We shall see later that the zero approximation over contains a nonconnected part, generating in the equation the rst order graphs, corresponding in the usual Fermi system theory to Hartree-Fock terms.The connected part of the zero approximation generates new type graphs absent in the Fermi system theory.Corrections of the rst and next orders generate vertex parts, more complicated in comparison with the usual Fermi systems 5,6].

Figure 4 .
Figure 4. Graphic representation of 1 Z 2 Z t 12 t 34 in zero approximation.Dasked lines with an arrow denotes the magnon Green function (2.9).

Figure 5 .
Figure 5.The third order variational derivative in zero approximation.

(5. 3 )
Taking into account in (5.1)only two systems of pairing, we can write 1

Figure 7 .
Figure 7.The forth order variational derivative in zero approximation.

Figure 8 .
Figure 8.A second order correction to magnon Green function.