A new self-consistent method to solve the SBO green's function equations

Volume 73A, number 2

PHYSICS LETTERS

3 September 1979

A NEW SELF-CONSISTENT METHOD TO SOLVE THE SHO GREEN’S FUNCTION EQUATIONS M. DRZAZGA Institute of Physics, Silesian University, 40-OO7Katowice, Poland Received 15 June 1979

A new self-consistent method to calculate the standard-basis-operator (SBO) Green’s functions desaibing “collective excitations” is obtained. The method is equivalent to an iterative scheme where RPA Green’s functions are used as a starting point.

The equation-of-motion method for temperature-dependent Green’s functions has been widely used in solid state statistical physics. Standard-basis-operator (SBO) Green’s functions seem to be the best because they lead to the possibility to take into account the full one-site operator algebra. Recently, Haley [1] had developed the random-phase-approximation RPA for that type of Green’s functions, valid for a wide class of hamiltonians. After that, decoupling the second-order equations of motion, a Hubbard-Ill-like [2] approximation has been obtained by Robinson [3] for hamiltonians with fermion-type interaction. It is the purpose of this paper to point out how, writing an infinite chain of equations of motion, one can get a self-consistent method to calculate SBO Green’s functions. This method is equivalent to an iterative scheme with RPA Green’s functions as a starting point. It can be used for a large class of hamiltonians which includes the electronic Hubbard-like model and the Heisenberg spin model. After a small modification the method can be ‘adopted for the s—d model. The notation used in this paper is in agreement with that of Hubbard-W [4] and Haley [I].

E~

~

H = fa

E

L~~

+ f,g

+ f,g ~

~4 + E i~~ f,g

(B5,> = flO + H1,

(1)

where ë~Ea+2 EI~~pj(B~)

(2)

g

are the single-site effective-field energies and A~ (B~>are the fluctuation operators. The set of operators {L~,B~J} {X~}forms a complete standard-basis-operator algebra on the ith site. and B~denote fermionlike and boson-like operators, respectively. They obey the following commutation rules: —

r vi

v/ 1

—

a

ía

vi

+

a

vi

\

L’~ 4~’ AgpJ ~ UjJ~U~3,~ ~cw _~~avtgj1~~ 0 in hamiltonian the + sign is used (1) describes when X~,~ the one-site are fermion-like energy, the operators, second term thea fermion-type sign fOr the interaction other cases.(electron The firsthopping term H between sites, for example), and the third term a boson-type interaction (exchange, for example). We shall consider the temperature Green’s functions: —

=

—

in(t)(E(t),~tj~ 1~.]+),

(4) 137

Volume 73A, number 2

PHYSICS LETTERS

3 September 1979

where both operators are either boson-type (—sign) or fermion-type (+sign). The first-order Green’s function equations of mo~ionhave the fo~n: [E_h°1io,jai((L’oiILJ~)>=Aja, +h’j~go1<(L~1IL~))+ 1~ ~ [E—hO]’°~”°’ ((h~1i~ê~>> =Aio~,J). ~ ~ ~>~jfr~,~1,ga2 ~ ~2 )>+T~1g02((L~,1L~2 h~)). Here the matrices T, T and I, Tarise from the commutation rules for X~and the fermion-type and boson.type 0 from commutation with H0. The superscripts denote the boson-type variables, interaction, respectively, and h and the subscripts the fermion-type, and the notation a = (o,o’) has been introduced. The matrices h’ and I’ appear in eq. (5) as a result of the substitution B~= + .If we had neglected the Green’s functions of second order we would obtain the random phase approximation. However, we make no approximation in the first-order equations in this paper. Let us note, that if one type of excitation has appeared at the first step in the ith site then it propagates as excitations ofvarious kinds to the neighboring sites “g” due to the interaction. If we are interested in the propagation of a collective excitation we should exactly commute the operators at the gth site in the second-order Green’s function equations. This leads to propagation of the excitation to the g’th sites which are the nearest neighbours of “g”. On the other hand, the commutation of the operators at the ith site again gives, as it were, a “second wave of perturbation” which propagates around the same ith site. As we are interested in collective excitation, and we are not able to take into account all “perturbation waves”, we take into consideration the “first wave” correctly and from the “second one” the h0 part and such terms which can be reduced to Green’s functions which have appeared earlier. Let us write, for example, the second-order equation of motion for one of the Green’s functions: [E—(h°)(~)] ((E~L~

1IL~)> h~ai,ga2((E~L~2I L’~)>+ T~,g’o2((E~~ L~2IL’X))

(6)

(,

+I%~jO~i,gg2

0, D from the reduction of that term with where follows from commutation of both [B~, operators with H h’ for g’(hO)(a) = i, and C from thatthe part of the commutator H] which can be reduced to the last term. The other

terms are neglected since they give rise tohigher-order corrections as was pointed out by Hubbard [2]. Similarly we deal with the nth order Green’s function (C~ >). We commute the last operator ~ exactly, the one before the last approximately, and do not commute the others at all. ~or example: ...

[E

(h0)(0~_1)] ~

+

•..B~fl ~L~I)) = ~

r:,~:,~+1~~+1 ~ •••~~~~‘i h~Lf1

+~

((X~...L~J’1I>) + ~


~1

I)) ~

~i

~ ~

I)) L~E~1~

(7)

(Ct41 ... L~I>),

and so on. In this way we obtain an infinite chain of equations of motions. Eqs. (6) and (7) are in their functional form like those obtained by Hubbard [2]. Thus we shall construct the solution similarly. Let us assume that we have eliminated all Green’s functions higher from 0)~’~ 1)of order (i~0)(’~n —i),than h’ nh’, andthe the dischain of equations. This leads to the renormalization of the matrices (h appearance of terms with Tand I in eq. (7). In these new equations we can again express the Green’s functions of nth order by those of(n— 1)th order, etc. Coming back to the first-order equations we get the solution: -+

~

[E—

138

—

~‘(k)] ~.

A~,xe/k(1 _I),

((b~I~)~-~ [E—h~ —7(k)] ~

-+

ej~i),

(8)

Volume 73A, number 2

PHYSICS LETTERS

3 September 1979

where: .

=

iO,W

to,1a2

2

=

+ T’°

(W .~(°)D’° . + Thor. (~gor,lcrci,i~(o1) ~k~i g~~,bu3,j’ ‘U3,1(2 W,lG~ ‘ /

b0,8~1 ‘

h~0,102+ T~,g01(Wgai,ia3,i)(°)C~3102+ =

h~’°’~°4 + j~,za1,gor2~

=

!~~1

~ba~03,1or4

(W~2”~3”)~1) C~,mr3,£24

+ j,i,g2

and:

(Wgor,ia1,~)(01) b~~~102 ~

+ T~1,ga2 (Wgoa +

ix3,i)@1) ~~lq3’

T !~,ga2(Wg02 lu3,i)~°’~ DiO~,I(i3

4Y’(Ggç,)~’°)]a,~’

(Wgu,ia’,g’)~= [(Ggi)~ (Gg,g~) k(Gg’,g’)~’

(9)

,

(10)

—

(Gga,io’)~=j~E[E_(h°)~

—1i’(k)];~.~_ik(g_l)

(11)

(hL 0.10101’)= (~~3 e~)+ —

FOci010’1

‘

(12)

and similarly for boson variables. The self-consistent set of equations (8)— (12) is equivalent0, to chain Green’s function equations of h’,an I’ infinite we obtain theofrandom-phase-approximation. motion. Inserting into these equations the unrenormalized h This can be considered as the first step in the iterative scheme. Next, substituting the RPA Green’s functions in eqs. (10) and (9), we get the renormalized ~ ~‘, I’, which give rise to the second-order approximation for the Green’s function, etc. Cutting the iteration at the nth order is equivalent to cutting the chain of equations at the nth order. For the Hubbard model, where there is no boson interaction, the results are equivalent to the Hubbard III approximation. However, Hubbard has obtained this approximation by cutting the chain of equations at second order and then replacing the resulting RPA Green’s function with the true one. As one can see this is equivalent to an infinite chain of equations of motion. From eqs. (8)—(l2) it is evident that the renormalization of a fermiontype Green’s function is connected with the renormalization of a boson-type Green’s function. This has not been considered by Hubbard. The energy spectrum following from eq. (8) can be complex in general. It leads to damping and a finite lifetime of pseudo-particles. This fact is related to the scattering of pseudo-particles at the fluctuations and other types of pseudo-particles. A more accurate description of the method together with applications will soon be published elsewhere. I should like to thank Professor Andrzej Pawlikowski for helpful discussions and valuable suggestions when writing this paper. The work is partially supported by the Ministry of Science, High Education and Technology. References [11 S.B. Haly, Phys. Rev. B17 (1978) 337. [2] J. Hubbard, Proc. Roy. Soc. A281 (1964) 401. [3] J.M. Robinson, Theory of electronic structure of rare earths and actinide metals using Hubbard’s alloy approximation, Indiana University-Purdue University at Fort Wayne (1978). [4] J. Hubbard, Proc. Roy. Soc. A285 (1965) 542.

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