IL NUOVO CIMENTO VOL. 30B, N. 1 l l Novembre 1975

Self-Consistent Formulation of Itinerant-EleetronFerromagnet.

M. ~ . SEAH

Department o/ Physics, University o/ Wisconsin-Milwaukee - Milwaukee, Wis.

G. VITIELLO

Istituto di Pisica dell' Universitd - Salerno

(ricevuto il 30 Dicembre 1974)

S u m m a r y . - - The self-consistent method in quantum field theory is applied to the itinerant-electron ferromagnet. The magnon as a bound state of electrons is studied by using Bethe-Salpeter equations in the pair approximation. The spin rotation of electrons is induced by an E~-symmetry transformation of magnons and quasi-electrons which is different from a rotation. A simple derivation of the low-energy theorem is discussed.

1. - I n t r o d u c t i o n .

In a recent paper (1), hereafter referred to as I , we showed tha t the magnons

in a ferromagnetic system behave, not according to the spin rotation symmetry ,

bu t according to the symmet ry of another group called E~. This was proved

without any approximation by using the path integral technique. Furthermore,

the arguments presented in I were completely general; no assumption was made

on the Lagrangian except its invariance under the spin rota t ion t ransformat ion

0.1) ~(i)(X) ~ ~( i ) (x ) - - Oi Ei~k S(k)(X) ,

(1) M. N. SH~, H. U•EzAwA and G. VITIt!ILLO: Phys. Rev. B, 10, 4724 (1974).

21

22 M.N. SHAH and G. VITIELLO

or, in terms of the electron field v2(x),

(1.2) ~(x) --> exp [iO, ~] ~(x) .

I n (1.1) ei~ = (-- 1) ~, where p is the number of permutat ions of 1, 2 and 3, and the spin densi ty operators S(~)(x) are made of the Heisenberg electron field v2(x )

( 1 . a ) =

The results found in I hold for the case of localized spin (Heisenberg ferromagnet) as well as for the continuous spin distribution (it inerant-electron ferromagnet). Wi thou t any approximation, we found that , when mat r ix elements of the spin densi ty operators are expressed in terms of quasi-particles and are summed over all the space points, then the spin matrices formed by these mat r ix elements satisfy the algebra used in the Holstein-Primakoff (~) approximation. This does not mean tha t the Holstein-Primakoff approximat ion is justified. In their article they considered a nonlinear representat ion of the ro ta t ion group where the spin operators are expressed in terms of the magnou Heisenberg operator a~. Fur ther , they used a linear approximat ion in order to diagonalize the Hamil tonian. Instead, we found that , if we express the spin densi ty operators in terms of quasi-particles, i.e. quasi-electron and quasi-boson B (magnon), then we are natura l ly led to the linear representat ion of the

B,-group. Although the pa th integral me thod presents various results which are in-

dependent of specific model and of any approximat ion, i t does not help us in comput ing model-dependent quantities. The main purpose of the present work is to present a model-dependent computat ion which is more interest ing from a praet ieal point of view. We will consider the case of the i t inerant-electron ferromagnetism.

I t is well known (8) t ha t the tIeisenberg model leads to difficulties when applied to metals where electrons cannot be regarded as (, localized ~). On the

other hand, not all the results obtained from the Heisenberg model for ferro- magnets are successfully recovered in the i t inerant-electron model (3.~). The approach we will use in this paper is the self-consistent method of quan tum

(a) T. HOLSTEIN and H. PRIMAKOFF: Phys. Rev., 58, 1098 (1940). (a) D. C. MXTTIS: The Theory o/ Magnetism (New York, N.Y., 1965); C. HERRING: Magnetism, Vol. 4 (New York, N.Y., 1966); D. WAGNER: Introduction to the Theory o] Magnetism (Oxford, 1972). (4) E . P . WO~ILF~-RTH: Ferromagnetism and exchange in metals, in Elements o] Theoretical Magnetism, edited by S. KRUPICKA and J. STERUBERK, Chap. 5 (London, 1968).

8ELF-CON$1STENT FORMULATION OF ITINERANT-ELECTRON FERROMAGNET 2 3

fieht theory if.s). We will perform our computa t ion in the pair approx imat ion , i.e. wc will consider only those processes which preserve the number of pairs.

Le t us briefly summar ize the self-consistent me thod in q u a n t u m field theory. In the self-consistent method one expresses the Heisenberg operators (in the present case, the Heisenberg electron field ~v(x)) in te rms of quasi-particle fields, which satisfy linear homogeneous (i.e. free field) equat ions; thus we write

(1.4) ~p(x) ---- ~ (Z(x) ) .

The relation (1.4) is called dynamica l m a p and mus t be unders tood as a (~ weak ,~ relation, i.e. as a relation among matrix elements:

(1.5) <a[~v(x) ]b> ---- <al~(X(x))Ib>.

Here ]a> and ]b> represent quasi-part icle s tates (b). The symbol Z(x) in (1.4)

and (1..5) denotes all the quasi-particle fields present in the theory. We will see t ha t the internal consistency of the formulat ion will require the existence

of a gapless boson B(x) which is a bound s ta te of electrons. This boson B(x) is the magnon. Denot ing the Hami l ton ian for quasi-particles as H0, we use (1.5) to obta in

(1 .(~) d d <a[[w(x), H]lb > = ih(a[ ~ ~,(x)lb> = ih<a] ~ ~P(g(x))Ib> =

= I b > .

This shows tha t the Hami l ton ian for quasi-part icles acts as the t ime t ranslat ion opera tor as H does. In other words we have

(1.7) (aIHlb > = (alHo[b >

when we chose the c-number up to which Ho is defined, in a convenient way. I t is known (5.,) t ha t the relat ion (1.7) does not mean tha t the quasi-particles do not interact . The relat ion (1.7) only means tha t the interact ion does not

create any energy; the to ta l energy of the sys tem (which conserves in the inter-

action process) is indeed given by the sum of the energies of the quasi-particles. When we express Heisenberg fields in te rms of quasi-particles, we also say tha t we use the quasi-part icle picture.

(s) H. UMF~ZAWA: ;~'UOVO Cimento, 40 A, 450 (1965); L. LEPLAE and H. UMEZAWX: NUOVO Cimento, 44 B, 410 (1966); L. L]~PLAE, :R. N. SEN" and H. UMEZAWA: Suppl. Prog. Theor. Phys., extra number, 637 (1965). For a self-consistent formulation of tho Heisenberg model for ferromagnet see A. K. BENSON: Phys. Rev. B. 7, 4158 (1973). (6) J. ~EST, V. SRINIVASAN and II. UMEZAWA: Phys. Rev. D. 3, 1890 (1971).

24 M..'q. SHAH and G. VITIELL(~

We find that , although we use the pair approximation, the results presented in this work agree with the rigorous relations obtained in I : the spin rotat ion transformations (1.1) (and (1.2)) of the Heisenberg fields are induced by the E,-group transformations of the quasi-particle fields r and B(x). This fact is expressed by saying tha t the original spin rotation symmetry is dynamically rearranged. In I we analyzed in detail the origin of the change of the symmet ry group and we found tha t it is due to the fact tha t in macroscopic observations one misses locally infinitesimal effects of order of magnitude 1/V (with the volume V -~ cr The missing effects are responsible for the change of the sym- met ry group. When they are taken into account by integrating over the whole volume of the system, the original spin rotat ion symmet ry group is recovered (1.~). I t is interesting tha t these results can be obtained also in the pair approximation when one uses the quasi-particle picture. One of the most important conse- quences is the well-known low-energy theorem, which states tha t reactions among magnons vanish in the low-momentum limit. This theorem, first proved by DYso]~ (s), is indeed based on the E, symmet ry of quasi-particle transfor- mations (1).

In the present paper we will restrict ourselves to T = 0 ~ and we will t reat the magnon, which is a bound state of fermions, by means of the Bethe- Salpeter equations.

2. - T h e q u a s i - f e r m i o n .

We consider the Hamil tonian (~.*)

f ~ ~ (2.1) H = h d 'x [~e (~)~-4 - ~P~e(O)Vr + ; t V ~ v 2 t - - / ~ ( ~ v ~ A- ~ v ~ ) ] ,

where ~.~ are the Heisenberg fields for the electrons:

\~ ,~(x ) ! ' (2.2)

and

1 (2.3) e(~) = -- 2--m (V' + g ) ,

where k r is the Fermi momentum and m is the electron mass. The last term in (2.1) is introduced to eliminate the self-energy of the electron.

(7) H. I-~ATSU~rOTO, H. UMEZAWA, G. VITIELLO and J. K. WYLY: Phys. Rev. D, 9, 2806 (1974). (s) F . J . DYSON: Phys. Rev., 102, 1217 (1956).

SELF-CONSISTENT FORMULATION OF ITINERANT-ELECTRON FERROMAGNET 2 5

Here we assume the equal-time ant icommutat ion relations for the Heisenberg electron field ?(x)

(2.4) [,p,(x), ~o~,(y)]+ O(t, -- t ) --(y~ (x) V~,(y) + yJ~,(y) ~o (x)) (~(t -- t,) = 5(x-- y) (~,~,;

s and s' s tand for f or ~. Use of (2.4) leads us to the field equat ion

(2.5) l ~ ~ ) e(~) + -~ ~ - - M a ~ ~o(x) ~ - - .s' - - ~la.~ v / (x) + #~o(x) ,

where a3 is the 3rd Pauli matrix, s is given by

() (2.6) s ~ s~ \'fl~flr '

and the term -- )~a3 added to both sides of (2.5) acts as energy counter- term. I ts presence is justified by the fundamenta l requirement tha t t he to t a l Hamil- tonian H must be equal, up to a c-number constant, to the free Hamfl tonian (cf. eq. (1.6)). In other words, the interaction should not create any self-energy for the quasi-electron. This is equivalent to requiring tha t the quasi-electron field

(2.7)

satisfies the free-field equation

= t '(xq t~,Cx)/

(2.8) (17 ~ + E(V~)) ~(x) = 0

with

(2.9) E(V 2) -- (s(~) -- Jl~ra3).

The ant icommutat ion relations for ~v(x) are

(2.10) [~v(x), t ( y ) ] + 5( t -- t ) = 6(x-- y) I ,

where I is the unit matrix. Let us note tha t the quasi-electron energy (cf. (2.9) and (2.3))

(2.11) E~,~ = e~ :F

with

1 (2.12) e~ -- 2 ~ (k' -- k~)

M. N. s H ~ and G. YITIF.LLO

can be either positive or negative. We then associate the annihilation operator ~ , ~ with the positive energy and the creation operator flt_k~,~ with the negative energy in order to ensure the positive definiteness of excitation energies. The ant icommutators are

~z.']+ [,Ok., = ~ ( k - - l) ~~ (2.13) [~k,, ~ * = fil.'l+

all other ant icommutators are zero. The quasi-electron field q(x) is then written as

_ [ d"k_ ' E (2.14) ~v~,~(x) - - j (2~)! [a~t'~O(Ekt'~) + fl-h~,~O(-- k~,~)] exp [ i k . x - iE~t, ~ t ] ,

where 0 is the step function. We now introduce the 2 • 2 matr ix

(2.15) S(x-- y) =--: <01T[~(x), ~*(y)]10>,

where T[...] means chronological product of the fields ~v. S(x -- y) is the Green's function of (2.8):

(2.16) i ( 1 ~-- + E(V~_)) S(x--y) = 5(x--y)I ~t~

We have 0

(2.17) S(x - -Y )= (S'(o--Y) S~(x--y))

with

�9 d~k (2.18) S,,(x--y) = S,(x--y) =-- <01T[~0V(x), ~0~(x)][0> = , j ( ~ ) , .

.[ ] O(k ~Q~)A- 1 O(Q~--k")] exp[ik(x--y)-- iE(t~-- t , )] , E -- E~, -4- ie E -- Ek+ -- ie

where the limit e-+ 0 is understood and

(2.19) Q~,; ~ k~ 4- 2m)J~ r .

Equat ion (2.18) with ~ replaced by ~ gives $22(x--y)- ,.~(x--y). In the self-consistent method one has to find the coefficients of the dynamical

map in order to find the dynamical solutions. In the present case our problem is to express the Heisenberg electron field in terms of the quasi-particles. We choose as candidates for quasi-particles the quasi-electrons. Since we are mainly interested in quantities bilinear in the Heisenberg electron field, we consider the

S E L F - C O N S I S T E N T F O R M U L A T I O N O F I T I N E R A N T - E L E C T R O N F E R R O M A G N E T 27

following dynamical map:

f f , (2.20) T[~(x), ~o~(y)] = )C(x -- y) + d~q d~p T")(p, q; x, y)aqto~e, ~ +

f f , + d3q dtp T(~)(p, q; x, y)%,fl~,, + .. . ,

where the dots denote other normal-product terms bilinear in ~ and fl plus higher- order normal-product terms. In (2.20) we exclude terms which do not conserve the spin of the quasi-electrons. The coefficients of the map, which are c-number 2 x2 matrices, ~re given by

(2.21) <0iT[~(x), vJ(y)]10> = Z ( x - - y ) ,

(2.22) <altlT[~(x), w*(y)]l~,> = T(,)(Z, Z; x, y), etc .

Our problem is now to calculate the coefficients of the map. We will derive the Bethe-Salpeter (B-S) equations and try to solve them in the pair approx- imation.

3 . - Be the -Sa lpater equat ion .

To find the B-S equation for g ( x - y) we consider

AiS.) )~(x - y ) A ( - ~) = A(~)<OIT[~(x), ~o*(y)] 10>A ( - ~ ) , (3.1)

where

(3.2) E(V:)]

A ( ~ ) )~(x -- y )A( - - ~ ) = iA(~) 5(x -- y ) I +

where

(3.4)

+ i S ( t - t,)<O][v2(x), Jt(y)]lO > A- <OlT[J(x), J*(y)]tO>,

J ( x ) = s + a a J T l ~ ( x ) - - ~ ( x ) .

In the pair approximation, the last term in (3.3) does not contribute; then using the canonical commutator (2.4) we get

)~(x -- y) = S(x -- y) -- i f d4~ S(x -- ~)[-~a3 -- ~tI -4- Z'] S(~ -- y) . k~3 ~ 5 ~

Here Z' denotes a 2 • matrix of elements

l, x~, -- ,~<01~(x) ~,(x)10>, (3.6) , ,~

! x,~ = - ~<01~(x) ~ ( x ) [ 0 > ,

r z:2 = ~<ol~'~(x) ~t(x)lo>.

28 M.N. SH~ and O. WT~LLO

Since the interaction does not create any self-energy for the quasi-electron, we should have

(3.7) <0[T[~(x), ~ (y) ] 10> ---- <01T[~(x), ~(y)]10>,

i.e. from (2.21) and {2.15)

(3.8) X(x -- y) ---- S(x -- y) .

The relation (3.5) then shows that

(3.9) ~l'aa -- ~tI + X '= O,

i.e.

2 t (3.1o) ~ = ~ <o1~ (x)~3~(x)]o>,

(3.11) g = ~ <o[~*(x)~(x)lO>.

Equat ion (3.10) gives us the magnetization 2~ r, which must be different from zero for a ferromagnetic system. (3.10) is the boundary condition under which the field equat ion (2.5) must be solved. (3.10) and (3.7) give

(3.12) j~r _-= 2 <O]q~*(x)a3q~(x)]O>,

i.e. by using (2.14) and the definition (2.19) we get

q~ ;t ~ d 'k

(3.J3) 1 = 2--~J (2~t) a" q~

(3.13) is called the gap equation. When (3.10) is satisfied, an energy differenco 22~ r appears between spin-up and spin-down quasi-electrons.

Let us now s tudy the B-S ampli tude

(3.14) G(x, y) = <jlT[v/(x) , ~ ( y ) ] l i > - <0[T[~(x), v2*(y)][0> (~,

with the states li> and I J> satisfying (i]j> = ~,j. We then consider the relation

(3.15) A ~ ) G(x, y)A(--~v) = A~){<jlT[~o(x), ~'(y)] Ii> --

- <OlT[~(x), ~*(y)] Io>/},} A(-- O,).

8 E L F - C O N S I S T F N T F O R M U L A T I O N O F I T I N E R A N T - E L E C T R O N F E R R O ] ~ A G N E T

By using the field equation (2.5) and (3.15) we have

(3.16)

29

A ~ . ) O(x, y) A ( - - ~ ) = i6(t. - t~)[ <jl[v2(x), J+(y)]+[i> -

- <01[~,(x), Y(y)]+I0> 6 . ] + <jlT[J(x), J ' ( y ) ] [ i > - <OIT[J(x), Jr(y)]]0> ~,,,

where J(x) is defined in eq. (3.4). In the pair approximation the last two terms in (3.16) do not contribute.

Using the commutator (2.4) we get

(3.17) G(x, y) = G~ y) -- i f d*~ 8 (x - - ~) F(~) S(~-- y) ,

where the 2 • 2 matr ix F($) is

(3.~8) ( - ~ G = ( ~ , ~) ~G~,(~, ~)~ F(~) =: ).a~,(~, ~) - - ~G,,(~, ~)] '

and G~ y) is defined by

(3.19) G~ y) = <j IT[ ~ ( x ) , ~o'(y)] ]i5 - <0]T[r ~0*(y)] [O> 6 , .

This shows tha t G~ y) is different from zero only in the following cases (s and s' are r or ~):

[J> Ii5

(3.20)

a) J0>

b) l~_p.., fi-t+p...>

c) l~-p,,>

d) Ifi-,+~,s>

I~,_p.,, ~p,,,>, !o5,

In these eases (3.19) gives

(3.21) G~ y) ~-

1 - - 0

(2re)3 G,,, (t, E) exp [iq. x] exp l ip .y] exp [-- iE~ t,] exp [-- i(E -- E~) t,] ,

where we have used the notation E ~ E , - - E j ,

(3.22) [G~t G ~ IG~I G~ G O G O ------

30 M. N. SHAH a n d 6. VITIELL(~

and

~ = E,_~,,, > 0 , q=t--p, E,, < 0 , E =.E,_,,,=+ IE~,,,, I

E'_= E_,.~,,< 0, q = l - - p , E_,,, , > 0, E = -- IE_,+.,.I- E_~,.,

E' . = Ez_~,, . > 0 , q = 1 - - p , E_~, , . , > 0 , E = Ez_~ , , - - E_I,,o,

E'_= E_ ,+ . . .<o , q=t--p, ~ , < o , E=--I~_,+.,,I + IE.,.,I

in the case a ) ;

in the case b);

in the ease e);

in the case d) .

If we write O(x, x) as 1

(3.23) O(x, x) = ~ G(l, E) exp [ i l .x - - le t] ,

(3.17) and (3.21) give

(3.24)

o,,(t, E) = oL(t, E) + Q , , q , E)O~,(t, E ) ,

0,,(~, E) = O~(t, ~) -- Q,,(t, E)Ol,(t, E ) ,

O~,(/, E) = O~,(1, E) -- Q,,(1, E)O,,(1, E) ,

0,~(t, .E) = O~,(t, E) +Q~,(t, E)O,,(t, E) ,

where the Q~J(l, E) are defined by

(3.25) Q~J(x) = i2S"~(x)S(~(--x) - - f dsl dEo"~l E)exp l

I l l . x - - lEt] (2rip 2n ~ ~ '

(i , j = 1, 2)

with Sin(x)= Sn(x), SCZ)(x)= S~(x) (el. (2.18)). We also introduce the no- ta t ion Gm--- G~2 and G ~2~ = - G~. (3.25) and (2.18) give

(3.26)

Q"(1, E) =

Q'2(1, E ) = --

Q~'(1, E) =

Q22(/, E) =

- +

_ i t dSk 1

3(2~) ~ E + E , - E + + '

- §

. ~ dSk 1 = Q,~( - l, - E ) ,

2 r d k 1 j~2-~ E+ E_+--E++'

e~

S E L F - C O N S I S T E N T F O R M U L A T I O N " O F I T I N F ~ R A N T - E L E C T R O N F E R R O M A G N ~ T

where E...t, + ~ El~,+.21t,. , and

l "~ 7~ COS 0 ,

(3.27) l

4Qt,~) ~ L ~ (~ cos ~ 0 - t ~ +

with Qt,~ as def ined in (2.19).

3 1

4 . - T h e s p i n c u r r e n t .

L e t us now tu rn our a t t e n t i o n to the <, spin cu r r en t

] t (4.]) j,,)(x) = ~ [~ (x) w , ~ ( x ) - (v~ ' (x ) ) ~ ( x ) ] =

1 - 2 m i [ ( V ~ - - V , ) T r [ a , ~ ( x ) ~ t ( y ) ] ] ~ , - ~ - c~ ,

and to the (, spin dens i ty ~)

(4.2) o")(x) = yjt(x) a, y~(x) -~ - - Tr [a, ~v(x) ~ot{y)]~., 4-. c~ } , i = 1, 2, 3,

-{~} a n d c "} are c - n u m b e r s c r ea t ed b y c o m m u t a t i o n s a m o n g ~ a n d y ; . where c~ q

Use of the field e q u a t i o n (2.5) leads to ,% C .

(4.3) V . j ( ' ~ ( x ) + ~ o ( ' ~ ( x ) = O , i = 1, 2, 3 .

We have , in any of the cases (3.20),

1 (4.4) (j!j'(x)ii) -: - - '2 m----i Tr [ ( V ~ - - V ~ ) a , G ~ y)]~_~ ~- ~ f ~ l ( x ) , i ~-- 1 ,2 , 3

wi th

i V (4.5) ~j'"(x)= ~m[( ~--V,)fd,~Tr[a,S(x_~)F(~)S(~_y)]]~., where F(~) is defined in (3.18) and can be wr i t t en as

1 (4.6) F ( x ) --~ ~2~) ~ F ( l , E ) exp [ i l . x - - i E t ] .

Simi la r ly 7 we h a v e

(4.7) ( j[o( ' ) (x) [i) ---- - - Tr [a, G~ y)]~. , d- ~ ( ' ) ( x ) , i = 1, 2, 3 ,

:32 M . N . SHAH a n d o. VITIELLO

with

(4.8) se,',(x)-- [ifd'# Tr [a~S(x - -~)F(~)S(~- - y) ] ] ,_ , .

Use of (2.18) and similar equa t ions for S , , ( x - y) gives

(4.9)

1 )~ ~j(~)(x) = exp [it. x -- iEt]"

(2n)~ m

' - - E + E _ ~ - - E + t ]

2 - - exp [il. x - - iEt]. ~J~'(~:) (2~)~ m

k. ( d.k "[~ (Z~)~ E + E-t--~+r +J(2~) ~ ~ :

exp [il. x - - lEt]. (2:~)~ m

[_3 (2~)~ E + E_ t - E+t J (2~)~ k

ac')(l,E) } E + E_~ ~ E + t '

Gn(l, E) } E § E _ ~ : E + ~ "

U s i n g eqs. (3.24) and (3.26) we can wri te

(4.10)

~j('(x) = - - i

3j(2)(x) ---- i

~j("'(x) = -

--l exp Ill" x - iEt] [ 2 2 ~ ( G ~ - G~2)--E(G(~'Q2'--G(')Q~2)] l ' (2~) 3

_ / e x p [il. x - - iEt] [22~(GO + G[~)+ E(GC2)Q2~+ G(1)Q,*)], 12 (2~)*

1 exp [i l .x - - lE t] E[G~2Q~ _ Gn Q~']. l = (2:~) 3

In the same way we can wri te

~(4.11)

exp [i l .x - - iE t] [G(1)Q~2 - G(~Q2~] ' ~e~')(x) = (2n)3

8~(~l(x) = i exp [il. x - - lEt] [G(1)QI~ + G(2)Q21] , (2n)3

exp [ i l . x - - i E t ] [ G , 1 Q 2 2 G~Q,I].

S E L F - C O N S I S T E N T F O R b l U L A T I O N OF I T I N E R A N T - E L E C T R O N FERI:r, O M A G N E T

~3,re in t roduce n e x t the (, f ree-sp in cu r r en t ,)

(4.12) l

1 - 2mi [ (v~- v , ) Tr [~-,~(x)~o'(y)]]~+ do'}, i=1,2,3,

and the (, f ree-sp in dens i ty ,>

(4.13)

Then

(4.14)

(4.15)

Thus

(4.16)

e~~ = Ct(x) ~, ~(x) - - - - T r [a,r ~+(y)],=y ~- ~(,I , Uo0

<j jfi"(x)li> - 1

2mi T r [ (V~- - V~)a,G~ y)]~=~,

<J 10 (x)ti> = - T r [a, G~ y)]~_,.

j("(x) "(o == 1o (x) + ~ j" ' ( x ) ,

o(,l(x) ~(,,, �9 eo tx) + ~q")(x),

i = 1, 2, 3.

where j"~, j~n, o,J, ~ ) m u s t be unde r s tood as the co r respond ing m a t r i x e lements

<J IJ'(x) Ii>, etc. L e t us no te t ha t , while

(4.17)

and s imi lar equa t i on for i = 2, for i = 3 we h a v e

(4.is) C. (3) vjo~3'(x) + ~: ~,o (x) = 0 ,

which is the c o n s e r v a t i o n law for the (( f ree-spin cu r r en t ,> in the 3rd direct ion. F r o m eq. (4.17) a n d (3.19) we see t h a t

(4.19)

C~ 0.__o_o ~--~ <j)Vj(o ~)-t ~t Ii) 2i~l(G~ -- GOD exp [ i l . x - - i E t ] (2n)~

", (2) <j!Vjj~) + c o u 'i3 -- 2_~r(G~,-k- G~ exp [ i l . x - - iEt]

?.t ~" (2~)~

Using eqs. (4.10), (4.11), (4.16) and (4.19) we can see t h a t the conse rva t ion law

Y .j"~(x) + ?~ o~'~(x) -- o

3 - II N u o v o C i m e ~ l o B .

34 ~. N. s K ~ and o. VITIF~LLO"

is still satisfied, although we consider the current and the density in the pa i r approximation. This gives a good reason to use the pair approximation.

We can show also tha t

( 4 . 2 0 ) o ' , ( x ) l,_o = o,~,(x)ls_o = ~ ( " ( x ) l , - o = o .

We observe that , for the momentum I different from zero, the gap equat ion (3.13) can be generalized as

_/" dak ] (4.2]) 1 = - ~ j (F~: o , + k . i / ~ - 2 ~ / '

where o~[~_ 0 = 0. Then, by using (3.26) we get

1 + Q~2(1, JE) = (JE + w,)A(1, E) ,

1 + Q~,q, E) = (E - 03 Bq, E) , (4.22)

where

(4.23) A(l, E) = - - B ( - - 1 , - - E) =

J (2~)~ -

1 1

o~z--k. l /m + �89 E - - k . l / m + 2J7I "

Use of (3.24) and (4.22) shows tha t 8j(l'(x), ~j(~'(x), 8q(~(x) and ~0(~)(x) have poles a t E = 4- e% (ef. (4.9) and (4.11)).

By solving (4.21) for eo~ we find w~oc l ~ for small 1 (~.4); indeed

(4.24) [5___~/~m ~ 1 3 ] 12 co, ---- -- (Q~ - Q~) - m (Qr + q~) 2 (Q~ - Q~)

for small 1.

5 . - T h e b o s o n c u r r e n t .

The fact tha t spin cnrrents and densities, for i = 1 and 2, have singularities at E = 4-w~ suggests tha t there exists a boson of energy o~. We have to modify then the set of the quasi-particles by introducing the magnon field B(x). Consequently, we must add terms containing the field B(x) to the dynamical map (2.20). The terms we add to the r ight-hand side of (2.20) are

(5.1) f dtl ~,1,(/, x, y)B~ + fd*l ~,2,(/, x, y)B, + ...,

SELF-CONSISTENT FORMULATION OF ITIN]~RANT-]~LECTRON F]~I~ROMAGNET ~

where B~ is the annihilation operator of the boson quantum (the ma~aon) and the dots mean higher-order normal-product terms. We are going to prove the existence of this bosom Since no Iteisenberg operator for such a boson is introduced, the boson must be a bound state of electrons.

Let us s tudy the B-S amplitude G(x~, x~) obtained by considering the states IJ~ = [0~ and the one-magnon state ]i~----IBm):

(5.2) G(x, y) ---- ~<~)(/, x, y) .

By a computat ion similar to the one for G(x, y) defined in (3.14), we find

(5.~) G(x, y) = -- i f d~ ~(x-- ~) ~(~) S(~-- Y)

with F(~) defined in (3.18). We can write

(5.4)

and obtain

(5.5)

1 G(x, x) = ~ G(1, E) exp [ i l . x - - i E t ] , ~ ) ~

G::(Z, E) = Q~(~, E) G::(Z, E) ,

G~(1, E ) ---- - - Q~2(1, E ) G,~(1, E ) = G ~> ,

G~(1, E ) ---- - - Q~l(l, E ) G~a(1, E ) ~-- - - G (~ ,

G,,(l , E ) -~ Q~( l , E ) g,~(1, E ) ,

which are the B-S equations corresponding to (3.24) (in (5.5) there are no in- homogeneous terms). (5.5) together with (4.22) gives

(5.6) (E + w~)G(1)(/, E) --~ 0,

(E -- o~) G(2)(1, E) = 0 ,

which are the wave equations for the magnon and show tha t the magnon energy is • co~.

The computat ion of the mag~aon current j~(x) and of the (~ magnon density ,~ e~)(x) is analogous to the one for the spin current and density. We find

] l (5.7) (O[j(BI'(x)[B~ = (2~) 3 ~ e x p [il.x--ioJzt]o~(Gr ~o~)- O(1)(l, ~ot)),

/ l (5.8) (01j~2~(x) IBm) -- (2~) 3 12 exp [il.x--ir ~o~) + O(~(l, w~)),

1 (5.9) ;01~o~)(x)lB~} = (2~)3 exp [il.x--i~o~t](G(2)(1, e ~ ) _ GCl)(l, w~)),

i (5.10) (0 [os(2)(x)lB~ } _ (2~) 3 exp [ i l . x - - iw~t](O('~(1, o~) + V(~)(l, w~))

36

with the conservation law

(5.11) v .j~'(x) + ~ e~'(x) = o,

M. 1~. SH~ and o. VITIF.LLO

We now write the field B(x) as

(5.12) B(x) = ( d3/ Bzexp [il.x--io~zt] j ( 2 = ) t

with the commutat ion relation

(5.1.3) [B(x), ~ B (y)], ,_,= ~(x--y) .

The free-field equations for B(x) and Bt(x) are then (cf. (5.6))

(5.14) (~ ~ - - w , ) B ( x ) = 0 , ( ~ + w,)B*(x)----0.

From (5.7)-(5.10) and (5.12) we see that

(5.15) jta~,(x) _ i _ e~Zg(iV)[VB(x)_VBt(x)] (2~)t 1 ~

l wt (5.16) j(,~'(x) = (2~t)! 12 g(iV)[VB(x) + VBt(x)] ,

1 (5.17) ~)(x) -- (2~t)tg(iV)[B(x) + B~(x)],

i (5.18) Q~)(x) = (2: t) ig(iV)[B(x)-- Bt(x)],

where 9(iV) is defined by

g(iV) exp [il.x --/~ott] -- g(l) exp [il.x -- ieozt] (5.19)

with

(5.20) Ig(Z) l = = (2=Pl<010"'(x) IB,>I = ,

I t can be shown (el. eq. (A.12)) tha t

(5.21) g(h = \~A(~,,--~o,)/

with A(l, oJ,) given in (4.23). tion law (5.11).

i : 1 , 2 .

i----1~2.

Note that (5.15)-(5.18) satisfy the conserva-

SELF-CONSISTENT FORMULATION OF ITINERANT-ELECTRON FERROMAGNET 37

We also have

(5.22) <olj~)(x)lB,> :

(5.23) <0]q@)(x)lB,> = - - -

H o w e v e r , since

(5.24)

from (5.5) we have tha t

1 l (27~) 3 ~ exp [ i l . x - ieo~t]o~(Gl~(l, eg~)- G2~l, o)~)),

(2~)~ exp [ i l " x - i~% t]( Gxl(1, o)~) - G2~(1, ~o~) ) .

{ 1 - Q ~ ( I , E) r at E = •

1 - Q~(1, E) # 0 at E = • ~ ,

(5.25) G~(1, wz) -= G~(1, eg~) = O,

which means tha t the r ight-hand sides of (5.22) and (5.23) are zero. This tells us tha t there are no linear terms in B or B* m:~ $~'(a) and ~ , i.e. j ~ and ~ have terms at least bilinear in B and B*. In the pair approximation, however, such terms cannot be computed since B is a bound state of electrons: terms bilinear in B and B* represent at least two pairs.

6. - Dynamica l rearrangement o f symmetry .

Let us now note tha t the Hamil ton;an (2.1) is invariant under the spin rotat ion

(6.1) y~(x) -->exp [iO~ 2~] ~p(x) , i = 1, 2, 3,

where 2~ = �89 Generator of this t ransformation is

(6.2) S'"= �89 ,,7 �89 etot(X) ---- d3x ~*(x) a~ ~(x) .

I t is well known tha t the generators S (i) are not well defined. Rigorously, there- fore we should proceed as in I, Sect. 3, by inserting in the definition (6.2) a square integrable funct ion/(x) , which is a solution of the magnon equation (3.14) and which is taken to be 1 after the integration is performed. Here, for simplicity, we do not write explicitly the function ](x) in the generators; the previous prescription is understood. When (I) ~tot(X) is writ ten in terms of quasi-electrons and magnons, we can use the notat ion

(6.3) •)(i)/X• to,~ J = ~ ) ( x ) + e ~ ( x ) ,

where the subscript F refers to quasi-electrons.

H . N, S H A H & n d G. VITIELLO

At l = 0, (4.20) (in which a subscript F should be understood) tells us tha t

(6.4)

(f) 0~,(x) = e~'(x) for i = 1 , 2 , at /----0,

a t / = 0 .

The generators can then be writ ten as (cf. also (5.17), (5.18) and (5.21))

(6.5)

r S")-- 2-2 d3x[B(x)t(x)+B*(x)J*(x)] '

S'~-'=--i (~ ) ' fd~x[B(x)l(x)--B*(x)l*(x)]

1 a

s,3, = f + f d x ,

where ](x) is a square integrable function which is a solution of the magnoa equation (5.14). We saw at the end of Sect. 5 tha t 0~ would have terms at least bilinear in B and B' . In I we showed (cf. eq. (3.36e) in (1)) tha t �89 is actually -- B*(x) B(x).

Using the generators (6.5) we can see tha t the free fields 9(x) and B(x) trans- form in the following way, where Ba(x ) and 9o(x) denote the transformed fields and we put ] ( x ) = 1:

(6.6)

Bo(x) = B(x) + iO, ~.

9o(x) = q~(x)

Bo(x)= B(x)--O~ 2-2

9dx) = 9(x)

Bo(x) = exp [--iO~]B(x) [

I qzo(x) = exp [iOa23]9(x)

0~= 0~= 0 ,

01 = 03 = 0 ,

01 = 02 ---- 0 .

These transformations are such tha t they induce the transformation (6.1) in the Heisenberg field ~(x). We observe that , due to dynamical effects, the original spin rotationM symmetry is changed into the E~-group (6.6). Since the gen- erators (6.5) are t ime independent, they cannot mix fields of different energies; then the difference of the energy 21);/ between spin-up and spin-down quasi- electrons forbids mixing among them: quasi-electrons are frozen under spin rotat ion around 1 and 2 directions. The spin property is instead carried by the

S E L F - C O N S I S T E N T F O R M U L A T I O N OF I T I N E R A N T - E L E C T R O N F E R R O M A G N E T 39

magnon, which undergoes the t ransformations in (6.6)(~): magnons (i.e. spin wave quanta) are condensed in the ground state, which thus acquires the charac- teristic s t ructure of ferromagnetism. The original invariance under spin rotat ion manifests itself in the invariance of the free-field equations under the quasi- particle t ransformations (6.6).

7 . - C o n c l u s i o n .

We have studied the i t inerant-electron ferromagnetism in the f ramework of the self-consistent method by considering a practical model. To summarize, we have presented in the pair approximat ion the computa t ion of the coef- ficients of the dynamical map by means of which the mat r ix elements of tteisen- berg electron fields are expressed as matr ix elements among states of quasi- particles. The self-consistency of the theory requires the existence of a gapless boson, the magnon, among the physical particles. The magnon, as a bound state of electrons, is studied by means of the Bethe-Salpeter equations. The spin operators are also writ ten in terms of quasi-fermion and bosom The nature of the quasi-particle t ransformations is found different from the original spin rotat ion t ransformat ion: this fact is expressed by saying tha t dynamical rear- rangement of symmet ry occurs. The change of the spin rota t ion group into the E~-group is caused by infra-red effects which are locally infinitesimal but give a finite contr ibut ion when integrated over the whole volume of the system (1.7). As observed in I, the E~ symmet ry is related to observable results since quasi- particles are related to observable energy levels. The magnons form an irre- ducible representat ion of the E~ symmet ry group. The low-energy theorem (8) also follows (1): Due to the invariance of the theory under the original spin rota t ion t ransformation, the S-matr ix is invariant under the t ransformat ion B - + B ~ c; thus the magnon operator B always appears with its derivatives in the S-matr ix expressed in terms of quasi-particles: consequently the magnon interact ion disappears in the zero-momentum limit. Although the pair approx- imat ion computat ion is not an improvement over the usual random-phase approximat ion (3), use of the self-consistent formulat ion gives a deeper under- s tanding of the symmet ry properties of a ferromagnetic system and a simple derivat ion of the low-energy theorem. I t will be interesting to use the formalism

of the present paper to extend the results for ferromagnetic systems at finite tempera ture . Since in our formulat ion the boson excitations are automatical ly t aken into account, it should be possible to recover the Tt-dependence of the

magnetizat ion also for the i t inerant-electron model of ferromagnetism. While there are reasonable theoretical arguments for the T t law in the case of the

(9) A transformation like B(x)--+ B(x)~-el(x) with c a c-number is called boson trans- formation (cf. rcf. (1.a)).

40 M. N. SH~ and G. V'ITIELLO

Heisenberg ferromagnet , there is no clear-cut proof for the i t inerant-electron case. In this respect it is interesting to note the presence of the B*B t e rm in the spin density S (3). Indeed, the vacuum expectat ion value of such a te rm at T =/= 0, for low temperature , immediately gives

<B*(x) B(x)>~.o oc T ! .

(Recall tha t the T ! te rm in the case of the Heisenberg ferromagnet comes from the a*a te rm in S (3) (2.3).)

Another theoretical improvement using the self-consistent me thod would be to ex tend the pair approximat ion to a be t te r approximat ion in such a way as to include graphs with many internal lines. Then it would be possible to consider magnon effects in the renormalization of magnetization. The s tudy of these problems is in our future plans.

I t is our pleasure to thank Prof. H. UM~ZAWA (University of Wisconsin- Milwaukee) for many enlightening discussions and constant guidance. We are also grateful to Prof. M. TACHIKI (Tohoku University) for m an y helpful

discussions.

A P P E N D I X

Normalization of the B-S amplitudes.

To determine the normalization of the Bethe-Salpeter ampli tudes (G (~), G (2) and G11, G22), we consider the following quant i ty :

(A.1) O~(x,x~) = <0 IT[~(x,) ~(x~) e(')(z)][0>.

In the pair approximat ion it can be shown tha t this satisfies the following B-S equat ion:

(A.2) G~Z)(xlx2) ---- S ( x l - - z ) a , S ( z - - x z ) - - i f d 4 ~ S(x,--~)F~(~)S(~--x~) ,

where the @~") are related to the G~ "~ in the same way as in eq. (3.18). at aro the Pauli matrices.

Recalling the definition of Q" (eq. (3.25)), and writing

(A.3) dSldE G 1 G~ (x, x) ---- J ~ ~( , E) exp [il. ( x - - z) - - iE ( t , - - t , ) ] ,

SELF-COnSISTENT fORMULATION O ~ Z T I N ~ R A ~ T - E L E C T R O ~ W R R O M A O ~ E ~

we have for i - - 1 (i.e. ~U)(z)~-o(~l(z) in (A.1))

GH~,~(I, E) = Q~(l, E)G~(, ( I , E ) ,

G22(;)(/, E ) : Q22(l, E)Gll(1)(l, E ) ,

1 (A.4) V,2(1)(l, E) -- - - o ~ l l E) E)Q]~ , - i ) . . ~ ~., - - G12 i I ) ( I , E )

l G~m,(1, E) == -~Q2'(l, E) - - G2m)(/, E)Q"'(1, E ) ,

and for i - - -2 G,l(2)(l, JE) :=- Q]'(l, E)G22(2)(l, E ) ,

G22(2)(1, E) = Q'-2(1, E)GI~(2)(I, E ) ,

[ (A.5) G~2(2)(1, E) ~- - - ~ Q'2(l, E) - - G~2(~(1, E)Q'2(I, E) ,

l G,(2)(1, E) --= ~Q~"(I, E ) - - G,(.~)(1, E)Q2'(I, E) .

Let us consider

(A.6) ~(,,)(x-- z) = (O]ei~,(x)e(,)(z)[O) .

For t~>t~ and i---- l we have

(A.7) ~( l i ) ( x_z ) ~(~) i x) ~(~) '~ x ) . - - ~.a 2 1 ( i ) ~3~'~ ~ - - ~ 1 2 ( 1 ) ~ ,

We can write 3 f d l d E

(h.8) (~(]~)(x--z) J (2~) ~ g ( n ) ( l , E ) e x p [ i l . ( x - - z ) - - i E ( t ~ - - t ~ ) ] .

Then, using eq. (A.4), we can write

(A.9) ~(1~)(/, E) = Q'2(l, E) Q2~(I, E) i~(l § Q'~(I, S)) i2(1 § Q2~(1, E)) "

Recalling eq. (4.22) we obtain

l (A.10) r d E s ( u ) ( l , E)exp [ i l . ( x - - z ) - - i E ( t ~ - - t ~ ) ] -- ) . A ( - - l , - - o ~ )

J 2~ " '

Writing (A.6) as

(A.I]) ~ ( " ) ( x - - z ) = (OJe(')(x)lB,)(B~lo(')(z)lO) J I ~)

we get ]

(A. 12) ](0 I~"'(x)In,> i '~ = )l A(--- l, -- wzi"

We can show tha t the same result holds for ](0LoI~l(x)[Bz)[-'.

41

4 2 M.N. SHAH and G. VITIELLO

�9 R I A S S U N T O (*)

Si appl ica il metodo autoconsis tcnte del la teor ia quant i s t i ca dei campi al f e r romagne te con e lc t t rone mobile . Si s tudia il magnone come s ta te legato di e le t t roni faeendo use de l l ' equaz ione di Be the .Sa lpe te r ne l l ' appross imazione delle coppie. La ro taz ione dello spin degli e le t t roni ~ in t rodo t t a da tma t ras formazione del la s immet r ia di E~ dei magnoni e dci quasi e le t t roni ehe A diversa da una rotazione. Si discute una semplice deduzione del t eo rcma di ba l sa onergia.

(*) Traduzione a eura della Redazione.

CaMocoraacoaaanau r ~la~q (~eppoMaraeTmca co << c~rl~.~rsymlllm4 >)

3JleKTpOHOM,

Pe3toMe (*). - - CaMOCOFnaCOBaHHbI~ MeTO~ B KBaHTOBOI~ Teopim IlonR npRMeHfleTCg K Hcc,.qe.aoBaHmo ~peppoMarHeTHKa co (<cTpaHCTBylOIIDIM)~, 3.rlewrpoHoM. HCllO.rlb3ylg ypaBHeHHe BeTe-CazmeTepa B ~apHoM rrpH6ng~eHm4, Hccne~yeTc~ MaFHOH, gaK CB~3amqoc COCTO~IHHe 3/Ielci'pOHOS. Bpamem~e e m m a 3nex-rpoHoB BBO]IHTC~I C nOMORI~O npeo6- pa3OBaHH~ E s CI42~MeTpHtl ~.rl~ MRrHOHOB I4 KBa3H3.rIeKTpOHOB, gOTOpOC OT.rlld~aCTCIt OT ~pal~CBHS. O6cy~laeTc~l IIpOCTOt~ Bl,mO~l TCOpeMbl ~J1H HH3KHX ~Heprral.

( ' ) Hepeaedeno pebax~ue~.