Phys. kondens. Materie 15, 46--60 (1972) �9 by Springer-Verlag 1972

Ferromagnetism of the Electron Gas

JoHn; LAM

Max-Planck-Institut fiir Physik und Astrophysik, Miinchen (Germany)

Received April 5, 1972

The ground-state energy of the ferromagnetic electron gas is calculated for the relative polarization ~ = 0 -- 1 and the interelectron separation rs = 5 -- 12. The method consists in describing the electron gas approximately by a quadratic boson Hamiltonian, and contains the random-phase approximation as a special case. Numerical studies show that in both the ran- dom-phase and the present approximations the paramagnetic state has the lowest energy: the energy increases with ~ for all values of rs considered. In the present approximation instabil- ities are found to occur for rs above a critical value, due to exchange processes of finite mo- mentum transfers. For ~ = 0 this critical value of rs is 9.4; it decreases with increasing $. However, the fully-polarized state (~ = 1), which lies above the rest, is always stable. The conclusions are as follows: (1) For rs < 9.4 the electron gas is paramagnetic. (2) At rs = 9.4 it goes over to the fully-polarized ferromagnetic state. (3) This phase transition requires an energy absorption of 0.03 rydberg per electron. (4) The fully-polarized state is not obtainable as the limit ~ -+ 1.

I. Introduction

At sufficiently high densi ty the ground state of the electron gas is a lmost certainly paramagnct ic . However , as the densi ty is gradual ly lowered, the question as to whether it will eventual ly become ferromagnetic has no t ye t found a con- clusive answer, despite its great impor tance to the theory of metals. I n 1929 Bloch [1] init iated the problem by not ing t h a t in the t Ia r t ree -Fock approximation, when correlation effects are neglected, the fully-polarized ferromagntic state lies below the paramagnet ic state for values of the specific interelectron separat ion rs >~ 5.5, which is inside the metallic densi ty range. The energy separation, however, is so small t h a t it is just as easy to expect a confirmation as a reversal of Bloch's finding when correlation effects are considered. On the other hand this value of rs corresponds to an intermediate coupling, and the correlation problem becomes very difficult. Consequently there have been m a n y investigations lead- ing to contradict ing conclusions.

I n a previous publicat ion [2] we presented a calculation of the correlation energy of the paramagnet ic electron gas at metallic densities, using a quadrat ic boson Hamil tonian. We found an instabi l i ty in the tr iplet state of the electron- hole pair for rs > 9.4. I n view of Bloch's finding it is t empt ing to interpret the instabi l i ty as a t ransi t ion to the ferromagnetic phase. We s tudy this possibili ty in the present work b y generalizing the previous method to calculate the energy of the ferromagnetic ground state. Such an invest igation is of interest to the i t inerant- electron theory of ferromagnetism.

Since this calculation is an extension of t ha t in Ref. [2], cumbersome details are f requent ly no t repeated. The reader is referred to the previous paper for consultation.

Ferromagnetism of the Electron Gas 47

II. Boson Hamiltonian

We consider a gas of N electrons in a volume ~ and at zero temperature. moving in a uniform background of neutralizing positive charge density and interacting with one another through the Coulomb potential. I t is described by the fermion Hamiltonian

1 a r . t ~ " (2.1) HF= Z e e a ; . a p ( , + ~ Z V(q) .+q~.p._qo.~F~.~...,

p a qpp" a ~ r

p2 8 p - - 2~rl~

4~ e~ V(q) = - - - , q . 0 q2

= 0 , q = O .

at and a are electron creation and annihilation operators; m and e are the electronic mass and charge. I t can be shown that the strength of the coupling is measured by the intereleetron separation r~ (in units of the Bohr radius) defined by

~ - = ~ 7~ (rs a0) 3 , a0 = - . (2.2) q?~ e 2

When the Coulomb interaction is switched off, the ground state is para- magnetic: the up-spin and down-spin electrons are equal in number and each fill a Fermi sphere of radius Pr given by

N 4 (2zt)a - ~ = 2 ~ z p~. (2.3)

From (2.2) and (2.3) we have the relation

1 ( 4 ~ lla - , ~ = ( 2 . 4 )

p F ~ r e a 0 \9=-/ "

Suppose we impose on the system the constraint that there be �89 (1 ~- ~) up-spin and �89 (1 -- ~) down-spin electrons, with the relative polarization ~ lying between 0 and 1. The system is now ferromagnetic with a net magnetic moment. The Fermi surface is split into two with radii given by

pFa = (l -~ ~)1/3 p F . (2.5)

The upper and lower signs will always refer to up-spin ((~ = ~) and down-spin (o = ~) electrons, respectively. These Fermi-sphere states are the vacuum states in a particle-hoIe description.

When the interaction is switched on, these states are no longer the ones of lowest energy. Instead, the true ground state for a given ~ contains a distribution of electrons above the Fermi levels. I t can be reached by operating on the Fermi spheres with a suitable combination of particle-hole-pair creation operators

atp~ ap~, P > pFa, P < PF~. (2.6)

48 J. Lain:

These operators describe the deviation of the system from the Fermi spheres. When the deviation in the ground state is small we can expand the Hamiltonian in terms of them, as was done by Sawada [3] for the paramagnetio state at high density. Itowcver, these operators obey neither Fermi nor Bose statistics, and are only approximately Bose-like in the sense that they commute but that the commutator with the Hermitian conjugate is not equal to a product of delta- functions. This difficulty is resolved by a method of Usui [4], which we employed in Ref. [2], in which the electron-hole pair is transformed into a boson.

In this method we consider a set of boson operators

Pt C ~ , P > T~a, p < pF~ (2.7)

satisfying the Bose commutation rules:

[0~ , P' C~,~,] = 0 , (2.s)

[0;,%, = %,. By applying these operators on the boson vacuum we generate a boson state- space. As was described in Ref. [4] or [2], we can set up a one-to-one correspon- dence between the fermion state-space and a so-called physical subspace of the boson state-space. For example, the Fermi-spheres are mapped into the boson vacuum, the one-electron-hole-pair states into the one-boson states, the two- pair states into the two-boson states, etc. Attention must be paid, however, to the fact tha t there are unphysical boson states. For example, a state containing two identical bosons has no correspondence in the fermion state-space by reason of the Pauli exclusion principle. The suppression of unphysical states leads to a certain kinematical interaction among the bosons.

Under this transformation the fermion Hamiltonian H~ is mapped into a boson Hamiltonian of the form

H~ = H0 -~ H2 ~- H3 -~ Hd, (2.9)

where H0 is a c-number, and H2, Ha and H4 are, respectively, quadratic, cubic and quartie in boson operators. The transitions effected by H a in the physical boson subspace reproduce in a one-to-one manner the fermion processes induced by H r . The reason there are no higher-order terms is that a process such as the Coulomb interaction, which involves the simultaneous creation or annihilation of four electrons or holes, can be regarded as the creation or annihilation of no more than four electron-hole pairs. We note tha t since there is a cubic part the boson number is not conserved. The transformation has been worked out in l~ef. [4] or [2] for the paramagnetie state. The generalization to the ferromagnetic state is obvious and we will just write down the results.

H0 is simply the ground-state energy in the Hartree-Fock approximation:

H0 = = Z c , v ( p - p ' ) . LP<~Fa ~0,p'<pF~

The contribution to the ground-state energy from the remaining parts of HB is therefore, by definition, the correlation energy. We first make a change of nota- tion:

Ferromagnetism of the Electron Gas 49

Fig. 1. Regions of summation in momentum space (q ~ PFa). All three spheres have radius PFa. The one centred at 0 is the Fermi sphere

where q = P -- p is the momentum transfer. Then the quadratic part is

q p a

1 ~ - ~ ~ ( 2 [ V ( q ) - ~aa' V(p--p')]C+p,(q)Cp,,,(q) (2.12)

q p p ' G G r

[V(q) -- ~ , W~(p + p', q)J[C~c,(q) G~e,~ ,, (-- q) --}- H.e.]},

_ 1

w~(p + p', q) = v (p § p' + q) § o ~ [ - (p § p'). q] [v (q) - v (p § p' § q)].

In H2 the momentum transfer q is summed over the whole momentum space. For a given q and a, because of (2.11), p and - - p ' are summed only over those regions inside the Fermi sphere for spin g such that p § q and - - p ' - - q lie outside of it. These are depicted as regions 1 a and 1' g, respectively, in Fig. 1 for the case q < PFa. They overlap in a region 2a. The function 0 2 a [ - - ( p ~ p ' ) . q] has the value 1 when p and --p ' are both inside 2~ and simultaneously its argument is positive; otherwise its value is 0. For q > 2pFa, all three regions i g, l ' a and 2 g coincide with the Fermi sphere.

~p~ in (2.12) is recognized to be the quasiparticle energy in the Hartree-Fock approximation. V(q) is the direct Coulomb potential. The other terms, pro- portional to Oaa', are the exchange potentials. The term in 02a arises from the kinematical interaction and measures the deviation of an electron-hole pair from a true boson. Its presence ensures that each fermion process is uniquely represented by a process in the physical boson subspace. At high density the most important processes are those of small q, in which case the volume of region 2 a vanishes like q3. The kinematical interaction then is negligible, in agreement with the well- known result that an electron-hole pair behaves practically like a boson in this region. Finally we notice that the coefficients of H2 are invariant under inversion in momentum space.

In the following calculations we shall for simplicity adopt the harmonic approximation in which H3 and Ha are discarded. The error thus incurred is

4 Phys. kondens. ~aterio, Vol. 15

50 J. Lam:

difficult to estimate, particularly in the intermediate coupling region. Roughly speaking, the approximation is good ff the boson density is low, that is, if the fraction of electrons outside of the Fermi spheres is small. Calculations of the momentum distribution function for the paramagnetie state at metallic densities [5] indicate that some one-fourth of the electrons are depleted from the Fermi spheres. A sizable portion of these electrons, however, are distributed far away from the origin of momentum space so that they can be expected to have little influence on processes of momentum transfers q ~ PFa which are the most important at these densities.

III. D iagona l i za t ion

We apply a canonical transformation to diagonalize H2 which is of the form

H2 = H t + H~ + Ht~, (3.1) where H t and He contain only up-spin and down-spin operators, respectively, and H?~ is a coupling term. For ~ = 0 this coupling can be first transformed away by introducing singlet and triplet operators, as was done in Ref. [2]. For general $, we consider the following linear transformation to a set of new boson operators A : 17

Al , (q) : ~ [a~p(q) Cpt(q ) + bzu,(q) C+--pt(-- q)] P 1~ (3.2)

+ ~ [c~p (q) opt(q) + ~k~ (q) o+-~(- q)]. P

The superscripts 1~ and 1 ~ indicate the regions of summation. For a given q we can take the set of indices k to be the same as the set p a. The coefficients a, b, c and d are taken to be real and invariant under inversion in momentum space. The transformation (3.2) is canonical, that is, A satisfies the canonical Bose commuta- tion relations

[A~(q), A~,(q')] = 0, (3.3)

[A~(q), A~,(q')] = ~ , ~qq,, ff the coeff• satisfy the following set of identities:

it [akp (q) b~,p (q) -- bkp (q) a~,p (q)]

P

~- ~ [c~p(q) dk,p(q) - - dl, p (q ) c~,p (q)] = 0, (3.4) P

it ~. [a~ (q) a~,p (q) -- b~p (q) b~,p (q)] P

+ ~. [chp (q) cl,,p(q) - - d~p(q) d~,p (q)] : ~ , .

The inverse transformation is given by q

Cpt(q) = ~. [aap (q) A k ( q ) - - bkp(q) At_.k( - q)], k

q (3.5) C~,r (q) = ~, [c~p (q) A~,(q) - - d~ , (q ) A+_k( - q)].

k

Ferromagnetism of the Electron Gas 51

We use the superscript q to indicate that the range of the k-summation is depen- dent on q. By (2.8) we also have the following set of identities:

q

[a~,p (q) bke" (q) -- b~p (q) akp, (q)] ----- 0, b

q

[a~p (q) a~e,(q ) -- bkp(q) bkp,(q)] = ~pp,, k

q

[ckp (q) clap, (q) -- dap (q) cap, (q)] = 0, (3.6) k

q

[cap (q) ~ap,(q) - dap (q) ~ap,(q)] = ~pp,, k

q

[ak~ (q) dap,(q) -- bkp (q) cap,(q)] = 0, k

q

[nap (q) cae'(q) -- bap (q) dap,(q)] = 0. k

Then H2 can be brought to the diagonal form

H2 ---- Ecorr + ~ ~ga(q)A~(q)An(q), (3.7) qk

with the eigenfrcquencies ~ga(q) = #2-a(-- q) determined from the eigcnvalue equation

#2a(q) Aa(q) = JAn(q), H2] . (3.8) I t can be shown that

Ecorr = -- ~ #24 (q) b~, (q) -{- ~ d~, (q) . (3.9) qk p

I f all the eigenvahies are positive, Ecorr is the correlation energy in the harmonic approximation.

Strictly speaking, the canonical transformation diagonalizes H2 in the entire boson state-space. The ground state thus obtained will have an unphysieal componenb. However, ff the harmonic approximation is valid, the contribution to Ecorr from the unphysical admixture can be expected to be small. Alternatively we can form the ground state from states in the physical subspace and solve the eigenvalue problem for the ground-state energy. This Ieads to an infinite set of equations which can only be treated approximately.

We write down the components of (3.8) as follows:

[#2a(q) -- cop (q)] aap(q) -- Na(q)

1 ~ [ V ( p - - p ' ) V(p--p ' -4-q)]aap(q) #2 ~'<w~

1 ~t =O ~ [V ( p - - p ' ) a a e , ( q ) - W~(p-t- p',q)bkp,(q)],

p"

4*

52 J. Lain:

[~2k(q) + ~op (q)] bkp(q) - - Nk(q) 1

= - - - - ~ , [ V ( p - - p ' ) - - V(p--p'-[-q)]b~u,(q) "Q P'<p~t

[v(p - F) b~p,(q) -- Wt(p + p', q) a~p,(q)]

_ 1 ~. [ V ( ~ ' - Y ) - V ( p - Y + q)]%,(q) ~ ~'<~F~

-- ~ - ~ [V(P -- Y) c~p'(q) -- W,(p + y , q) d~p,(q)], p'

(3.1o)

[gk(q) + %(q)] d~, (q) -- Nk(q)

= - - 1 ~ [V(p- -p ' ) - - V(p--p'+q)]d~,p(q) Q ~,<~

1 ~ + -~ ~ [V(p --p')dkp,(q) -- W~(p +p' , q) c~,~,,(q)],

p"

where

%(q) = e~+q - e~,

Nk(q) = V(q) {fit(q) [a~(q) -- b~(q)] ~-/~4 (q) [cA(q) -- dk(q)]},

~ (q) = ~ ~ 1,

1 it 1 it (3.11) a~(q) = ~ ( q ) ~ ~,~,(q),~ b~(q) = ~t(q)~ ~p b~p(q),

1 1~ 1 ~-~ c~(q) - ~(q) ~ ~ eke(q), dk(q) -- ~(q) Q /p dkp(q).

We have purposely gathered all the exchange terms on the right-hand sides. This set of equations can only be solved approximately. In the random-phase approximation the right-hand sides are set equal to zero. We ~11 use an approxi- mation in which they are replaced by ee~ain averages over I a. This procedure has been described in detail in Sects. IV and V of Ref. [2] for ~ = 0. The generaliza- tion to ~ ~= 0 is obvious and we will simply write down the results. For example, the right-hand side of the first equation in (3.10) is replaced by

2/zt(q) [/t (q) ak(q) -- gt(q) bk(q)], (3.12) where

1 la l~(q)- ~ (q )~ ~ /po(q),

M

1 ~ (3.13)

P

Ferromagnetism of the Electron Gas 53

and

1~, (q) = F [ V ( p - - p ' + q) - - V ( p - - p ' ) ] A- ~ V ( p - - p ' ) p '

1 ~ (3.14) w~(p + p', q).

gP~(q)-= 2#~(q)~2 p,

As a consequence (3.10) is simplified to the form

[t2k (q) -- • (q)3 ak~ (q) = #t (q) [Jr (q) a~ (q) -- K t (q) bk (q)] A- f~(q) V (q) [c~(q) -- d~(q)] ,

[t2h (q) § % (q)] bkp (q) = f t (q) [K+ (q) ak (q) -- J t (q) ba (q)] + f~(q) V(q) [ca(q) -- da(q)], (3.15)

[~93 (q) -- co r (q)] cap (q) = f , (q) [J~ (q) ca (q) -- K~ (q) da (q)]

-4- fit(q) V (q) [aa(q) -- ba(q)] , [t2k(q) § o~(q)] ds,~,(q) = f~(q) EKe(q) ck(q) -- Jr d~(q)]

"4- ft(q) V(q) Eaa(q) - ba(q)],

where J(~(q) = V(q) -4- 2/a(q) ,

K~(q) ---- V(q) -{- 2g~(q). (3.16)

These equations are what we would have obtained ff we had used for H2 the effec- tive expression

qpa

1 (3 .17) -{- ~ ~. {2[V(q) + ~ , , 2 / a ( q ) ] C~o(q)Gp,,,,(q)

qpp' (jcjt

+ IV(q) § ~,r 2 g~(q)] [C~,(q) C+p,o.(-- q) A- H.c.]}.

1a(q) and ga(q) play the role of effective exchange potentials. For q ~ r they both approach the limit --�89 V(q).

(3.15) can be reduced to a system of four homogeneous linear equations. The vanishing of the determinant yields the dispersion relation for the eigenfrequencies:

where e (q, ~l,( q) ) = O, (3.18)

e(q, co) = ~t(q, ~o) e4(q, ~o) -- ~t(q, co) ~ (q , co), (3.19) and

I ~ 2 a)~(q) ~(q, ~) = 1 - J~ (q ) ~

~0 2 CO 2 (q)

- - [ J~ (q ) - - K ~ ( q ) ] L ~ - ~ co - ~o~(q) J L ~ ~ o~ + ~o~(q) '

i ~- 2~o~(q) ~a (q, w) = - v (q) ~ ~.

CO 2 (q) t*

- - 2 V ( q ) [ J a ( q ) - - K a ( q ) ] [ - ~ P

(3.20)

] co - % (q) co + c% (q) "

54 J. Lain:

The dispersion funct ion (3.19) is normal ized to approach the l imit 1 as eo --> oo. F r o m (3.4), (3.6), (3.9) and (3.15) it can be shown t h a t

l a

This expresses Ecorr as the shif t in the zero-point energy of the diagonal pa r t of (3.17).

IV. Correlation Energy

2 (q) and zeros The dispersion funct ion s (q, ~o) is ana ly t ic in o~ 2, wi th poles a t (o~ a t ~ (q). B y evaluat ing the contour integral

1 ~ dzy~ ~__~lns(q, ~/z) (4.1) 2 ~ i

a round a p a t h enclosing all the poles and zeros and then mak ing a suitable change of variable, i t can be shown t h a t

r q l a 1 ( ~ [21,(q)- ~. o)p(q)-~ ~ j d u l n s ( q , i u ) . (4.2) It p a

- - c o

Using the iden t i ty oo

f ~%(q) 1_ du - - 1 a)p (q) > 0 (4.3)

u2 + ~ (q) '

we obta in f rom (3,21) co

~oo,r = - ~ d~ In s (q, iu) -- ~ J~ (q) ~ ~, u2 + ~ (q) - - o o

This expression enables us to calculate the correlat ion energy wi thout an explicit formula for Dk (q).

We take the l imit 2V, f2 -+ cr keeping the densi ty constant . I n this case the m o m e n t u m sums go over to integrals. We introduce dimensionless m o m e n t u m and energy var iables x and y, respectively, for the three different Fe rmi spheres as follows :

q ---- x pF ----- x t pF t ~ X$ pF~,

U --~ y EF • Yt EFt ----- y~EF$,

E]~ -- EFa -- 2m ' 2m "

F r o m (2.5) we have the relat ions

X

x~ - (1 • ~)1/3 '

In t e rms of x~ and ya we ob ta in

1 ~ ~ 1 mpF~ .(2 ~ i u -4- ~% (q) 8 ~2

Y Y~ = (1 • ~)2/3 "

[ • R (xa, ya) - - i I (x~, y~)],

(4.5)

(4.6)

(4.7)

Ferromagnetism of the Electron Gas 55

where

I ( x , y ) =

y2 + 4X 2 __ X 4 (2X ~- X2) 2 ~- y~ R (x, y) = 1 -4- 8x a In (2x - - x2) 2 -]- y2

y t a n -1 2 x + x 2 + t a n - 1 - 2 x y y '

y2-4-4x2--x4 ( 2x-4-x 9" 2 x - - x 2 ) 4 x a t a n - 1 t a n - 1 - -

Y Y

y [(2x + x2) 2 + y2] [(2x - - x2) ~' A- y2] y + In

y4 2x '

y2 -~- 4x2 -- x4 ( 2x --]- x 2 2x -- x 2) 4 x 3 t a n - 1 . ~- f an-1 _ _

Y Y

~ x (2x + x2) 9' + y2 y In (2X - - X2) 2 + y2 X2 , X > 2 .

(4.8)

0 < x < 2

The p o t e n t i a l s are scaled accord ing to t h e p re sc r ip t ions :

4~e 2 1 V ( q ) - / ~ V ( x ~ ) , V(x ) - x~ '

47g 62 / a (q ) - - p ~ , ] (xa ) , (4.9)

4g e 2 ga(q) - - p ~ g (xa ) .

The f u n c t i o n s / (x) a n d g (x) d e p e n d on t h e sp in s t a te o n l y t h r o u g h t he a r g u m e n t , a n d are in fac t t h e s ame as t h e ones def ined for t he p a r a m a g n e t i e s ta te . The l a t t e r have b e e n ca l cu la t ed a n d p l o t t ed in Sect. V of Ref. [2]. B u t for c o m p u t a t i o n a l pu rposes we can use t h e fo l lowing empi r i ca l fo rmulae [6] :

/ (x) --~ - - 0.6667 In x ~- 0.2677 A- 0.2500 x in x (4.10)

- - 0.3269 x + 0.0824 x 2 -]- 0.0036 xs, 0 < x < 2

2x 2 - - 1.0179 ' x > 2

g (x) = - - 0.8750 - - 0.2500 x In x -[- 0.3854 x

-~ 0.1596 x 2 - - 0.0402 x 3 , 0 < x < 2

1

2 x 2 - - 0.5436 x > 2 .

W e not ice t h a t [(x) is l o g a r i t h mi c a l l y d i v e rgen t whi le g (x) is f in i te as x--> 01.

1 In Ref. [2], at the end of Sect. V, it is incorrectly stated that f(0) is finite.

56 J. Lain:

1.0

0.5

--0.5

-1.0 L. 0

V(X)

g (X)

I x 2

Fig. 2. Effective potentials

These and related functions J(x) = V(x) + 21(x)

K(x) = V(x) § 2g(x), are plotted in Fig. 2.

Gathering all the results, we obtain

where

o o

C (x, y) ~- Ins (x, i y) -- �89 ~ ya J (xa) R (xa, ya),

Y Y'~ - (1 + ~')1/3 '

e 2 pp 2:r - - rs ~--- 0.332 rs,

(x, i y) = ~t (xt i YO ~r (x~, i y~) - ~t (+ , i YO ~ (x~, i y~)

ea (xa, i ya) ---- 1 § �89 ya J (xa) R (xa, ya)

§ ~ ~,~ [g2 (xa) -- K u (xa)] [R 2 (xa, ya) § 12 (xa, Ya)],

Va (xa, i Ya) ---- �89 ya V (xa) R (xa, ya)

§ �89 ~ V(xq) [J(x~) -- K (x,)] [R2 (xq, y~) + 12 (x~, y~)].

(4.11)

(4.12)

(4.13)

Ferromagnetism of the Electron Gas 57

When /(x) and g (x) are set equal to zero, we obtain the results in the random- phase approximation:

C gPA (x, y) -~ In e ~PA (x, i y) -- �89 ~ ~a V (xa) R (xa, Ya), (4.14)

~PA (z, i y) = 1 ~- �89 ~ ~a V (xa) R (xa, ya), a

which agree with the findings of Brueckner and Sawada [7] by perturbation theory. For the paramagnetic ease $ -- 0 we recover the results of Ref. [2] since ~ (x, i y )

becomes factorizable: (x, i y) ---- ~(1) (x, i y) e(8) (x, i y) , (4.15)

where the factors are the dispersion functions for the eigenfrequeneies of the singlet and triplet states.

The ground-state energy is given by

Egr ~- E ~ A ~- Ecorr �9 (4.16) From (2.10) we obtain

In units of rydbergs (--~ e2/2ao),

1 3.683 EF---- ~2r[ ---~ r~ (4.18)

V. Numerical Results

The two-dimensional integral for the correlation energy in (4.12) is evaluated numerically for ~ = 0 -- I and rs ~-- 5 - - 12. The correlation energy in the random- phase approximation is also evaluated for comparison. The ground-state energy is plotted in Fig. 3. The curves of Eg~r ~A for various values of $ lie at the top. There is an intricate pattern of over-crossing for rs between 5 and 6. For rs <~ 5.5 the curve for ~ = 0 is lowest, while for rs >~ 5.5 the one for ~ ~-- 1 is lowest. Beyond rs ~_ 6 the energy increases with decreasing ~.

The ground-state energy in the random-phase approximation lies lowest of all, and increases with ~. Ferromagnetism is therefore not favoured. Beyond rs ~ 8,

the curves for various $ converge noticeably but never cross over. Our findings are therefore in disagreement with those of Misawa [8], who reported a lower energy for the fully-polarized state than the paramagnetic state for rs > 7.4. This is probably due to his taking, in the evaluation ^c vR~A an approximation which t ) J . . [ 2 J c o r r ,

discards all contributions from processes with momentum transfers q > pp. In Fig. 3 most of the curves in the present approximation are terminated

before rs = 12. The end points represent the onset of instability when the correla- tion energy acquires an imaginary part. This is due to the fact that, for rs beyond a certain limit, e(x, i y ) in (4.12) can become negative within a certain range of values of x. These curves lie between those in the Hartree-Fock and random- phase approximations. They are raised with increasing $; and the spacings are much wider than in the other approximations. The critical value of rs decreases

58 J. Lam:

- 0.061 ~ i J ...-

I 0.0.- "I .-

.'~0.8 .-'I f-"

-~Io

7 z

-0.14

- 0.18

o.s / / ./~!~

/ ./// . / ' " �9

. I.U...... /' .

"-- I" / /" �9 / .

/

0.01, ' / /

T I

'6 8 ~0 rs 12

Fig. 3. Ground-state energy in various approximations: E~ Fi (broken line), E~ PA (dots and dashes), Egr (solid line). The number above each curve is the relative polarization

with ~ except that, to our surprise, the fully-polarized state ~ ~ 1 is stable throughout the range of rs considered. This latter state, however, lies above all the rest. We will examine this phenomenon in Sect. VI. Thus, ignoring the competition of other possible (for example, antiferromagnetic) ground states, we can say that the electron gas stays paramagnetie until rs ~ 9.4 when it goes over to the fully- polarized ferromagnetic phase. Such a transition requires an energy absorption of some 0.03 rydberg per electron.

u Stability Limits

We examine in some detail the question of stability. The condition for the correlation energy in (4.12) to be real is tha t s (x, iy) be positive. For a dispersion function of the type (3.19) and (3.20) it is well-known that we only need to require

s(x, 0) > 0. (6.1)

This ensures tha t all the eigenfrequencies are real. Replacing the sign in (6.1) by a equality and discarding two non-zero factors, we obtain a quadratic equation for determining the critical value of ~ at fixed values of x and ~:

A (x, ~) y~ -l- B (x, ~) 7 ~- I = 0, (6.2)

Ferromagnetism of the Electron Gas 59

3

Y

1 1 O. 1.0 1.5

X

Fig. 4. Determination of stability limits. The curves are labelled by the relative polarization

where

The solution is

A (x, ~) = u t(x, ~) u~(x, r - - v t(x, ~) v~(x, ~),

B (x, ~) = u t (x, ~) + u4 (x, ~),

[J (xa) + K (x~)] ua(x, $) = 4(1 ~- ~)1/3 R(xa, 0) ,

V (x,;) v~ (x, ~) - 2 (1 + r R (x~, 0) ,

4 - - x 2 In 2 + x R ( ~ , o ) = l + 4-~-- ~ - ~ - "

(6.3)

The other root is negative and hence unphysical. Fig. 4 shows a plot of (6.4) as a function of x for various $. Each curve exhibits

a pronounced minimum at a non-zero value of x. The ordinate of the minimum measures the critical value of 7 above which the ground state for a given ~ becomes unstable. We see tha t the minima are lowered for increasing ~. The instability is therefore pushed up to higher densities for larger relative polarizations.

The fully-polarized state, however, cannot be regarded as the limit ~ - + 1. The reason is as follows. We can show that , for the values of rs considered, the sum of the last two terms on the left-hand side of (6.2) is always positive. The equality, and hence instability, can only come about with a non-vanishing A (x, ~). In the random-phase approximation A is zero because of the absence of exchange potentials. For the fully-polarized state A is zero because of the absence of down- spin electrons. The stability of these two cases has been confirmed by Fig. 3. In the other cases the non-vanishing of A is due to the exchange potentials ] (x) and

B + g ~ - 4 a ~' = 2A (6.4)

60 J. Lam: Ferromagnetism of the Electron Gas

g (x). However, starting from ~ < 1, A does not approach zero as ~ --> 1. This is because A depends on ~ mainly through the combination

x Xa - - (l -I- ~)1/3 " (6.5)

For any value of ~, however close to 1, xa still ranges over all values between 0 and c~. A change of $ merely effects a change of scale and a shift of the minimum. As a consequence the fully-polarized state, with ~ --~ 1 exactly, is a case all by itself.

Acl~nowledgements. The author wishes to thank ProL Dr. W. GStze for his hospitality at the Max-Planck-Institut.

References

1. Bloch, F.: Z. Phys. 57, 545 (1929). 2. Lain, J.: Phys. Rev. BS, 1910 (1971). 3. Sawada, K.: Phys. Rev. 106, 372 (1957). 4. Usui, T. : Progr. theor. Phys. (Kyoto) 28, 787 (1960). 5. Lain, J.: Phys. Rev. B3, 3243 (1971). 6. -- Phys. Rev. BS, 1254 (1972). 7. Brueekner, K. A., Sawada, K.: Phys. Rev. 112, 328 (1958). 8. Misawa, S.: Phys. Rev. 140, A1645 (1965).

Dr. John Lam Max-Planck-Institut ffir Physik und Astrophysik D-8000 Miinchen 40 FShringer Ring 6 1%deral Republic of Germany