Spontaneous magnetization in the hubbard model

Volume 44A, number 6

PHYSICS LET1~ERS

16 July 1973

SPONTANEOUS MAGNETIZATION IN THE HUBBARD MODEL H. NISHIMURA Institute of Physics, College of General Education, Kyushu University, Ropponmatsu, Fukuoka, Japan Received 18 June 1873 By using the functional integral technique, the spontaneous magnetization is evaluated in a molecular field approximation. It is demonstrated that the ferromagnetic Curie temperature exists under a certain necessary condition.

Recently Evenson, Schrieffer and Wang [1] have applied the functional integral technique to the discussion of ferromagnetism. Cyrot [2] has discussed the Hubbard model along the same lines as Evenson et a!.. Their treatment is, however, restricted to the two center problem, and is unsatisfactory to obtain the spontaneous magnetization of the itinerant electron system. In this paper, we want to evaluate properly the spontaneous magnetization of the system. The partition function of a system with Hamiltonian H 0+H1,

tam an exact functional integral representation of the partition function: rv (v )i 1W (v )i Z q n q “ IZ({Vq(Pn)~Wq(P~)}) q,n ~/~3c J ‘../~3c

=f [J

-~

X exp

j— 2j3

~ q,n

—~—~

C

2] [IVq(P,~)I~ + Wq(l)n)1

—

Z({Vq(Pn)~Wq(Vn)})

—

z0 exp {Tr log [(K~Kt)_l

2K~g~O*Ktg~)] } (1

_

Z=Zo(T~exp[_f drH 1(r)]),

(1)

0

is evaluated by using the Stratonovich transformation [3] explal2 =f~2xexp[_~IxI2_vI~a*x+ax*)1 (2) In order to be~adaptedto eq. (2), the Hubbard Hamiltonian [4] of the system is written in the crystal momentum representation as H 0

=

~

~C~aCkO

‘

~k

=

ek+~U+a!JBh,

=

~

of the quantities with respect to the fictitious timer appearing in eq. (1), i~and i~are the matrices in the (q, i.’7,) space, Tr means the trace in this space, and is the zeroth order Green function. As a first approximation to eq. (4), we retain only the Gaussian random molecular fields

4 (3)

k,a

H1

K°= In the above, the integration variables play the role of a random magnetic fieldcoupling acting on the electron spins, 2is the constant, the frequency c~‘n= has (irU/3Nj3)~ been introduced in the Fourier transformation

(Sq• S_q)~

where ek is the band energy, cko the electron operator in the Bloch state (k, a), ~B the Bohr magneton, h an external magnetic field,N the total number of atomic sites U the Coulomb interaction on the same site, and is the Fourier transform of the electron spin operator on an atomic site. After some calculations, we ob-

vqo(vn 0), wq=o( ~“n= 0). In the case of no external magnetic field, eq. (4) is reduced to Z = ~J~ir

f d~

~2

exp [-F~)],

0 5

= ~

‘ ‘

2 ~ 3/2

~o~l3 ~1F3I2(n—u + 13c~)+ F312 (i~ u where the density of states per spin has been assumed —

—

—

405

Volume 44A, number 6

PHYSICS LETTERS

to have the form D(e) = ~i potential). u = ~U/2, and F

151 M

=

.

(2,

73

=

~ip (p the chemical

5(73) is the Fermi function The spontaneous magnetization is given as 32~1F

p,~CLd

u+!3c~) F

1~2(~

~

~

16 July 1973

where the relative magnetization

~‘

has been introduced,

~mbeing the number of electrons. Thus the spontaneous magnetization (6) is given as =

(12;

lilt ,,+~ Ii

,~

where 7... ~ defines the average:

I

O~~)I ~(r~ ~ ~ -

B

We can evaluate the integrals(S) and (7) by considerlog the extrenii contributions troni the exponen

by defining the exchange interaclion parameter k1~o= nU/3N, ~ being the Boltmann constant. Now the condition (9) or (10) corresponds to the situation nm which the temperature range is either above or below the Curie point respectively. The Curie point T1~,is thus defined as the temperature satisfying the relation.

F(s). The solution ~ = 0 of ~Fh~= 0. that is. 1a~ 2c~F ~= 112(73 u+~c~)F1 2(1) U

~r 2a~ c112(7] which is written as

=

~

[ ~

I

I~) I

7)

I

.

(~~) gives the dominant contribution 2E’ (~ u) > 0 . if the condition, I. 2 - 2a~/ 2 c

(9)

<0.

2 = ~

{l

+~-~

k

T 2

}

(--~._~)4

by using the expression

is fulfilled, and leads to the vanishing of the spontaneous magnetization (6). Instead, if we have the condilion. U)

k 0

0.

U)

I l.~)

~),

3aLj; tor the non-interacting system. According to eq. (10 or (14), the condilion ~UD(E1~)/N > I is necessary for ferromagnetism to Occur at all.

(tO)

the pararnagnetic solution ~ = 0 does Ilot contribute to the integrals, and therm there appears another nonzero solution for eq. (8). This solution ~ turns out 10 he the ferroniagnetic solution and gives the dominant 2F/a~2) to the integrals ii the condition contribution (a 11> 0, which is compatible with the condi-

References

tion (10), is fulfilled. We can approximate this solution

15! E.C. Stoner, Proc. Roy. Soc.A 165 (1938)372.

to

(lIt

406

(14)

ii =

1! W.L. Lvcnson, J.R. Schricfter and SQ. Wang, J. Appt Phys. 41(1970) 1199; Phys. Rev. 02 (1970) 2604. 121 N. Cyrit, Phys. Tev. LeO. 25 t 1970) 871. 3! R.L. Siratonovmch, Soviet Doklady (1958) 416. 4! J. hubbard, Proc. Roy. SocIThys. A276 (1963)2 283