POSSIBLE FERROMAGNETISM OF AN ELECTRON PAIR IN ONE DIMENSION

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POSSIBLE FERROMAGNETISM OF AN ELECTRON

PAIR IN ONE DIMENSION

S. Olszewski

To cite this version:

JOURNAL DE PHYSIQUE Collogue C6, supplkment au no

8,

Tome 39, aoiit 1978, page C6-7 13

P O S S I B L E FERROMAGNETISM O F AN ELECTRON P A I R I N ONE D I M E N S I O N S. Olszewski,

I n s t i t u t e of PhysicaZ Chemistry, PO l i s h Academy of Sciences, Warsaw, PO Zand.

R6sumd.- Dans le cas d'une rdpulsion Coulombienne l'ltat le plus bas d'une paire dt61ectrons libres n'est pas un singlet non-magndtique mais il est un triplet qui peut porter un moment magndtique pro- pre de spin. Ce rlsultat peut mettre en doute le thdorlme de Lieb et Mattis qui dit que 1'Qtat fon- damental de chaque systiime unidimensionnel de plusieurs dlectrons est compldment nonmagnltique. Abstract.- In the caseof the Coulomb-like repulsion the lowest state of a free-electron pair in a

I-dimensional potential box can be not a non-magnetic singlet, but a triplet, which may carry a net spin magnetic moment. This result may question the Lieb-Mattis theorem that the ground state of any I-dimensional many-electron system is fully non-magnetic.

Lieb and Mattis / l / have given a theorem that the ground state of any 1-dimensional many-electron system must be fully non-magnetic. The theorem ap- plies to a wide class of potentials V(x,, X,,

....

5)

between N electrons, since V is assumed only real and symmetric with respect to the variables

x1,x2,

...,%.

Otherwise, V is completely arbitrary. The theorem is an extension of a similar theorem for an electron pair : if V(x,,x,) is any real symmetric potential the ground state of the electron pair should be a non-magnetic singlet The aim of the present note is to show that for

where C is a constant, the theorem of /l/ does not hold : the ground state of an electron pair in a

I-dimensional potential box is a triplet, gA, which may carry a net spin magnetic moment. This is so on condition that any @ of the pair is continuous and has continuous first derivatives. The boundary con- ditions 0 ~ x 1 , x 2 ~ L and @ = 0 for xl,x, = 0 or L are the same as in /I/.

The above problem has been raised earlier /2/ but no calculations were done. The result expected in /2/ was the same as it was in /,l/. The main argu- ment in /2/ for as the ground state was that @A had to satisfy the condition $ (X X ) = 0 when

A 1 ' 2

X; = X,, whereas @s(xl,xp)was free from any cons- traint within the box ; the conditions satisfied by and

OS

at the box ends were the same. But it can be shown that VI$ (x,,x,) = 0 where the gra- dient operates along the line X, = a-X, perpendicu- lar to the line X, = x2 and is calculated at the in- tersection point of two lines, 0 < a <L. The result

is obtained because any is a symmetric combina- tion

(OA

is antisymmetric) of

2 mv

Y(m,n) =

-

L sin (- L xl) sin

(9

X,)

Hence and has each a different constraint within the box and the mutual position of energies E(@s) and E(@ ) cannot be predicted a priori. The

A

main argument for E($s)< E($ ) given in /l / was A

that Q of any ground state should be nodeless. This theorem is based on the calculus of variations which admits as solutions of the eigenproblem the functions having both continuous and discontinuous first derivatives /3/. It may therefore happen that a nodeless function of the ground state has a dis- continuous first derivative. This function cannot be accepted for physical reasons, since no sinks or sources occur in the system. We calculate

where V is a constant potential of a uniform po- sitive background which makes the system neutral ; the unperturbed wave functions are + O Y(1,l) ;

-

_[Y(l,2)

_-

_Y(2,1)]

_.

_{We obtain}

'+A

-

JZ

E ( ) = ( l ) 31n(0)

+

31nL

-

2 2 ~ 2 L

References

y is the Euler constant. Evidently, E(') ($s)>~(l) 0%)

/l/ Lieb, E.H. and Mattis, D.C., providing C>O. The calculation of E)'( for v

2

2 re- Phys. Rev.

125

(1962) 164

quires a potential box that has finite but very /2/ Slater, J.C., Statz, H. and Koster, G.F., small transversal dimensions. We assume the box to Phys. Rev.

2

(1953) 1323

be a cylinder whose radius R = I<<L, which is the /3/ Courant, R. and Hilbert, D., "Methods of Mathematical Physics" box length. )'(E is the combination of (Wiley, New York) 1953

v factors U -+

+I...w

. , . W -+

n P q'r s'n

p,q.. .# n

v-l factors

The summation runs over the both components of vec- -+

tors W = (wl, w2) = (w2, W ) that label the elec-

1

tron levels occupied in excited states $ O ; the

-P -+

ground state is n = (1.1) for $: and n =(1,2)=(2,1) for g; ;

-+

where i = S or A ; =xl and

]

r21 =x2. Putting

and taking into account in the perturbation series only Gell-Mann and Brueckner bubble graphs we obtain

QA and Q are constants independent of L. The series S

2

E

)'(E are convergent providing that l.'>1 and v=2

L>>1.