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Submitted on 1 Jan 1978

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RENORMALIZATION CALCULATION OF THE

GROUND STATE ENERGY OF SOME QUANTUM

MODELS

R. Dekeyser

To cite this version:

(2)

JOURNAL DE PHYSIQUE Colloque

C6,

supplPment ou no

8 , Tome 39, a061

1978,

page

C6-747

I

RENORMALIZATION CALCULATION OF THE GROUND STATE

ENERGY

OF SOME QUANTUM NODELS

R. Dekeyser

I n s t i t u t

voor

t h e o r e t i s c h e

f y s i k a ,

Celesti jnenlaan 200 D, 8-3030 Leuven (Belgium)

R6sumd.- Nous dvaluons par une mQthode de renormalisation, 1'Qnergie fondamentale du modlle

XY

1

spin 1/2 sur le r6seau triangulaire. L1itEration

ap~ropride

ne converge que pour un choix de

la partie perturbative de llHamiltonien.

Pour tester la mdthode, nous l'avons appliquee aussi

au modele

XY

et au modsle de Heisenberg antiferromagndtique,

B

une dimension.

Abstract.- The ground state energy of the spin 1/2 XY-model on the triangular lattice is esti-

mated by a renormalization calculation. This iterative procedure converges only after a redefi-

nition of the perturbative part of the Hamiltonian. In order to test this method, we applied it

also to the one-dimensional

XY-

and Heisenberg antiferromagnetic models.

The puzzling nature of the ground state of

turbation method gives then new ground states

the spin 112 XY-model in two dimensions has received

I@,(P)

>

,

and the equation for their energies can

considerable attention during the last years. By

be written in matrix form as

numerical calculations on finite lattices, Oitmaa

and Betts /l/ obtained an estimate for the ground

<@o (p)

IHI@,(~)>

= W O

6pq

+

<@,(P)\H'~@,(~)>

+

state energy, which was confirmed by a variational

calculation by Suzuki and Mi~ashita

/2/. There is,

<@o

(

p

W,-H, H'

'

I

~

'

+.

.(3)

A

however, a third method for looking at the ground

state structure and energy, inspired by the real-

with

space renormalization group techniques /3/. This

P

=

1

-

1

I@,(P)>

<@,(P)I

.

method has already been applied by Friedman 141 to

P

the Ising model and by Van de Braak et a1.151 to

the antiferromagnetic Heisenberg linear chain. The

application fo this method to the two-dimensional

X '

model might give an alternative insight in the

ground state, complementary to the one given by

Suzuki /2/.

The essential idea of this method, as ap-

plied to spin 1/2 systems, is to construct cells on

the lattice, in such a way that the ground state of

the Hamiltonian

-

if restricted to such a cell

-

is

a Kramers doublet. These cell ground states are

then interpreted as the new spin 1/2 doublet on the

cell. With these states, it is possible to construct

unperturbed ground states I@~(P)> on the cell latti-

ce, where p

P

1,2,.

. .

,2M. M being the number of cells

If the Hamiltonian is split in two parts

H, containing as usual the intracell and

H'

the

intercell interactions, the Rayleigh-Schr6dinger p e r

This matrix may then be interpreted as a renormali-

zed Hamiltonian between the cell spin variables. If

the original interactions in the Hamiltonian on the

lattice of N sites are denoted by

.l

(o), equation

j

(3)

may be written as

and from this, the ground state energy is derived by

iteration as

n

= N/M,

the number of lattice sites in a cell.

We first tried to apply this procedure in a

direct way to the

XY model on the triangular latti-

ce, with the usual triangular cell choice (See Refs,

3 and 4). Starting with the pure nearest neighbour

XY Hamiltonian, one generates also next and third

neighbour interactions, and a nearest neighbour

Ising interaction. The general form of the Hamilto-

nian after the k-th step is

:

(3)

E v a l u a t i n g t h e n e c e s s a r y m a t r i x e l e m e n t s , and ad- d i n g up a l l t h e c o n t r i b u t i o n s , we f i n d t h a t t h e r a t i o s J ~ ~ ) / J ( ~ ) and K ( ~ ) / J ( * ) converge r a p i d l y t o some f i x e d v a l u e s , b u t t h e r a t i o J ( ~ + ' ) / J ( ~ ) t e n d s t o t h e v a l u e 5.076. S i n c e E, i s p r o p o r t i o n a l t o J , and n=3, we i m m e d i a t e l y s e e t h a t t h e sum i n (5) d i v e r g e s . T h i s means t h a t t h i s method c a n n o t be a p p l i e d t o t h i s model i n a s t r a i g h t f o r w a r d way.

The fundamental r e a s o n why t h i s p e r t u r b a t i o n - r e n o r m a l i z a t i o n scheme d o e s n o t converge i s proba- b l y t h e f a c t t h a t t h e p e r t u r b a t i o n H' i s n o t s m a l l w i t h r e s p e c t t o H,. I n d e e d , on o u r t r i a n g u l a r l a t - t i c e w i t h N s i t e s , H, c o n t a i n s N n e a r e s t n e i g h b o u r n t e r a c t i o n s , whereas H' c o n t a i n s 2N o f them. A way

o u t o f t h i s problem i s t o i n t r o d u c e a new s e p a r a t i o n o f t h e H a m i l t o n i a n i n u n p e r t u r b e d and p e r t u r b e d p a r t s . I n s t e a d o f ( l ) , we may a l s o w r i t e where V = (1

-

a)Ho

+

H ' A n a t u r a l c h o i c e o n t h e t r i a n g u l a r l a t t i c e would t h e n be a = 3. Then V c o n t a i n s 2 N i n t e r a c t i o n s w i t h a p l u s s i g n and 2N w i t h a minus s i g n , and we may hope t o have minimized i t s e f f e c t . I n t h e p e r t u r b a - t i o n e q u a t i o n ( 3 ) , t h i s d o e s n o t change t h e f i r s t two terms, s i n c e t h e y compensate e a c h o t h e r , b u t t h e t h i r d term g e t s d i v i d e d by a .

I f we go olrer t h e c a l c u l a t i o n s once more, b u t now w i t h ( 7 ) , we f i n d f o r t h e l i m i t i n g v a l u e o f J ( k * ' ) / ~ ( k ) a v a l u e 1 . 9 3 9 , l e a d i n g i n (5) t o t h e c o n v e r g e n t r e s u l t E,, -1.62878 N J , which s h o u l d b e comparedwith S u z u k i ' s e s t i m a t e / 2 / E , =-l .56985 N J . The s e c o n d a r y i n t e r a c t i o n c o n s t a n t s t e n d t o t h e f o l - l o v i n g l i m i t s : ~ ( ~ ) + 0 . 1 4 2 1 ( * ) , ~ ~ ( ~ ) + 0 . 2 7 4 ~ ( ~ ) and

(k)

J 3 ( k ) ~ . 1 3 4 J

.

Ref .5. We o b t a i n e d f o r t h e ground s t a t e e n e r g i e s t h e f o l l o w i n g r e s u l t s : f o r t h e XY model, and EoAF(aal)

-

1.81007 N J

f o r t h e a n t i f e r r o m a g n e t . F o r t h e

XI

model, where the e x a c t ground s t a t e e n e r g y i s known / 6 / t o b e Eo=

-

2 / n N J =

-

0.63662 N J , t h e e s t i m a t e o b t a i n e d . i s 2.7% t o o low when we do n o t a p p l y t h e c o r r e c t i o n f o r o b t a i n i n g a "minimal p e r t u r b i n g p a r t " , and i t i s 1.9% t o o h i g h w i t h t h i s c o r r e c t i o n . A v e r y good e s t i m a t e f o r t h e l i n e a r a n t i f e r r o m a g n e t ground s t a t e may p r o b a b l y b e o b t a i n e d by i n t e r p o l a t i n g between t h e a = l and a r 1 . 5 r e s u l t s , w i t h t h e known e x a c t X I ground s t a t e e n e r g y a s g u i d a n c e . T h i s y i e l d s E = - 1 . 7 6 0 3 N J , which compares v e r y w e l l w i t h t h e 0 o l d r e s u l t o b t a i n e d by H u l t h e n / 7 / : E o =

-

1.7726NJ. R e f e r e n c e s / l / Oitmaa, J. and B e t t s , D.D., p r e p i n t . 1 2 1 Suzuki, M. and M i y a s h i t a , S., p r e p i n t . / 3 / ~ i e m e i j e r , Th. and Van Leeuwen, J.M.J.,Physica

71 (1974) 17.

-

1 4 1 ~ r i e d m a n , Z., Phys. Rev. L e t t .

2

(1976) 1326. /5/ Van d e Braak, H.P., C a s p e r s , W . J . , De Lange,

C . and Willemse, M.W.M., P h y s i c a (1977) 354. I 6 1 L i e b , E . , S c h u l t z , Th. and Mattis, D . , Ann.

Phys. (N.Y.)

16

(1961) 407.

/ 7 / HulthLn, L., A r k i v Mat. Astron. F y s i k (1938) n o 11.

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