The hubbard identity and the absence of phase transitions in some spin systems

THE HUBBARD IDENTITY AND THE ABSENCE OF PHASE TRANSITIONS IN SOME SPIN SYSTEMS H. FANCHIOTTI.

C.A. CARCiA

Depc~rtatnento de lT.~ica. Unicemidud

CANAL*

and H. VUCETICH

Nacionul de La Plata, La Pluta, Argentina

Received

26 April

1977

The Hubbard identity is used to show the absence of magnetic phase transitions spin systems in one and two dimensions. generalizing Mermin and Wagner’s alternative way as Thorpe has done.

in Heisenberg result in an

The central problem in statistical mechanics is the computation of partition functions. This problem could not be simple when spin operators are present in the hamiltonian of the system. For example, only for spin f the calculation of partition functions of spin systems with biquadratic interactions can be reduced to the traditional bilinear Heisenberg one. This is due to the fact that the biquadratic form of Pauli spin operators can be reduced to a linear combination up to a bilinear term

as a consequence of the two dimensionality of these operators. Whenever one considers spins of larger order this simplifying process is no longer valid since the biquadratic term is then a linearly independent one. Therefore in the general case of spins of any order it would certainly be very desirable to have some simplifying device similar to the one mentioned above. Such a device has been described some time ago by Hubbard’) as a method in which the so-called Stratonovich transformation is used to calculate partition functions of some many-body systems. Our purpose is to use this approach for the case of spin systems with biquadratic or more general interactions, proving the absence of ferromagnetic or antiferromagnetic phase transitions in any Heisenberg spin system in one and two dimensions with finite range interactions. This will be a generalization of Mermin and Wagner’s result’) that has already been obtained in an alternative way in ref. 3.

* Member

of the Consejo

National

de Investigaciones, 164

Argentina.

ABSENCE

We begin hamiltonian

H =

OF PHASE

by studying

CK;j(Si

.

Sj)"

+

the

TRANSITION

partition

function

165

IN SPIN SYSTEMS

of a system

of spins

h * T Si eiKRl,

with

(1)

Ii

where h is an external magnetic field and the exponential in the second term of the r.h.s. is to eliminate all possible kind of spiral or antiferromagnetic order. We will present in some detail the case corresponding to n = 2; the other cases are then a generalization that will be discussed later. To linearize the biquadratic term we use the identity’)

e

I

AL _

-

e -m--2a’/*Ax

dx,

(2)

-zc

that cannot be used directly due to involved unless we use the Feynman of a label t. This t-labeling procedure were c-numbers. In doing so we write

X

Tr { exp [ 2

x

ij

the noncommutativity of the operators ordering device4), i.e., the introduction allows us to handle operators as if they for the partition function

(Kijnn)“2 T Xijr(S * Silt] exp [hT &t

eiKRt]).

(3)

where we have choosen h to be in the z-direction. After calculation it will be necessary to rewrite results in the ordinary way, i.e. one disentangles the obtained expression. To this end there exists a series of theorems proved in ref. 4. In our case disentangling will not be necessary since the inequalities we need in the proof are obviously valid under t-integration. Perhaps it is more transparent to rewrite eq. (3) as a path integral making x = (/(7r)“*

Z=

:*Tr{exp[ldt(2~(Kp)“Zb,($.Sj)I+hCS.seiKR.)]}

(4)

0

in such a way that the linearization procedure could be understood as an average over an external field 5. The absence of spontaneous magnetization in this kind of system for one and two spatial dimensions is proved by showing that for a sufficiently small external field h the mean value of the spin of the system in a given direction vanishes. We compute (S,) = (ZiSz; eiKRi) by differentiation of eq. (4) with

H. FANCHIOTTI

166

respect

ET AI

to h. and obtain

x exp

II 0

dt 2 C (K;i)“‘tij,(S; ( ii

* Sj),

+ h C

Sz;f eiKRl

,

The trace operation, being linear, can be introduced into the t-integration and Mermin and Wagner’s proof of the absence of magnetic order based on Bogoliubov’s inequality follows immediately for every t and independently of the path integration. In other words, the linearizing method allowed the reduction of the problem to the original one studied in ref. 2. The generalization of the above discussion for spin systems with interactions of the type (S, - S,)” for any value of II follows the same line as above but now it is necessary to use the representation of a general functional given by Feynman [see eq. (6a) in the appendix of ref. 41 with a word of caution: as stated in this reference. the mathematical basis is much less established in this case. Nevertheless it is an appropriate way of proving that there is no long-range magnetic order in one and two dimensions for generalized Heisenberg symmetric interactions of the mentioned type. This is an alternative way of obtaining the results of Thorpe’) and we have presented it to show the power of the method of linearization of spin interactions through its simplicity and elegance. We would like to mention that eq. (4) indicates that it is possible to simulate interactions with external fields, i.e. phonon interactions, through a biquadratic term in the hamiltonian of the system. Work is in progress in that direction.

References I) .I. Hubbard, 2) N.D.

Phys.

Mermin

3) M.F. Thorpe. 4) R.P. Feynman,

Rev.

Letters

and H. Wagner, J. Appl. Phys.

Phys.

3 (1959) 77. Phys.

42 (1971)

Rev. 84 (1951)

Rev.

Letters

1410. 108.

17 (1966)

1133.