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HAL Id: jpa-00209769

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Submitted on 1 Jan 1984

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On the existence of long-range order in low-dimensional quantum spin systems with planar symmetry

A.V. Chubukov

To cite this version:

A.V. Chubukov. On the existence of long-range order in low-dimensional quantum spin systems with planar symmetry. Journal de Physique, 1984, 45 (3), pp.401-403. �10.1051/jphys:01984004503040100�.

�jpa-00209769�

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401

On the existence of long-range order in low-dimensional quantum spin systems with planar symmetry

A. V. Chubukov

Moscow State University, 117234, Moscow, U.S.S.R.

(Reçu le 7 octobre 1983, accepté le 21 novembre 1983)

Résumé.

2014

La question de savoir si les fluctuations quantiques sont capables d’éliminer l’ordre a longue distance (OLD) dans

un

systeme de spins quantiques de basse dimension

avec

sym6trie planaire à T

=

0 est examinée

en

utilisant l’approche récente du groupe de renormalisation. Les solutions des équations de renormalisation mettent

en

évidence l’existence d’ordre dans les systèmes 2D

avec un

spin S arbitraire dans le

cas

de symétrie planaire.

Abstract.

2014

The question of whether quantum fluctuations

are

able to eliminate long-range order (LRO) in low- dimensional quantum spin systems with planar symmetry at T

=

0, is studied using

a

recently developed

renor-

malization-group approach to the investigation of LRO at T

=

0. The solutions of the RG equations point to the

existence of order in 2D systems with planar symmetry and arbitrary S.

J. Physique 45 (1984) 401-403

MARS

1984,

Classification Physics Abstracts

75.10J201375.40

The question of whether there is long-range order (LRO) in low-dimensional quantum spin systems with planar symmetry at T

=

0 has been the subject

of interest in recent years (1) [1-8]. It is known that

such systems construct noncolinear spin structures,

so they contain quantum fluctuations, which are in principle able to exterminate LRO. The following

information is available about the problem : there

is no LRO in one dimension (1D) for any value of the spin S [1, 2], but in the 2D case the existence of order is an open question. For example, for the

S = 2 X Y model the extrapolation from finite cell data [4, 5] leads to some degree of order in the ground (1) For any finite T the Mermin-Wagner theorem [13]

excludes the existence of order; at T

=

0 this theorem is not valid.

state, whereas the two-level renormalization group

approach [3] points to the absence of order. In this connection the authors of [3] proposed that

a

phase

transition may exist in the ground state as a function

of the spin value S. The purpose of the present paper is to check the possibility of such a transition. We will take advantage of the recently developed [7] semi-

classical (S > 1) approach to the investigation of

LRO in quantum systems in D

=

1 +

g

dimensions with the renormalization-group (RG) method and will discuss the implications of the analysis for the possibility of LRO near S - 1 and E N l. Our results

point to the existence of order at T = 0 for 2D sys- tems with planar symmetry and arbitrary S; this is in

agreement with

a

general point of view, that quantum fluctuations are weaker than the related classical fluctuations and also agrees with the results of nume- rical analysis [4] and [5].

1. Model and formalism.

Let

us

focus on a spin system with the following model Hamiltonian :

(here A is the vector connecting the nearest neighbours, I, A, g > 0). This model includes an anisotropic Hei- senberg ferromagnet (D

--_

0) with the X Y model as a particular case ( g

-_-

1) and a single-axis ferromagnet

with single-ion

«

easy plane » anisotropy (FEP) ( g

--_

0).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004503040100

(3)

402

We will

use

the Holstein-Primakoff transformation (2) in order to change from the spin operator basis

to a set of boson spin deviation operators a: and ak ; then, after

a

change of coordinates, Fourier transformation to momentum space and diagonalization of the quadratic form in the Hamiltonian using the generalized Bogo-

lubov transformation (a/ - ck ),

we

obtain for small

wave

vectors :

where

Z is the number of nearest neighbours; (we introduced U into (2) in order to work out RG equations, see below).

It is clear

now

from (2) and (3), that in the S > 1 limit

our

model is equivalent to the XY model with the renormalized spin S (S -> S *), so further on we will consider the question of LRO only for the X Y model.

For the RG approach it is convenient to

use

the coherent states representation [lo], and to construct the effec-

tive classical Hamiltonian, which will lead to the same value of statistical sum, as the quantum one. Omitting the

details of calculations, we will write only the final result : such a transformation will formally lead to the following

substitution in (2) (3) :

Together with (4) we must determine the limit in the following way :

(lower index indicates the averaging with Gaussian Hamiltonian).

As a result we will obtain the classical effective Hamiltonian expression with a two-component order parameter. The renormalization procedure results in

the following equationg for the spin S and coupling (’) In the authors opinion this transformation is more convenient for non-colinear spin structures, than the alter- native Dyson-Maleev transformation, because the latter leads to

a

strongly non-Hermitian Hamiltonian,

see

[8].

(3) Such

a

construction is similar to

one

proposed by

Goldhirch [9] but the condition (5) enables

us

to make m unbounded and thus to avoid the additional integration

over

the

«

outer shell » in

m

space at the first step of RG transformation.

constant U (see Fig. 1) :

It can be seen from (6) and (7), that the X Y model does not preserve its form after RG transformation and so the change of the spin value is not the only

effect of renormalization (in contrast with the case of

a Heisenberg antiferromagnet [7]). But most likely

the question about LRO must be determined from

(4)

403

(6) : actually in one-dimension (c

=

0) it is reasonable to expect that starting from large S > 1 we will in the end scale to S = 2 limit, where there is no order [11J,

but at D

=

1 +

g

there is a fixed point of (6)

and if

we

begin scaling from S > So, then in the end

we scale to the classical (S

=

oo) limit, where order of

course

exists.

According to (8),

even

for S = t the LRO appears

when E ~ 0.34 1, which in some degree justifies

the first-order approximation in the RG equations 6

and 7 ; as to the application of

our

semiclassical results

Fig. 1.

-

Lowest-order diagrams for RG equations 6

and 7. Diagrams (a) and (b)

are

equal to

zero

because of the structure of the propagator in (k, T) space : G(k, T)

--_

0 when

/ r ,

Table I.

-

A comparison of semiclassical results for

transverse correlation junction ’1 (9) with the known

exact results for various models (D

=

1).

to systems with S = 1, most likely this approach will

lead to sufficiently accurate results. This can be seen

from table I, where semiclassical results for the trans- verse correlation function exponent 17 at D

=

1 [ l, 2J :

are compared with the exact results for the S = 2 X Y

model [11] and S

=

1 FEP [12].

2. Conclusion.

Our investigation points toward the existence of long-

range order at T

=

0 in 2D quantum systems with planar symmetry and arbitrary S (in agreement

with extrapolation from finite cell data [4, 5]).

The author is thankful to Prow M. I. Kaganov for

fruitful consultations.

References

[1] VILLAIN, J., J. Physique 35 (1974) 27.

[2] MIKESKA, H. J., PATZAK, E., Z. Phys. B 26 (1977) 253.

[3] PENSON, K. A., JULLIEN, R., PFEUTY, P., Phys. Rev.

B 22 (1980) 380.

[4] OITMAA, J., BETTS, D. D., Can. J. Phys. 56 (1978) 897.

[5] OITMAA, J., BETTS, D. D., J. Phys. A 14 (1981) L69.

[6] JULLIEN, R. PFEUTY, P., J. Phys. A 14 (1981) 3111.

[7] RIDGWAY, W. L., Phys. Rev. B 25 (1982) 1931.

[8] CHUBUKOV, A., JETP 85 (1983) 1319.

[9] GOLDHIRCH, I., J. Phys. C 12 (1979) 5345.

[10] CLAUDER, J. R., Ann. Phys. (N. Y.) 11 (1960) 123.

[11] LIEB, E., SHULTS, T., MATTIS, D. C., Ann. Phys. (N.Y.)

16 (1961) 407.

[12] KJEMS, J. K., STEINER, M., KAKURAI, K., J. Magn.

Magn. Mat. 31-34 (1983) 1133.

[13] MERMIN, N., VAGNER, H., Phys. Rev. Lett. 17 (1966)

1133.

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