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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�
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.
2014La question de savoir si les fluctuations quantiques sont capables d’éliminer l’ordre a longue distance (OLD) dans
unsysteme de spins quantiques de basse dimension
avecsym6trie planaire à T
=0 est examinée
enutilisant l’approche récente du groupe de renormalisation. Les solutions des équations de renormalisation mettent
enévidence l’existence d’ordre dans les systèmes 2D
avec unspin S arbitraire dans le
casde symétrie planaire.
Abstract.
2014The question of whether quantum fluctuations
areable to eliminate long-range order (LRO) in low- dimensional quantum spin systems with planar symmetry at T
=0, is studied using
arecently 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
MARS1984,
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
aphase
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 +
gdimensions 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
ageneral 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
usfocus 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
402
We will
usethe 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
achange of coordinates, Fourier transformation to momentum space and diagonalization of the quadratic form in the Hamiltonian using the generalized Bogo-
lubov transformation (a/ - ck ),
weobtain for small
wavevectors :
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
nowfrom (2) and (3), that in the S > 1 limit
ourmodel 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
usethe 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
astrongly non-Hermitian Hamiltonian,
see[8].
(3) Such
aconstruction is similar to
oneproposed by
Goldhirch [9] but the condition (5) enables
usto make m unbounded and thus to avoid the additional integration
over
the
«outer shell » in
mspace 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
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 +
gthere is a fixed point of (6)
and if
webegin scaling from S > So, then in the end
we scale to the classical (S
=oo) limit, where order of
course
exists.
According to (8),
evenfor 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
oursemiclassical results
Fig. 1.
-Lowest-order diagrams for RG equations 6
and 7. Diagrams (a) and (b)
areequal to
zerobecause of the structure of the propagator in (k, T) space : G(k, T)
--_0 when
/ r ,