diagram are studied using the method of Villain (1975). We consider classical X Y spins Si = (cos $J~, sin +i) at the sites i of a square lattice interacting via the ...
J. Phys. C: Solid St. Phys., 13 (1980) L887-93. Printed in Great Britain
LETTER TO THE EDITOR
On the critical behaviour of two-dimensional X Y helimagnets T Garel and S Doniacht Laboratoire de Physique des Solides (Laboratoire associe au CNRS), Universite Pans Sud, 91405 Orsay, France
Received 18 August 1980
Abstract. We study the nature of the excitations of a simple two-dimensional classical X Y model with anisotropic coupling yielding a helical state at zero temperature. We show that, at low temperatures in the region of the Lifshitz point, the system displays two types of topological defect corresponding to vortices and to domain walls between regions of opposing chirality of the helical state. As a result of the existence of these excitations we find two phase transitions, of X Y and Ising character respectively.
In two dimensions, the classical ferromagnetic Ising and isotropic X Y models both have a phase transition at finite temperature, associated respectively with the onset of longrange and topological order (for a review see Kosterlitz and Thouless 1978).Isotropic models with n 2 3 (where n denotes the number of spin components),are expected to have T, = 0. In this note, we report results of an investigation of a model displaying both discrete and continuous symmetries simultaneously, namely the X Y helimagnet, the helical structure of the spins being due to competing exchange interactions (Herpin 1968). Similar models have been considered by Villain (1977) from the symmetry point of view, and by Kaplan (1980) in the framework of the harmonic approximation. Moreover, this system is of interest in the study of Lifshitz points (Hornreich 1980 and references therein). In this Letter we present the model and briefly recall Villain’s symmetry analysis; a model Hamiltonian valid at low temperatures in which only phase fluctuations are considered is proposed, and its topological excitations and resulting phase diagram are studied using the method of Villain (1975). sin +i) at the sites i of a square lattice We consider classical X Y spins Si = (cos $J~, interacting via the following Hamiltonian:
where the sum in equation (1)is restricted to nearest neighbours in the x and y directions (J1) and to next-nearest neighbours in the x direction ( J J . We will furthermore assume that J,
> 0,
J,
< 0,
IJ,I
> J,/4
(2)
t
Also at Groupe de Physique des Solides, Universite Paris VII. Permanent address: Department of Applied Physics, Stanford University, CA 94305, U S A .
0022-3719/80/310887 + 07 $01.50 01980 The Institute of Physics
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Letter to the Editor
which ensures that the zero temperature configuration of equation (1) is a modulated structure, with wavevector
+ko = (+ki,O) cos kEa = - J,/4J,
(3)
where a is the lattice spacing. Throughout this paper we will restrict our discussion to the case of incommensurate wavevectors. This assumption should be reasonable in the vicinity of the Lifshitz point: (kgu 4 1) by virtue of the continuous XY nature of the model, in contrast with the case of anisotropic Ising models considered by Fisher and Selke (1980) where k: is forced to take commensurate values. From the point of view of critical phenomena, the model defined by equations (1H3) is described by an II = 4 order parameter (Mukamel 1975); moreover the appropriate Ginzburg-Landau (GL) functional will have some anisotropic quartic invariants (see below), which are known to be strongly relevant in two dimensions (Pelcovits and Nelson 1976, BrCzin et a1 1976, Nattermann 1978). It therefore seems appropriate to consider in more depth the symmetry properties of this model. Villain (1977) has shown that the XY helimagnet displays, for general values of k: not equal to 7(/2a, a special kind of low-temperature order which he refers to as ‘chiral’ order. Loosely speaking, this kind of order is due to the discrete (Ising) character of the helical wavevectors (+ k”,, and shows up in some four-spin correlation functions. Consequently, the continuous distortions of the order parameter associated with the X Y spin directions now coexist with fluctuations of an Ising-like order parameter measuring the local chirality of the antiferromagnetic helix, so that the nature of the orderdisorder transitions for the model requires further investigation. For a related example concerning He3, the reader is referred to Stein and Cross (1979). Using standard techniques (Hubbard 1959, Stratanovitch 1958), we construct a GL form from equation (l), i.e. we consider the long-wavelength approximation (k;a < 1) and get (Hubert 1974)
where I and U have their ‘usual’ GL meaning: r=T-To To
U =
(To = 25,
+ J,)
and with
(>O) -(J1 + 16J2)/24 (>O).
Cy = J,/2
bx =
As expected, equation (4) yields two mean field solutions:
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Letter to the Editor
where $o is the mean field amplitude and k: is given by the expansion of equation (3) close to the (uniaxial) Lifshitz point :
(k”,)’ = 6(J1
+ 43,)/(3, + 163,).
(7) Guided by the ferromagnetic case (Berezinskii 1970, 1971, Wegner 1967, Kosterlitz and Thouless 1973), we will consider only phase fluctuations and take $ o as a constant. This ‘phase-only approximation’ is expected to be accurate at low temperatures. Around the two minima (kk”,, one can then introduce two phase variables (e*) measuring local fluctuations relative to the zero temperature helices, which will be represented by two psuedo XY spins (Si).The harmonic approximation in equation (4) yields:
+ (terms of the same form for e-). The variables S* would be appropriate for a helical state of the DzyaloshinskiiMorya type (Liu 1973).To go beyond the harmonic approximation, we must now include a coupling term which allows for the possibility of coexistence of domains of opposite chirality. The simplest possible term of this form may be written (Garel and Pfeuty 1976)
i sd’x(S+*S-)’
=g
s
d2xcos2(8+ - e-)
(9)
where g = ~(11/1~1~/4). Note that this coupling term has two minima:
e+ - 8e+ - 8-
=0
(mod 2n)
=
(mod 274.
The constant g is therefore a measure of the barrier between domains of different chirality. In order to estimate g, we consider a domain wall running along the y axis, separating two domains of opposite chirality. We then identify the energy per unit length of the resulting domain wall for the model Hamiltonian equation (8) together with equation (9), with that calculated from the original Hamiltonian equation (1) (Hubert 1974). We then find $;g
(8/3u2)(J,
+ 43,)2/13, + 163,l.
(10)
Finally, the Hamiltonian obtained by combining equations (8) and (9) does not include the possibility of point defects (vortices) in the e* fields. We therefore generalise to a lattice Hamiltonian of the form
H
A ; ~[cos(e+ - 8;)
= a = x , y i, j
where
+ cos(e;
-e
~ +]
COS i
2(e+ -
e,:)
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for (i, j ) nearest neighbours and zero otherwise. This model Hamiltonian will describe all relevant fluctuations of the 2D X Y helimagnet at low temperatures, i.e. near the Lifshitz point, and will be used to investigate its phase transition behaviour in what follows. Formally, we have decomposed the initial n = 4 problem into two coupled n = 2 models. A somewhat similar model would be obtained by considering a biaxial Lifshitz point in Ising systems (Hornreich et al 1979). The form of equation (11) naturally includes the two types of topological excitation expected for the system at low temperatures : point excitations describing vortices in the phases e*, relative to the zero temperature helices, and line excitations describing domain walls between regions of opposing chirality of the helix. These excitations may be made explicit by rewriting the partition function in terms of continuous variables and integer variables, following Villain (1975). First we perform a rescaling of the underlying square lattice of equation (11) in order to express it as a pair of coupled isotropic X Y models with effective interaction A = 1)~[5,(415,1 - J1)]”’. In its Villain form, equation (11) now becomes at low temperatures
where n:j, p:j and Aiare integer fields. The advantage of this representation lies in the fact that the restriction to values 0 < 0: < 271 in the original partition function (11) is now removed. However, a disadvantage in the present case is that the evident twofold degeneracy of the model is no longer manifest. In order to ensure this twofold character, we make a further approximation by restricting the values of Aito 0 or 1, corresponding to domains in which el: N 0; or N el: + n respectively. At finite temperatures these domains will be separated by fluctuating line defects (domain walls). After some Fourier transforms, equation (12) can be cast into the following form:
H
=
H,
+ HV+ Ha
with
H, = k
(AK2
+ 29) tlk$ +
+
A K ~ 49 A K+ ~2g p k p ;
where ak = e;
p,
=
e;
+
1 2inAC Kolnka- 271gAk - 290; A K ~+ 2s a
+ AAKK22++ 42gg (A K2rg1k+ 2 g 4ing Kanka +AKz+2g?F)
+
2in C K,Pkor K2 ~
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We have used in equations (13) and (14) the notation of Villain (1975). We now observe that the above Hamiltonian decouples in terms of the following variables : 'ka 'ku
=
nka.
- 'ka
+ Pku
- Pka'
We then find
with
where
tk= Ak + 2i
Kavku/K2 a
As can be seen from equations (15) and (16), we now have a regular U vortex transition and in addition equation (15) represents, in the Villain form, a second X Y model of v vortices in the presence of a twofold (Ising) anisotropy, of strength proportional to g. The latter model has been studied by Kadanoff (1978) and by Jose et a1 (1977), who found this anisotropy to be relevant for the critical behaviour but did not study in detail the phase diagram for small g. We now argue that the transition temperature will be lower than the Kosterlitz-Thouless transition temperature and will be proportional to g for small finite g. The argument goes as follows (Einhorn et a1 1979). In the low-temperature phase where the symmetry is discrete, the dominant defects should be domain boundaries (strings) rather than vortices. A U vortex will therefore have strings attached to it, and the interaction energy between two vortices will now be proportional to the length of the string joining them rather than logarithmic as in the isotropic case. This string effect exists as soon as g is nonzero. The resulting confinement of vortex-antivortex pairs therefore suppresses the Kosterlitz-Thouless transition. The resulting Ising transition temperature may be estimated by setting vka = 0 in equation (16), leading to an Ising model (Ai= 0 , l ) with interaction of range a(A/g)'I2and strength 8n2g2/A.Thus, counting A/g nearest neighbours, Tfsing will be of order g while the U vortices will have a KosterlitzThouless transition temperature T,vortices = O(A). The resulting phase diagram? is sketched in figure 1, where for IJ21 < J1/4the ferromagnetic X Y transition temperature is of the order of @ J , ( J , - 4 IJz1)]li2(Kaplan 1980). We have shown that the Lifshitz point for the 2D X Y helimagnet is in fact a multicritical point at which two lines of Kosterlitz-Thouless character associated with
xi
t For a system with anisotropy term cos m(e; - 0;) with m > 4, one expects an additional phase transition following the results of Jose et al(1977) and Einhorn et al(1979). The case m = 1 has been considered by Parga and Van Himbergen (1980) and leads to suppression of the U vortex transition ( X Y model in a magnetic field).
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Letter to the Editor
1.0
t
L i f s h i t z point
L IJ2 I /J,
Figure 1. Sketch of the phase diagram of the 2D X Y helimagnet in the vicinity of the Lifshitz point.
vortices of the spin phase angle meet a line of Ising transitions associated with the disordering of the chiral order of the helimagnet. These results are based on a model (equation (10))in which amplitude fluctuations of the original spin order parameter are neglected, and hence are only expected to be reliable in the immediate vicinity of the Lifshitz point. As the pitch of the helix is decreased by increasing (4J,(relative to J,, we expect the interaction between defects of the ordered state (vortices and strings) to become important. In other words, one would then have to deal with a coupled (U, v, A) Hamiltonian. The resulting coupling terms may be relevant and may therefore lead to modification of the above conclusions as one goes away from the Lifshitz point into the antiferromagnetic region. We also expect effects of commensurability with the underlying lattice to become important as k:u becomes comparable to unity. An extreme case of the latter type of effect would, for instance, be that of the antiferromagnetic X Y model on a triangular lattice. On the experimental side, the transition of the quasi-two-dimensional compound BaCo,(AsO,), appears to be unique and quite abrupt (Regnault 1976, Regnault et ul 1977). However, this system appears to be more complex than the simple model treated here. We thank R Savit, W Selke and J E Van Himbergen for giving us copies of their work in advance of publication. T Garel would like to thank M Devoret, D Estbve, J Friedel and H J Schulz for useful discussions. S Doniach thanks the members of the ‘Physique des Solides’ groups at UniversitC de Paris, Jussieu and Orsay campuses for their kind hospitality.
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