Z. Phys. B - Condensed Matter 53, 221-231 (1983) Condensed f,?'äfliif;' Matter (Q Springer-Verlag 1983
Rectangular Lattice Gas Model with Competing Interactions and the Two-Dimensional ANNNI Model in a Field
P. Rujan Institute for Theoretical Physics, Eötvös University, Budapest, Hungary
W. Selke Institut für trestkörperforschung der Kernforschungsanlage Jülich, Federai Republic of Germany
G.V. Uimin L.D. Landau Institute for Theoretical Physics, Chernogolovka, USSR
Received July 18, 1983
A lattice gas model with short range competing interactions for adsorption on (110) surlaces of fcc crystals, in particular for O/Pd(110), as well as its Ising analog, the two- dimensional ANNNI model with antiferromagnetic axial nearest and next-nearest neighbour interactions in a field, are studied using the free lermion approximation and Monte Carlo techniques. The phase diagrams display dilferent commensurate phases and incommensurate regions. Static and dynamic aspects of topological defects (walls and dislocations) characterising the incommensurate structures are investigated.
1. Introduction
Currently commensurate (C) and incommensurate ANNNI model with competing axial antiferromag- (IC) structures are investigated extensively, both ex- netic interactions (Jr,Jr<0) in a lield (the three- perimentally and theoretically [1]. On the theoreti- dimensional version, -Ir>0, is considered in Ul). cal side it was found that rather simple models give The study is motivated by work of Ertl and Küppers rise to a remarkable richness of phenomena. For on the adsorbed overlayer O/Pd(110) [8]. Indeed, example, the three-dimensional axial next nearest the model, they suggested in that context, is just the neighbour Ising (or ANNNI) model [2] exhibits not lattice gas analog of our model. Experimentally, only an incommensurate ordering t3], but also a O/Pd(110) displays different commensurate phases, sequence of inlinitely many distinct commensurate (.2x1) aI coverages around Il2 and (3 x 1) at co- phases [2] and a special multicritical point, the Lif- verages around 1/3 ((3 x 1) denotes a situation, where shitz point. (In the d-dimensional ANNNI model the oxygen is adsorbed predominantly on sites of nearest neighbour interactions tn (d- 1) dimensional parallel rows separated by two mostly empty rows. planes, .Io, and between adjacent planes, ,.Ir, are aug- The adsorption lattice is a rectangular one in the mented by competing next-nearest neighbour inter- case of O/Pd(l10). In between these two C phases actions, Jr, acting parallel to a single axis.) Although IC structures are observed. In the corresponding the ANNNI model seems to be too simple to de- theoretical analysis [8], however, no attempt has scribe specific experiments quantitatively, it repro- been made to establish the phase diagram ol the duces crucial qualitative leatures observed in modu- lattice gas model: only a few Monte Carlo (MC) lated structure materials [4]; in particular, it pro- equilibrium configurations and MC data of the vides a useful framework to study complicated phase structure lactor at a lew temperatures and coverages diagrams in magnetic systems, such as the cerium were presented. In view of recent elaborate investi- monopnictide compounds CeSb [5] and CeBi [6]. gations on the two-dimensional ANNNI model In this article we shall study the two-dimensional without lield [9-12] as weli as on lattice gas models 222 P. Rujan et al.: Rectangular Lattice Gas Model with Competing lnteractions
: ("/1 with competing interactions [13] it seems to be fea- /{ -L s",rs"+ 1,y * /2s,.ys"1z,y sible to analyse that lattice gas model and its Ising analog, the ANNNI model in a field, in much more iJoS',rS,,r* 1+I1S",y) (1) detail. Our aim is to determine their phase diagrams Ising spins, are located on a and to relate the results to current concepts on two- where the S,,r: t1, rectangular lattice with the lattice constants a, in r- dimensional C and IC structures (of course, it is Jo refers coupling between hoped that the results of our study may also stimu- direction, and b. to the along the y-direction; and J, are nea- late further experiments on O/Pd(110)). In particu- neighbours J, neighbour exchange interac- lar, we shall pay attention to the existence and ex- rest and next nearest (i.e. tent of an IC phase with algebraic decay of cor- tions along the r-direction Jo, Jr, "/r denote interactions between up to geometrically third near- relations, the floating phase [14, 9] (in contrast to est neighbours appropriate choices of a and b). disordered IC structures with exponentially decaying lor In the following we set The analogous lattice correlations). It has been argued convincingly that c:1. gas has been considered some years ago there is no direct transition from the (2 x 1) phase to model [8] in order to describe experimental findings on the such a floating phase [14, 15]. Various, mutually adsorbate system O/Pd(110). As usual for (110) sur- exclusive, theoretical suggestions have been put for- faces of fcc substrate crystals the adsorption lattice ward on the melting of the (3 x 1) phase ((3 x 1) indeed, rectangular one; this case the two phases are realized in the three-state chiral Potts is, a in lattice constants are ax2.5A and bx3.6A. The cor- model [16-19] as well as in other adsorption sys- responding Hamiltonian is tems like H/Fe(110) [13]): Second order transitions of conventional 18] or "chiral" type from //' (to, [17, [20] : - c'.', c, t .r * uoz c^., c"- .:., the (3 x 1) phase into the disordered phase have been ,-I. (3 1) advocated. It has also been claimed that the x *€ro C*,rC*,r* r f fui*e) C".n). Q) phase should always melt into a floating phase [21], unless the transition is of first order. Even the exis- The occupation number C,,, takes on the values 0 tence of a commensurate (3 x 1) phase has been and 1 depending on whether the site I = (x, y) is questioned [22]. One might hope that these con- empty or occupied; hence flicting statements can be clarified in the lramework C*,r:(1 (3) of our fairly simple model. Indeed, we shall present -5,,)12 some evidence on these subtle questions, although e denotes the binding energy of the adatoms, O, to the main emphasis will be on the gross features of the substrate, Pd, and p is the chemical potential. the phase diagrams. The coverage ol the overlayer is given by the In Sect. 2 we shall define the models, both tL ANNNI model in a lield and its lattice gas analog. c:l t
223 P. Rujan et al.: Rectangular Lattice Gas Model with Competing Interactions
Using (3) one can easily transform the two Hamil- ground state under a change of the sign of the field; tonians, (1) and (2), into each other. In particular, (5) in particular (3 x 1)* goes over into (3 x 1) . - At corresponds to non-zero temperatures the model with Jt,Jr
1.0 I i
I l 08 I I . t I /Dt S0RDER 06 \\LINE / 04
02
0 01?31., 4 h =2(1-2x) (b) Fig.2a and b. Spin chain with competing nearest, Jr, and next nearest neighbour, .Ir, antiferromagnetic interactions in a fie1d, h, with rc: JzlJ t:0.3 at T1:kBTllJ rl:0.15 and b T1 :0.5. In both cases the wavenumber,^ the correlation length, (, and the mag- x 4, 0 05 1 netization, rrr, are plotted vs. h. The additional lines, shown in b, Fig. 1. Ground states of the ANNNI model in a fteld, h:HlJo are explained in the text P. Rujan e1 al.: Rectangular Lattice Gas Model with Competing Interactions creased, reflects the sharp transitions occuring at spin (MCS/S); because of very slow relaxation and zero temperature, as displayed in Fig. 1. But already fluctuation times in the IC region directly above the aI kuTllJrlryO.5 the steps are washed out. It is in- low temperature commensurate phases, it proved teresting to compare this behaviour to the experi- important to average over several long runs there. mentally observed two abrupt jumps of the field- We recorded the energy, E, the specific heat, C, the dependent magnetization in CoClr.2H2(o at magnetization per row kuT=lJrl 1241. That substance has, indeed, been de- interac- 1 scribed by an Ising chain with competing m(x\:- (t',") (1 1) tions and a ralio JrlJ, of about 0.3 [24]. Our result M T suggests that a coupling between chains has to be the coverage included to reproduce the experimental findings.
In contrast to the smooth behaviour of the magneti- I zation, we find cusps in the correlation length and (12) ?. the phase at certain points, Tp, Hp, in the tempera- ^/ ture-field plane. At these points the correlations and the structure factor change from antiferromagnetic type (q fixed at z) at low temperatures to incommensurate type, where q l_ .., - \ (S,S, ) exp(iq(r (1 3) varies continuously with temperature and fieid. Hen- /lql: lrl,M'-., ' " -r')) ceforth, the points (To,Ho) deline a "disorder 1ine" [25]. Similar lines have been found in a variety of where the sum is over all lattice sites I = (x, y) and different models [9, 13, 26, 211. Note that we find a i'-(.x',y'). disorder line only in connection with the antiferro- The phase diagrams for the parameters rc:JzlJ, magnetic ground state, but not with Ihe (2,1) and :eozlEot:0.3 and Jof Jr:erof t0r: - 1 are shown in the lerromagnetic structure. In those other cases the Figs.3 and 4. Extended low temperature commen- phase goes smoothly to its commensurate limit at surate phases of antiferromagnetic, (2 x 1), and zero temperature, i.e. the asymptotic correlations are <2,1>, (3 x 1), type as well as a narrow floating in- always ol incommensurate type at non-zero tem- commensurate phase adjacent to the <2,I>, (3 x 1), peratures due to easily excited symmetry-breaking phase are identified. The correlations in the disor- defects in the one-dimensional ANNNI model dered phase, as determined from the structure factor, in a field. - Additional characteristic lines may are of commensurate type below the disorder lines be defined by locating the maximum of the cor- and ol incommensurate type at higher temperatures; relation length in the region, where the ground state detaiis will be given in the later discussion. Note is (2, 1), l"-"*, and by identifying the fields and that the phase diagram of the lattice gas model is temperatures, at which the wavenumber of the symmetric in the (?10)-p1ane about the line 0:I, asymptotic correlations is c1--2nf3, Lr,,r, corre- because the Hamiltonian is invariant under the sponding to the (2,1) ground state. These lines are transformation 11, S,+-11, -S;, hence 0+1-0, as not identical and depend weakly on temperature, as has been observed before 123, 281. This symmetry depicted in trig. 2.
3. The Monte Carlo Study 2.0 RAM AG N ETIi 3.1. Phase Diagram 1.8 In this section results of the Monte Carlo study* on T-I-f\- the ANNNI model, (1), and its lattice gas analog, LOATIN6 (2), are reported. Lattices ol sizes N x M with lull r periodic boundary conditions were used; lü refers to \ the direction with competing interactions, x. In gen- <2,1> eral, 1{ was taken to be 60 (120 in a few runs), while M varied from 3 to 30. The typical observation time 20 3.0 ranged from 5 . 103 to 2. 104 Monte Carlo steps per Fig. 3. Phase diagram, using Monte Carlo techniques, of the two- * Some results of the Monte Carlo study have been reported at dimensional ANNNI model in a field, ft, with r:0.3 and Jo: the Conference on Phase Transformations in Solids. Crete, June Jr. I, denotes the disorder line. The transition from the float- 1983 [38] ing to the disordered phase is remarked by rectangles (!) P. Rujan et al.: Rectangular Lattrce Gas Model with Competing Interactions can be broken by triple interactions (indeed, the mainly destroyed by dislocations, which seem not to phase diagram ol O/Pd(l10) is not symmetric [8]; change the periodicity of the structure. Indeed, a thus the Hamiltonian, (2), is insufficient to explain detailed analysis, based on data for the structure the whole (! 0) plane. Nevertheless, it may describe factor and specific heat for lattices with lü:60 and the behavior for 0-<0.5 reasonably weli). M:I0,20 and 30, confirms that, at small fields, the The transition lines in the phase diagram are ob- C phase melts, driven by dislocations, into a disor- tained in various ways. For instance, the boundary lines of the C structures are estimated from linear 12x1) (2x1) r (2x1) lf M-extrapolations of the location in the turning point of X@). Lafiices of sizes lü:60 and M: 10, 20 and 30 were used. To identify the structures above the low temperature C phases and possibly further A transitions is far from trivial. To get a rough idea on t".ffi the relevant mechanism in the melting of the C phases one may study typtcal Monte Carlo equilib- ls ql rium configurations (of course, one should worry, whether equilibrium has been reached and whether the bound- typical configurations do exist) close to I ary of the C phase. Then extensive finite size ana- 26+ h=02,Tr=1 55, 0=0484 h=0.74,Tr=12, e=0441 lyses on appropriately chosen quantities, e.g. l@), (o ) (b) have to be done to get reliable quantitative infor- mation. (lx1) i {3x 1) We shall lirst discuss the situation on the antiferro- magnetic, (2xI), side of the phase diagram, i.e. for fields h:HAJrl<0.8. Typical Monte Carlo configu- rations near the melting transition of the C phase c are shown in Fig.5. For example, at h only slightly smaller than 0.8, 0.8 > h ä 0.6, a predominantly (2 x 1) or antiferromagnetic structure is observed, see Fig.5b, disturbed by a pair of dislocations (a disJo-. cation is characterised by a finite extent, in y-direc- Burgers' tion, of a row of occupied sites, i.e. the I vector does not vanish) and a wall of "+ + or (3 -" h = 1 0, T1 = 1 2, 0 = 0 356 h=17,T1 =1 5, e=0119 x 1) type, thermally activated by kinks. Due to such (c) (d ) walls, incommensurate correlations are quite likely (lx1) (3x1) to occur in contrast to the case of small fields, say, * h<0.4. There, as shown in Fig.5a, the C structure is '-f 1B lrr xx orsolno 16 dnro FL0ATTN6 l/"\ I ILOATING t.4 "\l i"") f 6""\ 1? I /= fl h=2.5, T1 =085, 0=0272 10 il i=i iei =\ (e) löl Fig. 5a -e. Typical equilibrium Monte Carlo configurations close 08 laf (2 (3 x phases. to the melting of the commensurate, x 1) and l), Occupied sites ("-" spins) are marked by small circles, walls by 02 04 06 08e arrows, kinks by large circles, and dislocations by rectangles. Fig. 4. Phase diagram, using Monte Carlo techniques, of the Only parts of the lattices (26 x 30 out of 60 x 30) are shown. a and lattice gas model of Ertl and Küppers [8] for O on Pd(110) with b refer to the melting of the (2 x 1) (or AF) phase; c, d, and e to eor:0.3eor and eor: -tro. The crosses (x) denote the disorder the melting of the (3 x 1) (or (2,1)) phase. The horizontal axis is 1ines. T' :kBTllJ[ 0 is the coverage the x-axis zzo P. Rujan et al.: Rectangular Lattice Gas Model with Competing Interactions dered structure with commensurate correiations, i.e. (2 x 1) type. The time evolution of walls will be the peak of x@) occurs at q:v" The critical behavior discussed below. Large fluctuations in the positions is expected to be Ising-like. Only at appreciably of the walls are observed during a MC run. In higher temperatures, above the disorder line, Ir, in- general, the walls are not straight, but are deformed commensurate correlations set in; the characteristic by kinks. At larger fields dislocations start to occur. wavenumber decreases. The disorder line (for 1(q), (ii) N 1.4 X,(q)/X,,(q) the C phase, we could easily be led to a wrong interpretation. (Indeed, the "disorder line" on the .'lxl Nd (2,1) side, e.g. at h:1.7, is shifted to appreciably lower temperatures, but is still above the melting transition, as one increases l{, e.g. lü:120, instead of 60. A more systematic study is certainly desir- able.) As before, the same comment applies to ex- periments on adsorption systems of terraces on the Q-=2ln/30 surlace of the substrate. At any rate, we cannot ex- OB 2n/) clude the existence of a, presumably very narrow, 0.7 floating phase or a transition to an incommensurate disordered phase, as would be the case in a chiral 06 1.0 12 1L 1.6 1.8 r, transition [20], in the thermodynamic limit. Nev- ertheless, we even tried to extract critical exponents Fig. 6. Plot of the function for the ANNNI model in a Xt@)l"Xr((l) data that region, especially at h field with h-0.9 on 60xl lattices for l:21'and the indicated 4- from our MC in values. The arrows locate the estimate of the melting transition :1.2, 1.5, and 2.0. We obtained a rather large ex- temperature of the (2,1) phase (q:2r13) and of the transition ponent lor the specific heat, at least 0.3. We also from the floating to the paramagnetic phase (11-23n13) delined an "order parameter" lor the (2, 1) phase by (M(T)-M(T,)llG-M(7,)), where M(T) is the to- ta1 magnetization (alternatively, one might use ((S" established finite [14]. But, because there is no well -S,* rS"*z)*(S"+ 1 -s,S,*r)-2(S,*2 -S"S,+ 1))/8 size scaling theory for the floating phase (e.g. trans- as the order parameter, where S, relers to a spin in fer matrix calculations on infinitely long strips ol row r). For lattices of size 60 x 30 we obtained in 2 1 finite widths encounter various problems 122, 311, the range 3.10 St:(7"-T)lT,<2'10 surpris- we refrained from such estimates. Results for h:0.84 ingly large effective exponents, f, for the "order pa- and h:0.9, as shown in Figs.3, 4, and 6, indicate rameter", []x0.2-0.4. Apparently B depends on h. that the floating phase is quite narrow and cannot However, these large values of $ may be just due to be resolved for larger fields within the accuracy ol a poor choice ol the order parameter or may show the Monte Carlo data. Therefore we have to rely that finite size effects obscured the results on the upon somewhat speculative arguments on that part melting of the C phase seriously. of the phase diagram, i.e. roughly regions (ü) and We did not explore region (iii) in detail, but, again. (lli), mentioned above. Our arguments are partly a floating phase would be very tiny, if it exists. based on simpie llM-extrapolations of the position of the maximum in the specific heat and the turning 3.2. Comparison to O/Pd(110) point in y(q): In region (i), h51.0, the positions do not coincide even in the thermodynamic limit (in The resuits of the previous paragraph were obtained contrast to the situation on the (2 x 1) side, discussed for the ratio aor/e ,o: -1, while the values suggested above). This behaviour provides additional evidence to give the best description of O/Pd(110) are JofJ, for a floating phase and its transition of XY-like or :stoleot: -0.3 [8]. However, the concrete choice Kosterlitz-Thouless character to the disordered of this ratio does not affect the results significantly: phase [9]. However, for larger fields 2.42h] 1.2, i.e. Indeed, lor the later value, we find that the melting mainly region (ii), Lhe two characteristic tempera- temperatures of the C phases, see Figs.3 and 4, are tures seem to merge, as M is increased. Thus it is just reduced by roughly a factor of two, and that the tempting to suggest a direct transition from the C general topology of the phase diagram does not phase to the disordered phase in region (ii). Data change much (the C phases in the (?lO)-plane be- for y,(q) in systems of size 60 x 20 seem to indicate a come a little bit slimmer). We can therelore con- melting into a commensurate disordered phase with clude that the model of Küppers and Ertl [8] pre- a disorder line deviating from the transition line by dicts a phase diagram with (a) the maximal tran- at most 1,5)', up Io h:1.7. However, several un- sition temperature of the (2 x 1) phase should be certainties obscure these observations. In particular, only slightly larger than the one of the (3 x 1) phase; the wavenumbers, q, consistent with the period (b ) the (3 x 1) phase at half of its maximal transition boundary consistent, are "quantised" by an incre- temperature should range from 0.275050.35. For ment of /r1:2nlN. lf the true wavenumber, in the the (2 x 1) phase one expects 0.4650-<0.54; (c) the thermodynamic limit, would deviate by a smaller floating incommensurate phase should be quite nar- amount from the C value, q:2n13, closely above row (and hard to detect experimentally); (d) there 228 P. Rujan et al.:Rectangular Lattice Gas Model with Competing Interactions f = I l'4tS,zS l0 I 0l 0 34 0 38 042 0 46 t +16+ Fig. 7. Wavenumber, q, vs. coverage, 0, see phase diagram Fig.4, at fixed temperature Tr:kuTl1J, in the incommensurate region. Lattices of size 60 x 30 were used t= 45 should be a disorder line for the (2 x 1) phase. - However, the model seems to have some insufficien- cies: one should include triple interactions to repro- duce the experimentally found asymmetry about 0 : j in the (0, T) plane. Also pair interactions to the geometrically third nearest neighbour (eo, refers to the fourth nearest neighbours), i.e. diagonal interac- tions, might be included. But first one should wait lor further measurements, including ones on the co- verage-dependence of the wavenumber in the IC re- gion in between the (2 x 1) and (3 x 1) phases. Our Monte Carlo data give evidence that it is not exactly (b) linear, at least in the mode1, in contrast to previous results [8], as shown in Fig. 7. j.3. Dynamical Aspects t Concluding this section on Monte Carlo results we will briefly discuss a few dynamical aspects, in par- I ticular on the emergence and fluctuations ol walls. In Figs.8a-c the evolution of (2 x 1) or "* -" walls 30 out of the (3 x 1) structure ar 0x0.375 (h:0.9) and Tt : knT llJ rl: 1.2 is depicted. As initial configu- ration of the Monte Carlo run we took the (3 x 1) structure. The nucleus of three "heavy", (2 x 1), walls (three in order to disturb the (3 x 1) pattern only I tl fl 1ocally; see also already appears after a few 28+ [9]) (c) MCS/S in form ol dislocations. Its size, measured by its linear extent in y-direction, increases quickly to Fig. 8. Evolution of walls of "+ " (2 x 1) type slightly above the about M12 and keeps this value for about 30MCS/S. melting temperature of the (3 x 1) phase (h:0.9, Tt:12,0:0.375 Then, within a few MCS/S it transforms into three in equilibrium) during a Monte Carlo run. The initial configu- ration, r:0MCS/S, is the perfect (3 x 1) structure. Only parts (16 adjacent straight walls. The following separation of x 30 or 28 x 30 out of 60 x 30) of the system are shown the walls occurs on a much larger time scale, a few hundred MCS/S. The magnetization (or average number of particles) per row, calculated during this process, exhibits the "squeezing eflect", observed be- able, see also current work on nucleation of walis fore in the three-dimensional ANNNI model [3], i.e. [32]. A more detailed investigation on dynamical during the evolution of the walls just one maximum properties (creation and annihilation of walls and or crest in the magnetization dies off. An analytic dislocations, motion of walls) should be done in the understanding of this phenomenon is certainly desir- luture. P. Rujan et al.: Rectangular Lattice Gas Model with Competing Interactrons 4. Analytic Approaches the one of Villain and Bak [14], see also [35], and we do not need to present any details. Minimizing 4.1. The Free Fermion Approximation the free energy, one obtains the average density of walls as a function of rc and The melting lines of In this section the phase diagrams of the models are T the phases, studied using analytic approximations. We mainly commensurate AF and <2,I>, are then determined by the condition that the wall density restrict ourselves to the case x:JzlJt:0.3, as vanishes. suggested in connection with the adsorption system * a(o- O/Pd(110), and "/o -Jr. Recall that in this case AF: -D+2K, - 4K.:2.e (t4) there ground one are three different states as (2,1) 4Ko, changes the reduced external field h:HlJo. For near Mr: (h-2Kr+4K)3:e (15) 1Kn 0<ä<0.8 one has antiferromagnetic (AF) ordering, (2,1) near Mz: -1i-ZXr-2K2)13:e (16) for 0.8 * (i-2K r-2K)p :0.7933 ... e 4Ko (1e) leads to some interesting complications. The "naive" lree fermion approximation, as discussed in the lirst The results for rc:0.3 and "/o - -J1 are shown in part ol this section, assumes a perfect commensurate Fig.9. At low temperatures, where the lree fermion phase with sharp maxima in the structure factor. quite approximation is valid, the floating phases are (For a related description of the (2,1) phase as a narrow, in agreement with the MC findings. As - dimerized phase ol "+ -" and "+" particles, see remarked before, the results become fairly poor at [36].) However, thermal excitations in the (2,1) higher temperatures. In the following we shall dis- phase may alter this picture. ln particular, lor larger cuss, in which way one might improve the approxi- values ol rc, point dislocations (or pairs of different mations to extend their range of validity. walls, namely ol "-i-" and type) are found to give rise to a broadening ol the structure 4.2. Modifications factor and to change the location of the boundary between the floating and the disordered phase. De- Let us lirst consider the boundary of the AF phase. tailed calculations will be published elsewhere So lar we included in the free lermion approxima- [37]. tion only walis of "+ + -" type and kinks, which shift parts of the wall by just one lattice constant, see trig.5b. The resulting melting line seems to be 5. Summary accurate near the multiphase point Mr, but occurs at much too high temperatures for small fields. In this article we studied a lattice gas model for the There, additional excitations become important, as adsorbed overlayer O/Pd(l10) and its Ising analog, observed in the MC study, see Fig.5a. In particular, the two-dimensional ANNNI model in a field. The walls of "+ + -" and "- - t" type are, of course, main features of the phase diagrams, including com- equally likely at ä:0. Also we allowed walls to shift mensurate (antiferromagnetic and (2,1) or (2xI) by two lattice constants and pairs of the two dif- and (3 x 1) phases). floating incommensurate and ferent types of walls (" + + - - ") were also included. disordered phases, were established doing linite size To evaluate the partition sum for these excitations analyses ol Monte Carlo data. Typical excitations we introduced again a fictitions system of walls as near the melting of the commensurate low tempera- was done before in the simple lree lermion approxi- ture phases were identified (walls, kinks, and dislo- mation [33, 14]. The calculation is then straightfor- cations). A simple analytic approach, the lree fer- ward, and the boundary line of the Atr phase is mion approximation, was found to locate accurately obtained from the phase boundaries at low temperatures near mul- tiphase points. Including additional excitations, it ti' : x 4 K 2 Ktr)t 4 Kät (20) 1z r - r - - could be improved to estimate the entire antiferro- with K$:e 2Ko. This expression agrees at low tem- magnetic boundary well. The precise description of peratures near Mr with (14) and at h:0 with the Ihe (2,1) phase and, especially, oi its melting de- result of the previous study (including walls of type serves lurther attention. "+ + - -") t141. The case rc:0.3 and Jo: -J, is shown in Fig.9. We now observe a good agreement The kind hospltality of the Institut für Festkörperforschung, with the MC results lor all fields, 0 4. Selke. W.: Modulated structure materials, Crete 1983. Tsaka- Haldane, F.D.M., Bak, P., Bohr, T.: Phys. Rev. B 28,2113 lakos, T. (ed,); Selke, W., Binder, K.: Ferroeleqtrics (to be ( 1983) published 1984); see also: Kulik, J., de Fontaine, D.: Preprint 72. Kinze| W.: Preprint (1983) (1e83) 23. Roelofs, L.D., Estrup, P.J.: Surf. Sci. 125, 51 (1983) 5. Pokrovsky, V.L., Uimin, G.V.: J. 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