Monte Carlo study of the Villain version of the fully frustrated XY model

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1 PHYSICAL REVIEW B VOLUME 55, NUMBER 6 1 FEBRUARY 1997-II Monte Calo study of the Villain vesion of the fully fustated XY model Pete Olsson* Depatment of Theoetical Physics, Umeå Univesity, Umeå, Sweden Received 12 Septembe 1996; evised manuscipt eceived 22 Octobe 1996 The fully fustated XY model with Villain inteaction on a squae lattice is studied by means of Monte Calo simulations. On the basis of the univesal jump condition it is agued that thee ae two distinct tansitions in the model, coesponding to the loss of XY ode and Z 2 ode, espectively. The Kostelitz-Thouless KT tansition is analyzed by finite-size scaling of the helicity modulus at lattices of size L32 though 128, giving T KT (1). The voticity-voticity coelation function is used to detemine two diffeent chaacteistic lengths, the Z 2 coelation length, and the sceening length, associated with the KT tansition and fee votices. The tempeatue dependence of is examined in ode to detemine T c and the coelation length exponent,. The exponent is found to be consistent with the two-dimensional 2D Ising value, 1, and the obtained citical tempeatue is T c (5). The deteminations of both and ae done caefully, fist applying the techniques to the 2D Ising model, which seves as a convenient testing gound. S I. INTRODUCTION The citical behavio of the fully fustated XY FFXY) model has eceived much attention duing the last decade. This is due to the two kinds of symmeties pesent in the systems and the associated possibility of new citical behavio. But in spite of the lage numbe of papes, thee is still no consensus about the phase tansitions in this model. The odinay fustated XY model with cosines inteaction is govened by the Hamiltonian HJ ij cos i j A ij, whee i and j enumeate the lattice sites, i is the angle at lattice point i, A ij is the quenched vecto potential, and the sum is ove neaest neighbos. The fustation is detemined by the sum of A ij aound a plaquette see below. In the fully fustated case examined in the pesent pape this sum is equal to. The peculiaities of the fully fustated models stem fom the two diffeent symmeties. Beside the otational symmety of the XY model, the model also has a Z 2 symmety associated with the chiality. The squae-lattice vesion in the gound state has a checkeboad patten of plaquettes with positive o negative chiality, coesponding to clockwise o counteclockwise otation of the spins. 1 This is the same symmety as in the antifeomagnetic Ising model. Some othe ealizations of FFXY models ae the antifeomagnetic XY model on a tiangula lattice, 2 the Coulomb gas with half-intege chages, 3 and the 19-vetex vesion of the FFXY model. 4 All these models ae geneally assumed to have simila citical behavio. This is also expected to be tue fo the XY Ising model 5 fo cetain choices of some paametes. In the fist Monte Calo MC simulations 6 Teitel and Jayapakash found a divegence in the specific heat, consistent with an Ising tansition, accompanied by a steep dop in the helicity modulus. But since the data did not allow fo any pecise detemination of the citical tempeatues, the authos suggested two possible scenaios: 1 At tempeatues closely below T c, the Ising excitations give ise to a steep dop in. When appoaches 2T/, the votex excitations take ove and poduce an univesal jump at a tempeatue T KT T c. This means two distinct tansitions in well-known univesality classes. 2 The Ising excitations give a tansition with an associated nonunivesal jump at the same tempeatue as the peak of the specific heat. This altenative is a single tansition in a new univesality class. While these often have been consideed the main options, seveal othe possibilities have also been suggested in the liteatue, as, e.g., an Ising tansition at a lowe tempeatue than the Kostelitz-Thouless KT tansition. 7 Ove the yeas thee have appeaed seveal epots of MC studies whee the losses of both XY and Z 2 odes have been studied. Fo the antifeomagnetic XY model on a tiangula lattice, 2 a study of the heat capacity and the XY susceptibility suggested two distinct tansitions, with T KT T c. The tempeatue diffeence was, howeve, quite small and the possibility of a single tansition could not be uled out. In a second study of the same model, 8 including somewhat lage lattices and with a caeful analysis of the Z 2 tansition, the tansitions wee found to be even close togethe. The tempeatue diffeence was in this study well below the statistical uncetainty. The esults wee theefoe suggestive of a single citical point. Likewise, thee ae conflicting esults fo the Coulomb gas with half-intege chages in the liteatue. A study by Thijssen and Knops 3 suggested coinciding tansitions wheeas Gest found two distinct tansitions. 9 The conflicting values of the tempeatue fo the loss of XY ode was appaently due to diffeent methods to locate the tansition. In the fist case T KT was detemined fom the maximum finitesize dependence in 1/, and in the second case fom the cossing of 1/ fo diffeent system sizes. The latte method gives a lowe value of the tansition tempeatue. The diffeent esults fo the Z 2 tansition seem to be due to diffeences /97/556/358518/$ The Ameican Physical Society

2 3586 PETER OLSSON 55 in the MC data. Wheeas the ealie study epoted a dift in position of the peak in the heat capacity to lowe tempeatues with inceasing lattice size, such a size dependence was not veified in the latte simulation. This discepancy gave highe values of T c. Simila conclusions wee also obtained fom ecent simulations of the Coulomb gas CG with halfintege chages. 10 The esults fom this study wee two distinct tansitions; at a lowe tempeatue a KT tansition with a nonunivesal jump, followed by a Z 2 tansition with non- Ising exponents see below. As discussed in Ref. 11 the dielectic constant at smallest possible wave vecto k2/l fom the CG simulations, is not an ideal quantity fo locating the KT tansition. With this kind of bounday conditions, thee ae two finite-size effects woking in the opposite diections. That this quantity is moe o less size independent only means that these two effects happen to nealy cancel each othe. This casts doubt on both the KT tempeatues and the nonunivesal jumps found in the above studies. In ode to cicumvent the difficulties associated with a pecise detemination of T KT it has been agued that a detemination of the citical exponents fo the Z 2 tansition by means of finite-size scaling, would be the best way to aive at some fim conclusions. This, at fist, seems a good idea since the study of finite-size effects ight at T c usually is the by fa most efficient way to extact the citical behavio by MC simulations. In this spiit the coelation length exponent, has been detemined in a faily lage numbe of studies by means of finite-size scaling at T c. In the MC simulations this exponent is extacted fom the tempeatue dependence of vaious kinds of measues of the distibution of the staggeed magnetization. The same exponent has also been obtained fom tansfe-matix calculations. 15 The esults ae geneally in favo of non-ising exponents, 0.85(3), , , 0.804, and the citical tempeatues T c /J 0.455(2), , , Deteminations of the same exponent in the 19-vetex vesion of the FFXY model, 4 the Coulomb gas with half-intege chages, 10 and the XY Ising model, 5,16 gave 0.77(3), 0.843, 0.853, and 0.79, espectively. It seems, howeve, to be the case that such finite-size scalings in many cases ae not quite satisfactoy and ae theefoe not conclusive. 5,15,10 With the steadily inceasing computational esouces it has been possible to obtain data with high pecision fo inceasingly lage lattices. A ecent pape epoted esults fo the helicity modulus at a L128 system. 13 These data have fa-eaching implications since it was shown that the helicity modulus cosses the univesal line, 2T/, at a supisingly low tempeatue, well below the tempeatues quoted above fo the Z 2 tansition. This must be consideed vey stong evidence that the XY ode is lost at a tempeatue below the T c obtained fom finite-size scaling; and thus excludes the single tansition scenaio. Howeve, with this position the non-ising exponents become poblematic. The non-ising exponents ae usually explained as an effect of the inteaction between XY and Ising citical excitations a easonable explanation only if the two kinds of ode ae lost at the same tempeatue. We theefoe have two pieces of evidence pointing in opposite diections. The pesence of non-ising exponents stongly suggests a single tansition, wheeas the ealy dop in the helicity modulus seems to exclude this possibility. A consistent view of these mattes was ecently suggested in Ref. 17. The key obsevation is that a consequence of a KT tansition below T c would be the pesence of a finite but lage sceening length, atthez 2 tansition tempeatue T c. Fo finite-size scaling to be valid, it is necessay that L be much lage than all othe finite length scales in the system, and in paticula. The lage value of could theefoe invalidate ealie finite-size scaling analyses. The condition fo a successful application of finite-size scaling at T c, L, may imply vey lage systems. In the most ambitious study so fa of the XY Ising model by means of Monte Calo tansfe-matix calculations on infinite stips with widths up to 30 lattice spacings, Nightingale et al. again found evidence fo non-ising exponents. 16 They did, howeve, also find an intenal inconsistency in two diffeent deteminations of the themal exponent y T, which led them to call in question the applicability of scaling theoy. This inconsistency is cetainly in line with the suggested failue of finite-size scaling due to the finite sceening length. The main esults in Ref. 17 wee a pecise detemination of T KT, togethe with a demonstation that the staggeed magnetization is, indeed, influenced by the sceening length, unless L, i.e., the helicity modulus 0. In ode to show that the behavio is consistent with the Ising exponent 1, the behavio of the coelation length was also examined. This pat of the study was, howeve, hampeed by two diffeent complications. Fist, the coelation function did only fit nicely to an exponential decay fo tempeatues petty fa away fom T c. Second, it was difficult to include the effect of the spin waves in an entiely convincing manne. While it is cetainly possible to ague in favo of the employed technique, 17 this is at best only an appoximative way to compensate fo the tempeatue-dependent effects of the spin waves. One of the aims of the pesent study is to impove on the poblematic points in the tempeatue dependence of the coelation length. The complications with tempeatuedependent effects of the spin waves is taken cae of by pefoming simulations in an FFXY model with Villain inteaction the model dual to the CG with half-intege chages. The point is that both the votex inteaction and the voticity (1/2) ae manifestly tempeatue independent in that model. To find a eliable technique fo deteminations of the coelation length, we compae with the behavio in the two-dimensional 2D Ising model. In that case we benefit fom the dual advantages of a fast cluste algoithm and exact knowledge of the citical behavio. Fom the simulations of the 2D Ising model we show that the egion whee the tue Ising exponent may be found is vey naow. A simila analysis of the FFXY model gives the same kind of conclusion, and it theefoe seems that the data points in Fig. 5 of Ref. 17 actually ae outside the citical egion. The main esult of the pesent analysis of the fully fustated 2D XY model with Villain inteaction is the existence of two distinct tansitions. An odinay Kostelitz-Thouless tansition at T KT /J0.8108(1) followed by an Ising tansition at T c /J0.8225(5), about 1.4% above. This is a faily small tempeatue diffeence but, as we will see below, the

3 55 MONTE CARLO STUDY OF THE VILLAIN VERSION OF conclusion of two distinct tansitions is not built on an estimate of the tempeatue diffeence between two sepaate tansitions. Section III C 2. gives a stong agument fo the existence of two tansitions which is not based on the deteminations of the two tansition tempeatues. The oganization of the pesent pape is as follows: In Sec. II we define the model, descibe the quantities measued in the simulations and some of the analyses to be pefomed on the data. Section III begins ou analyses of MC data. We shotly descibe the MC pocedue employed to obtain the data and some checks used to validate the esults. The majo pat of Sec. III gives the esults fom vaious analyses that take advantage of the finite-size dependence in the MC data. Among these ae the agument fo two distinct tansitions, the detemination of T KT though finite-size scaling of the helicity modulus, and an analysis of Binde s cumulant fo the staggeed magnetization. Section IV contains the deteminations of the chaacteistic lengths and fom the coelation function. In this pape denotes the coelation length associated with the Ising-like degees of feedom, wheeas is the sceening length associated with the KT tansition, which besides a constant facto is equivalent to the XY coelation length. Since the finite-size effects in this context ae unwanted complications, we take some pains to examine the appeaance of finite-size effects. In ode to test some techniques fo the analysis of coelation functions, and the citical behavio fom the coelation length, we make use of the 2D Ising model as a testing gound. Afte these peliminaies we employ these techniques to the coelation functions fom the FFXY model to detemine the tempeatue dependence of both and above T c, and at low tempeatues. Finally, in Sec. V we put ou esults in elation to some esults by Begé et al. fo a model with a vaiable coupling fo one link pe plaquette, 18 and summaize ou findings. II. BACKGROUND In this section we descibe the model, discuss some quantities measued in the MC simulations and thei elation to the moe convenient Coulomb gas quantities, and shotly descibe some analyses to be applied to the MC data. A. Model The model is defined though the patition function Z 0 2 i d i 2 eh, whee the Hamiltonian fo a fustated system is given by H ij U i j A ij. In the pesent case the Villain vesion of the FFXY model the spin inteaction U(), is given by e U n e J2n2 /2, whee is an angula diffeence between neaest neighbos. In the Hamiltonian above A ij is the vecto potential, and the fustation is given by the otation of A ij : f 1 2 DA 1 2 A y xˆ/2 y A xˆ/2 x x A ŷ/2 A ŷ/2 Full fustation, f 1/2, may, e.g., be obtained by setting A y 0 eveywhee and A x at evey second ow and zeo othewise. Hee we intoduce the discete diffeence opeato, D(D x,d y ), D f f ˆ /2f ˆ /2, and A x and A y fo the vecto potential at links in the x and y diections, espectively. Associated with the discete diffeence is k which is obtained fom ik xe ik D x e ik k x2sin k x 2, and also gives k 2 42cosk x 2cosk y. B. Measued quantities We now descibe some of the quantities which ae measued in the simulations, and ae of cental impotance fo the analyses in Secs. III and IV. 1. Helicity modulus The helicity modulus, is a measue of the quasi-longange ode in XY models. It is defined fom the incease in fee enegy due to a small twist acoss the system in one diection, 2 F 2. Witten in this way, and with cuentf/, the helicity modulus may be intepeted as the popotionality constant between the applied twist and the obtained macoscopic cuent. In MC simulations the helicity modulus is obtained fom the coelation function 19 J 0 L 2 U x 2, whee J 0 U(), and the sum in the second tem is ove all links in one diection, hee the x diection. 2. Voticity Beside the helicity modulus the main quantity measued in ou simulations is the Fouie tansfom of the voticity. The voticity is defined in tems of the otation of the cuent U( ij )U( i j A ij ) aound a plaquette, 2 v 1 2J U 12U 23 U 34 U The facto 2 in the denominato is chosen to give the zeotempeatue limit v1/2. This follows fom the angula diffeence /4 in the gound state. The steps in the simulations consist of measuing v at each plaquette, Fouie tansfoming,.

4 3588 PETER OLSSON 55 v k v e ik, H CG 4 2 J 1 2, m Gm, 3 and accumulating the Fouie components squaed, v k 2. It is also common to define the voticity in tems of the angula diffeences. That coesponds to the chiality, in the context of FFXY models. Howeve, an appealing featue of the voticity defined in Eq. 1 is that it is elated to some deivatives of the fee enegy. This is also the eason fo the existence of some exact elations between the measued votex coelations and the coelations in the 2D CG with halfintege chages. The chiality, on the othe hand, is somewhat peculia in that it jumps discontinuously as a function of the angula diffeences. 3. The staggeed magnetization Fo the study of the Z 2 degees of feedom associated with the symmety of the antifeomagnetic Ising model, a convenient quantity is the staggeed magnetization M 2 L 2 1 x yv, 2 whee the sum is ove all the plaquettes of the system, and the altenating sign is included to take cae of the checkeboad patten. The facto of 2/L 2 is chosen to give M1 in a well-odeed system. In an infinite system this quantity has a finite value in the low-tempeatue phase and goes to zeo as T c is appoached fom below, but this shap behavio is consideably smoothed in the finite systems of the MC simulations. Since (1) x ye i(,), M is diectly elated to the k(,) component of the voticity. Also useful ae some powes of the staggeed magnetization, M 2 p L 2 1 x yv p. Binde s cumulant in Sec. II E is defined fom M 2 and M 4. C. Duality elation and the coelation function It has been agued that both the votex inteaction and the aveage voticity at a plaquette ae tempeatue dependent in the FFXY model with cosines inteaction. 17 To avoid this kind of complicating factos one would athe have esults fom the CG with half-intege chages, since both the aveage voticity and the votex inteaction in that model ae manifestly tempeatue independent. Howeve, since simulations of that model ae consideably moe time-consuming, we instead pefom simulations of the spin model with the Villain inteaction, and make use of an exact elation between the measued voticity coelations and the coesponding coelations fo the CG half-intege chages. In this section we shotly discuss the duality tansfomation, define the Z 2 coelation function, g() and g(k), and deive the link between ou measued voticity coelations and g(k). In the Appendix we discuss the duality tansfomation 20,21 applied to the FFXY model. This gives the Hamiltonian whee m ae half-intege chages, m 1/2, 3/2,..., and G() the lattices Geen s function is the solution to D G(), with G(0)0 and pope bounday conditions. An excellent appoximation fo 1 is 2G()lnconst. It is now convenient to define the Z 2 coelation function in tems of the CG chages m. Fo the coelation function in odinay space we wite, g41 x ym 0 m, 4 whee the pefactos, again, ae fo nomalization and the checkeboad patten in the well-odeed gound state. In the low-tempeatue phase this quantity has a finite value in the limit, wheeas it appoaches zeo above T c. The appoaches to these limits ae exponential, govened by the coelation length. As discussed below, the coelation length is, howeve, bette detemined fom the Fouie components g(k). The Fouie expansion of the coelation function is g 1 1 L 2 gke ik 1 x y k L 2 gqe iq, q whee we intoduce q(,)k and g(q). Togethe with Eq. 4 this gives gk 4 L 2 m k m k. A link between the voticity coelation function and the coesponding coelation function of CG chages may be obtained by consideing two diffeent expessions fo the wave-vecto dependent helicity modulus (k), which is equivalent to the dielectic function J/(k). 22 In the Coulomb gas pictue, with the inteaction 4 2 JG()2Jln, (k) becomes kj 42 J 2 TL m km 2k 2 k, wheeas the same quantity in the XY vaiables is kj 0 42 J 2 TL 2k 2 v kv k. 7 In the Appendix we show that these two expessions ae the same deivative of the fee enegy, which means that they have to be equal. This gives the desied link between the coelation functions fo ou measued voticity and the halfintege vaiables of the CG model, m k m k v k v k TJJ 0 J 2 L 2 k Togethe with Eq. 5 this gives the desied expession fo the coelation function g(k) in tems of the measued coelations v k v k. This is the pocedue used to detemine g(k) which is analyzed in Sec. IV

5 55 MONTE CARLO STUDY OF THE VILLAIN VERSION OF The last tem in Eq. 8 is unde cetain conditions vey small beside the voticity coelation tem v k v k. This is especially the case at q0 fo tempeatues aound T c, which means that the deteminations of and in Sec. IV would be influenced only vey slightly by neglecting this coection. The elation between the CG coelations and the voticity coelations in odinay space has been discussed in Ref. 21. Fo the case with the Villain spin inteaction the esult was m 0 m v 0 v. Howeve, as seen in thei deivation this holds fo 1, only. Fo geneal the Fouie tansfom of Eq. 8 gives 0 /2J m 0 m v 0 v 4TJJ 2, if 0, TJJ 0 /2J 2, if 1, 0, othewise. 9 In Sec. III B the above elation fo 0 is used in a consistency check. D. Analysis of In this section we discuss the size dependence of the helicity modulus and its elation to the univesal jump and Kostelitz enomalization-goup RG equations. We will focus on the dimensionless quantity T/. Fo the finite-size scaling analysis of T/ one assumes that the size dependence in this quantity is elated to the behavio of a set of RG tajectoies. 23 The stating point in paamete space, and theeby the elevant tajectoy, is detemined by the tempeatue. These tajectoies behave diffeently in the low- and high-tempeatue phases. In the lowtempeatue phase they teminate at finite values of T/ wheeas they continue to infinity, coesponding to 0, in the high-tempeatue phase. The last tajectoy in the low-tempeatue phase ends at the univesal value T CG /T/(2)1/4. This means that the helicity modulus fo that vey tempeatue in an infinite system is 2T/. The jump of this quantity to zeo is the well-known univesal jump. 24,25 The abupt univesal jump of an infinite system is, of couse, not seen in finite systems. Since deceases with inceasing system size, the univesal jump conditions 2T/ may, howeve, be used to establish an uppe limit fo T KT. The tempeatue obtained in that way is a igoous uppe limit, since the univesal jump condition constitutes an absolute stability citeion. The appoach to the univesal value, /(2T)1, with inceasing system size, may also be used to examine the citical popeties. Fom Kostelitz RG equations 23 the finite-size scaling elation fo L becomes 26 L 2T 1 1 2lnLl Kostelitz RG equations ae expected to be valid only in the limit of low votex density. This means that the above finitesize scaling elation is expected to be valid only at low enomalized votex density. Accodingly, L /(2T) should be not too fa fom unity enomalized out to length scale L should not be too fa fom the fully enomalized out to infinity. This implies the dilute limit fo sizes bigge than L. The same idea may also be used both above and below T KT. A moe complete discussion is given in Ref. 27. Close to T KT we expect 22,27 L 2T 1ccoth2cl 0lnL, TT c, 11 L 2T 1ccot2cl 0lnL, TT c, 12 whee l 0 and c ae fee paametes to be detemined fom the fits. Equation 10 is the c 0 limit of Eq. 11. c vanishes as T KT is appoached fom below o above as 23 cbt/t KT In the high-tempeatue phase l 0 is identified with the logaithm of the sceening length,. In the immediate vicinity of T KT the tempeatue dependence of l 0 should theefoe be given by Kostelitz expession 23 whee 22 C l 0 T/T KT 1 const, C 2B. E. Binde s cumulant Bindes cumulant is a convenient quantity that, in most cases, facilitates deteminations of both the citical tempeatue and the coelation length exponent. Even though the quantity was oiginally pesented in tems of aveages ove blocks of diffeent size in a single simulation with a fixed total system size, 28 it may also be used with data fom systems of diffeent size. Binde s cumulant is obtained fom some moments of the ode paamete, U1 M 4 3M The cucial popety of U is its size independence pecisely at T c. Theefoe, plotting U vesus tempeatue fo seveal diffeent sizes is expected to give a unique cossing point at the citical tempeatue. Futhemoe, the coelation length exponent may be detemined by plotting the data against (TT c )L 1/. The coect value of is expected to give a collapse of that data onto a single cuve. In pactice thee ae, howeve, often coections to scaling which make the conclusions fom this kind of analysis less diect and pecise. F. Bounday conditions It has ecently been pointed out that peiodic bounday conditions PBC s in the XY model may be genealized by including twist fluctuations along the x and y diections in the system fluctuating bounday conditions FBC s. 29,11 Thee ae seveal advantages with consideing such a gene-

6 3590 PETER OLSSON 55 The diffeent methods to analyze MC data may, geneally speaking, be divided into two classes. The most obvious one is to calculate the coelation functions and detemine the coelation length and the associated exponents fom this kind of data. In this kind of analyses one is inteested in the behavio of an infinite system and, accodingly, the finitesize effects ae undesied complications. This kind of analyses is the subject of Sec. IV. The second class of methods instead take advantage of the finite-size dependence in the MC data. This is geneally a moe efficient appoach to analyzing the citical behavio. In this section we employ some techniques that make use of the finite-size dependence in vaious ways. Afte a shot desciption of the simulations and some checks employed to validate the esults, we focus on esults fom the univesal jump condition in Sec. III C. In Sec. III D we pefom finite-size scaling analyses of the helicity modulus both ight at T KT and in the immediate neighbohood aound T KT. With this detemined value fo T KT we then take a close look at the data fom the univesal jump condition in Sec. III E. To obtain a efeence tempeatue we then apply finitesize scaling analysis of Binde s cumulant at T c. Just as in the elated models this kind of analysis gives 1. As suggested in Ref. 17 this seems to be an atifact of the pesence of a finite sceening length associated with the neaby KT tansition. FIG. 1. The size dependence of T (L) KT. The coss is T KT fom Sec. III D and the dashed line is fom the analysis in Sec. III E. alization. Fist, it is with these bounday conditions that the Villain vesion of the XY model is exactly dual to the CG with peiodic bounday conditions. Second, the finite-size effects in seveal quantities wok in the opposite way afte the inclusion of these twists. Finally, with a self-consistently chosen amplitude of these twists, the finite-size effect on the coelation function tuns out to be vitually eliminated. The self-consistent bounday conditions do not seem to be applicable in the fully fustated case. This is possibly an effect of the Z 2 fluctuations. It is, howeve, possible to obtain the coelation function as in an infinite system by taking the aveage of data fo PBC s and FBC s, cf. Fig. 1 in Ref. 29. This technique woks up to, and possibly slightly above T KT. At highe tempeatues both sets of data go down with inceasing lattice size, which makes it consideably moe difficult to extact any esult fo the themodynamic limit. III. FINITE-SIZE ANALYSES A. Monte Calo simulations The Monte Calo simulations wee pefomed with the odinay Metopolis algoithm with sequential sweeps ove the lattices. One such sweep with one tial update pe spin is called a MC step. Fo most of the data thee wee fou MC steps between consecutive measuements. But since it was noted that a majo pat of the compute time, especially on the lage lattices, was used in the fast Fouie tansfom of the measued voticity v, the simulations fo L128 and 256 fo deteminations of wee pefomed with as much as 64 MC steps between consecutive measuements. Fo the latte data the numbe of MC steps ae given in Table II. In Sec. IV we make use of MC data fom the 2D Ising model as a convenient testing gound fo the methods used to analyze the FFXY model. These simulations wee pefomed with Wolff s cluste algoithm. 30 All the simulations wee done on a set of DEC-alpha wokstations. B. Monte Calo data A MC study is, of couse, neve moe eliable than the undelying data. It is theefoe essential to check that the pogam, indeed, does povide coect data. This may be done eithe by compaing with peviously published esults o by making use of some consistency tests. To the best of ou knowledge thee is no published MC data to compae with fo the Villain vesion of the fully fustated XY model with odinay PBC s. Fo the case with FBC s it is, howeve, possible to compae with data fo the half-intege CG. 10 Fo L8 and T/J0.82 ou simulations give Binde s cumulant, U This tempeatue coesponds to T CG , and as expected ou value fo U lies ight in-between the values fo U at T CG and in Fig. 3 of Ref. 10. As a second test we compae the values of (k2/l) fo L16 and T/J0.82. Again, the value obtained fom ou simulations, , is in good ageement with the coesponding values in Fig. 4 of Ref. 10. Fo the bulk of ou data, obtained with odinay PBC s we have to esot to intenal consistency tests. One such test is suggested by the analogy with the CG with half-intege chages. In that case, the chages m1/2 give m 2 1/4. Actually, the value 1/4 tuns out to be a lowe bound since the CG also includes chages of nonlowest ode, i.e., m3/2. 31 Fo ou measued quantity v 2, thee is no such simple esult, but as discussed above thee is an exact elation between these two quantities, Eq. 9. Wheeas v 2 deceases with inceasing tempeatue to about 2.5% below 1/4 at T/J0.9 we find that m 2 stays close to 1/4. Moe pecisely, the esults ae m at T/J0.4, at T/J0.77, and at T/J0.9. This constitutes a confimation of the coectness of the MC data. Fom the minute deviations of m 2 fom 1/4 it is also possible to obtain estimates of the faction of plaquettes with nonlowest ode chages in an equivalent CG. Fo the tempeatue inteval T/J0.77 though 0.9 ou data indicates that this faction is fom to

7 55 MONTE CARLO STUDY OF THE VILLAIN VERSION OF C. Univesal jump condition In this section we make use of the univesal jump condition to obtain both an uppe limit of T KT and a stong evidence in favo of two distinct tansitions. Since the univesal jump condition is an absolute stability equiement, we believe the agument of the pesent section to be especially fee of objections. Wheeas the moe pecise esults in the late sections ae obtained on the basis of additional assumptions, the diect use of the univesal jump condition is paticulaly clean. Fo the following discussion we intoduce the sizedependent tansition tempeatue T (L) KT, as the tempeatue whee the helicity modulus fo system size L is equal to the univesal value, L 2T (L) KT /. 1. Uppe limit of T KT The univesal jump condition was applied to the fully fustated XY model with cosines inteaction in Ref. 13 to establish an uppe limit fo the KT tempeatue. Fom the intesection of the MC data fo L128 with the univesal value, these authos found, as discussed in the Intoduction, T (128) KT /J0.449(1)J, clealy below the values of T c detemined with finite-size scaling. The same appoach with the data fo the Villain vesion gives T (128) KT 0.816J as the uppe limit fo L128. Figue 1 shows the size dependence of T (L) KT. The dashed line is fom the analysis in Sec. III E. 2. Two tansitions We now tun to the agument fo two distinct tansitions based solely on the univesal jump condition. 24,25 The agument has also been biefly pesented in Ref. 32 whee it was applied to the cosines vesion of the FFXY model. The stating point is that the staggeed magnetization fo an infinite system, M, vanishes at the Z 2 tansition tempeatue T c. To establish the existence of two distinct tansitions it is theefoe sufficient to examine M ight at T KT. A nonzeo value of M (T KT ) would be an unequivocal demonstation of Ising ode, which then implies that this ode is lost at a highe tempeatue, T c T KT. The detemination of M (T KT ) at fist seems vey difficult since, beside the usual poblem of appoaching the themodynamic limit, the value of T KT has to be known with high pecision. The univesal jump citeion used so fa, is only capable of yielding uppe limits. A way aound both these difficulties at the same time is to focus on M L (T (L) KT ), the staggeed magnetization of finite lattices at the size-dependent KT tempeatues, and examine the behavio of this quantity as a function of system size, L. The point is that the desied quantity M (T KT ) is the lage-l limit of M L (T (L) KT ), and that the staggeed magnetization is eadily detemined by simulations fo vaious system sizes, L. The esults fom this analysis ae shown in Fig. 2. The figue shows that the staggeed magnetization at T (L) KT is an inceasing function of lattice size. The figue gives as a lowe limit of M (T KT ), and a naive extapolation, that neglects the cuvatue, suggests M (T KT ) We conside this to be a vey stong agument that the Z 2 ode pesists at the KT tansition tempeatue. Fo the FIG. 2. Evidence fo two distinct tansitions. The plotted quantity is known to appoach M (T KT ) in the limit L. The dashed line shows a naive extapolation, wheeas the dot is this quantity fom Sec. IV E obtained with the value of T KT fom Sec. III D. opposite to be tue, this inceasing tend towad a finite value of M (T KT ) should change to a deceasing tend down to zeo. Even though this possibility could neve be uled out fom the data fo finite systems alone, such a change in tend seems vey unlikely. Futhemoe, the moe detailed analysis in the following sections yields M (T KT )0.783(2), entiely consistent with the inceasing tend in Fig. 2. It should also be noted that this line of evidence does not depend on the assumption of a univesal jump. The agument holds equally well with a jump /(2T)g, g1. (g1 is excluded by stability. D. Kostelitz-Thouless tansition The pupose of this section is fist to detemine the Kostelitz-Thouless tempeatue, T KT, and, second, to examine the behavio closely below and above this tempeatue. The method employed is finite-size scaling analysis of the helicity modulus as discussed in Sec. II D. The basic idea is that the size dependence of the helicity modulus, at and in the vicinity of T KT, may be obtained fom Kostelitz RG equations 23 as given by Eqs. 10, 11, and 12. The analysis of the helicity modulus, is based on a lage amount of MC data. In ode to make efficient use of the data and get L as continuous functions of T, we fist detemine the helicity modulus as second-ode polynomials, one fo each L, int/j0.8107: L 2T L L L These second-ode expansions ae only expected to be valid within athe naow tempeatue intevals. We theefoe only include data in the fits fo tempeatues T ange /J, whee T ange deceases with inceasing L. The paametes fom this analysis togethe with the size of the tempeatue intevals ae shown in Table I. 1. Detemination of T KT We now apply the finite-size scaling elations fo the helicity modulus as discussed in Sec. II D. Since the elations in this system only ae expected to be valid fo faily lage

8 3592 PETER OLSSON 55 TABLE I. Paametes fom fitting MC data fo the helicity modulus to Eq. 17. The data included in the fits ae esticted to T/J0.8107T ange /J. L L L L T ange /J lattices we fist follow the pocedue in Ref. 27 and pefom the analysis with systems of size LL min though 128 and vaious values fo L min. The eos in the fits ae shown in Fig. 3. On the basis of this analysis we conclude that L min 32 does give a good fit. This is the same choice as fo the FFXY model with cosines inteaction in Ref. 17. Figue 4 shows the good fit of the MC data to Eq. 10 obtained by adjusting T KT and l 0. The obtained value fo the KT tempeatue is T KT /J0.8108(1). We conside the good fit to Eq. 10 to be vey stong evidence fo an odinay KT tansition. Note that T KT /J is well below the uppe limit T/J0.816 fom the univesal jump citeion in Fig. 3. It is also slightly lowe than what a simple linea extapolation of the fou lowest points to 1/L0 in Fig. 1 would suggest. 2. Finite-size scaling aound T KT We now shotly discuss the citical behavio in the immediate vicinity of T KT. The appoach closely follows Ref. 27. In fitting ou MC data to Eqs. 11 and 12 we fix the tempeatue, calculate L /(2T) fom the paametes in Table I and adjust c and l 0 to get the best possible fit. In the high-tempeatue phase, l 0 is the logaithm of the sceening length,. Below T KT this quantity has no such diect intepetation. The paamete B in Eq. 13 may then be obtained FIG. 4. L vesus lattice size fo T KT J. The solid line is Eq. 10 with l The good fit is stong evidence fo an odinay KT tansition. fom the tempeatue dependence of c. Just as in the analysis of the odinay XY model the values of the paametes B obtained fom the low- and high-tempeatue data ae, within statistical eos, the same, B2.889 and 2.834, espectively. Though Eq. 15 this gives C0.544 and fo the slope of the logaithm of the chaacteistic length. The slope C may also be obtained diectly fom the tempeatue dependence of l 0. In this way we aive at C0.532, and ou final estimate fo the slope becomes C In Secs. III E and IV D we will obtain diffeent values fo C, but we conside the pesent detemination to be the moe eliable one fo two easons. Fist, it does build on excellent ageements with pedictions fom the Kostelitz RG equations and, second, in contast to othe deteminations, this method does pobe the behavio in the immediate vicinity of T KT. L E. Size dependence of T KT In Sec. III C we made use of the univesal jump condition to detemine a kind of size-dependent KT tempeatues T (L) KT as uppe bounds fo T KT. Fom Fig. 1, it seemed difficult to extapolate such data to the themodynamic limit. We now demonstate that the size dependence of T (L) KT has the same fom as the Kostelitz expession fo the tempeatue dependence of the coelation length, i.e., lnlconst T KT C L /T KT FIG. 3. The quality of the fit fom fitting L to Eq. 10. L min is the smallest size included in the fit. Fo a good fit one expects 2 /degee of feedom1. The figue indicates that the fit is not vey successful when including small lattices, but becomes acceptable fo L min 32. Figue 5 shows the appoach of T (L) KT to T KT J with inceasing L. The points do, indeed, fall on a staight line. This is the same dashed line as in Fig. 1. It should be noted that it does not seem possible to link this behavio diectly to the divegence of the sceening length. In the pesent appoach the slope is C0.265, wheeas the same constant in the tempeatue dependence of l 0 yielded C0.54 which is about twice as big. This size dependence of T (L) KT is also found in the odinay 2D XY model with no fustation. Hee the slope is found to

9 55 MONTE CARLO STUDY OF THE VILLAIN VERSION OF FIG. 5. The size dependence of T (L) KT fo the FFXY model, with lattice sizes L16, 32, 48, 64, 96, and 128. The value fo the tansition tempeatue is taken fom the detemination in Sec. III D. Note that the slope C is entiely diffeent fom the coesponding slopes fom the tempeatue dependence of the sceening length. be 0.85, and, again, the tempeatue dependence of the chaacteistic length gives a slope that is about two times as big. 33 F. Binde s cumulant As discussed in the Intoduction the existence of two tansitions close to each othe may give ise to poblems with odinay finite-size scaling. The evidence fo the two tansitions given in the pevious section stongly implies that this actually is the case fo the FFXY model. This means that the citical popeties ae not accessible with finite-size scaling at T c unless the systems employed ae consideably lage than the coelation length associated with the othe hee the KT tansition. The pupose of the pesent finite-size scaling analysis is theefoe not to extact the coect citical behavio, but athe to povide a efeence tempeatue and to veify that the FFXY model with Villain inteaction indeed does behave in a way that is simila to the moe studied FFXY model with cosines inteaction. Figue 6a shows Binde s cumulant vesus tempeatue fo L8, 16, 32, and 64. As discussed in Sec. II E, U is expected to be size-independent ight at the citical tempeatue. This is not quite bone out by the data. A close look eveals that the cossing points move slowly to lowe tempeatues fo lage system sizes. Fo the pais of lattice sizes L8, 16, L16, 32, and L32, 64, the cossing tempeatues ae 0.827, 0.825, and 0.824, espectively, though the two last tempeatues ae within the statistical uncetainties. This is in good ageement with (3)0.826(2) obtained fom simulations of the CG with half-intege chages on lattices with size L10 24 in Ref. 10. Since the odinay peiodic bounday conditions in that simulation coesponds to FBC s in spin models cf. Sec. II F, wheeas the pesent simulations ae pefomed with PBC s, the value of the cumulant at citicality is, howeve, not expected to be the same. 17 Figue 6b shows the data collapse. Following Ref. 10 we assume that U L (T)(tL 1/ ), whee tt/t c 1. We then expand (x) 0 1 x 2 x 2 fo small x and adjust FIG. 6. Binde s cumulant U L fo diffeent lattice sizes. a U is moe o less size independent fo L16 at T/J b An attempted data collapse. Just as in the FFXY model with cosines inteaction this kind of analysis suggests 1. these thee paametes togethe with and T c to get the best possible fit. With data close to T c, (0.605U0.640) fo system sizes L16, 32, and 64, we obtain T c and This value of is in good ageement with the published values, listed in the Intoduction. IV. CORRELATION LENGTHS In the pevious section we pefomed a numbe of analyses with methods that take advantage of the finite-size effects in the MC data. This is usually the most efficient way to detemine the citical behavio fom Monte Calo simulations. The altenative appoach is to detemine the coelation length fom the length dependence of some coelation functions and then extact the citical behavio fom its tempeatue dependence. In the pesent section we take this altenative oute. In ode to test some techniques fo detemining the coelation length and extacting the coelation length exponent and the citical tempeatue, we fist pesent an analysis of the 2D Ising model. Because of the dual advantage of a known citical behavio and a fast cluste update algoithm, this model seves as a vey convenient testing gound. One mino diffeence between the analysis of the 2D Ising model and the FFXY model is due to the antifeomagnetic odeing. Wheeas the citical behavio of the feomagnetic Ising model is obtained in the k 0 limit, the coesponding citi-

10 3594 PETER OLSSON 55 cal behavio in the FFXY model manifests itself at k(,). This is taken cae of by pefoming all the analyses in Sec. IV C in tems of q(,)k instead of k, cf. Sec. II C. In this section we detemine two diffeent chaacteistic lengths fom ou MC data fo v k v k. The eason that it is at all possible to define two diffeent chaacteistic lengths is elated to the above discussion. Wheeas the sceening length is detemined fom the k 0 limit of these coelations, the coelation length is detemined fom the limit q 0. Fo the detemination of the chaacteistic lengths one would like to have the coelation function fo an infinite system. It is theefoe of geat impotance to know when the undesied effects of the finite lattice size set in. Befoe applying the obtained techniques to the FFXY model we do a caeful analysis of the finite-size effects in this model. The esults cooboate the suggestion 17 that the coelation function is plagued by finite-size effects unless the system is lage enough that 0. Afte these peliminaies we then tun to deteminations of the coelation length, the citical exponent, and the citical tempeatue T c, in the FFXY model. Much as expected fom the evidence of two distinct tansitions, the behavio is found to be consistent with an odinay Ising tansition, 1. Howeve, fo this demonstation it tuns out to be necessay to examine the behavio faily close to T c, which coesponds to lage coelation lengths, 10. The sceening length associated with the KT tansition is also detemined, and its tempeatue dependence is found to be diffeent fom the behavio of, but in good ageement with Kostelitz esult, Eq We finally tun to the behavio of at TT c. At these tempeatues we have no data with 0, and if we had, thee might also be a poblem fulfilling L) and we theefoe need some othe methods to avoid the finite-size effects. The solution is to estict the analysis to TT KT, whee the diffeent bounday conditions of Sec. II F may be employed. Howeve, the tempeatue dependence of below T c and T KT ) does not seem to be useful fo assessing the citical behavio, possibly an effect of the pesence of the KT tansition between the obtained data and T c. A. The coelation length in the 2D Ising model The Hamiltonian of the Ising model is H I J ij s i s j, whee i and j numeate the lattice points, s i 1, and the summation is esticted to neaest neighbos. In two dimensions the coelation length exponent is 1, and at a squae lattice the citical tempeatue is known to be T I 2 c /J ln Since 1, a plot of 1/ vesus T is expected to yield a ectilinea behavio down to T c I. Howeve, the veification of this tuns out to equie data faily close to T c I, lage coelation lengths, and theefoe athe big lattices. Fo most puposes this execise is pointless, since beside being obtained fom the exact solution the value 1 may be veified fom MC simulations by means of finite-size scaling at T c I. But since this kind of finite-size scaling does not seem to wok in the FFXY models fo the accessible lattice sizes, we have to esot to analyses of the coelation functions. With that backgound, analyses of the coelation function fo the 2D Ising model seves as a help to develop techniques fo simila analyses of the FFXY model. Beside the benefit of the exactly known citical behavio, the analysis of the Ising model is geatly simplified by means of the cluste algoithm 30 that is instumental in obtaining MC data with small statistical eos. 1. Detemination of the coelation length At fist sight the obvious way to detemine the coelation length is to examine the exponential decease of the coelation function g() down to zeo. This amounts to adjusting the paametes A and to obtain the best possible fit to the expession, gae /. At tempeatues closely above T c, the above expession should be modified to take coelations acoss the whole system into account. This is customaily done by instead fitting to an expession with the peiodicity of the system, gae / e L/. 19 It is, howeve, difficult to obtain eliable values fo the coelation length with this pocedue. The main complication is that the optimum value of does depend on the ange in employed fo the fit. This is not too supising, since the pue exponential decay only is expected fo vey small values of g(). An altenative detemination of by means of g(k), the Fouie components of the coelation function, has been suggested in Ref. 34, 2 L g0 g2/l An advantage with this expession is that no fitting is needed, and that the abitainess involved in choosing the fitting inteval is eliminated. This expession may be deived fom 1 gk k The elation to an exponential decay is obtained since the Fouie tansfom of this function is the Bessel-K 0 function, gk 0 /, with the limiting behavio e /. In ode to check that the deteminations of fom Eq. 20 eally do pobe the long-distance limit, we have fitted g() fo min 2 min to Eq. 19, and examined the dependence of 1/ on min. It tuns out that, plotted vesus 1/ min, these points fall on close to a staight line, which as 1/ min 0 appoaches the value of 1/ obtained fom Eq.

11 55 MONTE CARLO STUDY OF THE VILLAIN VERSION OF FIG. 7. Coelation length in the 2D Ising model fo L64, 128, and 256. The solid line is fom fitting to Eq. 24 fo T/J2.33 with data that fulfills L/ On the basis of this compaison we believe that Eq. 20 indeed does give a eliable way to detemine the coelation length. A pecise detemination of with Eq. 20 equies faily long MC simulations. This is the case since g(2/l) and g(0) measue the amplitude of the lagest fluctuations in the system, with the coespondingly long decoelation times. A way to educe the effect of statistical eos is to include some moe k vectos in the analysis. This is motivated by the difficulty to obtain good accuacy fom Eq. 20 on data fom the FFXY model at lage lattices (L128, 256. Howeve, using g(k) in a too lage inteval will affect the coelation length. Assuming that the exponential dependence only holds fo, o, similaly, that the asymptotic k dependence only is valid fo k2/, we estict ouselves to making use of data fom wave vectos k/, only. Fo small values of this is not vey estictive, and since we ae inteested in the small-k limit we impose the additional condition k0.1. The pocedue to detemine is then to fist fit the data to FIG. 8. Deteminations of T c and in the 2D Ising model fom the coelation length data by fitting to Eqs. 24 and 25. The open cicles ae fom fits with the exponent kept fixed, 1, wheeas the solid squaes ae obtained with both T c and as fee paametes. The geneal tend is that both quantities appoach the exact values, indicated by the dashed lines, as the tempeatue inteval is deceased. T c I. Estimates of T c I may be obtained by fitting 1 gk g 0g 1 k 2 g 2 k 2 2, 22 1/ATT c, 24 whee we also include a second-ode tem in k 2 to take cae of the cuvatue in the data and then extact the coelation length though g 1 /g In the limiting case with only two wave vectos, this pocedue is eadily shown to be equivalent to Eq Detemination of the citical behavio Figue 7 shows the tempeatue dependence of the coelation length in the 2D Ising model fo thee diffeent system I sizes, L64, 128, and 256. In the vicinity of T c the data eveals some finite-size effects. The coelation length becomes smalle in a too small system. The pesent data seems to suggest that the deteminations ae eliable only if L/5. Also appaent in the figue is a slight cuvatue in the data. The expected linea behavio is found only ight above with A and T c as fee paametes. In these analyses we only make use of data fo L128 and 256. Due to the cuvatue in the data, the citical tempeatue obtained in this way does depend on the tempeatue inteval fo the fit. Fo tempeatues closely above T c we only include data points with I L/5. This gives L-dependent lowe limits fo the tempeatue inteval. The uppe limit of the tempeatue inteval is given by T max. The fit is then pefomed fo seveal diffeent values of T max. The dependence of T c on T max is I shown by open cicles in Fig. 8a. The dashed line is the exact value of T I c. Fo lage T max lage tempeatue intevals the analysis yields too low estimates of the citical tempeatue, but with deceasing T max the estimated T I c inceases towads the coect value. This linea fit pesumes a known value of the coelation length exponent. Since the value of in the FFXY model is highly disputed it is also of inteest to pefom the fit with as a fee paamete. That is done by fitting

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