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Phase diagram of generalized fully frustrated XY model in two dimensions Petter Minnhagen,1 Beom Jun Kim,2 Sebastian Bernhardsson,1 and Gerardo Cristofano3 1Dept. of Physics, Ume˚a University, 901 87 Ume˚a, Sweden 2Dept. ofPhysics,BK21PhysicsResearchDivisionandInstituteofBasicScience,SungkyunkwanUniv.,Suwon440-746,Korea 3Dipartimento di Scienze Fisiche, Universitá di Napoli “Federico II” and INFN, Sezione di Napoli, Via Cintia, Compl. universitario M. Sant’Angelo, 80126 Napoli, Italy Itisshownthatthephasediagramofthetwo-dimensionalgeneralizedfully-frustratedXY model 8 on a square lattice contains a crossing of the chirality transition and the Kosterlitz-Thouless (KT) 0 transition, as well as a stable phase characterized by a finite helicity modulus Υ and an unbroken 0 chirality symmetry. The crossing point itself is consistent with a critical point without any jump 2 in Υ, with the size (L) scaling Υ∼L−0.63 and the critical index ν ≈0.77. The KT transition line remains continuousbeyondthecrossing buteventuallyturnsintoafirst-orderline. Theresults are n established using Monte-Carlo simulations of the staggered magnetization, helicity modulus, and a J thefourth-orderhelicity modulus. 0 PACSnumbers: 64.60.Cn,75.10.Hk,74.50.+r 2 ] n I. INTRODUCTION II. THE GENERALIZED FFXY MODEL o c The XY model on a square lattice in the presence of - The phase transitions of the two-dimensional (2D) r anexternalmagneticfieldtransversaltothelatticeplane p fully-frustrated XY (FFXY) model on a square lattice is described by the action: u has been a subject of controversy.1,2,3,4,5 The emerging .s consensus is that the model, as the temperature is low- J at ered, first undergoes an Ising-like transition associated H =−k T cos(φij ≡θi−θj −Aij), (1) B m with the chirality. At a slightly lowertemperature it un- Xhiji dergoesauniversaljumpKosterlitz-Thouless(KT)tran- d- sition associated with the phase angles.1,4 Since the two where θi is the phase variable at the ith site, the sum is over nearest neighbors, J(> 0) is the coupling constant, n transitionsare extremely closeto eachother in tempera- o ture the questionwhether thereis onlyonemergedtran- T is the temperature, kB is the Boltzmann constant, [c sition or actually two separate transitions has been the and Aij =(2e/~c) ijA·dl is the line integral along the cause of the controversy. An argument for two separate bond between adjacent sites i and j. We consider the R 1 transitionswiththeKTtransitionalwaysatalowertem- case where the bond variables Aij are fixed, uniformly v perature than the chirality transition was given by Kor- quenched, out of equilibrium with the site variables and 70 shunovinRef.5intermsofakink-antikinkinstabilityof satisfythecondition pAij =2πf: herethesumisover the domain walls separating domains with different chi- eachsetofbondsofanelementaryplaquetteandf isthe 0 P rality. The2DgeneralizedfullyfrustratedXY(GFFXY) strength of frustration. We assume that the local mag- 3 . model has the same degrees of freedom and the same neticfieldinEq.(1)isequaltotheuniformappliedfield; 1 symmetries as the FFXY model. The argument by Ko- such an approximation is more valid the smaller is the 0 rshunov5 is quite general and appears to hinge only on sample size L compared with the transverse penetration 8 0 the combined U(1) and Z2 symmetry of the model and depthλ⊥.7 Inthecaseoffullfrustration,i.e.,f =1/2,of : the existence of Ising-like domain walls associated with interest to us here, such a model has a continuous U(1) v the broken Z symmetry.6 This strongly suggests that symmetry associated with the rotation of spins and an i 2 X alsothe generalizedmodelwiththe verysame degreesof extra discrete Z2 symmetry, as it has been shownby an- r freedom and symmetry should always have a KT transi- alyzing the degeneracy of the ground state.8,9 Choosing a tionatalowertemperaturethanthe chiralitytransition. theLandaugauge,suchthatvectorpotentialvanisheson As shown here, this is not the case: The two transitions all horizontal bonds and on alternating vertical bounds, can merge in a single critical point. The reason for this wegetalatticewhereeachplaquettedisplaysoneantifer- unexpected result is that the symmetry of the model al- romagneticandthreeferromagneticbonds. Suchachoice lows for a new phase 2D “quasi” phase-order. corresponds to switching the sign of the interaction. ThegeneralizedFFXY modelisobtainedbychanging The present paper is organized as follows: In Sec. II, the form of the interaction from Jcosφ to10,11 we introduce the generalizedfully-frustrated XY model. − Theresultsofournumericalsimulationsarepresentedin U(φ)= 2J 1 cos2p2(φ/2) . Sec. III for the staggered magnetization, in Sec. IV for p2 − the helicity modulus, and in Sec. V for the fourth-order h i modulus, respectively. Finally, Sec. VI is devoted to the This does notalterany symmetrypresentinthe original summary of the paper. FFXY model which corresponds to p = 1 since 2[1 − 2 1 T =0.100 L=8 16 0.8 24 32 0.6 m 0.4 0.2 (a) 0 1.34 1.345 1.35 1.355 1.36 1.365 p FIG. 1: The phase diagram of the 2D GFFXY model. The 0.8 phases are characterized by the helicity modulus Υ and the p=1 staggeredmagnetizationm. Thefourphasescorrespondtoall possible combinations of finite and vanishing Υ and m. The 0.6 pointscorrespondtodataobtainedfromthesimulations. The T =0.454 horizontal line p=1 corresponds to theusual FFXY model. ch The phase lines cross and merge in one point. m 0.4 B L=16 24 cos2(φ/2))]=1 cosφ. The essential point is that U(φ) 0.2 32 − is periodic in 2π and that the first term in an expansion 48 forsmallφissecondorder,i.e.,U(φ) Jφ2/2forφ 1. (b) 64 ≈ ≪ Figure 1 is the summary of the results present in this 0 paper and shows the phase diagram in the (p,T)-plane 0.4 0.42 0.44 0.46 0.48 0.5 as obtained from Monte-Carlo (MC) simulations. The T helicity modulus Υ, which relates to the continuous an- gular symmetry, and the staggered magnetizations m, FIG. 2: MC determination of the staggered magnetization. whichrelatestothediscretechiralitysymmetry,areused (a)Thetransitionacrossthehorizontalphaselinep ≈1.3479 c to detect phase boundaries. The phase diagram con- isconfirmedbyasharpdroptozeroforlargersizesL. (b)Size tains all four possible combinations of these two, i.e., scalingofBindercumulantB forp=1. Theuniquecrossing m (Υ,m)=(0,0),(0,=0),(=0,0),(=0,=0). The dashed of curves for different sizes yields the transition temperature horizontal line at p6 =1 co6 rrespond6 s to6 the usual FFXY Tch ≈0.454. This method is used to determine the m-phase model, for which the phase (Υ = 0,m = 0) is not real- line for p<pc. 6 ized. with different checker board patterns are separated by III. STAGGERED MAGNETIZATION an infinite energy barrier in the thermodynamic limit, the phasewiththe brokenchiralitysymmetrypersistsat WefirstpresentnumericalMCresultsofthestaggered low enough temperatures as long as the pattern, where magnetization m, defined as12 all links contribute the same energy, indeed corresponds to the ground state. However, this ceases to be true L2 1 when p becomes larger than p . In this new region the m= ( 1)xl+yls , c *(cid:12)(cid:12)L2 l=1 − l(cid:12)(cid:12)+ groundstate instead corresponds to a pattern consisting (cid:12) X (cid:12) of plaquettes with phase difference 0 on three sides and (cid:12) (cid:12) where is the en(cid:12)semble averageand(cid:12)the vorticity for π onthe remaining. Theenergyperlinkishenceinstead thelthhe·l·e·mientarypl(cid:12)aquetteat(xl,yl)i(cid:12)scomputedfrom U(π)/4. The critical value pc is easily computed to be sl ≡ (1/π) hiji∈lφij = ±1 with the sum taken in the pCcon≈se1q.u34en7t9lyf,rothmentheewcgornoduintdiosntattheaattUp(>π/p4)h=asUn(oπb)/ro4-. anti-clockwise around the given plaquette. c P kenchiralitysymmetry andhence correspondsto m=0. The ground states with the spontaneously broken chirality symmetry correspond to the two possible checker Figure 2(a) illustrates the vanishing of the staggered board patterns with alternating positive and negative magnetization m at p p for T = 0.1 (the tempera- c ≈ vorticity. The energy per link in these ground states ture is in units of J/k throughout the present work). B is given by U(π/4) which corresponds to all links con- Thishorizontalpartofthem-phaseboundaryeventually tributing the same energy. Since the two ground states bends down toward smaller p as T is increased. This 3 0.8 size L. These values are well approximated by a second p=1 b=1(universaljump) 0.5 0.6 b=2 p=1 order polynomial as shown in Fig. 3(b). The extrapola- tion to L = gives T = 0.447, which is very close KT ϒ 00..24 L=12346428 bb==112 TKT=0.4473 0.45TL1(/)KT mKtoTenthttresahnvoaswiltusieotnh0∞a.t4et4mt6hpeoebrmataetutinhreeod.dTignhiveRedseafa.tga4o.poodTinhetsesticimnloaFsteiega.og1frtefhoeer- (a)64 (b) p ≤ 1.32 are obtained by this method. One notes that 0 0.4 the m-phaseline andthe KT-line areextremelyclose for 0.2 0.3 0.4 0.5 0 0.04 0.08 0.12 thesep-valuesandonlythesmallerp-values,likep=0.5, T 1/L 0.14 0.02 display a clear separation within our accuracy. 0.12 b=1,21 b=2,1 ThedeterminationoftheKT-linerestsontheassump- 0.1 pb==21,1 p=1.5 0.015 tion that the KT-jump has the universal value 2/π. A 0.08 jump means that the crossing point between the Υ(T) T 0.01 T ∆ 0.06 ∆ and b(2/π)T should give the same TKT(L = ) for all ∞ 0.04 b 1. Figure 3(c) shows the difference ∆T(b,L) = 0.005 ≤ 0.02 (c) (d) TKT(b,L)−TKT(b/2,L) for b = 1. Our result is consistent with a universal jump KT transition, since ∆T(b= 0 0 0 0.04 0.08 0.12 0 0.02 0.04 0.06 1,L) is consistent with a vanishing for L = . On the 1/L 1/L ∞ other hand the jumps size is inconsistent with a double jump since ∆T(b = 2,L) approaches a finite value. For FIG.3: MCdeterminationoftheKTtransitionline. (a)The larger values of p > p , like p = 1.5, the jump is, on helicity modulus Υ as a function T for various sizes L. The c the other hand, consistent with a jump larger than the brokenlinescorrespondtothejumpcondition;b=1istheex- pecteduniversaljump. T isestimatedbyextrapolatingthe universal jump as illustrated in Fig. 3(d): For this value KT crossing point with theb=1 line and thedata to L=∞, as ∆T(b = 2,L) is consistent with a vanishing, suggest- shownin(b),whichshowsasecondorderpolynomialextrap- ing that the jump at the KT transition could be larger olation of the crossing points and TKT = 0.447 is obtained than the universal KT value. The transition at p = 1.5 in accord with the finding TKT = 0.446 in Ref. 4. (c) The shows no sign of any first order characterfrom which we difference∆T ≡|TKT(b,L)−TKT(b/2,L)| in crossing points conclude that it is continuous. In this case the jump is T [see (a)] for (b,b/2) = (1,1/2) and (2,1). It is clearly KT expected to have the universal value. Our data neither seen that the jump has to be less than twice the universal supportnorruleoutthisexpectation. Conversely,acon- jump consistent with the universal jump. (d) The same con- tinuousKTtransitionwith anonuniversaljumpcannei- struction for p = 1.5. We believe that a nonuniversal jump ther be ruled out. However, when p is increased further forp>p cannotbeentirelyruled out. (a)-(c)correspond to c p=1 while (d) is for p=1.5. the transition does eventually become first order as can be detected from the double well structure in the energy histogram. Forthefirstordertransitionthejumpshould partofthe m-phaseboundarywehavetracedoutby the be nonuniversal and larger than the universal jump.16 standard size scaling of Binder’s cumulant B for the In the limit of p = the model reduces to the infinite orderparameterm,13 as displayedin Fig.2(b)mforp=1. statePottsmodel,16∞whichis knowntohaveafirstorder T 0.454 is obtained, in a good agreement with the transition. ch ≈ earlier value 0.452 in Ref. 4. The complete m-phase line with marked data points are shown in Fig. 1. V. FOURTH-ORDER MODULUS IV. HELICITY MODULUS According to the argument given by Korshunov5 the KTtransitionalwaysoccursatalowertemperaturethan The quasi 2D phase ordering is measured by the he- thechiralitytransition. Thisisconsistentwiththeresult licity modulus Υ defined as the stiffness in response to wefindforp-valuesbelowthehorizontallineinthephase the twist δ of the phase variables across the system: diagram. Inthispartofthephasediagramtheargument Υ (∂2F/∂δ2) whereF isthefreeenergy. Thecondi- by Korshunov5 is valid and the chirality transition and δ=0 ≡ tionforaKTtransitionis characterizedby the universal the KT transitions are separated with the KT transi- jump in the helicity modulus, Υ(T )/T =2/π.14,17. tion always at a lower temperature. On the other hand, KT KT ThusaKTtransitioncanbelocatedbythecrossingpoint whenthehorizontalphaselinemeetsandcrossesthe KT between the line y = (2/π)T and the helicity modulus line, Korshunov’s argument5 is no longer valid and one curve y = Υ(T) as illustrated for p = 1 in Fig. 3(a). expects a merged character of the transition. The ar- In practice, a precise determination requires the difficult gument by Korshunov fails because it presumes the ex- taskofextrapolatingtoL= .1 Hereweuse thefollow- istence of Ising-like domain walls and such walls do not ∞ ing method: The values of theT atthe crossingpoint exist above the horizontal line. In order to monitor the KT with the line y =(2/π)T aredetermined asa functionof changeofcharacterofthetransitionwestudythefourth- 4 0 1 L=16 -0.04 0.01 ϒ4-0.08 L=1p6=1.0 0.02 p=1.31 0.9 2342 24 0.03 -0.12 32 L=16 0.8 (a) 0.04 2342 -0.16 0.35 0.40 0.45 0.50 0.55 (b) ϒ a L 0.7 a=0.63 -0.1 T =0.1675 0.6 c ϒ4 -0.2 p=1.346 p=1.35 (a) -0.3 L=16 L=16 0.5 24 24 (c) 32 (d) 32 0.15 0.16 0.17 0.18 0.19 -0.4 T -0.1 0.9 ϒ4 -0.2 p=1.38 p=√2 L=16 24 -0.3 L=16 L=16 24 24 32 (e) 32 (f) 32 -0.4 0.8 0.08 0.12 0.16 0.2 0.24 0.28 0.12 0.16 0.2 0.24 0.28 T T ϒ ν 0.77 a L ≈ FIG. 4: MC simulation of the fourth-order modulus Υ4.(a) 0.7 shows the typical Υ4-characteristic of a KT-transition for p = 1: The well minimum moves from higher T towards the transition temperature (vertical broken line) with increasing L. Thesizeoftheminimumextrapolatestoafinitevalue. (a) (b) and (b) show that this typical KT-feature remains intact as 0.6 pisincreases tothevicinityofpc. (c)and (d)showthatthis -1 -0.5 0 0.5 1 KT-featureisdramaticallychangedintheimmediatevicinity L1/ν(T T ) ofp ,while(e)and(f)showthattheKT-featureisrecovered c c − for p-valuesabove p c FIG. 5: MC-simulation of critical size scaling for the helicity modulus Υ at the crossing point of the phase lines. (a) The orderhelicitymodulus,15 definedbytheexpansionofthe size scaling Υ ∼ L−a is obeyed to very good approximation. free energy∆F =F(δ) F(0)=Υδ2/2!+Υ δ4/4!. The The collapsing point of the data determines the critical tem- 4 KobTsetrrvaantsioitniotnh1a5tlΥea4disstfio−ntitheeacnodncnleugsaiotnivethpartecΥisemlyakaetstahne pscearlaintugreΥT=cL≈−a0F.1[6L715/.ν((Tb)−WTci)t]hwtihthe νTc=fr0o.7m7(isa)bothrneofuultl.er abrupt jump at T in order to fulfill the requirement KT of ∆F 0 at any T. Thus Υ offers a way to verify a 4 disconti≥nuous jump at a KT transition.15 to have a jump at the transition, which opens up the possibilityofacontinuousvanishingofΥandthecritical Figure4(a)forp=1isconsistentwiththetypicalKT- scalingΥ L−a. Figure5(a)showsthatsuchasizescal- features for Υ : The minimum position well approaches 4 ∼ ingisindeedobtainedclosetop 1.3479. Furthermore, T fromabove(the verticallinesinFig.4marktheex- c KT ≈ Fig.5(b)showsthatthestandardcriticalscalingformfor pected positions of the transition) and the depth of the a continuous phase transition Υ = L−aF[L1/ν(T T )] well remains finite, as L is increased, confirming the ex- c − istence ofanabruptjump ofthe helicity modulus.15 The is also valid to very good approximation which suggests that the correlation length ξ diverges as ξ T T −ν. shift of the minimum position towards lower tempera- c ∼| − | The values obtained for the critical indices are a 0.63 tures constitutes the characteristics of a KT transition ≈ and ν 0.77. and is well established up to p = 1.31 [see Fig. 4(b)]. ≈ As p is increased through the crossing region around p 1.3479 there are dramatic changes but for larger p c the≈typical KT behavior of Υ reappears [see Fig. 4(e)]. VI. SUMMARY 4 This is consistent with a crossing where a KT transition disappears and reappears as p is increased. The charac- The main result of the present work is that in general teristics close to pc 1.3479 is instead consistent with a model with the same symmetry and degrees of free- ≈ Υ4 =0 as L [see Fig. 4(c) and (d)]. dom as the 2D fully frustrated XY model can have four →∞ If Υ = 0 then the helicity modulus Υ does not need stable phases. Only three of these phases are present in 4 5 theusualFFXY model. Thenewphaseallowedbysym- ment(MOEHRD)KRF-2005-005-J11903. P.M.andS.B. metry combines unbroken chirality with quasi 2D phase acknowledgesupportfromtheSwedishResearchCouncil ordering. Theexistenceofthisphasealsomeansthatthe grant 621-2002-4135. KT and chirality phase lines cross. The crossingpoint is a criticalpoint atwhichthe helicity modulus obeysscal- ing and vanishes smoothly without a universal jump. Acknowledgments B.J.K. acknowledges the support by the Korea Re- searchFoundation Grant funded by the KoreanGovern- 1 For a recent review, see, e.g., , M. Hasenbusch, A. 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