arX

iv:1

008.

0868

v1 [

cond

-mat

.sta

t-m

ech]

4 A

ug 2

010

New ordered phases in a class of generalized XY models

Fabio C. Poderoso, Jeferson J. Arenzon, and Yan LevinInstituto de Fısica, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil

(Dated: August 6, 2010)

It is well known that the 2d XY model exhibits an unusual infinite order phase transition whichbelongs to the Kosterlitz-Thouless (KT) universality class. Surprisingly, introduction of a nematiccoupling into the XY Hamiltonian leads to a new phase transition in the Ising universality class [1].In this paper, using a combination of extensive Monte Carlo simulations and finite size scaling, weshow that higher order harmonics lead to further ordered phases, with an even more complex phasediagrams. The new phase transitions belong to the 2d q-Potts or KT universality classes.

The low temperature behavior of two dimensional (2d)systems with continuous symmetries is controlled bytopological defects, such as vortices and domain walls.Although massless Goldstone excitations, such as spinwaves, destroy the long-range order of these systems, apseudo-long-range order with algebraically decaying cor-relation functions still remains possible. At low tem-peratures the topological defects which undermine thepseudo-long-range order are all paired up, while abovethe critical temperature these defects unbind, leading toexponentially decaying correlation functions and a lossof the pseudo-long-range order. A classical example ofsuch system is the XY model. At low temperature,the topological defects, in the form of integer valuedvertices, are joined in vertex-antivertex pairs, resultingin algebraically decaying spin-spin correlation functions.Above the critical temperature known as the Kosterlitz-Thouless transition (KT) [2, 3], these pairs unbind andthe correlation functions decay exponentially.

Unlike in 3d, for two dimensional systems the argu-ments based purely on symmetry considerations are notsufficient to fix the universality class of possible phasetransitions [4–7]. It is thus possible that two microscop-ically different systems with the same underlying sym-metries and which have the same coarse-grained Landau-Ginzburg-Wilson Hamiltonian, do not belong to the sameuniversality class. It is, therefore, natural to ask ifall the Hamiltonians invariant under the transformationθ → θ + 2π belong to the KT universality class. In thispaper we will show that in general this is not the case. Inparticular, we will study, through extensive Monte Carlosimulations and finite size scaling analysis, a large classof generalized XY models which, while preserving thesame θ → θ+2π symmetry, have very complex phase di-agram, with phase transitions belonging to the Ising andPotts universality classes, in addition to the usual KTtransition. In some of these models, transitions can beunderstood in terms of new topological defects, such asfractional vortices and domain walls [1, 8]. Apart fromthe fundamental considerations regarding the connectionbetween symmetry and universality, our purpose is to de-scribe new, previously unnoticed, ordered phases whichoccur in 2d systems with continuous symmetry.

The model considered here is a mixture of ferromag-netic and nematic-like interactions which favour parallel

and skewed spin alignment,

H = −∑

〈ij〉

[∆ cos(θi − θj) + (1−∆) cos(qθi − qθj)], (1)

where the sum is over the nearest neighbors spins on asquare lattice, 0 ≤ ∆ ≤ 1, and q is a positive integer.The first term is the usual ferromagnetic coupling (XYmodel), while the second one favors adjacent spins tohave a phase difference of 2kπ/q, where k ≤ q is an inte-ger. Independent of the value of ∆, the Hamiltonian (1)has the symmetry of the pure XY model, recovered when∆ = 1, and is invariant under rotations θj → θj+2π. For∆ = 0, we have a purely nematic-like Hamiltonian, whichis also invariant under the transformation θj → θj+2π/q.It is easy to show that in this case there will also be a KTphase transition at exactly the same critical temperatureas in the pure XY model. The low temperature phase for∆ = 0 will, therefore, have a pseudo-long-range nematic-like order. An interesting question concerns the thermo-dynamics of the model described by the Hamiltonian (1)for 0 < ∆ < 1, where both terms compete.For q = 2 the Hamiltonian, eq. (1), has been stud-

ied by many authors [1, 8–12]. In this case, the pres-ence of the second term in eq. (1) leads to metastablestates in which spins have antiparallel orientation. Themodel has new excitations not present in the ∆ = 1case: half-integer vertices connected by strings (domainwalls) [1], across which spins are anti-paralelly aligned.At low temperatures both half-integer and integer ver-tices are bound in vertex-antivertex pairs leading to apseudo-long-range ferromagnetic order. If ∆ < ∆mc, asthe temperature is raised, the string tension between half-integer vertex and antivertex pair vanishes, and the twounbind. This results in a pseudo-nematic-like order. Onfurther increase of temperature, integer vertices unbindand the system melts into a completely disordered para-magnetic phase. As expected, the transition between ne-matic and paramagnetic phases belongs to the KT uni-versality class [13]. Surprisingly, however, the transitionbetween the two pseudo-long-range ordered phases — ne-matic and ferromagnetic— is found to be in the Ising uni-versality class [1]. This behavior has been verified withsimulations performed both on square and on triangularlattices [10, 11]. On a triangular lattice the geometricfrustration introduces also a tiny chiral phase above the

2

KT line, but otherwise the phase diagram retains thesame topology as on the square lattice [11]. A relatedmodel, with a similar phase diagram, was also studied inRef. [14]. An interesting question is whether, for q > 2,the topology of the phase diagram remains unchanged.To answer this, we explore the equilibrium phase diagramof the system described by the Hamiltonian Eq. (1) forseveral values of q ranging from 2 to 10. We find that forq = 2, 3, 4 the topology of the phase diagram remains thesame as for q = 2, however, the phase transition betweenthe pseudo-ferromagnetic and pseudo-nematic phases be-longs to the q-state 2d Potts model universality class, seeFig. (1). For q ≥ 5, the topology of the phase diagramchanges completely and new phases, only present whenboth terms in Eq. (1) compete, with a pseudo-long rangeorder come into existence.The simulations were performed on a square lattice of

linear size L and periodic boundary conditions. BothMetropolis single-flip and the Wolff algorithm [15] wereused. In accordance with the symmetry of the Hamilto-nian, the possible order parameters are

mk =1

L2

∣

∣

∣

∣

∣

∑

i

eikθi

∣

∣

∣

∣

∣

(2)

where k = 1, . . . , q. The corresponding generalizedsusceptibilities are χk = βL2(〈m2

k〉 − 〈mk〉2) and the

Binder cumulants are Uk = 1 − 〈m4

k〉/3〈m2

k〉2 [16, 17].

If the transition is not KT, the usual finite size scalingcan be used to get the critical exponents β, γ, and ν:m = L−β/νf(tL1/ν) and χ = Lγ/νg(tL1/ν), where m isthe order parameter and χ its susceptibility, f and g areuniversal scaling functions, and t = T/Tc − 1 is the re-duced temperature. This finite size scaling, however, isnot valid at the KT transition for which all of the lowtemperature phase is critical and the correlation lengthand the susceptibility are infinite [3]. Nevertheless, it ispossible to show that at the KT transition and in thelow-temperature phase, the critical exponent ratios arewell defined and the order parameter and the general-ized susceptibility scale with the size of the system asm ∝ L−β/ν and χ ∝ Lγ/ν . Exactly at the transition,β/ν = 1/8 and γ/ν = 7/4, which are the same ratios asfor 2d Ising model. However, what distinguishes the KTtransition from the Ising one, is the behavior of the orderparameter and the susceptibility in the low temperaturephase where they also have finite size scaling, but withnon-universal critical exponents. Recall that for normalsecond order phase transition, finite size scaling existsonly at the critical point. This difference, can be used todistinguish the KT transition from the Ising one in thesimulations.Figure 1 shows the schematic phase diagram for q = 3,

which is topologically identical to the q = 2 case. Athigh temperatures, the equilibrium state is the disor-dered paramagnetic (P). As the temperature is lowered,the system enters either in the usual ferromagnetic (UF)or the generalized-nematic phase (N), depending on the

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

T

∆

FIG. 1: Schematic phase diagram for q = 3. There aretwo low temperature phases with a pseudo-long range or-der: generalized-nematic (three preferred spin orientations)and ferromagnetic (broken reflection symmetry). At both ex-tremes ∆ = 0 and 1, the order-disorder transition tempera-ture is Tc(∆ = 0) = Tc(∆ = 1) = 0.893. The lines are onlya guide to the eyes. Notice the existence of a multicriticalpoint around ∆mc ≃ 0.4. For ∆ < ∆mc, there are two transi-tions: paramagnetic to nematic, in the KT universality class;and nematic to ferromagnetic, in the q = 3 Potts universalityclass (as shown in Fig. 2).

value of ∆. Both these order-disorder transitions belongto the KT universality class. Up to the multicritical pointlocated at ∆ ≃ 0.4, there is a line of critical points sepa-rating the generalized-nematic from the UF phase. Theorder parameter m1 is used to distinguish between thegeneralized-nematic and ferromagnetic phases: m1 ≃ 0in the nematic phase and is ≈ 1 in the ferromagneticphase. Using finite size scaling we find that the criticalpoints along this line are in the 3-state Potts universalityclass. Fig. 2 shows the data collapse ofm1 (inset) and thecorresponding susceptibility χ1, for a critical point with∆ = 0.25. The collapse is excellent using the critical ex-ponents of the 2d, 3-state Potts model [18]: β = 1/9,γ = 13/9 and ν = 5/6.

The same topology of the phase diagram persists forq = 4, except that the transition from the generalized-nematic to the ferromagnetic phase is now in the uni-versality class of 4-state Potts model. For q ≥ 5, how-ever, the topology changes dramatically. Two new phasesemerge from the T = 0, ∆ = 0 fixed point, see Fig. 3. Thenew phases are pseudo-ferromagnetic and have a brokenreflection symmetry. We shall denote these phases as F1

and F2. The low-temperature phases F1 and F2 haveonly one preferred spin orientation, while in the phaseF2 there are four preferred spin orientations with differ-ent weights, see Figs. 3 and 4. It is interesting to notethat in the new topology there is no longer a transitionbetween the nematic and the UF phases. Fig. 4 exhibitshistograms of spin orientation across the low tempera-ture phases at fixed T = 0.16 and varying ∆, along witha pictoric representation of the possible orientations. All

3

-5 0 5 10 15

tL1/ν

0

0.1

0.2

0.3

χ 1L-γ

/ν

-4 -2 0 2 4

tL1/ν

0

0.5

1

1.5

m1L

β/ν

L = 45L = 60L = 75L = 90

FIG. 2: Ferromagnetic-Nematic transition for the q = 3 case.Finite size scaling analysis of the susceptibility χ1 and theorder parameter m1 (inset) for ∆ = 0.25, for different valuesof L. In both cases the data collapse is excellent using the3-state Potts critical exponents. The critical temperature atthis point is Tc ≃ 0.3655.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

T

∆

FIG. 3: Schematic phase diagram for q = 8. Besides theparamagnetic, the usual ferromagnetic (UF), and the nematicphases, there are two new ferromagnetic phases (F1 and F2)in which spins have half-plane preferred orientations. Foreach phase we have also pictorically depicted the possible spinorientations along with their relative weights. The lines areguides to the eye only. Notice that for ∆ 6= 0 or 1, there aremultiple transitions as the temperature changes, while thepure models (∆ = 0 and 1) have a singe one.

ferromagnetic phases have a quasi-long-range order anda broken reflection symmetry. For F2, the distributionfunction has four relevant peaks, while for F1 there isonly a single narrow peak. The usual ferromagnetic (UF)phase of XY model has a broad continuous distributionof spin orientations. The nematic phase is characterizedby eight congruent discrete spin orientations, separatedby π/4.

Besides the order-disorder KT transitions, several neworder-order transition lines appear in the q ≥ 5 phasediagrams fig. 3. The transition from the F1 to F2 phaseis well described by the m4 order parameter, which isclose to zero in the later phase. The data collapse for

P(θ

)

0

0.1

0 π/2 π 3π/2 2π

θ

∆ = 0

∆ = 0.25

∆ = 0.7

∆ = 1

FIG. 4: Angle distribution for the several phases along thefixed temperature line T = 0.16. All graphs have the samevertical scale.

-10 0 10 20 30 40

tL1/ν

0

0.1

0.2

χ 4L-γ

/ν-5 0 5 10

tL1/ν

0

0.5

1

m4L

β/ν

L = 56L = 64L = 80L = 96

FIG. 5: Data collapse for the susceptibility χ4 and the orderparameter m4 (inset) for q = 8 at ∆ = 0.5 as the systemcrosses the border between F1 and F2 phases. The data col-lapse was obtained using the critical exponents of the 2d Isingmodel, γ = 7/4, ν = 1 and β = 1/8.

several lattice sizes is shown in fig. 5 along with the datacollapse for the susceptibility χ4. Both collapses wereobtained using the set of critical exponents of 2d Isingmodel: γ = 7/4, ν = 1 and β = 1/8. The transition fromF2 to nematic is also continuous and in 2d Ising univer-sality class, as can be seen in fig. 6, where a good datacollapse of the order parameter and the correspondingsusceptibility are shown. For this transition, the relevantorder parameter is m1. Finally, the transition from F1

to UF and the transition from F2 to UF are both welldescribed by the the m8 order parameter. The two tran-sitions are found to belong to the KT universality classwith the critical exponent ratios, γ/ν = 7/4 and β/ν =1/8. Furthermore, Fig. 7 shows that both F1 and F2

are critical phases (with respect to m8 order parameter)and have the characteristic finite size scaling of a low-temperature KT phase.We have studied, through extensive numerical simula-

tion, the phase diagram of a generalized XY model in

4

-5 0 5 10 15 20 25 30 35

tL1/ν

0

0.05

0.1χ 1L

-γ/ν

0 5 10 15

tL1/ν

0

0.5

1

m1L

β/ν

L = 56L = 64L = 80L = 96

FIG. 6: Data collapse for the susceptibility χ1 and the orderparameter m1 (inset) for q = 8 at ∆ = 0.35 as the systemcrosses the border between the generalized-nematic and F2

phases at Tc ≃ 0.34. The data collapse was obtained usingthe critical exponents for the 2d Ising model, γ = 7/4, ν = 1and β = 1/8.

-0.30

-0.26

-0.22

-0.18

1.8 1.9 2 2.1 2.2 2.3 2.4

log 1

0mk

log10L

β/ν = 0.122

β/ν = 0.088

∆ = 0.6, T = 0.32∆ = 0.6, T = 0.36∆ = 0.85, T = 0.16∆ = 1, T = 0.893

FIG. 7: Scaling of the order parameter at the KT phaseboundaries (three bottonmost lines) and inside the criticalphase (topmost line) for q = 8. Notice that along the KTtransition the exponent β/ν has the Ising value (1/8), whileinside the critical phase the exponent is non universal.

which the Hamiltonian, eq. (1), has both ferromagneticand nematic-like coupling between the spins. Previousresults for q = 2 have shown that besides the usual KTtransition there is also a nematic to ferromagnetic tran-sition, belonging to the Ising universality class. We findthat for q = 3 and 4 the topology of the phase diagramremains unchanged, but the ferromagnetic-generalized-nematic transition belongs to the universality class ofthe q-state Potts model. For q = 5, the topology of thephase diagram changes dramatically and two new ferro-magnetic phases appear. After this, up to q = 10, themaximum value explored in this work, the topology of thephase diagram remains unchanged. It should be interest-ing to explore if other bifurcations in the phase diagramappear for larger values of q. One of the important con-clusions of this work is a significant lack of universalityof 2d systems. While all the Hamiltonians studied in thispaper have the same underlying symmetry, θ → θ + 2π,the transitions between the different phases belong toa variety of different universality classes. Furthermore,since the Hamiltonians discussed in this paper can bethought of as the leading orders in a Fourier expansionof a general microscopic spin-spin interaction V (θi − θj),the work raises a troubling question: How much can wereally deduce about the thermodynamics of 2d systemsfrom the form of their coarse-grained Landau-Ginzburg-Wilson (LGW) action? It is clear that the symmetry ar-guments alone are not sufficient to determine the phasediagram of these systems, and one needs to have a de-tailed knowledge of the microscopic interactions [4].

Acknowledgments

We are grateful to D. Stariolo for discussions related tothis paper. FP, JJA and YL are partially supported bythe Brazilian agency CNPq. JJA acknowledges supportfrom the INCT-Sistemas Complexos and CNPq/Prosul-490440/2007. YL would also like to acknowledge sup-port from INCT-FCx, and US-AFOSR under the grantFA9550-09-1-0283.

[1] D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541(1985).

[2] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181(1973).

[3] J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).[4] Y. Levin, Phys. Rev. Lett. 99, 228903 (2007).[5] E. Domany, M. Schick, and R. H. Swendsen, Phys. Rev.

Lett. 52, 1535 (1984).[6] R. L. C. Vinck, Phys. Rev. Lett. 98, 217801 (2007).[7] R. J. Baxter, Phys. Rev. Lett. 26, 832 (1971).[8] S. E. Korshunov, JETP 41, 263 (1985).[9] T. J. Sluckin and T. Ziman, J. Phys. France 49, 567

(1988).[10] D. B. Carpenter and J. T. Chalker, J. Phys.: Condens.

Matter 1, 4907 (1989).[11] J.-H. Park, S. Onoda, N. Nagaosa, and J. H. Han, Phys.

Rev. Lett. 101, 167202 (2008).[12] L. Komendova and R. Hlubina, Phys. Rev. B 81, 012505

(2010).[13] D. G. Barci and D. A. Stariolo, Phys. Rev. Lett. 98,

200604 (2007).[14] J. Geng and J. V. Selinger, Phys. Rev. E 80, 011707

(2009).[15] U. Wolff, Phys. Rev. Lett. 62, 361 (1989).[16] D. Loison, J. Phys.: Condens. Matter 11, L401 (1999).[17] M. Hasenbusch, J. Stat. Mech. p. P08003 (2008).[18] F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).