Kosterlitz-Thouless and Potts transitions in a generalized XY model

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by  Gabriel A. Canova

Kosterlitz-Thouless and Potts transitions in a generalized XY model
Gabriel A. Canova,1 Yan Levin,1 and Jeferson J. Arenzon1
1Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil
(Dated: January 20, 2014)
We present extensive numerical simulations of a generalized XY model with nematic-like terms
recently proposed by Poderoso et al [PRL 106(2011)067202]. Using finite size scaling and focusing
on the q = 3 case, we locate the transitions between the paramagnetic (P), the nematic-like (N)
4 andtheferromagnetic(F)phases. Theresultsarecomparedwiththerecentlyderivedlowerbounds
1 for the P-N and P-F transitions. While the P-N transition is found to be very close to the lower
0
bound,theP-Ftransition occurssignificantlyabovethebound. Finally,thetransition betweenthe
2
nematic-likeandtheferromagneticphasesisfoundtobelongtothe3-statesPottsuniversalityclass.

n
a
J I. INTRODUCTION istsfor allvalues of∆andextends downto zerotemper-

7 ature (because parallelspins minimize both terms of the
1 Twodimensionalsystemswithshort-rangeinteractions Hamiltonian,while nematic-likeorderingminimizes only
can not break continuous symmetry. This is the reason thesecondterm). Forsmallvaluesof∆,thereisaninter-
]
h why neither true crystals, ferromagnets,or nematics can mediate temperature phase with nematic-like (N) quasi
c exist in 2D [1]. Nevertheless, at low temperatures these long-rangeorder. A secondorder phase transitionline is
e systems can exhibit a quasi-long range order, with the found to separateF and N phases,ending in a multicrit-
m
correlation functions decaying with distance as a power ical point at ∆mult. Only recently the phase diagramfor
- law. The transition between the “ordered” and disor- q = 2 has been precisely obtained [19]. In a recent pa-
at dered phases is often found to be of infinite order and per,Poderosoetalhavestudied[21]theq-nematicNtoF
st to belong to the Kosterlitz-Thouless (KT) universality transitionfortheq =3model. Thistransitionwasfound
. class [2, 3]. The transition is driven by the unbinding tobelong tothe 3-statesPottsuniversalityclass. The P-
t
a of topological charges (vortices). At low temperature N and P-F transitions were expected to belong to the
m the chargesareboundindipolarvortex-antivortexpairs, KT class, however, neither the location nor the univer-
- while at high temperature the vortices unbind and lead sality class of these transitions was precisely determined
d
to the destruction of the quasi long-range order. Unlike in Ref. [21]. Using heuristic arguments Korshunov [25],
n
o the usual thermodynamic phases, the low temperature suggested that for q > 2 a q-nematic phase is impossi-
c KT phase is critical for all temperatures and is charac- ble[35]. This,however,clearlycontradictedtheresultsof
[ terized by a discontinuous jump of the helicity modulus, the simulations of Poderoso et al [21]. Furthermore, the

whichisthe orderparameterthatmeasureshowthe sys- mapping between ∆ = 0 and 1 shows that the N phase
1
v tem responds to a global twist [4–6]. exists at least for ∆=0. Using Ginibre’s inequality [27]
2 A generalization of the XY model, including nematic- it can then be shown [20] that the N-P transition must
4 like terms, has been recently introduced and studied by alsoextendtofinite∆. Ginibre’sinequalityalsoprovides
4 several authors for both q = 2 [7–19] and integers q > a rigorous lower bound [20] for the transition tempera-
4 2 [20, 21], ture between P and N phases, TKT(∆)≥(1−∆)TKT(0)
1. for ∆ < ∆mult. For ∆ > ∆mult, the KT transition is
0 H=− [∆cos(θ −θ )+(1−∆)cos(qθ −qθ )], (1) betweenPandFphases[20],withthelowerboundgiven
i j i j
4 Xhiji by TKT(∆) ≥ TKT(0)∆. Since at very low temperature
1 the system must be in the F phase, this proves the ex-
v: with0≤∆≤1,0≤θi ≤2πandnearestneighborsinter- istence of all three phases for small, but finite ∆. The
i actions. Similar models have recentlybeen consideredin objectiveofthe presentworkisto preciselycalculatethe
X the contexts of collective motion of active nematics [22] phase diagram for the q = 3 generalized XY model and
r and Hamiltonianmean field models [23]. For ∆=1, one tocomparethecriticaltemperaturesfortheP-NandP-F
a
recovers the usual XY model with the critical tempera- transitions with the bounds obtained in Ref. [20]. More-
ture TKT(1) ≃ 0.893. Changing variables in the parti- over, since the very existence of the N-P transition has
tion function, qθi → θ¯i, shows that the ∆ = 0 model is beencontestedforq >2[25],itisimportanttopresenta
also isomorphic to the XY model with the same critical broadsetofsolidevidencessupportingsuchatransition.
temperature (see also Ref. [24]), T (0) = T (1). In
KT KT
between, when0<∆<1, Eq.(1) describes the competi-
tionbetweendirectionalandnematic-likealignment(i.e.,
II. SIMULATIONS
2kπ/q with integer k ≤ q), with a line of critical points
T (∆). Ingeneral,besides the hightemperature, para-
KT
magnetic (P) phase, there are at least two other phases The simulations were performed on a square lattice of
thatareextensionsofthe phasesoccurringat∆=0and linear size L and periodic boundary conditions. Both
∆ = 1. A quasi long-range ferromagnetic (F) phase ex- Metropolis single-flip and Wolff cluster algorithms [28]
2
were used. The phase transitions are characterized by has an essential singularity, the correlation length grows
observables such as the generalized magnetizations and exponentially, and this FSS is no longer valid. Further-
the corresponding susceptibilities, more, the whole low temperature phase is critical and
both the correlationlength and the susceptibility are in-
1 finite in the thermodynamic limit [3, 32]. Nevertheless,
m = exp(ikθ ) (2)
k L2 (cid:12)(cid:12)(cid:12)Xi i (cid:12)(cid:12)(cid:12) ftoiorsaallretewmepllerdaetfiunreeds Tan≤d tThKeTm, tahgenectriiztaictaiolnexapnodntehnet raas--
χk =βL2(cid:12)(cid:12)(hm2ki−hmki(cid:12)(cid:12)2) (3) sociated susceptibility scale, respectively, as m ∝ L−β/ν
and χ ∝ Lγ/ν. Exactly at the transition, β/ν = 1/8
where k =1,...,q, and the Binder cumulants [29, 30],
and γ/ν = 7/4, which are the same ratios as for the 2d
hm2i2 Isingmodel. Belowthephasetransitiontemperaturethe
U = k . (4)
k hm4i critical exponents are non-universal.
k
Since there is no long-range order in 2d, the observables 1
m are not, strictly speaking, the order parameters for
k
the phase transition. The order parameter for KT tran- 0.6
sition is the helicity modulus Υ which, in the thermody- m m
1
m
namic limit, is zero in the disordered phase and remains 3
0.2
finite in the ordered phase. It is defined as the response
uponasmall,globaltwistalongoneparticulardirection.
6 L=64
Following Ref. [31], it can be written as Υ =e−L2βs2, 128
where e ≡ L−2 U′′(φ) and s ≡ L−2 U′ (φ) 4 256
hijix ij hijix ij i
(the sum is ovePr the nearest neighbors aloPng the hori- Υ 512
h 2 1024
zontal direction), φ=θ −θ and U (φ) is the potential
i j ij
between spins i and j. For the Hamiltonian Eq. (1) [19],
0
0.3 0.5 0.7 0.9
1
Υ= ∆cosφ+q2(1−∆)cos(qφ)
L2 T
hXijix(cid:2) (cid:3)
2
β FIG. 1: (Top) Average magnetization m1 and m3 [21] for
−  [∆sinφ+q(1−∆)sin(qφ)] . (5) ∆ = 0.25 and several system sizes L showing the N-F phase
L2
hXijix transitions at TPotts ≃ 0.36 and P-N transition at TKT ≃
  0.67. (Bottom) Average helicity modulus Υ versus T. The
For the XY model with ∆ = 1, the critical tempera- crossing of the helicity with the line 18T/π (see text for an
ture is determined by the condition Υ(TKT) = 2TKT/π explanation)atTKT(L)gives,forL→∞,TKT ≃0.67. Notice
[4, 5]. The isomorphism between the ferromagnetic XY also that the system sizes used here are considerably larger
model with ∆ = 1 and a purely q-nematic model with than those in Ref. [21].
∆=0requiresthatthe criticaltemperaturemustbe the
same for both models. The helicity modulus for the q- Results forthe averagemagnetizationsm andm are
1 3
nematic, however, contains an extra factor of q2 which shown in Fig. 1 (top) for ∆ = 1/4. For this ∆ the crit-
must be accounted for in the condition for criticality. ical temperature for the N-F transition was found to be
Following Ref. [19] the location of the KT transition T (0.25) ≃ 0.365 [21] at which m drops to very low
Potts 1
will be determined by the asymptotic crossing point of values. On the other hand, the nematic magnetization
Υ and the line 2TKT/λ2π. The factor λ is related to the m3 clearly shows the N-P transition. At the transition
charge of the topological excitation. For the F-P transi- temperature T , m behaves as m (T ) ∼ L−0.128
KT 3 3 KT
tion (q = 1), λ = 1 and the transition is driven by the (notshown). TheexponentisveryclosetotheKTvalue,
unbinding of integer vortices. For q = 2, λ = 1/2 in β/ν = 1/8. In the bottom part of Fig. 1, the averaged
the N phase, correspondingto half-integer vortices (con- dataforthehelicitymodulusΥ[4]for∆=0.25isshown
nected by domain walls). In general, λ = 1/q for the along with the line 18T/π. Following Refs. [19, 33], we
q-nematic-paramagnetic transition. firstfitthe helicitymodulusdataforafixedtemperature
For the N-F transition the usual finite size scaling assumingalogarithmicapproachtoitsasymptoticvalue:
(FSS) analysis provides the critical exponents β,γ, and
ν: m = L−β/νf(tL1/ν) and χ = Lγ/νg(tL1/ν), where m 2TA 1 1
Υ (L)= 1+ , (6)
is the order parameter and χ is its susceptibility, f and fit π (cid:18) 2logL+C(cid:19)
g are the scaling functions, and t = T/T −1 is the re-
c
duced temperature. Indeed, for q = 3 this transition is where A and C are fitting parameters. The constant A
in the universality class of the 3-states Potts model with is related to the vorticity of the system and is expected
ν = 5/6, β = 1/9 and γ = 13/9, and has been stud- to be A = 1/λ2 at the transition. This form, valid at
ied in detail in Ref. [21]. However, the KT transition T , was inspired by renormalizationgroupcalculations
KT
3
and was observed to be valid even for very small sys- maxima, χmax(∆), all points fall on the same universal
k
tems [33]. Away from T , the above expression is no curveclosetothe criticalregion(the line,aparabolicfit,
KT
longer valid, the data deviates from it and the fitting is just a guide to the eyes). A probable explanation is
error increases. Indeed, after repeating the process for that,despitecorrespondingtotransitionsbetweendiffer-
several temperatures close to the transition, the critical ent phases, P-N and P-F, they are all in the KT univer-
temperature corresponds to the one that minimizes the sality class.
normalized quadratic error,
2 1
hΥ(T,L )i−Υ (T,L )
i fit i
ε= , (7)
(cid:18) σ(T,L ) (cid:19)
Xi i 0.8
2
−
withσ(T,Li)= hΥ2i−hΥi2. Followingthisprocedure ηL 0.6
we obtain, for ∆p= 0.25, TKT ≃ 0.671 and A ≃ 8.97 )
x L=96
(λ ≃ 1/3). Within error bars, this value is the same mak 128
as the lower bound. For the F-P transition one expects /χ 0.4 160
λ = 1 and, repeating the same procedure one gets, for k 256
χ
384
∆=0.7, A≃0.99 and TKT ≃0.748. ( 0.2 512
768
L=256 1024
9000 ) 384 0
T 8 0.5 0.6 0.7 0.8 0.9 1
K 512
(T3 6 γ/ν 1706284 Uk
χ
) 6000 ln 4 FIG. 3: Collapse of therescaled susceptibility vs. theBinder
(T 4 5 6 7 cumulant, Uk = hm2ki2/hm4ki, for several values of ∆. With
χ3 lnL k=3wehave∆=0.1and0.25and,withk=1,∆=0.6,0.7,
3000 0.9 and 1. The linear size ranges from L = 96 to 1024. The
collapse is obtained with the KT value of the exponent, η =
1/4. In addition, by rescaling the curves with the maximum
valueofeachsusceptibility,χmkax(∆),theresultsforbothP-N
0 and P-F transitions all collapse on the same universal curve.
0.5 0.6 0.7 0.8
The solid line, showing a small deviation from the parabolic
T behavior, is just a guide to theeyes.
FIG.2: Thesusceptibilityχ3(T)associatedwithm3 nearthe The information above can be used to construct the
KT transition for ∆ = 0.25. Notice that χ3 grows with the phasediagramfortheq =3generalizedXYmodelshown
system size even far below the critical temperature. At the
transition,thepeakincreasesasχ3(TKT)∼L1.755,wherethe in Fig. 5. Remarkably, the transition line N-P is very
close to the lower bound calculated in Ref. [20] (except,
exponent was obtained from thefit shown in theinset.
close to the multicritical point). The F-P line, on the
other hand, is wellabove the lowerbound and, as a con-
Fig. 2 shows that the susceptibility χ diverges for
3 sequence, the multicritical point is located away from
all temperatures below T as L → ∞. The exponent
KT ∆ = 0.5. This phase diagram is qualitatively similar to
is clearly non universal and is larger in the critical re-
theoneobtainedusingasimplemean-fieldanalysisofthe
gion. For a KT phase transition, for increasing L, one
Hamiltonian Eq. (1). Within the mean-field approxima-
expects χ (T ) ∼ Lγ/ν, with γ/ν = 7/4. Indeed, we
3 KT tion allthe spins are connected andthe Mermin-Wagner
find χ (T ) ∼ L1.755, as can be seen in the inset of
3 KT theorem does not apply. The magnetizations, m and
Fig.2. Furtherevidenceofthetransitioncanbeobtained 1
m , become the true orderparameters,and are found to
from the Binder cumulant, Eq. (4), even without the 3
satisfy a set of coupled equations
knowledge of the critical temperature. Near the phase
transition the Binder cumulant scales as U = h(L/ξ),
I (2βm ∆,2βm (1−∆))
1 1 3
where h(x) is a scaling function, and ξ is the correlation m = (8)
1
I (2βm ∆,2βm (1−∆))
length. On the other hand χ = L2−ηg(L/ξ), so that 0 1 3
3
χ3Lη−2 = g[h−1(U)]. Therefore, by plotting χ3Lη−2 m = I3(2βm1∆,2βm3(1−∆)), (9)
3
(with η = 1/4 for KT transition) vs. the Binder cu- I (2βm ∆,2βm (1−∆))
0 1 3
mulant [29], all the susceptibilities for different system
sizes and temperatures should collapse onto one univer- where β = 1/T, I1(x,y) = ∂I0(x,y)/∂x, I3(x,y) =
sal curve. This is precisely what is found in our simula- ∂I0(x,y)/∂y and
tions. Fig. 3 shows the data collapse for U and χ for
1 1
severalvaluesof∆<0.5andU andχ for∆>0.5. Af- 1 2π
3 3 I (x,y)= dθexp[xcosθ+ycos(3θ)]. (10)
ter rescaling the collapsed curves by the height of their 0 2π Z
0
4
1
9.1 ∆=0.25 P
0.8
9
4
A 0.6
ε 2 T N
g
8.9 o
l 0 0.4
0.67 0.672
F
T
8.8 0.2
0.668 0.67 0.672
2d
T
1.04 0
2 0 0.2 0.4 0.6 0.8 1
ε 1 ∆
g
1.02 o
l 0
FIG. 5: Phase diagram for q =3. There are two KT phases,
0.747 0.749
A T N and F, both with quasi long-range order. The points were
1
obtained using the helicity modulus, while the lines are only
guides to the eye. The transition between N and F phases is
0.98 ∆=0.7 in the 3-states Potts universality class. The dashed lines are
the lower bounds for the order-disorder transitions obtained
in Ref. [20], T = TKT(0)(1−∆) and T = TKT(0)∆ for ∆ ≤
0.746 0.748 0.75
0.5 and ∆ ≥ 0.5, respectively, with a multicritical point at
T
∆mult =1/2 and TKT(1/2)=TKT(0)/2.
FIG.4: FitparameterAofthehelicitymodulusfor∆=0.25
1
(top) and 0.7 (bottom). Notice that A crosses the value 9
(top), that is, q2, and 1 (bottom) very close to the tempera-
P
turein which thefittingerror is minimum. 0.8
0.6
N
T
The free energy density is
0.4
F
f =−T lnI0(2βm1∆,2βm3(1−∆))+m21∆+(1−∆)m23. 0.2
(11) MF
Athightemperaturesthereisaparamagneticphasewith
0
m1 = m3 = 0. The phase transition between the P and 0 0.2 0.4 0.6 0.8 1
FphasesoccursatTc =∆for∆≥0.5andthetransition ∆
between the P and N phases happens at T = 1−∆ for
c
∆≤0.5,seeFig.6. Bothtransitionsareofsecondorder. FIG. 6: Mean field phase diagram for q = 3 generalized XY
Within the N phase, m is identically zero, while the model. The P-N and P-F transitions phase are continuous
1
nematic ordervanishes asm ∼(T −T)1/2 asthe phase (solid lines) while the transition N-F (dashed line) is of first
3 c
transition line is approached from below. On the other order.
hand, inside the F phase the order parameters vanish as
m ∼ (T − T)1/2 and m ∼ (T − T)3/2 as the F-P
1 c 3 c
III. DISCUSSION AND CONCLUSIONS
phase boundary is approached from below. Notice that
the critical temperature for ∆ = 0 (and 1) is T = 1
and differ from the (smaller) KT value. As expected In conclusion, we have studied, through extensive
the fluctuations decrease the critical temperature. At Monte Carlo simulations and FSS, a generalized XY
lower temperatures, and 0 ≤ ∆ ≤ 0.5, there is a second model with q = 3. Contrary to the early doubts re-
transition at which m jumps discontinuously from 0 to garding the existence of a nematic-like (N) phase in this
1
a finite value, corresponding to a first order transition model, as opposed to a simple crossover [25], we have
between the N and F phases. Thus, although the nature presented strong numerical evidence that the model has
of the phase transitions is not correctly captured by the P, N, and F phases. Curiously, the boundary between
meanfieldtheory,boththetopologyofthephasediagram P and N phases coincides closely with the lower bound
and the fact that the transition N-F is not in the same obtainedin Ref. [20] while, on the other hand, the phase
universalityclassasthetransitionsbetweenP-NandP-F transitionbetweenFandPphaseslieswellaboveit. The
phases is correctly predicted. transitionbetweentheNandFphases,whichforq =2is
5
inthe2dIsinguniversalityclass,forq =3belongstothe this model in 3d. In 2d it is important to explore the
3 states Potts universality class. The overalltopology of phase diagrams for larger q values, since there are indi-
theq =2and3casesareverysimilarandasimplemean- cationsofthechangeintopologythatoccursforlargerq.
fieldanalysisiscapabletograsptheexistence(albeitnot In particular, it was observed [21] that, differently from
the actual nature) of each phase. q = 2 and 3, new phases appear for q = 8. A detailed
With the possible exception of the region around the explorationof the nature of these phases is in order, not
multicritical point, the transition line N-P is very close onlyforq =8butforintermediatevaluesaswell. Finally,
to the lower bound (indicated in Fig. 6 by the dotted it has recently been proposed [18, 34], that for q = 2,
line) predicted by Romano [20]. That is, the declivity of there may exist a region close to the multicritical point
T (∆), within our precision,is −1. Onthe otherhand, in which the transition to the paramagnetic phase is not
KT
thelineF-Phasaslopesmallerthanunityandislocated KT,butIsing. TheveryexistenceofthisIsingtransition
far above the lower bound. As a consequence, the mul- region is still an open question [19] (and thus care must
ticritical point at which both lines merge is not located be taken with the use of the term multicritical). It will
at ∆=1/2 and is above the lower bound T /2. These be interesting to see whether this topology might also
KT
featurescanbeobservedinthephasediagramFig.5and extend to the larger q values.
in the diagram for q = 2 [19] as well. Moreover, for dif-
ferentvaluesofq,themulticriticalpoint∆(q) islocated
mult
atincreasingvaluesof∆: ∆(2) ≃0.32[19],∆(3) ≃0.4. Acknowledgments
mult mult
For q = 8, we do not yet have a precise location, but it
appears to be close to ∆ = 0.5. We observe that these We thank S. Romano for bringing Ref. [20] to our at-
points tend to the multicriticalpoint consistentwith the tentionandF.C.Poderosofor collaboratingatthe early
lower bound calculated by Romano [20], ∆ = 1/2. stages of this project. JJA acknowledges the warm hos-
mult
Thus, the F-P transition approaches, as q increases, the pitality of the LPTHE-Jussieu in Paris during his stay
lowerboundT (∆)=T ∆(or,possibly,theremayexist where part of this work was done and a discussion with
c KT
a critical value of q above which the lower bound might F. Zamponi on the mean field approach for this model.
be exact). ThisworkwassupportedbyCNPq,CAPES,FAPERGS,
There are several possible future extensions of this INCT-SC, INCT-FCx, and by US-AFOSR under the
work. We are presently performing a detailed study of grant FA9550-12-1-0438.
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