Ashkin-Teller criticality and pseudo first-order behavior in a frustrated Ising model on the square lattice

0.29 MB·English

by  Songbo Jin

Ashkin-Teller criticality and pseudo first-order behavior
in a frustrated Ising model on the square lattice
Songbo Jin, Arnab Sen, and Anders W. Sandvik
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
(Dated: January 30, 2012)
Westudythechallengingthermalphasetransition tostripeorderinthefrustrated square-lattice
Ising model with couplings J1 <0 (nearest-neighbor, ferromagnetic) and J2 >0 (second-neighbor,
2
antiferromagnetic) for g = J2/|J1| > 1/2. Using Monte Carlo simulations and known analytical
1 ∗
results, we demonstrate Ashkin-Tellercriticality for g≥g , i.e., thecritical exponentsvary contin-
0 ∗
uously between those of the 4-state Potts model at g =g and the Ising model for g →∞. Thus,
2
stripetransitionsofferaroutetorealizing arelated class ofconformal field theorieswith conformal
n charge c = 1 and varying exponents. The transition is first-order for g < g∗ = 0.67±0.01, much
a lower than previously believed, and exhibits pseudo first-order behavior for g∗≤g.1.
J

6 PACSnumbers: 64.60.De,64.60.F-,75.10.Hk,05.70.Ln
2

] Breakingafour-foldsymmetry(Z4)atathermalphase (ferromagnetic)andJ2 >0(antiferromagnetic)isdefined
h
transition in a two-dimensional (2D) system can lead to by the Hamiltonian
c
e averyspecialkindofcriticalbehavior,wherethecritical
m exponents depend on microscopic details, in contrast to H =J1 σiσj +J2 σiσj, (1)
- the universal exponents at normal transitions. Contin- Xhiji hXhijii
t
a uously varying exponents are possible in conformal field
wherefirstandsecond(diagonal)neighborsonthesquare
t theory (CFT) when the conformal charge c ≥ 1 [1, 2].
s lattice are denoted by hiji and hhijii, respectively, and
. A well-known microscopic Z4-symmetric model realizes
t σ = ±1. A stripe state obtains at low temperature (T)
a this behavior for c = 1; the Ashkin-Teller (AT) model i
m whenthe couplingratiog =J2/|J1|>1/2. Eventhough
[3, 4] consisting of two square-lattice Ising models cou-
this model represents one of the most natural and sim-
- pledbyafour-spininteraction. AZ4-symmetricorderpa-
d ple extensions of the standard 2D Ising model, its stripe
rameter also obtains in the standardferromagnetic Ising
n transition remains highly controversial. The most basic
o modelwhensufficientlystrongfrustrating(antiferromag- questionunderdebateiswhetherthetransitioniscontin-
c netic)second-neighborcouplingsareadded,inwhichcase
uous for all g >1/2, or there is a line of first-order tran-
[ the low-temperature ordered state exhibits stripes of al-
sitionsuptoapointg∗. Thenextissueistoestablishthe
2 ternating magnetization (which can be arranged in four
universalityclassofthecontinuoustransitions. Earlynu-
v different ways on the square lattice). While c = 1 AT
mericalandanalytic approachessupported the idea that
4 criticality has been suspected at the stripe transition,
7 the transition is always continuous, but with the criti-
it has not been possible to demonstrate this convinc-
8 cal exponents changing with g. Some variational stud-
ingly for a wide range of couplings (only at very strong
5 ies [13, 14] and recent MC studies [5, 6] have, however,
0. second-neighbor coupling, where the system reduces to found a line of first-order transitions for 1/2<g .1. A
twoweaklycoupledIsingmodels[5,6]). Ithasevenbeen
1 veryrecentMCstudy[6]usedadouble-peakstructurein
difficult to determine whether the transition is contin-
1 energyhistogramstoconcludethatthetransitionisfirst-
1 uous or first-order [5–14]. A convincing demonstration
order at least up to g = 0.9. For large g the transition
: of robust c =1 criticality with varying exponents would
v should be AT-like, based on an effective continuum the-
be important, since stripe ordering can be achieved in
i oryoftwoweaklycoupledIsingmodels[6]. Forsmallerg,
X many different systems [15] and represents a more re-
the AT behaviorcouldeithercontinue downto the point
r alistic route to c = 1 and varying exponents than the
a at which the transition turns first-order (with the criti-
somewhat artificial AT model.
calend-pointofthe AT line correspondingto the 4-state
Using MonteCarlo(MC)simulationsofthe frustrated Potts model), or some other behavior that is beyond the
IsingandATmodels,alongwithknownanalyticalresults AT description may apply instead. The MC results of
for the AT model, we here demonstrate c = 1 scaling Ref. 6 could not distinguish between these scenarios.
at the stripe transition, with continuously varying expo- Weshowherethatthestripetransitionisfirstorderin
nents matching those of the AT model. We also uncover amuchsmallerrangeofcouplingsthanclaimedinRef.6;
a remarkable pseudo first-order behavior associated with for1/2<g <g∗ withg∗ ≈0.67. Forg >g∗ itiscontinu-
this type of continuous phase transition, which had not ous and in the AT class. The exponents change continu-
been previously noticed and which led to wrong conclu- ously with g as in the AT model [4], with g∗ correspond-
sions regarding the nature of the stripe transition. ingtothe4-statePottsmodel[16,17]andg →∞tostan-
The 2D frustrated Ising model with couplings J1 < 0 dard Ising universality [4]. First-order indicators (neces-
2
soarrdyerb-uptarnaomtestueffircdiiesnttr)ib,ue.tgio.,nms,ulletaipdletopeoavkesr-iensteinmeargtiyonanodf 0.20 Cmax/L 0.20 χmax/L7/4
(a) (b)
the regionof discontinuous transitions. As we will show,
0.10
this is because the 4-state Potts model and neighboring
transitions in the AT model exhibit pseudo first-order
0.10
behavior, though these transitions are known to be con- g=0.60 0.05
tinuous [4]. The Potts point had not been identified in g=0.68
g=0.80
previous works, and the pseudo first-order behavior was
4-state Potts
interpreted as actual first-order transitions. 0.05 0.02
20 50 100 200 20 50 100 200
Stripe transition.—The striped phase is characterized L L
by a two-component order parameter (m ,m ) with
x y
FIG. 1: (Color online) Peak value of the specific heat and
N N stripesusceptibilityvsLfortheJ1-J2 and4-statePottsmod-
1 1
m = σ (−1)xi, m = σ (−1)yi, (2) els. Factors corresponding the asymptotic Potts scaling have
x i y i
N N beendividedout. In(a)thecurvesthroughtheg=0.68and
Xi=1 Xi=1 Potts data are fits to the Potts form C =aLln(L/b)−3/2. A
where (x ,y ) are the coordinates of site i on an L×L log-correction should also be present in (b) but the expected
i i
periodic lattice and N = L2. We define m2 = m2 +m2 Pottsformχ=aL7/4ln(L/b)−1/8[17]isnotseenforL≤256.
x y
and the stripe susceptibility χ = N(hm2i−h|m|i2)/T.
We employ the standard single-spin Metropolis MC al- The AT Hamiltonian can be written as
gorithm and use |J1|=1 as the unit of T.
WeanalyzethepeakvalueCmax(L)ofthespecificheat H =− (σiσj +τiτj +Kσiσjτiτj), (3)
and χmax(L) of the stripe susceptibility. By standard Xhiji
finite-size scaling arguments [18], Cmax(L) ∼ Lα/ν and
χmax(L) ∼ Lγ/ν. For first-order 2D transitions these wheretwoIsingvariables,σi,τi,resideoneachsiteiand
2 arecoupledto eachotherthroughK. The ferromagnetic
quantities shouldinsteaddivergeas L . Examplesofthe
phaseoftheATmodel(hστi=6 0andhσi=±hτi)breaks
scaling behavior are shown in Fig 1. For the L ≤ 256
systems studied, the exponent α/ν, estimated from the Z4 symmetry and the order parameter can again be ex-
slope on the log-logscale, decreaseswith increasing g (it pressed as a vector (mσ,mτ) with
iscloseto2forg =0.51),whileγ/ν remainscloseto7/4
N N
fortheg-valuesdisplayedhere(alsoapproaching2closer 1 1
m = σ , m = τ . (4)
to g = 1/2). These behaviors can be affected by scaling σ N i τ N i
Xi=1 Xi=1
corrections, and the slopes for moderate sizes may not
reflect the true exponents. Fig. 1 also shows results for The relevant Z4 transitions take place when K ∈ [0,1],
the 4-state Potts model, for which it is rigorouslyknown where K =0 correspondsto two decoupled Ising models
thatα/ν =1andγ/ν =7/4,buttherearemultiplicative and K = 1 can be mapped to the 4-state Potts model.
logarithmic scaling corrections that affect the behavior The T > 0 transitions for K ∈ [0,1] are all continuous,
strongly for lattices accessible in MC simulations [17]. withtheexponentsdependingonK (butη =1/4always)
For g in the range 0.66−0.70, the J1-J2 scaling agrees [4].
well (up to factors) with that of the Potts model. Binder cumulant.—To further understand the nature
According to CFT [1, 2], the exponents can change of the phase transitions, we probe the Binder cumulant
continuously when the ordered phase breaks Z4 symme- of the stripe order parameter. For a 2-componentvector
try. Therearetwoknownmicroscopicscenarios,exempli- order-parameter,the cumulant is defined as
fied by: (i) The XY model in a four-fold anisotropy field
h4. For h4 = 0 there is a Kosterlitz-Thouless transition 1 hm4i
U =2 1− , (5)
[19, 20], while |h4| → ∞ gives standard Ising universal- (cid:18) 2hm2i2(cid:19)
ity [21]. (ii) The line of fixed points connecting the Ising
and 4-state Potts points in the square-lattice AT model where the factors are chosen to make U → 0 in the dis-
[3,4]. Assumingthatoneofthesescenariosappliestothe ordered phase and U → 1 in the ordered phase in the
continuous transitions in the J1-J2 model, γ/ν = 7/4 is thermodynamic limit. We define U for the AT model in
fixedbecausethisholdsalwaysforboth(i)and(ii). This the same way, by using m2 =m2 +m2.
σ τ
explains why χ/L7/4 is almost constant for all cases in For continuous transitions, the cumulant typically
Fig.1(b). However,theobservedvariationofα/ν withg growsmonotonicallyuponloweringT andstaysbounded
in Fig. 1(a) (and at larger g) rules out scenario (i), since within [0,1]. It approaches a step function at T as
c
α = 0 for all the fixed points there. Then the natural L → ∞ [22]. For a first-order transition, it instead
scenario left to consider is that there is a line of fixed shows a non-monotonic behavior with T for large sys-
points corresponding the AT model. tems [23], developing a negative peak which approaches
3
1.0 (a) (b)
1.0 0.00
g=0.55 g=0.70
0.5
0.0
0.5
-0.20 4-state Potts (1)
-0.5 min 4-state Potts (2)
-1.0 0.0 U g=0.60
-0.40 g=0.65
0.6 0.8 1.0 1.2 1.2 1.3 1.4 1.5 g=0.66
1.0 L=8 1.0 g=0.68
L=16 -0.60 g=0.72
L=32
0.5 K=1.0 L=64 0.5 K=0.95
0.00 0.05 0.10
1/L
0.0
0.0
(c) (d) FIG.3: (Coloronline)PeakvalueoftheBindercumulantvsL
-0.5 fortheJ1-J2 modelatdifferentgvaluesandthe4-statePotts
3.4 3.6 3.8 4.0 4.2 3.2 3.4 3.6 3.8 4.0 4.2
model. For the latter (1) in the legend refers to the original
T T
data and (2) has rescaled size, L→aL, with a≈2.36.
FIG. 2: (Color online) Binder cumulant vs temperature for
(a,b) theJ1-J2 modelat g=0.55 and0.70, and(c,d) theAT a 4-state Potts-universalpoint.
model at K =1 (the4-state Potts point) and K =0.95.
We presume that the stripe transitionis first-orderfor
g < g∗, instead of some alternative (and unlikely) ex-
Tc andgrowsnarroweranddivergesasL2 (intwodimen- otic behavior outside known scenarios for Z4 symmetry-
sions)whenL→∞. Thenonmonotonicbehaviorcanbe breaking. Thediscontinuitiesarealwaysveryweak,how-
traced to the emergence of multiple peaks in the order- ever, so that the asymptotic first-order scaling behavior
parameter distribution (reflecting phase coexistence), as cannot be observed in practice. For instance, the nega-
discussed in the context of the J1-J2 model in [24]. tive U(T) peak for g < g∗ and the specific heat appear
As is clear from Figs. 2(a,b), U(T) of the J1-J2 model to diverge muchslowerthan the expected L2 form. Sim-
indeeddevelopsanegativepeakthatgrowswithincreas- ilar behavior can also be observed for the 5-state Potts
ing L for g = 0.55 and g = 0.70. As g increases, the model, which is a well-known prototypical example of
systemsizesneededtoobserveapeakalsoincrease,indi- a weak first-order transition [16] with anomalously large
cating a weakening discontinuity of m. The dependence correlationlengthξ[25]. Toobserveclearfirst-orderscal-
of the peak value Umin on L is shown in Fig. 3 for sev- ing, one has to use lattices with L≫ξ.
eral values of g (where Umin > 0 corresponds to a local Histograms—Wenextexamineinsomemoredetailthe
minimum). Interpolating these data, we can extract a pseudo first-order signals in the AT and J1-J2 models,
length L0(g) where Umin crosses zero for given g. L0(g) using the probability distribution of the order parame-
growswithincreasingg anddivergeswheng ≈0.82(i.e., ter and the energy. Some aspects of the full distribu-
for larger g there is no negative peak). The vanishing tion P(m ,m ) were discussed in [24]. Here we consider
x y
of the negative peak might be taken as an estimate of P(m2). It is well known that phase coexistence at a
the location g∗ of the multicritical point. One could first-order transition leads to a double-peak distribution
alsoexamine the L(g)at whicha minimum in U(T)first in a narrow window (of size ∼ 1/L2 in two dimensions)
forms (but is not yet negative). This gives a still higher around T . For L→∞, the distribution approaches two
c
value of g∗. However, such procedures, or ones based on delta-functions (at m = 0 and the value m¯ of the order-
multiple peaks in the order-parameter or energy distri- parameter just below T ), with weight shifting between
c
bution, over-estimate g∗ because these features can ap- the two across the narrow T window. A double peak in
c
pear also for continuous transitions. Indeed, as shownin the energy distribution corresponds to latent heat.
Figs 2(c,d) and 3, we observe negative cumulant peaks AswouldbeexpectedbasedonthenegativeBindercu-
alsointhePottsandATmodels,inspiteofthesemodels mulants,thepseudofirst-orderbehavioralsogivesriseto
being rigorously known to have continuous transitions. doublepeaksintheorder-parameterandenergydistribu-
The most natural scenario suggested by these data is tionsclosetothetransition. ThisisshowninFigs.4(a,b)
againthattheJ1-J2 modelatapointg =g∗ corresponds for the 4-state Potts model. Similar behavior can be
to the 4-state Potts model. It therefore exhibits pseudo seen in the J1-J2 model, for which results are shown in
first-order behavior for some range of g-values above g∗, Figs.4(c,d)atg =0.67≈g∗(wherethetransitionshould
just like the AT model does for K at and slightly be- beweaklyfirst-order). InRef.[6]adouble-peakedenergy
low the Potts point K = 1. The very similar behavior histogram was seen all the way up to g = 0.9 on large
of Cmax(L) for g = 0.68 and the 4-state Potts model lattices. This was taken as a first-order transition, while
in Fig. 1(a) already suggests g∗ ≈ 0.68. Below we will our results show that this is merely pseudo first-order
presentfurther evidenceofg∗ =0.67±0.01indeedbeing behavior [26]. Combining the results, we conclude that
4
5 P(m2) (a) 20 P(E) (b) U* (a) 0.83 U* (b)
4 L=128 0.84
L=256 15 g=0.66 0.82
3 g=0.68
2 10 0.82 4-state Potts 0.81
Potts
0.80 4-state Potts
1 Potts 5
0.80 0.79
0 0
0.0 0.1 0.2 0.3 0.4 0.5 -1.60-1.55-1.50-1.45-1.40 0.78
14
4 P(m2) (c) 12 P(E) (d) 0.708.00 0.02 0.04 0.06 0.77 0.65 0.70 0.75
3 10 1/L g
8
2 FIG. 5: (Color online ) (a) Binder cumulant crossing points
6
J-J for (L,2L) system pairs. The approach to U∗(L = ∞) is
1 J1-J2 24 1 2 expectedtobegovernedbyanonuniversalscalingcorrection.
The curves show fits of the form U = a+b/Lc. (b) L→ ∞
0 0
0.0 0.2 0.4 0.6 -1.10 -1.05 -1.00 -0.95 extrapolatedU∗ oftheJ1-J2 modelcomparedwiththePotts
2 E
m result (thelines indicate the estimate with error bar).
FIG. 4: (Color online) Histograms of the order parameter
m2 andtheenergyE for(a),(b)the4-statePottsmodel(the clearwhetherUmin heredivergesveryslowlyorconverges
K =1ATmodel)and(c),(d)fortheJ1-J2modelatg=0.67. to a finite value.
HereT is veryclose toT , chosen suchthat thetwopeaksin
c Conclusions.—By combining many mutually consis-
the energy histograms are of the same height; for the Potts
tent signals, we have demonstrated a point g = g∗ =
model T/K =3.64231 for L=128 and 3.64460 for L=256,
while for the J1-J2 model T/J1 = 1.2014 for L = 128 and 0.67 ± 0.01 at which the J1-J2 model is controlled by
1.2004 for L=256. the 4-state Potts fixed point. The scenario of the criti-
cal curve for g ∈ [g∗,∞) being in one-to-one correspon-
dence with that of the AT model for K ∈ [1,0) is the
there is pseudo first-order behavior in the J1-J2 model
from the Potts point g∗ ≈0.67 up to at least g =0.9. onlypossibilityconsistentwithknownCFTscenariosand
MC results. The pseudo first-order behavior uncovered
Potts point.—If the continuous transitions indeed be-
here, along with the very weak first-order transitions for
long to the AT class for g ∈ [g∗,∞], then a way to es-
g ∈ (1/2,g∗), is what made it so difficult to correctly
timate the Potts point g∗ more precisely is to use the
characterize the nature of the transitions until now.
universal Binder crossing value U∗ (see Fig. 2) of curves
With the nature of the transition now understood, it
U(T) for different L at fixed g. For the 4-state Potts
model(theK =1ATmodel),weestimateU∗ =0.792(4) will be interesting to study other aspect of the model,
i.e., the kinetics [28, 29] at the weakly first-order and
by extracting crossing points between data for pairs
pseudo first-order transitions. We also note that many
(L,2L) and extrapolating to L = ∞. Examples of the
interesting quantum problems involve the same kind of
finite-sizescalingofcrossingpointsareshowninFig5(a).
In the J1-J2 model, U∗(g) increases monotonically with order-parametersymmetryintwodimensions,e.g.,stripe
g, as shown in Fig 5(b). We can now estimate g∗ by states in models of high-Tc superconductors [30, 31] and
equating U∗(g) to the 4-state Potts value. This gives valence-bond-solid states in quantum antiferromagnets
g∗ = 0.67±0.01, in good agreement with the Potts-like [32]. The T > 0 transitions in these systems, as well,
maybeimpactedbytheissueswehavepointedouthere.
behavior of the specific heat shown in Fig. 1(a) and the
Acknowledgments.—We thankBillKleinandSidRedner
histograms in Fig. 4.
for useful discussions. This work was supported in part
Asafurthertest,weuseamethodinspiredbytheflow-
by the NSF under Grant No. DMR-1104708.
gram technique [27]: Since the order-parameterdistribu-
tion should be universal (up to scale factors) at T for a
c
continuoustransition,andthenegativepeakinU(T)rep-
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