U(1) Emergence versus Chiral Symmetry Restoration in the Ashkin Teller Model

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by  Soumyadeep Bhattacharya

U(1) Emergence versus Chiral Symmetry Restoration in the Ashkin Teller Model
Soumyadeep Bhattacharya∗
Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
(Dated: January 13, 2016)
We show that suppression of vortices in the Ashkin Teller ferromagnet on a square lattice splits
theorder-disordertransitionandopensupanintermediatephasewherethemacroscopicsymmetry
enhancestoU(1). Whenweselectivelysuppresstheformationofnon-chiralvortices,chiralvortices
proliferate and replace the U(1) phase with a new phase where chiral symmetry is restored. This
resultdemonstratesafascinatingphenomenoninwhichthesymmetryinformationencodedintopo-
6 logical defects manifests itself in the symmetry of the phase where the defects proliferate. We also
1 show that this phenomenon can occur in all Zn ferromagnets with even values of n.
0 0 0 1 1 1 1
2
Topologicaldefectsplayacrucialroleinenhancingthe
n
a microscopic Zn symmetry of discrete ferromagnets to a 0 2 2 1 2 1
J U(1)symmetryatthemacroscopicscale. Proliferationof

2 domainwalldefectsresultsintheformationofnumerous
1 domains and the spins are able to change their orienta- 1 2 3 0 3 2

tion by arbitrary amounts over large distances in a man-
]
ch nanergusluacrhfltuhcatutatthieonmsaucnriofoscrompliycaolrodnegrapllardairmecettieornesx[h1]i.biItns FIG.1. (Coloronline)AZ4spinconfigurationisshownwith
domain walls (gray), a non-chiral vortex (blue), a non-chiral
e
m some ferromagnets, however, discrete vortices proliferate antivortex (red) and a chiral vortex (green).
simultaneously with domain walls and disorder the sys-
-
t tem before the U(1) symmetry emerges. A forced sup-
a pressionofvortices,insuchcases,candelaytheirprolifer- aU(1)symmetryisreplacedbyrestorationofchiralsym-
t
s ation and allow the intermediate U(1) phase to manifest metry,i.e. aZ2 subgroupoftheZ4 Z2 Z2 symmetry
. → ×
t itself. is restored. In order to verify that this phenomenon is
a
m On the square lattice, Zn ferromagnets with n 5 notanartifactoftheZ4 symmetry,wedemonstrateU(1)
exhibit an intermediate U(1) phase without vortex s≥up- emergence versus chiral symmetry restoration via vortex
d- pression [1–8, 28]. The direct order-disorder phase tran- suppression in the Z6 ferromagnet as well. We conclude
n sition in the Z (three state Potts) ferromagnet, on the thatthisphenomenoncanoccurinallferromagnetswith
3
o otherhand,wasrecentlyshowntosplitunderstrongsup- even n.
[c pression of vortices and an intermediate U(1) phase was In a general Zn ferromagnet, a spin si is placed at
uncovered [9]. The Z4 Ashkin Teller ferromagnet also each vertex i of a lattice Λ (a square lattice in this case)
1 undergoesadirectorder-disordertransition[10]. Thein- and each spin can be in one of n different states si
v ∈
terplaybetweenvorticesanddomainwallsatthistransi- 0,1,...,n 1 (Fig.1). Intheorderedphaseamajority
5 { − }
5 tion is known to generate a line of continuously varying of the spins take up a common state while the states
7 critical exponents [11]. A decoupling of this interplay are taken up arbitrarily in the disordered phase. The
2 and demonstration of an extended phase where a micro- vectororderparameterwhichcapturesadirecttransition
0 scopicZ symmetryenhancestoU(1)hasinterestingim- between these two phases is defined for a system of L2
01. pnleitcsat[i1o2n–4s1f4o],rqmuealtnitnugmopfhcarysesttarlanfislmitison[1s]inanadntaifdesrorropmtaiogn- sapnidnsmays =m L≡−(2m(cid:80)xi,msiny)2πwshi/enre m[6x, 8=].LT−h2e(cid:80)micaocsro2sπcsoip/inc
6
of gas particles on metal surfaces [15–19]. Can the sup- symmetry of the system is reflected in the distribution
1
: pression of vortices open up a U(1) phase in this model P(mx,my) of this order parameter.
v as well? Domain walls and vortex defects reside on the dual
Xi In this article, we show that suppression of vortices in lattice Λ(cid:48), which in this case is another square lattice
r the Z4 ferromagnet on a square lattice indeed destroys shifted from Λ by half a lattice spacing along each direc-
a tion[1,3]. IftwoneighboringspinsonΛareindissimilar
the direct order-disorder transition and results in the
states, then a domain wall is placed on the edge of Λ(cid:48)
formation of an intermediate phase with emergent U(1)
separating the two spins (Fig. 1). Each vertex i(cid:48) in Λ(cid:48) is
symmetry. Interestingly, however, U(1) emergence is not
assigned a winding number
the only possibility for the intermediate phase. Due the
egvueisnhpbaertiwtyeeonfchnirianlvtohristicmesoadneld, nwoen-acrheiraalbvleorttoicedsi.stWine- ωi(cid:48) =(∆nb,a+∆nc,b+∆nd,c+∆na,d)/n (1)
show that a selective suppression of non-chiral vortices which is essentially the finite difference equivalent of a
leavesthechiralvorticestoproliferateintheintermediate circuitintegralincontinuumspacewheneachelementary
phase. As a result, enhancement of the Z symmetry to squareplaquetteinΛischosenasacircuitandthespins
4
2
at the four corners of each plaquette are in states a, b, Non-chiral vortices can be selectively suppressed by re-
c and d when traversed in an anticlockwise sequence. A placing the second term with λnc(cid:80) ωnc .
i(cid:48)∈Λ(cid:48)| i(cid:48) |
vortex is present at i(cid:48) if ωi(cid:48) = +1 and an anti-vortex is Wehavesimulatedboththesecasesacrossawiderange
present if ωi(cid:48) = 1 (Fig. 1). of suppression strength and temperature T for different
−
∆a,b represents the difference, modulo n, between system sizes. In our simulation, spins were initialized
states a and b. More precisely, this modular difference toacompletelyorderedconfigurationandupdatedusing
is calculated as theMetropolissinglespin-flipalgorithmastheplaquette
 basedtermforvortexsuppressioncannotbeincorporated
a b n, if a b>+n/2
 − − − into cluster update algorithms [27]. The autocorrelation
∆n = a b+n, if a b n/2 (2)
a,b a−b, othe−rwis≤e − tgilmeespfionr-floibpsearlvgaobriltehs,mwsh,iwchastmenedastuorbedeqautietaeclharpgaerfaomrestiner-
−
point. Around 104 uncorrelated configurations were dis-
The asymmetry in this calculation is evident from the
carded for equilibriation following which measurements
presence of an equality condition in a b n/2. If were made on 105 uncorrelated configurations.
− ≤ −
a spin state s is viewed as two dimensional unit vector
i For each configuration, we calculated the magnetiza-
orientedatanangleθi =2πsi/n,thenthiscalculationre- (cid:113)
tion m = m2 +m2, the angle of the order parameter
stricts the angle difference to lie in ( π,+π] [3, 26]. The | | x y
−
asymmetryinducedbytheinclusionof+π andexclusion φ=arctan(m /m ),thedensityofdomainwallsρ de-
y x dw
of π is the origin of chirality in vortices. Consider, for fined as the fraction of edges in Λ(cid:48) containing a domain
exa−mple,aconfigurationofZ spinswithstates0,1,2and wall and the density for the different types of vortex de-
4
2arrangedatthecornersofasquareplaquettewhenread fects, defined as the fraction of vertices in Λ(cid:48) containing
in an anticlockwise sense (Fig. 1). The winding number a defect of the given type. ρ represents the density of
vx
for this plaquette, according to the above calculation is standard vortices with ωi(cid:48) =0, ρncvx represents the den-
(cid:54)
+1, indicating the presence of a vortex. If, on the other sityofnon-chiralvorticesandρ representsthedensity
cvx
hand,wecalculatethewindingnumberbytraversingthe of chiral vortices.
plaquette in a clockwise sense, then the winding number In the pure Z clock ferromagnet (λ = 0), the mag-
4
turns out to be zero, indicating the absence of vorticity. netization decays to zero across the direct order-disorder
As we will show, this chirality that is built into the defi- phasetransitionatT 1.1(Fig.2). Thepatternofsym-
nition of vorticity can have fascinating consequences for metrybreakinginthe∼lowtemperaturephaseiscaptured
the phase diagram of the system. by cos4φ which takes a value +1 when the Z symme-
4
In order to highlight these consequences, we define a try(cid:104)is brok(cid:105)en. Both domain walls and vortices are ob-
differenttypeofvorticitywhichdoesnotexhibitthischi- served to proliferate simultaneously near this transition,
rality. A non-chiral version of the modular difference is as indicated by an increase in their densities. When the
formulated as formation of vortices is suppressed (λ = 2), the transi-
 tion splits into two as indicated by the appearance of a
a b n, if a b>+n/2
 − − − two step decay in the magnetization. The first decay is
∆n,nc = a b+n, if a b< n/2 (3)
a,b a−b, othe−rwise − athcceosmecpoanndieodnebycotrhreespproonlidfesrtaotitohneodfisdoormdeariinngwtarlalsnswithioilne
−
driven by the proliferation of vortices. This behavior is
Thecorrespondingcalculationforthewindingnumberis
similartotheoneobtainedwithvortexsuppressioninthe
Z ferromagnet[9]. Spinconfigurationsintheintermedi-
ωnc =(∆n,nc+∆n,nc+∆n,nc+∆n,nc)/n (4) 3
i(cid:48) b,a c,b d,c a,d ate phase (Fig. 3(a)) show a few bound pairs of vortices
The configuration of Z states 0,1,2 and 2, for example, and fragmented domains typical of quasi long range or-
4
der[9]. Thisintermediatephasecanbeextendedfurther
results in zero vorticity with this definition when calcu-
by pushing the disordering transition to higher tempera-
latedeitherinclockwiseorinanticlockwisesense. Vortex
defects with ωnc = 0 are termed as non-chiral vortices. turesviastrongsuppressionofvortices(λ=100). When
The special deif(cid:48)ect(cid:54)s which have ωin(cid:48)c = 0 but ωi(cid:48) (cid:54)= 0 are tohrdeefroirnmgattriaonnsoitfiovnortviacneissihsecsoamnpdlettheelysfyosrtbeimddiesn,letfhtewdiitsh-
termed as chiral vortices.
a single transition from the Z symmetry broken phase
The formation of vortices and antivortices can be sup- 4
to the emergent U(1) phase(Fig. 2). This result is in-
pressed by raising their core energy by an amount λ [9].
terestingbecauseathermodynamicallystableemergence
With the inclusion of such a term for suppressing stan-
dardvortices,theclockHamiltonianforaZ ferromagnet of U(1) symmetry, which is usually reported only at the
n
disorderingtransitionpoint,isshowntoappearthrough-
becomes
outanextendedphaseinthiscase. However,asweshow
(cid:88) (cid:88)
= cos2π(si sj)/n+λ ωi(cid:48) (5) next, U(1) emergence is not the only possibility for the
H − − | |
(cid:104)i,j(cid:105)∈Λ i(cid:48)∈Λ(cid:48) intermediate phase.
3
λ=0 λ=2 λ=100 1
1.0 1.0 1.0 0.5
m 0
h| |i 0.5 0.5 0.5
cos4φ 0.5
h i −
1
0.0 0.0 0.0 − 1 0.5 0 0.5 1
− −
1.0 1.0 1.0 1
0.5
ρ
dw
h i
ρ 0.5 0.5 0.5 0
vx
h i
ρ 0.5
cvx
h i −
0.0 0.0 0.0 1
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 − 1 0.5 0 0.5 1
− −
T T T
FIG.2. (Coloronline)Asplitintheorder-disordertransitionandtheappearanceofanintermediatephaseisshowntooccur
withincreasingsuppressionstrengthλ. ObservablevaluescorrespondtosystemsizesL=16(diamond),L=32(triangle)and
L=64(circle). Forλ=100,vorticesarenearlyabsentandonlythedomainwallsarelefttoproliferate. Theorderparameter
distribution P(mx,my) for a 322 system at this λ shows Z4 symmetry breaking in the ordered phase at T = 1.1 (top right)
and enhancement to U(1) symmetry at T =3.0 (bottom right).
liferation of non-chiral vortices is observed to weaken.
The crucial difference in the pattern of symmetry break-
ingonthehightemperaturesideiscapturedby cos4φ ,
(cid:104) (cid:105)
which changes its sign to a negative value. This repre-
sentstherestorationofZ chiralsymmetryandindicates
2
a transition from order to incomplete order [20]. Exam-
ples of similar Z symmetry restored phases are the σ
2
(cid:104) (cid:105)
phase of the Ashkin Teller model obtained beyond the
clock point in three dimensions [21] and the disordered
flat phase in restricted solid-on-solid models of crystal
(a) (b)
growth[22]. Typicalspinconfigurationsobtainedforthis
phase clearly reveal the proliferation of chiral vortices
FIG. 3. (Color online) Typical spin configurations for a 482 (Fig. 3(b)). Interestingly, the full Z symmetry is never
n
systemshow(a)numerousdomainsandafewvorticesinthe
restored at any value of temperature above the chiral
emergent U(1) phase when standard vortices are suppressed,
symmetry restoring transition. Instead, cos4φ appears
and(b)proliferationofchiralvorticesinthechiralsymmetry
(cid:104) (cid:105)
restored phase when non-chiral vortices are suppressed. Chi- tobecomemorenegativewithincreasingL(Fig.4),indi-
ralvortices(greendots)areoverlaidonthestandardvortices catingthepersistenceofathermodynamicallystablechi-
(redandblue)in(a)andshownalongsidenon-chiralvortices ral symmetry restored phase even at high temperatures.
in(b). BothconfigurationsweregeneratedatT =2.5witha This result suggests that proliferation of non-chiral vor-
suppression value of 4.
ticesisnotstrongenoughtodriveadisorderingtransition
andaccompaniesagradualcrossovertodisorderinstead.
We return to the pure Z ferromagnet and begin a se- The restoration of a Z symmetry raises an important
4 2
lectivesuppressionofnon-chiralvorticesusingaλncterm question: is this phenomenon a special feature restricted
for (5). When λnc = 0, both chiral and non-chiral vor- to the Z ferromagnet, possibly because it can be de-
4
tices proliferate simultaneously accompanied by the pro- composedintoaZ Z model? Whiledefiningthenon-
2 2
×
liferationofdomainwallsacrossthedirectorder-disorder chiral vortices, we had not restricted ourselves to n=4.
transition(Fig.4). Whenthenon-chiralvorticesaresup- The distinction between chiral and non-chiral vortices
pressed (λnc = 2), the transition appears to split again. was made possible using the asymmetry of modular dif-
In this case, however, the proliferation of domain walls ferencesfortheparticularcasewhen∆ = n/2in(3).
a,b
−
is closely followed by the proliferation of chiral vortices. Suchcasesappearwhenniseven. Itshould,therefore,be
With increasing λnc, the proliferation of both domain possibleto verifywhetherthe replacement ofU(1)emer-
walls and chiral vortices remains unchanged while pro- gence by chiral symmetry restoration occurs in models
4
λnc =0 λnc =2 λnc =100 1
1.0 1.0 1.0 0.5
m 0
h| |i
cos4φ 0.0 0.0 0.0 0.5
h i −
1
− 1 0.5 0 0.5 1
1.0 1.0 1.0 − −
− − −
1.0 1.0 1.0 1
0.5
ρ
dw
h i
ρ 0.5 0.5 0.5 0
ncvx
h i
ρ 0.5
cvx
h i −
0.0 0.0 0.0 1
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 − 1 0.5 0 0.5 1
− −
T T T
FIG. 4. (Color online) The order-disorder transition is replaced by a chiral symmetry restoring transition when non-chiral
vortices are suppressed using increasing values of λnc. The Z4 symmetry breaking at T =1.1 in the ordered phase (top right)
and chiral symmetry restoration at T = 3.0 (bottom right) are clearly reflected in the order parameter distribution obtained
for a 322 system with λnc =100. System sizes used to calculate observable statistics are the same as those in Fig. 2.
h|m|i hcos6φi hρcvxi This model exhibits a narrow intermediate U(1) phase
1.0 1.0 even without the suppression of vortices [3, 5, 8]. When
we suppress the formation of the standard vortices, the
U(1)phaseisobservedtoextendfurtheruptohighertem-
0.0 0.0
peratures and chiral vortices are nearly absent in the
λ=0 λnc=0 phase (Fig. 5). On suppression of non-chiral vortices,
1.0 1.0
− − however, the chiral vortices are left behind to prolifer-
1.0 1.0
ate and they drive a phase transition from the ordered
phase to a chiral symmetry restored phase. The symme-
0.0 0.0 tryrestorationisclearlyreflectedintheorderparameter
distribution and in cos6φ which changes sign across
λ=100 λnc=100 (cid:104) (cid:105)
1.0 1.0 the transition. This result confirms that chiral symme-
− 0.0 1.0 2T.0 3.0 4.0 − 0.0 1.0 2T.0 3.0 4.0 try restoration is realizable in general Zn ferromagnets
(a) (b) with even values of n.
Thesubtledifferencebetweentheproliferationofstan-
(c) (d) (e) (f)
dard vortices, which contains both chiral and non-chiral
FIG. 5. (Color online) The emergent U(1) phase in the Z6 defects, and the proliferation of chiral defects highlights
clock ferromagnet is (a) shown to extend upon suppression the crucial role played by these topological defects in de-
of standard vortices, and (b) shown to get replaced by a chi- termining the phase diagram of Zn ferromagnets. Our
ral symmetry restored phase upon suppression of non-chiral result can have interesting implications for phase tran-
vortices. TheorderparameterdistributionshowsthattheZ6 sitions in a variety of other systems like superfluids and
symmetryisbrokenintheorderedphaseinbothcasesofsup-
superconductors which are also governed by the prolifer-
pression(c)and(e),butgetsenhancedtoU(1)intheformer
ation of vortex defects [23, 24]. Moreover, our demon-
(d), and replaced by chiral symmetry restoration in the later
stration provides a simple example of how the symme-
(f).
try information contained inside topological defects can
determine the symmetry of the phase in which they pro-
liferate. It would, therefore, be interesting to apply our
with higher even values of n as well.
techniqueandopenupricherphasediagramsforsystems
We have simulated the Z clock ferromagnet on the likeliquidcrystals[25]inwhichthedefectsareassociated
6
square lattice with different types of vortex suppression. with higher symmetries.
5
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