Aspects of topological actions on the lattice

0.32 MB·

by  Oscar Akerlund

CERN-PH-TH-2015-309
Aspects of topological actions on the lattice
6
1 Oscar Åkerlund
0 InstitutfürTheoretischePhysik,ETHZurich,CH-8093Zürich,Switzerland
2
E-mail: [email protected]

y
a Philippe de Forcrand
∗
M
InstitutfürTheoretischePhysik,ETHZurich,CH-8093Zürich,Switzerland
1 CERN,PhysicsDepartment,THUnit,CH-1211Geneva23,Switzerland
2 E-mail: [email protected]

]
t
a Weconsideralatticeactionwhichforbidslargefields,andwhichremainsinvariantundersmooth
l
- deformations of the field. Such a “topological” action depends on one parameter, the field cut-
p
e off, but does not have a classical continuum limit as this cutoff approaches zero. We study the
h
properties of such an action in 4d compactU(1) lattice gauge theory, and compare them with
[

those of the Wilson action. In both cases, we find a weakly first-order transition separating a
2
v confining phase where monopoles condense, and a Coulomb phase where monopoles are expo-
5
nentiallysuppressed. Wealsofindadifferent,criticalvalueofthefieldcutoffwheremonopoles
0
9 completely disappear. Finally, we show that a topological action simplifies the measurement of
3
thefreeenergy.
0
.
1
0
6
1
:
v
i
X
r
a
The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Topologicalactions PhilippedeForcrand
1. Introduction
Atleasttwostrategiescanbeadoptedtoimprovetheapproachofalatticeactiontothecontin-
uumlimit. ThemostwidelyconsideredistheSymanzikstrategy[1],whereirrelevantoperatorsof
increasingdimensionareaddedtothestandardaction,withcoefficientsdeterminedperturbatively
or not. Another, less common, strategy is motivated by the observation that the approach to the
continuumbehaviorisfasterforweakerfields: suppressingstrongfieldswillimprovetheactionin
a non-parametric way. Indeed, the Manton action [2], or renormalization-group improved actions
likethe“perfect”action[3]allsuppresslargefields.
Aradicalimplementationofthissecondstrategyconsistsofimposingacutoffontheamountof
localexcitation,theearliestexamplebeingthepositive-plaquetteaction[4]. Recently,acombina-
tionofWilson-typeactionandcutoffconstrainthasbeenshowntodisplaymuchimprovedscaling
inaspinmodel[5]. Veryrecently,asimilarapproachhasbeenappliedtoagaugetheory[6].
This leaves open the question of the approach to the continuum limit of a constraint-only
action,namelyanactionSsuchthatexp( S)=1ifeverylocalconstraintissatisfied,0otherwise.
−
Such an action, which is invariant under small deformations of the field, is for this reason called
“topological”. While it is intuitively clear that the correlation length will keep increasing as the
constraint selects weaker and weaker fields, it is not at all obvious which continuum theory will
be approached: a gradient expansion of the lattice action is not possible, since the action is either
zero or infinite. Nevertheless, it has been shown numerically [7] that in a 2d XY spin model, the
sameCoulombphaseisobtainedaswiththestandardaction. Here,westudythecaseofa4dU(1)
gaugetheory,andcomparetheWilsonactionSW = β∑PcosθP withthetopologicalaction
−
exp( S )=∏Θ(δ θ ) (1.1)
top P
− −| |
P
whereθ istheangleofplaquettePandδ [0,π]isthecutoffonθ . Mostoftheresultspresented
P P
∈
herehaveappearedin[8].
2. Monopolesandergodicity
Itiswell-knownthat,usingtheWilsonaction,the4dU(1)latticegaugetheoryhastwophases:
Coulombatweakcoupling,confiningatstrongcoupling,withaweakfirst-ordertransitionatβ
c
≈
1.01 . Moreover, in the confining phase magnetic monopoles condense [9]. With a topological
···
action, the existence of monopoles is governed by the cutoff angle δ. As shown Fig. 1 (left), a
totalfluxof2π throughthe6facesofacube,whichcharacterizesamonopole,requiresδ tobeat
least π/3. Thus the monopole density vanishes as δ π+, and is strictly zero for δ < π. This
→ 3 3
non-analyticityofthemonopoledensitycausesaphasetransitionatδ =π/3,whichonemayguess
coincideswiththeCoulomb-confiningtransition. Thisguessiswrongaswewillshowbelow.
The numerical study of the regime δ <π requires care. As shown Fig. 1 (middle), the usual
single-link update will not succeed in changing the monopole flux by 2π, and thus is not ergodic.
Tomaintainergodicityrequiresupdatingatleasttwolinkssimultaneously,asshownFig.1(right).
2
π π
±
Topologicalactions PhilippedeForcrand
π
±
π π π
±
Figure 1: Left: A magnetic monopole is identified via its 2π magnetic flux exiting an elementary cube.
Thus,itrequiressomeplaquetteanglestoreachorexceeπdπ/3. Iftheactionforbidssuchplaquetteangles,
±
monopoles are absent. Middle: Creating or destroying a monopole requires changing the magnetic flux
π
exiting an elementary cube by 2π. A 2π flux change via a single-link update implies changing by π the
anglesoftwoplaquettes.Thus,iftheactionforbidsplaquetteangleslargerthanπ andasingle-linkupdateis
used,monopoleswillbefrozenandergodicitywillbelost. Right: Thisproblemdisappears,andergodicity
isrestored,iftwolinksareupdatedsimultaneously.
3. Helicitymodulusandconfining/Coulombphasetransition
Thehelicitymodulushmeasurestheresponseofthesystemtoatwistintheboundaryconditions:
(cid:12)
∂2f(φ )(cid:12)
h µν (cid:12) , (3.1)
≡ ∂φ2 (cid:12)
µν (cid:12)
φµν=0
where f isthefreeenergydensityandφ isanexternalfluxthroughthe(µ,ν)planes. Inamassive
µν
(confining) phase, h goes to zero exponentially with the system size. In a massless (Coulomb)
phase,itcanbenon-zerointhethermodynamiclimit. Thus,itcanbeusedasanorderparameterin
the4dU(1)theory,asshownin[10]. Whilehcanbeexpressedasasimpleexpectationvaluewith
the Wilson action, with the topological action it cannot. As in [11], we measured the distribution
ofallowedtwistanglesφ ,fittingthecurvatureofthecorrespondingeffectivepotentialtoobtain
µν
h.
In Fig. 2 (top), we show the measured values of the helicity modulus, as a function of β for
the Wilson action (left), and as a function of cosδ for the topological action (right). The lines
are fits assuming a first-order transition, where the free energy varies linearly with the control
parameter (but with different coefficients in the two phases) [12]. The similarity between the two
figures leaves no doubt that a first-order transition (confining Coulomb) takes place with the
↔
topologicalaction,atacutoffanglecosδ 0.368quitedifferentfromδ =π/3. Ameasurement
c
≈−
ofCreutzratiosinbothphasesisconsistentwithanarealawandaCoulomblaw,respectively.
Note that the helicity modulus in the Coulomb phase has almost the same value at the tran-
sition for both actions. On the other hand, the fitted latent heat is about half as large with the
topologicalaction. Also,themonopoledensitychangesdiscontinuouslyatthephasetransition,for
both actions, as illustrated Fig. 2 (bottom), but the topological action yields a smaller density and
associatedjump(right).
Inadditiontobeinganorderparameter,thehelicitymodulusleadsalsotoasimpledefinition
oftherenormalizedcharge[10]. Thefreeenergy f(φ )iswelldescribedbytheclassicalansatz
µν
f(φµν)= log∑e−β2R(φµν−2πk)2 (3.2)
−
k
3
Topologicalactions PhilippedeForcrand
L=6 0.45 L=6
0.5 L=8 L=8
L=12 0.4 L=12
0.4 0.35
0.3
0.3 0.25
h h
0.2
0.2
0.15
0.1
0.1
0.05
0 0
0.99 0.995 1 1.005 1.01 1.015 1.02 0.4 0.39 0.38 0.37 0.36 0.35
− − − − − −
β cosδmax
0.12 0.07
0.11
0.1 0.06
0.09
monop.i00..0078 monop.i0.05
nh nh0.04 L=6
0.06
L=8
0.05 L=6 0.03 L=12
0.04 L=8 L=16
L=12 L=24
0.03 0.02
0.99 0.995 1 1.005 1.01 1.015 1.02 0.39 0.385 0.38 0.375 0.37 0.365 0.36
− − − − − − −
β cosδmax
Figure2:Helicitymodulus(toprow)andmonopoledensity(bottomrow)fortheWilsonaction(leftcolumn)
andforthetopologicalaction(rightcolumn),acrossthephasetransition. Forbothactions,thetransitionis
first-order: the helicity modulus jumps at the transition and is well described by a finite-size first-order
ansatz[12];themonopoledensityalsojumps,butthejumpissmallerforthetopologicalaction.
fromwhichβ =1/e2 canbeextracted,foreithertypeofaction.
R R
4. MonopoledensityintheCoulombphase
In the confining phase, monopoles condense. In the Coulomb phase, monopoles are expo-
nentially suppressed: the density of monopole current behaves as exp( β M), where M can be
R
−
considered as the monopole mass. Fig. 3 (left) shows this exponential suppression, for both ac-
tions. With the topological action, monopoles are much heavier, i.e. more suppressed. This is
evidence that the topological action is improved: for a given value of the renormalized coupling,
latticeartifactsarereducedconsiderably.
Actually, as explained in Sec. 2, with the topological action monopoles cannot exist if the
cutoff δ on the plaquette angle is π/3 or less. Thus, the density of monopole current vanishes,
and is singular at δ =π/3. We tried to monitor this singular behavior. As δ approaches π/3, the
density of monopole current deviates from the exponential behavior of Fig. 3 (left) and seems to
obey a power law, being proportional to (cos(π/3) cos(δ))γ. See Fig. 3 (right). We measure a
−
largevalue 12forthe“criticalexponent”γ,whichcanbeexplainedheuristically.
∼
4
Topologicalactions PhilippedeForcrand
Consider first a 2d XY model, with topological action exp( Stop) = ∏ ij Θ(δ θi θj ).
− (cid:104) (cid:105) −| − |
Vortices in this model come from the spin orientation θ winding by 2π as one goes around an
i
elementary plaquette. Thus, vortices cannot exist if δ < π/2. When δ just exceeds π/2, each
angle θ θ aroundavortexmustbenearπ/2,withinanangle(δ π/2)ofit. Thephasespace
i j
| − | −
for this is ∝ (δ π/2). Since there are 4 links around a vortex, related by one constraint (the
−
sumofthefour(θ θ )is2π), thephasespaceforavortexis∝(δ π/2)3. Now, onaperiodic
i j
− −
latticeavortexisalwaysaccompaniedbyananti-vortex. Thus,theprobabilityoffindingavortex
anti-vortex pair should be ∝ (δ π/2)6. This is remarkably consistent with the measurements
−
shown Fig. 4 (right). We can then apply a similar argument to theU(1) monopole density. The
smallestmonopolecurrentloopisdualto4cubes. Eachcubecontains6plaquettes,whichtogether
obey a Bianchi constraint so that the total outgoing flux through the 6 plaquettes is 2π. Thus, the
probability of finding such a monopole loop is ∝(δ π/3)4 5. Our data, shown Fig. 4 (left), is
×
−
moreconsistentwithanexponent 12,indicatingthatourheuristicargumentistoocrude.
∼
0.1 0.1
Top. act. L=4
Top. act. L=8 0.01
Wil. act. L=4
Wil. act. L=8 0.001
0.01
monop.i monop 0.0001
n n 1e-05
h
0.001
1e-06 44
84
1e-07 124
exponential
0.0001 1e-08 0.12(0.5−cosδmax)11.7
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.3 -0.2 -0.1 0 0.1 0.2
βR cosδmax
Figure3:MonopoledensityintheCoulombphase.Left:Themonopoledensitydecreasesexponentiallyasa
functionoftheinverserenormalizedcouplingβ =1/e2,bothfortheWilsonactionandforthetopological
R R
action, with a steeper decrease for the latter. Right: using the topological action, the monopole density
vanisheswhentheplaquetteanglescannotexceedπ/3. Asoneapproachesthiscriticalvalue,themonopole
densityvanishesasapowerlawofthedistancetothecriticalconstraintangle,withacriticalexponent 12.
∼
5. Freeenergyforfree
With a topological action, the partition function Z(δ) simply counts the number of configu-
rations which satisfy the constraint. Therefore, Z(δ )/Z(δ ), with δ <δ , is the proportion of
2 1 2 1
configurations satisfying the constraint δ which also satisfy the tighter constraint δ . These con-
1 2
figurationscanbecountedastheMonteCarlosamplingofZ(δ )proceeds,givingthevariationof
1
thefreeenergy f(δ)“forfree”.
We have applied this simple strategy near the phase transition, in the 2d XY model and in
the 4d U(1) gauge theory. The phase transition is of infinite order (BKT) in the former case, of
first-order in the latter. The results for ∂f(δ)/∂δ, shown Fig. 5, make this difference clear: for
thefirst-ordertransition(left),thefirstderivativeofthefreeenergydevelopsadiscontinuityasthe
systemsizeincreases,whileitremainssmoothfortheinfinite-ordertransition.
5
Topologicalactions PhilippedeForcrand
0.1 10
MonteCarlo,322
0.01 1 2.16(−cosδmax)6.0
0.001 0.1
0.0001 0.01
monop 1e-05 monop 0.001
n n
1e-06 0.0001
1e-07 1e-05
44
1e-08 81424 1e-06
1e-09 0.12(cosπ/3−cosδmax)11.7 1e-07
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.8 0.5 0.3 0.2 0.1
− − − − − − − − − − − −
cosδmax−cosπ/3 cosδmax
Figure 4: Left: monopole density in the Coulomb phase, for the topological action, as a function of the
distance to the critical constraint: a power law is clearly visible on this log-log plot. Right: similar figure
showingthedensityofvorticesinthemasslessphaseofthe2d XYmodel,usingatopologicalaction. The
vortexdensityvanishesasapowerofthedistancetothecriticalconstraint,whichlimitsthedifferenceamong
neighboringspinanglestoπ/2. Thecriticalexponent6isconsistentwithaheuristicargument(seetext).
17 5.2
42
16 cosδc 4.58 16822
322
15 4.6
14 4.4
δmax 13 δmax 4.2 cosδc
/∂ /∂ 4
f f
∂ 12 ∂ 3.8
11 3.6
24 3.4
10 34
44 3.2
84
9 3
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
cosδmax cosδmax
Figure5: Derivativeofthefreeenergydensitywithrespecttotheconstraintangle,acrosstheconfinement-
Coulombphasetransitionmarkedbytheverticalline,forthe4dU(1)gaugetheory(left)andforthe2dXY
model(right). Asthesystemsizeisincreased,adiscontinuityisclearlyvisibleintheformercase,signaling
a first-order transition. In contrast, the derivative of the free energy remains smooth in the XY model, as
expectedfromatransitionofinfiniteorder. Withatopologicalaction,thefreeenergyisobtainedbysimply
countingthenumberofconfigurationswhichsurviveatighterconstraint(seetext).
6. Conclusion
Atopologicalactiontakesonlytwovalues: 0and+∞,wherethelatterremovesanyconfigura-
tionwhichviolatesalocalsmoothnessconstraint. Wehavestudiedthepropertiesofthetopological
actionS(δ)= ∑PlogΘ(δ θP )inthe4dU(1)gaugetheory,asafunctionoftheconstraintan-
− −| |
gle δ. As δ 0, all plaquettes P are forced to approach the identity. Yet the approach to the
→
continuum action (cid:82) d4xF2 is not clear a priori. Nevertheless, we have shown that the phase dia-
µν
gramofthislatticetheoryisverysimilartothatobtainedwiththeWilsonaction,withaconfining,
6
Topologicalactions PhilippedeForcrand
disordered phase at large δ, separated from a Coulomb, ordered phase at small δ by a first-order
transition. This transition is associated with a jump in the density of magnetic monopoles, which
condense in the confining phase and are exponentially suppressed in the Coulomb phase. With
the topological action, the monopole density is smaller in the Coulomb phase, which reflects the
improvement of the action. The helicity modulus serves both as an order parameter for the phase
transition,andasarenormalizedcouplingintheCoulombphase.
A feature specific to the topological action is the total suppression of monopoles, or of topo-
logical defects in general, for a particular value of the constraint (here, δ =π/3). This threshold
c
is analogous to Lüscher’s “admissibility condition”, which guarantees that topological sectors are
distinct, even on the lattice [13]. Here, the monopole density is singular at δ : we observe that it
c
decreases as a high power of (δ δ ) when δ δ+, then is identically zero for δ <δ . Other
c c c
− →
effectsofthisphasetransition,ifany,remaintobeelucidated.
Finally,atopologicalactionprovidesasimplemeanstomeasurethefreeenergy,asthefraction
of configurations in a given ensemble which continue to satisfy the constraint as this constraint is
tightened. Thismightbeusefulinsomeattemptstotacklethesignproblem,wherethefreeenergy
mustbeconvolutedwithacomplexphasefactor[14].
References
[1] K.Symanzik,Nucl.Phys.B226,187(1983).
[2] C.MichaelandM.Teper,Nucl.Phys.B305,453(1988).
[3] P.HasenfratzandF.Niedermayer,Nucl.Phys.B414,785(1994)[hep-lat/9308004].
[4] J.Fingberg,U.M.HellerandV.K.Mitrjushkin,Nucl.Phys.B435,311(1995)[hep-lat/9407011].
[5] J.Balog,F.Niedermayer,M.Pepe,P.WeiszandU.-J.Wiese,JHEP1211,140(2012)
[arXiv:1208.6232[hep-lat]].
[6] D.Banerjee,M.Bögli,K.Holland,F.Niedermayer,M.Pepe,U.WengerandU.J.Wiese,
arXiv:1512.04984[hep-lat].
[7] W.Bietenholz,M.Bögli,F.Niedermayer,M.Pepe,F.G.Rejon-BarreraandU.J.Wiese,JHEP1303,
141(2013)[arXiv:1212.0579[hep-lat]].
[8] O.AkerlundandP.deForcrand,JHEP1506,183(2015)[arXiv:1505.02666[hep-lat]].
[9] T.A.DeGrandandD.Toussaint,Phys.Rev.D22,2478(1980).
[10] M.VettorazzoandP.deForcrand,Nucl.Phys.B686,85(2004)[hep-lat/0311006].
[11] W.Bietenholz,U.GerberandF.G.Rejón-Barrera,J.Stat.Mech.2013,P12009(2013)
[arXiv:1307.0485[hep-lat]].
[12] C.BorgsandR.Kotecky,Phys.Rev.Lett.68,1734(1992).
[13] M.Luscher,Commun.Math.Phys.85,39(1982).
[14] K.LangfeldandB.Lucini,Phys.Rev.D90,no.9,094502(2014)[arXiv:1404.7187[hep-lat]].
7