Reweighted complex Langevin and its application to two-dimensional QCD

0.25 MB·

by  Jacques Bloch

Reweighted complex Langevin and its application to
two-dimensional QCD
Jacques Bloch∗
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1 UniversityofRegensburg
0 E-mail: [email protected]
2
n Johannes Meisinger
a UniversityofRegensburg
J
E-mail: [email protected]
5

Sebastian Schmalzbauer
]
t GoetheUniversityofFrankfurt
a
l E-mail: [email protected]
-
p
e
h WepresentthereweightedcomplexLangevinmethod,whichenlargestheapplicabilityrangeof
[
thecomplexLangevinmethodbyreweightingthecomplextrajectories. Inthisreweightingpro-

1 cedureboththeauxiliaryandtargetensembleshaveacomplexaction. Wevalidatethemethodby
v
8 applyingittotwo-dimensionalstrong-couplingQCDatnonzerochemicalpotential,andobserve
9 that it gives access to parameter regions that could otherwise not be reached with the complex
2
1 Langevinmethod.
0
SupportedbytheDeutscheForschungsgemeinschaft(SFB/TRR-55)
.
1
0
7
1
:
v
i
X
r
a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
1. Introduction
A major obstacle to simulate lattice QCD at nonzero chemical potential is formed by the
complex valued determinant of the Dirac operator. This sign problem prohibits the use of impor-
tance sampling algorithms to generate relevant configurations and most solutions to circumvent
this problem have a computational cost that grows exponentially in the volume and are restricted
totheregionofthephasediagramwhereµ/T <1,awayfromthecriticalregion.
An alternative which became increasingly popular in recent years is the complex Langevin
(CL)method. Althoughtheequationsemployedinthismethodarestraightforwardgeneralizations
oftherealLangevinequationsfromrealtocomplexactions,thevalidityofthecomplexifiedequa-
tions require a number of conditions to be met. It is important to investigate theories and models
with complex actions to deepen our understanding when the complex Langevin method can be
usedandtrusted. Recentinvestigationsinheavy-denseQCD[1]andfullQCD[2]showedthatthe
method breaks down in the transition region. Problems with the CL method were also uncovered
in low-dimensional strong-coupling QCD [3], as the method converges to wrong values for small
massesatlargecoupling.
InthispresentationweintroduceanovelideawherewecombinetheCLmethodandreweight-
ing of complex trajectories to form the reweighted complex Langevin (RCL) method [4]. As we
willshowintheresultsontwo-dimensionalQCDthismethodallowsustoreachregionsofparam-
eterspacethatarenotsimulatedcorrectlybytheCLmethodalone.
2. ComplexLangevinMethod
WefirstbrieflyintroducethecomplexLangevinmethod. Assumeapartitionfunction
(cid:90)
Z= dxe−S(x), (2.1)
with real degrees of freedom x and complex action S(x), where x can be assumed to a multidi-
mensionalvectorofvariables. WhenapplyingtheLangevinequationsonasystemwithacomplex
actiontherealvariablesareautomaticallydrivenintothecomplexplane,suchthatx→z=x+iy.
ThesecomplexvariablessatisfytheCLevolutionequation
∂S
z˙(t)=− +η(t). (2.2)
∂z
This equation can be solved numerically after proper discretization and the standard stochastic
EulerdiscretizationyieldsthediscreteLangevintimeevolution
√
z(t+1)=z(t)+εK+ εη, (2.3)
with drift K = −∂S/∂z, step size ε and independent Gaussian noise η (chosen real for better
convergence)withmean0andvariance2.
It is important to understand if and when the fixed point solution of the complex Langevin
equation reproduces the correct expectation values of the partition function (2.1). It was shown
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ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
that if the action S and the observable O are holomorphic in the complexified variables (up to
singularities)thecrucialequivalenceidentity
1(cid:90) (cid:90)
(cid:104)O(cid:105)≡ dxe−S(x)O(x)= dxdyP(x+iy)O(x+iy) (2.4)
Z
holdsiftheprobabilitydensityP(z)ofthecomplexifiedvariableszalongtheCLtrajectoriesissup-
pressedclosetosingularitiesofdriftandobservableanddecayssufficientlyrapidlyintheimaginary
direction of z [5, 6]. If the CL validity conditions are not satisfied for some parameter values the
CL method will fail and produce incorrect results. For QCD this depends on the values of the
parametersµ,m,β,seeSec.4.
3. ReweightingthecomplexLangevintrajectories
Below we introduce the reweighted complex Langevin (RCL) method [4]. The principle of
themethodistogenerateaCLtrajectoryforanauxiliaryensemblewheretheCLmethodisvalid
and to reweight this complex trajectory to compute observables in a target ensemble. The aim is
toextendtheapplicabilityrangeoftheCLmethodtoparameterregionsforwhichtheCLvalidity
conditionsmaynotbesatisfied.
Consider a target ensemble with parameters ξ and an auxiliary ensemble with parameters
ξ . The general reweighting formula to compute expectation values in the target ensemble using
0
configurationsfromtheauxiliaryensembleisgivenby
(cid:104)O(cid:105) = (cid:82) dxw(x;ξ)O(x;ξ) = (cid:82) dxw(x;ξ0)(cid:104)ww((xx;;ξξ0))O(x;ξ)(cid:105) = (cid:68)ww((xx;;ξξ0))O(x;ξ)(cid:69)ξ0. (3.1)
ξ (cid:82) dxw(x;ξ) (cid:82) dxw(x;ξ )(cid:104)w(x;ξ)(cid:105) (cid:68)w(x;ξ)(cid:69)
0 w(x;ξ0) w(x;ξ0) ξ0
The peculiarity of our reweighting method is that we consider an auxiliary ensemble at nonzero
chemical potential such that the auxiliary weights w(x;ξ ) are complex, whereas these are taken
0
realandpositiveinstandardreweightingprocedures. Thereforewecannotuseimportancesampling
to generate relevant configurations in the auxiliary ensemble, but will instead use the CL method
togenerateanauxiliarytrajectoryofcomplexvaluedconfigurations.
IftheCLmethodisvalidfortheparametersξ ,theCLequivalence(2.4)holdsfortheauxiliary
0
ensemble and can be applied to both (cid:104)···(cid:105) in the reweighting formula (3.1), which yields the
ξ0
followingRCLequation:
(cid:104) (cid:105)
(cid:82) dxdyP(z;ξ ) w(z;ξ)O(z;ξ)
(cid:104)O(cid:105) = 0 w(z;ξ0) . (3.2)
ξ (cid:82) dxdyP(z;ξ )(cid:104)w(z;ξ)(cid:105)
0 w(z;ξ0)
The expectation value (cid:104)O(cid:105) in the target ensemble is thus computed as a ratio of averages, both
ξ
evaluated along the auxiliary CL trajectory. As the CL method is valid in the auxiliary ensemble,
both expectation values in this ratio will be evaluated reliably, independently of the fact if the CL
method itself is valid or not in the target ensemble. Indeed, as we will see in the next section, the
RCLmethodalsoworkswelliftheCLvalidityconditionsareviolatedinthetargetensemble.
Asthisreweightingalongcomplextrajectoriesisanovelidea,itisimportanttoputthemethod
atatestandverifythatitworksasexpected. Inthenextsectionweshowresultsintwo-dimensional
QCD,butthemethodhasalsobeensuccessfullytestedonrandommatrixmodelsforQCD.
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ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
4. ReweightedcomplexLangevinfor1+1dQCD
4.1 ComplexLangevinforQCD
LetusconsiderthepartitionfunctionoflatticeQCD,
(cid:34) (cid:35)
V d−1(cid:90)
Z= ∏∏ dU exp[−S (β)]detD(m;µ), (4.1)
x,ν g
x=1ν=0
with SU(3) matricesU , Wilson gauge action S (β) and staggered Dirac operator D(m;µ) for a
x,ν g
quark of massmat chemical potential µ. For nonzero µ the Dirac determinant is complex, so we
have a complex action and a potential sign problem. In this case, the CL equations will drive the
linksfromSU(3)intoSL(3,C). WhenthecomplextrajectorieswanderofftoofarfromSU(3)the
CLmethodbecomesinvalid. Thisproblemisresolvedingaugetheoriesbyapplyinggaugecooling
[7],whichkeepsthetrajectoriesascloseaspossibletoSU(3)usingSL(3,C)gaugeinvariance.
Although gauge cooling can alter the CL trajectory such that the CL validity conditions are
satisfied, there is no guarantee that this can be achieved for all parameter values. This was in-
vestigated in detail for strong-coupling QCD in 1+1 dimensions [3] where we showed that, even
with gauge cooling, the CL results are only valid in a certain (m,µ) range. At small masses the
CLtrajectoriesstillpopulatethesingularityatdetD=0andCLirrecoverablygiveswrongresults.
ThesearecasesofinteresttoverifyiftheRCLmethodcanbeusedtorecoverthecorrectresults.
4.2 Reweightinginthemass
We first test the CL method by computing the chiral condensate and the number density on
a 4×4 lattice in the strong-coupling limit β =0 and choose µ =0.3 as the sign problem is most
2.5 1.8
pq-rew
1.6 CL
2 1.4 RCL - m0=0.4
1.2
1.5 1
4x4
S n 0.8 4x4
1 0.6
0.4
0.5
RCL - m0=0.4 0.2
CL
pq-rew 0
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
m m
2.5
pq-rew
2 CL
2 RCL - m0=0.3
1.5
1.5 6x6
S n 1 6x6
1
0.5
0.5
RCL - m0=0.3
CL
pq-rew 0
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
m m
Figure1: Chiralcondensate(left)andnumberdensity(right)versusmassatµ =0.3fora4×4lattice(top
row)anda6×6lattice(bottomrow): dataforCL(bluepoints)versusRCL(redpoints). Thegreybandare
benchmarkresultscomputedwithstandardphase-quenchedreweighting.
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ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
pronounced around this value. Even though the sign problem is still mild on such a small lattice,
the results of gauge cooled CL are wrong for small masses (m(cid:46)0.2), as is illustrated by the blue
pointsinthetoprowofFig.1,duetothesingulardriftproblem. Thebenchmarkresults(greyband)
were computed with phase-quenched reweighting simulations. We apply the RCL equation (3.2)
usinganauxiliaryCLtrajectorygeneratedatmassm =0.4forthesame µ andcomputetheRCL
0
resultsfortherangem∈[0,0.4]. Theresultsaregivenbytheredpointsinthefigure. ClearlyRCL
inmassworksoverthecompletemassrangeonthislatticesize,eveninthestrongcouplinglimit.
NextwetestedtheRCLmethodona6×6latticeatβ =0, againfor µ =0.3wherethesign
problem is strongest. The results are shown in the bottom row of Fig. 1. Again, gauge cooled CL
(blue points) is wrong for small masses (m(cid:46)0.2). The RCL method uses an auxiliary trajectory
generatedatm =0.4and µ =µ tocomputetheresultsform∈[0,0.4]. Onthissomewhatlarger
0 0
latticetheoverlapandsignproblemsbecomevisiblethroughlargererrorbars,butnevertheless,the
RCLmethoddoesworkdowntoverysmallmasses.
4.3 Reweightinginthechemicalpotential
ClearlywecanusetheRCLmethodtoreweightinanyrelevantparameterandwehereinves-
tigate the reweighting in the chemical potential µ. In Fig. 2 we show the strong-coupling (β =0)
results for small mass m=0.1 on 4×4 (top) and 6×6 (bottom) lattices as a function of µ. We
see that for such small masses the gauge cooled CL method fails to reproduce the correct results.
TheRCLresultsarecomputedusinganauxiliaryCLtrajectorygeneratedatµ =0.8forthe4×4
0
latticeandat µ =0.6forthe6×6lattice. InallcasestheRCLresultsagreewiththebenchmark
0
over the complete µ range within the error bars. The increasing errors for the 6×6 lattice in the
2.5 pq-rew 3
CL
2 RCL - m0=0.8 2.5
2 4x4
1.5
S n 1.5
1 4x4
1
0.5 0.5 pq-rew
CL
0 0 RCL - m0=0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
m m
2.5 pq-rew 3
CL
2 RCL - m0=0.6 2.5
2 6x6
1.5
S n 1.5
1 6x6
1
0.5 0.5 pq-rew
CL
0 0 RCL - m0=0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
m m
Figure2: Chiralcondensate(left)andquarknumber(right)form=0.1on4×4(top)and6×6(bottom)
lattices versus chemical potential µ. Results for CL (blue points) and RCL (red points). The grey band
benchmarkiscomputedusingphase-quenchedreweighting.
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ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
critical region point to an increasing sign problem. It is however surprising that RCL works well
for small µ, i.e., far from the auxiliary value, meaning that there is no serious overlap problem in
thiscase. Whenperformingthisreweightinginµ forevensmallermassesthereweightingmethod
willstarttobreakdownasthesignproblembecomesstronger.
4.4 Reweightinginthecoupling
EventhoughRCLinµ andmcouldjustaswellhavebeenperformedatnonzeroβ,awayfrom
thestrongcouplinglimit,wechosetoworkatβ =0asthispushestheCLmethodtoitslimits.
AsalasttestweuseRCLtoperformreweightinginβ. ItisknownthattheCLmethodisvalid
forlargeβ butbreaksdownforlowervalues. Simulationsatsuchβ valuesarehoweverneededto
reach the critical region in current lattice simulations. It would therefore be helpful if RCL could
beappliedtovalidCLtrajectoriesgeneratedatlargeβ toreachlowerβ values.
Fig.3showstheresultsforµ =0.3andm=0.1ona4×4lattice. TheCLmethodonlyworks
correctlyforlargeβ >6,soweinvestigateifRCLallowsustoreachlowerβ values. Unfortunately,
therangeofapplicationofRCLinβ seemsquiterestricted: startingfromanauxiliarytrajectoryat
β =10RCLworksdowntoβ ≈8.5andfromβ =8itworksdowntoβ ≈6.5. InbothcasesRCL
0 0
doesnotperformbetterthantheoriginalCL.Thisisduetotheextremelysharpprobabilitydensity
of the gauge action, where β is a multiplicative factor in the exponential. Even tiny changes in β
stronglyaffectthegaugeweightandreweightingisinefficient.
1.8
1.6 m=0.1
1.4
1.2
S 1
0.8
0.6 RRCCLL -- bb00==180
CL
pq-rew
0.4
0 2 4 6 8 10
b
Figure3:Chiralcondensateversusβ ona4×4latticeforµ=0.3andm=0.1.ResultsforCL(bluepoints)
versusRCL(redandpurplepoints).
5. Someadditionalremarks
Although RCL works correctly to reweight from one set of parameters to another, it suffers
fromtheusualoverlapandsignproblems. Apossibleadvantageoverotherreweightingprocedures
(phase-quenched, Glasgow and quenched reweightings) is that the auxiliary ensemble at µ (cid:54)= 0
couldbeclosertothetargetensemble,thusincreasingtheoverlapbetweenthetargetandauxiliary
ensembles. InGlasgowreweightingtheauxiliaryensembleistakenatµ =0. Clearly,reweighting
0
fromµ (cid:54)=0usingRCLstartsfromanauxiliaryensemblethatisclosertothetargetensemble.
0
Inphase-quenchedreweightingtheauxiliaryensembleusesthemagnitudeofthefermionde-
terminant as sampling weights. The auxiliary and target ensembles are in different phases when
µ >m /2 and there is therefore little overlap between the relevant configurations in both ensem-
π
bles. InRCL,however,theauxiliaryandtargetensemblesarebothtakeninfullQCDandhencethis
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ReweightedcomplexLangevinanditsapplicationtotwo-dimensionalQCD JacquesBloch
problemcouldbealleviated. Moreover,RCLusesasingleCLtrajectorytoreweighttoarangeof
target parameter values, whereas phase-quenched reweighting typically constructs a new Markov
chainforeachnewparametervalue.
6. Summaryandoutlook
For many theories with a complex action the complex Langevin method works correctly for
some range of parameters, but fails for other parameter values when the validity conditions are
violated. InthistalkwehavepresentedthereweightedcomplexLangevinmethod,whichcombines
complex Langevin and reweighting to compute observables in a target ensemble using complex
trajectoriesgeneratedforanauxiliaryensembleforwhichtheCLvalidityconditionsaremet.
AsaproofofprincipleweappliedRCLonQCDin1+1dimensionusingreweightinginm, µ
andβ atµ (cid:54)=0andverifiedthattheRCLprocedureworkscorrectly. WeobservedthatRCLworks
best when reweighting in the mass, while reweighting in µ works well as long as the mass is not
toosmall. Reweightinginβ hardlyworksasthegaugeprobabilityisnarrowandverysensitiveto
β. Clearly,themethodcouldbefurtheroptimizedbymakingamultiparameterRCLinµ,m,β [8].
Asthemethodsuffersfromtheusualoverlapandsignproblems,itsefficacyshouldbeinvestigated
further.
Asanoutlookforfutureworkwecanpinpointacoupleofavenues. Oneinterestingapplication
would be to test RCL on full four-dimensional QCD where it was shown that CL breaks down in
thecriticalregion. Asmassreweightingworksbestthestrategycouldbetochooseahighenough
m to get a valid CL trajectory for a particular (µ,β) and then reweight in m to get down to the
physicalmassregion. Alternativelyonecouldfollowalineinthe(m,µ)-planekeepingβ fixed.
Clearly, one still has to learn how to reweight most efficiently with the RCL method. A
useful exercise would be to make a validity map of the CL method in the (m,µ,β)-space for
two-dimensionalQCDanddevisethebestreweightingpathtocoverallparametervalues.
NotethatRCLopensanewavenueasreweightinginthechemicalpotentialcanbeextendedto
interpolateratherthenjustextrapolateifweuseauxiliaryensemblesat µ valuesaboveandbelow
0
thecriticalregion,whichcouldimprovethequalityoftheresults.
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