It is proven that ensembleaveragecomputedfrom a complexLangevin (CL) simulationwillnecessarilyconvergeto the correctvaluesif the ensembleaveragesbecometime independent.This is illustratedwith two modelproblemsdefinedon thecompactspacesU(1) and~2, aswellaswith a lattice fermion model. For all threeproblems,the CL method is foundto be, with fewexceptions,applicable.For the U(1) problem,this is demonstratedvia a semi-analyticsolutionfor the expectationvalues. The difficulties of obtaining accuratenumerical solutionsof thestochasticdifferentialequationsare discussed.
1. Introduction
In many physicalproblems,the quantitiesof interestarenaturallyexpressedasmultidimensionalintegral averagesovera complex-valuedweight function e ~. Insuch cases,standardMonte Carlo algorithmswhich rely on a real, positiveweight
function to define an importancesamplingareonly indirectly applicable.Althoughonemayuseonly the realpart of the complexaction to definea suitablestochasticprocess,the complexpart of the actionmay containessentialphysical information.Failure to incorporatethis information into the dynamicsof the simulation maylead to so-called “sign problems”, for which, in the exampleof the two-dimen-
sional Hubbardmodel,thereis still no solution.One important alternative that has been proposedis basedon the complex
Langevin(CL) equation{1 ,2]. Sincethereis no formal restriction to a real-valueddrift termin the Langevinequation,the CL methodusestheentirecomplexactionS to define a stochasticprocess,which may convergedirectly to the desireddistribution e~ Recentlyit hasbeenshownthat the CL method,in conjunctionwith a modified coherent-statepath-integralrepresentation,is a promisingcandi-date for treating two-dimensional lattice fermions [31.In this representation,fermionic degreesof freedomare mappedonto (c-number)variables(0, 4) livingon the unit sphereS2. Also of interestare problems,such as U(1) lattice gaugetheory, which are formulatedon the unit circle. Thus thereis a naturalinterestin
Unfortunately,CLpresentlysuffersfrom its own problems.Despitesomerecentprogress[4], thereis currentlyno completetheoryof the CL method.Moreseriousstill, in some casesnumericalsimulationsseemto show that long time averageseither convergeto the wrong result, or not at all [5,6]. For simple actions,anappropriatechoice of kernel in the Langevinequation may correct this errant
behavior[7,8], but for more generalproblems,it is unfortunatelynot clear whatchoiceof kernelis needed.
In this paperwe prove that ensembleaveragescomputedfrom a complexLangevinprocessnecessarilyconvergeto the correctvaluesif they becometimeindependent.Thus thereis a simpleandeffectivea posteriori testof the accuracy
of CL simulations.This is demonstratedwith threedifferentproblemson compactspaces.For one model problem, we will see that the CL processconvergescorrectlyeventhoughthe numericalsolutions to the stochasticdifferential equa-tion seemto indicateotherwise.This situationis discussed,andwe speculatethatthis may indicatean inherentlimitation of the CL method.
2. Generalconsiderations
The following discussionis restrictedto the one-dimensionalcase,since thegeneralizationto higher dimensionsis immediate. Furthermore,it is assumedthroughoutthe paper that the distributione~ is an infinitely smooth(analytic),integrable(L1) function. For problemsdefinedby a real action S, the Langevinmethodevaluatesintegralaveragesof the form
1(F(x))=— I dxF(x) ~
At)
‘V= f dx ~ (2.1)
by creatinga stochasticprocessvia a Langevin equationwhose unique stableequilibrium distribution is (1/.,v) e ~ Thus,ensembleaverages(or time aver-ages,sincesuch a processis ergodic)computedfrom the processwill relaxto thedesiredintegralaverages.The Langevinequationis givenby
1 asdX(r) = 2aX() dT+dW(T), (2.2)
wherewe havedefined
as asaX(r) aX x=X(~) (2.3)
S. Lee / ComplexLangevinsimulations 829
andW(’r) is a standardWiener processwith zeromeanandcovariance
(W(r1)W(T2)) =min(r1, r2). (2.4)
Associatedwith this processis a Fokker—Planckequation for the probabilitydensityP(x, T)
aP(x,T) 1 a a as(x)—+ P(x T) (2.5)2ax ax ax
whereensembleaveragesoverthe Wienermeasureareequal to integralaveragesoverP(x, T)
EF(X(r)) =J dxP(x, T)F(X). (2.6)
It can be shown (see,for example, ref. [6]) that any initial densityP(x, 0) will
convergein timeto the uniquestationarystate
1P(x T) —~ — e_S~. (2.7)
At
All of the abovecanbe shownrigorously for real S(x).For complex-valuedactions.5: R —~ C, the complexLangevinprescriptionlooks
superficiallysimilar. Since S hasa well-definedcomplexextensionS(z), onemayintroducethe complexLangevin(CL) equation
1 asdZ(r) = ~Jaz(T)dr+dW(r), (2.8)
whereZ(T) X(T) + IY(r) E C, but W(r) EI~asbefore. If oneassumesthat thereis an associatedcomplex valuedfunction P: l~—* C satisfyinga pseudoFokker—
Planck(FP)equation
a15(x,i-) 1 a a as(x) -
—+ P(x r) (2.9)aT 2ax ax ax
which also reproducesthe expectationvaluesof the process
EF(Z(’r)) =1 dxP(x, r)F(x), (2.10)
then, as in the real case,ensembleaveragesof the processmay convergeto thedesiredintegralaverage,since(2.9) hasthe samestationarysolution cx e—5(x)
830 S. Lee/ ComplexLangevinsimulations
Unfortunately,unlike in the real case,convergenceof solutionsof the pseudoFokker—Planckequation(2.9) to the desiredstationarystate is no longerguaran-teedwhen 5 is complex,but ratherdependson the eigenvaluespectrumof theoperator
1 a a asT~—— —+— . (2.11)
2ax ax ax
In particular,oneexpectsthat,when T hasa positiverealpart of an eigenvalue,
solutionsto (2.9) will divergeexponentiallyin time from the stationarystate.Thisbehaviorwill be clearly reflected in the numerical solutionsof the CL equation(2.8), which becomedivergent.
Fortunately,in practice,suchcasesare relativelyuncommon.Morecommonarethe unfortunatebut intriguingsituationsin which long-timeCL averagesappeartosimply convergeto the wrong results[5,61.Understandably,this hasbeena greatproblemfor CL asa practicalmethodfor computingintegralaverages,since it hasbeengenerallyneithera priori feasible nor a posterioripossibleto guaranteetheaccuracyof a given CL simulation.
In order to understandthis surprisingbehavior,one might begin by examiningthe relevanceof the pseudoFP equation(2.9). Indeed,thereis no fundamentalprinciple which guaranteesthe existenceof a complex-valuedfunction P describ-ing a CL processin twice as many dimensions.Rather,eq. (2.9) is a mathematical
conveniencewhich derives its ultimate validity from the nature of the trueprobabilitydensityP(x, y, r) of the process
1 35dX(T)=Re 23Z(T) dT+dW(T),
i asdY(r) = Im — ~ az(r) di-. (2.12)
In particular, this requires that for a given P(x, y)> 0, thereexists a P: l~l—~ Csuch that
This equationhasa formal solution,for which thereareseveralexpressions.Twosuch expressionsare [81
15(x) =f dy exp(—iya~)P(x,y) =f dyP(x—iy, y), (2.14)
S. Lee / ComplexLangevinsimulations 831
and
fi(x, T) = —~---- ~ E(ei~T)) ~ (2.15)
where the sum in the last expressionis taken to be continuous(discrete)if the
variablex lives in a noncompact(compact)space.Often this solution is indeedmerelyformal, as may be seenwith the simple
exampleof a gaussianprobabilitydensity
P(x, y) = exp(—x2—ay2), 0<a. (2.16)
For a> 1, we have
~(x) = ~~(aa_ 1) exp(—ax2/(a—1)), (2.17)
however,for a~ 1, P is divergent.Clearly, P(x, T) will be a smoothfunction onlyif the probability density P(x, y, T) satisfies rather stringent conditions. Forexample,from eq. (21.5) we see that this requiresthat the expectationvalues
E(e~~)decreaseto 0 “rapidly enough” in somesenseas 1k I —~ co•Nevertheless,it is not necessaryfor P to alwaysexist asa classicalfunction.As
an illustrativeexample,considerthe real-valuedaction S(x) = x2. For this action,the CL equationsare,in termsof its realandimaginarycomponents
dX(T) = —x dr + dW(T), dY(T) = —Ydt. (2.18)
The equilibrium distributionof this processis given by
1
Peq(X, y) = —~==- exp(—x2)~(y), (2.19)
independentof the initial distribution. If the initial distribution is given by (2.16),
with a <1, thenat a later time T,
~a e2TP(x, y, T) = exp(—x2) exp(—ae2Ty2). (2.20)
Although CL is completelyapplicableto thisproblem,for T ~ — ~ ln a,P is not afunction! However, P may alwaysbe understoodas a distribution whoseactionona function F(x) is definedas
P[F] f dx dy P(x, y)F(x + ly) (2.21)
832 5. Lee / ComplexLangevinsimulations
(Note that for T = — ~ ln a, P(x, r) = 6(x)). In such cases the notation EF(Z) =
f dx P(x)F(x) isunderstoodtobesuggestive.Likewise, thepseudoFokker—Planckequation,which is now meaningfulonly as anintegral equation
3P(x, T) 1 a a 3S(x) -J dxF(x) 3T =fdxF(x)~_(~_+ 3x )P(x~T)~(2.22)
maybe takenas a purelyformal expressionfor the identity
which follows directly from an applicationof the Ito calculusrules to the CLequation(2.8).
Note, however,that the stationarysolution to eq. (2.22) (eq. (2.23)) is not animproperdistribution. Namely, Peq(x) cx e S This may be easily demonstratedasfollows. Here we discussthe casein which x lives on a compact space.Thenoncorflpactcasex E P is discussedin appendixA.
Let x E [a, bl, and the action haveperiodicity S(x+ nA) = S(x), A = lb — al.First, supposeS(x) is repr~sentedby a finite Fourier series
By direct substitution,the unique time-independentsolution t9Ak/3r = 0, up to anormalizationfactor, is seento be,
= —f dx eikx eS. (2.27)
This solution is uniquebecausethe so-called“spurioussolution” [4]
dy eS~, (2.28)
S. Lee / ComplexLangevinsimulations 833
which is the other time-independentsolutionof the pseudoFokker—Planckequa-tion (2.9), doesnot, in fact, representa stationarysolutionof (2.22),sinceit doesnot satisfy the properperiodicboundaryconditionsnecessaryto relatethe formalexpression(2.22) to eq.(2.23) (see appendixB).
Now let S moregenerallybe representedby an infinite Fourier series.Definethe truncatedFourier series
fi2~r \SQ= ~ SqexP~~—qxJ (2.29)
IqI<Q /
and
AQk(T) E(exp(ikZQ(T))), (2.30)
whereZQ(T) is the solutionof the CL process(2.8) for the action (2.29).The unique stationaryA~are thengiven by
A~k—~~f dx eikx exp(—SQ). (2.31)
Since exp(— SQ) are smooth,boundedfunctions on a compactspace,it followsfrom the dominatedconvergencetheoremthat
limA~k=—Jdx e e5. (2.32)
We thus arrive at the following conclusion:If the expectationvaluescomputedfrom a CL processbecometime independent,theynecessarilyconvergeto the correctvalues.
In CL simulations,this is a necessaryand sufficient condition for determiningthe correctnessof the result. Another useful sufficient condition is given by thefact that, by the Riemann—Lebesguelemma,
lim (e11~>= 0. (2.33)
This behaviormustalso be reflectedin the expectationvaluesE(e”~~)for larger. In numerical simulations, it is found that these two criteria are excellentindicatorsof the accuracyof CL simulations.
3. U(1) model problem
Let usconsiderthe action
S(O)=—pcoso, j3EC, OE[0,2’w). (3.1)
834 S. Lee / ComplexLangevinsimulations
The CL equationin this caseis givenby
dZ(r) = —~-f3sin Z(T) d’r+dW(T). (3.2)
This model is exactlysolvablewith
Kcos 0) =I~(f3)/I~(f3), (3.3)
where I~are the modified Besselfunctionsof the first kind. To integrateeq. (3.2),an explicit Runge—Kuttaschemewhich is 0(h2) accuratein the time step h wasused.Forthe generalSDEwith additive noise
dX(r) =a(X) dT+dW(T), (3.4)
the algorithmis [91
a0 = a(X0),
a1 = a(X0 + a0h +
Xh—XO+h(aO+al)/2+~/kw, (3.5)
where w is a normal, randomvariablewith zeromeanandunit variance:Ew = 0,Ew
2 = 1. The simulation results were obtained by performing a running timeintegral over the Langevintime r and averagingoverseveralsamplepaths:
T E(cosZ) dT E~(cosZ(T)), T>> 1, (3.6)
where
1PE~(F)_~~1~ (3.7)
is the expectationvaluecomputedfrom the simulatedprocessby summingover Pindependentsamplepaths i = 1,..., P.
Fig. 1 comparestheconvergenceof the time averageT E(cos Z) for two values
of ~. For I 131 = 1.0, arg(J3)= 300, the CL simulation convergesto an incorrectvalue.From fig. 2 weseethat for this choiceof 13, the expectationvalue E~(cosZ)itself doesnot converge.Figs. 4 and5 comparethe expectationvaluesEs(e~~T))atlarge T for the samep-values.Again, we seea clear distinction betweencorrectand incorrectCL results.
Nevertheless,we will now seethat contraryto what numericalintegrationof theSDE(3.2) indicates,in almostall casestheCL equationis indeedappropriateto this
S. Lee / ComplexLangevinsimulations 835
1.0 I I I I
~
0.8 — [p]=5 arg=30° —
.—. 0.6— —N
&_0.4 -ó’ —
0
[p1=1 arg=30°
0.2 — —
I I I I20 40 60 80 100
TFig. 1. Realpart of the time averageT E(cosZ) versustime T from eq. (3.2) for two /3-values.With
TFig. 2. Real part of the ensembleaverageE(cos Z) versus time T from eq. (3.2) for 1/31=1.0,arg(/3)= 30°.With 100 independentsamplepathsand time steph = 0.005. Solid line indicatesexact
value.
836 S. Lee/ ComplexLangevinsimulations
1.0 I I I I I I I I I I I I I I I I I I I I I I
[/31=1 arg=30~
~ :: ____________ _____
I I I I I I I I I I I I I I I I I I I I I I I
20 40 60 80 100T
Fig. 3. Real part of the ensembleaverageE(cos Z) versus time T from eq. (3.2) for /3 I = 5.0,arg(/3)= 300. With 100 independentsamplepathsand time steph = 0.005. Solid line indicatesexact
value.
2.0 i I I I I I I I I I I I I I I I I I I rT ~
1.5 — [p]=i arg’=30° —
N 11I~Fig. 4. Realpart of the ensembleaverageE(eZ) versusk from eq. (3.2) for l~I= 1.0, arg(13)= 30°at
T = 100. With 100 independentsamplepathsandtime steph = 0.005.
S. Lee / ComplexLangevinsimulations 837
2.0 I I I I I I I I I I I I I I I I I I I I I I I I I
1.5 — [p]=5 arg=30° —
II I I I I I I I I I I I I I I I I I I I I—2 0 2 4
k
Fig. 5. Realpart of the ensembleaverageE(etIcZ) versusk from eq.(3.2) for I /3 I = 5.0, arg(/3)= 30°atT = 100. With 100 independentsamplepaths and time steph = 0.005.
problem,and ensembleaveragesfrom the stochasticprocessconvergeto the desiredintegral averages.
For the action(3.1), eq. (2.26) becomes
3Ak(r) k2 if3k
ar = — -~--Ak(r)— —~--(A~÷~Ak_i) ~MkqAq(T). (3.8)
This set of equationsmaybe solved numerically in the following way. First notethat, since the stationary solution is unique, the behaviorof solutions for largetimes is uniquely determinedby the eigenvaluespectrumof the matrix Mkq,independentof the initial values.Hencethe initial valuesAk(T = 0) = 6~omay bechosen. Furthermore,the stationary solutions 4 satisfy J 41—~ 0 as I k I —~ °°.
Therefore,we may effectively truncatethe infinite set of eqs.(3.8) to a finite set— K < k <K. Of course, in practical calculations it is necessary to take K largeenough so that the final results Ak(T — large) become independent of the trunca-tion. Note that this procedure is not unlike the method of coupled Greenfunctions.
Alternatively, we may simply solve for the eigenvaluesandzero eigenvectorofthe (truncated) matrix Mkq. If there are no positive eigenvalues,the stationarysolution (zero eigenvector)is stable,otherwise,solutions diverge.This was con-
838 5. Lee / ComplexLangevinsimulations
TABLE 1Comparisonof (cos0) fromthe deterministicequation(3.8) andfrom the RK simulationof theSDE
(3.2). For theSDEsimulationh = 0.005, T = 100, 100samplepaths
60° (0.9440,0.0996) no convergence no convergence90° (0.0000,1.8445) no convergence no convergence
~) h=o.0025.
firmed with the EISPACK routine CG, and is completely consistentwith the
numericalsolutionsto (3.8).Table 1 comparesthe resultsof solving eq.(3.8) with exact results (cos 0) and
resultsfrom the numericalsolutionsof the SDE(3.2). Surprisingly,we seethat CLaveragesconvergeto the propervaluesmore generally than integrationof (3.2)suggests.Indeed,of four different numericalalgorithms tested for solving (3.2),including implicit schemesanddynamical time steps,not oneof them leadsto thecorrect results for these13-values! Clearly, the tenacitywith which thesesimula-tions fail requiresconsideration.
We may speculateon the reasonfor the failed numericalsolutionsas follows:Note that the formal solution of eq.(2.13)may also be written as
P(x) = ~Jdk e~~Q(k, ik), (3.9)
where
Q(k,q)=JdxdyP(x, y)exp(ikx+iqy). (3.10)
S. Lee / ComplexLangevinsimulations 839
Clearly the probabilitydensityP(x, y) is not uniquelydeterminedby Q(k, —ik),andhencethe solutionP(x) to eq.(2.13)maynotbe unique.Thisisnot surprising,since P(x) only describesthe ensembleaveragesof functions of the specialvariable Z mX+ iY. Furthermore,due to the singular diffusion matrix, eq. (2.12)may not havea unique stationarysolution(seeref. [4] andreferenceswithin). Weconcludethat unique time-independentexpectationvaluesEF(Z) do not require auniquestationarystateof the CL equation! However, while this is true analytically,it is clear that a numericalsimulation of such a processwill generallynot reachequilibrium, and hencenot be ergodic. Indeed,precisely thosesimulationswhichappearto convergeto the incorrectvaluesare found to benonergodic.If the abovescenarioholds,then the potentialexistenceof a degeneratestationarystateof theprocess(2.12)may representa natural limitation of the CL method.
On the other hand, it is interesting to note that for the value 1131 = 5.0,arg(f3) = 60°,the CL simulation can appearconvergentfor reasonablelengthsoftime T beforesuddenlydiverging.Indeed,during thesetimes the simulationsyieldanaccurateestimateof the correctvalue (cos 0) = (0.9440, 0.0996).Examiningthe
eigenvaluesof Mkq for thiscase,we seethat thereis onesmallpositiverealpart ofan eigenvalue 0.47. With a fortuitouschoice of initial distribution, this eigen-value may be suppressedfor evenmoderatelylarge times, and the expectationvalueswill beapproximatelystationary.From the perspectiveof numericalsimula-tions,this will be sufficient to yield accurateestimatesof the expectationvalues.
4. Complex Langevin on S2
For many problems,one is interestedin computing integral averagesoversphericalcoordinates(0, 4) of the form
(F(0, ~)) = At~fdfl e~~°’~F(0,~), (4.1)
where dtl = (1/2~r)sin 0 dO d4. The Langevin equationsappropriate to thisproblem are [1,101
d~(r) = ~~3O() [S—ln(sin ~)} dr+dW1(r), (4.2)
1 a[s — ln(sin �~)]sin(�~)d1(r) = — ~ sin ~ 3’P(r) dr + dW2(T), (4.3)
whereW1, W2 are independent,standardWienerprocesses.
840 S. Lee / ComplexLangevinsimulations
Of special interest here is the fact that any spin- ~ system may be formulated onSO(3) via the spin coherent-state representation of the Pauli matrices [3]
~7X=}4IJ° d~xIf~Xf~Ix,
r~’=w~fd~QXflIy,
cizwf d~rxlflXfllz, (4.4)
where x = sin 0 cos 4, y = sin 0 sin 4, z= cos 0. The spin coherent states I (2),which aredefinedas [11]
As discussedin ref. [3], the overcompleteness of the coherent states implies thatthe function fW) is not unique,andmay bechosento suit the particular problemconsidered.
A generalspin hamiltonianmay thenbe written as
H= f d~IQXflIh(fl). (4.9)
S. Lee / ComplexLangevinsimulations 841
Factorizing e = (exp( — aoH))K, where a0 = /3/K, we obtain an approximateexpressionfor the partition function
Tr(e~’~)= JUl dQk e5 + O(a~K), (4.10)
wherethe action S is given by
K
~ [—ln(Qk+llQk)—ln f(~k) +aoh(Qk)] (4.11)k=i
with periodicboundaryconditionson the “thermal time” lattice
I~K+i)~~ (4.12)
Note that the action S is complex for any /3 due to the complexvaluednessof thecoherent-state overlap.
Wewish to test the appropriateness of the CL method for computing integralaverages of the form
1 K(0) = —ff1 dfl~ 0 e~ (4.13)
with the simple example of
h(i2k)m/3zk61k, f(~k)=1• (4.14)
This choice of hUlk) doesnot correspondto any hamiltonian,but doesprovide anexactly solvable toy model. In particular, CL was used to compute (Zk), which isgiven by
(Zk) = (coth /3 — l/P)61k- (4.15)
Fornumericalsimulations,it is convenientto considerthe aboveproblemin theembeddingspacecoordinatesx, y, z. In ref. [101it wasfound that eqs.(4.2) and(4.3) are stochastically equivalent to
dX(r)=D~dr+YdW3(r)—zdW2(r), (4.16)
dY(r)=D~ dr+ZdW1(r)—XdW3(r), (4.17)
dZ(r) =D~dr+X dW2(r) —YdW1(r), (4.18)
842 S. Lee / ComplexLangevinsimulations
wherethe drift terms aregivenby
D~=-x- ~_X2)3~) ~8Y() _XZaZ( )], (4.19)
D~=_Y_ ~ ) +(1_Y2)3~~) _YZaz~j~ (4.20)
as as asD~=—Z—~—XZ —YZ +(1—Z
2) . (4.21)c9X(r) 3Y(r) 8Z(r)
The appropriateCL equationsin x, y, z coordinatesare rather long andpresentedin detail in ref. [10]. Due to the presenceof multiplicativenoise in theproblem, we used an explicit algorithm due to Petersen[12]. For the generalmultivariate SDEwith multiplicativenoise
wherew~andw~are independent,random,normalvariableswith zeromeanandunit variance.The randommatrix A’~is given by
A’3 = — T’J)
and T’3 is of the form
T” = 1,
T13= —T3 (i�j),
S. Lee / ComplexLangevinsimulations 843
I I I I I I I I I I I I I I I I I I I I I
0.8 —
0.6 — —
NN
0.4—
0.2 —I I I I I I I I I I I I I I I I I I I I I
1 2 3 4 5
pFig. 6. Realpart of the time averageT EZ
1 versus/3 for the model (4.14) K = 4, h = 0.01, T = 10, 50independentsamplepaths.Solidline indicatesexactresults.
whereeachelementT” is itself anindependent,standardnormal randomvariable
as well. This algorithmis 0(h2) accuratefor all momentsEX°.
Fig. 6 showsthe resultsof CL simulationsfor several/3-values.As canbe seen,the CL methodprovidesexcellentaccuracyfor this problem.For all valuestested,
the ensembleaverage E~(Z1)convergedwithin statistical error to the correct
values.
5. Free-fermion model
We considernow a two-dimensionallattice free-fermionmodel with a finite
chemicalpotential jx, which is definedby the hamiltonian
H= _t~C1tC~_p~C1tC~, (5.1)(if)
wherethe sum ~~(ij) is overnearest-neighborslattice sites.
As discussedin refs. [13,3], two-dimensionallattice fermion operatorssatisfying
{Cn,C~}=6mn, (Cn,CmJ=0 (5.2)
844 5. Lee / ComplexLangevinsimulations
may be represented by spin operatorsvia the Jordan—Wignertransformation[14]
n—iC~=o~flo~, n~2, C1=o~, (5.3)
i= 1
wherea~= ~ — io~).This transformationpresumesan orderingof the two-di-mensional L X M lattice into a one-dimensionalchain n = 1,.. . , N= L xM.Given an orderedlattice, andthe spin coherent-staterepresentationof the Paulimatrices,bilinear fermion operators,as well as higher-orderproducts, have asimple c-nummerrepresentationin termsof the euclideancoordinatesx, y, z:
C~C~—~ ~(1 + w~z~),
~ + C~÷1C~—~ 4(w~x~+ix~+ w~y~+1y~),
m-i
C~Cm+C~Cn3~(w~xnxm+w~ynym) 1—I ~ m>n+ 1. (5.4)1=n+ i
The coherent-statemeasureis chosento be of theform
f(Q)=(1—a)(1+m) cosm0+a, (5.5)
where m is an even integer, and 0 <a ~ 1. With this measure, w~= iVY, and theweight factors are functions of a, m. The free parametersa, m allow the
potentially large proportionality constant W~M~ in the action to be favorablycontrolled.
In this representation,the appropriatecomplex action for the hamiltonianmodel is
N K/ k+i k\ k
S= ~ ~ —lnc~(x,y,z)~(x,y,z)~j—lnf(z~)n=1 k=i
1 21 k k+ k k—raotw
5~x~÷ix~y~÷1y~
— ~a0~v~w~ i( xfl+Mxfl + y~+Myfl) z~+f
— ~aoM(1+ w~z~).
The readeris referredto refs.[3,10] for details.
We measuredthe filling
uN \= ~ c~c~), (5.6)
S. Lee / ComplexLangevinsimulations 845
1.0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
/3=0.50.9 — —
Fig. 7. x2 fit of the filling p versusthe thermal lattice size K for the lattice model (5.1). For a 4x4spatiallattice,h = 0.005, 7500iterations.Dottedline indicatesasymptoticvalue.
which as a spin operator is given as
iN~ (1+(o~>), (5.7)
n=1
andcomparedthe simulation resultswith the known exactvalues.Although the model (5.1) is exactly solvable, the partition function (4.10) hasa
systematicerror of order0(1/K). Hence,to estimatethe true filling, for eachsetof values /3 and p., severalsimulationsfor different K-valueswereperformed. A
x2 fit basedon the model
constantK (5.8)
wascomputedto obtain the asymptoticvalue pa,, as K —~ co• The simulationswereperformedon a 4 X 4 spatial lattice, with h = 0.005, and 7500 iterations, with thefirst 2500 iterationsdiscarded to allow the systemto equilibrate. With 10 indepen-dent samplepaths,this required roughly 30 minutesCPU time on a DEC 5000workstation. Values of a = 0.2, m = 10 were chosen throughout, for which w~ 1.2.Fig. 7 shows the x2 fit for /3 = 0.5, p. = 4.0. The dotted line indicates theasymptoticestimateof 0.786±0.02, comparedto the exactvalue of p = 0.771.
846 S. Lee / ComplexLangevinsimulations
1.2 I I I I I I I I I I I I I I I I I I I I I I I I I I I I
1.0 — /3=0.5
0.4 I I I I I I I I I I I I I I I I I I I I I I I I I I I I
0 1 2 3 4 5
/iFig. 8. Filling p versus~sfor the lattice model (5.1) for ~/3= 0.5. 4 x4 spatiallattice h = 0.005, 7500
iterations.Solid line indicatesexactvalues.
Fig. 8 shows the resultsof the simulationsfor several p. values for /3 = 0.5. Asexpected,for all values tested, the expectationvalues convergedwell to theappropriatevalues,andthe characteristic-likefunctionsE(e’~)also displayedtheappropriatebehavior.
6. Conclusions
We haveshownthat a necessaryand sufficient conditionfor simulationsbasedon a complex Langevin equation to converge correctly is that the expectationvaluesbecometime independent.This condition, as well as the behaviorof thecharacteristic-likefunctions E(e~~)are excellent numerical indicators for theaccuracyof the simulation.This hasbeenillustratedwith threedifferent problems,and the CL method is seento be valid for almost all casesstudied.In particular,the CL method appearsto be quite appropriate for studying lattice fermionmodels. Unfortunately, for the U(1) problem studied, there were occasionaldifficulties in the numerical solution of the appropriatestochasticdifferentialequation.This maybeduethe existenceof a degeneratestationarystateof the CLprocess,which would representan almostinsurmountablenumericaldifficulty insuchcases.
S. Lee / ComplexLangevinsimulations 847
The author is grateful to John Klauder of the University of Florida for
invaluablediscussionsandhiscommentson themanuscript.The authorwould alsolike to thank Wes Petersenof the IPS-ETH Zurich for helpful suggestionsandtechnical support, and Phillipe de Forcrand of the IPS-ETH Zurich, HelmutGaustererof the University of Graz,and GajendraTulsian of the University ofFlorida for illuminatingcomments.
AppendixA
When x E I~,we are interestedin those actions for which e_S vanishesasxl —~ oo~Typically, S(x) is a polynomial, so that e_S ~5”, the Schwarzspaceof C
functions of rapid decrease.We may extendthe resultsof sect. 2 to this case asfollows: Define a smoothperiodicfunction 5A(x + nA) = 5A(x) so that SA(x) = 5(x)for lxi ~ A — A’. ForA — A’ <lxi ~ A, SA maybe any suitableC function chosensothat SA(x) is smoothat I x = A — A’. (Note that if A’ = 0, SA(x)cannotin generalbedifferentiableat I x = A). Since SA(x) is bounded,smoothandperiodic, it followsfrom sect. 2 that if ZA(T) is the solution to the Langevinprocess(2.8) with action5A’ and
AAk(T) E(exp(ikZA(T))), (A.1)
then
A~k=Atf dx eIkx exp(—SA). (A.2)
Since e S is of rapid decrease,by the•dominatedconvergencetheoremit followsthat
1lim A~k= —f dx eil~xe_S. (A.3)
At
Appendix B
First, considerthe compactcasex E [a, b]. In orderfor a weightfunction m(x)to representa stationarysolution to eq.(2.26)
= fb dx e”~m(x), (B.1)
848 5. Lee/ ComplexLangevinsimulations
m(x) must satisfy the stationary pseudo Fokker—Planckequation and satisfym(a)= m(b). However,for the “spurious” solution (2.28),1~(a)# 1~(b),since
jb dy ~ ~ 0. (B.2)
Whenx E l~,m(x) mustsatisfymorethanm( — cc) = m(oo)= 0. By direct substi-tution into eq.(2.26) it is seenthat the necessaryboundaryconditionis
aS(x)
a m(x) —p0, lxi —~co. (B.3)
However, as may seenby differentiation, for I x I>> 1,
x
f dy eS~ a5S(x)’ (B.4)
hence,as x I —‘ cc
aS(x) -
3x P~(x)—~1. (B.5)
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