PPerturbative Removal of a Sign Problem
Scott Lawrence ∗ Department of Physics, University of Colorado, Boulder, CO 80309, USA (Dated: October 27, 2020)This paper presents a method for alleviating sign problems in lattice path integrals, including thoseassociated with finite fermion density in relativistic systems. The method makes use of informationgained from some systematic expansion — such as perturbation theory — in order to acceleratethe Monte Carlo. The method is exact, in the sense that no approximation to the lattice pathintegral is introduced. Thanks to the underlying systematic expansion, the method is systematicallyimprovable, so that an arbitrary reduction in the sign problem can in principle be obtained. TheThirring model (in 0 + 1 and 1 + 1 dimensions) is used to demonstrate the ability of this method toreduce the finite-density sign problem.
I. INTRODUCTION
Lattice Monte Carlo methods are able to provide non-perturbative access to observables in quantum field theo-ries. They are unique in this respect for many stronglycoupled theories. Under certain circumstances, such asat finite density of relativistic fermions and the Hubbardmodel away from half-filling, lattice methods are madedramatically less efficient by the so-called sign problem.This sign problem is a central obstacle to first-principlescalculations in many regimes of strongly coupled theories,including ab initio studies of the nuclear equation of state.In lattice field theory, spacetime is treated as discreteand observables are obtained from the high-dimensionallattice path integral. Lattice field theory is ordinarilyused to study a system in thermal equilibrium, and thepartition function is written as Z = R DA e − S ( φ ) , where S is the (Euclidean) action and the integral is taken overall configurations of a field A . Observables are givenby various derivatives of the logarithm of the partitionfunction. These derivatives are ordinarily sampled by im-portance sampling, which hinges on the treatment of thenormalized Boltzmann factor e − S /Z as a probability dis-tribution. For some systems, including those with a finitedensity of relativistic fermions, the action S is complex,and this is not possible — this is the sign problem.Importance sampling commonly takes a polynomialamount of time in the spacetime volume being simulated(although this is proven only in a few cases [1–3]). Impor-tance sampling can be modified to work even where S iscomplex, but at the cost of efficiency. In this modification,the “quenched” Boltzmann factor | e − S | /Z is treated as aprobability with respect to which sampling is performed.Ordinary expectation values are obtained in terms ofquenched expectation values: hOi = hO e − iS I i Q / h e − iS I i Q .The loss of efficiency comes primarily from the denomi-nator. The average of the exponential of the imaginarypart of the action, often termed the “average phase”, isequal to the ratio of the physical to quenched partitionfunctions Z/Z Q , and characteristically scales like e − βV . ∗ [email protected] Resolving this exponentially small quantity, by averagingmany quantities of unit magnitude, requires ∼ e βV sam-ples; thus the reweighting procedure incurs an exponentialcost in the volume. This failure affects a wide variety ofmodels, and a correspondingly wide variety of methodshave been proposed to mitigate it: complex Langevin [4],the density of states method [5], canonical methods [6, 7],reweighting methods [8], series expansions in the chemicalpotential [9], fermion bags [10], field complexification [11],and analytic continuation from imaginary chemical po-tentials [12].In this paper we will examine a new method, inspired bytwo observations: first, that the partition function is un-changed if a function that integrates to zero is added to theBoltzmann factor, and second, that lattice methods canencounter a fatal sign problem even in regimes under goodcontrol by perturbation theory (or any other systematicexpansion). To any fixed order in perturbation theory, thesign problem can be (non-uniquely) identified with someoscillating part of the Bolzmann factor which integratesto zero, and this part can then be subtracted off, withoutchanging the partition function or any observables. Infact, we will see that this subtraction can be performed insuch a way that even the nonperturbative partition func-tion and observables remain unchanged. Where the modelis under good control by perturbation theory, meaningthat the partition function is well-approximated by theintegral of a perturbative expansion of the integrand, thissubtraction is nearly the entire sign problem. In regimeswhere perturbation theory is a poor approximation, wemay hope to isolate and remove a single component ofthe sign problem, thereby improving the efficiency of thenecessary nonperturbative calculation.The method described in this paper exhibits two favor-able characteristics worth noting before we begin. Firstly,it is an exact method, in the sense that the modified formof the partition function is precisely equal to the original,physical form. As a consequence, all observables retaintheir physical values, and the only errors are statisticalones associated to the sampling process. This is trueregardless of the quality of the systematic expansion used:the removal of the sign problem is approximate, but theobservables computed are exact. Secondly, although the a r X i v : . [ h e p - l a t ] O c t removal of the sign problem is approximate, it is sys-tematically improvable . If a certain order in perturbationtheory does not yield a sufficiently moderate sign problem,a higher order can in principle be used. As long as the ex-pansion converges (on the lattice), a sufficiently high orderis guaranteed to remove the sign problem to any desireddegree. Of course, an exponential cost is associated withgoing to higher orders in most expansions, and it is to beexpected that this property of systematic improvability isnot a practical way to solve many problems, as it merelytrades one exponential cost for another. Nevertheless,this is an unusual and promising combination.This paper uses the Thirring model [13] in 0 + 1 and1 + 1 dimensions as a testbed for the method of subtrac-tions. This model has frequently been used, in varyingdimensions, to test methods for treating the fermion signproblem in the past, including complexification [14, 15]and complex Langevin [16].In the next section, the general method of subtractionsis described in detail, with an emphasis on subtractionsthat are constructed via some systematic expansion. InSec. III, the heavy-dense limit is used to construct asubtraction for the Thirring model in 0 + 1 dimensions.This is extended in Sec. IV, where the 1 + 1-dimensionalThirring model is treated with a variety of expansions.A nonperturbative method of optimizing subtractionsis described in Sec. V. Finally we conclude in Sec. VI,discussing in particular a relation between this methodand the method of field complexification. II. GENERAL METHOD
For brevity, let us write the Boltzmann factor as f ( A ) ≡ e − S ( A ) , so that the unmodified form of the partitionfunction is Z = R D A f ( A ). If we let g ( A ) be somefunction which integrates to 0 (e.g. a total derivative ofa function with appropriate behavior on the boundaryof configuration space), then the numerical value of thepartition function is unmodified by the subtraction of g ( A ) from the Boltzmann factor: Z = Z D A f ( A ) = Z D A f ( A ) − g ( A ). (1)The quenched partition function, and therefore the av-erage phase h σ i ≡ Z/Z Q , is generically changed by thisoperation. Therefore, a suitable g ( A ) may improve thesign problem. In fact, a subtraction always exists whichremoves the sign problem entirely: g ideal ( A ) = f ( A ) − R D A f ( A ) R D A . (2)This particular subtraction is unusable in practice, ascomputing it requires exact knowledge of the partitionfunction. Indeed, using this subtraction is equivalent toperforming the entire computation analytically.Particularly in the case where g is constructed froma perturbative expansion (described below) this method can be thought of as splitting the path integrand into afew terms, and integrating some analytically. In the caseof the ideal subtraction of Eq. (2), the entire path integralis performed analytically.Once a subtraction is selected, it remains to compute anobservable. We must express hOi (an expectation valueover f ) as an expectation value taken over the distribution f − g . It is tempting to write hOi = R D A ( f ( A ) − g ( A )) O ( A ) f ( A ) f ( A ) − g ( A ) R D A f ( A ) − g ( A ) . (3)This equation is correct, but not useful for computing theexpectation value, as the measurement of the modifiedobservable encounters a signal-to-noise problem compa-rable to the original sign problem. This is particularlyclear in the case of O = 1, where the numerator is equalto R D A f ( A ), the highly oscillatory integral we wantedto avoid in the first place.Consider a conjugate variable ξ to O , such that hOi = ∂∂ξ log Z . The previous approach corresponds to treating g as constant in ξ . Instead, take g to vary with ξ , insuch a way that R g = 0 for any value of ξ . The desiredexpectation value is now hOi = R D A O ( A ) f ( A ) − ∂∂ξ g ( A ) R D A f ( A ) − g ( A ) , (4)which does not necessarily (and does not in practice, as wewill see) suffer from the same magnitude of signal-to-noiseproblem.We now discuss how to construct a suitable subtraction g ( A ) in a systematic manner. One strategy is to attemptto approximate Eq. (2) as closely as possible, with ananalytic expansion. For the purposes of removing the signproblem, however, it is sufficient to replace f ( · ) in Eq. (2)by just the part of the Boltzmann factor that oscillates.Removing the oscillations will cure the sign problem, evenif the rest of the partition function is not approximatedwell at all.To make this concrete, suppose a perturbative expan-sion of f ( A ) f ( A ) = f ( A ) + λf ( A ) + λ f ( A ) + · · · (5)is available, such that the partition functions at low or-der are readily (perhaps analytically) obtained. Defining Z n = R D A f n ( A ), we can construct a wide variety offunctions which integrate to 0 and approximate variousparts of the original Boltzmann factor. It is often conve-nient to pick (some linear combination of) g n ( A ) = f n ( A ) − f ( A ) Z Z n . (6)The factor of the free theory Boltzmann factor is some-what arbitrary — any function of A with unit integralwill do.This procedure does not depend on the precise natureof the systematic expansion. Our first application of thismethod (in Sec. III) will use the heavy-dense limit toconstruct a subtraction, instead of an expansion aroundfree field theory.Because the subtracton was constructed from a sys-tematic expansion, g n naturally depends on ξ . ApplyingEq. (4) to this construction, the physical expectationvalue of O is given by hOi = * O f − ∂∂ξ f n + f Z ∂∂ξ Z n f − g n + f − g n . (7)Note that it is not in general true that ∂∂ξ f n = O f n , nor isit generally true that the same derivative of log Z n yieldsa perturbative expectation value.In deriving this expression, we have chosen for conve-nience not to let f , and therefore Z , vary with ξ . This, like the precise manner of constructing the subtraction, isan arbitrary choice. We will not, in this paper, explore thequestion of what the optimal construction of a modifiedobservable is.Of course, even after the subtraction, a residual signproblem typically remains, which is addressed by reweight-ing. III. QUANTUM MECHANICS
In this section we demonstrate the method on a 0 + 1-dimensional variant of the Thirring model. Describedin [15, 17], this model is defined by the lattice action S = 12 g X t (1 − cos A ( t )) − log det K [ A ] . (8)The Dirac matrix K [ A ] is given by K [ A ] tt = 12 (cid:2) e µ + iA ( t ) δ ( t +1) t − e − µ − iA ( t ) δ ( t +1) t − e µ + iA ( t ) δ tN δ t + e − µ − iA ( t ) δ t δ t N (cid:3) + mδ tt . (9)Above, m is the bare mass and g a coupling constant; weare implicitly working in units where the lattice spacingis 1, so that the number of sites is equal to the inversetemperature β . The sign problem, created by the chemicalpotential µ , is portrayed in Fig. 1; the average phasedecays exponentially with the inverse temperature, andso the cost of calculations increases exponentially withthe same.A suitable subtraction is provided by the heavy-denselimit of µ → ∞ . The Dirac matrix can be expanded viathe polymer representation [18], and the dominant termof det K in the limit of large µ isdet K = e βµ (2 − β e i P t A ( t ) + O ( e − βµ )). (10)We will use the leading-order term as our subtraction: f ( A ) = e g P t cos A ( t ) × − β e βµ + i P t A ( t ) . (11)Integrating over all fields yields the leading-order partition function Z = e βµ (cid:2) πI (cid:0) / g (cid:1)(cid:3) β . (12)(Here and throughout, I ν ( · ) denotes the modified Besselfunction of the first kind, of order ν .)For the scaling factor f ( A ) /Z in Eq. (6) we couldsimply choose (2 π ) − β , but it is convenient in this case touse the bosonic part of the Boltzmann factor: f ( A ) Z = exp (cid:16) g P t (1 − cos A ( t )) (cid:17) [2 πI (1 / g )] β . (13)The observable we will focus on is the number density,defined as h n i = β − ∂∂µ log Z . In order to measure thisobservable with the subtraction method, we need the µ -derivatives of f and Z as per Eq. (7). Happily, inthis case they are particularly simple: ∂∂µ f = f and ∂∂µ Z = Z . This reflects the fact that, in the heavy-dense limit, the density is 1 regardless of temperature.To summarize, before performing the subtraction, thepartition function was written Z = R e − S with the action S defined by Eq. (8). The modified form of the partitionfunction is Z = Z D A exp g X t (1 − cos A ( t )) !| {z } f (cid:20) det K | {z } f/f − − β e βµ + i P t A ( t ) | {z } f /f + e βµ (cid:2) πI (1 / g ) (cid:3) β [2 πI (1 / g )] β | {z } Z /Z (cid:21) , (14)where the scaling factor f and its integral Z are defined by Eq. (13), and the subtraction is constructed from theheavy-dense term f and its integral Z , given in Eqs. (11)and (12).While numerically identical, this form is hoped to havea reduced sign problem. The density is given by theexpectation value, taken in the subtracted ensemble, β h n i = * Tr K − ∂K∂µ − f + f Z Z f − f + f Z Z + f − g . (15)The results of this procedure are shown in Fig. 1. Spe-cially in 0 + 1 dimensions, the sign problem is no longerexponential in the volume, but rather improves slightly as β is increased. This is not to be expected to hold true forhigher dimensional theories. In general, the exponentialdifficulty of the sign problem will not be removed by thesubtraction method, but merely ameliorated. (In the caseof the particular model at hand, it is possible to constructa subtraction that entirely removes the sign problem, butonly because the entire partition function is analyticallyknown.)Lastly, note that all data points in Fig. 1 are constructedfrom 10 samples. The data points calculated with thesubtraction have much smaller error bars (for µ = 2 .
0, theerror bar width is ∼ − ) even than the sign-free µ = 0data point without the subtraction; this procedure hasimproved the signal-to-noise ratio in addition to reducingthe sign problem. In the limit of the ideal subtraction ofEq. 2, there is no variance remaining in the observable,and a single measurement yields the exact answer. IV. FIELD THEORY
We now move to the 1 + 1-dimensional Thirring modelwith staggered fermions. The lattice action of this modelis [14] S = X x,ν =0 , g (1 − cos A ν ( x )) − log det K [ A ] (16)with the Dirac matrix now defined by K [ A ] xy = mδ xy + 12 X ν =0 , η ν e iA ν ( x )+ µδ ν, δ x + ν,y (17) − η ν e − iA ν ( y ) − µδ ν, δ y + ν,x ,where as before m is the bare mass, g the coupling, and µ the chemical potential. The staggered fermions aredefined by η = ( − δ x and η = ( − x . As in the0 + 1-dimensional model, a sign problem is created at µ = 0.The first subtraction procedes from the same heavy-dense limit we used for the quantum mechanical modelabove. As before, we define f = e g P x,ν cos A ν ( x ) . Theleading-order term in the heavy-dense expansion is f = e g P x,ν cos A ν ( x ) − βL e βLµ + i P x A ( x ) (18) which, when integrated over all fields, yields the partialpartition function Z = e βLµ − βL (cid:0) πI (2 /g ) I (2 /g ) (cid:1) βL . (19)At this order in the heavy-dense expansion, everythingtakes the form of L copies of the quantum mechanicalmodel above. In particular, the µ -derivatives of f and Z are Lf and LZ , respectively.The results of simulating with the leading-order heavy-dense subtraction, on a 12 × (cid:46) − ) by µ ≈ .
8; after thesubtraction, the sign problem is manageable from µ = 0through lattice saturation.At the next order in the heavy-dense limit, the numberof diagrams in the polymer representation is exponentialin β . Therefore, it is not practical (barring another wayof computing the NLO heavy-dense partition function) touse this expansion at higher orders. Another expansionto consider is the hopping expansion. However this ex-pansion is also not practical for the purpose of removinga sign problem, as the lowest-order term in the hoppingexpansion that has a sign problem is at order κ β .At small g , the auxiliary field is pegged to A ∼ A term in the action. As a result, it is possibleto construct a “weak-coupling” expansion for the lat-tice Thirring model described here by Taylor expandingdet K [ A ] in the fields A . The term first-order in A makesa particularly convenient subtraction: as it is odd in A ,it integrates to 0, and the corresponding partial partitionfunction Z vanishes. The subtracted integrand of thepartition function is f − g = f (cid:20) det K − det K Tr K − (cid:18) ∂K∂A (cid:19) A =0 A (cid:21) (20)where K is K evaluated at A = 0, and f = e g P x,ν cos A ν ( x ) as usual. Fig. 3 shows the magnitudeof the sign problem on a 6 × g ,the subtraction is no longer guaranteed to help. V. NONPERTURBATIVE OPTIMIZATION
So far, we have described how a suitable subtraction canbe engineered with the aid of a systematic expansion, suchas the weak coupling or heavy-dense limit. Subtractionsconstructed in this manner need not be optimal, and itmay be profitable to consider other possibilities. In thissection we will see that it is possible to efficiently performa nonperturbative optimization on a family of ansatzsubtractions to find the one with the largest averagephase. The method discussed here was used in a verysimilar form for optimizing manifolds of integration [20], − . − . . . . .
81 0 0 . . h n i µ OriginalSubtracted . . . .
81 0 0 . . h σ i µ − . . . . .
81 0 5 10 15 20 25 30 35 h σ i β FIG. 1. The subtraction method as applied to the 0 + 1-dimensional Thirring model. The leftmost plot shows the density as afuncton of chemical potential with β = 8, m = 1, and g = 0 .
2. The exact result is from [19]. The center plot shows the averagephase, again as a function of µ , for the same parameters. On the right is the average phase for µ = 1 . β . . . . .
81 0 0 . . . . . . h n i µ OriginalSubtracted . . . .
81 0 0 . . . . . . h σ i µ FIG. 2. Simulation of the 1 + 1-dimensional Thirring model on a 12 × m = 0 . g = 0 .
3. The left plot shows thedensity as a function of chemical potential, and on the right are the corresponding average phases. Each data point is backed by10 samples. . . . .
81 0 0 .
02 0 .
04 0 .
06 0 .
08 0 . .
12 0 .
14 0 .
16 0 .
18 0 . h σ i g OriginalSubtracted
FIG. 3. The sign problem on a 6 × m = 0 .
15 and chemical potential µ = 1, with and without thesubtraction of Eq. (20). and has been applied (in one form or another) to severaldifferent field theories [21–23].Suppose we have a continuous family of actions S α (the parameter α may have many components), such thatthe partition function Z = R e − S α does not depend on α . This is exactly the case if α defines a subtraction, or asin [20], a manifold of integration. Although the partitionfunction has no dependence on α , the quenched partitionfunction and therefore the sign problem may. In general,computing the sign problem for any fixed α is computa-tionally expensive. We would like to invest computationalresources efficiently, performing a simulation with thevalue of α that has the mildest sign problem. However,finding such a value appears hard: it certainly isn’t fea-sible to do a grid search, resolving the sign problem foreach value of α , in order to find the best one.Consider performing gradient ascent on the logarithmof the average phase. Arbitrarily picking some initial α ,we would like to calculate ∂∂α log ZZ Q ( α ) , which specifies thedirection in which we should move. If we were to calculatethis by finite differencing, we would need to resolve thesign problem at both α and α + (cid:15) , an expensive proposition.However, observe that ∂∂α log ZZ Q ( α ) = − ∂∂α log Z Q ( α ) (21)has the form of a derivative of the logarithm of the quenched partition function, and the contribution of the . . . .
81 0 0 . . . . . h σ i α FIG. 4. The magnitude of the sign problem for a 4 × m = 0 . g = 0 .
3, and µ = 1, as a function of thesubtraction coefficient α , using the subtraction of Eq. (22). physical Z cancels entirely. The direction which mostquickly alleviates the sign problem is a quenched expecta-tion value, which can be computed without encounteringa sign problem.With this observation in hand, we see that it is possibleto begin with a family of subtractions g α , and performan efficient, sign-free gradient descent to find the optimalsubtraction in that family. At this point, a (comparativelyexpensive) Monte Carlo can be performed, with highstatistics to counter the remaining sign problem.One motivation for this method stems from the “weak-coupling” subtraction of the previous method. The sub-traction g = f − Z defined by Eq. (20) can be multipliedby an arbitrary coefficient α , so that the integrand of thepartition function is modified by − g α = − αf det K Tr K − (cid:18) ∂K∂A (cid:19) A =0 A . (22)In the previous section, the coefficient used was implicitly1; as shown in Fig. 4, it turns out that this is not theoptimal coefficient. The optimization procedure describedabove can be used to optimize this coefficient at scale.Note that for the example shown here, the full-magnitudefirst-order subtraction makes the sign problem worse at g = 0 .
3. However, nonperturbative optimization canreverse this, making the first-order subtraction usefuleven at this relatively large coupling.
VI. DISCUSSION
The method of subtractions described in this paperallows practical mitigation of sign problems associatedto finite fermion density and real-time observables. Themethod is exact in the sense that it makes no additionalapproximations in the partition function. Furthermore,the removal of the sign problem, although only approxi-mate, is systematically improvable.This method is not unrelated to prior work. In particu-lar, the method of field complexification [11] may be seen as a specific strategy for constructing a subtraction . Inthat method, the original domain of the path integral — R N , say — is expanded to a complex space of twice the(real) dimension. In this case, the expanded space wouldbe C N . By Cauchy’s integral theorem, the path integralcan now be performed over any N -real-dimensional man-ifold M ⊂ C N obtained by a smooth deformation from R N (and with mild constraints at infinity, when the com-plex space is unbounded). Typically the new manifold isparameterized by the real plane via a function ˜ φ mappingfield configurations φ ∈ R N to field configurations on M ,so that the deformed path integral is written Z = Z R N D φ e − S [ φ ] = Z R N D φ e − S [ ˜ φ ( φ )] det ∂ ˜ φ∂φ . (23)The difference between the two integrands is zero, andso can be viewed as a subtraction. Of course, in thisview, every modification to the path integral that leavesthe integration domain unchanged is a special case of thesubtraction method.We can also go a step further and note that the dif-ference between the two integrands is a total derivative.Concretely, in one dimension, the difference between thetwo Boltzmann factors is e − S [ ˜ φ ( φ )] det ∂ ˜ φ∂φ − e − S [ φ ] = ∂∂φ Z ˜ φ ( φ ) φ e − S [ φ ] d φ . (24)It is notable that a well-chosen subtraction can resolvea sign problem even in cases where no manifold can. Asimple example of a sign problem unremovable by anychoice of manifold is the one-dimensional integral (whichis to be considered a mock partition function) Z = Z π − π d θ [cos( θ ) + (cid:15) ] . (25)The sign problem associated to this partition functionbecomes arbitrarily bad as (cid:15) is taken towards 0. Thissign problem was shown in [24] to be unremovable byany choice of integration contour. In fact, the originalintegration domain S has a more mild sign problem thanany other choice of domain. In this case, it’s particularlyeasy to see that a subtraction of cos θ completely removesthe sign problem, where no manifold can. Thus themethod of subtractions is strictly more powerful thanthat of complexification.The manifold used in [20, 21] to improve the sign prob-lem of the Thirring model in 1 + 1 and 2 + 1 dimensionswas motivated (post-hoc) by the leading-order term in theheavy-dense expansion. In [25] it was shown that a man-ifold of that form can entirely remove the sign problem In fact, the subtraction method was initially inspired by anattempt to extend the method of field complexification to thecase of path integrals with discrete domains of integration. coming from that leading-order term. This choice of man-ifold is therefore equivalent to a subtraction constructedfrom that term.The complexification method has been applied to real-time observables through the lattice Schwinger-Keldyshformalism [26]. The determination of real-time observ-ables on the lattice remains a largely unexplored area. Fu-ture work should be able to apply the subtraction methodto real-time calculations through the same formalism.The success of the method described in this paperdepends on the availabilty of a systematic expansion inwhich the sign problem can be seen. We have seen thatseveral options exist for the Thirring model. Examiningand making use of such expansions in other models is acritical next step.We noted in Sec. III that in addition to improving thesign problem, the signal-to-noise ratio associated with themodified observable was improved from the one associ-ated with the original observable. This was not explored further in this paper, but it suggests that the same ora similar method could be deployed explicitly for treat-ing expensive signal-to-noise problems. It is not entirelysurprising that this should be possible, as the closely re-lated complexification method has recently been appliedto noisy observables in Abelian gauge theory and complexscalar field theory [27].
ACKNOWLEDGMENTS
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