Avoiding the sign-problem in lattice field theory
AAvoiding the sign-problem in lattice field theory
Tobias Hartung, Karl Jansen, Hernan Le¨ovey, and Julia Volmer
Abstract
In lattice field theory, the interactions of elementary particles can be com-puted via high-dimensional integrals. Markov-chain Monte Carlo (MCMC) methodsbased on importance sampling are normally efficient to solve most of these integrals.But these methods give large errors for oscillatory integrands, exhibiting the so-calledsign-problem. We developed new quadrature rules using the symmetry of the con-sidered systems to avoid the sign-problem in physical one-dimensional models forthe resulting high-dimensional integrals. This article gives a short introduction tointegrals used in lattice QCD where the interactions of gluon and quark elementaryparticles are investigated, explains the alternative integration methods we developedand shows results of applying them to models with one physical dimension. Thenew quadrature rules avoid the sign-problem and can therefore be used to performsimulations at until now not reachable regions in parameter space, where the MCMCerrors are too big for affordable sample sizes. However, it is still a challenge todevelop these techniques further for applications with physical higher-dimensionalsystems.
Karl Jansen · Julia Volmer - Speakers,DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germanye-mail: [email protected], e-mail: [email protected] HartungDepartment of Mathematics, Kings College London, Strand, London WC2R 2LS, United Kingdome-mail: [email protected] Le¨oveyStructured Energy Management, Axpo Trading, Parkstrasse 23, 5400 Baden, Germanye-mail: [email protected] 1 a r X i v : . [ h e p - l a t ] F e b Tobias Hartung, Karl Jansen, Hernan Le¨ovey, and Julia Volmer
Monte Carlo (MC) methods are in general very efficient to solve high-dimensionalintegrals. They use the law of large numbers to approximate an integral with quadra-ture rules that use random sampling points. But MC methods are highly inefficient for oscillatory integrand functions, e.g. the function shown in Figure 1a. An exact integration of oscillatory functions would, of course, result in the cancellation oflarge negative and positive contributions to the integral - in the example in Figure 1athis would give an integral of zero. However, the random choice of sampling pointsin MC methods, shown as black points in Figure 1a, does lead only to approximatecancellation when the number of points is relatively big and hence, it is very difficultto obtain accurate results with affordable sample sizes. This non-perfect cancellationof negative and positive parts in the integration method, usually resulting in largequadrature rule errors, is called the sign-problem . The sign-problem is for examplethe reason why important physical interactions in the early universe cannot be simu-lated which could explain why there is more matter than anti-matter in our universetoday. To acquaint better knowledge of these fundamental phenomena, it is essentialto develop alternative quadrature rules to MC that avoid the sign-problem.In physical applications, the function to-be-integrated describes some characteris-tic in a given physical model. We investigated methods that use some symmetry ofthe physical model to result in the exact cancellation of positive and negative parts inthe quadrature rule. If the model behind the function in Figure 1b has a reflectionsymmetry, few MC sampling points - in black - can be chosen and together with theirreflected - white - points they form a set of sampling points that results in an exactquadrature rule. In this specific example even one MC point with its reflection pointwould give an exact result, for more complicated functions more sampling points areneeded.This article first gives a short introduction to the high-dimensional integrals thathave to be solved in particle physics, more precisely in lattice QCD. Readers that aremostly interested in the integration methods can easily skip this part. The main part ofthis article presents the methods we developed and tested to avoid the sign-problemfor high-dimensional integration in physical one-dimensional systems.We found that symmetrically chosen quadrature rules can avoid the sign-problemand can efficiently be applied also to high-dimensional integrals. These rules can helpto perform simulations in important, not-yet reachable regions in parameter space, atleast in physical one-dimensional systems so far. To apply them to higher physicaldimensions, in particular to physical four-dimensional systems in high energy physicsas lattice QCD, they clearly need to be developed further.
In theoretical physics the interaction between elemetary particles such as the elec-tron, is described by quantum field theories (QFT), see e.g. [12]. The mathematical
Page: 2 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 3 xf (a) MC sampling points xf (b) Symmetric sampling points Fig. 1: MC integration of an oscillatory function results in large errors, known as thesign-problem. This problem is due to the non-cancellation of positive and negativecontributions to the quadrature rule (a). Choosing sampling points by using thesymmetry of the underlying model results in an exact quadrature rule (b).formalism in QFT defines particles as classical fields that are functions in three spacedimensions and one time dimension, P ( x , y , z , t ) . Operators, O [ P ] , are functionals ofthese fields and describe the interactions between them. An expectation value A ofthis interaction or operator O [ P ] , also called amplitude, is computed via the pathintegral, A = (cid:82) O [ P ] B [ P ] d P (cid:82) B [ P ] d P . (1) (cid:82) d P is the infinite-dimensional integration over all possible states of the field P intime and space. The path integral becomes a well defined expression, if a Euclideanmetric is used and the fields are defined on a finite dimensional, discrete lattice . In(1), B [ P ] is called the Boltzmann-weight and provides a probability which weights theparticle (field) interactions. The denominator in (1) insures the proper normalizationof A . The expectation value A is interesting because physical observables can bederived from it and their numerical values can be compared with experimental resultsor can give new results that are not yet possible to reach with experiments.In lattice field theory , space-time and the involved functionals O [ P ] and B [ P ] arediscretized in Euclidean space, such that (1) can be computed numerically. Often, theBoltzmann-weight is a highly peaked function suggesting that this computation can bedone using importance sampling techniques. In most computations, this importancesampling is done by a Markov chain MC (MCMC) algorithm using a Markov chainthat leaves the distribution density B [ P ] (cid:82) B [ P ] d P invariant. To compute A numerically, four-dimensional space-time is discretized on a four-dimensional lattice with four direc-tions µ ∈ { , , , } , lattice sites nnn ∈ Λ = { ( n , n , n , n ) | n , n , n , n ∈ { , ..., d }} For an alternative definition using the ζ -regularization see [24, 25]. Page: 3 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41
Tobias Hartung, Karl Jansen, Hernan Le¨ovey, and Julia Volmer and discretized fields P . This results in an 4 d -dimensional integration over the Haarmeasure of the compact group S U ( ) . For real applications, d can be very large,reaching orders of magnitude of several thousands nowadays. Thus, we are left withan extremely high dimensional integration problem. Moreover, for some physicallyvery important questions MCMC methods cannot be applied succesfully. This con-cerns, for example, the very early universe or the matter anti-matter asymmetrywhich leads to our sheer existence. Thus, a number of interesting questions remaincompletely unanswered and it is exactly here where new high dimensional integrationmethods could be extremely helpfulStill, the MCM methods have led to very successful computations already. Byperforming numerical computations on massively parallel super computers a veryimpressive result of such a lattice MCMC can be obtained: namely, the mass spec-trum of the lightest composite particles made out of quarks and gluons that agreescompletely with the experimental values, see Figure 2. To get similar precise resultsfor other, more error-prone observables, research is going on to develop new methodsto make this high-dimensional integration faster and the results more precise.Fig. 2: The via lattice QCD computed masses of different composite particles (dotswith vertical error bars) agree with the experimentally measured values (horizontallines with error boxes) [10]. The masses of π , K and ∑ (dots without error bars) wereinput values to the computation.A more detailed introduction to lattice QCD is for example given in the text-books [13, 14, 15]. Page: 4 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 5
In lattice QCD, the amplitude of interactions between quarks and gluons in physicalfour-dimensional space-time can be computed via a high-dimensional integral usingthe Haar-measure over the compact group
S U ( ) , see section 2. This integral istypically solved numerically using MCMC methods. If the integrand is an oscillatoryfunction, this method results in the sign-problem that gives large errors and avoidsphysical insights in important processes. We developed alternative methods that avoidthe sign-problem and at the same time are efficient for high-dimensional integrationover compact groups. Due to various complications with physical four-dimensionallattice QCD, we developed and tested the methods for physical one-dimensionalmodels that involve low-dimensional and high-dimensional integration over compactgroups. As suggested in section 1, we developed quadrature rules using the symmetryof the models.This section is structured from low-dimensional to high-dimensional integration:First, it introduces symmetric quadrature rules for one-dimensional integration overcompact groups to avoid the sign-problem here. Then, it presents the recursivenumerical integration (RNI), a method to reduce high-dimensional integrals to nestedone-dimensional integrals. Finally, it shows how to combine both methods to avoidthe sign-problem for high-dimensional integration over compact groups. For all threepresented methods, the section shows results of applying them to simple physical,one-dimensional models. More detailed explanations of the methods and applicationscan be found in [22]. The sign-problem can already arise in a one-dimensional integration, solving I ( f ) = (cid:90) G f ( U ) d U (2)with MC methods over the Haar-measure of G ∈ { U ( N ) , S U ( N ) } . Finding analternative suitable quadrature rule Q ( f ) ad-hoc to approximate this integral is notstraightforward. The articles [8, 9] suggest that using symmetrically distributedsampling points can be beneficial for avoiding the sign-problem, possibly resulting inan exact cancellation of positive and negative contributions to the integral, as stated insection 1. The article of Genz [16] gives efficient quadrature rules for integrations overspheres, choosing the sampling points symmetrically on the spheres. We searchedfor measure preserving homeomorphisms to apply the symmetric quadrature ruleson spheres to the integration over compact groups. This section describes the twosteps to create the symmetric quadrature rules Q ( f ) for (2):Sym 1. Rewrite the integral I ( f ) over the compact group G into an integral over spheres.We restricted ourselves to G ∈ { U ( ) , U ( ) , U ( ) , S U ( ) , S U ( ) } . Page: 5 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41
Tobias Hartung, Karl Jansen, Hernan Le¨ovey, and Julia Volmer
Sym 2. Approximate each integral over one spheres by a symmetric quadrature rule asproposed in Genz [16], and combine them to a product rule Q ( f ) .Finally, this section shows results of applying Q ( f ) to the one-dimensional QCDmodel with a sign-problem. A more detailed explanation of the method can be foundin [5, 3].By finding measure preserving homeomorphisms between the compact groupsand products of spheres we created polynomially exact quadrature rules for compactgroups. The application of these rules to the one-dimensional QCD model gaveresults on machine precision where the standard MC method shows a sign-problem.Therefore the symmetric quadrature rules avoid the sign-problem and give rise tosolve integrals in beforehand non-reachable parameter regions. The symmetric quadrature rules of Genz [16] are designed for the integration over k -dimensional spheres S k . To use them for the integration over the compact groups U ( N ) and S U ( N ) with N ∈ { , } in (2), the compact groups have to be associatedwith spheres. The facts that U ( N ) is isomorphic to the semidirect product of S U ( N ) acting on U ( ) (cid:0) U ( N ) ∼ = SU ( N ) (cid:111) U ( ) (cid:1) , that U ( ) is isomorphic to S (cid:0) U ( ) ∼ = S (cid:1) and that S U ( N ) is a principal S U ( N − ) bundle over S N − result in S U ( N ) (cid:39) S × S × ... × S N − , (3) U ( N ) (cid:39) S × S × ... × S N − . (4)Then, the integral over the Haar-measure of G in (2) can be rewritten as the integralover products of spheres, (cid:90) G d U f ( U ) = (cid:90) S N − (cid:32) (cid:90) S N − (cid:18) · · · (cid:90) S n + (cid:16) (cid:90) S n f (cid:0) Φ ( xxx S N − , xxx S N − , . . . , xxx S n + , xxx S n ) (cid:1) (5)d xxx S n (cid:17) d xxx S n + · · · (cid:17) d xxx S N − (cid:17) d xxx S N − , with n = U ( N ) and n = S U ( N ) [4]. Here, xxx S k is an element on the k -sphere and Φ : × j S j − → G with G ∈ { U ( N ) , S U ( N ) } is a measure preservinghomeomorphism. We found the homeomorphisms Φ G ≡ Φ for the compact groups G ∈ { U ( ) , U ( ) , U ( ) , S U ( ) , S U ( ) } : • For
S U ( ) , Φ is an isomorphism, given by Φ S U ( ) : S → S U ( ) , xxx (cid:55)→ (cid:18) x + ix − ( x + ix ) ∗ x + ix ( x + ix ) ∗ (cid:19) . (6) Page: 6 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 7 • For
S U ( ) , spherical coordinates of S are needed, Ψ : [ , π ) × [ , π ) → S , ( α , α , α , φ , φ ) (cid:55)→ cos α sin φ sin α sin φ sin α cos φ sin φ cos α cos φ sin φ sin α cos φ cos φ cos α cos φ cos φ . (7)Then, Φ is given by Φ S U ( ) : S × S → S U ( ) , ( xxx , yyy ) (cid:55)→ A ( Ψ − ( xxx )) · B ( yyy ) , (8)with the matrices A ( Ψ − ( xxx )) = e i α cos φ i α sin φ − e i α sin φ sin φ e − i ( α + α ) cos φ e i α cos φ sin φ − e i α sin φ cos φ − e − i ( α + α ) sin φ e i α cos φ cos φ , (9) B ( yyy ) = x + ix − ( x + ix ) ∗ x + ix ( x + ix ) ∗
00 0 1 . (10) Ψ − ( xxx ) is the inverse transformation of (7) from Euclidean to spherical coordi-nates. S denotes S without its poles, φ = φ =
0, because at these pointsthe inverse transformation is not unique. The therefore excluded set is a null set,thus Φ S U ( ) can still be used in (6). • For U ( ) , Φ is an isomorphism, Φ U ( ) : S → U ( ) , α (cid:55)→ e i α , (11)with α ∈ [ , π ) . • For U ( ) , Φ is an isormophism, Φ U ( ) : S × S → U ( ) , ( xxx , α ) (cid:55)→ Φ S U ( ) ( xxx ) · diag ( e i α , ) . (12) • For U ( ) , Φ is given by Φ S U ( ) : S × S × S → U ( ) , ( xxx , yyy , α ) (cid:55)→ Φ S U ( ) ( xxx , yyy ) · diag ( e i α , , ) . (13) Page: 7 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41
Tobias Hartung, Karl Jansen, Hernan Le¨ovey, and Julia Volmer
With the measure preserving homeomorphism Φ in section 3.1.1, the integral (2) canbe written as an integral over a product of spheres as in (6). To approximate the fullintegral numerically, one can use a product quadrature rule with quadratures Q S k ( g ) that are specifically designed for integrations over spheres. The full integral can becomputed efficiently if the number of involved spheres is small. As pointed out inthe last subsection, in practice we are interested to build product rules for at most S × S × S . The quadratures over each sphere can be built in many ways. Since weare aiming for resulting quadratures that exhibit some symmetry characteristics tohopefully overcome the sign-problem, it seems that quadrature rules given in [16]exhibit all requiered properties, i.e. high accuracy due to polynomial exactness overspheres, numerical stability of the resulting weights, and beeing fully symmetric.The quadratures over each sphere take the form Q S k ( g ) = N sym ∑ γ = w γ g ( ttt γ ) . (14)The sampling points ttt ∈ S k are chosen symmetrically on the k -sphere and are weightedvia w ∈ R . The specific definitions of ttt , w and N sym for different k are given in [16].(Note that in this reference, the notation U k is equivalent to the here used S k − .) Itis possible to randomize these quadrature rules, such that an error estimate for eachquadrature rule can be computed via independent replication [16].The final quadrature rule Q ( f ) of the full integral in (6) is a combination ofdifferent single-sphere quadrature rules given in (14). Due to the symmetric choice ofthe sampling points on spheres, the rule Q ( f ) is in the following called symmetrizedquadrature rule . A more detailed description of Q S k ( g ) and Q ( f ) is given in [22],section 6.1. We applied these constructed quadrature rules to physical one-dimensional QCDproblems [7], which is a simplified model of strong interactions in elementary particlephysics. This model is a good test model because it can be solved analytically,giving a well defined measure for the uncertainties computed by different numericalintegration methods. This model has one integration variable U ∈ G and three realinput parameters: a mass m , a chemical potential µ and a length scale d . A smallmass ( m (cid:28) d µ ) introduces a sign-problem which makes it very hard for standardmethods as MC to compute amplitudes as in (1) numerically.We computed the chiral condensate in this model, given by χ = (cid:82) G ∂ m B [ U ] d U (cid:82) G B [ U ] d U , (15) Page: 8 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 9 with the Boltzmann-weight B [ U ] = det (cid:0) c ( m ) + c ( d , µ ) U † + c ( d , µ ) U (cid:1) , (16)expressed via the parameters c ( m ) = L ∏ j = ˜ m j , ˜ m = m , ˜ m j = m +
14 ˜ m j − ∀ j ∈ { , , ..., d − } , ˜ m d = m +
14 ˜ m d − + d − ∑ j = ( − ) j + − j ˜ m j ∏ j − k = ˜ m k , (17) c ( d , µ ) = − d e − d µ , (18) c ( d , µ ) = ( − ) d − d e d µ . (19)For brevity, the dependencies of these parameters are in the following only writtenwhen needed.In all numerical calculations, we first computed both numerator and denominatorof (15) separately and then divided them. We computed the numerator by symboli-cally differentiating B [ U ] and computing the integral over the result numerically.We compared the results for χ using the symmetrized quadrature rules that aredescribed in 3.1.2, with a standard integration method, ordinary MC sampling. Thelatter quadrature rule is given by Q ( f ) = N MC N MC ∑ γ = f ( V γ ) , (20)where the V are matrices that are chosen randomly from a uniform distribution. Wechose N MC to be as large as the number of used symmetric sampling points.Because the analytic results of χ can be calculated straightforwardly, we computedthe error estimates of the numerical solutions - MC and symmetrized quadraturerules - directly via the relative deviation from the analytic value, ∆ χ = | χ numerical − χ analytic || χ analytic | (21)and derived the standard deviation of this error by repeatedly using on the one handthe MC quadrature rules with different random matrices V ’s and on the other handthe randomized symmetrized quadrature rules as indicated in section 3.1.2.The results for ∆ χ of both MC and symmetrized quadrature rule can be roughlysplit into a small m ( m < − ), a large m ( m > . ) and a transition region, shownin Figure 3 for constant µ = d =
8, extended 1024-bit machine precision anddifferent compact groups. For both quadrature rules we used the sampling sizes
Page: 9 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 -4 SU(2) -310 precision numbers ∆χ MCsymmetrized rule 10 -310 -300 MCsymmetrized rule
SU(3) -4 U(1) ∆χ -310 -300 -10 -5 m U(2) -10 -5 m U(3) -10 -5 m Fig. 3: The sign-problem arises for MC results with small m constants, giving errorsof the order of one. On the contrast, the symmetrized quadrature rules avoid thesign-problem in this region completely, giving errors approximately at machineprecision for all shown groups. N ≡ N sym = N MC = S U ( ) , N =
96 for
S U ( ) , N = U ( ) , N = U ( ) and N =
384 for U ( ) .First, we describe the MC results: In the small m region, ∆ χ for all groups arelarge - equal or larger than one. It can be shown that in this region the numerator of χ is such small that the MC evaluation cannot resolve these values for affordablesample sizes, resulting in large errors [22]. This is the manifestation of the sign-problem, making it almost impossible to compute reasonable values of χ with MCin the small m region. On the other side, for large m all groups have a smaller MCerror estimate than in the small m region. Here the numerator of χ tends to be largerand especially the denominator becomes very large, both resulting in a slightly bettererror estimate for the MC results.Opposed to MC results, the symmetrized quadrature rules give error estimatesapproximately at machine precision up to very small m values, see Figure 3. Thesenumerical results show that the symmetrized quadrature rules give significant resultsin the sign-problem region in practice, where MC simulations have error estimates oforder one. Page: 10 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 11
The previous section shows efficient quadrature rules for physical one-dimensionalintegration to avoid the sign-problem. Most physical models have more than oneintegration variable. In general, it is not straightforward to find an efficient quadraturerule, and usually restricted Monte Carlo methods are applied to high-dimensional in-tegrals. As a first alternative, we investigated the recursive numerical integration(RNI)method. This method reduces the d -dimensional integral I ( f ) = (cid:90) D d f [ ϕ ] d ϕ (22)with d ϕ = ∏ di = d ϕ i and D = [ , π ) into many recursive one-dimensional integrals,and can be applied for several physical models of interest.This is done by utilizing the typical structure of the integrand f [ ϕ ] . This sectionfocuses on the RNI method and how to find an efficient quadrature rule for a high-dimensional integral. It does not discuss the sign-problem which is investigatedfurther in section 3.3. More specifically, this section describes the two steps to createan efficient quadrature rules Q ( f ) for the integral I ( f ) in (22):RNI 1. Use the structure of the integrand of the high-dimensional integral to rewrite itinto recursive one-dimensional integrals.RNI 2. Choose an efficient quadrature rule to compute each one-dimensional integralnumerically. Recursively doing this results in the full quadrature rule Q ( f ) .Finally, this section shows results of applying the method to a physical model calledthe topological osciallator. A more detailed explanation of the method and the resultscan be found in [2, 1]. Many models in one physical dimensional have integrands with the structure f [ ϕ ] = d ∏ i = f i ( ϕ i + , ϕ i ) , (23)with periodic boundary conditions ϕ d + = ϕ . These models have only next-neighborcouplings.The integral of (23) can be rewritten using recursive integration as described in[17, 19]: Because of next-neighbor couplings, each variable ϕ i appears only twice in f [ ϕ ] , in f i and f i − , and therefore the integral can be written as d nested one-variableintegrals I i , Page: 11 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 I ( f ) = (cid:90) D ... (cid:90) D d ∏ i = f i ( ϕ i , ϕ i + ) d ϕ d · · · d ϕ (24) = (cid:90) D (cid:32) ... (cid:18) (cid:90) D f d − ( ϕ d − , ϕ d − ) · (cid:18) (cid:90) D f d − ( ϕ d − , ϕ d ) · f d ( ϕ d , ϕ d + ) d ϕ d (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) I d d ϕ d − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) I d − ··· (cid:33) d ϕ (cid:124) (cid:123)(cid:122) (cid:125) I . This full integral can be computed recursively: I d integrates out ϕ d first, then I d − integrates out ϕ d − and so on until finally I = I ( f ) integrates out ϕ .To avoid under- and overflow of the single quadrature rule results, we actually usedquadrature rules to approximate I ∗ i = c i I i with c i > I = (cid:0) ∏ di = c i (cid:1) I ∗ . For brevity, the method is describedin the following without this trick.Each integral is approximated by using an N quad -point quadrature rule. The firstintegrand in (24) (last from the right) depends on three variables ϕ d − , ϕ d and ϕ d + .The variable ϕ d is integrated out, therefore the quadrature rule Q d ( f d − · f d ) ≡ Q d of I d depends on two variables, Q d ( ϕ d − , ϕ d + ) = N quad ∑ γ = w γ f d − ( ϕ d − , t γ ) f d ( t γ , ϕ d + ) , (25)with sampling points t and weights w . The next integral I d − is approximated by thequadrature rule Q d − ( ϕ d − , ϕ d + ) = N quad ∑ γ = w γ f d − ( ϕ d − , t γ ) Q d ( t γ , ϕ d + ) , (26)and includes the quadrature rule Q d given in (25). The quadrature rules Q d − , ..., Q are created analogically to (26). Using the same sampling points w γ and weights t γ , γ ∈ { , ..., N quad } in all quadrature rules Q i results in the full quadrature rule for (24), Q = Q = N quad ∑ γ = w γ Q ( t γ , t γ ) = tr (cid:34) d ∏ i = (cid:16) M i · diag ( w , . . . , w N quad ) (cid:17)(cid:35) , (27)with M i beeing an N quad × N quad matrix with entries ( M i ) αβ = f i ( t α , t β ) . We used the Gaussian-Legendre N quad -point quadrature rule, see [21] to define thesampling points t and weights w . For this rule, the error scales asymptotically (forlarge N quad ) as σ ∼ O (cid:16) ( N quad ) ! (cid:17) (For Legendre polynomials the correct asymptotic Page: 12 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 13 error scaling is ( N quad ! ) (( N quad ) ! ) [20] which is slightly improved over ( N quad ) ! .). The Stirlingformula ( N quad ! ≈ (cid:112) π N quad (cid:16) N quad e (cid:17) N quad asymptotically) approximates the factorialto give σ ∼ O (cid:32) exp ( − N quad ln N quad ) (cid:112) N quad (cid:33) (28)asymptotically. This is a huge improvement over the MC error scaling 1 / √ N MC . We applied the RNI method to the topological oscillator [6], also called quantumrotor, which is a simple, physically one-dimensional model that has non-trivialcharacteristics which are also present in more complex models. It has d variables φ i ∈ [ , π ) , a length scale T and a coupling constant c . We investigated the topologicalcharge susceptibility of this model, χ top = (cid:82) O [ ϕ ] B [ ϕ ] d ϕ (cid:82) B [ ϕ ] d ϕ , (29)with Boltzmann-weight B [ ϕ ] = exp (cid:32) − c d ∑ i = ( − cos ( ϕ i + − ϕ i )) (cid:33) , (30)and a squared topological charge O [ ϕ ] = T (cid:32) π d ∑ i = ( ϕ i + − ϕ i ) mod 2 π (cid:33) . (31)With RNI, we computed both numerator and denominator of χ top separately, bothdiffering in the factorization (23) of their integrands. Straightforwardly, the denom-inator integrand consists out of local exponential factors. The numerator consistsout of summands with varying factorization schemes, each of these summands iscomputed separately with RNI and they are presented in more detail in [22], section5.2. We estimated the error of χ top by choosing a large number of samples N g quad in(25), (26) and similar ones for which we assumed that χ top ( N g quad ) has converged tothe actual value and computed the difference of χ top ( N quad ) for N quad < N g quad to thisvalue, ∆ χ top ( N quad ) = | χ top ( N quad ) − χ top ( N g quad ) | . (32) Page: 13 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41
We tested beforehand that this truncation error behaves exponentially for large N quad in practice, as expected from (28), [2].We compared the results of the RNI method with results using the Cluster algo-rithm [23], which we found is an optimal MCMC method for the application to thetopological oscillator [2]. Due to the exponential error scaling of the Gauss-Legendrerule, the new method advances MCMC for large enough N quad . We found that theRNI method is also advantageous for lower N quad -values: our simulations showedthat the RNI method needs orders of magnitude less runtime than the Cluster algo-rithm to result in a specified error estimate on an observable, compare Figure 4 for c = . T = d = ∆ χ ( t op ) runtime in s Cluster algorithmGaussian quadrature
Fig. 4: The runtime to arrive at a given error estimate is orders of magnitudes smallerwhen using the RNI method with Gauss-Legendre points than using the ClusterMCMC algorithm.estimate that decreases proportional to t − / for runtime t , consistent with the typicalMC error scaling [11]. We used between 10 and 10 sampling points here. TheRNI method, using between 10 and 300 sampling points with N g quad = Page: 14 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 15 estimate, even for a number of sampling points where the RNI error does not yetscale exponentially.
Section 3.1 shows that the sign-problem can be avoided for one-dimensional in-tegrals using symmetric quadrature rules. But what about the sign-problem forhigh-dimensional integrals? A quadrature rule for high-dimensional integrals overcompact groups, I ( f ) = (cid:90) G d f [ U ] d U , (33)with d U = ∏ di = d U i is needed that also avoids the sign-problem. We combined bothalready presented methods, the symmetric quadrature rules in section 3.1 and the RNIin section 3.2 to find an efficient quadrature rule Q ( f ) for I ( f ) in (33). An alternativeattempt to generalize the symmetrized quadrature rules to high-dimensional integralsis discussed in [18]. RNI can be used to transform the high-dimensional integral I ( f ) in (33) into one-dimensional integrals. These one-dimensional integrals can be approximated re-cursively, using the symmetric quadrature rules. In the following, these steps aredescribed in more detail:RNI 1. Find the structure, i.e. all f i , of the integrand f [ U ] = d ∏ i = f i ( U i + , U i ) , (34)to be able to write the full integral as nested one-dimensional integrals, similar to(24).RNI 2. Apply symmetric quadrature rules to each one-dimensional integration over U i .Here is an example how to do this for the innermost integral I d , integrating over U d :Sym 1. Rewrite the integral over U d into an integral over the products of spheres asdone in (6).Sym 2. Approximate each iterated integral I d ( g ) by a product rule of quadratures overspheres parametrising the group U d to be integrated. Note that the group U d isparametrised at most as the product of S , S and S . Page: 15 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41
We applied this combined method again to the topological oscillator discussed in3.2.3. This time we transformed the variables ϕ i to new variables U j = e i ϕ j ∈ U ( ) .Additionally, we added a sign-problem to the model by using an additional factor ∏ dj = U − θ j in the Boltzmann-weight, B [ U ] = exp (cid:32) − c d ∑ i = ℜ ( − U i + U ∗ i ) (cid:33) · d ∏ j = U − θ j , (35)with a new parameter θ ∈ R . If this parameter is larger than zero, the sign-problemarises and is most severe for θ = π .In this model we computed the plaquette, plaquette = (cid:82) O [ U ] B [ U ] d U (cid:82) B [ U ] d U , (36)with O [ U ] = d ℜ (cid:32) d ∑ i = U i + U ∗ i (cid:33) . (37)For the combined method, we computed both numerator and denominator of (36)separately and divided the values. We used a truncation error, similar to the one givenin (32). We compared the method with a standard MC method as used in 3.1.3. TheMC error is computed via the standard deviation.For θ = π we found that the combined method avoids the sign-problem that isvisible with the MC computation, compare Figure 5. It gives orders of magnitudesmaller errors that shrink the more symmetrization points are used. Therefore thecombination of RNI and symmetric quadrature rules is suitable to avoid the sign-problem for high-dimensional integration. In this contribution we have demonstrated that through symmetric quadrature rulesexact symmetrization and recursive numerical integration techniques problems inhigh energy physics can be solved which constitute a major, if not unsurmountableobstacle for standard Markov chain Monte Carlo methods. The examples we haveconsidered here invole only a time lattice and are hence 0+1-dimensional in space-time, where as real physical problem include spatials dimensions of up to 3. Weare presently investigating whether the methods we have presented here can beextended to higher, i.e. including also spacial, dimensions. While for the recursive
Page: 16 job: author macro: svmult.cls date/time: 18-Feb-2020/1:41 voiding the sign-problem in lattice field theory 17
Fig. 5: The combined method avoids the sign-problem that exists when using theMC method.numerical integration technique we have first results which are promising, for thefull symmetrization method we were so far not successful.Also combining the symmetrized quadrature rules with MC methods did not leadto a practically feasible method in higher dimensions. However, we are followinga path to combine Quasi Monte Carlo, recursive numerical integration and a fullsymmetrization to overcome this problem and hope to report about these attempts inthe future.
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