Thimble regularization as a solution to the sign problem has been
successfully put at work for a few toy models. Given the non trivial nature of
the method (also from the algorithmic point of view) it is compelling to
provide evidence that it works for realistic models. A Chiral Random Matrix
theory has been studied in detail. The known analytical solution shows that the
model is non-trivial as for the sign problem (in particular, phase quenched
results can be very far away from the exact solution). This study gave us the
chance to address a couple of key issues: how many thimbles contribute to the
solution of a realistic problem? Can one devise algorithms which are robust as
for staying on the correct manifold? The obvious step forward consists of
applications to gauge theories.Comment: 7 pages, 1 figure. Talk given at the Lattice2015 Conferenc

Di Renzo, F.

Eruzzi, G.

English

arXiv

ERUZZI, Giovanni

DI RENZO, Francesco

Archivio istituzionale della Ricerca -  Università degli Studi di Parma

20 November 2022University of Parma Research RepositoryThimble regularization at work besides toy models: from Random Matrix Theory to Gauge Theories /Eruzzi, Giovanni; DI RENZO, Francesco. - In: POS PROCEEDINGS OF SCIENCE. - ISSN 1824-8039. -251:(2015), pp. 188.1-188.7. ((Intervento presentato al convegno The 33rd International Symposium onLattice Field Theory Lattice 2015 tenutosi a Kobe International Conference Center, Kobe, Japan nel 14 -18July 2015 [10.22323/1.251.0188].OriginalThimble regularization at work besides toy models: from Random Matrix Theory to Gauge TheoriesPublisher:PublishedDOI:10.22323/1.251.0188Terms of use:openAccessPublisher copyright(Article begins on next page)Anyone can freely access the full text of works made available as "Open Access". Works made availableAvailability:This version is available at: 11381/2814638 since: 2022-09-27T16:41:34ZThis is the peer reviewd version of the followng article:PoS(LATTICE 2015)188Thimble regularization at work besides toy models:from Random Matrix Theory to Gauge Theories.Giovanni Eruzzi∗ and Francesco Di RenzoUniversity of Parma and INFNE-mail: giovanni.eruzzi@pr.infn.it, francesco.direnzo@pr.infn.itThimble regularization as a solution to the sign problem has been successfully put at work for afew toy models. Given the non trivial nature of the method (also from the algorithmic point ofview) it is compelling to provide evidence that it works for realistic models. A Chiral RandomMatrix theory has been studied in detail. The known analytical solution shows that the model isnon-trivial as for the sign problem (in particular, phase quenched results can be very far away fromthe exact solution). This study gave us the chance to address a couple of key issues: how manythimbles contribute to the solution of a realistic problem? Can one devise algorithms which arerobust as for staying on the correct manifold? The obvious step forward consists of applicationsto gauge theories.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni Eruzzi1. MotivationWhen a field theory features complex terms in its action, we say that it is plagued by theso-called sign problem (e.g. it usually occurs at finite chemical potential); complex terms preventnumerical simulation of such theories by means of Monte Carlo methods. Several wayouts havebeen proposed so far, but a universal and rigorous prescription to circumvent the sign problem isstill missing. The power of the thimble approach lies in the fact that integrating along the thimbleautomatically keeps SI = ℑ(S) constant, thus avoiding the problem of sampling complex actions.A few multi-dimensional models have been successfully investigated with this approach ([1, 2]).Quite interestingly, the simple ones (e.g. the 0-dimensional φ 4 toy-model [3]) showed a non-trivialthimble structure. It is therefore interesting to apply the thimble approach to a simple, yet multi-dimensional model (the CRM theory) to check if more than one thimble is needed to recover theexpected results. At the same time, a new method to sample on the thimble will be presented. Thedetails of the following presentation and its application to the CRM theory can be found in [4].2. Essential results from Morse theoryIn this section we collect main results from Picard-Lefschetz (i.e. complex Morse) theory, fol-lowing the notation of [5]. For simplicity of notation, let us consider a scalar field theory consistingof a collection of n real degrees of freedom {xi}i=1···n defined on a certain domain C (we will takeC = Rn). The path-integral expression for the expectation value of an observable O is〈O〉= Z−1∫CdnxO [x]e−S[x] (2.1)with the partition function Z given by the same integral without O [x]. The (complex) actionS = SR + iSI and the observable O are assumed to be some holomorphic functions of the fields.Now we complexify the fields, letting xi 7→ zi = xi + i yi ∈C; Morse theory states that the followingdecomposition holdsZ = ∑σmσ∫Jσdnze−S(z) = ∑σmσ e−iSI(zσ )∫Jσdnze−SR(z) (2.2)where σ labels each critical point zσ of the action S (z) in the complexified domain andJσ ⊂C n is the (stable) Lefschetz thimble associated with the critical point zσ . mσ ∈Z are integercoefficients whose computation is potentially non-trivial (they will be discussed later). An analo-gous result holds for the integral in eq. (2.1). As previously stated, SI is constant along the thimbleand therefore can be computed once and for all at the critical point: all is left is a real weight e−SRwhich can be used for Monte Carlo simulations.We now give the definition of stable thimble: it is a manifold of real dimension n consistingof the union of all those curves of steepest-ascent (SA) for the “action” SR which fall into zσ atτ →−∞, that is solutions ofdxidτ=∂SR (x,y)∂xidyidτ=∂SR (x,y)∂yi(2.3)2PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni EruzziIt is easy to check that, starting from the critical point at τ→−∞, along these SA curves SR isalways increasing, while SI stays constant.As for the coefficients mσ , they are defined as the intersection numbers between Kσ and theoriginal domain of integration C : mσ = 〈C ,Kσ 〉, with Kσ the unstable thimble associated with zσ .The unstable thimble is given by the union of all the curves z(τ) that are solutions to (2.3) whichfall into zσ for τ →+∞. The problem of knowing which thimbles contribute to the decomposition(2.2) is a highly non trivial one (see [6] for a detailed discussion). In the present work, we haveanalytical results available and so will discuss the relevance of different thimbles a-posteriori.Let us now consider the integration measure appearing in (2.2). The canonical basis of Cnwhose duals appear in dnz = dz1 ∧ ·· · ∧ dzn is in general not parallel to the tangent space basisof the thimble at a generic point z (we will name such tangent space TzJσ ). Let {U (i)}i=1···n bethe tangent space basis at z (these are n complex vectors with n components forming the columnsof an n× n unitary matrix U). Now we make a change of coordinates from the canonical onesof Cn to the ones of TzJσ (these are n real numbers {yi}i=1···n). The integration measure thusbecomes dnz = detU dny, where the term detU = eiω is what is referred to as the residual phase,which accounts for the non-parallelism of TzJσ with the canonical basis of Cn [6, 2, 7, 4]. Thisis a complex term which must be taken into account by using reweighting; this could in principlecreate a residual sign problem. We expect this not to be the case, as ω varies smoothly along SAcurves. Moreover the regions in which it could get large are likely to be far from the critical point,that is regions where the action SR is large. However, this is something which should always bechecked and so far the results have been very encouraging (see [2] and [7] for a more thoroughdiscussion). In applying the thimble approach to the CRM theory, we also found that the residualphase can be safely taken into account with reweighting.3. An algorithm to sample on the thimbleIn this section a new algorithm to sample field configurations on a thimble is sketched (thisalgorithm was first proposed in [8]), the interested reader can find all the details in [4].We now make a change of notation and work with 2n real fields {φi} which consist in boththe real and the imaginary parts of z. Let us begin by describing the tangent space to Jσ in thevicinity of the critical point φσ . In this region, we can expand the actionSR (φ) = SR (φσ )+12ΦT HΦ+O(φ3) (3.1)where the components of the 2n-dimensional real vector Φ are Φi = φi− φσ ,i and H is the2n× 2n hessian matrix of SR evaluated at the critical point. The hessian can be put into diag-onal form by the transformation H = WΛW T , with W an orthogonal matrix whose columns arethe eigenvectors of H. Because of holomorphicity of the action, the spectrum features n positiveeigenvalues {λi}i=1···n and their negative counterparts (equal in modulus). The positive eigenvaluescorrespond to eigenvectors which span the tangent space to the stable thimble at the critical point:any combination of these gives a direction along which SR grows (we shall label this set of eigen-vectors {v(i)}i=1···n). A particular SA curve on the thimble can be identified defining the direction3PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni Eruzzialong which one leaves the critical point; this direction is given by an n-dimensional normalised1vector n̂. Therefore a point Φ ∈Jσ can be unambiguously identified by a choice of n̂ and the timet of integration of the SA equations (2.3). The mapping is thusSn−1×R 3 (n̂, t)↔Φ ∈Jσ (3.2)If one explores whole SA curves by integrating in t ∈ R, the thimble can then be sampled bypicking up all the possible SA curves. This is the essence of the algorithm which is described inthis work.In the following we need to compute the tangent space to the thimble at a generic point Φ, forwhich we lack a local description. We shall resort to parallel-transporting the well-known tangentspace at the critical point along the SA curve to the point Φ. This is accomplished by solving aparallel-transport equation for each basis vector ([6, 2, 7])dV (i)jdt=2n∑k=1∂ 2SR∂φ j∂φkV (i)k i = 1 · · ·n j = 1 · · ·2n (3.3)The asymptotic (t→−∞) solution to (2.3) and (3.3) is given byt 1φ j (t)≈ φσ , j +n∑i=1v(i)j eλitni j = 1 · · ·2n |~n|2 = 1V (i)j (t)≈ v(i)j eλit j = 1 · · ·2n i = 1 · · ·n(3.4)If we know a reference time t0 1 at which (3.4) holds, the latter can be used to provide aninitial condition for a SA on the thimble.We now rewrite the partition functions as a sum of contributions from all the possibile SAcurves. We will take a few shortcuts (further details can be found in [4]). The partition functionfunction can be cast into the formZ = ∑σmσ e−iSI(φσ )∫Jσn∏i=1dyi e−SReiω = ∑σmσ e−iSI(φσ )∫ n∏i=1dni δ(|~n|2−1)Zn̂〈eiω〉n̂ (3.5)by employing a Faddeev-Popov-like trick. The quantity 〈 f 〉n̂ may be regarded as the “expec-tation value” of f on a single SA curve〈 f 〉n̂ =+∞∫−∞dt ∆n̂ (t)e−SR(n̂,t) f (t)+∞∫−∞dt ∆n̂ (t)e−SR(n̂,t)(3.6)where the quantity in the denominator can be regarded as a “partial” partition function Zn̂. Anexpression for ∆n̂ (t) is worked out in [4] in terms of the local (parallel-transported) basis vectors.The same vectors are then used to compute the complex n×n matrix U whose determinant is theresidual phase. The expectation value of an observable O can be computed as1Being normalised, the vector n̂ effectively encodes n− 1 degrees of freedom, which, along with the time t ofintegration of SA equations, is precisely what is needed to recover the original n degrees of freedom.4PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni Eruzzi〈O〉= 1Z ∑σmσ e−iSI(φσ )∫ n∏i=1dni δ(|~n|2−1)Zn̂〈Oeiω〉n̂ (3.7)The t integration which is evident in (3.6) is carried out numerically while integrating (2.3).We are then left with integration in n̂-space. As each SA contributes to the total partition functionwith a weight Zn̂/Z, it would be reasonable to do some kind of importance sampling with respectto this measure. This is difficult, since computing Zn̂ requires integration along all the SA curveidentified by n̂. As a first try for the algorithm we have therefore used a static, crude Monte Carlo,that is we sampled the n̂-space uniformly. This method can easily become very inefficient (e.g. forlarge systems) and is the cause of the larger error bars in Figure 1; anyway, this crude approachproved to be enough to show the effectiveness of the thimble regularization for the CRM model inthe range of parameters we explored.4. The CRM theoryThe Chiral Random Matrix theory we refer to is described in [9] and is related to QCD in theε-regime of Chiral Perturbation Theory. The sign problem comes from the presence of a chemicalpotential µ . In [9] complex Langevin was used to compute the expectation value of the chiralcondensate for different values of the quark mass and failures occured at low masses, where thealgorithm seemed to follow the predictions of the phase-quenched theory instead of the exact one.In [10] a different parametrization was proposed which made complex Langevin work fine. In thiswork we make use of the first parametrization and show that the thimble approach has no problemsin computing expectation values of the condensate. The partition function of the CRM model isZN fN (m) =∫dΦdΨdetN f (D(µ)+m)exp(−N Tr[Ψ†Ψ+Φ†Φ]), (4.1)withD(µ)+m =(m icosh(µ)Φ+ sinh(µ)Ψicosh(µ)Φ† + sinh(µ)Ψ† m)(4.2)Φ and Ψ are two N ×N complex matrices, that is Φ = a + ib and Ψ = α + iβ . We havetherefore 4N2 degrees of freedom which will be complexified, the thimble being a manifold of realdimension 4N2 embedded in an euclidean space of dimension 8N2. The observable we will beconcerned with is the chiral condensate defined as1N〈η̄η〉= 1N∂∂mlogZ (4.3)In Figure 1 we see the results of the thimble simulation for the real part of the condensate atfixed N f = 2 and µ̃ =√Nµ = 2 as a function of m̃ = Nm for N = 1,2,3,4. The exact results arerecovered within errors, even in regions where the sign problem is severe (the exact result is verydifferent from the phase-quenched one). All the simulations were carried out considering only thethimble attached to the trivial vacuum a = b = α = β = 0. We found no evidence of the relevance ofmore than this thimble, which was quite unexpected, especially at low dimensions. As a check, we5PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni Eruzzitried to identify other critical points: we found two classes of them (there is a symmetry involved -see [4]) but they all hadSR (φσ 6= 0) < SR (φσ = 0) = minφ∈R4(N×N)SR (φ) (4.4)and by Morse theory they are all irrelevant in the decomposition (2.2) [6, 5, 2].2 4 6 8 10 12 14 16 18 20 2200.511.522.5 m̃1 N〈η̄η〉N=1N=2N=3N=4Figure 1: Exact (solid red line), phase-quenched (dashed blue line) and thimble simulation results for thechiral condensate at fixed N f = 2, µ̃ = 2.5. Gauge theoriesGauge theories are the ultimate goal for the thimble regularization. Before tackling them, testson low-dimensional toy models featuring gauge invariance are mandatory, being the approach stillnew and in need of some tuning. The details of the thimble formalism applied to theories featuringgauge invariance (as well as some examples) are discussed in [11]. Here we just mention threefundamental steps. The first step is the complexification of the gauge group (which we assume tobe SU(N))SU(N) 3U = eixaT a → eizaT a = ei(xa+iya)T a ∈ SL(N,C) (5.1)then we have to rewrite the SA equations (2.3) in a gauge-covariant form and the final ingre-dient is the parallel-transport equation for each basis vector, generalized to a non-abelian gaugegroup. For further details, the reader may consult [11].6. ConclusionsWe have discussed the thimble approach applied to the Chiral Random Matrix theory, yieldingcorrect results in a region where the sign problem is quite severe. On this occasion we also tested6PoS(LATTICE 2015)188Thimble regularization from Random Matrix Theory to Gauge Theories. Giovanni Eruzzifor the first time on a multi-dimensional model the algorithm initially proposed in [8]. The wholemethod proved to be quite effective, even in its crudest implementation, that is the static MonteCarlo for the extraction of the SA curves in n̂-space. Moreover, we found no evidence for thepresence of relevant thimbles beyond the one associated to the trivial vacuum (at least, in theregion of parameter space we explored). This is of course something which is hard to generaliseto a realistic field theory, nevertheless these first results are very encouraging with respect to morerealistic applications of the thimble regularisation.AcknowledgementsWe deeply thank L. Scorzato for the many useful and stimulating discussions. We also thankC. Torrero, K. Splittorff, Y. Kikukawa and H. Fujii.References[1] M. Cristoforetti, F. Di Renzo, Abhishek Mukherjee, L. Scorzato, Monte Carlo simulations on theLefschetz thimble: taming the sign problem, Phys. Rev. D 88, 051501(R) (2013),[arXiv:1303.7204v2] [hep-lat].[2] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu, T. Sano, Hybrid Monte Carlo on Lefschetzthimbles - A study of the residual sign problem, JHEP 1310 (2013) 147 [arXiv:1309.4371v2][hep-lat].[3] G. Eruzzi, Solution of simple toy models via thimble regularization of lattice field theory, inproceedings of The 32nd International Symposium on Lattice Field Theory,PoS(LATTICE2014)202.[4] F. Di Renzo and G. Eruzzi, Thimble regularization at work: from toy models to chiral random matrixtheories, Phys. Rev. D 92, 085030 (2015) [arXiv:1507.03858v3] [hep-lat].[5] E. Witten, Analytic continuation of Chern-Simons Theory, arXiv:1001.2933v4 [hep-th].[6] M. Cristoforetti, F. Di Renzo, L. Scorzato, New approach to the sign problem in quantum fieldtheories: High density QCD on a Lefschetz thimble, Phys. Rev. D 86, 074506 (2012)[arXiv:1205.3996v3] [hep-lat].[7] M. Cristoforetti, F. Di Renzo, G. Eruzzi, A. Mukherjee, C. Schmidt, L. Scorzato, and C. Torrero, Anefficient method to compute the residual phase on a Lefschetz thimble, Phys. Rev. D 89, 114505(2014) [arXiv:1403.5637v2] [hep-lat].[8] F. Di Renzo, An algorithm for thimble regularization of lattice field theories (and possibly not only forthat), in proceedings of The 32nd International Symposium on Lattice Field Theory,PoS(LATTICE2014)046.[9] A. Mollgaard and K. Splittorff, Complex Langevin Dynamics for chiral Random Matrix Theory, Phys.Rev. D 88, 116007 (2013) [arXiv:1309.4335] [hep-lat].[10] A. Mollgaard and K. Splittorff, Full simulation of chiral random matrix theory at nonzero chemicalpotential by complex Langevin, Phys. Rev. D 91, 036007 (2015) [arXiv:1412.2729] [hep-lat].[11] F. Di Renzo, Thimble regularization at work for Gauge Theories: from toy models onwards, inproceedings of The 33rd International Symposium on Lattice Field Theory,PoS(LATTICE2015)189.7

Thimble regularization at work besides toy models: from Random Matrix Theory to Gauge Theories

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