Diagrammatic Monte Carlo for the Weak-Coupling Expansion of Non-Abelian Lattice Field Theories: Large-Nuðnþ × Uðnþ Principal Chiral Model

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by University of Regensburg Publication Server PHYSICAL REVIEW D 96, 114512 (2017) Diagrammatic Monte Carlo for the weak-coupling expansion of non-Abelian lattice field theories: Large-NUðNÞ × UðNÞ principal chiral model

† P. V. Buividovich* and A. Davody Institute of Theoretical Physics, University of Regensburg, Universitätsstrasse 31, D-93053 Regensburg, Germany (Received 18 May 2017; revised manuscript received 20 October 2017; published 29 December 2017) We develop numerical tools for diagrammatic Monte Carlo simulations of non-Abelian lattice field theories in the t’Hooft large-N limit based on the weak-coupling expansion. First, we note that the path integral measure of such theories contributes a bare mass term in the effective action which is proportional to the bare coupling constant. This mass term renders the perturbative expansion infrared-finite and allows us to study it directly in the large-N and infinite-volume limits using the diagrammatic Monte Carlo approach. On the exactly solvable example of a large-NOðNÞ sigma model in D ¼ 2 dimensions we show that this infrared-finite weak-coupling expansion contains, in addition to powers of bare coupling, also powers of its logarithm, reminiscent of resummed perturbation theory in thermal field theory and resurgent trans-series without exponential terms. We numerically demonstrate the convergence of these double series to the manifestly nonperturbative dynamical mass gap. We then develop a diagrammatic Monte Carlo algorithm for sampling planar diagrams in the large-N matrix field theory, and apply it to study this infrared-finite weak-coupling expansion for large-NUðNÞ × UðNÞ nonlinear sigma model (principal chiral model) in D ¼ 2. We sample up to 12 leading orders of the weak-coupling expansion, which is the practical limit set by the increasingly strong sign problem at high orders. Comparing diagrammatic Monte Carlo with conventional Monte Carlo simulations extrapolated to infinite N, we find a good agreement for the energy density as well as for the critical temperature of the “deconfinement” transition. Finally, we comment on the applicability of our approach to planar QCD at zero and finite density.

DOI: 10.1103/PhysRevD.96.114512

I. INTRODUCTION to OðNÞ and CPN−1 nonlinear sigma models [15–18] and An infamous fermionic sign problem in Monte Carlo Abelian gauge theories with fermionic and bosonic matter – simulations based on sampling field configurations in the fields [19 21], which are more similar to QCD. This path integral is currently one of the main obstacles for selection of references is by no means exhaustive. systematic first-principle studies of the phase diagram This success has motivated various attempts to apply and equation of state of dense strongly interacting matter DiagMC to simulations of non-Abelian lattice gauge theories – as described by quantum chromodynamics (QCD). This at finite density [22 28] or the effective models thereof – problem has motivated the search for alternative simulation [29 31]. Despite significant progress, these attempts have algorithms for non-Abelian gauge theories, which would not so far resulted in a first-principle, systematically improv- hopefully avoid the sign problem or make it milder. able simulation strategy. At the qualitative level, one of the One of the alternative first-principle simulation strategies most successful strategies is based on the strong-coupling is the so-called Diagrammatic approach [1], conventionally expansion of lattice QCD, which captures such nonpertur- abbreviated as DiagMC, which stochastically samples the bative features of QCD as confinement, chiral symmetry diagrams of the weak- or strong-coupling expansions, breaking and the baryon and meson bound states already at rather than field configurations. DiagMC algorithms, com- the leading order [32,33]. DiagMC simulations of lattice plemented with the “worm” algorithm techniques [2], QCD with staggered fermions at infinitely strong coupling turned out to be very helpful for eliminating sign problem are free of the sign problem in the massless limit, and allow in physical models with relatively simple weak- or strong- one to reproduce the expected qualitative features of the coupling expansions. The physical models and theories QCD phase diagram [23,24]. Unfortunately, an extension of which were successfully studied using DiagMC range from this approach to the finite values of gauge coupling meets strongly interacting fermionic systems [3–9] relevant in conceptual difficulties, as the next orders of the strong- condensed matter physics, via scalar field theories [10–14], coupling expansion coming from the bosonic part of the action should be incorporated in the DiagMC algorithm *[email protected] manually [24], which becomes more and more complicated † On leave of absence from IPM, Teheran, Iran. for higher orders [26]. Ideally, one would also like to sample [email protected] the strong-coupling expansion in powers of inverse gauge

2470-0010=2017=96(11)=114512(32) 114512-1 © 2017 American Physical Society P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) coupling using DiagMC algorithm. Since on finite-volume [39–43], in which the number of Feynman diagrams grows lattices the strong-coupling expansion is convergent for all only exponentially with order [44]. It is known that if one values of coupling, simulating sufficiently high orders stops the running of the coupling at some artificially should allow to access even the weak-coupling scaling introduced infrared cutoff scale (e.g., by considering the region where lattice theory approaches continuum QCD. theory in a finite volume), this factorial growth disappears Recent attempts to extend the DiagMC simulations to finite [45,46], thus, in a sense it is a remainder of IR divergences gauge coupling using the method of auxiliary Hubbard- in perturbative expansion. Stratonovich variables [25] or the Abelian color cycles [27] Recently infrared renormalon divergences have been are very promising, but still not quite practical. Development given a physical interpretation in terms of the action of of practical tools for generating high orders of strong- unstable or complex-valued saddles of the path integral, coupling expansion is, thus, currently an important unsolved along with a resummation prescription based on the problem. mathematical notion of resurgent trans-series [47–52]. However, in the infinite-volume limit strong coupling Trans-series is a generalization of the ordinary power series expansion is known to have only a finite radius of to a more general form convergence (see, e.g., [34]). One can, thus, expect that X Xmi Xþ∞ as one approaches the weak-coupling regime, the conver- − λ O λ Si= λ l λp gence of the strong-coupling series will increasingly h ið Þ¼ e log ð Þ ci;l;p ; ð1Þ stronger depend on the lattice size, which might well make i l¼0 p¼0 the continuum and infinite-volume extrapolation difficult where hOiðλÞ is the expectation value of some physical in practice, especially for the infrared-sensitive physical observable as a function of the coupling constant λ, c are observables such as hadron masses and transport coeffi- i;l;p the coefficients of perturbative expansion of path integral cients. Moreover, in the t’Hooft large-N limit one even around the i-th saddle point of the path integral with the expects a phase transition separating the strong-coupling action S and l labels flat directions of the action in the and the weak-coupling phases [35–37] even on the finite- i vicinity of this saddle. It turns out that non-Borel- volume lattice. These properties of the strong-coupling summable factorial divergences cancel between perturba- expansion might significantly limit the ability of DiagMC tive expansions around different saddle points i in (1). to work directly in the thermodynamic limit, which is one Unfortunately, currently no method is available to generate of the most important advantages of DiagMC over conven- the trans-series (1) for strongly coupled field theories in a tional Monte Carlo methods [6,7]. systematic way. The resurgent analysis of perturbative These considerations suggest that the extrapolation to expansions has been done so far almost exclusively in the continuum and thermodynamic limit might be easier the regime where gauge theories or sigma models are for DiagMC simulations based on the weak-coupling artificially driven to the weak-coupling limit by a special expansion. However, the weak-coupling expansion for compactification of space-time to D ¼ 1 þ 0 dimensions asymptotically free non-Abelian field theories, such as with twisted boundary conditions [47–52]. four-dimensional non-Abelian gauge theories and two- The aim of this work is to develop DiagMC methods dimensional nonlinear sigma models, has two closely suitable for the bosonic sector of non-Abelian lattice field related problems, which make a direct application of a theories, which is currently a challenge for DiagMC DiagMC approach conceptually difficult at first sight. The approach. We develop two practically independent tools, first problem are infrared divergences due to massless which are finally combined together to perform practical gluon propagators, which cancel only in the final expres- simulations. sions for physical observables. The second problem is the The first tool is the resummation prescription which non-sign-alternating factorial growth of the weak-coupling renders the bare weak-coupling expansion in non-Abelian expansion coefficients, which is commonly referred to as field theories infrared-finite and, thus, suitable for DiagMC the infrared renormalon [38] and cannot be dealt with using simulations which sample Feynman diagrams. This pre- the Borel resummation techniques. These factorial diver- scription is introduced in Sec. II and tested in Sec. III on gences in the perturbative expansions of physical observ- some exactly solvable examples. The basic idea is to work ables originate in the renormalization-group running of the with massive bare propagators, where the bare mass term coupling constant, and have nothing to do with the purely comes from the nontrivial path integral measure for non- combinatorial factorial growth of the number of Feynman 1 Abelian groups and is proportional to the bare coupling. diagrams with diagram order. In particular, infrared By analogy with Fujikawa’s approach to axial anomaly renormalon behavior persists also in the large-N limit [53], this bare mass serves as a seed for dynamical mass generation and conformal anomaly in non-Abelian field 1The recent work [9] suggests that renormalon divergences theories which are scale invariant at the classical level. might be absent in two-dimensional asymptotically free fer- A similar partial resummation of the perturbative series mionic theories. with the help of the bare mass term is also often used in

114512-2 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) finite-temperature quantum field theory [54,55], in particu- simpler to implement. We discuss the extension of our lar, in the contexts of hard thermal loop approximation [56] approach to large-N gauge theories and illustrate the and screened perturbation theory [57]. Upon this resum- expected structure of infrared-finite weak-coupling expan- mation, the weak-coupling expansion becomes formally sion for this case in Sec. VI. similar to the trans-series (1), but contains only the powers of coupling and of the logs of coupling only, while the II. INFRARED-FINITE WEAK-COUPLING − λ exponential factors e Si= seem to be absent. This expan- EXPANSION FOR THE UðNÞ × UðNÞ sion appears to be manifestly infrared-finite and free of PRINCIPAL CHIRAL MODEL factorial divergences at least in the large-N limit. An exactly solvable example of the OðNÞ nonlinear sigma The first step in the construction of lattice perturbation model demonstrates that our expansion reproduces non- theory is to parametrize the small fluctuations of lattice fields around some perturbative vacuum state. For nonperturbative quantities such as the dynamically generated ∈ mass gap (see subsection III B). Abelian lattice fields gx SUðNÞ the most popular para- The second tool, described in Sec. IV, is the general metrization is the exponential mapping from the space of traceless Hermitian matrices ϕx to SUðNÞ group: method for devising DiagMC algorithms from the full αϕ α untruncated hierarchy of Schwinger-Dyson equations gx ¼ expði xÞ, where is related to the coupling constant 1 [58,59], bypassing any explicit construction of a diagram- and gx ¼ is the perturbative vacuum. While all the results matic representation. Here the basic idea is to generate a in this paper could be also obtained for exponential series expansion by stochastic iterations of Schwinger- mapping, for our purposes it will be more convenient to Dyson equations. We believe the method to be advanta- use the Cayley map geous for sampling expansions for which the complicated Xþ∞ 1 þ iαϕ form of diagrammatic rules makes the straightforward g ¼ x ¼ 1 þ 2 ðiαÞkϕk; ð3Þ x 1 − iαϕ x application of e.g., worm algorithm hardly possible, with x k¼1 strong-coupling expansion in non-Abelian gauge theories being one particular example. In this work, we apply the where ϕx are again Hermitian matrices and the value of α method to sample the (resummed) weak-coupling expan- will be specified later. Since in this work we will be sion of Sec. II, which for non-Abelian lattice field theories working in the large-N limit in which the UðNÞ and SUðNÞ also has complicated form due to the multitude of higher- groups are indistinguishable, we omit here the zero trace order interaction vertices. condition for ϕx and work with UðNÞ group. This will As a practical application which takes advantage of both significantly simplify the derivation of Schwinger-Dyson tools, in this work we consider the large-NUðNÞ × UðNÞ equations in terms of ϕx fields; see subsection IVA. principal chiral model on the lattice, which is defined by the One of the advantages of the Cayley map is the following partition function: particularly simple form of the UðNÞ Haar measure (see Z X Appendix B1 for the derivation): N † Z Z Z ¼ dgx exp − 2ReTrðgxgyÞ ; ð2Þ UðNÞ λ 2 2 −N hx;yi dgx ¼ dϕx det ð1 þ α ϕxÞ UðNÞ Z where integration in the path integral is performed over 2 2 ¼ dϕx expð−NTr lnð1 þ α ϕxÞÞ unitary N × N matrices gx living on the sites of the square λ ’ Z two-dimensional lattice, is the t Hooft coupling constant α4 which is kept fixed when taking the large-N limit, and ϕ − α2 ϕ2 ϕ4 P ¼ d x exp N Tr x þ 2 Tr x þ ; hx;yi denotes summation over all pairs of neighboring lattice sites x and y. Numerical results for this model ð4Þ obtained by combining the methods of Secs. II and IV are presented in Sec. V. where the integral on the right-hand side is over all While we expect that our approach can be extended to Hermitian matrices ϕx and … represent the terms of order 6 6 large-N pure gauge theory without conceptual difficulties, Oðα ϕxÞ or higher. Thus, the Cayley map provides a unique in this proof-of-concept study we prefer to work with the one-to-one mapping between the UðNÞ group manifold and principal chiral model (2). The reason is that while this the space of all Hermitian matrices. Strictly speaking, one model exhibits nonperturbative features very similar to should exclude a single point gx ¼ −1 from this mapping, those of non-Abelian gauge theories, including dynamical which, however, has zero measure in the path integral. In mass gap generation, dimensional transmutation and infra- contrast, for exponential mapping the integration over ϕx red renormalons [41,42], it has a much simpler structure should be restricted to a certain region within the space of of the perturbative expansion than non-Abelian gauge Hermitian matrices in order to avoid multiple covering of theories. Correspondingly, the DiagMC algorithm is also UðNÞ group. In addition, for exponential mapping the

114512-3 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) Z exponentiated Jacobian contains double-trace terms [60], Z ¼ dϕ expð−NS½ϕÞ: ð8Þ which make the planar perturbation theory technically x somewhat more complicated (see Appendix B2). The Cayley map was also used in the context of lattice QCD The observation which is central for this work is that the in the Landau gauge [61–64] and for studying unitary Jacobian (4) contributes a term matrix models [65]. λ X 2 X The classical action of the principal chiral model (2) can ϕ2 m0 ϕ2 ϕ 8 x ¼ 2 x ð9Þ be rewritten in terms of the fields x as x x X −1 † to the quadratic part S0 ¼ λ ΔxyTrðgxgyÞ x;y 1 X λ Xþ∞ X S ½ϕ¼ Δ þ δ Trðϕ ϕ Þ −1 k l k l F 2 xy 4 xy x y ¼ 4λ ð−iαÞ ðiαÞ ΔxyTrðϕxϕyÞ x;y k;l¼1 x;y X 1 2 Xþ∞ X ≡ ðΔxy þ m0δxyÞTrðϕxϕyÞ k−l 2 −1 2 kþl k l ¼ 4λ ð−1Þ α ΔxyTrðϕxϕyÞ; ð5Þ x;y k;l¼1 x;y 1 X kþl¼2n ≡ 0 −1 ϕ ϕ 2 ððG Þ ÞxyTrð x yÞð10Þ x;y where we have introduced the lattice Laplacian operator in

D-dimensional Euclidean space, of the action (7), which endows the field ϕx with the bare 2 λ mass m0 ¼ 4. Thus, the effect of the Jacobian (4) is to break XD−1 XD−1 the shift symmetry ϕ → ϕ þ cI of the classical action, Δ 2 δ − δ − δ x x x;y ¼ D x;y x;y−μˆ x;yþμˆ ; ð6Þ serving as a seed for dynamical scale generation and μ¼0 μ¼0 conformal anomaly in the principal chiral model (2). A conventional approach in lattice perturbation theory, and used the unitarity of gx to add the diagonal terms which constructs weak-coupling expansion as formal series † TrðgxgxÞ¼N to the action. Here μ ¼ 0, 1 labels the two in powers of coupling λ, is to treat the quadratic term (9) in lattice coordinates, and μˆ denotes the unit lattice vector in the action (7) as a perturbation [66] on top of the kinetic the direction μ. We see now that in order to get the term. Perturbative expansion then involves massless propa- Δ−1 canonicalP normalization of the kinetic term in the action gators ð Þxy coming from the scale-invariant classical 1 ϕ Δ ϕ … S0 ¼ 2 x;y x x;y y þ, where denotes the terms part of the action, which leads to infrared divergences in 2 λ with more than two fields ϕ, we have to set α ¼ 8. We also the contributions of individual diagrams. These infrared note that the classical action (5) is invariant under the shifts divergences, however, cancel in the final results for physical observables [66]. Furthermore, the resulting of the field variable ϕx → ϕx þ cI, where I is the N × N identity matrix. This symmetry is a consequence of the power series in the coupling constant λ contain factorially scale invariance of the classical action of the principal divergent terms, some of which have nonalternating signs chiral model. and, thus, cannot be removed by Borel resummation Rewriting the path integral in the partition function (2) in techniques. The only recently proposed way to deal with terms of the new fields ϕ , we also need to include the these divergences is to use the mathematical apparatus of x – Jacobian (4). Adding the expansion of the Jacobian (4) in resurgent trans-series [47 52]. powers of λ and ϕ to the classical action (5), we obtain the In this work, we consider a different approach to the weak-coupling expansion based on the action (7), some- full action for the fields ϕx: what similar in spirit to the screened perturbation theory 1 X λ [54,55,57] and to hard thermal loop perturbation theory ϕ Δ δ ϕ ϕ m2 λ S½ ¼2 xy þ 4 xy Trð x yÞ [56,67]. Namely, we include the mass 0 ¼ 4 coming from x;y 0 Δ 2 −1 (9) into the bare lattice propagators Gxy ¼ð þ m0Þxy Xþ∞ λk X which are used to construct the weights of Feynman þ ð−1Þk−1 Trϕ2k k8k x diagrams. This corresponds to a trivial resummation of k¼2 x an infinite number of chain-like diagrams with “two-leg” ∞ kþl−2 Xþ 2 X k−l 4λ vertices. Since this prescription puts the coupling λ both in −1 2 Δ ϕkϕl þ ð Þ kþl xyTrð x yÞ; ð7Þ 8 2 the propagators and vertices, we need to define the formal k;l¼1 x;y kþl¼2n;kþl>2 power counting scheme for our expansion. To this end, we group together the terms with the products of ð2v þ 2Þ so that the partition function (2) reads fields, v ¼ 1; 2; … þ ∞, which correspond to interaction

114512-4 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) vertices with ð2v þ 2Þ legs, and multiply them with the the weights of the vertices which in general depend on ξv ξ … … powers of an auxiliary parameter which should be set q1 qf−1. The momenta Q1 Ql are in general not to ξ ¼ 1 to obtain the physical result: independent and can be expressed as some linear combinations of q1; …;qf−1. All momenta belong to the Brillouin Xþ∞ λ v ϕ − ξv ð2vþ2Þ ϕ zone Qμ ∈ ½−π; π of the square lattice. Thus, any possible SI½ x¼ 8 SI ½ x; ð11Þ v¼1 ultraviolet divergences in the integral (14) are regulated by the lattice ultraviolet cutoff Λ ∼ 1, and we only have to 2 X UV 2 2 m ð vþ Þ ϕ 0 ϕ2vþ2 care about infrared divergences. Each vertex in (14) SI ½ x¼2 1 Tr x 2 ðv þ Þ x contributes either a power of m0 or a square of some combination of momenta q1…q −1, coming from the first 1 2Xvþ1 X f −1 l−1 Δ ϕ2vþ2−lϕl or the second terms in (12), respectively. If our planar þ 2 ð Þ xyTrð x yÞ; ð12Þ l¼1 x;y Feynman diagram is considered as a planar graph drawn on the sphere, the number f is the number of faces of where, in the last equation, we have used our definition this planar graph. We now take into account the identity 2 λ m0 ¼ 4 in order to formally eliminate the coupling λ from f − l þ v ¼ 2 for the Euler characteristic of a planar graph. ð2vþ2Þ ϕ Applying now the standard dimensional analysis, we see interaction vertices SI ½ x. This construction is similar to the “shifted action” of [68]. that the above relation between f, l and v implies that WK Λ2 2 Perturbative expansion of any physical observable can only contain the terms proportional to UV, m0, 4 Λ2 hO½ϕxi is then organized as a formal power series m0= UV and so on, probably times logarithmic terms 2 expansion in the auxiliary parameter ξ in (11): ∼ logðm0Þ. In subsection III B, we will explicitly illustrate the importance of these logarithmic terms. No negative XM λ m m powers of m0 can ever appear. hO½ϕxi ¼ lim lim ξ − Omðm0Þ; ð13Þ 2 λ ξ→1 M→∞ 8 Remembering now the relation m0 ¼ 4, and taking into m¼0 account the powers of coupling associated with vertices, where M is the maximal order of the expansion. This means we immediately conclude that the weights of Feynman 2 λ that we ignore the relation between the mass m0 ¼ 4 and the diagrams in our perturbation theory with the bare mass 2 2 ∼ λ λ coupling constant λ and treat the terms Sð vþ Þ½ϕ as bare m0 are proportional to positive powers of , times I x λ m λ m 0 vertices with 2v þ 2 legs, counting only those powers λv of possible logarithmic terms log ð Þ ,logðlogð ÞÞ (m> ) the t’Hooft coupling constant which come from the vertex andsoon.From(12) we see that the combinatorial pre- 2 factor of vertex with 2v þ 2 legs grows at most linearly pre-factors in (11). The bare mass m0 in the expansion coefficients in (13) is substituted with λ=4 at the very end of with v. Since the number of planar diagrams which the calculation. contribute to the expansion (13) at order m grows at most Let us discuss the general structure of an expansion (13) exponentially with m, and the weight of any diagram is both UV and IR finite, contributions which grow facto- for the free energy of the two-dimensional principal chiral 2 model (2) in the large-N limit, in which only connected rially with m cannot appear in the expansion (13) in D ¼ planar Feynman diagrams with no external legs contribute. dimensions. Consider such a planar diagram with f − 1 independent The numerical results which we present below suggest that (13) is a convergent weak-coupling expansion. At internal loop momenta q1; …;q −1, and with l bare f first sight this contradicts the common lore that pertur- propagators and v vertices. The kinematic weight of such bative expansion cannot capture the nonperturbative a diagram can be written as physics of the model (2). Let us remember, however, Z … that our expansion is no longer a strict power series in λ, ∼ 2 … 2 V1 Vv WK d q1 d qf−1 2 2 ; which is known to be factorially divergent. Rather, the ðΔðQ1Þþm0Þ…ðΔðQlÞþm0Þ 2 mass term m0 ∼ λ in bare propagators allows for loga- ð14Þ rithms of coupling in the series, in close analogy with resummation of infrared divergences in hard thermal loop … where Q1 Ql are momenta flowing through each diagram perturbation theory [67]. It is easy to convince oneself line, that the formal expansion which contains powers of logs of coupling along with the powers of coupling can XD−1 2 Qμ incorporate the nonperturbative scaling of the dynamical ΔðQÞ¼ 4 sin ð15Þ 2 mass gap μ¼0 1 is the lattice Laplacian (6) in momentum space which m2 ∼ exp − ; ð16Þ 2 β0λ behaves as ΔðQÞ ∼ Q at small momenta and V1…Vv are

114512-5 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) where β0 is the first coefficient in the perturbative where Δx;y is again the lattice Laplacian (6). The well- β-function. Indeed, we can rewrite the function (16) as known saddle-point solution shows that the two-point λ a convergent expansion in powers of z ¼ logð Þ: correlation function Gxy ¼hnaxnayi is proportional to the propagator of a free massive scalar field, with the mass 1 −1 −z m being the solution of the gap equation: exp − ¼ expð−β0 e Þ β0λ ∞ Z Xþ −β −k D 1 ð 0Þ −kz d p ¼ e λI0ðmÞ¼1;I0ðmÞ ≡ : ð19Þ k! 2π D Δ 2 k¼0 ð Þ ðpÞþm Xþ∞ l Xþ∞ −k l ð−zÞ β0 k ¼ : ð17Þ 1 l! k! In D ¼ , 2, 3 dimensions, the function I0ðmÞ is l¼0 k¼0 8 In what follows, we apply this resummation prescription > 1 1 m2 −1=2 1; < 2m ð þ 4 Þ ;D¼ to construct the infrared-finite weak-coupling expansion 2 I0ðmÞ¼ − 1 m 1 2 2; ð20Þ for several models: exactly solvable large-NOðNÞ sigma > 4π logð32Þð þ Oðm ÞÞ;D¼ :> model in D ¼ 1, 2, 3 dimensions, exactly solvable large-N A þ Bm þ;D¼ 3; principal chiral model on one-dimensional lattices with L0 ¼ 2, 3, 4 lattice sites, and, most importantly, the 0 252731 −0 0795775 principal chiral model in D ¼ 2 dimensions at zero and where in the last line A ¼ . and B ¼ . . finite temperature. If the mass gap m is small enough, the corresponding solutions for m read

III. EXACTLY SOLVABLE EXAMPLES AND 8 ffiffiffiffiffiffiffiffiffiffiffiffiffi > p 2 COUNTEREXAMPLES <> λ þ 4 − 2;D¼ 1; 2 4π A. Large-NOðNÞ sigma model m ðλÞ¼ 32 expð− Þ;D¼ 2; ð21Þ > λ in D = 1, 2, 3 dimensions : −1 m ¼jB jðA − 1=λÞ;D¼ 3. Large-NOðNÞ sigma model provides one of the simplest examples of a nonperturbative quantum field theory which can be exactly solved by using saddle point approximation. The goal of this section is to construct the infrared-finite In D ¼ 2 dimensions, this model has dynamically generated weak-coupling expansion following the prescription of nonperturbative mass gap, similarly to the more complicated Sec. II. In order to parametrize the fluctuations of the 1 principal chiral models and non-Abelian gauge theories. nax field around the classical vacuum with n0x ¼ ,we On the other hand, for this model one can also explicitly use the stereographic projection from the unit sphere − 1 construct the formal weak-coupling perturbative expansion, in N-dimensions to N -dimensional real space of fields ϕ 1 … − 1 which is dominated by the “cactus”-type diagrams, sche- ix, i ¼ ; ;N , which is the analogue of the Cayley matically shown in Fig. 1. These properties make this model map (3): an ideal testing ground for the infrared-finite weak-coupling pffiffiffi expansion described in Sec. II. 1 − λ ϕ2 4 x λϕix The partition function of the OðNÞ sigma model is n0x ¼ ;nix ¼ ; ð22Þ 1 þ λ ϕ2 1 þ λ ϕ2 given by the path integral over N-component unit vectors 4 x 4 x 0… − 1 nax, a ¼ N attached to the sites of D-dimensional P ϕ2 ≡ ϕ ϕ square lattice, which we label by x: where x i ix ix.Thisisauniquemapbetweentheunit Z sphere embedded in N-dimensional space and the N − 1- N X Z Dn exp − Δ n n ; 18 dimensional real space, in which only a single point with ¼ x 2λ xy ax ay ð Þ −1 0 x;y;a n0x ¼ , nix ¼ is excluded. The integration measure on the unit sphere can be written in terms of the stereographic coordinates ϕix as [60] (see also Appendix B3): λ −N D Dϕ 1 ϕ2 nx ¼ x þ 4 x : ð23Þ

FIG. 1. Feynman diagrams which contribute to the two-point Following the construction of Sec. II, we now include this correlator in the large-N limit of the OðNÞ sigma model. Jacobian in the action for the fields ϕix, which reads:

114512-6 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) 1 X λ ϕ Δ δ ϕ ϕ On the other hand, we can use the Eq. (28) for an S½ ¼2 xy þ 2 xy ð x · yÞ iterative calculation of the coefficients of the formal weak- x;y coupling expansion Xþ∞ −1 kþlλkþl X ð Þ 2 k 2 l þ ΔxyðϕxÞ ðϕyÞ ðϕx · ϕyÞ 2 2 2 2 2 4kþl m ¼ m0 þ λξσ1ðm0Þþλ ξ σ2ðm0Þþ; k;l¼0 · x;y kþl≠0 2 2 z ¼ 1 þ λξz1ðm0Þþλ ξ z2ðm0Þþ; ð29Þ Xþ∞ −1 X ð−1Þk λk þ ðϕ2Þk; ð24Þ 4k x 2 k x which is also the formal power series in the auxiliary k¼ ξ P parameter in (28). In practice, we continue the iterations ϕ ϕ ≡ ϕ ϕ up to some finite order M, and calculate m2 and z by where x · y i ix · iy.WehavealsotakenintoP Δ ϕ2 k ϕ2 l summing up all the terms in the series (29) with powers of ξ account that the terms of the form x;y xyð xÞ ð yÞ vanish less than or equal to M. After such a truncation of series, we at large N due to factorization property hðϕx · ϕyÞðϕz · ϕtÞi ¼ substitute ξ 1 and m0 → λ=2. hϕ · ϕ ihϕ · ϕ i and translational invariance. In full analogy ¼ x y z t In D ¼ 1 dimensions, our prescriptions yields the with the derivation of the action (7) in Sec. II, the Jacobian of series with summands of the form λkð8 þ λÞ−M−2 and the transformation from compact to noncompact field vari- 1 2 − −3 2 2 λkþ = 8 λ M = , k 1; 2; …, times some integer- ables results in the bare “mass term” m ¼ λ=2. ð þ Þ ¼ 0 valued coefficients which depend on M. The expansion, A convenient way to arrive at the perturbative expansion thus, behaves regularly in the limit λ → 0, where the mass for the action (24) is to iterate the corresponding goes to zero linearly in λ. Schwinger-Dyson equations In D ¼ 2 dimensions, we get the expansion which η 2 η involves both powers of λ and logðλÞ (see also [60]): −1 Δ 2 − Δ ð þ Þ G ðpÞ¼ ðpÞþm0 ðpÞ 2 ð1 þ ηÞ 1 Z XM λ l MþXminðl; Þ D 2 k λ η λ 1 d q m ðλ;MÞ¼ − log σl;kλ ; ð30Þ − − ΔðqÞGðqÞ; 64 2 1 þ η 2 ð1 þ ηÞ3 ð2πÞD l¼0 k¼minðlþ2;MÞ Z λ λ D 2 d p and similarly for zðλ;MÞ. As discussed above, the expan- η ≡ hϕxi¼ GðpÞ; ð25Þ 4 4 ð2πÞD sion involving the logs of λ can reproduce the nonperturbative scaling (21) of the mass gap in D ¼ 2. Finally, in which in the large-N limit involve only the two-point D 3 dimensions we obtain the expansion in positive ¼ pffiffiffi function powers of λ. Z In Fig. 2, we compare the convergence of the expansions D d p − 1 hϕ · ϕ i¼ eipμðx yÞμ GðpÞ: ð26Þ in D ¼ , 2, 3 dimensions, plotting the relative error x y ð2πÞD mðλ;MÞ − mðλÞ ϵðmðλ;MÞÞ ¼ ð31Þ The form of Eq. (25) suggests that the solution has the form mðλÞ of the free scalar field propagator with the mass m, times some wave function renormalization factor z: of the truncated series (29) with respect to the exact result mðλÞ given by (21). The values of the coupling λ are fixed z G p : in such a way that the exact nonperturbative mass gap is ð Þ¼Δ 2 ð27Þ ðpÞþm equal to m ¼ 0.4 for all dimensions D ¼ 1, 2, 3. The series (29) exhibit the fastest convergence at D ¼ 1.AtD ¼ 2, Substituting this ansatz in (25), we obtain the following the convergence is not monotonic, and also slower than in equations for m and z: D ¼ 1.InD ¼ 3 the convergence is slowest, and from the λξ λξ convergence plot in Fig. 2 it is difficult to say whether 2 2 2 − 2 2 the series converge or not, even with up to 500 terms in the m ¼ m0z 2 z þ 2 zm I0ðmÞ; expansion. These results suggest that for the large-NOðNÞ λξ 2 sigma model D 2 seems to be the critical dimension z ¼ 1 þ z I0ðmÞ; ð28Þ ¼ 4 separating the convergent and nonconvergent expansions. where we have again introduced the auxiliary parameter ξ, to be set to ξ ¼ 1, in order to define the formal power B. Nonperturbative mass gap in the 2 two-dimensional OðNÞ sigma model counting scheme. Substituting ξ ¼ 1 and m0 ¼ λ=2,we immediately arrive at the exact solutions (21) for the mass We now turn to the physically most interesting example m.Forz the exact solution is simply z ¼ 2. of the two-dimensional OðNÞ sigma model in the large-N

114512-7 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) ⋆ D=1 minimum as M1 , the position of the next maximum of this D=2 — M⋆ 0.2 D=3 dependence as 2 , and so on. These positions depend on λ and shift towards larger M at smaller λ. We can think of ⋆ ⋆ the extrema positions M1 ;M2 ; … as of some characteristic scales which determine the convergence rate of the trun- ,M)) 0.1

λ ⋆ cated series. In particular, at M>M1 the convergence to (m( ε the exact value is rather fast. Remarkably, we have found ⋆ λ 0 that M1 as a function of with good precision appears to be proportional to the square of correlation length (in lattice units): -0.1 0 0.01 0.02 0.03 0.04 4π ⋆ ∼ 2 ∼ −2 ∼ 1/M M1 lc m exp λ : ð32Þ FIG. 2. Relative error (31) of the mass gap for the OðNÞ sigma ⋆ λ 1 λ model from the truncated expansion (29) as a function of For illustration, in Fig. 5 we plot logðM1;2ð ÞÞ versus = , truncation order M for D ¼ 1, 2, 3. The coupling is fixed to together with the linear fits of the form logðM⋆ðλÞÞ ¼ λ 0 4 ⋆ yield the mass mð Þ¼ . in (21) for all D. const − β=λ.ForM1 the best fit yields the coefficient β ¼ ⋆ 12.49 which is very close to 4π ¼ 12.57.ForM2 we get limit, where the mass gap (21) exhibits nonperturbative the best fit value β ¼ 9.33, which is close to 3π ¼ 9.42 2 1 ⋆ 3=2 −3=2 scaling m ∼ exp − similar to the one found also in and suggests the scaling M2 ∼ lc ∼ m . This might ð β0λÞ imply that at sufficiently small λ the two extrema might two-dimensional principal chiral model and in non-Abelian ⋆ gauge theories. In Fig. 3, we illustrate the convergence of merge. However, for M2,wehavefewerdatapoints,and the series (30) for the mass m and the renormalization those exhibit some tendency for faster growth at larger 1 λ ⋆ factor z to the exact results (21) and z ¼ 2 for different values of = , thus, the scaling exponent for M2 might be values of the coupling constant λ. To make the convergence different at asymptotically small λ. at large truncation orders M more obvious, in Fig. 4 we also The data presented in Figs. 4 and 5 provide a clear show the relative error (31) of the truncated series (30) as a numerical evidence that the infrared-finite weak-coupling function of M in logarithmic scale for several values of λ. expansion (30) indeed incorporates the nonperturbative 2 4π While at large values of λ the series converge with the dynamically generated mass gap m ¼ 32 expð− λ Þ.Ifwe precision of 10−16 at M ∼ 500, the convergence becomes were to sample the series (30) using DiagMC algorithm, ⋆ slower at smaller values of λ—and, correspondingly, at the linear scaling of M1 ðλÞ with correlation length would smaller values of the mass m. also result in the expectable critical slowing down of An interesting feature of the dependence of mðλ;MÞ on simulations near the continuum limit. M is that it is not monotonous, but rather has several The series which we obtain from the field-theoretical extrema with respect to M. We denote the smallest value weak-coupling expansion (29) based on Schwinger-Dyson of M at which the dependence of mðλ; MÞ on M has a equations (28) are in this respect drastically different from

2.1 1.4 2 λ=4.6 λ=3.2 λ λ 1.2 1.9 =4.2 =2.8 λ=3.8 λ=2.4 1 1.8 λ=3.4 λ=2.0 0.8 1.7 ,M) ,M) λ λ 1.6 z(

m( 0.6 1.5 0.4 1.4 0.2 λ=4.6 λ=3.8 λ=3.2 λ=2.4 λ=4.2 λ=3.4 λ=2.8 λ=2.0 1.3 0 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1/M 1/M

FIG. 3. Convergence of the truncated series (29) to the exact values (21) for the effective mass term m (on the left) and the renormalization factor z (on the right), which are plotted as functions of 1=M. The data points at 1=M ¼ 0 correspond to the exact results (21).

114512-8 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) -2 8 7 M* -4 *1 M 2 6 λ -6 12.49/ -0.72 5 9.33/λ + 2.60 , M))) ) * λ -8 4 (m( ε

( 3

-10 log(M 10 2 log -12 λ=4.8 λ=4.2 1 -14 λ=3.4 0 -16 -1 150 200 250 300 350 400 450 500 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 M 1/λ

ϵ λ ⋆ λ FIG. 4. Relative error ðmð ;MÞÞ of the effective mass m FIG. 5. The expansion truncation orders M1;2ð Þ at which the obtained from the truncated series (29) at large values of M. truncated series (30) have local extrema, plotted as functions of the inverse coupling constant λ−1. Solid lines are the fits of the form logðM⋆ðλÞÞ ¼ const − β=λ. the simplest formal expansion (17) of the exact answer (21) in powers of logs. We have explicitly checked that for this case the positions of the extrema of m with respect to M are 1 λ independent of λ. In this respect, our infrared-finite weak- † Trðg1g0Þ ¼ 1 − ; ðL0 ¼ 2; λc ¼ 4Þð33Þ coupling expansion (29) seems to be closer to the notion of N 8 trans-series. Let us stress, however, that the series (30) are not trans- 3 1 1 1 λ 1 λ 2 series yet, despite formal similarity with (1).Ina † − − 1− 3 λ 3 Trðg1g0Þ ¼ λ þ2 8 λ 3 ; ðL0 ¼ ; c ¼ Þ: mathematically strict sense, resurgent trans-series (1) N should give a “minimal” representation of a function ð34Þ with optimal convergence. For the large-NOðNÞ sigma model, this optimal trans-series representation for the For L0 ¼ 4, the weak-coupling solution is only available in dynamically generated mass is given by a single term 2 4π the implicit form: m ¼ 32 expð− λ Þ [see the exact solution (21)]. Thus, the true trans-series for this model consist of only one term, 1 2 λ 2 8 which is enough for convergence! The expansion (30) † − − δ2 λ Trðg1g0Þ ¼ λ 8 λ ; c ¼ π ; ð35Þ also converges to the exact result (21), but at a non- N optimal rate.2 8 2 where δ is the solution of the equation πλ ðEð1 − δ Þ− 2 2 C. Exactly solvable counter-example: δ Fð1 − δ ÞÞ ¼ 1 and E and F are the complete elliptic Finite chiral chain models integrals of second and first type, respectively. These analytic expressions offer the possibility to check The U N × U N principal chiral model (2) can be ð Þ ð Þ the convergence of the infrared-finite weak-coupling solved exactly in D 1 dimensions for lattice sizes ¼ expansion of Sec. II for small lattice sizes. Of course, L0 2, 3, 4 and L0 ∞ [69]. The first and the last ¼ ¼þ for finite lattices the dimensional analysis following cases can be reduced to the Gross-Witten unitary matrix Eq. (14) would not work, and loop integrals will only model [35]. For other two cases, solutions can be found in produce some rational functions of λ instead of logs. [69]. For all values of L0 the one-dimensional principal Nevertheless it is interesting to check whether these rational chiral model exhibits a third-order Gross-Witten phase functions provide a good approximation to the exact results transition at some critical coupling λ λ L0 , which ¼ cð Þ (33), (35) and (35). To this end, we have performed exact separates the strong-coupling and the weak-coupling 1 † calculations of the coefficients of the weak-coupling regimes. In particular, for the mean link hN Trðg1g0Þi the expansion (13), generating them recursively up to some λ λ exact results in the weak-coupling regime < c are maximal order M from the Schwinger-Dyson equations (50), to be presented in subsection IVA. Such an exact calculation is possible for L0 ¼ 2 up to M ¼ 16 and 4 8 2We thank Ovidiu Costin for pointing out these properties of for L0 ¼ up to M ¼ . Explicit expression for the mean “ ” 1 † ϕ trans-series to us at the Resurgence 2016 workshop in Lisbon. link hN Trðg1g0Þi in terms of the correlators of x matrices

114512-9 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) 0 IV. SCHWINGER-DYSON EQUATIONS AND DIAGMC ALGORITHM FOR UðNÞ × UðNÞ

)) -1

M PRINCIPAL CHIRAL MODEL )> 0

g -2

1 The aim of this section is to construct a DiagMC + algorithm for sampling the weak-coupling expansion (13). -3 While in this work we will only consider the weak-coupling

(<1/N tr(g L =2, λ/λ =0.5 expansion (13) based on the action (7) of the large-N

ε -4 0 c ( λ λ L0=3, / c=0.5 principal chiral model (2), our construction of the 10 λ λ L0=4, / c=0.5

log λ λ DiagMC algorithm is quite general and can be applied to -5 L0=2, / c=0.8 L =3, λ/λ =0.8 0 λ λc both to the weak and strong-coupling expansions in general L0=4, / c=0.8 -6 large-N matrix field theories. We will briefly outline the 0 0.2 0.4 0.6 0.8 1 extension to non-Abelian gauge theories in Sec. VI. 1/M

FIG. 6. Relative error of the weak-coupling expansion (13) for A. Schwinger-Dyson equations for the large-N 1 † the mean link hN Trðg1g0ÞiM of the principal chiral model (2) on UðNÞ × UðNÞ principal chiral model with Cayley the one-dimensional lattices with L0 ¼ 2, 3, 4 sites, as a function parametrization of UðNÞ group of truncation order M for fixed λ=λ ¼ 0.5 and λ=λ ¼ 0.8. c c As in [4,5,8,13,14], the starting point for the construction of our DiagMC algorithm are the Schwinger-Dyson equa- and the prescription for truncating the expansion at some tions, which we derive for the action (7) of the principal finite order M are given in subsection VA. chiral model (2). In contrast to the works [13,14] on the In Fig. 6, we illustrate the relative error of the bosonic DiagMC, here we consider the full untruncated truncated weak-coupling expansion (13) for the mean hierarchy of Schwinger-Dyson equations for multitrace, 1 † in general disconnected, correlators of the ϕ fields defined link h Trðg1g0ÞiM as a function of inverse truncation x N in (3): order 1=M for L0 ¼ 2,3,4,fixingλ=λc ¼ 0.5 and λ=λc ¼ 0.8. The relative error is defined in full analogy 1 1 2 2 r r h½x1;…;x ½x1;…;x …½x1;…;x i with (31). At the first sight, the convergence seems to n1 n2 nr be rather fast; however, a detailed inspection of 1 1 1 ≡ ϕ 1 …ϕ 1 ϕ 2 …ϕ 2 … ϕ r …ϕ r numerical results reveals a tiny discrepancy of order Trð x x Þ Trð x x Þ Trð x xn Þ : N 1 n1 N 1 n2 N 1 r of 0.1% which cannot be ascribed to numerical round- off errors, and which seems to stabilize at some finite ð36Þ value as M is increased (at least for L0 ¼ 2). An analytic calculation of the few first orders revealed To derive the Schwinger-Dyson equations for the that the weak-coupling expansion (13) correctly repro- field correlators (36), let’s consider the full derivative of duces only the three leading coefficients of the standard the form perturbative expansion in powers of λ, which is con- Z vergent for finite L0. δikδjl ∂ Dϕ ððϕ 1 …ϕ 1 Þ Trðϕ 2 …ϕ 2 Þ… r x2 xn1 kl x1 xn2 It seems, thus, that for finite-size one-dimensional N ∂ϕxij lattices we observe the misleading convergence of the ×Trðϕxr …ϕxr Þ expð−SF½ϕ − SI½ϕÞÞ ¼ 0; ð37Þ resummed perturbative expansion to some unphysical 1 nr branch of the solutions of Schwinger-Dyson equations. This phenomenon might happen even in zero-dimensional where we have explicitly separated the action (7) into the models [68] and was first noticed in Bold DiagMC quadratic and interacting parts given by (10) and (11), (12) simulations of interacting fermions [70]. Remembering to simplify the derivation and notation. We now expand the fact that for the large-NOðNÞ sigma model in the the derivative according to the Leibniz product rule, infinite space the expansion (13) converged with much contract matrix indices and finally contract the result with 0 2 −1 higher precision than the discrepancy which we observe the bare propagator G 1 ¼ðΔ þ m0Þ coming from the x1x xy here for finite-size chiral chains (compare Figs. 6 and 4), quadratic part SF½ϕx of the action (7). This yields the we conclude that infinite lattice size and the presence of following infinite system of equations for the multitrace λ logð Þ terms seem to be crucial for the convergence to the observables (36): correct physical result. An important extension of our analysis, which we relegate for future work, would be to X 0 0 ∂SI h½x1;x2i ¼ G þ G ;x2 ; ð38Þ extend the sufficient conditions of [68] for the convergence x1x2 x1x ∂ϕ to the physical result to the weak-coupling expansion (13). x x

114512-10 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) X ∂S 1 1 2 … 2 … r … r 0 2 … 2 … r … r 0 I 1 2 … 2 … r … r h½x1;x2½x1; ;xn2 ½x1; ;xn i ¼ G 1 1 h½x1; ;xn2 ½x1; ;xn i þ G 1 ;x2 ½x1; ;xn2 ½x1; ;xn r x1x2 r x1x ∂ϕ r x x 1 Xr Xns 0 2 … 2 … s … s 1 s … s … r … r þ 2 G 1 s h½x1; ;xn2 ½x1; ;x −1;x2;x 1; ;xn ½x1; ;xn i ð39Þ x1xp p pþ s r N s¼2 p¼1

1 … 1 2 … 2 … r … r h½x1; ;xn1 ½x1; ;xn2 ½x1; ;xnr i 0 1 1 2 2 r r 0 1 1 2 2 r r 1 1 … … … … 1 1 … … … … ¼ G h½x3; ;xn1 ½x1; ;xn2 ½x1; ;xn i þ G h½x2; ;xn1−1½x1; ;xn2 ½x1; ;xn i x1x2 r x1xn1 r Xn−2 0 1 … 1 1 … 1 2 … 2 … r … r þ G 1 1 h½x2; ;x −1½x 1; ;xn1 ½x1; ;xn2 ½x1; ;xn i x1xu u uþ r u¼3 X ∂S 0 I 1 … 1 2 … 2 … r … r þ G 1 ;x2; ;xn1 ½x1; ;xn2 ½x1; ;xn x1x ∂ϕ r x x 1 Xr Xns 0 2 … 2 … s … s 1 … 1 s … s … r … r þ 2 G 1 s h½x1; ;xn2 ½x1; ;x −1;x2; ;xn1 ;x 1; ;xn ½x1; ;xn i; ð40Þ x1xp p pþ s r N s¼2 p¼1

∂ Here by SI we have schematically denoted the terms which x1; …;x1 … xr; …;xr x1;…;x1 … xr; …;xr ; ∂ϕx h½ 1 n1 ½ 1 nr i ¼ h½ 1 n1 i h½ 1 nr i are produced by applying the derivative operator in (37) to 42 the interacting part (11) of the action (7): ð Þ

with single-trace correlators h½x1; …;xni obeying the much Xþ∞ ð2vþ2Þ simpler nonlinear equation ∂S ½ϕ λ v ∂S ½ϕ I ¼ ξv − I ; ∂ϕ 8 ∂ϕ 0 0 x v 1 x … … … ¼ h½x1; ;xni ¼ Gx1x2 h½x3; ;xni þ Gx1xn h½x2; ;xn−1i ð2vþ2Þ 2Xvþ1 ∂S ½ϕ Xn−2 I 2ϕ2vþ1 −1 l−1 0 ¼ m0 x þ ð Þ G x2; …;x −1 x 1; …;x ∂ϕ þ x1xu h½ u ih½ uþ ni x l¼1 u¼3 2vXþ1−l X X ∂ 2 1− − 0 SI Δ ϕmϕl ϕ vþ l m þ G ;x2; …;x : ð43Þ × xy x y x : ð41Þ x1x ∂ϕ n m¼0 y x x

The form of the Schwinger-Dyson equation (38) for the These terms contain three or more ϕ fields and should be two-point correlator does not change. inserted into the single-trace expectation values (36) within Schwinger-Dyson equations for large-N quantum field ∂SI ½ϕ 1 ∂SI ½ϕ theories are almost always considered in the nonlinear the Eqs. (38), (39), and (40): h½ ∂ϕ yi ¼ h Trð ∂ϕ ϕyÞi, x N x form (43), which is much more compact than the linear and so on. For the sake of brevity, we omit the matrix ϕ ϕ equations (40). One of the well-known examples of such indices of x, so that all products of x are understood as N ϕ ϕ nonlinear Schwinger-Dyson equations in the large- limit matrix products. In particular, this implies that x and y do are the Migdal-Makeenko loop equations for non-Abelian ≠ not commute for x y. gauge theories [71,72]. Likewise, even at finite N one ’ In the t Hooft large-N limit, the last summands in the can also arrive at a more compact nonlinear representa- Eqs. (39) and (40) can be omitted, since they are suppressed tions of the Schwinger-Dyson equations (38), (39), and by a factor 1=N2 and all multitrace correlators (36) are of 0 (40) by rewriting them in terms of connected or one- order N . Since in this limit the correlators (36) with odd particle-irreducible correlators. This strategy is numeri- ϕ numbers of x fields within some of the traces, such as cally very efficient for approximate solutions based on ≡ 1 ϕ 1 ϕ 1 2 h½x½yi hN Trð xÞ N Trð yÞi, are suppressed as =N ,we some truncations of the full set of equations (especially can also limit the equations to the space of multitrace for correlators with small numbers of fields), and is correlator which contain an even number of ϕ fields within commonly used for numerical solutions of QCD each trace. Furthermore, in the t’Hooft large-N limit the Schwinger-Dyson equations [73,74] and in condensed Eqs. (38), (39) and (40) without the 1=N2-suppressed terms matter physics [1,14,75]. However, Schwinger-Dyson admit the large-N factorized solution, equations are always linear equations when written in

114512-11 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) ð2vþ2Þ X ð2vþ2Þ terms of disconnected correlators of the form (36). This ∂S ½ϕ 1 ∂S ½ϕ I ≡ eipx I linear form is very convenient for the construction of ∂ϕ V ∂ϕ DiagMC algorithms, which we consider in the next p Xx x δ − − − subsection. ¼ ðp q1 q2vþ1Þ … To proceed towards the DiagMC algorithm, we go to the q1 q2vþ1 momentum space, and define the momentum-space field × Vðq1; …;q2 1Þϕ …ϕ : ð46Þ operators and correlators as vþ q1 q2vþ1 An important property of the vertex functions (45) is that … 1 X X they are positive for all values of q1; ;q2vþ1.Whilewe ipx −ipx ’ ϕp ¼ e ϕx; ϕx ¼ e ϕp: ð44Þ don t have an explicit proof of this statement, in our V 14 x p Monte Carlo simulations with over 10 Metropolis updates in total we have explicitly checked that the values of vertex function were always positive. Furthermore, in Appendix A, Note that we have assumed the finite lattice volume V, we demonstrate that if all the momenta in (45) are small and which will be convenient for further analysis. In particu- the lattice Laplacian can be approximated as ΔðpÞ¼p2,the lar, this allows us to interpret all the delta-functions with vertex functions take the simple form momentum-valued arguments as Kronecker deltas on the … 2 2 discrete set of lattice momenta. For brevity, we will Vðq1; ;q2vþ1Þ¼m0 þðq1 þ q3 þþq2vþ1Þ ; ð47Þ denote δðpÞ ≡ δp;0. We will see in what follows that the algorithm performance is practically volume-independent, which manifestly demonstrates their positivity. thus, one can make a smooth transition from discrete to Furthermore, in order to define the power counting continuous momenta if necessary. scheme to be used in our DiagMC simulations, we expand It is also convenient to introduce the momentum-space the momentum-space correlators of ϕ fields in formal vertex functions power series in our auxiliary parameter ξ: 1 1 Trðϕ 1 …ϕ 1 Þ… Trðϕ r …ϕ r Þ … p1 pn p1 pn Vðq1; ;q2vþ1Þ N 1 N r 2Xvþ1 2vXþ1−l Xþ∞ λ m 2 l−1 m 1 1 r r ¼ m0 þ ð−1Þ Δðq 1 þþq Þ; ð45Þ ¼ ξ − h½p1; …;p …½p1; …;p i : ð48Þ mþ mþl 8 n1 nr m l¼1 m¼0 m¼0

In terms of the correlators (48) of the momentum-space so that the Fourier transform of the derivative (41) of the fields ϕp, the large-N limit of the Schwinger-Dyson interacting part of the action can be compactly represented as equations (38), (39), and (40) read

δðp1 þ p2Þ h½p1;p2i ¼ δ 0 G0ðp1Þð49aÞ m m; V

Xm X − δ − − − … … G0ðp1Þ ðp1 q1 q2vþ1ÞVðq1; ;q2vþ1Þh½q1; ;q2vþ1;p2im−v; ð49bÞ … v¼1 q1; ;q2vþ1

δ 1 1 1 1 2 2 r r ðp1 þ p2Þ 1 2 2 r r h½p1;p2½p1; …;p …½p1; …;p i ¼ G0ðp1Þh½p1; …;p …½p1; …;p i ð49cÞ n2 nr m V n2 nr m

Xm X − 1 δ 1 − − − … … 1 2 … 2 … r … r G0ðp1Þ ðp1 q1 q2vþ1ÞVðq1; ;q2vþ1Þh½q1; ;q2vþ1;p2½p1; ;pn2 ½p1; ;pnr im−v; ð49dÞ … v¼1 q1; ;q2vþ1

δ 1 1 1 1 1 2 2 r r ðp1 þ p2Þ 1 1 1 2 2 r r h½p1;p2; …;p ½p1; …;p …½p1; …;p i ¼ G0ðp1Þh½p3; …;p ½p1; …;p …½p1; …;p i ð49eÞ n1 n2 nr m V n1 n2 nr m

114512-12 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) δ 1 1 ðp1 þ pn1 Þ 1 1 1 1 2 2 r r þ G0ðp1Þh½p2;p3; …;p −1½p1; …;p …½p1; …;p i ð49fÞ V n1 n2 nr m

nX1−2 δ 1 1 ðp1 þ pAÞ 1 1 1 1 1 2 2 r r þ G0ðp1Þh½p2; …;p −1½p 1; …;p ½p1; …;p …½p1; …;p i ð49gÞ V A Aþ n1 n2 nr m A¼4

Xm X − 1 δ 1 − − − … G0ðp1Þ ðp1 q1 q2vþ1ÞVðq1; ;q2vþ1Þ … v¼1 q1; ;q2vþ1 … 1 1 … 1 2 … 2 … r … r ×h½q1; ;q2vþ1;p2;p3; ;pn1 ½p1; ;pn2 ½p1; ;pnr im−v; ð49hÞ

We have also introduced the bare propagator in the order of the expansion all diagram weights are still momentum space infrared-finite.

1 2 λ G0ðpÞ¼ ;m0 ≡ : ð50Þ B. Metropolis algorithm for stochastic solution Δ p m2 4 ð Þþ 0 of linear equations It is easy to convince oneself that recursive solution of As discussed in the previous subsection, Schwinger- the Schwinger-Dyson equations (49) generates the con- Dyson equations in terms of disconnected field correlators ventional weak-coupling expansion, with the auxiliary always form an infinite system of linear equations, which parameter ξ formally serving as a coupling. For example, can be written in the following general form to find the order m ¼ 2 contribution to the two-point X correlator h½p1;p2i, we use the first equation in (49) to ϕðXÞ¼bðXÞþ AðXjYÞϕðYÞ; ð53Þ express it in terms of four-point correlator of order m ¼ 1 Y and the six-point correlator of order m ¼ 0. The four- point correlator at order m ¼ 1 can be related to two-point where AðXjYÞ is some theory-dependent linear operator, correlator at order m ¼ 1 and the six-point correlator at bðXÞ represents the “source” terms in the Schwinger- m ¼ 0 using the last equation in (49). Finally, the two- Dyson equations (typically, the contact delta function term point correlator at order m ¼ 1 is again related to four- in the lowest-order equation, like in (38) and (49), and X point correlator at m ¼ 0 by virtue of the first equation in and Y are generalized indices incorporating all arguments (49). This chain of relations is schematically illustrated of field correlators which enter the Schwinger-Dyson in Fig. 7. equations. For the Schwinger-Dyson equations (49) It is instructive to explicitly calculate the first perturbative correction to the two-point correlator, which can be readily found from the Schwinger-Dyson equations (49) for the two-point correlator:

2 δðp1 þ p2Þ h½p1p2i1 ¼ G ðp1Þ Σ1ðp1Þ; ð51Þ 0 V where 1 X Σ1ðpÞ¼− ðVðq;−q;pÞþVðp;q;−qÞÞG0ðqÞ V q 2 X λ − 2Δ 2Δ − Δ − ¼ G0ðqÞ 4 þ ðpÞþ ðqÞ ðp qÞ : V q ð52Þ

In particular, Σ1ðp ¼ 0Þ¼−2. If we interpret this quan- tity as the first-order correction to the self-energy, then at zero momentum this correction, being multiplied by FIG. 7. An example of the recursive calculation of diagrams ð−λ=8Þ and ξ ¼ 1, exactly cancels the bare mass term contributing to order m ¼ 2 two-point correlator from Schwinger- λ=4! Of course, this cancellation requires resummation of Dyson equations (49). Combinatorial diagram weights are not the infinite chain of bubble diagrams, and at any finite explicitly shown in order not to clutter the plot.

114512-13 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) discussed in the previous subsection, X and Y are sequen- momenta p1 and p2 whenever X contains only two of them. 1 … 1 … r … r Thus, no field correlators should be explicitly saved in ces of momenta ffp1; ;pn1 g; ; fp1; ;pnr gg which parametrize the multitrace correlators (48) with an addi- computer memory, and the memory required by the tional order-counting integer variable m. Note that the set of algorithm does not directly scale with system volume X indices is in most cases infinite, which excludes the (however, it does scale with the maximal expansion order, application of standard numerical linear algebra methods which might be correlated with the volume via required to the equation (53) (see, however, [76] for an attempt in statistical error). this direction). Bringing the Schwinger-Dyson equations In order to simplify the notation in what follows, let us into the form (53) can be subtle for field theories for also define the quantities which the free (quadratic) part of the action has flat X X directions or vanishes, but these subtleties are not relevant N ðYÞ¼ jAðXjYÞj; N b ¼ jbðXÞj: ð56Þ for the present work. X X ϕ The solution ðXÞ of (53) can be written as a formal S geometric series in A: In this work, we propose to sample the sequences with probability (55) by using the Metropolis-Hastings algo- Xþ∞ X X rithm with the following updates: ϕðXÞ¼ … δðX; XnÞAðXnjXn−1Þ… n¼0 X0 Xn Algorithm IV. B. 1

× AðX1jX0ÞbðX0Þ: ð54Þ Add index: With probability pþ a new generalized index Xnþ1 is added to the sequence fXn; …;X0g, where the probability π N If the diagrammatic expansion which is obtained by distribution of Xnþ1 is ðXnþ1jXnÞ¼jAðXnþ1jXnÞj= ðXnÞ. σ iterating Schwinger-Dyson equations is truncated at some The sign variable is multiplied by signðAðXnþ1jXnÞÞ. Remove index:If the sequence contains more than one finite order, there is only a finite number of terms in these 1 − generalized index, with probability pþ the last generalized series, which correspond to a finite number of diagrams index X is removed from the sequence, which, thus, changes contributing at a given expansion order. n from fXn;Xn−1; …;X0g to fXn−1; …;X0g. The sign variable σ The key idea of the DiagMC algorithm which we is multiplied by signðAðXnjXn−1ÞÞ. devise in this work is to evaluate the series (54) by Restart:If the sequence contains only one generalized index X0, 0 1 − stochastically sampling the sequences of generalized it is replaced by X0 with probability pþ. The probability 0 π 0 0 N σ indices S ¼fXn; …;X0g with arbitrary n in (54) with distribution of X0 is ðX0Þ¼jbðX0Þj= b. The sign variable 0 probability is set to signðbðX0ÞÞ. These updates should be accepted or rejected with the … probability [77] S jAðXnjXn−1Þj jAðX1jX0ÞjjbðX0Þj wð Þ¼ N ; ð55Þ w wðS0ÞπðS0 → SÞ αðS → S0Þ¼min 1; : ð57Þ S π S → S0 where N w is the appropriate normalization factor. wð Þ ð Þ The solution ϕðXÞ to the system (53) can be found by histogramming the last generalized index Xn in the sequence Explicit calculation yields the following acceptance prob- fXn; …;X0g, where each occurrence of Xn is weighted abilities for the three updates introduced above: with the sign σðXn;…;X0Þ¼signðAðXnjXn−1ÞÞ… N ðX Þð1 − p Þ signðAðX1jX0ÞÞsignðbðX0ÞÞ. This sign reweighting puts α ¼ min 1; n þ ; add p certain limitations on the use of the method, since we are þ effectively sampling the series in which the linear operator α 1 pþ AðXjYÞ is replaced by jAðXjYÞj and which typically have a remove ¼ min ; ; N ðX −1Þð1 − p Þ smaller radius of convergence. In the worst case, the n þ α 1: 58 expectation value of the reweighting sign σðXn; …;X0Þ restart ¼ ð Þ can decrease exponentially or faster with n, thus, limiting the maximal expansion order which can be sampled by the After some algebraic manipulations the overall average algorithm. acceptance rate can be expressed as Let us also note that since for Schwinger-Dyson equa- N tions the generalized indices X take infinitely many values, α 1 − b 2 1 − N h i¼ð pþÞ N þ hminðpþ; ð pþÞ ðXnÞÞi; in practice histogramming of all possible values of X is not w possible, but also not necessary. For example, if one is ð59Þ interested only in the momentum-space two-point correla- 1 ϕ ϕ tor hN Trð p1 p2 Þi¼h½p1;p2i which corresponds to X of where in the last line the expectation value is taken with the form X ¼ffp1;p2gg, one can histogram only the respect to the stationary probability distribution of Xn.

114512-14 … DIAGRAMMATIC MONTE CARLO FOR THE WEAK- PHYSICALX REVIEW D 96, 114512 (2017) For the optimal performance of the algorithm it is desirable N w ¼ N w wðXÞ X to choose pþ (which is the only free parameter in our X X algorithm) in such a way that the acceptance rate is ¼ N w jAðXjYÞjwðYÞþ jbðXÞj maximally close to unity. To this end, we first perform XX;Y X Metropolis updates with some initial value of pþ, estimate N N N 0 1 − 0 N ¼ w ðYÞwðYÞþ b; ð61Þ the expectation values hminðpþ; ð pþÞ ðXnÞÞi in (59) 0 Y for several trial values pþ and then select the value for which the acceptance is maximal. This process is repeated where wðXÞ is the probability distribution of the last several times until the values of the average acceptance and element Xn in the sequences S, with any sign σ. The pþ stabilize. solution of this equation can be written as Sometimes the transformations X → X 1 can be n nþ N joined into groups of very similar transformations. For N b w ¼ 1 − N ; ð62Þ example, if we have to generate a random loop momentum h ðXnÞi when sampling the Feynman diagrams in momentum where again the expectation value is taken with respect to space, in the notation of algorithm IV. B. 1 generating the stationary probability distribution of the Metropolis every single possible momentum would be a separate algorithm IV. B. 1. transformation, but it is more convenient to describe them The general implementation of the Metropolis algorithm as a single group of transformations. Denoting possible IV. B. 1 for which the user-defined matrix elements AðXjYÞ outcomes of such transformations as X 1 ∈ T ðX Þ, where nþ n and the source vector bðXÞ should be provided is available T ðX Þ labels the group of transformations, we can express n as a part of GitHub repository [78]. the probability distribution of the new element Xnþ1 in the “Add index” update in (IV. B. 1) as the product of the π T C. DiagMC algorithm for the large-NUðNÞ × UðNÞ conditional probability ðXnþ1j ðXnÞÞ to generate the new T principal chiral model element Xnþ1 by one of the transformations in ðXnÞ, times the overall probability πðT jXnÞ to perform any In this subsection, we adapt the general Metropolis transformation from T : algorithm described in the previous subsection IV B for solving the linear Schwinger-Dyson equations (49) π π T π T N ðXnþ1jXnÞ¼ ðXnþ1j ðXnÞÞ ð jXnÞ; obtained from the action (7) of the large- principal chiral model (2). The generalized indices X will take values in the πðX 1jT ðX ÞÞ ¼ AðX 1jX Þ=N T ðX Þ; nþ n Xnþ n n space of partitioned sequences of momenta N T ðX Þ¼ AðX 1jX Þ; n nþ n X p1; …;p1 ; …; pr; …;pr ; ∈T ¼ff 1 n1 g f 1 nr ggm ð63Þ Xnþ1

πðT jXnÞ¼N T ðXnÞ=N ðXnÞ; additionally labelled with an expansion order m ¼ X 0; 1; 2; …. These sequences will be generated with the N ðXnÞ¼ N T ðXnÞ: ð60Þ 1 1 probability proportional to the correlators h½p1; …;p … T n1 ðXnÞ r … r ½p1; ;pnr im defined in (48), up to the sign re- σ 1 Such grouping of transformations will be very con- weighting with the sign variable ¼ , introduced in venient for a compact description of our DiagMC algorithm IV. B. 1. algorithm in what follows. Sometimes we will also omit The system of equations (49) already has the form (53). The only nonzero elements of the source vector bðXÞ the arguments of N T ðX Þ and T ðX Þ for the sake of n n correspond to the delta function term [in the right-hand side brevity. of line (49a)] in the first, inhomogeneous Schwinger-Dyson Finally, let us calculate the normalization factor equation in (49): N w relating the (sign weighted) histogram of the last generalized index X in the sequences S ¼fX ; …;X0g n n bðffp; −pggÞ ¼ δm;0G0ðpÞ=V: ð64Þ and the solution ϕðXÞ of the linear equations (53). Combining (55) and (53), we obtain a linear equation From Eq. (49), we can also directly read off the following for N w: groups of nonzero elements of the AðXjYÞ matrix:

A p1;p1 ; p2; …;p2 ; …; pr ; …;pr p2; …;p2 ; …; pr ; …;pr ðff 1 2g f 1 n2 g f 1 nr ggmjff 1 n2 g f 1 nr ggmÞ 1 1 1 G0ðp1Þδðp1 þ p2Þ ¼ ; ½from the term ð49cÞ ð65aÞ V

114512-15 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) A p1;p1;p1; …;p1 ; … pr; …;pr p1; …;p1 ; … pr ; …;pr ðff 1 2 3 n1 g f 1 nr ggmjff 3 n1 g f 1 nr ggmÞ 1 1 1 G0ðp1Þδðp1 þ p2Þ ¼ ;n1 ≥ 4; ½from the term ð49eÞ ð65bÞ V

Aðffp1;p1; …;p1 ;p1 g; …fpr; …;pr gg jffp1; …;p1 g; …; fpr; …;pr gg Þ 1 2 n1−1 n1 1 nr m 2 n1−1 1 nr m 1 δ 1 1 G0ðp1Þ ðp1 þ pn1 Þ ¼ ;n1 ≥ 4; ½from the term ð49fÞ ð65cÞ V

A p1; …;p1 ; …;p1 ; p2; …;p2 ; … pr; …;pr ðff 1 A n1 g f 1 n2 g f 1 nr ggmj p1; …;p1 ; p1 ; …;p1 ; p2; …;p2 ; … pr; …;pr jff 2 A−1g f Aþ1 n1 g f 1 n2 g f 1 nr ggmÞ 1 δ 1 1 G0ðp1Þ ðp1 þ pAÞ ¼ ;A¼ 4; 6; …;n1 − 2;n1 ≥ 6; ½from the term ð49gÞ ð65dÞ V

1 1 1 r r 1 1 r r A p ;p ; …;p ; …; p ; …;p q1; …;q2 1;p ; …;p ; …; p ; …;p ðff 1 2 n1 g f 1 nr ggmjff vþ 2 n1 gg f 1 nr gm−vÞ − 1 δ 1 − − − … 49 49 49 ¼ G0ðp1Þ ðp1 q1 q2vþ1ÞVðq1; ;q2vþ1Þ: ½from the terms ð bÞ; ð dÞ; ð hÞ: ð65eÞ

In the above representation, we write down all nonzero elements of the matrix AðXjYÞ for the fixed first index X. However, “ ” for the Add index update in the algorithm IV. B. 1 we need to know all nonzero elements AðXnþ1jXnÞ for the fixed second index Xn. In order to obtain a suitable representation of the matrix AðXjYÞ, we relabel the matrix elements (65), assuming Y A X Y Y p1; …;p1 ; …; pr; …;pr that the second index in ð j Þ has the most general form ¼ff 1 n1 g f 1 nr ggm, and express the a momenta in the first index X in terms of these momenta pi . This gives

1 1 r r 1 1 r r G0ðpÞ Aðffp; −pg; fp1; …;p g; …; fp1; …;p gg jffp1; …;p g; …; fp1; …;p gg Þ¼ ; ½from the term ð49cÞ n1 nr m n1 nr m V ð66aÞ

1 1 r r 1 1 r r G0ðpÞ Aðffp; −p; p1; …;p g; …fp1; …;p gg jffp1; …;p g; …fp1; …;p gg Þ¼ ; ½from the term ð49eÞ n1 nr m n1 nr m V ð66bÞ

1 1 r r 1 1 r r G0ðpÞ Aðffp; p1; …;p ; −pg; …fp1; …;p gg jffp1; …;p g; …; fp1; …;p gg Þ¼ ; ½from the term ð49fÞ n1 nr m n1 nr m V ð66cÞ A p; p1; …;p1 ; −p; p2; …;p2 ; p3; …;p3 ; … pr; …;pr ðff 1 n1 1 n2 g f 1 n3 g f 1 nr ggmj

1 1 2 2 3 3 r r G0ðpÞ jffp1; …;p g; fp1; …;p g;fp1; …;p g; …; fp1; …;p gg Þ¼ ;r≥ 2; ½from the term ð49gÞ ð66dÞ n1 n2 n3 nr m V

A p1 p1 p1 ;p1 ; …;p1 ; …; pr; …;pr ðff 1 þ 2 þþ 2vþ1 2vþ2 n1 g f 1 nr ggmþvj p1; …;p1 ; …; pr; …;pr jff 1 n1 g f 1 nr ggmÞ − 1 1 1 1 … 1 2 1 49 49 49 ¼ G0ðp1 þ p2 þþp2vþ1ÞVðp1; ;p2vþ1Þ; v þ

114512-16 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) the generalized indices X0; …;Xn which enter the series (54). index Xn in the series (54), returns only nonzero values of However, this is not necessary in practice, since the matrix AðXnþ1jXnÞ and the corresponding set of Metropolis → C AðXjYÞ for the Schwinger-Dyson equations (49) is very proposals Xn Xnþ1. The code of the Metropolis sparse, as one can see from the equations (66). algorithm IV. B. 1 available at [78] relies exactly on such “ ” Also, only a few momenta in the generalized index Xn ¼ a functional implementation of the matrix AðXnþ1jXnÞ. 1 1 r r ffp1; …;p g; …; fp1; …;p gg are different between the With all these considerations in mind, from (66) we can n1 nr → first and the second generalized indices of the matrix finally deduce the following set of transformations Xn X 1 in the “Add index” update of algorithm IV. B. 1: AðXnþ1jXnÞ. Thus, only a few momenta will be changed nþ Transformation set IV. C. 1: when generating the new generalized index Xnþ1 from Xn. Therefore it is sufficient to keep in computer memory only the topmost sequence X ,aswellasthesequenceof Transformation set IV.C.1 n Push a new pair of momenta to the stack, from matrix transformations X → X 1 which led to the current n nþ elements (66a): sequence, along with a few parameters which characterize (i) A new subsequence containing a pair of momenta these transformations. In fact, only the information which fp; −pg is pushed to the top of the momentum is necessary to recover X from X 1 in the “Remove n nþ stack. The probability distribution for p is π0ðpÞ, index” update in algorithm IV. B. 1 should be stored. → defined in (68). We note also that the transformations Xn Xnþ1 which (ii) The order m and the sign σ do not change. correspond to matrix elements (66a), (65b), (65c), (66e) (iii) The corresponding term in the Schwinger-Dyson 1 … 1 involve only the first momentum subsequence fp1; ;pn1 g equations is (49c). in (63), and the transformation corresponding to (66d) (iv) The overall weight for theseP transformations, summed 2 … 2 involves also the second subsequence fp1; ;pn2 g. This over all p,isN T ¼ ∈T jAðX 1jX Þj ¼ Σ0, Xnþ1 nþ n naturally endows the partitioned sequences (63) with the using the same notation as in (60). stack structure, in the algorithmic sense. Correspondingly, (v) To recover Xn back from Xn 1, the topmost 1 … 1 þ we will refer to the subsequence fp1; ;pn1 g as the subsequence should be removed from the stack “ ” “ ” topmost or first subsequence, the subsequence configuration Xnþ1. No additional information 2 … 2 — “ ” fp1; ;pn2 g as the second sequence and so on. should be saved for this backward step. Interestingly, the fact that the random sampling process Add new momenta to the topmost subsequence, from matrix operates on the stack implies the factorization of proba- elements (66b), (66c): bilities for subsequences in the stack [58,79,80], which (i) A new pair of momenta is inserted into the topmost 1 … 1 corresponds to factorization of single-trace correlators in subsequence fp1; ;pn1 g, either a) at the beginn- 2 − 1 … 1 the large-N limit. If one proceeds keeping the 1=N terms ing as fp; p;p1; ;pn1 g, or b) at the beginning and 1 … 1 − in the equations (39) and (40), our partitioned momentum at the end as fp; p1; ;pn1 ; pg. The probability sequences will no longer have stack structure. In particular, distribution for p is again π0ðpÞ. the fourth line of the Schwinger-Dyson equations (40) (ii) The order m and the sign σ do not change. would correspond to a transformation where the topmost (iii) The corresponding terms in the Schwinger-Dyson momentum subsequence is split into two subsequences, equations (49) are (49e), (49f). one of which is inserted into arbitrary subsequence in the (iv) The overall weights for each of the transformations partitioned sequence (63) [81]. Obviously, the number a) and b) are N T ¼ Σ0. of ways to do such a splitting and insertion grows (v) To recover Xn back from Xnþ1, either the two first polynomially both in the number of subsequences and momenta, or the first and the last momenta should their lengths. This polynomial growth of the number of be removed from the topmost subsequence in the terms in the Schwinger-Dyson equations results in the stack. Correspondingly, one should remember factorial growth of the number of nonplanar Feynman whether the transformation a) or b) was chosen. diagrams with order [59,81]. In contrast, the matrix Merge two topmost subsequences, from matrix elements (66d): elements (66) do not depend on the number and lengths (i) If there are two or more momentum subsequences in of subsequences, so that the algorithm samples a much X, they are merged and interleaved with a pair of smaller space of planar diagrams, the number of which momenta p; −p, where p is generated with proba- π grows only exponentially with expansion order. This is the bility distribution 0ðpÞ: reason why diagrammatic Monte Carlo should be, in general, much more efficient in the large-N limit. 1 1 2 2 ffp1; …;p g; fp1; …;p g; …g Since the generalized indices X take an infinite number n1 n2 → 1 … 1 − 2 … 2 … of values, it is impossible but also not necessary to keep the ffp; p1; ;pn1 ; p; p1; pn2 g; g: ð69Þ matrix elements AðXjYÞ in computer memory. Rather, they (Table continued) can be implemented as a function which, given the topmost

114512-17 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) → (Continued) The sequence of updates Xn Xnþ1 in the Metropolis algorithm IV. B. 1 with transformations IV. C. 1 can be (ii) The order m and the sign σ do not change. visually interpreted as a step-by-step process of drawing (iii) The corresponding term in the Schwinger-Dyson planar, in general disconnected Feynman diagrams with equations (49) is (49g). arbitrary number of external legs. The transformations (iv) The overall weight for this transformation, summed IV.C.1areusedinthe “Add index” update and corre- over all p,isN T ¼ Σ0. spond to adding a randomly chosen new element to the (v) To recover Xn back from Xnþ1, one should split the diagram. The first three transformations in the set IV. C. 1 topmost subsequence in the stack, which now has correspond to drawing free propagator lines with different the form of the right-hand side of (69), at the positions of endpoints. The last transformation in the set position n1 þ 1, and remove the first momenta from IV. C. 1 draws an interaction vertex with 2v þ 2 legs by the resulting subsequences. Correspondingly, the joining 2v þ 1 external lines into a single external line. It length n1 of the topmost subsequence in Xn should is also this transformation which turns external lines into → be stored before the transformation Xn Xnþ1. internal ones. The “Remove index” update in the Create ð2v þ 2Þ-leg vertex, from matrix elements (66e): Metropolis algorithm IV. B. 1 erases the diagram element (i) If there are four or more momenta in the topmost which was drawn last, and can be thought of as an “Undo” momentum subsequence in Xn, replace the first operation. The initial state of the algorithm is just a single 2v þ 1, v ¼ 1…n1=2 − 1 momenta by a single free propagator line, which is the simplest Feynman momentum equal to their sum: diagram one can ever draw. Simultaneous generation of random momenta distributed with the probability π0ðpÞ p1;p1; …;p1 ;p1 ; …;p1 ; … ff 1 2 2vþ1 2vþ2 n1 g g proportional to the free propagator G0ðpÞ implements the → 1 1 1 1 … 1 … Monte Carlo integration over internal loop momenta in ffp1 þ p2 þþp2vþ1;p2vþ2; ;pn1 g; g: Feynman diagrams. In this way, the diagram weights are ð70Þ represented as products of propagators with freely generated momenta times vertex functions and propagators (ii) The order m is increased by v, and the sign σ is −1 which contain some linear combinations thereof. Let us multiplied by . also note that the momentum conservation is automati- (iii) The corresponding terms in the Schwinger-Dyson cally valid for all the transformations in (IV. C. 1), which equations (49) are (49b), (49d) and (49h). always lead to zero total momentum for any momentum (iv) The weight of this transformation for each particular subsequence in the stack. v is Computationally, the most expensive operation in our − 1 1 DiagMC algorithm is the calculation of vertex functions AðXnþ1jXnÞ¼ Gðp1 þþp2vþ1Þ 1 1 Vðp1; …;p2 1Þ for all v ¼ 1…n1=2 − 1, which is neces- 1 … 1 vþ × Vðp1; ;p2vþ1Þ: ð71Þ sary to obtain acceptance probabilities for all possible “Create vertex” transformations in the set IV. C. 1. From (v) To recover Xn back from Xnþ1, one should replace (45) one can infer that for each v the calculation of the the first momentum in the topmost subsequence in vertex function involves ∼v2 floating-point operations. As 1 … 1 the stack by the momenta p1; ;p2vþ1.Tothisend, a result, the CPU time needed to calculate all vertex 2 1 … 1 3 the vertex order v and the v momenta p1; ;p2v functions scales as n1. This CPU time is dominating the 1 should be stored. The momentum p2vþ1 can be performance of the algorithm in the regime sufficiently recovered from momentum conservation constraint. close to the continuum limit, where subsequences with Restart, from the source vector (64): rather large lengths n1 can appear. It is not inconceivable (i) If the “Restart” update is selected in algorithm IV. that this scaling can be made milder by carefully optimizing B. 1, the new value of X0 has a single subsequence the calculation of vertex functions and reusing some data 0 − with just two momenta: X0 ¼ffp; pgg, where p from previous Metropolis updates. We leave such develop- is distributed with probability π0ðpÞ. ments for future work. (ii) The order m is set to m ¼ 0, and the sign variable to After calculation of vertex functions, the next most σ ¼þ1. This state is also the initial state for the expensive and frequently used part of the algorithm is algorithm. the generation of random momenta with probability π0ðpÞ. (iii) The corresponding term in the Schwinger-Dyson To this end, one can implement any method for generating equations (49) is the term on the right-hand side of continuous variables with definite probability distribution, line (49a). P for example, the biased Metropolis algorithm [82]. Note 0 (iv) The norm of the source vector is N ¼ 0 bðX Þ¼ b X0 0 that this allows us to work directly in the infinite volume Σ 0, which can be used to recover the overall norma- limit, which is one of the attractive features of DiagMC lization factor for field correlators from (62). approach in general [1].

114512-18 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) In this work, we have still chosen to work with finite but of data collection and processing. The production code sufficiently large spatial lattice size L0 × L1, which is much with the transformations IV. C. 1 in the Metropolis larger than the correlation length for all values of λ which algorithm IV.B.1 is available as a part of GitHub we have used. In this case, we generate the momenta p by repository [78]. randomly choosing among a finite number of discrete values with given (in general, unequal) probabilities. By V. NUMERICAL RESULTS OF THE DIAGMC creating an indexed “lookup table” for the discrete values of SIMULATIONS OF THE LARGE-NUðNÞ × UðNÞ momenta this procedure can be performed in CPU time PRINCIPAL CHIRAL MODEL which scales only logarithmically with volume. The only reason for working in a finite volume is that this A. Correlators of gx variables in terms of ϕp fields simplifies the histogramming of correlators of gx variables While the DiagMC algorithm described in subsection IV C [original UðNÞ-valued variables in the partition function stochastically samples the correlators of ϕp fields in momen- (2)]; see subsection VA for a more detailed discussion. The tum space, physically one is typically interested in correlators performance of the algorithm, however, practically does not of UðNÞ-valued gx variables which enter the partition depend on the lattice size as long as it is much larger than function (2). the correlation length. Rather, the dynamically generated The first nontrivial correlator is the expectation value mass gap in the principal chiral model (2) effectively sets 1 h Trg i of the trace of g . It should vanish if the UðNÞ × the infrared scale for the algorithm. Needless to say, finite N x x UðNÞ symmetry of the principal chiral model (2) is not lattice size L0 in the Euclidean time direction μ ¼ 0 is also broken. By virtue of the Mermin-Wagner theorem which necessary to study the theory at finite temperature (see 2 subsection VD). prohibits spontaneous symmetry breaking in D ¼ The nonconvergence of the resummed perturbation dimensions, this should be the case at any finite value → ∞ theory of Sec. II for principal chiral model on finite-size of N. Exact solution of the OðNÞ sigma model at N one-dimensional lattice suggests that the series obtained on suggests also that spontaneous symmetry breaking does 2 finite-size lattices might have worse convergence properties not happen for nonlinear sigma models in D ¼ in the than in the infinite-volume limit. We expect that these large-N limit. problems will appear only at very high orders of perturba- However, the infrared-finite weak-coupling expansion tion theory. Indeed, replacing continuum integrals over sampled by our DiagMC algorithm is built around the vacuum state with g ¼ 1, h1 Trg i¼1, which explicitly Brillouin zone in the infrared-finite weak-couplingP expan- x N x 1 breaks the UðNÞ × UðNÞ symmetry of the principal chiral sion of Sec. II by discrete sums of the form L2 p with 2πmμ model. To ensure that our DiagMC algorithm produces pμ , mμ 0…L − 1 results in corrections to the ¼ L ¼ physically consistent results, it is, thus, important to check continuum integrals which decrease with L approximately 1 that the expectation value h Trg i approaches zero as the as e−mL, but increase with diagram order. For example, for a N x maximal order of the weak-coupling expansion M is chain of l one-loop diagrams, such corrections depend on extrapolated to infinity. We demonstrate this in Fig. 10 l L 1 ae−mL l order and lattice size as ð þ Þ , with some below. a constant . Thus, if the expansion is limited to some Using the definition (3), it is straightforward to expand maximal order M which also limits the maximal number of the expectation value h1 Trg i in powers of the fields ϕ . loop momenta, finite-volume corrections can always be N x x made smaller by using sufficiently large lattice volume V. Since our perturbative series (13) was obtained from the same expansion, it is natural to extend our formal power In subsection VC, we explicitly demonstrate the smallness ξ of finite-volume corrections by comparing the results of counting scheme based on the auxiliary parameter also to simulations at 108 × 108 and 256 × 256 lattices. correlators of gx variables, which leads to It seems also that our DiagMC algorithm does not allow 1 1 X 1 for a straightforward parallelization. The only potential for Trgx ¼ Trgx parallelization which we currently see is to relegate the N V x N generation of random momenta with probability distribu- Xþ∞ λ k 1 X 1 k 2k tion π0ðpÞ to a separate MPI or OpenMP thread. However, ¼ 1þ2 − ξ Trϕ ; ð72Þ 8 V N x as our algorithm can operate directly in the infinite-volume k¼1 x limit, parallelization might not be really necessary. Having a multi-CPU computer at hand, one can speed up simu- where in the first line we have used translational invariance ξ ξ 1 lations by gathering statistics independently on differ- and should be set to ¼ at the end of the calculation. ent CPUs. Yet another observable which we consider in this work is 1 † In Sec. V, we discuss in detail the numerical results the two-point correlator hN TrðgxgyÞi of gx variables. This is which we have obtained using the algorithm IV. B. 1 the simplest nontrivial correlator from which one can infer with transformation set IV. C. 1, as well as the details the mean energy, the mass gap and the static screening

114512-19 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) ≤ length in the large-N principal chiral model (2). Similarly to k þ m M, we set to zero all matrix elements AðXnþ1jXnÞ (72), we can use the definition (3) to express this two-point for transformations which lead to k þ m>M in the correlator in terms of ϕ fields as Metropolis algorithm IV. B. 1. In our simulations we have been able to measure the first M ¼ 12 coefficients of the 1 † infrared-finite weak-coupling expansion of Sec. II with Trðg g Þ N x y reasonably small statistical error. Note also that once the field correlators h½p1; …;p2 i are measured for all values Xþ∞ kþl k m 1 k−l λ 2 kþl 1 ¼2 Trg −1þ4 ð−1Þ 2 ξ 2 Trðϕkϕl Þ of k þ m

114512-20 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) the normalization factor (62). While equation (62) suggests set is, thus, around 1013, which takes around 104 that this normalization factor can be calculated by meas- CPU-hours (single core). The speed of our algorithm uring the expectation value hN ðXnÞi over the Monte Carlo can be probably increased by an order of magnitude by history, in practice we have found that N ðXnÞ is a very carefully optimizing the calculation of vertex functions noisy observable which is likely to have a heavy-tailed and generation of random momenta. Let us also note probability distribution. Thus, directly using the equa- that achieving a comparable precision for the large-N tion (62) to obtain the normalization factor N w results extrapolation of the results of conventional Monte Carlo in large statistical errors. We have used a more convenient simulations takes a comparable amount of CPU practical strategy and fixed N w to be the ratio of the time [84]. weighted probability in the histogram bin with k ¼ 1, In each of the statistically independent simulations, 4 m ¼ 1 andP the first perturbative correction to the two-point during the first 5 × 10 Metropolis updates we tune the “ ” function p0;p1 hp0p1i1, which we have explicitly calcu- probability pþ of the Add index update in the Metropolis lated using (51). As a cross-check, we made sure that using algorithm IV. B. 1 in order to maximize the average N the value of w calculated in this way leadsP to the correct acceptance rate. The optimal value of pþ and the resulting ≃ 0 4…0 6 estimate of the next perturbative correction hp0p1i2, acceptance rate take values in the range pþ . . , p0;p1 α ≃ 0 5…0 65 ’ which we have also explicitly calculated. h i . . and depend rather weakly on the t Hooft coupling and temperature. The average length of the sequence fXn; …;X0g of generalized indices in the B. Simulation setup, sign problem, and restoration Metropolis algorithm IV.B.1 is in the range between hni¼ of UðNÞ × UðNÞ symmetry 17.3 (for λ ¼ 3.012) and hni¼18.1 (for λ ¼ 5.0). The In our simulations, we perform two scans over param- average number of subsequences in the momentum stack eters of the principal chiral model (2). In the first scan, on which the transformations IV. C. 1 operate is in the discussed in subsection VC, we fix the lattice size to range between hri¼1.2 (for λ ¼ 5.0) and hri¼1.4 (for L0 × L1 ¼ 108 × 108 and study the dependence of physi- λ ¼ 3.012). The dependence of both hni and hri on cal observables on the t’Hooft coupling λ in the range temperature is much weaker, and is completely within λ ¼ 3.012…5.0, which includes the scaling region of the the ranges given above. principal chiral model (2) where the continuum physics sets As the most pessimistic estimate for the autocorrelation in, as well as the point of the large-N phase transition [35] time of our DiagMC algorithm, we can use the average between the weak- and the strong-coupling phases at λc ¼ number of Monte Carlo steps between the “Restart” 3.27 [37]. To check the finite-volume effects, we have also updates in the Metropolis algorithm IV. B. 1. Indeed, performed a single simulation with L0 × L1 ¼ 256 × 256 the “Restart” update completely resets the state of the lattice at λ ¼ 3.1. In the second scan, discussed in algorithm, and the sequence of generalized indices subsection VD, we fix the t’Hooft coupling to λ ¼ generated after the “Restart” update is completely sta- 3.012 and study the temperature dependence of physical tistically independent from the sequence which was observables by varying the temporal lattice size L0 in the generated before that. In Fig. 8, we illustrate the range L0 ¼ 4…90. As discussed above, we work on finite- dependence of this “mean return time” on the t’Hooft volume lattices only to simplify the histogramming pro- coupling λ. It appears that the “Restart” updates are very cedure, and the speed of our DiagMC algorithm depends on rare for all values of λ. On average, they are separated by the lattice size at most logarithmically. ∼104…105 Monte Carlo steps. The mean return time We perform histogramming of observables every time strongly increases towards larger values of the t’Hooft when our DiagMC algorithm reaches a configuration coupling λ. Together with the λ dependence of hni, this with only one momentum subsequence in the stack. One observation shows that for larger λ higher orders in can also explicitly take into account the large-N factori- the infrared-finite weak-coupling expansion of Sec. II zation (42) and sample from multiple subsequences in the become more important, and the algorithm spends more … stack; however, this does not give a large advantage since time sampling correlators hp1 p2kim with larger values in this case the samples appear to be rather strongly of k þ m. Correspondingly, returns to the initial state correlated. In order to avoid potential problems with of the algorithm with k ¼ 1, m ¼ 0 are less frequent. autocorrelations, for each parameter set we run around In the scan over finite temperatures, the mean return 6 × 104 simulations of 2 × 108 Metropolis updates each time practically does not depend on temperature, with statistically independent seeds for the ranlux and only at the highest temperatures which correspond random number generator [83]. The outcome of each of to L0 ¼ 4 and L0 ¼ 8 does it exhibit a quick increase these simulations is treated as statistically independent from approximately 3.5 × 104 to 5.3 × 104. data point, and the statistical error for physical observ- In Fig. 9, we illustrate the probability distribution of ables is calculated as the standard error for this data set. the length k of the topmost subsequence in the stack and The total number of Metropolis updates for each data the order counter m in our DiagMC algorithm for λ ¼ 3.1.

114512-21 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) 5.5 1 In Fig. 10, we show the dependence of h TrgxiM on 5.4 N 1=M for several values of t’Hooft coupling and several 5.3 1 temperatures. In all cases, hN TrgxiM exhibits a nonlinear 5.2 but monotonically decreasing dependence on 1=M. For the 5.1 largest t’Hooft couplings λ ¼ 5.0, 4.0 and for the highest 5 1 4 1 temperature =T ¼ L0 ¼ the tendency of hN TrgxiM to 4.9 vanish is rather clear. In contrast, it seems that at small λ the (Mean return time) 4.8 10 dependence on M should become very nonlinear at small 1 log 4.7 1 =M for hN TrgxiM to extrapolate to zero. Indeed, the slope 4.6 1 of hN TrgxiM becomes larger towards smaller values of 4.5 1 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 =M. These observations support the expectation that 1 → ∞ λ hN TrgxiM vanishes at M . As already mentioned in subsection IV C, not all FIG. 8. The dependence of the average number of Monte Carlo coefficients AðXnþ1jXnÞ are positive for the transforma- steps between the “Restart” updates in the Metropolis algorithm tion set IV. C. 1. We, thus, have to use the sign reweight- ’ λ IV. B. 1 on the t Hooft coupling . ing, as discussed in subsection IV B.Therearetwo potential sources of the sign problem which might hinder our DiagMC simulations: sign cancellations within The largest fraction of time (almost 20%) is spent on … the expectation values of the correlators h½p1; ;p2kim sampling the two-point correlator hp0p1im with the − 1 11 of ϕ variables, and sign cancellations within the observ- maximal value m ¼ Mmax (¼ in our case). The 0 12 ables (74) and (75) themselves. In order to quantify sign least probable configuration is m ¼ , k ¼ ,whichis … cancellations in the correlators hp1; ;p2kim, in Fig. 11 the disconnected correlator with the maximal number of σ external legs which contributes to expectation values (74) we plot the expectation value of the sign variable in the Metropolis algorithm IV.B.1 as a function of the t’Hooft and (75). In total, the algorithm spends most time λ λ σ sampling the higher-order diagrams. This is a completely coupling . For all values of , the mean sign h i appears 10−4…10−5 reasonable behavior, as Monte Carlo integration over to be rather small, around ; thus, the sign loop momenta at high values of m requires a lot of problem is indeed important in our simulations. A some- statistics, and also sign cancellations are most important what encouraging feature is that the sign problem for higher-order Feynman diagrams. becomes milder towards the weak-coupling limit in Let us now present our numerical results. We start which the theory approaches the continuum limit. by demonstrating that the expectation value h1 Trg i However, from Fig. 10, as well as from the exactly N x M solvable example of the two-dimensional OðNÞ sigma indeed tends to zero as the maximal order M in the model (see subsection III B), we can conclude that at the truncated weak-coupling expansion (74) becomes larger. same time the infrared-finite weak-coupling expansion of This is an indication that in the limit of large M the global Sec. II converges less rapidly at small λ, so that more terms UðNÞ × UðNÞ symmetry of the principal chiral model (2) in the series should be sampled. The mean sign hσi also is restored.

-0.5 0.8

-1 0.7 -1.5 0.6 -2

M 0.5 -2.5 > x -3 0.4 (p(k, m))

10 k=1 4x108, λ=3.012 -3.5 0.3

<1/N tr g λ

log k=3 12x108, =3.012 -4 k=5 108x108, λ=3.012 0.2 λ -4.5 k=7 108x108, =3.230 k=9 108x108, λ=3.570 -5 k=11 0.1 108x108, λ=4.000 k=12 108x108, λ=5.000 -5.5 0 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 m 1/M

1 FIG. 9. The probability pðk; mÞ to obtain the topmost momen- FIG. 10. Convergence of the expectation value hN TrgxiM of the tum subsequence with 2k momenta and order counter m for single field variable to zero as the truncation order M in the weak- λ ¼ 3.1. coupling expansion (74) is increased.

114512-22 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) -4.2 1 † logðMÞ C Trðg1g0Þ ¼ A − B þ ; ð77Þ -4.4 N M M M -4.6 1 † B C >) Trðg1g0Þ ¼ A þ pffiffiffiffiffi þ : ð78Þ σ -4.8 N M M

(< M

10 -5 log We perform extrapolations by fitting the numerical data -5.2 with these functions using the χ2 weight which includes the -5.4 data covariance matrix to account for correlations between numerical values at different M. The fit parameters are A, B -5.6 and C, with A obviously corresponding to the result of 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 → ∞ λ extrapolation to M . Upon fitting, both extrapolating functions (77) and (78) lead to roughly the same values of 2 FIG. 11. Expectation value of the reweighting sign σ in the χ over the number of degrees of freedom in the fit. These Metropolis algorithm IV. B. 1 as a function of the t’Hooft extrapolations are shown in Fig. 12 as red and blue strips coupling λ. for the functions (77) and (78), respectively. The strip width corresponds to the confidence interval set by the statistical depends rather weakly on temperature, and exhibits a uncertainty in the optimal values of A, B and C. The −5 −5 difference between the two extrapolations (77) and (78) can rapid drop from hσi ∼ 5 × 10 to hσi ∼ 3 × 10 only at be used as an estimate of a systematic error of our highest temperatures (L0 ¼ 4 and L0 ¼ 8). In Fig. 14,in the next subsection, we will also illustrate the sign extrapolation procedure in the absence of other prior problem for a particular physical observable (mean link). information which helps to choose between the two fits (see below). To compare our data with the results of conventional C. Two-point correlators at zero temperature Monte Carlo simulations, in Fig. 12, we also show the In this subsection, we consider the simplest nontrivial mean link values obtained in Monte Carlo simulations of 1 † two-point correlation function hN Trðgxg0Þi of UðNÞ- both SUðNÞ × SUðNÞ and UðNÞ × UðNÞ principal chiral valued fields. From this correlator one can extract such models as functions of 1=N2 (rescaled as 10=N2 for better important physical properties of the principal chiral model readability of the plots). These data are taken from (2) as the energy density and the mass gap in the principal [37,85,86] for t’Hooft couplings λ ¼ 3.100, 3.230, 4.000 chiral model (2). It has been also extensively studied using and N ¼ 6, 9, 15, 21 and from [84] for λ ¼ 3.012 and conventional Monte Carlo simulations of both SUðNÞ × N ¼ 6, 9, 12, 18. The finite-N data are extrapolated to SUðNÞ and UðNÞ × UðNÞ principal chiral models at the large-N limit using the linear fits of the form N – 1 † 2 several large values of [37,84 86]. Weak-coupling h Trðg g0Þi ¼ A þ B=N , which are shown in Fig. 12 1 † N 1 expansion of hN Trðgxg0Þi was given up to three first orders as straight solid green lines. We have to note that the in λ at arbitrary values of N in [86]. In this subsection, we Monte Carlo data for λ ¼ 3.100, 3.230, 4.000 show will compare the large-N extrapolations of these results noticeable deviations from the expected A þ B=N2 behav- with the results of our DiagMC simulations. This allows us ior which significantly exceed the very small statistical to compare the performance of our algorithm with conven- errors in these data. Thus, our N → ∞ extrapolation might tional Monte Carlo simulations, as well as the convergence be also not completely accurate. Also, the Monte Carlo data of our infrared-finite weak-coupling expansion with that of for the finite-NUðNÞ × UðNÞ principal chiral model, taken the standard lattice perturbation theory. from [37], do not seem to follow the 1=N2 scaling of finite- A particularly simple and important observable is the 1 † N corrections. mean link hN Trðg1g0Þi, which is also related to the energy For the three values λ ¼ 3.0120, 3.100 and λ ¼ 3.230 the density of the principal chiral model. In Fig. 12,we extrapolation of the finite-N data for the SUðNÞ × SUðNÞ illustrate the convergence of the truncated series (75) for principal chiral model clearly approaches the result of the 1 † → ∞ the mean link by plotting hN Trðg1g0ÞiM as a function of M extrapolation with the extrapolating function (77). inverse truncation order 1=M. The dependence on 1=M is While the differences of the N → ∞ and M → ∞ extrap- nonlinear with the curvature increasing towards larger M, olations lie within the statistical uncertainty of the M → ∞ which requires some nonlinear extrapolation procedure to extrapolation of the DiagMC data, extrapolating function reach the limit M → ∞ (1=M → 0). We have found that the (77) has a small but noticeable tendency for over-estimating dependence of the truncated series for the mean link on the mean link. On the other hand, the second extrapolating 1=M is well described by the following two extrapolating function (78) underestimates the mean link rather signifi- functions: cantly, and we, thus, conclude that the first extrapolating

114512-23 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) 0.72 0.7 0.7 λ = 3.0120 λ = 3.1000 0.68 0.65 0.66 )> )>

0 0.64 0 g g 0.6 + + 1 0.62 1 0.6 0.55 0.58 MC,SU(N) MC,SU(N)

<1/N tr(g Standard PT <1/N tr(g Standard PT 0.56 DiagMC DiagMC 0.54 Log ext 0.5 Log ext 0.52 Sqrt ext Sqrt ext 0.5 0.45 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1/M, 10/N2 1/M, 10/N2

0.7 0.65 λ λ = 3.2300 0.6 = 4.0000 0.65 0.55

)> 0.6 )> 0 0

g g 0.5 + + 1 1 0.55 0.45

MC,SU(N) 0.4 MC,SU(N) 0.5 <1/N tr(g MC,U(N) <1/N tr(g Standard PT Standard PT 0.35 DiagMC 0.45 DiagMC Log ext Log ext 0.3 Sqrt ext Sqrt ext 0.4 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1/M, 10/N2 1/M, 10/N2

1 † FIG. 12. A comparison of the mean link values hN Trðg1g0Þi as obtained from DiagMC simulations of the infrared-finite weak- coupling expansion (13), from standard Monte Carlo simulations at finite N (“MC,SU(N)” and “MC,U(N)”), and from the three lowest orders of the standard lattice perturbation theory (“Standard PT”). Perturbative results are plotted as the functions of the inverse expansion truncation order 1=M. DiagMC data are extrapolated to 1=M → 0 by fitting the 1=M dependence to the functions (77) (red strip) and (78) (blue strip). Strip width as well as the error bars on the extrapolated values (“Log ext” and “Sqrt ext”) are the confidence intervals for the extrapolating functions. The results of standard Monte Carlo simulations are plotted as functions of 10=N2, and extrapolated to 1=N2 → 0 using linear fits for SUðNÞ data (solid green lines). function (77) yields more realistic results in the M → ∞ 0.68 limit.

M 0.66

To illustrate the ability of our DiagMC algorithm to work > x at arbitrarily large volumes without significant slow-down, 0.64 in Fig. 13 we also compare the convergence of the truncated 0.62 1 † expansions (74) and (75) for the mean link hN Trðg1g0Þi and , <1/N tr g 0.6 1 M the single-field expectation value h Trgxi for lattice sizes )> N 0 0.58 g

L0 × L1 ¼ 108 × 108 and L0 × L1 ¼ 256 × 256. Both + 1 0.56 simulations took approximately the same CPU time, and 0.54 Link,108x108 had the same strength of the sign problem. Link,256x256

In order to compare the convergence of our massive <1/N tr(g 0.52 gx,108x108 gx,256x256 lattice perturbation theory constructed in Sec. II with the 0.5 convergence of the standard lattice perturbation theory 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 constructed using massless propagators, in Fig. 12, we also 1/M 1 show the one-loop, two-loop and three-loop (M ¼ ,2,3, FIG. 13. A comparison of truncated expansions (74) and (75) respectively) results of [86] for the mean link at N → ∞.It 1 † for the mean link hN Trðg1g0Þi and the single-field expectation appears that conventional perturbation theory yields the 1 108 108 value hN Trgxi for lattice sizes L0 × L1 ¼ × and results which are, at least for the first three orders, closer to L0 × L1 ¼ 256 × 256. CPU time is roughly the same for both the physical result and also seem to converge faster. Thus, simulations.

114512-24 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) the price which we have to pay for the absence of infrared 0 divergences is the slower convergence of our infrared-finite -1 weak-coupling expansion. λ 4 00 For the largest value ¼ . which we consider our -2 ) n

massive perturbation theory with both extrapolations (77) > σ

and (78) yields the result which is significantly larger (< -3 than the large-N extrapolation of the Monte Carlo data. 10 Also the standard lattice perturbation theory [86] seems to log -4 converge to a larger value. This disagreement between λ = 3.012 weak-coupling expansions and the physical result is an -5 λ = 3.100 λ = 3.230 indication of the large-N transition between the weak- λ = 4.000 -6 coupling and the strong-coupling regimes [35], which in 0 2 4 6 8 10 12 λ 3 27 the principal chiral model (2) happens at c ¼ . [37]. n After this transition, the physical values should be rather σ calculated using the strong-coupling expansion. FIG. 14. Average sign h in of the infrared-finite weak-coupling As we have already discussed in subsection VBabove, expansion diagrams which contribute to the link expectation our simulations are limited to around M ∼ 101 first orders value (75) at order n ¼ k þ m as a function of n for different λ of the weak-coupling expansion due to more and more values of . important sign cancellations between different Feynman diagrams. To illustrate this sign problem in more detail, we the first two orders in coordinate space were given in [86]. have considered the sign cancellations within contributions These results are shown in Fig. 15 as solid black (for the λ mþk of order ð− 8Þ with fixed m þ k ¼ n to the mean link first order) and grey (for the second order) lines. As in the 1 † case of mean plaquette, we observe that at short distances hN Trðg1g0ÞiM given by (75). We have characterized these þ− − In In þ the standard perturbative expansion is closer to the first- σ − I sign cancellations by the ratio h in ¼hIþþI i, where n and − n n principle Monte Carlo data than those from our infrared- In are the sums of the absolute values of diagrams which finite weak-coupling expansion. The results of standard contribute at order n ¼ k þ m to the mean link (75) with perturbative expansion show, however, a completely differ- σ positive and negative re-weighting signs , respectively, ent behavior at large distances. At distance x ∼ 10, they … and h i denotes averaging over Monte Carlo history. In become negative, and further grow logarithmically with σ Fig. 14, we illustrate that h in decays seemingly exponen- distance. In this sense, in the far infrared, they behave very ’ tially with n for several values of the t Hooft coupling differently from the exponentially decaying physical result. λ . Since our algorithm spends most time sampling Furthermore, the second-order result of [86] starts growing diagrams with large n ∼ M, this explains the overall strength of the sign problem which we have previously illustrated in Fig. 11. DiagMC,M=1 Std.PT,NLO Finally, in Fig. 15, we compare the x dependence of 1 DiagMC,M=12 MC,SU(6) 1 † λ 3 1 Std.PT,LO MC,SU(9) the two-point correlator hN Trðgxg0ÞiM at ¼ . with 0.8 Monte Carlo data obtained in [84] at N ¼ 6 and N ¼ 9. )> 0

For DiagMC results the truncation order M is encoded in g 0.6 + color, from pure blue for M ¼ 1 to pure red for M ¼ 12. x While at distances of a few lattice spacings the DiagMC 0.4

data clearly approach the Monte Carlo data, at large <1/N tr(g 0.2 distances the convergence of our massive perturbation theory becomes slower. Since the expression (75) for 0 1 † λ = 3.1000 h Trðgxg0Þi contains the constant x-independent contri- N M -0.2 bution, it is not surprising that it decays down to some finite 0 5 10 15 20 25 “plateau” value, rather than exponentially decaying to zero x as the full correlators for SUð6Þ and SUð9Þ do. The height “ ” FIG. 15. Coordinate dependence of the two-point correlator of this plateau is, however, monotonically decreasing 1 † Tr g g0 given by (75) at various truncation orders M of the with M. It seems that one has to go to much higher values hN ð x ÞiM infrared-finite weak-coupling expansion at λ ¼ 3.1. The values of of M than we use to really see how the exponential decay of 1 1 † M are encoded in color, from pure blue for M ¼ to pure red for the hN Trðgxg0Þi emerges. M ¼ 12. For comparison, we also show the data from the It is also interesting to compare the correlators obtained standard Monte Carlo simulations with N ¼ 6 and N ¼ 9, and from our infrared-finite weak-coupling expansion with the the first two orders of the standard perturbative expansion of 1 † results of the standard lattice perturbation theory, for which hN Trðgxg0Þi.

114512-25 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) 3 2.5 logarithmically towards positive infinity at x ≳ 3 × 10 . M=10 While such large distances are certainly not interesting for 2 M=6 M M=2 )>

numerical analysis, this discussion illustrates the infrared 0 1.5 g divergences of the standard lattice perturbation theory + /2

1 1 which become only stronger at higher orders. L 0.5 D. Finite-temperature phase transition 0 <1/N tr(g δ

As a further application of our DiagMC algorithm, in this 2 -0.5 subsection we consider the principal chiral model (2) at 10 -1 finite temperature. Not much is known about the finite- -1.5 temperature physics of this model. As for other asymp- 0 10 20 30 40 50 60 70 80 90 100 totically free quantum field theories, on general grounds 1/T (L0) one can expect that at sufficiently high temperature the SUðNÞ-singlet degrees of freedom are effectively “decon- FIG. 16. Temperature dependence of the correlator ” 1 † 0 2 λ 3 012 fined. However, for the principal chiral models with hN Trðgxg0ÞiM at x ¼ð ;L1= Þ (midpoint), ¼ . and differ- N>2, this expectation has never been explicitly tested. ent truncation orders M of the infrared-finite weak-coupling 108 One of the subtle points which make such tests difficult is expansion. The zero-temperature values at L0 ¼ and corre- the absence of a local order parameter for this “deconfine- sponding truncation order M are subtracted. ment” transition in the principal chiral model. From general physical intuition in such situations one expects either a shown in Fig. 16 for the truncation orders M ¼ 2, 6, 10. first-order phase transition or a crossover. Understanding The spatial lattice size is again L1 ¼ 108. the “deconfinement transition” in the principal chiral model As the truncation order M is increased, the difference can be important for understanding the physics of one- between the finite-temperature and zero-temperature corre- dimensional compactifications of this model, which have lator seems to grow. However, the statistical errors also grow been actively used in recent years for constructing resurgent with M and for most values of L0 the differences lie within trans-series [48]. An explicit analytic calculation [87] of the statistical errors. However, for the smallest values L0 ≃ 4; 8 correlation functions in the large-NSUðNÞ × SUðNÞ the difference clearly decreases, indicating the decrease of principal chiral model based on the Bethe ansatz tech- correlation length at very high temperatures which is niques, unfortunately, does not allow us, so far, to study the probably related to Debye screening. Furthermore, for L0 thermodynamics of this model. roughly between 35 and 45 the difference seems to grow with In a Monte Carlo study [84] coauthored by one of us, the M beyond statistical uncertainty, resulting in a sort of peak numerical results for the energy density and the correlation structure around L0 ≃ 40. This enhancement of correlations length at different temperatures are presented for the principal was also reported in [84] at the same value of L0. It can be chiral model (2) at quite large but finite N,forthesamet’Hooft interpreted is an indication of a weak finite-temperature coupling λ ¼ 3.012 and the spatial lattice size L1 ¼ 108 as in phase transition. Since in DiagMC simulations the correlator the present study. The study [84] indicates a weak enhance- 1 † hN Trðgxg0ÞiM quickly grows with M, from our data we ment of correlation length at some critical temperature, which cannot exclude the scenario offinite-order phase transition at very slowly tends to become sharper towards larger N.No which the correlation length would diverge. jump or hysteresis of the free energy indicative of a first-order phase transition was observed. VI. EXTENSION TO GAUGE THEORIES In order to get further insights into the nature of this finite-temperature phase transition or crossover in the In this section, we discuss possible extension of the principal chiral model, we have performed DiagMC sim- approaches outlined in this work to non-Abelian gauge ulations at several different temperatures, which correspond theories at finite density, which is the ultimate motivation of to different lattice sizes L0 ¼ 4…108 in the Euclidean time this work. The most straightforward extension would be to direction at fixed t’Hooft coupling λ ¼ 3.012 (and hence construct the infrared-finite weak-coupling expansion of fixed lattice spacing, which only depends on λ). As a Sec. II for pure UðNÞ lattice gauge theory with the partition physical observable which is most sensitive to long-range function Z correlations of gx fields we have chosen the value of the X 1 † N † † two-point correlator h Trðgxg0Þi at x ¼ð0;L1=2Þ, which Z dg μ exp − ReTr g μg μˆ νg g ν : N M ¼ x; 4λ ð x; xþ ; xþνˆ;μ x; Þ is the point most distant from the source at (0, 0) on a UðNÞ x;μ;ν periodic lattice of spatial size L1. This point is also the ð79Þ “midpoint” of the correlator where it takes the minimal value. To get a clearer signal, we subtract the zero-temper- Using the Cayley map parametrization (3) and the UðNÞ ature data from the finite-temperature data. The result is integration measure (4) in terms of N × N Hermitian

114512-26 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) matrices Ax;μ, one can calculate the few first orders of the dynamical quarks are included and the Veneziano large-N expansion of the action in powers of the t’Hooft coupling λ limit is taken. Inside the convergence radius of the strong- Z X coupling and hopping expansions, their truncation error 2 2 should scale down exponentially with truncation order. At Z ¼ dAx;μ exp −N α TrAx;μ x;μ the same time, in the large-N limit the number of strong- α2 X coupling expansion diagrams should grow not faster than − N 2 ∇2 α4 exponentially with order [89]. Furthermore, for hopping λ TrðFx;μνÞþOð ; Þ ; ð80Þ x;μ;ν expansion adding chemical potential does not introduce any additional sign cancellations between the diagrams, ∇ − ∇ − 2 α where Fx;μν ¼ μAx;ν νAx;μ i ½Ax;μ;Ax;ν is the lat- but merely modifies the hopping coefficients in the time tice field strength tensor constructed from the noncompact direction [23,24,26,28]. Thus, even despite the sign can- ∇ field Ax;μ and μ is the forward lattice derivative. Again we cellations which happen between strong-coupling expan- 2 have to choose α ∼ λ to arrive at the canonical normalization sion diagrams at higher orders, DiagMC algorithms based of the kinetic terms in the action. Higher-order terms in the on strong-coupling expansion can be expected to solve, at action of the Ax;μ gauge fields are either suppressed by higher least formally, the computational complexity problem (in powers of α or contain higher lattice derivatives which are the terminology of [6]) for finite-density gauge theories in suppressed in the naive continuum limit. Again we see that the the Veneziano large-N limit. “gluon” field Ax;μ has acquired the mass term proportional to The Metropolis algorithm IV.B. 1 allows us to system- the t’Hooft coupling λ. The action (80) is of course still atically sample the strong-coupling expansion for large-N → Ω Ω† non-Abelian gauge theories. To this end, it should be applied invariant under gauge transformations gx;μ xgx;μ xþμˆ , to solve the gauge-invariant Schwinger-Dyson equations for which, however, now have quite a nontrivial nonlinear form 1 lattice Wilson loops h Trðg μ g μ …g μ Þi (Migdal- in terms of Ax;μ variables. In particular, they are different N x1; 1 x2; 2 xn; n from the conventional continuum gauge transformations Makeenko loop equations [71,72]). These equations are very similar in structure to equations (40) in subsection IVA. Ax;μ → Ax;μ þ ∇μϕx − i½Ax;μ; ϕx. This is why the nonzero gluon mass in (80) does not contradict the gauge invariance of In fact, we have already tried to solve the Migdal- the theory. It is interesting here to draw parallels with the Makeenko loop equations in large-N pure gauge theory gluon mass which one can obtain from nonperturbative using the algorithm IV.B.1, sampling the loops on the lattice with probabilities proportional to Wilson loops solution of (truncated) Schwinger-Dyson equations and 1 Tr g μ g μ …g μ . Unfortunately, this straightfor- Ward identities of continuum theory [88]. hN ð x1; 1 x2; 2 xn; n Þi The inclusion of finite-density quarks would require ward attempt did not allow to go beyond the first few orders sampling of both fermionic and bosonic correlators. Since of strong-coupling expansion, which could anyway be fermionic propagators at finite density are in general computed manually. The reason is the “zigzag symmetry” complex-valued, this would require additional re-weighting [90,91] between Wilson loops which differ by the insertion and make the sign problem worse. For perturbative expan- of arbitrary path traversed forward and immediately back- sions with truncation error which scales exponentially ward (see Fig. 17), which stems from unitarity of link M ≡ † 1 ϵM ∼ α , jαj < 1 with the truncation order M this still matrices gx;μgxþμ;−μ gx;μgx;μ ¼ . Most of the computer allows us to obtain the result in a computational time which time goes into sampling numerous redundant loops with scales as a polynomial of the required error, provided one multiple “zigzag” insertions, and nontrivial loops which can sample Feynman diagrams in a time which grows only contribute to higher orders of strong-coupling expansion exponentially with expansion order [6]. In the large-N are entropically disfavored. Possible solution to this prob- limit, the number of Feynman diagrams grows exponen- lem is to rewrite the Migdal-Makeenko loop equations in tially with their order, thus, even the direct summation will terms of “irreducible” loops without zigzag insertions, satisfy this requirement. However, for our infrared-finite which makes the equation structure significantly more weak-coupling expansion (13), the truncation error scales nontrivial. We hope that this “gauge-fixing” in loop space −γ as ϵM ∼ M , γ > 0 [see (77), (78)]; thus, computational would allow to go to sufficiently high orders of strong- time will grow faster then polynomial in the required error, coupling expansion. Work in this direction is in progress. similar to what happens for conventional Monte Carlo Another challenge is to extend this approach to finite N, simulations with a sign problem. An important direction for where the combinatorial structures involved in the strong- further work is, thus, to find the resummation strategy coupling expansion coefficients become more complicated which would lead to exponentially fast convergence of [26,92], and 1=N expansion might not work properly [92]. the series, with bold diagrammatic expansions being one of Importantly, the baryon bound states only exist at finite N. the options. Nevertheless, even the finite-density gauge theory in the A less straightforward extension of this work to non- large-N Veneziano limit might exhibit some universal Abelian gauge theories might be based on the lattice strong- features for densities below the baryon condensation coupling expansion, combined with hopping expansion if threshold and is therefore physically interesting [93,94].

114512-27 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) example, the planar ϕ4 model with negative coupling constant or the Weingarten model of random planar surfaces on the lattice [95]. The resummation strategy used in this work also seems to be quite general for non-Abelian field theories. Its FIG. 17. Zigzag symmetry of Wilson loops in gauge theory. particular advantages are: (i) The absence of infrared divergences which are Needless to say, large-N limit is also important in the characteristic for the standard weak-coupling ex- context of AdS=CFT holographic duality between gauge pansion of massless quantum field theories. While theories and gravity. these divergences cancel in physical observables, they should be regulated at intermediate steps, which enhances the sign problem upon removing the VII. CONCLUSIONS infrared regulator due to cancellations between In this work, we have explored the advantages and numerically large summands of opposite signs. In disadvantages of the diagrammatic Monte Carlo (DiagMC) more conventional approaches to lattice perturbation approach for studying asymptotically free non-Abelian theory such as numerical stochastic perturbation lattice field theories in the weak-coupling regime. To this theory (NSPT) [45,46] dealing with infrared diver- end, we have combined two largely independent tools: the gences also requires additional infrared regulators. Metropolis algorithm for solving infinitely-dimensional (ii) The absence of factorial divergences in the weak- Schwinger-Dyson equations, and the partial resummation coupling expansion in the large-N limit, which of lattice perturbation theory which leads to massive bare arise at sufficiently high orders in conventional propagators. lattice perturbation theory. The general arguments While our primary motivation are non-Abelian lattice of Sec. II, however, do not exclude the possibility gauge theories, eventually coupled to finite-density fer- of exponential divergences. Our arguments in mions, in this exploratory study we have considered an favor of convergence of our infrared-finite weak- example of the large-NUðNÞ × UðNÞ principal chiral coupling expansion are only based on the numerical model. This model has significantly simpler weak-coupling evidence—for up to M ¼ 500 orders for the large-N expansion, while having most of the nonperturbative OðNÞ sigma model, and for up to M ¼ 12 orders for features of the large-N pure gauge theories on the lattice. the principal chiral model. The construction of Metropolis algorithm IV. B. 1 from Of course, these advantageous features come at a certain Schwinger-Dyson equations is very general, and can be price, and in some respects the resummed perturbative applied to practically any quantum field theory. Its main expansion behaves worse than the bare one: conceptual advantages are the following: (i) Sampling our weak-coupling expansion using a (i) The applicability of algorithm to any expansion, DiagMC algorithm leads to the sign problem due even without explicit diagrammatic rules. In par- to in general nonpositive weights of Feynman ticular, it can be used to sample both weak- and diagrams, even though the conventional Monte Carlo strong-coupling expansions in lattice gauge theory. simulations do not have any sign problem. Further- (ii) The ability to work directly in the infinite-volume more, for lattice gauge theory this sign problem is limit. Of course, this does not imply the absence of expected to become worse if finite-density fermions slowing down towards the continuum limit, since the are added. number of expansion coefficients which should be (ii) The convergence is slower than for the conventional sampled to reach a given precision might grow with lattice perturbation theory, see Figs. 12 and 15. correlation length (see, e.g., subsection III B), and (iii) The expansion starts from the wrong vacuum state sign problem might make this growth faster (see with explicitly broken UðNÞ × UðNÞ symmetry of subsection VB). the principal chiral model. While this symmetry (iii) The ability to work directly in the large-N limit, where seems to be gradually restored at high orders, at the number of diagrams contributing to either strong- finite expansion order this explicit symmetry break- or weak-coupling expansions becomes significantly ing results in constant contribution to the correlators smaller. Systematic calculation of the coefficients of of gx variables, which in reality are exponentially 1=N-expansion is also possible [81].Large-N limit is decaying. not directly accessible in most other DiagMC simu- Summarizing these advantages and disadvantages, it lations of non-Abelian gauge theories [23–28]. seems that the DiagMC approach based on the weak- Our algorithm is particularly suitable for exploring the coupling expansion is comparable in terms of computa- critical behavior in large-N quantum field theories with tional time to conventional simulation methods as applied manifestly positive weights of Feynman diagrams, for to non-Abelian lattice field theories in the large-N limit. In

114512-28 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) particular, for Fig. 12, the finite-N data from conventional which is the only term in the true trans-series for the mass Monte Carlo and the DiagMC data at infinite N were gap in the large-NOðNÞ sigma model. One could, thus, produced within the comparable amount of CPU time. It is expect that if such an approach works also for the principal also conceivable, for example, that applying Numerical chiral model, it might resum many terms in our expansion Stochastic Perturbation Theory (NSPT) [45,46] to finite-N into exponentials, thus, turning it into the true trans-series principal chiral model and then extrapolating to large- of the form (1) with optimal convergence properties and volume and large-N limits would lead to the comparable reducing the sign problem due to cancellations between quality of numerical data within the comparable CPU time. Feynman diagrams. Implementing this approach in practice It remains to be seen how efficient the sampling of strong- turns out to be nontrivial, and would be explored in coupling expansion might be. future work. Probably the most promising direction for further improvement is the Bold Diagrammatic Monte Carlo ACKNOWLEDGMENTS (BDMC) approach, which can efficiently capture nonperturbative information by combining perturbative expan- This work was supported by the S. Kowalevskaja award sion with a self-consistent mean-field-like approximation from the Alexander von Humboldt Foundation. The [5,13,14,96]. Unfortunately, currently most of BDMC authors are grateful to the Mainz Institute for Theoretical algorithms for bosonic fields are based on certain trunca- Physics (MITP) for its hospitality and its partial support tions of Schwinger-Dyson equations even for the simplest during the completion of this work. The calculations were case of scalar field theory with ϕ4 interactions [13,14]. performed on “iDataCool” at Regensburg University, on One possibility to construct the bold diagrammatic the ITEP cluster in Moscow and on the LRZ cluster in expansion would be to work with Schwinger-Dyson Garching. We acknowledge valuable discussions with S. equations for n-particle-irreducible field correlators [74]. Valgushev, F. Werner, B. Svistunov, T. Sulejmanpasic, Another possibility, more specific for non-Abelian field L. von Smekal, N. Prokof’ev, G. Dunne, O. Costin, A. theories, is to introduce the matrix-valued Lagrange multi- Cherman and G. Bali. † plier enforcing the unitarity constraint gxgx ¼ 1, and integrate out the gx fields, which would automatically ensure the UðNÞ × UðNÞ-invariant vacuum state with APPENDIX A: VERTEX FUNCTIONS IN THE 1 0 SMALL MOMENTUM LIMIT hN Trgxi¼ . For the large-NOðNÞ sigma model, a similar treatment immediately yields the exact nonperturbative The vertex functions (45) can be drastically simplified if solution, with the mass being equal to the saddle-point all the momenta in their arguments are much smaller than value of the Lagrange multiplier field. In fact, this mean- the lattice cutoff scale. In this limit the lattice Laplacian field solution resums our double series (30) with coupling ΔðpÞ behaves as ΔðpÞ ≈ p2, and the vertex function − 4π … and logs of coupling into a single exponential expð λ Þ, Vðq1; ;q2nþ1Þ can be represented as

2Xnþ1 Xk−1 … −1 k−1 Δ Vðq1; ;q2nþ1Þ¼ ð Þ ðqmþ1 þþqmþ2nþ2−kÞ k¼1 m¼0 2Xnþ1 Xk−1 −1 k−1 2 ¼ ð Þ ðqmþ1 þþqmþ2nþ2−kÞ k¼1 m¼0 2Xnþ1 Xk−1 2nXþ2−k 2Xnþ1 Xk−1 2nXþ2−k Xi−1 −1 k−1 2 2 −1 k−1 ¼ ð Þ qmþi þ ð Þ qmþiqmþj ðA1Þ k¼1 m¼0 i¼1 k¼1 m¼0 i¼1 j¼1

The first term in the last line of the above equation can be simplified as

2Xnþ1 Xk−1 2nXþ2−k X2n 2Xnþ1 Xk−1 2Xr−1 −1 k−1 2 2 −1 k−1 δ θ 2 2 − − 2 1 ð Þ qmþi ¼ q2r ð Þ m;l ð n þ k r þ l þ Þ k¼1 m¼0 i¼1 r¼1 k¼1 m¼0 l¼0 X2n 2Xnþ1 Xk−1 X2r 2 −1 k−1 δ θ 2 2 − − 2 þ q2rþ1 ð Þ m;l ð n þ k r þ lÞ r¼0 k¼1 m¼0 l¼0 X2n 2Xr−1 2nþX2−2rþl X2n X2r 2nþX1−2rþl X2n 2 −1 k−1 2 −1 k−1 2 ¼ q2r ð Þ þ q2rþ1 ð Þ ¼ q2rþ1; ðA2Þ r¼1 l¼0 k¼lþ1 r¼0 l¼0 k¼lþ1 r¼0

114512-29 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) θ 1 ϕ2 −1 where is the unit step function. Similar algebra gives for where we have denoted fi ¼ð þ i Þ . Combining the second term in the last line of (A1): everything together, we arrive at Z Z 2Xnþ1 Xk−1 2nXþ2−k Xi−1 2 −1 k−1 ϕ 1 ϕ2 −N ð Þ qmþiqmþj dg ¼ d det ð þ Þ ; ðB4Þ H k¼1 m¼0 i¼1 j¼1 UðNÞ N×N X2n Xr−1 2 where on the right hand side the integration goes over ¼ q2rþ1q2sþ1 ðA3Þ all N × N Hermitian matrices. This expression is iden- 0 0 r¼ s¼ tical to (4) in the main text, up to the trivial rescaling Combining (A2) and (A3), we get ϕ → αϕ.

… 2 Vðq1; ;q2nþ1Þ¼ðq1 þ q3 þþq2nþ1Þ ; ðA4Þ 2. Exponential map for UðNÞ group Proceeding in the same way as for the Cayley map, for which is the formula (47) in the main text. the exponential map g ¼ eiϕ we can relate the differentials of ϕ and g as APPENDIX B: INTEGRATION MEASURES Z 1 ON S AND UðNÞ MANIFOLDS ϕ 1− ϕ N dg ¼ dzeiz dϕeið zÞ : ðB5Þ 1. Cayley map for UðNÞ group 0 1þiϕ 2 − 1 The group-invariant distance element takes the form With the Cayley mapping g ¼ 1−iϕ ¼ 1−iϕ , the group-invariant distance element ds2 ¼ Trðdgdg†Þ on ds2 ¼ Trðdgdg†Þ UðNÞ can be expressed as Z Z 1 1 iðz2−z1Þϕ ϕ −iðz2−z1Þϕ ϕ −2i 1 2i 1 ¼ dz1dz2Trðe d e d Þ: ðB6Þ ds2 ¼ Tr dϕ dϕ 0 0 1 − iϕ 1 − iϕ 1 þ iϕ 1 þ iϕ 1 1 Again, we can now use the invariance of the measure ¼ 4Tr dϕ dϕ ; ðB1Þ under similarity transformations g → ugu†, ϕ → ugu†, 1 þ ϕ2 1 þ ϕ2 u ∈ UðNÞ to diagonalize g and ϕ, and express the distance The invariance of ds2 under shifts g → ug, u ∈ UðNÞ element (B6) as Z Z implies, in particular, the invariance of the measure under X 1 1 † − ϕ −ϕ 2 → iðz2 z1Þð i jÞ similarity transformations g ugu which allow to diag- dz1dz2e jdϕijj onalize g and hence the matrix ϕ in (B1). We, thus, assume i;j 0 0 Z Z that the matrix ϕ has diagonal form ϕ ϕ δ . The X X 1 1 ij ij ¼ i ij 2 2 ϕ 2 − distance element ds can be then written as a convolution of ¼ d ii þ dz1dz2 cosððz2 z1Þ i i>j 0 0 the coordinate differentials dϕij with the diagonal metrics: 2 × ðϕi − ϕjÞÞjdϕijj 2 ϕ ϕ ds ¼ gijkld ijd kl X X 2 ϕ − ϕ 2 2 sin ðð i jÞ= Þ 2 X 2 X 2 2 dϕ 8 dϕ : dϕ ðdReϕ Þ þðdImϕ Þ ¼ ii þ ϕ − ϕ 2 j ijj ðB7Þ ii 2 ij ij ð i jÞ ¼ 1 ϕ2 þ 1 ϕ2 1 ϕ2 ; ðB2Þ i i>j i þ i j>i ð þ i Þð þ j Þ We see that the metric tensor is again diagonal with our where we have taken into account that the off-diagonal choice of group coordinates, and we finally get for the components ϕij are not all independent, but rather satisfy integration measure (up to an overall normalization factor) ¯ ϕ ¼ ϕ . We can now express the integration measure, up Z Z ij ji R R pffiffiffi Y 2 ϕ − ϕ 2 ϕ sin ðð i jÞ= Þ to an irrelevant normalization factor, as dg ¼ d g, dg dϕ ; ¼ ϕ − ϕ 2 ðB8Þ where g is the determinant of the metric tensor which is UðNÞ M i>j ð i jÞ equal to the product of all the diagonal metric elements appearing in (B2): where M is some closed domain in the N2-dimensional H Y Y Y space N×N of the Hermitian N × N matrices. The UðNÞ g f2f2 f f f group manifold is uniquely covered by the integration in ¼ i j i ¼ i j M 2 − 1 j>i i i;j (B8) if is bounded by the N -dimensional manifolds ϕ Y 2N at which at least two eigenvalues of coincide. E.g., in f det 1 ϕ2 −2N; B3 the case of Uð2Þ group, M is the direct product of the ¼ i i ¼ ð þ Þ ð Þ 3-dimensional ball and the one-dimensional circle S1.

114512-30 DIAGRAMMATIC MONTE CARLO FOR THE WEAK- … PHYSICAL REVIEW D 96, 114512 (2017) 2 2 2 In order to compare with the formula (4) in the main text, ds ¼ dn0 þ dni it is instructive to expand the Jacobian (B8) in powers of ϕ: 16 ϕ ϕ 2 2 ϕ 4ϕ dϕ ϕ 2 ð id iÞ d i − j j i Z ¼ 2 4 þ 2 2 2 X sin2 ϕ − ϕ =2 ð1 þ ϕ Þ 1 þ ϕ ð1 þ ϕ Þ ϕ ðð i jÞ Þ d exp log 2 2 M ðϕ − ϕ Þ 4dϕ i;j i j i : B10 Z ¼ 1 ϕ2 2 ð Þ X ϕ − ϕ 2 ð þ Þ ϕ − ð i jÞ λ4 ¼ d exp 12 þ Oð Þ M i;j The metric tensor is again diagonal in the stereographic Z coordinates ϕ (which is not surprising, as stereographic NTrϕ2 − TrϕTrϕ i ¼ dϕ exp − þ Oðϕ4Þ ; ðB9Þ mapping is conformal), and the integration measure (up to a M 6 normalization constant) can be written in terms of the square root of the metric determinant as which illustrates the appearance of double-trace terms Z Z already in the lowest order of the expansion. dn ¼ dN−1ϕð1 þ ϕ2Þ−ðN−1Þ; ðB11Þ N−1 SN R 3. Stereographic map for the sphere SN In order to find the integration measure on the N-sphere where the integration is over the whole N − 1 dimensional ϕ 1 − ϕ2 RN−1 SN parametrized by the coordinates Pi as n0 ¼ð Þ real space .Inthelarge-N limit one can also replace 1 ϕ2 −1 2ϕ 1 ϕ2 −1 ϕ2 ≡ ϕ2 − − 1 − ð þ Þ , ni ¼ ið þ Þ , i i , we again start the power of ðN Þ in (B11) by N, which leads to by first expressing the OðNÞ-invariant distance element the expression ffiffiffi(23) in the main text after the trivial re- 2 p ds ¼ dnadna on SN in terms of the coordinates ϕi: scaling ϕ → λ=2ϕ.

[1] K. Van Houcke, E. Kozik, N. Prokof’ev, and B. Svistunov, [18] T. Rindlisbacher and P. de Forcrand, Nucl. Phys. B918, 178 Phys. Procedia 6, 95 (2010). (2017). [2] N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 87, 160601 [19] T. Korzec and U. Wolff, Proc. Sci., LAT2010 (2010) 029. (2001). [20] C. Gattringer, T. Kloiber, and V. Sazonov, Nucl. Phys. B897, [3] E. Burovski, N. Prokof’ev, B. Svistunov, and M. Troyer, 732 (2015). New J. Phys. 8, 153 (2006). [21] Y. Delgado, C. Gattringer, and A. Schmidt, Comput. Phys. [4] N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 99, 250201 Commun. 184, 1535 (2013). (2007). [22] K. Miura, T. Z. Nakano, A. Ohnishi, and N. Kawamoto, [5] K. van Houcke, F. Werner, E. Kozik, N. Prokofev, B. arXiv:1106.1219. Svistunov, M. J. H. Ku, A. T. Sommer, L. W. Cheuk, [23] W. Unger and P. de Forcrand, J. Phys. G 38, 124190 (2011). A. Schirotzek, and M. W. Zwierlein, Nat. Phys. 8, 366 [24] P. de Forcrand, J. Langelage, O. Philipsen, and W. Unger, (2012). Phys. Rev. Lett. 113, 152002 (2014). [6] R. Rossi, N. Prokof’ev, B. Svistunov, K. Van Houcke, and F. [25] H. Vairinhos and P. de Forcrand, J. High Energy Phys. 12 Werner, Europhys. Lett. 118, 10004 (2017). (2014) 038. [7] R. Rossi, Phys. Rev. Lett. 119, 045701 (2017). [26] W. Unger, Proc. Sci., LATTICE2014 (2014) 192. [8] I. Tupitsyn and N. Prokof’ev, Phys. Rev. Lett. 118, 026403 [27] C. Gattringer and C. Marchis, Nucl. Phys. B916, 627 (2017). (2017). [9] V. Ayyar and S. Chandrasekharan, Phys. Rev. D 96, 114506 [28] P. de Forcrand, P. Romatschke, W. Unger, and H. Vairinhos, (2017). Proc. Sci., LATTICE2016 (2016) 086. [10] U. Wolff, Phys. Rev. D 79, 105002 (2009). [29] M. Fromm, J. Langelage, S. Lottini, M. Neuman, and O. [11] T. Korzec, I. Vierhaus, and U. Wolff, Comput. Phys. Philipsen, Phys. Rev. Lett. 110, 122001 (2013). Commun. 182, 1477 (2011). [30] Y. D. Mercado, H. G. Evertz, and C. Gattringer, Phys. Rev. [12] C. Gattringer and T. Kloiber, Nucl. Phys. B869, 56 (2013). Lett. 106, 222001 (2011). [13] A. Davody, Phys. Rev. D 88, 125023 (2013). [31] C. Gattringer, Nucl. Phys. B850, 242 (2011). [14] T. Pfeffer and L. Pollet, New J. Phys. 19, 043005 [32] P. Rossi and U. Wolff, Nucl. Phys. B248, 105 (1984). (2017). [33] U. Wolff, Phys. Lett. 153B, 92 (1985). [15] U. Wolff, Nucl. Phys. B824, 254 (2010). [34] J. Langelage, G. Münster, and O. Philipsen, J. High Energy [16] U. Wolff, Nucl. Phys. B832, 520 (2010). Phys. 07 (2008) 036. [17] F. Bruckmann, C. Gattringer, T. Kloiber, and T. Sulejman- [35] D. J. Gross and E. Witten, Phys. Rev. D 21, 446 (1980). pasic, Phys. Lett. B 749, 495 (2015). [36] F. Green and S. Samuel, Phys. Lett. 103B, 48 (1981).

114512-31 P. V. BUIVIDOVICH and A. DAVODY PHYSICAL REVIEW D 96, 114512 (2017) [37] M. Campostrini, P. Rossi, and E. Vicari, Phys. Rev. D 52, [66] Y. Brihaye and P. Rossi, Nucl. Phys. B235, 226 (1984). 395 (1995). [67] A. K. Rebhan, Nucl. Phys. B430, 319 (1994). [38] M. Beneke, Phys. Rep. 317, 1 (1999). [68] R. Rossi, F. Werner, N. Prokof’ev, and B. Svistunov, Phys. [39] F. David, Commun. Math. Phys. 81, 149 (1981). Rev. B 93, 161102 (2016). [40] F. David, Nucl. Phys. B209, 433 (1982). [69] P. Rossi, M. Campostrini, and E. Vicari, Phys. Rep. 302, 143 [41] V. A. Fateev, V. A. Kazakov, and P. B. Wiegmann, Nucl. (1998). Phys. B424, 505 (1994). [70] E. Kozik, M. Ferrero, and A. Georges, Phys. Rev. Lett. 114, [42] V. A. Fateev, V. A. Kazakov, and P. B. Wiegmann, Phys. 156402 (2015). Rev. Lett. 73, 1750 (1994). [71] Y. Makeenko and A. A. Migdal, Nucl. Phys. B188, 269 [43] M. Beneke, V. M. Braun, and N. Kivel, Phys. Lett. B 443, (1981). 308 (1998). [72] T. Eguchi and H. Kawai, Phys.Rev.Lett.48,1063 [44] E. Br´ezin, C. Itzykson, G. Parisi, and J. B. Zuber, Commun. (1982). Math. Phys. 59, 35 (1978). [73] R. Alkofer and L. von Smekal, Phys. Rep. 353,281 [45] G. S. Bali, C. Bauer, and A. Pineda, Phys. Rev. D 89, (2001). 054505 (2014). [74] J. Berges, Phys. Rev. D 70, 105010 (2004). [46] G. S. Bali and A. Pineda, AIP Conf. Proc. 1701, 030010 [75] N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. Lett. 81, (2016). 2514 (1998). [47] G. V. Dunne and M. Unsal, Phys. Rev. D 87, 025015 (2013). [76] P. D. Anderson and M. Kruczenski, Nucl. Phys. B 921, 702 [48] A. Cherman, D. Dorigoni, G. V. Dunne, and M. Unsal, (2017). Phys. Rev. Lett. 112, 021601 (2014). [77] W. K. Hastings, Biometrika 57, 97 (1970). [49] G. V. Dunne and M. Unsal, Phys. Rev. D 89, 041701 (2014). [78] P. V. Buividovich, SchwingerDysonMetropolis, GitHub [50] G. V. Dunne and M. Unsal, Phys. Rev. D 89, 105009 (2014). repository (2017). [51] G. V. Dunne and M. Unsal, J. High Energy Phys. 09 (2015) [79] K. Etessami and M. Yannakakis, Lect. Notes Comput. Sci. 199. 3404, 340 (2005). [52] A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic, and [80] H. P. McKean, Jr., in Stochastic differential equations, M. Unsal, Annals of Mathematical Sciences and Applica- Lecture Series in Differential Equations (Catholic Univ., tions 2, 95 (2017). 1967), pp. 41–57. [53] K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979). [81] P. V. Buividovich, Proc. Sci., LATTICE2016 (2016) 315. [54] J. R. Espinosa, M. Quirosn, and F. Zwirner, Phys. Lett. B [82] A. Bazavov and B. A. Berg, Phys. Rev. D 71, 114506 (2005). 314, 206 (1993). [83] M. Lüscher, Comput. Phys. Commun. 79, 100 (1994). [55] P. Arnold and C. Zhai, Phys. Rev. D 50, 7603 (1994). [84] P. V. Buividovich and S. N. Valgushev, arXiv:1706.08954. [56] E. Braaten and R. D. Pisarski, Phys. Rev. D 45, R1827 [85] P. Rossi and E. Vicari, Phys. Rev. D 49, 1621 (1994). (1992). [86] P. Rossi and E. Vicari, Phys. Rev. D 49, 6072 (1994). [57] F. Karsch, A. Patkos, and P. Petreczky, Phys. Lett. B 401,69 [87] A. C. Cubero, Phys. Rev. D 91, 105025 (2015). (1997). [88] J. M. Cornwall, Phys. Rev. D 26, 1453 (1982). [58] P. V. Buividovich, Phys. Rev. D 83, 045021 (2011). [89] K. H. O’Brien and J. Zuber, Nucl. Phys. B253,621 [59] P. V. Buividovich, Nucl. Phys. B853, 688 (2011). (1985). [60] P. V. Buividovich, Proc. Sci., LATTICE2015 (2015) 293. [90] A. M. Polyakov, Nucl. Phys. B68, 1 (1998). [61] L. von Smekal, D. Mehta, A. Sternbeck, and A. G. Williams, [91] N. Drukker, D. J. Gross, and H. Ooguri, Phys. Rev. D 60, Proc. Sci., LAT2007 (2007) 382. 125006 (1999). [62] L. von Smekal, A. Jorkowski, D. Mehta, and A. Sternbeck, [92] S. Samuel, J. Math. Phys. (N.Y.) 21, 2695 (1980). Proc. Sci., Confinement8 (2008) 048. [93] T. Alho, M. Jarvinen, K. Kajantie, E. Kiritsis, C. Rosen, and [63] A. Sternbeck and L. von Smekal, Eur. Phys. J. C 68, 487 K. Tuominen, J. High Energy Phys. 04 (2014) 124. (2010). [94] U. Gursoy, arXiv:1612.00899. [64] L. von Smekal and M. Bischoff, Proc. Sci., ConfinementX [95] D. Weingarten, Phys. Lett. 90B, 280 (1980). (2012) 068. [96] L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. B [65] S. Mizoguchi, Nucl. Phys. B716, 462 (2005). 83, 161103 (2011).

114512-32