arXiv:1705.03368v2 [hep-lat] 1 Dec 2017 atrpyis i clrfil hois[10–14] theories field scalar via physics, in- strongly matter [3–9] from systems range suc- fermionic were DiagMC teracting which using theories expan- studied and cessfully strong-coupling models physical or models The weak- physical sions. in simple problem relatively sign with eliminating for helpful rtos igCagrtm,cmlmne ihthe with [2] complemented techniques config- algorithm algorithms, “worm” field DiagMC than weak- rather the urations. expansions, of strong-coupling diagrams the or samples stochastically which isi h ocle igamtcMneCroap- Monte-Carlo Diagrammatic so-called [1] the proach is gies milder. it make or which problem sign theories, the simu- avoid gauge hopefully alternative would non-Abelian for for search algorithms the lation motivated has This (QCD). problem Chromodynamics matter Quantum diagram interacting by strongly described for phase dense as of obstacles the state main of of equation the and studies of first-principle one currently systematic the is in integral configurations field path sampling on based simulations ∗ ‡ † igamtcMneCrofrwa-opigepnino n of expansion weak-coupling for Monte-Carlo Diagrammatic P,Thrn Iran Teheran, IPM, absenc of leave On [email protected]; [email protected] hove information. 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U N N Dtd oebr1t 2017) 1st, November (Dated: ii ae ntewa-opigepnin is enote we First expansion. weak-coupling the on based limit ( N N U N > from e ) arxfil hoy n pl tt td hsinfrared- this study to it apply and theory, field matrix to , the r × ( N U ) † ( × N n .Davody A. and n eo on ttsaraya h edn order leading the at already states 33] baryon [32, bound the and meson breaking and symmetry as chiral QCD, QCD of confinement, lattice features of non-perturbative is such expansion strategies captures which strong-coupling successful first-principle, the most the on the a based At of in one level, resulted strategy. qualitative far simulation improvable so systematically not have attempts hoisa nt est [22–28] gauge density lattice finite non-Abelian at of theories simulations to DiagMC ply exhaustive. means no by is references of selection ae o ihrodr [26] orders higher for cated rdc h xetdqaiaiefaue fteQCD the re- of 24] to features [23, one qualitative diagram allow phase and expected limit, the massless the produce the of in free are problem coupling sign strong infinitely at fermions gered bla ag hoiswt emoi n ooi mat- bosonic [19–21] and fields fermionic ter with theories gauge Abelian opigsaigrgo hr atc hoyapproaches theory lattice weak- where the even region access suffi- scaling to simulating allow coupling coupling, should orders of high values expansion ciently all strong-coupling for Since convergent the is lattices algorithm. finite-volume DiagMC using on of coupling powers in gauge expansion inverse strong-coupling the sample to like algorithm DiagMC the the [24] in of manually incorporated part strong- be bosonic the should the of action from orders coming next expansion the meets coupling coupling as gauge difficulties, of values conceptual finite the to approach this [29–31] thereof els O U ) ( hssceshsmtvtdvrosatmt oap- to attempts various motivated has success This ( N N rnia hrlmodel chiral principal and ) olna im oe picplchiral (principal model sigma nonlinear ) thg res oprn Diagrammatic Comparing orders. high at rpltdt infinite to trapolated > aems emi h ffcieaction effective the in term mass bare a h large- the igCsmltoso atc C ihstag- with QCD lattice of simulations DiagMC . praht lnrQDa eoand zero at QCD planar to approach fislgrtm eiicn fre- of reminiscent logarithm, its of s ettassre ihu exponential without trans-series gent obesre otemnfsl non- manifestly the to series double meaueo h “deconfinement” the of emperature CP str edr h perturbative the renders term ss ovbeeapeo large- a of example solvable y uain fnnAeinlattice non-Abelian of mulations > N tcMneCroagrtmfor algorithm Monte-Carlo atic hc eoe oeadmr compli- more and more becomes which , ‡ − > N 1 hc r oesmlrt C.This QCD. to similar more are which , > olna im oes[15–18] models sigma nonlinear n nnt-ouelimits infinite-volume and ept infiatpors,these progress, significant Despite . > nAeinltiefield lattice on-Abelian g, notntl,a xeso of extension an Unfortunately, . nin hc sthe is which ansion, N > efidagood a find we , dal,oewudalso would one Ideally, . gexpansion ng > ∗ rteeetv mod- effective the or N > and 2 continuum QCD. Recent attempts to extend the DiagMC divergences in perturbative expansion. simulations to finite gauge coupling using the method Recently infrared renormalon divergences have been of auxiliary Hubbard-Stratonovich variables [25]> or the given a physical interpretation in terms of the action Abelian color cycles [27]> are very promising, but still of unstable or complex-valued saddles of the path inte- not quite practical. Development of practical tools for gral, along with a resummation prescription based on the generating high orders of strong-coupling expansion is mathematical notion of resurgent trans-series [47–52]>. thus currently an important unsolved problem. Trans-series is a generalization of the ordinary power se- However, in the infinite-volume limit strong coupling ries to a more general form expansion is known to have only a finite radius of conver- gence (see e.g. [34]>). One can thus expect that as one mi +∞ (λ)= e−Si/λ log (λ)l c λp, (1) approaches the weak-coupling regime, the convergence h O i i,l,p i p=0 of the strong-coupling series will increasingly stronger X Xl=0 X depend on the lattice size, which might well make the where (λ) is the expectation value of some physical continuum and infinite-volume extrapolation difficult in observableh O i as a function of the coupling constant λ, c practice, especially for the infrared-sensitive physical ob- i,l,p are the coefficients of perturbative expansion of path inte- servables such as hadron masses and transport coeffi- gral around the i-th saddle point of the path integral with cients. Moreover, in the t’Hooft large-N limit one even the action S and l labels flat directions of the action in expects a phase transition separating the strong-coupling i the vicinity of this saddle. It turns out that non-Borel- and the weak-coupling phases [35–37]> even on the finite- summable factorial divergences cancel between pertur- volume lattice. These properties of the strong-coupling bative expansions around different saddle points i in (1). expansion might significantly limit the ability of DiagMC Unfortunately, currently no method is available to gener- to work directly in the thermodynamic limit, which is one ate the trans-series (1) for strongly coupled field theories of the most important advantages of DiagMC over con- in a systematic way. The resurgent analysis of perturba- ventional Monte-Carlo methods [6, 7]>. tive expansions has been done so far almost exclusively These considerations suggest that the extrapolation to in the regime where gauge theories or sigma models are the continuum and thermodynamic limit might be eas- artificially driven to the weak-coupling limit by a special ier for DiagMC simulations based on the weak-coupling compactification of space-time to D = 1 + 0 dimensions expansion. However, the weak-coupling expansion for with twisted boundary conditions [47–52]>. asymptotically free non-Abelian field theories, such as The aim of this work is to develop DiagMC methods four-dimensional non-Abelian gauge theories and two- suitable for the bosonic sector of non-Abelian lattice field dimensional nonlinear sigma models, has two closely re- theories, which is currently a challenge for DiagMC ap- lated problems, which make a direct application of a proach. We develop two practically independent tools, DiagMC approach conceptually difficult at first sight. which are finally combined together to perform practical The first problem are infrared divergences due to mass- simulations. less gluon propagators, which cancel only in the final The first tool is the resummation prescription which expressions for physical observables. The second prob- lem is the non-sign-alternating factorial growth of the renders the bare weak-coupling expansion in non-Abelian field theories infrared-finite and thus suitable for DiagMC weak-coupling expansion coefficients, which is commonly referred to as the infrared renormalon [38]> and can- simulations which sample Feynman diagrams. This pre- scription is introduced in Section II and tested in Sec- not be dealt with using the Borel resummation tech- tion III on some exactly solvable examples. The basic niques. These factorial divergences in the perturba- tive expansions of physical observables originate in the idea is to work with massive bare propagators, where the bare mass term comes from the nontrivial path integral renormalization-group running of the coupling constant, and have nothing to do with the purely combinatorial fac- measure for non-Abelian groups and is proportional to the bare coupling. By analogy with Fujikawa’s approach torial growth of the number of Feynman diagrams with > diagram order1. In particular, infrared renormalon be- to axial anomaly [53] , this bare mass serves as a seed for dynamical mass generation and conformal anomaly haviour persists also in the large-N limit [39–43]>, in which the number of Feynman diagrams grows only ex- in non-Abelian field theories which are scale invariant at the classical level. A similar partial resummation of the ponentially with order [44]>. It is known that if one stops the running of the coupling at some artificially in- perturbative series with the help of the bare mass term is also often used in finite-temperature quantum field the- troduced infrared cutoff scale (e.g. by considering the > theory in a finite volume), this factorial growth disap- ory [54, 55] , in particular, in the contexts of hard ther- mal loop approximation [56]> and screened perturbation pears [45, 46]>, thus in a sense it is a remainder of IR theory [57]>. Upon this resummation, the weak-coupling expansion becomes formally similar to the trans-series (1), but contains only the powers of coupling and of the −S /λ 1 The recent work [9]> suggests that renormalon divergences might logs of coupling only, while the exponential factors e i be absent in two-dimensional asymptotically free fermionic the- seem to be absent. This expansion appears to be man- ories. ifestly infrared-finite and free of factorial divergences at 3 least in the large-N limit. An exactly solvable exam- II. INFRARED-FINITE WEAK-COUPLING ple of the O (N) nonlinear sigma model demonstrates EXPANSION FOR THE U (N) × U (N) PRINCIPAL that our expansion reproduces non-perturbative quan- CHIRAL MODEL tities such as the dynamically generated mass gap (see Subsection IIIB). The first step in the construction of lattice perturba- The second tool, described in Section IV, is the gen- tion theory is to parameterize the small fluctuations of eral method for devising DiagMC algorithms from the lattice fields around some perturbative vacuum state. For non-Abelian lattice fields g SU (N) the most pop- full untruncated hierarchy of Schwinger-Dyson equa- x ∈ tions [58, 59]>, bypassing any explicit construction of ular parametrization is the exponential mapping from a diagrammatic representation. Here the basic idea the space of traceless Hermitian matrices φx to SU (N) is to generate a series expansion by stochastic itera- group: gx = exp(iαφx), where α is related to the cou- tions of Schwinger-Dyson equations. We believe the pling constant and gx = 1 is the perturbative vacuum. method to be advantageous for sampling expansions for While all the results in this paper could be also obtained which the complicated form of diagrammatic rules makes for exponential mapping, for our purposes it will be more the straightforward application of e.g. worm algorithm convenient to use the Cayley map hardly possible, with strong-coupling expansion in non- +∞ Abelian gauge theories being one particular example. 1+ iαφx k k gx = =1+2 (iα) φx, (3) In this work we apply the method to sample the (re- 1 iαφx summed) weak-coupling expansion of Section II, which − Xk=1 for non-Abelian lattice field theories also has complicated form due to the multitude of higher-order interaction ver- where φx are again Hermitian matrices and the value of α tices. will be specified later. Since in this work we will be work- ing in the large-N limit in which the U (N) and SU (N) As a practical application which takes advantage of groups are indistinguishable, we omit here the zero trace both tools, in this work we consider the large-NU (N) condition for φ and work with U (N) group. This will × x U (N) principal chiral model on the lattice, which is de- significantly simplify the derivation of Schwinger-Dyson fined by the following partition function: equations in terms of φx fields, see Subsection IV A. One of the advantages of the Cayley map is the par- ticularly simple form of the U (N) Haar measure (see Appendix B1 for the derivation): N = dg exp 2Re Tr g† g , (2) Z x − λ x y
−1 † S0 = λ ∆xyTr gxgy = x,y X +∞ −1 k l k l 1 λ =4λ ( iα) (iα) ∆xyTr φx φy = S [φ]= ∆ + δ Tr (φ φ ) − F xy xy x y k,l=1 x,y 2 4 ≡ X X x,y +∞ X k−l 1 2 −1 2 k+l k l ∆xy + m δxy Tr (φxφy) =4λ ( 1) α ∆xyTr φx φy , (5) ≡ 2 0 ≡ − x,y k,l=1 x,y X k+Xl=2n X 1 0 −1 G Tr (φxφy) (10) ≡ 2 xy where we have introduced the lattice Laplacian operator x,y X in D-dimensional Euclidean space
D−1 D−1 ∆ =2Dδ δ − δ , (6) x,y x,y − x,y µˆ − x,y+ˆµ µ=0 µ=0 of the action (7), which endows the field φx with the bare X X 2 λ mass m0 = 4 . Thus the effect of the Jacobian (4) is to and used the unitarity of gx to add the diagonal terms break the shift symmetry φx φx + c I of the classical † → Tr gxgx = N to the action. Here µ =0, 1 labels the two action, serving as a seed for dynamical scale generation lattice coordinates, andµ ˆ denotes the unit lattice vector and conformal anomaly in the principal chiral model (2). in the direction µ. We see now that in order to get the canonical normalization of the kinetic term in the action A conventional approach in lattice perturbation the- 1 S0 = 2 φx∆x,yφy + ..., where ... denotes the terms ory, which constructs weak-coupling expansion as formal x,y series in powers of coupling λ, is to treat the quadratic P 2 λ > with more than two fields φ, we have to set α = 8 . term (9) in the action (7) as a perturbation [66] on We also note that the classical action (5) is invariant top of the kinetic term. Perturbative expansion then in- under the shifts of the field variable φx φx + cI, where volves massless propagators ∆−1 coming from the → xy I is the N N identity matrix. This symmetry is a scale-invariant classical part of the action, which leads × consequence of the scale invariance of the classical action to infrared divergences in the contributions of individual of the principal chiral model. diagrams. These infrared divergences, however, cancel Rewriting the path integral in the partition function in the final results for physical observables [66]>. Fur- (2) in terms of the new fields φx, we also need to include thermore, the resulting power series in the coupling con- the Jacobian (4). Adding the expansion of the Jacobian stant λ contain factorially divergent terms, some of which (4) in powers of λ and φ to the classical action (5), we have non-alternating signs and thus cannot be removed obtain the full action for the fields φx: by Borel resummation techniques. The only recently proposed way to deal with these divergences is to use 1 λ S [φ]= ∆ + δ Tr (φ φ )+ the mathematical apparatus of resurgent trans-series [47– 2 xy 4 xy x y > x,y 52] . X +∞ λk + ( 1)k−1 Tr φ2k + In this work we consider a different approach to the − k 8k x x weak-coupling expansion based on the action (7), some- Xk=2 X +∞ k+l−2 what similar in spirit to the screened perturbation the- k−l 4 λ 2 > 2 k l ory [54, 55, 57] and to hard thermal loop perturba- + ( 1) k+l ∆xyTr φx φy (7), > − 2 tion theory [56, 67] . Namely, we include the mass k,l=1 8 x,y X X 2 λ k+l=2n,k+l>2 m0 = 4 coming from (9) into the bare lattice propa- −1 gators G0 = ∆+ m2 which are used to construct so that the partition function (2) reads xy 0 xy the weights of Feynman diagrams. This corresponds to a trivial resummation of an infinite number of chain-like = dφ exp( NS [φ]). (8) diagrams with “two-leg” vertices. Since this prescription Z x − Z puts the coupling λ both in the propagators and vertices, we need to define the formal power counting scheme for The observation which is central for this work is that our expansion. To this end we group together the terms the Jacobian (4) contributes a term with the products of (2v + 2) fields, v = 1, 2,... + , ∞ 2 which correspond to interaction vertices with (2v + 2) λ m v φ2 = 0 φ2 (9) legs, and multiply them with the powers ξ of an auxil- 8 x 2 x x x iary parameter ξ which should be set to ξ = 1 to obtain X X 5 the physical result: regulated by the lattice ultraviolet cutoff ΛUV 1, and we only have to care about infrared divergences.∼ Each +∞ v 2 λ v (2v+2) vertex in (14) contributes either a power of m0 or a square SI [φx]= ξ SI [φx] , (11) − 8 of some combination of momenta q1 ...qf−1, coming from v=1 X the first or the second terms in (12), respectively. If our m2 S(2v+2) [φ ]= 0 Tr φ2v+2 + planar Feynman diagram is considered as a planar graph I x 2 (v + 1) x x drawn on the sphere, the number f is the number of faces X 2v+1 of this planar graph. We now take into account the iden- 1 + ( 1)l−1 ∆ Tr φ2v+2−lφl , (12) tity f l + v = 2 for the Euler characteristic of a planar 2 − xy x y − l=1 x,y graph. Applying now the standard dimensional analysis, X X we see that the above relation between f, l and v im- where in the last equation we have used our definition plies that WK can only contain the terms proportional 2 λ 2 2 4 2 m0 = 4 in order to formally eliminate the coupling λ to ΛUV , m0, m0/ΛUV and so on, probably times loga- (2v+2) 2 from interaction vertices S [φ ]. This construction rithmic terms log m0 . In Subsection IIIB we will I x ∼ is similar to the “shifted action” of [68]>. explicitly illustrate the importance of these logarithmic Perturbative expansion of any physical observable terms. No negative powers of m0 can ever appear. 2 λ [φx] is then organized as a formal power series ex- Remembering now the relation m0 = 4 , and taking hpansion O i in the auxiliary parameter ξ in (11): into account the powers of coupling associated with ver- tices, we immediately conclude that the weights of Feyn- M m man diagrams in our perturbation theory with the bare m λ [φx] = lim lim ξ m (m0) ,(13) mass m λ are proportional to positive powers of λ, h O i ξ→1 M→∞ − 8 O 0 m=0 ∼ m m X times possible logarithmic terms log (λ) , log (log (λ)) where M is the maximal order of the expansion. This (m> 0) and so on. From (12) we see that the combina- means that we ignore the relation between the mass torial pre-factor of vertex with 2v +2 legs grows at most 2 λ linearly with v. Since the number of planar diagrams m0 = 4 and the coupling constant λ and treat the terms (2v+2) which contribute to the expansion (13) at order m grows SI [φx] as bare vertices with 2v + 2 legs, counting v at most exponentially with m, and the weight of any dia- only those powers λ of the t’Hooft coupling constant gram is both UV and IR finite, contributions which grow which come from the vertex pre-factors in (11). The bare 2 factorially with m cannot appear in the expansion (13) mass m0 in the expansion coefficients in (13) is substi- in D = 2 dimensions. tuted with λ/4 at the very end of the calculation. The numerical results which we present below suggest Let us discuss the general structure of an expansion that (13) is a convergent weak-coupling expansion. At (13) for the free energy of the two-dimensional principal first sight this contradicts the common lore that pertur- chiral model (2) in the large-N limit, in which only con- bative expansion cannot capture the non-perturbative nected planar Feynman diagrams with no external legs physics of the model (2). Let us remember, however, contribute. Consider such a planar diagram with f 1 in- − that our expansion is no longer a strict power series in λ, dependent internal loop momenta q1,...,qf−1, and with which is known to be factorially divergent. Rather, the l bare propagators and v vertices. The kinematic weight 2 mass term m0 λ in bare propagators allows for log- of such a diagram can be written as arithms of coupling∼ in the series, in close analogy with resummation of infrared divergences in hard thermal loop 2 2 W d q ...d q − > K ∼ 1 f 1 perturbation theory [67] . It is easy to convince oneself Z that the formal expansion which contains powers of logs V1 ...Vv 2 2 , (14) of coupling along with the powers of coupling can in- (∆ (Q1)+ m0) ... (∆ (Ql)+ m0) corporate the non-perturbative scaling of the dynamical mass gap where Q1 ...Ql are momenta flowing through each dia- gram line, 1 m2 exp , (16) ∼ −β λ D−1 0 2 Qµ ∆ (Q)= 4 sin (15) where β0 is the first coefficient in the perturbative β- 2 µ=0 function. Indeed, we can rewrite the function (16) as a X convergent expansion in powers of z = log (λ): is the lattice Laplacian (6) in momentum space which 2 1 −1 −z behaves as ∆ (Q) Q at small momenta and V1 ...Vv exp = exp β e = ∼ −β λ − 0 are the weights of the vertices which in general depend 0 +∞ −k +∞ l +∞ on q1 ...qf−1. The momenta Q1 ...Ql are in general not ( β ) ( z) β−k kl = − 0 e−kz = − 0 . (17) independent and can be expressed as some linear combi- k! l! k! nations of q1,...,qf−1. All momenta belong to the Bril- Xk=0 Xl=0 Xk=0 louin zone Qµ [ π, π] of the square lattice. Thus any In what follows we apply this resummation prescrip- possible ultraviolet∈ − divergences in the integral (14) are tion to construct the infrared-finite weak-coupling expan- 6 sion for several models: exactly solvable large-N O (N) where in the last line A =0.252731 and B = 0.0795775. sigma model in D = 1, 2, 3 dimensions, exactly solvable If the mass gap m is small enough, the corresponding− large-N principal chiral model on one-dimensional lat- solutions for m read tices with L0 = 2, 3, 4 lattice sites, and, most impor- tantly, the principal chiral model in D = 2 dimensions at √λ2 +4 2, D = 1; zero and finite temperature. m2 (λ)= 32 exp −4π , D = 2; (21) λ m = B−1 (A 1/λ) , D = 3. | | − III. EXACTLY SOLVABLE EXAMPLES AND COUNTEREXAMPLES The goal of this Section is to construct the infrared- finite weak-coupling expansion following the prescription of Section II. In order to parameterize the fluctuations of A. Large-N O (N) sigma model in D = 1, 2, 3 dimensions the nax field around the classical vacuum with n0 x = 1, we use the stereographic projection from the unit sphere in N-dimensions to N 1-dimensional real space of fields − φix, i =1,...,N 1, which is the analogue of the Cayley map (3): −
1 λ φ2 √λ φ n = − 4 x , n = ix , (22) 0 x λ 2 ix λ 2 1+ 4 φx 1+ 4 φx
FIG. 1. Feynman diagrams which contribute to the two-point 2 correlator in the large-N limit of the O (N) sigma model. where φx φixφix. This is a unique map between ≡ i the unit sphereP embedded in N-dimensional space and Large-N O (N) sigma model provides one of the sim- the N 1-dimensional real space, in which only a single − plest examples of a non-perturbative quantum field the- point with n0 x = 1, nix = 0 is excluded. The integra- − ory which can be exactly solved by using saddle point tion measure on the unit sphere can be written in terms > approximation. In D = 2 dimensions this model has of the stereographic coordinates φix as [60] (see also dynamically generated non-perturbative mass gap, simi- Appendix B3): larly to the more complicated principal chiral models and non-Abelian gauge theories. On the other hand, for this λ −N n = φ 1+ φ2 . (23) model one can also explicitly construct the formal weak- D x D x 4 x coupling perturbative expansion, which is dominated by the “cactus”-type diagrams, schematically shown on Following the construction of Section II, we now include Fig. 1. These properties make this model an ideal test- this Jacobian in the action for the fields φ , which reads: ing ground for the infrared-finite weak-coupling expan- ix sion described in Section II. 1 λ The partition function of the O (N) sigma model is S [φ]= ∆ + δ (φ φ )+ 2 xy 2 xy x · y given by the path integral over N-component unit vec- x,y X tors nax, a = 0 ...N 1 attached to the sites of D- +∞ k+l k+l − ( 1) λ k l dimensional square lattice, which we label by x: + − ∆ φ2 φ2 (φ φ )+ 2 4k+l xy x y x · y k,l=0 · x,y N k+Xl=06 X = nx exp ∆xynax nay , (18) Z D −2λ +∞ k−1 k Z x,y,a ! ( 1) λ k X + − φ2 , (24) where ∆ is again the lattice Laplacian (6). The well- 4k k x x,y k=2 x known saddle-point solution shows that the two-point X X correlation function Gxy = nax nay is proportional to h i where φx φy φix φiy. We have also taken into the propagator of a free massive scalar field, with the · ≡ i · mass m being the solution of the gap equation: P 2 k 2 l account that the terms of the form ∆xy φx φy dDp 1 x,y λI (m)=1, I (m) . (19) vanish at large N due to factorizationP property 0 0 ≡ D ∆ (p)+ m2 Z (2π) (φx φy) (φz φt) = φx φy φz φt and transla- tionalh · invariance.· Ini fullh analogy· ih with· thei derivation of In D =1, 2, 3 dimensions the function I0 (m) is the action (7) in Section II, the Jacobian of the trans- −1/2 1 m2 formation from compact to non-compact field variables 2m 1+ 4 , D = 1; 2 2 results in the bare “mass term” m0 = λ/2. I0 (m)= 1 log m 1+ O m2 , D = 2; (20) 4π 32 A convenient way to arrive at the perturbative expan- − A + Bm+ , D = 3 , sion for the action (24) is to iterate the corresponding ··· 7
Schwinger-Dyson equations which is also the formal power series in the auxiliary pa- η (2 + η) rameter ξ in (28). In practice, we continue the iterations G−1 (p)=∆(p)+ m2 ∆ (p) up to some finite order M, and calculate m2 and z by 0 − 2 − (1 + η) summing up all the terms in the series (29) with powers λ η λ 1 dDq of ξ less than or equal to M. After such a truncation of ∆ (q) G (q) , − 2 1+ η − 2 (1 + η)3 D series, we substitute ξ = 1 and m0 λ/2. Z (2π) → D In D = 1 dimensions, our prescriptions yields the λ λ d p −M−2 2 series with summands of the form λk (8 + λ) and η φx = D G (p) , (25) ≡ 4 h i 4 −M−3/2 Z (2π) λk+1/2 (8 + λ) , k = 1, 2,..., times some integer- which in the large-N limit involve only the two-point valued coefficients which depend on M. The expansion function thus behaves regularly in the limit λ 0, where the mass → D goes to zero linearly in λ. d p ipµ(x−y) φx φy = e µ G (p) . (26) In D = 2 dimensions, we get the expansion which in- h · i D Z (2π) volves both powers of λ and log (λ) (see also [60]>): The form of equation (25) suggests that the solution has M l the form of the free scalar field propagator with the mass λ m2 (λ, M)= log m, times some wave function renormalization factor z: − 64 × z Xl=0 G (p)= . (27) M+min(l,1) ∆ (p)+ m2 σ λk, (30) × l,k Substituting this ansatz in (25), we obtain the following k=min(l+2,M) equations for m and z: X and similarly for z (λ, M). As discussed above, the ex- 2 2 2 λξ 2 λξ 2 m = m0z z + zm I0 (m) , pansion involving the logs of λ can reproduce the non- − 2 2 perturbative scaling (21) of the mass gap in D = 2. Fi- λξ 2 z =1+ z I0 (m) , (28) nally, in D = 3 dimensions we obtain the expansion in 4 positive powers of √λ. where we have again introduced the auxiliary parameter On Fig. 2 we compare the convergence of the expan- ξ, to be set to ξ = 1, in order to define the formal power sions in D =1, 2, 3 dimensions, plotting the relative er- 2 counting scheme. Substituting ξ = 1 and m0 = λ/2, ror we immediately arrive at the exact solutions (21) for the m (λ, M) m (λ) mass m. For z the exact solution is simply z = 2. ǫ (m (λ, M)) = − (31) m (λ) of the truncated series (29) with respect to the exact re- D=1 D=2 sult m (λ) given by (21). The values of the coupling λ are 0.2 D=3 fixed in such a way that the exact non-perturbative mass gap is equal to m = 0.4 for all dimensions D = 1, 2, 3. The series (29) exhibit the fastest convergence at D = 1. At D = 2, the convergence is not monotonic, and also ,M)) 0.1 λ slower than in D = 1. In D = 3 the convergence is (m(
ε slowest, and from the convergence plot in Fig. 2 it is dif- 0 ficult to say whether the series converge or not, even with up to 500 terms in the expansion. These results suggest that for the large-N O (N) sigma model D = 2 seems to be the critical dimension separating the convergent and -0.1 0 0.01 0.02 0.03 0.04 non-convergent expansions. 1/M
FIG. 2. Relative error (31) of the mass gap for the O (N) B. Non-perturbative mass gap in the 2D O (N) sigma model from the truncated expansion (29) as a function sigma model of truncation order M for D = 1, 2, 3. The coupling is fixed to yield the mass m (λ) = 0.4 in (21) for all D. We now turn to the physically most interesting ex- ample of the two-dimensional O (N) sigma model in the On the other hand, we can use the equations (28) for large-N limit, where the mass gap (21) exhibits non- an iterative calculation of the coefficients of the formal perturbative scaling m2 exp 1 similar to the one weak-coupling expansion ∼ − β0λ found also in two-dimensional principal chiral model and 2 2 2 2 m = m0 + λ ξ σ1 (m0)+ λ ξ σ2 (m0)+ ..., in non-Abelian gauge theories. On Fig. 3 we illustrate 2 2 z =1+ λξz1 (m0)+ λ ξ z2 (m0)+ ..., (29) the convergence of the series (30) for the mass m and 8
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 = 0 correspond to the exact results (21).
-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 1 λ=4.2 -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
⋆ FIG. 4. Relative error ǫ (m (λ, M)) of the effective mass m FIG. 5. The expansion truncation orders M1,2 (λ) at which obtained from the truncated series (29) at large values of M. the truncated series (30) have local extrema, plotted as func- tions of the inverse coupling constant λ−1. Solid lines are the fits of the form log (M ⋆ (λ)) = const − β/λ. the renormalization factor z to the exact results (21) and z = 2 for different values of the coupling constant λ. vergence rate of the truncated series. In particular, at To make the convergence at large truncation orders M M >M ⋆ the convergence to the exact value is rather more obvious, on Fig. 4 we also show the relative error 1 fast. Remarkably, we have found that M ⋆ as a function (31) of the truncated series (30) as a function of M in 1 of λ with good precision appears to be proportional to logarithmic scale for several values of λ. While at large the square of correlation length (in lattice units): values of λ the series converge with the precision of 10−16 at M 500, the convergence becomes slower at smaller 4π ∼ M ⋆ l2 m−2 exp . (32) values of λ - and, correspondingly, at smaller values of 1 ∼ c ∼ ∼ λ the mass m. ⋆ An interesting feature of the dependence of m (λ, M) For illustration, on Fig. 5 we plot log M1,2 (λ) ver- on M is that it is not monotonous, but rather has sev- sus 1/λ, together with the linear fits of the form log (M ⋆ (λ)) = const β/λ. For M ⋆ the best fit yields the eral extrema with respect to M. We denote the smallest − 1 value of M at which the dependence of m (λ; M) on M coefficient β = 12.49 which is very close to 4π = 12.57. ⋆ For M ⋆ we get the best fit value β = 9.33, which is has a minimum as M1 , the position of the next maximum 2 ⋆ ⋆ 3/2 of this dependence - as M2 , and so on. These positions close to 3π =9.42 and suggests the scaling M2 lc depend on λ and shift towards larger M at smaller λ. m−3/2. This might imply that at sufficiently small∼ λ the∼ ⋆ ⋆ ⋆ We can think of the extrema positions M1 ,M2 ,... as two extrema might merge. However, for M2 we have of some characteristic scales which determine the con- fewer data points, and those exhibit some tendency for 9 faster growth at larger values of 1/λ, thus the scaling ex- 0 ⋆ ponent for M2 might be different at asymptotically small λ. )) -1 M
The data presented on Figs. 4 and 5 provide a clear )> 0
g -2 numerical evidence that the infrared-finite weak-coupling 1 + expansion (30) indeed incorporates the non-perturbative 2 4π -3 dynamically generated mass gap m = 32exp λ . If we were to sample the series (30) using DiagMC− algo- (<1/N tr(g L =2, λ/λ =0.5
ε -4 0 c ⋆ ( λ λ rithm, the linear scaling of M1 (λ) with correlation length L0=3, / c=0.5 10 λ λ would also result in the expectable critical slowing down L0=4, / c=0.5
log λ λ -5 L0=2, / c=0.8 of simulations near the continuum limit. L =3, λ/λ =0.8 0 λ λc L0=4, / c=0.8 The series which we obtain from the field-theoretical -6 weak-coupling expansion (29) based on Schwinger-Dyson 0 0.2 0.4 0.6 0.8 1 equations (28) are in this respect drastically different 1/M from the simplest formal expansion (17) of the exact an- swer (21) in powers of logs. We have explicitly checked FIG. 6. Relative error of the weak-coupling expansion (13) for 1 † 0 that for this case the positions of the extrema of m with the mean link h N Tr g1g iM of the principal chiral model respect to M are independent of λ. In this respect our (2) on the one-dimensional lattices with L0 = 2, 3, 4 sites, as infrared-finite weak-coupling expansion (29) seems to be a function of truncation order M for fixed λ/λc = 0.5 and closer to the notion of trans-series. λ/λc = 0.8. Let us stress, however, that the series (30) are not trans-series yet, despite formal similarity with (1). In regime λ < λc are: a mathematically strict sense, resurgent trans-series (1) should give a “minimal” representation of a function 1 † λ Tr g g0 =1 , (L0 =2, λc = 4) (33) with optimal convergence. For the large-N O (N) sigma h N 1 i − 8 1 1 1 λ model, this optimal trans-series representation for the Tr g†g = + dynamically generated mass is given by a single term h N 1 0 i λ 2 − 8 − 2 4π 3 m = 32exp (see the exact solution (21)). Thus 2 − λ 1 λ the true trans-series for this model consist of only one 1 , (L0 =3, λc = 3). (34) −λ − 3 term, which is enough for convergence! The expansion (30) also converges to the exact result (21), but at a non- For L0 = 4 the weak-coupling solution is only available optimal rate 2. in the implicit form: 1 2 λ 2 8 Tr g†g = δ2, (λ = ), (35) h N 1 0 i λ − 8 − λ c π where δ is the solution of the equation C. Exactly solvable counter-example: finite chiral 8 2 2 2 πλ E 1 δ δ F 1 δ = 1 and E and F chain models are the complete− − elliptic− 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 the convergence of the infrared-finite weak-coupling ex- solved exactly× in D = 1 dimensions for lattice sizes pansion of Section II for small lattice sizes. Of course, L = 2, 3, 4 and L = + [69]>. The first and 0 0 for finite lattices the dimensional analysis following the the last cases can be reduced∞ to the Gross-Witten uni- equation (14) would not work, and loop integrals will only tary matrix model [35]>. For other two cases, solutions produce some rational functions of λ instead of logs. Nev- can be found in [69]>. For all values of L the one- 0 ertheless it is interesting to check whether these rational dimensional principal chiral model exhibits a third-order functions provide a good approximation to the exact re- Gross-Witten phase transition at some critical coupling sults (33), (34) and (35). To this end we have performed λ = λ (L ), which separates the strong-coupling and c 0 exact calculations of the coefficients of the weak-coupling the weak-coupling regimes. In particular, for the mean expansion (13), generating them recursively up to some link 1 Tr g†g the exact results in the weak-coupling h N 1 0 i maximal order M from the Schwinger-Dyson equations (49), to be presented in Subsection IV A. Such an exact calculation is possible for L0 = 2 up to M = 16 and for L0 = 4 up to M = 8. Explicit expression for the mean 1 † link N Tr g1g0 in terms of the correlators of φx ma- 2 h i We thank Ovidiu Costin for pointing out these properties of trices and the prescription for truncating the expansion trans-series to us at the “Resurgence 2016” workshop in Lisbon. at some finite order M are given in Subsection V A. 10
On Fig. 6 we illustrate the relative error of the trun- While in this work we will only consider the weak- cated weak-coupling expansion (13) for the mean link coupling expansion (13) based on the action (7) of the 1 † large-N principal chiral model (2), our construction of Tr g g0 M as a function of inverse truncation order h N 1 i the DiagMC algorithm is quite general and can be ap- 1/M for L0 =2, 3, 4, fixing λ/λc =0.5 and λ/λc =0.8. plied to both to the weak and strong-coupling expansions The relative error is defined in full analogy with (31). At in general large-N matrix field theories. We will briefly the first sight the convergence seems to be rather fast, outline the extension to non-Abelian gauge theories in however, a detailed inspection of numerical results re- Section VI. veals a tiny discrepancy of order of 0.1% which cannot be ascribed to numerical round-off errors, and which seems to stabilize at some finite value as M is increased (at least for L0 = 2). An analytic calculation of the few first orders revealed that the weak-coupling expansion (13) correctly reproduces only the three leading coefficients of the standard perturbative expansion in powers of λ, which is convergent for finite L . 0 A. Schwinger-Dyson equations for the large-N It seems thus that for finite-size one-dimensional lat- U (N) × U (N) principal chiral model with Cayley tices we observe the misleading convergence of the parameterization of U (N) group re-summed perturbative expansion to some unphysical branch of the solutions of Schwinger-Dyson equations. This phenomenon might happen even in zero-dimensional As in [4, 5, 8, 13, 14]>, the starting point for the con- models [68]> and was first noticed in Bold DiagMC sim- struction of our DiagMC algorithm are the Schwinger- ulations of interacting fermions [70]>. Remembering the Dyson equations, which we derive for the action (7) of fact that for the large-N O (N) sigma model in the infi- the principal chiral model (2). In contrast to the works nite space the expansion (13) converged with much higher [13, 14]> on the bosonic DiagMC, here we consider the precision than the discrepancy which we observe here for full untruncated hierarchy of Schwinger-Dyson equations finite-size chiral chains (compare Figs. 6 and 4), we con- for multi-trace, in general disconnected, correlators of the clude that infinite lattice size and the presence of log (λ) φ fields defined in (3): terms seem to be crucial for the convergence to the cor- x rect physical result. An important extension of our anal- ysis, which we relegate for future work, would be to ex- tend the sufficient conditions of [68]> for the convergence to the physical result to the weak-coupling expansion 1 1 2 2 r r x1,...,xn1 x1,...,xn2 ... x1,...,xnr (13). h i≡ 1 1 Tr φx1 ...φx1 Tr φx2 ...φx2 ... ≡ N 1 n1 N 1 n2 IV. SCHWINGER-DYSON EQUATIONS AND 1 ... Tr φxr ...φxr .(36) DIAGMC ALGORITHM FOR U (N) × U (N) N 1 nr PRINCIPAL CHIRAL MODEL
The aim of this Section is to construct a DiagMC al- gorithm for sampling the weak-coupling expansion (13).
To derive the Schwinger-Dyson equations for the field correlators (36), let’s consider the full derivative of the form
δikδjl ∂ φ φ 1 ...φ 1 Tr φ 2 ...φ 2 ... Tr φ r ...φ r exp( S [φ] S [φ]) =0, (37) r x2 xn x1 xn x1 xnr F I N D ∂φx ij 1 kl 2 − − Z where we have explicitly separated the action (7) into the quadratic and interacting parts given by (10) and (11), (12) to simplify the derivation and notation. We now expand the derivative according to the Leibniz product rule, 0 2 −1 contract matrix indices and finally contract the result with the bare propagator G 1 = ∆+ m coming from x1 x 0 xy the quadratic part SF [φx] of the action (7). This yields the following infinite system of equations for the multi-trace observables (36):
∂S [x , x ] = G0 + G0 I , x , (38) h 1 2 i x1x2 x1xh ∂φ 2 i x x X 11
1 1 2 2 r r 0 2 2 r r x1, x2 x1,...,xn ... x1,...,xn = G 1 1 x1,...,xn ... x1,...,xn + h 2 r i x1x2 h 2 r i 0 ∂SI 1 2 2 r r + G 1 , x2 x1,...,xn ... x1,...,xn + x1xh ∂φ 2 r i x x X r ns 1 0 2 2 s s 1 s s r r + G 1 s x1,...,xn ... x1,...,xp−1, x2, xp+1,...,xn ... x1,...,xn (39) N 2 x1xp h 2 s r i s=2 p=1 X X
x1,...,x1 x2,...,x2 ... xr,...,xr = h 1 n1 1 n2 1 nr i 0 1 1 2 2 r r = G 1 1 x3,...,xn x1,...,xn ... x1,...,xn + x1x2 h 1 2 r i 0 1 1 2 2 r r 1 1 +Gx x x2,...,xn1−1 x1,...,xn2 ... x1,...,xnr + 1 n1 h i n−2 0 1 1 1 1 2 2 r r + G 1 1 x2,...,xu−1 xu+1,...,xn x1,...,xn ... x1,...,xn + x1xu h 1 2 r i u=3 X 0 ∂SI 1 1 2 2 r r + G 1 , x2,...,xn x1,...,xn ... x1,...,xn + x1xh ∂φ 1 2 r i x x r ns X 1 0 2 2 s s 1 1 s s r r + G 1 s x1,...,xn ... x1,...,xp−1, x2,...,xn , xp+1,...,xn ... x1,...,xn , (40) N 2 x1xp h 2 1 s r i s=2 p=1 X X where we have used red and violet colors to highlight the field arguments x and the single-trace terms which are different on the left- and the right-hand sides. By ∂SI we have schematically denoted the terms which are produced ∂φx by applying the derivative operator in (37) to the interacting part (11) of the action (7):
∞ ∂S [φ] + λ v ∂S(2v+2) [φ] I = ξv I , ∂φ − 8 ∂φ x v=1 x X − ∂S(2v+2) [φ] 2v+1 2v+1 l I = m2φ2v+1 + ( 1)l−1 ∆ φmφl φ2v+1−l−m. (41) ∂φ 0 x − xy x y x x m=0 y Xl=1 X X
These terms contain three or more φ fields and should with single-trace correlators [x1,...,xn] obeying the be inserted into the single-trace expectation values (36) much simpler non-linear equationh i within the equations (38), (39) and (40): ∂SI [φ] y = ∂φx [x ,...,x ] = h i h 1 n i 1 ∂SI [φ] h i 0 0 Tr φy , and so on. For the sake of brevity, = Gx1x2 [x3,...,xn] + Gx1x [x2,...,xn−1] + h N ∂φx i h i n h i n−2 we omit the matrix indices of φx, so that all products 0 of φ are understood as matrix products. In particular, + Gx x [x2,...,xu−1] [xu+1,...,xn] + x 1 u h ih i this implies that φ and φ do not commute for x = y. u=3 x y 6 X In the t’Hooft large-N limit, the last summands in the 0 ∂SI + G , x2,...,xn (43). equations (39) and (40) can be omitted, since they are x1xh ∂φ i x x suppressed by a factor 1/N 2 and all multi-trace correla- X tors (36) are of order N 0. Since in this limit the corre- The form of the Schwinger-Dyson equation (38) for the lators (36) with odd numbers of φx fields within some of two-point correlator does not change. the traces, such as [x][y] 1 Tr (φ ) 1 Tr (φ ) , are Schwinger-Dyson equations for large-N quantum field h i ≡ h N x N y i suppressed as 1/N 2, we can also limit the equations to theories are almost always considered in the nonlinear the space of multi-trace correlator which contain an even form (43), which is much more compact than the lin- number of φ fields within each trace. Furthermore, in the ear equations (40). One of the well-known examples of t’Hooft large-N limit the equations (38), (39) and (40) such non-linear Schwinger-Dyson equations in the large- 2 N limit are the Migdal-Makeenko loop equations for without the 1/N -suppressed terms admit the large-N > factorized solution non-Abelian gauge theories [71, 72] . Likewise, even at finite N one can also arrive at a more compact non- linear representations of the Schwinger-Dyson equations x1,...,x1 ... xr,...,xr = h 1 n1 1 nr i (38), (39) and (40) by rewriting them in terms of con- = x1,...,x1 ... xr,...,xr (42) nected or one-particle-irreducible correlators. This strat- h 1 n1 i h 1 nr i 12 egy is numerically very efficient for approximate solutions resented as based on some truncations of the full set of equations ∂S(2v+2) [φ] 1 ∂S(2v+2) [φ] (especially for correlators with small numbers of fields), I eipx I = ∂φ ≡ V ∂φ and is commonly used for numerical solutions of QCD p x x Schwinger-Dyson equations [73, 74]> and in condensed X > = δ (p q ... q ) matter physics [1, 14, 75] . However, Schwinger-Dyson − 1 − − 2v+1 × q ...q equations are always linear equations when written in 1 X2v+1 terms of disconnected correlators of the form (36). This V (q1,...,q2v+1) φq1 ...φq2v+1 . (46) × linear form is very convenient for the construction of Di- agMC algorithms, which we consider in the next Subsec- An important property of the vertex functions (45) tion. is that they are positive for all values of q1,...,q2v+1. While we don’t have an explicit proof of this statement, To proceed towards the DiagMC algorithm, we go to in our Monte-Carlo simulations with over 1014 Metropolis the momentum space, and define the momentum-space updates in total we have explicitly checked that the val- field operators and correlators as ues of vertex function were always positive. Furthermore, 1 in Appendix A we demonstrate that if all the momenta φ = eipxφ , φ = e−ipxφ . (44) p V x x p in (45) are small and the lattice Laplacian can be ap- x p X X proximated as ∆ (p) = p2, the vertex functions take the Note that we have assumed the finite lattice volume V , simple form which will be convenient for further analysis. In particu- V (q ,...,q )= m2 + (q + q + ... + q )2 (47), lar, this allows us to interpret all the delta-functions with 1 2v+1 0 1 3 2v+1 momentum-valued arguments as Kronecker deltas on the which manifestly demonstrates their positivity. discrete set of lattice momenta. For brevity, we will de- Furthermore, in order to define the power counting note δ (p) δp,0. We will see in what follows that the al- scheme to be used in our DiagMC simulations, we ex- ≡ gorithm performance is practically volume-independent, pand the momentum-space correlators of φ fields in for- thus one can make a smooth transition from discrete to mal power series in our auxiliary parameter ξ: continuous momenta if necessary. 1 1 It is also convenient to introduce the momentum-space Tr φp1 ...φp1 ... Tr φpr ...φpr = vertex functions h N 1 n1 N 1 nr i +∞ m λ V (q ,...,q )= m2 + = ξm 1 2v+1 0 − 8 × 2v+1 2v+1−l m=0 l−1 1 1 X r r + ( 1) ∆ (q + ... + q ) , (45) p ,...,p ... p ,...,p m.(48) − m+1 m+l ×h 1 n1 1 nr i l=1 m=0 X X In terms of the correlators (48) of the momentum-space so that the Fourier transform of the derivative (41) of fields φp, the large-N limit of the Schwinger-Dyson equa- the interacting part of the action can be compactly rep- tions (38), (39) and (40) read:
δ (p + p ) [p ,p ] = δ 1 2 G (p ) (49a) h 1 2 im m,0 V 0 1 − m G (p ) δ (p q ... q ) V (q ,...,q ) [q ,...,q , p ] − , (49b) − 0 1 1 − 1 − − 2v+1 1 2v+1 h 1 2v+1 2 im v v=1 q1,...,q2 +1 X Xv
p1,p1 p2,...,p2 ... pr,...,pr = h 1 2 1 n2 1 nr im 1 1 δ p1 + p2 1 2 2 r r = G0 p1 p1,...,pn2 ... p1,...,pnr m (49c) V h i − m G p1 δ p1 q ... q V (q ,...,q ) − 0 1 1 − 1 − − 2v+1 1 2v+1 × v=1 q1,...,q2 +1 X Xv 1 2 2 r r q ,...,q , p p ,...,p ... p ,...,p − , (49d) ×h 1 2v+1 2 1 n2 1 nr im v 13
p1,p1,...,p1 p2,...,p2 ... pr,...,pr = h 1 2 n1 1 n2 1 nr im 1 1 δ p1 + p2 1 1 1 2 2 r r = G0 p1 p3,...,pn1 p1,...,pn2 ... p1,...,pnr m + (49e) V h i 1 1 δ p1 + pn1 1 1 1 1 2 2 r r + G0 p1 p2,p3,...,pn1−1 p1,...,pn2 ... p1,...,pnr m + (49f) V h i n1−2 δ p1 + p1 + 1 A G p1 p1,...,p1 p1 ,...,p1 p2,...,p2 ... pr,...,pr (49g) V 0 1 h 2 A−1 A+1 n1 1 n2 1 nr im − A=4 X m G p1 δ p1 q ... q V (q ,...,q ) − 0 1 1 − 1 − − 2v+1 1 2v+1 × v=1 q1,...,q2 +1 X Xv 1 1 1 2 2 r r q ,...,q , p ,p ,...,p p ,...,p ... p ,...,p − , (49h) ×h 1 2v+1 2 3 n1 1 n2 1 nr im v where we have again used red and violet colors to highlight the individual momenta, order labels and momentum subsequences in correlators (48) which are different on the right- and left-hand sides. We have also introduced the bare propagator in momentum space
1 2 λ G0 (p)= 2 , m0 . (50) ∆ (p)+ m0 ≡ 4
ically illustrated on Fig. 7. It is instructive to explicitly calculate the first per- turbative correction to the two-point correlator, which can be readily found from the Schwinger-Dyson equa- tions (49) for the two-point correlator:
δ (p + p ) [p p ] = G2 (p ) 1 2 Σ (p ) , (51) h 1 2 i1 0 1 V 1 1 where
Σ1 (p)= 1 = (V (q, q,p)+ V (p, q, q)) G (q)= −V − − 0 q X 2 = G (q) FIG. 7. An example of the recursive calculation of dia- −V 0 × q grams contributing to order m = 2 two-point correlator X from Schwinger-Dyson equations (49). Combinatorial dia- λ +2∆(p)+2∆(q) ∆ (p q) . (52) gram weights are not explicitly shown in order not to clutter × 4 − − the plot. In particular, Σ1 (p =0)= 2. If we interpret this quan- tity as the first-order correction− to the self-energy, then It is easy to convince oneself that recursive solution of at zero momentum this correction, being multiplied by ( λ/8) and ξ = 1, exactly cancels the bare mass term the Schwinger-Dyson equations (49) generates the con- − ventional weak-coupling expansion, with the auxiliary λ/4! Of course, this cancellation requires re-summation parameter ξ formally serving as a coupling. For exam- of the infinite chain of bubble diagrams, and at any fi- ple, to find the order m = 2 contribution to the two- nite order of the expansion all diagram weights are still infrared-finite. point correlator [p1,p2] , we use the first equation in (49) to express ith in termsi of four-point correlator of or- der m = 1 and the six-point correlator of order m = 0. The four-point correlator at order m = 1 can be related B. Metropolis algorithm for stochastic solution of to two-point correlator at order m = 1 and the six-point linear equations correlator at m = 0 using the last equation in (49). Fi- nally, the two-point correlator at order m = 1 is again As discussed in the previous Subsection, Schwinger- related to four-point correlator at m = 0 by virtue of the Dyson equations in terms of disconnected field correlators first equation in (49). This chain of relations is schemat- always form an infinite system of linear equations, which 14 can be written in the following general form Let us also note that since for Schwinger-Dyson equa- tions the generalized indices X take infinitely many val- φ (X)= b (X)+ A (X Y ) φ (Y ) , (53) ues, in practice histogramming of all possible values of | Y X is not possible, but also not necessary. E.g. if one is X interested only in the momentum-space two-point corre- where A (X Y ) is some theory-dependent linear op- lator 1 Tr (φ φ ) = [p ,p ] which corresponds to erator, b (X|) represents the “source” terms in the N p1 p2 1 2 X ofh the form X = i p h,p , onei can histogram only Schwinger-Dyson equations (typically, the contact delta- 1 2 the momenta p and{{p whenever}} X contains only two function term in the lowest-order equation, like in 1 2 of them. Thus no field correlators should be explicitly (38) and (49), and X and Y are generalized in- saved in computer memory, and the memory required by dices incorporating all arguments of field correlators the algorithm does not directly scale with system volume which enter the Schwinger-Dyson equations. For the (however, it does scale with the maximal expansion order, Schwinger-Dyson equations (49) discussed in the pre- which might be correlated with the volume via required vious Subsection, X and Y are sequences of momenta statistical error). p1,...,p1 ,..., pr,...,pr which parameterize 1 n1 1 nr In order to simplify the notation in what follows, let the multi-trace correlators (48) with an additional order- us also define the quantities counting integer variable m. Note that the set of X in- dices is in most cases infinite, which excludes the applica- (Y )= A (X Y ) , = b (X) . (56) tion of standard numerical linear algebra methods to the N | | | Nb | | equation (53) (see however [76]> for an attempt in this XX XX direction). Bringing the Schwinger-Dyson equations into In this work we propose to sample the sequences the form (53) can be subtle for field theories for which with probability (55) by using the Metropolis-HastingsS the free (quadratic) part of the action has flat directions algorithm with the following updates: or vanishes, but these subtleties are not relevant for the present work. Algorithm IV.B.1 The solution φ (X) of (53) can be written as formal geometric series in A: Add index: With probability p+ a new generalized in- dex Xn+1 is added to the sequence Xn,...,X0 , +∞ { } where the probability distribution of Xn+1 is φ (X)= ... δ (X,Xn) × π (Xn+1 Xn) = A (Xn+1 Xn) / (Xn). The sign n=0 X0 Xn | | | | N X X X variable σ is multiplied by sign (A (Xn+1 Xn)). A (X X − ) ...A (X X ) b (X ) . (54) | × n| n 1 1| 0 0 Remove index: If the sequence contains more than one If the diagrammatic expansion which is obtained by it- generalized index, with probability 1 p+ the last erating Schwinger-Dyson equations is truncated at some generalized index X is removed from− the sequence, finite order, there is only a finite number of terms in these n which thus changes from X ,X − ,...,X to series, which correspond to a finite number of diagrams { n n 1 0} Xn−1,...,X0 . The sign variable σ is multiplied contributing at a given expansion order. { } by sign (A (Xn Xn−1)). The key idea of the DiagMC algorithm which we devise | in this work is to evaluate the series (54) by stochasti- Restart: If the sequence contains only one generalized cally sampling the sequences of generalized indices = ′ S index X0, it is replaced by X0 with probability Xn,...,X0 with arbitrary n in (54) with probability ′ 1 p+. The probability distribution of X0 is { } − ′ ′ π (X0) = b (X0) / b. The sign variable σ is set A (Xn Xn−1) ... A (X1 X0) b (X0) | ′ | N w ( )= | | | | | || |,(55) to sign (b (X )). S 0 Nw These updates should be accepted or rejected with the where w is the appropriate normalization factor. > TheN solution φ (X) to the system (53) can be found probability [77] by histogramming the last generalized index Xn in ′ ′ ′ w ( ) π ( ) the sequence Xn,...,X0 , where each occurrence α ( ) = min 1, S S → S . (57) { } S → S w ( ) π ( ′) of Xn is weighted with the sign σ (Xn,...,X0) = S S → S sign (A (X X − )) ... sign (A (X X )) sign (b (X )). n| n 1 1| 0 0 Explicit calculation yields the following acceptance prob- This sign reweighting puts certain limitations on the abilities for the three updates introduced above: use of the method, since we are effectively sampling the series in which the linear operator A (X Y ) is replaced (Xn) (1 p+) by A (X Y ) and which typically have a| smaller radius αadd = min 1, N − , p+ of convergence.| | | In the worst case, the expectation value p+ of the reweighting sign σ (Xn,...,X0) can decrease αremove = min 1, , (X − ) (1 p ) exponentially or faster with n, thus limiting the maximal N n 1 − + expansion order which can be sampled by the algorithm. αrestart =1. (58) 15
After some algebraic manipulations the overall average the solution φ (X) of the linear equations (53). Combin- acceptance rate can be expressed as ing (55) and (53), we obtain a linear equation for : Nw b α = (1 p+) N + w = w w (X)= h i − w N N N XX +2 min (p+, (1 p+) (Xn)) , (59) = A (X Y ) w (Y )+ b (X) = h − N i Nw | | | | | where in the last line the expectation value is taken X,YX XX with respect to the stationary probability distribution = w (Y ) w (Y )+ b, (61) of Xn. For the optimal performance of the algorithm N N N XY it is desirable to choose p+ (which is the only free pa- rameter in our algorithm) in such a way that the ac- where w (X) is the probability distribution of the last element X in the sequences , with any sign σ. The ceptance rate is maximally close to unity. To this n S end we first perform Metropolis updates with some solution of this equation can be written as initial value of p+, estimate the expectation values ′ ′ b min p+, 1 p+ (Xn) in (59) for several trial val- w = N , (62) h ′ − N i N 1 (Xn) ues p and then select the value for which the acceptance − h N i + is maximal. This process is repeated several times until where again the expectation value is taken with respect to the values of the average acceptance and p+ stabilize. the stationary probability distribution of the Metropolis Sometimes the transformations Xn Xn+1 can be algorithm IV.B.1. → joined into groups of very similar transformations. For The general implementation of the Metropolis algo- example, if we have to generate a random loop momen- rithm IV.B.1 for which the user-defined matrix elements tum when sampling the Feynman diagrams in momen- A (X Y ) and the source vector b (X) should be provided tum space, in the notation of Algorithm IV.B.1 gener- is available| as a part of GitHub repository [78]>. ating every single possible momentum would be a sep- arate transformation, but it is more convenient to de- scribe them as a single group of transformations. De- C. DiagMC algorithm for the large-N U (N) × U (N) noting possible outcomes of such transformations as principal chiral model X (X ), where (X ) labels the group of trans- n+1 ∈ T n T n formations, we can express the probability distribution In this Subsection we adapt the general Metropolis al- of the new element Xn+1 in the “Add index” update gorithm described in the previous Subsection IVB for in (IV.B.1) as the product of the conditional probabil- solving the linear Schwinger-Dyson equations (49) ob- ity π (Xn+1 (Xn)) to generate the new element Xn+1 tained from the action (7) of the large-N principal chiral |T by one of the transformations in (Xn), times the over- model (2). The generalized indices X will take values in T all probability π ( Xn) to perform any transformation the space of partitioned sequences of momenta from : T | T X = p1,...,p1 ,..., pr,...,pr , (63) 1 n1 1 nr m π (Xn+1 Xn)= π (Xn+1 (Xn)) π ( Xn) , | |T T | additionally labelled with an expansion order m = π (X (X )) = A (X X ) / T (X ) , n+1|T n n+1| n N n 0, 1, 2,.... These sequences will be generated T (Xn)= A (Xn+1 Xn) , with the probability proportional to the correlators N | 1 1 r r Xn+1∈T p ,...,p ... p ,...,p defined in (48), up to X h 1 n1 1 nr im π ( Xn)= T (Xn) / (Xn) , the sign re-weighting with the sign variable σ = 1, in- T | N N troduced in algorithm IV.B.1. ± (X )= T (X ) . (60) N n N n The system of equations (49) already has the form (53). TX(Xn) The only non-zero elements of the source vector b (X) Such grouping of transformations will be very convenient correspond the delta-function term (in the right-hand for a compact description of our DiagMC algorithm in side of line (49a)) in the first, inhomogeneous Schwinger- what follows. Sometimes we will also omit the arguments Dyson equation in (49): of T (X ) and (X ) for the sake of brevity. N n T n b ( p, p )= δm,0G0 (p) /V. (64) Finally, let us calculate the normalization factor w {{ − }} relating the (sign weighted) histogram of the last gener-N From equations (49) we can also directly read off the fol- alized index X in the sequences = X ,...,X and lowing groups of nonzero elements of the A (X Y ) matrix: n S { n 0} |
A p1,p1 , p2,...,p2 ,..., pr,...,pr p2,...,p2 ,..., pr,...,pr = 1 2 1 n2 1 nr m | 1 n2 1 nr m 1 1 1 G0 p δ p + p = 1 1 2 , (from the term (49c)) (65a) V 16
A p1,p1,p1,...,p1 ,... pr,...,pr p1,...,p1 ,... pr,...,pr = 1 2 3 n1 1 nr m | 3 n1 1 nr m 1 1 1 G0p1 δ p1 + p2 = , n1 4, (from the term (49e)) (65b) V ≥ A p1,p1,...,p1 , p1 ,... pr,...,pr p1,...,p1 ,..., pr,...,pr = 1 2 n1−1 n1 1 nr m | 2 n1−1 1 nr m 1 1 1 G 0 p1 δ p1 + pn1 = , n1 4, (from the term (49f)) (65c) V ≥ A p1, ..., p1 , ...,p1 , p2,...,p2 ,... pr,...,pr 1 A n1 1 n2 1 nr m | p1,...,p1 , p1 ,...,p1 , p2,...,p2 ,... pr,...,pr = | 2 A− 1 A+1 n1 1 n2 1 nr m 1 1 1 G0 p1 δ p1 + pA = , A =4, 6,...,n1 2, n1 6, (from the term (49g)) (65d) V − ≥ 1 1 1 r r 1 1 r r A p ,p ,...,p ,..., p ,...,p q ,...,q ,p ,...,p ,..., p ,...,p − = 1 2 n1 1 nr m | 1 2v+1 2 n1 1 nr m v = G p1 δ p1 q ... q V (q ,...,q ) . (from the terms (49b), (49d),(49h)) (65e) − 0 1 1 − 1 − − 2v+1 1 2v+1 Here we have used red and violet colors to highlight the individual momenta and order labels and momentum subse- quences which are different for the first and the second indices of the A (X Y ) matrix. |
In the above representation we write down all nonzero obtain a suitable representation of the matrix A (X Y ), elements of the matrix A (X Y ) for the fixed first in- we re-label the matrix elements (65), assuming that| the dex X. However, for the “Add| index” update in the second index Y in A (X Y ) has the most general form Algorithm IV.B.1 we need to know all nonzero elements Y = p1,...,p1 ,...,| pr,...,pr , and express 1 n1 1 nr m A (Xn+1 Xn) for the fixed second index Xn. In order to the momenta in the first index X in terms of these mo- | a menta pi . This gives
A p, p , p1,...,p1 ,..., pr,...,pr p1,...,p1 ,..., pr,...,pr = { − } 1 n1 1 nr m | 1 n1 1 nr m G (p) = 0 , (from the term (49c)) (66a) V
A p, p,p1,...,p1 ,... pr,...,pr p1,...,p1 ,... pr,...,pr = − 1 n1 1 nr m | 1 n1 1 nr m G (p) = 0 , (from the term (49e)) (66b) V
A p,p1,...,p1 , p ,... pr,...,pr p1,...,p1 ,..., pr,...,pr = 1 n1 − 1 nr m | 1 n1 1 nr m G (p) = 0 , (from the term (49f)) (66c) V
A p,p1,...,p1 , p,p2,...,p2 , p3,...,p3 ,... pr,...,pr 1 n1 − 1 n2 1 n3 1 nr m | p1,...,p1 , p2,...,p2 , p3,...,p3 ,..., pr,...,pr = | 1 n1 1 n2 1 n3 1 nr m G (p) = 0 , r 2, (from the term (49g)) (66d) V ≥
A p1 + p1 + ... + p1 , p1 ,...,p1 ,..., pr,...,pr 1 2 2v+1 2v+2 n1 1 nr m+v | p1, ...,p1 ,...,pr,...,pr = | 1 n1 1 nr m = G p1 + p1 + ... + p1 V p1,...,p1 , 2v +1 The matrix elements (66a) - (66d) form sets of elements parameterized by the momentum variable p. We denote 17 1 1 the normalization factor T (X ) in (60) for these sets p ,...,p as the “topmost” or “first” subsequence, N n 1 n1 as the subsequence p2,...,p2 - as the “second” sequence 1 n2 1 and so on. Σ0 = G0 (p) , (67) Interestingly, the fact that the random sampling pro- V p X cess operates on the stack implies the factorization of probabilities for subsequences in the stack [58, 79, 80]>, where Σ has an obvious interpretation of the one-loop 0 which corresponds to factorization of single-trace corre- self-energy. Correspondingly, the probability distribution lators in the large-N limit. If one proceeds keeping the of the new momenta p is 1/N 2 terms in the equations (39) and (40), our parti- π0 (p) G0 (p) / (V Σ0) . (68) tioned momentum sequences will no longer have stack ≡ structure. In particular, the fourth line of the Schwinger- At the first sight it might seem that the Metropolis Dyson equations (40) would correspond to a transforma- algorithm (IV.B.1) would require keeping in memory all tion where the topmost momentum subsequence is split the generalized indices X0,...,Xn which enter the series into two subsequences, one of which is inserted into arbi- (54). However, this is not necessary in practice, since trary subsequence in the partitioned sequence (63) [81]>. the matrix A (X Y ) for the Schwinger-Dyson equations Obviously, the number of ways to do such a splitting (49) is very sparse,| as one can see from the equations and insertion grows polynomially both in the number of (66). Also, only a few momenta in the generalized in- subsequences and their lengths. This polynomial growth 1 1 r r dex Xn = p1,...,pn1 ,..., p1,...,pnr are differ- of the number of terms in the Schwinger-Dyson equa- ent between the first and the second generalized indices of tions results in the factorial growth of the number of > the matrix A (Xn+1 Xn). Thus only a few momenta will non-planar Feynman diagrams with order [59, 81] . In be changed when generating| the new generalized index contrast, the matrix elements (66) do not depend on the Xn+1 from Xn. Therefore it is sufficient to keep in com- number and lengths of subsequences, so that the algo- puter memory only the topmost sequence Xn, as well as rithm samples a much smaller space of planar diagrams, the sequence of transformations Xn Xn+1 which led to the number of which grows only exponentially with ex- the current sequence, along with a few→ parameters which pansion order. This is the reason why Diagrammatic characterize these transformations. In fact, only the in- Monte Carlo should be in general much more efficient in formation which is necessary to recover Xn from Xn+1 in the large-N limit. the “Remove index” update in Algorithm IV.B.1 should Since the generalized indices X take an infinite num- be stored. ber of values, it is impossible but also not necessary to We note also that the transformations Xn Xn+1 keep the matrix elements A (X Y ) in computer memory. which correspond to matrix elements (66a), (65b→), (65c), Rather, they can be implemented| as a function which, (66e) involve only the first momentum subsequence given the topmost index Xn in the series (54), returns p1,...,p1 in (63), and the transformation corre- only nonzero values of A (X X ) and the correspond- 1 n1 n+1| n sponding to (66d) involves also the second subsequence ing set of Metropolis proposals Xn Xn+1. The C code 2 2 → > p1,...,pn2 . This naturally endows the partitioned se- of the Metropolis algorithm IV.B.1 available at [78] re- quences (63) with the stack structure, in the algorithmic lies exactly on such a “functional” implementation of the sense. Correspondingly, we will refer to the subsequence matrix A (X X ). n+1| n With all these considerations in mind, from (66) we can finally deduce the following set of transformations X X in the “Add index” update of Algorithm IV.B.1: n → n+1 Transformation set IV.C.1 Push a new pair of momenta to the stack, from matrix elements (66a): A new subsequence containing a pair of momenta p, p is pushed to the top of the momentum stack. • { − } The probability distribution for p is π0 (p), defined in (68). The order m and the sign σ do not change. • The corresponding term in the Schwinger-Dyson equations is (49c). • The overall weight for these transformations, summed over all p, is T = A (Xn+1 Xn) =Σ0, • N Xn+1∈T | | | using the same notation as in (60). P To recover X back from X , the topmost subsequence should be removed from the stack configuration • n n+1 Xn+1. No additional information should be saved for this backward step. Add new momenta to the topmost subsequence, from matrix elements (66b), (66c): 18 A new pair of momenta is inserted into the topmost subsequence p1,...,p1 , either a) at the beginning • 1 n1 as p, p, p1,...,p1 , or b) at the beginning and at the end as p, p1,...,p1 , p . The probability − 1 n1 1 n1 − distribution for p is again π0 (p). The order m and the sign σ do not change. • The corresponding terms in the Schwinger-Dyson equations (49) are (49e), (49f). • The overall weights for each of the transformations a) and b) are T =Σ . • N 0 To recover Xn back from Xn+1, either the two first momenta, or the first and the last momenta should be • removed from the topmost subsequence in the stack. Correspondingly, one should remember whether the transformation a) or b) was chosen. Merge two topmost subsequences, from matrix elements (66d): If there are two or more momentum subsequences in X, they are merged and interleaved with a pair of • momenta p, p, where p is generated with probability distribution π (p): − 0 p1,...,p1 , p2,...,p2 ,... p, p1,...,p1 , p,p2,...p2 ,... . (69) 1 n1 1 n2 → 1 n1 − 1 n2 The order m and the sign σ do not change. • The corresponding term in the Schwinger-Dyson equations (49) is (49g). • The overall weight for this transformation, summed over all p, is T =Σ . • N 0 To recover X back from X , one should split the topmost subsequence in the stack, which now has the • n n+1 form of the right-hand side of (69), at the position n1 + 1, and remove the first momenta from the resulting subsequences. Correspondingly, the length n1 of the topmost subsequence in Xn should be stored before the transformation X X . n → n+1 Create (2v + 2)-leg vertex, from matrix elements (66e): If there are four or more momenta in the topmost momentum subsequence in Xn, replace the first 2v + 1, • v =1 ...n /2 1 momenta by a single momentum equal to their sum: 1 − p1,p1,...,p1 , p1 ,...,p1 ,... p1 + p1 + ... + p1 , p1 ,...,p1 ,... . (70) 1 2 2v+1 2v+2 n1 → 1 2 2v+1 2v+2 n1 The order m is increased by v, and the sign σ is multiplied by 1. • − The corresponding terms in the Schwinger-Dyson equations (49) are (49b), (49d) and (49h). • The weight of this transformation for each particular v is • A (X X )= G p1 + ... + p1 V p1,...,p1 . (71) n+1| n − 1 2v+1 1 2v+1 To recover Xn back from Xn+1, one should replace the first momentum in the topmost subsequence in • 1 1 1 1 the stack by the momenta p1,...,p2v+1. To this end, the vertex order v and the 2v momenta p1,...,p2v 1 should be stored. The momentum p2v+1 can be recovered from momentum conservation constraint. Restart, from the source vector (64): If the “Restart” update is selected in Algorithm IV.B.1, the new value of X0 has a single subsequence with • just two momenta: X′ = p, p , where p is distributed with probability π (p). 0 {{ − }} 0 The order m is set to m = 0, and the sign variable to σ = +1. This state is also the initial state for the • algorithm. The corresponding term in the Schwinger-Dyson equations (49) is the term on the right-hand side of line • (49a). ′ The norm of the source vector is b = b (X0)=Σ0, which can be used to recover the overall normalization • N ′ X0 factor for field correlators from (62). P The sequence of updates Xn Xn+1 in the Metropo- ing planar, in general disconnected Feynman diagrams lis algorithm IV.B.1 with transformations→ IV.C.1 can be with arbitrary number of external legs. The transforma- visually interpreted as a step-by-step process of draw- tions IV.C.1 are used in the “Add index” update and 19 correspond to adding a randomly chosen new element with volume. to the diagram. The first three transformations in the set IV.C.1 correspond to drawing free propagator lines The only reason for working in a finite volume is that with different positions of endpoints. The last transfor- this simplifies the histogramming of correlators of gx vari- mation in the set IV.C.1 draws an interaction vertex with ables (original U (N)-valued variables in the partition 2v + 2 legs by joining 2v + 1 external lines into a single function (2)), see Subsection V A for a more detailed external line. It is also this transformation which turns discussion. The performance of the algorithm, however, “Remove index” external lines into internal ones. The practically does not depend on the lattice size as long update in the Metropolis Algorithm IV.B.1 erases the di- as it is much larger than the correlation length. Rather, agram element which was drawn last, and can be thought the dynamically generated mass gap in the principal chi- of as an “Undo” operation. The initial state of the algo- ral model (2) effectively sets the infrared scale for the rithm is just a single free propagator line, which is the algorithm. Needless to say, finite lattice size L0 in the simplest Feynman diagram one can ever draw. Simulta- Euclidean time direction µ = 0 is also necessary to study neous generation of random momenta distributed with the theory at finite temperature (see Subsection V D). the probability π0 (p) proportional to the free propaga- tor G0 (p) implements the Monte-Carlo integration over internal loop momenta in Feynman diagrams. In this The non-convergence of the re-summed perturbation way the diagram weights are represented as products of theory of Section II for principal chiral model on finite- propagators with freely generated momenta times ver- size one-dimensional lattice suggests that the series ob- tex functions and propagators which contain some linear tained on finite-size lattices might have worse conver- combinations thereof. Let us also note that the momen- gence properties than in the infinite-volume limit. We tum conservation is automatically valid for all the trans- expect that these problems will appear only at very high formations in (IV.C.1), which always lead to zero total orders of perturbation theory. Indeed, replacing contin- momentum for any momentum subsequence in the stack. uum integrals over Brillouin zone in the infrared-finite Computationally, the most expensive operation in our weak-coupling expansion of Section II by discrete sums 1 2πmµ DiagMC algorithm is the calculation of vertex functions of the form L2 with pµ = L , mµ = 0 ...L 1 re- p − V p1,...,p1 for all v =1 ...n /2 1, which is nec- 1 2v+1 1 − sults in correctionsP to the continuum integrals which de- essary to obtain acceptance probabilities for all possible −mL crease with L approximately as e , but increase with “Create vertex” transformations in the set IV.C.1. From diagram order. For example, for a chain of l one-loop (45) one can infer that for each v the calculation of the 2 diagrams, such corrections depend on order l and lattice vertex function involves v floating-point operations. l size L as 1+ ae−mL , with some constant a. Thus if As a result, the CPU time∼ needed to calculate all vertex the expansion is limited to some maximal order M which functions scales as n3. This CPU time is dominating the 1 also limits the maximal number of loop momenta, finite- performance of the algorithm in the regime sufficiently volume corrections can always be made smaller by using close to the continuum limit, where subsequences with sufficiently large lattice volume V . In Subsection VC rather large lengths n can appear. It is not inconceiv- 1 we explicitly demonstrate the smallness of finite-volume able that this scaling can be made milder by carefully op- corrections by comparing the results of simulations at timizing the calculation of vertex functions and re-using 108 108 and 256 256 lattices. some data from previous Metropolis updates. We leave × × such developments for future work. After calculation of vertex functions, the next most ex- It seems also that our DiagMC algorithm does not al- pensive and frequently used part of the algorithm is the low for a straightforward parallelization. The only po- generation of random momenta with probability π0 (p). tential for parallelization which we currently see is to To this end one can implement any method for gener- relegate the generation of random momenta with proba- ating continuous variables with definite probability dis- bility distribution π0 (p) to a separate MPI or OpenMP tribution, for example, the biased Metropolis algorithm thread. However, as our algorithm can operate directly [82]>. Note that this allows to work directly in the infi- in the infinite-volume limit, parallelization might not be nite volume limit, which is one of the attractive features really necessary. Having a multi-CPU computer at hand, of DiagMC approach in general [1]>. one can speed up simulations by gathering statistics in- In this work we have still chosen to work with finite dependently on different CPUs. but sufficiently large spatial lattice size L0 L1, which is much larger than the correlation length for× all values In the next Section V we discuss in detail the nu- of λ which we have used. In this case we generate the merical results which we have obtained using the algo- momenta p by randomly choosing among a finite number rithm IV.B.1 with transformation set IV.C.1, as well of discrete values with given (in general, unequal) prob- as the details of data collection and processing. The abilities. By creating an indexed “lookup table” for the production code with the transformations IV.C.1 in the discrete values of momenta this procedure can be per- Metropolis algorithm IV.B.1 is available as a part of formed in CPU time which scales only logarithmically GitHub repository [78]>. 20 V. NUMERICAL RESULTS OF THE DIAGMC express this two-point correlator in terms of φ fields as SIMULATIONS OF THE LARGE-N U (N) × U (N) 1 1 PRINCIPAL CHIRAL MODEL Tr g† g =2 Tr g 1+ h N x y i h N x i− ∞ k+l + 2 k−l λ k+l 1 A. Correlators of gx variables in terms of φp fields +4 ( 1) 2 ξ 2 Tr φkφl = − 8 h N x y i k,l=1 X While the DiagMC algorithm described in Subsec- 1 = 2 Tr gx 1 + tion IVC stochastically samples the correlators of φ h N i− p fields in momentum space, physically one is typically in- +∞ k 2k−1 λ k l 1 l 2k−l terested in correlators of U (N)-valued gx variables which +4 ξ ( 1) Tr φ φ .(73) − 8 − h N x y i enter the partition function (2). k=1 l=1 X X The first non-trivial correlator is the expectation value In order to perform a controllable extrapolation of 1 Tr gx of the trace of gx. It should vanish if the the expansion order to infinity, we now limit our expan- h N i U (N) U (N) symmetry of the principal chiral model sion in powers of ξ to some maximal order M, which × (2) is not broken. By virtue of the Mermin-Wagner the- will serve as extrapolation parameter. Furthermore, we orem which prohibits spontaneous symmetry breaking in use the momentum-space field correlators (48) which are D = 2 dimensions, this should be the case at any finite sampled in our DiagMC simulations and explicitly take value of N. Exact solution of the O (N) sigma model at into account the momentum conservation, which implies N suggests also that spontaneous symmetry break- translational invariance for the correlators of g variables. → ∞ x ing does not happen for nonlinear sigma models in D =2 This allows us to express the expectation values (72) and in the large-N limit. (73) as However, the infrared-finite weak-coupling expansion 1 sampled by our DiagMC algorithm is built around the Tr gx M =1+ 1 h N i vacuum state with gx = 1, Tr gx = 1, which explic- M M−k h N i λ k+m itly breaks the U (N) U (N) symmetry of the principal +2 ξk+m chiral model. To ensure× that our DiagMC algorithm pro- − 8 × k=1 m=0 duces physically consistent results, it is thus important X X to check that the expectation value 1 Tr g approaches [p1,...,p2k] m. (74) N x × h i h i p ,...,p zero as the maximal order of the weak-coupling expan- 1 X 2k sion M is extrapolated to infinity. We demonstrate this on Fig. 10 below. 1 † 1 Tr gxg0 M =2 Tr gx M 1+ Using the definition (3), it is straightforward to expand h N i h N i − 1 M M−k the expectation value Tr gx in powers of the fields φx. h N i +4 ( λ/8)k+m ξk+m Since our perturbative series (13) was obtained from the − × m=0 same expansion, it is natural to extend our formal power Xk=1 X counting scheme based on the auxiliary parameter ξ also Γ (x; p ,...,p ) [p ,...,p ] × 1 2k h 1 2k im to correlators of g variables, which leads to p ,...,p x 1 X 2k Γ (x; p1,...,p2k)= 2k−1 1 1 1 l Tr g = Tr g = = ( 1) cos ((p1 + ... + pl) x) . (75) h N x i V h N x i − x l=1 X X +∞ k λ k 1 1 2k In these expressions, we use the subscript M for the cor- =1+2 ξ Tr φx , (72) − 8 V h N i relators of gx variables to denote that our weak-coupling x Xk=1 X expansion (as defined in Section II) was truncated at or- der M. While the algorithm IV.B.1 with the transforma- where in the first line we have used translational invari- tion set IV.C.1 can in principle sample correlators ance and ξ should be set to ξ = 1 at the end of the [p ,...,p ] to arbitrarily high orders m, we have calculation. 1 2k m hfound that ini practice one can achieve better precision Yet another observable which we consider in this work by sampling only field correlators [p1,...,p2k] m with 1 † h i is the two-point correlator N Tr gx gy of gx variables. k + m M. Indeed, only these field correlators con- This is the simplest nontrivialh correlatori from which one tribute≤ to the expectation values (74) and (75). In or- can infer the mean energy, the mass gap and the static der to limit Monte-Carlo sampling to k + m M, we ≤ screening length in the large-N principal chiral model set to zero all matrix elements A (Xn+1 Xn) for trans- (2). Similarly to (72), we can use the definition (3) to formations which lead to k + m>M in| the Metropolis 21 algorithm IV.B.1. In our simulations we have been able a very noisy observable which is likely to have a heavy- to measure the first M = 12 coefficients of the infrared- tailed probability distribution. Thus directly using the finite weak-coupling expansion of Section II with reason- equation (62) to obtain the normalization factor w re- ably small statistical error. Note also that once the field sults in large statistical errors. We have used aN more correlators [p ,...,p ] are measured for all values convenient practical strategy and fixed to be the ra- h 1 2k im Nw of k + m (Mean return time) -2.5 4.8 10 -3 (p(k, m)) log 4.7 10 -3.5 k=1 4.6 log k=3 -4 4.5 k=5 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 -4.5 k=7 k=9 λ -5 k=11 k=12 -5.5 FIG. 8. The dependence of the average number of Monte- 0 2 4 6 8 10 “Restart” Carlo steps between the updates in the Metropolis m algorithm IV.B.1 on the t’Hooft coupling λ. FIG. 9. The probability p (k,m) to obtain the topmost mo- As the most pessimistic estimate for the autocor- mentum subsequence with 2k momenta and order counter m relation time of our DiagMC algorithm, we can use for λ = 3.1. the average number of Monte-Carlo steps between the “Restart” updates in the Metropolis algorithm IV.B.1. Let us now present our numerical results. We start 1 Indeed, the “Restart” update completely resets the by demonstrating that the expectation value N Tr gx M state of the algorithm, and the sequence of generalized indeed tends to zero as the maximal order M hin the trun-i indices generated after the “Restart” update is com- cated weak-coupling expansion (74) becomes larger. This pletely statistically independent from the sequence which is an indication that in the limit of large M the global was generated before that. On Fig. 8 we illustrate the de- U (N) U (N) symmetry of the principal chiral model pendence of this “mean return time” on the t’Hooft cou- (2) is restored.× 1 pling λ. It appears that the “Restart” updates are very On Fig. 10 we show the dependence of N Tr gx M on rare for all values of λ. On average, they are separated by 1/M for several values of t’Hooft couplingh and severali 4 5 1 10 ... 10 Monte-Carlo steps. The mean return time temperatures. In all cases N Tr gx M exhibits a nonlin- ∼strongly increases towards larger values of the t’Hooft ear but monotonically decreasingh dependencei on 1/M. coupling λ. Together with the λ dependence of n , this For the largest t’Hooft couplings λ = 5.0, 4.0 and for h i observation shows that for larger λ higher orders in the the highest temperature 1/T = L0 = 4 the tendency of 1 infrared-finite weak-coupling expansion of Section II be- N Tr gx M to vanish is rather clear. In contrast, it seems come more important, and the algorithm spends more thath at smalli λ the dependence on M should become very 23 0.8 -4.2 0.7 -4.4 0.6 -4.6 M 0.5 > >) x σ -4.8 0.4 (< 10 4x108, λ=3.012 -5 0.3 log <1/N tr g 12x108, λ=3.012 108x108, λ=3.012 -5.2 0.2 108x108, λ=3.230 λ 108x108, =3.570 -5.4 0.1 108x108, λ=4.000 108x108, λ=5.000 0 -5.6 0 0.2 0.4 0.6 0.8 1 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 1/M λ 1 FIG. 11. Expectation value of the reweighting sign σ in the FIG. 10. Convergence of the expectation value h N Tr gx iM of the single field variable to zero as the truncation order M Metropolis algorithm IV.B.1 as a function of the t’Hooft cou- in the weak-coupling expansion (74) is increased. pling λ. C. Two-point correlators at zero temperature In this Subsection we consider the simplest nontrivial 1 † 1 two-point correlation function Tr gxg0 of U (N)- nonlinear at small 1/M for N Tr gx M to extrapolate to h N i h1 i valued fields. From this correlator one can extract zero. Indeed, the slope of N Tr gx M becomes larger to- wards smaller values of 1/Mh . Thesei observations support such important physical properties of the principal chiral the expectation that 1 Tr g vanishes at M . model (2) as the energy density and the mass gap in the h N x iM → ∞ principal chiral model (2). It has been also extensively studied using conventional Monte-Carlo simulations of As already mentioned in Subsection IVC, not all coef- both SU (N) SU (N) and U (N) U (N) principal chi- × × > ficients A (Xn+1 Xn) are positive for the transformation ral models at several large values of N [37, 84–86] . | 1 † set IV.C.1. We thus have to use the sign reweighting, Weak-coupling expansion of Tr g g0 was given up h N x i as discussed in Subsection IVB. There are two poten- to three first orders in λ at arbitrary values of N in [86]>. tial sources of the sign problem which might hinder our In this Subsection we will compare the large-N extrap- DiagMC simulations: sign cancellations within the ex- olations of these results with the results of our DiagMC pectation values of the correlators [p1,...,p2k] m of φ simulations. This allows us to compare the performance variables, and sign cancellations withinh the observablesi of our algorithm with conventional Monte-Carlo simu- (74) and (75) themselves. In order to quantify sign can- lations, as well as the convergence of our infrared-finite cellations in the correlators p1,...,p2k m, on Fig. 11 we weak-coupling expansion with that of the standard lat- plot the expectation valueh of the sign variablei σ in the tice perturbation theory. Metropolis algorithm IV.B.1 as a function of the t’Hooft A particularly simple and important observable is the coupling λ. For all values of λ the mean sign σ ap- 1 † mean link N Tr g1g0 , which is also related to the pears to be rather small, around 10−4 ... 10−5, thush i sign h i energy density of the principal chiral model. On Fig. 12 problem is indeed important in our simulations. A some- we illustrate the convergence of the truncated series (75) what encouraging feature is that the sign problem be- for the mean link by plotting 1 Tr g†g as a func- comes milder towards the weak-coupling limit in which h N 1 0 iM the theory approaches the continuum limit. However, tion of inverse truncation order 1/M. The dependence on from Fig. 10, as well as from the exactly solvable exam- 1/M is nonlinear with the curvature increasing towards ple of the 2D O (N) sigma model (see Subsection IIIB) larger M, which requires some non-linear extrapolation we can conclude that at the same time the infrared- procedure to reach the limit M (1/M 0). We finite weak-coupling expansion of Section II converges have found that the dependence→ of ∞the truncated→ series less rapidly at small λ, so that more terms in the se- for the mean link on 1/M is well described by the follow- ries should be sampled. The mean sign σ also depends ing two extrapolating functions: rather weakly on temperature, and exhibitsh i a rapid drop − − from σ 5 10 5 to σ 3 10 5 only at highest 1 † log (M) C h i ∼ · h i ∼ · Tr g1g0 M = A B + , (77) temperatures (L0 = 4 and L0 = 8). On Fig. 14 in the h N i − M M next Subsection we will also illustrate the sign problem 1 † B C Tr g g0 M = A + + . (78) for a particular physical observable (mean link). h N 1 i √M M 24 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 † 0 FIG. 12. A comparison of the mean link values h N Tr g1g i 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 is 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/N 2, and extrapolated to 1/N 2 → 0 using linear fits for SU (N) data (solid green lines). We perform extrapolations by fitting the numerical data Monte-Carlo simulations, on Fig. 12 we also show the with these functions using the χ2 weight which includes mean link values obtained in Monte-Carlo simulations of the data covariance matrix to account for correlations both SU (N) SU (N) and U (N) U (N) principal chi- between numerical values at different M. The fit param- ral models as× functions of 1/N 2 (rescaled× as 10/N 2 for eters are A, B and C, with A obviously corresponding better readability of the plots). These data is taken from to the result of extrapolation to M . Upon fit- [37, 85, 86]> for t’Hooft couplings λ =3.100, 3.230, 4.000 ting, both extrapolating functions (77)→ and ∞ (78) lead to and N = 6, 9, 15, 21 and from [84]> for λ = 3.012 roughly the same values of χ2 over the number of degrees and N = 6, 9, 12, 18. The finite-N data is extrapo- of freedom in the fit. These extrapolations are shown on lated to the large-N limit using the linear fits of the form Fig. 12 as red and blue strips for the functions (77) and 1 † 2 Tr g1g0 = A + B/N , which are shown on Fig. 12 (78), respectively. The strip width corresponds to the h N i as straight solid green lines. We have to note that the confidence interval set by the statistical uncertainty in Monte-Carlo data for λ = 3.100, 3.230, 4.000 shows no- the optimal values of A, B and C. The difference be- ticeable deviations from the expected A+B/N 2 behavior tween the two extrapolations (77) and (78) can be used which significantly exceed the very small statistical errors as an estimate of a systematic error of our extrapolation in these data. Thus our N extrapolation might be procedure in the absence of other prior information which also not completely accurate.→ Also ∞ the Monte-Carlo data helps to choose between the two fits (see below). for the finite-NU (N) U (N) principal chiral model, > × 2 To compare our data with the results of conventional taken from [37] , does not seem to follow the 1/N scal- 25 ing of finite-N corrections. expansion. For the three values λ = 3.0120, 3.100 and λ = 3.230 For the largest value λ = 4.00 which we consider our the extrapolation of the finite-N data for the SU (N) massive perturbation theory with both extrapolations SU (N) principal chiral model clearly approaches the re-× (77) and (78) yields the result which is significantly larger sult of the M extrapolation with the extrapolat- than the large-N extrapolation of the Monte-Carlo data. ing function (77→). ∞ While the differences of the N Also the standard lattice perturbation theory [86]> seems and M extrapolations lie within the statistical→ un- ∞ to converge to a larger value. This disagreement between certainty→ of ∞ the M extrapolation of the DiagMC weak-coupling expansions and the physical result is an data, extrapolating function→ ∞ (77) has a small but notice- indication of the large-N transition between the weak- able tendency for over-estimating the mean link. On the coupling and the strong-coupling regimes [35]>, which in > other hand, the second extrapolating function (78) un- the principal chiral model (2) happens at λc =3.27 [37] . derestimates the mean link rather significantly, and we After this transition, the physical values should be rather thus conclude that the first extrapolating function (77) calculated using the strong-coupling expansion. yields more realistic results in the M limit. → ∞ To illustrate the ability of our DiagMC algorithm 0 to work at arbitrarily large volumes without significant slow-down, on Fig. 13 we also compare the convergence -1 of the truncated expansions (74) and (75) for the mean 1 † -2 link Tr g g and the single-field expectation value ) N 1 0 n h i > 1 σ Tr gx for lattice sizes L0 L1 = 108 108 and h N i × × (< -3 L L = 256 256. Both simulations took approxi- 10 0 × 1 × mately the same CPU time, and had the same strength log -4 of the sign problem. λ = 3.012 -5 λ = 3.100 λ = 3.230 0.68 λ = 4.000 -6 M 0.66 0 2 4 6 8 10 12 > x 0.64 n 0.62 FIG. 14. Average sign h σ in of the infrared-finite weak- , <1/N tr g 0.6 coupling expansion diagrams which contribute to the link ex- M )> pectation value (75) at order n = k + m as a function of n for 0 0.58 g + different values of λ. 1 0.56 0.54 Link,108x108 As we have already discussed in Subsection VB above, Link,256x256 1 <1/N tr(g our simulations are limited to around M 10 first 0.52 gx,108x108 ∼ gx,256x256 orders of the weak-coupling expansion due to more 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 and more important sign cancellations between differ- 1/M ent Feynman diagrams. To illustrate this sign problem in more detail, we have considered the sign cancella- FIG. 13. A comparison of truncated expansions (74) and (75) λ m+k tions within contributions of order 8 with fixed 1 † − for the mean link h Tr g1g0 i and the single-field expec- 1 † N m + k = n to the mean link Tr g g0 M given by 1 N 1 tation value h Tr gx i for lattice sizes L0 × L1 = 108 × 108 h i N (75). We have characterized these sign cancellations by and L0 × L1 = 256 × 256. CPU time is roughly the same for + − In −In + − both simulations. the ratio σ n = + − , where In and In are the h i h In +In i sums of the absolute values of diagrams which contribute In order to compare the convergence of our massive at order n = k + m to the mean link (75) with positive lattice perturbation theory constructed in Section II and negative re-weighting signs σ, respectively, and ... with the convergence of the standard lattice perturba- denotes averaging over Monte-Carlo history. On Fig.h 14i tion theory constructed using massless propagators, on we illustrate that σ n decays seemingly exponentially Fig. 12 we also show the one-loop, two-loop and three- with n for several valuesh i of the t’Hooft coupling λ. Since loop (M = 1, 2, 3, respectively) results of [86]> for the our algorithm spends most time sampling diagrams with mean link at N . It appears that conventional per- large n M, this explains the overall strength of the sign turbation theory→ yields ∞ the results which are, at least for problem∼ which we have previously illustrated on Fig. 11. the first three orders, closer to the physical result and Finally, on Fig. 15 we compare the x dependence of 1 † also seem to converge faster. Thus the price which we the two-point correlator N Tr gxg0 M at λ =3.1 with have to pay for the absence of infrared divergences is the Monte-Carlo data obtainedh in [84]> atiN = 6 and N = 9. slower convergence of our infrared-finite weak-coupling For DiagMC results the truncation order M is encoded 26 divergences of the standard lattice perturbation theory DiagMC,M=1 Std.PT,NLO which become only stronger at higher orders. 1 DiagMC,M=12 MC,SU(6) Std.PT,LO MC,SU(9) 0.8 )> D. Finite-temperature phase transition 0 g 0.6 + x 0.4 2.5 <1/N tr(g 0.2 M=10 2 M=6 M M=2 0 )> λ 0 1.5 = 3.1000 g + /2 -0.2 1 1 0 5 10 15 20 25 L 0.5 x 0 <1/N tr(g FIG. 15. Coordinate dependence of the two-point correlator δ 1 † 2 -0.5 0 h N Tr gxg iM given by (75) at various truncation orders M of the infrared-finite weak-coupling expansion at λ = 3.1. The 10 -1 values of M are encoded in color, from pure blue for M =1to -1.5 pure red for M = 12. For comparison, we also show the data 0 10 20 30 40 50 60 70 80 90 100 from the standard Monte-Carlo simulations with N = 6 and 1/T (L0) N = 9, and the first two orders of the standard perturbative 1 † expansion of h Tr g g0 i. N x FIG. 16. Temperature dependence of the correlator 1 † 0 1 h N Tr gxg iM at x = (0, L /2) (midpoint), λ = 3.012 and different truncation orders M of the infrared-finite weak- in color, from pure blue for M = 1 to pure red for coupling expansion. The zero-temperature values at L0 = 108 M = 12. While at distances of a few lattice spacings the and corresponding truncation order M are subtracted. DiagMC data clearly approaches the Monte-Carlo data, at large distances the convergence of our massive pertur- As a further application of our DiagMC algorithm, in bation theory becomes slower. Since the expression (75) this Subsection we consider the principal chiral model 1 † for N Tr gxg0 M contains the constant x-independent (2) at finite temperature. Not much is known about the contribution,h it isi not surprising that it decays down to finite-temperature physics of this model. As for other some finite “plateau” value, rather than exponentially asymptotically free quantum field theories, on general decaying to zero as the full correlators for SU (6) and grounds one can expect that at sufficiently high tem- SU (9) do. The height of this “plateau” is, however, perature the SU (N)-singlet degrees of freedom are ef- monotonically decreasing with M. It seems that one has fectively “deconfined”. However, for the principal chiral to go to much higher values of M than we use to really see models with N > 2 this expectation has never been ex- how the exponential decay of the 1 Tr g† g emerges. h N x 0 i plicitly tested. One of the subtle points which make such It is also interesting to compare the correlators ob- tests difficult is the absence of a local order parameter tained from our infrared-finite weak-coupling expansion for this “deconfinement” transition in the principal chi- with the results of the standard lattice perturbation the- ral model. From general physical intuition in such situa- ory, for which the first two orders in coordinate space tions one expects either a first-order phase transition or were given in [86]>. These results are shown on Fig. 15 a crossover. Understanding the “deconfinement transi- as solid black (for the first order) and grey (for the sec- tion” in the principal chiral model can be important for ond order) lines. As in the case of mean plaquette, we understanding the physics of one-dimensional compacti- observe that at short distances the standard perturba- fications of this model, which have been actively used in tive expansion is closer to the first-principle Monte-Carlo recent years for constructing resurgent trans-series [48]>. data than those from our infrared-finite weak-coupling An explicit analytic calculation [87]> of the correlation expansion. The results of standard perturbative expan- functions in the large-NSU (N) SU (N) principal chi- sion show, however, a completely different behavior at ral model based on the Bethe ansatz× techniques, unfortu- large distances. At distance x 10 they become nega- nately, does not allow so far to study the thermodynamics tive, and further grow logarithmically∼ with distance. In of this model. this sense in the far infrared they behave very differently In a Monte-Carlo study [84]> co-authored by one of from the exponentially decaying physical result. Further- us the numerical results for the energy density and the more, the second-order result of [86]> starts growing log- correlation length at different temperatures are presented arithmically towards positive infinity at x & 3 103. While for the principal chiral model (2) at quite large but finite such large distances are certainly not interesting· for nu- N, for the same t’Hooft coupling λ =3.012 and the spa- merical analysis, this discussion illustrates the infrared tial lattice size L1 = 108 as in the present study. The 27 study [84]> indicates a weak enhancement of correlation Using the Cayley map parameterization (3) and the length at some critical temperature, which very slowly U (N) integration measure (4) in terms of N N Hermi- × tends to become sharper towards larger N. No jump tian matrices Ax,µ, one can calculate the few first orders or hysteresis of the free energy indicative of a first-order of the expansion of the action in powers of the t’Hooft phase transition was observed. coupling λ: In order to get further insights into the nature of this finite-temperature phase transition or crossover in the = dA exp N α2Tr A2 principal chiral model, we have performed DiagMC sim- Z x,µ − x,µ− x,µ ulations at several different temperatures, which corre- Z X 2 spond to different lattice sizes L0 = 4 ... 108 in the Eu- Nα Tr F 2 + O 2, α4 , (80) clidean time direction at fixed t’Hooft coupling λ =3.012 − λ x,µν ∇ x,µ,ν ! (and hence fixed lattice spacing, which only depends X on λ). As a physical observable which is most sensi- where Fx,µν = µAx,ν ν Ax,µ 2iα [Ax,µ, Ax,ν ] is the tive to long-range correlations of gx fields we have cho- ∇ − ∇ − 1 † lattice field strength tensor constructed from the non- sen the value of the two-point correlator Tr gxg0 M h N i compact field Ax,µ and µ is the forward lattice deriva- at x = (0,L1/2), which is the point most distant from ∇ 2 tive. Again we have to choose α λ to arrive at the source at (0, 0) on a periodic lattice of spatial size the canonical normalization of the kinetic∼ terms in the L1. This point is also the “midpoint” of the correlator action. Higher-order terms in the action of the Ax,µ where it takes the minimal value. To get a clearer signal, gauge fields are either suppressed by higher powers of we subtract the zero-temperature data from the finite- α or contain higher lattice derivatives which are sup- temperature data. The result is shown on Fig. 16 for the pressed in the naive continuum limit. Again we see that truncation orders M = 2, 6, 10. The spatial lattice size the “gluon” field Ax,µ has acquired the mass term pro- is again L1 = 108. portional to the t’Hooft coupling λ. The action (80) As the truncation order M is increased, the difference is of course still invariant under gauge transformations between the finite-temperature and zero-temperature † g Ω g Ω , which, however, now have quite a correlator seems to grow. However, the statistical er- x,µ x x,µ x+ˆµ non-trivial→ nonlinear form in terms of A variables. In rors also grow with M and for most values of L the x,µ 0 particular, they are different from the conventional con- differences lie within statistical errors. However, for the tinuum gauge transformations A A + φ smallest values L 4, 8 the difference clearly decreases, x,µ x,µ µ x 0 i [A , φ ]. This is why the nonzero→ gluon mass∇ in (80−) indicating the decrease≃ of correlation length at very high x,µ x does not contradict the gauge invariance of the theory. It temperatures which is probably related to Debye screen- is interesting here to draw parallels with the gluon mass ing. Furthermore, for L roughly between 35 and 45 the 0 which one can obtain from non-perturbative solution of difference seems to grow with M beyond statistical un- (truncated) Schwinger-Dyson equations and Ward iden- certainty, resulting in a sort of peak structure around tities of continuum theory [88]>. L 40. This enhancement of correlations was also re- 0 The inclusion of finite-density quarks would require ported≃ in [84]> at the same value of L . It can be in- 0 sampling of both fermionic and bosonic correlators. terpreted is an indication of a weak finite-temperature Since fermionic propagators at finite density are in gen- phase transition. Since in DiagMC simulations the cor- eral complex-valued, this would require additional re- relator 1 Tr g† g quickly grows with M, from our N x 0 M weighting and make the sign problem worse. For per- data weh cannot excludei the scenario of finite-order phase turbative expansions with truncation error which scales transition at which the correlation length would diverge. M exponentially ǫM α , α < 1 with the truncation order M this still∼ allows to| obtain| the result in a com- VI. EXTENSION TO GAUGE THEORIES putational time which scales as a polynomial of the re- quired error, provided one can sample Feynman diagrams in a time which grows only exponentially with expansion In this Section we discuss possible extension of the ap- order [6]>. In the large-N limit, the number of Feyn- proaches outlined in this work to non-Abelian gauge the- man diagrams grows exponentially with their order, thus ories at finite density, which is the ultimate motivation of even the direct summation will satisfy this requirement. this work. The most straightforward extension would be However, for our infrared-finite weak-coupling expansion to construct the infrared-finite weak-coupling expansion −γ (13), the truncation error scales as ǫM M , γ > 0 of Section II for pure U (N) lattice gauge theory with the (see (77), (78)), thus computational time∼ will grow faster partition function then polynomial in the required error, similarly to what happens for conventional Monte-Carlo simulations with = dg Z x,µ sign problem. An important direction for further work is U(ZN) thus to find the re-summation strategy which would lead to exponentially fast convergence of the series, with bold N exp Re Tr g g g† g† . (79) diagrammatic expansions being one of the options. −4λ x,µ x+ˆµ,ν x+ˆν,µ x,ν x,µ,ν ! A less straightforward extension of this work to non- X 28 sion. Work in this direction is in progress. Another challenge is to extend this approach to fi- nite N, where the combinatorial structures involved in the strong-coupling expansion coefficients become more complicated [26, 92]>, and 1/N expansion might not work properly [92]>. Importantly, the baryon bound FIG. 17. Zigzag symmetry of Wilson loops in gauge theory. states only exist at finite N. Nevertheless, even the finite-density gauge theory in the large-N Veneziano limit might exhibit some universal features for densities below Abelian gauge theories might be based on the lat- the baryon condensation threshold and is therefore phys- tice strong-coupling expansion, combined with hopping ically interesting [93, 94]>. Needless to say, large-N limit expansion if dynamical quarks are included and the is also important in the context of AdS/CFT holographic Veneziano large-N limit is taken. Inside the convergence duality between gauge theories and gravity. radius of the strong-coupling and hopping expansions, their truncation error should scale down exponentially with truncation order. At the same time, in the large-N VII. CONCLUSIONS limit the number of strong-coupling expansion diagrams should grow not faster than exponentially with order [89]>. Furthermore, for hopping expansion adding chem- In this work we have explored the advantages and dis- ical potential does not introduce any additional sign can- advantages of the Diagrammatic Monte-Carlo (DiagMC) cellations between the diagrams, but merely modifies the approach for studying asymptotically free non-Abelian hopping coefficients in the time direction [23, 24, 26, 28]>. lattice field theories in the weak-coupling regime. To this Thus even despite the sign cancellations which happen end we have combined two largely independent tools: the between strong-coupling expansion diagrams at higher Metropolis algorithm for solving infinitely-dimensional orders, DiagMC algorithms based on strong-coupling ex- Schwinger-Dyson equations, and the partial resumma- pansion can be expected to solve, at least formally, the tion of lattice perturbation theory which leads to massive computational complexity problem (in the terminology bare propagators. of [6]>) for finite-density gauge theories in the Veneziano While our primary motivation are non-Abelian lat- large-N limit. tice gauge theories, eventually coupled to finite-density The Metropolis algorithm IV.B.1 allows to sys- fermions, in this exploratory study we have consid- ered an example of the large-NU (N) U (N) princi- tematically sample the strong-coupling expansion for × large-N non-Abelian gauge theories. To this end pal chiral model. This model has significantly simpler it should be applied to solve the gauge-invariant weak-coupling expansion, while having most of the non- Schwinger-Dyson equations for lattice Wilson loops perturbative features of the large-N pure gauge theories 1 on the lattice. N Tr (gx1,µ1 gx2,µ2 ...gxn,µn ) (Migdal-Makeenko loop equationsh [71, 72]>). These equationsi are very similar The construction of Metropolis algorithm IV.B.1 from in structure to equations (40) in Subsection IV A. Schwinger-Dyson equations is very general, and can be In fact, we have already tried to solve the Migdal- applied to practically any quantum field theory. Its main Makeenko loop equations in large-N pure gauge the- conceptual advantages are: ory using the algorithm IV.B.1, sampling the loops The applicability of algorithm to any expansion, on the lattice with probabilities proportional to Wil- • even without explicit diagrammatic rules. In son loops 1 Tr (g g ...g ) . Unfortunately, N x1,µ1 x2,µ2 xn,µn particular, it can be used to sample both weak- and this straightforwardh attempt did not allowi to go beyond strong-coupling expansions in lattice gauge theory. the first few orders of strong-coupling expansion, which could anyway be computed manually. The reason is the The ability to work directly in the infinite- > “zigzag symmetry” [90, 91] between Wilson loops which • volume limit. Of course, this does not imply the differ by the insertion of arbitrary path traversed forward absence of slowing down towards the continuum and immediately backward (see Fig. 17), which stems limit, since the number of expansion coefficients † from unitarity of link matrices g g − g g = x,µ x+µ, µ ≡ x,µ x,µ which should be sampled to reach a given precision 1. Most of the computer time goes into sampling nu- might grow with correlation length (see e.g. Sub- merous redundant loops with multiple “zigzag” inser- section IIIB), and sign problem might make this tions, and nontrivial loops which contribute to higher growth faster (see Subsection VB). orders of strong-coupling expansion are entropically dis- favoured. Possible solution to this problem is to rewrite The ability to work directly in the large-N limit, the Migdal-Makeenko loop equations in terms of “irre- • where the number of diagrams contributing to ei- ducible” loops without zigzag insertions, which makes ther strong- or weak-coupling expansions becomes the equation structure significantly more nontrivial. We significantly smaller. Systematic calculation of the hope that this “gauge-fixing” in loop space would allow coefficients of 1/N-expansion is also possible [81]>. to go to sufficiently high orders of strong-coupling expan- Large-N limit is not directly accessible in most 29 other DiagMC simulations of non-Abelian gauge orders, at finite expansion order this explicit sym- theories [23–28]>. metry breaking results in constant contribution to the correlators of gx variables, which in reality are Our algorithm is particularly suitable for exploring the exponentially decaying. critical behavior in large-N quantum field theories with manifestly positive weights of Feynman diagrams, for ex- ample, the planar φ4 model with negative coupling con- stant or the Weingarten model of random planar surfaces Summarizing these advantages and disadvantages, it > on the lattice [95] . seems that the DiagMC approach based on the weak- The resummation strategy used in this work also seems coupling expansion is comparable in terms of computa- to be quite general for non-Abelian field theories. Its tional time to conventional simulation methods as ap- particular advantages are: plied to non-Abelian lattice field theories in the large-N limit. In particular, for Fig. 12 the finite-N data from The absence of infrared divergences which are • conventional Monte-Carlo and the DiagMC data at infi- characteristic for the standard weak-coupling ex- nite N were produced within the comparable amount of pansion of massless quantum field theories. While CPU time. It is also conceivable, for example, that apply- these divergences cancel in physical observables, ing Numerical Stochastic Perturbation Theory (NSPT) they should be regulated at intermediate steps, [45, 46]> to finite-N principal chiral model and then which enhances the sign problem upon removing extrapolating to large-volume and large-N limits would the infrared regulator due to cancellations between lead to the comparable quality of numerical data within numerically large summands of opposite signs. In the comparable CPU time. It remains to be seen how ef- more conventional approaches to lattice perturba- ficient the sampling of strong-coupling expansion might tion theory such as Numerical Stochastic Perturba- be. tion Theory (NSPT) [45, 46]> dealing with infrared divergences also requires additional infrared regu- Probably the most promising direction for further lators. improvement is the Bold Diagrammatic Monte-Carlo (BDMC) approach, which can efficiently capture non- The absence of factorial divergences in the • perturbative information by combining perturbative ex- weak-coupling expansion in the large-N limit, pansion with a self-consistent mean-field-like approxima- which arise at sufficiently high orders in conven- tion [5, 13, 14, 96]>. Unfortunately, currently most of tional lattice perturbation theory. The general ar- BDMC algorithms for bosonic fields are based on certain guments of Section II, however, does not exclude truncations of Schwinger-Dyson equations even for the the possibility of exponential divergences. Our ar- simplest case of scalar field theory with φ4 interactions guments in favour of convergence of our infrared- [13, 14]>. finite weak-coupling expansion are only based on the numerical evidence - for up to M = 500 orders One possibility to construct the bold diagrammatic ex- for the large-N O (N) sigma model, and for up to pansion would be to work with Schwinger-Dyson equa- M = 12 orders for the principal chiral model. tions for n-particle-irreducible field correlators [74]>. Another possibility, more specific for non-Abelian field Of course, these advantageous features come at a certain theories, is to introduce the matrix-valued Lagrange mul- price, and in some respects the resummed perturbative tiplier enforcing the unitarity constraint g† g = 1, and expansion behaves worse than the bare one: x x integrate out the gx fields, which would automatically Sampling our weak-coupling expansion using a Di- ensure the U (N) U (N)-invariant vacuum state with • 1 Tr g = 0. For× the large-N O (N) sigma model, agMC algorithm leads to the sign problem due h N x i to in general non-positive weights of Feynman dia- a similar treatment immediately yields the exact non- grams, even though the conventional Monte-Carlo perturbative solution, with the mass being equal to the simulations do not have any sign problem. Further- saddle-point value of the Lagrange multiplier field. In more, for lattice gauge theory this sign problem is fact, this mean-field solution resums our double series expected to become worse if finite-density fermions (30) with coupling and logs of coupling into a single exponential exp 4π , which is the only term in the are added. λ true trans-series for− the mass gap in the large-N O (N) The convergence is slower than for the sigma model. One could thus expect that if such ap- • conventional lattice perturbation theory, see proach works also for the principal chiral model, it might Figs. 12 and 15. re-sum many terms in our expansion into exponentials, thus turning it into the true trans-series of the form (1) The expansion starts from the wrong vac- with optimal convergence properties and reducing the • uum state with explicitly broken U (N) U (N) sign problem due to cancellations between Feynman dia- symmetry of the principal chiral model. While× this grams. Implementing this approach in practice turns out symmetry seems to be gradually restored at high to be non-trivial, and would be explored in future work. 30 ACKNOWLEDGMENTS calculations were performed on “iDataCool” at Regens- burg University, on the ITEP cluster in Moscow and This work was supported by the S. Kowalevskaja on the LRZ cluster in Garching. We acknowledge valu- award from the Alexander von Humboldt Foundation. able discussions with S. Valgushev, F. Werner, B. Svis- The authors are grateful to the Mainz Institute for The- tunov, T. Sulejmanpasic, L. von Smekal, N. Prokof’ev, oretical Physics (MITP) for its hospitality and its par- G. Dunne, O. Costin, A. Cherman and G. Bali. tial support during the completion of this work. The [1] K. Van Houcke, E. Kozik, N. Prokof’ev, and B. Svis- [26] W. Unger, PoS LATTICE2014, 192 (2014), 1411.4493. tunov, Phys.Procedia 6, 95 (2010), 0802.2923. [27] C. Gattringer and C. Marchis, [2] N. Prokof’ev and B. Svistunov, Nucl. Phys. 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In this limit the lattice [78] P. V. Buividovich, “SchwingerDysonMetropolis,” GitHub Laplacian ∆ (p) behaves as ∆ (p) p2, and the vertex repository (2017). function V (q , , q ) can be represented≈ as 1 ··· 2n+1 2n+1 k−1 k−1 V (q , , q )= ( 1) ∆(q + + q − )= 1 ··· 2n+1 − m+1 ··· m+2n+2 k m=0 Xk=1 X 2n+1 k−1 k−1 2 = ( 1) (q + + q − ) = − m+1 ··· m+2n+2 k m=0 Xk=1 X 2n+1 k−1 2n+2−k 2n+1 k−1 2n+2−k i−1 = ( 1)k−1 q2 +2 ( 1)k−1 q q (A1) − m+i − m+i m+j m=0 i=1 m=0 i=1 j=1 Xk=1 X X kX=1 X X X 32 The first term in the last line of the above equation can be simplified as 2n+1 k−1 2n+2−k 2n 2n+1 k−1 2r−1 ( 1)k−1 q2 = q2 ( 1)k−1 δ θ(2n +2 k 2r + l +1)+ − m+i 2r − m,l − − m=0 i=1 r=1 m=0 kX=1 X X X Xk=1 X Xl=0 2n 2n+1 k−1 2r + q2 ( 1)k−1 δ θ(2n +2 k 2r + l)= 2r+1 − m,l − − r=0 m=0 X kX=1 X Xl=0 2n 2r−1 2n+2−2r+l 2n 2r 2n+1−2r+l 2n = q2 ( 1)k−1 + q2 ( 1)k−1 = q2 , (A2) 2r − 2r+1 − 2r+1 r=1 r=0 r=0 X Xl=0 kX=l+1 X Xl=0 k=Xl+1 X where θ is the unit step function. Similar algebra gives where we have taken into account that the off-diagonal for the second term in the last line of (A1): components φij are not all independent, but rather sat- isfy φij = φ¯ji. We can now express the integration 2n+1 k−1 2n+2−k i−1 measure, up to an irrelevant normalization factor, as 2 ( 1)k−1 q q = − m+i m+j dg = dφ√g, where g is the determinant of the metric m=0 i=1 j=1 kX=1 X X X tensor which is equal to the product of all the diagonal 2n r−1 metricR elementsR appearing in (B2): =2 q2r+1q2s+1 (A3) r=0 s=0 X X g = f 2f 2 f = f f = Combining (A2) and (A3), we get i j i i j j>i i i,j Y Y Y 2 2N V (q1,...,q2n+1) = (q1 + q3 + + q2n+1) , (A4) 2 −2N ··· = fi = det 1+ φ , (B3) which is the formula (47) in the main text. i ! Y 2 −1 where we have denoted fi = 1+ φi . Combining ev- Appendix B: Integration measures on SN and U (N) erything together, we arrive at manifolds −N dg = dφ det 1+ φ2 , (B4) 1. Cayley map for U (N) group Z H Z U(N) N×N 1+iφ 2 where on the right hand side the integration goes over all With the Cayley mapping g = − = − 1, the 1 iφ 1 iφ − N N Hermitian matrices. This expression is identical to group-invariant distance element ds2 = Tr dg dg† on (4)× in the main text, up to the trivial rescaling φ αφ. U (N) can be expressed as → ds2 = 2i 1 2i 1 2. Exponential map for U (N) group = Tr − dφ dφ = 1 iφ 1 iφ 1+ iφ 1+ iφ − − 1 1 Proceeding in the same way as for the Cayley map, for = 4Tr dφ dφ , (B1) the exponential map g = eiφ we can relate the differen- 1+ φ2 1+ φ2 tials of φ and g as The invariance of ds2 under shifts g ug, u U (N) im- → ∈ 1 plies, in particular, the invariance of the measure under dg = dz eizφdφei(1−z)φ. (B5) † similarity transformations g ugu which allow to diag- Z0 onalize g and hence the matrix→φ in (B1). We thus assume The group-invariant distance element takes the form that the matrix φij has diagonal form φij = φiδij . The 2 distance element ds can be then written as a convolu- ds2 = Tr dg dg† = tion of the coordinate differentials dφij with the diagonal 1 1 metrics: i(z2−z1)φ −i(z2−z1)φ = dz1dz2 Tr e dφ e dφ .(B6) 2 ds = gij kldφij dφkl = Z0 Z0 dφ2 (dRe φ )2 + (dIm φ )2 = ii +2 ij ij , (B2) Again, we can now use the invariance of the measure 1+ φ2 (1 + φ2) 1+ φ2 † † i i j>i i j under similarity transformations g ugu , φ ugu , X X → → 33 u U (N) to diagonalize g and φ, and express the dis- powers of φ: tance∈ element (B6) as sin2 ((φ φ ) /2) dφ exp log i − j = 2 i,j (φi φj ) ! MZ X − 2 1 1 (φ φ ) = dφ exp i − j + O λ4 = dz dz ei(z2−z1)(φi−φj ) dφ 2 = dφ2 + − 12 1 2 ij ii MZ i,j | | X i,j Z0 Z0 i 2 X X NTr φ Tr φ Tr φ 4 1 1 = dφ exp − + O φ , (B9) − 6 +2 dz dz cos ((z z ) (φ φ )) dφ 2 = MZ 1 2 2 − 1 i − j | ij | i>j X Z0 Z0 which illustrates the appearance of double-trace terms sin2 ((φ φ ) /2) already in the lowest order of the expansion. = dφ2 +8 i − j dφ 2. (B7) ii 2 | ij | i i>j (φi φj ) X X − 3. Stereographic map for the sphere SN In order to find the integration measure on the N- sphere SN parameterized by the coordinates φi as n0 = We see that the metric tensor is again diagonal with our −1 −1 1 φ2 1+ φ2 , n = 2φ 1+ φ2 , φ2 φ2, choice of group coordinates, and we finally get for the in- − i i ≡ i i tegration measure (up to an overall normalization factor) we again start by first expressing the O (N)-invariant dis- 2 P tance element ds = dna dna on SN in terms of the co- ordinates φi: 16 (φ dφ )2 ds2 = dn2 + dn2 = i i + 2 0 i 2 4 sin ((φi φj ) /2) (1 + φ ) dg = dφ − 2 , (B8) 2 2 i>j (φi φj ) 2dφ 4φ dφ φ 4 dφ U(ZN) MZ Y − i j j i i + 2 2 = 2 . (B10) 1+ φ − (1 + φ2) ! (1 + φ2) The metric tensor is again diagonal in the stereographic coordinates φi (which is not surprising, as stereographic mapping is conformal), and the integration measure (up where M is some closed domain in the N 2-dimensional to a normalization constant) can be written in terms of space HN×N of the Hermitian N N matrices. The the square root of the metric determinant as U (N) group manifold is uniquely covered× by the integra- −(N−1) tion in (B8) if M is bounded by the N 2 1-dimensional dn = dN−1φ 1+ φ2 , (B11) manifolds at which at least two eigenvalues− of φ coincide. SZ RNZ−1 E.g. in the case of U (2) group, M is the direct product N of the 3-dimensional ball and the one-dimensional circle where the integration is over the whole N 1 dimensional N−1 − S1. real space R . Inthe large-N limit one can also replace the power of (N 1) in (B11) by N, which leads to − − − In order to compare with the formula (4) in the main the expression (23) in the main text after the trivial re- text, it is instructive to expand the Jacobian (B8) in scaling φ √λ/2 φ. →