Shifted-Action Expansion and Applicability of Dressed Diagrammatic Schemes
Riccardo Rossi, Felix Werner, Nikolay Prokof'ev, Boris Svistunov
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Shifted-Action Expansion and Applicability of Dressed Diagrammatic Schemes
Riccardo Rossi and F´elix Werner Laboratoire de Physique Statistique and Laboratoire Kastler Brossel ,Ecole Normale Sup´erieure, UPMC, CNRS, Universit´e Paris Diderot ,Coll`ege de France , 24 rue Lhomond, 75005 Paris, France Nikolay Prokof’ev , and Boris Svistunov , , Department of Physics, University of Massachusetts, Amherst, MA 01003, USA Russian Research Center “Kurchatov Institute,” 123182 Moscow, Russia and Wilczek Quantum Center, Zhejiang University of Technology, Hangzhou 310014, China
While bare diagrammatic series are merely Taylor expansions in powers of interaction strength,dressed diagrammatic series, built on fully or partially dressed lines and vertices, are usually con-structed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can evenlead to convergence to an unphysical result [Kozik, Ferrero, and Georges, PRL , 156402 (2015)].Here we show that for a broad class of partially dressed diagrammatic schemes, there exists anaction S ( ξ ) depending analytically on an auxiliary complex parameter ξ , such that the Taylor ex-pansion in ξ of correlation functions reproduces the original diagrammatic series. The resultingapplicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, ana-lyticity of S ( ξ ) is not granted, and we formulate a sufficient condition for converging to the correctresult. PACS numbers: 71.10.Fd, 02.70.Ss
Much of theoretical physics is formulated in the lan-guage of Feynman diagrams, in various fields such ascondensed matter, nuclear physics, and QCD. A pow-erful feature of the diagrammatic technique, used in eachof the above fields, is the possibility to build diagramson partially or fully dressed propagators or vertices, see, e.g. , Refs. [1–5]. In quantum many-body physics, no-table examples include dilute gases, whose descriptionis radically improved if ladder diagrams are summed upso that the expansion is done in terms of the scatteringamplitude instead of the bare interaction potential, andCoulomb interactions, which one has to screen to have ameaningful diagrammatic technique.With the development of Diagrammatic Monte Carlo,it becomes possible to compute Feynman diagrammaticexpansions to high order for fermionic strongly corre-lated quantum many-body problems [6–11]. The numberof diagrams grows factorially with the order, even for afully irreducible skeleton scheme [12]. Nevertheless, forfermionic systems on a lattice at finite temperature, dia-grammatic series (of the form P n a n with a n the sum ofall order- n diagrams) are typically convergent in a broadrange of parameters, due to a nearly perfect cancellationof contributions of different sign within each order, asproven mathematically [13] and seen numerically [6, 7, 9–11].One might think that partial or full renormalization ofdiagrammatic elements (propagators, interactions, ver-tices, etc.) always leads to more compact and better be-having diagrammatic expansion. However, such a dresseddiagrammatic series cannot be used blindly: Even whenit converges, the result is not guaranteed to be correct,since it is a priori not allowed to reorder the terms of a series that is not absolutely convergent (the sum of theabsolute values of individual diagrams is typically infi-nite, due to factorial scaling of the number of diagramswith the order). And indeed, for a skeleton series, i.e. , aseries built on the fully dressed propagator, convergenceto a wrong result does occur in the case of the Hub-bard model in the strongly correlated regime near halffilling [14], and preliminary results suggest that the cor-responding self-consistent skeleton scheme converges to awrong result as a function of the maximal self-energy di-agram order N [15]. Both of these phenomena are clearlyseen in the exactly solvable zero space-time dimensionalcase [16, 17].In this work, we establish a condition that is neces-sarily violated in the event of convergence to a wrongresult of the self-consistent skeleton scheme. Further-more, we show that this convergence issue is absent fora broad class of partially dressed schemes. In particular,we propose a simple scheme based on the truncated skele-ton series. The underlying idea is to construct an action S ( ξ ) that depends on an auxiliary complex parameter ξ such that the Taylor series in ξ of correlation functionsreproduces the dressed diagrammatic series built on agiven partially or fully dressed propagator. This makesthe dressed scheme as mathematically justified as a barescheme, provided S ( ξ ) is analytic with respect to ξ and S ( ξ =1) coincides with the physical action; these condi-tions hold automatically in the partially dressed case,while in the fully dressed case they hold under a simplesufficient condition which we provide. Our constructionapplies to a general class of diagrammatic schemes builton dressed lines and vertices, including two-particle lad-ders and screened long-ranged potentials. Partially dressed single-particle propagator.
We con-sider a generic fermionic many-body problem describedby an action S [ ψ, ¯ ψ ] = h ψ | G − | ψ i + S int [ ψ, ¯ ψ ] (1)where ψ, ¯ ψ are Grassmann fields [18], and we use bra-ket notations to suppress space, imaginary time, possi-ble internal quantum numbers, and integrals/sums overthem, i.e. , h ψ | G − | ψ i denotes the integral/sum over r , τ and σ of ¯ ψ σ ( r , τ ) ( G − ,σ ψ σ )( r , τ ). G − stands for theinverse, in the sense of operators, of the free propaga-tor. The full propagator G and the self-energy Σ arerelated through the Dyson equation G − = G − − Σ.The bare Feynman diagrammatic expansion correspondsto perturbation theory in S int . In order to generate a di-agrammatic expansion built on a partially dressed single-particle propagator ˜ G N , we introduce an auxiliary actionof the form S ( ξ ) N [ ψ, ¯ ψ ] = h ψ | G − , N ( ξ ) | ψ i + ξS int [ ψ, ¯ ψ ] , (2)where G − , N ( ξ ) = ˜ G − N + ξ Λ + . . . + ξ N Λ N , (3) ξ is an auxiliary complex parameter, and Λ , . . . , Λ N areappropriate operators. ˜ G N is the single particle prop-agator for S ( ξ =0) N . At ξ = 0, one can still view ˜ G N asthe free propagator, provided one includes in the interac-tion terms not only ξS int , but also the quadratic terms h ψ | ξ n Λ n | ψ i . Accordingly, ξ is interpreted as a couplingconstant, and the ξ n Λ n acquire the meaning of counter-terms. These counter-terms can be tuned to cancel outreducible diagrams, thereby enforcing the dressed char-acter of the diagrammatic expansion. A natural require-ment is that S ( ξ =1) N coincides with the physical action S , i.e. , that ˜ G − N + N X n =1 Λ n = G − . (4)For given G , this should be viewed as an equation tobe solved for ˜ G N (it is non-linear if the Λ n ’s depend on˜ G N ). The unperturbed action for the dressed expansion, h ψ | ˜ G − N | ψ i , is shifted by the Λ n ’s with respect to theunperturbed action for the bare expansion, h ψ | G − | ψ i .We can then use any action of the generic class (2) forproducing physical answers in the form of Taylor expan-sion in powers of ξ , provided the propagator ˜ G N and theshifts Λ n satisfy Eq. (4). More precisely, consider the fullsingle-particle propagator G N ( ξ ) of the action S ( ξ ) N , andthe corresponding self-energyΣ N ( ξ ) := G − , N ( ξ ) − G − N ( ξ ) . (5)Note that since S ( ξ =1) N = S , we have G N ( ξ =1) = G andhence also Σ N ( ξ =1) = Σ. We assume for simplicity that Σ N ( ξ ) is analytic at ξ = 0, and that its Taylor series P ∞ n =1 Σ ( n ) N [ ˜ G N ] ξ n , converges at ξ = 1. We expect theseassumptions to hold for fermionic lattice models at finitetemperature in a broad parameter regime, given that theaction S ( ξ ) N is analytic in ξ [6, 7, 9–11, 13, 19]. Then,since S ( ξ ) N is an entire function of ξ , we can conclude thatΣ = ∞ X n =1 Σ ( n ) N [ ˜ G N ] , (6) i.e. , the physical self-energy is equal to the dressed dia-grammatic series.This last step of the reasoning can be justified usingthe following presumption: Let D be a connected openregion of the complex plane containing . Assume that S ( ξ ) is analytic in D , that the corresponding self-energy Σ( ξ ) is analytic at ξ = 0 , and that Σ( ξ ) admits an an-alytic continuation ˜Σ( ξ ) in D . Then, Σ and ˜Σ coincideon D . This presumption is based on the following ar-gument: Since S ( ξ ) is analytical, if no phase transitionoccurs when varying ξ in D , then Σ( ξ ) is analytical on D ,and by the identity theorem for analytic functions, Σ and˜Σ coincide on D . If a phase transition would be crossedas a function of ξ in D , analytic continuation through thephase transition would not be possible [20], contradictingthe above assumption on the existence of ˜Σ. Applyingthis presumption to ˜Σ( ξ ) := P ∞ n =1 Σ ( n ) N [ ˜ G N ] ξ n , whichhas a radius of congergence R ≥ D the open disc ofradius R , we directly obtain Eq. (6) provided R >
1. If R =1, we can still derive Eq. (6), using Abel’s theoremand assuming that Σ N ( ξ ) is continuous at ξ =1, which,given that the action is entire in ξ , is generically expected(except for physical parameters fined-tuned precisely toa first-order phase transition, where Σ is not uniquelydefined). Semi-bold scheme.
We first focus on the choiceΛ n = Σ ( n )bold [ ˜ G N ] (1 ≤ n ≤ N ) , (7)where Σ ( n )bold [ G ] is the sum of all skeleton diagrams of or-der n , built with the propagator G and the bare in-teraction vertex corresponding to S int , that remain con-nected when cutting two G lines. This means that ˜ G N is the solution of the bold scheme for maximal order N ,cf. Eq. (4). For a given N , higher-order dressed graphscan then be built on ˜ G N . The numerical protocol corre-sponding to this ‘semi-bold’ scheme consists of two inde-pendent parts: Part I is the Bold Diagrammatic MonteCarlo simulation of the truncated order- N skeleton sumemployed to solve iteratively for ˜ G N satisfying Eqs. (4,7);Part II is the diagrammatic Monte Carlo simulation ofhigher-order terms, Σ ( n ) N [ ˜ G N ] , n > N , that uses ˜ G N asthe bare propagator. Note that here N is fixed (con-trarily to the conventional skeleton scheme discussed be-low), and the infinite-order extrapolation is done only inPart II.The Feynman rules for this scheme are as follows:Σ ( n ) N [ ˜ G N ] = Σ ( n )bold [ ˜ G N ] for n ≤ N ; (8)while for n ≥ N + 1, Σ ( n ) N [ ˜ G N ] is the sum of all barediagrams, built with ˜ G N as free propagator and the bareinteraction vertex corresponding to S int , which do notcontain any insertion of a subdiagram contributing toΣ ( n )bold [ ˜ G N ] with n ≤ N . Indeed, each such insertion isexactly compensated by the corresponding counterterm.To derive Eq. (8), we will use the relationΣ N ( ξ ) ˆ= ∞ X n =1 Σ ( n )bold [ G N ( ξ )] ξ n (9)where ˆ= stands for equality in the sense of formal powerseries in ξ , and we will show the propositionΣ N ( ξ ) ˆ= k X n =1 Σ ( n )bold [ ˜ G N ] ξ n + O ( ξ k +1 ) ( P k )for any k ∈ { , . . . , N + 1 } , by recursion over k . ( P k =0 )clearly holds. If ( P k ) holds for some k ≤ N , then wehave G N ( ξ ) ˆ= ˜ G N + O ( ξ k +1 ), as follows from Eqs. (5),(3) and (7). Substitution into Eq. (9) then yields ( P k +1 ).Alternatively to the semi-bold scheme Eq. (7), otherchoices are possible for the shifts Λ , . . . , Λ N and thedressed propagator ˜ G N . For example, the shifts can bebased on diagrams containing the original bare propaga-tor G instead of ˜ G N . In the absence of exact cancella-tion, all diagrams should be simulated in Part II of thenumerical protocol, and Λ n will enter the theory explic-itly. This flexibility of choosing the form of Λ n ’s, alongwith the obvious option of exploring different N ’s, pro-vides a tool for controlling systematic errors coming fromtruncation of the ξ -series [23]. Skeleton scheme.
We turn to the conventional schemein which diagrams are built on the fully dressed single-particle propagator. The corresponding numerical pro-tocol is identical to Part I of the above one, with theadditional step of extrapolating N to infinity, as donein [8–11, 21]. Accordingly, we assume that the ‘skele-ton sequence’ ˜ G N converges to a limit ˜ G when N → ∞ .The crucial question is under what conditions one canbe confident that ˜ G is the genuine propagator G of theoriginal model. The answer comes from the properties ofthe sequence of functions L ( ξ ) N := N X n =1 Σ ( n )bold [ ˜ G N ] ξ n . (10)Let us show that ˜ G = G holds under the following suffi-cient condition:(i) for any ξ in a disc D = {| ξ | < R } of radius R > p , τ ), L ( ξ ) N ( p , τ ) converges for N →∞ ; more-over this sequence is uniformly bounded, i.e. , there ex-ists a function C ( p , τ ) such that ∀ ξ ∈ D , ∀ ( N , p , τ ), | L ( ξ ) N ( p , τ ) | ≤ C ( p , τ ); and(ii) ˜ G N ( p , τ ) is uniformly bounded, i.e. , there exists aconstant C such that for all ( N , p , τ ), | ˜ G N ( p , τ ) | ≤ C .Our derivation is based on the action S ( ξ ) ∞ := lim N →∞ S ( ξ ) N . (11)Clearly, S ( ξ ) ∞ = h ψ | ˜ G − + L ( ξ ) | ψ i + ξS int (12)with L ( ξ ) ( p , τ ) := lim N →∞ L ( ξ ) N ( p , τ ) . (13)Since S ( ξ =1) N = S , we have S ( ξ =1) ∞ = S , and thus G ∞ ( ξ =1) = G where G ∞ ( ξ ) is the full propagator ofthe action S ( ξ ) ∞ .We first observe that L ( ξ ) ( p , τ ) is an analytic functionof ξ ∈ D for all ( p , τ ), and that1 n ! ∂ n ∂ξ n L ( ξ ) ( p , τ ) (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = Σ ( n )bold [ ˜ G ]( p , τ ) . (14)This follows from conditions (i,ii), given that momentaare bounded for lattice models. Indeed, for any trian-gle T included in D , H T dξ L ( ξ ) N ( p , τ ) = 0. Thanks tocondition (i), the dominated convergence theorem is ap-plicable, yielding H T dξ L ( ξ ) ( p , τ ) = 0. The analyticity of ξ L ( ξ ) ( p , τ ) follows by Morera’s theorem. To deriveEq. (14) we start from1 n ! ∂ n ∂ξ n L ( ξ ) N ( p , τ ) (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = Σ ( n )bold [ ˜ G N ]( p , τ ) . (15)By Cauchy’s integral formula, the l.h.s. of Eq. (15)equals 1 / (2 iπ ) H C dξ L ( ξ ) N ( p , τ ) /ξ n +1 where C is the unitcircle. Using again condition (i) and the domi-nated convergence theorem, when N →∞ , this tendsto 1 / (2 iπ ) H C dξ L ( ξ ) ( p , τ ) /ξ n +1 , which equals the l.h.s.of Eq. (14). To show that Σ ( n )bold [ ˜ G N ]( p , τ ) tends toΣ ( n )bold [ ˜ G ]( p , τ ), we consider each Feynman diagram sepa-rately; the dominated convergence theorem is applicablethanks to condition (ii), the boundedness of the integra-tion domain for internal momenta and imaginary times,and assuming that interactions decay sufficiently quicklyat large distances for the bare interaction vertex to bebounded in momentum representation.Hence L ( ξ ) = ∞ X n =1 Σ ( n )bold [ ˜ G ] ξ n . (16)As a consequence, the action S ( ξ ) ∞ generates the fullydressed skeleton series built on ˜ G , i.e. , its self-energyΣ ∞ ( ξ ) has the Taylor expansion P ∞ n =1 Σ ( n )bold [ ˜ G ] ξ n , andthe Taylor series of G ∞ ( ξ ) reduces to the ξ -independentterm ˜ G . This can be derived in the same way as Eq. (8),by showing by recursion over k that for any k ≥ ∞ ( ξ ) = P kn =1 Σ ( n )bold [ ˜ G ] ξ n + O ( ξ k +1 ). Furthermore,having shown above the analiticity of L ( ξ ) , i.e. , of S ( ξ ) ∞ ,we again expect that G ∞ ( ξ ) is analytic at ξ = 0 (forfermions on a lattice at finite temperature), and wecan use again the above presumption to conclude that G ∞ ( ξ = 1) = G = ˜ G . Dressed pair propagator.
So far we have discusseddressing of the single-particle propagator while keepingthe bare interaction vertices. We turn to diagrammaticschemes built on dressed pair propagators. We restrictto spin-1 / S int [ ψ, ¯ ψ ] = U X r Z β dτ ( ¯ ψ ↑ ¯ ψ ↓ ψ ↓ ψ ↑ )( r , τ ) , (17)where U is the bare interaction strength. For simplicitywe discuss dressing of the pair propagator while keep-ing the bare G . It is necessary to perform a Hubbard-Stratonovich transformation in order to construct the ap-propriate auxiliary action. Introducing a complex scalarHubbard-Stratonovich field η leads to the action S [ ψ, ¯ ψ, η, ¯ η ] = h ψ | G − | ψ i − h η | Γ − | η i − h η | Π | η i + h η | ψ ↓ ψ ↑ i + h ψ ↓ ψ ↑ | η i , (18)where Π ( r , τ ) = − ( G , ↑ G , ↓ )( r , τ ) and Γ is the sum ofthe ladder diagrams, Γ − ( p , Ω n ) = U − − Π ( p , Ω n ) withΩ n the bosonic Matsubara frequencies.We first consider the diagrammatic scheme built on G and Γ . We denote by Σ ( n )lad [ G , Γ ] the sum of all self-energy diagrams of order n , i.e. containing n Γ -lines.This diagrammatic series is generated by the shifted ac-tion S ( ξ )lad [ ψ, ¯ ψ, η, ¯ η ] = h ψ | G − | ψ i − h η | Γ − | η i − ξ h η | Π | η i + ξ (cid:0) h η | ψ ↓ ψ ↑ i + h ψ ↓ ψ ↑ | η i (cid:1) , (19)in the sense that self-energy Σ lad ( ξ ) corresponding to thisaction has the Taylor series P ∞ n =1 Σ ( n )lad [ G , Γ ] ξ n . In-deed, the counter-term ξ Π cancels out the reduciblediagrams contatining G G bubbles. Therefore, if thisdiagrammatic series converges, then it yields the physi-cal self-energy. This follows from the same reasoning asbelow Eq. (5). The same applies to the series for the pairself-energy Π in terms of [ G , Γ ]. Here Π is defined byΓ − = Γ − − Π, where Γ denotes the fully dressed pairpropagator, used in [8, 11].More complex schemes, built on other dressed pairpropagators than Γ , can be generated by the shiftedaction S ( ξ ) N [ ψ, ¯ ψ, η, ¯ η ] = h ψ | G − | ψ i−h η | Γ − , N ( ξ ) | η i− ξ h η | Π | η i + ξ (cid:0) h η | ψ ↓ ψ ↑ i + h ψ ↓ ψ ↑ | η i (cid:1) , (20) where Γ − , N ( ξ ) = ˜Γ − N + ξ Ω + . . . + ξ N Ω N (21)and one imposes Γ , N ( ξ = 1) = Γ . In particular, thesemi-bold scheme is defined byΩ n = Π ( n )bold [˜Γ N ] , (22)where Π ( n )bold [ γ ] is the sum of all skeleton diagrams oforder n built with the pair-propagator γ that remainconnected when cutting two γ -lines. As usual, Π (1)bold = − GG + G G . This scheme was introduced previouslyfor N =1 [22].Finally, we consider the skeleton scheme built on G and Γ. Assuming that the skeleton sequence ˜Γ N con-verges to some ˜Γ, one can show analogously to the abovereasoning that ˜Γ is equal to the exact Γ under the fol-lowing sufficient condition:(i) for any ξ in a disc D = {| ξ | < R } of radius R >
1, andfor all ( p , Ω n ), M ( ξ ) N ( p , Ω n ) := P N n =1 Π ( n )bold [˜Γ N ]( p , Ω n ) ξ n converges for N →∞ ; moreover this sequence is uni-formly bounded, i.e. , there exists C ( p , Ω n ) such that ∀ ξ ∈ D , ∀ ( N , p , Ω n ) , | M ( ξ ) N ( p , Ω n ) | ≤ C ( p , Ω n ); and(ii) ˜Γ N ( p , Ω n ) is uniformly bounded. Screened interaction potential.
Finally, we briefly ad-dress the procedure of dressing the interaction line, whichis particularly important for long-range interaction po-tentials. Restricting for simplicity to a spin-independentinteraction potential V ( r ), the interaction part of the ac-tion writes12 X σ,σ ′ X r , r ′ Z β dτ ( ¯ ψ σ ψ σ )( r , τ ) V ( r − r ′ ) ( ¯ ψ σ ′ ψ σ ′ )( r ′ , τ ) . (23)We again keep the bare G for simplicity and considerdressing of V only. Introducing a real scalar Hubbard-Stratonovich field χ leads to the action S [ ψ, ¯ ψ, χ ] = h ψ | G − | ψ i + 12 h χ | V − | χ i + i X σ h χ | ¯ ψ σ ψ σ i . (24)Here we assume that the Fourier transform V ( q ) of theinteraction potential is positive, so that the quadraticform h χ | V − | χ i = (2 π ) − d R β dτ R d d q | χ ( q , τ ) | /V ( q ) ispositive definite. The auxiliary action takes the form S ( ξ ) N [ ψ, ¯ ψ, χ ] = h ψ | G − | ψ i + 12 h χ | ˜ V − N + ξ Ω + . . . + ξ N Ω N | χ i + iξ X σ h χ | ¯ ψ σ ψ σ i . (25)The semi-bold scheme corresponds to Ω n = Π ( n )bold [ ˜ V N ]where Π now stands for the polarization. In particular,˜ V is the RPA screened interaction.Summarizing, we have revealed an analytic structurebehind dressed-line diagrammatics. More precisely, wehave exhibited the function which analytically continuesa dressed diagrammatic series. This function originatesfrom an action that depends on an auxiliary parame-ter ξ . When the action is a polynomial in ξ , the situ-ation reduces to the one of a bare expansion. Withinthis category, a particular case well suited for numericalimplementation is the semi-bold scheme for which thebare propagator is taken from the truncated bold self-consistent equation. For the fully bold scheme, we con-struct an appropriate auxiliary action, but only undera certain condition. If this condition is verified numeri-cally, it is safe to use the fully bold scheme. If not, thesemi-bold scheme remains applicable.Furthermore we have demonstrated the generality ofthe shifted-action construction by treating the case ofa dressed pair propagator and of a screened long-rangeinteraction. Further extensions left for future work aredressing of three-point vertices, as well as justifying re-summation of divergent diagrammatic series by consid-ering non disc-shaped analyticity domains D .We are grateful to Youjin Deng, Evgeny Kozik, KrisVan Houcke, for valuable exchanges. This work wassupported by the National Science Foundation underthe grant PHY-1314735, The Simons Collaboration onthe Many Electron Problem, the MURI Program “NewQuantum Phases of Matter” from AFOSR, CNRS, ERCand IFRAF-NanoK (grants PICS 06220, Thermody-namix , and
Atomix ). R.R. and F.W. are affiliated to
Paris Sciences et Lettres, Sorbonne Universit´es , and
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