Physical and unphysical regimes of self-consistent many-body perturbation theory
SSciPost Physics Submission
Physical and unphysical regimesof self-consistent many-body perturbation theory
K. Van Houcke , E. Kozik , R. Rossi † , Y. Deng , and F. Werner Laboratoire de Physique de l’´Ecole normale sup´erieure, ENS - Universit´e PSL, CNRS,Sorbonne Universit´e, Universit´e de Paris, 75005 Paris, France Physics Department, King’s College, London WC2R 2LS, United Kingdom Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Shanghai Research Center for Quantum Science, Shanghai 201315, China Laboratoire Kastler Brossel, ´Ecole Normale Sup´erieure - Universit´e PSL, CNRS,Sorbonne Universit´e, Coll`ege de France, 75005 Paris, France † Present address: Institute of Physics, EPFL, 1015 Lausanne, Switzerland * [email protected] February 10, 2021
Abstract
In the standard framework of self-consistent many-body perturbation theory, theskeleton series for the self-energy is truncated at a finite order N and pluggedinto the Dyson equation, which is then solved for the propagator G N . For twosimple examples of fermionic models – the Hubbard atom at half filling and itszero space-time dimensional simplified version – we find that G N converges when N → ∞ to a limit G ∞ , which coincides with the exact physical propagator G exact atsmall enough coupling, while G ∞ (cid:54) = G exact at strong coupling. We also demonstratethat it is possible to discriminate between these two regimes thanks to a criterionwhich does not require the knowledge of G exact , as proposed in [1]. Self-consistent perturbation theory is a particularly elegant and powerful approach in quantummany-body physics [2, 3]. The single-particle propagator G is expressed through the Dysonequation G − = G − − Σ (1)in terms of the non-interacting propagator G and the self-energy Σ, which itself is formallyexpressed in terms of G through the skeleton seriesΣ = Σ bold [ G ] = ∞ (cid:88) n =1 Σ ( n )bold [ G ] (2)where Σ ( n )bold [ G ] is the sum of all two-particle irreducible Feynman diagrams of order n (builtwith bold propagator lines representing G ). 1 a r X i v : . [ c ond - m a t . s t r- e l ] F e b ciPost Physics Submission The standard procedure for solving Eqs. (1,2) is to truncate the skeleton series at a finiteorder N , and to look for the solution G N of the self-consistency equation G − N = G − − Σ ( ≤ N )bold [ G N ] (3)with Σ ( ≤ N )bold := (cid:80) Nn =1 Σ ( n )bold . The natural expectation is that one obtains the exact propagatorby sending the truncation order to infinity: G N → G exact for N → ∞ .However, as was discovered in [4], the series Σ ( ≤ N )bold [ G exact ] can converge when N → ∞ to aresult which differs from the exact physical self-energy Σ exact = G − − G − . This misleadingconvergence phenomenon was observed for three fermionic textbook models —Hubbard atom,Anderson impurity model, and half-filled 2D Hubbard model— in a region of the parameterspace (at and around half filling, at strong interaction and low temperature). G exact wascomputed with a numerically exact quantum Monte Carlo method, and the skeleton serieswas evaluated up to N = 6 or 8 by diagrammatic Monte Carlo [5]. Numerous works [1, 6–14]have studied various aspects of the problem found in [4], as well as the related divergences ofirreducible vertices ([8, 10, 11, 13, 15–18] and Refs. therein). In particular, Ref. [7] introducedan exactly solvable toy model, which has the structure of a fermionic model in zero space-timedimensions, and features the misleading convergence problem of [4].In this article, we study the consequences of this problem for the sequence G N , which is thecrucial question in the most relevant cases where G exact is unknown. For the toy model of [7],we find that G N converges when N → ∞ to a limit G ∞ which differs from G exact at strongcoupling. For the Hubbard atom, our numerical data strongly indicate that such misleadingconvergence of the sequence G N also occurs at large coupling and half filling. Moreover, wedemonstrate that a criterion proposed in [1] allows to discriminate between the G ∞ (cid:54) = G exact and G ∞ = G exact regimes without using the knowledge of G exact .We note that although we restrict here to the scheme (1,2) where G is the only bold ele-ment, our findings may also be relevant to other schemes containing additional bold elements,such as a bold interaction line W , or a bold pair propagator line Γ. The scheme built with G and W is natural for Coulomb interactions, and is widely used for solids and molecules witha truncation order N = 1 (the GW approximation) and sometimes with N = 2 (see, e.g. ,Refs. [19–22]), while for several paradigmatic lattice models, bold diagrammatic Monte Carlo(BDMC) made it possible to reach larger N and claim a small residual trunction error [23–26].The scheme built with G and Γ is natural for contact interactions; truncation at order N = 1then corresponds to the self-consistent T-matrix approximation [27–29], and precise large- N results were obtained by BDMC in the normal phase of the Hubbard model [30, 31] and ofthe unitary Fermi gas [32–34]. Other BDMC results were obtained for models of coupledelectrons and phonons, where it is natural to introduce a bold phonon propagator [23, 35],and for frustrated spins [36–38]. Schemes containing three- or four-point bold vertices werealso employed, to construct extensions of dynamical mean-field theory [17, 39]. We assume that the solution G N of (3) is unique, or at least that there is no difficulty in identifying aunique physical solution ( e.g. , by starting from the weakly interacting limit where G N → G , and followingthe solution as a function of interaction strength). ciPost Physics Submission We begin with some reminders from [7]. While fermionic many-body problems can be repre-sented by a functional integral over Grassmann fields , which depend on d space coordinatesand one imaginary time coordinate [40], in this simplified toy model the Grassmann fields arereplaced with Grassmann numbers ϕ s and ¯ ϕ s that do not depend on anything, apart from aspin index s ∈ {↑ , ↓} . The partition function, the action and the propagator are then definedby Z = (cid:90) (cid:32)(cid:89) s dϕ s d ¯ ϕ s (cid:33) e − S [ ¯ ϕ s ,ϕ s ] S [ ¯ ϕ s , ϕ s ] = − µ (cid:88) s ¯ ϕ s ϕ s + U ¯ ϕ ↑ ϕ ↑ ¯ ϕ ↓ ϕ ↓ G = − Z (cid:90) (cid:32)(cid:89) s dϕ s d ¯ ϕ s (cid:33) e − S [ ¯ ϕ s ,ϕ s ] ϕ s ¯ ϕ s , the dimensionless parameters µ and U being the analogs of chemical potential and interactionstrength. We restrict for convenience to µ > µ essentially amounts tothe change of variables ϕ ↔ ¯ ϕ ) and to U < bold [ G ] = ∞ (cid:88) n =1 a n G n − U n with a n = ( − n − (2 n − n !( n − . It is convenient to work with rescaled variables, multiplying propagators with (cid:112) | U | anddividing self-energies with the same factor, g := G (cid:112) | U | , σ := Σ / (cid:112) | U | . (4)The rescaled skeleton series is then given by σ bold ( g ) = ∞ (cid:88) n =1 σ ( n )bold ( g ) with σ ( n )bold ( g ) = a n ( − n g n − and accordingly σ ( ≤ N )bold ( g ) ≡ (cid:80) Nn =1 σ ( n )bold ( g ).The exact self-energy and propagator are given by σ exact ( g ) = − g g exact ( g ) = g g in terms of the rescaled free propagator g := (cid:112) | U | G = (cid:112) | U | /µ .If one evaluates the bold series at the exact G , one obtains the correct physical self-energyfor | U | < µ and an incorrect result for | U | > µ . More precisely, the self-energy functional(which reduces to a function in this toy model) has the two branches σ ( ± ) ( g ) = − ± (cid:112) − g g (5)3 ciPost Physics Submission Figure 1: The two branches of the self-energy as a function of the full propagator, for thetoy model in zero space-time dimensions. The skeleton series converges up to g = 1 / σ bold ( g ) = σ (+) ( g ) for g ≤ / g <
1, and the ( − )branch for g > i.e. , σ exact ( g ) = σ (sign(1 − g )) ( g exact ( g )). On the other hand, the boldseries, evaluated at the exact physical propagator, always converges to the (+) branch; i.e. , σ bold ( g exact ( g )) = σ (+) ( g exact ( g )) for all g > σ bold ( g ) is the expansion of σ (+) ( g ) in powers of g , and thus from (5) theconvergence radius of the series σ bold ( g ) is 1/2. We will refer to the sequence G N as the skeleton sequence. Rescaling variables as in (4), inparticular setting g N := G N (cid:112) | U | , the self-consistency equation (3) becomes1 g N = 1 g − σ ( ≤ N )bold ( g N ) . (6)This equation is readily solved for g N numerically: The solutions are roots of a polynomialof order 2 N , and we observe that there is a unique real positive root, which we take to be g N (recall that the exact g is always real and positive); alternatively, we solved Eq. (6) byiterations (with a damping procedure described in the next Section), and we found convergenceto this same g N . We find that • for g < g N −→ N →∞ g exact ( g ) • for g > g N −→ N →∞ g ∞ (cid:54) = g exact ( g ) i.e. , the skeleton sequence converges to the correct physical result below a critical couplingstrength, and to an unphysical result above it.4 ciPost Physics Submission Let us focus on the regime g >
1, where the convergence to an unphysical result takesplace (as demonstrated in Fig. 2). The fact that the skeleton sequence converges at all in thisregime is non-trivial. The value of the unphysical limit g ∞ = 1 / g N is equal to the radius of convergence of the skeleton series σ bold ( g ). This is not a coincidence,and the reason for this self-tuning towards the convergence radius becomes clear from Fig. 3:For a large truncation order, the curve representing the truncated bold series as a functionof g becomes an almost vertical line above the position of the convergence radius ( g = 1 / g . It also becomes clear thatwe are in an unusual situation wherelim N →∞ σ ( ≤ N )bold ( g N ) (cid:54) = lim N →∞ σ ( ≤ N )bold ( g ∞ ) ≡ σ bold ( g ∞ ) . In the general case where G exact is unknown, when one observes numerically that G N convergesto some limit, one needs a way to tell whether or not this limit is equal to G exact . Assumingthat G N → G ∞ for N → ∞ , the following criterion [1] is a sufficient condition for G ∞ to beequal to G exact : There exists (cid:15) > such that for any ξ in the disc D = {| ξ | < (cid:15) } , Σ N,ξ converges for N → ∞ ;moreover, this sequence is uniformly bounded for ξ ∈ D where Σ N,ξ := N (cid:88) n =1 Σ ( n )bold [ G N ] ξ n . (7)For all practical purpose, we expect this criterion to be essentially equivalent to the followingsimpler one: There exists ξ > such that Σ N,ξ converges for N → ∞ . (8)In what follows we will use this simplified criterion. We also introduce an extra factor 1 /ξ N in the definition (7) of Σ N,ξ , where the value of N will be conveniently chosen; such an N -independent factor does not matter for the criterion (it does not change whether or not thesequence Σ N,ξ converges).For the toy-model, this means that assuming g N → g ∞ for N → ∞ , a sufficient conditionfor g ∞ to be equal to the correct physical g exact ( g ) is that there exists ξ > σ N,ξ := N (cid:88) n =1 σ ( n )bold ( g N ) ξ n − = σ ( ≤ N )bold ( g N (cid:112) ξ ) / (cid:112) ξ (9)converges for N → ∞ . As illustrated in Fig. 4, this criterion indeed allows to detect themisleading convergence for g >
1, and to trust the result for g < ciPost Physics Submission g N Figure 2:
Illustrative example of misleading convergence of the skeleton sequence for the toymodel.
The rescaled propagator g N , obtained from the self-consistency equation with theskeleton series truncated at order N , converges for N → ∞ to the limit 0.5, which differsfrom the exact result (dashed line). This happens when the rescaled free propagator g > g = 1 . Explanation for the misleading convergence.
The two branches of the self-energy σ ( ± ) ( g ), together with the partial sums of the skeleton series σ ( ≤ N )bold ( g ) for different valuesof the truncation order N . Also shown is the curve corresponding to the Dyson equation, − σ = 1 /g − /g . This Dyson-equation curve intersects σ ( ≤ N )bold ( g ) at g = g N , whereas theexact propagator g = g exact is given by the intersection of the Dyson-equation curve with thephysical branch σ (sign(1 − g )) ( g ). It appears clearly that for g < g N converges to the exact g , while for g > g N always tends to 1 /
2, the convergence radius of the skeleton series.6 ciPost Physics Submission
Figure 4:
Detecting the misleading convergence for the toy model.
Introducing a finite ξ , thesequence becomes divergent which allows to detect the problem (left), or remains convergentwhich allows to trust the result (right). We turn to the single-site Hubbard model, defined by the grand-canonical Hamiltonian − µ (cid:80) s n s + U n ↑ n ↓ . The propagator can be expressed as a functional integral over β -antiperiodicGrassmann fields [40], G s ( τ ) = − (cid:82) D ϕ D ¯ ϕ ϕ s ( τ ) ¯ ϕ s (0) e − S (cid:82) D ϕ D ¯ ϕ e − S (10)with the action S = (cid:90) β dτ (cid:34) − (cid:88) s ¯ ϕ s ( τ )( G − ϕ s )( τ ) + U ( ¯ ϕ ↑ ¯ ϕ ↓ ϕ ↓ ϕ ↑ )( τ ) (cid:35) (11)and G − = µ − ddτ . (12)We restrict for simplicity to the half-filled case µ = U/
2, which should be the most danger-ous case, since it is at and around half-filling that the misleading convergence of Σ bold [ G exact ]was discovered in [4]. We use the BDMC method [5, 32, 41, 42] to sum all skeleton diagramsand solve the self-consistency equation (3) for truncation orders N ≤ ( n )bold = 0 for all odd n > G N can also converge to an unphysicalresult, or equivalently, whether Σ ( ≤ N )bold [ G N ] =: Σ N can converge to an unphysical result. Letus first consider the double occupancy D = (cid:104) n ↑ n ↓ (cid:105) = U − tr(Σ G ) (13)and the corresponding sequence D N := U − tr(Σ N G N ). At large enough U , our data stronglyindicate that this sequence does converge (albeit slowly) towards an unphysical result, see leftpanel of Fig. 5. For small enough U , there is a fast convergence to the correct result, see rightpanel of Fig. 5. 7 ciPost Physics Submission D N β U=8 0.188 0.19 0.192 0 0.1 0.2 0.3 0.4 0.5 D N β U=1
Figure 5: For the Hubbard atom at half filling, the double occupancy, as obtained from theskeleton sequence, converges to an unphysical result for large U (left) and to the correct resultfor small enough U (right) when the truncation order N → ∞ (dashed line: exact result).The next question is whether the criterion (8) allows us to discriminate between these twosituations. We therefore plot the sequence Σ N,ξ in Figs. 6 and 7. We only show the imaginarypart because in the considered half-filled case, the real part of Σ N ( ω n ) automatically equals U/
2; moreover we focus for simplicity on the lowest Matsubara frequency ω = π/β , and wechoose N = 2.For ξ = 1, Σ N,ξ reduces to the original skeleton sequence Σ N , and the behavior is similar tothe double occupancy: The sequence appears to converge, albeit slowly, towards an unphysicalresult for βU = 8 (Fig. 6), while fast convergence to the correct physical result takes place for βU = 1 (Fig. 7). For ξ >
1, the sequence does not appear to converge any more for βU = 8,see Fig. 6: The criterion correctly indicates that the results should not be trusted in this case.In contrast, for βU = 1, the criterion allows to validate the results, since the sequence remainsconvergent at ξ >
1, see Fig. 7. Regarding the choice of ξ , it should be neither too small inorder to have an effect at the accessible orders, nor too large to avoid making the criteriontoo conservative; here we see that ξ = 1 . N → ∞ occurs, convergence as a function of iterations at fixed N only takes place if we use a damping procedure, where G N at iteration ( i + 1) is obtained as G ( i +1) N = [ G − − Σ ( i ) ] − with Σ ( i ) a weighted average of Σ ( ≤ N )bold ( G ( i ) N ) and Σ ( i − [while the fixedpoint is unstable for the undamped iterative procedure Σ ( i ) := Σ ( ≤ N )bold ( G ( i ) N ) ]. Such a dampingprocedure is commonly used in BDMC where it also reduces the statistical error [42, 43]. Inthe toy model, one can easily show that an increasingly strong damping is required when N isincreased, because for N → ∞ , the slope [ dσ ( ≤ N )bold ( g ) /dg ] g = g N diverges, making the undampediterative procedure unstable. This observation could also be useful for misleading-convergencedetection. 8 ciPost Physics Submission -3-2-1 0 0.1 0.2 0.3 0.4 0.5 I m Σ N , ξ ( ω ) β U = 8 ξ =1exact ξ =1.1 ξ =1.2 Figure 6: For the half-filled Hubbard atom at large coupling, the original skeleton sequence( ξ = 1) converges to an unphysical result. At ξ >
1, the sequence does not converge any more:The criterion allows to detect the misleading convergence. -0.042-0.04-0.038-0.036 0 0.1 0.2 0.3 0.4 0.5 I m Σ N , ξ ( ω ) β U = 1 ξ =1exact ξ =1.1 ξ =1.2 Figure 7: For the half-filled Hubbard atom at small enough coupling, the original skeletonsequence ( ξ = 1) converges to the correct physical result. At ξ >
1, the sequence remainsconvergent: The criterion allows to trust the result.9 ciPost Physics Submission
Finally, we comment on the link with the multivaluedness of the self-energy functional Σ[ G ]( i.e. , of the Luttinger-Ward functional). In [4], the misleading convergence of the skeletonseries was found to be towards an unphysical branch of the self-energy functional, in the sensethat if Eqs. (10,11) are viewed as a mapping G (cid:55)→ G [ G ], then there exists G , unphys suchthat Σ bold [ G exact ] = G − , unphys − G − and G [ G , unphys ] = G exact ≡ G [ G ]. As noted in [4],this G , unphys does not belong to the set of physical bare propagators which are of the form(12) for some value of chemical potential; therefore, by looking at G , unphys , one can tell thatthe result is on an unphysical branch, and hence detect the misleading convergence of theskeleton series. In contrast, the misleading convergence of the skeleton sequence found herecannot be detected in this way. Indeed, the self-consistency equation (3) is enforced with theoriginal physical G . We have observed that there is a regime where the solution of self-consistent many-body per-turbation theory converges to an unphysical result in the limit of infinite truncation orderof the skeleton series. This surprising breakdown of the standard framework results from asubtle mathematical mechanism which we have clarified by analyzing the zero space-space di-mensional model. In this problematic regime, lowest order calculations can be off by one orderof magnitude, but access to higher orders allows to detect the problem numerically throughthe divergence of a slightly modified sequence, whereas seeing convergence of this modifiedsequence allows to rule out misleading convergence and to trust the result, as proposed in [1]and demonstrated here for the Hubbard atom. Such a proof of principle is relevant formany-body problems in regimes where, in spite of important progress with non self-consistentframeworks [44–54] (for which misleading convergence generically does not occur [1]), self-consistent BDMC remains the state of the art to date. In particular, the present findingsserved as a basis to discriminate between physical and unphysical BDMC results for thedoped two-dimensional Hubbard model at strong coupling in a non-Fermi liquid regime [55].
Acknowledgements
We thank N. Prokof’ev, B. Svistunov and L. Reining for useful discussions and comments.
Funding information
F.W. acknowledges support from H2020/ERC Advanced grant Critisup2(No. 743159), E.K. from the Simons Foundation through the Simons Collaboration on theMany Electron Problem and from EPSRC (grant No. EP/P003052/1), and Y.D. from theNational Natural Science Foundation of China (grant No. 11625522) and the Science andTechnology Committee of Shanghai (grant No. 20DZ2210100). The Flatiron Institute is adivision of the Simons Foundation. 10 ciPost Physics Submission
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