Shifted-action expansion and applicability of dressed diagrammatic schemes

Abstract

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 constructed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can even lead to convergence to an unphysical result [Kozik, Ferrero, and Georges, PRL 114, 156402 (2015)]. Here we show that for a broad class of partially dressed diagrammatic schemes, there exists an action S (ξ) depending analytically on an auxiliary complex parameter ξ, such that the Taylor expansion in ξ of correlation functions reproduces the original diagrammatic series. The resulting applicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, analyticity of S (ξ) is not granted, and we formulate a sufficient condition for converging to the correct result. Much of theoretical physics is formulated in the language of Feynman diagrams, in various fields such as condensed matter, nuclear physics, and QCD. A powerful feature of the diagrammatic technique, used in each of the above fields, is the possibility to build diagrams on partially or fully dressed propagators or vertices, see, e.g., Refs. [1][2][3][4][5]. In quantum many-body physics, notable examples include dilute gases, whose description is radically improved if ladder diagrams are summed up so that the expansion is done in terms of the scattering amplitude instead of the bare interaction potential, and Coulomb interactions, which one has to screen to have a meaningful diagrammatic technique.With the development of Diagrammatic Monte Carlo, it becomes possible to compute Feynman diagrammatic expansions to high order for fermionic strongly correlated quantum many-body problems [6][7][8][9][10][11]. The number of diagrams grows factorially with the order, even for a fully irreducible skeleton scheme [12]. Nevertheless, for fermionic systems on a lattice at finite temperature, diagrammatic series (of the form n a n with a n the sum of all order-n diagrams) are typically convergent in a broad range of parameters, due to a nearly perfect cancellation of contributions of different sign within each order, as proven mathematically [13] and seen numerically [6,7,[9][10][11].One might think that partial or full renormalization of diagrammatic elements (propagators, interactions, vertices, etc.) always leads to more compact and better behaving diagrammatic expansion. However, such a dressed diagrammatic series cannot be used blindly: Even when it 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 the absolute values of individual diagrams is typically infinite, due to factorial scaling of the number of diagrams with the order). And indeed, for a skeleton series, i.e., a series built on the fully dressed propagator, convergence to a wrong result does occur in the case of the Hubbard model in the strongly correlated regime near ...

Hypergeometric resummation of self-consistent sunset diagrams for steady-state electron-boson quantum many-body systems out of equilibrium

“…Also, very recent studies have shown that self-consistent Φ-derivable approximations can converge to unphysical solutions; [34][35][36][37][38][39][40][41][42][43][44]47,48 similar issues have been noted in the context of time-dependent simulations [40][41][42][43][44][47][48][49] using the Kadanoff-Baym equations, but can be remedied by introducing self-consistent contributions in controlled fashion. 43 Nevertheless, the use of self-consistent MBPT may be justified to some extent.…”

Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

“…First specific reports about unexpected theoretical consequences induced by the breakdown of the many-body perturbation expansion have been recently presented by several groups. Such effects range from the occurrence [24][25][26][27] of low-frequency divergences of two-particle irreducible vertex functions in the Hubbard 28 and Falicov-Kimball 29 models to the multivaluedness [30][31][32][33][34][35][36] of the electronic self-energy expressed as a functional of the (interacting) Green's function or, equivalently, of the Luttinger-Ward functional. All these manifestations of non-perturbativeness are interconnected and represent different aspects of the same problem, as suggested in some of the abovementioned works.…”

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

“…Finally, we note that it is possible to generalize the method to more general diagrammatic schemes by the use of a shifted action [26], where one has to consider additional interaction vertices that act as counterterms.…”

“…All our error bars correspond to one standard deviation. We resum all bare tadpoles diagrams, whose effect is to shift the chemical potential µ(U ) = µ 0 + U n 0 /2, where µ 0 is the chemical potential needed to get the density n 0 in the absence of interactions (this corresponds to the first-order semi-bold scheme introduced in [26]). This is useful because one has a smaller density shift as a function of U .…”