Distribution of solutions of the fastest apparent convergence condition in optimized perturbation theory and its relation to anti-Stokes lines

Abstract

We discuss fundamental properties of the fastest apparent convergence (FAC) condition which is used as a variational criterion in optimized perturbation theory (OPT). We examine an integral representation of the FAC condition and a distribution of the zeros of the integral in a complex artificial parameter space on the basis of theory of Lefschetz thimbles. We find that the zeros accumulate on a certain line segment so-called anti-Stokes line in the limit K → ∞, where K is a truncation order of a perturbation series. This phenomenon gives an underlying mechanism that physical quantities calculated by OPT can be insensitive to the choice of the artificial parameter. * stsutsui@post.kek.jp 1 arXiv:1901.04353v1 [hep-th]

From weak to strong: constrained extrapolation of perturbation series with applications to dilute Fermi systems

“…In the original ODM method [15], this is implemented by fixing α(N) via the FAC ("fastest apparent convergence") criterion h N = 0 (see also Refs. [19,24]). Another heuristic prescription is the "principle of minimal sensitivity" (PMS) [36,41], meaning that α(N) should be chosen such that Z N (g; α) is least sensitive to variations of α about its chosen value (see also Refs.…”

“…This yields several possibilities for α(N), of which one is selected according to an additional criterion, e.g., smallest h N−1 . For the 0D model this approach converges to the exact solution as N → ∞ (the imaginary part of Z N (g) converges to zero for g ∈ [0, ∞]); the FAC criterion is not crucial for this, but suitable N dependence of α is [15,19,24,36,42,43].…”

“…0D model. A well-known benchmark for resummation methods is the 0D field theory model [2,15,[19][20][21][22][23][24]…”

Constrained Extrapolation Problem and Order‐Dependent Mappings

“…Two commonly used prescriptions for choosing (or , in optimized PT) are: 1) the criterion of “fastest apparent convergence (FAC)” [ 28,56 ] that fixes such that the order N coefficient in the re‐expanded series vanishes; and 2) the “principle of minimal sensitivity (PMS)” [ 29,57 ] meaning should be chosen such that the ODM approximant is least sensitive to variations of α about its chosen value.…”

Constrained extrapolation problem and order-dependent mappings

“…• the criterion of "fastest apparent convergence" or "FAC" [26,55] that fixes α(N ) such that the order N coefficient in the re-expanded series vanishes;…”