Reliability of the optimized perturbation theory in the 0-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.gif" display="inline" overflow="scroll"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> scalar field model

Rosa, D. S.; Farias, Ricardo L. S.; Ramos, Rudnei O.

doi:10.1016/j.physa.2016.07.067

Cited by 16 publications

(9 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…O(N ) models this initial condition is in the family of UV potentials discussed qualitatively at length and to some extent even quantitatively in Refs. [43,51,57]. In Fig.…”

Section: Rg Flow Of the N-point Vertex Functionsmentioning

confidence: 96%

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The $O(N)$ model

Koenigstein,

Steil,

Wink

et al. 2021

Preprint

View full text Add to dashboard Cite

The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows to reliably treat the emergence of non-analytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higherdimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.

show abstract

“…O(N ) models this initial condition is in the family of UV potentials discussed qualitatively at length and to some extent even quantitatively in Refs. [43,51,57]. In Fig.…”

Section: Rg Flow Of the N-point Vertex Functionsmentioning

confidence: 96%

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The $O(N)$ model

Koenigstein,

Steil,

Wink

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For related works on this toy model QFT, we again refer to Refs. [3,24,[33][34][35][36][37][38][39][40][41][42][43][44][45][48][49][50][51][52][53][54].…”

Section: The Zero Dimensional O(n )-Modelmentioning

confidence: 99%

“…[53] and Ref. [55], we also discuss at length and test the explicit implementation of the extremely important spatial (field space) boundary conditions for RG flow equations of type (39), which is not repeated at this point.…”

Section: B the Rg And (Numerical) Fluid Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part III: Shock and rarefaction waves in RG flows reveal limitations of the $N \rightarrow \infty$ limit in $O(N)$-type models

Steil,

Koenigstein

2021

Preprint

View full text Add to dashboard Cite

Using an O(N )-symmetric toy model QFT in zero space-time dimensions we discuss several aspects and limitations of the 1 N -expansion. We demonstrate, how slight modifications in a classical UV action can lead the 1 N -expansion astray and how the infinite-N limit may alter fundamental properties of a QFT. Thereby we present the problem of calculating correlation functions from two totally different perspectives: First, we explicitly analyze our model within an 1 N -saddle-point expansion and show its limitations. Secondly, we picture the same problem within the framework of the Functional Renormalization Group. Applying novel analogies between (F)RG flow equations and numerical fluid dynamics from parts I and II of this series of publications, we recast the calculation of expectation values of our toy model into solving a highly non-linear but exact advection(-diffusion) equation. In doing so, we find that the applicability of the 1 N -expansion to our toy model is linked to freezing shock waves in field space in the FRG-fluid dynamic picture, while the failure of the 1 N -expansion in this context is related to the annihilation of two opposing shock waves in field space.

show abstract

“…Optimized perturbation theory has been used for considering several field-theory models. Among them: the O(N)-symmetric ϕ 4 theory [51,[100][101][102][103][104][105][106][107][108][109][110], charged scalar field in an external field [111], Walecka nuclear-matter model [112], massive Schwinger model on a lattice [113], U(2) theory on a lattice [114], and Gross-Neveu model [115][116][117][118][119]. Note that the so-called screened perturbation theory and hard-thermal-loop resummation, used in field theory are just variants of optimized perturbation theory [120].…”

Section: Field-theory Modelsmentioning

confidence: 99%

Interplay between Approximation Theory and Renormalization Group

Yukalov

2019

Phys. Part. Nuclei

View full text Add to dashboard Cite

The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is Optimized Perturbation Theory now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is Self-Similar Approximation Theory allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.

show abstract

Reliability of the optimized perturbation theory in the 0-dimensional O(N) scalar field model

Cited by 16 publications

References 51 publications

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The $O(N)$ model

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The $O(N)$ model

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part III: Shock and rarefaction waves in RG flows reveal limitations of the $N \rightarrow \infty$ limit in $O(N)$-type models

Interplay between Approximation Theory and Renormalization Group

Contact Info

Product

Resources

About