Solving functional flow equations with pseudospectral methods

Borchardt, Julia; Knorr, Benjamin

doi:10.1103/physrevd.94.025027

Cited by 58 publications

(55 citation statements)

References 103 publications

(116 reference statements)

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“…In [225], several different methods for high accuracy calculations are discussed, which also include solution methods for global descriptions of the potential. More recently, pseudo-spectral methods have been used as a high-accuracy method making use of Chebyshev polynomials as basis functions for a global description of the potential in field space [226,227], successfully benchmarked against known high-accuracy results, and among other applications employed to solve RG flows for O(N ) models in d = 3 and fractal d = 2.4 dimensions.…”

Section: Functional Renormalization Group and Wetterich Equationmentioning

confidence: 99%

Modeling finite-volume effects and chiral symmetry breaking in two-flavor QCD thermodynamics

Klein

2017

Physics Reports

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Finite-volume effects in Quantum Chromodynamics (QCD) have been a subject of much theoretical interest for more than two decades. They are in particular important for the analysis and interpretation of QCD simulations on a finite, discrete space-time lattice. Most of these effects are closely related to the phenomenon of spontaneous breaking of the chiral flavor symmetry and the emergence of pions as light Goldstone bosons. These long-range fluctuations are strongly affected by putting the system into a finite box, and an analysis with different methods can be organized according to the interplay between pion mass and box size. The finite volume also affects critical behavior at the chiral phase transition in QCD. In the present review, I will be mainly concerned with modeling such finite volume effects as they affect the thermodynamics of the chiral phase transition for two quark flavors.I review recent work on the analysis of finite-volume effects which makes use of the quark-meson model for dynamical chiral symmetry breaking. To account for the effects of critical long-range fluctuations close to the phase transition, most of the calculations have been performed using non-perturbative Renormalization Group (RG) methods. I give an overview over the application of these methods to a finite volume. The method, the model and the results are put into the context of related work in random matrix theory for very small volumes, chiral perturbation theory for larger volumes, and related methods and approaches. They are applied towards the analysis of finite-volume effects in lattice QCD simulations and their interpretation, mainly in the context of the chiral phase transition for two quark flavors.

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Section: Functional Renormalization Group and Wetterich Equationmentioning

confidence: 99%

Modeling finite-volume effects and chiral symmetry breaking in two-flavor QCD thermodynamics

Klein

2017

Physics Reports

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“…64, the fixed point structure of the O(N ) ⊕ O(M )-model was studied pseudo-spectrally between two and three dimensions. Lately, the method was extended to the integration of flows 65,66 . In contrast to the situation in perturbation theory, not much is known about the convergence properties of approximations to the exact renormalisation group from first principles.…”

Section: Introductionmentioning

confidence: 99%

Ising and Gross-Neveu model in next-to-leading order

Knorr

2016

Phys. Rev. B

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We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalisation group. Their critical behaviour is of special interest in condensed matter systems, in particular graphene. To derive the beta functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudo-spectral methods to guarantee high accuracy. New estimates on critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalisation group studies of the same level of approximation. By contrast, the approximation for the Gross-Neveu model retains many more operators than all earlier studies. For two Dirac fermions, the results agree with both lattice and large-N f calculations, but for a single flavour, different methods disagree quantitatively, and further studies are necessary.

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“…This is a manifestation of the standard hierarchy problem in the present setting. Computing such a functional flow for a full potential over many orders of magnitude represents a viable challenge for modern FRG PDE solvers [91].…”

Section: Ir Flows and Mass Spectrummentioning

confidence: 99%

Non-Abelian Higgs models: Paving the way for asymptotic freedom

Gies¹,

Zambelli²

2017

Phys. Rev. D

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Asymptotically free renormalization group trajectories can be constructed in nonabelian Higgs models with the aid of generalized boundary conditions imposed on the renormalized action. We detail this construction within the languages of simple low-order perturbation theory, effective field theory, as well as modern functional renormalization group equations. We construct a family of explicit scaling solutions using a controlled weak-coupling expansion in the ultraviolet, and obtain a standard Wilsonian RG relevance classification of perturbations about scaling solutions. We obtain global information about the quasi-fixed function for the scalar potential by means of analytic asymptotic expansions and numerical shooting methods. Further analytical evidence for such asymptotically free theories is provided in the large-N limit. We estimate the long-range properties of these theories, and identify initial/boundary conditions giving rise to a conventional Higgs phase.

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Solving functional flow equations with pseudospectral methods

Cited by 58 publications

References 103 publications

Modeling finite-volume effects and chiral symmetry breaking in two-flavor QCD thermodynamics

Modeling finite-volume effects and chiral symmetry breaking in two-flavor QCD thermodynamics

Ising and Gross-Neveu model in next-to-leading order

Non-Abelian Higgs models: Paving the way for asymptotic freedom

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