Gentle introduction to rigorous Renormalization Group: a worked fermionic example

A bstract Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed poin...

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Veröffentlicht in:The journal of high energy physics Jg. 2021; H. 1; S. 1 - 112
Hauptverfasser: Giuliani, Alessandro, Mastropietro, Vieri, Rychkov, Slava
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2021
Springer Nature B.V
Springer
SpringerOpen
Schlagworte:
Approximation
Banach spaces
Classical and Quantum Gravitation
Critical phenomena
Elementary Particles
Fermions
General Physics
High energy physics
High Energy Physics - Theory
Ising model
Mathematical Physics
Nonperturbative Effects
Parameter identification
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Quantum Physics
Regular Article - Theoretical Physics
Relativity Theory
Renormalization Group
String Theory
Renormalization Group
Nonperturbative Effects
nonlocal
nonperturbative
fermion: symplectic
renormalization group: fixed point
critical phenomena
Ising model
kinetic
ISSN:1029-8479, 1126-6708, 1029-8479
Online-Zugang:Volltext
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Zusammenfassung:A bstract Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non- perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal (“long-range”) kinetic term depending on a parameter ε and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with ϕ 4 or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in ε , a somewhat surprising fact relying on the fermionic nature of the problem.
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ISSN:1029-8479
1126-6708
1029-8479
DOI:10.1007/JHEP01(2021)026