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

Abstract

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 us to reliably treat the emergence of nonanalytic 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 higher-dimensional 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.

arXiv comment68 pages (main text 56, appendix 12), 4 tables, 39 figures (plot data included in arXiv source file); Updated, published version

References in corpus (98)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Exact evolution equation for the effective potential
• Modern Theory of Nuclear Forces
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• Nonperturbative Evolution Equation for Quantum Gravity
• The Exact Renormalisation Group and Approximate Solutions
• Aspects of the Functional Renormalisation Group
• On Renormalization Group Flows in Four Dimensions
• The nonperturbative functional renormalization group and its applications
• Optimisation of the exact renormalisation group
• Quantum Field Theory in the Large N Limit: a review
• Renormalization Group Flow of Quantum Gravity in the Einstein-Hilbert Truncation
• The QCD phase structure at finite temperature and density
• Critical Exponents from the Effective Average Action
• Effective Field Theory Approach to High-Temperature Thermodynamics
• Flow Equations for N Point Functions and Bound States
• Fundamentals of the Exact Renormalization Group
• QCD at finite temperature and chemical potential from Dyson-Schwinger equations
• Chiral symmetry breaking in continuum QCD
• Landau gauge Yang-Mills correlation functions
• Non-perturbative quark, gluon and meson correlators of unquenched QCD
• A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations
• Spectral Methods for Numerical Relativity
• Critical exponents from optimised renormalisation group flows
• Optimization of the derivative expansion in the nonperturbative renormalization group
• Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order ${\partial }^4$
• The QCD equation of state from the lattice
• GRHydro: A new open source general-relativistic magnetohydrodynamics code for the Einstein Toolkit
• Convergence of Non-Perturbative Approximations to the Renormalization Group
• Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory
• Quark-meson-diquark model for two-color QCD
• Ising exponents from the functional renormalisation group
• Improved Polyakov-loop potential for effective models from functional calculations
• Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group
• Thermodynamics of QCD at vanishing density
• Higher order quark-mesonic scattering processes and the phase structure of QCD
• Non-perturbative renormalization group preserving full-momentum dependence: implementation and quantitative evaluation
• Analytical Solutions of Exact Renormalization Group Equations
• Renormalization Group Flow and Equation of State of Quarks and Mesons
• Solutions of renormalization group flow equations with full momentum dependence
• Solving non-perturbative flow equations
• Effective average action in statistical physics and quantum field theory
• Towards an accurate determination of the critical exponents with the Renormalization Group flow equations
• Fluctuation-induced modifications of the phase structure in (2+1)-flavor QCD
• Large N and the renormalization group
• Fluctuations and the axial anomaly with three quark flavors
• Non-perturbative finite-temperature Yang-Mills theory
• Two loop results from one loop computations and non perturbative solutions of exact evolution equations
• Global solutions of functional fixed point equations via pseudo-spectral methods
• Fierz-complete NJL model study III: Emergence from quark-gluon dynamics
• A proof of the irreversibility of renormalization group flows in four dimensions
• High-accuracy scaling exponents in the local potential approximation
• On the Convergence of the Expansion of Renormalization Group Flow Equation
• Universality and the Renormalisation Group
• Solving functional flow equations with pseudo-spectral methods
• Finite temperature spectral functions in the O(N)-model
• Linking the Quark Meson Model with QCD at High Temperature
• Fierz-complete NJL model study: fixed points and phase structure at finite temperature and density
• Spontaneous Symmetry Breaking and Proper-Time Flow Equations
• Fierz-complete NJL model study II: towards the fixed-point and phase structure of hot and dense two-flavor QCD
• Second-order and Fluctuation-induced First-order Phase Transitions with Functional Renormalization Group Equations
• Gradient Flows from an Approximation to the Exact Renormalization Group
• Complex Langevin simulations and the QCD phase diagram: Recent developments
• Shocks and quark-meson scatterings at large density
• RG flows, cycles, and c-theorem folklore
• Asymptotic safety of scalar field theories
• Why Might the Standard Large $N$ Analysis Fail in the O($N$) Model: The Role of Cusps in the Fixed Point Potentials
• Functional Renormalization Group and Kohn-Sham scheme in Density Functional Theory
• Functional renormalization group and 2PI effective action formalism
• Renormalization Group Studies of Dense Relativistic Systems
• An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach
• The non-perturbative renormalization group in the ordered phase
• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows
• Two-particle irreducible functional renormalization group schemes---a comparative study
• Finite-Temperature Gluon Condensate with Renormalization Group Flow Equations
• Phase structure of NJL model with weak renormalization group
• Towards a Renormalization Group Approach to Density Functional Theory - General Formalism and Case Studies -
• Towards a $C$-function in 4D quantum gravity
• Low-energy limit of the $O(4)$ quark-meson model from the functional renormalization group approach
• Remarks on Exact RG Equations
• Onset of symmetry breaking by the functional RG method
• Competition of inhomogeneous chiral phases and two-flavor color superconductivity in the NJL model
• C Function Representation of the Local Potential Approximation
• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. III. Shock and rarefaction waves in RG flows reveal limitations of the $N\to \infty$ limit in $O(N)$-type models
• QCD-inspired determination of NJL model parameters
• Visualising quantum effective action calculations in zero dimensions
• Reliability of the optimized perturbation theory in the 0-dimensional $O(N)$ scalar field model
• The ordered phase of $O(N)$ model within the Non-Perturbative Renormalization Group
• Dynamical generation of low-energy couplings from quark-meson fluctuations
• A functional RG equation for the c-function
• Structure of the broken phase of the sine-Gordon model using functional renormalization
• Alternative flow equation for the functional renormalization group
• The zero-dimensional O(N) vector model as a benchmark for perturbation theory, the large-N expansion and the functional renormalization group
• Functional and Local Renormalization Groups
• Large N limit of O(N) vector models
• Fluctuation-induced higher-derivative couplings and infrared dynamics of the Quark-Meson-Diquark Model
• Field dependence of the Yukawa coupling in the three flavor quark-meson model
• The $O(4)$ ${\varphi }^4$ model as an effective light meson theory: A lattice-continuum comparison

Cited by in corpus (15)

• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows
• Numerical RG-time integration of the effective potential: Analysis and Benchmark
• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. III. Shock and rarefaction waves in RG flows reveal limitations of the $N\to \infty$ limit in $O(N)$-type models
• Functional flows for complex effective actions
• Dissipation dynamics of a scalar field
• Regulator scheme dependence of the chiral phase transition at high densities
• Ordinary and exotic mesons in the extended Linear Sigma Model
• Field Theory Approaches to Relativistic Hydrodynamics
• Addressing energy density functionals in the language of path-integrals I: Comparative study of diagrammatic techniques applied to the (0+0)-D $O(N)$-symmetric ${\phi }^4$-theory
• Addressing energy density functionals in the language of path-integrals II: Comparative study of functional renormalization group techniques applied to the (0+0)-D $O(N)$-symmetric ${\phi }^4$-theory
• Tunneling with physics-informed RG flows in the anharmonic oscillator
• Renormalization Flow of Nonlinear Electrodynamics
• Vertex functions and their flow equations from the 2PI effective action
• Physics-informed operator flows and observables
• Fluctuation-induced first-order superfluid transition in unitary $\mathrm{SU}(N)$ Fermi gases