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The Nonperturbative Functional Renormalization Group and Its Applications

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The Nonperturbative Functional Renormalization Group and Its Applications The nonperturbative functional renormalization group and its applications N. Dupuisa, L. Canetb, A. Eichhornc,d, W. Metznere, J. M. Pawlowskid,f, M. Tissiera, N. Wscheborg aSorbonne Universit´e,CNRS, Laboratoire de Physique Th´eoriquede la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France bUniversit´eGrenoble Alpes and CNRS, Laboratoire de Physique et Mod´elisationdes Milieux Condens´es,LPMMC, 38000 Grenoble, France cCP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark dInstitut f¨urTheoretische Physik, Universit¨atHeidelberg, Philosophenweg 16, 69120 Heidelberg, Germany eMax-Planck-Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany fExtreMe Matter Institute EMMI, GSI, Planckstr. 1, D-64291 Darmstadt, Germany gInstituto de F´ısica,Facultad de Ingenier´ıa,Universidad de la Rep´ublica,J.H.y Reissig 565, 11000 Montevideo, Uruguay Abstract The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on dif- ferent energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementa- tion of Wilson’s RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity. Contents 1 Introduction 3 1.1 Wilson’s renormalization group . .4 1.2 The functional renormalization group . .4 1.3 Scope of the review . .5 2 The FRG in a nutshell 6 2.1 The scale-dependent effective action . .6 2.1.1 The regulator function Rk .....................................7 2.1.2 General properties of the effective action Γk ...........................7 2.2 The exact flow equation . .8 2.3 The derivative expansion (DE) . .9 2.3.1 Local-potential approximation: LPA and LPA0 ......................... 10 2.3.2 Second order of the DE . 11 2.3.3 Validity of the DE . 13 2.3.4 Further results obtained from the DE . 13 2.4 Computing momentum-dependent correlation functions . 14 2.5 Lattice models and realistic microscopic actions . 15 arXiv:2006.04853v3 [cond-mat.stat-mech] 7 May 2021 2.6 Quantum models . 15 3 Statistical mechanics 15 3.1 Why and when is the FRG useful? . 15 3.2 Equilibrium statistical mechanics . 16 3.2.1 Frustrated magnets as an example . 16 Preprint submitted to Elsevier May 10, 2021 3.2.2 Critical phenomena and universal long distance regime . 19 3.2.3 Nonuniversal properties . 19 3.2.4 Conformal invariance and c-theorem . 20 3.3 Disordered systems . 20 3.3.1 Replica formalism and FRG . 20 3.3.2 The Random Field Ising Model . 21 3.3.3 Other disordered systems . 23 3.4 Classical non-equilibrium systems and FRG . 23 3.4.1 Langevin stochastic dynamics . 24 3.4.2 Master equations: reaction-diffusion processes . 30 3.4.3 Non-equilibrium quantum systems . 32 4 Quantum many-particle systems 33 4.1 Bosons . 33 4.1.1 Superfluidity in a dilute Bose gas . 33 4.1.2 Superfluid to Mott-insulator transition . 34 4.1.3 Relativistic bosons and quantum O(N) model . 35 4.2 Fermions . 36 4.2.1 Fermion flow equation . 37 4.2.2 Competing instabilities . 38 4.2.3 Spontaneous symmetry breaking . 41 4.2.4 Quantum transport . 42 4.2.5 Leap to strong coupling . 43 4.2.6 Omissions . 44 5 High-energy physics 45 5.1 Introduction: High-energy physics . 45 5.2 The functional renormalization group for gauge theories . 45 5.2.1 Gauge-fixed flows and modified Slavnov-Taylor identities . 46 5.2.2 Gauge invariance, locality & confinement . 50 5.2.3 Gauge-invariant flows & the quest for simplicity . 52 5.3 QCD.................................................... 53 5.3.1 Confinement and the flow of composite operators . 54 5.3.2 Chiral symmetry breaking . 56 5.3.3 Vacuum QCD and hadronic bound states . 57 5.4 Phase structure and dynamics of QCD . 60 5.4.1 The unreasonable effectiveness of low-energy effective theories . 60 5.4.2 The phase structure of QCD from low-energy effective models . 61 5.4.3 Towards the QCD phase structure from first principles . 63 5.5 Electroweak phase transition, BSM physics & Supersymmetry . 64 5.6 Summary . 65 6 Gravity 65 6.1 Introduction: Quantum gravity - why and what? . 65 6.2 Path integral for quantum gravity and asymptotic safety . 67 6.3 The FRG for quantum gravity: a brief manual . 68 6.4 Status and open questions of asymptotically safe gravity . 70 6.4.1 Indications for the Reuter fixed point . 70 6.4.2 Background field and dynamical field . 72 6.4.3 Key challenges for asymptotically safe gravity . 73 6.4.4 Understanding spacetime structure . 74 6.4.5 Asymptotically safe gravity and particle physics . 75 2 6.5 The FRG in other approaches to quantum gravity . 77 6.5.1 Lorentz-symmetry violating quantum gravity . 78 6.5.2 Discrete quantum-gravity models . 78 7 Acknowledgments 79 Appendix A The effective action formalism 79 Appendix A.1 Effective action . 80 Appendix A.2 1PI vertices . 80 Appendix A.3 Loop expansion . 81 Appendix B The FRG at work: the case of the derivative expansion 82 Appendix B.1 The exact flow equation . 82 Appendix B.2 The local potential approximation . 83 Appendix B.2.1 Scaling form of the LPA flow equation . 84 Appendix B.2.2 Fixed-point solutions and critical exponents . 84 Appendix B.2.3 Upper and lower critical dimensions . 87 Appendix B.2.4 Spontaneous symmetry breaking and approach to convexity . 87 Appendix B.3 Improving the LPA: the LPA0 ............................... 88 Appendix B.3.1 Critical behavior: the limits d ! 4, d ! 2 and N ! 1 ............... 89 Appendix B.3.2 Low-temperature phase . 90 Appendix B.4 Second-order of the derivative expansion . 91 Appendix B.4.1 Choice of the regulator . 91 Appendix B.4.2 Estimate of the error . 92 Appendix B.5 Fourth- and sixth-order of the derivative expansion . 93 1. Introduction Linking descriptions of physics at various scales and relating the macroscopic physical properties of systems to the microscopic interactions and degrees of freedom is the primary goal of research in many areas of physics, from condensed matter and cold atoms to high-energy physics and quantum gravity. The aim of this review is to describe a theoretical approach, the nonperturbative functional renormalization group (FRG), which provides us with an efficient and versatile tool to bridge the gap between micro- and macroscopic scales and thus determine the physical properties of a wide variety of systems. Strongly correlated systems, such as electrons in solids interacting via the Coulomb interaction or quarks in nucle- ons subjected to the strong interaction, although different in some respects share nevertheless a number of properties. When external parameters (temperature, density, etc.) are varied, they often exhibit rich phase diagrams due to com- peting collective phenomena. The theoretical study of their properties faces two major difficulties. First, there is often no small parameter that would allow for a systematic perturbative expansion. Second, whenever the degrees of free- dom are correlated over distances much larger than the microscopic scales, collective effects become important at low energies. This implies that these systems may exhibit very distinct behavior on different energy scales and the relevant degrees of freedom which permit a simple formulation of the low-energy (macroscopic) properties may be different from the microscopic ones. This diversity of scales explains the difficulty of a straightforward numerical solution of microscopic models, since the interesting phenomena emerge only at low energies and in large-size.
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