The nonperturbative functional renormalization group and its applications (2006.04853v3)

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 different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation 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.

• The paper establishes the FRG as a nonperturbative approach that accurately captures phase transitions and critical scaling in strongly correlated systems.
• It details advanced methods such as the derivative and vertex expansions to enhance predictions of critical exponents and momentum-dependent correlation functions.
• The study explores diverse applications in statistical mechanics, quantum many-body systems, and quantum gravity, highlighting practical insights and future prospects.

The Nonperturbative Functional Renormalization Group and its Applications

The paper "The nonperturbative functional renormalization group and its applications" provides a comprehensive paper of the functional renormalization group (FRG) and its various applications across different fields of physics. The FRG is an advanced technique that extends the traditional renormalization group approaches by offering nonperturbative tools that go beyond standard perturbative methods. It resolves theoretical challenges when dealing with strongly correlated systems, statistical physics, quantum many-particle systems, high-energy physics, and quantum gravity.

Overview of the Functional Renormalization Group (FRG)

The FRG offers a framework to paper systems at varying energy scales by progressively integrating fluctuations from short to long distances. It efficiently addresses the consistent description of system behaviors across different scales. The formalism involves a scale-dependent effective action $\Gamma_k[\phi]$, governed by the exact Wetterich equation, which serves as a one-loop improved equation for renormalizing effective actions. The FRG is characterized by the functional form of its flow equations, enabling it to tackle nonperturbative problems.

Key Features and Approximations

• Derivative Expansion (DE): The DE is a central nonperturbative expansion method predominately used within the FRG. It expresses the effective action in terms of field derivatives up to a finite order. The local potential approximation (LPA) is the lowest order, capturing phase transitions and critical behaviors in models like the O$(N)$ models. Higher-order expansions provide improved accuracy for critical exponents and universal scaling functions.
• Vertex Expansion: This approach involves truncating the hierarchy of equations for vertices to capture momentum dependencies accurately. It helps in determining momentum-dependent correlation functions, which are crucial for understanding systems beyond mere static properties.
• Nonperturbative Aspects: The FRG effectively encompasses phenomena such as collective behaviors in strongly interacting systems, critical scaling, and phase transitions without relying on small expansion parameters typical for perturbation theories.

Applications Across Physics Domains

1. Statistical Mechanics:
   • The FRG successfully addresses critical exponents and scaling behaviors, outperforming perturbative RG in complex systems such as frustrated magnets.
   • It handles nonuniversal properties in disordered systems and provides insights into universality classes near phase transitions.
2. Quantum Many-Body Systems:
   • Models of strongly correlated bosons, such as superfluidity in Bose gases, have been studied extensively using FRG to derive excitation spectra and transition temperatures.
   • Quantum antiferromagnets, relativistic Bosons, and the Bose-Hubbard model demonstrate FRG's capability in tackling quantum phase transitions and emergent phenomena.
3. High Energy Physics and Quantum Gravity:
   • FRG enables research into quantum chromodynamics (QCD), addressing issues such as dynamic mass generation and the confinement-deconfinement transition.
   • Further, it investigates asymptotic safety in quantum gravity, providing a potential UV complete theory as an alternative to perturbatively nonrenormalizable gravitational theories.

Practical and Theoretical Implications

The paper elaborates on significant aspects and technicalities of FRG, offering a robust alternative to tackling some of the most pressing problems in modern theoretical physics. Its applicability to systems with strong quantum fluctuations and nontrivial phase structures makes it invaluable for theoretical and computational physics. The FRG's predictions on universal properties and its detailed characterization of phase transitions signify a strategic tool capable of rivaling conventional methods that are either computationally expensive or inherently limited by perturbative constraints.

Future Prospects

The versatility and growing computational refinement of FRG methodologies promise substantial advancement in numerous domains, including condensed matter physics, cosmology, and beyond. More sophisticated approximations and implementations will synergistically improve its predictive power, deepen insights into strongly correlated phenomena, and potentially unveil new physics.