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 school 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 nucleons 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 competing 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 freedom 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 systems. It is also responsible for the fact that perturbation theories are often plagued with infrared divergences and may be inapplicable even at weak coupling. Conversely, in fundamental physics one is often confronted with the opposite problem: While the low-energy description is known, one searches for a consistent underlying microphysics. Many of the technical challenges and conceptual insights — maybe surprisingly — resemble those of the previous examples.
Wilson’s renormalization group
The renormalization group (RG) is a natural framework to study systems with many degrees of freedom correlated over long distances. In Wilson’s modern formulation, fluctuations at short distances and high energies are progressively integrated out to obtain an effective (coarse-grained) description at long distances and low energies. The RG not only gives us an explanation of cooperative behavior and universality but also provides us with a practical tool to study systems where correlations and fluctuations play an important role, the prime example being systems in the vicinity of a second-order phase transition. In high-energy physics, the RG provides us with a powerful conceptual understanding of fundamental interactions, as it allows to distinguish effective theories (which break down in the UV limit, e.g., due to the triviality problem) from fundamental theories which hold over an infinite range of scales due to asymptotic freedom or safety.
In its standard formulation however, the RG usually relies on perturbation theory and is therefore restricted to weakly interacting systems where a small expansion parameter allows one to systematically compute the effects of fluctuations beyond the noninteracting limit or the mean-field theory. For example in one of the most studied models, the O(N) model (φ4 theory with O(N) symmetry), the critical exponents are computed either from a Ginzburg-Landau-Wilson functional in an ε=4-d expansion or from the nonlinear sigma model in an ε=d-2 expansion (with d being the space dimension). In the latter case the perturbative RG series is usually considered as useless due to the lack of Borel summability. Thus even high-order perturbative RG expansions cannot relate the two expansions, except in the large-N limit. This is not crucial for the O(N) model where the critical behavior does not change qualitatively for 2<d<4 but in other models forbids a completely coherent picture of the physics between d=2 and d=4 (or, more generally, between the lower and upper critical dimensions).
Moreover, even when the field theoretical perturbative RG yields an accurate determination of universal quantities (e.g., the critical exponents or universal scaling functions), it is often not clear how to compute nonuniversal quantities such as a transition temperature, the phase diagram of equilibrium and out-of-equilibrium systems, the spectrum of bound states in strongly correlated theories, etc., since these properties depend on the underlying microscopic models.
Last, the perturbative RG is useless for genuinely nonperturbative problems such as the Berezinskii-Kosterliz-Thouless (BKT) transition in the two-dimensional O(2) model, the growth of stochastically growing interfaces, turbulent flows, the confinement of quarks in quantum chromodynamics (QCD), gravity at the Planck scale, etc. In the latter case, a perturbative analysis indicates perturbative non-renormalizability, rendering non-perturbative tools necessary to describe quantum gravity beyond the Planck scale.
The functional renormalization group
Although various systems may be characterized by different microscopic energy scales (e.g., 10-7 meV for a ultracold atomic gas, 1 eV for conduction electrons in a solid, 1 GeV for QCD, 125 GeV for the Higgs particle, 1019 GeV for the Planck scale in quantum gravity), from a theoretical point of view there is no fundamental difference between relativistic quantum field theory (that describes elementary particles and their interactions) and statistical field theory (that describes the statistical properties of quantum or classical systems where the degrees of freedom are represented by fields). In a modern language, both are formulated as functional integrals in d space dimensions or d+1 spacetime dimensions. The primary goal of these functional approaches is to compute correlation functions as well as the free energy of the system.
The FRG combines the functional approach with the Wilson RG idea of integrating out fluctuations not all at once but progressively from high- to low-energy scales. The expression functional RG stems from the fact that one naturally deals with (possibly singular) functions of the field rather than a finite number of coupling constants. In the literature, the (nonperturbative) FRG is sometimes merely referred to as the nonperturbative RG. Both aspects (nonperturbative and functional) are actually crucial features of the method. Note however that the RG approach can be functional and perturbative, or nonperturbative and nonfunctional. The FRG is also referred to as the Exact RG due to its one-loop exact (closed) functional form. This epithet does not imply that the flow equation can be solved exactly (except for models that can be solved more easily with other methods): Most nontrivial applications rely on an approximate solution.
Different versions of exact functional RG equations, such as the functional Callan-Symanzik, Wilson-Polchinski and Wegner-Houghton formulations, have already a long history. However the application of these methods has been hindered for a long time by the complexity of functional differential equations and the difficulty to devise nonperturbative and reliable approximation schemes. Early works were mainly based on the so-called local potential approximation and neglected the momentum dependence of the interaction vertices, with apparently no possibility of a systematic improvement. The FRG has nevertheless been useful in its perturbative formulation for the study of disordered systems. The necessity of a functional approach in this context is due to an infinite number of operators being marginal at the upper or lower (whichever the case of interest) critical dimension. It is therefore not possible, in some disordered systems, to restrict oneself to a finite number of coupling constants and one must consider functions of the field.
The formulation of the FRG based on a formally exact flow equation for a scale-dependent « effective action » Γk[ϕ] (or Gibbs free energy in the language of statistical physics), the generating functional of one-particle irreducible vertices, has proven successful in devising nonperturbative approximation schemes. The functional Γk[ϕ] may be seen as a coarse-grained free energy that includes only fluctuations with momenta or energies larger than a scale k. The field ɸ(r) or ɸ(r,t) represents the relevant microscopic degrees of freedom, e.g., the local magnetization in a solid. If necessary, one may include additional fields corresponding to emerging low-energy (collective) degrees of freedom: a pairing field in a superconductor, a meson field in QCD, etc. The functional Γk=0[ϕ] includes fluctuations on all scales, and allows us to obtain the free energy and the one-particle irreducible vertices. In principle, all correlation functions can be deduced from Γk=0[ϕ]. Thus the FRG replaces the difficult determination of Γk=0[ϕ] from a direct calculation of the functional integral (that defines the partition function) by the solving of a functional differential equation ∂k Γk[ϕ] with an initial condition ΓΛ[ϕ], which often (but not always) corresponds to the bare action or the mean-field solution of the model, at some microscopic scale Λ. The flow equation ∂k Γk[ϕ] closely resembles a renormalization-group improved one-loop equation, but is exact. This close connection to perturbation theory, for which we have an intuitive understanding, is an important key for devising meaningful nonperturbative approximations.
For further reading, see Dupuis, N.; Canet, L.; Eichhorn, A.; Metzner, W.; Pawlowski, J.; Tissier, M. & Wschebor, N., The nonperturbative functional renormalization group and its applications, Phys. Rep., 2021, 910, 1.