Functional renormalization group

In theoretical physics, functional renormalization group (FRG) is an implementation of the renormalization group (RG) concept which is used in quantum and statistical field theory, especially when dealing with strongly interacting systems. The method combines functional methods of quantum field theory with the intuitive renormalization group idea of Kenneth G. Wilson. This technique allows to interpolate smoothly between the known microscopic laws and the complicated macroscopic phenomena in physical systems. In this sense, it bridges the transition from simplicity of microphysics to complexity of macrophysics. Figuratively speaking, FRG acts as a microscope with a variable resolution. One starts with a high-resolution picture of the known microphysical laws and subsequently decreases the resolution to obtain a coarse-grained picture of macroscopic collective phenomena. The method is nonperturbative, meaning that it does not rely on an expansion in a small coupling constant. Mathematically, FRG is based on an exact functional differential equation for a scale-dependent effective action.

The flow equation for the effective action

In quantum field theory, the effective action \Gamma is an analogue of the classical action functional S and depends on the fields of a given theory. It includes all quantum and thermal fluctuations. Variation of \Gamma yields exact quantum field equations, for example for cosmology or the electrodynamics of superconductors. Mathematically, \Gamma is the generating functional of the one-particle irreducible Feynman diagrams. Interesting physics, as propagators and effective couplings for interactions, can be straightforwardly extracted from it. In a generic interacting field theory the effective action \Gamma, however, is difficult to obtain. FRG provides a practical tool to calculate \Gamma employing the renormalization group concept.

The central object in FRG is a scale-dependent effective action functional \Gamma_{k} often called average action or flowing action. The dependence on the RG sliding scale k is introduced by adding a regulator (infrared cutoff) R_{k} to the full inverse propagator \Gamma^{(2)}_{k}. Roughly speaking, the regulator R_k decouples slow modes with momenta q\lesssim k by giving them a large mass, while high momentum modes are not affected. Thus, \Gamma_{k} includes all quantum and statistical fluctuations with momenta q\gtrsim k. The flowing action \Gamma_k obeys the exact functional flow equation

\partial_k \Gamma_k = \frac{1}{2} \text{STr} \, \partial_k R_k \, (\Gamma^{(2)}_k + R_k)^{-1},

derived by Christof Wetterich in 1993 and Tim R. Morris in 1994. Here \partial_k denotes a derivative with respect to the RG scale k at fixed values of the fields. The functional differential equation for \Gamma_{k} must be supplemented with the initial condition \Gamma_{k\to\Lambda}=S, where the "classical action" S describes the physics at the microscopic ultraviolet scale k=\Lambda. Importantly, in the infrared limit k\to 0 the full effective action \Gamma=\Gamma_{k\to 0} is obtained. In the Wetterich equation \text{STr} denotes a supertrace which sums over momenta, frequencies, internal indices, and fields (taking bosons with a plus and fermions with a minus sign). The exact flow equation for \Gamma_k has a one-loop structure. This is an important simplification compared to perturbation theory, where multi-loop diagrams must be included. The second functional derivative \Gamma^{(2)}_{k} is the full inverse field propagator modified by the presence of the regulator R_k.

The renormalization group evolution of \Gamma_k can be illustrated in the theory space, which is a multi-dimensional space of all possible running couplings \{c_{n} \} allowed by the symmetries of the problem. As schematically shown in the figure, at the microscopic ultraviolet scale k=\Lambda one starts with the initial condition \Gamma_{k=\Lambda}=S.

As the sliding scale k is lowered, the flowing action \Gamma_k evolves in the theory space according to the functional flow equation. The choice of the regulator R_k is not unique, which introduces some scheme dependence into the renormalization group flow. For this reason, different choices of the regulator R_k correspond to the different paths in the figure. At the infrared scale k=0, however, the full effective action \Gamma_{k=0}=\Gamma is recovered for every choice of the cut-off R_k, and all trajectories meet at the same point in the theory space.

In most cases of interest the Wetterich equation can only be solved approximately. Usually some type of expansion of \Gamma_{k} is performed, which is then truncated at finite order leading to a finite system of ordinary differential equations. Different systematic expansion schemes (such as the derivative expansion, vertex expansion, etc.) were developed. The choice of the suitable scheme should be physically motivated and depends on a given problem. The expansions do not necessarily involve a small parameter (like an interaction coupling constant) and thus they are, in general, of nonperturbative nature.

Aspects of functional renormalization

Functional renormalization-group for Wick-ordered effective interaction

Contrary to the flow equation for the effective action, this scheme is formulated for the effective interaction

\mathcal{V}[\eta ,\eta ^{+}] =-\ln Z[G_{0}^{-1} \eta , G_{0}^{-1}\eta ^{+}]-\eta G_{0}^{-1}\eta ^{+}

which generates n-particle interaction vertices, amputated by the bare propagators G_{0}; Z[\eta ,\eta ^{+}] is the "standard" generating functional for the n-particle Green functions.

The Wick ordering of effective interaction with respect to Green function D can be defined by

\mathcal{W}[\eta ,\eta ^{+}]=\exp(-\Delta _D)\mathcal{V}[\eta ,\eta ^{+}].

where \Delta=D \delta^2 /(\delta \eta \delta \eta^ {+}) is the Laplacian in the field space. This operation is similar to Normal order and excludes from the interaction all possible terms, formed by a convolution of source fields with respective Green function D. Introducing some cutoff \Lambda the Polchinskii equation

\frac{\partial }{{\partial \Lambda }}{{V}_\Lambda }(\psi ) = -{\dot \Delta _{{G_{0,\Lambda }}}}{{V}_\Lambda }(\psi ) + \Delta _{{{\dot G}_{0,\Lambda }}}^{12}\mathcal {V}_\Lambda ^{(1)}\mathcal {V}_\Lambda ^{(2)}

takes the form of the Wick-ordered equation

{\partial _\Lambda }{\mathcal {W}_\Lambda } = -{\Delta _{{{\dot D}_\Lambda } + {{\dot G}_{0,\Lambda }}}}{\mathcal { W}_\Lambda } + {e^{-\Delta _{{D_\Lambda }}^{12}}}\Delta _{{{\dot G}_{0,\Lambda }}}^{12}\mathcal {W}_\Lambda ^{(1)}\mathcal {W}_\Lambda ^{(2)}

where

\Delta _{{{\dot G}_{0,\Lambda }}}^{12}\mathcal {V}_\Lambda ^{(1)}\mathcal {V}_\Lambda ^{(2)}=\frac{1}{2}\left( {\frac{{\delta {{V}_\Lambda }(\psi )}}{{\delta \psi }},{{\dot G}_{0,\Lambda }}\frac{{\delta {{V}_\Lambda }(\psi )}}{{\delta \psi }}} \right)

Applications

The method was applied to numerous problems in physics, e.g.:

See also

References

Papers

    Pedagogic reviews

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