Note on general functional flows in equilibrium systems

TL;DR

This paper develops a systematic formalism for deriving functional flow equations in equilibrium quantum field theory and statistical mechanics, enabling analysis of parameter responses and reproducing known flow equations.

Contribution

It introduces a general response framework for generating functionals in equilibrium systems, deriving hierarchical flow equations for vertices and connecting to Callan-Symanzik equations.

Findings

Derived general functional flow equations for generating functionals.

Established hierarchical equations for 1PI vertices.

Reproduced known functional flow equations in examples.

Abstract

We study the response of generating functionals to a variation of parameters (couplings) in equilibrium systems i.e. in quantum field theory (QFT) and equilibrium statistical mechanics. These parameters can be either physical ones such as coupling constants or artificial ones which are intentionally introduced such as the renormalization scale in field theories. We first derive general functional flow equations for the generating functional (grand-canonical potential) $W[J]$ of the connected diagrams. Then, we obtain functional flow equations for the one-particle irreducible ($1$PI) vertex functional (canonical potential) $\Gamma[\phi]$ by performing the Legendre transformation. By taking the functional derivatives of the flow equations, we can obtain an infinite hierarchical equations for the $1$PI vertices. We also point out that a Callan-Symanzik type equation holds among the vertices when partition function is invariant under some changes of the parameters. After discussing general aspects of parameter response, we apply our formalism to several examples and reproduce the well-known functional flow equations. Our response theory provides us a systematic and general way to obtain various functional flow equations in equilibrium systems.