Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The $O(N)$ model

Abstract

The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows to reliably treat the emergence of non-analytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higherdimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.

Bosonic fluctuations in the $( 1 + 1 )$-dimensional Gross-Neveu(-Yukawa) model at varying $μ$ and $T$ and finite $N$

“…( 15). However, only recently it was found by some of the authors and their collaborators [110,[142][143][144][145]154] that the RG flow equation for the scale dependent effective potential for a large class of models from QFT can be recast as a conservation law in terms of a (advection-)diffusion-source/sink equation and be entirely understood in terms of fluid dynamic notions 15 . A fluid dynamic reinterpretation is as advantageous for several reasons.…”

“…For further discussions on these and related issues, we refer to Refs. [110,120,[143][144][145]178].…”

“…For a detailed discussion on field space boundary conditions for RG flow equations (and their explicit numeric implementation), we refer to Refs. [110,144] and Sub.Sec. IV E.…”

“…Refs. [110,142,144,145,154], the huge gradients introduced via the chemical potential induce a shock wave that propagates towards smaller |σ|.…”

Fluctuating relativistic dissipative hydrodynamics as a gauge theory

“…This means that a continuum of equivalent theories is possible where fluctuation and dissipation "conspire" to produce the same macroscopic dynamics, in a way that is hidden in how the two microscopic scales interact. The different frames of hydrodynamics is a manifestation of this, where the analogy between renormalization group flow and hydrodynamics [48] is used to evolve one coarse-graining scale in the space of theories, and the other as a time-scale. Unlike gauge theory, hydrodynamics is neither unitary nor renormalizable; Therefore, one cannot hope gauge dependence will cancel order-by-order when diagrams are summed (a "tree-level" example is Fig.…”