RG Flows and Bifurcations

TL;DR

This work treats renormalization group (RG) flows as dynamical systems on theory space and develops a topological framework, via the Conley index, to constrain and infer interior fixed points and flow connections from boundary information. It introduces a two-field holographic model to realize marginality crossing as a waterfall transition between AdS vacua, linking RG behavior to phase-transition-like events in the dual theory. The analysis combines Conley index techniques with bifurcation theory to make concrete predictions for scaling dimensions near the lower end of conformal windows in models such as the $O(N)$ theory, QED$_3$, and QCD$_4$, including possible square-root, quadratic, or linear dependences on critical parameters. The approach promises topological and dynamical-system insights that complement perturbative, large-N, and holographic methods, with potential experimental or lattice tests for strongly coupled regimes.

Abstract

Interpreting RG flows as dynamical systems in the space of couplings we produce a variety of constraints, global (topological) as well as local. These constraints, in turn, rule out some of the proposed RG flows and also predict new phases and fixed points, surprisingly, even in familiar theories such as O(N) model, QED-3, or QCD-4.

Figures (9)

• Figure 1: Various bounds on scaling dimensions for scalar operators in CFT$_d$.
• Figure 2: In holography, marginality crossing is realized by a model solution that "rolls" between two saddle points of a potential function $V(\phi_1, \phi_2)$.
• Figure 3: By knowing the flow at the boundary of a region $N$ one can deduce whether the RG flow should have any fixed points in the interior of $N$. For example, is it possible that the RG flow shown in this figure has no fixed points in $\text{Int} (N)$?
• Figure 4: An isolated invariant set $S$, an isolating neighborhood $N$, and the corresponding exit set $L$, all shown on the plot (from Cardy:1996xt) of the RG flow in the $O(N)$ model. There are four fixed points: ($G$) Gaussian, ($H$) Wilson-Fisher, ($I$) Ising, and ($C$) Cubic with $\mu = 2$, $1$, $1$, and $0$, respectively.
• Figure 5: Computing the Conley index of the RG flow shown on the left involves identifying the points of the exit set $L$. The resulting pointed space $N/L$ has the homotopy type of the wedge sum of two circles (shown on the right) MR518546.