IR-Stable Fixed Points

• IR-stable fixed points are invariant solutions of renormalization group flows that attract systems toward robust low-energy behavior.
• They organize phase diagrams by classifying universality and critical phenomena in quantum field theory, statistical mechanics, and complex systems.
• Analytical, numerical, and computer-assisted methods rigorously establish their stability, revealing implications for emergent collective behavior.

An IR-stable fixed point (infrared-stable fixed point) is a solution of the renormalization group (RG) flow equations or an analogous evolutionary dynamic that remains invariant under scale transformations and to which the system is attracted as one proceeds to lower energies or longer length scales. IR-stable fixed points play a key role across quantum field theory, statistical mechanics, mathematical dynamics, and complex systems, underpinning phenomena such as universality, emergent collective behavior, robust long-term equilibria, and the stability of computational and physical algorithms. Their mathematical structure, physical realization, and role in organizing parameter space vary widely by context, as illustrated by a broad array of models and techniques in the literature.

1. Renormalization Group Structure and Fixed Point Classification

The RG formalism systematically analyzes the scale dependence of couplings and observables. A fixed point is defined by $\beta_i(\vec{g}_*) = 0$ for all couplings $g_i$, where $\beta_i$ is the RG beta function. Stability is determined by the eigenvalues of the stability (Jacobian) matrix $\partial\beta_i/\partial g_j|_{\vec{g}_*}$: IR-stable fixed points have all relevant (unstable) directions associated with negative eigenvalues, so that perturbations in those directions flow away, while irrelevant directions with positive eigenvalues flow toward the fixed point as energy decreases (Aristov et al., 2011, Hartnoll et al., 2014, Ishikawa et al., 2015, Aharony et al., 4 Sep 2025).

In lattice gauge theories, Chern–Simons-matter models, Luttinger liquids, and tensor field theories, this structure provides the apparatus for identifying phases, computing anomalous dimensions, and classifying universality classes. In high-dimensional dynamical systems, analogous stability analysis for nonlinear operators, transfer operators, or neural network dynamics underpins both theoretical and algorithmic IR stability (1909.02947, Castorrini et al., 16 Aug 2024, Berlyand et al., 7 Jan 2025).

2. Fixed Points in Specific Physical and Mathematical Systems

Context	Example Fixed Points / Characteristics	Analytical/Numerical Signature
Luttinger liquid Y-junctions (Aristov et al., 2011)	FP $N$, $A$, $M$, $\chi^{\pm}$; lines of fixed points; stability controlled by $g$, $g_3$ and flux	RG beta functions for $(a,b,c)$ conductance parameters; FP stability from sign of linearization
Disordered holographic CFTs (Hartnoll et al., 2014)	Disorder-driven IR fixed point with Lifshitz exponent $z>1$	Resummation of disorder perturbation; metric scaling symmetry; $z$ from disorder strength
$SU(3)$ gauge theories (Ishikawa et al., 2015)	Conformal window IR fixed points for various $N_f$	Overlap of scaled effective mass curves for different $N$; scaling of propagators
Chern–Simons-matter (Aharony et al., 4 Sep 2025)	Quasi-bosonic theory 3-coupling cubic beta functions; IR-stable FPs for some $N_f$, $\lambda$	Merging/disappearance of FPs as parameters are tuned; cubic RG equations
Tensor field theories (Harribey, 2022)	Melonic FPs; lines and isolated FPs in large $N$ limit	Dominance of melonic Feynman diagrams; vanishing beta functions
Self-consistent transfer operators (Castorrini et al., 16 Aug 2024)	Multiple stable density FPs; stability via cone contractions	Order-preserving Birkhoff cone contraction; explicit Hilbert metric estimates
Deep neural networks (Berlyand et al., 7 Jan 2025)	Unique or multiple FPs (robust attractors); non-monotonic $Q(N, L)$	Stability from contraction mapping and ESD of Jacobian; number of FPs varies with depth

Contextual significance varies: in quantum field theory, IR-stable FPs govern emergent conformal invariance and scale universality; in dynamical systems, they organize long-term statistical behavior; in learning algorithms, they delineate convergence and the structure of memory or class representatives.

3. Mathematical Mechanisms and Methods of FP Analysis

The existence and stability of IR-stable fixed points are established through various analytical, computational, and rigorous-numerical frameworks:

• RG Equations and Beta Functions: Linearization of RG flow at the fixed point (eigenvalues of the stability matrix) distinguishes IR-stable from IR-unstable points in both perturbative (e.g., Banks-Zaks, Chern–Simons) and nonperturbative (melonic, large-$N$) regimes (Aristov et al., 2011, Aharony et al., 4 Sep 2025, Harribey, 2022).
• Operator Theory and Functional Analysis: For transfer operators or self-consistent nonlinear operators, contraction mapping principles, Birkhoff cone theory, and order preservation under operator differentials provide stability criteria (Castorrini et al., 16 Aug 2024, Galatolo, 2015).
• Spectral Theory and Empirical Spectra: Deep neural network fixed point stability is deduced from the spectral radius of the input-output Jacobian; contraction mapping and empirical spectral distributions (ESDs) substantiate global attraction properties (Berlyand et al., 7 Jan 2025).
• Center Manifold Theory: For non-hyperbolic fixed points (e.g., in invariant-preserving time integration), center manifold reduction yields rigorous local stability and convergence results (Izgin et al., 2022).
• Computer-Assisted Proofs: In high-precision studies (e.g., Feigenbaum–Cvitanović universality), Chebyshev series and interval arithmetic are employed to ensure validated bounds on fixed point existence, uniqueness, and universality constants (Breden et al., 30 Sep 2024).

These tools, often adapted to the regularity and dimensionality of the problem, are widely transferable to allied models.

4. Physical Consequences: Universality, Robustness, and Emergence

IR-stable fixed points underpin a spectrum of robust phenomena:

• Universality: RG flow to IR-stable fixed points causes different microscopic models to exhibit identical critical exponents/scaling, explaining the remarkable universality observed in statistical physics and quantum critical systems (Dermisek et al., 2018, Harribey, 2022).
• Robustness to Perturbations: The attraction basin of an IR-stable fixed point ensures that small deterministic or random perturbations do not alter the qualitative physics or statistical behavior, a property rigorously connected to decay of correlations and statistical stability (Galatolo, 2015, Hartnoll et al., 2014).
• Memory and Pattern Completion: In recurrent neural architectures and combinatorial threshold-linear networks, stable fixed points encode memory patterns; universally, only certain subgraphs (target-free cliques) possess IR stability, which bounds the capacity and retrieval fidelity (1909.02947).
• Basin Structure and Non-monotonicity: In deep learning, IR-stable fixed points dictate class structure, and architectural choices directly affect their number and properties—for example, the non-monotonic $Q(N, L)$ fixed point count signals an optimal depth for memory complexity (Berlyand et al., 7 Jan 2025).
• Non-uniqueness and Coexistence: Some systems (self-consistent transfer operators, Y-junctions of Luttinger liquids, tensor models) demonstrate continuous lines or multiple coexisting IR-stable fixed points, parametrized by system symmetries or fine-tuning (Aristov et al., 2011, Harribey, 2022, Castorrini et al., 16 Aug 2024).

These outcomes extend to the emergence of disordered universality classes, screening of symmetry-breaking fields (e.g., magnetic flux in transport), and phase diagrams shaped by fixed-point structure and merging/splitting events (Hartnoll et al., 2014, Aharony et al., 4 Sep 2025).

5. Extensions: Non-hyperbolic FPs, Structural Stability, and Cohomology

Not all IR-stable fixed points are isolated; in infinite-dimensional or constrained systems, they may reside in subspaces or manifolds, often possessing non-hyperbolic (marginal) directions:

• Conservation Laws and Invariants: For invariant-preserving time integration, the steady state forms a subspace in which the Jacobian has eigenvalue 1, and stability must be assessed by center manifold theory (rather than linear spectral criteria alone) (Izgin et al., 2022).
• Structural (Statistical) Stability: Maps with indifferent (neutral) fixed points possess FPs (invariant measures) that vary Hölder-continuously with system perturbations, reflecting a quantitative connection between convergence to equilibrium and stability of global system statistics (Galatolo, 2015).
• Cohomological Criteria: In Dirac geometry, the persistence of fixed points under deformation is controlled by vanishing of (obstruction) cohomology groups associated to the moduli problem, ensuring that all nearby structures admit gauge-equivalent fixed points (i.e., IR-stability in deformation theory) (Singh et al., 2023).

Such extensions connect IR-stability in the classical RG sense with deeper mathematical notions of persistence, deformation invariance, and (in dynamical systems) measure-theoretic stability.

6. Special Cases: Lines of Fixed Points and Marginal Behavior

Some parameter regimes admit not isolated fixed points but continuous lines or even higher-dimensional sets of stable fixed points:

• Line of Stable FPs in Luttinger Liquid Y-Junctions: Special values of interaction parameters $g, g_3$ yield entire lines of stable fixed points, signaled by the merging of FP basins and continuous variation of conductance parameters (Aristov et al., 2011).
• Multicriticality and Marginality in CS–Matter Theories: The occurrence, merging, and disappearance of IR-stable fixed points for flavor multi-critical Chern–Simons models are controlled by the structure of higher-order polynomial beta functions in the marginal couplings—paralleling classical bifurcation phenomena (Aharony et al., 4 Sep 2025).
• Non-uniqueness in STO Dynamics: Multiple IR-stable fixed points (e.g., Lebesgue and Dirac measures) can coexist for certain self-consistent transfer operators, with stability determined by cone contraction, but with some fixed points representing only long metastable transients in finite systems (Castorrini et al., 16 Aug 2024).

This multistability and the transitions governed by parameter change exemplify the rich phase diagrams and bifurcation structures possible for RG flows and nonlinear operator dynamics.

7. Broader Impact and Future Directions

IR-stable fixed points function as organizing centers for low-energy, long-time, or large-scale physics across a diverse set of disciplines:

• Field Theory and Holography: Structuring the landscape of conformal phases, critical exponents, and the interplay of disorder, symmetry, or topological effects (Hartnoll et al., 2014, Horowitz et al., 2022).
• Complex Systems and Algorithms: Enabling robust design and analysis of nonlinear iterative algorithms, neural networks with controllable attractor structure, and scalable time-integration schemes (Berlyand et al., 7 Jan 2025, Izgin et al., 2022, Breden et al., 30 Sep 2024).
• Mathematical Structures and Deformation Theory: Connecting IR-stability to moduli problems, cohomological obstructions, and structural persistence in geometry or analysis (Singh et al., 2023).

Contemporary research further addresses open problems including the full classification of possible IR-stable fixed point structures (e.g., in strongly coupled systems, higher rank tensor models), the effect of disorder and non-uniformity in large networks, and the algorithmic exploitation of lines or families of stable FPs for optimization and learning (Aharony et al., 4 Sep 2025, Harribey, 2022, Castorrini et al., 16 Aug 2024).

---

In summary, IR-stable fixed points serve as the fundamental organizing principle for a wide variety of physical, dynamical, and computational systems, enabling universality, robustness, and emergent order through their dominance in the infrared regime. Their discovery, analysis, and application leverage a broad mathematical toolkit, ranging from RG theory and spectral analysis to cohomology and computer-assisted rigorous numerics.