[Paper Review] The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3

P. K. Mitter, Scoppola, B.|Sep 25, 2007

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the existence of a globally bounded renormalization group (RG) trajectory in a critical supersymmetric field theory on the Z³ lattice, with interactions uniformly bounded away from zero across all scales. Using a rigorous Wilsonian RG approach with Banach space norms and polymer expansions, it proves the existence of a non-Gaussian fixed point for weakly self-avoiding Lévy walks in three dimensions, laying the foundation for computing critical exponents.

ABSTRACT

We consider an Euclidean supersymmetric field theory in $Z^3$ given by a supersymmetric $Φ^4$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) Lévy random walk in $Z^3$. The Green's function depends on the Lévy-Khintchine parameter $α={3+ε\over 2}$ with $00$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding Lévy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in $Z^3$.

Motivation & Objective

• To rigorously construct a non-Gaussian supersymmetric field theory on the Z³ lattice for weakly self-avoiding Lévy walks.
• To establish the existence of a globally bounded renormalization group trajectory across all lattice scales.
• To prove the existence of a critical (stable) manifold in the RG flow for small ε > 0.
• To provide a foundation for computing critical exponents of weakly self-avoiding Lévy walks in three dimensions.

Proposed method

• Formulates a supersymmetric Φ⁴ perturbation of a massless Gaussian measure with covariance given by the Green’s function of a stable Lévy walk on Z³.
• Uses a Wilsonian RG approach with iterative coarse-graining maps on increasingly fine lattices, tracking coupling constants and polymer activities.
• Defines Banach spaces of interactions with norms that control the RG flow uniformly in scale, avoiding Griffiths singularity pathologies.
• Employs polymer expansions and cluster expansion techniques to handle irrelevant interactions and ensure convergence.
• Applies the implicit function theorem in Banach spaces to prove uniqueness of the critical mass at the unit lattice scale.
• Establishes uniform boundedness of effective coupling constants gn away from zero for all RG scales, indicating a non-Gaussian fixed point.

Experimental results

Research questions

• RQ1 Does a globally bounded renormalization group trajectory exist for the critical supersymmetric Φ⁴ theory on Z³ with α = (3+ε)/2 and small ε > 0?
• RQ2 Can the critical (stable) manifold be rigorously constructed in this lattice supersymmetric field theory?
• RQ3 Are the effective coupling constants uniformly bounded away from zero across all RG scales, indicating a non-Gaussian fixed point?
• RQ4 Can the RG trajectory be extended to the infinite volume limit while preserving uniformity in lattice scale?

Key findings

• The RG trajectory is globally bounded and uniformly controlled across all renormalization group scales, even as the lattice becomes arbitrarily fine.
• The effective coupling constant gn remains uniformly bounded away from zero for all n ≥ 0, satisfying (1 − 1/(4ν))¯g < gn < (1 + 1/(4ν))¯g with 0 < ν < 1/2.
• The critical mass µn0 is uniquely determined as a C¹ function of the initial coupling ˜g0, proving uniqueness of the critical trajectory.
• The existence of a non-Gaussian fixed point is established via uniform positivity of the effective coupling across all scales.
• The RG flow is analytic in Banach space norms, enabling rigorous application of the implicit function theorem to construct the critical manifold.
• The construction provides a rigorous foundation for computing critical exponents of weakly self-avoiding Lévy walks in three dimensions.