Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their $\epsilon$-Expansions

TL;DR

This paper develops a numerical method to compute nontrivial fixed points in hierarchical renormalization group models, compares eigenvalues with epsilon-expansion, and avoids fine-tuning of parameters.

Contribution

It introduces a numerical approach for fixed point computation that bypasses fine-tuning and compares results with epsilon-expansion for validation.

Findings

Numerical fixed points match epsilon-expansion predictions.

Eigenvalues of the linearized transformation are computed.

Method avoids fine-tuning of relevant parameters.

Abstract

Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with $\epsilon$-expansion.