Universality, marginal operators, and limit cycles

TL;DR

This paper demonstrates the universality of limit cycle behavior in a simple Hamiltonian model, revealing the roles of marginal and irrelevant operators and how tuning can improve convergence of bound state energies.

Contribution

It provides an analytical solution for a non-perturbative renormalization group equation exhibiting limit cycle behavior, highlighting the role of marginal and irrelevant operators and their impact on bound state spectra.

Findings

The model exhibits a limit cycle with a marginal operator and infinite irrelevant operators.

Wegner's eigenvalues for irrelevant operators are independent of limit cycle position.

Tuning the Hamiltonian accelerates convergence of bound state energies to a geometric series.

Abstract

The universality of renormalization group limit cycle behavior is illustrated with a simple discrete Hamiltonian model. A non-perturbative renormalization group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero - also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound state eigenvalues converge much more quickly than otherwise to an exact geometric series.