The impact of bound states on similarity renormalization group transformations

TL;DR

This paper investigates how different unitary renormalization group transformations, governed by a parameter, affect the treatment of bound states in Hamiltonian flows, revealing divergent behaviors near bound state thresholds.

Contribution

It introduces and compares a family of RG transformations parameterized by f, analyzing their effects on bound states and divergence issues in Hamiltonian diagonalization.

Findings

f=0 transformation diagonalizes Hamiltonian at bound-state scales

f=1 transformation diverges at low momentum scales

Analytical shift mechanism explained in a 3x3 matrix model

Abstract

We study a simple class of unitary renormalization group transformations governed by a parameter f in the range [0,1]. For f=0, the transformation is one introduced by Wegner in condensed matter physics, and for f=1 it is a simpler transformation that is being used in nuclear theory. The transformation with f=0 diagonalizes the Hamiltonian but in the transformations with f near 1 divergent couplings arise as bound state thresholds emerge. To illustrate and diagnose this behavior, we numerically study Hamiltonian flows in two simple models with bound states: one with asymptotic freedom and a related one with a limit cycle. The f=0 transformation places bound-state eigenvalues on the diagonal at their natural scale, after which the bound states decouple from the dynamics at much smaller momentum scales. At the other extreme, the f=1 transformation tries to move bound-state eigenvalues to the part of the diagonal corresponding to the lowest momentum scales available and inevitably diverges when this scale is taken to zero. We analyze the shift mechanism analytically in a 3x3 matrix model, which displays the essense of this RG behavior.