Impact of bound states on similarity renormalization group transformations

Abstract

We study a simple class of unitary renormalization group (RG) 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 withf near 1 divergent couplings arise as bound state thresholds emerge. To illustrate and diagnose this behavior, we numerically study Hamiltonian ows 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. Intermediate values of f cause intermediate shifts of bound state eigenvalues down the diagonal and produce increasingly large coupling constants to do this. In discrete models, there is a critical value, fc, below which bound state eigenvalues appear at their natural scale and the entire ow to the diagonal is well-behaved. We analyze the shift mechanism analytically in a 3x3 matrix model,