Impact of bound states on similarity renormalization group transformations

We study a simple class Of unitary renormalization group trans format ions governed by a parameter f in the range [0, 1]. For f = 0, the trans formation 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 I 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 = I 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 f(c) below which bound-state eigenvalues appear at their natural scale, and the entire flow to the diagonal is well behaved. We analyze the shift mechanism analytically in a 3 X 3 matrix model, which displays the essence of this renormalization group behavior, and we compute f(c) for this model.