Abstract
We study a simple class of unitary renormalization group transformations governed by a parameter in the range [0, 1]. For , the transformation is one introduced by Wegner in condensed matter physics, and for it is a simpler transformation that is being used in nuclear theory. The transformation with diagonalizes the Hamiltonian but in the transformations with 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 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 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 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 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 matrix model, which displays the essence of this renormalization group behavior, and we compute for this model.
- Received 8 April 2008
DOI:https://doi.org/10.1103/PhysRevD.78.045011
©2008 American Physical Society