The impact of bound states on similarity renormalization group transformations

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.

27 pages, 3 figures

References in corpus (11)

• Renormalization of the Three-Body System with Short-Range Interactions
• Model-independent low momentum nucleon interaction from phase shift equivalence
• Similarity Renormalization Group for Nucleon-Nucleon Interactions
• The Three-Boson System with Short-Range Interactions
• Effective Theory of the Triton
• An Infrared Renormalization Group Limit Cycle in QCD
• Precise numerical results for limit cycles in the quantum three-body problem
• Asymptotic Freedom and Bound States in Hamiltonian Dynamics
• Universality, marginal operators, and limit cycles
• Are low-energy nuclear observables sensitive to high-energy phase shifts?
• Limit cycles of effective theories

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