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Anderson Orthogonality and the Numerical Renormalization Group

Anderson Orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say |G_I> and |G_F>, become orthogonal in the thermodynamic limit of large particle number N, in that |<G_I|G_F>| ~ N^(- Delta_AO^2 /2) for N->infinity. We show that the numerical renormalization group (NRG) offers a simple and precise way to calculate the exponent Delta_AO: the overlap, calculated as function of Wilson chain length k, decays exponentially, ~ exp(-k alpha), and Delta_AO can be extracted directly from the exponent alpha. The results for Delta_AO so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground state properties change upon switching from |G_I> to |G_F>: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.