Finite-size scaling of classical long-ranged Ising chains and the criticality of dissipative quantum impurity models

Motivated in part by quantum criticality in dissipative Kondo systems, we revisit the finite-size scaling of a classical Ising chain with 1/r;{2-} interactions. For 1/2<<1, the scaling of the dynamical spin susceptibility is sensitive to the degree of "winding" of the interaction under periodic boundary conditions. Infinite winding yields the expected mean-field behavior, whereas without any winding the scaling is of an interacting omega/T form. The contrast with the behavior of the Bose-Fermi Kondo model suggests a breakdown of a mapping from the quantum model to a classical one due to the smearing of the Kondo spin flips by the continuum limit taken in this mapping.