Finite Size Scaling of Classical Long-Ranged Ising Chains and the Criticality of Dissipative Quantum Impurity Models

TL;DR

This paper investigates the finite-size scaling behavior of classical long-range Ising chains with 1/r^{2-epsilon} interactions, revealing how winding affects critical scaling and highlighting differences from quantum impurity models.

Contribution

It demonstrates the impact of interaction winding on the scaling of classical Ising chains and discusses implications for quantum-to-classical mappings in dissipative impurity models.

Findings

Winding influences the dynamical spin susceptibility scaling.

Infinite winding leads to mean-field behavior.

Absence of winding results in an interacting omega/T scaling.

Abstract

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-epsilon} interactions. For 1/2<epsilon<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.