Discrete Scale Invariance of Ising Mesons

Abstract

Discrete scale invariance often arises from resonant few-body or singular scale-invariant dynamics. We show that it can instead emerge in the internal motion of a composite excitation in an ordered quantum magnet. In the marginal $1/r^4$ transverse-field Ising chain, long-range bonds generate an asymptotic inverse-square attraction between dressed domain walls, with strength fixed by the spontaneous magnetization. The dressed two-kink threshold curvature sets the relative kinetic scale. Together, these independently accessible quantities determine the scale-anomaly exponent and hence the meson hierarchy. In the supercritical regime, the same exponent governs geometric binding-energy and size ratios, logarithmic level accumulation, and log-periodic threshold scattering. Full-spin exact diagonalization and a fourth-order weak-field Hamiltonian support the coupled energy--size scaling in the finite window between core and ring effects. This provides a microscopic many-body realization of quantum limit-cycle physics, with its universal scaling fixed by magnetic order and kink mobility.