On the landscape of scale invariance in quantum mechanics

TL;DR

This paper explores the effects of scale invariance in quantum mechanics, revealing a quantum phase transition characterized by a limit cycle and other complex behaviors in the renormalisation group flow.

Contribution

It introduces a comprehensive renormalisation group analysis of scale invariant radial Hamiltonians, identifying novel phase transition phenomena and their relation to wave function power laws.

Findings

Existence of a quantum phase transition from scale invariant to discrete scale invariant phase.

Identification of a limit cycle in the renormalisation group flow at the critical point.

Relation between the critical point, RG flow, and wave function power laws.

Abstract

We consider the most general scale invariant radial Hamiltonian allowing for anisotropic scaling between space and time. We formulate a renormalisation group analysis of this system and demonstrate the existence of a quantum phase transition from a continuous scale invariant phase to a discrete scale invariant phase. Close to the critical point, the discrete scale invariant phase is characterised by an isolated, closed, attracting trajectory in renomalisation group space (a limit cycle). Moving in appropriate directions in the parameter space of couplings this picture is altered to one controlled by a quasi periodic attracting trajectory (a limit torus) or fixed points. We identify a direct relation between the critical point, the renormalisation group picture and the power laws characterising the zero energy wave functions.