Entropy from Scaling Symmetry Breaking

TL;DR

This paper explores how breaking and restoring scaling symmetry in conformal quantum mechanics relates to the von Neumann entropy, providing explicit formulas and linking symmetry restoration to quantum information measures.

Contribution

It introduces a method to restore broken scaling symmetry in CQM via mixed states and derives an explicit entropy formula, connecting symmetry and quantum entropy.

Findings

Scaling symmetry can be broken by boundary conditions in CQM.

A mixed state can restore the broken symmetry.

Explicit von Neumann entropy formula for CQM is provided.

Abstract

The scaling symmetry in conformal quantum mechanics (CQM) can be broken due to the boundary conditions that follow from the requirement of a unitary time evolution of the Hamiltonian. We show that the scaling symmetry of CQM can be restored by introducing a suitable mixed state, which is associated with a nonvanishing von Neumann entropy. We give an explicit formula for the entropy arising from the mixed state in CQM. Our work provides a direct link between the restoration of a broken symmetry and von Neumann entropy.