Entropy from Scaling Symmetry Breaking
  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. 
Neymar Cavalcante, Saulo Diles, Kumar S. Gupta and Amilcar R. de   Queiroz
hep-th cond-mat.str-el quant-ph