Dirac's time-dependent variational principle to phase transition of ground state in the quantum double-well model

The ground-state phase transition of the quantum double-well model is studied by virtue of Dirac's time-dependent variational principle together with a Hartree-type many-body trial wave function for the particles. The single-particle state is taken to be a Jackiw–Kerman wave function. We derive an effective classical Hamiltonian for the system, and coupled equations of particle's expectation values and quantum fluctuations for the ground state. It is shown clearly that the ground state transits from broken symmetric phase to restored symmetric phase as the quantum fluctuations renormalize the substrate potential from effective double-well potential to effective single-well potential. We obtain quantitatively the phase diagram, the quantum fluctuations and the energy of the ground state, respectively.