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Quantum phase transitions in an effective Hamiltonian

We study the effects of quantum phase transitions in a generic Hamiltonian describing interaction between a fast and a slow systems. We obtain and analyze the effective potential
for the slow system. The effective Hamiltonians in the thermodynamic limit describes a bifurcation of the ground state leading to the effect of the Quantum Phase Transition in the slow system. We treat specific examples, atom-field and
atom-atom interactions.
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