The Critical Properties of Two-Level Models of the Fundamental Hamiltonians
It has been shown  recently that the well-advertised “phase-transition” implied by the so-called Dicke Hamiltonian fails to occur if the two-level model of the minimal-coupling atomic Hamiltonian includes the A2 terms in the interaction, and if the Thomas-Reiche-Kuhn sum rule is invoked. However, the question arises: since the “phase transition” is due to the properties of the p · A term of the minimal coupling Hamiltonian, and since those properties are shared by the d · E term in the dipole Hamiltonian, can we not recover the “phase transition” simply by making the unitary transformation from minimal coupling to dipole Hamiltonian at the outset, and then working entirely with d · E thereafter? The answer is that we cannot, and the reason why not is connected with a much-neglected term in the dipole Hamiltonian, and with the nature of two-level models.
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