First order dipolar phase transition in the Dicke model with infinitely coordinated frustrating interaction

We found analytically a first order quantum phase transition in the Cooper pair box array of $N$ low-capacitance Josephson junctions capacitively coupled to a resonant photon in a microwave cavity. The Hamiltonian of the system maps on the extended Dicke Hamiltonian of $N$ spins one-half with infinitely coordinated antiferromagnetic (frustrating) interaction. This interaction arises from the gauge-invariant coupling of the Josephson junctions phases to the vector potential of the resonant photon field. In $N \gg 1$ semiclassical limit, we found a critical coupling at which ground state of the system switches to the one with a net collective electric dipole moment of the Cooper pair boxes coupled to superradiant equilibrium photonic condensate. This phase transition changes from the first to second order if the frustrating interaction is switched off. A self-consistently `rotating' Holstein-Primakoff representation for the Cartesian components of the total superspin is proposed, that enables to trace both the first and the second order quantum phase transitions in the extended and standard Dicke models respectively.Comment: 12 pages, 10 figure

Bragg's law for X-ray scattering by quantum thermodynamic time crystals, Q-balls, as manifestation of the mechanism of High-Tc_c

Proposed by the author Q-ball mechanism of the pseudogap state and high-Tc superconductivity in cuprates was recently supported by micro X-ray diffraction data in HgBa$_2$CuO$_{4+y}$. This provides a remarkable opportunity to investigate X-ray diffraction produced by the quantum thermodynamic time crystals, a direct embodiment of those are just the Euclidean Q-balls considered in the aforementioned theory. For this purpose the X-ray 'photon' Green's function scattered by the Q-balls in the pseudogap phase of high-Tc superconductors is calculated using the same , but adjusted, Feynman diagrammatic technique that was previously used for random impurity scattering in metal alloys. The Bragg's peaks intensity, provided by the imaginary part of the retarded 'photon' Green's function, is calculated. Thus obtained results describe X-ray scattering on the finite size Q-ball of CDW/SDW density, that oscillates with bosonic frequency $\Omega=2\pi nT$ in Matsubara time, i.e. the quantum thermodynamic time crystal. It is found, that theoretical results are in good correspondence with X-ray diffraction data in HgBa$_2$CuO$_{4+y}$ reported recently.Comment: 19 pages, 8 figure