We study the d -dimensional Bose gas at finite temperature using the renormalization group method. The flow -equations and the free energy have been obtained for dimension d, and the cases d < 2 and d = 2 have been analysed in the limit of low and high temperatures. The critical temperature, the coherence length and the specific heat of a two dimensional Bose gas have been obtained using a solution for the coupling constant which does not present a singular behavior.
In-plane, inter-carrier correlations in hole doped cuprates are investigated by ab initio multiconfiguration calculations. The dressed carriers display features that are reminiscent of both Zhang-Rice (ZR) CuO4 singlet states and Jahn-Teller polarons. The interaction between these quasiparticles is repulsive. At doping levels that are high enough, the interplay between long-range unscreened Coulomb interactions and long-range phase coherence among the O-ion half-breathing vibrations on the ZR plaquettes may lead to a strong reduction of the effective adiabatic energy barrier associated to each polaronic state. Tunneling effects cannot be neglected for a relatively flat, multi-well energy landscape. We suggest that the coherent, superconducting quantum state is the result of such coherent quantum lattice fluctuations involving the in-plane O ions. Our findings appear to support models where the superconductivity is related to a lowering of the in-plane kinetic energy.The parent compounds of the high-temperature cuprate superconductors are Mott insulating antiferromagnets. Superconductivity (SC) occurs in these systems upon hole or electron doping. While the magnetic interactions are remarkably strong, there is increasing evidence that many properties cannot be understood without taking into account the lattice degrees of freedom. Experiments that indicate strong electron-phonon (EP) couplings are the inelastic neutron scattering [1, 2], angle-resolved photoemission (PE) [3,4], X-ray absorption fine structure (XAFS) [5], and electron paramagnetic resonance [6]. Anomalies in the phonon, PE, and XAFS spectra were previously addressed with t -J or Hubbard-like models supplemented with EP interaction terms [7,8,9,10]. Strong and anomalous electron-lattice couplings were also found by ab initio, explicitly correlated calculations [11,12]. In the case of hole doping [13], multiconfiguration (MC) calculations on clusters of few CuO 6 octahedra show that the doped holes enter O 2p x and 2p y orbitals that form σ-bonds with the open-shell Cu 3d x 2 −y 2 orbitals [11] and give rise to singlet states similar to the kind of configuration proposed by Zhang and Rice (ZR) long time ago [14]. A major deviation from the original ZR picture is that the formation of such a singlet is associated with significant lattice deformations. The most stable configuration corresponds to Cu-O distances that are shorter by 5-6%. Most remarkably, O-ion displacements that restore the translational symmetry induce strong charge redistribution. The 2p hole, which for a distorted CuO 4 plaquette is equally distributed over the four anions, can be partially transferred onto a single ligand to give an electronic wavefunction (WF) with a dominant contribution from a ...-Cu d 9 -O p 5 -Cu d 9 -... configuration. The ZR-like singlet polaron (ZRP) can hop thus within the CuO 2 plane via such d 9 -p 5 -d 9 states through coupling to the oxygen vibrations. The energy barrier associated with the hopping process is few hundreds meV, for an isolated 2p hole [11...
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