This article reviews the current status of precursor superconducting phase fluctuations as a possible mechanism for pseudogap formation in high-temperature superconductors. In particular we compare this approach which relies on the twodimensional nature of the superconductivity to the often used T -matrix approach. Starting from simple pairing Hamiltonians we present a broad pedagogical introduction to the BCS-Bose crossover problem. The finite temperature extension of these models naturally leads to a discussion of the Berezinskii-Kosterlitz-Thouless superconducting transition and the related phase diagram including the effects of quantum phase fluctuations and impurities. We stress the differences between simple Bose-BCS crossover theories and the current approach where one can have a large pseudogap region even at high carrier density where the Fermi surface is welldefined. The Green's function and its associated spectral function, which explicitly show non-Fermi liquid behaviour, is constructed in the presence of vortices. Finally different mechanisms including quasi-particle-vortex and vortex-vortex interactions for the filling of the gap above T c are considered. 5The superconducting transition as a BKT transition and the temperature scale for the opening of the pseudogap 52 5.1 Phase diagram based on the classical phase fluctuations in the absence of Coulomb repulsion 52 5.2 The peculiarities of the phase diagram for the lattice model 67 5.3 The effects of Coulomb repulsion and quantum phase fluctuations 70 5.4 The effect of non-magnetic impurities 77 6 The Green's function in modulus-phase representation and non-Fermi liquid behaviour 86 6.1 The modulus-phase representation for the fermion Green function 87 6.2 The correlation function for the phase fluctuations 88 6.3 The Fourier transform of D(r) 92 6.4 The derivation of the fermion Green's function in Matsubara representation and its analytical continuation 93 6.5 The branch cut structure of G(ω, k) and non-Fermi liquid behaviour 97 7 The spectral function in the modulus-phase representation and filling of the gap 99 7.1 Absence of gap filling for the Green's function calculated for the static phase fluctuations in the absence of spin-charge coupling 99 7.2 Gap filling by static phase fluctuations due to quasi-particle vortex interactions. The phenomenology of ARPES 103 7.3 Gap filling due to dynamical phase fluctuations without quasi-particle vortex interactions 106 8 Concluding remarks 109 A Calculation of the effective potential 111 B Another representation for the retarded Green's function 113 References 114particle weight, or equivalently a pseudogap, below T * . Until recently most ARPES measurements were performed for the photon energy range between 19 and 25eV . The most recent data [10], which was taken with a higher photon energy of 33eV , is in contradiction with the previous data. This question is now under intensive investigation [11], particularly since ohmic losses may alter the ARPES spectra [12].An explanation of the pseudogap phenomen...

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We investigate the Rabi Hamiltonian ͑or the Jaynes-Cummings model without the rotating-wave approxi-mation͒, describing the coupling of a single mode of electromagnetic radiation to a two-level system, by means of the coupled cluster method. We find strong evidence for a second-order quantum phase transition.

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We present simple two-and three-parameter variational calculations for the Rabi Hamiltonian. The importance of symmetry in the anstitze is stressed. The numerical results indicate that our anstitze provide accurate approximations both to the groundstate energy and wavefunction and to the first excited state if the two-boson energy significantly exceeds the level splitting.