On the phase diagram of the attractive Hubbard model: Crossover and quantum critical phenomena
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We explore the phase diagram of the attractive Hubbard model from the point of view of phase transition theory. Using the quantum Monte-Carlo technique, the scaling theory of quantum critical phenomena and the mapping to the XY model we investigate the critical properties along the temperature, band filling and coupling strength axes of the phase diagram. Particular emphasis is devoted to the insulator to superconductor and superconductor to normal metal transitions driven by the variation of the coupling strength U. We also discuss the particular similarities between this simple lattice model and the high-Tc cuprates: both systems exhibit a phase transition along a certain critical line as a function of a particular control parameter, which is e.g. the strength of the attractive coupling |U| in the case of the Hubbard model and the electron doping in the copper oxide planes for the cuprates; both show a remarkable crossover along this phase transition line with similar consequences. At one end point (overdoped/small-U regime) we find a behavior similar to a conventional BCS-type superconductor, the system undergoes a normal metal to superconductor transition (NS), whereas at the other end (underdoped/large-U limit) a description in terms of Bose-Einstein condensation (BEC) of preformed pairs is more adequate, with a superconductor to insulator critical endpoint (SI). There superconductivity occurs when the phases of the pairs become coherent, not when the pairs are initially formed. In addition to the fixed electron density, interaction driven crossover the Hubbard model undergoes a further T=0 quantum transition from a (super-) conductor to an insulator for \(\rho \to 0\).
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