Quantum Monte Carlo study of the three-dimensional attractive Hubbard model

The existence of a smooth crossover between the two paradigms of quantum superfluidity, the BardeenCooper-Schrieffer (BCS) superconductivity and the Bose-Einstein condensation (BEC) is firmly established [1, 2]. In this context, the attractive Hubbard model (AHM) has appeared as an ideal presentation of the whole evolution between the BCS and BEC physics [3]. A concrete property of this Hamiltonian is the existence of two (not always) distinct energy scales: one associated with the formation of Cooper pairs (T � ) and another with the onset of long-range order in the system (Tc) [4]. Although their qualitative behavior is well-known, a quantitative determination is still missing, due to the fact that it is hard to access the intermediate regime by a controlled approximation scheme. In this respect the Determinant Quantum Monte Carlo (DQMC) method [5, 6] is a powerful tool as it provides results free of systematic errors. A detailed finite-size analysis is however necessary in order to extract the thermodynamic limit properties, which can then be compared with the outputs of other methods recently applied to the same problem [7, 8]. At this point we should stress the role of dimensionality that determines the nature of the superconducting phase transition at Tc; the strictly 2D realization of the model is characterized by a Berezinskii-Kosterlitz-Thouless type phase transition, whereas the 3D case displays a “normal” second-order one, which is more easily accessible by DQMC. Since the intermediate regime of the AHM constitutes the simplest model for a short-coherence-length superconductor, the considerations presented hereafter may as well help to clarify the influence of the dimensionality on some properties exhibited by the 3D strongly anisotropic high-Tc superconductors. In this Letter, we present the results of extensive DQMC simulations for the finite-temperature properties of the AHM in three dimensions. In spite of finite-size effects, we show that it is possible by a scaling analysis to quantitatively establish the phase diagram of Tc(U, n) as a function of the interaction strength and density of a model that exhibits a genuine second-order phase transition (unlike its 2D version). Furthermore, the pair for