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Quantum Monte Carlo methods 23QUANTUM MONTE CARLO METHODSQuantum Monte Carlo methods are particularly well suited theoretical tools toinvestigate the properties of physical systems in which correlations and fluctuationsplay a major role. An important example is provided by a two-component Fermi gaswhere the interaction strength is tuned by means of a Feshbach resonance. At lowtemperatures the state of the gas changes in a continous way from a BCS superfluid oflarge Cooper pairs to a Bose-Einstein condensate of tightly bound molecules (BCS-<strong>BEC</strong>crossover). The equation of state along the crossover and the properties of correlationfunctions, such as the momentum distribution, the two-body density matrix, the paircorrelation function and the static structure factor, have been calculated by the Trentogroup in a number of publications using the fixed-node diffusion Monte Carlo (FN-DMC) method [1]. This technique is applicable at zero temperature and is based on anansatz for the nodal surface of the many-body wavefunction which provides a rigorousupper bound for the ground-state energy of the gas. Even though the FN-DMC methodis not “exact”, as it relies on the fixed-node approximation to overcome the fermionicsign problem, it is known to be highly accurate to describe the properties of gas-likestates.A direction of research which in the last couple of years has become extremely interestingbecause of the experimental activity carried out first at MIT and Rice Universityand now in many other laboratories, concerns two-component gases with imbalancedFermi energies, either by a different mass or a different population of the two species.In particular, in the case of polarized systems, one predicts a quantum phase transitionfrom a normal to a superfluid state by changing the interaction strength for fixedpolarization. The Trento group has carried out an extensive study, using the FN-DMCmethod, of the phase diagram of polarized Fermi gases along the BCS-<strong>BEC</strong> crossoverand of the nature of the superfluid to normal quantum phase transition.Another example of a class of many-body problems that can only be addressedusing numerical techniques concerns the critical behavior and the determination ofthe superfluid transition temperature. In this case critical fluctuations dominate thephysical properties of the system and, in general, do not allow for a treatment in termsof mean-field or perturbation theory. We have addressed, using the exact Path IntegralMonte Carlo (PIMC) method, the problem of the superfluid transition temperatureof a Bose gas in three and two spatial dimensions. In 3D we improved on previouscalculations by using a recently proposed algorithm [2] which allows for an efficientcalculation of the superfluid density of very large systems. In 2D we investigated
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