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This chapter discusses the quantum phase transition using the example of superconductor-insulator transition. For the 3D case, a version of the Ginzburg-Landau formalism is formulated from both normal and anomalous diffusion. The description of such transition in the case of 2D superconductors is very specific and strongly differs from the 3D case. The ideas of boson and fermion mechanisms of the T_c suppression in 2D cases are presented, and their predictions are compared with the experimental conditions.

Superconductor-insulator (SI) transitions of homogeneously disordered ultrathin quench-condensed films in many instances appear to be direct, without any intervening metallic regime. This is in contrast with what has been found in some other systems. These direct transitions have been analyzed using finite size scaling. The products of the dynamical critical exponent and the coherence length exponent found vary, depending upon the tuning parameter. They are approximately 1.3 for the thickness tuned SI transition, and approximately 0.7 for perpendicular and parallel magnetic field tuning. Charge tuning also yields 0.7. Assuming that the dynamical critical exponent is unity as is anticipated for systems with long range interactions, all of the transitions, except the thickness-tuned transition would appear to belong to the 3D XY universality class. This behavior is different from that observed for magnetic field tuned transitions of compounds such as InO­_x or TiN, or other metallic systems. The source of these differences is not known but may be due to differences in carrier density or structural or chemical disorder on a mesoscopic scale.

The superconductor-insulator quantum phase transitions tuned by three independent parameters: disorder, paramagnetic impurity and perpendicular magnetic field, have been directly compared in the same homogeneous ultrathin a-Pb films. The magnetic impurity-tuned transition is characterized by progressive suppression of the critical temperature to zero and a continuous transition to a weakly insulating normal state with increasing impurity density. In all key aspects, the disorder-tuned transition closely resembles the magnetic impurity-tuned transition. The magnetic field-tuned transition appears qualitatively different, and in many aspects resembles the behavior of the superconductor-insulator transition in granular films which is a canonical example of a bosonic transition. In the critical region, the field-tuned transition exhibits transport behavior that suggests magnetic field induced mesoscale phase separation and presence of Cooper pairing in the insulating state.

Transport and spin-resolved electron tunneling measurements on ultra-thin Al and Be films are used to extract the spin-mediated behavior on either side of the zero-field superconductor-insulator transition. The primary focus of these studies is the behavior of the Zeeman-mediated superconductor-insulator transition in Al films with sheet resistances well below the quantum resistance and the nature of the correlated insulator phase in very high resistance Be films. By applying magnetic fields in the plane of the films one can control the Zeeman splitting in these low spin-orbit scattering systems without introducing an orbital response. The data give clear evidence for incoherent pairing effects in both the Zeeman-limited normal state of low resistance Al films and in the correlated insulator phase of very high resistance Be films.

This chapter reviews Determinant Quantum Monte Carlo studies of the disordered Hubbard Hamiltonian. The effect of the interplay of interactions and a variety of realizations of the randomness, including spatially varying hopping and chemical potentials, on the conductivity, magnetism, Mott gap formation, and s-wave superconductivity is evaluated. Advances in algorithms and in computer hardware have made possible an order of magnitude increase in system sizes over the last several years, and optical lattice emulators of the Hubbard Hamiltonian are providing a new experimental realization of disordered and interacting fermions. This suggests the possibility of new efforts in this field.