Quantum Monte Carlo Studies of Anderson and Kondo Systems.

In this thesis, we used a recently developed quantum Monte Carlo algorithm to study Anderson and Kondo Hamiltonians, models of magnetic ions interacting with a conduction or conduction-like band. These Hamiltonians are particularly relevant to "heavy fermion" materials, materials with anomalously high low-temperature susceptibilities and specific heats. One of the main purposes of our thesis is to gain some understanding of the magnetic properties of such systems. We begin in the first two chapters by analyzing the one systematic approximation in the algorithm we use, the so-called "Trotter approximation." We show the vanishing of the leading error term under quite general conditions, and derive the form of the next-order term, enabling one to extrapolate in a controlled way to the exact limit. We then simulate systems of increasing complexity: first, a single magnetic ion impurity interacting with a conduction band; then, two impurities; and, finally, a lattice of regularly spaced magnetic ions, interacting with a conduction band. We investigate in particular the competition between the "Kondo effect," which tends to quench the magnetic moments of the ions, and "RKKY" interactions between the ions, which can lead to long-range magnetic order in a lattice. We parameterize the general magnetic behavior of the different Hamiltonians we study, and discuss the implications of our results of heavy fermion materials.