Applications of Effective Field Theories to the Many-Body Nuclear Problem and Frustrated Spin Chains

Modern effective-theory techniques are applied to the many-body nuclear problem and frustrated quantum spin chains. A novel approach is proposed for the renormalization of nucleon-nucleon operators in a manner consistent with the construction of the effective potential. To test this approach a one-dimesional, yet realistic, nucleon-nucleon potential is introduced. An effective potential is then constructed by tuning its parameters to reproduce the exact effective range expansion and a variety of bare operators are renormalized in a fashion compatible with this construction. Predictions for the expectation values of these operators in the ground state reproduce the results of the exact theory with remarkable accuracy (at the 0.5% level). We illustrate the main ideas of this work using the elastic form factor of the deuteron as an example. We also apply th COntractor REnormalizator technique to the study of frustrated anti-ferromagnetic zig-zag spin chains with arbitrary half-integer spin. A basis is employed in which three neighboring spins are coupled to a well-defined value of the total angular momentum. The basis is then truncated to retain only the lowest lying energy states, and the Hamiltonian renormalized to reproduce the low-lying spectrum of the original system. We prove the necessity of retaining two, rather than one, lowest energy eigenstates as frustration is increased. A finite size scaling approach is used to extract ground state energy densities in good agreement with DMRG calculations and spin gaps in qualitative agreement with the disappearance of the Haldane phase around a=0.3. Moreover, we are able to develop a renormalization group equation that predicts accurately the ground state energy density of the chain in the thermodynamic limit.