2003

Ph.D

University of Madras

The low energy properties of many strongly correlated electronic systems can be described by an effective theory in terms of just spin degrees of freedom. One of the simplest interacting model, Heisenberg model has been successfully used to describe many such systems. The Heisenberg antiferromagnet (HAFM) on various lattices and in different dimensions are studied in this thesis. It is known that in one dimension the ground state is always disordered. But some times can have a 'quasi long - range order', in which case the spin correlations fall off as a power law in the asymptotic limit and the system is gapless. In one dimensional systems, though the quantum system does not have a broken symmetry, it has some short range order which can distinguish between various 'phases'. The existence of ground state in three phases viz., Neel, Spiral and Colinear phases are studied and a phase diagram is obtained from the short range correlations of the spin. The (J1 - J2 - (delta)) model has been studied numerically for extreme quantum cases. In this thesis the demerised spin chain is studied with a view to understand O(3) NLSM at strong coupling. A real space renormalization group(RG) scheme has been developed for the demerised spin chain from which the anticipated RG flow is obtained. It is possible to correctly describe the edge physics of the sigma model, by analyzing a semi-infinite chain. In two dimensions all unfrustrated systems have long range order. Even triangular lattice HAFM, which is frustrated is ordered, and requires stronger frustration to obtain a disordered ground state. Shastry-Sutherland(SSM) Model is HAFM on the square lattice with an additional frustrating coupling which pairs all spins. A quadramerized version of SSM is described which has a rich phase diagram. For the classical ground state the two models are connected by a transformation; For strong frustration, dimer-singlet is the ground state. For smaller frustration the ground state is a plaquette ordered singlet. As the frustration increased from 0 to infinity the model goes from decoupled plaquettes to decoupled dimers. A generalised SSM for arbitrary dimensions has been constructed. Also the arbitrary dimensional generalized Shastry Sutherland Model in the presence of an external magnetic field is studied. The possibility of two different mechanisms for the plateau formation in different parameter regimes is explored. The dimerised spin chain and its connections with O(3) sigma model are studied; The Grassmann sigma models are derived from SU(N) spin chains. Models with exact ground states are also studied.