The density matrix renormalization group (\DMRG") discovered by White has shown to be a powerful method to understand the properties of many one dimensional quantum systems. In the case where renormalization eventually converges to a xed point we show that quantum states in the thermodynamic limit with periodic boundary conditions can be simply represented by a special type of product ground state with a natural description of Bloch states of elementary excitations that are spin-1 solitons. We then observe that these states can be rederived through a simple variational ansatz making no reference to a renormalization construction. The method is tested on the spin-1 Heisenberg model. Typeset using REVT E X

We investigate the density matrix renormalization group (DMRG) discovered by White and show that in the case where the renormalization eventually converges to a fixed point the DMRG ground state can be simply written as a "matrix product" form. This ground state can also be rederived through a simple variational ansatz making no reference to the DMRG construction. We also show how to construct the "matrix product" states and how to calculate their properties, including the excitation spectrum. This paper provides details of many results announced in an earlier letter. 75.10.Jm, 75.40.Mg

The two-dimensional classical XY model in a random p-fold symmetry-breaking field is studied using the replica method. For p &2v 2 an XY-like phase exists at intermediate values of temperature and weak field. For p (4 we are able to describe the transition into the low-temperature glassy continuation of the paramagnetic phase, while for p & 4 the results suggest the transition may be first order, driven by the unbinding of vortices.Several new fixed points and lines are found in the replicated Kosterlitz-Thouless-type recursion relations corresponding to these various transitions. The method we use considers n coupled XY models from which we construct a Coulomb gas with n (n -1)/2 types of (n -1)-dimensional vector charges.

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