Using the density matrix renormalization group technique, we study the ground state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-half chain with both dimerization and frustration, i.e., an alternation δ of the nearest neighbor exchanges and a next-nearest-neighbor exchange J 2 . For δ = 0, the system is gapless for J 2 < J 2c and has a gap for J 2 > J 2c where J 2c is about 0.241.For J 2 = J 2c , the gap above the ground state grows as δ to the power 0.667 ± 0.001. In the J 2 − δ plane, there is a disorder line 2J 2 + δ = 1. To the left of this line, the peak in the static structure factor S(q) is at q max = π (Neel phase), while to the right of the line, q max decreases from π to π/2 as J 2 is increased to large values (spiral phase). For δ = 1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.

An interesting feature of spin-1 2 chains with a gap is that they undergo a commensurate-incommensurate transition in the presence of an external magnetic field H. The system is in a gapless incommensurate regime for all values of the magnetic field between the lower critical field H c1 and an upper critical field H c2 , where it is gapless and has power law correlations. We calculate the critical exponents for such a generic gapped system in the incommensurate regime at the critical field H c1 and in its vicinity. Our analysis also applies to the spin-1 2 ladder. We compute the full dynamical susceptibilities at finite temperature. We use the same to discuss the thermal broadening of various modes and obtain the low temperature behaviour of the nuclear spin relaxation rate T −1 1 . We discuss the results obtained here for the special cases of the dimerised chain, frustrated chain and the spin-

We construct a new functional for the single particle Green's function, which is a variant of the standard Baym Kadanoff functional. The stability of the stationary solutions to the new functional is directly related to aspects of the irreducible particle hole interaction through the Bethe Salpeter equation. A startling aspect of this functional is that it allows a simple and rigorous derivation of both the standard and extended dynamical mean field (DMFT) equations as stationary conditions. Though the DMFT equations were formerly obtained only in the limit of infinite lattice coordination, the new functional described in the work, presents a way of directly extending DMFT to finite dimensional systems, both on a lattice and in a continuum. Instabilities of the stationary solution at the bifurcation point of the functional, signal the appearance of a zero mode at the Mott transition which then couples to physical quantities resulting in divergences at the transition.

We reconsider the Mott transition problem in the presence of long range Coulomb interactions. Using an extended dynamical mean field theory (DMFT) that sums an important class of diagrams absent in ordinary DMFT, we show that in the presence of Coulomb interactions, the zero temperature Mott transition is, as envisioned by Mott, discontinuous in two and three dimensions.