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The ground state energy and the singlet-triplet energy gap of the antiferromagnetic Heisenberg model on a ladder is investigated using a mean field theory and the density matrix renormalization group. Spin wave theory shows that the corrections to the local magnetization are infinite. This indicates that no long range order occurs in this system. A flux-phase state is used to calculate the energy gap as a function of the transverse coupling, J ⊥ , in the ladder. It is found that the gap is linear in J ⊥ for J ⊥ ≫ 1 and goes to zero for J ⊥ → 0. The mean field theory agrees well with the numerical results.PACS number(s): 75.30. Ee, 75.30.Kz, 75.10.Jm Typeset using REVT E X

We analyze a possibility of quantum criticality (gaplessness) in dimerized antiferromagnetic twoand three-leg spin-1 2 ladders. Contrary to earlier studies of these models, we examine different dimerization patterns in the ladder. We find that ladders with the columnar dimerization order have lower zero-temperature energies and they are always gapped. For the staggered dimerization order, we find the quantum critical lines, in agreement with earlier analyses. The bond meanfield theory we apply, demonstrates its quantitative accuracy and agrees with available numerical results. We conclude that unless some mechanism for locking dimerization into the energetically less favorable staggered configuration is provided, the dimerized ladders do not order into the phase where the quantum criticality occurs.

In order to address the issue of the puzzling properties of high-temperature superconductors in the underdoped regime, we investigated the effect of on-site Coulomb repulsion on superconductivity by proposing a two-dimensional disordered quantum state that is characterized by rotating antiferromagnetism. The underdoped and overdoped phases are found to be separated by a quantum critical point where the rotating order parameter vanishes. This point is practically equal to the optimal doping density. Our calculations of the doping dependence of the ratio Rϭ2E d /k B T C shows a sharp change at the quantum critical point and accounts for the experimental variations of R in YBCO systems for example. Several other quantities are evaluated and compared to experimental data with fairly good agreement.