| Strong electron correlations in condensed matter systems give rise to a wide range of striking physical properties, producing phenomena as varied as high temperature superconductivity, metal-insulator transitions and the integer and fractional quantum Hall effects. Quantum critical systems also exhibit strong correlations between a large number of degrees of freedom. In this thesis we study these complicated systems using a combination of analytical and numerical approaches. We perform systematic investigations, which adds to the robustness of our results.; We develop a new method, based on the density-matrix renormalization-group (DMRG) algorithm combined with finite-size scaling analysis, to study critical behavior in quantum spin chains and extract critical exponents. Accurate results are obtained for spin-1/2 antiferromagnetic chains and the spin-1 chain at the critical point separating the Haldane and the dimerized phases.; Disorder in a system can change its properties drastically. Plateau transitions in the integer quantum Hall effect provide the clearest example of quantum critical behavior in a disordered system. We provide analytical proof that the Chalker-Coddington model, which is used to describe the plateau transitions, is quantum critical. Starting from a field theory based on this model, equivalent to a non-Hermitian supersymmetric spin chain, we prove quantum criticality by a Lieb-Schultz-Mattis type theorem. This approach was motivated by numerical results obtained using the DMRG/finite-size scaling method. Our generalized LSM theorem also applies to the spin quantum Hall effect, which can appear in disordered d-wave superconductors with broken time-reversal symmetry.; The last part of the thesis is a renormalization-group study of two dimensional interacting electron systems. We obtain results relevant to high-temperature superconductors and also to the family of κ - (BEDT - TTF)2X organic superconductors. At half filling, the fully nested square lattice Hubbard model has antiferromagnetic spin density wave instability. In agreement with the observed behavior of the high-Tc cuprates, when the system is doped slightly away from half-filling, there is a crossover to a superconducting instability. We also find superconducting instabilities when antiferromagnetic couplings are frustrated on an anisotropic triangular lattice. |
