Quantum Phase Transitions in d-wave Superconductors and Antiferromagnetic Kagome Lattices

Strongly correlated systems are of interest due to their exotic collective behavior. In this thesis we study low energy effective theory and quantum phase transitions of d-wave superconductors and spin liquids.;First we examine the quantum theory of the spontaneous breaking of lattice rotation symmetry in d-wave superconductors on the square lattice. This is described by a field theory of an Ising nematic order parameter coupled to the gapless fermionic quasiparticles. We determine the structure of the renormalization group to all orders in a 1/Nf expansion, where Nf is the number of fermion spin components. Asymptotically exact results are obtained for the quantum critical theory in which, as in the large Nf theory, the nematic order has a large anomalous dimension, and the fermion spectral functions are highly anisotropic.;Next we study quantum phase transitions in antiferromagnetic kagome lattices. Due to the high geometric frustration, this system poses as a good candidate for a spin liquid with exotic excitations. Here we look at physics of the spinon and vison sector.;In the spinon sector, we investigate the zero-temperature phase diagram of the nearest-neighbor kagome antiferromagnet in the presence of Dzyaloshinksii-Moriya interaction. We develop a theory for the transition between Z 2 spin liquids with bosonic spinons and a phase with antiferromagnetic long-range order. Connections to recent numerical studies and experiments are discussed.;Finally in the vison sector, we present a projective symmetry group (PSG) analysis of the spinless excitations of Z2 spin liquids on the kagome lattice. In the simplest case, vortices carrying Z2 magnetic flux ('visons') are shown to transform under the 48 element group GL(2, Z3 ). Alternative exchange couplings can also lead to a second case with visons transforming under 288 element group GL(2, Z3 ) x D3. We study the quantum phase transition in which visons condense into confining states with valence bond solid order. The critical field theories and confining states are classified using the vison PSGs.