Competing orders in the cuprate superconductors

When the superconducting order in a doped cuprate system is suppressed, for example by magnetic field, impurities, finite temperature or less optimal doping condition, various competing orders are revealed. This thesis contains the theoretical analysis of these competing orders, especially the analysis of the quantum phase transitions from a d-wave superconductor to a superconductor featuring a certain other order. Such orders can be divided into two types according to whether or not they carry non-zero net momentum Q. We classify and analyze all order parameters with Q = 0 on the basis of group-theoretic and symmetry arguments. Thereafter, by a renormalization group analysis, we identify that among all these orders the fluctuations of d_{x2 − y2} + id_xy paring order might strongly damp the nodal quasiparticles of the cuprate superconductor, while leaving antinodal quasiparticles undamped. We also study order parameters with Q ≠ 0, which include spin and charge density wave orders. For exact nesting case, fluctuations of these two orders also give rise to nodal quasiparticle damping. We also examine the influence of a magnetic field on the spin and charge density waves. We argue that the underdoped cuprates might undergo a second order phase transition into a region where superconducting order coexists with long-range magnetic order (and hence charge order). The transition might be driven by lowering the doping or increasing the magnetic field, all of which decrease the spin gap and affect the excitation spectrum. Our quantum field theory for the transition takes into account the quantum fluctuation of spin order, while keeping superconducting order a mean-field quantity that serves as a tuning parameter by coupling to spin order. Complete numerical solutions of a self-consistent large-N theory provide detailed information on the phase diagram and the spatial structure of the dynamic spin spectrum. We also compute the pinning of charge density order by the vortices where the spin order dynamically fluctuates. Emerging experimental evidence has strongly supported our theory and the subsequent predictions.