Competing orders and quantum criticality in doped antiferromagnets

We use a number of large-N limits to explore the competition between ground states of square lattice doped antiferromagnets which break electromagnetic $U(1),$ time-reversal, or square lattice space-group symmetries. Among the states we find are d-, ${(s}^{*}+id)\ensuremath{-},$ and ${(d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}})$-wave superconductors, Wigner crystals, Wigner crystals of hole pairs, orbital antiferromagnets (or staggered-flux states), and states with spin-Peierls and bond-centered charge stripe order. In the vicinity of second-order quantum phase transitions between the states, we go beyond the large-N limit by identifying the universal quantum field theories for the critical points, and computing the finite temperature, quantum critical damping of fermion spectral functions. We identify candidate critical points for the recently observed quantum critical behavior in photoemission experiments on ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Ca}\mathrm{}\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ by Valla et al. [Science 285, 2110 (1999)]. These involve onset of a charge-density wave, or of broken time-reversal symmetry with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}}$ or ${s}^{*}+id$ pairing, in a d-wave superconductor. It is not required (although it is allowed) that the stable state in the doped cuprates be anything other than the d-wave superconductor\char22{}the other states need only be stable nearby in parameter space. At finite temperatures, fluctuations associated with these nearby states lead to the observed fermion damping in the vicinity of the nodal points in the Brillouin zone. The cases with broken time-reversal symmetry are appealing because the order parameter is not required to satisfy any special commensurability conditions. The observed absence of inelastic damping of quasiparticles with momenta $(\ensuremath{\pi},k),$ $(k,\ensuremath{\pi})$ (with $0l~kl~\ensuremath{\pi})$ also appears very naturally for the case of fluctuations to ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}}$ order.