Translational symmetry breaking in the superconducting state of the cuprates: Analysis of the quasiparticle density of states

Motivated by recent scanning tuneling microscopy (STM) experiments on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ [J. E. Hoffman et al., Science $295,$ 466 (2002); C. Howald et al., cond-mat/0201546 (unpublished); J. E. Hoffman et al., Science $297,$ 1149 (2002): K. McElroy et al. (unpublished); C. Howald et al., cond-mat/0208442 (unpublished)], we study the effects of weak translational symmetry breaking on the quasiparticle spectrum of a d-wave superconductor. We develop a general formalism to discuss periodic charge order, as well as quasiparticle scattering off localized defects. We argue that the STM experiments in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ cannot be explained using a simple charge density wave order parameter, but are consistent with the presence of a periodic modulation in the electron hopping or pairing amplitude. We review the effects of randomness and pinning of the charge order and compare it to the impurity scattering of quasiparticles. We also discuss implications of weak translational symmetry breaking for angle resolved photoemission spectroscopy experiments.