Translational Symmetry Breaking in the Superconducting State of the Cuprates: Analysis of the Quasiparticle Density of States

Motivated by the recent STM experiments of J.E. Hoffman et.al. and C. Howald et.al., we study the effects of weak translational symmetry breaking on the quasiparticle spectrum of a d-wave superconductor. We derive an explicit expression for the Fourier component of the local density of states at the ordering wavevector, and demonstrate that various patterns of translational symmetry breaking exhibit different energy dependence. We argue that the STM experiments in $Bi_2Sr_2CaCu_2O_{8+\delta}$ 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. We also discuss implications of weak translational symmetry breaking for the ARPES experiments.