Roton Induced Modulations in Underdoped Cuprates as a Signature of   Incipient Electronic Order
  In the last years there have been measurements of energy independent modulations in underdoped cuprates via scanning tunelling microscopy. These modulations of around 4 lattice constants occur near vortex cores in samples with and without superconductivity, and in the pseudogap regime of some superconducting samples. Recently, it was proposed that the origin of the modulations is electronic charge ordered phases intrinsic to doped Mott insulators. We show that even if the system has not charge ordered, proximity to such an order will induce modulations at the same ordering wavevector. Such modulations are a signature of the real or incipient order. We construct ground state wavefunctions that include strong fluctuations of the competing order without having ordered. They exhibit deep roton minima at wavevectors of the reciprocal lattice of the incipient order due to proximity to the phase transition into the charge ordered state. Such minima, in turn, produce strong modulations. Moreover, the effects are generic and are present whenever an electronic system is near a crystalline phase. This leads to experimental signatures even if the electronic system has not solidified. The signatures will be ubiquitous near the dopings that stabilize the electronic order. 
Zaira Nazario, David I. Santiago