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In underdoped cuprates, both stripes and spiral states may account for the incommensurate spin response observed by elastic neutron scattering experiments. Here, we investigate the respective stability of both textures within the framework of the three-band Hubbard model which we treat within the unrestricted Gutzwiller approximation. Our calculations indicate that for parameter sets appropriate for lanthanum cuprates and small doping nor purely longitudinal stripes nor uniform spirals are stable but stripes with significant spin canting. Indeed at small doping uniform spirals are unstable toward nanoscale phase separation.
It is shown that the magnetic structure of high-T c superconductors is strongly influenced by the next-nearest neighbor hopping parameter t' which distinguishes different families of cuprates. Our investigations indicate that uniform spirals get favored by a large t'=t ratio but are unstable at small doping towards stripes with spin canting. For large /t'/t/ spirals can be stabilized under certain conditions in the overdoped regime which may explain the elastic incommensurate magnetic response recently observed in iron-co-doped Bi2201 materials.
Besides significant electronic correlations, high-temperature superconductors also show a strong coupling of electrons to a number of lattice modes. Combined with the experimental detection of electronic inhomogeneities and ordering phenomena in many high-Tc compounds, these features raise the question as to what extent phonons are involved in the associated instabilities. Here we address this problem based on the Hubbard model including a coupling to phonons in order to capture several salient features of the phase diagram of hole-doped cuprates. Charge degrees of freedom, which are suppressed by the large Hubbard U near half-filling, are found to become active at a fairly low doping level. We find that possible charge order is mainly driven by Fermi surface nesting, with competition between a near-$(\pi ,\pi )$ order at low doping and antinodal nesting at higher doping, very similar to the momentum structure of magnetic fluctuations. The resulting nesting vectors are generally consistent with photoemission and tunneling observations, evidence for charge density wave order in YBa2Cu3O$_{7-\delta }$ including Kohn anomalies, and suggestions of competition between one- and two-q-vector nesting.