Competing order, Fermi surface reconstruction, and quantum oscillations in underdoped high-temperature superconductors

We consider incommensurate order parameters for electrons on a square lattice which reduce to d -density wave order when the ordering wave vector Q is close to Q0 = (π/a,π/a), a being the lattice spacing and describe the associated charge and current distributions within a single-harmonic approximation that conserves current to lowest order. Such incommensurate orders can arise at the mean-field level in extended Hubbard models, but the main goal here is to explore thoroughly the consequences within a Hartree-Fock approximation. We find that Fermi surface reconstruction in the underdoped regime can correctly capture the phenomenology of the recent quantum oscillation experiments that suggest incommensurate order, in particular the de Haas-van Alphen oscillations of the magnetization in high fields and very low temperatures in presumably the mixed state of these superconductors. For 10% hole doping in YBa2 Cu3 O6+δ, we find in addition to the main frequency around 530 T arising from the electron pocket and a hole frequency at around 1650 T, a new low frequency from a smaller hole pocket at 250 T for which there are some indications that require further investigations. The oscillation corresponding to the electron pocket will be further split due to bilayer coupling, but the splitting is sufficiently small to require more refined measurements. The truly incommensurate d -density wave breaks both time reversal and inversion, but the product of these two symmetry operations is preserved. The resulting Fermi surface splits into spin-up and spin-down sectors that are inversion conjugates. Each of the spin sectors results in a band structure that violates reflection symmetry, which can be determined in spin and angle-resolved photoemission spectroscopies. For those experiments such as the current photoemission experiments or the quantum oscillation measurements that cannot resolve the spin components, the bands will appear to be symmetric because of the equal mixture of the two spin sectors. There is some similarity of our results with the spiral spin-density wave order which, as pointed out by Overhauser, also breaks time reversal and inversion. Calculations corresponding to higher order commensuration produce results similar to antiphase spin stripes but appear to us to be an unlikely explanation of the experiments. The analysis of the Gorkov equation in the mixed state shows that the oscillation frequencies are unshifted from the putative normal state and the additional Dingle factor arising from the presence of the mixed state can provide a subtle distinction between the spiral spin-density wave and the d -density wave, although this is very difficult to establish precisely.