Unusual electronic ordering in the pseudogap phase of underdoped cuprate superconductors

The pseudogap phase of the underdoped cuprate superconductors harbours diverse manifestations of different ordered electronic-states, and then these ordered electronic-states coexist or compete with superconductivity. Here starting from the microscopic electron propagator, the nature of the ordered electronic-states in the pseudogap phase is investigated within the $T$-matrix approach. This $T$-matrix is derived in terms of the inverse of matrix for various kinds of a single impurity, and then is used to evaluate the local density of states (LDOS) by the involvement of all the quasiparticle excitations and scattering processes. It is shown that a number of the anomalous properties in the underdoped cuprate superconductors is directly correlated to the opening of the normal-state pseudogap: (i) the structure of the microscopic octet scattering model generated by the normal-state pseudogap is essentially the same both in the superconducting (SC)-state and pseudogap phase, which naturally leads to that the quasiparticle scattering interference octet phenomenology observed in the SC-state exists in the pseudogap phase; (ii) however, the spectral weight at around the antinodal region in the SC-state is gapped out completely by both the SC gap and normal-state pseudogap, while it in the pseudogap phase is suppressed partially by the normal-state pseudogap, this directly leads to that the non-dispersive checkerboard charge ordering with a finite wave vector ${\bf Q}$ appears in the pseudogap phase only. The theory therefore also shows that the electronic-states affected by the normal-state pseudogap exhibit the LDOS modulation spectrum organization.