  We introduce and study an XY-type model of thermal and quantum phase
fluctuations in a two-dimensional correlated lattice d-wave superconductor
based on the QED3 effective theory of high temperature superconductors. General
features of and selected results obtained within this model were reported
earlier in an abbreviated format (Z. Tesanovic, cond-mat/0405235). The model is
geared toward describing not only the long distance but also the intermediate
lengthscale physics of underdoped cuprates. In particular, we elucidate the
dynamical origin and investigate specific features of the Cooper pair
charge-density-wave (CPCDW), which we argue is the state behind the periodic
charge density modulation discovered in recent STM experiments. We illustrate
how Mott-Hubbard correlations near half-filling suppress superfluid density and
favor an incompressible state which breaks translational symmetry of the atomic
lattice. We show how the formation of the CPCDW in such a strongly quantum
fluctuating superconductor can be understood as an Abrikosov-Hofstadter problem
in a type-II dual superconductor, with the role of the dual magnetic field
played by the electron density. The resulting Abrikosov lattice of dual
vortices translates into the periodic modulation of the Bogoliubov-deGennes gap
function and the electronic density. We compute detailed signatures of various
Abrikosov-Hofstadter dual vortex arrays in the single-particle local tunneling
density of states. A 4x4 checkerboard-type modulation pattern naturally arises
as an energetically favored ground state at and near the x = 1/8 doping and
produces good agreement with experimental observations.
