  Unconventional strongly correlated phases of the repulsive Fermi-Hubbard
model, which could be emulated by ultracold vapors loaded in optical lattices,
are investigated by means of energy minimizations with quantum number
projection before variation and without any assumed order parameter. In a
tube-like geometry of optical plaquettes to realize the four-leg ladder Hubbard
Hamiltonian, we highlight the intertwining of spin-, charge-, and pair-density
waves embedded in a uniform d-wave superfluid background. As the lattice
filling increases, this phase emerges from homogenous states exhibiting spiral
magnetism and evolves towards a doped antiferromagnet. A concomitant
enhancement of long-ranged d-wave pairing correlations is also found. Numerical
tests of the approach for two-dimensional clusters are carried out, too.

Quantum Gases;Strongly Correlated Electrons