  We present a study of the effects of simultaneous charge- and
spin-frustration on the two-dimensional strongly correlated quarter-filled band
on an anisotropic triangular lattice. The broken-symmetry states that dominate
in the weakly frustrated region near the rectangular lattice limit are the well
known antiferromagnetic state with in-phase lattice dimerization along one
direction, and the Wigner crystal state with the checkerboard charge order. For
moderate to strong frustration, however, the dominant phase is a novel
spin-singlet paired-electron crystal (PEC), consisting of pairs of charge-rich
sites separated by pairs of charge-poor sites. The PEC, with coexisting
charge-order and spin-gap in two dimension, is the quarter-filled band
equivalent of the valence bond solid (VBS) that can appear in the frustrated
half-filled band within antiferromagnetic spin Hamiltonians. We discuss the
phase diagram as a function of on-site and intersite Coulomb interactions as
well as electron-phonon coupling strength. We speculate that the spin-bonded
pairs of the PEC can become mobile for even stronger frustration, giving rise
to a paired-electron liquid. We discuss the implications of the PEC concept for
understanding several classes of quarter-filled band materials that display
unconventional superconductivity, focusing in particular on organic charge
transfer solids. Our work points out the need to go beyond quantum spin liquid
(QSL) concepts for highly frustrated organic charge-transfer solids such as
kappa-(BEDT-TTF)_2Cu_2(CN)_3 and EtMe_3Sb[Pd(dmit)_2]_2, which we believe show
frustration-induced charge disproportionation at low temperatures. We discuss
possible application to layered cobaltates and 1/4-filled band spinels.
