  We study a spinor (two-component) Bose gas confined in a one-dimensional
double-valley optical lattice which has a double-well structure in momentum
space. Based on field theory analysis, it is found that spinor bosons in the
double-valley band may form a spin-charge mixed chiral spin quasicondensate
under certain conditions. Our numerical calculations in a concrete $\pi$-flux
triangular ladder system confirm the robustness of the chiral spin order
against interactions and quantum fluctuations. This exotic atomic Bose-Einstein
condensate exhibits spatially staggered spin loop currents without any charge
dynamics despite the complete absence of spin-orbit coupling in the system,
creating an interesting approach to atom spintronics. The entanglement entropy
scaling allows us to extract conformal-field-theory central charge and
establish the low-energy effective field theory for the chiral spin condensate
as a two-component Luttinger liquid. Our predictions should be detectable in
atomic experiments through spin-resolved time-of-flight techniques.
