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Motivated by the close analogy with the fractional quantum Hall states(FQHSs), fractional Chern insulators (FCIs) are envisioned as stronglycorrelated, incompressible states emerging in a fractionally filled, (nearly)flat band with non-trivial Chern number. Built upon this vision, fractionaltopological insulators (FTIs) have been proposed as being composed of twoindependent copies of the FCI with opposite Chern numbers for different spins,preserving the time-reversal symmetry as a whole. An important question is ifthe correlation between electrons with different spins can be really ignored.To address this question, we investigate the effects of correlation in thepresence of spin-dependent holomorphicity, i.e., electrons of one spin speciesreside in the holomorphic lowest Landau level, while those of the other in theantiholomorphic counterpart. By constructing and performing exactdiagonalization of an appropriate model Hamiltonian, here, we show thatgeneric, strongly correlated, fractionally filled states with spin-dependentholomorphicity cannot be described as two independent copies of the FQHS,suggesting that FTIs in the lattice cannot be described as those of the FCIeither. Fractionally filled states in this system are generally compressibleexcept at half filling, where an insulating state called the half-filledspin-holomorphic FTI occurs. It is predicted that the half-filledspin-holomorphic FTI is susceptible to an inherent spontaneous symmetrybreaking, leading to the spatial separation of spins.

Spin separation due to an inherent spontaneous symmetry breaking of the fractional topological insulator