Quantum spin liquids are long-range entangled phases whose magnetic correlations are determined by strong quantum fluctuations. While an overarching principle specifying the precise microscopic coupling scenarios for which quantum spin-liquid behavior arises is unknown, it is well-established that they are preferably found in spin systems where the corresponding classical limit of spin magnitudes $Srightarrowinfty$ exhibits a macroscopic ground state degeneracy, so-called classical spin liquids. Spiral spin liquids represent a special family of classical spin liquids where degenerate manifolds of spin spirals form closed contours or surfaces in momentum space. Here, we investigate the potential of spiral spin liquids to evoke quantum spin-liquid behavior when the spin magnitude is tuned from the classical $Srightarrowinfty$ limit to the quantum $S=1/2$ case. To this end, we first use the Luttinger-Tisza method to formulate a general scheme which allows one to construct new spiral spin liquids based on bipartite lattices. We apply this approach to the two-dimensional square lattice and the three-dimensional hcp lattice to design classical spiral spin-liquid phases which have not been previously studied. By employing the pseudofermion functional renormalization group (PFFRG) technique we investigate the effects of quantum fluctuations when the classical spins are replaced by quantum $S=1/2$ spins. We indeed find that extended spiral spin-liquid regimes change into paramagnetic quantum phases possibly realizing quantum spin liquids. Remnants of the degenerate spiral surfaces are still discernible in the momentum-resolved susceptibility, even in the quantum $S=1/2$ case. In total, this corroborates the potential of classical spiral spin liquids to induce more complex non-magnetic quantum phases.
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