We investigate the ground state and critical temperature (Tc) phase diagrams of the classical and quantum S=12 pyrochlore lattice with nearest-neighbor Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider ferromagnetic and antiferromagnetic Heisenberg exchange interaction as well as direct and indirect DMI. At the classical level, three ground states are found: all-in/all-out, ferromagnetic, and a locally ordered XY phase, known as Γ5, which displays an accidental classical U(1) degeneracy at the mean-field level. Quantum zero-point energy fluctuations computed to order 1/S are found to lift the classical ground-state degeneracy and select the so-called ψ3 state out of the degenerate manifold in most parts of the Γ5 regime. Likewise, thermal fluctuations treated classically at the Gaussian level entropically select the ψ3 state at T=0+. In contrast to this low-temperature state-selection behavior, classical Monte Carlo simulations find that the system orders at Tc in the noncoplanar ψ2 state of Γ5 for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from ψ2 to ψ3 at a temperature TΓ5<Tc. The same method finds that the system orders via a single transition at Tc directly into the ψ3 state for most of the region with ferromagnetic Heisenberg exchange and indirect DMI. Such ordering behavior at Tc for the S=12 quantum model is corroborated by high-temperature series expansion. To investigate the T=0 quantum ground state of the model, we apply the pseudo-fermion functional renormalization group (PFFRG). The quantum paramagnetic phase of the pure antiferromagnetic S=12 Heisenberg model is found to persist over a finite region in the phase diagram for both direct or indirect DMI. Interestingly, we find that a combined ferromagnetic Heisenberg and indirect DMI, near the boundary of ferromagnetism and Γ5 antiferromagnetism, may potentially realize a T=0 quantum ground state lacking conventional magnetic order. Otherwise, for the largest portion of the phase diagram, PFFRG finds the same long-range ordered phases (all-in/all-out, ferromagnetic, and Γ5) as in the classical model.
