Classical and quantum phases of the pyrochlore S=21 magnet with Heisenberg and Dzyaloshinskii-Moriya interactions

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 ⠂5, 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 & psi;3 state out of the degenerate manifold in most parts of the ⠂5 regime. Likewise, thermal fluctuations treated classically at the Gaussian level entropically select the & psi;3 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 & psi;2 state of ⠂5 for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from & psi;2 to & psi;3 at a temperature T ⠂5 < Tc. The same method finds that the system orders via a single transition at Tc directly into the & psi;3 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 ⠂5 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 ⠂5) as in the classical model.