Order-by-disorder without quantum zero-point fluctuations in the pyrochlore Heisenberg ferromagnet with Dzyaloshinskii-Moriya interactions

Order-by-disorder, whereby fluctuations lift an accidental classical ground state degeneracy to stabilize a subset of ordered states, is a recurrent and prominent theme in the field of frustrated magnetism where magnetic moments are subject to competing interactions. Thus far, such a phenomenon has been discussed in systems where the quantum ground state is not a classical product state. In such a case, both thermal and quantum fluctuations act to lift the accidental classical degeneracy, begging the question of whether one mechanism of order-by-disorder is possible without the other. In this paper, we present results exposing an uncharted route to order-by-disorder, one without quantum zero-point fluctuations, in the ferromagnetic pyrochlore Heisenberg system with the Dzyaloshinskii-Moriya (DM) interaction as the leading perturbation. We prove that any colinear ferromagnetic state is an exact eigenstate while thermal fluctuations give rise to a preference in the magnetization direction. Using linear spin wave theory, we find that the anisotropy appears at lowest order as a sub-leading term in the low-temperature expansion of the free energy, proportional to $T^{7/2}$. Our results thus show that the phenomenon of thermal order-by-disorder can occur even in the absence of quantum zero-point fluctuations driving quantum order-by-disorder, this being so in particular when the accidentally degenerate ground state of the classical model turns out to be an exact eigenstate of the quantum version of the model. In addition, we find that when the DM interaction is large, the fully polarized ferromagnetic ground state becomes unstable for a spin-$\frac{1}{2}$ system within the framework of non-linear spin wave theory, a result that is presumably closely related to the recent report of a quantum spin liquid in this spin-$\frac{1}{2}$ model reported in [https://doi.org/10.1073/pnas.2403487121].