Pseudo-Goldstone modes and dynamical gap generation from order-by-thermal-disorder

Accidental ground state degeneracies -- those not a consequence of symmetry -- are inevitably lifted by fluctuations, a phenomenon known as ``order-by-disorder’’ (ObD). In two dimensions at non-zero temperature, this presents a puzzle: while ObD suggests that thermal fluctuations will stabilize long-range order, for continuous degeneracies, divergence of long-wavelength fluctuations is expected to destroy it in accordance with the Mermin-Wagner-Hohenberg theorem. The resolution of this paradox is tied to the fate of the expected Goldstone modes and how a pseudo-Goldstone gap is dynamically generated by thermal fluctuations that drive the ObD. We study the properties and consequences of such pseudo-Goldstone modes in a minimal two-dimensional model: the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap and show that it scales with temperature T as T^1/2, and that it is well-defined, with a thermally induced linewidth scaling as T^3/2. We show explicitly how this pseudo-Goldstone mode eliminates the infrared-divergent fluctuations, ensuring true long-range order at T>0 in the thermodynamic limit. Finally, we show that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.