Figure 1: Phonon-induced spin relaxation for an S=1 system.

(a) Displays a Lorentzian and a Dirac-like phonons’ density of states. The spin energy barrier profile is pictured in b, where the ground state of energy E₀ is separated from the spin-flip state of energy E₁ by a barrier U₀=E₂−E₀. The spin relaxation occurs by exciting the spin system to the state of energy E₂ via absorption of a phonon. In the standard Orbach’s process, the phonon is resonant with the spin excitation energy and its spectral function, A(E), is a δ-function (black bar in a). In contrast, when one considers a finite phonon lifetime (red curve in a), the phonon does not need to be resonant with the spin levels. In c, we report the logarithm of the relaxation time τ against the temperature T scaled by the excitation energy U₀/k_B. The inset reports the qualitative behaviour of the phonon linewidth, Δ, as function of the temperature, where T* represents the temperature above which the anharmonic effects start to be important. The black symbols describe the Arrhenius behaviour expected from the standard Orbach process, while the solid lines represent the expected behaviour for anharmonic crystals in three different regimes.