We explain the anomalous temperature dependence of spin diffusion in liquid $^{3}\mathrm{\ensuremath{\uparrow}}$ and $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures. The anomaly is an experimental manifestation of a unique zero-temperature attenuation in the Fermi liquid theory. We extended our microscopic theory of spin dynamics in spin-polarized Fermi liquids to finite temperatures. The zero-temperature attenuation changes the behavior of the spectrum near the singular point. The data indicate that the superfluid transition temperature for 3He in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is much lower than the current estimates.
Based on the stochastic decoupling framework and auxiliary function averaging technique, a functional integral equation for the zero-temperature population dynamics of the spin-boson model is established. This equation naturally displays the hierarchical structure of the dissipative dynamics and is amenable to nonperturbative approximations. It is shown that at strong dissipation the spin-boson model displays exponential decay with the rate decreasing from the maximum at the coherence-incoherence crossover to zero at the critical dissipation, a manifestation of quantum phase transition. Besides, the scaled time does not change in the whole regime of strong dissipation.
The zero-temperature dynamics of the spin-boson model with strong dissipation has been a challenging problem for more than 20 years. To solve this and quantum dynamics of dissipative systems at large, we recently proposed a mixed random-deterministic method. This scheme has been successfully used to simulate the time evolution of the spin-boson model at zero temperature for weak to moderate dissipation. For a better numerical performance, the approach is further modified so that it is flexible to convert a certain part of the random treatment to a deterministic one a la hierarchical equations. Applying the new method to the strong dissipated spin-boson model at zero temperature, we observe that the population in the localized state obeys a simple decay dynamics and the time scale is proportional to the reciprocal of the cutoff frequency.