Variational dynamics of the sub-Ohmic spin-boson model on the basis of multiple Davydov $\mathrm{D}_1$ states

Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D$_1$ trial states, also known as the multi-D$_1$ Ansatz. Accuracy in dynamics simulations is improved significantly over the single D$_1$ Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D$_1$ Ansatz for various coupling strengths and initial conditions are also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D$_1$ Ansatz. The phase diagram is obtained with a critical point $s_{c}=0.4$. For $s_{c}<s<1$, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For $s<s_{c}$, only the coherent phase exists.