Equilibrium and Nonequilibrium Dynamics of the Sub-Ohmic Spin-Boson Model

Employing the nonperturbative numerical renormalization group method, we study the dynamics of the spin-boson model, which describes a two-level system coupled to a bosonic bath with a spectral density $J(\omega )\propto {\omega }^s$. We show that, in contrast with the case of Ohmic damping, the delocalized phase of the sub-Ohmic model cannot be characterized by a single energy scale only, due to the presence of a nontrivial quantum phase transition. In the strongly sub-Ohmic regime, $s\ll 1$, weakly damped coherent oscillations on short time scales are possible even in the localized phase—this is of crucial relevance, e.g., for qubits subject to electromagnetic noise.

Figure 1

Zero-temperature phase diagram of the spin-boson model from NRG [23], for fixed $\Delta =0.1$ and ${\omega }_c=1$. The axes denote the bath exponent $s$ and the dissipation strength $\alpha$. The circles indicate the parameter values for which results will be presented in this Letter.

Figure 2

Schematic crossover diagrams for the spin-boson model for different bath exponents $s$, deduced from NRG, with regimes of localized, delocalized, and QC behavior. The open dots denote QPTs, for details see text. (Note that the $\alpha$ scale is different for the three panels.)

Figure 3

Equilibrium spin correlation function $C(\omega )$ for (a) $s=0.5$ and $\alpha =0.02$, 0.06, 0.09, 0.10, 0.107, 0.11; (b) $s=0.1$ and $\alpha =0.002$, 0.004, 0.006, 0.007, 0.01; the thick curves are very close to the critical point (for $s=0.5$ we find ${\alpha }_c=0.1065$ and $s=0.1$, ${\alpha }_c=0.0071$). The curves correspond to the parameter values shown as circles in Fig. 1. (The peak width at small $\alpha$ is an artifact of the NRG broadening.)

Figure 4

Equilibrium spin correlation function $C(\omega )$ for $s=0$ and $\alpha =0.003$, 0.005, 0.007, 0.01, 0.02. Both panels show the same data, but the right one has a linear $\omega$ scale. The $A\delta (\omega )$ contribution is not shown.

Figure 5

Time evolution of $P(t)$ for $s=0.5$, $\Delta /{\omega }_c=0.1$, and various values of $\alpha$ (${\alpha }_c=0.1065$).

Figure 6

Time evolution of $P(t)$ for $s=0$, $\Delta /{\omega }_c=0.1$, and various values of $\alpha$ (${\alpha }_c=0$ here).