Ohmic two-state system from the perspective of the interacting resonant level model: Thermodynamics and transient dynamics

Abstract

We investigate the thermodynamics and transient dynamics of the (unbiased) Ohmic two-state system by exploiting the equivalence of this model to the interacting resonant level model. For the thermodynamics, we show, by using the numerical renormalization group (NRG) method, how the universal specific heat and susceptibility curves evolve with increasing dissipation strength, $\alpha$, from those of an isolated two-level system at vanishingly small dissipation strength, with the characteristic activated-like behavior in this limit, to those of the isotropic Kondo model in the limit $\alpha\to 1^{-}$. For the transient dynamics of the two-level system, $P(t)=\langle \sigma_{z}(t)\rangle$, with initial-state preparation $P(t\leq0)=+1$, we apply the time-dependent extension of the NRG (TDNRG) to the interacting resonant level model, and compare the results obtained with those from the noninteracting-blip approximation (NIBA), the functional renormalization group (FRG), and the time-dependent density matrix renormalization group (TD-DMRG). We demonstrate excellent agreement on short to intermediate time scales between TDNRG and TD-DMRG for $0\lesssim\alpha \lesssim 0.9$ for $P(t)$, and between TDNRG and FRG in the vicinity of $\alpha=1/2$. Furthermore, we quantify the error in the NIBA for a range of $\alpha$, finding significant errors in the latter even for $0.1\leq \alpha\leq 0.4$. We also briefly discuss why the long-time errors in the present formulation of the TDNRG prevents an investigation of the crossover between coherent and incoherent dynamics. Our results for $P(t)$ at short to intermediate times could act as useful benchmarks for the development of new techniques to simulate the transient dynamics of spin-boson problems.