Generalized Polaron Ansatz for the Ground State of the Sub-Ohmic Spin-Boson Model: An Analytic Theory of the Localization Transition

The sub-Ohmic spin-boson model possesses a quantum phase transition at zero temperature between a localized and a delocalized phase, whose properties have so far only been extracted by numerical approaches. Here we present an extension of the Silbey-Harris variational polaron ansatz which allows us to develop an analytical theory which correctly describes a continuous transition with mean-field exponents for $0<s<0.5$. The critical properties, couplings, and observables we obtain show excellent agreement with existing numerical results, and we give an intuitive microscopic description of the changing correlations between the system and bath which suppress the spin coherence and drive the transition.

Figure 1

Magnetization $M$ as a function of $(\alpha -{\alpha }_c)/{\alpha }_c$ for $\alpha >{\alpha }_c$, $\Delta =1$, and ${\omega }_c=10$. For visibility, the curves have been multiplied by 1, 2, 4, and 8 for $s=0.1$, 0.2, 0.3, and 0.4, respectively.

Figure 2

Expectation value $⟨{\sigma }_x⟩$ as a function of $\alpha$ for $\Delta =1$ and ${\omega }_c=10$. Cusps appear at the critical couplings ${\alpha }_c$. The corresponding scaling limit critical couplings are indicated by arrows.

Figure 3

Entanglement between the spin and bosonic environment as a function of $\alpha$ for $\Delta =1$ and ${\omega }_c=10$. The entanglement is defined here as the von Neumann entropy of the reduced density matrix of the TLS. Maxima occur at ${\alpha }_c$, and the predicted scaling limit critical couplings are indicated by arrows.

Figure 4

Fidelity of the variational ansatz with the ground state determined by DMRG as a function of $\alpha$ for $\Delta =1$ and ${\omega }_c=10$. The DMRG simulation used 100 sites; $N_b=15$ and 10 Schmidt coefficients were retained. At the critical coupling, the fidelity drops suddenly from finite values to zero. The inset shows the typically dramatic decrease in the fidelity with system size above the transition. Inset data correspond to $s=0.3$ and $\alpha =0.0328$ (${\alpha }_c=0.0316$).