Numerical Renormalization Group for Bosonic Systems and Application to the Sub-Ohmic Spin-Boson Model

We describe the generalization of Wilson’s numerical renormalization group method to quantum impurity models with a bosonic bath, providing a general nonperturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, $J(\omega )\propto {\omega }^s$. We find clear evidence for a line of continuous quantum phase transitions for sub-Ohmic bath exponents $0<s<1$; the line terminates in the well-known Kosterlitz-Thouless transition at $s=1$. Contact is made with results from perturbative renormalization group, and various other applications are outlined.

Figure 1

(a) Phase diagram for the transition between delocalized ($\alpha <{\alpha }_c$) and localized phases ($\alpha >{\alpha }_c$) of the spin-boson model (1) for bias $ϵ=0$ and various values of $\Delta$, deduced from the NRG flow. (b) $\Delta$ dependence of the critical coupling ${\alpha }_c$ for various values of the bath exponent $s$. The dashed lines are guides to the eye; the solid lines are fits to Eq. (10) using the $\Delta <{10}^{-7}$ points only. For $s$ close to 1 the asymptotic regime is reached only for very small $\Delta$. NRG parameters here and in Figs. 3, 4, 5 are $\Lambda =2$, $N_b=8$, and $N_s=100$.

Figure 2

$\Lambda$ dependence of the critical coupling ${\alpha }_c$ at $s=0.9$, $\Delta ={10}^{-3}$, for various NRG parameters $N_s$ and $N_b$ [16]. The dashed line is a linear fit to the $N_s=120$, $N_b=8$ data in the range $2\le \Lambda \le 3$.

Figure 3

Sample NRG flow diagrams for the sub-Ohmic (upper panel, $s=0.6$) and the Ohmic case (lower panel, $s=1$), close to the phase transition (except for the lower right-hand panel).

Figure 4

Crossover scale $T^{*}$ in the vicinity of the phase transition for $\alpha <{\alpha }_c$, defined here through the first excited NRG level ${\Lambda }^n{E}_{n1}(T^{*})=0.3$. (a) Sub-Ohmic case $s=0.8$, with power-law fits, (b) Ohmic case $s=1$, with exponential fits.

Figure 5

(a) NRG results for the critical exponent $\nu z$, characterizing the vanishing of the crossover energy $T^{*}$ near the critical point, as a function of the bath exponent $s$. Inset: data near $s=1$ together with the function $1/\sqrt{2(1-s)}+0.5$, Eq. (9). (b) Spin autocorrelation function $C(\omega )$ for $s=0.6$, $\Delta ={10}^{-2}$, and $\alpha =0.19\lesssim {\alpha }_c$ close to the transition.