Numerical renormalization group for quantum impurities in a bosonic bath

We present a detailed description of the recently proposed numerical renormalization group method for models of quantum impurities coupled to a bosonic bath. Specifically, the method is applied to the spin-boson model, both in the Ohmic and sub-Ohmic cases. We present various results for static as well as dynamic quantities and discuss details of the numerical implementation, e.g., the discretization of a bosonic bath with arbitrary continuous spectral density, the suitable choice of a finite basis in the bosonic Hilbert space, and questions of convergence with respect to truncation parameters. The method is shown to provide high-accuracy data over the whole range of model parameters and temperatures, which are in agreement with exact results and other numerical data from the literature.

Figure 1

Logarithmic discretization of the bath spectral function in intervals $[{\Lambda }^{-(n+1)},{\Lambda }^{-n}]$ $(n=0,1,2,\dots )$; typical values of the NRG discretization parameter $\Lambda$ as used in the bosonic NRG are $\Lambda =1.5\dots 3.0$.

Figure 2

(a) Structure of the spin-boson model corresponding to Eq. (14) as used for the chain-NRG and (b) to Eq. (11) as used for the star-NRG; the boxes indicate the iterative diagonalization scheme for both cases.

Figure 3

Dependence of the expectation value $n_{N+1}=⟨b_{N+1}^{†}{b}_{N+1}⟩$ on the number of basis states $N_b$ for a chain-NRG calculation. The values of $n_{N+1}\left(N_b\right)$ quickly saturate for $\alpha =0.05<{\alpha }_c$ and ${\alpha }_c=0.06113$ whereas no saturation is observed for $\alpha =0.1>{\alpha }_c$. (Parameters are $s=0.6$, $N=20$, $\Lambda =2.0$, $N_s=60$, $\Delta =0.01$.)

Figure 4

Dependence of the energies of the lowest eigenstates of $H_{N+1}$ on the displacement $\theta$ used for constructing the basis for the degree of freedom added in each iteration step. All levels shown here have their minimum at the same value $\theta =\theta *$ (indicated by the vertical line) which is the optimal value for setting up the basis. The parameters for this calculation are: $s=0.2$, $\Delta =1.0$, $\alpha =0.25>{\alpha }_c$.

Figure 5

Flow diagrams calculated with the chain-NRG for the parameters $s=1$ (Ohmic case), ${\omega }_c=1$, $ϵ=0$, $\Delta =0.01$, and $\alpha =0.6$ in (a) and $\alpha =1.4$ in (b). The NRG parameters are $N_s=100$, $N_b=8$, and $\Lambda =2.0$.

Figure 6

Comparison between the fixed-point spectra for the delocalized fixed point calculated with the chain-NRG (circles) for various values of $N_b$ and the fixed point spectra constructed from the single-particle levels ${\overline{\omega }}_n$ in Eq. (50) (solid lines) for a selection of states $E_N\left(r\right)$. The parameters are $s=1$, $\Delta =1.0$, and $\alpha <{\alpha }_c$. The NRG parameters are $N_s=100$, and $\Lambda =2.0$.

Figure 7

Dependence of the critical coupling ${\alpha }_c$ on the NRG parameters $N_s$, $N_b$, and $\Lambda$ for the Ohmic case; (a) dependence on $N_s$ for fixed $N_b=10$; (b) dependence on $N_b$ for fixed $N_s=100$ [$\Lambda =2.0$ for both (a) and (b)]; (c) $\Lambda$ dependence of ${\alpha }_c$ for two values of $\Delta$, and various NRG parameters $N_b$ and $N_s$. The dashed lines are linear fits to the $N_b=8$ and $N_s=100$ data in the range $1.8⩽\Lambda ⩽3$.

Figure 8

Dependence of the extrapolated value ${\alpha }_c(\Lambda \to 1)$ on the parameter $\Delta$ for the Ohmic case $s=1$. The crosses are the numerical data and the solid line is a linear fit which gives ${\alpha }_c\left(\Delta \right)=0.99+0.53\Delta$. The NRG parameters are $N_s=100$ and $N_b=8$.

Figure 9

Scaling of the flow of the many-particle levels $E_N\left(r\right)$ for fixed $\alpha =0.6$, $s=1$, and various values of $\Delta$. The NRG parameters are $N_s=100$, $N_b=8$, and $\Lambda =2.0$.

Figure 10

Dependence of the crossover temperature $T*$ on $\Delta$ for $s=1$ and fixed values of $\alpha$. The exponents in $T*\propto {\Delta }^x$ are $x\approx 1.55$ for $\alpha =0.4$, $x\approx 2.15$ for $\alpha =0.6$, and $x\approx 3.49$ for $\alpha =0.8$. The NRG parameters are $N_s=100$, $N_b=8$, and $\Lambda =2.0$.

Figure 11

Dependence of the crossover temperature $T*$ on $\alpha$ for $s=1$ and fixed values of $\Delta$ [data for $\Delta ={10}^{-3}$ and $\Delta ={10}^{-4}$ same as in Fig. 4(b) of Ref. 13]. The values for the critical coupling are ${\alpha }_c=1.162$ for $\Delta ={10}^{-2}$, ${\alpha }_c=1.150$ for $\Delta ={10}^{-3}$, and ${\alpha }_c=1.147$ for $\Delta ={10}^{-4}$. The NRG parameters are $N_s=100$, $N_b=8$, and $\Lambda =2.0$.

Figure 12

(Color online) Flow diagram of the lowest lying many-particle energies calculated with the star-NRG for the sub-Ohmic case ($s=0.8$, $\Delta =0.1$), using displaced oscillators as optimized basis. The critical value is ${\alpha }_c=0.40294$. The NRG parameters are $N_s=80$, $N_b=8$, and $\Lambda =2.0$.

Figure 13

Dependence of the expectation value $n_N=⟨b_N^{†}{b}_N⟩$ on the iteration number $N$ for $s=0.8$, $\Delta =0.01$, $\alpha =0.2<{\alpha }_c$ (squares), $\alpha ={\alpha }_c=0.21488785$ (diamonds), and $\alpha =0.4>{\alpha }_c$ (circles), and two different values of $N_b$, calculated with the star-NRG.

Figure 14

Temperature dependence of the impurity contribution to the entropy, $S_{\mathrm{imp}}\left(T\right)$, for $\alpha =1∕3$, $s=1$ (Ohmic case), and various values of $\Delta$.

Figure 15

(a) Scaling curves of the impurity contribution to the entropy, $S_{\mathrm{imp}}\left(T\right)$, for $s=1$ (Ohmic case), and various values of $\alpha$; (b) Scaling curves of the impurity contribution to the specific heat, $C_{\mathrm{imp}}\left(T\right)∕(T∕T*)$, for the same parameters as in (a).

Figure 16

Temperature dependence of the impurity contribution to the entropy, $S_{\mathrm{imp}}\left(T\right)$, in the sub-Ohmic case for various values of $\alpha$ and $s=0.8$ (main panel) and various values of $s$ (inset). The coupling $\alpha$ is below ${\alpha }_c$ so that the flow is to the delocalized phase for all parameters in this figure. Lines with symbols in the inset are data from the bosonic NRG and solid lines are fits assuming a power-law, $S_{\mathrm{imp}}\left(T\right)\propto T^s$.

Figure 17

Spin-spin correlation function $C\left(\omega \right)$ calculated for the Ohmic case, $s=1$, $\Delta =0.01$, and various values of $\alpha <{\alpha }_c$ close to the transition. For small frequencies, $C\left(\omega \right)\propto \omega$, whereas for higher frequencies we observe a divergence, $C\left(\omega \right)\propto {\omega }^{-1}$, with logarithmic corrections.

Figure 18

Spin-spin correlation function $S\left(\omega \right)=2C\left(\omega \right)∕{\omega }^{\delta }$ at the Toulouse point $\alpha =\frac{1}{2}$ for the Ohmic case $s=1$ and various values of $\Delta$. The inset shows the scaling of these curves [$S\left(\omega \right)∕S\left(0\right)$ plotted versus $\omega ∕{\Delta }_r$] together with the exact result (thick dashed line).

Figure 19

NRG results for ${\chi }_{\mathrm{NRG}}$ calculated at the Toulouse point $\alpha =\frac{1}{2}$ for the Ohmic case $s=1$ and various values of $\Lambda$ and $\Delta$ (circles: $\Delta =0.0125$, squares: $\Delta =0.025$, diamonds: $\Delta =0.05$, triangles: $\Delta =0.1$). Dashed lines show the exact values ${\chi }_e=16∕\left({\pi }^2{\Delta }^2\right)$ and thin solid lines are fits to the numerical results.

Figure 20

Scaling spectra for the spin-spin correlation function $S\left(\omega \right)=2C\left(\omega \right)∕{\omega }^{\delta }$ for various values of $\alpha$ in the Ohmic case $s=1$.

Figure 21

(a) Energy of the first excited state $E\left(1\right)$ versus iteration number for $s=1.0$, $\Delta =0.01$, and various values of $\alpha$. (b) matrix element ${\mid ⟨0\mid {\sigma }_z\mid 1⟩\mid }^2$ vs iteration number for the same parameters as in (a).