Density matrix numerical renormalization group for non-Abelian symmetries

We generalize the spectral sum rule preserving density matrix numerical renormalization group (DM-NRG) method in such a way that it can make use of an arbitrary number of not necessarily Abelian local symmetries present in the quantum impurity system. We illustrate the benefits of using non-Abelian symmetries by the example of calculations for the $T$ matrix of the two-channel Kondo model in the presence of magnetic field, for which conventional NRG methods produce large errors and/or take a long run-time.

Figure 1

Construction of the Hamiltonian on the Wilson chain of length $N$. The impurity sits at the square-shaped site labeled by 0; dots represent further sites. At the zeroth NRG iteration $H_0$ is identical with ${\mathcal{H}}_0$. As the $n\text{th}$ site is added to the chain, $H_n$ is constructed recursively from the hopping term $\left({\tau }_{n-1,n}\right)$, from the on-site energy $\left({\mathcal{H}}_n\right)$, and from $H_{n-1}$.

Figure 2

(Color online) A complete basis of a Wilson chain represented as the exponentially increasing number of energy levels belonging to the successive iterations. Continuous/dashed lines represent kept, low-energy/discarded, high-energy levels, respectively. For the consecutive iteration steps the distances between the levels illustrate how the energy resolution of NRG gets exponentially refined.

Figure 3

(Color online) Dimensionless spectral function of $f_{0,1,\uparrow }$ normalized by $D$, the bandwidth cutoff, as a function of $\omega /T_K$ in the absence of magnetic field obtained with DM-NRG and with NRG using the symmetries: ${\text{U}}_S\left(1\right)\times {\text{SU}}_{C1}\left(2\right)\times {\text{SU}}_{C2}\left(2\right)$ and ${\text{U}}_S\left(1\right)\times {\text{U}}_{C1}\left(1\right)\times {\text{U}}_{C2}\left(1\right)$. (a) Comparison between the spectral weights of the DM-NRG and NRG results: DM-NRG fulfills the sum rule entirely even when the used symmetry group and therefore the number of kept states is largely reduced. NRG violates the sum rule to over 15% if the number of kept states is $\approx 7\times {10}^4$ in each iteration. (b) The same spectral functions as a function of $\sqrt{\omega /T_K}$. If a sufficient number of states is kept, i.e., when using larger symmetry groups, the expected $\sqrt{\omega }$ behavior around the 2CK fixed point is nicely recovered. (c) The same spectral functions on a logarithmic scale.

Figure 4

(Color online) Dimensionless spectral function of $f_{0,1,\uparrow }$ normalized by $D$, the bandwidth cutoff, as a function of $\omega /T_K$ in the presence of magnetic field obtained with DM-NRG and with NRG using the symmetries: ${\text{U}}_S\left(1\right)\times {\text{SU}}_{C1}\left(2\right)\times {\text{SU}}_{C2}\left(2\right)$ and ${\text{U}}_S\left(1\right)\times {\text{U}}_{C1}\left(1\right)\times {\text{U}}_{C2}\left(1\right)$. (b) On a smaller scale at $\omega =0$ we show the smoothness of the DM-NRG data using both groups and the jump at $\omega =0$ in the NRG results using the larger group.

Figure 5

(Color online) Dimensionless spectral function of [(a) and (c)] the $\downarrow$-and [(b) and (d)] the $\uparrow$-spin components of the local composite fermion operator normalized by $D$, the bandwidth cutoff, for sufficiently small values of $B$ as a function of $\omega /T_h$ scaled on top of each other using NRG and DM-NRG together with the group ${\text{U}}_S\left(1\right)\times {\text{SU}}_{C1}\left(2\right)\times {\text{SU}}_{C2}\left(2\right)$.

Figure 6

(Color online) Imaginary part of the on-shell $T$ matrix using DM-NRG and NRG together with the group ${\text{U}}_S\left(1\right)\times {\text{SU}}_{C1}\left(2\right)\times {\text{SU}}_{C2}\left(2\right)$ for various magnetic-field values as a function of $\omega /T_K$.

Figure 7

(Color online) Polarization in the 2CKM as a function of $\omega /T_K$ computed by (a) DM-NRG and by (b) NRG, and the same quantity as a function of $\omega /T_h$ showing universal behavior for small $\omega$, but completely contradicting each other as computed by (c) DM-NRG and by (d) NRG.