Variational Numerical Renormalization Group: Bridging the Gap between NRG and Density Matrix Renormalization Group

The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows us to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.

Figure 1

Tensor network representation of the (weighted) cost function (4) in the optimization of a tensor ${\underline{A}}^{[j]}$ (circles) under the Hamiltonian matrix product operator (squares).

Figure 2

Accuracy of the low-energy spectral levels for the tilted quantum Ising chain (5) on 200 sites with $h_x=h_z=1$, obtained by the DMRG (red) and optimized by the variational method (blue), for various bond dimensions $D$.

Figure 3

Accuracy of the eigenstates for noninteracting SIAM chain (6) on 70 sites ($N=68$) for NRG, DMRG, and the variational NRG, for $D\in \{100,150\}$. We have set $\Lambda =1.7$, ${\xi }_0=0.1$, ${ϵ}_f=-0.1$.

Figure 4

Accuracy of the lowest 1000 states for the interacting SIAM chain (6) on 40 sites ($N=38$), for NRG and the variational NRG with symmetries. Subsector bond dimensions are taken $D\in \{64,100,128\}$, truncated to $\{6000,8000,10000\}$ states of lowest energy. We have set $\Lambda =2$, ${\xi }_0=0.01$, ${ϵ}_f=-0.05$, and $U=0.1$.