Numerical renormalization

Iztok Pižorn, Frank Verstraete

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.

Heinz‐Otto Peitgen, Péter Richter

Hamed Saberi, Andreas Weichselbaum, Jan von Delft

Wilson's numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White's density-matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model with a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an ``unfolded'' Wilson chain, which brings about a significant reduction in numerical costs compared to those of NRG. We show that all NRG eigenstates (kept and discarded) can be reproduced using VMPS, and compare the difference in truncation criteria, sharp vs smooth in energy space, of the two approaches. Finally, we demonstrate that NRG results can be improved upon systematically by performing a variational optimization in the space of variational matrix product states, using the states produced by NRG as input.

Alexander Altland, Benjamin D. Simons

Pedro D Sacramento

Roman Krčmár, Michael Weyrauch

Luttinger-liquid parameters are extracted from ground-state energy calculations using a fermionic functional renormalization group (fRG) scheme similar to that proposed by Andergassen et al. [Phys. Rev. B 73, 045125 (2006)]. A different way of interaction renormalization is proposed and compared to the method in this reference. The proposed new interaction renormalization produces superior numerical results for chemical potential as well as compressibility, while the Luttinger parameter $K$ is obtained quantitatively similar in both interaction renormalization schemes. It is demonstrated that a consistent extraction of Luttinger-liquid parameters from the numerically calculated ground state within fRG meets several severe numerical issues, partly related to the fact that the truncated fRG is not a conserving approximation.