Restoring the continuum limit in the time-dependent numerical renormalization group approach

TL;DR

This paper introduces a hybrid time-dependent numerical renormalization group method that combines NRG with a Bloch-Redfield formalism to accurately restore the continuum limit in quantum impurity problems, improving non-equilibrium dynamics simulations.

Contribution

It presents a novel hybrid approach integrating NRG and Bloch-Redfield formalism to address bath discretization issues in time-dependent NRG for quantum impurity systems.

Findings

Successfully restores the continuum limit in time-dependent NRG.

Accurately models non-equilibrium dynamics in quantum impurity systems.

Validated against analytical solutions and applied to complex models.

Abstract

The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the non-equilibrium dynamics on the finite size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $\Lambda > 1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic quantum-impurity systems. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to non-trivial models, such as the interacting resonant-level model and the single impurity Anderson model.