Time-dependent numerical renormalization group method for multiple quenches: Towards exact results for the long-time limit of thermodynamic observables and spectral functions

Abstract

We develop an alternative time-dependent numerical renormalization group (TDNRG) formalism for multiple quenches and implement it to study the response of a quantum impurity system to a general pulse. Within this approach, we reduce the contribution of the NRG approximation to numerical errors in the time evolution of observables by a formulation that avoids the use of the generalized overlap matrix elements in our previous multiple-quench TDNRG formalism [Nghiem et al., Phys. Rev. B 89, 075118 (2014); Phys. Rev. B 90, 035129 (2014)]. We demonstrate that the formalism yields a smaller cumulative error in the trace of the projected density matrix as a function of time and a smaller discontinuity of local observables between quenches than in our previous approach. Moreover, by increasing the switch-on time, the time between the first and last quench of the discretized pulse, the long-time limit of observables systematically converges to its expected value in the final state, i.e., the more adiabatic the switching, the more accurately is the long-time limit recovered. The present formalism can be straightforwardly extended to infinite switch-on times. We show that this yields highly accurate results for the long-time limit of both thermodynamic observables and spectral functions, and overcomes the significant errors within the single quench formalism [Anders et al., Phys. Rev. Lett. 95, 196801 (2005); Nghiem et al., Phys. Rev. Lett. 119, 156601 (2017)]. This improvement provides a first step towards an accurate description of nonequilibrium steady states of quantum impurity systems, e.g., within the scattering states NRG approach [Anders, Phys. Rev. Lett. 101, 066804 (2008)].

Quench dynamics of spin in quantum dots coupled to spin-polarized leads

“…As we are interested in the charge and spin dynamics, we use the extension of NRG introduced by Anders and Schiller, namely the time-dependent numerical renormalization group (tNRG) method [48,49]. This method was subsequently generalized by Nghiem and Costi to finite temperatures, multiple quenches and possibility to study time evolution in response to general pulses and periodic driving [50][51][52]. While tNRG has already provided a valuable insight into the dynamics of Kondo-correlated molecules and quantum dots attached to nonmagnetic leads [43,[53][54][55], the time-dependent transport properties of correlated impurities with spinpolarized contacts remain rather unexplored.…”

Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy

“…Such an imperfect thermalization at long positive times is expected within the single-quench TDNRG approach 30 . A more precise description of the thermalization at infinite time can be achieved within the multiple-quench TDNRG approach 31 .…”

“…In addition, solving for the nonequilibrium dynamics of quantum impurity systems is a prerequisite for applications to the nonequilibrium dynamical mean field theory 7 of correlated materials, with relevance to interpreting timeresolved photoemission experiments 8,9 . While there are many studies investigating the time-dependent dynamics of quantum impurity systems, including functional and realtime renormalization group methods [10][11][12] , flow equation 13,14 , quantum Monte Carlo 15,16 , density matrix renormalization group methods [17][18][19] , hierarchical quantum master equation approach 20,21 , and, the time-dependent numerical renormalization group (TDNRG) method 5,[22][23][24][25][26][27][28][29][30][31] , there are fewer studies devoted to investigating the nature of the time-dependent spectral function in nonequilibrium situations 16,24,[30][31][32][33][34][35] .…”

Time-dependent spintronic anisotropy in magnetic molecules

“…In pursuance of the precise analysis of the system's response to the considered quench in the strong coupling regime, we resort to the Wilson's numerical renormalization group (NRG) method [67][68][69]. We use the extended implementation allowing for studying the time evolution of the system, namely, the time-dependent numerical renormalization group (tNRG) [70][71][72][73][74]. This method allows for taking into account all the correlations in a fully non-perturbative manner and, thus, generating reliable predictions for the dynamics of the system under investigation.…”