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Register a new userSelf-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach
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The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green functions and allows for a more accurate calculation of equilibrium spectral functions, than is possible directly from the one-particle Green function [Bulla {it et al.} Journal of Physics: Condensed Matter {bf 10}, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model . We use the equation of motion method to obtain closed expressions for the local Green function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a two- and a one-particle time-dependent Green function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and, the closed form expressions for the Green function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous time quantum Monte Carlo techniques.
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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 {em et al.,} Phys. Rev. B {bf 89}, 075118 (2014); Phys. Rev. B {bf 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 {em et al.}, Phys. Rev. Lett. {bf 95}, 196801 (2005); Nghiem {em et al.}, Phys. Rev. Lett. {bf 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. {bf 101}, 066804 (2008)].
We investigate several definitions of the time-dependent spectral function $A(omega,t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group method. In terms of the two-time retarded Green function $G^r(t_1,t_2)$, the definitions differ in the choice of the time variable $t$ with respect to $t_1$ and/or $t_2$. In a previous study [Nghiem {it et al.} Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function, obtained from the Fourier transform of ${rm Im}[G^r(t_1,t_2)]$ w.r.t. the time difference $t=t_1-t_2$, with $t=t_2$. Here, we derivie expressions for the retarded Green function for the choices $t=t_1$ and the average time $t=(t_1+t_2)/2$, within the TDNRG approach. We compare and contrast the resulting $A(omega,t)$ for the different choices of time reference. Expressions for the lesser, greater and advanced Green functions are also derived within TDNRG for all choices of time reference. The average time lesser Green function $G^<(omega,t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(omega,t)=G^<(omega,t)/(2pi i)$, a quantity that determines the photoemission current in time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(omega,t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy peaks. We also discuss the issue of thermalization at long times for $N(omega,t)$. Finally, we use the results for $N(omega,t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.
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.
The time-dependent numerical renormalization group method (TDNRG) [Anders et al., Phys. Rev. Lett. {bf 95}, 196801 (2005)] was recently generalized to multiple quenches and arbitrary finite temperatures [Nghiem et al., Phys. Rev. B {bf 89}, 075118 (2014)] by using the full density matrix approach [Weichselbaum et al., Phys. Rev. Lett. {bf 99}, 076402 (2007)]. In this paper, we numerically implement this formalism to study the response of a quantum impurity system to a general pulse and periodic driving which are approximated by a sufficient number of quenches. We show how the NRG approximation affects the trace of the projected density matrices and the continuity of the time-evolution of a local observable. For the general pulse case, the local observable in the long-time limit exhibits a dependence on the switch-on time, the time interval between the first and last quenches, as well as on the pulse shape. In particular, the long-time limit is improved for longer switch-on times and smoother pulses. This lends support to our earlier suggestion that the long-time limit of observables can be improved by replacing a sudden large quench by a sequence of smaller ones acting over a finite time-interval: longer switch-on times and smoother pulses, i.e., increased adiabaticity, favor relaxation of the system to its correct thermodynamic long-time limit. For the case of periodic driving, we compare the TDNRG results to exact analytic ones for the non-interacting resonant level model, finding better agreement at short to intermediate time scales in the case of smoother driving. Finally, we demonstrate the validity of the multiple-quench TDNRG formalism for arbitrary temperatures by studying the time-evolution of the occupation number in the Anderson impurity model in response to a periodic switching of the local level from the mixed valence to the Kondo regime at finite temperatures.
We show how the density-matrix numerical renormalization group (DM-NRG) method can be used in combination with non-Abelian symmetries such as SU(N), where the decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.
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