Time-dependent numerical renormalization group method for multiple quenches: Application to general pulses and periodic driving

Abstract

The time-dependent numerical renormalization group method (TDNRG) [Anders et al.,, Phys. Rev. Lett. 95, 196801 (2005)] was recently generalized to multiple quenches and arbitrary finite temperatures [Nghiem et al., Phys. Rev. B 89, 075118 (2014)] by using the full density matrix approach [Weichselbaum et al., Phys. Rev. Lett. 99, 076402 (2007)]. The formalism rests solely on the numerical renormalization group (NRG) approximation. In this paper, we numerically implement this formalism to study the response of a quantum impurity system to a general pulse and to periodic driving, in which a smooth pulse or a periodic train of pulses is 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. We also investigate the long-time limit of a local observable upon switching from a given initial state to a given final state as a function of both the pulse shape and the switch-on time, finding that this limit is improved for smoother pulse shapes and longer switch-on times. This lends support to our earlier suggestion that the long-time limit of observables, following a quench between a given initial state and a given final state, can be improved by replacing a sudden large and instantaneous 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 the 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 fields. 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 interacting Anderson impurity model in response to a periodic switching of the local level from the mixed valence to the Kondo regime at low, intermediate, and high temperatures.

Quantum sweeps, synchronization, and Kibble-Zurek physics in dissipative quantum spin systems

“…Several theoretical methods have been devised to study the dissipative spin dynamics for one spin in an ohmic bath such as the non-interacting blip approximation 2,3 , Quantum Monte Carlo (QMC) methods on the Keldysh contour [19][20][21][22] , and the time-dependent (TD) Numerical Renormalization Group (NRG) approach [23][24][25][26] , with recent progress done concerning the treatment of driving and quenches 27 . Stochastic approaches have been developed both in the context of stochastic wavefunction approaches 28 or Stochastic Schrödinger Equation (SSE) methods on the density matrix [29][30][31] .…”

Quench dynamics of Anderson impurity model using configuration interaction method

“…The numerical renormalization group method [22][23][24][25][26][27] provides a direct access to the wave function and has a very high energy resolution in the energy window around the Fermi energy, but is quite limited for the energy distribution of bath orbitals. The quantum Monte-Carlo methods, which first formulate the quantity of interest (typically Green's functions) as a weighted summation of infinite terms, and subsequently perform the summation using the standard Monte-Carlo technique (such as heat bath or Metropolis algorithm), are formally exact.…”

The nonequilibrium quantum many-body problem as a paradigm for extreme data science

“…For current transport problems, it is important to formulate the system in terms of the scattering states of the noninteracting system, which is somewhat different from the conventional way of developing the numerical renormalization group because the impurity operators are part of the scattering states operators. Generalizations to include multiple quenches have also been made 72 . One of the issues with this approach is that the exponential decay of the hopping matrix elements projects the system to lower and lower energy shells of the final Hamiltonian, and it isn't clear that the nonequilibrium system is projected onto those low energy states.…”