The time-dependent numerical renormalization-group approach (TD-NRG), originally devised for tracking the real-time dynamics of quantum-impurity systems following a single quantum quench, is extended to multiple switching events. This generalization of the TD-NRG encompasses the possibility of periodic switching, allowing for coherent control of strongly correlated systems by an external time-dependent field. To this end, we have embedded the TD-NRG in a hybrid framework that combines the outstanding capabilities of the numerical renormalization group to systematically construct the effective low-energy Hamiltonian of the system with the prowess of complementary approaches for calculating the real-time dynamics derived from this Hamiltonian. We demonstrate the power of our approach by hybridizing the TD-NRG with the Chebyshev expansion technique in order to investigate periodic switching in the interacting resonant-level model. Although the interacting model shares the same low-energy fixed point as its noninteracting counterpart, we surprisingly find the gradual emergence of damped oscillations as the interaction strength is increased. Focusing on a single quantum quench and using a strong-coupling analysis, we reveal the origin of these interaction-induced oscillations and provide an analytical estimate for their frequency. The latter agrees well with the numerical results.
The time-dependent numerical renormalization-group approach (TD-NRG), originally devised for tracking the real-time dynamics of quantum-impurity systems following a single quantum quench, is extended to multiple switching events. This generalization of the TD-NRG encompasses the possibility of periodic switching, allowing for coherent control of strongly correlated systems by an external time-dependent field. To this end, we have embedded the TD-NRG in a hybrid framework that combines the outstanding capabilities of the numerical renormalization group to systematically construct the effective low-energy Hamiltonian of the system with the prowess of complementary approaches for calculating the real-time dynamics derived from this Hamiltonian. We demonstrate the power of our approach by hybridizing the TD-NRG with the Chebyshev expansion technique in order to investigate periodic switching in the interacting resonant-level model. Although the interacting model shares the same low-energy fixed point as its noninteracting counterpart, we surprisingly find the gradual emergence of damped oscillations as the interaction strength is increased. Focusing on a single quantum quench and using a strong-coupling analysis, we reveal the origin of these interaction-induced oscillations and provide an analytical estimate for their frequency. The latter agrees well with the numerical results.
A hybrid approach to nonequilibrium dynamics of quantum impurity systems is presented. The numerical renormalization group serves as a means to generate a suitable low-energy Hamiltonian, allowing for an accurate evaluation of the real-time dynamics of the problem up to exponentially long times using primarily the time-adaptive density-matrix renormalization group. In particular, by constructing a suitable hybrid chain, discretization errors are essentially eliminated on all time scales of interest. We extract the decay time of the interaction-enhanced oscillations in the interacting resonant-level model and show their quadratic divergence with the interaction strength U. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in 1/U.
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 noninteracting 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.
A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum-impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. Comparisons with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establish the accuracy of the method. The real-time dynamics of a single spin coupled to each type of bath is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short- and long-time dynamics are studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling J. In the ferromagnetic regime, the transient spin decay is described by the scaling variable $x=2{\ensuremath{\rho}}_{F}\ensuremath{\mid}J(T)\ensuremath{\mid}Tt$. In the antiferromagnetic regime, the long-time decay is governed for T<TK by the Kondo time scale 1∕TK. Here ${\ensuremath{\rho}}_{F}$ is the conduction-electron density of states, TK is the Kondo temperature, and J(T) is the effective exchange coupling at temperature T. Results for spin precession are obtained by rotating the external magnetic field from the x axis to the z axis.
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)].
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 nonequilibrium 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 $\mathrm{\ensuremath{\Lambda}}>1$. 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 QISs. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interacting resonant-level model and the single-impurity Anderson model.
Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.