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arXiv:0810.2900 · Physical Review B

Density matrix renormalization group study of a quantum impurity model with Landau-Zener time-dependent Hamiltonian

Cheng Guo, Andreas Weichselbaum, Stefan K. Kehrein, Tao Xiang, Jan von Delft

Published Mar 30, 2009• 20 citations
arXivHTMLPDF

Abstract

We use the adaptive time-dependent density matrix renormalization group method (t-DMRG) to study the nonequilibrium dynamics of a benchmark quantum impurity system which has a time-dependent Hamiltonian. This model is a resonant-level model, obtained by a mapping from a certain Ohmic spin-boson model describing the dissipative Landau-Zener transition. We map the resonant-level model onto a Wilson chain, then calculate the time-dependent occupation ${n}_{d}(t)$ of the resonant level. We compare t-DMRG results with exact results at zero temperature and find very good agreement. We also give a physical interpretation of the numerical results.

DOI
10.1103/physrevb.79.115137

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We use the adaptive time-dependent density matrix renormalization group method (t-DMRG) to study the nonequilibrium dynamics of a benchmark quantum impurity system which has a time-dependent Hamiltonian. This model is a resonant-level model, obtained by a mapping from a certain Ohmic spin-boson model describing the dissipative Landau-Zener transition. We map the resonant-level model onto a Wilson chain, then calculate the time-dependent occupation nd(t){n}_{d}(t)nd​(t) of the resonant level. We compare t-DMRG results with exact results at zero temperature and find very good agreement. We also give a physical interpretation of 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 UUU. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in 1/U1/U1/U.
This thesis is concerned with open quantum systems, and more specifically, quantum impurity models. By this we mean a small local quantum system in contact with a macroscopic non-interacting environment. This can be used to model individual impurities in metals and quantum information systems where the influence from the surrounding environment is not negligible. The numerical renormalization group (NRG) is the traditional method to study quantum impurity models. However its application is limited when dealing with real-time dynamics and bosonic systems. In recent years some of NRG's techniques have been introduced to the density matrix renormalization group method (DMRG), which itself is the most powerful numerical method to study one-dimensional quantum systems. The resulting method shows great potential, and this thesis explores and extends the power of the NRG+DMRG combination in treating open quantum systems. We focus mainly on two types of problem. The first is an open quantum system with a time-dependent Hamiltonian, which for example could be the theoretical description of various problems encountered in qubit manipulation. We combine NRG discretization and adaptive time-dependent DMRG (t-DMRG) to study the dissipative Landau-Zener problem. We also use this method to study the quantum decoherence process and the dynamical properties of the telegraph noise model. The results show that the NRG and t-DMRG combination is a fast, accurate and versatile method for such problems. The second type of problem we study involves the quantum critical properties of one- and two-bath spin-boson models. Unlike fermion and spin models, models with bosons are difficult to treat numerically as each boson basis has an infinite number of dimensions. By introducing the optimal boson basis into the variational matrix product state method (VMPS), which is a variant of DMRG, we can deal with an effective local boson basis as large as 101010^{10}1010. This is the crucial improvement over NRG which can only deal with at most a few dozen local basis states. With this powerful tool we have settled a controversy about the nature of the quantum phase transition in a sub-Ohmic spin-boson model regarding quantum to classical mapping. There, NRG fails to yield the right result due to the highly truncated local boson basis. We also explore the phase diagram of the two-bath spin-boson model and find a new critical phase. We demonstrate that NRG+VMPS with optimal boson basis represents a powerful setting to study quantum impurity models with a bosonic environment.
Resonant tunneling through quantum dot under a finite bias voltage at zero temperature is investigated by using the adaptive time-dependent density matrix renormalization group(TdDMRG) method. Quantum dot is modeled by the Anderson Hamiltonian with the 1-D nearest-neighbor tight-binding leads. Initially the ground state wave function is calculated with the usual DMRG method. Then the time evolution of the wave function due to the slowly changing bias voltage between the two leads is calculated by using the TdDMRG technique. Even though the system size is finite, the expectation values of current operator show steady-like behavior for a finite time interval, in which the system is expected to resemble the real nonequilibrium steady state of the infinitely long system. We show that from the time intervals one can obtain quantitatively correct results for differential conductance in a wide range of bias voltage. Finally we observe an anomalous behavior in the expectation value of the double occupation operator at the dot $$ as a function of bias voltage.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
In this work we develop a new numerical method, the adaptive time-dependent density-matrix renormalization-group (adaptive t-DMRG), which turns out to be very well suited to investigate time-dependent phenomena in one-dimensional strongly correlated systems. The method is based on the original density-matrix renormalization group method by White and the time evolving block-decimation procedure by Vidal. We further show the applicability of the new method to different physical systems: bosonic, fermionic, and spin chains. We first discuss the consequences of the presence of an external trapping potential on the static properties of trapped cold bosonic atoms in an optical lattice using the Bose-Hubbard model. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. We note that the superfluid to Mott-insulating transition is seen most directly in the half width of the interference peak. Then we turn to the evolution of density perturbations in these systems. In particular, we discuss the dependence of the velocity and the decay of the amplitude on density, interaction strength, and the extent and height of the perturbation in a numerically exact way, covering a wide range of interaction and perturbation strengths. By comparing our results for the sound velocity to theoretical predictions, we determine the limits of a Gross-Pitaevskii or Bogoliubov type description and the regime where repulsive one-dimensional Bose gases exhibit fermionic behaviour. As a second application of the adaptive t-DMRG, we investigate the phenomenon of spin-charge separation in the Hubbard model and propose an experimental setup for its observation in cold Fermi gases. We show the robustness of this separation beyond the low-energy regime by studying the time evolution of density wave packets of finite strength and at length scales down to a few lattice spacings. A striking signature of spin-charge separation is found in 1D cold Fermi gases in a harmonic trap using the different propagation properties of the liquid and Mott-insulating phases. The third application relates to the area of spintronics. A simplified model for the magnetization transport between two coupled reservoirs with opposite spin polarization is studied. We investigate whether a simple long time limit for the spin transport exists and if so, how it depends on the properties of the system such as integrability and criticality. Time-scales accessible to us are of the order of 100 units of time measured in 1/J while maintaining insignificant error in the observables.
By extending White's density-matrix renormalization group (DMRG) technique to finite temperatures, we calculate the impurity susceptibility for the following systems: (a) a spin-1/2 impurity coupled to a single site of an antiferromagnetic Heisenberg spin-1/2 chain and (b) an impurity coupled symmetrically to the two spins at the open ends of the chain. We also perform a Monte Carlo simulation and show that the DMRG adaptation gives reliable results. The calculated impurity susceptibility shows Pauli-like behavior at low temperatures in the former case while a ln T divergence is found in the latter case, in agreement with predictions based on conformal field theory.