Nanoscale physics and dynamical mean-field theory have both generated increased interest in complex quantum impurity problems and so have focused attention on the need for flexible quantum impurity solvers. Here we demonstrate that the mapping of single-quantum impurity problems onto spin chains can be exploited to yield a powerful and extremely flexible impurity solver. We implement this cluster algorithm explicitly for the Anderson and Kondo Hamiltonians, and illustrate its use in the ``mesoscopic Kondo problem.'' To study universal Kondo physics, a large ratio between the effective bandwidth
Deff and the temperature
T is required; our cluster algorithm treats the mesoscopic fluctuations exactly while being able to approach the large
Deff∕T limit with ease. We emphasize that the flexibility of our method allows it to tackle a wide variety of quantum impurity problems; thus, it may also be relevant to the dynamical mean-field theory of lattice problems.
The system of several
(N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel
N-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the
N-impurity
S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin
S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the
N-impurity Anderson model to the exact Bethe ansatz results of a
S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective
S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities
N. We demonstrate the robustness of the spin
N∕2 ground state as well as of the associated
S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.
Quantum walk (QW) on a disordered lattice leads to a multitude of interesting phenomena, such as Anderson localization. While QW has been realized in various optical and atomic systems, its implementation with superconducting qubits still remains pending. The major challenge in simulating QW with superconducting qubits emerges from the fact that on-chip superconducting qubits cannot hop between two adjacent lattice sites. Here we overcome this barrier and develop a gate-based scheme to realize the discrete time QW by placing a pair of qubits on each site of a one-dimensional (1D) lattice and treating an excitation as a walker. It is also shown that various lattice disorders can be introduced and fully controlled by tuning the qubit parameters in our quantum walk circuit. We observe a distinct signature of transition from the ballistic regime to a localized QW with an increasing strength of disorder. Finally, an eight-qubit experiment is proposed where the signatures of such localized and delocalized regimes can be detected with existing superconducting technology. Our proposal opens up the possibility of exploring various quantum transport processes with promising superconducting qubits.
This thesis discusses mainly two Fermionic lattice systems, first a Kondo lattice with additional Hubbard interaction and second a Hubbard Hamiltonian augmented with additional spin and charge interactions. We first introduce the Quantum Monte Carlo technique, which is then employed to study the two respective systems. We present an innovation that allows to calculate time displaced Greens functions more efficiently. Compared with previously used numerically stable algorithms the new method gains an order of magnitude in speed, but is just as precise, and very simple to implement. In the second chapter we consider the Kondo lattice model in two dimensions at half filling. In addition to the Fermionic hopping integral t and the superexchange coupling J the role of a Coulomb repulsion U in the conduction band is investigated. We find the model to display a magnetic order-disorder transition in the U-J plane with a critical value of Jc which is decreasing as a function of U. The single-particle spectral function A(k,ω) is computed across this transition. We conclude that (i) the local screening of impurity spins determines the low-energy behavior of the spectral function and (ii) one cannot deform continuously the spectral function of the half-filled Hubbard model at J=0 to that of the Kondo insulator at J>Jc. In the third chapter we investigate the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.
We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
We demonstrate that the problem of coupled two-level systems (``qubits'') which are also subject to a generic (sub-)Ohmic dissipative environment belongs to the same class of models as those describing (non)magnetic impurities embedded in strongly correlated systems. A further insight into the generalized single- and two-impurity Bose/Fermi Kondo models enables one to make specific recommendations towards a systematic engineering of highly coherent multiqubit assemblies for potential applications in quantum information processing.