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 $10^{10}$. 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.