DMRG studies of Chebyshev-expanded spectral functions and quantum impurity models

This thesis is concerned with two main topics: first, the
advancement of the density matrix renormalization group (DMRG) and,
second, its applications. In the first project of this thesis we
exploit the common mathematical structure of the numerical
renormalization group and the DMRG, namely, matrix product states
(MPS), to implement an efficient numerical treatment of a two-lead,
multi-level Anderson impurity model. By adopting a star-like
geometry, where each species (spin and lead) of conduction
electrons is described by its own so-called Wilson chain, instead
of a single Wilson chain we achieve a very significant reduction in
the numerical resources required to obtain reliable results.
Moreover, we show that it is possible to find an "optimal" chain
basis, in which chain degrees of freedom of different Wilson chains
become effectively decoupled from each other further out on the
Wilson chains. This basis turns out to also diagonalize the model's
chain-to-chain scattering matrix. In the second project we show
that Chebychev expansions offer numerically efficient
representations for calculating spectral functions of
one-dimensional lattice models using MPS methods. The main features
of this Chebychev matrix product state (CheMPS) approach are: (i)
it achieves uniform resolution over the spectral function's entire
spectral width; (ii) it offers a well-controlled broadening scheme;
(iii) it is based on using MPS tools to recursively calculate a
succession of Chebychev vectors, (iv) whose entanglement entropies
were found to remain bounded with increasing recursion order for
all cases analyzed here. We present CheMPS results for the
structure factor of spin-1/2 antiferromagnetic Heisenberg chains
and perform a detailed finite-size analysis. Making comparisons to
benchmark methods, we find that CheMPS yields results comparable in
quality to those of correction vector DMRG, at dramatically reduced
numerical cost and agrees well with Bethe Ansatz results for an
infinite system, within the limitations expected for numerics on
finite systems. Following these technologically focused projects we
study the so-called Kondo cloud by means of the DMRG in the third
project. The Kondo cloud describes the effect of spatially extended
spin-spin correlations of a magnetic moment and the conduction
electrons which screen the magnetic moment through the Kondo effect
at low temperatures. We focus on the question whether the Kondo
screening length, typically assumed to be proportional to the
inverse Kondo temperature, can be extracted from the spin-spin
correlations. We investigate how perturbations which destroy the
Kondo effect, like an applied gate potential or a magnetic field,
affect the formation of the screening cloud. In a forth project we
address the impact of Quantum (anti-)Zeno physics resulting from
repeated single-site resolved observations on the many-body
dynamics. We use time-dependent DMRG to obtain the time evolution
of the full many-body wave function that is then periodically
projected in order to simulate realizations of stroboscopic
measurements. For the example of a 1-D lattice of spin-polarized
fermions with nearest-neighbor interactions, we find regimes for
which many-particle configurations are stabilized and destabilized
depending on the interaction strength and the time between
observations.