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<p>The simulation of interacting quantum matter remains challenging. The Hilbert space
dimension required to describe the physics grows exponentially with the system size,
yet many interesting collective phenomena emerge only for large enough systems. Tensor
network methods allow for the simulation of quantum many-body systems by reducing
the effective number of degrees of freedom in controlled approximations. For one-dimensional
lattice models, algorithms employing matrix product states (MPS) are currently regarded
as the most powerful numerical techniques. For example, density matrix renormalization
group (DMRG) algorithms can be used to efficiently compute precise approximations
for ground states of local Hamiltonians. The computation of finite-temperature properties
and dynamical response functions is more challenging, yet crucial for a complete understanding
of the physics and for comparisons with experiments.</p><p>In the first part of this
dissertation, we introduce and demonstrate novel matrix product state techniques.
First, we present an improved version of the minimally entangled typical thermal states
(METTS) algorithm, a sampling approach for the simulation of thermal equilibrium.
Our modification allows the use of symmetries in the MPS operations, which renders
the algorithm significantly more efficient and makes the finite-temperature simulation
of previously inaccessible models possible. Then, we introduce a new technique utilizing
infinite matrix product states (iMPS) with infinite boundary conditions for the computation
of response functions. These quantities are of great importance as their Fourier transform
yields spectral functions or dynamic structure factors, which give detailed insights
into the low-lying excitations of a model and can be directly compared to experimental
data. Our improved algorithm allows to significantly reduce the number of required
time-evolution runs in the simulations.</p><p>In the second part of this dissertation,
we study the physics of the bilinear-biquadratic spin-1 chain in detail. Our new scheme
for the simulation of response functions enables us to compute high-resolution dynamic
structure factors for the model, which we use as a starting point to explore the low-lying
excitations in all quantum phases of the rich phase diagram. Comparing our numerical
data to exact results and field-theory approximations, we gain insights into the nature
of the relevant excitations. In the Haldane phase, the model can be mapped to a continuum
field theory, the non-linear sigma model (NLSM). We find that the NLSM does not capture
the influence of the biquadratic term correctly and gives only unsatisfactory predictions
for the relevant physical quantities. However, several features in the Haldane phase
can be explained by a non-interacting approximation for two- and three-magnon states.
Moving into the extended critical phase, we explain the observed contraction of the
multi-soliton continua from the Uimin-Lai-Sutherland point by comparison with a field-theory
description. In addition, we discover new excitations at higher energies and find
that their dispersions are described by simple cosine-functions in the purely biquadratic
limit. We characterize them as elementary one-particle excitations and relate them
to the integrable Temperley-Lieb chain. The Temperley-Lieb chain can also be used
to describe the physics at the opposite biquadratic point, which places the model
in the gapped dimerized phase. Here, the excitation spectrum is related to that of
an anisotropic spin-1/2 chain. In the ferromagnetic phase, the two-magnon excitations
can be computed exactly and contain bound and resonant states in addition to two-particle
continua.</p><p>Finally, we address the extraction of spectral functions or dynamic
structure factors from real-time response functions computed with MPS techniques.
In the time evolution, the computation costs grow with time, hence the response function
can only be evaluated up to some maximum time. As the spectral functions are obtained
by Fourier transform from the time to the frequency domain, this limits the frequency
resolution of the result. Here, we introduce and discuss new approaches for the extraction
of dynamic structure factors from the limited response-function data.</p>
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