Tensor networks for the simulation of strongly correlated systems

This thesis treats the classical simulation of strongly-interacting
many-body quantum-mechanical systems in more than one dimension
using matrix product states and the more general tensor product
states. Contrary to classical systems, quantum many-body systems
possess an exponentially larger number of degrees of freedom,
thereby significantly complicating their numerical treatment on a
classical computer. For this thesis two different representations
of quantum many-body states were employed. The first, the so-called
matrix product states (MPS) form the basis for the extremely
successful density matrix renormalization group (DMRG) algorithm.
While originally conceived for one-dimensional systems, MPS are in
principle capable of describing arbitrary quantum many-body states.
Using concepts from quantum information theory it is possible to
show that MPS provide a representation of one-dimensional quantum
systems that scales polynomially in the number of particles,
therefore allowing an efficient simulation of one-dimensional
systems on a classical computer. One of the key results of this
thesis is that MPS representations are indeed efficient enough to
describe even large systems in two dimensions, thereby enabling the
simulation of such systems using DMRG. As a demonstration of the
power of the DMRG algorithm, it is applied to the Heisenberg
antiferromagnet with spin $S = 1/2$ on the kagome lattice. This
model's ground state has long been under debate, with proposals
ranging from static spin configurations to so-called quantum spin
liquids, states where quantum fluctuations destroy conventional
order and give rise to exotic quantum orders. Using a fully
$SU(2)$-symmetric implementation allowed us to handle the
exponential growth of entanglement and to perform a large-scale
study of this system, finding the ground state for cylinders of up
to 700 sites. Despite employing a one-dimensional algorithm for a
two-dimensional system, we were able to compute the spin gap (i.e.
the energy gap to the first spinful excitation) and study the
ground state properties, such as the decay of correlation
functions, the static spin structure factors, and the structure and
distribution of the nearest-neighbor spin-spin correlations.
Additionally, by applying a new tool from quantum information
theory, the topological entanglement entropy, we could also with
high confidence demonstrate the ground state of this model to be
the elusive gapped $Z_2$ quantum spin liquid with topological
order. To complement this study, we also considered the extension
of MPS to higher dimensions, known as tensor product states (TPS).
We implemented an optimization algorithm exploiting symmetries for
this class of states and applied it to the
bilinear-biquadratic-bicubic Heisenberg model with spin $S=3/2$ on
the $z=3$ Bethe lattice. By carefully analyzing the simulation data
we were able to determine the presence of both conventional and
symmetry-protected topological order in this model, thereby
demonstrating the analytically predicted existence of the Haldane
phase in higher dimensions within an extended region of the phase
diagram. Key properties of this symmetry-protected topological
order include a doubling of the levels in the entanglement spectrum
and the presence of edge spins, both of which were confirmed in our
simulations. This finding simultaneously validated the
applicability of the novel TPS algorithms to the search for exotic
order.