Algorithms for Hamiltonian quantum field theories

Quantum field theories and in particular gauge theories are at the base of our understanding of modern physics. In the Standard Model of Particle Physics, they explain three of the four fundamental interaction known in nature. The evaluation of quantum field theories (QFT), however, is quite involved. While weakly coupled QFT can be studied with perturbation theory in the continuum, strongly coupled theories like quantum chromodynamics are usually discretized. Even after a discretization on a lattice, certain regions of the phase diagram of the standard model cannot be explored with traditional Monte Carlo methods. The path to the finite-density regime of the standard model is blocked due to the sign problem in Monte Carlo and time dynamics cannot be explored since the action formalism has no explicit notion of time. Here, we explore new algorithms for QFT in the continuum and on the lattice. Instead of the action formalism, we focus on the Hamiltonian formulation of QFT. The combination of Hamiltonian formalism and new algorithms enables us to explore which were inaccessible with prior methods. The thesis is structured into four main parts which focus on different combinations of algorithms and systems. All projects are grouped along two axes: the type of the algorithm and the discretization of the system. The first two chapters give an increasingly technical introduction into the topic and present the methods which are used throughout. We review recent advances and show where we connect to the known state of the field. The first project investigates a pure $Z_3$ lattice gauge theory with a tensor network based ansatz, gauged Gaussian projected entangled pair states (GGPEPS) in two spatial dimensions. While the gauge field can be integrated out in one dimension, the two-dimensional case is more involved due to the self-interactions of the gauge fields. GGPEPS are locally gauge invariant by design and explore only the physically relevant part of the Hilbert space. In a first explorative, numerical study, we benchmark the performance of the states and investigate their viability. In the second project, we change from tensor networks to a periodic Gaussian Ansatz to simulate compact quantum electrodynamics (cQED) on the lattice. We stay with the lattice formulation in two spatial dimensions, while adapting the states. Extending previous work to complex periodic Gaussian states, we simulate the first time-evolution after quenches in two-dimensional cQED. In this thesis, we focus on the formulation of the Ansatz and computational challenges. The third project changes the focus from lattice gauge theories to QFT in the continuum while returning to tensor network based states. We choose to work with Gaussian continuous tensor network states (GCTNS), a restriction of general continuous tensor networks. The Gaussian character lets us treat most of the calculations analytically, such that the numerical investigation can focus on the actual match of the states to the true ground state. GCTNS capture the ground states of Gaussian theories excellently, and we can explore their limitations with quartic theories like the Lieb-Liniger model. We prove the numerical viability of CTNS as a numerical tool to investigate theories iii directly in the continuum. Finally, we turn to the last remaining combination: a study of QFT (without discretization) without tensor networks. The computation of entanglement entropies in quantum field theories directly in the continuum is challenging since the usually used momentum basis does not allow for a trivial bipartition in real space. In the framework of Hamiltonian truncation, we devise the first algorithm to numerically investigate arbitrary entanglement measures for quantum field theories. By splitting the system into two subsystems, and mapping the full fields onto fields on the partitions, we are able to explicitly compute the reduced density matrix, giving access to many entanglement related quantities. We verify the procedure on the massive Klein-Gordon and obtain interesting new results on the interacting sine-Gordon model.