We analyze and discuss convergence properties of a numerically exact algorithm tailored to study the dynamics of interacting two-dimensional lattice systems. The method is based on the application of the time-dependent variational principle in a manifold of binary and quaternary Tree Tensor Network States. The approach is found to be competitive with existing matrix product state approaches. We discuss issues related to the convergence of the method, which could be relevant to a broader set of numerical techniques used for the study of two-dimensional systems.
We analyze and discuss convergence properties of a numerically exact algorithm tailored to study the dynamics of interacting two-dimensional lattice systems. The method is based on the application of the time-dependent variational principle in a manifold of binary and quaternary Tree Tensor Network States. The approach is found to be competitive with existing matrix product state approaches. We discuss issues related to the convergence of the method, which could be relevant to a broader set of numerical techniques used for the study of two-dimensional systems.
In this thesis I present contributions to the development, analysis and application of tensor network state methods for numerically exact quantum dynamics in one and two-dimensional lattice systems. The setting of numerically exact quantum dynamics is introduced in Chapter 2. This includes a discussion of exact diagonalization approaches and massively parallel implementations thereof as well as a brief introduction of tensor network states. In Chapter 3, I perform a detailed analysis of the performance of n-ary tree tensor network states for simulating the dynamics of two-dimensional lattices. This constitutes the first application of this class of tensor network to dynamics in two spatial dimensions, a long-standing challenge, and the method is found to perform on par with existing state-of-the-art approaches. Chapter 4 showcases the efficacy of a novel tensor network format I developed, tailored to electron-phonon coupled problems in their single-electron sector, through an application to the Holstein model. The applicability of the approach to a broad range of parameters of the model allows to reveal the strong influence of the spread of the electron distribution on the initial state of the phonons at the site where the electron is introduced, for which a simple physical picture is offered. I depart from method development in Chapter 5 and analyse the prospects of using tensor network states evolved using the time-dependent variational principle as an approximate approach to determine asymptotic transport properties with a finite, moderate computational effort. The method is shown to not yield the correct asymptotics in a clean, non-integrable system and can thus not be expected to work in generic systems, outside of finely tuned parameter regimes of certain models. Chapters 6 and 7 are concerned with studies of spin transport in long-range interacting systems using tensor network state methods. For the clean case, discussed in Chapter 6, we find that for sufficiently short-ranged interactions, the spreading of the bulk of the excitation is diffusive and thus dominated by the local part of the interaction, while the tail of the excitation decays with a powerlaw that is twice as large as the powerlaw of the interaction. Similarly, in the disordered case, analysed in Chapter 7, we find subdiffusive transport of spin and sub-linear growth of entanglement entropy. This behaviour is in agreement with the behaviour of systems with local interactions at intermediate disorder strength, but provides evidence against the phenomelogical Griffith picture of rare, strongly disordered insulating regions. We generalize the latter to long-ranged interactions and show that it predicts to diffusion, in contrast to the local case where it results in subdiffusive behaviour.
We study the applicability of the time-dependent variational principle in matrix-product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case, the apparent convergence of this approach is reliable. Our study indicates that, while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.
The numerical simulation of two-dimensional quantum many-body systems away from equilibrium constitutes a major challenge for all known computational methods. We investigate the utility of tree tensor network (TTN) states to solve the dynamics of the quantum Ising model in two dimensions. Within the perturbative regime of small transverse fields, TTNs faithfully reproduce analytically known, but nontrivial and physically interesting dynamics of magnetic domains, for lattices up to 16 × 16 sites. Limitations of the method related to the rapid growth of entanglement entropy are explored within more general, paradigmatic quench settings. We provide and discuss comprehensive benchmarks regarding the benefit of GPU acceleration and the impact of using local operator sums on the performance.
In this thesis, we use the new numerical method multiscale entanglement
renormalization ansatz (MERA) to study the one dimensional spin system. This
method, MERA, utilizes a network of tensors to represent quantum many-body
systems on a lattice. Other examples of tensor network state are matrix product
states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a
tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be-
yond. The three structures dier in the graph that denes how the tensors are
interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are,
respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor
networks the expectation value of local observables can be computed eciently.
Developing algorithms to simulate quantum many-body systems is important
for understanding and studying the physical principles. And these may become
powerful tools for the analysis of quantum computation and quantum information
theory. But in general ecient simulation is dicult, due to the fundamental
quantum physical laws.
The main method we have studied and worked on is multiscale entangle-
ment renormalization ansatz (MERA) which is a variational ansatz for many-
body states. MERA is a tree tensor network for 1D, 2D systems and beyond.
We have developed code of ternary-MERA to compute the ground state energy,
expected values of local observables, and correlators for 1D lattice systems. In
a nutshell, MERA is an algorithm for entanglement renormalization which is a
numerical technique based on locally reorganizing the Hilbert space of a quantum
many-body system with the aim to reduce the amount of entanglement in its wave
function. We use this method to study 1D Ising model, XXZ model, and J1 - J2
model, and then comparing the numerical results with exact diagonalization (ED)
and density matrix renormalization group (DMRG).
Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 86. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with 1/r2 interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.
The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.