The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure

A Classical Analysis of Double Ionization of Helium in Ultra Short Laser Pulses

Experiments of double ionization in noble gases [58, 64, 68, 84] were the catalyst for extensive theoretical investigations [9, 11, 13, 21, 39, 80, 87]. The measurement of the momenta of outgoing electrons in non-sequential strong field double ionization exposed the correlated nature of their escape [66, 67, 88, 90]. A (1+1)-dimensional model for helium, introduced in [25, 73], has been the foundation of ongoing research into non-sequential double ionization [24, 26, 27, 71, 74]. The model reproduces the re-scattering scenario, the correlation between the outgoing electrons, and the interference patterns in the momentum distribution [72]. The observed interference patterns depend on the amplitude of the external field, pulse duration, and carrier envelope phase. Guided by the semi-classical idea that many paths contribute to the double ionization events and the interference between these paths could cause the patterns, a rigorous analysis of the classical trajectories depicting double ionization was undertaken. Applying few-cycle pulses, the effects from multiple re-scattering are intrinsically minimized. In classical calculations, field parameters were varied and configurations yielding trajectories of reduced complexity were targeted. The classical trajectories allow a connection between the initial conditions in phase space and the final states to be established. A link between the external field strength and the electrons initial conditions was found. In the single-cycle limit, the electrons mutual repulsion ensures that anti-parallel double ionization is the only double ionization mechanism at intensities above the threshold. Stable and symmetric back-to-back double ionization trajectories are identified. Parallel non-symmetric double ionization with same final momentum was generated from two-cycle fields. The extent of the frequency and field strength dependency on classical non-sequential double ionization was determined

Oriented trees and paths in digraphs

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area

SHEEP: Signed Hamiltonian Eigenvector Embedding for Proximity

We introduce a spectral embedding algorithm for finding proximal relationships between nodes in signed graphs, where edges can take either positive or negative weights. Adopting a physical perspective, we construct a Hamiltonian which is dependent on the distance between nodes, such that relative embedding distance results in a similarity metric between nodes. The Hamiltonian admits a global minimum energy configuration, which can be reconfigured as an eigenvector problem, and therefore is computationally efficient to compute. We use matrix perturbation theory to show that the embedding generates a ground state energy, which can be used as a statistical test for the presence of strong balance, and to develop an energy-based approach for locating the optimal embedding dimension. Finally, we show through a series of experiments on synthetic and empirical networks, that the resulting position in the embedding can be used to recover certain continuous node attributes, and that the distance to the origin in the optimal embedding gives a measure of node extremism.Comment: 20 pages, 36 figure