Multi-Layer Potfit: An Accurate Potential Representation for Efficient High-Dimensional Quantum Dynamics

The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control. This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used Potfit representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.Comment: Copyright (2014) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physic

On Groupoids and Hypergraphs

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure.Comment: Explicit completion of H in HxI (Section 2) is unstable (incompatible with restrictions), hence does not support inductive construction towards Prop. 2.17 based on Lem 2.16 as claimed. For corresponding technical result, now see arxiv:1806.08664; for discussion of main applications first announced here, now see arxiv:1709.0003

On annealed elliptic Green function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb{Z}^d$. The distribution $\langle \cdot \rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ resp. $L^1$ in probability, i.e. they obtained bounds on $\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(t,x,y)| \rangle$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\nabla\phi$ interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function $G(x,y)$ satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. $L^p$ in probability for all $p<\infty$, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for $\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(x,y)| \rangle$.Comment: 15 page

Differential ttˉt\bar{t} Cross Section Measurements as a Function of Variables other than Kinematics

An overview of cross section measurements as a function of jet multiplicities and jet kinematics in association with $t\bar{t}$ production is presented. Both the ATLAS and the CMS collaborations performed a large number of measurements at different center-of-mass energies of the LHC using various $t\bar{t}$ decay channels. Theoretical predictions of these quantities usually rely on parton shower simulations that strongly depends on tunable parameters and come with large uncertainties. The measurements are compared to various theoretical descriptions based on different combinations of matrix-element calculations and parton-shower models

Cross-over in scaling laws: A simple example from micromagnetics

Scaling laws for characteristic length scales (in time or in the model parameters) are both experimentally robust and accessible for rigorous analysis. In multiscale situations cross--overs between different scaling laws are observed. We give a simple example from micromagnetics. In soft ferromagnetic films, the geometric character of a wall separating two magnetic domains depends on the film thickness. We identify this transition from a N\'eel wall to an Asymmetric Bloch wall by rigorously establishing a cross--over in the specific wall energy