Density Matrix Renormalisation Group Calculations for Two-Dimensional Lattices: An Application to the Spin-Half and Spin-One Square-Lattice Heisenberg Models

A new density matrix renormalisation group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multi-chain-type 2D DMRG approach which utilises previously determined system and environment blocks {\it at all points}. One firstly builds up effective quasi-1D system and environment blocks of width $L$ and these quasi-1D blocks are then used to as the initial building-blocks of a new 2D infinite-lattice algorithm. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N = 20 \times 20$ lattice using this treatment are given by $E_g/N_B = -0.3321$ and $E_g/N_B = -1.1525$, respectively.Comment: 7 Figures. Accepted for publication in Phys. Rev.

Ground-state ordering of the J1 - J2 model on the simple cubic and body-centered cubic lattices

The J1−J2 Heisenberg model is a “canonical” model in the field of quantum magnetism in order to study the interplay between frustration and quantum fluctuations as well as quantum phase transitions driven by frustration. Here we apply the coupled cluster method (CCM) to study the spin-half J1−J2 model with antiferromagnetic nearest-neighbor bonds J1>0 and next-nearest-neighbor bonds J2>0 for the simple cubic (sc) and body-centered cubic (bcc) lattices. In particular, we wish to study the ground-state ordering of these systems as a function of the frustration parameter p=z2J2/z1J1, where z1 (z2) is the number of nearest (next-nearest) neighbors. We wish to determine the positions of the phase transitions using the CCM and we aim to resolve the nature of the phase transition points. We consider the ground-state energy, order parameters, spin-spin correlation functions, as well as the spin stiffness in order to determine the ground-state phase diagrams of these models. We find a direct first-order phase transition at a value of p=0.528 from a state of nearest-neighbor Néel order to next-nearest-neighbor Néel order for the bcc lattice. For the sc lattice the situation is more subtle. CCM results for the energy, the order parameter, the spin-spin correlation functions, and the spin stiffness indicate that there is no direct first-order transition between ground-state phases with magnetic long-range order, rather it is more likely that two phases with antiferromagnetic long range are separated by a narrow region of a spin-liquid-like quantum phase around p=0.55. Thus the strong frustration present in the J1−J2 Heisenberg model on the sc lattice may open a window for an unconventional quantum ground state in this three-dimensional spin model

An exploration of pathologies of multilevel principal components analysis in statistical models of shape

3D facial surface imaging is a useful tool in dentistry and in terms of diagnostics and treatment planning. Between-group PCA (bgPCA) is a method that has been used to analyse shapes in biological morphometrics, although various “pathologies” of bgPCA have recently been proposed. Monte Carlo (MC) simulated datasets were created here in order to explore “pathologies” of multilevel PCA (mPCA), where mPCA with two levels is equivalent to bgPCA. The first set of MC experiments involved 300 uncorrelated normally distributed variables, whereas the second set of MC experiments used correlated multivariate MC data describing 3D facial shape. We confirmed results of numerical experiments from other researchers that indicated that bgPCA (and so also mPCA) can give a false impression of strong differences in component scores between groups when there is none in reality. These spurious differences in component scores via mPCA decreased significantly as the sample sizes per group were increased. Eigenvalues via mPCA were also found to be strongly affected by imbalances in sample sizes per group, although this problem was removed by using weighted forms of covariance matrices suggested by the maximum likelihood solution of the two-level model. However, this did not solve problems of spurious differences between groups in these simulations, which was driven by very small sample sizes in one group. As a “rule of thumb” only, all of our experiments indicate that reasonable results are obtained when sample sizes per group in all groups are at least equal to the number of variables. Interestingly, the sum of all eigenvalues over both levels via mPCA scaled approximately linearly with the inverse of the sample size per group in all experiments. Finally, between-group variation was added explicitly to the MC data generation model in two experiments considered here. Results for the sum of all eigenvalues via mPCA predicted the asymptotic amount for the total amount of variance correctly in this case, whereas standard “single-level” PCA underestimated this quantity

Initial steps towards a multilevel functional principal components analysis model of dynamical shape changes

In this article, multilevel principal components analysis (mPCA) is used to treat dynamical changes in shape. Results of standard (single-level) PCA are also presented here as a comparison. Monte Carlo (MC) simulation is used to create univariate data (i.e., a single “outcome” variable) that contain two distinct classes of trajectory with time. MC simulation is also used to create multivariate data of sixteen 2D points that (broadly) represent an eye; these data also have two distinct classes of trajectory (an eye blinking and an eye widening in surprise). This is followed by an application of mPCA and single-level PCA to “real” data consisting of twelve 3D landmarks outlining the mouth that are tracked over all phases of a smile. By consideration of eigenvalues, results for the MC datasets find correctly that variation due to differences in groups between the two classes of trajectories are larger than variation within each group. In both cases, differences in standardized component scores between the two groups are observed as expected. Modes of variation are shown to model the univariate MC data correctly, and good model fits are found for both the “blinking” and “surprised” trajectories for the MC “eye” data. Results for the “smile” data show that the smile trajectory is modelled correctly; that is, the corners of the mouth are drawn backwards and wider during a smile. Furthermore, the first mode of variation at level 1 of the mPCA model shows only subtle and minor changes in mouth shape due to sex; whereas the first mode of variation at level 2 of the mPCA model governs whether the mouth is upturned or downturned. These results are all an excellent test of mPCA, showing that mPCA presents a viable method of modeling dynamical changes in shape

Violence in England and Wales in 2017: An Accident and Emergency perspective

Executive Summary • Serious violence levels and trends in England and Wales were studied based on data from a structured sample of 94 Emergency Departments (EDs), Minor Injury Units (MIUs) and Walk-in Centres. All are certified members of the National Violence Surveillance Network (NVSN). • Overall, an estimated 190,747 people attended EDs in England and Wales for treatment following violence in 2017, 1942 more than in 2016; a 1% increase. Falls or no change in overall violence levels in England and Wales according to this public health measure over the past decade were maintained in 2017. • In 2017, males (4.6 per 1,000 residents) were more than twice as likely as females (1.9 per 1,000 residents) to be treated in EDs following injury in violence. • Increases in violent injury among those aged 0-10 years (11%), 31-50 years (4.6%) and those aged 51 years and over (2.1%) were offset by the 1.8% decrease in violence among those aged 18-30 years. Due to small numbers, NVSN is unable to provide reliable violence trends for those aged 0-10 years. • Implementation of the new Emergency Care Data Set (ECDS) in Type 1 EDs in England led to increases in violence recording in the three months, October to December 2017. • Those most at risk of violence-related injury were males and those aged 18 to 30. Violence-related ED attendance was most frequent on Saturdays and Sundays