Monte Carlo Optimization of Trial Wave Functions in Quantum Mechanics and Statistical Mechanics

This review covers applications of quantum Monte Carlo methods to quantum mechanical problems in the study of electronic and atomic structure, as well as applications to statistical mechanical problems both of static and dynamic nature. The common thread in all these applications is optimization of many-parameter trial states, which is done by minimization of the variance of the local or, more generally for arbitrary eigenvalue problems, minimization of the variance of the configurational eigenvalue.Comment: 27 pages to appear in " Recent Advances in Quantum Monte Carlo Methods" edited by W.A. Leste

Universal Dynamics of Independent Critical Relaxation Modes

Scaling behavior is studied of several dominant eigenvalues of spectra of Markov matrices and the associated correlation times governing critical slowing down in models in the universality class of the two-dimensional Ising model. A scheme is developed to optimize variational approximants of progressively rapid, independent relaxation modes. These approximants are used to reduce the variance of results obtained by means of an adaptation of a quantum Monte Carlo method to compute eigenvalues subject to errors predominantly of statistical nature. The resulting spectra and correlation times are found to be universal up to a single, non-universal time scale for each model

Wave speeds and Green’s tensors for shear wave propagation in incompressible, hyperelastic materials with uniaxial stretch

Assessing elastic material properties from shear wave propagation following an acoustic radiation force impulse (ARFI) excitation is difficult in anisotropic materials because of the complex relations among the propagation direction, shear wave polarizations, and material symmetries. In this paper, we describe a method to calculate shear wave signals using Green's tensor methods in an incompressible, hyperelastic material with uniaxial stretch. Phase and group velocities are determined for SH and SV propagation modes as a function of stretch by constructing the equation of motion from the Cauchy stress tensor determined from the strain energy density. The Green's tensor is expressed as the sum of contributions from the SH and SV propagation modes with the SH contribution determined using a closed-form expression and the SV contribution determined by numerical integration. Results are presented for a Mooney-Rivlin material model with a tall Gaussian excitation similar to an ARFI excitation. For an experimental configuration with a tilted material symmetry axis, results show that shear wave signals exhibit complex structures such as shear splitting that are characteristic of both the SH and SV propagation modes

Knitting a Frame [16mm film]

The films ‘Knitting a Frame’ and ‘Knitting pattern no.1’, employ common systems of production to reflect issues of time and pattern, exploring analogies between artists’ film/animation and knitting. These films extend the notion that construction of the film/and or fabric can be reliant on programming, controls and human gestures outside of their respective conventions. In ‘Knitting a Frame’ a single frame animation method is used, the act of exposure is instigated by the knitter (themselves), who we see in the image. The wool, also seen, is looped from the knitter over the single frame release of the camera and back to the knitting; when a knitted stitch is made the wool is pulled taught and because of this a single frame exposure is made. The unit of construction, a ‘frame’, is also the product of the knitting, this knitted frame of fabric is held up to the camera signalling the completion of the object and the end of the film

Monte Carlo computation of correlation times of independent relaxation modes at criticality

We investigate aspects of universality of Glauber critical dynamics in two dimensions. We compute the critical exponent $z$ and numerically corroborate its universality for three different models in the static Ising universality class and for five independent relaxation modes. We also present evidence for universality of amplitude ratios, which shows that, as far as dynamic behavior is concerned, each model in a given universality class is characterized by a single non-universal metric factor which determines the overall time scale. This paper also discusses in detail the variational and projection methods that are used to compute relaxation times with high accuracy

Transfer-Matrix Monte Carlo Estimates of Critical Points in the Simple Cubic Ising, Planar and Heisenberg Models

The principle and the efficiency of the Monte Carlo transfer-matrix algorithm are discussed. Enhancements of this algorithm are illustrated by applications to several phase transitions in lattice spin models. We demonstrate how the statistical noise can be reduced considerably by a similarity transformation of the transfer matrix using a variational estimate of its leading eigenvector, in analogy with a common practice in various quantum Monte Carlo techniques. Here we take the two-dimensional coupled $XY$-Ising model as an example. Furthermore, we calculate interface free energies of finite three-dimensional O($n$) models, for the three cases $n=1$, 2 and 3. Application of finite-size scaling to the numerical results yields estimates of the critical points of these three models. The statistical precision of the estimates is satisfactory for the modest amount of computer time spent

Cognitive rehearsal, cognitive bias and the development of fear in high trait-anxious children

Previous research has shown that high trait-anxious children, relative to low trait-anxious children, are at an increased risk of developing fear due to threatening information (Field, 2006b; Field and Price-Evans, 2009). However, the mechanism that underlies this relationship remains unknown. Cognitive models of vulnerability to anxiety propose that biases in the processing of threat-relevant material play a part in the aetiology and maintenance of anxiety disorders (Beck and Clark, 1997; Eysenck, 1992) and as such could potentially explain the relationship between trait-anxiety and fear development in the face of ambiguous information in children. For example, high-anxious children tend to interpret ambiguous information in a more negative manner (interpretation bias) and remember ambiguous information as being more threatening than it was originally (memory bias) (see Hadwin and Field, 2010, for a review). Additionally, high-anxious children have been found to engage in negative cognitive rehearsal (Comer, Kendall, Franklin, Hudson, and Pimental, 2004). The experiments in this thesis investigated whether these cognitive biases underlie the relationship between trait anxiety and fear development in non-clinical children. In a series of three experiments, children (aged 8-11 years) were presented with some ambiguous information regarding two novel animals (the quoll and the cuscus) and before completing a cognitive rehearsal task were told that they would soon be asked to approach the animals. There were several findings: 1) High trait-anxious children were not significantly more likely than low trait-anxious children to display any of the cognitive biases tested (i.e., interpretation bias, memory bias or cognitive rehearsal). However, tentative evidence suggested that interpretation bias exacerbated the relationship between trait anxiety and fear; 2) Whether children cognitively rehearsed the ambiguous information or not had no significant impact on their fear for the animals, nor did the valence of their thoughts; 3) Children who interpreted the ambiguous information more negatively were more likely to become fearful of the animals and were also more likely to remember more negatively-biased and less positively-biased pieces of ambiguous information; 4) It was the lack of positively-biased memories not the increased number of negatively-biased memories that led children who interpreted the information more negatively to become more fearful of the animals as a result. The findings are discussed with reference to their implications for the theory and prevention of childhood fear: that positive interpretation and memory bias training may act to decrease or even help to prevent fear development in children

Excitation Spectrum at the Yang-Lee Edge Singularity of 2D Ising Model on the Strip

At the Yang-Lee edge singularity, finite-size scaling behavior is used to measure the low-lying excitation spectrum of the Ising quantum spin chain for free boundary conditions. The measured spectrum is used to identify the CFT that describes the Yang-Lee edge singularity of the 2D Ising model for free boundary conditions.Comment: 7 pages, 1 figur

Accuracy of Electronic Wave Functions in Quantum Monte Carlo: the Effect of High-Order Correlations

Compact and accurate wave functions can be constructed by quantum Monte Carlo methods. Typically, these wave functions consist of a sum of a small number of Slater determinants multiplied by a Jastrow factor. In this paper we study the importance of including high-order, nucleus-three-electron correlations in the Jastrow factor. An efficient algorithm based on the theory of invariants is used to compute the high-body correlations. We observe significant improvements in the variational Monte Carlo energy and in the fluctuations of the local energies but not in the fixed-node diffusion Monte Carlo energies. Improvements for the ground states of physical, fermionic atoms are found to be smaller than those for the ground states of fictitious, bosonic atoms, indicating that errors in the nodal surfaces of the fermionic wave functions are a limiting factor.Comment: 9 pages, no figures, Late