arXiv subjects

Ciaran Hughes

Publications and source records attributed to Ciaran Hughes.

22 records · Page 2Linked to original sources

arXiv2014

Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points

The application of numerical techniques to the study of energy landscapes of large systems relies on sufficient sampling of the stationary points. Since the number of stationary points is believed to grow exponentially with system size, we can only sample a small fraction. We investigate the interplay between this restricted sample size and the physical features of the potential energy landscape for the two-dimensional $XY$ model in the absence of disorder with up to $N=100$ spins. Using an eigenvector-following technique, we numerically compute stationary points with a given Hessian index $I$ for all possible values of $I$. We investigate the number of stationary points, their energy and index distributions, and other related quantities, with particular focus on the scaling with $N$. The results are used to test a number of conjectures and approximate analytic results for the general properties of energy landscapes.

cond-mat.stat-mech↗

arXiv2014

An Inversion-Relaxation Approach for Sampling Stationary Points of Spin Model Hamiltonians

Sampling the stationary points of a complicated potential energy landscape is a challenging problem. Here we introduce a sampling method based on relaxation from stationary points of the highest index of the Hessian matrix. We illustrate how this approach can find all the stationary points for potentials or Hamiltonians bounded from above, which includes a large class of important spin models, and we show that it is far more efficient than previous methods. For potentials unbounded from above, the relaxation part of the method is still efficient in finding minima and transition states, which are usually the primary focus of attention for atomistic systems.

cond-mat.stat-mech↗

arXiv2013

Potential Energy Landscapes for the 2D XY Model: Minima, Transition States and Pathways

We describe a numerical study of the potential energy landscape for the two-dimensional XY model (with no disorder), considering up to 100 spins and CPU and GPU implementations of local optimization, focusing on minima and saddles of index one (transition states). We examine both periodic and anti-periodic boundary conditions, and show that the number of stationary points located increases exponentially with increasing lattice size. The corresponding disconnectivity graphs exhibit funneled landscapes; the global minima are readily located because they exhibit relatively large basins of attraction compared to the higher energy minima as the lattice size increases.

cond-mat.stat-mech↗

arXiv2013

Enumerating Gribov copies on the lattice

In the modern formulation of lattice gauge-fixing, the gauge fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions called Gribov copies. The dependence of the number of Gribov copies, n[U] on the different gauge orbits plays an important role in constructing the Faddeev-Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: 1. analytical results in lower dimensions, and some lower bounds on n[U] in higher dimensions, 2. the numerical polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and 3. numerical minimisation ("brute force"), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU(N_c)/U(1) of an SU(N_c) theory is orbit-independent, we concentrate on the residual compact U(1) case in this article and establish that n is orbit-dependent for the minimal lattice Landau gauge and orbit-independent for the absolute lattice Landau gauge. We also observe that contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.

hep-lat↗