@article{hughes_mehta_wales_2014, title={An inversion-relaxation approach for sampling stationary points of spin model Hamiltonians}, volume={140}, ISSN={["1089-7690"]}, DOI={10.1063/1.4875697}, abstractNote={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.}, number={19}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Hughes, Ciaran and Mehta, Dhagash and Wales, David J.}, year={2014}, month={May} }
 @article{mehta_hauenstein_wales_2014, title={Certification and the potential energy landscape}, volume={140}, ISSN={["1089-7690"]}, DOI={10.1063/1.4881638}, abstractNote={Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when further optimization is attempted. In some cases, these non-solutions could be misleading. Proving that a numerical approximation will quadratically converge to a stationary point is termed certification. In this report, we provide details of how Smale's α-theory can be used to certify numerically obtained stationary points of a potential energy landscape, providing a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed.}, number={22}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Mehta, Dhagash and Hauenstein, Jonathan D. and Wales, David J.}, year={2014}, month={Jun} }
 @article{mehta_schroeck_2014, title={Enumerating copies in the first Gribov region on the lattice in up to four dimensions}, volume={89}, ISSN={["1550-2368"]}, DOI={10.1103/physrevd.89.094512}, abstractNote={The covariant gauges are known to suffer from the Gribov problem: even after fixing a gauge non-perturbatively, there may still exist residual copies which are physically equivalent to each other, called Gribov copies. While the influence of Gribov copies in the relevant quantities such as gluon propagators has been heavily debated in recent studies, the significance of the role they play in the Faddeev--Popov procedure is hardly doubted. We concentrate on Gribov copies in the first Gribov region, i.e., the space of Gribov copies at which the Faddeev--Popov operator is strictly positive (semi)definite. We investigate compact U($1$) as the prototypical model of the more complicated standard model group SU($N_{c}$). With our Graphical Processing Unit (GPU) implementation of the relaxation method we collect up to a few million Gribov copies per orbit. We show that the numbers of Gribov copies even in the first Gribov region increase exponentially in two, three and four dimensions. Furthermore, we provide strong indication that the number of Gribov copies is gauge orbit dependent.}, number={9}, journal={PHYSICAL REVIEW D}, author={Mehta, Dhagash and Schroeck, Mario}, year={2014}, month={May} }
 @article{maniatis_mehta_2014, title={On exact minimization of Higgs potentials}, volume={129}, ISSN={["2190-5444"]}, DOI={10.1140/epjp/i2014-14109-0}, abstractNote={Minimizing the Higgs potential is an essential task in any model involving Higgs bosons. Exact minimization methods proposed in the literature are based on the polynomial form of the potential. These methods will in general no longer work if loop contributions to the potential are taken into account. We present a method to keep the tree level global minimum unchanged in passing to the effective potential. We illustrate the method for the case of the Minimal Supersymmetric Model (MSSM).}, number={6}, journal={EUROPEAN PHYSICAL JOURNAL PLUS}, author={Maniatis, Markos and Mehta, Dhagash}, year={2014}, month={Jun} }
 @article{mehta_hughes_kastner_wales_2014, title={Potential energy landscape of the two-dimensional XY model: Higher-index stationary points}, volume={140}, ISSN={["1089-7690"]}, DOI={10.1063/1.4880417}, abstractNote={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.}, number={22}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Mehta, D. and Hughes, C. and Kastner, M. and Wales, D. J.}, year={2014}, month={Jun} }
 @article{mehta_hughes_schroeck_wales_2013, title={Potential energy landscapes for the 2D XY model: Minima, transition states, and pathways}, volume={139}, ISSN={["1089-7690"]}, DOI={10.1063/1.4830400}, abstractNote={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 central processing unit and graphics processing unit 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.}, number={19}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Mehta, Dhagash and Hughes, Ciaran and Schroeck, Mario and Wales, David J.}, year={2013}, month={Nov} }