A modern framework for classical spin models and lattice field theory on general topologies


Dr. Florian Goth
Institut für Theoretische Physik
Universität Würzburg

Project Overview

The present grant proposal focuses on the development of an adaptable framework for the simulation and analysis of equilibrium lattice spin models and its optimization on massively parallel architectures. As this area of research, with origins in statistical physics, particularly aims at improving our understanding of critical phenomena, there exists considerable overlap with other research fields, such as solid-state physics, material science, and high-energy physics. In general, phase transitions are characterized by the emergence of long-range collective behavior of the individual microscopic degrees of freedom. This gives rise to the important phenomenon of universality, in which the character of a phase transition is solely determined by a few fundamental properties of the system (such as global symmetries and interaction ranges). Theoretically, the emergence of universality can be motivated by renormalization group approaches, where at criticality the system forgets about its microscopic realization under repeated scale transformations. Quenched spatial disorder, however, can present a relevant perturbation to the phase transition, leaving the character of the transition unchanged in some scenarios but leading to entirely different critical behavior in others . We note that spatial disorder is not only of theoretical interest, but intrinsic to most real systems, arising from impurities, defects, and other inhomogeneities. Although the role of uncorrelated randomness (such as in diluted regular lattices) was successfully explained by Harris’ seminal criterion, the role of topological randomness, i.e., lattices with random connectivity, is yet to be fully understood. Recent contributions in this field still strongly rely on numerical simulations, as topological disorder has so far not been within the reach of analytical methods. Specifically, we recently used the Ising model for probing disordered geometries and determine whether the universal character of the transition is broken with respect to the clean model without disorder. Finally, Ising-like systems are also studied on abstract topologies like scale-free and small-world networks as well as in curved geometries such as the hyperbolic plane. Apart from being used as a probe to investigate aspects of topology and disorder relevance, the Ising model, due to its conceptual simplicity, is widely applied in different areas of research. Besides
uniaxial magnets, it describes, e.g., experimental liquid-vapor transitions and binary mixtures, certain transitions in high-energy physics, voter dynamics and can even be found in fairly unexpected applications such as the dynamics of traffic signals or pistachio trees. Generalizing the spin-1/2 Ising model to general O (N) symmetric spin vectors, one obtains the XY model (N = 2), featuring a Kosterlitz–Thouless transition and the Heisenberg model (N = 3), modeling certain real magnetic materials. Even higher symmetries (N = 4, 5) become, for instance, again relevant in particle physics and high-temperature superconductor transitions.
This large scope of applications justifies the continued relevance of classical spin models through decades of research. To our surprise a comprehensive software package for those models does not exist to this date. Hence, our goal is to fill this gap, offering a software framework combining modern HPC-optimized numerical simulation techniques and a state-of-the-art statistical analysis. However, we intend to not only provide a convenient workflow, but also focus on quenched disordered systems, where parallelization and automatization aspects become particularly important.