Frustrating disorder for continuous spins: low-temperature properties of spin glasses and random-field systems

• Applicant: Professor Dr. Martin Weigel
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2007 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 55457478

• Final Report Year: 2015

Final Report Abstract

According to the classical theories of condensed matter physics, substances surrounding us come in only three distinct states of aggregation: gaseous, liquid and solid (or mixtures thereof such as emulsions, for example in butter). Solids are characterised by the regular arrangement of atoms in a lattice structure, whereas liquids consist of closely interacting molecules which are able to slide relative to each other. Finally, gases microscopically appear as almost non-interacting molecules in random motion. As external parameters such as temperature are varied, matter undergoes transformations between these states known as phase transitions (such as, e.g., the melting of ice to water). The microscopical structure of these phases and the mechanisms of phase transitions are rather well understood. It is only in the last couple of decades, however, that scientists gradually realise that this trichotomy of phases is not complete, and a number of exciting, “exotic” phases have been discovered. Many of them seem to be triggered by the combined effect of disorder and frustration. Disorder is present in all everyday substances which naturally contain impurities or imperfections in lattice structure. These imperfections may (at a certain temperature) destroy the perfect order of a solid lattice, forcing the substance into a liquid state, but it does not make it leave the trichotomy of phases. Frustration arises when a locally optimal arrangement of particles forces sub-optimal arrangement elsewhere due to inherent restrictions — one may imagine, for instance, a jigsaw puzzle made from a set of identical shapes: a puzzle of squares or triangles can be assembled without gaps, but a puzzle of pentagons cannot, it is frustrated. Due to such restrictions, a frustrated substance is prevented from arranging in its natural, ordered state. In a class of magnetic materials known as spin glasses, a disordered and frustrated state is the ultimately stable configuration. In the present project, the role of the symmetry of the spin variables for the shape of the “energy landscape” has been elucidated. As more and more spin components are added, the barriers in the landscape are smoothed out and, in the limit of an infinite number of components, the metastability that is at the heart of the observed slow relaxation and computational complexity of these systems disappears. Thus, the system becomes much more tractable in this limit. Suprisingly, for finite systems the loss of metastability happens for a finite number of spin components, a phenomenon akin to Bose-Einstein condensation in a superfluid or superconductor. Still, certain aspects of glassy behavior remain even in this extremal case. For studying systems in two dimensions, new highly efficient algorithms have been developed that allow to find exact ground states for systems composed of up to 108 spin variables. These are based on a new mapping of the ground-state problem onto a graph-theoretic problem known as minimum-weight perfect matching. Extensions of these approaches allow for a treatment of systems with fully periodic boundary conditions as well as for the uniform sampling of ground states in the case of degeneracies. These problems have recently moved into the focus of attention of researchers interested in the adiabatic quantum computing paradigm as realized in the quantum annealing devices implemented by D-Wave Systems Inc. Methodologically, there have been significant advances in the context of the present project in the form of utilizing graphics processing units (GPUs) for simulating spin models including spin glasses and random-field systems. By tailoring the simulation scheme to the massively parallel architecture with a specific hierarchy of memories, it is possible to achieve speed-ups of GPU simulation codes of two to three orders of magnitude as compared to scalar codes running on CPUs. These techniques are now widely used in statistical physics and other communities using spin models for instance for image segmentation. Some reports on the research performed in the framework of the Emmy Noether Group appeared in media outlets with a broader reach than specific topical journals. These included an article in “Physik Journal”, with more than 50.000 monthly copies the largest physics journal in German language, on the rugged free-energy landscapes characteristic of spin-glasses and other glassy systems. The relevant references are listed below. [1] M. Weigel, The GPU revolution at work, Computing in Science and Engineering 13 (5), 5 (2011). [2] A. Heuer, M. Kastner, A. Hartmann and M. Weigel, Wanderungen in Energielandschaften, Physik Journal 12/2010, 35 (2010). [3] M. Weigel and T. Schilling, Der Komplexität auf den Grund gehen, Natur & Geist 2/2009, 11 (2009)

Publications

• Zero-temperature phase of the XY spin glass in two dimensions: Genetic embedded matching heuristic, Phys. Rev. B 77, 104437 (2008)
  M. Weigel and M. J. P. Gingras
  
• Cross Correlations in Scaling Aanalyses of Phase Transitions, Phys. Rev. Lett. 102, 100601 (2009)
  M. Weigel and W. Janke
  
• Connected component identification and cluster update on GPU, Phys. Rev. E 84, 036709 (2011)
  M. Weigel
  
• Domain walls and Schramm-Loewner evolution in the random-field Ising model, Europhys. Lett. 95, 40001 (2011)
  J. D. Stevenson and M. Weigel
  
• Non-reversible Monte Carlo simulations of spin models, Comput. Phys. Commun. 182, 1856 (2011)
  H. C. M. Fernandes and M. Weigel
  
• Simulating spin models on GPU, Comput. Phys. Commun. 182, 1833 (2011)
  M. Weigel
  
• Spin stiffness of vector spin glasses, Comput. Phys. Commun. 182, 1883 (2011)
  F. Beyer and M. Weigel
  
• One-dimensional infinite component vector spin glass with long-range interactions, Phys. Rev. B 86, 014431 (2012)
  F. Beyer, M. Weigel, and M. A. Moore
  
• Optimized GPU simulation of continuous-spin glass models, Eur. Phys. J. Spec. Topics 210, 159 (2012)
  T. Yavors’kii and M. Weigel
  
• Random number generators for massively parallel simulations on GPU, Eur. Phys. J. Spec. Topics 210, 53 (2012)
  M. Manssen, M. Weigel, and A. K. Hartmann