Finding and proving the exact ground state of a generalized Ising model by convex optimization and MAX-SAT
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PhysRevB.94.134424.pdf
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Author(s)
Urban, Alexander
Luo, Chuan
Huang, Wenxuan
Kitchaev, Daniil Andreevich
Dacek, Stephen Thomas
Cao, Shan
Ceder, Gerbrand
Rong, Ziqin
Date Issued
October 2016
Journal
Physical Review B
Publisher
American Physical Society
Citation
Huang, Wenxuan et al. “Finding and Proving the Exact Ground State of a Generalized Ising Model by Convex Optimization and MAX-SAT.” Physical Review B 94.13 (2016): n. pag. ©2016 American Physical Society
Version
Final published version
Abstract
Lattice models, also known as generalized Ising models or cluster expansions, are widely used in many areas of science and are routinely applied to the study of alloy thermodynamics, solid-solid phase transitions, magnetic and thermal properties of solids, fluid mechanics, and others. However, the problem of finding and proving the global ground state of a lattice model, which is essential for all of the aforementioned applications, has remained unresolved for relatively complex practical systems, with only a limited number of results for highly simplified systems known. In this paper, we present a practical and general algorithm that provides a provable periodically constrained ground state of a complex lattice model up to a given unit cell size and in many cases is able to prove global optimality over all other choices of unit cell. We transform the infinite-discrete-optimization problem into a pair of combinatorial optimization (MAX-SAT) and nonsmooth convex optimization (MAX-MIN) problems, which provide upper and lower bounds on the ground state energy, respectively. By systematically converging these bounds to each other, we may find and prove the exact ground state of realistic Hamiltonians whose exact solutions are difficult, if not impossible, to obtain via traditional methods. Considering that currently such practical Hamiltonians are solved using simulated annealing and genetic algorithms that are often unable to find the true global energy minimum and inherently cannot prove the optimality of their result, our paper opens the door to resolving longstanding uncertainties in lattice models of physical phenomena. An implementation of the algorithm is available at https://github.com/dkitch/maxsat-ising.
MIT Department
Massachusetts Institute of Technology. Department of Materials Science and Engineering
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DOI of Published Version
https://doi.org/10.1103/PhysRevB.94.134424
