Finding and proving the exact ground state of a generalized Ising model by convex optimization and MAX-SAT
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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
Huang , W , Kitchaev , D A , Dacek , S T , Rong , Z , Urban , A , Cao , S , Luo , C & Ceder , G 2016 , ' Finding and proving the exact ground state of a generalized Ising model by convex optimization and MAX-SAT ' Physical Review B , vol 94 , no. 13 , 134424 . DOI: 10.1103/PhysRevB.94.134424
Physical Review B
© 2016 Publisher / the Author(s). Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
This paper was supported primarily by the US Department of Energy (DOE) under Contract No. DE-FG02-96ER45571. In addition, some of the test cases for ground states were supported by the Office of Naval Research under contract N00014-14-1-0444.
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