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Phys Rev E. 2019 Jan;99(1-1):010105. doi: 10.1103/PhysRevE.99.010105.

Clock Monte Carlo methods.

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Centre de Mathématiques Appliquées, UMR 7641, École Polytechnique, Palaiseau, France.
Orange Labs, 44 Avenue de la République, CS 50010, 92326 Châtillon Cedex, France.
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.


We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of O(N^{κ}) (0≤κ≤1) can be achieved compared to the Metropolis method, where N is the system size and the κ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O(n) spin models in a wide parameter regime: for n=1,2,3, with disordered, algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yosida-type interactions and with and without external fields, and in spatial dimensions from d=1,2,3 to the mean field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.


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