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WORM ALGORITHM FOR CLASSICAL AND QUANTUM STATISTICAL MODELS

WORM ALGORITHM FOR CLASSICAL AND QUANTUM STATISTICAL MODELS. Nikolay Prokofiev, Umass, Amherst. Many thanks to collaborators on major algorithm developments. Boris Svistunov, Umass, Amherst. Igor Tupitsyn, PITP. Vladimir Kashurnikov, MEPI, Moscow. Evgeni Burovski, Umass, Amherst.

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WORM ALGORITHM FOR CLASSICAL AND QUANTUM STATISTICAL MODELS

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  1. WORM ALGORITHM FOR CLASSICAL AND QUANTUM STATISTICAL MODELS Nikolay Prokofiev, Umass, Amherst Many thanks to collaborators on major algorithm developments Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP Vladimir Kashurnikov, MEPI, Moscow Evgeni Burovski, Umass, Amherst Massimo Boninsegni, UAlberta, Edmonton NASA Les Houches, June 2006

  2. Worm algorithm idea Consider: - configuration space = arbitrary closed loops - each cnf. has a weight factor - quantity of interest

  3. “conventional” sampling scheme: localshape change Add/delete small loops No sampling of topological classes can not evolve to dynamical critical exponent in many cases Critical slowing down

  4. Worm algorithm idea draw and erase: Masha Ira Ira + Masha Masha Masha keep drawing or Topological classes are (whatever you can draw!) No critical slowing down in most cases Disconnected loops relate to important physics (correlation functions) and are not merely an algorithm trick!

  5. High-T expansion for the Ising model where 3 4 2 1 4 4 2 number of lines; enter/exit rule

  6. Spin-spin correlation function: Worm algorithm cnf. space = Same as for generalized partition 1 I 3 4 M 4 2

  7. Getting more practical: since Complete algorithm: -If , select a new site for at random - select direction to move , let it be bond - If accept with prob.

  8. I=M I M M M M Correlation function: Magnetization fluctuations: Energy: either or

  9. Ising lattice field theory expand if closed oriented loops where tabulated numbers

  10. Flux in = Flux out closed oriented loops of integer N-currents I (one open loop) M Z-configurations have

  11. Same algorithm: I=M sectors, prob. to accept M M draw M M erase Keep drawing/erasing …

  12. Multi-component gauge field-theory (deconfined criticality, XY-VBS and Neel-VBS quantum phase transitions… XY-VBS transition; understood (?) no DCP, always first-order Neel-VBS transition, unknown !

  13. Winding numbers Homogeneous gauge in x-direction: Ceperley Pollock ‘86

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