1. WORM ALGORITHM FOR CLASSICAL AND QUANTUM STATISTICAL MODELS Nikolay Prokofiev, Umass, Amherst Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP Vladimir Kashurnikov, MEPI, Moscow Evgeni Burovski, Umass, Amherst Massimo Boninsegni, UAlberta, Edmonton Many thanks to collaborators on major algorithm developments 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: local shape changeAdd/delete small loops can not evolve to No sampling of topological classes Critical slowing down dynamical critical exponent in many cases

4. Worm algorithm idea draw and erase: Masha Ira or Masha Ira + keep drawing Masha 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 1 3 2 number of lines; enter/exit rule 2 4 4 4

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

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 M I M M M Correlation function: Energy: either Magnetization fluctuations: or

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

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

11. Same algorithm: sectors, prob. to accept I=M draw erase M M M M 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 Ceperley Pollock ‘86 Homogeneous gauge in x-direction: