1. WORM ALGORITHM: LIQUID & SOLID HE-4 Nikolay Prokofiev, Umass, Amherst NASA RMBT14, Barcelona July 2007 Boris Svistunov, Umass, Amherst Massimo Boninsegni, UAlberta Matthias Troyer, ETH Lode Pollet, ETH Anatoly Kuklov, CSI CUNY Masha Ira

2. Why bother with worm algorithm? PhD while still young New quantities to address physics Grand canonical ensemble Off-diagonal correlations condensate wave functions Winding numbers and Examples from: helium liquid & solid lattice bosons/spins, classical stat. mech. disordered systems, deconfined criticality, resonant fermions, polarons … Efficiency PhD while still young Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams

3. Worm algorithm idea Consider: - configuration space = closed loops - each cnf. has a weight factor - quantity of interest NP, B. Svistunov, I. Tupitsyn, ‘97 P 1 2 P Feynman path integrals for What is the best updating strategy?

4. “conventional” sampling scheme: local shape changeAdd/delete small loops can not evolve to No sampling of topological classes (non-ergodic) Critical slowing down (large loops are related to critical modes) dynamical critical exponent in many cases

5. Worm algorithm idea draw and erase: Masha Ira or Masha Ira + keep drawing Masha All topologies are sampled (whatever you can draw!) No critical slowing down in most cases Disconnected loop is related to the off-diagonal correlation function and is not merely an algorithm trick! NP, B. Svistunov, I. Tupitsyn, ‘97 GC ensemble Green function winding numbers condensate wave func.,etc.

6. (open/close update)

7. (insert/remove update)

8. (advance/recede update)

9. (swap update)

10. Path integrals + Feynman diagrams for ignore : stat. weight 1 Account for : stat. weight p statistical interpretation 10 times faster than conventional scheme, scalable (size independent) updates with exact account of interactions between all particles (no truncation radius)

11. Grand-canonical calculations:, compressibility, phase separation, disordered/inhomogeneous systems, etc. Matsubara Green function: Probability density of Ira-Masha distance in space time Energy gaps/spectrum, quasi-particle Z-factors One-body density matrix, Cond. density particle “wave funct.” at Winding numbers: superfluid density Winding number exchange cycles maps of local superfluid response At the same CPU price as energy in conventional schemes!

12. Ceperley, Pollock ‘89 “Vortex diameter” 2D He-4 superfluid density & critical temperature Critical temp.

13. 3D He-4 at P=0 superfluid density & critical temperature 64 2048 experiment Pollock, Runge ‘92 ?

14. N=64 N=2048 3D He-4 at P=0 Density matrix & condensate fraction (Bogoliubov)

15. 3D He-4 liquid near the freezing point, T=0.25 K, N=800 Calculated from

16. Weakly interacting Bose gas, pair product approximation; ( example) Ceperley, Laloe ‘97 Nho, Landau ‘04 discrepancy ! wrong number of slices (5 vs 15) underestimated error bars + too small system size Worm algorithm: Pilati, Giorgini, NP 100,000

17. Solid (hcp) He-4 Density matrix near melting InsulatorExponential decay

18. Solid (hcp) He-4 Green function melting density Large vacancy / interstitial gaps at all P InsulatorExponential decay in the solid phase Energy subtraction is not required!

19. Supersolid He-4 “… ice cream” “… transparent honey”, … GB Ridge He-3 SF/SG A network of SF grain boundaries, dislocations, and ridges with superglass/superfluid pockets (if any). Dislocations network (Shevchenko state) at where All “ice cream ingredients” are confirmed to have superfluid properties Disl He-3 Frozen vortex tangle; relaxation time vs exp. timescale

20. Supersolid phase of He-4 Is due to extended defects: metastable liquid grain boundaries screw dislocation, etc. Pinned atoms “physical” particles screw dislocation axis

21. Supersolid phase of He-4 Is due to extended defects: metastable liquid grain boundaries screw dislocation, etc. Screw dislocation has a superfluid core: Maps of exchange cycles with non-zero winding number Top (z-axis) view Side (x-axis) view

22. + superfluid glass phase (metastable) anisotropic stress domain walls superfluid grain boundaries

23. Lattice path-integrals for bosons/spins (continuous time) imaginary time lattice site imaginary time lattice site

24. M I I I I M At one can simulate cold atom experimental system “as is” for as many as atoms!

25. Classical models: Ising, XY, closed loops Ising model (WA is the best possible algorithm) Ira Masha

26. I=M M I M M M Complete algorithm: - If, select a new site for at random - otherwise, propose to move in randomly selected direction Easier to implement then single-flip!

27. Conclusions no critical slowing down Grand Canonical ensemble off-diagonal correlators superfluid density Worm Algorithm = extended configuration space Z+G all updated are local & through end points exclusively At no extra cost you get Continuous space path integrals Lattice systems of bosons/spins Classical stat. mech. ( the best method for the Ising model ! ) Diagrammatic MC ( cnfig. space of Feynman diagrams ) Disordered systems A method of choice for

28. GB GB (periodic BC) 3a XY-view XZ-view Superfluid grain boundaries in He-4 Maps of exchange-cycles with non-zero winding numbers two cuboids atoms each

29. ODLRO’ Superfluid grain boundaries in He-4 Continuation of the -line to solid densities