WORM ALGORITHM: LIQUID & SOLID HE-4

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

2. Why bother with worm algorithm? Efficiency New quantities to address physics PhD while still young PhD while still young Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams 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 …

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

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

5. Worm algorithm idea NP, B. Svistunov, I. Tupitsyn, ‘97 draw and erase: Masha Ira Ira + Masha Masha Masha keep drawing or All topologies are sampled (whatever you can draw!) Disconnected loop is related to the off-diagonal correlation function and is not merely an algorithm trick! No critical slowing down in most cases 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 statistical interpretation ignore : stat.weight 1 Account for : stat. weight p 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 ofIra-Masha distance in space time particle “wave funct.” at Energy gaps/spectrum, quasi-particle Z-factors One-body density matrix, Cond. density Winding numbers: superfluid density Winding number exchange cycles maps of local superfluid response At the same CPU price as energy in conventional schemes!

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

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

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

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 Worm algorithm: Pilati, Giorgini, NP wrong number of slices (5 vs 15) underestimated error bars + too small system size discrepancy ! 100,000

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

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

19. Supersolid He-4 “… ice cream” “… transparent honey”, … A network of SFgrain boundaries, dislocations, and ridges with superglass/superfluid pockets (if any). GB SF/SG Ridge He-3 Disl He-3 All “ice cream ingredients” are confirmed to have superfluid properties Dislocations network (Shevchenko state) at where 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: Top (z-axis) view Side (x-axis) view Maps of exchange cycles with non-zero winding number

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

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

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

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

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

27. Conclusions extended configuration space Z+G Worm Algorithm = all updated are local & through end points exclusively no critical slowing down Grand Canonical ensemble off-diagonal correlators superfluid density 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. Superfluid grain boundaries in He-4 GB (periodic BC) two cuboids GB atoms each Maps ofexchange-cycles with non-zero winding numbers 3a XY-view XZ-view

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