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WORM ALGORITHM: LIQUID & SOLID HE-4 Nikolay Prokofiev, Umass, Amherst Boris Svistunov, Umass, Amherst Masha Massimo Boninsegni, UAlberta Matthias Troyer, ETH Ira Lode Pollet, ETH Anatoly Kuklov, CSI CUNY NASA RMBT14, Barcelona July 2007 Why bother with worm algorithm? Efficiency New quantities to address physics PhD while still young Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams Grand canonical ensemble N ( ) Off-diagonal correlations G ( r , ) condensate wave functions ( r ) Winding numbers and S Examples from: helium liquid & solid lattice bosons/spins, classical stat. mech. disordered systems, deconfined criticality, resonant fermions, polarons … NP, B. Svistunov, I. Tupitsyn, ‘97 Worm algorithm idea Feynman path integrals for Consider: H RPMBT 14 pi2 V (ri rj ) i 2m i j - configuration space = closed loops P / Z dR ...dR ( R , Ri ,) P 1 - each cnf. has1 a weight factor i W i 1 cnf P A W W cnf - quantity of interest A cnf cnf cnf cnf P ri ,1 ri ,2 Ri (ri ,1 ,ri ,2 ,...,ri , N ) 2 What is the best updating strategy? 1 P “conventional” sampling scheme: local shape change No sampling of topological classes can not evolve to (non-ergodic) Critical slowing down (large loops are related to critical modes) Add/delete small loops auto Nupdates z L d L dynamical critical exponent z 2 in many cases NP, B. Svistunov, I. Tupitsyn, ‘97 Worm algorithm idea draw and erase: Masha Ira Ira Masha + Masha Masha or keep drawing 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! GC ensemble N ( ) Green function G ( r , ) winding numbers S condensate wave func.( r ),etc. Z G (r ', t ') (r , t ) (open/close update) Z G Ira Masha (insert/remove update) G Ira Ira Masha (advance/recede update) G Masha Ira (swap update) Ira Path integrals + Feynman diagrams for V (r ) 0 e V ( rij ) 1 (e V ( rij ) 1) 1 pij Masha statistical interpretation ignore Ira V ( rij ) Account for : stat. weight 1 V ( rij ) : stat. weight p p ij i j 10 times faster than conventional scheme, scalable (size independent) updates with exact account of interactions between all particles (no truncation radius) Grand-canonical calculations: n( , T ) , compressibility N 2 TV , phase separation, disordered/inhomogeneous systems, etc. Matsubara Green function: G (r , r ', ') T † (r , ) (r ', ') Probability density of Ira-Masha distance in space time lim G( p, ) Z p e E ( p ) G (r , 0) n(r ) Energy gaps/spectrum, quasi-particle Z-factors One-body density matrix, Cond. density n0 n(r ) Winding numbers: ns m W2 d 2 dTL Winding number exchange cycles lim G(r , r ', / 2) (r ) (r ') |r r '| particle “wave funct.” at superfluid density maps of local superfluid response At the same CPU price as energy in conventional schemes! 2D He-4 superfluid density & critical temperature (n 0.0432 A2 ) TC 0.65(1) Critical temp. “Vortex diameter” d 9 A Ceperley, Pollock ‘89 TC 0.72(2), d 3.5 A 3D He-4 at P=0 superfluid density & critical temperature 64 experiment 2048 Pollock, Runge ‘92 ? TCAziz 2.193vs TCexp 2.177 3D He-4 at P=0 Density matrix & condensate fraction N=64 N=64 (Bogoliubov) N=2048 n(r ) n0 e mT / 4 rns n0 0.024 N=2048 3D He-4 liquid near the freezing point, T=0.25 K, N=800 Calculated from Weakly interacting Bose gas, pair product approximation; TC (V ) Ceperley, Laloe ‘97 3 3 ( na 5 10 example) Nho, Landau ‘04 TC / T0 1.057(2)? T / T0 TC / T0 1.078(1)? Worm algorithm: Pilati, Giorgini, NP underestimated wrong oferror slices ! bars 20number discrepancy + too small (5 vs system 15) size 100,000 Solid (hcp) He-4 Density matrix T 0.2K ,N 800 Exponential decay Insulator near melting o 3 n 0.0292 A o 3 n 0.0359 A Solid (hcp) He-4 Green function Exponential decay Insulator T 0.25K ,N 800 E i,v is not required! EN E G( p,| | ) Z e 1 N Energy subtraction melting density in the solid phase Large vacancy / interstitial gaps at all P Supersolid He-4 “… ice cream” “… transparent honey”, … A network of SF grain boundaries, dislocations, and ridges with superglass/superfluid pockets (if any). GB SF/SG Disl All “ice cream ingredients” are confirmed to have superfluid properties Dislocations network (Shevchenko state) at Ridge He-3 He-3 a T T TC where TC ~ T T l 1 ~ T T / T 8 K 1 Frozen vortex tangle; relaxation time vs exp. timescale Supersolid phase of He-4 Is due to extended defects: metastable liquid grain boundaries screw dislocation, etc. Pinned atoms T (0.25 K ,n 0.0287 N 384 1536 “physical” particles screw dislocation axis Supersolid phase of He-4 Is due to extended defects: metastable liquid grain boundaries screw dislocation, etc. Screw dislocation has a superfluid core: nS 1 A1,gLutt.Liq. 5 T (0.25 K ,n 0.0287 N 384 1536 Side (x-axis) view Top (z-axis) view T (nliquid nsolid ) 1.5(1)K Maps of exchange cycles with non-zero winding number superfluid grain boundaries anisotropic stress T (@ solid densities) domain walls + superfluid glass phase (metastable) Lattice path-integrals for bosons/spins (continuous time) H H 0 H1 U ij ni n j i ni t (ni , n j )b j bi ij ij i Z = Tr e- H imaginary time imaginary time GIM =Tr T bM† ( M ) bI ( I ) e- H Masha Ira 0 0 lattice site lattice site I I M M I I At T ~ t one can simulate cold atom experimental 6 system “as is” for as many as N 10 atoms! Classical models: Ising, XY, 4 Ising model (WA is the best possible algorithm) H K i j ( 1) T ij Z e H / T GIM e H / T I M {i } {i } closed loops Ira Masha I=M I M M M M Complete algorithm: - If I M , select a new site for I M at random - otherwise, propose to move M in randomly selected direction R min(1, tanh( J / T ))for nbond 0nbond 1 min(1, tanh 1 ( J / T ))for nbond 1nbond 0 Easier to implement then single-flip! Conclusions extended configuration space Z+G Worm Algorithm = all updated are local & through end points exclusively At no extra cost you get A method of choice for no critical slowing down Grand Canonical ensemble off-diagonal correlators superfluid density 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 Superfluid grain boundaries in He-4 GB (periodic BC) GB Maps of exchange-cycles with non-zero winding numbers nS two cuboids N 12 12 7 atoms each mT W 2 dL Lz Ly 3a XY-view Lx XZ-view Lx Superfluid grain boundaries in He-4 12 12 7 7 ODLRO’ (TC )GB max 1.5K TKT 0.6K Continuation of the -line to solid densities