WORM ALGORITHM APPLICATIONS Nikolay Prokofiev, Umass, Amherst Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP Vladimir Kashurnikov, MEPI, Moscow Massimo Boninsegni, UAlberta, Edmonton Many thanks to collaborators NASA Les Houches, June 2006 Matthias Troyer, ETH Lode Pollet, ETH Anatoly Kuklov, CSI, CUNY

Worm Algorithm No critical slowing down (efficiency) Better accuracy Large system size Finite-size scaling Critical phenomena Phase diagrams Reliably! New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations “Single-particle” and/or condensate wave functions Winding numbers and Examples from: superfluid-insulator transition, spin chains, helium solid & glass, deconfined criticality, holes in the t-J model, resonant fermions, …

Superfluid-insulator transition in disordered bosonic system For any finite the sequence is always SF - Bose glass - Mott insulator Fisher, Weichman,Grinstein ‘89 Not found in helium films “Disproved” in numerical simulations (many, 1D and 2D) New theories to support direct SF-MI transition have emerged ?

- The data look as a perfect direct SF-MI transition ( ) - Up to the data look as a direct SF-MI transition, but … For small the Bose glass state is dominated by rare (exponentially) statistical fluctuations resulting in hole-rich and particle-rich regions

“Wave function” of the added particle Complete phase diagram Gap in the Ideal system It is a theorem that for the compressibility is finite

Quantum spin chains magnetization curves, gaps, spin wave spectra S=1/2 Heisenberg chain Bethe ansatz MC data

Line is for the effective fermion theory with spectrum Lou, Qin, Ng, Su, Affleck ‘99 deviations are due to magnon-magnon interactions

One dimensional S=1 chain with Spin gap Z -factor Energy gaps:

Spin waves spectrum: One dimensional S=1 Heisenberg chain

Kosterlitz-Thouless scaling: Is (red curve) an exact answer ? Superfluid (XY) – insulator transition in the one dimensional S=1 Heisenberg chain

Density matrix close to First principles simulations of helium: Superfluid hydrodynamics (Bogoliubov) Finte-size scaling

Better then 1% agreement at all T after finite-size scaling calculated experiment

Exponential decay of the single-particle density matrix Insulating hcp crystals of He-4 near melting

Activation energies for vacancies and interstitials:, of course Melting density, N=800, T=0.2 K E(N+1)-E(N) can not be done with this accuracy Large activation energies at all Pressures (thermodynamic limit) In fact, the vacancy gas, even if introduced “by hand”, is absolutely unstable and phase Separates (grand canonical simulations with )

Superglass state of He-4 Single-particle density matrixdensity-density correlator ODLRO, Monte Carlo temperature quench from normal liquid

Condensate wave function maps reveal broken translation symmetry 10 slices across the z-axis A rough estimate of metastability: Superglass state of He-4 density of points

Condensate maps simulation box the 4 slices across x-axis across y-axis across z-axis Each of the 8 cubes is a randomly oriented crystallite (24 interfaces) Superfluid ridges and interfaces in He-4

Worm Algorithm No critical slowing down (efficiency) Better accuracy Large system size Finite-size scaling Critical phenomena Phase diagrams Reliably! New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations “Single-particle” and/or condensate wave functions Winding numbers and classical stat. mech. models [Ising, lattice field theories, polymers], quantum lattice spin and particle systems, continuous space quantum particle systems (high-T series, Feynman diagrams in either momentum or real space, path-integrals, whatever loop-like …) - Extended configuration space for local moves of source/drain or etc. operators - All updates exclusively through