Worm Algorithm Path Integral Monte Carlo for Quantum Fluids and Gases. The Needs for GRID computing

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Worm Algorithm Path Integral Monte Carlo for
Quantum Fluids and Gases. The Needs for GRID
computing

• Asaad R. Sakhel
• Al-Balqa Applied University, JORDAN
• (EPIKH 2011 Amman JORDAN)

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• Projects undertaken in the condensed matter group
  of
• Prof. Humam B. Ghassib (University of
  Jordan, Amman) using Quantum Monte Carlo
  techniques
• Tunneling of Bosons in optical lattices (VMC
  VPI)
• (A. R. Sakhel et al., Phys. Rev. A 77,
  043627 (2008))
• Microscopic investigations of the 3He- 4He
  sandwich system on a graphite substrate (PIMC
  Worm)
• (Ghassib et al., submitted to Phys. Rev. E)
• Dynamics of Bose-Einstein condensates (future)
• (R. R. Sakhel et al., Phys. Rev. A 84, 033634
  (2011))

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Broad goals using other numerical methods

• VPI effects of interactions (repulsive or
  attractive) on the properties of tunneling in
  optical lattices, double wells.
• WAPIMC thermodynamic properties of low
  dimensional quantum fluids entropy, pressure,
  internal energy, specific heat capacity.
• Crank Nicolson dynamical evolution of
  Bose-Einstein condensates

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Microscopic Investigations of the 3He-4He
sandwich system on a graphite substrate

• The 3He-4He sandwich system consists of a 4He
  layer ( 3.6 Å thick) , a 3He-4He mixture layer
  (7- 11 Å) and a bulk 3He ( 11-12 Å) layer

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Worm Algorithm (PIMCWorm)

• Worm Algorithm Monte Carlo (WAMC) is Path
  Integral Monte Carlo (PIMC) using Worm updates
  (Boninsegni et al., Phys. Rev. E 74, 036701
  (2006)). Reference for PIMC is Ceperleys Rev.
  Mod. Phys. 67, 279 (1995)
• WAMC and PIMC particle is a trajectory in
  space-time which closes upon itself after a
  time ß1/kT. Each position in space-time
  represented by a bead and the trajectory is
  composed of a number of M beads
• Particle trajectories are updated using open
  configurations called worms.

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Visualizing the Trajectories

• Imaginary time versus position (see also Ceperley
  1995)

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Technical details

• Each bead is given an index and a name.
• Simulation space is divided into a number of
  boxes (cells) labeled by arrays. The purpose is
  then to identify the particles which lie within
  these cells.
• Extensive use of large arrays, e.g.
• hash(icell,it,ind)
• hash() gives particle name in cell icell, at
  time it and for bead index ind.
• Several subroutines which update the trajectories
  of each particle Insert, Remove, Cut, Glue, Move
  Forward Masha, Move Backward Masha, Move Forward
  Ira, Move Backward Ira, etc.
• Hard to parallelize for high performance computing

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Resources needed for WAPIMC

• WAPIMC was written by Boninsegni et al. in
  Fortran 90 and needs ifort compiler
• No libraries needed. Code is self-contained
• High speed, 16 24 CPU Core 64 bit Workstations
  for massive computation
• Large RAM 32 GB (large arrays)
• Large storage gt 500 GB
• Long CPU computational time gt 3 months for each
  WAPIMC run

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Density matrix

• PIMC thermal density matrix (Ceperley 1995)
• is a set of
  particle positions
• PIMC is concerned with the evaluation of
  statistical mechanical averages using Feynmans
  path integral approach

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• Integral of is divided into
  time slices
• Using
• Can then write density matrix using kinetic and
  potential energy

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• Simple action and interaction
• Action used in Metropolis algorithm (acceptance
  criteria)

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Conclusions

• The 3He-4He sandwich system on graphite is a
  system which has been rarely investigated to the
  best of our knowledge. Hence it would be
  interesting to get GRID computing to explore
  larger N. Currently use N 360.
• GRID computing is severely needed to perform WAMC
  calculations for many-body systems with N of
  order 1000 and beyond.

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Our collaborators and friends

• Nikolay Prokofev, UMass, Amherst MA, USA
• William J. Mullin, UMass, Amherst MA, USA
• Saverio Moroni, National Institute for the
  Physics of Matter (INFM), Trieste, ITALY