Afternoon Session: Quantum Monte Carlo

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Talks

  • Topic, more specifically: Path Integral Quantum Monte Carlo
  • Chair: Boris Svistunov and Synge Todo
  • Speakers
    • Synge Todo: Parallel Monte Carlo
      • Why large load imbalance for parallel tempering?
        • Dependence of CPU time on temperature requires fine-tuning
      • Multi-canonical ensemble
        • Weight depends on global properties of the system
    • Simon Trebst: Extended statistical ensembles
      • More than one variable? Possible, but no examples where it helps significantly
        • Energy/temperature are good choices for thermal phase transitions
        • Ising: energy allone suffices, cannot get better than N^2 behaviour
    • Anatoly Kuklov: Flowgram Method in QM
      • Apply to known problems!
    • Matthias Troyer: The Sign Problem
      • Where does it appear?
        • Bosonic systems: no
        • Fermions: exchanging fermions leads to sign problems
        • Bosons in a gauge field: phases
        • Frustrated magnets: exchange leads to sign problem
      • Can it be solved?
        • Basis-dependent!
        • In general: NP-hard & QMA-hard
        • Scaling with system size: using an appropriate method away from criticality, the scaling should not be exponential
        • In some cases, symmetries help to solve sign problem completely: Meron cluster algorithm
        • In these cases, phases are close to classical phases
      • Where does it come from?
        • Sampling with absolute value of the weights is equivalent to sampling bosons to learn about fermions
    • Boris Svistunov

Discussion

  • What models? What system sizes/temperatures?
  • QMC: unfrustrated spin systems
  • Why spin chains and Haldane conjecture?
    • Because I can!
    • Because Petaflop computers are there and this is an application: do what you can do
    • First purpose of the petaflop machine is to be used
  • Create a list of open problems in physics?
    • Quantum magnets: beta * Vol = 10^8 on standard clusters
      • 10^7 spins or 10^5 lattice bosons or 10^4 continuous bosons on a single node
      • 10^8 on a MPP: memory constraints
      • Disorder etc: up to a few 100,000 CPUs
    • Bosonic models without frustration (hopping matrix cannot trivially be mapped to positive definite matrix)
    • No real time dynamics (only equilibrium statistics) (short time with diagrammatic MC)
  • Challenges
    • Better representations, better analytical/conceptual ideas are needed
    • General challenges: solve the sign problem
    • More specific ones: find a good representation for specific problems -> diagrammatic Monte Carlo
    • Finite-size analysis
    • First-order phase transitions
    • Disrodered systems
    • Correctly distinguishing second order from weakly first order transitions: flowgram technique
  • Methods
    • Loop algorithm
    • Worm algorithm
    • Directed loop
    • Worm spin-offs
    • Determinant Monte Carlo
      • Under control?
    • All of the above: path integral representation (and SSE)
      • Discrete time (cont. space), continuous time (discr. space), SSE
    • Treat representation and update strategy separately
    • Are there more or less reliable methods?
      • Some people call variational/fixed-node/... Monte Carlo QMC
      • Reliable == controllable error (no systematic errors)
      • Who's doing it?: Sufficient equilibration
      • Diagrammatic MC: is it clear ow controlled it is?
      • Path Integral MC is well-established
      • Defining well-established
        • Depend on mapping to classical system: d -> d+1
        • Everything depends on having sufficiently good statistics
        • Everything said about classical statistics maps into some quantum counterpart
        • Many quantum problems map into well-behaved classical problems
          • Slowly equilibrating problems are rarely considered in QMC calculations
          • Counterexample: melting of solid Helium