Afternoon Session: Quantum Monte Carlo

From ALPS

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