Difference between revisions of "Afternoon Session: Quantum Monte Carlo"

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== Talks ==
 +
* Topic, more specifically: Path Integral Quantum Monte Carlo
 +
 
* Chair: Boris Svistunov and Synge Todo
 
* Chair: Boris Svistunov and Synge Todo
 
* Speakers
 
* Speakers
Line 10: Line 13:
 
**** Energy/temperature are good choices for thermal phase transitions
 
**** Energy/temperature are good choices for thermal phase transitions
 
**** Ising: energy allone suffices, cannot get better than N^2 behaviour
 
**** Ising: energy allone suffices, cannot get better than N^2 behaviour
** Anatoly Kuklov
+
** Anatoly Kuklov: ''Flowgram Method in QM''
***  
+
*** Apply to known problems!
** Matthias Troyer
+
** 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
 
** 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

Latest revision as of 04:50, 4 May 2009

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