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Classical and Quantum Monte Carlo Methods

Classical and Quantum Monte Carlo Methods. Or: Why we know as little as we do about interacting fermions. Erez Berg Student/Postdoc Journal Club, Oct. 2007. Outline. Introduction to MC Quantum and classical statistical mechanics Classical Monte Carlo algorithm for the Ising model

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Classical and Quantum Monte Carlo Methods

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  1. Classical and Quantum Monte Carlo Methods Or: Why we know as little as we do about interacting fermions Erez Berg Student/Postdoc Journal Club, Oct. 2007

  2. Outline • Introduction to MC • Quantum and classical statistical mechanics • Classical Monte Carlo algorithm for the Ising model • Quantum Monte Carlo algorithm for the Hubbard model • “Sign problems”

  3. Introduction: Monte Carlo Monte Carlo, Monaco www.wikipedia.org

  4. “Monte Carlo” solution: i = Introduction: Monte Carlo Suppose we are given the problem of calculating And have nothing but a pen and paper. … And we may need to sum much fewer numbers.

  5. Statistical mechanics Thermodynamic quantities Correlation functions

  6. Problem: calculate Statistical mechanics Example: Classical Ising model 2D lattice with 10x10 sites: Number of terms = 2100=1030 On a supercomputer that does 1015 summations/sec, this takes 107 years…

  7. Trick: write Is an arbitrary probability distribution Pick N configurations randomly with probability Calculate Stochastic summation

  8. So… for anychoice of P. Stochastic summation (cont.) Mean and standard deviation: (Central limit theorem) How to choose P?

  9. We should choose P such thatis minimized. For example, if , then ! Importance Sampling … This is a cheat, because to normalize P we need to sum over f. But it shows the correct trend: choose P which is large where f is large.

  10. A natural choice of P: How to choose random configurations with probability ? Solution: Generate a Markov process that converges to Sampling Technique Back to the Ising model:

  11. Start from a random configuration • Pick a spin j. Propose a new configuration that differs by one spin flip • If ,accept the new configuration: • If , accept the new configuration with probability • And back to step 2… The Metropolis Algorithm “Random walk” in configuration space:

  12. Outline • Introduction to MC • Quantum and classical statistical mechanics • Classical Monte Carlo algorithm for the Ising model • Quantum Monte Carlo algorithm for the Hubbard model • “Sign problems”

  13. Quantum statistical mechanics …But now, H is an operator. In general, we don’t even know how to calculate exp(-H). Example:Single particle Schrodinger equation

  14. Quantum statistical mechanics Discrete time version: P 2 1 P Path integral formulation:

  15. The Hubbard Model • “Prototype” model for correlated electrons • Relation to real materials: HTC, organic SC,… • No exact (or even approximate) solution for D>1 How to formulate QMC algorithm?

  16. Trotter-Suzuki decomposition: Determinantal MC Blankenbecler, Scalapino, Sugar (1981)

  17. The term is quadratic, and can be handled exactly. What to do with the term? Hubbard-Stratonovich transformation: Determinantal MC (2) Note that this works only for U>0

  18. Determinantal MC (3) Hubbard-Stratonovich transformation for any U: U<0 U>0

  19. Determinantal MC (4) For the U>0 case, the partition function becomes: Here  k+1 sik k k-1 i i+1

  20. Determinantal MC (5) Now, since the action is quadratic, the fermions can be traced out.

  21. Monte Carlo: interpret as a probability P{s} Monte Carlo Evaluation And, by a variation of Wick’s theorem, How to calculate this sum?

  22. Solution: Probability distribution: Sign Problem Problem: is not necessarily positive. And evaluate the numerator and denominator by MC!

  23. At low temperatures and large U, the denominator becomes extremely small, causing large errors in . Sign Problem (2) But… 4x4 Hubbard model (Loh et al., 1990)

  24. Sign Problem (3) Note that for U<0, Therefore And there is no sign problem!

  25. Summary • “Sign problem free” models can be considered as essentially solved! • In models with sign problems, in many cases, the low temperature physics is still unclear. • Unfortunately, many interesting models belong to the second type.

  26. Summary Quoting M. Troyer: “If you want you can try your luck: the person who finds a general solution to the sign problem will surely get a Nobel prize!”

  27. References • M. Troyer, “Quantum and classical monte carlo algorithms, www.itp.phys.ethz.ch/staff/troyer/publications/troyerP27.pdf • N. Prokofiev, lecture notes on “Worm algorithms for classical and quantum statistical models”, Les Houchessummer school on quantum magnetism (2006). • R. R. Dos Santos, Braz. J. Phys. 33, 36 (2003). • R. T. Scalettar, “How to write a determinant QMC code”, http://leopard.physics.ucdavis.edu/rts/p210/howto1.pdf • E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). • J. W. Negele and H. Orland, “Quantum many particle systems”, Addison-Wesley (1988).

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