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 www.wikipedia.org Monte Carlo, Monaco

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. Statistical mechanics Example: Classical Ising model Problem: calculate Number of terms = 2 100 =10 30 On a supercomputer that does 10 15 summations/sec, this takes 10 7 years… 2D lattice with 10x10 sites:

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

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

9. Importance Sampling We should choose P such that is minimized.For example, if, then ! … 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. Sampling Technique Back to the Ising model: A natural choice of P: How to choose random configurations with probability ? Solution: Generate a Markov process that converges to

11. The Metropolis Algorithm 1.Start from a random configuration 2.Pick a spin j. Propose a new configuration that differs by one spin flip 3.If, accept the new configuration: 4.If, accept the new configuration with probability 5.And back to step 2… “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 Path integral formulation: Discrete time version: P 1 2 P

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

17. Determinantal MC (2) The term is quadratic, and can be handled exactly. What to do with the term? Hubbard-Stratonovich transformation: 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 i  s ik kk  k+1  k-1 i+1

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

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

22. Sign Problem Problem: is not necessarily positive. Solution: Probability distribution: 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 No sign problemSign problem Hubbard model (U<0)Hubbard model (U>0): generic filling Hubbard model (U>0): half filling Heisenberg model, triangular lattice “Bose-Hubbard” model (any U)Most “frustrated” spin models Heisenberg model, square lattice “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 www.itp.phys.ethz.ch/staff/troyer/publications/troyerP27.pdf N. Prokofiev, lecture notes on “Worm algorithms for classical and quantum statistical models”, Les Houches summer 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 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).