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Concepts and implementation of CT-QMC

Concepts and implementation of CT-QMC. Markus Dutschke Dec. 6th 2013 (St. Nicholas` Day). impurity modell. This is where the magic happens !. solver. solver. G. DMFT. lattice modell. CT-QMC solver. Most flexible solver Restricted to finite temperature. Content. Motivation

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Concepts and implementation of CT-QMC

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  1. Concepts and implementation of CT-QMC Markus DutschkeDec. 6th 2013 (St. Nicholas` Day)

  2. impurity modell This is where the magic happens ! solver solver G DMFT lattice modell

  3. CT-QMC solver • Most flexible solver • Restricted to finite temperature

  4. Content • Motivation • Analytic foundations • Monte Carlo algorithm • Implementation and problems • Results

  5. Analytic foundations

  6. Spinless non interacting single impurity Anderson model NOT the Fermi energy

  7. Hybridisation expansion

  8. Wick‘s theorem Determinant: very costly

  9. Impurity Green function Werner, Comanac, Medici, Troyer and Millis, PRL 97, 076405 (2006): Matrix inversion: also very costly

  10. Segment picture Werner et al., PRL, 2006

  11. Operator representation of SIAM: Segment picture: L: sum of the lengths of all segments

  12. Interacting SIAM

  13. Spinnless noninteracting SIAM: Interacting SIAM with spin:

  14. Interaction in the Segment picture

  15. Monte Carlo

  16. Metropolis-Hasting algorithm Detailed Balance Condition: Metropolis choice:

  17. Detailed Balance Condition: Metropolis choice:

  18. Phase space

  19. Phase space for one spin channel

  20. Update processes Start configuration:

  21. How do we implement those processes? How do we implement those processes?

  22. Example: Metropolis-Hasting acceptance probability for add process Algorithm Metropolis-Hasting: Physical problem Discretisation of configurations:

  23. Add process • Add process: • decide to add a segment • take a random meshpoint (start of the segment) from the intervall(if an existing segment is hit -> weight = 0) • Take a random meshpoint between startpoint and start of the next segment

  24. Remove process • remove process: • Decide to remove a segment • choose a random segment to remove

  25. Weight prefactors add the discretisation factor to the weights

  26. Metropolis-Hasting in the Segment picture

  27. This is beautiful ... ... But some things are not as pretty as they look like!

  28. Note: half open segments Remember:

  29. Quick example: half open segments

  30. Numerical precision

  31. Now some results ...

  32. CT-QMC vs. analytic solution

  33. Computational limits:

  34. Summary Segment picture: quick and simple Agreement with analytic solution

  35. Outlook DMFT for the Hubbard model with magnetic Field Vollhardt, Ann. Phys, 524:1-19, doi: 10.1002/andp.201100250

  36. Acknowledgements: Junya Otsuki Liviu Chioncel Michael Sekania Jaromir Panas Christian Gramsch

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