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

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

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

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

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

7. Hybridisation expansion 7

8. Wick‘s theorem 8

9. Impurity Green function Werner, Comanac, Medici, Troyer and Millis, PRL 97, 076405 (2006): 9

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

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

12. Interacting SIAM 12

13. Spinnless noninteracting SIAM: Interacting SIAM with spin: 13

14. Interaction in the Segment picture 14

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

17. Metropolis choice:Detailed Balance Condition: 17

18. Phase space 18

19. Phase space for one spin channel 19

20. Start configuration: Update processes 20

21. How do we implement those processes? 21

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

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 23

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

25. Weight prefactors add the discretisation factor to the weights 25

26. Metropolis-Hasting in the Segment picture processprobability Add segment Remove segment Add antisegment Remove antisegment 26

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

28. Note: half open segments Remember: 28

29. Quick example: half open segments 29

30. Numerical precision 30

31. Now some results... 31

32. CT-QMC vs. analytic solution 32

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35. Computational limits: 35

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38. Summary Segment picture: quick and simple Agreement with analytic solution 38

39. Outlook Spin up Spin down with magnetic FieldDMFT for the Hubbard model 39 Vollhardt, Ann. Phys, 524:1-19, doi: 10.1002/andp.201100250

40. Acknowledgements: Junya Otsuki Liviu Chioncel Michael Sekania Jaromir Panas Christian Gramsch 40