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

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

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

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

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

Hybridisation expansion 7

Wick‘s theorem 8

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

Segment picture Werner et al., PRL,

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

Interacting SIAM 12

Spinnless noninteracting SIAM: Interacting SIAM with spin: 13

Interaction in the Segment picture 14

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

Metropolis choice:Detailed Balance Condition: 17

Phase space 18

Phase space for one spin channel 19

Start configuration: Update processes 20

How do we implement those processes? 21

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

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

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

Weight prefactors add the discretisation factor to the weights 25

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

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

Note: half open segments Remember: 28

Quick example: half open segments 29

Numerical precision 30

Now some results... 31

CT-QMC vs. analytic solution 32

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Computational limits: 35

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

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

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