Concepts and implementation of CT-QMC Markus Dutschke Dec. 6th 2013 (St. Nicholas` Day) 1 impurity modell This is where the magic happens ! Gimp G DMFT Glattice lattice modell 2 CT-QMC solver • Most flexible solver • Restricted to finite temperature 3 Content • • • • • Motivation Analytic foundations Monte Carlo algorithm Implementation and problems Results 4 5 Spinless non interacting single impurity Anderson model NOT the Fermi energy 6 Hybridisation expansion 7 Wick‘s theorem 8 Impurity Green function Werner, Comanac, Medici, Troyer and Millis, PRL 97, 076405 (2006): 9 Segment picture Werner et al., PRL, 2006 10 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 15 Metropolis-Hasting algorithm Detailed Balance Condition: Metropolis choice: 16 Detailed Balance Condition: Metropolis choice: 17 Phase space 18 Phase space for one spin channel 19 Update processes Start configuration: 20 21 Example: Metropolis-Hasting acceptance probability for add process Algorithm Metropolis-Hasting: Physical problem Discretisation of configurations: 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 process probability 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 33 34 Computational limits: 35 36 37 Summary Segment picture: quick and simple Agreement with analytic solution 38 Outlook DMFT for the Hubbard model with magnetic Field Spin up Spin down Vollhardt, Ann. Phys, 524:1-19, doi: 10.1002/andp.201100250 39 Acknowledgements: Junya Otsuki Liviu Chioncel Michael Sekania Jaromir Panas Christian Gramsch 40

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# How do we implement those processes?