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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 Markus DutschkeDec. 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 • Analytic foundations • Monte Carlo algorithm • Implementation and problems • Results
Spinless non interacting single impurity Anderson model NOT the Fermi energy
Wick‘s theorem Determinant: very costly
Impurity Green function Werner, Comanac, Medici, Troyer and Millis, PRL 97, 076405 (2006): Matrix inversion: also very costly
Segment picture Werner et al., PRL, 2006
Operator representation of SIAM: Segment picture: L: sum of the lengths of all segments
Spinnless noninteracting SIAM: Interacting SIAM with spin:
Metropolis-Hasting algorithm Detailed Balance Condition: Metropolis choice:
Detailed Balance Condition: Metropolis choice:
Update processes Start configuration:
How do we implement those processes? How do we implement those processes?
Example: Metropolis-Hasting acceptance probability for add process Algorithm Metropolis-Hasting: Physical problem Discretisation of configurations:
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
Remove process • remove process: • Decide to remove a segment • choose a random segment to remove
Weight prefactors add the discretisation factor to the weights
This is beautiful ... ... But some things are not as pretty as they look like!
Note: half open segments Remember:
Summary Segment picture: quick and simple Agreement with analytic solution
Outlook DMFT for the Hubbard model with magnetic Field Vollhardt, Ann. Phys, 524:1-19, doi: 10.1002/andp.201100250
Acknowledgements: Junya Otsuki Liviu Chioncel Michael Sekania Jaromir Panas Christian Gramsch