Concepts and implementation of
CT-QMC
Markus Dutschke
Dec. 6th 2013
(St. Nicholas` Day)
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impurity modell
This is where the
magic happens !
Gimp
G
DMFT

Glattice
lattice modell
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CT-QMC solver
• Most flexible solver
• Restricted to finite temperature
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Content
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Motivation
Analytic foundations
Monte Carlo algorithm
Implementation and problems
Results
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Spinless non interacting single
impurity Anderson model
NOT the Fermi energy
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Hybridisation expansion
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Wick‘s theorem
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Impurity Green function
Werner, Comanac, Medici, Troyer and Millis, PRL 97, 076405 (2006):
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Segment picture
Werner et al., PRL, 2006
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Operator representation of SIAM:
Segment picture:
L: sum of the lengths of all segments
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Interacting SIAM
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Spinnless noninteracting SIAM:
Interacting SIAM with spin:
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Interaction in the Segment picture
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Metropolis-Hasting algorithm
Detailed Balance Condition:
Metropolis choice:
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Detailed Balance Condition:
Metropolis choice:
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Phase space
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Phase space for one spin channel
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Update processes
Start configuration:
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Example: Metropolis-Hasting
acceptance probability for add process
Algorithm
Metropolis-Hasting:
Physical problem
Discretisation of configurations:
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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
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Remove process
remove process:
•Decide to remove a segment
•choose a random segment to remove
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Weight prefactors
add the discretisation factor to the weights

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Metropolis-Hasting in the Segment
picture
process
probability
Add segment
Remove segment
Add antisegment
Remove antisegment
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This is beautiful ...
... But some things are not as pretty as they look like!
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Note: half open segments
Remember:
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Quick example: half open segments
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Numerical precision
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Now some results ...
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CT-QMC vs. analytic solution
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Computational limits:
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Summary
Segment picture:
quick and simple
Agreement with
analytic solution
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Outlook
DMFT for the Hubbard model with magnetic Field
Spin up
Spin down
Vollhardt, Ann. Phys, 524:1-19, doi: 10.1002/andp.201100250
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Acknowledgements:
Junya Otsuki
Liviu Chioncel
Michael Sekania
Jaromir Panas
Christian Gramsch
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How do we implement those processes?