In computational solid state physics, Continuous-time quantum Monte Carlo (CT-QMC) is a family of stochastic algorithms for solving the Anderson impurity model at finite temperature.
[1][2][3][4][5] These methods first expand the full partition function as a series of Feynman diagrams, employ Wick's theorem to group diagrams into determinants, and finally use Markov chain Monte Carlo to stochastically sum up the resulting series.
To distinguish it from other Monte Carlo methods for such systems that also work in continuous time, the method is then usually referred to as Diagrammatic determinantal quantum Monte Carlo (DDQMC or DDMC).
collects the spin index and possibly other quantum numbers such as orbital (in the case of a multi-orbital impurity) and cluster site (in the case of multi-site impurity).
Common choices are: Step 2 is to switch to the interaction picture and expand the partition function in terms of a Dyson series: where
The presence of a (zero-dimensional) lattice regularises the series and the finite size and temperature of the system makes renormalisation unnecessary.
[2] The Dyson series generates a factorial number of identical diagrams per order, which makes sampling more difficult and possibly worsen the sign problem.
Thus, as step 3, one uses Wick's theorem to group identical diagrams into determinants.
This leads to the expressions:[1] In a final step, one notes that this is nothing but an integral over a large domain and performs it using a Monte Carlo method, usually the Metropolis–Hastings algorithm.