Résumé
Diagrammatic Monte Carlo (DiagMC) is a versatile numerical technique capable of solving strongly correlated fermion systems in a controlled and accurate way. The core idea behind the technique is to sum all connected Feynman diagrams in a systematic way up to high order. In this talk, I will present some key results that have been obtained for the BEC-BCS crossover and the attractive Hubbard model. These results are directly relevant to experiments on ultra-cold fermionic atoms.
In particular, I will discuss the unitary Fermi gas, a model of spin-1/2 fermions in three-dimensional continuous space and a prototypical example of a strongly correlated fermionic system. Despite the fact that the diagrammatic series has a vanishing radius of convergence, key properties such as the equation of state and the contact can still be calculated in an unbiased and accurate way [1,2,3]. When the Fermi gas is highly spin-polarised, the problem reduces to the so-called Fermi polaron problem: a single mobile impurity immersed in an ideal Fermi sea. I will discuss how the connected determinant (CDet) algorithm simplifies in this case and enables the polaron properties to be calculated with unprecedented accuracy [4]. Finally I will present results for the 3D attractive Hubbard model in the superfluid phase obtained with a CDet algorithm that sums diagrams starting from a BCS hamiltonian [5]. Here, we observe convergence of the diagrammatic series. Moreover, our study includes the polarized regime, where conventional quantum Monte Carlo methods suffer from the fermion sign problem. In the limit of high spin polarization the model corresponds to a Fermi polaron on the lattice, for which we recently found that the polaron-dimeron transition at zero temperature disappears at some critical value of the filling fraction [6].
