STAGE Study of strongly correlated fermions via stochastic evaluation of Feynman diagrams

Strongly correlated fermion problems are faced in several areas of physics: condensed
matter, notably high Tc superconductors, but also ultracold atomic gases, nuclear physics, or
QCD. Numerous predictions are being obtained from approximate methods, with
uncontrolled systematic errors. Conventional quantum Monte Carlo methods are exact, but
generally limited to small systems, because the anti-symmetry of the fermionic wave
function leads to a computational time growing exponentially with system volume.
We follow a different approach: Feynman diagrams, formulated directly for infinite volume.
In contrast to usual diagrammatic calculations, we control the series-truncation error by
going to high orders. To this end we develop new Monte Carlo algorithms allowing to
sample diagrammatic series. Often, however, diagrammatic series are divergent. One then
has to give a mathematical meaning to the series, and then to construct a resummation
method capable of transforming the divergent series into a result that converges towards the
exact physical value in the limit of infinite truncation-order. The appropriate resummation
method can be found by studying the large-order asymptotic behavior.
We have recently realized this program for the first time for strongly correlated fermions in
continuous space, obtaining the large-order asymptotics analytically using functionalintegral and saddle-point techniques. Specifically, we studied spin-1/2 non-relativistic
fermions in 3D with contact interactions of infinite scattering length, the so-called unitary
Fermi gas. Apart from being qualitatively relevant for high Tc superconductors and neutron
matter, the unitary gas model accurately describes experiments on ultracold atomic gases
conducted at LKB, which allows for direct theory-experiment comparison.
For the internship we propose to study the properties of the strongly spin-imbalanced Fermi
gas – the so-called Fermi polaron, a single impurity in a Fermi sea – for which new
experimental data are available. Another possibility is to study the large-order asymptotics,
either for the superfluid phase of the unitary Fermi gas, or for the electron gas with
Coulomb interactions, a central problem in material science for which we are also
developing diagrammatic Monte Carlo algorithms.