Bold Diagrammatic Monte Carlo Technique: When the Sign Problem Is Welcome

We introduce a Monte Carlo scheme for sampling a bold-line diagrammatic series specifying an unknown function in terms of itself. The range of convergence of this bold(-line) diagrammatic Monte Carlo (BMC) technique is significantly broader than that of a simple iterative scheme for solving integral equations. With the BMC technique, a moderate “sign problem” turns out to be an advantage in terms of the convergence of the process. For an illustrative purpose, we solve the one-particle $s$-scattering problem. As an important application, we obtain the $T$ matrix for a Fermi polaron (one spin-down particle interacting with the spin-up fermionic sea).

Figure 1

Scattering wave function at zero energy (solid line) and scattering potential (dashed line) for the attractive potential well with one bound state ($U_0=-3$).

Figure 2

The diagrammatic equation for the $T$ matrix (heavy dashed line) in terms of the vacuum $T$ matrix (light dashed line), spin-down vacuum propagator (straight solid line), and truncated (to the momenta less than Fermi momentum) spin-up propagator (solid arc).