Fermi-polaron problem: Diagrammatic Monte Carlo method for divergent sign-alternating series

We use the diagrammatic Monte Carlo approach to solve the problem of a single spin-down fermion resonantly interacting with a Fermi gas of spin-up particles. Our solution is important for understanding the phase diagram and properties of the crossover from the BCS regime to the Bose-Einstein condensate in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem-tolerant method for exact numerical solution of polaron-type models. This is a characteristic example of how Monte Carlo methods can be used to simulate divergent sign-alternating diagrammatic series.

Figure 1

First-order diagram for ${\Sigma }_{\downarrow }$, consisting of the $T$-matrix propagator $\Gamma$ (heavy dashed line) and the spin-up propagator (solid arc).

Figure 2

First diagram for the molecular self-energy $K$, consisting of one $T$-matrix propagator (heavy dashed line), two spin-up propagators (solid arcs), and two spin-down propagators (solid straight lines).

Figure 3

Molecule energy (at $k_{F}a=1$) as a function of truncation parameter $N_{*}$ for different summation techniques: Cesàro (open squares), Riesz $\delta =2$ (filled circles, fitted with the parabola $y=-2.6164+0.28013x+0.01638x^2$), Riesz $\delta =4$ (open circles, fitted with the parabola $y=-2.6190+0.61635x-0.3515x^2$), Eq. (8) (stars fitted with the horizontal dashed line).

Figure 4

Polaron energy (at the unitarity point $a^{-1}=0$) as a function of truncation parameter $N_{*}$ using Eq. (8). The asymptotic behavior at $1∕N_{*}\to 0$ is perfectly fitted by a straight line.

Figure 5

(Color online) Polaron (black circles) and molecule [red (gray) triangles] energies (in units of ${ϵ}_F$) as functions of $k_{F}a$. The dashed line on the lower panel corresponds to Eq. (9).

Figure 6

(Color online) Polaron (black circles) and molecule [red (gray) triangles] effective mass as functions of $k_{F}a$. The vertical dotted line stands for $(k_{F}a{)}_c=1.11$. The dashed line is the contribution from the first diagram (Ref. 11).