Bold diagrammatic Monte Carlo: A generic sign-problem tolerant technique for polaron models and possibly interacting many-body problems

We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy, which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign alternating and divergent, one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series resummation techniques and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e., making the entire scheme self-consistent. Our solution is important for understanding the phase diagram and properties of the Bardeen-Cooper-Schrieffer–Bose-Einstein Condensation crossover 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 and, possibly, of the interacting many-body Hamiltonians.

Figure 1

Dyson equation for the single-particle Green’s function.

Figure 2

Defining functions $Q$ and $\tilde{K}$ in the two-particle channel.

Figure 3

The backbone of the cyclic diagram.

Figure 4

Forward connection. The arc represents $-G_{\uparrow }(\tau ={\tau }_c+{\tau }_d+{\tau }_e)$.

Figure 5

Backward connection. The pair of $\Gamma$ ends being connected is precisely the same as in Fig. 4, but the direction is opposite, and the arc represents $-G_{\uparrow }(\tau =-{\tau }_f-{\tau }_a-{\tau }_b)$.

Figure 6

A diagram with two worms, $\mathcal{I}$ and $\mathcal{M}$.

Figure 7

“Normalization” diagram. It is the simplest diagram with the worm; its weight is a product of $G_{\downarrow }^{\left(0\right)}\left({\tau }_a\right)$ and $\Gamma \left({\tau }_b\right)$.

Figure 8

A $G\Sigma$ diagram contributing to the polaron self-energy. It factorizes into a product of $G_{\downarrow }^{\left(0\right)}\left({\tau }_a\right)$ and $\Sigma (\tau ={\tau }_b+{\tau }_c+{\tau }_d+{\tau }_e+{\tau }_f)$.

Figure 9

A $\Gamma \tilde{K}$ diagram contributing to the molecule self-energy. It factorizes into a product of $\Gamma \left({\tau }_b\right)$ and $\tilde{K}(\tau ={\tau }_c+{\tau }_d+{\tau }_e)$. The vertical dashed lines cut (for the sake of better visual perception) the same $\Gamma \left({\tau }_b\right)$ line.

Figure 10

Diagrammatic equation for the $T$ matrix $\Gamma (\tau ,p)$. The arc is the vacuum spin-up propagator with the constraint $q<k_F$ on its momentum $q$. The meaning of this equation is nothing but correcting the vacuum result ${\Gamma }_0$ by subtracting contributions from the spin-up fermions with momenta $q<k_F$.

Figure 11

The molecule energy (at $k_{F}a=1$) as a function of the maximum diagram order $N_{*}$ 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 $y=-2.6190+0.61635x-0.3515x^2$), and Eq. (61) (stars fitted with the horizontal dashed line). Reproduced from Ref. 14.

Figure 12

(Color online) The polaron energy (at the unitarity point $a^{-1}=0$) as a function of the maximum diagram order $N_{*}$ using Eq. (61). The data are fitted using linear $-0.618+0.033∕N_{*}$ (red) and exponential $-0.6151+0.026e^{-0.39N_{*}}$ (black) functions to have an estimate of systematic errors introduced by the extrapolation procedure.

Figure 13

(Color online) Polaron (black circles) and molecule (red triangles) energies (in units of ${\epsilon }_F$) as functions of $k_{F}a$. The dashed line on the lower panel corresponds to Eq. (62). Reproduced from Ref. 14.

Figure 14

(Color online) Polaron (black circles) and molecule (red triangles) effective masses as functions of $k_{F}a$. The vertical dotted line stands for ${\left(k_{F}a\right)}_c=1.11$. The dashed line is the contribution from the first diagram (Ref. 39). Reproduced from Ref. 14.

Figure 15

(Color online) Polaron energy as a function of the maximum diagram order $N_{*}$ at the unitary point $k_{F}a=\infty$ within the bold-line approach. Black squares show results when the bold-line approach was implemented only for the polaron propagator. When both polaron and molecule propagators in diagrams are given by $G_{\downarrow }$ and $Q$, one obtains results shown by red circles.