Diagrammatic Monte Carlo algorithm for the resonant Fermi gas

We provide a description of a diagrammatic Monte Carlo algorithm for the resonant Fermi gas in the normal phase. Details are given on the diagrammatic framework, Monte Carlo moves, and incorporation of ultraviolet asymptotics. Apart from the self-consistent bold scheme, we also describe a non-self-consistent scheme, for which the ultraviolet treatment is more involved.

Figure 1

Examples of diagrammatic topologies from the different sectors of the Monte Carlo configuration space: (a) $\mathrm{\Sigma }$ sector, (b) $\mathrm{\Pi }$ sector, (c) worm sector, (d) normalization sector. The dashed black line is the measuring line, which has the structure of a one-body propagator in the $\mathrm{\Sigma }$ sector, and of a pair propagator in the $\mathrm{\Pi }$ sector. In the worm sector, the worm ends are represented by red dots connected with an extra unphysical thread (dashed red line).

Figure 2

Graphical representation of the complementary pair of updates Create-Delete. Only relevant (i.e., updated) parts of the Feynman diagrams are drawn. An unphysical thread (red dashed line) running from Ira to Masha and carrying momentum $\mathbf{\delta }$ is added to the graph in Create, and is removed in the inverse update. Momentum conservation is maintained when taking into account both the physical propagators and the unphysical thread.

Figure 3

Graphical representation of the self-complementary update Move. In this particular case, Ira is moved from one vertex to another along a $G_{\sigma }$-propagator line. To preserve momentum conservation, the momentum of this line is changed.

Figure 4

Graphical representation of the self-complementary update Shift. The time $\tau$ of a three-point vertex is shifted to ${\tau }^{\prime }$.

Figure 5

Graphical representation of the self-complementary update Reconnect. In this particular example, the two propagators incoming to the Ira and Masha vertices are interchanged. The momentum carried by the unphysical thread is changed from $\mathbf{\delta }$ to $\mathbf{\delta }+\mathbf{p}-\mathbf{q}$.

Figure 6

Graphical representation of the Swap measuring line update.

Figure 7

Graphical representation of the complementary pair of moves Add-Remove. In this particular example, the $G_{\sigma }$ propagator in Eq. (53) corresponds to the spin-$\downarrow$ line carrying momentum ${\mathbf{q}}_{\downarrow }$, and $G_{-\sigma }$ is the spin-$\uparrow$ line with momentum ${\mathbf{q}}_{\uparrow }$.

Figure 8

Graphical representation of the updates Add-Remove loop.

Figure 9

Lowest-order bold self-energy diagram, expressing ${\mathrm{\Sigma }}^{\left(1\right)}$ in terms of $G$ and $\mathrm{\Gamma }$. This diagram contains the dominant contributions to the self-energy at large momentum.

Figure 10

Leading diagrammatic contribution to the momentum distribution $n_{\sigma }\left(\mathbf{k}\right)$ at large $k$. The imaginary time is running from right to left. The single-particle lines propagate forward in time and can be replaced with the vacuum propagators. The pair propagator runs backwards in time and is fully dressed.