Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

We present a simple trick that allows us to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional diagrammatic Monte Carlo algorithm, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increases only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model.

Figure 1

A twelve-order disconnected contribution for the four-point correlation function. The big arrows represent the external lines, while the points with small arrows represent two-particle interaction vertices. The ellipses represent connected parts. The green ellipse represents all the diagrams contributing to the connected part with four interaction vertices at fixed space-time position. The black ellipses can be considered collectively as arising from $\text{connected}+\text{disconnected}$ diagrams with no external lines.

Figure 2

Double occupancy as a function of truncation order, for $\beta =8$, $U=2$ and $n=0.87500(2)$. The DiagMC number is taken from Ref. [9] and it refers to infinite-order extrapolation of the bold series for $n=0.875$.

Figure 3

Pressure as a function of truncation order, for $\beta =8$, $U=2$, and $n=0.87500(2)$.

Figure 4

$|P_N{U}_s^N|$ as a function of the expansion order $N$, where $U_s=-5.1$ and $P_N$ is the coefficient of the series for the pressure, $P(U)=\sum _{N=0}^{\infty }{P}_N{U}^N$. The blue circle points correspond to positive values of $P_N{U}_S^N$, while the red square point corresponds to a negative value. We see that starting from order $\sim 5$ we are in the high-order asymptotic regime. This implies that the radius of convergence of the series is $R=5.1(1)$.