Determinant Monte Carlo for irreducible Feynman diagrams in the strongly correlated regime

We develop a numerically exact method for the summation of irreducible Feynman diagrams for fermionic self-energy in the thermodynamic limit. The technique, based on the diagrammatic determinant Monte Carlo and its recent extension to connected diagrams, allows us to reach high $(\sim 10)$ orders of the weak-coupling expansion for the self-energy of the two-dimensional Hubbard model. Access to high orders reveals a nontrivial analytic structure of the self-energy and enables its controlled reconstruction with arbitrary momentum resolution in the nonperturbative regime of essentially strong correlations, which has recently been reached with ultracold atoms in optical lattices.

Figure 1

Partial sum ${\mathrm{\Sigma }}_{\mathrm{k}\sigma }^{\left(n\right)}$ of the original expansion (7) at $\xi =1$ (circles) and that in terms of the transformed variable $w\left(\xi \right)$ (squares) at $T=0.2t,U=7t,\mu =2t,\omega ={\omega }_0,\mathbf{k}=(\pi /8,\pi )$ as a function of inverse truncation order. The dashed line is an extrapolation by the series for $g(w\left(1\right))$, Eq. (9). The horizontal bands are the claimed result, Fig. 3.

Figure 2

The ratio of successive coefficients of the original, $a_{n-1}/a_n$ (circles), and transformed by the map $w\left(\xi \right),b_{n-1}/b_n$ (squares), series for the parameters of Fig. 1. The first singularity is observed at $U{\xi }_{s_1}\approx -5$ and the second one at $w_{s_2}\approx 0.27+i0.03$ (corresponding to $U{\xi }_{s_2}\approx -9)$, while $w(\xi =1)=0.23$.

Figure 3

Evaluation of ${\mathrm{\Sigma }}_{\mathrm{k}\sigma }$ for the parameters of Fig. 1 using the IA method (9) applied to the series (7) (labeled IA) and the shifted series for $\mathrm{\Sigma }/{\xi }^2$ (sIA) for various choices of $[L,M,N]$. The horizontal bands show the claimed result with the error bar.

Figure 4

(a) ${\mathrm{\Sigma }}_{\mathbf{k}\sigma }$ at $\omega ={\omega }_0$. Inset: ${\mathrm{\Sigma }}_{\mathbf{k}\sigma }\left(i{\omega }_n\right)$ at $\mathbf{k}=(\pi /2,\pi /2),(\pi /8,\pi )$. (b) The corresponding momentum distribution $n\left(\mathbf{k}\right)$ at $T=0.2t,U=7t,\mu =2t$ $[n=0.950(6\left)\right]$ and $n_0\left(\mathbf{k}\right)$ of the ideal Fermi gas, $U=0$.