Controlling Feynman diagrammatic expansions: Physical nature of the pseudogap in the two-dimensional Hubbard model

We introduce a method for summing Feynman's perturbation series based on diagrammatic Monte Carlo that significantly improves its convergence properties. This allows us to investigate in a controllable manner the pseudogap regime of the Hubbard model and to study the nodal/antinodal dichotomy at low doping and intermediate coupling. Marked differences from the weak-coupling scenario are manifest, such as a higher degree of incoherence at the antinodes than at the “hot spots”. Our results show that the pseudogap and reduction of quasiparticle coherence at the antinode is due to antiferromagnetic spin correlations centered around the commensurate $(\pi ,\pi )$ wave vector. In contrast, the dominant source of scattering at the node is associated with incommensurate momentum transfer. Umklapp scattering is found to play a key role in the nodal/antinodal dichotomy.

Figure 1

Left column: Imaginary part of the Hubbard model self-energy at the first four Matsubara frequencies obtained as a sum of the first $K$ perturbation orders. The parameters are $U=4.0, t^{\prime }=-0.3, \mu =0, n\sim 0.725, T=0.5$. The dashed lines are a benchmark from determinantal QMC simulations on a $16\times 16$ lattice (the discrete time interval is $\mathrm{\Delta }\tau =0.0375$, where the Trotter error is negligible). (a) Standard series with Hartree diagrams included in bare Green's function. (b) $\alpha$-shifted case with $\alpha =0.6$. (c) The optimal case $\alpha =1.53$. Right column: Results for the atomic limit $t=0, U=4, T=0.5, \mu =0.138, n\sim 0.725$. (e) Modulus/phase (displayed as saturation/hue) map of the self-energy in the complex $\xi$ plane for $\alpha =0.6$. The physical solution is at the cross $\xi =1$ on the unit circle. (d) The arrows and dots show trajectories of three poles as $\alpha$ is changed from 0.5 to 1.8. (f) Optimal $\alpha$ and corresponding maximal convergence radius as a function of the density $\langle n\rangle$ in the Hubbard atom.

Figure 2

Imaginary part of the self-energy at the node, hot spot, and antinode at $U=5.6, t^{\prime }=-0.3, n=0.96, T=0.2$. Inset: DCA results with cluster size $N_c=8$, 16, 32, 52 extrapolate to the DiagMC-summed result (plain horizontal lines) at different frequencies.

Figure 3

(a) Imaginary part of the Green's functions at node and antinode as a function of frequency, in cases with and without umklapp processes. (b) Decomposition of the self-energy using the Dyson-Schwinger equation of motion [cf. Eq. (8)]. (c) Illustration of a typical weak-coupling umklapp process with $G=(2\pi ,2\pi )$.

Figure 4

Maps of the transfer momentum $\mathbf{q}$ contributions to $\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)+\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i{\omega }_1)$ in the spin (sp), charge (ch), and particle-particle (pp) channels, both at the antinode, $\mathbf{k}=(3.04,0.49)$ (left column), and node, $\mathbf{k}=(1.47,1.47)$ (right column). The two panels on the top are results at perturbation order 2 while the rest are obtained up to order 7.