Self-energy corrections to anisotropic Fermi surfaces

The electron-electron interactions affect the low-energy excitations of an electronic system and induce deformations of the Fermi surface. These effects are especially important in anisotropic materials with strong correlations, such as copper-oxide superconductors or ruthenates. Here we analyze the deformations produced by electronic correlations in the Fermi surface of anisotropic two-dimensional systems, treating the regular and singular regions of the Fermi surface on the same footing. Simple analytical expressions are obtained for the corrections, based on local features of the Fermi surface. It is shown that, even for weak local interactions, the behavior of the self-energy is nontrivial, showing a momentum dependence and a self-consistent interplay with the Fermi surface topology. Results are compared to experimental observations and to other theoretical results.

Figure 1

(Color online) Qualitative picture of the evolution of the FS with filling from almost isotropic to convex, going through a FS exhibiting inflexion points, and one with Van Hove singularities (top panel, $t^{\prime }=-0.3t$). A region with almost perfect nesting is shown in the bottom panel $(t^{\prime }=0.3t)$.

Figure 2

Low-order self-energy diagrams. (a) Hartree diagram. (b) Two loop correction.

Figure 3

(Color online) Real and imaginary parts of the self-energy as a function of the energy $\omega$: full red line $-\mathrm{Re}\Sigma$, dashed blue line $\mathrm{Im}\Sigma$

Figure 4

(Color online) Deformations induced by the interactions on the FS of the $t-t^{\prime }$ Hubbard model. The axes are labeled in units of $a^{-1}$, where $a$ is the lattice constant. The black line represents the unperturbed FS while the red (gray) line represents the FS corrected by the interaction. For $t^{\prime }∕t=-0.3$ (a): high doping range and (b) close to half-filling. For $t^{\prime }∕t=+0.3$ (c) close to half-filling.

Figure 5

(Color online) Real part of the self-energy for $t^{\prime }=-0.3t$ represented in the square Brilluoin zone.

Figure 6

(Color online) Left: Band structure of ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ obtained from Eq. (27) using the parameter values given in Ref. 54. The blue (outer) sheet corresponds to the $\alpha$ band, the red (inner) sheet is the $\beta$ band, and the green sheet corresponds to the $\gamma$ band. Right: Corresponding Fermi surface.

Figure 7

(Color online) Fermi velocity of the noninteracting bands: The left graph corresponds to the $\alpha$ band, center graph corresponds to $\beta$ band, and right graph corresponds to $\gamma$ band of ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$.

Figure 8

(Color online) Real part of the self-energy corrections for the three bands of ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ in the first square BZ: The left graph corresponds to the $\alpha$ band, middle graph to the $\beta$ band, and right graph corresponds to $\gamma$ band. We have used the parameter values $U=0.01$ and $\Lambda =1$. Notice the different scales in the three graphs.

Figure 9

(Color online) Bare (continuous blue) and interacting (dashed red) Fermi surface of ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$.