Interaction-driven Lifshitz transition with dipolar fermions in optical lattices

Anisotropic dipole-dipole interactions between ultracold dipolar fermions break the symmetry of the Fermi surface and thereby deform it. Here we demonstrate that such a Fermi surface deformation induces a topological phase transition—the so-called Lifshitz transition—in the regime accessible to present-day experiments. We describe the impact of the Lifshitz transition on observable quantities such as the Fermi surface topology, the density-density correlation function, and the excitation spectrum of the system. The Lifshitz transition in ultracold atoms can be controlled by tuning the dipole orientation and, in contrast to the transition studied in crystalline solids, is completely interaction driven.

Figure 1

Single-component dipolar fermions on a square optical lattice. Due to the symmetry, the orientation of the dipoles is given by two angles, $\varphi \in [0,0.25\pi ]$ and $\theta \in [0,0.50\pi ]$.

Figure 2

Fermi surfaces. The red curve corresponds to $\varphi =0.25\pi$, where the Fermi surface is independent of the angle $\theta$. At $\varphi =0,\theta =0.5\pi$ (blue line), the Fermi surface is deformed anisotropically due to the DDI. This breaks the symmetry between the $X$ and $Y$ points in the Brillouin zone. In the $k_x$ direction, the Fermi surfaces of neighboring Brillouin zones are connected and the VHS (black dots) is enclosed by the Fermi surface. In the $k_y$ direction, the Fermi surfaces are not connected and the VHS is outside of the Fermi surface.

Figure 3

Spectral function at $\varphi =0.25\pi$ (left) and $\varphi =0$ (right), both for $\theta =0.50\pi$. The sum of the spectral function along the $\mathrm{\Gamma }\text{−}X\text{−}M$ and $\mathrm{\Gamma }\text{−}Y\text{−}M$ paths (see Fig. 2) is shown. At $\varphi =0.25\pi$, the spectral functions along these paths are identical, whereas at $\varphi =0$ the $X$ and $Y$ points are distinguishable due to the anisotropic interaction. This corresponds to the splitting into two bands.

Figure 4

Static density-density correlation function in momentum space. The dipolar angles correspond to those of Fig. 1.

Figure 5

(a) $q_x=0$ cross section of the static susceptibility. The arrows indicate the momenta corresponding to the transitions illustrated on the right. (b) Fermi surface (only the top right quadrant of the Brillouin zone is shown), the green Fermi surface corresponds to Fig. 2. Different lines correspond to the densities $n=0.37,0.38,0.40,0.42$, and $0.44$. The dipole orientation is fixed to $\varphi =0,\theta =0.5\pi$.

Figure 6

Feynman diagrams employed in the dual boson approach. Diagram (a) renormalizes the fermion propagator and diagram (b) renormalizes the susceptibility. The Fermi surface deformation occurs due to the anisotropy of diagram (a), coming from the anisotropic DDI.

Figure 7

For a noninteracting, half-filled system, the red part of the Brillouin zone lies within the Fermi surface. The black and gray dots denote the Van Hove singularities (stationary points and end points of the dispersion respectively), and we define the points $\mathrm{\Gamma }=(0,0),X=(\pi ,0),Y=(0,\pi )$, and $M=(\pi ,\pi )$.