Second-order interaction corrections to the Fermi surface and the quasiparticle properties of dipolar fermions in three dimensions

We calculate the renormalized Fermi surface and the quasiparticle properties in the Fermi liquid phase of three-dimensional dipolar fermions to second order in the dipole-dipole interaction. Using parameters relevant to an ultracold gas of erbium atoms, we find that the second-order corrections typically renormalize the Hartree-Fock results by less than 1%. On the other hand, if we use the second-order correction to the compressibility to estimate the regime of stability of the system, the point of instability is already reached for a significantly smaller interaction strength than in the Hartree-Fock approximation.

Figure 1

For the explicit evaluation of the self-energy we choose our coordinate system such that the $z$ axis points in the direction $\widehat{\mathit{d}}$ of the dipoles and the $x$ axis lies in the plane spanned by $\widehat{\mathit{d}}$ and $\widehat{\mathit{k}}$. We call $\widehat{\mathit{d}}·\widehat{\mathit{k}}=cos\alpha ,{\mathit{k}}_{\perp }=sin\alpha \widehat{\mathit{x}}$, and parametrize the integration vector $\mathit{p}$ in Eq. (12) as $\mathit{p}=p[cos\theta \widehat{\mathit{d}}+sin\theta (cos\phi \widehat{\mathit{x}}+sin\phi \widehat{\mathit{y}})]$.

Figure 2

Relevant Feynman diagrams: (a) first-order Hartree-Fock diagrams; (b) second-order diagrams generated by the self-consistent Hartree-Fock approximation; (c) additional (frequency dependent) second-order diagrams. Here solid lines denote the bare propagator while wavy lines denote the dipole-dipole interaction.

Figure 3

Fermi surface of dipolar fermions for $u=1.5$ to first order (solid green), to second order in the self-consistent Hartree-Fock approximation (dashed blue), and to full second order (dot-dashed red) in the interaction. The spherical Fermi surface of the noninteracting system is given as a reference (solid black line). Note that the deformed Fermi surface still has the azimuthal symmetry around the $z$ axis.

Figure 4

Aspect ratio $k_F\left(0\right)/k_F\left(\frac{\pi }{2}\right)$ of the deformed Fermi surface for values of $u$ which have recently been reached experimentally [5]. The lines correspond to our results to first order (solid green), to second order in the self-consistent Hartree-Fock approximation (dashed blue), and to full second order (dot-dashed red) in the interaction. While the second-order corrections taken into account in the self-consistent Hartree-Fock approximation have practically no effect, the full second-order calculation changes the first-order result for the aspect ratio by about 1 per mille which is of the same order of magnitude as the experimental uncertainty [5].

Figure 5

Quasiparticle residue $Z_{{\mathit{k}}_F}$ of dipolar fermions to second order in the interaction; see Eq. (47). In the upper panel we show $Z_{{\mathit{k}}_F}$ as a function of the dimensionless interaction $u=8\pi \nu d^2/3$ and the angle $\alpha$ between ${\mathit{k}}_F$ and $\widehat{\mathit{d}}$. In the lower panel we have fixed $u=0.15$ and show the angular dependence of the quasiparticle residue. Note that the value $u=0.15$ lies in the currently accessible experimental range [5].

Figure 6

Modulus of the renormalized Fermi velocity ${\mathit{v}}_{{\mathit{k}}_F}$ in units of the bare Fermi velocity to second order in the dimensionless interaction $u$; see Eq. (48). The upper picture shows the behavior for different interaction strengths, while the lower picture shows the velocity for fixed $u=0.15$ as a function of the angle $\alpha$ between ${\mathit{k}}_F$ and $\widehat{\mathit{d}}$.

Figure 7

Spectral function $\rho (\mathit{k},\omega )$ for $u=0.15$ obtained by inserting our numerical results of the second-order self-energy into Eq. (40). The upper pictures show the spectral function for excitations with wave vectors below the renormalized Fermi surface, while the lower pictures give the spectrum for excitations with wave vectors above the renormalized Fermi surface. Note that in the latter case the spectral line shape exhibits a much stronger angular dependence.