Sound velocity and dimensional crossover in a superfluid Fermi gas in an optical lattice

We study the sound velocity in cubic and noncubic three-dimensional optical lattices. We show how the van Hove singularity of the free Fermi gas is smoothed by interactions and eventually vanishes when interactions are strong enough. For noncubic lattices, we show that the speed of sound (Bogoliubov-Anderson phonon) shows clear signatures of dimensional crossover both in the one- and two-dimensional limits.

Figure 1

The Anderson-Bogoliubov phonon in a cubic lattice. The $x$ axis is the $x$ component of the momentum vector in the units of inverse lattice spacing, the $y$ axis is energy in the units of the pairing gap divided by $\hslash$, and the $z$ axis shows the dynamical structure factor from Eq. (28), in arbitrary units.

Figure 2

The dispersion of the Anderson-Bogoliubov phonon. The $x$ axis is the $x$ component of the momentum vector in the units of inverse lattice spacing and the $y$ axis is the energy in the units of the pairing gap divided by $\hslash$.

Figure 3

Pair size compared with the system size. The $x$ axis is the number of the lattice sites in the $x$ direction and the $y$ axis is the absolute value of the scattering length, in Bohr radii. For the density response to be experimentally observable, the coherence length, i.e., the size of a Cooper pair, must be smaller than the system. The linear sound dispersion is clearly visible above the solid line.

Figure 4

The speed of sound as a function of the lattice filling fraction, for the noninteracting Fermi gas (top curve) and for interacting Fermi gases with different scattering lenghts: (in Bohr radii, from top to bottom) $-750$, $-1000$, $-1200$, $-1500$, and $-2000$. The speed has a local minimum at a point where $E_F=4J$, but this minimum is smoothed out as interactions become stronger.

Figure 5

The Fermi surface of the lattice with different densities, as seen from the $z$ direction. The axis are the $x$ and $y$ components of the momentum vector. The surfaces above were plotted with filling fractions $0.2$, $0.44$, $0.47$, and $0.8$, from left to right. Color coding: white corresponds to the Fermi surface reaching the edge of the Brillouin zone in the $z$ direction, whereas the black area is outside the surface. Note that the presence of the excitation gap spreads the Fermi surface, wheras for a noninteracting system, the edges of the Brillouin zone are not reached until the filling fraction exceeds 0.42.

Figure 6

The speed of sound as a function of the scattering length $a$. Here $d$ is the lattice spacing, the filling fraction is $0.5$, and the lattice is cubic, with $s_x,s_y,s_z=2.5$.

Figure 7

The energy gap as a function of the filling fraction with $s_x=s_y<s_z$, corresponding to quasi-two-dimensional pancakes along the $xy$ plane. The different curves are for different ratios of $s_x∕s_z$: (from top to bottom) $0.5$, $0.62$, $0.71$, $0.83$, and $1.0$ (cubic case). There is no qualitative difference in the different curves. Here the scattering length $a$ is all the time $-1000a_0$.

Figure 8

The speed of sound as a function of the lattice filling fraction in noncubic lattices with $s_x=s_y<s_z$, corresponding to quasi-two-dimensional pancakes along the $xy$ plane. The left figure shows the speed in the plane, i.e., parallel to the pancakes, and the one on the right shows the speed orthogonal to the them, i.e., along the $z$ axis. The different curves are for different ratios of $s_x∕s_z$: (from top to bottom) $1.0$ (cubic case), $0.83$, $0.71$, $0.62$, and $0.5$. Here the scattering length $a$ is all the time $-1000a_0$.

Figure 9

The speed of sound as a function of the lattice filling fraction in noncubic lattices with $s_x=s_y>s_z$, corresponding to quasi-one-dimensional tubes along the $z$ axis. The figure on the left shows the speed in the $z$ direction, i.e., parallel to the tubes, and the right one the speed orthogonal to them, i.e., in the $xy$ plane. The different curves are for different ratios of $s_x∕s_z$: (from top to bottom) $1.0$ (cubic case), $1.2$, $1.4$, $1.6$, $1.8$, and $2.0$. Here the background scattering length $a$ is all the time $-1000a_0$.