Heavy fermions in an optical lattice

We employ a mean-field theory to study ground-state properties and transport of a two-dimensional gas of ultracold alkaline-earth-metal atoms governed by the Kondo lattice Hamiltonian plus a parabolic confining potential. In a homogenous system, this mean-field theory is believed to give a qualitatively correct description of heavy-fermion metals and Kondo insulators: It reproduces the Kondo-like scaling of the quasiparticle mass in the former and the same scaling of the excitation gap in the latter. In order to understand ground-state properties in a trap, we extend this mean-field theory via local-density approximation. We find that the Kondo insulator gap manifests as a shell structure in the trapped density profile. In addition, a strong signature of the large Fermi surface expected for heavy-fermion systems survives the confinement and could be probed in time-of-flight experiments. From a full self-consistent diagonalization of the mean-field theory, we are able to study dynamics in the trap. We find that the mass enhancement of quasiparticle excitations in the heavy-Fermi liquid phase manifests as slowing of the dipole oscillations that result from a sudden displacement of the trap center.

Figure 1

(a) In the KLM, the conduction electrons (red, upper lattice) can hop from site to site, and they interact with localized spins (blue, lower lattice) via a Heisenberg exchange. (b) Two-dimensional mean-field ground-state phase diagram as constructed in Ref. [19], $n_g$ being the conduction electron density and $v$ being a dimensionless measure of the interaction strength. PM is a paramagnetic phase, in which the heavy-Fermi liquid (HFL) behavior is expected. The FM (AFM) phase is where RKKY interactions [20, 21, 22] generate ferromagnetic (antiferromagnetic) order among the localized spins. Exactly at $n_g=1$, the AFM phase gives way to a nonmagnetic insulating state for sufficiently large $v$ [23].

Figure 2

Spectrum of the translationally invariant MFT for (a) $n_g<1$ and for (b) $n_g=1$, shown in 1D for clarity. The blue dashed line is the $e$ atom band ($E={\mu }_e$), and the red solid curve is the $g$ atom band [$E=-2J_g\cos (k)$]. The purple dotted lines are the hybridized bands ${\mathcal{E}}_{\pm }$, with the thicker solid section covering the Fermi volume. In (b), the lower band is filled and separated from the upper band by the hybridization gap ${\Delta }_{\mathrm{H}}$ (green slashed region), causing the system to be insulating for $n_g=1$.

Figure 3

Schematic ground-state mean-field phase diagram of the KLM as a function of chemical potential ${\mu }_g$ and dimensionless coupling $v$. The black dotted lines are given by ${\mu }_g=\pm {\Delta }_{\mathrm{H}}/2$, which crosses over from Kondo-like scaling (${\Delta }_{\mathrm{H}}~J_g{e}^{-1/v}$) at small $v$ to linear scaling (${\Delta }_{\mathrm{H}}=J_{g}v$) at large $v$.

Figure 4

(a) The formation of a Kondo plateau between $r_1$ and $r_2$ at high trap fillings, obtained within LDA. (b) The hybridization also displays the plateau, but as the density increases past $n_g=1$, the hybridization goes back down. In both (a) and (b), the parameters used are $N_g\approx 550$, $\Omega /J_g=45/1000$, and $v=8$.

Figure 5

(a) The first Brillouin zone of a 2D square lattice. The central white region (within the dotted circle) is the small Fermi sea of the translationally invariant KLM with a conduction atom density $n_g\approx 0.19$. The shaded green region (within the solid red line) fills out the large Fermi sea (the red solid line is the large Fermi surface, as defined in Sec. 4b). The red dotted line (just outside of the red solid line) is the large Fermi surface corresponding to a uniform filling equal to the density at the center of the trap. Even when there are no conduction atoms, the large Fermi sea occupies half of the zone (within the dotted diamond), since there is still one localized atom per site contributing to the volume. (b) Quasimomentum distribution of the conduction atoms plotted over one quadrant of the Brillouin zone, showing a prominent feature at the large Fermi surface (red ribbon). (c) A 1D cut along the line $k_y=k_x$ in (b), ${\mathrm{ñ}}_{\mathrm{LDA}}[k_x,k_x]$ is the blue solid line, and the large Fermi surface is the red dotted line. For all plots, the parameters used are $N_g\approx 116$, $\Omega /J_g=5/1000$, and $v=8$.

Figure 6

(a) Here we plot the time evolution of the c.m. after displacement of the trap by one lattice site (red solid line). The blue dashed line is the noninteracting dynamics for comparison. The parameters used in this plot are $q=235,v=4$, $L_t=8,N_g=40$, and $J_l/J_t=2$. (b) The effective mass scaling is $\mathrm{m̃}/m^{*}~(\mathrm{\tau ̃}/{\tau }^{*}){}^2$ (see the vertical axis). By extracting the period of oscillation from several traces [such as the one in (a)] at different values of $v$, we find significant enhancement of the effective mass as $v\rightsquigarrow 1$. The red circles are from self-consistent diagonalization of the MFT, and the blue line is a guide to the eye.

Figure 7

c.m. oscillations for decreasing values of $v$, showing how the fast free-particle dynamics emerge on top of the slow quasiparticle dynamics. The red solid line is from MFT dynamics, and the blue dotted line is the noninteracting solution. The parameters used in this plot are $q=235$, $L_t=1$, $N_g=6$, and $J_l/J_t=2$.