Heavy-Fermion Valence-Bond Liquids in Ultracold Atoms: Cooperation of the Kondo Effect and Geometric Frustration

We analyze a microscopic mechanism behind the coexistence of a heavy Fermi liquid and geometric frustration in Kondo lattices. We consider a geometrically frustrated periodic Anderson model and demonstrate how orbital fluctuations lead to a Kondo-screened phase in the limit of extreme strong frustration when only local singlet states participate in the low-energy physics. We also propose a setup to realize and study this exotic state with $SU(3)$-symmetric alkaline-earth cold atoms.

Figure 1

(a) The triangular tube lattice. Black and blue circles denote local ($d_{ia\sigma }^{†}$) and itinerant ($c_{ka\sigma }^{†}$) electrons. Grey ellipses are VB singlets (2) (empty circles indicate holes). Itinerant fermions propagate (by hopping between triangles with an amplitude $t$) in the leads with momentum $k$. (b) Valence fluctuations away from the two-electron singlet GS of a triangle, leading to VB flips. (c) Schematic plot of the VB delocalization and heavy-fermion formation due to Kondo screening. Shaded regions are Fermi surfaces in the Brillouin zone. (d) Magic optical lattice that implements the frustrated KL model (3) [band 1 (2) is localized (itinerant)]. Red and blue circles are AEAs in ${{}^{3}P}_0$ ($e$) and ${{}^{1}S}_0$ ($g$) clock states. Grey ellipses show local $e-g$ entangled states (with energy $-V_{eg}$). $V_{eg}^{\prime }$ and $V_{gg}$ are the interband exchange interactions ($V_{eg}\gg V_{eg}^{\prime },|V_{gg}|$). (e) Mapping from VB singlets on a triangle to lowest-energy $e-g$ pair states for $SU(3)$ AEAs. (f) Two different scattering lengths for spin symmetric $|t⟩$ and antisymmetric $|s⟩$ channels. The three levels indicate nuclear-spin states for each atom (the black circle marks a populated state).

Figure 2

$T=0$ phase diagram of the KL model, Eq. (3), with $N=4900$ sites and electron density $n^c=0.8$. $m_{\text{tot}}=[(m_3^c+m_3^f{)}^2+(m_8^c+m_8^f{)}^2{]}^{1/2}$ plays a role of the total magnetization. The black circle at $V_{\parallel }/t\sim 0.76$ marks the first order transition between Kondo-screened and normal phases. Inset: The HF band structure. The band splitting at the Fermi level is $\sim V{\chi }_{1,2}$. The table shows symmetries broken by different OPs and corresponding VB liquid phases.

Figure 3

Generic $T=0$ phase diagram of Eq. (3) [or Eq. (7) with $U_{gg}=0$] with $N=3600$ sites and $n^c=0.8$. All phase transitions are first order. Inside the nematic phase there is a “metamagnetic” transition between states with $m_1^{c,f}\ne 0$, $m_{3,8}^{c,f}=0$ (at smaller $V_{\parallel }$) and $m_1^{c,f}=0$, $m_{3,8}^{c,f}\ne 0$ (for larger $V_{\parallel }$) which are separated by a continuation of the dashed line. Inset: Noninteracting itinerant DOS $g(\epsilon )=(1/8{\pi }^{2}t)K[\sqrt{1-(\epsilon /4t{)}^2}]$ [$K(x)$ is an elliptic integral of the first kind] used to compute the phase diagram.