Spontaneous interlayer coherence in bilayer Kondo systems

Bilayer Kondo systems present interesting models to illustrate the competition between the Kondo effect and intermoment exchange. Such bilayers can exhibit two sharply distinct Fermi liquid phases that are distinguished by whether or not the local moments participate in the Fermi sea. We study these phases and the evolution from one to the other upon changing Kondo coupling. We argue that an ordered state with spontaneous interlayer phase coherence generically intervenes between the two Fermi liquids. Such a condensate phase breaks a U(1) symmetry and is bounded by a finite-temperature Kosterlitz-Thouless transition. Based on general arguments and mean-field calculations we investigate the phase diagram and associated quantum phase transitions.

Figure 1

Schematic phase diagram of the bilayer Kondo lattice model (see text for details). At $T=0$ the interlayer coherence phase (COH) breaks a U(1) symmetry; it is bounded by a finite-temperature Kosterlitz-Thouless transition. The dashed lines denote crossovers below which the local moments get screened either by the Kondo effect or by interlayer singlet formation. (The in-plane exchange $I$ is assumed to be zero, and magnetic ordering tendencies due to Ruderman-Kittel-Kasuya-Yosida interaction, which occur for small $J_K∕t$, are ignored.)

Figure 2

Mean-field phase diagram, obtained from Eq. (7) on a bilayer square lattice, with parameters of near-neighbor hopping $t=0.6$, Kondo coupling $J_K=1$, conduction band filling ${\rho }_c=0.3$, and a unit cell of one conduction electron site per layer. Thin (thick) lines denote second- (first-) order transitions. Additional transitions within the COH phase, where a Fermi surface sheet (dis)appears (see text and Fig. 3) are not shown. Only the phase boundaries of the COH phase will survive upon inclusion of fluctuations around the saddle point.

Figure 3

Schematic evolution of the Fermi surfaces (in two-dimensional momentum space) along the horizontal axis of Fig. 1, for a conduction band filling ${\rho }_c<1∕2$, and $n_{\mathit{\ell }}=1$. In both the FL-L and FL-S phases there are two identical Fermi surfaces for both layers, with volumes $1+{\rho }_c$ and ${\rho }_c$, respectively. Upon entering the COH phase, this degeneracy is lifted and a spontaneous bilayer splitting develops. Within the COH phase there exist additional transitions, where one Fermi surface sheet (dis)appears; the total Fermi volume (mod 2) is always given by $2{\rho }_c$.