Fractionalized Fermi liquid in a frustrated Kondo lattice model

We consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a ${\mathbb{Z}}_2$ spin-liquid and a ferromagnetic phase. The model is amenable to sign free auxiliary-field quantum Monte Carlo simulations. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the ${\mathbb{Z}}_2$ spin-liquid phase. Since our model has an odd number of spins per unit cell, this phase is a non-Fermi liquid that violates the conventional Luttinger theorem which relates the Fermi surface volume to the particle density in a Fermi liquid. This non-Fermi liquid is a specific realization of the so-called fractionalized Fermi liquid proposed in the context of heavy fermions. We probe the violation of the Luttinger sum rule in this non-Fermi liquid phase via conventional observables such as the spectral function, and also by studying the mutual information between the electrons and the spins.

Figure 1

(a) Schematic phase diagram of the BFG model in the absence of Kondo coupling. (b) Schematic phase diagram of the BFG model in the presence of Kondo coupling. The stability of the $\mathrm{FL}$* phase is set by the gap in the BFG model that scales as ${\left(J^{\perp }\right)}^2/J^z$ in the strong coupling limit.

Figure 2

Left: The model—conduction ($c$) electrons hop, with matrix element $t$, between nearest-neighbor sites of the honeycomb lattice denoted by the red and blue circles. The kagome lattice (black) supports impurity spins described by the Balents-Fisher-Girvin model with nearest-neighbor spin-flip $J^{\perp }$ and interactions on hexagons of strength $J^z$ (green). The two systems are Kondo coupled with strength $J_K$ for each bond in the elemental triangles (thick red and blue bonds). For details, see Eq. (1). Right: Various patches $\mathrm{\Gamma }$ used to extract the Renyi mutual information. Subsets (b) and (c) belong to the triangle sequence, (d) and (e) are built out of unit cells.

Figure 3

We consider lattices $L=3\times 3$ and $L=3\times 6$ unit cells at an inverse temperature $\beta =12$ and at $J^z=7.5$ (a) Spin-spin correlations $S_{\text{AFM}}$ (see text), (b) Renyi mutual informations $I_2({\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_f)$ per site of the patch ${\mathrm{\Gamma }}_c\cup {\mathrm{\Gamma }}_f$ for $L=3\times 6$. Here we consider the patches listed in Figs. 2–2. (c) Conduction electron spectral function at the Dirac point $\mathit{K}$ for the $3\times 6$ lattice. (d) Same as (c), but at the $\mathrm{\Gamma }$-point. The imaginary time data from which panels (c) and (d) stem are presented in Appendix pp3.

Figure 4

We consider lattices $L=3\times 3$ and $L=3\times 6$ unit cells at an inverse temperature $\beta =12$ and at $J_K=1$ (a) Spin-spin correlations $S_{\text{AFM}}$ (see text), (b) Renyi mutual information $I_2({\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_f)$ per site of the patch ${\mathrm{\Gamma }}_c\cup {\mathrm{\Gamma }}_f$ for $L=3\times 6$. Here we consider the patches listed in Figs. 2–2. (c) Conduction electron spectral function at the Dirac point $\mathit{K}$ for the $3\times 6$ lattice. (d) same as (c), but at the $\mathrm{\Gamma }$-point. The imaginary time data from which panels (c) and (d) stem are presented in the Appendices.

Figure 5

Cut of the spectrum from $K^{\prime }$ to $K$ with the Fermi energy marked by the dashed, orange line constraint to the half-filled case and Fermi surface (blue, dashed) of a heavy Fermi Liquid state with $\mathrm{\Delta }=0.4$ and $t^{\prime }=0.2$. The shaded area marks the occupied part of the BZ.

Figure 6

The simulations were performed on the $L=3\times 6$ lattice at an inverse temperature of $\beta =12$. Left panels corresponds to the $J_K$ scan at $J^z=7.5$ and the right to the $J^z$ scan at $J_K=1.0$. For large $J_K$ or small $J^z$, we restricted the time domain in (c) and (d) to $\tau <3.5$ and $\tau <3.75$, respectively, since beyond this scale, the data becomes very noisy.