Mutual information in heavy-fermion systems

A key notion in heavy-fermion systems is the entanglement between conduction electrons and localized spin degrees of freedom. To study these systems from this point of view, we compute the mutual information in a ferromagnetic and antiferromagnetic Kondo lattice model in the presence of geometrical frustration. Here the interplay between the Kondo effect, the Ruderman-Kittel-Kasuya-Yosida interaction, and geometrical frustration leads to partial Kondo screened, conventional Kondo insulating, and antiferromagnetic phases. In each of these states the mutual information follows an area law, the coefficient of which shows sharp crossovers (on our finite lattices) across phase transitions. Deep in the respective phases, the area law coefficient can be understood in terms of simple direct product wave functions thereby yielding an accurate measure of the entanglement in each phase. The above-mentioned results stem from approximation-free auxiliary field quantum Monte Carlo simulations.

Figure 1

(a) Ground-state phase diagram with antiferromagnetic (AFM), out-of-plane PKS $(z$-PKS), spin-rotation symmetry breaking PKS $(xyz$-PKS), and KI phases from QMC simulations [22]. Dashed lines connects transition points and the dotted line sketches the expected boundary between the KI and $z$-PKS phases. The three thick lines indicate the scans of the phase diagram considered here. (b) Mean-field schematic picture of $S=1/2$ and $S=1$ AFM, KI, and $z$-PKS phases.

Figure 2

Coefficient $\alpha$ of the area law for the mutual information along the line (1) on Fig. 1 and also for $J_{\text{K}}\le -1,J_z=0$. We consider two inverse temperatures $\beta =40$, 60, and for four values of the minimum subsystem size taken into account in the fits.

Figure 3

Coefficient $\alpha$ of the area law for the mutual information along line (2) (above) and line (3) (below) on Fig. 1, for inverse temperatures $\beta =30$, 40, and three different fits (see main text).

Figure 4

Partition of lattice sites used to compute the mutual information $I_2({\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_S)$. For each illustrated subset of sites, ${\mathrm{\Gamma }}_c$ is the corresponding set of conduction electron sites, and ${\mathrm{\Gamma }}_S$ the set of localized spin sites coupled to ${\mathrm{\Gamma }}_c$. When $J_z>0$, due to the enlarged unit cell we have computed $I_2({\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_S)$ using only subsystems no. 3, no. 7, and no. 9.

Figure 5

Mutual information $I_2({\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_S)$ at $J_{\text{K}}=2.7,J_z=0,\beta =60$ for the choices of ${\mathrm{\Gamma }}_c,{\mathrm{\Gamma }}_S$ shown in Fig. 4, as a function of the size $N$ of ${\mathrm{\Gamma }}_c$ and ${\mathrm{\Gamma }}_S$. We compare with a linear fit to the right-hand side of Eq. (3), for different choices of the minimum subsystem size $N_{\text{min}}$ taken into account.

Figure 6

Same as Fig. 5 for $J_{\text{K}}=1.8,J_z=1.8,\beta =40$, in the $z-\mathrm{PKS}$ phase, and for subsystems no. 3, no. 7, and no. 9, shown in Fig. 4. We compare with three linear fits, obtained by using all data $\left(N_{\text{min}}=6\right)$, disregarding the smallest size $\left(N_{\text{min}}=13\right)$, and disregarding the largest size $\left(N=6,13\right)$.

Figure 7

Coefficient $\alpha$ of the mutual information area law for the wave function ansatz of Eq. (6).