Coherence and metamagnetism in the two-dimensional Kondo lattice model

We report the results of dynamical mean-field calculations for the metallic Kondo lattice model subject to an applied magnetic field. High-quality spectral functions reveal that the picture of rigid, hybridized bands, which are Zeeman shifted in proportion to the field strength, is qualitatively correct. We find evidence of a zero-temperature magnetization plateau whose onset coincides with the chemical potential entering the spin up hybridization gap. The plateau appears at the field scale predicted by a (static) large-$N$ mean-field theory and has a magnetization value consistent with that of $x=1-n_c$ spin-polarized heavy holes, where $n_c<1$ is the conduction band filling of the noninteracting system. We argue that the emergence of the plateau at a low temperature marks the onset of quasiparticle coherence.

Figure 1

The upper and lower hybridized bands $E_{\mathbf{k}}^{\pm }$ are plotted over a trajectory bounding one octant of the Brillouin zone. The gray line width is proportional to the $c$-electron spectral weight. The indirect gap $2\Delta$ and the optical gap $2V$ are indicated. The corresponding $c$-electron density of states ${\rho }_c\left(\omega \right)$ is plotted to the right on the same energy scale. In the magnification, the region around the gap is illustrated with the filled Fermi sea shaded in gray, revealing a small amount of headroom ${\epsilon }_t⪡2\Delta$ in the lower band. In the lower two panels, the small, noninteracting Fermi surface is contrasted to the large interacting one.

Figure 2

(Color online) (Upper panel) As a function of the bare filling $n_c$, the inverse quasiparticle DOS $1/\rho \left({\mu }_f\right)$ and the indirect gap $2\Delta$ are compared to the energy scales defined by the Kondo temperature $T_K$ (with $k_B=1$) and the width ${\epsilon }_t$ of unfilled states at the top of the lower band. (Lower panel) The dashed lines at $2{ϵ}_t$ and $2\left({ϵ}_t+\Delta \right)$ mark the plateau end points that arise from a purely rigid shifting of the bands (in units where ${\mu }_{B}g=1$). The shaded areas between $B^{\prime }$ and $B^{″}$ indicate the extent of the locked region as calculated by using the fully self-consistent mean-field equations. The discrepancy is a result of higher-order changes in the mean-field parameters. We expect that the inclusion of $V$ phase fluctuations restricts plateau formation to between $B^{\prime }$ and $B^{\prime }+{\Delta }^{\prime }$ (roughly) and leads to a slow disintegration of the heavy fermion state between $B^{\prime }+{\Delta }^{\prime }$ and $B^{″}$. The vertical dashed (red) lines between $B^{\prime }$ and $B^{\prime }+{\Delta }^{\prime }$ at $n_c=0.65$ and $n_c=0.85$ appear again as horizontal dashed (red) lines in Fig. 6.

Figure 3

The magnetization of conduction $m_c$ and local moments $m_f$, which are computed via the DMFT, are plotted as a function of magnetic field at various temperatures. In the upper panel, the lines running through the $f$-electron data points are fitted to the forms $\text{tanh}\left(S^{\text{eff}}g{\mu }_{B}B\beta \right)$. At $\beta t=5$, $2S^{\text{eff}}=1.03$ and at $\beta t=10$, $2S^{\text{eff}}=0.65$. In the lower panel, the magnetization profile is marked for the three regions. A sketch of the spin-resolved density of states as deduced from the single particle spectral function (see Fig. 5) is given in each region.

Figure 4

(Color online) The $c$-electron spectral function $A\left(\mathbf{k},\omega \right)$ in zero field is plotted for three progressively lower temperatures. The legend at the top indicates the false color values.

Figure 5

(Color online) Low temperature $\beta t=40$ spectral function in the up (left) and down (right) spin sector as a function of magnetic field. We use the same scale as in Fig. 4.

Figure 6

(Color online) Total magnetization $M=m_c+m_f$ as function of temperature and band filling. The plateaus at $x=1-n_c$, which are predicted by the mean-field theory, are indicated with horizontal dashed (red) lines and correspond to the vertical dashed (red) lines in the lower panel of Fig. 2.