Lifshitz phase transitions in the ferromagnetic regime of the Kondo lattice model

We establish the low-temperature phase diagrams of the spin-$1/2$ and spin-1 Kondo lattice models as a function of the conduction-band filling $n$ and the exchange coupling strength $J$ in the regime of ferromagnetic effective exchange interactions ($n\lesssim 0.5$, $J/D\lesssim 2$). We show that both models have several distinct ferromagnetic phases separated by continuous Lifshitz transitions of the Fermi-pocket vanishing or emergence type: one of the phases has a true gap in the minority band (half metal with magnetization rigidity), the others only a pseudogap. The spin-$1/2$ model has the half-metal phase, two topologically different pseudogap phases, and a paramagnetic state for very large $J$. The spin-1 model has the half-metal phase and a single pseudogap phase with an electron pocket. We find that, quite generically, ferromagnetism and Kondo screening coexist rather than compete both in spin-$1/2$ and spin-1 models. We establish the hysteretic behavior of the systems in an external magnetic field: spin-flop transitions preempt further Lifshitz transitions at finite magnetic field. We establish a “ferromagnetic Doniach diagram”: in the spin-$1/2$ model the Curie temperature peaks near $J/D\approx 1.2$ and goes to zero, while in the spin-1 model it saturates.

Figure 1

Phase diagrams of spin-$1/2$ and spin-1 Kondo lattice models for $n<0.5$. Phase A is a ferromagnetic half-metal phase with strong Kondo effect where the minority band is gapped. Phases B and B${}^{\prime }$ are itinerant ferromagnetic phases with a pseudogap. Phase C for the spin-1/2 model indicates the region with charge order (Ref. 49). For very small $n$, the calculations fail to converge.

Figure 2

Spin-resolved conduction-band local spectral functions $A_{c,\sigma }$ for the spin-1 KLM in the ferromagnetic half-metal phase (A) and in the itinerant ferromagnetic phase (B). The arrow indicates the main effect of decreasing interaction $J$: the lower edge of the upper hybridized band shifts to lower frequencies. The left insets in both panels show the $f$-level spectral functions $A_{f,\sigma }$ defined through the imaginary part of the scattering $T$ matrix. The right inset in the upper panel shows the spectral functions in the full frequency interval.

Figure 3

Total, conduction-band $c$-level, and localized $f$-level magnetizations (top panels) and the spin-dependent quasiparticle renormalization factors $Z_{\sigma }$ (bottom panels) across the phase transition, indicated by the vertical dashed lines. The magnetization is here defined as the expectation value of the spin operator without the $-g{\mu }_B$ factor: $m_f=\langle S_z\rangle$, $m_c=(n_{\uparrow }-n_{\downarrow })/2$, and $m_{\mathrm{total}}=m_f+m_c$. In the plots, $m_{\mathrm{total}}$ and $m_f$ are shifted by $m_S$ defined in Eq. (7).

Figure 4

Hysteresis loops: magnetization in a longitudinal external magnetic field. The dashed lines indicate the value of the frozen magnetization $m_S$. The $g$ factors are assumed equal for $c$ and $f$ levels, $g_c=g_f=g$. Occupancy is $n=0.3$.

Figure 5

Temperature dependence of the magnetization, resistivity, and heat capacity for the spin-$1/2$ Kondo lattice model in phases A and B. The horizontal axis is rescaled by the Curie temperature $T_C$. Resistivity is in units of ${\rho }_0=2\pi e^2\Phi \left(0\right)/\hslash D$, where $\Phi$ is the transport integral. The heat capacity curve was obtained by differentiating a piecewise interpolation of the numerical results for the total energy.

Figure 6

“Ferromagnetic Doniach diagram” for spin-$1/2$ and spin-1 Kondo lattice models.