Fine structure of spectra in the antiferromagnetic phase of the Kondo lattice model

We study the antiferromagnetic phase of the Kondo lattice model on bipartite lattices at half-filling using the dynamical mean-field theory with numerical renormalization group as the impurity solver, focusing on the detailed structure of the spectral function, self-energy, and optical conductivity. We discuss the deviations from the simple hybridization picture, which adequately describes the overall band structure of the system (four quasiparticle branches in the reduced Brillouin zone), but neglects all effects of the inelastic-scattering processes. These lead to additional structure inside the bands, in particular asymmetric resonances or dips that become more pronounced in the strong-coupling regime close to the antiferromagnet-paramagnetic Kondo insulator quantum phase transition. These features, which we name “spin resonances,” appear generically in all models where the $f$-orbital electrons are itinerant (large Fermi surface) and there is Néel antiferromagnetic order (staggered magnetization), such as periodic Anderson model and Kondo lattice model with antiferromagnetic Kondo coupling, but are absent in antiferromagnetic phases with localized $f$-orbital electrons (small Fermi surface), such as the Kondo lattice model with ferromagnetic Kondo coupling. The origin of the spin resonances is in the shifts of the resonance in the self-energy function in an order-parameter dependent way. We show that with increasing temperature and external magnetic-field the spin resonances become suppressed at the same time as the staggered magnetization is reduced. The optical conductivity $\sigma \left(\Omega \right)$ has a threshold associated with the indirect gap, followed by a plateau of low conductivity and the main peak associated with the direct gap, while the spin resonances are reflected as a secondary peak or a hump close to the main optical peak. This work demonstrates the utility of high-spectral-resolution impurity solvers to study the dynamical properties of strongly correlated fermion systems.

Figure 1

Local spectral functions $A_{\sigma }\left(\omega \right)$ (top row) computed for self-energies ${\Sigma }_{\sigma }\left(\omega \right)$ (bottom row) approximated by a single pole on or near the real axis for bipartite lattice that allows AFM order. In the Kondo insulator, there is no spin splitting and the pole is located in the center of the gap at $\omega =0$ on the real axis. In the antiferromagnetic state with “quasilocal compensation,” the poles are located symmetrically on the real axis at $\omega =\pm H$ and the real parts of $\Sigma$ are additionally shifted by $\pm h=\mp H.$ In generic Kondo antiferromagnet, the values of $h$ and $-H$ are not exactly the same and the poles are shifted away from the real axis (i.e., $\mathrm{Im}\Sigma$ is nonzero).

Figure 2

Multiple manifestations of the “spin resonance” (fine structure inside the bands) in the dynamical properties of the Kondo lattice model in the antiferromagnetic phase. (a) Peaks and dips in the spin-resolved local spectral function $A_{\sigma }\left(\omega \right)$ of the conduction band. (b) Peaks in the spin-averaged local spectral function $A\left(\omega \right)=A_{\uparrow }\left(\omega \right)+A_{\downarrow }\left(\omega \right).$ (c) Momentum-resolved spectral function $A(\epsilon ,\omega )$ and close-ups on the regions at $\epsilon =0$ showing (e) an enhanced density of states in the empty band (associated with the resonance) and (f) a reduced density of states, i.e., avoided crossing, in the occupied band (associated with the dip). (d) Optical conductivity exhibiting a hump for energies slightly above the main maximum. Labels (A), (B), and (C) are discussed in the text.

Figure 3

From weak-coupling to strong-coupling antiferromagnetism in the Kondo lattice model as a function of the Kondo exchange coupling $J.$ (a) Reduction of kinetic or potential energy with respect to the reference paramagnetic state, indicating different mechanisms of magnetic ordering for weak and strong coupling regimes. (b) Staggered magnetization of the conduction band electrons $\left(m_c\right)$ and local moments $\left(m_f\right)$. (c) Spectral gap $\Delta$ and spin-resonance position ${\omega }_{sr}$. ${\omega }_{sr}$ is not well defined for $J/D\lesssim 0.3$.

Figure 4

Spin-resolved spectral functions of the conduction band for a range of $J$: (a) weak-coupling Slater regime with inverse square root divergence at gap edges $(J/D=0.1$, $0.25)$, (b) crossover regime with a complex form of the spectra near the gap edges $\left(0.375\right)$, (c) strong-coupling regime with smooth gap edges and well-developed spin resonances $(0.5$, $0.55)$, and (d) Kondo insulator with no spin resonance $\left(0.6\right)$.

Figure 5

Optical conductivity for a range of $J/D$ (as indicated in the plot next to the corresponding curves). Weak-coupling antiferromagnetism is characterized by threshold behavior with a peak at $2\Delta$, while strong-coupling antiferromagnetism exhibits more complex behavior with a threshold at $2\Delta$ and the main peak at $2{\omega }^{*}\approx 2{\omega }_{\mathrm{sr}}$, the spin resonance appearing as a secondary maximum or as a hump on the flank of the main peak. ${\sigma }_0=e^2/h$.

Figure 6

Comparison of antiferromagnetic and paramagnetic DMFT solutions for equal values of $J$ reveals the fine details in the ordered phase.

Figure 7

Reduction of the spin-resonance structure (peak weight) with increasing temperature. Inset: temperature variation of the staggered magnetization across the thermal AFM-PM phase transition.

Figure 8

Effect of an applied magnetic field on the magnetic order in the antiferromagnetic phase of the Kondo lattice model. (Top) Uniform magnetization components $m_{\parallel }$ of $c$ and $f$ orbitals in the direction of the field. (Bottom) Staggered magnetization components $m_{\perp }$ of $c$ and $f$ orbitals perpendicular to the field. The following defining relations hold: $m_{x,A}=m_{x,B}=m_{\parallel }$ and $m_{z,A}=-m_{z,B}=m_{\perp }$.

Figure 9

Evolution of (a) local spectral function $A_{\sigma }\left(\omega \right)$ and (b) momentum-resolved spectra $A_{\sigma }({\epsilon }_{\mathbf{k}},\omega )$ for increasing external magnetic field $B$ in the strong-coupling AFM phase $(J/D=0.5)$. Labels L, H, and M are explained in the text. The cross-cut of the momentum-resolved spectrum at $\epsilon =0$ is shown in (c).

Figure 10

Spectral functions of the Kondo lattice model on four different lattices: Bethe lattice, 2D cubic (square lattice), 3D cubic, and infinite-D cubic (Gaussian DOS) lattices. Note that the figure axes are scaled in terms of the effective bandwidth $D_{\mathrm{eff}}$ and that the same rescaled parameter $J/D_{\mathrm{eff}}$ has been used in all four DMFT calculations.

Figure 11

In periodic Anderson model (PAM), if $J\propto t^2/U$ is kept constant while $t$ and $U$ increase, the AFM order is suppressed, the spin resonance disappears, yet the gap remains constant. Here, $t/D=0.49$.

Figure 12

Kondo antiferromagnet in the hybridization picture with spin polarization: parameters for the single-pole ansatz for the self-energy ${\Sigma }_{\sigma }\left(\omega \right)$ as a function of the Kondo coupling $J$.

Figure 13

Self-energy functions and fits to the hybridization-picture ansatz with a single-pole. The fit quality is mediocre in the weak-coupling regime $(J/D=0.1)$, but improves in the strong-coupling regime $(J/D=0.5)$. (b) $\mathrm{Im}\Sigma$ has some fine structure in addition to the dominant poles.

Figure 14

Analytical structure leading to the spin resonance. (a) Nearby solutions of $\omega =\mathrm{Re}{\Sigma }_{\sigma }\left(\omega \right)$ for $\sigma =\uparrow$ and $\downarrow .$ (b) $\mathrm{Im}{\Sigma }_{\sigma }\left(\omega \right)$ are nearly constant for $\omega \approx {\omega }_{\mathrm{sr}}.$ (c) Argument of the noninteracting Green's functions in the DMFT expression [Eq. (17)] for local Green's functions. (d) Resulting local spectral functions featuring enhancement or suppression at $\omega \approx {\omega }_{\mathrm{sr}}$.

Figure 15

(a) Parameters for the hybridization-picture ansatz for the Kondo antiferromagnet in external magnetic field. (b) Momentum-resolved spectral function at finite magnetic field.