Distinct effect of Kondo physics on crystal field splitting in electron and spin spectroscopies

TL;DR

This study dissects how Kondo physics renormalizes magnetic anisotropy in a minimal $SU(4)$ ($J=3/2$) Anderson impurity with crystal-field splitting. Using numerical renormalization group, it reveals two distinct signatures: an electronic renormalized crystal-field scale $\Delta^* = \omega_\Delta-\omega_K$ visible in $A_{\alpha\alpha}(\omega)$ (PES/RIXS/STM) and a spin-scale $\omega_\perp$ seen in the INS structure factor $S(\omega)$, linked to transitions between ground and excited Kondo singlets and related to $\omega_E=\Delta^*-\omega_{Ke}$. Valence fluctuations can enhance $\Delta^*$ by up to ~40% over the bare $\Delta$, while $\omega_\perp$ generally grows with temperature near $T_K$ and exhibits a distinct dependence on $\Delta/T_K$, making the two spectroscopic probes respond differently. The results explain why RIXS/STM and INS can measure different aspects of magnetic anisotropy in rare-earth systems and offer a framework for interpreting spectroscopic data across Ce/Yb/Sm-based materials, including potential magnet performance implications where anisotropy is operational at elevated temperatures. The work underscores that Kondo physics can differentially renormalize electronic versus spin signatures of crystal-field splitting, with clear experimental relevance for complex correlated materials.

Abstract

Magnetic anisotropy is a key feature of rare earth materials from permanent magnets to heavy fermions. We explore the complex interplay of Kondo physics and anisotropy, and their effect on different experimental probes of magnetic anisotropy in a minimal J = 3/2 Anderson impurity model using numerical renormalization group. While anisotropy suppresses Kondo screening, virtual valence fluctuations enhance the anisotropy. We find distinct renormalization of the magnetic anisotropy measured via dynamical spin response (inelastic neutron scattering) versus electronic excitations in the impurity spectral function (resonant inelastic x-rays and scanning tunneling spectroscopy). The two measurement types have different responses and dependences on the temperature and Kondo scales.

Figures (9)

• Figure 1: (a) Example impurity spectral function, with the ground ($A_{11}$) and excited state ($A_{22}$) spectral functions in blue and orange, for $|\epsilon_f|/\Gamma_0=8$, $\Delta/\Gamma_0=1$, $T/\Gamma_0 = 0.0001$, with $T=0$ empty state occupancy, $n_0 = .06$. (Inset) The $J = 3/2$ infinite-$U$ Anderson model has two $f^1$ doublets split by $\Delta$, $\ket{f^{\dagger }_{\alpha\sigma}} = \ket{J_z = \pm \frac{3}{2}}, \ket{J_z = \pm \frac{1}{2}}$, and an empty state ($f^0$) with energy cost $|\epsilon_f|$; other atomic configurations are forbidden. The doublets hybridize, $\Gamma_{1,2}$ through two conduction channels, allowing for perfect screening. (b) The four spectral function peaks capture charged excitations ($N\rightarrow N\pm1$) out of the ground state Kondo singlet, $|\phi_0\rangle$Bickers1987. The sharp central peak, $\omega_{K}$ is the Kondo resonance for breaking the Kondo singlet by adding an $f_1$; $\omega_\Delta = \omega_K + \Delta^*$ corresponds to adding an $f_2$; $\omega_{f0} = |\epsilon_f^*|$ is the empty state; and $\omega_{E} = \Delta^*-\omega_{EK}$ corresponds to removing an $f_1$ into an energy overlapping the excited Kondo singlet, $|\phi_e\rangle$, with energy lowered from the uncorrelated excited doublet by $\omega_{Ke}$.
• Figure 2: (a) Relevant energy levels for neutral magnetic excitations out of the ground state Kondo singlet, $|\phi_0\rangle$. Low-lying magnetic excitations to broken Kondo singlet states, $|f_{\alpha\sigma}^{\dagger }\rangle$ manifest as peaks in $\chi_{\alpha\alpha}"(\omega)$, while $\chi_\perp"(\omega)$ appears to be dominated by excitations between the ground and excited state Kondo singlets. (b) Temperature dependence of the INS structure factor, $S(\omega)$ for $\Delta/\Gamma_0 = 0.1$, $|\epsilon_f|/\Gamma_0 = 8$. Temperatures are $0.003\Gamma_0$ (blue), $0.006\Gamma_0$ (green), $0.01\Gamma_0$ (orange), $0.02\Gamma_0$ (red), all at or above $T_K \sim 0.003\Gamma_0$, defined as where the empty state occupancy, $n_0(T)$ saturates ($n_0 = .08$). The lower peak is associated with the Kondo resonance ($\propto \omega_K$), and shifts and broadens significantly with $T$, while the upper CEF peak ($\propto \omega_E$) shifts slightly without much broadening.
• Figure 3: Temperature dependence of the peaks in $A(\omega)$ [$\omega_K$, $\omega_\Delta$, $\omega_E$] and $\chi"(\omega)/\omega$ [$\omega_{11}$, $\omega_{22}$, $\omega_\perp$], for (a) $\Delta/\Gamma_0 = 1$, $\epsilon_f /\Gamma_0= 6$, $n_0 = .13$; (b) $\Delta/\Gamma_0 = 1$, $|\epsilon_f|/\Gamma_0 = 8$, $n_0 = .06$. The peaks in $\chi"(\omega)/\omega$ ($\chi$ = $\chi_{\alpha\alpha}$, $\chi_\perp$) are non-zero when inelastic and zero when quasi-elastic. When inelastic, the spectral and spin susceptibility peaks are proportional; the Kondo resonances, $\omega_K$ and $\omega_{11}$ (blue) are $\times 20$ for clarity. The monotonic in $T$ decrease of $\omega_\perp$ and $\omega_E$ (green) is similar, and distinct from the other two more dome-like sets, but $\omega_\perp/\omega_E$ varies markedly with $\epsilon_f$, likely due to increasing $n_0$.
• Figure 4: Temperature dependence of two measures of CEF splitting: $\Delta^*(T)$ (dashed lines) and $\omega_\perp$ (solid lines), both in units of the bare $\Delta$. Colors indicate different $|\epsilon_f|/\Gamma_0 = 6$ (black), $8$ (gray). (a) shows $\Delta/\Gamma_0 = 0.1$, where the Kondo scales are $\omega_K/\Gamma_0 = 0.08, 0.002$. The temperature scale cuts off at $T = \Delta$, where the peaks are very broad and $\chi_\perp$ becomes quasi-elastic. $|\epsilon_f|/\Gamma_0 = 6$ exemplifies the behavior for $\Delta \lesssim \omega_K$, where $\omega_\perp$ is substantially enhanced by valence fluctuations ($n_0 = .17$), particularly compared to $\Delta^*$. (b) shows $\Delta/\Gamma_0 = 1$, where $\omega_K/\Gamma_0 = 0.005, 0.0001$. Higher $T_K$'s lead to larger enhancement of $\omega_\perp$ at low $T$, agreeing with scaling predictions, while lower $T_K$'s lead to more $\Delta^*$ enhancement.
• Figure 5: Error estimates for $\Delta = .1 \Gamma_0$, $\epsilon_f = -8\Gamma_0$. Five spectral function curves vs $\omega$ (in units of $\Gamma_0$) are shown in each sub-figure, calculated for $\Lambda = 4$, $m$ = 1500, as in the main text (black); $\Lambda = 3.5$, $m$ = 1500 (dark blue); $\Lambda = 3.5$, $m$ = 2000 (light blue); $\Lambda = 2.5$, $m$ = 1500 (dark red); $\Lambda = 2.5$, $m$ = 2000 (light red). The top row is $T=.0001\Gamma_0$: (a) shows the peak in $A_{22}$, with average peak location and standard deviation: $\omega_\Delta = .110 \pm .001$; (b) shows the Kondo resonance peak in $A_{11}$, with $\omega_K = .0030\pm.0001$. The bottom row shows $T = .17\Gamma_0$. (a) is the peak in $A_{22}$, with $\omega_\Delta = .163\pm.009$; (b) is the Kondo resonance peak in $A_{11}$, with $\omega_K = .0325\pm.008$.