Revealing altermagnetic Fermi surfaces with two Kondo impurities

Abstract

Motivated by recent advances in the study of altermagnetism, or unconventional magnetism, and in the realization and manipulation of two-impurity Kondo physics in real materials, we propose a phase-sensitive method to explore unconventional magnetic symmetries. Our method can be implemented with spin-resolved scanning tunneling microscopy to study two-impurity Kondo phenomena on altermagnetic metals by varying the distance and orientation between magnetic impurities. Using quantum Monte Carlo simulations, we analyze the spin splitting of the Kondo resonance, whose spatial distribution sensitively captures the symmetry of the underlying altermagnetic order. Furthermore, the impurity spin correlations reflects the anisotropy of the RKKY interaction due to the altermagnetic Fermi surface splitting. This work provides a framework for studying the competition between the Kondo effect, the RKKY interaction and altermagnetism, in the simplest possible system.

Figures (13)

• Figure 1: Illustration of the two-impurity Kondo model, with the right panel depicting the Fermi surface of the underlying AM metal with $t'=0.4$.
• Figure 2: $d_{xy}$ AM symmetry at $J_{K} = 2$ and $t^{\prime}=0.4$. (a) Imaginary-time Green function of the composite fermion, $\tilde{G}_{\bm{R}}(\tau)$, for varying impurity separations $\bm{R} = (\delta x, \delta y)$, where $\bm{R}$ denotes the distance between the two magnetic impurities. (b) Corresponding spectral function $\tilde{N}_{\boldsymbol{R}}(\omega)$ for different $\bm{R}$. (c) Real-space distribution of the difference $\Delta \tilde{G}_{\boldsymbol{R}}$ between spin-up and spin-down Green functions, together with its Fourier transform. (d) Local magnetic moment $\langle \hat{S}_{\bm{R}}^z\rangle$ and the real-space equal-time spin correlation functions between the two impurities, showing the anisotropy $\Delta C_{\bm{R}} = C^{x}_{\bm{R}}(0^+) - C^{z}_{\bm{R}}(0^+)$. Temperatures are $T = 0.05$ for (a)(b), and $T = 0.10$ for (c)(d).
• Figure 3: The RKKY interaction at $J_{K}=2$ and $t^{\prime}=0.4$. (a) Spatial Fourier transform of the equal-time spin correlation functions $C^{x,z}(0^+)$ between two impurities for both the $d_{xy}$ and $d_{x^2 - y^2}$ AM cases. (b) Distance dependence of the composite fermion Green function difference $\Delta \tilde{G}_{\boldsymbol{R}}$ for the $d_{xy}$ and $d_{x^2 - y^2}$ cases at $T=0.05,~0.10$. (c) Distance dependence of the spin correlation $C_{\bm{R}}^{x,z}(0^+)$ between impurities and $C_{\bm{R}}^{t}(0^+)=\frac{1}{3}[2C_{\bm{R}}^{x}(0^+)+C_{\bm{R}}^{z}(0^+)]$. The direction analyzed for $d_{xy}$ case is along the diagonal $x = y$ direction, whereas for the $d_{x^2 - y^2}$ case it is along the $y$-axis. We note $T=0.1,~0.05$ for (a), (c), respectively.
• Figure 4: Temperature evolutions for the RKKY (a, b, c,d) and for Kondo splitting (e, f) at $t^{\prime} = 0.4$ for $J_K = 1.2,~1.6,~2.0$. (a,b) The $x$-component equal-time spin correlation function $C_{\bm{R}}^{x}(0^{+})$ as a function of temperature for the $d_{xy}$ and $d_{x^2 - y^2}$ cases. (c,d) Spin correlation along $z$ direction $C_{\bm{R}}^{z}(0^{+})$ as a function of temperature for the $d_{xy}$ and $d_{x^2 - y^2}$ symmetry. The arrows denote the $T_\chi$ scale, see main text and End Matter. (e,f) Temperature dependence of the composite fermion Green function difference $\Delta \tilde{G}_{\boldsymbol{R}}$ at $\bm{R} = (1,1)$ for the $d_{xy}$ case and at $\bm{R} = (0,1)$ for the $d_{x^2 - y^2}$ case.
• Figure 5: $d_{x^2 - y^2}$ altermagnetic symmetry at $J_{K} = 2$ and $t^{\prime}=0.4$. (a) Imaginary-time Green function of the composite fermion $\tilde{G}_{\boldsymbol{R}}(\tau)$ at different displacement vector $\bm{R} = (\delta x, \delta y)$ between two impurities. (b) Spectral function $\tilde{N}_{\boldsymbol{R}}(\omega)$ at different values of $\bm{R}$. (c) Real-space distribution of $\Delta \tilde{G}_{\mathbf{R}}$ defined in Eq. (\ref{['eq:DeltaG']}) along with its Fourier transform. (d)The local magnetic moment $\langle \hat{S}_{\bm{R}}^{z} \rangle$ and the real-space equal-time spin correlation functions between two impurities exhibit anisotropy, characterized by $\Delta C_{\bm{R}} = C^{x}_{\bm{R}}(0^+) - C^{z}_{\bm{R}}(0^+)$. Temperatures are set to $T = 0.05$ for panels (a) and (b), and $T = 0.10$ for panels (c) and (d).