Evolution from the Kondo phase to the RKKY phase in the small impurity spacing regime of the two-impurity Anderson model

Understanding the quantum critical phenomena is one of the most important and challenging tasks in condensed matter physics and the two-impurity Anderson model (TIAM) is a good starting point for this exploration. To this end, we employ the algebraic equation of motion approach to calculate the TIAM and analytically obtain the explicit single-particle impurity Green function under the soft cut-off approximation (SCA). This approach effectively incorporates the impurity spacing as an intrinsic parameter. By solving the pole equations of the Green function, we have, for the first time, qualitatively calculated the spectral weight functions of the corresponding low-energy excitations. We find that when the impurity spacing is less than one lattice distance, the dynamic Rudermann–Kittel–Kasuya–Yosida (RKKY) interaction effectively enters, resulting in a rapid increase in the spectral weights of the RKKY phase, which ultimately surpass those of the Kondo phase; while the spectral weights of the Kondo phase are strongly suppressed. From the perspective of spectral weights, we further confirm the existence of a crossover from the Kondo phase to the RKKY phase in the TIAM. Based on these results, the reasons for the phenomenon of the Kondo resonance splitting are also discussed.