Suppression of impurity magnetization by the saddle points

Abstract

We study the localized magnetic states in the semi-Dirac-like system, and find that due to the existence of the saddle point, the magnetic region diminishes greatly with the energy of the saddle point approaching the impurity energy, and reaches a minimum at the energy of the saddle point equal to the impurity energy. A similar feature is observed in the magnetic moment of the impurity. This suppression behavior for the magnetization of the impurity can be understood from the saddle point induced mitigation of asymmetry on the density of state at impurity energy. In contrast, when the saddle point vanishes, due to the opening of the gap, the magnetic region exhibits a monotonous decay when the conduction band edge goes up through the impurity energy. The combined effect of the saddle point and the Coulomb interaction at the impurity on the local magnetization is also investigated.

Fingerprint of parity anomaly for localized magnetic states in quantum anomalous Hall systems

“…For simplicity, here, we invoke the broad band approximation, and assume V f to be a constant [42]. This treatment can capture the primary physical characteristics of the impurity magnetization in the embedded system [43][44][45]. Within the mean field approximation, we have…”

“…In equations ( 9) and ( 10), E m is the high energy cutoff determined according to Debye's prescription, which ensures that the number of states in the Brillouin zone is conserved [43,44]. Similar to the semi-Dirac-like system, DOS is not simply Lorentz [45], but has a narrow peak around the quasi particle energy, ω ∼ ϵ σ . f(ω) is a function of ω, and changes very slowly near ω ∼ ϵ σ .…”

“…f(ω) is a function of ω, and changes very slowly near ω ∼ ϵ σ . Therefore, in numerical calculations [45,46], we approximate f(ω) as f(ϵ σ ), resulting in only a minor deviation from the exact result. By substituting equation (13) into equation (12), we can determine the occupation of the impurity, ⟨n σ ⟩, and subsequently establish the local magnetic states in the QAH system.…”

Localized magnetic states in a tilted Dirac cone system

“…For instance, the anisotropy of the Dirac cone can cause the electrons at the impurity level to delocalize, leading to a depletion of its magnetic region [39]. Additionally, when two Dirac cones approach each other, the presence of a saddle point significantly reduces the impurity magnetization [40]. Therefore, the tilt of the Dirac cone can induce novel characteristics in the magnetic states of impurities, which is the subject addressed here.…”

“…where n σ = f † σ f σ is the occupation number operator for each of two spin states, the Hubbard-like term U n ↓ n ↑ represents the Coulomb interaction between impurity electrons, and ε 0 is the energy level of the impurity. Within the mean-field approximation [39,40], we rewrite…”