Energy spectrum for a atrongly correlated network and local magnetism

Abstract

In this work, we consider a quantum strongly correlated network described by an Anderson s – d mixing model. By introducing the Green function on the projected formalism of the Schrieffer and Wolf transformation, the energy spectrum of the system can be obtained. Using this result we calculate the survivability distribution of the network and discuss the local magnetism in the network, which shows that the survivability is an important statistical characteristic quantity not just to reflect the network topological property but also dynamics.

Keywords

strongly correlated system / quantum network / Green function

1 Introduction

Nowadays, studying strongly correlated systems is an important direction that crosses many physical domains. The energy spectrum analysis for the strongly correlated system is crucial for studying the physical properties of these systems. However, the traditional perturbative method is not easy to use for getting the energy spectrum of these systems because these systems are strongly correlated. Using the retarded Green function approach, the spectrum analysis for the Anderson s – d mixed model has been achieved [1], but the energy spectrum for its Schrieffer and Wolf transformation is still unclear. In this work, we introduce a quantum strongly correlated network described by an Anderson s – d mixed model [2]. By using the Green function approach on the projected formalism of the Schrieffer and Wolf transformation, the energy spectrum of the network can be obtained based on the average field approach. This allows one to analyze some physical properties of the quantum network, such as average energy and the local magnetism in the nodes. As a demonstration, we calculate the connectivity distribution of the network based on the concept of the fermionic network [3], and discuss the local magnetism affected by the difference of the energy on the nodes.

2 Physical model

We consider a quantum network described by the Anderson s – d mixed model which mixes the impure atoms in a disordered matrix [2], i.e. the model is generally defined as a graph which is a collection of vertices or sites and unordered edges or bonds connecting certain distinct pairs of vertices [5]. Concretely speaking, in this network, a node of the network is composed of an electron confined in a quantum dot as an impurity. These nodes of the network are arranged as a topological disorder system without long distance order. For simplicity, this local electron is still called as d electron (as the original Anderson meaning). The free Hamiltonian of the nodes is described by {\sum }_{\sigma }{E}_d{n}_{d\sigma }, where n_{d\sigma }=d_{\sigma }^{+}{d}_{\sigma }, d_{\sigma }^{+} is a creation operator of the d electron with spin σ. The strong correlation between each node (quantum dot) is considered as the link of the network, and the corresponding Hamiltonian is supposed as U{n}_{d\uparrow }{n}_{d\downarrow }={\displaystyle \frac{U}{2}}{\sum }_{\sigma }{n}_{d\sigma }{n}_{d\overline{\sigma }}, where U\gg 1 is defined as a correlated energy, and the spins are given by \sigma =\uparrow ,\downarrow and \overline{\sigma }=-\sigma. The environmental (or control) field of the network is composed of many conductive electrons between quantum dots. For simplicity, we call them as s electrons, and their free Hamiltonian is given by {\sum }_{k,\sigma }{E}_k{n}_{k\sigma }, n_{k\sigma }=C_{k\sigma }^{+}{C}_{k\sigma }, where C_{k\sigma }^{+} is a creation operator of the s electron with wave vector k and spin σ. The interaction taking place between d electron confined in the quantum dot and s electron moving outside the quantum dot is described by {\sum }_{k,\sigma }{V}_{kd}\left(C_{k\sigma }^{+}{d}_{\sigma }+d_{\sigma }^{+}{C}_{k\sigma }\right). Considering there is no effect of external magnetic field and ignoring the s – d reciprocity, we define V_{kd}=V. Hence, the total Hamiltonian operator for the system can be expressed by

\begin{array}{}\text{(1)}& H=H_0+H_1\end{array}

with

\begin{array}{}\text{(2)}& \begin{array}{ccc}{H}_0\hfill & =H_d+H_k+H_u\hfill & \\ & ={\displaystyle \sum _{\sigma }}{E}_d{d}_{\sigma }^{+}{d}_{\sigma }+{\displaystyle \sum _{k,\sigma }}{E}_k{C}_{k\sigma }^{+}{C}_{k\sigma }+{\displaystyle \frac{U}{2}}{\displaystyle \sum _{\sigma }}{n}_{d\sigma }{n}_{d\overline{\sigma }}\hfill & \end{array}\end{array}

where

\begin{array}{}\text{(3)}& H_1={\displaystyle \sum _{k,\sigma }}V\left(C_k^{+}{d}_{\sigma }+d_{\sigma }^{+}{C}_k\right)\end{array}

3 Projection transformation

In Anderson mixed model, we notice that as a result of d electron not coupling with the s electron, when V=0, there are two energy levels of the impurity respectively, E_d and E_d+U, which correspond on three kinds of different electronic configurations in this network. This allows the nodes of the network to be possible in three combined status: 0(E_0=0), 1(\sigma =\uparrow or \downarrow, E_{1\sigma }=E_d), and 2 (\sigma =\uparrow and \downarrow, 2E_d+U) occupations. Following the Schrieffer and Wolf transformation [6, 7], we put the Anderson model into a Hilbert space and introduce three projection operators to divide the total Hilbert space as three subspaces:

\begin{array}{}\text{(4)}& P_0=\left(1-n_{d\uparrow }\right)\left(1-n_{d\downarrow }\right)\end{array}

\begin{array}{}\text{(5)}& P_1=n_{d\uparrow }\left(1-n_{d\downarrow }\right)+n_{d\downarrow }\left(1-n_{d\uparrow }\right)\end{array}

\begin{array}{}\text{(6)}& P_2=n_{d\uparrow }{n}_{d\downarrow }\end{array}

where P_n, n=0,1,2 corresponds upon 0,1,2 occupations, and satisfies the fundamental rule of the projection operator:

\begin{array}{}\text{(7)}& P_n^2=P_n\end{array}

\begin{array}{}\text{(8)}& P_n{P}_{n^{\prime }}=0(n\ne n^{\prime })\end{array}

Assume that the total wave function \Psi in the Hilbert space is composed by {\psi }_0, {\psi }_1, {\psi }_2 in these three subspaces, respectively, then the Schrödinger equation H\Psi =E\Psi can be written as:

\begin{array}{}\text{(9)}& \left(\begin{array}{ccc}{H}_{00}& H_{01}& H_{02}\\ H_{10}& H_{11}& H_{12}\\ H_{20}& H_{21}& H_{22}\end{array}\right)\left(\begin{array}{c}{\psi }_0\\ {\psi }_1\\ {\psi }_2\end{array}\right)=E\left(\begin{array}{c}{\psi }_0\\ {\psi }_1\\ {\psi }_2\end{array}\right)\end{array}

where H_{n{n}^{\prime }}=P_{n}H{P}_{n^{\prime }}, in terms of the Anderson model, it is impossible to produce and eliminate two electronic interactions simultaneously, hence

\begin{array}{}\text{(10)}& H_{02}=H_{20}=0\end{array}

Other off-diagonal matrix elements (such as H_{01} or H_{12}) can be obtained using the Pauli principle and the Fermionic operator equality n_{d\sigma }^2=n_{d\sigma },

\begin{array}{}\text{(11)}& H_{01}=P_{0}H{P}_1=V{\displaystyle \sum _{k,\sigma }}{C}_{k\sigma }^{+}\left(1-n_{d\sigma }\right)d_{\sigma }\end{array}

\begin{array}{}\text{(12)}& H_{12}=P_{1}H{P}_2=V{\displaystyle \sum _{k,\sigma }}{C}_{k\sigma }^{+}{d}_{\sigma }{n}_{d\sigma }\end{array}

At the same time the matrix is Hermitian, so

\begin{array}{}\text{(13)}& H_{10}=H_{01}^{+},H_{21}=H_{12}^{+}\end{array}

When the nodes are in the 1 occupation, the local extra spin appear, which allow the network to show the type of local magnetic effect, as a self-organized system. Thus, by cancelling {\psi }_0 and {\psi }_2 from Eq. (9) and considering Eqs. (11) and (12), a kinetic equation is given by

\begin{array}{}\text{(14)}& \Theta {\psi }_1=E_1{\psi }_1\end{array}

where \Theta is introduced by

\begin{array}{}\text{(15)}& \Theta =H_{11}+H_{12}{\left(E-H_{22}\right)}^{-1}{H}_{21}+H_{10}{\left(E-H_{00}\right)}^{-1}{H}_{01}\end{array}

which represents the effective Hamiltonian of single occupation. So far, we have not made any approximation. When V=0, we can replace \left(E_{k^{\prime }}-E_d-U\right) with \left(E-H_{22}\right), and \left(E_d-E_k\right) with \left(E-H_{00}\right). In the second order of approximation with respect to V, we have

\begin{array}{}\text{(16)}& H_{12}{(E-H_{22})}^{-1}{H}_{21}={\displaystyle \sum _{k,k^{\prime },\sigma ,{\sigma }^{\prime }}}{\displaystyle \frac{V^2}{U+E_d-E_{d^{\prime }}}}\cdot \left(-1\right)C_k^{+}{C}_{k^{\prime }}{d}_{\sigma }{n}_{d\sigma }{d}_{{\sigma }^{\prime }}^{+}{n}_{d{\sigma }^{\prime }}\end{array}

\begin{array}{}\text{(17)}& H_{10}{\left(E-H_{00}\right)}^{-1}{H}_{01}={\displaystyle \sum _{k,k^{\prime },\sigma ,{\sigma }^{\prime }}}{\displaystyle \frac{V^2}{E_d-E_k}}\cdot (-1)C_k^{+}{C}_{k^{\prime }}{d}_{{\sigma }^{\prime }}^{+}\left(1-n_{d{\sigma }^{\prime }}\right)d_{\sigma }\left(1-n_{d{\sigma }^{\prime }}\right)\end{array}

Considering n_d=n_{d\sigma }+n_{d\overline{\sigma }}=1, and introducing notation for the spinning of a half:

\begin{array}{}\text{(18)}& {\widehat{S}}^z={\displaystyle \frac{1}{2}}\left(d_{\sigma }^{+}{d}_{\sigma }-d_{\overline{\sigma }}^{+}{d}_{\overline{\sigma }}\right)\end{array}

\begin{array}{}\text{(19)}& {\widehat{S}}^{+}=d_{\sigma }^{+}{d}_{\overline{\sigma }}\end{array}

\begin{array}{}\text{(20)}& {\widehat{S}}^{-}=d_{\overline{\sigma }}^{+}{d}_{\sigma }\end{array}

Eqs. (16) and (17) are written as:

\begin{array}{}\text{(21)}& \begin{array}{c}{H}_{12}{\left(E-H_{22}\right)}^{-1}{H}_{21}={\displaystyle \sum _{k,k^{\prime }}}{\displaystyle \frac{V^2}{U+E_d-E_{k^{\prime }}}}\hfill \\ \left[\left({\widehat{S}}^z{C}_k^{+}{C}_{k^{\prime }}+{\widehat{S}}^{-}{C}_k^{+}{C}_{k^{\prime }}\right)-{\displaystyle \frac{1}{2}}{C}_k^{+}{C}_{k^{\prime }}\right]\hfill \end{array}\end{array}

and

\begin{array}{}\text{(22)}& H_{10}{\left(E-H_{00}\right)}^{-1}{H}_{01}={\displaystyle \sum _{k,k^{\prime }}}{\displaystyle \frac{V^2}{E_k-E_d}}\cdot [({\widehat{S}}^z{C}_k^{+}{C}_{k^{\prime }}+{\widehat{S}}^{-}{C}_k^{+}{C}_{k^{\prime }})+{\displaystyle \frac{1}{2}}{C}_k^{+}{C}_{k^{\prime }}]\end{array}

By considering Eqs. (21), (22), and simplifying H_{11} as H_{11}\approx H_0={\sum }_k{E}_k{C}_k^{+}{C}_k, Eq. (15) can be written as:

\begin{array}{}\text{(23)}& \Theta =H_0+H_p+H_{sd}\end{array}

where H_p is the scattering energy of the impurity,

\begin{array}{}\text{(24)}& H_p={\displaystyle \sum _{k,k^{\prime }}}{J}_{k{k}^{\prime }}{C}_k^{+}{C}_{k^{\prime }}\stackrel{U\to \mathrm{\infty }}{⟶}J{\displaystyle \sum _{k,k^{\prime }}}{C}_k^{+}{C}_{k^{\prime }}\end{array}

and H_{sd} represents the interaction with the impurity relating spinning degree-of-freedom,

\begin{array}{}\text{(25)}& \begin{array}{cc}{H}_{sd}=\hfill & -{\displaystyle \sum _{k,k^{\prime }}}{J}_{k{k}^{\prime }}\left({\widehat{S}}^z{C}_k^{+}{C}_{k^{\prime }}+{\widehat{S}}^{-}{C}_k^{+}{C}_{k^{\prime }}\right)\stackrel{U\to \mathrm{\infty }}{⟶}\hfill \\ & -J{\displaystyle \sum _{k,k^{\prime }}}\left({\widehat{S}}^z{C}_k^{+}{C}_{k^{\prime }}+{\widehat{S}}^{-}{C}_k^{+}{C}_{k^{\prime }}\right)\hfill \end{array}\end{array}

This is so because the s electron locates around the Fermi surface, and can replace E_F with E_k and E_{k^{\prime }}. Concerning U+E_d>E_F>E_d, one has , where in the strong correlation condition, U\to \mathrm{\infty }, the interaction has no relation with k,k^{\prime }, which results in

Therefore, by introducing projected operator P and Q=1-P, using the above projected formalism (the Schrieffer and Wolf transformation) and neglecting the scattering energy H_p, an effective Hamiltonian (the intermediate [8, 9]) \Theta can be described by

\begin{array}{}\text{(27)}& \begin{array}{cc}\Theta \hfill & ={\displaystyle \sum _{k,\sigma }}{E}_k{C}_{k\sigma }^{+}{C}_{k\sigma }-\{J{\displaystyle \sum _{k,k^{\prime }}}{\widehat{S}}^z(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})\hfill \\ & +J{\displaystyle \sum _{k,k^{\prime }}}{\widehat{S}}^{+}{C}_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }+J{\displaystyle \sum _{k,k^{\prime }}}{\widehat{S}}^{-}{C}_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}\}\hfill \\ & =H_0-(H_1+H_2+H_3)\hfill \end{array}\end{array}

4 Green function formalism

Introducing an equation of motion for two-time retarded Green function, one gets

\begin{array}{}\text{(28)}& ⟨⟨A(t);B(t^{\prime })⟩⟩=-\mathrm{i}\theta (t-t^{\prime })⟨[A(t),B(t^{\prime })]⟩\end{array}

By differentiating the quotient of time, we get

\begin{array}{}\text{(29)}& \mathrm{i}{\displaystyle \frac{\text{d}}{\text{d}t}}⟨⟨A(t);B(t^{\prime })⟩⟩=\delta (t-t^{\prime })⟨[A,B]⟩+⟨⟨[A,H]\mid B(t^{\prime })⟩⟩\end{array}

Taking the Fourier transformation

\begin{array}{}\text{(30)}& {⟨⟨A\mid B⟩⟩}_{\omega }={\displaystyle \int dt{\mathrm{e}}^{\mathrm{i}\omega (t-t^{\prime })}⟨⟨A(t);B(t^{\prime })⟩⟩}\end{array}

one gets a general formula of the Green function as:

\begin{array}{}\text{(31)}& \omega {⟨⟨A\mid B⟩⟩}_{\omega }=⟨{[A,B]}_{+}⟩+{⟨⟨[A,H]\mid B⟩⟩}_{\omega }\end{array}

This allows one to obtain an equation of motion for the Green function ⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩ by

\begin{array}{}\text{(32)}& \omega {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[d_{\sigma },d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[d_{\sigma },\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

where d electron is a fermion, which satisfies anticommutation relation with {[d_{\sigma },d_{\sigma }^{+}]}_{+}=1. Now considering the second term in the Eq. (32), using the Bosons and the Fermions commutation relation in quantum mechanics, and some related operations, one gets

\begin{array}{}\text{(33)}& \begin{array}{cc}[d_{\sigma },\Theta ]\hfill & =[d_{\sigma },H_0]-[d_{\sigma },H_1]-[d_{\sigma },H_2]-[d_{\sigma },H_3]\hfill \\ & =-{\displaystyle \frac{J}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})d_{\sigma }\hfill \\ & -J{\displaystyle \sum _{k,k^{\prime }}}{C}_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }{d}_{\overline{\sigma }}\hfill \end{array}\end{array}

This allows Eq. (32) to be written as:

\begin{array}{}\text{(34)}& \begin{array}{cc}& \omega {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & =1-⟨⟨{\displaystyle \frac{J}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})d_{\sigma }\mid d_{\sigma }^{+}⟩{⟩}_{\omega }\hfill \\ & -⟨⟨J{\displaystyle \sum _{k,k^{\prime }}}{C}_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩{⟩}_{\omega }\hfill \end{array}\end{array}

Introducing new operators:

\begin{array}{}\text{(35)}& \begin{array}{c}{\widehat{A}}^z={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})\hfill \\ {\widehat{A}}^{+}={\displaystyle \sum _{k,k^{\prime }}}{C}_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}\hfill \\ {\widehat{A}}^{-}={\displaystyle \sum _{k,k^{\prime }}}{C}_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }\hfill \end{array}\end{array}

Eq. (34) becomes

\begin{array}{}\text{(36)}& \omega {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }=1-J{⟨⟨{\widehat{A}}^z{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }-J{⟨⟨{\widehat{A}}^{-}{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

where [{\widehat{A}}^{+},{\widehat{A}}^{-}], [{\widehat{A}}^{+},{\widehat{A}}^z] and [{\widehat{A}}^{-},{\widehat{A}}^z] are calculated by commutation relation:

\begin{array}{}\text{(37)}& [{\widehat{A}}^{+},{\widehat{A}}^{-}]={\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }}){\delta }_{k{k}^{\prime }}=2{\widehat{A}}^z{\delta }_{k{k}^{\prime }}\end{array}

and

\begin{array}{}\text{(38)}& [{\widehat{A}}^{+},{\widehat{A}}^z]=-{\widehat{A}}^{+}{\delta }_{k{k}^{\prime }}\end{array}

\begin{array}{}\text{(39)}& [{\widehat{A}}^{-},{\widehat{A}}^z]={\widehat{A}}^{-}{\delta }_{k{k}^{\prime }}\end{array}

Therefore, the above Eq. (36) can be solved by the following procedure:

(1) The second term in Eq. (36) can be expanded as:

\begin{array}{}\text{(40)}& \omega {⟨⟨{\widehat{A}}^z{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[{\widehat{A}}^z{d}_{\sigma },d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[{\widehat{A}}^z{d}_{\sigma },\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

Furthermore,

\begin{array}{}\text{(41)}& \begin{array}{cc}[{\widehat{A}}^z{d}_{\sigma },\mathrm{\Theta }]=\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }}{k\ne k^{\prime }}}}{E}_k(C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }}-C_{k\sigma }^{+}{C}_{k^{\prime }\sigma })d_{\sigma }\hfill \\ & -2J{({\widehat{A}}^z)}^2{d}_{\sigma }\hfill \end{array}\end{array}

Eq. (40) is written as:

\begin{array}{}\text{(42)}& \begin{array}{cc}\omega {⟨⟨{\widehat{A}}^z{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }=\hfill & ⟨{\widehat{A}}^z⟩+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }}{k\ne k^{\prime }}}}{E}_k⟨C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }}\hfill \\ & -C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩{}_{\omega }\hfill \\ & -2J{⟨{\widehat{A}}^z⟩}^2{⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \end{array}\end{array}

(2) The third term in Eq. (36) can be expanded as:

\begin{array}{}\text{(43)}& \omega {⟨⟨{\widehat{A}}^{-}{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[{\widehat{A}}^{-}{d}_{\overline{\sigma }},d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[{\widehat{A}}^{-}{d}_{\overline{\sigma }},\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

where denoting

\begin{array}{}\text{(44)}& [{\widehat{A}}^{-}{d}_{\overline{\sigma }},\Theta ]=-{\displaystyle \sum _{k,k^{\prime }\left(k\ne k^{\prime }\right)}}{E}_k{C}_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }\end{array}

Using the same way, Eq. (43) can be written as:

\begin{array}{}\text{(45)}& \begin{array}{cc}& \omega {⟨⟨{\widehat{A}}^{-}{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & =-{\displaystyle \sum _{k,k^{\prime }\left(k\ne k^{\prime }\right)}}{E}_k⟨C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }⟩{⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \end{array}\end{array}

which is related to the Green function ⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩ and enable us to establish the ⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩ equation of motion.

(3) The Green function {⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }can be calculated by

\begin{array}{}\text{(46)}& \omega {⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[d_{\overline{\sigma }},d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[d_{\overline{\sigma }},\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

where [d_{\overline{\sigma }},\Theta ] can be calculated:

\begin{array}{}\text{(47)}& \begin{array}{cc}[d_{\overline{\sigma }},\Theta ]=\hfill & {\displaystyle \frac{J}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})d_{\overline{\sigma }}\hfill \\ & -J{\displaystyle \sum _{k,k^{\prime }}}{C}_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}{d}_{\sigma }\hfill \end{array}\end{array}

which enables Eq. (46) to become

\begin{array}{}\text{(48)}& \omega {⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }=J{⟨⟨{\widehat{A}}^z{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }-J{⟨⟨{\widehat{A}}^{+}{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

(4) Using the same way as Eq. (42), the third term {⟨⟨{\widehat{A}}^z{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega } in Eq. (48) can be written as:

\begin{array}{}\text{(49)}& \omega {⟨⟨{\widehat{A}}^z{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[{\widehat{A}}^z{d}_{\overline{\sigma }},d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[{\widehat{A}}^z{d}_{\overline{\sigma }},\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

where

\begin{array}{}\text{(50)}& [{\widehat{A}}^z{d}_{\overline{\sigma }},\Theta ]={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime },\left(k\ne k^{\prime }\right)}}{E}_k(C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }}\overline{\sigma }-C_{k\sigma }^{+}{C}_{k^{\prime }\sigma })d_{\overline{\sigma }}+2J{\left({\widehat{A}}^z\right)}^2{d}_{\overline{\sigma }}\end{array}

This allows Eq. (49) to become

\begin{array}{}\text{(51)}& \begin{array}{cc}& \omega {⟨⟨{\widehat{A}}^z{d}_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime },\left(k\ne k^{\prime }\right)}}{E}_k⟨C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }}-C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩{⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & +2J{⟨{\widehat{A}}^z⟩}^2{⟨⟨d_{\overline{\sigma }}\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \end{array}\end{array}

The second term in the Eq. (48) can also be expanded as:

\begin{array}{}\text{(52)}& \omega {⟨⟨{\widehat{A}}^{+}{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }=⟨{[{\widehat{A}}^{+}{d}_{\sigma },d_{\sigma }^{+}]}_{+}⟩+{⟨⟨[{\widehat{A}}^{+}{d}_{\sigma },\Theta ]\mid d_{\sigma }^{+}⟩⟩}_{\omega }\end{array}

it is just

\begin{array}{}\text{(53)}& [{\widehat{A}}^{+}{d}_{\sigma },\Theta ]=-{\displaystyle \sum _{k,k^{\prime },(k\ne k^{\prime })}}{E}_k{C}_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}{d}_{\sigma }\end{array}

Therefore Eq. (51) can be written as:

\begin{array}{}\text{(54)}& \begin{array}{cc}& \omega {⟨⟨{\widehat{A}}^{+}{d}_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & =⟨{\widehat{A}}^{+}⟩-⟨⟨{\displaystyle \sum _k}{E}_k{C}_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}{d}_{\sigma }\mid d_{\sigma }^{+}⟩{⟩}_{\omega }\hfill \\ & =⟨{\widehat{A}}^{+}⟩-{\displaystyle \sum _{k,k^{\prime }\left(k\ne k^{\prime }\right)}}{E}_k⟨C_{k\sigma }^{+}{C}_{k^{\prime }\overline{\sigma }}⟩⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩{⟩}_{\omega }\hfill \end{array}\end{array}

Now, combining Eqs. (36), (42), (45), (48), (51) and (54), one can obtain the Green function {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }:

\begin{array}{}\text{(55)}& \begin{array}{cc}& {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }\hfill \\ & ={\displaystyle \frac{(\omega -J⟨{\widehat{A}}^z⟩)({\omega }^2-M)-N}{{({\omega }^2-2J^2{⟨{\widehat{A}}^z⟩}^2)}^2-{\displaystyle \frac{J^2}{4}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime },\sigma }{k\ne k^{\prime }}}}{E}_k^2⟨C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩}}\hfill \end{array}\end{array}

where we introduce symbols:

\begin{array}{}\text{(56)}& M=2J^2⟨{\widehat{A}}^z⟩+{\displaystyle \frac{J}{2}}{\displaystyle \sum _{k,k^{\prime },\left(k\ne k^{\prime }\right)}}{E}_k⟨C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }}-C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩\end{array}

and

\begin{array}{}\text{(57)}& N=J^2{\displaystyle \sum _{k,k^{\prime },\left(k\ne k^{\prime }\right)}}{E}_k^2⟨C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\sigma }⟩⟨{\widehat{A}}^{+}⟩\end{array}

In order to obtain the function pole, the denominator of {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega } must be equal to zero, which allows one to have the function pole as:

\begin{array}{}\text{(58)}& \omega ={\left|2J^2{⟨{\widehat{A}}^z⟩}^2\pm {\displaystyle \frac{J}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime },\sigma }{k\ne k^{\prime }}}}{E}_k⟨C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩\right|}^{\frac{1}{2}}\end{array}

This allows the energy of d electrons to be given by

\begin{array}{}\text{(59)}& \begin{array}{cc}{E}_{d\sigma }=\hfill & |2J^2{⟨{\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})⟩}^2\hfill \\ & \pm {\displaystyle \frac{J}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime },\sigma }{k\ne k^{\prime }}}}{E}_k⟨C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩{|}^{\frac{1}{2}}\hfill \end{array}\end{array}

The obtained energy formula shows that the energy spectrum E_{d\sigma } is related to the strong correlation coupling J which is a kind of function related to E_d, and interaction between d electrons and s electrons. However, E_{d\sigma } is independent on individual k, k^{\prime }, and only changes with σ since {\displaystyle \frac{1}{2}}{\sum }_{k,k^{\prime }}{(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})}^2 change’s with σ.

5 Spectral function of Green function

We now turn into the expression {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega } following standard form, which gives

\begin{array}{}\text{(60)}& {⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega }={\displaystyle \frac{A_1}{\omega -E_0}}+{\displaystyle \frac{A_2}{\omega +E_0}}+{\displaystyle \frac{A_3\omega +A_4}{{\omega }^2-E_1^2}}\end{array}

where \omega =E_0 or {\omega }^2= E_1^2 is a pole of Eq. (55). According to the Green function’s spectral theorem,

\begin{array}{}\text{(61)}& ⟨AB⟩={\displaystyle \frac{\text{i}}{2\text{π}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{⟨⟨A\mid B⟩⟩}_{\omega +\mathrm{i}\eta }-{⟨⟨A\mid B⟩⟩}_{\omega -\mathrm{i}\eta }}{{\mathrm{e}}^{\beta \omega }+1}}\mathrm{d}\omega \end{array}

the average value ⟨d_{\sigma }{d}_{\sigma }^{+}⟩ represents the number of s electrons with spin \sigma in the nodes and can be calculated by

\begin{array}{}\text{(62)}& \begin{array}{ccc}& ⟨d_{\sigma }{d}_{\sigma }^{+}⟩\hfill & \\ & ={\displaystyle \frac{\text{i}}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{\displaystyle \frac{{⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega +\mathrm{i}\eta }-{⟨⟨d_{\sigma }\mid d_{\sigma }^{+}⟩⟩}_{\omega -\mathrm{i}\eta }}{{\mathrm{e}}^{\beta \omega }+1}}\mathrm{d}\omega \hfill & \\ & ={\displaystyle \frac{\text{i}}{2\text{π}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}[{\displaystyle \frac{-2\mathrm{i}\eta A_1}{{(\omega -E_0)}^2+{\eta }^2}}+{\displaystyle \frac{-2\mathrm{i}\eta A_2}{{(\omega +E_0)}^2+{\eta }^2}}\hfill & \\ & +{\displaystyle \frac{A_3(\omega +\mathrm{i}\eta )+A_4}{{(\omega +\mathrm{i}\eta )}^2-E_1^2}}-{\displaystyle \frac{A_3(\omega -\mathrm{i}\eta )+A_4}{{(\omega -\mathrm{i}\eta )}^2-E_1^2}}]{\displaystyle \frac{1}{\mathrm{e}{}^{\beta \omega }+1}}\mathrm{d}\omega \hfill & \end{array}\end{array}

where A₁, A₂, A₃, and A₄ are defined by

\begin{array}{}\text{(63)}& \begin{array}{c}{A}_1={\displaystyle \frac{(E_0^2-M)(E_0-J⟨{\widehat{A}}^z⟩)-N}{2(E_0^2-E_1^2)E_0}}\hfill \\ A_2={\displaystyle \frac{(E_0^2-M)(E_0-J⟨{\widehat{A}}^z⟩)-N}{2(E_0^2-E_1^2)E_0}}\hfill \\ A_3={\displaystyle \frac{M-E_1^2}{E_0^2-E_1^2}}\hfill \\ A_4={\displaystyle \frac{N-J⟨{\widehat{A}}^z⟩(M+E_1^2)}{E_0^2-E_1^2}}\hfill \\ E_0={\left|2J^2{⟨{\widehat{A}}^z⟩}^2+{\displaystyle \frac{J}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime },\sigma }{k\ne k^{\prime }}}}{E}_k⟨C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩\right|}^{\frac{1}{2}}\hfill \\ E_1={\left|2J^2{⟨{\widehat{A}}^z⟩}^2-{\displaystyle \frac{J}{2}}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime },\sigma }{k\ne k^{\prime }}}}{E}_k⟨C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }⟩\right|}^{\frac{1}{2}}\hfill \end{array}\end{array}

and

\begin{array}{}\text{(64)}& {\widehat{A}}^z={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime }}}(C_{k\sigma }^{+}{C}_{k^{\prime }\sigma }-C_{k\overline{\sigma }}^{+}{C}_{k^{\prime }\overline{\sigma }})\end{array}

Using a theorem of residues in the complex function {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(\omega )d\omega =2\text{π}\mathrm{i}{\displaystyle \sum _k}\mathrm{Re}sf({\omega }_k), through long term calculation, and taking the limit, finally we can obtain

\begin{array}{}\text{(65)}& \begin{array}{cccc}& ⟨d_{\sigma }{d}_{\sigma }^{+}⟩=\underset{\eta \to 0}{lim}{A}_1{[{\mathrm{e}}^{\beta (E_0+\mathrm{i}\eta )}+1]}^{-1}\hfill & & \\ & +\underset{\eta \to 0}{lim}{A}_2{[{\mathrm{e}}^{\beta (-E_0+\mathrm{i}\eta )}+1]}^{-1}\hfill & & \\ & +\underset{\eta \to 0}{lim}(-A_3{E}_1-A_4)[{\displaystyle \frac{{[{\mathrm{e}}^{\beta (E_1-\mathrm{i}\eta )}+1]}^{-1}}{2E_1}}\hfill & & \\ & +{\displaystyle \frac{{[{\mathrm{e}}^{\beta (E_1+\mathrm{i}\eta )}+1]}^{-1}}{2E_1}}]\hfill & & \\ & +\underset{\eta \to 0}{lim}(-A_3{E}_1+A_4)[{\displaystyle \frac{{\mathrm{e}}^{\beta (E_1+\mathrm{i}\eta )}{[{\mathrm{e}}^{\beta (E_1+\mathrm{i}\eta )}+1]}^{-1}}{2E_1}}\hfill & & \\ & +{\displaystyle \frac{{\mathrm{e}}^{\beta (E_1-\mathrm{i}\eta )}{[{\mathrm{e}}^{\beta (E_1-\mathrm{i}\eta )}+1]}^{-1}}{2E_1}}]\hfill & & \\ & ={({\mathrm{e}}^{\beta E_0}+1)}^{-1}\left(A_1+A_2{E}_0\right)-A_3{E}_0{E}_1\hfill & & \\ & +{\displaystyle \frac{{({\mathrm{e}}^{\beta E_1}+1)}^{-1}}{E_0}}{A}_4\left({\mathrm{e}}^{\beta E_1}-1\right)\hfill & & \end{array}\end{array}

From this we can see that the mean number of up or down spin for d electrons is related with that of c electrons. When the spin is up \left(\sigma =\uparrow \right), the above formula can be calculated as:

\begin{array}{}\text{(66)}& ⟨d_{\uparrow }{d}_{\uparrow }^{+}⟩={({\mathrm{e}}^{\beta E_0}+1)}^{-1}(A_{1\uparrow }+A_{2\uparrow }{E}_0)-A_{3\uparrow }{E}_0{E}_1+{\displaystyle \frac{{({\mathrm{e}}^{\beta E_1}+1)}^{-1}}{E_0\uparrow }}{A}_{4\uparrow }({\mathrm{e}}^{\beta E_1}-1)\end{array}

where the relevant parameter is given by

\begin{array}{}\text{(67)}& \begin{array}{c}{A}_{1\uparrow }={\displaystyle \frac{(E_0^2-M_{\uparrow })(E_0-J⟨{\widehat{A}}_{\uparrow \downarrow }^z⟩)-N_{\uparrow }}{2(E_0^2-E_1^2)E_0}}\hfill \\ A_{2\uparrow }={\displaystyle \frac{(E_0^2-M_{\uparrow })(E_0-J⟨{\widehat{A}}_{\uparrow \downarrow }^z⟩)-N_{\uparrow }}{2(E_0^2-E_1^2)E_0}}\hfill \\ A_{3\uparrow }={\displaystyle \frac{M_{\uparrow }-E_1^2}{E_0^2-E_1^2}}\hfill \\ A_{4\uparrow }={\displaystyle \frac{N_{\uparrow }-J⟨{\widehat{A}}_{\uparrow \downarrow }^z⟩(M_{\uparrow }+E_1^2)}{E_0^2-E_1^2}}\hfill \end{array}\end{array}

\begin{array}{}\text{(68)}& \begin{array}{c}{M}_{\uparrow }=2J^2⟨{\widehat{A}}_{\uparrow \downarrow }^z⟩+{\displaystyle \frac{J}{2}}{\displaystyle \sum _{k,k^{\prime }(k\ne k^{\prime })}}{E}_k(⟨C_{k\downarrow }^{+}{C}_{k^{\prime }\downarrow }⟩-⟨C_{k\uparrow }^{+}{C}_{k^{\prime }\uparrow }⟩)\hfill \\ N_{\uparrow }=J^2{\displaystyle \sum _{k,k^{\prime }(k\ne k^{\prime })}}{E}_k^2⟨C_{k\downarrow }^{+}{C}_{k^{\prime }\downarrow }⟩⟨{\widehat{A}}_{\uparrow \downarrow }^{+}⟩\hfill \end{array}\end{array}

and

\begin{array}{}\text{(69)}& \begin{array}{c}⟨{\widehat{A}}_{\uparrow \downarrow }^z⟩={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k,k^{\prime }}}\left(⟨C_{k\uparrow }^{+}{C}_{k^{\prime }\uparrow }⟩-⟨C_{k\downarrow }^{+}{C}_{k^{\prime }\downarrow }⟩\right)\hfill \\ ⟨{\widehat{A}}_{\uparrow \downarrow }^{+}⟩={\displaystyle \sum _{k,k^{\prime }}}⟨C_{k\uparrow }^{+}{C}_{k^{\prime }}\downarrow ⟩\hfill \end{array}\end{array}

In the same way, while the spin is down \left(\sigma =\downarrow \right),⟨d_{\downarrow }{d}_{\downarrow }^{+}⟩ is calculated as

\begin{array}{}\text{(70)}& ⟨d_{\downarrow }{d}_{\downarrow }^{+}⟩={({\mathrm{e}}^{\beta E_0}+1)}^{-1}(A_{1\downarrow }+A_{2\downarrow }{E}_0)-A_{3\downarrow }{E}_0{E}_1+{\displaystyle \frac{{({\mathrm{e}}^{\beta E_1}+1)}^{-1}}{E_0}}{A}_{4\downarrow }({\mathrm{e}}^{\beta E_1}-1)\end{array}

Both cases show that the electronic number of up spin is not equal to that of down spin, ⟨d_{\uparrow }{d}_{\uparrow }^{+}⟩\ne ⟨d_{\downarrow }{d}_{\downarrow }^{+}⟩, hence, the local magnetism can appear under this condition.

6 Application in network

The above model constructed iteratively by the creation and destruction of the d electrons (nodes) with interaction of the environmental s electrons can be specialized as a kind of fermionic network. Generally speaking, in this model, at each time step, a new node (d electron confined in an impurity) with m (or m^{\prime }) links ( strongly correlated to m confined d electrons) are added (or cancelled) to the network. In terms of the idea of fermionic network in the Refs. [3, 4], the new node has an energy and a new link is attached, with probability

\begin{array}{}\text{(70)}& P_j\propto {\mathrm{e}}^{\beta E_j}k\left(t|E_j,t_j\right)\end{array}

to node j arriving at the interface network at time t_j, with energy E_j and survivability k\left(t|E_j,t_j\right) at time t. Therefore, the continuous equation describing the time evolution of the survivability k\left(t|E_j,t_j\right) of the nodes is expressed by

\begin{array}{}\text{(71)}& {\displaystyle \frac{\partial k\left(t\right|E_j,t_j)}{\partial t}}=m{\displaystyle \frac{{\mathrm{e}}^{\beta E_j}k\left(t\right|E_j,t_j)}{{\displaystyle \sum _l}{\mathrm{e}}^{\beta E_j}k\left(t\right|E_l,t_l)}}\end{array}

with the initial condition

\begin{array}{}\text{(72)}& k_0=k\left(t|E_j,t\right)\end{array}

By using the asymptotic behavior in the thermodynamic limit

\begin{array}{}\text{(73)}& {\displaystyle \sum _k}{\mathrm{e}}^{\beta E_j}k\left(t|E_k,t_k\right)=Z⟶⟨Z⟩⟶mt{\mathrm{e}}^{\beta \mu }\end{array}

Eq. (71) can be reduced as

\begin{array}{}\text{(74)}& {\displaystyle \frac{\partial k\left(t\right|E_j,t_j)}{\partial t}}=m{\mathrm{e}}^{\beta \left(E_j-\mu \right)}{t}_j\frac{k\left(t|E_j,t_j\right)}{t}\end{array}

whose solution is found as

\begin{array}{}\text{(75)}& k\left(t|E_j,t_j\right)=k_{0}m{\left({\displaystyle \frac{t}{t_j}}\right)}^{f\left(E_j\right)}\end{array}

with

\begin{array}{}\text{(76)}& f\left(E_j\right)={\mathrm{e}}^{\beta \left(E_j-{\mu }_i\right)}\end{array}

where f\left(E_i\right) is the dynamic exponent which depends on the energy E_i, {\mu }_i is a chemical potential and β is the inverse temperature β. Therefore, the degree distribution is given by

\begin{array}{}\text{(77)}& P(k)={\displaystyle \int dEp(E){\displaystyle \frac{1}{{\displaystyle \frac{\partial k}{\partial t}}}}}\sim B{k}^{-1}\end{array}

where

\begin{array}{}\text{(78)}& B={\displaystyle \int dEp\left(E\right)}\end{array}

which demonstrates that the fermionic network system is scale-free if B is irrelevant to k or quite small. This allows one to find a novel way to study the local magnetism existence for the network. In fact, using the obtained survivability k\left(t|E_{d\uparrow },t_{d\uparrow }\right) for up spin and k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right) for down spin, one can find that the local magnetism existence can be judged by k\left(t|E_{d\uparrow },t_{d\uparrow }\right)\stackrel{\mathrm{?}}{=}k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right), since the survivability reflects the number of the up spins or down spins in the interface of the network. This interface evolves with time to construct the part of the network, hence the number of the up spins or down spins in the interface can reflect the total number of the up spins or down spins as the nodes in the network. That means if k\left(t|E_{d\uparrow },t_{d\uparrow }\right)=k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right), there is no local magnetism, otherwise, k\left(t|E_{d\uparrow },t_{d\uparrow }\right)\ne k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right), there exist the local magnetism. The local magnetism exists if the number of the up spins and down spins are not equal in the interface of the network [4],

\begin{array}{}\text{(79)}& N_{\uparrow }={\displaystyle \sum _{t_{d\uparrow }}}k\left(t|E_{d\uparrow },t_{d\uparrow }\right)\ne N_{\downarrow }={\displaystyle \sum _{t_{d^{\prime }\downarrow }}}k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right)\end{array}

Both cases can be expressed by

\begin{array}{}\text{(80)}& \begin{array}{c}k(t|E_{d\uparrow },t_{d\uparrow })=k_{0}m{\left({\displaystyle \frac{t}{t_{d\uparrow }}}\right)}^{f\left(E_{d\uparrow }\right)}\hfill \\ \{\begin{array}{cccc}\ne k(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow })=k_{0}m{\left({\displaystyle \frac{t}{t_{d^{\prime }\downarrow }}}\right)}^{f\left(E_{d^{\prime }\downarrow }\right)},\hfill & & & \\ \mathrm{for}{E}_{d\uparrow }\ne E_{d^{\prime }\downarrow },t_{d\uparrow }=t_{d^{\prime }\downarrow }\hfill & & & \\ =k(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow })=k_{0}m{\left({\displaystyle \frac{t}{t_{d^{\prime }\downarrow }}}\right)}^{f\left(E_{d\downarrow }\right)},\hfill & & & \\ \mathrm{for}{E}_{d\uparrow }=E_{d^{\prime }\downarrow },t_{d\uparrow }=t_{d^{\prime }\downarrow }\hfill & & & \end{array}\hfill \end{array}\end{array}

The above result shows that the origin of the local magnetism existence actually depends on k\left(t|E_{d\uparrow },t_{d\uparrow }\right)\ne k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right) which is related to the energy of the local nodes of the network. It is the difference of d electron energy for up spin and down spin, E_{d\uparrow }\ne E_{d^{\prime }\downarrow }, distributed in the local region, to result in the difference of k\left(t|E_{d\uparrow },t_{d\uparrow }\right) and k\left(t|E_{d^{\prime }\downarrow },t_{d^{\prime }\downarrow }\right), that introduces the local magnetism for the system.

7 Conclusion

The energy spectrum for the d electrons in the Schrieffer and Wolf transformation for the quantum network described by the Anderson d – s model is calculated using the Green function approach. The obtained energy formula shows that it is related to the strongly coupling number J and the interaction between d and s electrons through spins. The local magnetism of the model can be studied by using the spectral function of Green function, which shows how the energy of the system influence the number of up or down spin. Moreover, the given survivability of the fermionic network model can be used to study the local magnetism. The local magnetism existence actually depends on the difference of d electron energy for up spin and down spin. E_{d\uparrow }\ne E_{d^{\prime }\downarrow } distributed in the local nodes of the network can introduce the difference of the survivability for the up or down spins. This results in local magnetism in the network, which shows that survivability is an important statistical characteristic quantity not only to reflect the fermionic network topological property but also dynamics.

References

[1] P. W. Anderson, Phys. Rev., 1961, 124: 41

[2] J. R. Schrieffer and P. A. Wolff, Phys. Rev., 1966, 149: 491

[3] G. Bianconi, arXiv: cond-mat/0204506v2

[4] G. Bianconi, Phys. Rev. E, 2002, 66: 056123

[5] D. Baeriswyl, D. K. Campbell, J. M. P. Carmelo, F. Guinea, and E. Louis, The Hubbard Model, New York: Pienum Press, 1995

[6] J. R. Schrieffer and P. A. Wolff, Phys. Rev., 1966, 149: 491

[7] L. Z. Zheng, Solid State Theory, Beijing: Higher Education Press, 2002 (in Chinese)

[8] I. Antoniou, Y. Melnikov, and Q. Bi, Physica A, 1997, 246: 97

[9] Q. Bi, H. E. Ruda, M. S. Zhang, and X. H. Zeng, Physica A, 2003, 322: 345

Rights & permissions

©Higher Education Press and Springer-Verlag 2009