# 非对称性Anderson晶格模型中的磁性相变研究Magnetic Transition in Non-Symmetric Anderson Lattice Model

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Anderson晶格模型是描述重费米子系统的重要理论模型，而隶玻色子方法是求解该模型的常用方法。然而，通常的Kotliar-Ruckenstein隶玻色子方法只能处理该模型的粒子–空穴对称情形。本文将Kotliar-Ruckenstein方法加以推广，使之能够处理偏离粒子–空穴对称的非对称情形，并用之研究了非对称性Anderson晶格模型中的磁性相变，展示了晶格维度、电子跳迁系数、库仑关联强度、电子杂化强度、局域电子能级等多种参数对磁性相变边界的影响。

Anderson lattice model, which can be solved by the widely used slave-boson method, is a standard model to describe the heavy-fermion systems. However, the normal Kotliar-Ruckenstein slave-boson method is limited to the case with particle-hole symmetry, particular in investigating the magnetic phase in Anderson lattice model. In this paper, the Kotliar-Ruckenstein method is generalized to include the non-symmetric case which violates the particle-hole symmetry. Then, the magnetic phase boundary in Anderson lattice model is studied as a function of lattice dimension, electron-hopping amplitudes, Coulomb correlation, electron hybridization and local energy level.

1. 引言

2. 模型与计算方法

$\begin{array}{c}H=\underset{i,j,\sigma }{\sum }\left({t}_{ij}^{d}{d}_{i\sigma }^{†}{d}_{j\sigma }+{t}_{ij}^{f}{f}_{i\sigma }^{†}{f}_{j\sigma }\right)+{ϵ}_{f}\underset{i,\sigma }{\sum }\text{\hspace{0.17em}}{f}_{i\sigma }^{†}{f}_{i\sigma }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+U\underset{i}{\sum }\text{\hspace{0.17em}}{n}_{i↑}^{f}{n}_{i↓}^{f}\text{+}V\underset{i,\sigma }{\sum }\left({d}_{i\sigma }^{†}{f}_{\sigma }+h.c.\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mu \underset{i,\sigma }{\sum }\left({d}_{i\sigma }^{†}{d}_{i\sigma }+{f}_{i\sigma }^{†}{f}_{i\sigma }\right)\end{array}$ (1)

$\begin{array}{l}H=\underset{i}{\sum }\left(U{\delta }_{i}+{h}_{i}{m}_{i}^{f}-{\eta }_{i}{n}_{i}^{f}\right)+\underset{i,j,\sigma }{\sum }\left({t}_{ij}^{d}-\mu {\delta }_{ij}\right){d}_{i\sigma }^{†}{d}_{j\sigma }+\underset{i,\sigma }{\sum }\left({ϵ}_{f}+{\eta }_{i}-\sigma {h}_{i}-\mu \right){f}_{i\sigma }^{†}{f}_{i\sigma }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i,j,\sigma }{\sum }\text{\hspace{0.17em}}{t}_{ij}^{f}{Z}_{i\sigma }{Z}_{j\sigma }{f}_{i\sigma }^{†}{f}_{j\sigma }+V\underset{i,\sigma }{\sum }\left({Z}_{i\sigma }{d}_{i\sigma }^{†}{f}_{i\sigma }+h.c.\right)\end{array}$ (2)

$\begin{array}{l}{Z}_{1}=\frac{\sqrt{2}\left(\sqrt{\left({n}_{f}-2\delta +{m}_{f}\right)\left(1-{n}_{f}+\delta \right)}+\sqrt{\delta \left({n}_{f}-2\delta -{m}_{f}\right)}\right)}{\sqrt{\left({n}_{f}+{m}_{f}\right)\left(2-{n}_{f}-{m}_{f}\right)}},\\ {Z}_{2}=\frac{\sqrt{2}\left(\sqrt{\left({n}_{f}-2\delta -{m}_{f}\right)\left(1-{n}_{f}+\delta \right)}+\sqrt{\delta \left({n}_{f}-2\delta +{m}_{f}\right)}\right)}{\sqrt{\left({n}_{f}-{m}_{f}\right)\left(2-{n}_{f}+{m}_{f}\right)}}\end{array}$ (3)

$H=N\left(U\delta +h{m}_{f}-\eta {n}_{f}\right)+\underset{k\in \text{MBZ}}{\sum }\text{\hspace{0.17em}}{\Psi }_{k}^{†}{H}_{k}{\Psi }_{k},$ (4)

${\Psi }_{k}={\left({d}_{kA↑},{d}_{kA↓},{d}_{kB↑},{d}_{kB↓},{f}_{kA↑},{f}_{kA↓},{f}_{kB↑},{f}_{kB↓}\right)}^{\text{T}}$ ，哈密顿矩阵为

${H}_{k}=\left(\begin{array}{cc}{H}_{k}^{d}& {V}_{k}\\ {V}_{k}^{+}& {H}_{k}^{f}\end{array}\right),$ (5)

${V}_{k}=V\left(\begin{array}{cccc}{Z}_{1}& 0& 0& 0\\ 0& {Z}_{2}& 0& 0\\ 0& 0& {Z}_{2}& 0\\ 0& 0& 0& {Z}_{1}\end{array}\right),$

${H}_{k}^{f}=\left(\begin{array}{cccc}{e}_{1k}& 0& {Z}_{1}{Z}_{2}{t}_{f}{\lambda }_{k}& 0\\ 0& {e}_{2k}& 0& {Z}_{1}{Z}_{2}{t}_{f}{\lambda }_{k}\\ {Z}_{1}{Z}_{2}{t}_{f}{\lambda }_{k}& 0& {e}_{2k}& 0\\ 0& {Z}_{1}{Z}_{2}{t}_{f}{\lambda }_{k}& 0& {e}_{1k}\end{array}\right),$

3. 粒子–空穴对称情况

$H=N\left(U\delta +h{m}_{f}-U/2\right)+\underset{k\in \text{MBZ,}\sigma }{\sum }\text{\hspace{0.17em}}{\Psi }_{k\sigma }^{†}{H}_{k\sigma }{\Psi }_{k\sigma },$ (6)

$Q=\left(\pi ,\pi ,\pi \right)$ (三维)或 $Q=\left(\pi ,\pi \right)$ (二维)，哈密顿量矩阵为

${H}_{k\sigma }=\left(\begin{array}{cccc}{\epsilon }_{k}& 0& VZ& 0\\ 0& -{\epsilon }_{k}& 0& VZ\\ VZ& 0& \alpha {Z}^{2}{\epsilon }_{k}& 0\\ 0& VZ& 0& -\alpha {Z}^{2}{\epsilon }_{k}\end{array}\right),$ (7)

$\begin{array}{l}{E}_{k}^{±}=\frac{1}{\sqrt{2}}\sqrt{{E}_{1k}±\sqrt{{E}_{1k}^{2}-{E}_{2k}^{2}}},\\ {E}_{1k}=\left(\text{1+}{\alpha }^{2}{Z}^{4}\right){\epsilon }_{k}^{2}+{h}^{2}+2{V}^{2}{Z}^{2},\\ {E}_{2k}=\sqrt{4{\epsilon }_{k}^{2}{h}^{2}+4{Z}^{4}{\left({V}^{2}-\alpha {\epsilon }_{k}^{2}\right)}^{2}}\end{array}$ (8)

$\begin{array}{l}U-\frac{1}{N}\underset{k\in \text{BZ}}{\sum }\frac{2Z}{\sqrt{{E}_{1k}+{E}_{2k}}}\left[{V}^{2}+{t}_{f}^{2}{Z}^{2}{\epsilon }_{k}^{2}+\frac{2{Z}^{2}{\left({V}^{2}-{t}_{f}{\epsilon }_{k}^{2}\right)}^{2}}{{E}_{2k}}\right]\cdot \frac{\partial Z}{\partial \delta }=0,\\ {m}_{f}-\frac{1}{N}\underset{k\in \text{BZ}}{\sum }\frac{h}{\sqrt{{E}_{1k}+{E}_{2k}}}\left(1+\frac{2{\epsilon }_{k}^{2}}{{E}_{2k}}\right)=0,\\ h=U\cdot \frac{\partial Z}{\partial {m}_{f}}/\frac{\partial Z}{\partial \delta }\end{array}$ (9)

Figure 1. The staggered magnetization ${m}_{f}$ of the symmetric Anderson lattice model vs d-f hybridization V in the cubic lattice. At critical ${V}_{c}$ , an antiferromagnetic transition takes place. Model parameters: ${t}_{f}=-0.2$ , $U=4$

Figure 2. Antiferromagnetic transition point of the symmetric Anderson lattice model vs Coulomb correlation energy U (left) or ${t}_{f}$ (right) in square (thin lines) and cubic lattice (thick lines). Parameters: ${t}_{f}=-0.2$ for left and $U=4$ for right

4. 非粒子–空穴对称情况

${E}_{g}=N\left(U\delta +h{m}_{f}-\eta {n}_{f}+\mu {n}_{t}\right)+\underset{k\in \text{MBZ}}{\sum }{\left({H}_{k}\right)}_{nm}{〈nm〉}_{k},$ (10)

$U+\frac{1}{N}\underset{n,m,k\in \text{MBZ}}{\sum }\frac{\partial {\left({H}_{k}\right)}_{nm}}{\partial \delta }{〈nm〉}_{k}=0,$ (11)

$\begin{array}{c}{〈nm〉}_{k}\equiv 〈{\left({\Psi }_{k}^{†}\right)}_{n}{\left({\Psi }_{k}\right)}_{m}〉\\ =〈\underset{i}{\sum }{\left({\Phi }_{k}^{†}\right)}_{i}{\left({U}_{k}\right)}_{ni}^{\ast }\underset{j}{\sum }{\left({U}_{k}\right)}_{mj}{\left({\Phi }_{k}\right)}_{j}〉\\ =\underset{i=1}{\overset{8}{\sum }}{\left({U}_{k}\right)}_{ni}^{\ast }{\left({U}_{k}\right)}_{mi}\Theta \left(-{E}_{k}^{\left(i\right)}\right)\end{array}$ (12)

Figure 3. The antiferromagnetic transition point ${V}_{c}$ of the non-symmetric Anderson lattice model vs Coulomb correlation energy U in the square (square dots) and cubic lattice (circle dots). Parameters: ${\epsilon }_{f}=-3$ , ${t}_{d}=0.15$ , ${t}_{f}=-0.2$ , ${{t}^{\prime }}_{f}=-0.02$

Figure 4. The antiferromagnetic transition point ${V}_{c}$ of the non-symmetric Anderson lattice model in large U limit vs f level ${\epsilon }_{f}$ (left) or ${t}_{f}$ (right) in the square lattice (square dots) and cubic lattice (circle dots). Parameters:, ${t}_{f}=-0.2$ , ${{t}^{\prime }}_{f}=-0.02$ for left; ${\epsilon }_{f}=-3$ , ${t}_{d}=0.15$ , ${{t}^{\prime }}_{f}=0.1{t}_{f}$ for right

5. 结论

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