Magnetic properties of the lattice Anderson model - Department of ...

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FIG. 1. The ground state energy per site as function of U for the symmetric FIG. 2. Square of the f orbital local moment as function of U for the same Anderson lattice model with r=l, V=OS. The dashed line is the weak case as in Fig. 1. coupling perturbation result from Eq. (7), and the solid line is the strong coupling perturbation result from Eq. (9). The squares are the exact diagonalixation results and the uncertainties in the infinite system extrapolation are within the size of the symbol. 2-1 mfz-2 4u +JqT (12) According to perturbation theory,““’ for the symmetric case in the weak coupling limit where U/A is small, the ground state energy per site is with E(U)=; 2 E;-; k -N3 U2 ui=$l+ei/(ei+4V2)1’2] , vi=gl-ei/(ei+4V2)“2] . In the strong coupling limit where U/A is large, the ground state energy per site is E(U)=-; +; c e&e,,-? T ;;;:;j 7 k (10) with f(ek) being the zero temperature Fermi factor. We show weak coupling results as the dashed line and strong coupling results as the solid line in Fig. 1. Note that there is a singularity in strong coupling limit as ZJ+O [roughly In(U)] but it is hard to see from Fig. 1. As U increases from 0 to ~0, electrons tend to localize in the f orbitals and form local moments. We measure these moments by m& defined by 4z=((nfT-nf1)2) , (11) which varies from 1 when U =O, to 1 when U=m. coupling we have (7) (8) (9) For weak while for strong coupling we have l-f(ek) m +-q c k (U/2+ek)’ * (13) We show these results as the dashed line and solid line in Fig. 2, along with data points obtained from our exact diagonalization studies. Again, estimated errors in extrapolations are within the size of the symbol. From Fig. 2, it is evident that weak coupling perturbation is invalid for U/A>8 and strong coupling perturbation gives bad answers for U/A
0.2 0 ,,,I I I I I I1 I I I I* I I I 0 1 z 3 4 FIG. 3. The ratio of the hybridization matrix element (c!,fi,+hc) to its lJ=O value as a function of U for the same case as in Fig. 1. changes between 1 and 2. (3) The Kondo region, in which one has nf- 1, n,-1. Its range is roughly -U- t
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