Effective interactions in the periodic Anderson model in the regime of mixed valency with strong correlations

For the periodic Anderson model in the strong correlation regime, we construct the effective Hamiltonian $H_{\mathrm{eff}}$ up to terms of the fourth order in the parameter $V/U$, where $V$ is the hybridization interaction intensity and $U$ is the intra-atom Coulomb repulsion strength. This Hamiltonian contains interactions inducing both magnetic ordering and Cooper instability under conditions of a mixed valency of rare-earth ions. Based on numerical calculations, we obtain information about the dependences of the effective interaction parameters on the distance between crystal lattice sites. We demonstrate that realizing exchange interactions corresponds to a strongly frustrated system of localized spin moments and facilitates the suppression of the antiferromagnetic order parameter with a possible transition to the state of a quantum spin liquid. It is essential that among the terms in $H_{\mathrm{eff}}$ inducing the transition to the superconductivity phase, there are terms resulting in the $d$-type symmetry of the superconductivity order parameter; such a symmetry is realized in many heavy-fermion compounds.