Effective Hamiltonian of a three-orbital Hubbard model on the pyrochlore lattice: Application toLiV2O4

Abstract

To investigate heavy fermion behavior in the vanadium spinel ${\text{LiV}}_{2}{\text{O}}_{4}$, we start from a three-orbital Hubbard model on the pyrochlore lattice and derive its low-energy effective Hamiltonian by an approach of real-space renormalization-group type. We first derive the effective Hamiltonian numerically and then succeed in representing the results into an analytic form with physical operators for low-energy degrees of freedom in tetrahedron unit. The effective Hamiltonian is defined on the coarse-grained lattice, i.e., face-centered-cubic (fcc) lattice, and it operates in a restricted Hilbert space defined in terms of a specific molecular orbital ${T}_{2}$ in the unit. One important tetrahedron configuration has a threefold orbital degeneracy and spin $S=1$, and correspondingly, the effective Hamiltonian has spin and orbital exchange interactions of Kugel-Khomskii type as well as correlated electron hoppings. The coupling constants in the effective Hamiltonian are determined from the numerically obtained renormalized Hamiltonian and also by means of perturbation. We calculate and analyze low-energy states of the effective Hamiltonian for the unit of four coupled tetrahedra both analytically and numerically. Effective hopping elements in the effective Hamiltonian are renormalized to about 1/10 of the original hopping integral. It is important that different virtual processes make opposite contributions to the exchange term, and consequently the coupling constant is given by a remaining small value. This is particularly prominent in the spin-spin channel, where ferromagnetic double-exchange processes compete with antiferromagnetic superexchange processes. Another important point is that various spin and orbital exchange processes are competing to each other. Together with geometrical frustration of the effective fcc lattice, these two features result in nearly degenerate three lowest-energy states of different types in the four coupled tetrahedra, and each of the three has a finite degeneracy in spin and/or orbital. We also calculate spatial correlations of spin and orbital and found that short-range spin-spin correlations are strongly entangled with orbital configurations. This indicates that large remaining entropy at low temperature is related to slow coupled fluctuations of spin and orbital. These results suggest the absence of phase transition in spin and orbital spaces down to very low temperatures and their large fluctuations in the low-energy sector, which are key issues for understanding the heavy fermion behavior in ${\text{LiV}}_{2}{\text{O}}_{4}$.