  We study three-body systems in a mass-imbalanced two-component cold-atom
mixture, and investigate the three-body parameter of their Efimov states for
both bosonic and fermionic systems, with a major focus on the Er-Er-Li Efimov
states. For a system interacting solely via van der Waals interactions, the van
der Waals universality of the three-body parameter is analytically derived
using the quantum defect theory. With the addition of a perturbative dipole
interaction between the heavy atoms, the three-body parameters of the bosonic
and fermionic Efimov states are found to behave differently. When the dipole
interaction is as strong as the van der Waals interaction, corresponding to
realistic Er-Er-Li Efimov states, we show that the van der Waals universality
persists once the effects of the non-perturbative dipole interaction are
renormalized into the s-wave and p-wave scattering parameters between the heavy
atoms. For a dipole interaction much stronger than the van der Waals
interaction, we find that the universality of the Efimov states can be
alternatively characterized by a quasi-one-dimensional scattering parameter due
to a strong anisotropic deformation of the Efimov wavefunctions. Our work thus
clarifies the interplay of isotropic and anisotropic forces in the universality
of the Efimov states. Based on the renormalized van der Waals universality, the
three-body parameter is estimated for specific isotopes of Er-Li cold-atom
mixtures.
