Effective Low-Energy Hamiltonian for the Frustrated Tb$_2$Ti$_2$O$_7$ Pyrochlore Antiferromagnet

The antiferromagnetic pyrochlore Tb$_2$Ti$_2$O$_7$ presents a challenging puzzle to experimentalists and theorists studying frustrated magnets. Results from muon spin resonance and neutron scattering experiments for Tb$_2$Ti$_2$O$_7$ reveal a paramagnetic structure down to $50$mK despite an antiferromagnetic Curie-Weiss temperature, $\theta_{\rm CW}=-20 $K. Crystal field calculations show that the Tb$^{3+}$ ion in Tb$_2$Ti$_2$O$_7$ is a ground state doublet with local $\langle 111 \rangle$ anisotropy and is separated from the first excited doublet state by a gap of 20K. To understand the quantum nature of Tb$_2$Ti$_2$O$_7$, we apply the Rayleigh- Schrodinger method to map the four state problem with exchange and dipole-dipole interactions onto an effective Hamiltonian with two states per ion. We give some properties of this effective hamiltonian and compare the neutron scattering results with the experimental data.