In this thesis we investigate the frustrated spin liquid Tb$_2$Ti$_2$O$_7$ theoretically. The low-energy effective Hamiltonian of this compound is derived by integrating out the excited crystal field states. It is shown that the pairwise interaction in the effective Hamiltonian is renormalized by all the other Tb$_2$Ti$_2$O$_7$ ions in the system and dynamically generate frustration. The phase diagram of Tb$_2$Ti$_2$O$_7$ in the single tetrahedron approximation is calculated. It is shown that Tb$_2$Ti$_2$O$_7$ is in a singlet state which is a linear combination of all frustrated two-in/two-out states. Motivated by experimental results, the diffuse neutron scattering of Tb$_2$Ti$_2$O$_7$ is obtained within the single tetrahedron approximation. This diffuse neutron scattering captures semi-quantitatively most of the experimental neutron scattering features. The magnetization of Tb$_2$Ti$_2$O$_7$ in the single tetrahedron approximation is calculated. Two experiments based on diffuse neutron scattering and magnetization in high symmetry directions are proposed to verify the spin ice like ground state of Tb$_2$Ti$_2$O$_7$.

In this thesis we investigate the frustrated spin liquid Tb$_2$Ti$_2$O$_7$ theoretically. The low-energy effective Hamiltonian of this compound is derived by integrating out the excited crystal field states. It is shown that the pairwise interaction in the effective Hamiltonian is renormalized by all the other Tb$_2$Ti$_2$O$_7$ ions in the system and dynamically generate frustration. The phase diagram of Tb$_2$Ti$_2$O$_7$ in the single tetrahedron approximation is calculated. It is shown that Tb$_2$Ti$_2$O$_7$ is in a singlet state which is a linear combination of all frustrated two-in/two-out states. Motivated by experimental results, the diffuse neutron scattering of Tb$_2$Ti$_2$O$_7$ is obtained within the single tetrahedron approximation. This diffuse neutron scattering captures semi-quantitatively most of the experimental neutron scattering features. The magnetization of Tb$_2$Ti$_2$O$_7$ in the single tetrahedron approximation is calculated. Two experiments based on diffuse neutron scattering and magnetization in high symmetry directions are proposed to verify the spin ice like ground state of Tb$_2$Ti$_2$O$_7$.