t-J model ($J=4t^2/U$)

An effective low-energy lattice model obtained from a strongly repulsive Hubbard model near one electron per site. The retained Hilbert space excludes double occupancy. Its usual nearest-neighbor Hamiltonian has projected hopping $-t{\sum }_{⟨ij⟩,\sigma }({\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+h.c.)$ and antiferromagnetic superexchange $J{\sum }_{⟨ij⟩}({\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4)$. Virtual intermediate states with one double occupancy cost energy $U$, giving $J=4t^2/U$ to leading order. Three-site hopping terms of the same expansion order are omitted in this simplified version.

At half filling the projected hopping is blocked. Doped holes restore charge motion coupled to spin correlations. Nearest-neighbor singlet pairing can support a d-wave bond order ansatz on a square lattice, but the Hamiltonian alone is not a proof of a unique superconducting ground state for all parameters.

An electron annihilation operator restricted to the no-double-occupancy sector of the t-J model. It removes a spin-$\sigma$ electron only when the opposite-spin site occupation vanishes. The corresponding creation operator cannot create a doubly occupied site. These projected operators do not obey the unrestricted canonical on-site fermion algebra on all states.