t-J model

The t-J model was first derived in 1977 from the Hubbard model by Józef Spałek. The model describes strongly correlated electron systems. It is used to calculate high temperature superconductivity states in doped antiferromagnets.

The t-J Hamiltonian is:

\hat H = -t\sum_{<ij>\sigma}\left(\hat a^\dagger_{i\sigma} \hat a_{j\sigma} + \hat a^\dagger_{j\sigma} \hat a_{i\sigma}\right)
+
J\sum_{<ij>}(\vec S_{i}\cdot \vec S_{j}-n_in_j/4)

where

Connection to the high-temperature superconductivity

The Hamiltonian of the  t_{1}-t_{2}-J model in terms of  CP^{1} generalized model reads [1]


\mathbf{H} = t_1 \sum\limits_{<i,j>} \bigg( c_{i\sigma}^{\dagger} c_{j\sigma} + h.c. \bigg) \ + \ t_2 \sum\limits_{<<i,j>>} \bigg( c_{i\sigma}^{\dagger} c_{j\sigma} + h.c. \bigg) \ + \ J \sum\limits_{<i,j>} \bigg( \mathbf{S}_{i} \cdot \mathbf{S}_{j} - \frac{1}{4} n_{i} n_{j} \bigg) 
  - \ \mu\sum\limits_{i} n_{i} ,

where fermionic operators  c_{i\sigma} ,  c_{i\sigma} , the spin operators  \mathbf{S}_i and  \mathbf{S}_j , number operators  n_{i} and  n_{j} act on restricted Hilbert space and the doubly-occupied states are excluded. The sums in above mentioned equation are over all sites of a (2-d) square lattice, where  <...> and  <<...>> denote nearest and next-to-the-nearest neighbors, respectively.

References


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