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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:
\( {\displaystyle {\hat {H}}=-t\sum _{<ij>\sigma }\left({\hat {a}}_{i\sigma }^{\dagger }{\hat {a}}_{j\sigma }+{\hat {a}}_{j\sigma }^{\dagger }{\hat {a}}_{i\sigma }\right)+J\sum _{<ij>}({\vec {S}}_{i}\cdot {\vec {S}}_{j}-n_{i}n_{j}/4)} \)
where
∑\( {\displaystyle \sum _{<ij>}} \) - sum over nearest-neighbor sites i and j,
\( {\displaystyle {\hat {a}}_{i\sigma }^{\dagger },{\hat {a}}_{j\sigma }} \) - fermionic creation and annihilation operators,
\( {\displaystyle \sigma } \) - spin polarization,
\( {\displaystyle t}\) - hopping integral
\( {\displaystyle J}\) - coupling constant J = 4 t 2 / U {\displaystyle J=4t^{2}/U} ,
\( {\displaystyle U}\) - coulomb repulsion,
\( {\displaystyle n_{i}=\sum _{\sigma }{\hat {a}}_{i\sigma }^{\dagger }{\hat {a}}_{i\sigma }}\) - particle number at the site i, and
\( {\displaystyle {\vec {S}}_{i},{\vec {S}}_{j}}\) - spins on the sites i and j.
Connection to the high-temperature superconductivity
The Hamiltonian of the \( {\displaystyle t_{1}-t_{2}-J} \) model in terms of \( {\displaystyle CP^{1}} \) generalized model reads [1]
\( {\displaystyle \mathbf {H} =t_{1}\sum \limits _{<i,j>}{\bigg (}c_{i\sigma }^{\dagger }c_{j\sigma }+h.c.{\bigg )}\ +\ t_{2}\sum \limits _{<<i,j>>}{\bigg (}c_{i\sigma }^{\dagger }c_{j\sigma }+h.c.{\bigg )}\ +\ J\sum \limits _{<i,j>}{\bigg (}\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {1}{4}}n_{i}n_{j}{\bigg )}-\ \mu \sum \limits _{i}n_{i},}\)
where fermionic operators \( {\displaystyle c_{i\sigma }} \) , \( {\displaystyle c_{i\sigma }} \) , the spin operators \( {\displaystyle \mathbf {S} _{i}} \) and \( {\displaystyle \mathbf {S} _{j}}\) , number operators \( {\displaystyle n_{i}}\) and \( {\displaystyle n_{j}}\) 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 \( {\displaystyle <...>} \) and \( {\displaystyle <<...>>} \) denote nearest and next-to-the-nearest neighbors, respectively.
References
N. Karchev, Generalized \( {\displaystyle \mathbf {\mathit {CP^{1}}} } \) model from the \( {\displaystyle t_{1}-t_{2}-J} \) model
Lectures on Correlation and Magnetism, [Patrik Fazekas],[page no. :199]
t-J model then and now: A personal perspective from the pioneering times, Józef Spałek, arXiv:0706.4236
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