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:

${\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle \sigma }\left({\hat {a}}_{i\sigma }^{\dagger }{\hat {a}}_{j\sigma }+{\hat {a}}_{j\sigma }^{\dagger }{\hat {a}}_{i\sigma }\right)+J\sum _{\langle ij\rangle }\left({\vec {S}}_{i}\cdot {\vec {S}}_{j}-{\frac {n_{i}n_{j}}{4}}\right)}$

where

• is the sum over nearest-neighbor sites i and j,
• â†
  iσ, â
  iσ are the fermionic creation and annihilation operators,
• σ is the spin polarization,
• t is the hopping integral,
• J is the coupling constant, J = 4t²/U,
• U is the coulombic repulsion,
• n_i = â†
  iσâ
  iσ is the particle number at site i, and
• S→_i, S→_j are the spins on sites i and j.

Connection to the high-temperature superconductivity

The Hamiltonian of the t₁-t₂-J model in terms of the CP¹ generalized model is:^[1]

${\displaystyle \mathbf {H} =t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}$

where the fermionic operators c†
iσ and c
iσ, the spin operators S_i and S_j, and the number operators n_i and n_j all act on restricted Hilbert space and the doubly-occupied states are excluded. The sums in the above mentioned equation are over all sites of a 2-dimensional square lattice, where ⟨…⟩ and ⟨⟨…⟩⟩ denote the nearest and next-nearest neighbors, respectively.

References

1. Karchev, N. (1998). "Generalized CP¹ model from the t₁-t₂-J model". Phys. Rev. B. 57: 10913. doi:10.1103/PhysRevB.57.10913.

• Fazekas, Patrik, Lectures on Correlation and Magnetism, p. 199
• Spałek, Józef (2007). "t-J model then and now: A personal perspective from the pioneering times". Acta Phys. Polon. A. 111: 409–424. arXiv:0706.4236.