t-J model

In quantum physics system's models are usually based on the Hamiltonian operator ${\displaystyle {\hat {H}}}$, corresponding to the total energy of that system, including both kinetic energy and potential energy.

The t-J Hamiltonian can be derived from the ${\displaystyle {\hat {H}}}$ of the Hubbard model using the Schrieffer–Wolff transformation, with the transformation generator depending on t/U and excluding the possibility for electrons to doubly occupy a lattice's site,[6] which results in:[7]

${\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}$

where the term in t corresponds to the kinetic energy and is equal to the one in the Hubbard model. The second one is the potential energy approximated at the second order, because this is an approximation of the Hubbard model in the limit U >> t developed in power of t. Terms at higher order can be added.[1]

The parameters are:

• Σ⟨ij⟩ is the sum over nearest-neighbor sites i and j, for all sites, typically on a two-dimensional square lattice,
• c^†
  _iσ, c
  _iσ are the fermionic creation and annihilation operators at site i,
• σ is the spin polarization,
• t is the hopping integral,
• J is the antiferromagnetic exchange coupling, J = 4t²/U,
• U is the on-site coulombic repulsion, that must satisfy the condition for U >> t,
• n_i = Σσc^†
  _iσc
  _iσ is the particle number at site i and can be maximum 1, so that double occupancy is forbidden (in the Hubbard model is possible),
• S_i and S_j are the spins on sites i and j,
• h. c. stands for Hermitian conjugate,

If n_i = 1, that is when in the ground state, there is just one electron per lattice's site (half-filling), the model reduces to the Heisenberg model and the ground state reproduce a dielectric antiferromagnets (Mott insulator).[8]

The model can be further extended considering also the next-nearest-neighbor sites and the chemical potential to set the ground state in function of the total number of particles:[9][10]

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

where ⟨...⟩ and ⟨⟨...⟩⟩ denote the nearest and next-nearest neighbors, respectively, with two different values for the hopping integral (t₁ and t₂) and μ is the chemical potential.

• Fazekas, Patrik (1999). Lectures on Correlation and Magnetism. Series in Modern Condensed Matter Physics: Volume 5. Vol. 5. World Scientific. p. 199. doi:10.1142/2945. ISBN 978-981-4499-62-0.
• Spałek, Józef (2007). "t-J model then and now: A personal perspective from the pioneering times". Acta Phys. Pol. A. 111 (4): 409–424. arXiv:0706.4236. Bibcode:2007AcPPA.111..409S. doi:10.12693/APhysPolA.111.409. S2CID 53117123.
• Dr Mitchell, Electron interactions and the Hubbard model, retrieved 2022-08-29