Physical explanation of energy shift in Heisenberg term in t-J model

[randomquestion]

Homework Statement

Why Does the t-J Model Include a Density-Density Term?

Relevant Equations

$$H_{t-J} = -t \sum_{\langle i,j \rangle, \sigma} \mathcal{P} (c_{i\sigma}^\dagger c_{j\sigma} + \text{h.c.}) \mathcal{P} + J \sum_{\langle i,j \rangle} \left( \mathbf{S}_i \cdot \mathbf{S}_j - \frac{1}{4} n_i n_j \right)$$

$$H_{\text{AF}} = J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j$$

I'm reading "Lecture Notes on Electron Correlation and Magnetism" by P. Fazekas, and I came across a question regarding the form of the Heisenberg term in the t-J model.

In Chapter 5.1.4, the t-J model is written where there is an additional density-density term in the exchange interaction.

However, in Chapter 5.1.5, the antiferromagnetic Heisenberg model is written without this extra term.

I understand that the density-density term arises naturally when deriving the t-J model from the Hubbard model in the large-U limit. However, I’m trying to get an intuitive explanation of why this term is included in the t-J model but not in the standard Heisenberg model.

Does it relate to the presence of hole doping in the t-J model, or is it more of a reference energy shift? How should I think about its physical meaning?

Would appreciate any insights!

 

[techsingularity2042]

For the t-J model, The extra -1/4n_in_j term is essential because it reflects the fact that the system can have holes (i.e., deviations from one electron per site). Its presence is crucial for capturing the correct energetics and interactions in a doped Mott insulator, affecting how spins and holes interact.

For the Heisenberg model, the density is fixed at half-filling, so the density–density term merely contributes a constant energy offset. Since only energy differences matter for the physics, this constant is irrelevant to the spin dynamics, and hence the term is omitted.

 

[randomquestion]

For the t-J model, The extra -1/4n_in_j term is essential because it reflects the fact that the system can have holes (i.e., deviations from one electron per site). Its presence is crucial for capturing the correct energetics and interactions in a doped Mott insulator, affecting how spins and holes interact.

For the Heisenberg model, the density is fixed at half-filling, so the density–density term merely contributes a constant energy offset. Since only energy differences matter for the physics, this constant is irrelevant to the spin dynamics, and hence the term is omitted.

Thank you very much for your reply!
When you say 'capturing the correct energetics,' do you mean ensuring the proper singlet-triplet splitting (where the singlet is lower in energy)?

 

[techsingularity2042]

Yes, capturing the correct energetics means ensuring that the effective Hamiltonian reproduces the singlet-triplet splitting from the underlying Hubbard model, with the singlet state lowered by the exchange energy while the triplet remains at a higher energy.