Hubbard Model: Interaction and Double Occupancy

The Interaction Term and the Number Operator

The on-site repulsion term singles out a specific configuration: two electrons residing on the same lattice site. To build an operator that recognizes this configuration, we first need an operator that can count electrons. As you'll recall from our work on second quantization, this is the job of the number operator.

The following short video provides a clear refresher on the creation, annihilation, and number operators, culminating in the construction of the Hubbard interaction term.

As the video explains, the number operator for an electron of spin $\sigma$ at site $i$ is defined as:

The eigenvalues of this operator are 0 or 1, corresponding to the absence or presence of such an electron. This "counting" property is exactly what we need.

To build an operator that identifies a doubly-occupied site, we simply need to check for the simultaneous presence of a spin-up electron and a spin-down electron. The product of their respective number operators, ${\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }$, achieves this perfectly:

• If site $i$ is empty, ${\hat{n}}_{i\uparrow }=0$ and ${\hat{n}}_{i\downarrow }=0$. The product is 0.
• If site $i$ is singly occupied (either spin up or spin down), one number operator gives 1 and the other gives 0. The product is 0.
• If site $i$ is doubly occupied, ${\hat{n}}_{i\uparrow }=1$ and ${\hat{n}}_{i\downarrow }=1$. The product is 1.

This product acts as a "double occupancy detector". Multiplying it by the energy cost $U$ and summing over all sites gives us the full interaction term of the Hubbard Hamiltonian:

This term adds an energy $U$ to the system for every site that is doubly occupied and adds nothing otherwise.

For a concise reinforcement of this concept, Prof. James Freericks' lecture offers another excellent summary.

The Physical Meaning of U and Double Occupancy

We've established the mathematical form, but it's vital to connect it back to the underlying physics. The parameter $U$ is not just an abstract number; it is a direct measure of the Coulomb repulsion energy between two electrons confined to the same atomic-like orbital.

Recall from the derivation of the tight-binding model that we use a basis of Wannier functions, ${\varphi }_i(\mathbf{r})$, which are localized around each lattice site $i$. The full electron-electron interaction is a two-body operator. When we express this operator in the Wannier basis, we get a large number of terms. The Hubbard model makes the crucial approximation that the most significant of these is the term where all four Wannier states are on the same site, $i$.

The following reading formally defines $U$ as this specific matrix element and presents the final, simplified Hubbard Hamiltonian.

This reading confirms that $U$ is the energy cost associated with the state depicted below, where two electrons with opposite spins are forced to occupy the same localized region of space.

The Double Occupancy as an Observable

Since avoiding double occupancy is central to the physics of strong correlation, it is useful to define the total double occupancy, $D$, as a physical observable. Its operator is simply the sum of the local "detectors":

where $N$ is the number of sites. The expectation value, $D=⟨\hat{D}⟩$, gives the average fraction of doubly occupied sites in a given many-body state.

This quantity is more than just a theoretical convenience. It is a direct measure of correlation. For $U=0$, at half-filling (one electron per site on average), random hopping would lead to a certain statistical probability of finding two electrons on one site and another site empty. As $U\to \infty$, we expect $D\to 0$, because any configuration with a doubly occupied site would have infinite energy and be projected out of the low-energy space. Tracking how $D$ changes with $U/t$ is a primary way to study the metal-insulator transition.

The following text introduces this concept and connects it to another important physical quantity: the local magnetic moment.

The relation $⟨m^2⟩=\rho -2D$ (where $\rho$ is the electron density) is insightful. It shows that suppressing double occupancy ($D\to 0$) at fixed density necessarily enhances the formation of local magnetic moments ($⟨m^2⟩\to \rho$). This is our first hint at the deep connection between Mott physics and magnetism, a central theme of this course.

Finally, it's worth noting that the interaction term is sometimes written in a slightly different, but equivalent, form:

Expanding this expression shows that it is the same as our original ${\hat{H}}_U$ up to a shift in the chemical potential and an overall constant energy. This alternative form makes the particle-hole symmetry of the model at half-filling more apparent, a topic we will explore in a future lesson.

Conclusion

In this lesson, we have taken a deep dive into the Hubbard interaction term, ${\hat{H}}_U$. We have solidified its mathematical form and connected it directly to the physical reality of Coulomb repulsion.

Key Takeaways:

• The Hubbard interaction term is written as ${\hat{H}}_U=U{\sum }_i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }$.
• The operator product ${\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }$ acts as a "double occupancy detector," yielding 1 if a site is doubly occupied and 0 otherwise.
• The parameter $U$ is the potential energy cost of double occupancy, originating from the intra-atomic Coulomb repulsion integral in a localized Wannier basis.
• The average double occupancy, $D=⟨{\sum }_i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }/N⟩$, is a key observable for quantifying electron correlation. Suppressing $D$ is directly linked to the formation of local magnetic moments.

With a firm grasp of both the hopping term and the interaction term, we are now ready to analyze the model's behavior. In the next lesson, we will begin this exploration by examining the limiting cases of weak coupling ($U\ll t$) and strong coupling ($U\gg t$), which will give us our first glimpse into the metal-insulator transition and the emergence of magnetism.