Double Occupancy in the Ground State: U/t Dependence

Defining and Calculating Double Occupancy

The operator that counts the number of doubly occupied sites is a natural extension of the number operators we've used before. For a given site $i$, the operator $n_{i\uparrow }{n}_{i\downarrow }$ gives a value of 1 if the site is occupied by both a spin-up and a spin-down electron, and 0 otherwise. For our two-site system, the total double occupancy operator is:

Our goal is to compute the expectation value of this operator in the ground state $|{\Psi }_0⟩$ that we found previously:

Let's recall the ground state and the basis states:

where $|1+⟩=\frac{1}{\sqrt{2}}(c_{1\uparrow }^{†}{c}_{2\downarrow }^{†}+c_{2\uparrow }^{†}{c}_{1\downarrow }^{†})|0⟩$ represents single occupancy on each site, and $|2+⟩=\frac{1}{\sqrt{2}}(c_{1\uparrow }^{†}{c}_{1\downarrow }^{†}+c_{2\uparrow }^{†}{c}_{2\downarrow }^{†})|0⟩$ is a superposition of doubly occupied configurations.

To calculate the expectation value, we first need to see how the operator $D$ acts on our basis states.

• The state $|1+⟩$ contains no doubly occupied sites. Therefore, applying $D$ to it must yield zero: $D|1+⟩=0$.
• The state $|2+⟩$ is composed entirely of doubly occupied configurations. Applying the operator $D$ to it gives: $$D|2+⟩=(n_{1\uparrow }{n}_{1\downarrow }+n_{2\uparrow }{n}_{2\downarrow })\frac{1}{\sqrt{2}}(c_{1\uparrow }^{†}{c}_{1\downarrow }^{†}|0⟩+c_{2\uparrow }^{†}{c}_{2\downarrow }^{†}|0⟩)$$ The term $n_{1\uparrow }{n}_{1\downarrow }$ acts as 1 on the first part of the superposition and 0 on the second. The term $n_{2\uparrow }{n}_{2\downarrow }$ does the opposite. The result is: $$D|2+⟩=\frac{1}{\sqrt{2}}(1\cdot c_{1\uparrow }^{†}{c}_{1\downarrow }^{†}|0⟩+1\cdot c_{2\uparrow }^{†}{c}_{2\downarrow }^{†}|0⟩)=|2+⟩$$ So, $|2+⟩$ is an eigenstate of the double occupancy operator with an eigenvalue of 1.

Now, we can compute the expectation value:

Using the orthogonality of the basis states ($⟨1+|2+⟩=0$) and their normalization ($⟨2+|2+⟩=1$), we arrive at a remarkably simple result:

The total probability of finding a double occupancy in the ground state is simply $v^2$, the squared-magnitude of the coefficient of the doubly-occupied basis state $|2+⟩$.

Double Occupancy as a Function of U/t

Our final step is to find an explicit expression for $v^2$ in terms of the model parameters $U$ and $t$. We already have all the necessary components from solving the model. The ground state must be normalized, meaning $u^2+v^2=1$. From the eigenvalue problem, we also have the relation $u/v=-E_0/(2t)$. Combining these gives:

Substituting our known expression for the ground state energy, $E_0=\frac{U}{2}-\sqrt{(\frac{U}{2}{)}^2+4t^2}$, yields the double occupancy as a function of $U$ and $t$.

The following document provides the explicit normalized coefficients for the ground state wavefunction, which directly gives us the expression for $v^2$.

From the provided expression for $v$, we can write the double occupancy $d\equiv ⟨D⟩=v^2$. To make the physics clear, it's best to express this in terms of the dimensionless ratio $x=U/t$:

This elegant formula quantitatively describes the suppression of double occupancy as the on-site repulsion $U$ increases relative to the hopping $t$.

Analyzing the Result

Let's examine the behavior of $d(x)$ in the two physical limits we discussed.

1. Weak-Coupling Limit ($x=U/t\to 0$):
   As $x\to 0$, the fraction in the expression vanishes, and we get:

   $$d(0)=\frac{1}{2}$$

   This is the non-interacting result. When $U=0$, the two electrons are completely delocalized and move independently. The probability of finding them on the same site is exactly equal to the probability of finding them on different sites. The ground state has an equal mix of singly and doubly occupied configurations.

2. Strong-Coupling Limit ($x=U/t\to \infty$):
   As $x\to \infty$, the fraction approaches 1. We can see this by factoring $x/2$ out of the square root: $\sqrt{(x/2{)}^2+4}=(x/2)\sqrt{1+16/x^2}$. For large $x$, using the Taylor expansion $\sqrt{1+ϵ}\approx 1+ϵ/2$, the fraction becomes $\frac{1}{\sqrt{1+16/x^2}}\approx 1-8/x^2$.
   The double occupancy is then:

   $$d(x)\approx \frac{1}{2}(1-(1-8/x^2))=\frac{4}{x^2}=4{\left(\frac{t}{U}\right)}^2$$

   In the strong-coupling limit, the double occupancy is severely suppressed, vanishing as $(t/U{)}^2$. The electrons strongly avoid each other to minimize the large Coulomb energy penalty, leading to a state that is almost purely composed of singly-occupied sites. This confirms our qualitative picture of electron localization.

The behavior of double occupancy provides a quantitative signature of the crossover from a delocalized state to a localized one.

The plot above (note that it shows double occupancy per site, which is $d/2$) beautifully illustrates this crossover. It shows a smooth decrease from the metallic-like value of $d=1/2$ to the insulating-like suppression at large $U/t$.

Conclusion

In this lesson, we have quantified the physics of localization in the two-site Hubbard model by calculating the ground state double occupancy.

Key Takeaways:

• The expectation value of the total double occupancy operator, $⟨D⟩=⟨n_{1\uparrow }{n}_{1\downarrow }+n_{2\uparrow }{n}_{2\downarrow }⟩$, is given by $v^2$, where $v$ is the coefficient of the doubly-occupied component of the ground state wavefunction.
• The double occupancy $d=⟨D⟩$ is a function of the ratio $U/t$ only: $$d(U/t)=\frac{1}{2}\left(1-\frac{U/2t}{\sqrt{(U/2t{)}^2+4}}\right)$$
• In the weak-coupling limit ($U/t\to 0$), $d\to 1/2$, corresponding to a delocalized, metallic-like state with significant charge fluctuations.
• In the strong-coupling limit ($U/t\to \infty$), $d\to 4(t/U{)}^2$, showing that double occupancy is strongly suppressed as electrons localize to avoid the Coulomb penalty.

We have now thoroughly analyzed the ground state's charge properties. However, the strong-coupling limit, where electrons are localized to individual sites, has profound implications for their spin degrees of freedom. In the next lesson, we will investigate the energy splitting between the ground state and the lowest-lying excited state, which will reveal the emergence of an effective magnetic interaction between the localized electron spins.