Ground State Crossover: Delocalization to Localization

Welcome back. In our previous lesson, we accomplished the significant task of finding the complete set of exact energy eigenvalues and eigenstates for the two-site Hubbard model at half-filling. By leveraging symmetries, we turned a complex operator problem into a manageable matrix diagonalization, culminating in a precise expression for the ground state energy and its corresponding wavefunction.

Now, we will move from calculation to interpretation. The true power of this exact solution lies in what it reveals about the underlying physics. Our goal is to analyze the ground state as a function of the ratio $U / t$ , the parameter that encapsulates the central conflict of the model: the competition between electron delocalization (driven by hopping, $t$ ) and Coulomb repulsion (driven by the on-site interaction, $U$ ). This analysis will demonstrate a smooth crossover between two fundamentally different physical regimes—a delocalized, "metallic-like" state and a localized, "insulating-like" state—giving us a first, clear look at the physics of strong correlation.

Anatomy of the Ground State

Let's recall the results from the last lesson. The ground state, which has even parity and total $S_{z} = 0$ , is a superposition of a state with one electron per site and a state with double occupancy:

∣ Ψ_{0} ⟩ = u ∣1 + ⟩ + v ∣2 + ⟩

where

$∣1 + ⟩ = \frac{1}{2} (c_{1 ↑}^{†} c_{2 ↓}^{†} + c_{2 ↑}^{†} c_{1 ↓}^{†}) ∣0 ⟩$ represents singly-occupied sites.
$∣2 + ⟩ = \frac{1}{2} (c_{1 ↑}^{†} c_{1 ↓}^{†} + c_{2 ↑}^{†} c_{2 ↓}^{†}) ∣0 ⟩$ represents doubly-occupied sites.

The energy of this state is:

E_{0} = \frac{U}{2} - (\frac{U}{2})^{2} + 4 t^{2}

The character of the ground state is entirely determined by the relative weight of the two components, which can be found from the ratio of the coefficients, $v / u$ . From the eigenvalue equation we solved, $- 2 t v = E_{0} u$ , which gives $v / u = - E_{0} / (2 t)$ .

By examining how $E_{0}$ and the ratio $v / u$ behave in the limits of small and large $U / t$ , we can understand the system's crossover from delocalized to localized behavior. The following reading provides a concise analysis of these two crucial limits.

Electronic Hamiltonians - Part III 1 The 2-site Hubbard model

The document by Mona Berciu provides an elegant analysis of the two-site model's exact solution. Note that the notation in the document is slightly different from ours: it uses |\Psi\rangle= \alpha (\dots \text{double occupancy}) + \beta (\dots \text{single occupancy}). This means the coefficient \alpha in the text corresponds to our v, and \beta corresponds to our u.

Please read the subsection "1.3 Exact solution". Focus on the expression for the ground-state energy (which is identical to our E_0) and the resulting ratio of coefficients. Then, carefully follow the analysis of the two limits: the weak-coupling limit (U/t → 0) and the strong-coupling limit (U/t → ∞).

Interpreting the Two Regimes

The analysis in the reading reveals two distinct physical pictures. Let's consolidate and expand on those findings.

1. The Weak-Coupling Limit ( $U ≪ t$ )

In this regime, kinetic energy dominates. The electrons prefer to delocalize to lower their energy.

Energy: As shown in the reading, a Taylor expansion gives $E_{0} \approx - 2 t + U /2$ . The dominant term, $- 2 t$ , is precisely the ground state energy of two non-interacting electrons in the bonding molecular orbital ( $ϵ = - t$ ). The interaction $U$ provides a small positive correction.
Wavefunction: The ratio of coefficients approaches 1. In our notation, $v / u \to 1$ . This implies that $u \approx v$ , so the ground state is an approximately equal superposition of the singly-occupied and doubly-occupied configurations. $∣ Ψ_{0} ⟩ \propto ∣1 + ⟩ + ∣2 + ⟩$ If we substitute the definitions of these basis states, we find: $∣ Ψ_{0} ⟩ \propto (c_{1 ↑}^{†} c_{2 ↓}^{†} + c_{2 ↑}^{†} c_{1 ↓}^{†}) + (c_{1 ↑}^{†} c_{1 ↓}^{†} + c_{2 ↑}^{†} c_{2 ↓}^{†}) = (c_{1 ↑}^{†} + c_{2 ↑}^{†}) (c_{1 ↓}^{†} + c_{2 ↓}^{†}) ∣0 ⟩$ This is exactly the ground state of the non-interacting tight-binding model: two electrons, one spin-up and one spin-down, occupying the bonding orbital $c_{+, σ}^{†} = \frac{1}{2} (c_{1 σ}^{†} + c_{2 σ}^{†})$ . The electrons are completely delocalized over the two sites, and there is a 50% probability of finding both on the same site. This is a delocalized, "metallic-like" state.

2. The Strong-Coupling Limit ( $U ≫ t$ )

In this regime, Coulomb repulsion dominates. The system pays a large energy penalty $U$ for any double occupancy.

Energy: The expansion yields $E_{0} \approx - \frac{4 t ^{2}}{U}$ . The energy is close to zero, which is the energy of two non-interacting electrons on different sites. The small negative correction arises from a quantum mechanical process called "virtual hopping," where an electron can briefly hop to the neighboring site (costing energy $U$ ) and then hop back. This process, which we will later identify as superexchange, stabilizes the state slightly.
Wavefunction: The ratio of coefficients $v / u \approx 2 t / U$ , which approaches 0 as $U / t \to \infty$ . This means the coefficient $v$ of the doubly-occupied state $∣2 + ⟩$ becomes vanishingly small. The ground state is now overwhelmingly dominated by the singly-occupied configuration: $∣ Ψ_{0} ⟩ \approx ∣1 + ⟩$ The probability of finding a double occupancy, which is proportional to $∣ v ∣^{2}$ , is suppressed and scales as $(t / U)^{2}$ . To avoid the large energy cost, the electrons become localized, one to each site.

This crossover is the central result of our analysis. The same document you just read provides the standard physical terminology for these two regimes.

Electronic Hamiltonians - Part III 1 The 2-site Hubbard model

The analysis of the two limits reveals two distinct physical regimes. The document by Berciu provides the standard terminology for them.

Read the first paragraph of the next section, which starts with the phrase "So we see that HF gets the trends right". It explicitly describes the system as "metallic" at small U/t and "insulating" at large U/t, connecting our findings to the broader concepts of electron transport.

The Crossover and the Mott Transition

The smooth evolution of the ground state from a delocalized, metallic-like state to a localized, insulating-like state as $U / t$ increases is a hallmark of the Hubbard model. In our two-site system, this is a smooth crossover. However, in a macroscopic lattice (in dimensions greater than one), this crossover can become a sharp, bona fide quantum phase transition at zero temperature, known as the Mott metal-insulator transition. This is a profound phenomenon where electron-electron interactions alone, without any change in the crystal structure, can turn a metal into an insulator.

Our simple two-site model, therefore, captures the essential physics of this transition. It shows that even a system that should be a metal according to simple band theory (like a chain of hydrogen atoms, each contributing one electron to a half-filled band) can become an insulator if the Coulomb repulsion $U$ is strong enough to localize the electrons.

To place our specific results in a wider context, the following video segment discusses these two fundamental limits for a general Hubbard model on a lattice.

lecture38 The Hubbard Model

The concepts of weak and strong coupling are central to the entire theory of the Hubbard model, far beyond our two-site example. Professor James Freericks explains these limits in the context of a general lattice.

Watch the segment of the lecture that discusses the limits of U=0 and U=infinity. Notice how the U=0 limit corresponds to the delocalized band picture ("metal") and the U=\infty limit at half-filling corresponds to a state with no double occupancies and frozen electrons—a "Mott insulator."

Conclusion

In this lesson, we have moved beyond finding the solution to interpreting it. By analyzing the exact ground state of the two-site Hubbard model, we have uncovered the fundamental competition between delocalization and localization.

Key Takeaways:

The ratio $U / t$ governs the physics of the Hubbard model, determining whether kinetic energy or Coulomb repulsion dominates.
In the weak-coupling limit ( $U ≪ t$ ), the ground state is delocalized and resembles the non-interacting tight-binding solution. The electrons are mobile, a "metallic-like" behavior.
In the strong-coupling limit ( $U ≫ t$ ), double occupancy is strongly suppressed to avoid the large energy penalty $U$ . The ground state becomes localized, with one electron per site. This is an "insulating-like" behavior, the precursor to a Mott insulator.
The crossover between these two regimes in our simple model captures the essence of the Mott metal-insulator transition, a key phenomenon in strongly correlated electron systems.

We have now seen how the ground state energy and wavefunction evolve with $U / t$ . Next, we will continue our analysis by calculating other physical observables, such as the amount of double occupancy in the ground state. This will provide another quantitative measure of the localization crossover and set the stage for understanding magnetism in this system.

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