Hubbard model equation

Explore the Hubbard model, a fundamental tool in condensed matter physics, understanding its equation and practical calculations.

The Hubbard Model: Understanding its Equation

The Hubbard model is a quintessential tool in condensed matter physics, particularly useful for describing electron interactions in narrow energy bands. It serves as the simplest model of interacting particles in a lattice, providing profound insights into the study of strongly correlated materials, such as high-temperature superconductors and Mott insulators.

The Equation

The Hubbard model is described by the following Hamiltonian:

H = -t Σ_⟨ij⟩,σ (c_iσ† c_jσ + h.c.) + U Σ_i n_i↑ n_i↓

The first term describes ‘hopping’ of an electron with spin σ from site j to site i with a tunnelling amplitude t. The term ‘h.c.’ stands for the Hermitian conjugate of the preceding expression, referring to the reverse ‘hopping’. The symbol ‘Σ_⟨ij⟩,σ‘ indicates a sum over all neighboring sites i and j in the lattice for each spin direction.

The second term, denoted by ‘U’, stands for the on-site electron-electron interaction. Here, ‘n’ is the number operator, and ‘i↑’ and ‘i↓’ denote up and down spins respectively at site ‘i’. The symbol ‘Σ_i‘ indicates a sum over all sites ‘i’.

Importance of the Hubbard Model

Despite its apparent simplicity, the Hubbard model captures two essential aspects of electron behavior in solids: kinetic energy that drives electron motion and potential energy stemming from electron-electron interaction. This dichotomy exemplifies the challenge in understanding many-body quantum systems.

It’s important to note that while the Hubbard model is solvable in one dimension using Bethe ansatz, it becomes highly nontrivial in two and three dimensions, contributing to the notorious “many-body problem” in physics. This makes the model a topic of ongoing research interest, shedding light on a range of phenomena in condensed matter physics.

Conclusion

In conclusion, the Hubbard model is a cornerstone of theoretical condensed matter physics, offering a simplified yet powerful lens to explore complex electronic behaviors in materials. Its equation elegantly encapsulates the dual forces acting on electrons, forming a basis for understanding a variety of intriguing quantum phenomena.

Example of a Calculation Using the Hubbard Model

Let’s consider a simple example of a calculation using the Hubbard model. We will use a one-dimensional Hubbard model with two sites and two electrons, each with opposite spin. This is the simplest non-trivial example of the Hubbard model.

The Hamiltonian for this system is:

H = -t(c_1↑† c_2↑ + c_1↓† c_2↓ + h.c.) + U(n_1↑n_1↓ + n_2↑n_2↓)

Where the first term represents hopping between the sites and the second term represents on-site interaction. We have four possible states for this system:

1. |↑↓, 0⟩, both electrons on site 1, no electrons on site 2
2. |↑, ↓⟩, one electron on each site
3. |↓, ↑⟩, one electron on each site, but with swapped spins
4. |0, ↑↓⟩, no electrons on site 1, both electrons on site 2

Our goal is to find the energies of these states using the Hamiltonian. For state 1 and state 4, there is no hopping term, so only the interaction term contributes to the energy. Therefore, the energy of these states is U. For state 2 and state 3, both electrons can hop to the other site, so we get contributions from both the hopping and the interaction term. The total energy is -t + U.

In conclusion, the Hubbard model can be used to determine the energies of different states in a system. In this simple example, we can see how the interplay between hopping and interaction terms influences these energies.