Hopping term

Definition

The hopping term is a mathematical representation of the probability amplitude for a particle, such as an electron, to 'hop' or transfer between adjacent lattice sites in a solid. This term is essential in models that describe electron interactions and movement, particularly within the framework of the Hubbard model, where it helps to capture the effects of electron correlations and localization.

5 Must Know Facts For Your Next Test

1. In the Hubbard model, the hopping term is typically represented as a matrix element that quantifies how easily an electron can move from one site to a neighboring site.
2. The strength of the hopping term affects the bandwidth of the electronic states, influencing whether the system behaves like a conductor or an insulator.
3. When interactions are strong compared to the hopping term, the system can exhibit phenomena like Mott insulator behavior, where electrons localize despite having kinetic energy to move.
4. The hopping term can be modified to include effects like next-nearest neighbor hopping, which helps describe more complex lattice structures and interactions.
5. In many-body physics, the hopping term plays a crucial role in determining phase transitions and collective behaviors in strongly correlated electron systems.

Review Questions

• How does the hopping term influence electron dynamics within the Hubbard model?
  • The hopping term is fundamental in determining how electrons can move between adjacent sites in a lattice within the Hubbard model. It represents the probability amplitude for this movement and influences the system's kinetic energy. A strong hopping term leads to delocalized states, while a weak hopping term can lead to localization and more complex behaviors like those seen in Mott insulators.
• Discuss how variations in the hopping term can impact electronic properties like conductivity in materials described by the tight-binding model.
  • Variations in the hopping term directly affect the bandwidth of electronic states calculated in the tight-binding model. A larger hopping term generally results in a wider bandwidth, leading to increased conductivity as electrons can more easily hop between sites. Conversely, if the hopping is reduced, it can lead to insulating behavior due to localized electron states, highlighting how critical this term is in determining whether a material behaves as a conductor or an insulator.
• Evaluate the significance of the hopping term in understanding phase transitions in strongly correlated electron systems.
  • The hopping term is pivotal in understanding phase transitions in strongly correlated electron systems because it dictates how electrons interact and move through a lattice. As parameters change, such as increasing interaction strength relative to hopping, systems can transition from metallic to insulating states. This interplay can lead to various phenomena including superconductivity or magnetism, showcasing how crucial it is for explaining emergent behaviors in condensed matter physics.

Related terms

Hubbard model: A simplified model used in condensed matter physics to describe interacting electrons in a lattice, focusing on the competition between kinetic energy and potential energy due to electron-electron interactions.

tight-binding model: A theoretical framework that describes the behavior of electrons in a lattice where their wave functions are localized to specific sites, allowing for a simplified treatment of hopping terms.

band theory: A theoretical model that explains the electronic properties of solids by considering the allowed and forbidden energy levels that electrons can occupy in a crystal lattice.