Hubbard model

Tiny bits called electrons move in solid things.

They like to hop from one spot to another. But these bits also push each other away. This push can be very strong.

When the bits hop easily, they can carry power. This is like a metal. But if the push is too strong, they stay still. This makes the object an insulator.

Scientists use a special idea to study this. It is named after John Hubbard. It helps us see how bits move. This helps us learn about new materials.

It is a way to see how things work.

Scientists use a special idea to study how tiny bits move. This idea is called the Hubbard model. It is named after a man named John Hubbard.

Inside solid things, tiny bits called electrons move around. These electrons live on a lattice. A lattice is a set of spots arranged in a pattern. The model looks at two main forces. First, electrons like to hop. This means they jump from one spot to another. This hopping lets them move through the material. This is how metals work.

Second, electrons push each other away. This push is called repulsion. If the push is very strong, the electrons stay still. They cannot hop to a new spot if another electron is there. When this happens, the material becomes an insulator. An insulator does not let power flow easily.

The Hubbard model helps us see this change. It shows how a material can switch from a metal to an insulator. For example, heating some metal oxides can change them. This happens because the spots on the lattice move further apart. This makes it harder for electrons to hop. The Hubbard model is a very useful tool for science.

Scientists use the Hubbard model to understand how tiny particles move. This idea helps explain why some materials let electricity flow. Other materials act as insulators and stop electricity. This model is very important in a field called solid-state physics. It focuses on how electrons behave inside a solid object.

Imagine electrons living on a lattice. A lattice is a pattern of many fixed spots. The model looks at two different forces acting on electrons. The first is called hopping, which is a type of kinetic energy. This force lets an electron jump from one spot to another. The second force is the on-site interaction, which is a repulsive push. This push happens when two electrons try to occupy the same spot.

This idea was first shared in 1963. A scientist named John Hubbard proposed it to describe electrons in solids. Another scientist named Martin Gutzwiller also proposed it at the same time. Since then, people have used this model for many things. It helps us study how magnets work at a tiny scale. It is also used to study high-temperature superconductivity.

The model works by looking at the fight between these two forces. If hopping is strong, electrons move easily like a metal. If the repulsive push is strong, electrons get stuck. This creates something called a Mott insulator. Even if a material should conduct, the push keeps electrons still. For example, heating metal oxides can change how they work. As they heat up, the spots on the lattice move further apart. This makes it much harder for the electrons to hop.

Scientists even use modern tools to simulate this model. They use special materials called transition metal dichalcogenides to act like a lattice. By stacking these materials, they create a pattern called a moiré superlattice. This acts like a giant version of a tiny atom lattice. Researchers have even simulated up to eight electrons in one cell. This helps them see how the switch from metal to insulator happens.

The Hubbard model is a fundamental theoretical framework used in solid-state physics. It describes how particles move within a periodic potential, often called a lattice. This model is essential for understanding the transition between conducting and insulating systems. It helps scientists predict whether a material will allow electricity to flow or block it entirely. By focusing on the behavior of electrons, the model explains complex quantum phenomena.

At its core, the model describes a competition between two distinct physical forces. The first is the kinetic energy, which is represented by a "hopping" term. This term allows particles to tunnel or move between neighboring lattice sites. The second is the potential energy, known as the on-site interaction. This term reflects the repulsion that occurs when two particles occupy the same site. The physics of the system is determined by the ratio between these two strengths. If hopping dominates, the material acts as a metal. If the on-site repulsion is stronger, the particles become stuck.

There are several variations of this model depending on the particles involved. The original Hubbard model focuses on fermions, which are particles like electrons. If the model is applied to bosons, it is called the Bose-Hubbard model. Another variation is the extended Hubbard model. This version includes interactions between particles located at different lattice sites. While the standard model only looks at immediate neighbors, the extended version adds more complexity. These different versions allow scientists to study a wide range of physical systems.

History shows that the model emerged from a need to describe electrons in solids. It was proposed independently in 1963 by John Hubbard and Martin Gutzwiller. Since its introduction, it has become a vital tool for many branches of physics. Scientists use it to study high-temperature superconductivity and quantum magnetism. It also helps explain charge density waves. The model provides a bridge between simple theories and the complex reality of real materials.

One of the most significant achievements of the model is predicting Mott insulators. In conventional band theory, a material with an odd number of electrons per unit cell should be a conductor. However, the Hubbard model shows that strong electron repulsion can stop this movement. This creates a Mott insulator, where electrons are localized due to their own interactions. This effect is seen in metal oxides when they are heated. As temperature rises, the distance between atoms increases. This change reduces the hopping integral, allowing the on-site repulsion to dominate.

To understand this, consider a one-dimensional chain of hydrogen atoms. Each atom has one electron in its 1s orbital. Under standard band theory, this chain should conduct electricity easily. However, if you increase the spacing between the atoms, the hopping strength decreases. Eventually, the repulsion between electrons becomes the most important factor. At this point, the chain transitions from a conductor to an insulator. This simple example demonstrates how the ratio of interaction to hopping controls the entire system.

Modern researchers use advanced technology to simulate these complex interactions. They use stacks of transition metal dichalcogenides to create a moiré superlattice. This superlattice acts like a giant version of a tiny atomic lattice. In these simulations, the distance between supercells is about 100 times larger than the atoms within them. This helps reduce electron tunneling and makes the system easier to study. Scientists have even simulated up to eight electrons per supercell. These experiments help confirm how the transition from metal to insulator works in the real world.