Topological order

Matter can be in many forms. Some things are hard like a rock. Some flow like water. Most things are organized in simple ways. Tiny bits form patterns to make these forms. But some things have a new kind of order. This order comes from long links between bits. These links are spread out and far apart. This special order can help us build new tools. It might even help us make super fast computers. This new way of being is very special.

Matter can exist in many forms. We often see solids, liquids, or gases. These forms happen because of how tiny particles are organized. For a long time, scientists thought they understood all these patterns. They used a set of rules called Landau theory. This theory says that different forms come from changes in symmetry. For example, a crystal has a very strict pattern. A liquid has a much looser pattern.

But scientists found a new kind of pattern. It is called topological order. This order does not come from simple shapes or patterns. Instead, it comes from long-range entanglement. Entanglement is a way that tiny particles stay linked across far distances. This link is spread out among many particles.

This special state is very useful. It can help us build a topological quantum computer. These computers would be very strong and work well. We also see this order in things like superconductors. Scientists are still studying how these long links work. They hope to find many more new materials in the future.

Matter can exist in many different forms, which scientists call phases. Most phases, like solids or liquids, happen because of how particles are arranged in a pattern. For a long time, a set of rules called Landau symmetry-breaking theory explained this. This theory says that different phases come from changes in symmetry. For example, atoms in a liquid are spread out randomly. A crystal is different because its atoms sit in a strict, regular grid. This change from a random liquid to a structured crystal is a phase transition.

However, scientists discovered a new kind of pattern called topological order. This order does not come from simple shapes or local patterns. Instead, it comes from long-range quantum entanglement. Entanglement is a way that particles stay linked across far distances. In topological order, these links are spread out among many particles at once. This means the pattern is non-local, which is a fancy way to say it is not in just one spot. This special state usually happens at zero temperature.

People began to realize this new order existed in the late 1980s. Physicists were trying to explain a thing called the chiral spin state. They found that symmetry alone could not explain all the different types of these states. In 1989, Xiao-Gang Wen proposed the name "topological order" for this discovery. The name comes from a special kind of math called topological quantum field theory. This helped scientists describe patterns that the old Landau theory could not see.

We have already seen examples of this order in real life. The superconductor was discovered in 1911, and it is the first known topologically ordered state. Another example is the fractional quantum Hall state, which was found in 1982. Scientists also study spin liquids and the quantum Hall effect to find these orders. Some researchers even look at how strings or membranes can create these states. These different states have unique properties, like ground state degeneracy.

This new science could change how we use technology. Most tools we use today rely on older types of order. For example, hard drives use magnetic materials to store data. Crystals help us make transistors for computers. Topological order could lead to something even more amazing called topological quantum computing. These computers would use the links between particles to work in a very stable way. This could make them much more powerful than the computers we have now.

In physics, topological order describes a unique phase of matter. This phase arises from non-local interactions within a system. One example of this is entanglement in quantum mechanics. Another example involves floppy modes in elastic systems. Most common phases of matter, like solids or gases, rely on short-range interactions. These interactions create microscopic patterns in how particles are arranged in space. In contrast, topological orders are defined by patterns of long-range quantum entanglement. This means the connections between particles are spread out across the system. States with different topological orders cannot change into one another without a phase transition. Technically, this specific type of order occurs at zero temperature.

To understand this, we must look at how matter is organized. Matter is made of atoms that can take different forms. These forms are called states of matter or phases. In condensed matter physics, these properties emerge from how atoms are organized. This organization is known as the order in the material. For a long time, scientists used Landau symmetry-breaking theory to explain these orders. This theory states that different orders correspond to different symmetries. For instance, a liquid has continuous translation symmetry because atoms are randomly distributed. A crystal has discrete translation symmetry because atoms form a regular lattice. A phase transition occurs when the symmetry of the organization changes.

However, the discovery of new states showed that Landau theory was not complete. In the late 1980s, physicists studied the chiral spin state. They tried to use symmetry-breaking theory to explain it. They found the state broke time reversal and parity symmetries. Yet, they discovered many different chiral spin states shared the exact same symmetry. This meant symmetry alone could not characterize these states. There was a new kind of order present. In 1989, Xiao-Gang Wen proposed the name "topological order." This name was inspired by topological quantum field theory, or TQFT. This new framework allowed scientists to describe patterns that symmetry-breaking theory could not.

Topological order has several distinct and fascinating properties. One property is ground state degeneracy. This refers to having multiple states with the same lowest energy. These states can be defined on closed spaces or open spaces with gapped boundaries. Another property involves fractional statistics or non-Abelian group statistics. These are different from the standard statistics of bosons and fermions. There are also perfect conducting edge states. These states can be used in various device applications. Furthermore, topological order can lead to emergent gauge fields. This suggests that elementary particles might have a quantum information origin.

We can find real examples of these states in nature and experiments. The superconductor, discovered in 1911, is the first known topologically ordered state. It possesses what is called Z2 topological order. The fractional quantum Hall state was discovered in 1982. While it was found before the term "topological order" existed, it fits the description perfectly. Different quantum Hall states share the same symmetry but have different topological orders. Scientists also study spin liquids to understand these systems. Even in free fermion systems, a simple kind of topological order can appear. This is seen in the integral quantum Hall state, which is characterized by a Chern number.

One way these states form is through string-net condensation. In this mechanism, a large class of 2+1D topological orders is realized. This process can generate infinitely many different types of topological orders. The collective motions of these condensed strings create excitations. These excitations act as gauge bosons. The ends of the strings are considered defects. These defects act as gauge charges and can carry fractional statistics. Other complex objects like membranes or fractals can also lead to these phases. Researchers are even looking at 3+1 dimensional spacetime. There, they look for loop and string-like excitations to identify topological orders.

This science has massive potential for future technology. Most current technology relies on symmetry-breaking orders. Ferromagnetic materials allow us to store gigabytes of data on hard drives. Crystals allow us to create transistors for semiconducting devices. Topological order offers something even more advanced. It could enable topological quantum computing. This technique would use topologically ordered states as a medium. Because the entanglement is non-local, it is distributed among many particles. This non-locality could make quantum computers much more stable and powerful.