Topological order

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.