Topological order

Definition

Topological order refers to a type of quantum phase where the ground state of a system cannot be described by a local order parameter, but instead exhibits global entanglement that is robust against local perturbations. This concept is crucial in understanding certain phases of matter, particularly in systems exhibiting non-abelian anyons and topological quantum computing. Topological order allows for the classification of quantum states that possess inherent symmetries and invariances, leading to phenomena like fractional statistics and protected edge states.

5 Must Know Facts For Your Next Test

1. Topological order is not characterized by symmetry breaking, unlike conventional phases of matter, making it a unique feature in quantum physics.
2. The concept of topological order can be used to understand phases that are resilient to disorder, leading to robust quantum states that are ideal for fault-tolerant quantum computing.
3. In systems with topological order, excitations may have anyonic statistics, which are neither fermionic nor bosonic, leading to fractional quantum numbers.
4. Topological entanglement entropy is a measure used to characterize topological order, distinguishing it from other types of quantum entanglement.
5. The study of topological order has implications for emerging technologies like topological quantum computing, where information can be stored and manipulated using non-local properties.

Review Questions

• How does topological order differ from traditional forms of order seen in phase transitions?
  • Topological order differs from traditional forms of order by not relying on local order parameters or symmetry breaking. Instead, it is characterized by global properties and entanglement that remain stable under local disturbances. This results in unique phases of matter where excitations can exhibit non-trivial braiding statistics, setting it apart from conventional ordered phases like ferromagnetism or superconductivity.
• Discuss the significance of non-abelian anyons within the framework of topological order and their potential applications in quantum computing.
  • Non-abelian anyons are crucial for understanding topological order because they provide a mechanism for quantum information processing through their exchange properties. Their non-trivial braiding statistics allow for the creation of robust qubits that are less susceptible to local noise, which is essential for fault-tolerant quantum computing. By leveraging these anyons, researchers aim to develop systems that perform calculations using the principles of topology, enhancing the stability and efficiency of quantum computation.
• Evaluate the role of topological entanglement entropy in identifying phases with topological order and its implications for theoretical physics.
  • Topological entanglement entropy serves as a powerful tool for identifying phases with topological order by providing a distinct signature that differentiates them from conventional phases. It captures non-local correlations between degrees of freedom in a system and can reveal underlying symmetries associated with the topological phase. The implications for theoretical physics are significant as they deepen our understanding of quantum states and foster advancements in fields like condensed matter physics and quantum information theory.

Related terms

Quantum Hall Effect: A phenomenon observed in two-dimensional electron systems under low temperatures and strong magnetic fields, characterized by quantized Hall conductance and the emergence of topologically protected edge states.

Non-abelian anyons: Exotic quasiparticles that arise in two-dimensional systems exhibiting topological order, where their exchange leads to a change in the state of the system that depends on the order of exchange.

Topological insulator: A material that behaves as an insulator in its interior but supports conducting states on its surface due to topological properties, resulting from strong spin-orbit coupling.