Topological Order

Definition

Topological order is a type of quantum state that describes the global properties of a many-body system, characterized by non-local order parameters. It goes beyond traditional symmetry breaking and is particularly significant in systems with fractional excitations, making it a fundamental concept in understanding phenomena like the Quantum Hall effect and the fractional quantum Hall effect. This order implies that the ground state has a degenerate structure that is robust against local perturbations, leading to unique edge states and anyonic statistics.

5 Must Know Facts For Your Next Test

1. Topological order is distinct from conventional orders like ferromagnetism or superconductivity, as it does not rely on local order parameters.
2. In topologically ordered systems, ground states are degenerate and can transform into one another through non-local operations without changing their global properties.
3. Topological order enables the existence of excitations with fractional charge and statistics, allowing for phenomena like anyon formation.
4. The robustness of topological order makes it resilient to local disturbances, making systems that exhibit this order interesting for fault-tolerant quantum computing.
5. The fractional quantum Hall effect showcases topological order by revealing plateaus in Hall conductance at fractional values, indicating the presence of anyonic excitations.

Review Questions

• How does topological order differ from conventional orders found in quantum systems?
  • Topological order differs from conventional orders in that it is defined by non-local properties rather than local symmetries. While traditional phases, like magnetism or superconductivity, rely on breaking symmetry in local interactions, topological order features degenerate ground states with global properties that remain stable against local perturbations. This unique characteristic allows for fractional excitations and edge states, which aren't found in conventional ordered systems.
• Discuss how the presence of edge states relates to the concept of topological order in quantum systems.
  • Edge states are a direct consequence of topological order in a system. In a topologically ordered phase, these edge states appear at the boundaries and are protected by the topology of the bulk state. They carry current without dissipation and are robust against local disturbances due to their topological nature. This relationship highlights how topological features influence physical properties at the edges of materials exhibiting quantum phenomena like the Quantum Hall effect.
• Evaluate the implications of topological order for future technologies, particularly in quantum computing.
  • Topological order has significant implications for future technologies, especially quantum computing. Its resilience to local noise makes topologically ordered systems promising candidates for fault-tolerant quantum computation. The ability to host anyons—quasiparticles with fractional statistics—can be harnessed for topological quantum computation, where information is stored non-locally and manipulated through braiding operations. This approach potentially offers a more stable alternative to traditional qubit systems, aiming to address some of the key challenges faced in building practical quantum computers.

Related terms

Anyons:

Quasiparticles that exist in two-dimensional systems and exhibit statistics that are neither fermionic nor bosonic, crucial for topological order.

Quantum Hall Effect:

A quantum phenomenon observed in two-dimensional electron systems under strong magnetic fields, leading to quantized Hall conductivity and exhibiting topological order.

Edge States:

States that exist at the boundaries of a topologically ordered system, which are protected by the topology of the bulk state and can carry current without dissipation.