Topological order

Definition

Topological order refers to a type of order that goes beyond traditional symmetry-breaking scenarios in quantum systems, representing states that are robust against local perturbations. This concept highlights a unique form of long-range entanglement, where the properties of the system can remain invariant under smooth deformations, making it particularly relevant in the study of exotic phases of matter like those observed in the quantum Hall effect. Topological order provides insight into the non-local quantum correlations that characterize these states.

5 Must Know Facts For Your Next Test

1. Topological order is characterized by ground states that cannot be distinguished by local measurements alone, indicating the presence of global properties.
2. In systems with topological order, excitations can exhibit anyonic statistics, which are neither fermionic nor bosonic.
3. Topological order is often associated with robust features such as fractionalization of quantum numbers and emergence of degenerate ground states.
4. The quantum Hall effect exemplifies topological order, demonstrating how quantized conductance plateaus arise due to topological invariants.
5. Topologically ordered phases are resilient to local perturbations, making them potential candidates for fault-tolerant quantum computing applications.

Review Questions

• How does topological order differ from conventional orders observed in phase transitions?
  • Topological order differs from conventional orders in that it does not rely on symmetry breaking and is characterized by long-range entanglement. While traditional phase transitions involve changes in local order parameters, topological order reflects a deeper, global property of the system. This means that local measurements cannot fully capture the state of a system exhibiting topological order, highlighting its unique robustness and stability under local perturbations.
• Discuss the implications of anyons in relation to topological order and the quantum Hall effect.
  • Anyons are crucial to understanding topological order because they arise as excitations within such phases. Unlike conventional particles, anyons can exhibit fractional statistics, which can lead to unique braiding properties that are essential for topologically protected quantum computations. In the context of the quantum Hall effect, anyons emerge as excitations on the edge states, reinforcing the notion that these systems possess underlying topological characteristics that facilitate stable, quantized conductance.
• Evaluate how topological field theories contribute to our understanding of phases of matter with topological order.
  • Topological field theories provide a mathematical framework for classifying and analyzing phases of matter that exhibit topological order. By focusing on topological invariants rather than local properties, these theories allow researchers to identify distinct phases based on their global features and robust characteristics. This perspective not only deepens our understanding of phenomena like the quantum Hall effect but also opens avenues for exploring new materials and potential applications in quantum computing through their fault-tolerant properties.

Related terms

Quantum Hall effect: A phenomenon observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, characterized by quantized Hall conductance and topological invariance.

Anyons:

Exotic particles that exist in two dimensions and exhibit fractional statistics, playing a crucial role in the theoretical framework of topological order.

Topological field theory: A theoretical framework that describes quantum field theories with topological invariance, often used to understand and classify phases of matter and their associated order parameters.