Laughlin States

Definition

Laughlin states are specific quantum states of matter that occur in two-dimensional electron systems under strong magnetic fields, characterized by fractional quantized Hall conductance. They are a manifestation of the fractional quantum Hall effect, which reveals the topological order of the system and the emergence of anyonic excitations. These states are named after Robert Laughlin, who proposed their existence in 1983 as a way to explain the observed phenomena in experiments on the quantum Hall effect.

5 Must Know Facts For Your Next Test

1. Laughlin states arise from the many-body wave function formulated by Robert Laughlin, which describes a system of electrons in a strong magnetic field as a function of their coordinates.
2. These states are characterized by a filling factor $ u = rac{1}{m}$, where $m$ is an odd integer, leading to fractional quantization of the Hall conductance.
3. The excitation spectrum of Laughlin states includes anyonic quasiparticles, which exhibit fractional statistics and play a significant role in topological quantum computing.
4. Laughlin states are highly robust against disorder, making them ideal candidates for experimental realizations in systems like semiconductor heterostructures.
5. The discovery of Laughlin states advanced our understanding of topological phases of matter, linking condensed matter physics with ideas from mathematics and topology.

Review Questions

• How do Laughlin states relate to the concept of fractional quantum Hall effect and what is their significance?
  • Laughlin states are integral to understanding the fractional quantum Hall effect, as they provide a theoretical framework to explain the phenomenon where Hall conductance takes on fractional values. These states represent a new phase of matter that reveals unique properties such as topological order and anyonic excitations. Their significance lies in how they showcase the connection between quantum mechanics and topology, leading to potential applications in quantum computing.
• Discuss the role of anyons in Laughlin states and how they differ from traditional particles like bosons and fermions.
  • In Laughlin states, anyons emerge as quasiparticles that possess fractional statistics, unlike traditional particles such as bosons and fermions which obey integer statistics. The unique statistical behavior of anyons allows for braiding operations that can be used for fault-tolerant quantum computation. This property sets them apart from ordinary particles and highlights the exotic nature of two-dimensional systems where Laughlin states exist.
• Evaluate the implications of Laughlin states for advancements in quantum computing and condensed matter physics.
  • Laughlin states have significant implications for advancements in quantum computing because they provide a platform for realizing topologically protected qubits through anyonic braiding. This robustness against local perturbations suggests a pathway towards error-resistant quantum computation. Furthermore, studying Laughlin states deepens our understanding of topological phases in condensed matter physics, revealing new connections between physics and mathematical concepts that could lead to novel materials and technologies.

Related terms

Fractional Quantum Hall Effect: A quantum phenomenon observed in two-dimensional electron systems at low temperatures and high magnetic fields, where the Hall conductance takes on fractional values.

Topological Order: A type of order in a quantum state that is characterized by global properties rather than local ones, leading to robust excitations that are resistant to local perturbations.

Anyons: Exotic quasiparticles that arise in two-dimensional systems, which can have fractional statistics distinct from bosons and fermions.