[Paper Review] Taking stock of the quantum Hall effects: Thirty years on

Tapash Chakraborty, K. von Klitzing|arXiv (Cornell University)|Feb 25, 2011

Quantum and electron transport phenomena42 references3 citations

TL;DR

This paper reviews the quantum Hall effects (QHE) over the past 30 years, tracing the discovery of the integer and fractional QHE in two-dimensional electron systems, highlighting the role of Landau quantization, electron interactions, and topological order. It emphasizes the precision of the von Klitzing constant $ R_K = h/e^2 \approx 25812.807\,\Omega $, the emergence of composite fermions and non-Abelian quasiparticles at $ \nu = 5/2 $, and the QHE's influence on quantum gravity, string theory, and topological quantum computing.

ABSTRACT

The quantum Hall effects, discovered about thirty years ago have remained one of the most spectacular discoveries in condensed matter physics in the past century. Those discoveries triggered huge expansion in the field of low-dimensional electronic systems, the area grew at an unprecedented rate and continues to expand. Novel and challenging observations, be it theoretical or experimental, have been reported since then on a regular basis. Additionally, the effects have inspired physicists to find analogous situations in far-flung fields as disparate as string theory or black hole physics.

Motivation & Objective

• To summarize the historical development and key milestones in the discovery and understanding of the quantum Hall effects since their inception in 1980.
• To explain the theoretical and experimental foundations of the integer and fractional quantum Hall effects, including the role of Landau levels, electron-electron interactions, and topological invariants.
• To examine the implications of the QHE for fundamental physics, including the definition of the von Klitzing constant, the fine structure constant, and metrology.
• To explore the broader impact of QHE on theoretical physics, including analogies with string theory, black hole physics, and topological insulators.
• To discuss recent advances in graphene and the potential for fault-tolerant quantum computation via non-Abelian anyons at $ \nu = 5/2 $.

Proposed method

• Analysis of experimental data from silicon MOSFETs and GaAs heterostructures, focusing on Hall resistance plateaus at quantized values $ \rho_{xy} = h / (n e^2) $ under strong magnetic fields and low temperatures.
• Application of Landau quantization theory to describe the formation of discrete energy levels in 2D electron systems under perpendicular magnetic fields.
• Use of the filling factor $ \nu = n_s \Phi_0 / B $ to classify integer and fractional QHE states, with $ \Phi_0 = h/e $ as the magnetic flux quantum.
• Employment of Laughlin's theory of incompressible quantum fluids to explain the fractional QHE at odd-denominator filling factors, such as $ \nu = 1/3, 2/5 $, etc.
• Adoption of the composite fermion model (Jain theory) to describe higher-order FQHE states and explain the observed sequence of fractional plateaus.
• Use of the fermion-Chern-Simons theory to describe the compressible state at $ \nu = 1/2 $, where the system behaves as a Fermi gas of composite fermions.

Experimental results

Research questions

• RQ1 What causes the precise quantization of the Hall resistance to $ h/e^2 $, and why is it independent of material and geometry?
• RQ2 How do electron-electron interactions give rise to the fractional quantum Hall effect at rational filling factors?
• RQ3 Why is the $ \nu = 5/2 $ state particularly significant, and what evidence supports its non-Abelian anyonic nature?
• RQ4 How do the quantum Hall effects in graphene differ from conventional 2DEGs, and why is quantization observed at room temperature?
• RQ5 What are the implications of the QHE for fundamental physics, including quantum gravity and topological order?

Key findings

• The integer quantum Hall effect was discovered on February 5, 1980, at 2 a.m. in Grenoble, France, with Hall resistance quantized at $ h/e^2 \approx 25812.807\,\Omega $, independent of material and geometry.
• The fractional quantum Hall effect was discovered on October 7, 1981, at the Francis Bitter Magnet Laboratory, with quantized Hall resistance at rational filling factors such as $ \nu = 1/3, 2/5 $, etc.
• The von Klitzing constant $ R_K = 25812.807449 \pm 0.000086\,\Omega $ was adopted as a standard for resistance calibration in 1990, with a precision of one part in a billion.
• The $ \nu = 5/2 $ state exhibits evidence of fractionally charged quasiparticles with charge $ e^* = e/4 $, supporting a paired, non-Abelian state relevant to topological quantum computation.
• In graphene, the Hall resistance shows a half-integer quantization $ \rho_{xy} = h / (4e^2) $ due to massless Dirac fermions, with quantized plateaus observed even at room temperature.
• Theoretical models such as the composite fermion picture and the fermion-Chern-Simons theory successfully explain the FQHE at $ \nu = 1/2 $ and other filling factors, with the latter predicting a compressible state at $ \nu = 1/2 $.