Quantum Hall Effect - Casey Farren-Colloty

Overview

The Quantum Hall Effect represents one of the most remarkable manifestations of quantum mechanics in condensed matter systems. This phenomenon occurs when a two-dimensional electron gas is subjected to a strong magnetic field at low temperatures, leading to the quantization of the Hall conductivity in precise multiples of fundamental constants.

Hamilton Trust Internship 2025

This research is being conducted as part of the Hamilton Trust Internship program, working collaboratively with Daniel Barron and Matthew Blakeney. Our team is exploring the theoretical foundations of quantum many-body physics through the lens of the Quantum Hall Effect, combining our individual strengths to develop a comprehensive understanding of this fascinating phenomenon.

Key Discoveries

The discovery of the Integer Quantum Hall Effect by Klaus von Klitzing in 1980 earned him the Nobel Prize in Physics in 1985. Later, the Fractional Quantum Hall Effect, discovered by Tsui, Stormer, and Laughlin, revealed even more exotic quantum many-body states.

Our collaborative study focuses on understanding the theoretical foundations of this effect, particularly following the comprehensive treatment by Steven M. Girvin in "The Quantum Hall Effect: Novel Excitations and Broken Symmetries".

Theoretical Framework

The Hall Conductivity

The hallmark of the Quantum Hall Effect is the precise quantization of the Hall conductivity:

\[ \sigma_{xy} = \nu \frac{e^2}{h} \]

where ν is the filling factor, e is the elementary charge, and h is Planck's constant. The remarkable precision of this quantization has made it the basis for the modern definition of electrical resistance.

Landau Levels

In a strong magnetic field, the energy spectrum of electrons becomes highly degenerate, forming discrete Landau levels:

\[ E_n = \hbar \omega_c \left(n + \frac{1}{2}\right) \]

where ωc = eB/m is the cyclotron frequency. This quantization is fundamental to understanding the plateaus observed in the Hall resistance.

Laughlin Wavefunction

For the fractional quantum Hall states, Laughlin proposed a many-body wavefunction that captures the incompressible nature of these quantum fluids:

\[ \Psi_m = \prod_{i \lt j} (z_i - z_j)^p e^{-\frac{1}{4} \sum_i |z_i|^2 / l_B^2} \]

where $p$ is an odd integer and $l_B$ is the magnetic length.

Research Poster

Quantum Hall Effect Study

A comprehensive exploration of quantum many-body physics through the lens of the Quantum Hall Effect. This collaborative research, conducted under the Hamilton Trust Internship with Daniel Barron and Matthew Blakeney, showcases the theoretical framework, key equations, and physical insights gained from studying this remarkable quantum phenomenon.

Poster presentation coming soon - currently developing visual representations of the theoretical concepts and computational results.

This research poster will present our collaborative study of the Quantum Hall Effect under the Hamilton Trust Internship, including theoretical analysis, key mathematical formulations, and insights into the physics of quantum many-body systems. The work represents a joint effort between Casey Farren-Colloty, Daniel Barron, and Matthew Blakeney, based on extensive study of the literature and development of comprehensive working notes.

Simulations

Click here to see simulations.

Resources & References

Primary References

Masaki, Yusuke, Mizushima, Takeshi, and Nitta, Muneto. "Non-Abelian anyons and non-Abelian vortices in topological superconductors." In Encyclopedia of Condensed Matter Physics, pp. 755–794. Elsevier, 2024. ISBN: 9780323914086. DOI: 10.1016/b978-0-323-90800-9.00225-0
ResearchGate. "Typical measurement of the integer quantum Hall effect: The Hall resistivity exhibits plateaux." Available at: https://www.researchgate.net/figure/Typical-measurement-of-the-integer-quantum-Hall-effect-The-Hall-resistivity-exhibits_fig27_51949561
Tsui, D. C., Stormer, H. L., & Gossard, A. C. "Two-Dimensional Magnetotransport in the Extreme Quantum Limit." Physical Review Letters, vol. 48, pp. 1559–1562, 1982. DOI: 10.1103/PhysRevLett.48.1559
Girvin, Steven M. "The Quantum Hall Effect: Novel Excitations and Broken Symmetries." arXiv:cond-mat/9907002, 1999. https://arxiv.org/abs/cond-mat/9907002
Greiter, Martin. "Quantum Hall quarks." Physica E: Low-dimensional Systems and Nanostructures, vol. 1, no. 1, pp. 1–6, 1997. DOI: 10.1016/S1386-9477(97)00002-7. Full text
Tong, David. "The Quantum Hall Effect." TIFR Infosys Lectures, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 2016. http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf
Schwartz, Matthew D. "Chapter 12." In Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. ISBN: 978-1-107-03473-0.
Wilczek, Frank. "Quantum Mechanics of Fractional-Spin Particles." Physical Review Letters, vol. 49, no. 14, pp. 957–959, 1982. DOI: 10.1103/PhysRevLett.49.957. Full text

Key Concepts Explored

Landau Quantization
Energy levels in magnetic fields

Fractional States
Incompressible quantum fluids

Topological Order
Novel quantum phases of matter

Anyonic Excitations
Neither bosons nor fermions

Ongoing Investigation

This collaborative research continues to evolve as our team delves deeper into the mathematical formalism and physical insights of quantum many-body systems. The Hamilton Trust Internship provides an excellent framework for our joint investigation, which encompasses both the integer and fractional quantum Hall effects, exploring the emergence of topological order and exotic quasiparticle excitations. Working alongside Daniel Barron and Matthew Blakeney allows us to approach this complex topic from multiple perspectives, strengthening our overall understanding of these fundamental quantum phenomena.