From Abelian And Non Abelian Quantum Hall States To Exact Models Of Critical

The study of Quantum Hall states has revolutionized our understanding of condensed matter physics. Originally discovered in two-dimensional electron gas systems subjected to a strong magnetic field, Quantum Hall states are unique phenomena that exhibit fractional quantum numbers, robust topological properties, and fascinating edge states. These states have opened up new avenues for exploring emergent phenomena and have paved the way for the exploration of exotic states of matter.

Abelian Quantum Hall States

Abelian Quantum Hall states, also known as Laughlin states, were the first class of Quantum Hall states to be discovered. They are characterized by their fractionalized charges, which are anyonic in nature. These states arise due to strong electron-electron interactions in a two-dimensional system and are described by effective fractional statistics. The fractional charge of these states is given by the famous formula discovered by Robert B. Laughlin - e/m*3, where e is the elementary charge and m is an odd integer.

The fractional statistics of these states leads to phenomena such as the fractional quantum Hall effect, where the Hall conductance exhibits quantized plateaus at fractional multiples of the elementary conductance quantum. This effect has been experimentally observed in various systems, providing strong evidence for the existence of fractionalized excitations in these states.

Mapping of Parent Hamiltonians: From Abelian and non-Abelian Quantum Hall States to Exact Models of Critical Spin Chains (Springer Tracts in Modern Physics Book 244)

by Martin Greiter(2011th Edition, Kindle Edition)

4.4 out of 5

Language	:	English
File size	:	58028 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	390 pages
Screen Reader	:	Supported
Paperback	:	24 pages
Item Weight	:	4 ounces
Dimensions	:	8.27 x 0.06 x 11.69 inches

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Non-Abelian Quantum Hall States and Their Significance

Non-Abelian Quantum Hall states are an extension of the Abelian Quantum Hall states and represent even more exotic phases of matter. These states are characterized by non-Abelian anyons, which possess non-trivial braiding statistics. These anyons have the remarkable property that their fusion rules are topologically protected, giving rise to the potential for fault-tolerant quantum computation.

The discovery of non-Abelian states has had a profound impact on the field of topological quantum computation. These states provide a promising avenue for building fault-tolerant quantum computers as they are highly resistant to errors caused by decoherence. The braiding of anyons in these states can be used as a universal set of quantum gates, making them ideal building blocks for quantum computation.

Exact Models of Criticality and Their Connection to Quantum Hall States

Exact models of criticality have also emerged as an important area of research in condensed matter physics. These models describe critical points in phase transitions, where the system undergoes a dramatic change in its properties. The discovery of the connection between Quantum Hall states and exact models of criticality has opened up new ways of understanding the behavior of strongly correlated systems and has provided insights into the nature of emergent phenomena.

By mapping Quantum Hall states to exact models of criticality, researchers have gained a deeper understanding of the underlying physics behind these states. This mapping allows for the calculation of various critical exponents and scaling dimensions, shedding light on the universality of these states. It also provides a bridge between the fields of quantum Hall physics and conformal field theory, further enhancing our theoretical toolbox for studying these systems.

The Future of Quantum Hall Physics

The study of Quantum Hall states has only scratched the surface of the fascinating phenomena that can arise in condensed matter systems. Researchers are currently exploring new directions and investigating exotic states that exhibit even more intriguing properties. These include states with non-Abelian anyons that possess non-Abelian statistics, topologically protected quantum states, and even potential applications in quantum computing.

As our understanding of Quantum Hall states continues to deepen, it is likely that we will make further breakthroughs in our understanding of emergent phenomena and condensed matter physics. The field holds great promise for the development of new materials with unique properties and the exploration of novel quantum states that may revolutionize technology. The possibilities are endless, and the journey to unravel the mysteries of Quantum Hall physics is just beginning.

Mapping of Parent Hamiltonians: From Abelian and non-Abelian Quantum Hall States to Exact Models of Critical Spin Chains (Springer Tracts in Modern Physics Book 244)

by Martin Greiter(2011th Edition, Kindle Edition)

4.4 out of 5

Language	:	English
File size	:	58028 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	390 pages
Screen Reader	:	Supported
Paperback	:	24 pages
Item Weight	:	4 ounces
Dimensions	:	8.27 x 0.06 x 11.69 inches

FREE

This monograph introduces an exact model for a critical spin chain with arbitrary spin S, which includes the Haldane--Shastry model as the special case S=1/2.  While spinons in the Haldane-Shastry model obey abelian half-fermi statistics, the spinons in the general model introduced here obey non-abelian statistics.  This manifests itself through topological choices for the fractional momentum spacings.  The general model is derived by mapping exact models of quantized Hall states onto spin chains.  The book begins with pedagogical review of all the relevant models including the non-abelian statistics in the Pfaffian Hall state, and is understandable to every student with a graduate course in quantum mechanics.