What is the difference between hierarchy picture and composite Fermion in explaining FQHE

The fractional quantum Hall effect (FQHE) is a fascinating phenomenon in condensed matter physics, and understanding the differences between hierarchy picture and composite fermion theory is crucial for grasping how it works. Both concepts aim to explain the behavior of electrons in a two-dimensional system subjected to a strong magnetic field, but they approach the problem from different angles.

Hierarchy Picture

The hierarchy picture, proposed by Robert Laughlin, suggests that the FQHE arises from the formation of a series of fractional quantum Hall states, which can be thought of as a hierarchy of excitations. In this model, electrons condense into collective states characterized by fractional filling factors, such as 1/3, 2/5, and so on. Each of these states can be viewed as a new ground state formed by the interactions between electrons and the magnetic field.

• Fractional Filling Factors: The filling factor, denoted as ν, represents the ratio of the number of electrons to the number of magnetic flux quanta. In the hierarchy picture, each state corresponds to a different ν, leading to a rich structure of quantum states.
• Excitations: The excitations in this framework are quasiparticles that carry fractional charge and statistics, which can be understood as a result of the underlying electron interactions.

Composite Fermion Theory

On the other hand, the composite fermion theory, introduced by Jainendra Jain, provides a different perspective by treating electrons in a strong magnetic field as composite particles. In this model, each electron is thought to be bound to an even number of quantized vortices, effectively transforming the problem into one involving non-interacting particles in a reduced magnetic field.

• Composite Particles: The composite fermions are formed by attaching an even number of magnetic flux quanta to the electrons. This allows them to experience a weaker effective magnetic field, making their behavior more similar to that of non-interacting particles.
• Filling Factors: The filling factors for composite fermions are integer values, which correspond to the integer quantum Hall effect. This approach simplifies the understanding of the FQHE by relating it to the more familiar integer quantum Hall states.

Key Differences

To summarize the distinctions between the two theories:

• Approach: The hierarchy picture focuses on the interactions between electrons and the resulting fractional states, while composite fermion theory emphasizes the transformation of electrons into composite particles to simplify the problem.
• Nature of Particles: In the hierarchy picture, quasiparticles are fundamental excitations with fractional charge, whereas composite fermions are constructed from electrons and magnetic flux quanta.
• Filling Factors: The hierarchy picture deals with fractional filling factors, while composite fermion theory relates to integer filling factors in an effective magnetic field.

Real-World Implications

Both theories have been instrumental in advancing our understanding of the FQHE and have led to significant experimental predictions and observations. The hierarchy picture provides insight into the rich structure of fractional states, while composite fermion theory offers a more intuitive framework for understanding the behavior of electrons in strong magnetic fields.

In essence, while both approaches contribute valuable perspectives to the study of the fractional quantum Hall effect, they do so through different conceptual lenses, each illuminating unique aspects of this complex phenomenon.