1. Quantum Hall Fluids
By Andrew York
12/5/2008

2. Purpose and Outline
Purpose: To explain field theory behind quantum Hall fluids, which leads to a description of their properties
Outline
Integer Hall effect
Fractional Hall effect
Guiding assumptions
Developing the Lagrangian
Interpreting the Lagrangian
Understanding quasiparticles
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3. A Quantum Hall System B
Collection of electrons bound to a plane, under influence of magnetic field B
Magnetic field normal to plane, strong enough to align all electrons (treat as spinless)
Simple descriptions of motion:
Classical:
QM: after setting r
Electrons move in Larmor circles described by
Clearly, something interesting when
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4. Integer Hall Effect
Textbook case of a spinless electron in a magnetic field
Wave equation:
Energy levels:
Degeneracy:
Define “filling factor:”
Clearly for integer values of , system will be incompressible
Attempts to compress will reduce area and degeneracy, forcing electrons into higher energy level (keep in mind energy spacing of )
This is known as “integer Hall effect”
12/5/2008

5. Fractional Hall Effect
Surprising experimental discovery that Hall fluids prove incompressible for filling factors with odd-denominator fractions
Better description of system starts from following premises:
Still confined to plane (2+1 dimensions)
E-M current conserved,
Descriptive field theory will follow from good local Lagrangian
Important physics occurs at large distances and large time (lowest dimension terms will dominate the Lagrangian)
Parity and time reversal are broken by the external magnetic field
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6. Describing Current
Premises 1)* and 2)** are enough to conclude This is because Divergence = 0 implies current is the curl of something else
Note gauge invariance, constant under
normalization
* - 1) requires 2+1 dimensions
** - 2) requires
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7. Guessing the Lagrangian
3)* and 4)** motivate us to guess the local Lagrangian with lowest order gauge invariant terms
Self-interaction term:
Coupling to external potential A:
Quasiparticle*** interaction:
Lagrangian:
* ) states field theory follows from good local Lagrangian
** - 4) implies only lowest order terms matter in Lagrangian
*** - “quasiparticle” effects are effects due to many body interaction
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8. Simplifying the Lagrangian
Reorder coupling term (after integrating by parts and dropping the surface term)
Defining we get
Integrate using identity
Re-expand using
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9. Interpreting the Lagrangian
Field coupling
Quasiparticle coupling
Quasiparticle coupling to field
From field coupling term, we deduce
implies excess density of electrons related to variation in magnetic field by
implies electric field produces current in the orthogonal direction, so
From quasiparticle coupling to field, we deduce charge of quasiparticle is 1/k
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10. Interpreting the Quasiparticle Term
Quasiparticle effects are effects due to many-body interaction (structure in the system )
In this case, quasiparticles are groups of electrons or electron holes
Partially filled states can in principle be filled any way, but in practice will fall into evenly-distributed ground state due to electron repulsion
System ground state can be described by distribution of quasiparticles
Comparing quasiparticle term to Hopf term
Statistics parameter describes the exchange of individual quasiparticles
Phase factor describes the exchange of groups of k quasiparticles, and must be an odd integer for fermions
Thus, k must be an odd integer, and from earlier indicates system must have odd-denominator filling
12/5/2008