Abstract
We saw in my previous lecture on collective excitations that it was quite useful to study analogies between the fractional quantum Hall effect (FQHE) and superfluidity and superconductivity in films. Superfluids and superconductors are characterized by a spontaneously broken gauge symmetry and an associated order parameter. The purpose of this lecture is to investigate the question of whether or not there exists an order parameter which describes the FQHE state, possibly associated with some type of symmetry breaking1–8.
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References
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S. Kivelson (private communication) has pointed out that the statement in refs. 5 and 26 that this gauge field corresponds to λ (instead of λ/2) flux quanta per particle is incorrect. One must be careful to count the phase change associated with the motion of the flux tube in the presence of the other charge as well as the motion of the charge in the presence of the other flux tube. This minor error has no effect on the conclusions of either ref. 5 or 26.
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Polynomial terms of the form (|Ψ|2–1)2 which enforce the non-zero value of Ψ have not been explicitly included in the action in eq. (25). These will not be necessary in the FQHE where |Ψ|2 is defined by the total particle density42.
In a superconductor we do not envision that |Ψ|2 represents the total particle density since only a tiny fraction of the particles (those near the Fermi energy) actually take part in the pairing. We see from Laughlin’s wave function however that the FQHE ground state is very much different. Every electron is coupled to every other on an equal footing. Every electron is a source of m zeros seen by all the others. This is at least a hand-waving justification for the interpretation of |Ψ|2 as the total charge density.
For simplicity I am only considering the parent Laughlin states at filling factor v = 1/m and ignore the hierarchy of rational fractional states8.
The value of the coefficient in the linear term can be obtained from the previous argument using the singular gauge transformation. There is a unique value of the gauge flux per particle which leads to ODLRO (namely the value which allows the gauge field e to cancel on the average the vector potential of the applied field).
This is where the handedness described earlier appears.
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Girvin, S.M. (1988). Off-Diagonal Long-Range Order in the Quantum Hall Effect. In: Leavens, C.R., Taylor, R. (eds) Interfaces, Quantum Wells, and Superlattices. NATO ASI Series, vol 179. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1045-7_18
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DOI: https://doi.org/10.1007/978-1-4613-1045-7_18
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