Fractional Quantum Hall Effect in TGD framework

The generalization of the imbedding space discussed in previous posting allows to understand fractional quantum Hall effect (see this and this).

The formula for the quantized Hall conductance is given by

σ= ν× e²/h,ν=m/n.

Series of fractions in ν=1/3, 2/5 3/7, 4/9, 5/11, 6/13, 7/15..., 2/3, 3/5, 4/7 5/9, 6/11, 7/13..., 5/3, 8/5, 11/7, 14/9... 4/3 7/5, 10/7, 13/9... , 1/5, 2/9, 3/13..., 2/7 3/11..., 1/7.... with odd denominator have bee observed as are also ν=1/2 and ν=5/2 state with even denominator.

The model of Laughlin [Laughlin] cannot explain all aspects of FQHE. The best existing model proposed originally by Jain [Jain] is based on composite fermions resulting as bound states of electron and even number of magnetic flux quanta. Electrons remain integer charged but due to the effective magnetic field electrons appear to have fractional charges. Composite fermion picture predicts all the observed fractions and also their relative intensities and the order in which they appear as the quality of sample improves.

I have considered earlier a possible TGD based model of FQHE not involving hierarchy of Planck constants. The generalization of the notion of imbedding space suggests the interpretation of these states in terms of fractionized charge and electron number.

1. The easiest manner to understand the observed fractions is by assuming that both M⁴ an CP₂ correspond to covering spaces so that both spin and electric charge and fermion number are quantized. With this assumption the expression for the Planck constant becomes hbar/hbar₀ =n_b/n_a and charge and spin units are equal to 1/n_b and 1/n_a respectively. This gives ν =nn_a/n_b². The values n=2,3,5,7,.. are observed. Planck constant can have arbitrarily large values. There are general arguments stating that also spin is fractionized in FQHE and for n_a=kn_b required by the observed values of ν charge fractionization occurs in units of k/n_b and forces also spin fractionization. For factor space option in M⁴ degrees of freedom one would have ν= n/n_an_b².

2. The appearance of n_b=2 would suggest that also Z₂ appears as the homotopy group of the covering space: filling fraction 1/2 corresponds in the composite fermion model and also experimentally to the limit of zero magnetic fiel [Jain]. Also ν=5/2 has been observed.

3. A possible problematic aspect of the TGD based model is the experimental absence of even values of n_b except n_b=2. A possible explanation is that by some symmetry condition possibly related to fermionic statistics kn/n_b must reduce to a rational with an odd denominator for n_b>2. In other words, one has k propto 2^r, where 2^r the largest power of 2 divisor of n_b smaller than n_b.

4. Large values of n_b emerge as B increases. This can be understood from flux quantization. One has eBS= nhbar= n(n_b/n_a)hbar₀. The interpretation is that each of the n_b sheets contributes n/n_a units to the flux. As n_b increases also the flux increases for a fixed value of n_a and area S: note that magnetic field strength remains more or less constant so that kind of saturation effect for magnetic field strength would be in question. For n_a=kn_b one obtains eBS/hbar₀= n/k so that a fractionization of magnetic flux results and each sheet contributes 1/kn_b units to the flux. ν=1/2 correspond to k=1,n_b=2 and to a non-vanishing magnetic flux unlike in the case of composite fermion model.

5. The understanding of the thermal stability is not trivial. The original FQHE was observed in 80 mK temperature corresponding roughly to a thermal energy of T≈ 10^-5 eV. For graphene the effect is observed at room temperature. Cyclotron energy for electron is (from f_e= 6× 10⁵ Hz at B=.2 Gauss) of order thermal energy at room temperature in a magnetic field varying in the range 1-10 Tesla. This raises the question why the original FQHE requires so low a temperature? The magnetic energy of a flux tube of length L is by flux quantization roughly e²B²S≈ E_c(e)m_eL(hbar₀=c=1) and exceeds cyclotron energy roughly by factor L/L_e, L_e electron Compton length so that thermal stability of magnetic flux quanta is not the explanation.

   A possible explanation is that since FQHE involves several values of Planck constant, it is quantum critical phenomenon and is characterized by a critical temperature. The differences of the energies associated with the phase with ordinary Planck constant and phases with different Planck constant would characterize the transition temperature. Saturation of magnetic field strength would be energetically favored.

References

[Laughlin] R. B. Laughlin (1983), Phys. Rev. Lett. 50, 1395.
[Jain] J. K. Jain (1989), Phys. Rev. Lett. 63, 199.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-Matrix".