fqhe-268pt2 lecture - What about the spin degree of freedom Other explanations for the fractional quantized Hall effect Fractions other than 1\/m or

# fqhe-268pt2 lecture - What about the spin degree of freedom...

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What about the spin degree of freedom?
Other explanations for the fractional quantized Hall effect Fractions other than 1/m or their particle-hole conjugates
Unitary transformation (Generalized gauge transformation) Ψ tr { r j } = Ψ elec { r j } × Chern Simons magnetic field: b i ( r ) = × a i ( r ) = 2 π ρ i ( r )
Mean-Field Approximation (Hartree Approximation) Replace true Chern-Simons magnetic field b by its average value <b> = 4 π n e . Replace Coulomb interaction by the average electrostatic potential , which is just a constant for a homogeneous system. (We can choose constant = 0). Get free fermions in an effective magnetic field Δ B = B - 4 π n e . Define effective filling p 2 π n e / Δ B . Compare to true filling factor f = 2 π n e / B . Find p -1 =f -1 -2 , or: f = p / (2p+1).
Examples
What about the spin degree of freedom?
If the electron filling fraction is f=1/2 Mean-field ground state = filled Fermi Sea k F = (4 π n e ) 1/2 If this is correct, then there is no energy gap , no QHE . Should be able to calculate all properties of ν =1/2 state using perturbation theory , starting from the mean field state. Perturbations include effects of v( r i - r j ) and fluctuations in the Chern Simons field Δ b i b i - < b > . Fluctuations are crucial in calculating transport and dynamic properties, as well as for understanding the energy scale for excitations 2 flux quanta of actual magnetic field per electron. Effective magnetic field = B - <b> = 0.
Why should perturbation theory work? For the quantized Hall states , where the mean field theory predicts an energy gap between the ground state and excited states, it is reasonable that perturbation theory should converge: if perturbation is not too strong, will not destroy the energy gap or change the character of the ground state. We find that fluctuations reduce the energy gap significantly, but do not generally drive it to zero. But what about ν =1/2, where there is no energy gap?
Landau s theory of Fermi liquids ZhETF 30 , 1058 ( 1956 ); 32 , 59 ( 1957 ); 35 , 97 ( 1958 ).

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