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We shall denote the quasielectron position with η̄ rather than η for reasons that will be clear later.
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We shall denote the quasielectron position with η̄ rather than η for reasons that will be clear later.
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There a systematic way of improving the wave functions involves increasing the Hilbert space by including excitons, i.e., states with one or more composite fermions excited to a higher (unfilled) effective Landau level while conserving the total angular momentum.
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There a systematic way of improving the wave functions involves increasing the Hilbert space by including excitons, i.e., states with one or more composite fermions excited to a higher (unfilled) effective Landau level while conserving the total angular momentum.
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One is free to use representations with other linear combinations of the boson fields or additional auxiliary boson fields, as long as one does not change the correlations of the electron and hole operators, namely, □ Vα (z) Vβ (w) □ ∼ (z-w) Kαβ and □ Vα (z) Hβ (w) □ ∼ (z-w) δαβ where K is the K matrix of the state (Ref.).
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One is free to use representations with other linear combinations of the boson fields or additional auxiliary boson fields, as long as one does not change the correlations of the electron and hole operators, namely, □ Vα (z) Vβ (w) □ ∼ (z-w) Kαβ and □ Vα (z) Hβ (w) □ ∼ (z-w) δαβ where K is the K matrix of the state (Ref.).
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It is not obvious that this method will result in wave functions describing N electrons forming a quantum Hall droplet-i.e., a state that is homogenous up to distances on the order of the magnetic length away from the edge. That this is in fact the case can be understood by mapping the compactified plane to the sphere, where a homogeneous background charge can be introduced without breaking the conformal symmetry, by coupling it to the Gaussian curvature (Ref.). One can then show that the resulting correlators, which by construction describe a homogenous system, reproduce the result from the plane with the ad hoc prescription of taking a single compensating background point charge to infinity.
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It is not obvious that this method will result in wave functions describing N electrons forming a quantum Hall droplet-i.e., a state that is homogenous up to distances on the order of the magnetic length away from the edge. That this is in fact the case can be understood by mapping the compactified plane to the sphere, where a homogeneous background charge can be introduced without breaking the conformal symmetry, by coupling it to the Gaussian curvature (Ref.). One can then show that the resulting correlators, which by construction describe a homogenous system, reproduce the result from the plane with the ad hoc prescription of taking a single compensating background point charge to infinity.
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But, and this is very important, this conclusion is only valid provided there are no degeneracies in the multihole states, i.e., the state is completely determined by the position of the quasiparticles. This is not true for the Pfaffian state and this is precisely what opens the possibility for non-Abelian statistics. In CFT language, the fusion rules are now more complicated since there is an underlying SU(2) current algebra, which allows for two different splitting channels of the Laughlin hole.
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But, and this is very important, this conclusion is only valid provided there are no degeneracies in the multihole states, i.e., the state is completely determined by the position of the quasiparticles. This is not true for the Pfaffian state and this is precisely what opens the possibility for non-Abelian statistics. In CFT language, the fusion rules are now more complicated since there is an underlying SU(2) current algebra, which allows for two different splitting channels of the Laughlin hole.
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We want to stress that this technical difficulty does not affect the construction of the hierarchy ground states where there is a finite density of quasielectrons and where the charge is naturally neutralized by a new constant U(1) charge distribution that contributes to the physical background electric charge.
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We want to stress that this technical difficulty does not affect the construction of the hierarchy ground states where there is a finite density of quasielectrons and where the charge is naturally neutralized by a new constant U(1) charge distribution that contributes to the physical background electric charge.
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As explained in Ref., the operator Vα+1 is not unique. Technically speaking, there is a freedom in picking the coefficient of φα+1 when bosonizing the hole operator Hα since any even-integer braiding statistics of Hb is permitted, cf. Eq. -this leads to different Vα+1 ∼ ∂H bα -1 Vα. Physically, these different choices correspond to how closely one can "pack" the quasiparticles in the condensate. For instance, starting from 1/3, the "minimal" choice 5/3 in Eq. takes us to 2/5. Choosing the coefficient 5/3+2 = 11/3 instead, gives the less dense state 4/11 etc. The relation is independent of this choice.
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As explained in Ref., the operator Vα+1 is not unique. Technically speaking, there is a freedom in picking the coefficient of φα+1 when bosonizing the hole operator Hα since any even-integer braiding statistics of Hb is permitted, cf. Eq. -this leads to different Vα+1 ∼ ∂H bα -1 Vα. Physically, these different choices correspond to how closely one can "pack" the quasiparticles in the condensate. For instance, starting from 1/3, the "minimal" choice 5/3 in Eq. takes us to 2/5. Choosing the coefficient 5/3+2 = 11/3 instead, gives the less dense state 4/11 etc. The relation is independent of this choice.
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This was the reason for including the gaussian factor in the definition of P (η̄ k).
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This was the reason for including the gaussian factor in the definition of P (η̄ k).
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In fact, they do not correspond to a specific fusion channel but rather a linear combination of the two.
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See, e.g., chapter 9 in Ref..
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