The Laughlin wave function at filling fraction ν=1mis
Ψm=∏i<j(zi−zj)me−∑|zi|2/4l2B
It is claimed in section 7.2.3 of Wen's book that the wave function of a quasi-hole excitation on the top of this state is
Ψh(ξ,ξ∗)=√C(ξ,ξ∗)∏i(ξ−zi)Ψm
I am wondering why this is a wave function of a quasi-hole, it looks more like the wave function of a quasi-particle to me. If this is indeed the wave function for a quasi-hole instead of a quasi-particle, what is the wave function of a quasi-particle?
Later in section 7.2.4, the author discusses a way to construct generalization of Laughlin states by adding quasi-holes or quasi-particles on the top of Laughlin states, and when the density of quasi-holes or quasi-particles reaches certain value, they form a Laughlin state by itself. And there are two examples: by condensing quasi-holes of ν=1/3 Laughlin state, we get a ν=2/7 FQH state with wave function:
Ψ=∫∏id2ξiΨ3∏(ξi−zj)(ξ∗i−ξ∗j)e−14l2B13|ξi|2
and by condensing quasi-particles of ν=1/3 Laughlin state, we get a ν=2/5 FQH state with wave function:
Ψ=∫∏id2ξi∏i<j(ξi−ξj)2(ξ∗i−2∂zi)Ψ3
My understanding is, the original electrons and added excitations form a Laughlin state with a common filling fraction, then by charge conservation, we should have
original filling+resulting filling×charge of excitation=resulting filling
However, these two examples are telling me (notice the factor 12):
original filling+resulting filling×charge of excitation×12=resulting filling
(The first example fits as
13−13×27×12=27
and the second example fits as
13+13×25×12=25.)
My questions are:
1) Why should there by a 12 factor?
2) Is it related to the power of (ξi−ξj)2, namely 2?
3) If it is related to that 2, can I change that 2 into another integer to get another FQH state?