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  Quasi-particle and quasi-hole excitations of Laughlin states and generalization of Laughlin states

+ 3 like - 0 dislike

The Laughlin wave function at filling fraction \(\nu=\frac{1}{m}\)is

\Psi_m=\prod_{i<j}(z_i-z_j)^m e^{-\sum|z_i|^2/4l_B^2}

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


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 $\nu=1/3$ Laughlin state, we get a $\nu=2/7$ FQH state with wave function:


and by condensing quasi-particles of $\nu=1/3$ Laughlin state, we get a $\nu=2/5$ FQH state with wave function:


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

{\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}={\rm resulting\ filling}

However, these two examples are telling me (notice the factor $\frac{1}{2}$):

{\rm original\ filling}+{\rm resulting\ filling}\times{\rm charge\ of\ excitation}\times\frac{1}{2}={\rm resulting\ filling}

(The first example fits as
and the second example fits as

My questions are: 

1) Why should there by a $\frac{1}{2}$ factor? 

2) Is it related to the power of $(\xi_i-\xi_j)^2$, namely $2$? 

3) If it is related to that $2$, can I change that $2$ into another integer to get another FQH state?

asked Jun 5, 2014 in Theoretical Physics by Mr. Gentleman (270 points) [ no revision ]

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