# Pair correlation function of the QHE “plasma”

I am trying to teach myself the theory of quantum Hall effect, and realized that I can not reproduce a basic textbook result. Let me closely follow Girvin's Les Houches lectures (http://arxiv.org/abs/cond-mat/9907002 ).

Consider wave function of the $\nu=1$ filled state (formulas 1.83-1.88): $$\psi(z_1,\dots z_N)=\prod\limits_{i<j}(z_i-z_j)e^{-\frac14\sum\limits_k |z_k|^2},$$

According to Laughlin's suggestion we can treat the norm of the wave function as a partition function of a classical two-dimensional Coulomb plasma (1.89): $$Z=\int d^2z_1\dots d^2z_N|\psi(z_1,\dots z_N)|^2,$$ with a potential of the following form (1.90-1.91): \begin{eqnarray} |\psi(z_1,\dots z_N)|^2&=&e^{-2 U}, \\ U&=&-\sum\limits_{i<j}\ln |z_i-z_j|+\frac{1}{4}\sum_k |z_k|^2. \end{eqnarray} Now we define the two-point correlator of this classical plasma as an integral over all but two arguments of the squared wave function (1.111) \begin{eqnarray} g(r)=\frac{N(N-1)}{n^2Z}\int dz_3\dots dz_N |\psi(0,r,z_3,\dots z_N)|^2, \end{eqnarray} where $n$ is the uniform density of particles, which, according to Girvin, can be evaluated exactly and gives (1.112): $$g(r)=1-e^{-\frac{r^2}{2}}.$$

This integration is what I'm puzzled about. I tried to do it for few particles ($N=3,4,5$) analytically using Mathematica, analytically by hands and numerically, and I always obtained something like $$g(r)=e^{-\frac{r^2}{2}}\cdot r^2P(r),$$ where $P(r)$ is some low order polynomial with coefficients depending on $N$.

So, while the "correct" pair correlator behaves like $\lim g(r)\rightarrow 1$ as $r\rightarrow \infty$, my correlator dies off at infinity.

Does anyone know how to reproduce the classical result, or a good textbook/paper where this calculation is done in details?

Girvin himself gives some hint on how to do it (Ex. 1.15 on p.37 in his lectures), but to be honest I do not understand it.

Thanks a lot!

• Note that we expect $g(r)$ to vanish for large $r$ except in the thermodynamic limit $N\rightarrow\infty$. You should be able to calculate $g(r)$ for arbitrary $N$ and then take $N\rightarrow\infty$. I'll try to provide details in an answer soon. – d_b Oct 25 '15 at 1:50
• Thank you, @user37496 ! It is a very good point indeed. Looking forward to your detailed answer. – user10998 Oct 25 '15 at 21:02