Answer to Question #191936 in Quantum Mechanics for Svend

Question

I have just recently been introduced to the Kalmeyer-Laughling wavefunction

$   \psi(s_1,\dots,s_N) \propto \delta_s \prod_{i<j} (z_i-z_j)^{\frac{s_is_j}{2}-\frac{1}{2}}\prod_k e^{\frac{i\pi}{2}(k-1)(s_k+1)}$

where we consider a lattice in two dimensions, and on each lattice site there is a spin-1/2 particles in either state up or state down, $s_i = \pm 1$. $\delta_s$ is 1 if the sum of the spin is zero, zero otherwise. I am bit confused about the $k$, but i believe it is simply a product over every lattice site?

I want to rewrite the wavefunction using occupation numbers instead, so $s_j = 2n_j-1$ where $n_j$ is the number of particles on site $j$. I should be able to get the following form:

$  \psi(n_1,\dots,n_N) \propto \delta_n \prod_{i<j}(z_i-z_j)^{2n_in_j}\prod_{i\neq j}(z_i-z_j)^{\alpha n_i}$

I tried rewriting the exponential function in the original form:

$\prod_ke^{\frac{i\pi}{2}(k-1)(s_k+1)} = \prod_ke^{\frac{i\pi}{2}(k-1)(2n_k)} = \prod_ke^{i\pi(k-1)n_k} $

Answer 1

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