# Spin-squeezing scaling with the number of particles in one-axis twisting hamiltonian

I am exploring the one axis twisting (OAT) hamiltonian $$\hat{H}=\chi S_z^2$$ with $$S_z=\sum_{i=1}^N\frac{\sigma_z^i}{2}$$ and considering the initial state to be $$\left|\psi(0) \right>=\left|+x\right>^{\otimes N}$$, where +x means that the state is pointing along the x direction in the Bloch sphere, i denotes the index of the ith spin and $$\chi$$ is the coupling strength. Define the spin-squeezing $$\xi^2$$ as https://en.wikipedia.org/wiki/Spin_squeezing. For a large number of particles $$N$$ we can find the scaling of the minimum spin-squeezing ($$\xi^2_{min}$$) with the number of particles, a simple model for such scaling is $$\xi^2_{min}(N)=C/N^{\alpha}$$ (this can be done in general but the fitting improves at the large number of particles). My question is, is there any reference where this calculation or a numerical simulation has been done such that we have an estimation of $$\alpha$$ in OAT?. How can we include possible corrections beyond $$N^{\alpha}$$? Thanks.