I'm working on a project studying the transverse field Ising model in 1D with periodic boundary conditions, given by $$H = - J \sum_{i=1}^{N} \sigma^z_i \sigma^z_{i+1} - h \sum_{i=1}^N \sigma^x_i$$ I have written a code to construct and diagonalize the Hamiltonian. Taking the ground state eigenvector, I have written some code to compute the entanglement entropy between two halves of system $S(x)$, where in the left half of the system (subsystem A) there are $x$ spins and in the right half of the system (subsystem B) there are $N-x$ spins. I've plotted the results for various values of $h$ here:
The fit is the expected result from conformal field theory at the critical point $h_c=1$, $$S(x) = A + \frac{c}{3}\log\left[\frac{N}{\pi} \sin \frac{\pi x}{N}\right]$$ where $c$ is the central charge. From my fit I got $c \approx 0.510$, pretty close to the exact value of $c = 0.5$.
First I note that when $h=0$ we have a purely classical theory, the groundstate is simply all spins up or all spins down, and there is no entanglement entropy. In the limit $h\to\infty$, we again have a classical theory and all spins point along the field direction. Again the entanglement entropy is zero.
My questions is in the interpretation of these results in the regions $h<1$ and $h>1$. How do I understand why for $h<1$ the entanglement is large and relatively uniform, while for $h>1$ the entanglement is small and relatively uniform? And how do I understand why at $h=1$ the entaglement is "least flat", displaying the largest variation (small regions display low entanglement, larger regions display high entanglement)?