Magnetization ($z$-basis) of a 1D Transverse Ising Model

Question

I'm trying to find the magnetization $\langle\sigma_{z} \rangle$ of a 1D transverse Ising chain and plot it as a function of the transverse field $\lambda$. More specifically, I want to plot this for different values of $N$ (number of spins) for instance, $N=4,8,16,32$ etc. This is at zero temperature. The Hamiltonian of the system is this $$ H = \sum_{i=1}^{n} \sigma_{i}^{x} \sigma_{i+1}^{x} + \lambda \sum_{i=1}^{n} \sigma_{i}^{z} . $$ Introducing periodic boundary conditions I have the Hamiltonian as $$ H = \sum_{i=1}^{n} \sigma_{i}^{x} \sigma_{i+1}^{x} + \lambda \sum_{i=1}^{n} \sigma_{i}^{z} + \sigma_{1}^{y} \sigma_{2}^{z}... \sigma_{n-1}^{z} \sigma_{n}^{y} . $$ In particular I want to express the value of the magnetization solely as a function of $\lambda$ (if possible) and see how it changes with $N$. I'm expecting to see a kind of step function which becomes smoother as $N$ increases. Initially the spins starting in a paramagnetic phase and then moving to a ferromagnetic one.

Answer 1

In the thermodynamic limit, the magnetisation of the one-dimensional transverse field Ising model is $m^2(\lambda) + c$, where $c\propto|r-r’|$. However, this is only when $N\to\infty$. For finite $N$, there is no magnetisation ever.