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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.

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    $\begingroup$ Sorry why are there $\sigma^y$'s? $\endgroup$
    – jacob1729
    Jan 27 '21 at 18:48
  • $\begingroup$ There will be no step function. Also, what is your question? $\endgroup$ Jan 27 '21 at 20:07
  • $\begingroup$ @jacob1729 Presumably the boundary term couples to the global parity in the z basis. However, this is what I know from mapping to free fermions - then you get this effect in the fermion model. I would suspect that this is the kind of Ising model you get when transforming back a periodic free fermion chain. $\endgroup$ Jan 27 '21 at 20:14
  • $\begingroup$ @NorbertSchuch What is the equation that relates $\lambda$ and $\langle \sigma_{z} \rangle$? $\endgroup$
    – Alto_1254
    Jan 27 '21 at 20:51
  • $\begingroup$ Giving an equation would be a valid answer? (And it is $\langle\sigma_z\rangle$, not $\langle\sigma_x\rangle$?) In any case, this can be solved by mapping the system to free fermions via the Jordan-Wigner transformation. $\endgroup$ Jan 27 '21 at 21:45
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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.

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    $\begingroup$ The question asks for the magnetization along the field. $\endgroup$ Jan 28 '21 at 7:27

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