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Consider the transverse field Ising model, with

$H=-J\sum_i\left(\sigma^x_i\sigma^x_{i+1}+g\sigma^z_i\right)$

What happens to the expectation value of the magnetization $\langle\sigma_z\rangle$ at finite temperature? Can anyone give me a ref?

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This model can be solved exactly by Jordan-Wigner transformation and the expectation value of $\sigma_z$ can be calculated rather straightforwardly(it maps to a local quantity in the fermion model). Physically, when $g>1$ the spins are ordered, so $\sigma_z$ has a finite expectation value at $T=0$ which will be reduced at finite temperature. For $g<1$, spins are disordered ($\sigma_z=0$) at $T=0$. Around $g=1$ there is a "quantum critical fan" in which various physical quantity has a scaling behavior as a function of $T$.

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