Question: 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?

Answer: 
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 $T$.
Although your question is about the 1D Transverse Field Ising Model (TFIM), some people might be interested in the 3D TFIM. In this case, the normalized transverse magnetization $m$ evolves as follows with an applied transverse field:
$$ 
m = 
\begin{cases}
h & \text{if } h<h\_c \\
\tanh(h/t) & \text{if } h>h\_c \\
\end{cases}
$$
where:

* $h = \frac{H}{H\_c(T=0)}$ is the transverse field normalized to the critical field at $T=0$
* $t = \frac{T}{T\_c(T=0)}$ is the temperature normalized to the transition temperature at $H=0$
* $h\_c = \frac{H\_c(T)}{H\_c(T=0)}$ is the critical field at temperature $T$ normalized to the critical field at $T=0$

[![Transverse magnetization in the 3D TFIM at three temperatures](https://i.stack.imgur.com/f8ys6.png)](https://i.stack.imgur.com/f8ys6.png)

See section 2 of [this paper](http://stacks.iop.org/0022-3719/6/i=15/a=009) for more details. Note that in the paper, the labels $x$ and $z$ are inverted with respect to those in your equation.