Finite temperature transverse magnetization in transverse Ising model

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 1

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 $T$.

Answer 2

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 $T=0$

See section 2 of this paper for more details. Note that in the paper, the labels $x$ and $z$ are inverted with respect to those in your equation.