Spontaneous magnetization

Remarks

The limit over boxes exists, and $h \mapsto m(\beta, h)$ is nondecreasing, so the limit as $h \to 0^+$ exists. The value $m^*(\beta)$ equals $\langle \sigma_0 \rangle^+_\beta$, the magnetization of the infinite-volume state with plus boundary condition. It is the order parameter of the transition: for $d \ge 2$ there is a critical $\beta_c \in (0, \infty)$ with $m^*(\beta) = 0$ for $\beta < \beta_c$ and $m^*(\beta) > 0$ for $\beta > \beta_c$.

Example: on $\Z^2$, Yang proved in 1952 that $m^*(\beta) = \big(1 - \sinh(2\beta J)^{-4}\big)^{1/8}$ for $\sinh(2\beta J) > 1$, and $m^* = 0$ otherwise. Near $\beta_c$ this gives $m^* \sim (\beta - \beta_c)^{1/8}$, so the critical exponent is $\beta = 1/8$.

Non-example: the magnetization $m(\beta, h)$ at a fixed $h > 0$ is positive at every temperature. It is a response to the field, not spontaneous, because the limit $h \to 0^+$ has not been taken. In dimension $d = 1$, $m^*(\beta) = 0$ for every $\beta$, so there is no spontaneous magnetization at any positive temperature.