Curie–Weiss magnetization bifurcation at the mean-field critical point

Write $q(+1)=(1+m)\mathrm{/}2$ and $q(-1)=(1-m)\mathrm{/}2$. Substituting into [F1] with $p(+1)=p(-1)=1\mathrm{/}2$ gives $D(q\mathrm{\parallel }p)=\frac{1+m}{2}\ln (1+m)+\frac{1-m}{2}\ln (1-m),$ with the zero-mass endpoint term omitted. Since $q$ depends continuously on $m\in [-1,1]$ and [F8] makes $D(q\mathrm{\parallel }p)$ continuous on the whole simplex, adding $\beta h(q)=-\beta J{m}^2\mathrm{/}2$ shows that ${\mathrm{\Phi }}_{\beta }$ is continuous on $[-1,1]$; [F8] supplies the boundary limit $t\ln t\to 0$ behind the convention $0\ln 0=0$. The formula is even in $m$.

For $\mathrm{\mid }m\mathrm{\mid }<1$ define $a(m):=\mathrm{atanh}(m):=\frac{1}{2}\ln \frac{1+m}{1-m}$. By [F3]–[F5] and direct differentiation, $a^{\mathrm{\prime }}(m)=\frac{1}{1-m^2},{\mathrm{\Phi }}_{\beta }^{\mathrm{\prime }}(m)=a(m)-\beta Jm\mathrm{.}$ For $0<m<1$ put $g(m):=a(m)\mathrm{/}m$ and define $r:[0,1)\to \mathbb{R}$ by $r(m):=m\mathrm{/}(1-m^2)-a(m)$. Then $g^{\mathrm{\prime }}(m)=r(m)\mathrm{/}{m}^2$, $r(0)=0$, and $r^{\mathrm{\prime }}(m)=2m^2\mathrm{/}(1-m^2{)}^2>0$ for $0<m<1$. The mean value theorem [F6] applies to $r$ on $[0,m]$ because $r$ is continuous there and differentiable on $(0,m)$; it gives $r(m)>0$, hence $g^{\mathrm{\prime }}(m)>0$.

The same theorem [F6] applied to $a$ on $[0,m]$ gives $g(m)=a^{\mathrm{\prime }}(c_m)$ for some $c_m\in (0,m)$; as $m\downarrow 0$, $c_m\downarrow 0$ and $a^{\mathrm{\prime }}(c_m)\to 1$, so ${\lim }_{m\downarrow 0}g(m)=1$. As $m\uparrow 1$, $(1+m)\mathrm{/}(1-m)\to +\mathrm{\infty }$, and the increasing onto logarithm [F3] gives $a(m)\to +\mathrm{\infty }$, hence $g(m)\to +\mathrm{\infty }$. Thus $g$ is continuous, strictly increasing from limiting value $1$ to $+\mathrm{\infty }$ on $(0,1)$.

If $\beta J\le 1$, then for $0<m<1$, ${\mathrm{\Phi }}_{\beta }^{\mathrm{\prime }}(m)=m(g(m)-\beta J)>0$ by step 2.1. The mean value theorem [F6] on each subinterval of $[0,1]$ makes ${\mathrm{\Phi }}_{\beta }$ strictly increasing there; continuity includes the endpoint $m=1$. Evenness then shows ${\mathrm{\Phi }}_{\beta }(m)>{\mathrm{\Phi }}_{\beta }(0)$ for every $m\mathrm{\ne }0$, so $0$ is the unique global minimizer.

If $\beta J>1$, the limits in step 2.1 give $a,b$ with $0<a<b<1$, $g(a)<\beta J<g(b)$. Continuity of $g$ on $[a,b]$ and the intermediate value theorem [F7] give an $m_{\beta }\in (a,b)$ with $g(m_{\beta })=\beta J$; strict increase of $g$ makes it unique on $(0,1)$. The derivative formula in step 1.2 is negative for $0<m<m_{\beta }$ and positive for $m_{\beta }<m<1$. By [F6], ${\mathrm{\Phi }}_{\beta }$ strictly decreases on $[0,m_{\beta }]$ and strictly increases on $[m_{\beta },1]$, including the endpoint by continuity. Evenness proves that the only global minimizers on $[-1,1]$ are $\pm m_{\beta }$; in particular the stationary point $m=0$ is not a minimizer.

The equation $g(m_{\beta })=\beta J$ is $\mathrm{atanh}(m_{\beta })=\beta J{m}_{\beta }$. If $r=(1+m_{\beta })\mathrm{/}(1-m_{\beta })$, then $\exp (2\mathrm{atanh}(m_{\beta }))=\exp (\ln r)=r$ by [F3], so the stated formula for $\tanh$ gives $\tanh (\mathrm{atanh}(m_{\beta }))=(r-1)\mathrm{/}(r+1)=m_{\beta }$. Hence $m_{\beta }=\tanh (\beta J{m}_{\beta })$. Conversely, any $m\in (0,1)$ solving this fixed-point equation yields $\exp (2\beta Jm)=(1+m)\mathrm{/}(1-m)$ by rearrangement of the stated $\tanh$ formula; applying $\ln$ and [F3] gives $g(m)=\beta J$. Strict increase of $g$ therefore proves uniqueness of the positive fixed point. By [F2], the global minimizers just classified are exactly the canonical equilibrium magnetizations.