= Symmetry-breaking order of limits
{title2=$m_+=\lim_{h\downarrow0}\lim_{N\to\infty}\langle m\rangle_{N,h}$}
At zero field and with <spin inversion symmetry> in its <boundary conditions>, a finite <Ising model> pairs every configuration with its spin-reversed configuration. Their equal <Boltzmann weights> force the <expected value> of its <spin magnetization> $m=N^{-1}\sum_i s_i$ to vanish. Below a model's ordering <critical temperature>, an arbitrarily small positive field can instead select the positive <spontaneous magnetization> if size is taken to infinity first:
$$
\lim_{h\downarrow0}\lim_{N\to\infty}\langle m\rangle_{N,h}=m_+>0,
\qquad
\lim_{N\to\infty}\lim_{h\to0}\langle m\rangle_{N,h}=0.
$$
The distinction is the order of limits and the stated symmetric <boundary conditions>; <plus boundary conditions> already break <spin inversion symmetry>. A zero-field symmetric mixture of the two ordered states can retain zero mean <spin magnetization> even in the <thermodynamic limit>. Also, $\langle|m|\rangle>0$ in a finite system is not by itself proof of <spontaneous symmetry breaking>.<Statistical Field Theory by David Tong/1 From Spins to Fields>[§1.2.1, Spontaneous Symmetry Breaking: displayed nested limit and the immediately following explanation]{ref}
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