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I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino.

I have a question about page 3 of the paper. For a magnetic system the correlation length $\xi$ exhibits the behaviour around the critical point ($\tau\approx T-T_c =0$ and $h=0$): $$\xi \approx \xi_0 \tau^{-\nu} \qquad \tau>0, h=0$$ $$\xi \approx \xi_0' (-\tau)^{-\nu'} \qquad \tau<0, h=0$$ In order to obtain the amplitudes $\xi_0$ and $\xi_0'$ Delfino says that Due to the invariance under spin reversal at $h = 0$, the magnetisation operator couples only to the states with odd (even) number of particles for $\tau > 0$ ($\tau < 0$). When $h \neq 0$, $\sigma$ couples to any intermediate state. It follows: $$\xi_0=2 \xi_0'$$

What does it mean? In which sense the operator $\sigma$ couples with odd/even number of particles? How an amplitude is two-times the other one?

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