Given the following recursion relations:

$$K^\prime = 2K \Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big)^2$$


$$h^\prime = 3h\Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big),$$

where $K = J/k_B T$ and $h = H/k_B T$ with $H$ the external magnetic field.

I want to reconstruct Figure 9.7 in Goldenfeld Lectures on phase transitions and renormalization group. In order to do this I want to determine the fixed points but they do not coincide with the RG flow diagram sketched in the book. I thought the fixed points (defined by $K^\prime = K$ and $h^\prime =h$) were:

(1) $K=0$, $h \rightarrow \pm \infty$

(2) $K=h=0$

(3) $K \rightarrow \infty$, $h=0$

(4) $K \rightarrow \infty$, $h \rightarrow \pm \infty$

(5) $K \approx 0.34$, $h=0$ or $h \rightarrow \pm \infty.$

Where (5) describes the only non-trivial point. However in the plot I only see the points (1), (2), (3). And for (5) only the solution $K \approx 0.34$, $h = 0$. Why am I overcounting the fixed points in this problem? And how do the fixed points relate to the temperature e.g. in case (3), there we should have $T \rightarrow 0$ but how can we then find $h = 0$?


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