Can transition from ferromagnetism exist in 1D chain of spins?
I know that: $$\Delta F = 2J - kT\ln(N)$$
in the spin $1/2$ Ising model, where $N$ is the number of ways in which we can create the domain wall. For large $N$ and $T\neq 0$, second term will dominate and $\Delta F$ will be negative resulting in no long range ordering of spins and hence $0$ magnetization. But for $T = 0$, first term dominates and we have all spins up or all down in the ground state resulting in non zero magnetization. That means we can have a phase transition from ferromag/antiferromag to paramagnetism only at $T =0$. Is this argument correct for non existence of Ferromagnetism in 1D?