Phase transition 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?
 A: [NB: The correct relation is $\Delta F = 2J - kT\ln(N-1)$]

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.

Yes. The first term ($2J$), which represents the energy penalty in creating a "domain wall" (i.e. in flipping all the spins on one side of a randomly chosen one), is completely negligible when $N$ is large. The second (entropic) term will result in a decrease in $F$, therefore favoring the creation of more and more domain walls and preventing long range order from being established.

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?

Yes, this is correct. In the 1D Ising model (and actually in any 1D model), phase transitions can only occur at $T=0$.
For a discussion of this, see K. Huang, Statistical Mechanics, $14.3$.
