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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?

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  • $\begingroup$ Are you asking "Is there long range order at finite temperature in the 1D Ising model?" or "Is this a valid argument to show that there is no long range order in 1D?". If it is the first then long range order in 1D is prevented by the Mermin-Wagner Theorem. If it is the second then I would have to say, based solely on your argument, what prevents a phase transition at a small, but non-zero temperature? $\endgroup$ – By Symmetry Feb 7 '18 at 13:26
  • $\begingroup$ At finite non zero T also, based on J, $\Delta F$ can be +ve and hence there can be phase transition. But I have never come across such an argument and have read that Ferromagnetism can't exist in 1D. $\endgroup$ – Draco_1125 Feb 7 '18 at 13:31
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    $\begingroup$ @BySymmetry : Mermin-Wagner theorem has nothing to do with the absence of discrete symmetry breaking in one-dimensional systems (it applies to systems with a continuous symmetry). Nevertheless, there are very general theorems showing that one-dimensional systems (at least with bounded spins and interactions decaying not too slowly) cannot undergo a phase transition at positive temperature. See, for example, Section 6.5.5 in this book. $\endgroup$ – Yvan Velenik Feb 7 '18 at 18:51
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[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, $14.3$.

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