# Order parameter of first-order phase transition in the Ising model

Let's consider the $d-$dimensional Ising model, with $d>1$.

If we reverse the external magnetic field ($h \to -h$) at $T<T_c$ , there is a first-order phase transition where the magnetization $m$ -the order parameter- changes discontinuously (e.g. from $1$ to $-1$ if we go from positive to negative $h$).

What I don't understand is wether $m$ can be considered as the order parameter of this first order first transition. My doubt comes from the fact that the usual definition of order parameter is "a quantity that is non-zero in the ordered phase and zero in the disordered phase" (1).

During this first order phase transition, $m$ goes from a positive value to a negative value (or vice versa), but it is never $0$. Does this mean that we have to consider a different order parameter for this phase transition, for example $\eta=(m+1)/2$?

Also, there is no clear "ordered" and "disordered" phase here: the final states are statistically equivalent, as we can see from the fact that the Hamiltonian

$$H(\{\sigma_i\}) = -\sum_{\langle i,j\rangle} \sigma_i \sigma_j-h\sum_{i=1}^N \sigma_i$$

is invariant under the simultaneous reversal of all the spins and of the external field

$$\sigma_i \to -\sigma_i \ \ \forall i \\ h \to -h$$

References

(1) K. Binder, Theory of first-order phase transitions, 1987 Rep. Prog. Phys. 50 783

• An order parameter is just something which distinguishes two phases, whether or not it is zero on one side is just a matter of convention. In case there is a continuous transition, it is useful to work with an order parameter which is defined as to be zero in one of the two phases, since then by continuity, the order parameter is small in the neighbourhood of the transition, allowing one to expand the action in powers of it and hopefully derive some insightful low-energy effective description. This is not an option for first order transitions anyway. Feb 5, 2018 at 9:46
• Correction: expanding the action/free energy in terms of powers of the order parameter can still be useful for first order transitions as well (as I explained here: physics.stackexchange.com/questions/197784/… ) but it doesn't make much sense to derive an action in terms of the fluctuations of the order parameter, as is common (and very powerful) for continuous transitions. Feb 5, 2018 at 9:51
• @RubenVerresen So the conclusion is that $m$ is the correct order parameter for the 1st order transition of the Ising model? Feb 5, 2018 at 10:19