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We often learn that the order parameter is a perfect tool for the study of a phase transition (assuming Landau-Ginzburg theory is applicable). The order parameter is finite in the ordered phase, and perfectly zero in the disordered phase. Indeed, this is how we define the order parameter; if a physical observable is finite in the disordered phase, then it wouldn't be an order parameter by definition.

However, experimentally speaking, such a perfect order parameter seems unlikely. Are there any experimental, real-world examples (in condensed matter, high-energy, statistical physics, etc.) where the order parameter is nearly zero but still finite in the disordered phase, just above the critical point?

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    $\begingroup$ One limitation here is the experimental resolution. Another thing to consider is that the theory of phase transitions is developed for the thermodynamic limit: $N\rightarrow\infty, V\rightarrow\infty, N/V = const$, which is never the case for a finite experimental sample. $\endgroup$ – Vadim Apr 24 '20 at 15:07
  • $\begingroup$ @Vadim The inability to reach a "true" thermodynamic limit in experiment is an interesting point. Do you know of any experiments which explicitly show finite-size effects on the order parameter above/below the critical point? $\endgroup$ – Joshuah Heath Apr 24 '20 at 16:18
  • $\begingroup$ Any object you can see is effectively of infinite size for most statistical mechanics purposes. Roughly, you could say that the finite size effects are related to $L/\xi$, where $\xi$ is the correlation length. There are probably finite-size effects in mesoscopic systems (bigger than nano, smaller than microscopic). I believe you may be able to find papers discussing finite size effects in cold atom systems. $\endgroup$ – taciteloquence Apr 26 '20 at 8:00
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The version you learn in class is the simplest version. In practice, the order parameter doesn't need to be zero, it just needs to have a sharp change. For example: the liquid gas transition, the order parameter (density) is finite on both sides of the first order transition, but there is a sharp change.

If you're coming down through the critical point (like lowering $T$ with $h=0$ in a FM), there won't be a sharp change, but the density is still finite the whole time. You could define a modified density $\tilde \rho \equiv \rho - \rho_c$ where $\rho_c$ is the density at the critical point.

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