# Scale invariance beyond the critical point

Using Anderson localization as an example, I understand how scale invariance comes into play at a critical point - at a critical point, the localization length $$\xi$$ (the average "radius" of the eigenstates) diverges, so that if one "zooms out" then the system looks the same, because any finite rescaling of infinity is still infinity.

What I don't understand yet is why the same could not be said beyond the critical point. For example, suppose that below the critical disorder $$W>W_c$$ my system is localized, and above the critical disorder $$W my system is extended. The localization length $$\xi$$ is finite for $$W>W_c$$ and infinite for $$W. Why don't we say that the entire extended phase $$W is scale invariant? After all, $$\xi$$ is still infinite, so that the system will still look the same if we zoom out.

I would like to understand why the $$W case is not considered scale invariant in hopes that it will help me to understand what happens in finite systems. I've seen simulations of finite systems with two parameters, disorder $$W$$ and system size $$N$$, for which the critical parameter such as $$\xi$$ is independent of system size $$N$$ exactly at $$W=W_c$$ but neither for $$W>W_c$$ nor for $$W. Since scale invariance occurs at $$W=W_c$$, the point where the critical parameter is independent of $$N$$ is identified as $$W_c$$.

• Why do you think correlation length is infinite below the critical temperature? Aug 9, 2021 at 16:55
• I overgeneralized and used a bad example, in hopes that it would be familiar to more people. Would it be better to edit or delete and repost the question? The phenomenon I am really interested in is Anderson localization, where disorder drives a system from an extended state to a localized state. In this example, the localization length (the "size" of the quantum state) truly is infinite below the critical disorder. Aug 9, 2021 at 17:01
• Edit away! $\$ Aug 9, 2021 at 17:06
• Fixed; thank you! Aug 9, 2021 at 18:21

• I agree - the localization length controls the correlation length when in the localized phase, and some other physics controls the correlation length in the extended phase. However, I still have a localization length that is scale invariant at $W=W_c$ and nowhere else, even though the localization length diverges for all $W<W_c$. I would like to understand why this is. Aug 9, 2021 at 19:17