1
$\begingroup$

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 $W<W_c$ my system is extended. The localization length $\xi$ is finite for $W>W_c$ and infinite for $W<W_c$. Why don't we say that the entire extended phase $W<W_c$ 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 $W<W_c$ 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 $W<W_c$. Since scale invariance occurs at $W=W_c$, the point where the critical parameter is independent of $N$ is identified as $W_c$.

$\endgroup$
4
  • 1
    $\begingroup$ Why do you think correlation length is infinite below the critical temperature? $\endgroup$ Aug 9 at 16:55
  • $\begingroup$ 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. $\endgroup$
    – BGreen
    Aug 9 at 17:01
  • 1
    $\begingroup$ Edit away! $\ $ $\endgroup$ Aug 9 at 17:06
  • 1
    $\begingroup$ Fixed; thank you! $\endgroup$
    – BGreen
    Aug 9 at 18:21
0
$\begingroup$

Above the critical disorder you are no longer in the Anderson localized phase, so what happens will depend on what sort of phase you are in. However, regardless of what that phase may be you will need to describe the system in terms an appropriate set of parameters, which will typically include some sort of (finite) correlation length.

In the Anderson localized phase the physics is dominated by the localization of the individual eigenstates, so the correlation length is controlled by the localization length. Whatever phase the system transitions into at the critical point will be dominated by different physics and that physics will set the correlation length, for example in ferromagnetic phase the coherence length is controlled by the magnetic ordering of the material.

Since outside the Anderson localized phase the correlation length is not longer related to the (infinite) localization length the system will typically not be scale invariant.

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – BGreen
    Aug 9 at 19:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.