# Quantum fluctuations at critical point

1. Apr 22, 2015

### AcidRainLiTE

According to wikipedia:

"As for a classical second order transition, a quantum second order transition has a quantum critical point (QCP) where the quantum fluctuations driving the transition diverge and become scale invariant in space and time."

I am confused about what this means. Why do the fluctuations diverge? The quantum fluctuations become infinitely large at the critical point? That does not seem correct. Can someone clarify for me?

2. Apr 22, 2015

### atyy

Infinitely large is just shorthand for scale invariant. If the correlations decay exponentially, there is a finite length scale set by the exponential decay. There is no such single length scale for scale invariant correlations, and so the correlation length is said to diverge.

3. Apr 23, 2015

### AcidRainLiTE

Ok, that makes sense. It brings up another question for me though. Why do people call a system "scale invariant" when the correlation length diverges? The correlations still drop off (with distance) via a power law, right? So if I zoom out they will change, and so don't seem "invariant".

4. Apr 23, 2015

### atyy

If you zoom out, they will change, but only by an overall constant factor, so the correlations will still obey a power law with the same exponent

5. Apr 23, 2015

### AcidRainLiTE

Interpreting "diverging thermal fluctuations" as equivalent to "infinite correlation length" doesn't make sense with another article I am reading which says:
"What drives the correlation length to infinity are thermal fluctuations, which become very large close to criticality."

If large "large thermal fluctuations" is a synonym for diverging correlation length, than the above sentence is a tautology, simply stating "What drives the correlation length to infinity is the diverging correlation length." Instead, it seems to be saying that the two concepts are distinct, and one causes the other (large thermal fluctuations causing the diverging correlation length)

Wouldn't large thermal fluctuations do exactly the opposite? That is, wouldn't thermal fluctuations tend to increase the randomness in the system, not order it.

6. Apr 23, 2015

### atyy

I think the sentence is a tautology. My understanding is that in classical statistical mechanics, it is the thermal fluctuations that do anything by definition since we are using the canonical ensemble. Then in particular systems, there is a critical point at which the correlation length and other quantities diverge.

Here is a picture of the Ising model below, at and above criticality: http://www.nature.com/nphys/journal/v6/n10/box/nphys1803_BX1.html (just look at the picture, ignore the commentary about the brain, which I don't know whether is correct). Another example is

.

In some sense there clearly are "large fluctuations" at criticality, corresponding to there being "fluctuations on all length scales".

Last edited: Apr 23, 2015