Revisiting the Theory of Finite Size Scaling in Disordered Systems: \nu Can Be Less Than 2/d

TL;DR

This paper challenges the traditional bound on the critical exponent in disordered systems by introducing a new length scale caused by averaging noise, allowing for the true exponent  to be less than 2/d, and proposes a method to accurately measure it.

Contribution

The paper introduces a new perspective on finite size scaling in disordered systems, highlighting a noise-induced length scale and proposing an improved averaging method.

Findings

The standard bound   2/d can be violated in disordered systems.

A new diverging length scale  is identified, independent of the correlation length.

A novel averaging method reduces noise and accurately captures the true critical exponents.

Abstract

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: \nu_{FS}<= 2/d. It is believed that the true critical exponent \nu of a disorder induced phase transition satisfies the same bound. We argue that in disordered systems the standard averaging introduces a noise, and a corresponding new diverging length scale, characterized by \nu_{FS}=2/d. This length scale, however, is independent of the system's own correlation length \xi. Therefore \nu can be less than 2/d. We illustrate these ideas on two exact examples, with \nu < 2/d. We propose a new method of disorder averaging, which achieves a remarkable noise reduction, and thus is able to capture the true exponents.