We found a critical point in our model, separating smooth hysteresis loops from loops with a macroscopic jump or burst. At large disorder, every domain flips independently; at disorder small compared to the coupling between domains, a single domain flip can send its neighbors flipping until most of the system transforms in one burst. At the critical disorder, we get universal power laws and scaling.

**Phase transition as we vary disorder.**
*Three H(M) curves for different levels of disorder: above,
below, and near the critical disorderR_chen the burst disappears.
(Select it for an animation!)
For R > R_c the hysteresis loop is macroscopically smooth,
although of course microscopically it is a sequence of sizable avalanches.
The jump in the magnetization scales as (R-R_c)^beta, and the
magnetization at R_c has a power-law form M - M_c ~ (H-H_c)^{1/delta}.*

This is a simulation of a two-dimensional somewhat above the critical point, where none of the avalanches are very large.

This is a simulation closer to the transition. Notice the big avalanche which goes from one end of the system to the other! We're not sure yet whether there is a real transition in two dimensions: it's conceivable that no matter how weak the disorder, in a big enough system all avalanches would come to a halt at finite sizes. (We're sure that it happens that way in one dimension: there a single stuck spin can block the avalanche!)

**Avalanches come in various sizes:**
*Log-Log Plot of the avalanche-size distribution D(s) vs.
avalanche size s, integrated over one sweep of the magnetic field from
-infinity to +infinity, averaged over 5 systems of size
120^3. Notice the power-law region
D(s) ~ s^{-(tau + sigma beta delta)} and
the cutoff at s_{max} ~ (R-R_c)^{-1/sigma}.*

The critical point is where an infinite avalanche first occurs. Naturally, near the critical point, some of the avalanches get rather large - indeed, one gets avalanches of all scales. At least in our model, one gets rather large avalanches even pretty far from the critical point: we believe the noise in hysteresis loops is due to this critical point.

- Authors
- Hysteresis
- What is Hysteresis?
- Noise in Hysteresis
- Return Point Memory
- The Critical Point
- The Epsilon Expansion

- Sethna's Research 90-94
- Entertaining Science done at
- LASSP.

Last modified: December 12, 1994James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).