Extrapolation to the Critical Point
	Statistical Mechanics: Entropy, Order Parameters, and Complexity
		James P. Sethna, Cornell University, Spring 2020
				Lecture 24

Good morning! Last night and today we are spending on the low-temperature expansion of the Ising model. In later chapters, we will perturb systems and see how they respond -- roughly keeping the first term in a Taylor series inside a phase. The formula to x^54 is amazing, and really illustrates how phases are regions where perturbation theory works. But why would our colleagues work so hard to extract so many higher-order terms? We can understand this by figuring out how to transcend the limitations of the theory.

How can we study the critical point xc = e^(2 J/Tc) by creeping up to it from below?  Our study of continuous phase transitions in Chapter 12 (last week of class) will tell us that magnetization m~(Tc-T)^beta~(xc-x)^beta, where beta = 0.32641 is a universal constant. Our colleagues used these amazing series to estimate beta, before our field theory colleagues developed the conformal bootstrap method.

How did they do this? They used the series to generate a new series, for d log(m) / dx ~ beta/(x-xc). Even then, the series has a radius of convergence no larger than xc. (Maybe even smaller!) But now the singularity is a simple pole. They then sneakily rewrote their series as a ratio of polynomials (a Pad'e approximant). We can do so to order x^6 by using a ratio of two third-order polynomials, (a [3,3] Pad'e approximant) and matching coefficients. I did a [9,10] Pad'e approximant to make my figures in the text.

Then the pole just becomes one of the factors in the denominator! Which root of the denominator is our approximate xc?
 
Even to x^6, the predicted Tc is off by less than a couple of percent. And the rest of the formula gives us our estimate of beta, 0.32, which is also only 2% off from the now much better known value. 

We turn now to the in-class activity -- the 2D Ising model and the 
linked--cluster theorem.