Linked Cluster Theorem
	Statistical Mechanics: Entropy, Order Parameters, and Complexity
		James P. Sethna, Cornell University, Spring 2020
				Lecture 23

In class today, you basically rederived the linked cluster theorem in its simplest form (the dilute gas approximation). 

The idea is that if m spin flips are spread thinly among N sites, then you can to lowest order in m/N ignore any interactions between them (apart from the m! because they are indistinguishable). Adding over m, we find that the partition function multiplies by the Nth power of the contribution of a single spin flip at the origin, and the free energy thus adds N times the contribution of a single site at the origin.

This linked cluster theorem is a remarkably convenient tool for all kinds of diagrammatic expansion. The Feynman diagrams needed to calculate the motion of electrons in a dirty metal start out as a sum over 'gases' of diagrams like these, in all possible combinations filling space.  By using the linked cluster expansion, we can take this gas of diagrams and write it as an exponential of the sum of the connected diagrams.

By the way, just like we used Pad'e approximants to get around the singularity at xc in the Ising model, we can sum an infinite series of these diagrams to understand the localization of electrons in semiconductors. (Perturbation theory fails at the phase boundary, but going to infinite order can fix that.) Localization explains why diamond isn't a pretty good metal, as I understand it. The theory was invented by Phillip W. Anderson, my former PhD advisor, when I was three. He got the Nobel Prize for the discovery my first year as a grad student, and died last Sunday at the age of 96.

See you Friday.