Topological Theory of Walls
	Statistical Mechanics: Entropy, Order Parameters, and Complexity
		James P. Sethna, Cornell University, Spring 2020
				Lecture 25

Welcome to order parameters! The pre-class questions asked you to discuss the
topological order parameter space for a nematic liquid crystal, and for 
the Ising model. In class today we shall continue to work on the nematic
order parameter. But first, let us use the Ising model as a preview of 
our general topological theory of defects. How does the Ising model
order parameter space give us a topological theory of walls?

First, one should recognize that there are at least two commonly used
order parameters for the Ising model. Landau would tell us that the
order parameter should be a real number M(x) as a function of position,
measuring the fraction of spins up in the region around x. Below Tc, the
local free energy density has two minima, at the equilibrium
magnetization for that temperature (say, as described by the series we
studied Wednesday). Landau theory is useful for studying spatially
nonuniform states, interfaces, defect cores, and phase transitions where the
order parameter vanishes or changes symmetry. It is often only a
qualitative theory, ignoring fluctuations and higher
order gradients and nonlinearities. 

The topological theory just cares about the manifold of possible ground
states -- which here is just two points. We would usually call this S^0,
the unit sphere in one dimension (just the two points +-1). How is it
useful? Remember how the Ising model looked if you suddenly changed the
temperature to below Tc? It separated into up-spin and down-spin
regions. The natural defect in the Ising model is the wall between
up-spins and down-spins. The order parameter field for that defect takes
the left-hand side of the figure to the up state (+1 on S^0) and the
right-hand side of the figure to the down-state (-1 on S^0).

What about a more complicated order parameter space? Suppose we had a
system with twelve possible ground states (a clock model or a Potts model). 
What would our order parameter space be? How would you classify the walls?
Think about it for a minute or so.

Surely, each pair of ground states will have its own wall -- we need to 
study the possible ways of choosing two out of the total. The homotopy
theory of walls will tell us to surround the suspected defect with a 
sphere of suitable dimension, and to study how the order parameter varies
on that sphere. That is, a point s on the sphere maps to x(s) surrounding
the defect, and then to O(x(s)) in the order parameter space. We can
surround our clock-model wall with a sphere S^0 with -1 going to one
side of the wall and +1 going to the other side. Then O(x(s)) takes -1
into one of the clock states and +1 into another clock state. The walls
are then classified by the number of different ways of choosing pairs
of clock states -- the right answer, even if it seems a complicated way
of saying something simple.

We turn now to the in-class activity -- understanding the projective plane
RP2, the order parameter space for the nematic liquid crystal.