diff --git a/_thesis/2.1.2_AMK_Intro.html b/_thesis/2.1.2_AMK_Intro.html index 37ae7e4..878fa69 100644 --- a/_thesis/2.1.2_AMK_Intro.html +++ b/_thesis/2.1.2_AMK_Intro.html @@ -415,15 +415,15 @@ composition rule extends to arbitrary numbers of vortices which gives a discrete version of Stoke’s theorem. -

Wilson loops can always be decomposed into products of -plaquettes operators unless they are non-contractable

+

Takeaway: Wilson loops can always be decomposed into products of +plaquettes operators unless they are non-contractable.

Gauge Degeneracy and the Euler Equation

Figure 5: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left.