DOC

doc/discretization.md

@@ -49,16 +49,28 @@ the symmetry.
 
 ## Derivative term
 
+We discretize the derivatives a
+```math
+  D_{ij}
+  :=
+  \frac{1}{h}[Q(i) U_{ij} - U_{ij} Q(j)]
+  \,,
+  \qquad
+  \Rightarrow
+  Q(i) D_{ij} = - D_{ij} Q(j)
+  \,,
+```
+which satisfies an anticommutation relation analogous to continuum derivative.
+
 We then discretize
 ```math
 S_2
 =
 \mathrm{Tr} D(\hat{\nabla} Q)^2
 \mapsto
-D
-\frac{2d}{|\mathrm{neigh}|} \frac{h^d}{h^2} \sum_{\mathrm{neigh}(i,j)}\mathrm{Tr}[Q(i) - U_{ij}Q(j)U_{ji}]^2
+-D \frac{2d}{|\mathrm{neigh}|} h^d \sum_{\mathrm{neigh}(i,j)}\mathrm{Tr} D_{ij} D_{ji}
 ```
-where $`d`$ is the space dimension.
+where $`d`$ is the space dimension and `|neigh|` the number of neighbors on the grid.
 
 Expanding in small $`h`$ we have
 ```math
@@ -112,21 +124,7 @@ which then allows expressing $`F_{ij}`$ in terms of the link matrices.
 
 ## Hall term
 
-When discretizing the $`Q\hat{\nabla}_iQ\hat{\nabla}_jQ`$ part, we can
-consider a discrete derivative
-```math
-  D_{ij}
-  :=
-  Q(i) U_{ij} - U_{ij} Q(j)
-  \,,
-  \qquad
-  \Rightarrow
-  Q(i) D_{ij} = - D_{ij} Q(j)
-  \,,
-```
-which satisfies a discrete gauge-invariant version of the $`Q (\hat{\nabla}Q) =
--(\hat{\nabla}Q) Q`$ relation.  Then, in the plaquette of Fig. 1 we
-have
+In the plaquette of Fig. 1 we have
 ```math
   \mathrm{tr}
   F_{\mu\nu}Q \hat{\nabla}_\mu Q \hat{\nabla}_\nu Q