From 0171d1401bfa57c79eabd0920686bae6f87e604b Mon Sep 17 00:00:00 2001
From: patavirt <pauli.t.virtanen@jyu.fi>
Date: Tue, 30 Aug 2022 16:42:33 +0300
Subject: [PATCH] DOC
---
doc/discretization.md | 34 ++++++++++++++++------------------
1 file changed, 16 insertions(+), 18 deletions(-)
diff --git a/doc/discretization.md b/doc/discretization.md
index 10f0dfc..792e769 100644
--- a/doc/discretization.md
+++ b/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
--
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