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doc/c-code.tex

@@ -196,6 +196,27 @@ \subsection{$\gamma$ matrices}
 \end{split}
 \]
 
+\subsection{Pauli matrices}
+\[
+\begin{split}
+  \tau^1 =
+  \begin{pmatrix}
+    0 & 1 \\
+    1 & 0 \\
+  \end{pmatrix},\quad
+  \tau^2 = 
+  \begin{pmatrix}
+    0 & -i \\
+    i & 0 \\
+  \end{pmatrix},\quad
+  \tau^3 = 
+  \begin{pmatrix}
+    1 & 0 \\
+    0 & -1 \\
+  \end{pmatrix}
+\end{split}
+\]
+
 \subsection{Flavour Split Doublet Operator}
 
 The convention we use internally for the flavour split doublet Dirac
@@ -210,8 +231,20 @@ \subsection{Flavour Split Doublet Operator}
 \]
 The relation between the two is given by
 \[
-D_h' = (1+i\tau^2)D_h(1-i\tau^2)\, .
+D_h' = \frac{1}{\sqrt{2}}(1+i\tau^2)\ D_h\ \frac{1}{\sqrt{2}}(1-i\tau^2)\, .
+\]
+The implementation is then such that first the source $\xi$ is multiplied
+with
+\[
+\xi\to\xi=\frac{1}{\sqrt{2}}(1-i\tau^2)\xi
+\]
+on which $D_h$ is inverted and the solution $\phi$ is obtained. The
+solution is then multiplied
+\[
+\phi \to \phi=\frac{1}{\sqrt{2}}(1+i\tau^2)\phi
 \]
+to obtain the result for $D_h'$.
+
 The convention $D_h'$ is the one of \cite{Chiarappa:2006ae} with
 $\bar\mu = \mu_\sigma$ and $\bar\epsilon = \mu_\delta$. {\ttfamily
   invert\_doublet} inverts with the source first set for the upper
@@ -221,6 +254,71 @@ \subsection{Flavour Split Doublet Operator}
 lower, upper, lower flavour. The former two correspond to the source
 in the upper flavour, the latter to the source in the lower flavour.
 
+\subsection{Stochastic Volume Sources}
+
+In order to compute disconnected contributions volume (all spin,
+colour, space and time) sources are implemented. In this case only one
+inversion is required. The volume sources are generated with gaussian
+noise ($\sigma=1$) in real and imaginary part of the whole source
+spinor. Note that the normalisation with $1/\sqrt{2}$ is \emph{not}
+done and needs to be taken care off in the analysis.
+
+For the hopping parameter noise reduction method the following is
+needed: Following the notation in Ref.~\cite{Boucaud:2008xu} the
+operator can be written as
+\[
+D_h' = A + H = (1+H\cdot B)\cdot A,\qquad B=1/A\, .
+\]
+where $H^\dagger = \gamma_5 H \gamma_5$ and
+\[
+A = 1 + i\gamma_5\tau^1\tilde\mu_\sigma + \tau^3\tilde\mu_\delta
+\]
+where in the hopping parameter representation $\tilde\mu_\sigma =
+2\kappa\mu_\sigma$ and $\tilde\mu_\delta=2\kappa\mu_\delta$. $A$ can
+be inverted easily to
+\[
+A^{-1}=\frac{1-i\gamma_5 \tau^1 \tilde\mu_\sigma - \tau_3
+  \tilde\mu_\delta}{1+\tilde\mu_\sigma^2-\tilde\mu_\delta^2} 
+\]
+It follows that
+\[
+1/D_h' = B-BHB+B(HB)^2-B(HB)^3+1/D_h'(HB)^4\ .
+\]
+Then, since $\gamma_5$ commutes with $B$ one can evaluate the last
+term stochastically for any $\gamma$ and or colour matrix $X$ like 
+\[
+X (1/D_h')(HB)^4 = \lim_{R\to\infty}\left[(\gamma_5 (B^{\dagger} H )^4
+  \gamma_5 \xi)^* X \phi\right]_R
+\]
+where
+\[
+\phi = (D_h')^{-1}\xi
+\]
+and
+\[
+B(\tilde\mu_\sigma)^\dagger \equiv B(-\tilde\mu_\sigma)\ .
+\]
+The remaining terms can be computed exactly. For any source $\xi$ we 
+therefore have to generate
+\[
+\xi_r = \gamma_5 (B^{\dagger} H )^4\gamma_5 \xi.
+\]
+
+
+%(ii)--------------------------------------------
+%Use source in all flavour spin colour space time (Z2xZ2)
+%
+%Let u,v be flavour indices (c,s) equivalent
+%
+%M \phi = \xi  with flavour explicit  M_{uv) \phi_v = \xi_u
+%
+% Then required quantity is
+%    Tr (X M^{-1} )_{uv} where X is diagonal in flavour
+% and Tr is sum over colour, spin, space (at a given time)
+%    = Tr( \xi^*_v X \ph_u )
+%    = Tr ( {(g_5 (B^{\dag} H)^4 g_5 \xi)_v}^* X \phi_u)
+% for uv element
+
 
 
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