diff --git a/Benfatto_Gallavotti_Jauslin_2015.tex b/Benfatto_Gallavotti_Jauslin_2015.tex
index 6c9de92..c92e26c 100644
--- a/Benfatto_Gallavotti_Jauslin_2015.tex
+++ b/Benfatto_Gallavotti_Jauslin_2015.tex
@@ -70,7 +70,7 @@
 H_0=&\sum_{\alpha\in\{\uparrow,\downarrow\}}\sum_{x=-{L}/2}^{{L}/2-1} c^+_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c^-_\alpha(x)\\[0.75cm]
 H_K=&H_0+V_0+V_h:= H_0+V\\[0.25cm]
 V_0=&-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^+_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c^-_{\alpha_2}(0)\, d^+_{\alpha_3}\sigma^j_{\alpha_3,\alpha_4}d^-_{\alpha_4}\\[0.75cm]
-V_h=& -h \, \sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}d^+_\alpha\sigma^3_{\alpha,\alpha'} d_{\alpha'}^-
+V_h=& -h\sum_{j=1,2,3}\bm\omega_j \, \sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}d^+_\alpha\sigma^j_{\alpha,\alpha'} d_{\alpha'}^-
 \label{eqhamkondo}\end{array}\end{equation}
 where $\lambda_0,h$ are the interaction and magnetic field strengths and
 \begin{enumerate}[\ \ (1)\ \ ]
@@ -157,7 +157,7 @@ g_{\psi,\alpha}(x-x',t-t'):=&
 \frac{\mathrm{Tr}\, e^{-\beta H_0}c^-_{\alpha}(x,t)c^+_{\alpha}(x',t')}{\mathrm{Tr}\,e^{-\beta H_0}}&\mathrm{\ if\ } t>t'\\[0.5cm]
 -\frac{\mathrm{Tr}\,e^{-\beta H_0} c^+_{\alpha}(x',t')c^-_{\alpha}(x,t)}{\mathrm{Tr}\,e^{-\beta H_0}}&\mathrm{\ if\ } t\le t'
 \end{array}\right.\\[1.5cm]
-g_{\varphi,\alpha}:=&
+g_{\varphi,\alpha}(t-t'):=&
 \left\{\begin{array}{>{\displaystyle}ll}
 \mathrm{Tr}\,d^-_{\alpha}(t)d^+_{\alpha}(t') &\mathrm{\ if\ } t>t'\\[0.5cm]
 -\mathrm{Tr}\,d^+_{\alpha}(t')d^-_{\alpha}(t) &\mathrm{\ if\ }t\le t'