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Fix typos

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Ian Jauslin 2015-07-04 16:23:58 +00:00
parent 16992e42e1
commit bd5280f10c
1 changed files with 2 additions and 2 deletions
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Benfatto_Gallavotti_Jauslin_2015.tex
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@ -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_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] 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_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} \label{eqhamkondo}\end{array}\end{equation}
where $\lambda_0,h$ are the interaction and magnetic field strengths and where $\lambda_0,h$ are the interaction and magnetic field strengths and
\begin{enumerate}[\ \ (1)\ \ ] \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>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' -\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] \end{array}\right.\\[1.5cm]
g_{\varphi,\alpha}:=& g_{\varphi,\alpha}(t-t'):=&
\left\{\begin{array}{>{\displaystyle}ll} \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>t'\\[0.5cm]
-\mathrm{Tr}\,d^+_{\alpha}(t')d^-_{\alpha}(t) &\mathrm{\ if\ }t\le t' -\mathrm{Tr}\,d^+_{\alpha}(t')d^-_{\alpha}(t) &\mathrm{\ if\ }t\le t'