Fix typo in V_h

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\in\uparrow,\downarrow}d^+_\alpha\sigma^3_{\alpha,\alpha} d_\alpha^-
+V_h=& -h \, \sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}d^+_\alpha\sigma^3_{\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)\ \ ]
@@ -175,7 +175,7 @@ If $\beta,L$ are finite, $\int\,\frac{dk_0 dk}{(2\pi)^2}$ in~(\ref{eqpropk}) has
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
 V(\psi,\varphi)=&
 -\lambda_0 \sum_{{j\in\{1,2,3\}}\atop{\alpha_1,\alpha'_1,\alpha_2,\alpha_2'}}\int dt \,(\psi^+_{\alpha_1}(0,t)\sigma^j_{\alpha_1,\alpha'_1} \psi^-_{\alpha'_1}(0,t)) (\varphi^+_{\alpha_2}(t)\sigma^j_{\alpha_2,\alpha_2'} \varphi^-_{\alpha_2'}(t))\\
-&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{\alpha\in\uparrow,\downarrow}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha} \varphi^-_{\alpha}(t)
+&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t)
 \end{array}\label{eqpotgrass}\end{equation}
 Notice that $V$ only depends on the fields located at the site $x=0$. This is important because it will allow us to reduce the problem to a 1-dimensional one [\cite{aySN}, \cite{ayhSeZ}].
 
@@ -322,7 +322,7 @@ from which we see that the hierarchical model boils down to neglecting the $m'$
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
 V(\psi,\varphi)=&
 -\lambda_0 \sum_{{j\in\{1,2,3\}}\atop{\alpha_1,\alpha'_1,\alpha_2,\alpha_2'}}\int dt \,(\psi^+_{\alpha_1}(0,t)\sigma^j_{\alpha_1,\alpha'_1} \psi^-_{\alpha'_1}(0,t)) (\varphi^+_{\alpha_2}(t)\sigma^j_{\alpha_2,\alpha_2'} \varphi^-_{\alpha_2'}(t))\\
-&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{\alpha\in\uparrow,\downarrow}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha} \varphi^-_{\alpha}(t)
+&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t)
 \end{array}\label{eqpothier}\end{equation}
 in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~(\ref{eqfieldhier}).
 \medskip