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Fix typo in V_h

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Ian Jauslin 2015-06-30 10:52:28 +00:00
parent 121c2ba1a3
commit 16992e42e1
1 changed files with 3 additions and 3 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\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} \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)\ \ ]
@ -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} \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
V(\psi,\varphi)=& 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))\\ -\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} \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}]. 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} \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
V(\psi,\varphi)=& 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))\\ -\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} \end{array}\label{eqpothier}\end{equation}
in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~(\ref{eqfieldhier}). in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~(\ref{eqfieldhier}).
\medskip \medskip