Add direction for magnetic field

Benfatto_Gallavotti_Jauslin_2015.tex

@@ -66,10 +66,11 @@ V_h=& -h \, \sum_{\alpha\in\uparrow,\downarrow}d^+_\alpha\sigma^3_{\alpha,\alpha
 \label{e1.1}\end{array}\end{equation}
 where $\lambda_0,h$ are the interaction and magnetic field strengths and
 \begin{enumerate}[\ \ (1)\ \ ]
-\item$c_\alpha^\pm(x),d^\pm_\alpha, \,\alpha=\uparrow,\downarrow$ are creation and annihilation operators corresponding respectively to electrons and the impurity
-\item$\sigma^j,\,j=1,2,3$, are the Pauli matrices
-\item$x$ is on the unit lattice and $-{L}/2$, ${L}/2$ are identified (periodic boundary)
-\item$\Delta f(x)= f(x+1)-2f(x)+f(x-1)$ is the discrete Laplacian.
+\item$c_\alpha^\pm(x),d^\pm_\alpha, \,\alpha=\uparrow,\downarrow$ are creation and annihilation operators corresponding respectively to electrons and the impurity,
+\item$\sigma^j,\,j=1,2,3$, are the Pauli matrices,
+\item$x$ is on the unit lattice and $-{L}/2$, ${L}/2$ are identified (periodic boundary),
+\item$\Delta f(x)= f(x+1)-2f(x)+f(x-1)$ is the discrete Laplacian,
+\item$\bm\omega=(\bm\omega_1,\bm\omega_2,\bm\omega_3)$ is the direction of the field, which is a norm-1 vector.
 \end{enumerate}
 \medskip
 
@@ -167,7 +168,7 @@ If $\beta,L$ are finite, $\int\,\frac{dk_0 dk}{(2\pi)^2}$ in Eq.(\ref{e2.5}) 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 \int dt\sum_{\alpha\in\uparrow,\downarrow}\varphi^+_{\alpha}(t)\sigma^3_{\alpha,\alpha} \varphi^-_{\alpha}(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)
 \end{array}\label{e2.6}\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}].
 
@@ -314,7 +315,7 @@ 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 \int dt\sum_{\alpha\in\uparrow,\downarrow}\varphi^+_{\alpha}(t)\sigma^3_{\alpha,\alpha} \varphi^-_{\alpha}(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)
 \end{array}\label{e3.9}\end{equation}
 in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in Eq.(\ref{e3.5}).
 \medskip