To break a line in a LaTeX environment I need to use \\\\ instead of \\.
I guess here \ plays a role of escape character. However, it is inconsistent with other places in formulas where \ does not need to be escaped.
E.g.:
\begin{align}
c_{ij} &= - \langle \psi | \sigma^x_i \sigma^x_j | \psi \rangle \\\\
\sum_{i \neq j } c_{ij} &= 1
\end{align}
works but
\begin{align}
c_{ij} &= - \langle \psi | \sigma^x_i \sigma^x_j | \psi \rangle \\
\sum_{i \neq j } c_{ij} &= 1
\end{align}
not (i.e. there is no line break).