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&=& 2 \pi^2 \partial_x \left( \partial_x \phi \Pi + \Pi \partial_x \phi \right) \\  j_Q(x) & \propto & \partial_x \phi \Pi + \Pi \partial_x \phi   \end{align}  $$ \ Notice that this expression for the thermal current is (up to the term proportional to $k_F$) similar to the $T^{tx}$ component, as opposed to the $T^{xt}$ component which we agree is the right one.