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This is a follow up on "Why are the eigenvalues of a linearized RG transformation real?".

My question is simple: Is there some physical (or mathematical) reason for the stability matrix of Renormalisation Group flows close to fixed points to be diagnoalisable? What is it? If there isn't: Are there known counter examples? How do we deal with them?

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I don't believe there is a mathematical reason, especially if there is latitude in reverse-engineering the field theory or stat mech system to evince such a behavior. Indeed, if Lorentz-nonivariant systems are examined, things like limit cycles , e.g. this one are not hard to concoct. As for physical reasons, they might well be easy to bypass/moot if one argued for them. I don't know of any systems, however, with this property, which might not say much.

As a mathematical wisecrack, I could manufacture a simple toy system with two couplings, x and y and logarithmic scale variable t : $$ \dot{x}=-x + ay, \qquad \dot{y}= -y , $$ with evident solutions stable around the fixed point (0,0), $$ y= e^{-t}, \qquad x= (c +at) e^{-t} . $$ The stability matrix of the ODE system is $$ \left( \begin{array}{cc} -1 & a \\ 0 & -1 \\ \end{array} \right) $$ which is not diagonalizable, with only one eigenvector, $$ \left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right) $$ of eigenvalue -1. This is not to say the system is not stable, however, if one could solve the ODE, somehow, as here.

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