# Is the stability matrix of a linearised RG flow always diagonalisable?

+ 6 like - 0 dislike
529 views

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?

This post imported from StackExchange Physics at 2015-08-04 09:52 (UTC), posted by SE-user Steven Mathey

edited Oct 16, 2017

There is nothing that forces diagonalizability. I believe there are even counterexamples, where the Jordan structure (which replaces the diagonal structure in the nondiagonalizable case) produces logarithmic terms in the corresponding expansion. This should be discussed somewhere in Fisher's work around 1974.

Read "A renormalisation-group treatment of two-body scattering". After equation 9 : If, for a particular system, we find that the rescaled potential tends towards this fixed point as we lower the cut-off towards zero, then etc ... and the citations 11/13/18/19 with the conclusion.

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.