Consider a renormalization-group flow for a set of quantities $(x_1, ..., x_N) \equiv \bf x$, which can be written in the form ${\bf x}_{t+1} = {\bf F}[{\bf x}_t, T]$,where $T$ is the temperature. At each step, half of the degrees of freedom of the system are decimated.

Let us imagine that the structure of the flow are the ones, say, for the Ising model, i.e., the is an unstable, critical fixed point to which the iteration converges if $T = T_c$, and two stable high- and low-temperature fixed points.

In the specific problem that I am studying, I have come across two different methods to determine the critical exponent $\nu$ associated to the divergence of the correlation length:

  1. Iterate the transformation many times and adjusting $T$ at each step, in such a way that ${\bf x}_{t+1}$ is as close as possible to ${\bf x}_t$. After a large enough number of steps, ${\bf x}_t$ and $T$ settle to fixed values ${\bf x}_\ast$ and $T_c$, which are the critical fixed point and temperature. Compute $M_{ij} = \left. \frac{\partial F_i[{\bf x},T_c]}{\partial x_j } \right|_{{\bf x}_\ast}$, its largest eigenvalue $\lambda$, and extract $\nu$ from $\lambda = 2^{1/\nu}$.
  2. Iterate the transformation at fixed $T$, observe that there is a $T_c$ such that ${\bf x}_t$ flows away to two different fixed points if $T \gtrless T_c$, respectively. Iterate the RG transformation at two temperatures $T_1, T_2$ somewhat close to $T_c$, obtain the flows ${\bf x}^1_t, {\bf x}^2_t$. Extract nu by fitting $[{\bf x}^1_t/T_1 - {\bf x}^2_t/T_2]_1 = A 2^{t/\nu}$, where $[]_1$ stands for the first component of the vector.

I observe that 1) and 2) give different values for $\nu$. Which one would you trust?


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