= Correlation-length scaling from a thermal direction
{title2=$\nu=1/y_t$}
Add a thermal scaling coordinate $t$, with its critical value subtracted, satisfying
$$
\frac{dt}{d\ell}=y_t t,\qquad t(\ell)=b^{y_t}t_0,\qquad y_t>0.
$$
This is a <relevant direction of a fixed point>. A thermal scaling coordinate agrees to leading order with a nonzero multiple of <reduced temperature>; in a general model it can mix several bare parameters. An ordering <conjugate field> is held at its critical value. The combined <stability matrix> at $(g,t)=(\epsilon,0)$ has <eigenvalues> $-\epsilon$ and $y_t$. Attraction along $g$ therefore does not make the critical point attractive against thermal detuning. The <critical surface> in this example is $t=0$.
Let $X=\xi/a$ denote the <correlation length> in microscopic lattice units. A <renormalization-group transformation> of length factor $b$ gives
$$
X(t_0,g_0)=b\,X(b^{y_t}t_0,g(\log b)).
$$
Indeed, a tail with exponential envelope $\exp(-r/\xi)$ becomes $\exp[-r'/(\xi/b)]$ under $r'=r/b$. Choose the stopping scale $b_*=|t_0|^{-1/y_t}$ so that the transformed thermal coordinate is $+1$ or $-1$. If the stopped <correlation length> tends to a finite nonzero value $C_\pm$ as the irrelevant coupling approaches $g_*$, then
$$
\frac{\xi}{a}\sim C_\pm |t_0|^{-1/y_t},\qquad \nu=\frac1{y_t}.
$$
The limit requires $|t_0|\ll1$ and enough room between $a$, $\xi$, and the sample size. Singular dependence on a <dangerously irrelevant coupling> or a <marginal operator> needs separate analysis. On an ordered branch, use the <connected correlation function> in a <pure thermodynamic phase>.
For a completely specified illustrative scaling model choose $y_t=3/2$ and $X=C_+t^{-2/3}$ for $t>0$. Then $t=10^{-3}$ gives $X=100C_+$, whereas $t=10^{-6}$ gives $X=10^4C_+$. With $b=4$, $t'=8t$ and $X(t')=X(t)/4$. Models with $C_+=1$ and $C_+=2$ have different lengths but identical curves after plotting $X/C_+$ against $t$. This collapse checks the chosen scaling law; agreement of real data would test that law, rather than follow from this example. Neither $\epsilon=1/5$ nor $y_t=3/2$ is being fitted to a three-dimensional <Ising model>.
\Image[/renormalization-flow-and-correlation-length.png]
{title=A chosen <renormalization-group flow> loses coupling detail while a <correlation length> grows}
{description=Left: exact positive trajectories of $dg/d\ell=g/5-g^2$ approach $g_*=1/5$ for increasing log length. Right: the chosen $y_t=3/2$ gives $\xi/a=C_+t^{-2/3}$; different amplitudes collapse after division by $C_+$. These are analytic model curves, not experimental data or precision three-dimensional <Ising model> predictions.}
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