 Correlation-length scaling from a thermal direction

Add a thermal scaling coordinate $t$, with its critical value subtracted, satisfyingThis 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 $(g,t)=(ϵ,0)$ has eigenvalues $-ϵ$ 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 $b$ givesIndeed, a tail with exponential envelope $\exp (-r/\xi )$ becomes $\exp [-r^{\prime }/(\xi /b)]$ under $r^{\prime }=r/b$. Choose the stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ 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 $g_{\ast }$, thenThe limit requires $|t_0|\ll 1$ 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 $X=10^4{C}_{+}$. With $b=4$, $t^{\prime }=8t$ and $X(t^{\prime })=X(t)/4$. 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 $ϵ=1/5$ nor $y_t=3/2$ is being fitted to a three-dimensional Ising model.