= Domain-wall scaling coordinate of an Ising chain
{title2=$w=e^{-2K},\quad w'=2w/(1+w^2)$}
A reversed bond costs $2J$, so its <Boltzmann factor> $w=e^{-2J/(k_BT)}$ is a natural thermal coordinate near the zero-temperature chain fixed point. Exact length-two decimation gives $w'\simeq2w$, and the <One-dimensional Ising correlation function> gives $\xi/a=-1/\log\tanh K\sim1/(2w)$. This is a <length> exponent one with respect to $w$. The <correlation length> instead diverges exponentially in inverse physical <temperature>; claiming a conventional finite power $T^{-1}$ would confuse scaling coordinates. Choosing $e^{-K}$ gives multiplier $\sqrt2$ and power two in that alternative coordinate, the same physical <length>.
Back to article page
