\label{eq:ddgtoB}$$ This can be simplified, by Eq. (\[eq:mtorho\]), to $$\frac{g^{\prime \prime }\left( \rho _0\right) }{k_BT}=-mB^{\prime \prime
}(\rho _0)/B^{\prime }(\rho _0)+m^2,  \label{eq:ddgrho}$$ the quantity needed for computing the stiffness as a function of $m$. While straightforward, computing the derivatives with the general form for $B$ (Eq. (\[eq:defineB\]) with $\rho =i\mu $) is quite tedious. A slight simplification emerges if we specialize to the physically relevant case $U=D$. Then, with $S\equiv H+2D$, we have $$\begin{aligned}
B(\rho ) &=&1+\frac{e^{2D}}{e^{S+\rho }-1}+\frac{e^{2D}}{e^{S-\rho }-1} 
\nonumber \\
&=&1-e^{2D}+\frac{e^{2D}\sinh S}{\cosh S-\cosh \rho } \nonumber \\
& \equiv &1-e^{2 D}+\frac{e^{2D} \sinh S}{C(S,\rho)},\end{aligned}$$ so that $$B^{\prime }(\rho )=e^{2D}\sinh S\frac{\sinh \rho }{C^2(S,\rho)},$$ and $$B^{\prime \prime }(\rho )=e^{2D}\sinh S\left[ \frac{\cosh \rho }{%
C^2(S,\rho)}+\frac{2\sinh ^2\rho }{C^3(S,\rho)}%
\right] .$$ Inserting these expressions into Eq. (\[eq:mtorho\]), we have $$m=\frac{\sinh \rho _0\sinh S}{C(S,\rho_0)\left[ \sinh S-C(S,\rho_0)\left( 1-e^{-2D}\right) \right] }.$$ Similarly, with Eq. (\[eq:ddgrho\]), we find $$\frac{g^{\prime \prime }\left( \rho _0\right) }{k_BT}=-m\left[ \ \frac{%
2\sinh \rho _0}{C(S,\rho_0)}+\coth \rho _0\right] \ +m^2.$$

Acknowledgment {#acknowledgment .unnumbered}
==============

Work at the University of Maryland was supported by the NSF-MRSEC, Grant DMR 00-80008. One of us (RKPZ) acknowledges support by NSF Grants DMR 00-88451 and 04-14122. TLE acknowledges partial support of collaboration with ISG-3 at FZ-Jülich via a Humboldt U.S. Senior Scientist 