$\Delta $. Carrying out the summation in Eq. (\[eq:fourier\]) gives $$\frac{g\left( i\mu \right) }{k_BT}\equiv -\frac 1L\ln W(i \mu )=V+U+D-\ln
B(i\mu ),  \label{eq:gdefine}$$ where $$B(i\mu )\equiv 1+\frac{e^{2D}}{e^{H+U+D+i\mu }-1}+\frac{e^{2U}}{%
e^{H+U+D-i\mu }-1}.  \label{eq:defineB}$$ Thus, the original partition function $Z(Y)$ is: $$\begin{aligned}
Z\left( Y\right) &=&\frac 1{2\pi }\int_{-\infty }^\infty d\mu ~e^{-i\mu
Y}W\left( \mu \right)  \label{eq:ZtoW} \\
&=&\frac 1{2\pi }\int_{-\infty }^\infty d\mu ~\exp \left[ L\left( -i\mu \tan
\theta -\frac{g(i\mu )}{k_BT}\right) \right]  \nonumber\end{aligned}$$ For $L\gg 1$, we can evaluate this inverse transform by steepest decent approximation. The saddle point occurs on the imaginary axis ($\mu
=-i\rho $), at the value $\rho _0$ given by the stationary-phase condition: $$-\frac{g^{\prime }\left( \rho _0\right) }{k_BT}=m\equiv \tan \theta .$$ Calculating the derivative from Eqs. (\[eq:gdefine\]) and (\[eq:defineB\]), we find $$m=B^{\prime }(\rho _0)/B(\rho _0),  \label{eq:mtorho}$$ where prime stands for $\partial _\rho $. The leading contribution to this integral (\[eq:ZtoW\]) is just the integrand evaluated at this point: $$Z(Y)\approx \exp \left[ -L\left( m\rho _0+\frac{g\left( \rho _0\right) }{k_BT%
}\right) \right] .  \label{eq:ZtoWapproximationmu}$$

Analysis of $g^{\prime \prime }(\rho )$ and specialization to $%
U=D$
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From Eqs. (\[eq:gdefine\]), we find $$\frac{g^{\prime }\left( \rho \right) }{k_BT}=-B^{\prime }(\rho )/B(\rho )
\label{eq:dgtoB}$$ and $$\frac{g^{\prime \prime }\left( \rho \right) }{k_BT}=-B^{\prime \prime }(\rho
)/B(\rho )+\left[ B^{\prime }(\rho )/B(\rho )\right] ^2.  