(425) $M^{\rm sc}_{300}(3\alpha)=M^{\rm sc}_{030}(3\alpha)=M^{\rm
sc}_{003}(3\alpha)=-\frac{13}{\alpha}+\frac{35}{2}\,M^{\rm
sc}_{000}(3\alpha)+\frac{21}{\pi^{2}\alpha^{2}M^{\rm sc}_{000}(3\alpha)}\>.$
(426)

All other $M^{\rm sc}_{nmk}(3\alpha)$ can be obtained from the recurrence
relations [93, 96]. For example,

$M^{\rm sc}_{110}(3\alpha)=M^{\rm sc}_{101}(3\alpha)=M^{\rm
sc}_{011}(3\alpha)=\frac{1}{4}\left\\{5M^{\rm sc}_{000}(3\alpha)-M^{\rm
sc}_{200}(3\alpha)-\frac{2}{\alpha}\right\\}.$ (427)

###  D.2  Green’s functions, Eq. (219), in the tetragonal case,
$\alpha=\beta\neq\gamma$

The tetragonal Green’s functions are defined as

$M^{\rm
tg}_{nmk}=\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\frac{{\rm
d}q_{x}\,{\rm d}q_{y}\,{\rm
d}q_{z}}{(2\pi)^{3}}\frac{\cos{nq_{x}}\cos{mq_{y}}\cos{kq_{z}}}{|E|-\alpha(\cos{q_{x}}+\cos{q_{y}})-\gamma\cos{q_{z}}}\>.$
(428)

As far as we are aware, only the basic integral $M^{\rm tg}_{000}$ has been
evaluated analytically for arbitrary $|E|$, $\alpha$, and $\gamma$ by Delves,
Joyce and Zucker [94, 95, 97]:

$M^{\rm
tg}_{000}=\frac{1}{|E|}\frac{2}{\left[\sqrt{1-\frac{\alpha^{2}}{|E|^{2}}\left(2-\frac{\gamma}{\alpha}\right)^{2}}+\sqrt{1-\frac{\alpha^{2}}{|E|^{2}}\left(2+\frac{\gamma}{\alpha}\right)^{2}}\right]\sqrt{1-\kappa^{2}_{+}}\sqrt{1-\kappa^{2}_{-}}}\\!\left[\frac{2}{\pi}{\bf
K}\left(\sqrt{\frac{-\kappa^{2}_{+}}{1-\kappa^{2}_{+}}}\right)\\!\right]\\!\\!\left[\frac{2}{\pi}{\bf
K}\left(\sqrt{\frac{-\kappa^{2}_{-}}{1-\kappa^{2}_{-}}}\right)\\!\right]$
(429)

where

$\displaystyle\kappa^{2}_{\pm}$
$\displaystyle=\frac{1}{2}-\frac{1}{2}\frac{\left[\sqrt{1+\frac{\alpha}{|E|}\left(2-\frac{\gamma}{\alpha}\right)}\sqrt{1-\frac{\alpha}{|E|}\left(2+\frac{\gamma}{\alpha}\right)}+\sqrt{1-\frac{\alpha}{|E|}\left(2-\frac{\gamma}{\alpha}\right)}\sqrt{1+\frac{\alpha}{|E|}\left(2+\frac{\gamma}{\alpha}\right)}\right]}{\left[\sqrt{1-\frac{\alpha^{2}}{|E|^{2}}\left(2-\frac{\gamma}{\alpha}\right)^{2}}+\sqrt{1-\frac{\alpha^{2}}{|E|^{2}}\left(2+\frac{\gamma}{\alpha}\right)^{2}}\right]^{3}}\times$
(430)
$\displaystyle\times\left\\{\pm\frac{16\alpha^{2}}{|E|^{2}}+\sqrt{1-\frac{\gamma^{2}}{|E|^{2}}}\left[\sqrt{1+\frac{\alpha}{|E|}\left(2-\frac{\gamma}{\alpha}\right)}\sqrt{1+\frac{\alpha}{|E|}\left(2+\frac{\gamma}{\alpha}\right)}+\sqrt{1-\frac{\alpha}{|E|}\left(2-\frac{\gamma}{\alpha}\right)}\sqrt{1-\frac{\alpha}{|E|}\left(2+\frac{\gamma}{\alpha}\right)}\right]^{2}\right\\}.$

Compared with the original papers, we have applied the identity

${\bf K}(i\kappa)=\frac{1}{\sqrt{1-\kappa^{2}}}\>{\bf
K}\\!\left(\sqrt{\frac{-\kappa^{2}}{1-\kappa^{2}}}\right),$ (431)

since $\kappa^{2}_{\pm}<0$. Other integrals $M^{\rm tr}_{nmk}$ can be
calculated numerically using the recipes of D.3.

###  D.3  Green’s functions, Eq. (219), in the orthorhombic case,
$\alpha\neq\beta\neq\gamma$

The general integrals,

$M_{nmk}=\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\frac{{\rm
d}q_{x}\,{\rm d}q_{y}\,{\rm
d}q_{z}}{(2\pi)^{3}}\frac{\cos{nq_{x}}\cos{mq_{y}}\cos{kq_{z}}}{|E|-\alpha\cos{q_{x}}-\beta\cos{q_{y}}-\gamma\cos{q_{z}}}\>,$
(432)

apply to the simple cubic $UV$ model at arbitrary ${\bf P}$, tetragonal $UV$
model ($t_{x}=t_{y}\neq t_{z}$) at arbitrary ${\bf P}$, and orthorhombic $UV$
model ($t_{x}\neq t_{y}\neq t_{z}$, not studied in this paper) also at
arbitrary ${\bf P}$. They cannot be evaluated analytically. However, two out
of three integrations can be carried out analytically. Thus, only one
integration remains to be done numerically which leads to an efficient
numerical scheme. The first integration in Eq. (432) is elementary while the
second produces complete elliptic integrals. First, we introduce three
elliptic moduli:

$\displaystyle\kappa_{x}$
$\displaystyle\equiv\sqrt{\frac{4\beta\gamma}{(|E|-\alpha\cos{q_{x}})^{2}-(\beta-\gamma)^{2}}}\>,$
(433) $\displaystyle\kappa_{y}$
$\displaystyle\equiv\sqrt{\frac{4\alpha\gamma}{(|E|-\beta\cos{q_{y}})^{2}-(\alpha-\gamma)^{2}}}\>,$
(434) $\displaystyle\kappa_{z}$
$\displaystyle\equiv\sqrt{\frac{4\alpha\beta}{(|E|-\gamma\cos{q_{z}})^{2}-(\alpha-\beta)^{2}}}\>.$
(435)

Next, we apply Eqs. (316) and (327) to perform the double integration. For
$M_{000}$, three equivalent representations are possible, depending on which
two variables are integrated over:

$\displaystyle M_{000}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{x}}{\pi}\frac{2\,{\bf
K}(\kappa_{x})}{\pi\sqrt{(|E|-\alpha\cos{q_{x}})^{2}-(\beta-\gamma)^{2}}}\makebox[25.6073pt]{}$
(436) $\displaystyle=\int^{\pi}_{0}\frac{{\rm d}q_{y}}{\pi}\frac{2\,{\bf
K}(\kappa_{y})}{\pi\sqrt{(|E|-\beta\cos{q_{y}})^{2}-(\alpha-\gamma)^{2}}}$
(437) $\displaystyle=\int^{\pi}_{0}\frac{{\rm d}q_{z}}{\pi}\frac{2\,{\bf
K}(\kappa_{z})}{\pi\sqrt{(|E|-\gamma\cos{q_{z}})^{2}-(\alpha-\beta)^{2}}}\>.$
(438)

This equivalence can be used to validate the numerical method. Other integrals
are

$\displaystyle M_{100}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{x}}{\pi}\frac{2\cos{q_{x}}\,{\bf
K}(\kappa_{x})}{\pi\sqrt{(|E|-\alpha\cos{q_{x}})^{2}-(\beta-\gamma)^{2}}}\>,\makebox[22.76228pt]{}$
(439) $\displaystyle M_{010}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{y}}{\pi}\frac{2\cos{q_{y}}\,{\bf
K}(\kappa_{y})}{\pi\sqrt{(|E|-\beta\cos{q_{y}})^{2}-(\alpha-\gamma)^{2}}}\>,$
(440) $\displaystyle M_{001}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{z}}{\pi}\frac{2\cos{q_{z}}\,{\bf
K}(\kappa_{z})}{\pi\sqrt{(|E|-\gamma\cos{q_{z}})^{2}-(\alpha-\beta)^{2}}}\>.$
(441) $\displaystyle M_{200}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{x}}{\pi}\frac{2\cos{(2q_{x})}\,{\bf
K}(\kappa_{x})}{\pi\sqrt{(|E|-\alpha\cos{q_{x}})^{2}-(\beta-\gamma)^{2}}}\>,\makebox[22.76228pt]{}$
(442) $\displaystyle M_{020}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{y}}{\pi}\frac{2\cos{(2q_{y})}\,{\bf
K}(\kappa_{y})}{\pi\sqrt{(|E|-\beta\cos{q_{y}})^{2}-(\alpha-\gamma)^{2}}}\>,$
(443) $\displaystyle M_{002}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{z}}{\pi}\frac{2\cos{(2q_{z})}\,{\bf
K}(\kappa_{z})}{\pi\sqrt{(|E|-\gamma\cos{q_{z}})^{2}-(\alpha-\beta)^{2}}}\>.$
(444) $\displaystyle M_{110}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{z}}{\pi}\frac{(2-\kappa^{2}_{z}){\bf K}(\kappa_{z})-2\,{\bf
E}(\kappa_{z})}{\pi\kappa_{z}\sqrt{\alpha\beta}}\>,\makebox[28.45274pt]{}$
(445) $\displaystyle M_{101}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{y}}{\pi}\frac{(2-\kappa^{2}_{y}){\bf K}(\kappa_{y})-2\,{\bf
E}(\kappa_{y})}{\pi\kappa_{y}\sqrt{\alpha\gamma}}\>,$ (446) $\displaystyle
M_{011}$ $\displaystyle=\int^{\pi}_{0}\frac{{\rm
d}q_{x}}{\pi}\frac{(2-\kappa^{2}_{x}){\bf K}(\kappa_{x})-2\,{\bf
E}(\kappa_{x})}{\pi\kappa_{x}\sqrt{\beta\gamma}}\>.$ (447)

###  D.4  Determinant of Eq. (234)

We expand the determinant of Eq. (234) in powers of $U$, $V_{xy}$, and
$V_{z}$:

$\displaystyle D$ $\displaystyle=$ $\displaystyle
U\cdot|V_{xy}|\cdot|V_{z}|\cdot\left|\begin{array}[]{ccc}M_{000}&-2M_{100}&-2M_{001}\\\
2M_{100}&-(M_{000}+M_{200}+2M_{110})&-4M_{101}\\\
M_{001}&-2M_{101}&-(M_{000}+M_{002})\end{array}\right|$ (460)
$\displaystyle+\,U\cdot|V_{xy}|\cdot\left|\begin{array}[]{cc}M_{000}&-2M_{100}\\\
2M_{100}&-(M_{000}+M_{200}+2M_{110})\end{array}\right|+\,U\cdot|V_{z}|\cdot\left|\begin{array}[]{cc}M_{000}&-2M_{001}\\\
M_{001}&-(M_{000}+M_{002})\end{array}\right|$
$\displaystyle+\,|V_{xy}|\cdot|V_{z}|\cdot\left|\begin{array}[]{cc}-(M_{000}+M_{200}+2M_{110})&-4M_{101}\\\
-2M_{101}&-(M_{000}+M_{002})\end{array}\right|$
$\displaystyle+U\cdot\left[M_{000}\right]+|V_{xy}|\cdot\left[-(M_{000}+M_{200}+2M_{110})\right]+|V_{z}|\cdot\left[-(M_{000}+M_{002})\right]+1\>.$

From here, one potential can be expressed via the other two.

###  D.5  Derivation of Eq. (237)

For convenience, we introduce an anisotropy parameter, $\sigma\equiv t_{z}/t$.
Starting with Eq. (231), we wish to evaluate

$M_{000}-M_{002}=\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\\!\\!\int\limits^{\pi}_{-\pi}\frac{{\rm
d}q_{x}\,{\rm d}q_{y}\,{\rm
d}q_{z}}{(2\pi)^{3}}\frac{1-\cos{2q_{z}}}{8t+4t_{z}-4t\cos{q_{x}}-4t\cos{q_{y}}-4t_{z}\cos{q_{z}}}\>,$
(461)

in the limit $t_{z}\rightarrow 0$. Application of Eqs. (438) and (444) yields

$M_{000}-M_{002}=\frac{1}{2\pi t}\int^{\pi}_{0}\frac{{\rm
d}q_{z}}{\pi}\frac{1-\cos{2q_{z}}}{2-\sigma(1-\cos{q_{z}})}\>{\bf
K}\left[\frac{2}{2+\sigma(1-\cos{q_{z}})}\right].$ (462)

At $\sigma\rightarrow 0$, ${\bf K}$ logarithmically diverges. In the main
nonvanishing order, one can neglect the $\sigma$ term in the denominator
outside of ${\bf K}[\ldots]$. Using the asymptote, ${\bf K}(z\rightarrow
1)=\frac{1}{2}\ln{\frac{16}{1-z^{2}}}$, and changing variables, Eq. (462) is
brought to the form

$M_{000}-M_{002}\approx\frac{1}{8\pi
t}\left\\{\ln{\frac{16}{\sigma}-\frac{2}{\pi}}\int^{2}_{0}{\rm
d}u\,\sqrt{u(2-u)}\>\ln{(2-u)}\right\\}\equiv\frac{1}{8\pi
t}\left\\{\ln{\frac{16}{\sigma}}-J\right\\}.$ (463)

The remaining integral is expressible via $\Gamma$-function ([106],
#2.6.10.25)

$J=\frac{2}{\pi}\>4\frac{\Gamma(\frac{3}{2})\Gamma(\frac{3}{2})}{\Gamma(3)}\left[\ln{2}+\frac{\Gamma^{\prime}(\frac{3}{2})}{\Gamma(\frac{3}{2})}-\frac{\Gamma^{\prime}(3)}{\Gamma(3)}\right]=\frac{1}{2}-\ln{2}\>.$
(464)

Finally,

$M_{000}-M_{002}\approx\frac{1}{8\pi
t}\ln{\frac{32}{\sigma\sqrt{e}}}=\frac{1}{8\pi
t}\ln{\frac{32\,t}{\sqrt{e}\>t_{z}}}\>,$ (465)

from where Eq. (237) follows by inversion.

##  Appendix E  $UV$ model on the BCC lattice

###  E.1  Derivation of Eq. (102)

Introducing potential increment: $|U|=|U^{\rm bcc}_{\rm cr}|+u$, where
$u\ll|U^{\rm bcc}_{\rm cr}|$, and binding energy $E_{0}=-16t(1+\Delta)$, where
$\Delta\ll 1$, and expanding Eq. (100) for small $u$ and $\Delta$, one obtains

$\sqrt{\Delta}=\frac{u}{|U^{\rm bcc}_{\rm cr}|}\frac{{\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)}{{\bf
K}^{\prime}\\!\\!\left(\frac{1}{\sqrt{2}}\right)}\>,$ (466)

where the prime denotes derivative with respect to ${\bf K}$’s argument.
Applying Eq. (76) of the main text, the last expression is rewritten as

$\sqrt{\Delta}=\frac{u}{\sqrt{2}|U^{\rm bcc}_{\rm cr}|}\cdot\frac{{\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)}{2{\bf
E}\\!\\!\left(\frac{1}{\sqrt{2}}\right)-{\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)}\>.$ (467)

It is known that both ${\bf K}(2^{-1/2})$ and ${\bf E}(2^{-1/2})$ can be
expressed via gamma function, $\Gamma(1/4)$. Whittaker and Watson [171] derive
the following identities:

${\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{4\sqrt{\pi}}\left[\Gamma\left(\frac{1}{4}\right)\right]^{2},$
(468) $2{\bf E}\\!\\!\left(\frac{1}{\sqrt{2}}\right)-{\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)=4\pi^{\frac{3}{2}}\left[\Gamma\left(\frac{1}{4}\right)\right]^{-2}.$
(469)

Substituting them into Eq. (467) and using $|U^{\rm bcc}_{\rm cr}|$ from Eq.
(101), Eq. (466) becomes

$\sqrt{\Delta}=\frac{u}{t}\cdot\frac{1}{2^{9}\sqrt{2}\pi^{5}}\left[\Gamma\left(\frac{1}{4}\right)\right]^{8}.$
(470)

Finally, applying definitions of $\Delta$ and $u$, total pair energy $E_{0}$
can be expressed as Eq. (102) with the $u^{2}$ coefficient given by

$\frac{\left[\Gamma\\!\\!\left(\frac{1}{4}\right)\right]^{16}}{2^{15}\pi^{10}}=\frac{2}{\pi^{6}}\left[{\bf
K}\\!\\!\left(\frac{1}{\sqrt{2}}\right)\right]^{8}=0.29050160675...\>.$ (471)

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