$\displaystyle\widetilde{M}_{T,\mathbf{A}}\alpha(X,r)\coloneqq\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\,\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}_{h}(X)(r+s)}\,(\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\alpha)(X,s),$
(4.76)

where

$k_{T}(Z,r)\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}g_{0}^{\mathrm{i}\omega_{n}}\bigl{(}Z-\frac{r}{2}\bigr{)}\,g_{0}^{-\mathrm{i}\omega_{n}}\bigl{(}Z+\frac{r}{2}\bigr{)}.$
(4.77)

By $(D\mathbf{A})_{ij}\coloneqq\partial_{j}\mathbf{A}_{i}$ we denote the
Jacobi matrix of $\mathbf{A}$. Here and in the following we use the notation
that $D\mathbf{A}_{h}(X)(r+s)$ denotes the matrix $D\mathbf{A}_{h}(X)$ applied
to $r+s$. Accordingly, $\frac{r-s}{4}\cdot D\mathbf{A}_{h}(X)(r+s)$ is the
inner product of this vector with $\frac{r-s}{4}$.

The following proposition allows us to replace $\widetilde{L}_{T,\mathbf{A}}$
by $\widetilde{M}_{T,\mathbf{A}}$ in our computations.

######  Proposition 4.11.

Assume that $\mathbf{A}=\mathbf{A}_{e_{3}}+A$ with a periodic vector potential
$A\in W^{2,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$, let
$|\cdot|^{k}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1,2\\}$,
$\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$
as in (3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any
$0<h\leqslant h_{0}$ and any $T\geqslant T_{0}$ we have

$\displaystyle\|\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta\|_{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant
C\;h^{8}\;\max_{k=0,1,2}\|\,|\cdot|^{k}V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.78)

Before we give the proof of Proposition 4.11, we state and prove two
preparatory lemmas. The first lemma allows us to extract the term
$\Phi_{2A_{h}}(X,X+Z)$ and to replace
$\mathrm{e}^{\mathrm{i}Z\cdot(-\mathrm{i}\nabla)}$ by
$\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}}$ in the definition of
$\widetilde{L}_{T,\mathbf{A}}$. In [19, 16], where only the constant magnetic
field is present, this approximation holds as an algebraic identity. A version
of the first bound in (4.79) has, in case of a periodic vector potential, been
proved in [17]. In this reference $Z$ does not appear on the right side, which
is why this bound appears to be wrong.

######  Lemma 4.12.

Assume that $A\in W^{2,\infty}(\mathbb{R}^{3};\mathbb{R}^{3})$. We then have

$\displaystyle\sup_{X\in\mathbb{R}^{3}}\bigl{|}\widetilde{\Phi}_{A}(X,Z,r,s)-\Phi_{2A}(X,X+Z)+\frac{1}{4}(r-s)\cdot
DA(X)(r+s)\bigr{|}$ $\displaystyle\leqslant
C\;\|D^{2}A\|_{\infty}\;\bigl{(}|Z|+|r-s|\bigr{)}\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)},$
(4.79)

where $\Phi_{A}$ is defined in (4.2) and $\widetilde{\Phi}_{A}$ is defined in
(4.47). We also have

$\displaystyle\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{X}\widetilde{\Phi}_{A}(X,Z,r,s)-\nabla_{X}\Phi_{2A}(X,X+Z)\bigr{|}$
$\displaystyle\leqslant
C\;\|D^{2}A\|_{\infty}\;\bigl{(}|Z|+|r-s|\bigr{)}\bigl{(}|s|+|r-s|\bigr{)}$
(4.80)

as well as

$\displaystyle\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{X}(r-s)\cdot
DA(X)\cdot(r+s)\bigr{|}\leqslant
C\,\|D^{2}A\|_{\infty}\,|r-s|\,\bigl{(}|s|+|r-s|\bigr{)}$ (4.81)

and

$\displaystyle\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{r}\widetilde{\Phi}_{A}(X,Z,r,s)+\frac{1}{4}\nabla_{r}(r-s)\cdot
DA(X)(r+s)\bigr{|}$ $\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,\bigl{(}|s|^{2}+|r-s|^{2}+|Z|^{2}\bigr{)}.$ (4.82)

The same bounds hold if $A$ is replaced by $\mathbf{A}_{e_{3}}+A$.

###### Proof of Lemma 4.12.

We use the notation $\zeta_{X}^{r}\coloneqq X+\frac{r}{2}$ and start by
writing

$\displaystyle\widetilde{\Phi}_{A}(X,Z,r,s)$
$\displaystyle=\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}+A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}\cdot
Z$ $\displaystyle\hskip
50.0pt-\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}-A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}\cdot\frac{r-s}{2}.$
(4.83)

A second order Taylor expansion in the variable $\pm\frac{1}{2}(s+t(r-s))$
allows us to show that

$\displaystyle\Bigl{|}A\bigl{(}X+Z-tZ\pm\frac{s+t(r-s)}{2}\bigr{)}-A(X+Z-tZ)\mp\frac{1}{2}DA(X+Z-tZ)(s+t(r-s))\Bigr{|}$
$\displaystyle\leqslant
C\;\|D^{2}A\|_{\infty}\;\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)}.$ (4.84)

For the first term on the right side of (4.83), this implies

$\displaystyle\Bigl{|}\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}+A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}\cdot
Z-\Phi_{2A}(X,X+Z)\Bigr{|}$ $\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,|Z|\,\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)},$ (4.85)

and we find

$\displaystyle\Bigl{|}-\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}-A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}\cdot\frac{r-s}{2}+\frac{1}{4}(r-s)\cdot
DA(X)(r+s)\Bigr{|}$
$\displaystyle\leqslant\Bigl{|}-\frac{r-s}{2}\cdot\int_{0}^{1}\mathrm{d}t\;\bigl{[}DA(X+Z-tZ)-DA(X)\bigr{]}(s+t(r-s))\Bigr{|}$
$\displaystyle+C\,\|D^{2}A\|_{\infty}\,\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)}\,|r-s|$
$\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,|r-s|\,\bigl{[}|s|^{2}+|r-s|^{2}+\bigl{(}|s|+|r-s|\bigr{)}|Z|\bigr{]}$
(4.86)

for the second term. Adding up (4.85) and (4.86) proves (4.79). The proof of
(4.80) results from a first order Taylor expansion and that of (4.81) is a
straightforward computation. It remains to prove (4.82).

When we differentiate (4.83) with respect to $r$ this yields

$\displaystyle\nabla_{r}\widetilde{\Phi}_{A}(X,Z,r,s)$
$\displaystyle=\int_{0}^{1}\mathrm{d}t\;\frac{t}{2}\
Z\cdot\bigl{[}DA\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}-DA\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}$
$\displaystyle\hskip 50.0pt-\int_{0}^{1}\mathrm{d}t\;\frac{t}{2}\
\frac{r-s}{2}\cdot\bigl{[}DA\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}+DA\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}$
$\displaystyle\hskip
50.0pt-\frac{1}{2}\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}-A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}.$
(4.87)

A first order Taylor expansion shows that the absolute value of the first term
on the right side of (4.87) is bounded by
$C\|D^{2}A\|_{\infty}(|s|+|r-s|)|Z|$. We also note that

$\displaystyle\frac{1}{4}\nabla_{r}(r-s)\cdot DA(X)(r+s)=\frac{1}{4}(r-s)\cdot
DA(X)+\frac{1}{4}DA(X)(r+s).$ (4.88)

The second term on the right side of (4.87) obeys

$\displaystyle\Bigl{|}\int_{0}^{1}\mathrm{d}t\;t\;\frac{r-s}{4}\cdot\bigl{[}DA\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}+DA\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}-\frac{1}{4}(r-s)\cdot
DA(X)\bigr{|}$
$\displaystyle\leqslant\Bigl{|}\int_{0}^{1}\mathrm{d}t\;\frac{t}{2}\;(r-s)\cdot\bigl{[}DA(X+Z-tZ)-DA(X)\bigr{]}\Bigr{|}+C\,\|D^{2}A\|_{\infty}\bigl{(}|s|+|r-s|\bigr{)}|r-s|$
$\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,\bigl{[}|r-s|\,|Z|+\bigl{(}|s|+|r-s|\bigr{)}|r-s|\bigr{]}.$
(4.89)

For the third term on the right side of (4.87), we use the bound

$\displaystyle\Bigl{|}\frac{1}{2}\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{s+t(r-s)}\bigr{)}-A\bigl{(}\zeta_{X+Z-tZ}^{-s-t(r-s)}\bigr{)}\bigr{]}-\frac{1}{4}DA(X)(r+s)\Bigr{|}\leqslant\mathcal{T}_{+}+\mathcal{T}_{-}+\mathcal{T},$

where

$\displaystyle\mathcal{T}_{\pm}$
$\displaystyle\coloneqq\Bigl{|}\frac{1}{2}\int_{0}^{1}\mathrm{d}t\;\bigl{[}A\bigl{(}\zeta_{X+Z-tZ}^{\pm
s\pm t(r-s)}\bigr{)}\mp
A(X+Z-tZ)-\frac{1}{2}DA(X+Z-tZ)(s+t(r-s))\bigr{]}\Bigr{|}$

and

$\displaystyle\mathcal{T}$
$\displaystyle\coloneqq\Bigl{|}\frac{1}{2}\int_{0}^{1}\mathrm{d}t\;DA(X+Z-tZ)(s+t(r-s))-\frac{1}{4}DA(X)(r+s)\Bigr{|}.$

By (4.84), we have

$\displaystyle\mathcal{T}_{\pm}$ $\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)},$
$\displaystyle\mathcal{T}$ $\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\,\bigl{(}|s|+|r-s|\bigr{)}\,|Z|.$

In combination with (4.89), these considerations imply (4.82). The proof for
$\mathbf{A}_{e_{3}}+A$ is literally the same. ∎

The next lemma is a substitute for the identity

$\displaystyle\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(X\wedge
Z)}\mathrm{e}^{\mathrm{i}Z\cdot P_{X}}=\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{X}}$
(4.90)

in the case of a general magnetic field. It holds because
$\mathbf{B}\cdot(X\wedge Z)=Z\cdot(\mathbf{B}\wedge X)$ and the latter
commutes with $Z\cdot\Pi_{X}$. Here, we used the notations
$P=-\mathrm{i}\nabla$ for the momentum operator and
$\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$ for its magnetic counterpart.
Sometimes when several variables appear in an equation we write, e.g.,
$P_{X}$, $\Pi_{X}$, etc. to indicate on which variable $P$, $\Pi$, etc. is
acting.

######  Lemma 4.13.

Assume that $A\in L^{\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ is a periodic
function. Then

$\displaystyle\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\,\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{X}}=\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}},$
(4.91)

where $\Pi_{\mathbf{A}_{h}}=P_{X}+2\mathbf{A}_{h}(X)$ is understood to act on
the $X$ coordinate.

The above lemma is a consequence of the following more abstract proposition,
whose proof can be found in [37, p. 290]. For the sake of completeness we
repeat it here.

######  Proposition 4.14.

Let $\mathcal{H}$ be a separable Hilbert space, let
$P\colon\mathcal{D}(P)\rightarrow\mathcal{H}$ be a densely defined self-
adjoint operator, and assume that $Q$ is bounded and self-adjoint. Assume
further that
$[\mathrm{e}^{\mathrm{i}tP}\,Q\,\mathrm{e}^{-\mathrm{i}tP},\mathrm{e}^{\mathrm{i}sP}\,Q\,\mathrm{e}^{-\mathrm{i}sP}]=0$
for every $t,s\in[0,1]$. Then, we have

$\displaystyle\exp\Bigl{(}\mathrm{i}\int_{0}^{1}\mathrm{d}t\;\mathrm{e}^{\mathrm{i}tP}\,Q\,\mathrm{e}^{-\mathrm{i}tP}\Bigr{)}\,\mathrm{e}^{\mathrm{i}P}=\mathrm{e}^{\mathrm{i}(P+Q)}.$

###### Proof.

For $s\in\mathbb{R}$ we define
$Q(s)\coloneqq\mathrm{e}^{\mathrm{i}sP}\,Q\,\mathrm{e}^{-\mathrm{i}sP}$ and
$W(s)\coloneqq\mathrm{e}^{\mathrm{i}s(P+Q)}\,\mathrm{e}^{-\mathrm{i}sP}$. On
$\mathcal{D}(P)$, we may differentiate $W$ to get

$\displaystyle-\mathrm{i}W^{\prime}(s)=\mathrm{e}^{\mathrm{i}s(P+Q)}(P+Q)\,\mathrm{e}^{-\mathrm{i}sP}-\mathrm{e}^{\mathrm{i}s(P+Q)}\,P\,\mathrm{e}^{-\mathrm{i}sP}=\mathrm{e}^{\mathrm{i}s(P+Q)}\,Q\,\mathrm{e}^{-\mathrm{i}sP}=W(s)\,Q(s).$

Using that $Q$ is bounded we conclude that this identity also holds on
$\mathcal{H}$. Hence, $W$ satisfies the linear differential equation
$W^{\prime}(s)=\mathrm{i}W(s)\,Q(s)$. Since by assumption $[Q(s),Q(t)]=0$
holds for all $s,t\in[0,1]$, we conclude that the unique solution to this
equation can be written as

$\displaystyle\widetilde{W}(s)\coloneqq\exp\Bigl{(}\mathrm{i}\int_{0}^{s}\mathrm{d}t\;Q(t)\Bigr{)},$

and hence

$\displaystyle\exp\Bigl{(}\mathrm{i}\int_{0}^{s}\mathrm{d}t\;Q(t)\Bigr{)}=\mathrm{e}^{\mathrm{i}s(P+Q)}\mathrm{e}^{-\mathrm{i}sP}.$

With the choice $s=1$ this equation proves the claim. ∎

###### Proof of Lemma 4.13.

We first show that

$\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\,\mathrm{e}^{\mathrm{i}Z\cdot
P_{X}}=\mathrm{e}^{\mathrm{i}Z\cdot(P_{X}+2A_{h}(X))}$ (4.92)

holds. To that end, we apply Proposition 4.14 with the choices $P=Z\cdot
P_{X}$, where $P_{X}=-\mathrm{i}\nabla_{X}$, and $Q=2Z\cdot A_{h}(X)$ and find

$\displaystyle\exp\Bigl{(}\mathrm{i}\int_{0}^{1}\mathrm{d}t\;\mathrm{e}^{\mathrm{i}tZ\cdot
P_{X}}\,2Z\cdot A_{h}(X)\,\mathrm{e}^{-\mathrm{i}tZ\cdot
P_{X}}\Bigr{)}\,\mathrm{e}^{\mathrm{i}Z\cdot
P_{X}}=\mathrm{e}^{\mathrm{i}(Z\cdot(P_{X}+2A(X)))}.$

It remains to compute the integral in the exponential, which reads

$\int_{0}^{1}\mathrm{d}t\;\mathrm{e}^{\mathrm{i}tZ\cdot P_{X}}\,2Z\cdot
A_{h}(X)\,\mathrm{e}^{-\mathrm{i}tZ\cdot
P_{X}}=2\int_{0}^{1}\mathrm{d}t\;Z\cdot A_{h}(X+tZ)=\Phi_{2A_{h}}(X,X+Z).$

To obtain the last equality we applied the coordinate transformation $t\mapsto
1-t$. In combination, these considerations prove (4.92).

Next, we use (4.92) and (4.90) to see that

$\displaystyle\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\,\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{X}}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\,\mathrm{e}^{\mathrm{i}Z\cdot
P_{X}}\;\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(X\wedge
Z)}=\mathrm{e}^{\mathrm{i}Z\cdot(P_{X}+2A_{h}(X))}\;\mathrm{e}^{\mathrm{i}\mathbf{B}\cdot(X\wedge
Z)}$ $\displaystyle=\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}}$

holds. This proves Lemma 4.13. ∎

For $a\in\mathbb{N}_{0}$ we define the functions

$\displaystyle
F_{T}^{a}\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\sum_{m=0}^{a}\sum_{b=0}^{m}\binom{m}{b}\;\bigl{(}|\cdot|^{b}\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|^{m-b}\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
(4.93)

and

$\displaystyle G_{T}^{a}$
$\displaystyle\coloneqq\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\sum_{m=0}^{a}\sum_{b=0}^{m}\binom{m}{b}}\;\bigl{(}|\cdot|^{b}\,|\nabla
g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|^{m-b}\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
$\displaystyle\hskip
150.0pt+\bigl{(}|\cdot|^{b}\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|^{m-b}\,|\nabla
g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)},$ (4.94)

which play an prominent role in the proof of Proposition 4.11. An application
of Lemma 4.4, (4.56), and (4.57) shows that for $T\geqslant T_{0}>0$ and
$a\in\mathbb{N}_{0}$, we have

$\displaystyle\|F_{T}^{a}\|_{1}+\|G_{T}^{a}\|_{1}$ $\displaystyle\leqslant
C_{a}.$ (4.95)

We are now prepared to give the proof of Proposition 4.11.

###### Proof of Proposition 4.11.

We use (4.90) and
$\Phi_{\mathbf{A}_{\mathbf{B}}}(x,y)=\frac{\mathbb{B}}{2}\cdot(x\wedge y)$ to
write the operator $\widetilde{L}_{T,\mathbf{A}}$ as

$\displaystyle\widetilde{L}_{T,\mathbf{A}}\alpha(X,r)$
$\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge
s)}k_{T}(Z,r-s)\,\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)}\,(\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\alpha)(X,s),$

where $\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$. With the identities

$\displaystyle-\frac{r-s}{4}\cdot D\mathbf{A}_{\mathbf{B}}(r+s)$
$\displaystyle=\frac{\mathbf{B}}{4}\cdot(r\wedge s),$
$\displaystyle\Phi_{2\mathbf{A}_{\mathbf{B}}}(X,X+Z)$
$\displaystyle=Z\cdot(\mathbf{B}\wedge X),$

and (4.47) we check that

$\displaystyle\widetilde{\Phi}_{\mathbf{A}_{\mathbf{B}}}(X,Z,r,s)$
$\displaystyle=-\frac{r-s}{4}\cdot
D\mathbf{A}_{\mathbf{B}}(X)(r+s)+\Phi_{2\mathbf{A}_{\mathbf{B}}}(X,X+Z).$

In combination with Lemma 4.13 and the fact that the integrand in the
definition of $\widetilde{M}_{T,\mathbf{A}}$ is an even function of $Z$, this
allows us to write $\widetilde{M}_{T,\mathbf{A}}$ as

$\displaystyle\widetilde{M}_{T,\mathbf{A}}\alpha(X,r)$
$\displaystyle=\smash{\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}}\mathrm{d}Z\mathrm{d}s\;\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge
s)}k_{T}(Z,r-s)\,\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\,\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}$ $\displaystyle\hskip
230.0pt\times(\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\alpha)(X,s),$

and consequently,

$\displaystyle\bigl{(}\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta\bigr{)}(X,r)$
$\displaystyle=-2\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge
s)}k_{T}(Z,r-s)\,V\alpha_{*}(s)\,(\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\Psi)(X)$
$\displaystyle\times\bigl{[}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)}-\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}\bigr{]}.$ (4.96)

We claim that

$\displaystyle\|\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
C\,\|\Psi\|_{2}^{2}\;\|D^{2}A_{h}\|_{\infty}^{2}\;\|F_{T}^{3}*|V\alpha_{*}|+F_{T}^{1}*|\cdot|^{2}\,|V\alpha_{*}|\,\|_{2}^{2}$
$\displaystyle\leqslant
C\;h^{8}\;\max_{k=0,1,2}\|\,|\cdot|^{k}V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.97)

The second bound in the above equation follows from Young’s inequality, (2.7),
and (4.95). To prove the first bound in (4.97), we use (4.61) and find

$\displaystyle\|\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
4\,\|\Psi\|_{2}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|$
$\displaystyle\times\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)-\mathrm{i}\Phi_{2A_{h}}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}-1\bigr{|}\Bigr{|}^{2}.$ (4.98)

Furthermore, an application of Lemma 4.12 implies

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|Z|^{a}\;|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)-\mathrm{i}\Phi_{2A}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}-1\bigr{|}$ $\displaystyle\leqslant
C\;\|D^{2}A\|_{\infty}\bigl{[}F_{T}^{3+a}(r-s)+F_{T}^{1+a}(r-s)\;|s|^{2}\bigr{]}$
(4.99)

with $F_{T}^{a}$ in (4.93). We need this bound here only for the case $a=0$
but state it for general $a\geqslant 0$ for later reference. In combination
with (4.98), this proves (4.97).

We claim that the term involving
$\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$, which is understood to act
on the center-of-mass coordinate, is bounded by

$\displaystyle\|\Pi(\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\,\|D^{2}A_{h}\|_{\infty}^{2}\,\bigl{(}1+\|DA_{h}\|_{\infty}^{2}\bigr{)}$
$\displaystyle\times\bigl{[}\|F_{T}^{4}*|V\alpha_{*}|\,\|_{2}^{2}+\|F_{T}^{1}*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}+\|F_{T}^{2}*|\cdot|^{2}|V\alpha_{*}|\,\|_{2}^{2}\bigr{]}.$
(4.100)

If this holds, the desired bound for this term follows from Young’s inequality
and (4.95). To prove (4.100), we use (4.96) and argue as in the proof of
(4.69) to see that

$\displaystyle\|\Pi(\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta)\|_{2}^{2}\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Big{(}\;\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\,|V\alpha_{*}(s)|$
$\displaystyle\hskip
2.84544pt\times|k_{T}(Z,r-s)|\Bigl{[}h\,(1+|Z|)\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)-\mathrm{i}\Phi_{2A_{h}}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}-1\bigr{|}$ $\displaystyle\hskip
45.52458pt+\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{X}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)}-\nabla_{X}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}\bigr{|}\Bigr{]}\Bigr{|}^{2}\Big{)}.$ (4.101)

The difference of the phases involving a gradient can be estimated as

$\displaystyle\bigl{|}\nabla_{X}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)}-\nabla_{X}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}\bigr{|}$ $\displaystyle\hskip
28.45274pt\leqslant\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)-\mathrm{i}\Phi_{2A}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}-1\bigr{|}|\nabla_{X}\widetilde{\Phi}_{A}(X,Z,r,s)|$
$\displaystyle\hskip
42.67912pt+\bigl{|}\nabla_{X}\widetilde{\Phi}_{A}(X,Z,r,s)-\nabla_{X}\Phi_{2A}(X,X+Z)\bigr{|}+\bigl{|}\nabla_{X}(r-s)\cdot
DA(X)(r+s)\bigr{|},$

which, by (4.70) and Lemma 4.12, is bounded by

$\displaystyle C\,\|D^{2}A\|_{\infty}\,\bigl{(}1+\|DA\|_{\infty}\bigr{)}$
$\displaystyle\hskip
50.0pt\times\bigl{[}\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)}\bigl{(}1+|Z|^{2}+|r-s|^{2}\bigr{)}+\bigl{(}|s|+|r-s|\bigr{)}\bigl{(}|Z|+|r-s|\bigr{)}\bigr{]}.$

Accordingly,

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{X}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)}-\nabla_{X}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}\bigr{|}$ $\displaystyle\hskip 40.0pt\leqslant
C\,\|D^{2}A\|_{\infty}\bigl{(}1+\|DA\|_{\infty}\bigr{)}$ $\displaystyle\hskip
60.0pt\times\bigl{[}F_{T}^{4}(r-s)+F_{T}^{1}(r-s)\,|s|+F_{T}^{2}(r-s)\,|s|^{2}\bigr{]}.$

Using (4.99), we also find

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|k_{T}(Z,r-s)|\,h\,(1+|Z|)\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)-\mathrm{i}\Phi_{2A}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}-1\bigr{|}$ $\displaystyle\hskip 40.0pt\leqslant
C\,\|D^{2}A\|_{\infty}\;\bigl{[}F_{T}^{4}(r-s)+F_{T}^{2}(r-s)\,|s|^{2}\bigr{]}.$

In combination with (4.101), this proves (4.100).

We claim that the term involving
$\widetilde{\pi}=-\mathrm{i}\nabla+\frac{1}{2}\mathbf{A}_{\mathbf{B}}$, which
is understood to act on the relative coordinate, is bounded by

$\displaystyle\|\widetilde{\pi}(\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\,\|D^{2}A_{h}\|_{\infty}^{2}\,\bigl{(}1+\|A_{h}\|_{\infty}^{2}+\|DA_{h}\|_{\infty}^{2}\bigr{)}$
$\displaystyle\hskip
28.45274pt\times\bigl{\|}(F_{T}^{4}+G_{T}^{2})*(1+|\cdot|^{2})|V\alpha_{*}|\bigr{\|}_{2}^{2}$
$\displaystyle\leqslant
C\;h^{8}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\;\|(1+|\cdot|^{2})V\alpha_{*}\|_{2}^{2}.$
(4.102)

The second inequality is a consequence of Young’s inequality and (4.95). To
prove the first inequality in (4.102), we first use (4.96) and argue as in the
proof of (4.69) to see that

$\displaystyle\|\widetilde{\pi}(\widetilde{L}_{T,\mathbf{A}}\Delta-\widetilde{M}_{T,\mathbf{A}}\Delta)\|_{2}^{2}\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\,\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Big{(}\;\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|V\alpha_{*}(s)|$
$\displaystyle\hskip
28.45274pt\times\bigl{|}\widetilde{\pi}k_{T}(Z,r-s)\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge
s)}\bigr{|}\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)-\mathrm{i}\Phi_{2A_{h}}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}-1\bigr{|}$ $\displaystyle\hskip
14.22636pt+|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{r}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A_{h}}(X,Z,r,s)}-\nabla_{r}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA_{h}(X)(r+s)}\bigr{|}\Bigr{|}^{2}\Big{)}.$

A brief computation shows that the operator $\widetilde{\pi}$ obeys the
following intertwining relation with respect to
$\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge s)}$:

$\displaystyle\widetilde{\pi}_{r}\,\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}=\mathrm{e}^{\frac{\mathrm{i}}{4}\mathbf{B}\cdot(r\wedge
s)}\bigl{(}-\mathrm{i}\nabla_{r}+\frac{1}{4}\mathbf{B}\wedge(r-s)\bigr{)}.$
(4.103)

The notation $\widetilde{\pi}_{r}$ in the above equation highlights on which
of the two variables the operator $\widetilde{\pi}$ is acting. An application
of this identity shows

$\displaystyle|\widetilde{\pi}k_{T}(Z,r-s)\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}(r\wedge
s)}|$
$\displaystyle\leqslant|\nabla_{r}k_{T}(Z,r-s)|+\frac{|\mathbf{B}|}{4}\,|r-s|\,|k_{T}(Z,r-s)|.$
(4.104)

Hence, a computation similar to that leading to (4.99) shows that

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|\widetilde{\pi}k_{T}(Z,r-s)\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{4}\cdot(r\wedge
s)}|$
$\displaystyle\times\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)-\mathrm{i}\Phi_{2A}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}-1\bigr{|}$ $\displaystyle\leqslant
C\;\|D^{2}A\|_{\infty}\;\bigl{[}(F_{T}^{4}+G_{T}^{3})(r-s)+(F_{T}^{2}+G_{T}^{1})(r-s)\;|s|^{2}\bigr{]}$
(4.105)

with the function $F_{T}^{a}$ in (4.93) and $G_{T}^{a}$ in (4.94). Let us also
note that we estimate the factor $|\mathbf{B}|$ coming from the second term in
(4.104) by $1$.

We also have

$\displaystyle\bigl{|}\nabla_{r}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)}-\nabla_{r}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)-\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}\bigr{|}$ $\displaystyle\hskip
85.35826pt\leqslant\bigl{|}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)-\mathrm{i}\Phi_{2A}(X,X+Z)+\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}-1\bigr{|}|\nabla_{r}\widetilde{\Phi}_{A}(X,Z,r,s)|$
$\displaystyle\hskip
99.58464pt+\bigl{|}\nabla_{r}\widetilde{\Phi}_{A}(X,Z,r,s)+\nabla_{r}\frac{r-s}{4}\cdot
DA(X)(r+s)\bigr{|}.$ (4.106)

We use

$\displaystyle|\nabla_{r}\widetilde{\Phi}_{A}(X,Z,r,s)|$
$\displaystyle\leqslant
C\;\bigl{(}\|A\|_{\infty}+\|DA\|_{\infty}\bigr{)}\,\bigl{(}\bigl{|}Z+\frac{r-s}{2}\bigr{|}+\bigl{|}Z-\frac{r-s}{2}\bigr{|}\bigr{)}$
(4.107)

and Lemma 4.12 to see that the right side of (4.106) is bounded by

$\displaystyle
C\,\|D^{2}A\|_{\infty}\bigl{(}1+\|A\|_{\infty}+\|DA\|_{\infty}\bigr{)}\;\bigl{[}|s|^{2}+|r-s|^{2}+|Z|^{2}$
$\displaystyle\hskip
40.0pt+\bigl{(}|s|+|r-s|\bigr{)}\bigl{(}|Z|+|r-s|\bigr{)}\bigl{(}\bigl{|}Z+\frac{r-s}{2}\bigr{|}+\bigl{|}Z-\frac{r-s}{2}\bigr{|}\bigr{)}\bigr{]}.$

In combination with (4.74) these considerations imply

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{r}\mathrm{e}^{\mathrm{i}\widetilde{\Phi}_{A}(X,Z,r,s)}-\nabla_{r}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
DA(X)(r+s)}\bigr{|}$ $\displaystyle\leqslant
C\,\|D^{2}A\|_{\infty}\bigl{(}1+\|A\|_{\infty}+\|DA\|_{\infty}\bigr{)}$
$\displaystyle\times\bigl{[}F_{T}^{3}(r-s)+F_{T}^{2}(r-s)\,|s|+F_{T}^{0}(r-s)\,|s|^{2}\bigr{]}.$

When we combine this with (4.105), we obtain (4.102). ∎

#####  4.5.3.3 The operator $M_{T,\mathbf{A}}$.

We define the operator $M_{T,\mathbf{A}}$ by

$\displaystyle
M_{T,\mathbf{A}}\alpha(X,r)\coloneqq\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;(\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\alpha)(X,s),$
(4.108)

where $k_{T}$ is defined below (4.76). In our calculation, we may replace
$\widetilde{M}_{T,\mathbf{A}}$ by $M_{T,\mathbf{A}}$ due to the following
error bound.

######  Proposition 4.15.

Assume that $\mathbf{A}=\mathbf{A}_{e_{3}}+A$ with a $A\in
W^{2,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ periodic, let
$|\cdot|^{k}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1,2\\}$,
$\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$
as in (3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any
$0<h\leqslant h_{0}$ and any $T\geqslant T_{0}$ we have

$\displaystyle\|\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\|_{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant
C\;h^{6}\;\bigl{(}\max_{k\in\\{0,1,2\\}}\|\,|\cdot|^{k}V\alpha_{*}\|_{2}^{2}\bigr{)}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.109)

Furthermore,

$\displaystyle|\langle\Delta,\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\rangle|$ $\displaystyle\leqslant
C\;h^{6}\;\bigl{(}\|V\alpha_{*}\|_{2}^{2}+\|\,|\cdot|^{2}V\alpha_{*}\|_{2}^{2}\bigr{)}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.110)

######  Remark 4.16.

The two bounds in (4.109) and (4.110) are needed for the proof of Proposition
3.2 and Theorem 3.6, respectively. We highlight that the bound in (4.109) is
not strong enough to be useful in the proof of Theorem 3.6. More precisely, if
we apply Cauchy–Schwarz and use Lemma 4.1 as well as (4.109) to estimate
$\|\Delta\|_{2}$, we obtain a bound that is only of the order $h^{4}$. This is
not good enough because $h^{4}$ is the order of the Ginzburg–Landau energy. To
obtain (4.110), we exploit the fact that $V\alpha_{*}$ is real-valued, which
allows us to replace $\exp(-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}_{h}(X)(r+s))$ in the definition of $\widetilde{M}_{T,\mathbf{A}}$
by $\cos(\frac{r-s}{4}\cdot D\mathbf{A}_{h}(X)(r+s))$ and to win an additional
factor $h^{2}$.

###### Proof of Proposition 4.15.

The proof is similar to that of Proposition 4.9. We begin by proving (4.109)
and claim that

$\displaystyle\|\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\|_{2}^{2}$ $\displaystyle\leqslant
C\,\|\Psi\|_{2}^{2}\,\|D\mathbf{A}_{h}\|_{\infty}^{2}\,\|F_{T}^{2}*|V\alpha_{*}|+F_{T}^{1}*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}$
(4.111)

with the function $F_{T}^{a}$ in (4.93). If this holds, the desired bound for
this term follows from Young’s inequality, (2.7), and the $L^{1}$-norm
estimate on $F_{T}^{a}$ in (4.95). To prove (4.111), we argue as in
(4.60)–(4.62) and obtain

$\displaystyle\|\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\|_{2}^{2}$ $\displaystyle\leqslant 4\,\|\Psi\|_{2}^{2}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}_{h}(X)(r+s)}-1\bigr{|}\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$ (4.112)

When we combine the bound

$\displaystyle\bigl{|}(r-s)\cdot D\mathbf{A}(X)(r+s)\bigr{|}$
$\displaystyle\leqslant\|D\mathbf{A}\|_{\infty}\,|r-s|\,\bigl{(}2|s|+|r-s|\bigr{)}$
(4.113)

and the estimate for $|r-s|$ in (4.74), we obtain

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|k_{T}(Z,r-s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}(X)(r+s)}-1\bigr{|}$ $\displaystyle\leqslant
C\,\|D\mathbf{A}\|_{\infty}\bigl{[}F_{T}^{2}(r-s)+F_{T}^{1}(r-s)\;|s|\bigr{]}.$
(4.114)

In combination with (4.112), this proves (4.111).

We also claim that the term involving $\Pi$ is bounded by

$\displaystyle\|\Pi(\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta)\|_{2}^{2}$ $\displaystyle\leqslant
C\,h^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$
$\displaystyle\times\bigl{[}(|\mathbf{B}|^{2}+\|D^{2}A_{h}\|_{\infty}^{2})(1+|\mathbf{B}|^{2}+\|D^{2}A_{h}\|_{\infty}^{2})\,\|F_{T}^{3}*|V\alpha_{*}|+F_{T}^{2}*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}$
$\displaystyle+\|D^{2}\mathbf{A}_{h}\|_{\infty}^{2}\,\|F_{T}^{2}*|V\alpha_{*}|+F_{T}^{1}*|\cdot|\,|V\alpha_{*}|\,\|_{2}^{2}\bigr{]}.$
(4.115)

If this is correct, an application Young’s inequality and (4.95) shows the
desired bound for this term. To prove (4.115), we first show that

$\displaystyle\|\Pi\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\Psi\|_{2}\leqslant
C\,h^{2}\,(1+|Z|)\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}$ (4.116)

holds. From Lemma 4.13 we know that

$\displaystyle\mathrm{e}^{\pm\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}}=\mathrm{e}^{\pm\mathrm{i}Z\cdot\Pi}\mathrm{e}^{-\mathrm{i}\Phi_{2A_{h}}(X,X\mp
Z)}$

holds. An application of the intertwining relation in (4.67) for $\Pi_{X}$ and
$\mathrm{e}^{\mathrm{i}Z\cdot\Pi_{X}}$ therefore shows

$\displaystyle[\Pi,\mathrm{e}^{\pm\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}}]=\mathrm{e}^{\pm\mathrm{i}Z\cdot\Pi_{\mathbf{A}_{h}}}\bigl{[}\mp
2\,\mathbf{B}\wedge Z-\nabla_{X}\Phi_{2A_{h}}(X,X\mp Z)\bigr{]}.$

We highlight that $\Pi$ and $\Pi_{\mathbf{A}_{h}}$ in the two equations above
act on the coordinate $X$. Using Lemma 4.5, we check that

$\displaystyle\nabla_{X}\Phi_{2A}(X,X\mp Z)=2A(X\mp
Z)-2A(X)+2\widetilde{A}(X,X\mp Z)-2\widetilde{A}(X\mp Z,X)$

holds. Accordingly, $|\nabla_{X}\Phi_{2A}(X,X\mp Z)|\leqslant
C\,\|DA\|_{\infty}|Z|$, which implies

$\displaystyle\|[\Pi,\cos(Z\cdot\Pi_{\mathbf{A}_{h}})]\Psi\|_{2}\leqslant
C\,(|\mathbf{B}|+\|DA_{h}\|_{\infty})\,|Z|\,\|\Psi\|_{2}$ (4.117)

as well as (4.116). Finally, a computation similar to the one leading to
(4.69), Lemma 4.12, (4.114), (4.116), and the above considerations prove
(4.115). It remains to consider the term proportional to $\widetilde{\pi}$.

With an argument that is similar to the one leading to (4.72), we see that

$\displaystyle\|\widetilde{\pi}(\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta)\|_{2}^{2}$ $\displaystyle\leqslant
4\,\|\Psi\|_{2}^{2}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;\bigl{|}\widetilde{\pi}k_{T}(Z,r-s)\bigl{[}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}_{h}(X)\cdot(r+s)}-1\bigr{]}\bigr{|}\;|V\alpha_{*}(s)|\Bigr{|}^{2}.$

We also note that

$\displaystyle\bigl{|}\nabla_{r}(r-s)D\mathbf{A}(X)(r+s)\bigr{|}\leqslant
C\,\|D\mathbf{A}\|_{\infty}\bigl{(}|s|+|r-s|\bigr{)}.$

In combination with (4.113) and
$|\mathbf{A}_{\mathbf{B}}(r)|\leqslant|\mathbf{B}|\ (|r-s|+|s|)$, this implies

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;\bigl{|}\widetilde{\pi}k_{T}(Z,r-s)\bigl{[}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}(X)\cdot(r+s)}-1\bigr{]}\bigr{|}$ $\displaystyle\hskip
56.9055pt\leqslant
C\,(|\mathbf{B}|+\|DA\|_{\infty})\;\bigl{(}(F_{T}^{2}+G_{T}^{2})(r-s)+(F_{T}^{0}+G_{T}^{1})(r-s)\;|s|\bigr{)}$

with the function $F_{T}^{a}$ in (4.93) and $G_{T}^{a}$ in (4.94). When we
apply Young’s inequality and (4.95), we obtain (4.109). It remains to prove
(4.110).

To that end, we need to consider

$\displaystyle\langle\Delta,\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\rangle$
$\displaystyle=4\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}r\mathrm{d}s\;\bigl{(}\mathrm{e}^{-\mathrm{i}\frac{r-s}{4}\cdot
D\mathbf{A}_{h}(X)\cdot(r+s)}-1\bigr{)}V\alpha_{*}(r)V\alpha_{*}(s)$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}Z\;k_{T}(Z,r-s)\fint_{Q_{h}}\mathrm{d}X\;\overline{\Psi(X)}\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\Psi(X).$
(4.118)

The left side of this equation is real-valued. It therefore equals $1/2$ times
the right side plus $1/2$ times the complex conjugate of the right side. When
we use that $V\alpha_{*}$ is real-valued, that the Matsubara frequencies in
(3.10) satisfy $-\omega_{n}=\omega_{-(n+1)}$, and the transformation
$n\mapsto-n-1$ in the sum in the definition of $k_{T}(Z,r-s)$, we see that the
complex conjugate of the right side equals the same expression with
$\exp(-\mathrm{i}\frac{r-s}{4}\cdot D\mathbf{A}_{h}(X)(r+s))$ replaced by its
complex conjugate. Using this and the identity
$\cos(x)-1=-2\sin^{2}(\frac{x}{2})$ we find

$\displaystyle\langle\Delta,\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\rangle$
$\displaystyle=-8\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}r\mathrm{d}s\;\sin^{2}\left(\frac{(r-s)}{8}\cdot
D\mathbf{A}_{h}(X)\cdot(r+s)\right)$ $\displaystyle\times
V\alpha_{*}(r)V\alpha_{*}(s)\int_{\mathbb{R}^{3}}\mathrm{d}Z\;k_{T}(Z,r-s)\fint_{Q_{h}}\mathrm{d}X\;\overline{\Psi(X)}\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\Psi(X).$
(4.119)

Furthermore, (4.113) implies

$\displaystyle\sin^{2}\bigl{(}\frac{1}{8}(r-s)\cdot
D\mathbf{A}_{h}(X)(r+s)\bigr{)}\leqslant
C\,\|D\mathbf{A}_{h}\|_{\infty}^{2}\,|r-s|^{2}\,\bigl{(}|s|^{2}+|r-s|^{2}\bigr{)}.$

We use this bound, (4.119), and
$\|\cos(Z\cdot\Pi_{\mathbf{A}_{h}})\|_{\infty}\leqslant 1$ to see that

$\displaystyle|\langle\Delta,\widetilde{M}_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\rangle|$
$\displaystyle\leqslant\|D\mathbf{A}_{h}\|_{\infty}^{2}\;\|\Psi\|_{2}^{2}\;\bigl{\|}|V\alpha_{*}|\;\bigl{(}F_{T}^{4}*|V\alpha_{*}|+F_{T}^{2}*|\cdot|^{2}|V\alpha_{*}|\bigr{)}\bigr{\|}_{1}.$

The desired bound in (4.110) follows when we apply Young’s inequality and use
(2.7) as well as the $L^{1}$-norm estimate for $F_{T}^{a}$ in (4.95). This
completes the proof of Proposition 4.15. ∎

####  4.5.4 Analysis of $M_{T,\mathbf{A}}$ and calculation of two quadratic
terms in the Ginzburg–Landau functional

We decompose as
$M_{T,\mathbf{A}}=M_{T}^{(1)}+M_{T,\mathbf{A}}^{(2)}+M_{T,\mathbf{A}}^{(3)}$,
where

$\displaystyle M_{T}^{(1)}\alpha(X,r)$
$\displaystyle\coloneqq\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;\alpha(X,s),$
(4.120) $\displaystyle M_{T,\mathbf{A}}^{(2)}\alpha(X,r)$
$\displaystyle\coloneqq-\frac{1}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;(Z\cdot\Pi_{\mathbf{A}_{h}})^{2}\;\alpha(X,s),$
(4.121) $\displaystyle M_{T,\mathbf{A}}^{(3)}\alpha(X,r)$
$\displaystyle\coloneqq\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;\mathcal{R}(Z\cdot\Pi_{\mathbf{A}_{h}})\;\alpha(X,s),$
(4.122)

and $\mathcal{R}(x)=\cos(x)-1+\frac{1}{2}x^{2}$.

#####  4.5.4.1 The operator $M_{T}^{(1)}$.

From the quadratic form $\langle\Delta,M_{T}^{(1)}\Delta\rangle$ we extract
the quadratic term without external fields or a gradient in the
Ginzburg–Landau functional in (1.18). We also obtain a term that cancels the
last term on the left side of (3.6). The relevant computation can be found in
[16, Proposition 4.11]. For the sake of completeness, we state the result also
here. We recall that $\Delta\equiv\Delta_{\Psi}=-2V\alpha_{*}\Psi$ as in
(3.1).

######  Proposition 4.17.

Let $V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$, $\Psi\in
L_{\mathrm{mag}}^{2}(Q_{h})$, and $\Delta\equiv\Delta_{\Psi}$ as in (3.1).
Then the following statements hold.

  1. (a)

We have $M_{{T_{\mathrm{c}}}}^{(1)}\Delta(X,r)=-2\,\alpha_{*}(r)\Psi(X)$.

  2. (b)

For any $T_{0}>0$ there is a constant $c>0$ such that for $T_{0}\leqslant
T\leqslant{T_{\mathrm{c}}}$ we have

$\displaystyle\langle\Delta,M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\rangle\geqslant
c\,\frac{{T_{\mathrm{c}}}-T}{{T_{\mathrm{c}}}}\;\|\Psi\|_{2}^{2}.$

  3. (c)

Given $D\in\mathbb{R}$ there is $h_{0}>0$ such that for $0<h\leqslant h_{0}$,
and $T={T_{\mathrm{c}}}(1-Dh^{2})$ we have

$\displaystyle\langle\Delta,M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\rangle=4\;Dh^{2}\;\Lambda_{2}\;\|\Psi\|_{2}^{2}+R(\Delta)$

with the coefficient $\Lambda_{2}$ in (3.22), and

$\displaystyle|R(\Delta)|$ $\displaystyle\leqslant
C\;h^{6}\;\|V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

  4. (d)

Assume additionally that $|\cdot|V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$. Then,
there is $h_{0}>0$ such that for any $0<h\leqslant h_{0}$, any $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and any $T\geqslant T_{0}>0$ we have

$\displaystyle\|M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\|_{{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}$
$\displaystyle\leqslant
C\,h^{2}\,|T-{T_{\mathrm{c}}}|^{2}\,\bigl{(}\|V\alpha_{*}\|_{2}^{2}+\|\,|\cdot|V\alpha_{*}\|_{2}^{2}\bigr{)}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

######  Remark 4.18.

Parts (a) and (c) of the above proposition are needed for the proof of Theorem
3.6, part (b) is needed for the proof of Proposition 3.7, and part (d) for the
proof of Proposition 3.2.

#####  4.5.4.2 The operator $M_{T,\mathbf{A}}^{(2)}$.

The kinetic term in the Ginzburg–Landau functional in (1.18) is contained in
$\langle\Delta,M_{T,\mathbf{A}}^{(2)}\Delta\rangle$ with
$M_{T,\mathbf{A}}^{(2)}$ defined in (4.121). The following proposition allows
us to extract this term.

######  Proposition 4.19.

Let $V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ be a radial function, let $A\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be periodic, assume $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). We then have

$\displaystyle\langle\Delta,M_{{T_{\mathrm{c}}},\mathbf{A}}^{(2)}\Delta\rangle=-4\;\Lambda_{0}\;\|\Pi_{\mathbf{A}_{h}}\Psi\|_{2}^{2}$
(4.123)

with $\Lambda_{0}$ in (3.20). Moreover, for any $T\geqslant T_{0}>0$ we have

$\displaystyle|\langle\Delta,M_{T,\mathbf{A}}^{(2)}\Delta-
M_{{T_{\mathrm{c}}},\mathbf{A}}^{(2)}\Delta\rangle|\leqslant
C\;h^{4}\;|T-{T_{\mathrm{c}}}|\;\|V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.124)

###### Proof.

The proof is analogous to the proof of [16, Proposition 4.13] with the obvious
replacements, and is therefore omitted. ∎

#####  4.5.4.3 The operator $M_{T,\mathbf{A}}^{(3)}$.

The term $\langle\Delta,M_{T,\mathbf{A}}^{(3)}\Delta\rangle$ with
$M_{T,\mathbf{A}}^{(3)}$ in (4.122) does not contribute to the Ginzburg–Landau
energy. To obtain a bound for it, we need, as in [16], the
$H_{\mathrm{mag}}^{2}(Q_{h})$-norm of $\Psi$.

######  Proposition 4.20.

Let $V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$, let $A\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be periodic, assume that $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle|\langle\Delta,M_{T,\mathbf{A}}^{(3)}\Delta\rangle|$
$\displaystyle\leqslant
C\;h^{6}\;\|V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}.$

Before we give the proof of Proposition 4.20, we state and prove the following
lemma.

######  Lemma 4.21.

Assume that $\mathbf{A}=\mathbf{A}_{e_{3}}+A$ with a periodic function $A\in
W^{2,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$.

  1. (a)

For $\varepsilon>0$ we have

$\displaystyle|Z\cdot\Pi_{\mathbf{A}}|^{4}$ $\displaystyle\leqslant
C\;|Z|^{4}\;\bigl{[}\Pi_{\mathbf{A}}^{4}+\varepsilon\,\Pi_{\mathbf{A}}^{2}+\,|\operatorname{curl}\mathbf{A}|^{2}+\varepsilon^{-1}\,\bigl{(}|\operatorname{curl}(\operatorname{curl}\mathbf{A})|^{2}+|\nabla(\operatorname{curl}\mathbf{A})|^{2}\bigr{)}\bigr{]}.$
(4.125)

The gradient in the last term is understood to act on each component of
$\operatorname{curl}A$ separately. The result is a vector field with nine
components.

  2. (b)

Assume that $\Psi\in H_{\mathrm{mag}}^{2}(Q_{h})$. There is a constant
$h_{0}>0$ such that for any $0<h\leqslant h_{0}$, we have

$\displaystyle\langle\Psi,\,|Z\cdot\Pi_{\mathbf{A}_{h}}|^{4}\,\Psi\rangle$
$\displaystyle\leqslant
C\;h^{6}\;|Z|^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}.$

###### Proof.

We first give the proof of part (a) and start by noting that

$\displaystyle\bigl{[}\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(j)}\bigr{]}=-\mathrm{i}\sum_{k=1}^{3}\varepsilon_{ijk}\;(\operatorname{curl}\mathbf{A})_{k}$
(4.126)

with the Levi–Civita symbol $\varepsilon_{ijk}$, which is defined as $1$ if
$(i,j,k)$ is a cyclic permutation of $\\{1,2,3\\}$, as $-1$ if it is an
anticyclic permutation, and zero if at least two indices coincide. We claim
that

$\displaystyle\Pi_{\mathbf{A}}\;\Pi_{\mathbf{A}}^{2}\;\Pi_{\mathbf{A}}=\Pi_{\mathbf{A}}^{4}+2\,|\operatorname{curl}\mathbf{A}|^{2}-\operatorname{curl}(\operatorname{curl}\mathbf{A})\cdot\Pi_{\mathbf{A}}.$
(4.127)

If this holds, then we can use the fact that all terms in the above equation
except for the last are self-adjoint, to see that

$\displaystyle\bigl{[}\operatorname{curl}(\operatorname{curl}\mathbf{A}),\Pi_{\mathbf{A}}\bigr{]}=0.$
(4.128)

To prove (4.127), we first note that

$\displaystyle\Pi_{\mathbf{A}}\;\Pi_{\mathbf{A}}^{2}$
$\displaystyle=\Pi_{\mathbf{A}}^{2}\;\Pi_{\mathbf{A}}+2\sum_{i=1}^{3}[\Pi_{\mathbf{A}},\Pi_{\mathbf{A}}^{(i)}]\,\Pi_{\mathbf{A}}^{(i)}+\sum_{i=1}^{3}\bigl{[}\Pi_{\mathbf{A}}^{(i)},[\Pi_{\mathbf{A}},\Pi_{\mathbf{A}}^{(i)}]\bigr{]}.$
(4.129)

Moreover, an application of (4.126) shows that

$\displaystyle\sum_{i=1}^{3}[\Pi_{\mathbf{A}},\Pi_{\mathbf{A}}^{(i)}]\,\Pi_{\mathbf{A}}^{(i)}=\mathrm{i}(\operatorname{curl}\mathbf{A})\wedge\Pi_{\mathbf{A}}$
(4.130)

and

$\displaystyle\sum_{i=1}^{3}\bigl{[}\Pi_{\mathbf{A}}^{(i)},[\Pi_{\mathbf{A}},\Pi_{\mathbf{A}}^{(i)}]\bigr{]}=-\operatorname{curl}(\operatorname{curl}\mathbf{A}).$
(4.131)

We combine (4.129)-(4.131) and find

$\displaystyle\Pi_{\mathbf{A}}\;\Pi_{\mathbf{A}}^{2}\;\Pi_{\mathbf{A}}=\Pi_{\mathbf{A}}^{4}+2\mathrm{i}\bigl{(}(\operatorname{curl}\mathbf{A})\wedge\Pi_{\mathbf{A}}\bigr{)}\cdot\Pi_{\mathbf{A}}-\operatorname{curl}(\operatorname{curl}\mathbf{A})\cdot\Pi_{\mathbf{A}}$

Using additionally the identity
$\mathrm{i}((\operatorname{curl}\mathbf{A})\wedge\Pi_{\mathbf{A}})\cdot\Pi_{\mathbf{A}}=|\operatorname{curl}\mathbf{A}|^{2}$,
we conclude that (4.127) holds.

Our next goal is to prove the formula

$\displaystyle(Z\cdot\Pi_{\mathbf{A}})\,\Pi_{\mathbf{A}}^{2}\,(Z\cdot\Pi_{\mathbf{A}})$
$\displaystyle=\Pi_{\mathbf{A}}\,(Z\cdot\Pi_{\mathbf{A}})^{2}\,\Pi_{\mathbf{A}}+(Z\cdot\operatorname{curl}(\operatorname{curl}\mathbf{A}))\;(Z\cdot\Pi_{\mathbf{A}})$
$\displaystyle\hskip
100.0pt+(Z\wedge\Pi_{\mathbf{A}})\cdot\bigl{(}(Z\cdot\nabla)(\operatorname{curl}\mathbf{A})\bigr{)}.$
(4.132)

We note that

$\displaystyle(Z\cdot\Pi_{\mathbf{A}})\,\Pi_{\mathbf{A}}^{2}\,(Z\cdot\Pi_{\mathbf{A}})=\sum_{i,j=1}^{3}Z_{i}Z_{j}\;\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{2}\,\Pi_{\mathbf{A}}^{(j)}$
(4.133)

and

$\displaystyle\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{2}\,\Pi_{\mathbf{A}}^{(j)}=\sum_{m=1}^{3}\bigl{(}\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{(j)}\,\Pi_{\mathbf{A}}^{(m)}+\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(i)}\,[\Pi_{\mathbf{A}}^{(m)},\Pi_{\mathbf{A}}^{(j)}]+[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}]\,\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(j)}\bigr{)}.$

The sum in (4.133) is left unchanged when we exchange the indices $i$ and $j$.
Motivated by this, we combine $\frac{1}{2}$ times the original term and
$\frac{1}{2}$ times the term with $i$ and $j$ interchanged and find

$\displaystyle\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{2}\,\Pi_{\mathbf{A}}^{(j)}+\Pi_{\mathbf{A}}^{(j)}\,\Pi_{\mathbf{A}}^{2}\,\Pi_{\mathbf{A}}^{(i)}$
$\displaystyle=\sum_{m=1}^{3}\bigl{(}\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{(j)}\,\Pi_{\mathbf{A}}^{(m)}+\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(j)}\,\Pi_{\mathbf{A}}^{(i)}\,\Pi_{\mathbf{A}}^{(m)}$
$\displaystyle+\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(j)}\bigr{]}+\bigl{[}[\Pi_{\mathbf{A}}^{(j)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(i)}\bigr{]}\bigr{)}.$
(4.134)

Using the commutator identity $[A,BC]=B\,[A,C]+[A,B]\,C$ we write the third
term as

$\displaystyle\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(m)}\,\Pi_{\mathbf{A}}^{(j)}\bigr{]}$
$\displaystyle=\Pi_{\mathbf{A}}^{(m)}\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(j)}\bigr{]}+\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(m)}\bigr{]}\,\Pi_{\mathbf{A}}^{(j)}$
(4.135)

and likewise for the term with $i$ and $j$ interchanged. We also use (4.131)
to see that

$\displaystyle\frac{1}{2}\sum_{i,j=1}^{3}Z_{i}Z_{j}\Bigl{(}\bigl{[}[\Pi_{\mathbf{A}}^{(j)},\Pi_{\mathbf{A}}],\Pi_{\mathbf{A}}\bigr{]}\Pi_{\mathbf{A}}^{(i)}+\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}],\Pi_{\mathbf{A}}\bigr{]}\Pi_{\mathbf{A}}^{(j)}\Bigr{)}=(Z\cdot\operatorname{curl}(\operatorname{curl}\mathbf{A}))\;(Z\cdot\Pi_{\mathbf{A}})$

holds. Concerning the first term on the right side of (4.135), (4.126) can be
used to show

$\displaystyle\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}^{(m)}],\Pi_{\mathbf{A}}^{(j)}\bigr{]}=\sum_{k=1}^{3}\varepsilon_{imk}\,\partial_{j}(\operatorname{curl}\mathbf{A})_{k}),$

which implies

$\displaystyle\frac{1}{2}\sum_{i,j=1}^{3}Z_{i}Z_{j}\bigl{(}\Pi_{\mathbf{A}}\bigl{[}[\Pi_{\mathbf{A}}^{(i)},\Pi_{\mathbf{A}}],\Pi_{\mathbf{A}}^{(j)}\bigr{]}+\Pi_{\mathbf{A}}\bigl{[}[\Pi_{\mathbf{A}}^{(j)},\Pi_{\mathbf{A}}],\Pi_{\mathbf{A}}^{(i)}\bigr{]}\bigr{)}=(Z\wedge\Pi_{\mathbf{A}})\cdot\bigl{(}(Z\cdot\nabla)(\operatorname{curl}\mathbf{A})\bigr{)}.$
(4.136)

When we combine (4.133)-(4.136), this proves proves (4.132). We are now
prepared to give the proof of (4.125).

We start by noting that $|A+B+C|^{2}\leqslant 3(|A|^{2}+|B|^{2}+|C|^{2})$
holds for three linear operators $A,B,C$, which implies

$\displaystyle(Z\cdot\Pi_{\mathbf{A}})^{2}$ $\displaystyle\leqslant
3\;\bigl{(}Z_{1}^{2}\;(\Pi_{\mathbf{A}}^{(1)})^{2}+Z_{2}^{2}\;(\Pi_{\mathbf{A}}^{(2)})^{2}+Z_{3}^{2}\;(\Pi_{\mathbf{A}}^{(3)})^{2}\bigr{)}\leqslant
3\;|Z|^{2}\;\Pi_{\mathbf{A}}^{2}.$ (4.137)

We use (4.127), (4.132), and (4.137) to show

$\displaystyle(Z\cdot\Pi_{\mathbf{A}})\,\Pi_{\mathbf{A}}^{2}\,(Z\cdot\Pi_{\mathbf{A}})$
$\displaystyle\leqslant
3\,|Z|^{2}\,\bigl{(}\Pi_{\mathbf{A}}^{4}+2\,|\operatorname{curl}\mathbf{A}|^{2}-\operatorname{curl}(\operatorname{curl}\mathbf{A})\cdot\Pi_{\mathbf{A}}\bigr{)}$
$\displaystyle\hskip
10.0pt+(Z\cdot\operatorname{curl}(\operatorname{curl}\mathbf{A}))\,(Z\cdot\Pi_{\mathbf{A}})+(Z\wedge\Pi_{\mathbf{A}})\cdot\bigl{(}(Z\cdot\nabla)(\operatorname{curl}\mathbf{A})\bigr{)}.$

Next, we write
$|Z\cdot\Pi_{\mathbf{A}}|^{4}=(Z\cdot\Pi_{\mathbf{A}})(Z\cdot\Pi_{\mathbf{A}})^{2}(Z\cdot\Pi_{\mathbf{A}})$,
apply (4.137) to the term in the middle, and find

$\displaystyle|Z\cdot\Pi_{\mathbf{A}}|^{4}$ $\displaystyle\leqslant
9\,|Z|^{4}\bigl{(}\Pi_{\mathbf{A}}^{4}+2\,|\operatorname{curl}\mathbf{A}|^{2}-\operatorname{curl}(\operatorname{curl}\mathbf{A})\cdot\Pi_{\mathbf{A}}\bigr{)}$
$\displaystyle\hskip
50.0pt+3\,|Z|^{2}\bigl{[}(Z\cdot\operatorname{curl}(\operatorname{curl}\mathbf{A}))\,(Z\cdot\Pi_{\mathbf{A}})+(Z\wedge\Pi_{\mathbf{A}})\cdot\bigl{(}(Z\cdot\nabla)(\operatorname{curl}\mathbf{A})\bigr{)}\bigr{]}.$

Moreover, $AB+BA\leqslant\varepsilon\,A^{2}+\frac{1}{4\varepsilon}\,B^{2}$ for
$\varepsilon>0$ and self-adjoint operators $A$ and $B$, (4.137), and
$|Z\wedge\Pi_{\mathbf{A}}|^{2}\leqslant 3|Z|^{2}\Pi_{\mathbf{A}}^{2}$ imply
that the right side of the above equation is bounded by

$\displaystyle
C\,|Z|^{4}\bigl{[}\Pi_{\mathbf{A}}^{4}+\,|\operatorname{curl}\mathbf{A}|^{2}+\varepsilon\,\Pi_{\mathbf{A}}^{2}+\varepsilon^{-1}\bigl{(}|\operatorname{curl}(\operatorname{curl}\mathbf{A})|^{2}+|\nabla(\operatorname{curl}\mathbf{A})|^{2}\bigr{)}\bigr{]}.$

This proves part (a).

The proof of part (b) is a direct consequence of part (a) with the choice
$\varepsilon=h^{2}$, and is therefore left to the reader. This proves Lemma
4.21. ∎

###### Proof of Proposition 4.20.

We use the definition of $M_{T,\mathbf{A}}^{(3)}$ to write

$\displaystyle\langle\Delta,M_{T,\mathbf{A}}^{(3)}\Delta\rangle$
$\displaystyle=4\iiint_{\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}r\mathrm{d}s\mathrm{d}Z\;V\alpha_{*}(r)V\alpha_{*}(s)\,k_{T}(Z,r-s)\;\langle\Psi,\mathcal{R}(Z\cdot\Pi_{\mathbf{A}_{h}})\Psi\rangle.$
(4.138)

The function $\mathcal{R}(x)=\cos(x)-1+\frac{x^{2}}{2}$ satisfies the bound
$0\leqslant\mathcal{R}(x)\leqslant\frac{1}{24}x^{4}$, and hence an application
of Lemma 4.21 shows

$\displaystyle\langle\Psi,\mathcal{R}(Z\cdot\Pi_{\mathbf{A}_{h}})\Psi\rangle$
$\displaystyle\leqslant
C\;h^{6}\;|Z|^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}.$ (4.139)

When we apply the estimate
$|Z|^{4}\leqslant|Z+\frac{r}{2}|^{4}+|Z-\frac{r}{2}|^{4}$, we see that

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|Z|^{4}\;|k_{T}(Z,r)|$
$\displaystyle\leqslant F_{T}^{4}(r)$ (4.140)

holds with $F_{T}^{4}$ defined in (4.93). But this also shows

$|\langle\Delta,M_{T,\mathbf{A}}^{(3)}\Delta\rangle|\leqslant
C\;h^{6}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}\;\|(V\alpha_{*})F_{T}^{4}\ast(V\alpha_{*})\|_{1}.$

In combination with the $L^{1}(\mathbb{R}^{3})$-norm bound for $F_{T}^{4}$ in
(4.95), this proves the claim. ∎

####  4.5.5 A representation formula for the operator
$\mathcal{W}_{T,\mathbf{A}}$

In the next five subsections we study the operator
$\mathcal{W}_{T,\mathbf{A}}$ in (4.43). In particular, we extract the term in
the GL functional that is proportional to $W$ from
$\langle\Delta,\mathcal{W}_{T,\mathbf{A}}\Delta\rangle$. The operator
$\mathcal{W}_{T,0}$ has previously been studied in [18]. After the magnetic
field has been removed, our analysis mostly follows ideas in this reference.
Because of this and because several ideas of the previous sections appear
again, we keep our presentation rather short and only mention the main ideas.
As in the case of $L_{T,\mathbf{A}}$, we start our analysis with a
representation formula for $\mathcal{W}_{T,\mathbf{A}}$ in terms of relative
and center-of-mass coordinates.

######  Lemma 4.22.

The operator
$\mathcal{W}_{T,\mathbf{A}}\colon{L^{2}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}\rightarrow{L^{2}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}$
in (4.43) acts as

$\displaystyle\mathcal{W}_{T,\mathbf{A}}\alpha(X,r)$
$\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T,\mathbf{A},W}(X,Z,r,s)\;(\mathrm{e}^{\mathrm{i}Z\cdot(-\mathrm{i}\nabla_{X})}\alpha)(X,s),$
(4.141)

where

$\displaystyle k_{T,\mathbf{A},W}(X,Z,r,s)$
$\displaystyle\coloneqq\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\mathrm{d}Y}\;W_{h}(X+Y)\bigl{[}k_{T,\mathbf{A},+}^{n}(X,Y,Z,r,s)\;\mathrm{e}^{\mathrm{i}\Theta_{\mathbf{A}_{h}}^{+}(Y,Z,r,s)}$
$\displaystyle\hskip
80.0pt+k_{T,\mathbf{A},-}^{n}(X,Y,Z,r,s)\;\mathrm{e}^{\mathrm{i}\Theta_{\mathbf{A}_{h}}^{-}(Y,Z,r,s)}\bigr{]},$
(4.142)

where

$\displaystyle k_{T,\mathbf{A},\pm}^{n}(X,Y,Z,r,s)$ $\displaystyle\coloneqq
g_{h}^{\pm\mathrm{i}\omega_{n}}\bigl{(}X\pm\frac{r}{2},X+Y\bigr{)}\;g_{h}^{\pm\mathrm{i}\omega_{n}}\bigl{(}X+Y,X+Z\pm\frac{s}{2}\bigr{)}$
$\displaystyle\hskip 120.0pt\times
g_{h}^{\mp\mathrm{i}\omega_{n}}\bigl{(}X\mp\frac{r}{2},X+Z\mp\frac{s}{2}\bigr{)}.$
(4.143)

The function $g_{h}^{z}$ is defined in (4.3) and

$\displaystyle\Theta_{\mathbf{A}}^{\pm}(Y,Z,r,s)$
$\displaystyle\coloneqq\Phi_{\mathbf{A}}\bigl{(}X\pm\frac{r}{2},X+Y\bigr{)}+\Phi_{\mathbf{A}}\bigl{(}X+Y,X+Z\pm\frac{s}{2}\bigr{)}$
$\displaystyle\hskip
120.0pt+\Phi_{\mathbf{A}}\bigl{(}X\mp\frac{r}{2},X+Z\mp\frac{s}{2}\bigr{)}.$
(4.144)

###### Proof.

The proof that $\mathcal{W}_{T,\mathbf{A}}$ is a bounded linear map on
${L^{2}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}$ goes along the same lines as
that of Lemma 3.4. The proof of the representation formula is analogous to the
proof of Lemma 4.7. We use (4.40) for $W=0$ to write

$\displaystyle\mathcal{W}_{T,\mathbf{A}}\alpha(x,y)$
$\displaystyle=\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\iiint_{\mathbb{R}^{9}}\mathrm{d}u\mathrm{d}v\mathrm{d}w\;\bigl{[}G_{h}^{\mathrm{i}\omega_{n}}(x,u)\,W_{h}(u)\,G_{h}^{\mathrm{i}\omega_{n}}(u,v)\,\alpha(v,w)\,G_{h}^{-\mathrm{i}\omega_{n}}(y,w)$
$\displaystyle\hskip
60.0pt+G_{\mathbf{A}_{h}}^{\mathrm{i}\omega_{n}}(x,v)\,\alpha(v,w)\,G_{h}^{-\mathrm{i}\omega_{n}}(u,w)\,W_{h}(u)\,G_{h}^{-\mathrm{i}\omega_{n}}(y,u)\bigr{]}.$
(4.145)

When we define the coordinates $X=\frac{x+y}{2}$ and $r=x-y$, apply the change
of variables

$\displaystyle u$ $\displaystyle=X+Y,$ $\displaystyle v$
$\displaystyle=X+Z+\frac{s}{2},$ $\displaystyle w$
$\displaystyle=X+Z-\frac{s}{2},$

and use (4.3), this yields (4.141). We highlight that, by a slight abuse of
notation, we denoted the function $\alpha$ depending on the original
coordinates in (4.145) and the function depending on relative and center-of-
mass coordinates in (4.141) by the same symbol. ∎

####  4.5.6 Approximation of the operator $\mathcal{W}_{T,\mathbf{A}}$

The operator $\mathcal{W}_{T,\mathbf{A}}$ will be analyzed in three steps.
More precisely, we write

$\displaystyle\mathcal{W}_{T,\mathbf{A}}=\bigl{(}\mathcal{W}_{T,\mathbf{A}}-\widetilde{\mathcal{W}}_{T,\mathbf{A}}\bigr{)}+\bigl{(}\widetilde{\mathcal{W}}_{T,\mathbf{A}}-\mathcal{W}_{T}\bigr{)}+\mathcal{W}_{T},$
(4.146)

where $\widetilde{\mathcal{W}}_{T,\mathbf{A}}$ and $\mathcal{W}_{T}$ are
operators of increasing simplicity in their dependence on $W$ and
$\mathbf{A}$. They are defined below in (4.147) and (4.150), respectively. The
term in the Ginzburg–Landau functional that is proportional to $W$ will be
extracted from the expectation of the operator $\mathcal{W}_{T}$ with respect
to $\Delta$. The expectation of the first two terms in (4.146) will be shown
to be negligible.

#####  4.5.6.1 The operator $\widetilde{\mathcal{W}}_{T,\mathbf{A}}$.

We define the operator

$\displaystyle\widetilde{\mathcal{W}}_{T,A}\alpha(X,r)$
$\displaystyle\coloneqq
W_{h}(X)\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T,\mathbf{A},0}(X,Z,r,s)\;(\mathrm{e}^{\mathrm{i}Z\cdot(-\mathrm{i}\nabla_{X})}\alpha)(X,s),$
(4.147)

where $k_{T,\mathbf{A},W}$ is defined in (4.142). The following proposition
allows us to estimate the expectation of the first term in (4.146) with
respect to $\Delta$.

######  Proposition 4.23.

Let $|\cdot|^{k}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1\\}$, let
$A\in W^{3,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $W\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R})$ be periodic, assume $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle|\langle\Delta,\mathcal{W}_{T,\mathbf{A}}\Delta-\widetilde{\mathcal{W}}_{T,\mathbf{A}}\Delta\rangle|\leqslant
C\,h^{5}\,\max_{k=0,1}\|\,|\cdot|^{k}\
V\alpha_{*}\|_{2}^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

###### Proof.

We write

$\displaystyle|\langle\Delta,\mathcal{W}_{T,\mathbf{A}}\Delta-\widetilde{\mathcal{W}}_{T,\mathbf{A}}\Delta\rangle|$
$\displaystyle\leqslant
4\,\|\Psi\|_{2}^{2}\iiint_{\mathbb{R}^{9}}\mathrm{d}Z\mathrm{d}r\mathrm{d}s\;\operatorname*{ess\,sup}_{X\in\mathbb{R}^{3}}|(k_{T,\mathbf{A},W}-k_{T,\mathbf{A},0})(X,Z,r,s)|\;|V\alpha_{*}(r)|\,|V\alpha_{*}(s)|$
(4.148)

and note that

$\displaystyle|(k_{T,\mathbf{A},W}-k_{T,\mathbf{A},W})(X,Z,r,s)|$
$\displaystyle\leqslant\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\mathrm{d}Y\;|W_{h}(X+Y)-W_{h}(X)|\;\bigl{(}|k_{T,\mathbf{A},+}^{n}|+|k_{T,\mathbf{A},-}^{n}|\bigr{)}(X,Y,Z,r,s).$
(4.149)

When we use (4.149), $|W(X+Y)-W(X)|\leqslant\|\nabla W\|_{\infty}\,|Y|$,
$|Y|\leqslant|Y\pm\frac{r}{2}|+|\frac{r}{2}|$, and Proposition 4.2, we see
that the right side of (4.148) is bounded by a constant times

$\displaystyle\|\Psi\|_{2}^{2}\;\|\nabla
W_{h}\|_{\infty}\;\|(1+|\cdot|)(V\alpha_{*})\widetilde{F}_{T}^{1}\ast|V\alpha_{*}|\
\|_{1},$

where

$\displaystyle\widetilde{F}_{T}^{a}\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\sum_{b=0}^{a}\left[\bigl{(}|\cdot|^{b}\,\rho^{\mathrm{i}\omega_{n}}\bigr{)}*\rho^{\mathrm{i}\omega_{n}}*\rho^{-\mathrm{i}\omega_{n}}+\bigl{(}|\cdot|^{b}\,\rho^{-\mathrm{i}\omega_{n}}\bigr{)}*\rho^{-\mathrm{i}\omega_{n}}*\rho^{\mathrm{i}\omega_{n}}\right].$

With Proposition 4.2 and (4.56) we show that
$\|\widetilde{F}_{T}^{a}\|_{1}\leqslant C$. In combination with the bound
$\|\nabla W_{h}\|_{\infty}\leqslant Ch^{3}$, this proves the claim. ∎

#####  4.5.6.2 The operator $\mathcal{W}_{T}$.

We define the operator $\mathcal{W}_{T}$ by

$\displaystyle\mathcal{W}_{T}\alpha(X,r)$ $\displaystyle\coloneqq
W_{h}(X)\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;\alpha(X,s)$
(4.150)

where

$\displaystyle k_{T}(Z,r)$
$\displaystyle\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\bigl{(}k_{T,+}^{n}(Z,r)+k_{T,-}^{n}(Z,r)\bigr{)}$
(4.151)

and

$\displaystyle k_{T,\pm}^{n}(Z,r)$
$\displaystyle\coloneqq(g_{0}^{\pm\mathrm{i}\omega_{n}}*g_{0}^{\pm\mathrm{i}\omega_{n}})\bigl{(}Z\mp\frac{r}{2}\bigr{)}\;g_{0}^{\mp\mathrm{i}\omega_{n}}\bigl{(}Z\pm\frac{r}{2}\bigr{)}.$
(4.152)

######  Proposition 4.24.

Let $|\cdot|^{k}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1\\}$, let
$A\in W^{3,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $W\in
L^{\infty}(\mathbb{R}^{3},\mathbb{R})$ be periodic, assume $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle|\langle\Delta,\widetilde{\mathcal{W}}_{T,\mathbf{A}}\Delta-\mathcal{W}_{T}\Delta\rangle|\leqslant
C\,h^{5}\,\max_{k=0,1}\|\,|\cdot|^{k}\
V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

###### Sketch of proof.

The proof goes along the same lines as that of Propositions 4.9, 4.11, and
4.15 with the notable simplification that we only need to prove bounds for the
quadratic form. We therefore only mention the main steps that need to be
carried out and leave the details to the reader. In the first step $g_{h}^{z}$
is replaced by $g_{0}^{z}$. In the second step the part of the magnetic phase
coming from $\mathbf{A}_{\mathbf{B}}$ is split off. A careful analysis shows
that

$\displaystyle\Theta_{\mathbf{A}_{\mathbf{B}}}^{\pm}(X,Y,Z,r,s)=Z\cdot(\mathbf{B}\wedge
X)+\frac{\mathbf{B}}{2}\cdot\theta_{\pm}(Y,Z,r,s),$

where

$\displaystyle\theta_{\pm}(Y,Z,r,s)$
$\displaystyle\coloneqq\pm\frac{r}{2}\wedge\bigl{(}Y\mp\frac{r}{2}\bigr{)}+\bigl{(}Y\mp\frac{r}{2}\bigr{)}\wedge\bigl{(}Z-Y\pm\frac{s}{2}\bigr{)}$
$\displaystyle\hskip
60.0pt\pm\frac{r}{2}\wedge\bigl{(}Z-Y\pm\frac{s}{2}\bigr{)}\mp\frac{r}{2}\wedge\bigl{(}Z\pm\frac{r-s}{2}\bigr{)}.$
(4.153)

The phase $\exp(\mathrm{i}Z\cdot(\mathbf{B}\wedge X))$ and
$\exp(\mathrm{i}Z\cdot(-\mathrm{i}\nabla_{X}))$ are combined and give
$\exp(\mathrm{i}Z\cdot\Pi)$, see (4.90). In the third step the magnetic phases
coming from $\theta_{\pm}$ and from the periodic vector potential $A$, that
is, from $\Theta_{A}$, are removed. Afterwards, the emergent symmetry of the
integrand under the transformation $Z\rightarrow-Z$ is used to replace the
operator $\exp(\mathrm{i}Z\cdot\Pi)$ by $\cos(Z\cdot\Pi)$. In the final step,
we apply the estimate $1-\cos(Z\cdot\Pi)\leqslant C|Z|^{2}\,\Pi^{2}$. This
ends our sketch of proof. ∎

####  4.5.7 Analysis of $\mathcal{W}_{T}$ and calculation of the quadratic
$W$-term

######  Proposition 4.25.

Let $V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$, let $W\in
L^{\infty}(\mathbb{R}^{3},\mathbb{R})$ be periodic, assume that $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). There is $h_{0}>0$ such that for any $0<h\leqslant h_{0}$ we have

$\displaystyle\langle\Delta,\mathcal{W}_{T_{\mathrm{c}}}\Delta\rangle=-4\;\Lambda_{1}\;\langle\Psi,W_{h}\Psi\rangle$
(4.154)

with $\Lambda_{1}$ in (3.21). Moreover, for any $T\geqslant T_{0}>0$ we have

$\displaystyle|\langle\Delta,\mathcal{W}_{T}\Delta-\mathcal{W}_{T_{\mathrm{c}}}\Delta\rangle|\leqslant
C\;h^{4}\;|T-{T_{\mathrm{c}}}|\;\|V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.155)

###### Proof.

Using the definition of $k_{T}(Z,r)$ in (4.151), we write

$\displaystyle k_{T}(Z,r)$
$\displaystyle=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}q}{(2\pi)^{3}}\;\Bigl{[}\frac{\mathrm{e}^{\mathrm{i}Z\cdot(p+q)}\mathrm{e}^{\mathrm{i}\frac{r}{2}\cdot(q-p)}}{(\mathrm{i}\omega_{n}+\mu-p^{2})^{2}(\mathrm{i}\omega_{n}-\mu+p^{2})}$
$\displaystyle\hskip
150.0pt-\frac{\mathrm{e}^{\mathrm{i}Z\cdot(p+q)}\mathrm{e}^{\mathrm{i}\frac{r}{2}\cdot(p-q)}}{(\mathrm{i}\omega_{n}-\mu+p^{2})^{2}(\mathrm{i}\omega_{n}+\mu-p^{2})}\Bigr{]}$

as well as

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;k_{T}(Z,r)$
$\displaystyle=-\frac{4}{\beta}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;\mathrm{e}^{\mathrm{i}r\cdot
p}\,\frac{p^{2}-\mu}{(\mathrm{i}\omega_{n}+\mu-p^{2})^{2}(\mathrm{i}\omega_{n}-\mu+p^{2})^{2}}.$

With the Mittag-Leffler series expansion in (4.35), we also check that

$\displaystyle\frac{\beta}{2}\frac{1}{\cosh^{2}(\frac{\beta}{2}z)}=\frac{\mathrm{d}}{\mathrm{d}z}\tanh\bigl{(}\frac{\beta}{2}z\bigr{)}=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{(\mathrm{i}\omega_{n}-z)^{2}}$
(4.156)

holds. We use (4.156) and the partial fraction expansion

$\displaystyle\frac{1}{(\mathrm{i}\omega_{n}-E)^{2}(\mathrm{i}\omega_{n}+E)^{2}}=\frac{1}{4E^{2}}\Bigl{[}\frac{1}{(\mathrm{i}\omega_{n}-E)^{2}}+\frac{1}{(\mathrm{i}\omega_{n}+E)^{2}}\Bigr{]}-\frac{1}{4E^{3}}\Bigl{[}\frac{1}{\mathrm{i}\omega_{n}-E}-\frac{1}{\mathrm{i}\omega_{n}+E}\Bigr{]}$

to see that

$\displaystyle\frac{4}{\beta}\sum_{n\in\mathbb{Z}}\frac{E}{(\mathrm{i}\omega_{n}-E)^{2}(\mathrm{i}\omega_{n}+E)^{2}}=\beta^{2}\;g_{1}(\beta
E)$

with the function $g_{1}$ in (3.19). Therefore,

$\displaystyle\langle\Delta,M_{T_{\mathrm{c}}}^{W}\Delta\rangle$
$\displaystyle=-{\beta_{\mathrm{c}}}^{2}\int_{\mathbb{R}^{3}}\frac{\mathrm{d}p}{(2\pi)^{3}}\;|(-2)\widehat{V\alpha_{*}}(p)|^{2}\,g_{1}({\beta_{\mathrm{c}}}(p^{2}-\mu))\;\langle\Psi,W_{h}\Psi\rangle$
$\displaystyle=-4\,\Lambda_{1}\,\langle\Psi,W_{h}\Psi\rangle.$

This proves (4.154).

To obtain the bound in (4.155), we argue as in the proof of (4.124). ∎

####  4.5.8 Summary: The quadratic terms

In this section, we summarize our results concerning the quadratic terms (in
$\Delta$) that are relevant for the proof of Theorem 3.6. We also use our
results to prove another statement (Proposition 4.26 below), which will later
be used in the proof of Proposition 3.7. We start by summarizing our findings.

Let the assumptions of Theorem 3.6 hold and recall the definition of
$\mathcal{R}_{T,\mathbf{A},W}^{(2)}$ in (4.44). An application of Hölder’s
inequality in (2.1) and the bound
$\|(\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}})^{-1}\|_{\infty}\leqslant|\omega_{n}|^{-1}$
show that

$|\langle\Delta,\mathcal{R}_{T,\mathbf{A},W}^{(2)}\Delta\rangle|\leqslant
C\;\|\Delta\|_{2}^{2}\;\|W_{h}\|^{2}_{\infty}\leqslant
C\;h^{4}\;\|\Psi\|_{2}^{2}\leqslant
C\;h^{6}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}.$

We combine (4.42), this bound, and the results of Propositions 4.17, 4.19,
4.20, 4.23, 4.24, and 4.25, to see that for $T={T_{\mathrm{c}}}(1-Dh^{2})$
with $D\in\mathbb{R}$ the identity

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A},W}\Delta\rangle+\|\Psi\|_{2}^{2}\,\langle\alpha_{*},V\alpha_{*}\rangle$
$\displaystyle=\Lambda_{0}\;\|\Pi_{\mathbf{A}_{h}}\Psi\|_{2}^{2}+\Lambda_{1}\;\langle\Psi,W_{h}\Psi\rangle-
Dh^{2}\;\Lambda_{2}\;\|\Psi\|_{2}^{2}+R_{2}(\Delta)$ (4.157)

holds. The remainder term $R_{2}(\Delta)$ obeys the estimate

$\displaystyle|R_{2}(\Delta)|\leqslant
C\;\bigl{(}h^{5}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}+h^{6}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}\bigr{)}.$

This concludes the computation of the quadratic terms in the Ginzburg–Landau
functional. It remains to compute the term that is proportional to
$|\Psi|^{4}$, which is the content of the remaining part of Section 4.5.

Before we continue with the proof of Theorem 3.6, we state and prove the
following statement, which will later be used in the proof of Proposition 3.7.
It is a straightforward consequence of our results for the quadratic terms,
and we therefore prove it here.

######  Proposition 4.26.

Let $|\cdot|^{k}V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1,2\\}$,
let $A\in W^{3,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $W\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R})$ be periodic, assume $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A},W}\Delta\rangle+\|\Psi\|_{2}^{2}\;\langle\alpha_{*},V\alpha_{*}\rangle$
$\displaystyle\leqslant
c\,\frac{T-{T_{\mathrm{c}}}}{{T_{\mathrm{c}}}}\,\|\Psi\|_{2}^{2}+Ch^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$
(4.158)

###### Proof.

We write

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A},W}\Delta\rangle=-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A}}\Delta\rangle-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A},W}\Delta-
L_{T,\mathbf{A}}\Delta\rangle$ (4.159)

and use the resolvent identity in (4.36) to write one of the operators on the
right side as

$\displaystyle L_{T,\mathbf{A},W}\Delta-L_{T,\mathbf{A}}\Delta$
$\displaystyle=-\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}W_{h}\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\Delta\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}$
$\displaystyle\hskip
80.0pt-\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\Delta\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}W_{h}\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A}}}}.$
(4.160)

An application of Hölder’s inequality therefore implies the bound

$\displaystyle|\langle\Delta,L_{T,\mathbf{A},W}\Delta-
L_{T,\mathbf{A}}\Delta\rangle|\leqslant
C\,h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

When we additionally use the decomposition of $L_{T,\mathbf{A}}$ in (4.49),
Propositions 4.9, 4.11, and 4.15, we find

$\displaystyle-\frac{1}{4}\langle\Delta,L_{T,\mathbf{A}}\Delta\rangle+\|\Psi\|_{2}^{2}\;\langle\alpha_{*},V\alpha_{*}\rangle$
$\displaystyle=-\frac{1}{4}\langle\Delta,M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\rangle-\frac{1}{4}\langle\Delta,M_{T,\mathbf{A}}\Delta-
M_{T}^{(1)}\Delta\rangle+R_{1}(\Delta),$ (4.161)

with a remainder $R_{1}(\Delta)$ obeying the bound

$\displaystyle|R_{1}(\Delta)|\leqslant
C\;h^{5}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

From Proposition 4.17 we know that

$\displaystyle-\frac{1}{4}\langle\Delta,M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\rangle\leqslant
c\;\frac{T-{T_{\mathrm{c}}}}{{T_{\mathrm{c}}}}\;\|\Psi\|_{2}^{2}.$

We also claim that the bound

$\displaystyle|\langle\Delta,M_{T,\mathbf{A}}\Delta-M_{T}^{(1)}\Delta\rangle|$
$\displaystyle\leqslant
C\;h^{4}\;\|V\alpha_{*}\|_{2}^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$
(4.162)

holds. Its proof can be achieved with the same methods that have been used to
prove Proposition 4.15. The main point is that we have to use the bound

$\displaystyle|\langle\Psi,[\cos(Z\cdot\Pi_{\mathbf{A}})-1]\Psi\rangle|$
$\displaystyle\leqslant
C\;h^{4}\;|Z|^{2}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2},$ (4.163)

as well as the operator inequality in (4.137) for
$(Z\cdot\Pi_{\mathbf{A}})^{2}$. Since no additional difficulties occur, we
leave the details to the reader. This completes the proof of (4.158). ∎

####  4.5.9 A representation formula for the operator $N_{T,\mathbf{A}}$

In this and the following sections we investigate the nonlinear operator

$\displaystyle N_{T,\mathbf{A}}\coloneqq N_{T,\mathbf{A},0}$ (4.164)

with $N_{T,\mathbf{A},W}$ in (3.12). In particular, we show that the quartic
term in the GL functional emerges from
$\langle\Delta,N_{T,\mathbf{A}}(\Delta)\rangle$. In Section 4.5.12 we show
that
$\langle\Delta,N_{T,\mathbf{A}}(\Delta)-N_{T,\mathbf{A},W}(\Delta)\rangle$
yields a negligible contribution. In this section we will also collect all
previous results and finish the proof of Theorem 3.6.

Before we state a representation formula for $N_{T,\mathbf{A}}$ in terms of
relative and center-of-mass coordinates, we introduce the notation
$\mathbf{Z}$ to denote the vector $(Z_{1},Z_{2},Z_{3})$ with
$Z_{1},Z_{2},Z_{3}\in\mathbb{R}^{3}$ as well as
$\mathrm{d}\mathbf{Z}=\mathrm{d}Z_{1}\mathrm{d}Z_{2}\mathrm{d}Z_{3}$.

######  Lemma 4.27.

The operator $N_{T,\mathbf{A}}$ in (3.12) acts as

$\displaystyle N_{T,\mathbf{A}}(\alpha)(X,r)$
$\displaystyle=\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;\ell_{T,\mathbf{A}}(X,\mathbf{Z},r,\mathbf{s})\;\mathcal{A}(X,\mathbf{Z},\mathbf{s}),$

where

$\displaystyle\mathcal{A}(X,\mathbf{Z},\mathbf{s})$
$\displaystyle\coloneqq\mathrm{e}^{\mathrm{i}Z_{1}\cdot(-\mathrm{i}\nabla_{X})}\alpha(X,s_{1})\;\overline{\mathrm{e}^{\mathrm{i}Z_{2}\cdot(-\mathrm{i}\nabla_{X})}\alpha(X,s_{2})}\;\mathrm{e}^{\mathrm{i}Z_{3}\cdot(-\mathrm{i}\nabla_{X})}\alpha(X,s_{3})$
(4.165)

and

$\displaystyle\ell_{T,\mathbf{A}}(X,\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\ell_{T,\mathbf{A}}^{n}(X,\mathbf{Z},r,\mathbf{s})\;\mathrm{e}^{\mathrm{i}\Upsilon_{\mathbf{A}_{h}}(X,\mathbf{Z},r,s)}.$
(4.166)

Here,

$\displaystyle\ell_{T,\mathbf{A}}^{n}(X,\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq
g_{h}^{\mathrm{i}\omega_{n}}\bigl{(}X+\frac{r}{2}\,,\,X+Z_{1}+\frac{s_{1}}{2}\bigr{)}\,g_{h}^{-\mathrm{i}\omega_{n}}\bigl{(}X+Z_{2}+\frac{s_{2}}{2}\,,\,X+Z_{1}-\frac{s_{1}}{2}\bigr{)}$
$\displaystyle\times
g_{h}^{\mathrm{i}\omega_{n}}\bigl{(}X+Z_{2}-\frac{s_{2}}{2}\,,\,X+Z_{3}+\frac{s_{3}}{2}\bigr{)}\,g_{h}^{-\mathrm{i}\omega_{n}}\bigl{(}X-\frac{r}{2}\,,\,X+Z_{3}-\frac{s_{3}}{2}\bigr{)},$
(4.167)

with $g_{h}^{z}$ in (4.3) and

$\displaystyle\Upsilon_{\mathbf{A}}(X,\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq\Phi_{\mathbf{A}}\bigl{(}X+\frac{r}{2}\,,\,X+Z_{1}+\frac{s_{1}}{2}\bigr{)}+\Phi_{\mathbf{A}}\bigl{(}X+Z_{2}+\frac{s_{2}}{2}\,,\,X+Z_{1}-\frac{s_{1}}{2}\bigr{)}$
$\displaystyle+\Phi_{\mathbf{A}}\bigl{(}X+Z_{2}-\frac{s_{2}}{2}\,,\,X+Z_{3}+\frac{s_{3}}{2}\bigr{)}+\Phi_{\mathbf{A}}\bigl{(}X-\frac{r}{2}\,,\,X+Z_{3}-\frac{s_{3}}{2}\bigr{)}.$
(4.168)

######  Remark 4.28.

The above representation formula for $N_{T,\mathbf{A}}$ should be compared to
that in the case of a constant magnetic field in [16, Lemma 4.16] and to the
representation formula for $L_{T,\mathbf{A}}$ in 4.7. The following two
properties are relevant for us: (a) The functions $\alpha$ are multiplied by
translation operators that can later be completed with appropriate phase
factors to give magnetic translation operators. (b) The coordinates appearing
in $\Upsilon_{\mathbf{A}}$ in (4.168) equal those in the different factors in
the definition of $\ell_{T,\mathbf{A}}^{n}(X,\mathbf{Z},r,\mathbf{s})$ in
(4.167). When proving bounds, this allows us to find a similar structure of
nested convolutions as the one we already encountered in the analysis of
$L_{T,\mathbf{A}}$. The center-of-mass part of $\alpha$ never participates in
these convolutions.

###### Proof of Lemma 4.27.

When we compute the integral kernel of $N_{T,\mathbf{A}}$ using (4.40) in the
case $W=0$, we get

$\displaystyle N_{T,\mathbf{A}}(\alpha)(x,y)$
$\displaystyle=\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{u}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{v}}\;G_{h}^{\mathrm{i}\omega_{n}}(x,u_{1})\,\alpha(u_{1},v_{1})\,G_{h}^{-\mathrm{i}\omega_{n}}(u_{2},v_{1})\,\overline{\alpha(u_{2},v_{2})}$
$\displaystyle\hskip 120.0pt\times
G_{h}^{\mathrm{i}\omega_{n}}(v_{2},u_{3})\,\alpha(u_{3},v_{3})\,G_{h}^{-\mathrm{i}\omega_{n}}(y,v_{3}),$
(4.169)

We highlight that, by a slight abuse of notation, $\alpha$ and
$N_{T,\mathbf{A}}(\alpha)$ in the above equation are functions of the original
coordinates, while they are functions of relative and center-of-mass
coordinates in (4.165). Let us denote $X=\frac{x+y}{2}$, $r=x-y$ and let us
also introduce the relative coordinate $\mathbf{s}$ and the center-of-mass
coordinate $\mathbf{Z}$ by

$\displaystyle u_{i}$ $\displaystyle=X+Z_{i}+\frac{s_{i}}{2},$ $\displaystyle
v_{i}$ $\displaystyle=X+Z_{i}-\frac{s_{i}}{2},$ $\displaystyle i$
$\displaystyle=1,2,3.$

When we express the integration in (4.169) in terms of these coordinates and
use (4.3), we see that the claimed formula holds. ∎

As the operator $L_{T,\mathbf{A}}$, we analyze the operator $N_{T,\mathbf{A}}$
in four steps. More precisely, we decompose $N_{T,\mathbf{A}}$ as

$\displaystyle
N_{T,\mathbf{A}}=(N_{T,\mathbf{A}}-\widetilde{N}_{T,\mathbf{A}})+(\widetilde{N}_{T,\mathbf{A}}-N_{T,\mathbf{B}}^{(1)})+(N_{T,\mathbf{B}}^{(1)}-N_{T}^{(2)})+N_{T}^{(2)}$
(4.170)

with $\widetilde{N}_{T,\mathbf{A}}$ defined below in (4.171),
$N_{T,\mathbf{B}}^{(1)}$ in (4.180), and $N_{T}^{(2)}$ in (4.189). To obtain
the map $\widetilde{N}_{T,\mathbf{A}}$ from $N_{T,\mathbf{A}}$ we need to
replace $g_{h}^{z}$ by $g_{0}^{z}$. The operator $N_{T,\mathbf{B}}^{(1)}$
emerges when we use a part of the phase
$\exp(\mathrm{i}\Upsilon_{\mathbf{A}_{h}}(X,\mathbf{Z},r,s))$ in the
definition of $\ell_{T,\mathbf{A}}$ in (4.166) to replace the translation
operators $\exp(\mathrm{i}Z\cdot P_{X})$ in front the $\alpha$ factors by
magnetic translation operators. The part of the phase factor that is not
needed during this procedure is shown to yield a negligible contribution.
Finally, the operator $N_{T}^{(2)}$ is obtained when we replace the just found
magnetic translations by $1$. The above decomposition of $N_{T,\mathbf{A}}$
should be compared to that in [16, Eq. (4.120)]. In Section 4.5.10 we show
that the terms in brackets in (4.170) only yield negligible contributions.
Afterwards, we extract in Section 4.5.11 the quartic term in the
Ginzburg–Landau functional from $\langle\Delta,N_{T}^{(2)}(\Delta)\rangle$. In
Section 4.5.12 we summarize our findings.

####  4.5.10 Approximation of $N_{T,\mathbf{A}}$

#####  4.5.10.1 The operator $\widetilde{N}_{T,\mathbf{A}}$.

We define the operator $\widetilde{N}_{T,\mathbf{A}}$ by

$\displaystyle\widetilde{N}_{T,\mathbf{A}}(\alpha)(X,r)$
$\displaystyle\coloneqq\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;\widetilde{\ell}_{T,\mathbf{A}}(X,\mathbf{Z},r,\mathbf{s})\;\mathcal{A}(X,\mathbf{Z},\mathbf{s})$
(4.171)

with $\mathcal{A}$ in (4.165) and

$\displaystyle\widetilde{\ell}_{T,\mathbf{A}}(X,\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\ell_{T}^{n}(\mathbf{Z},r,\mathbf{s})\;\mathrm{e}^{\mathrm{i}\Upsilon_{\mathbf{A}_{h}}(X,\mathbf{Z},r,\mathbf{s})},$

where $\Upsilon_{\mathbf{A}}$ has been defined in (4.168), and

$\displaystyle\ell_{T}^{n}(\mathbf{Z},r,\mathbf{s})$ $\displaystyle\coloneqq
g_{0}^{\mathrm{i}\omega_{n}}\bigl{(}Z_{1}-\frac{r-s_{1}}{2}\bigr{)}\;g_{0}^{-\mathrm{i}\omega_{n}}\bigl{(}Z_{1}-Z_{2}-\frac{s_{1}+s_{2}}{2}\bigr{)}$
$\displaystyle\hskip 50.0pt\times
g_{0}^{\mathrm{i}\omega_{n}}\bigl{(}Z_{2}-Z_{3}-\frac{s_{2}+s_{3}}{2}\bigr{)}\;g_{0}^{-\mathrm{i}\omega_{n}}\bigl{(}Z_{3}+\frac{r-s_{3}}{2}\bigr{)}.$
(4.172)

In our calculation of the BCS energy we can replace $N_{T,\mathbf{A}}(\Delta)$
by $\widetilde{N}_{T,\mathbf{A}}(\Delta)$ because of the following error
bound.

######  Proposition 4.29.

Let $V\alpha_{*}\in L^{\nicefrac{{4}}{{3}}}(\mathbb{R}^{3})$, let $A\in
W^{3,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be periodic, assume that $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle|\langle\Delta,N_{T,\mathbf{A}}(\Delta)-\widetilde{N}_{T,\mathbf{A}}(\Delta)\rangle|$
$\displaystyle\leqslant
C\;h^{6}\;\|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}.$

The function

$\displaystyle J_{T,\mathbf{A}}$
$\displaystyle\coloneqq\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\;\tau^{\mathrm{i}\omega_{n}}*\rho^{-\mathrm{i}\omega_{n}}*\rho^{\mathrm{i}\omega_{n}}*\rho^{-\mathrm{i}\omega_{n}}+|g_{0}^{\mathrm{i}\omega_{n}}|*\tau^{-\mathrm{i}\omega_{n}}*\rho^{\mathrm{i}\omega_{n}}*\rho^{-\mathrm{i}\omega_{n}}$
$\displaystyle\hskip
50.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|*\tau^{\mathrm{i}\omega_{n}}*\rho^{-\mathrm{i}\omega_{n}}+|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|*|g_{0}^{\mathrm{i}\omega_{n}}|*\tau^{-\mathrm{i}\omega_{n}}.$
(4.173)

plays a prominent role in the proof of Proposition 4.29. Using Lemmas 4.2 and
4.4 as well as (4.56), we see that for any $T\geqslant T_{0}>0$ there is a
constant $C>0$ such that

$\displaystyle\|J_{T,\mathbf{A}_{h}}\|_{1}\leqslant C\;h^{3}$ (4.174)

holds.

###### Proof of Proposition 4.29.

The function $|\Psi|$ is periodic, and hence (2.9) implies

$\displaystyle\|\mathrm{e}^{\mathrm{i}Z\cdot(-\mathrm{i}\nabla_{X})}\Psi\|_{6}^{2}=\|\Psi\|_{6}^{2}\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$ (4.175)

In particular, we have

$\displaystyle\fint_{Q_{h}}\mathrm{d}X\;|\Psi(X)|\;\prod_{i=1}^{3}|\mathrm{e}^{\mathrm{i}Z_{i}\cdot(-\mathrm{i}\nabla_{X})}\Psi(X)|$
$\displaystyle\leqslant\|\Psi\|_{2}\;\prod_{i=1}^{3}\|\mathrm{e}^{\mathrm{i}Z_{i}\cdot(-\mathrm{i}\nabla_{X})}\Psi\|_{6}\leqslant
C\,h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}$ (4.176)

as well as

$\displaystyle|\langle\Delta,N_{T,\mathbf{A}}(\Delta)-\widetilde{N}_{T,\mathbf{A}}(\Delta)\rangle|$
$\displaystyle\leqslant
C\,h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h}))}^{4}\int_{\mathbb{R}^{3}}\mathrm{d}r\,\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;|V\alpha_{*}(r)|\;|V\alpha_{*}(s_{1})|\;|V\alpha_{*}(s_{2})|\;|V\alpha_{*}(s_{3})|$
$\displaystyle\times\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\;\operatorname*{ess\,sup}_{X\in\mathbb{R}^{3}}\bigl{|}(\ell_{T,\mathbf{A}}-\widetilde{\ell}_{T,\mathbf{A}})(X,\mathbf{Z},r,\mathbf{s})\bigr{|}.$
(4.177)

Next, we define the variables $Z_{1}^{\prime},Z_{2}^{\prime},Z_{3}^{\prime}$
via the equation

$\displaystyle Z_{1}^{\prime}-Z_{2}^{\prime}$ $\displaystyle\coloneqq
Z_{1}-Z_{2}-\frac{s_{1}+s_{2}}{2},$ $\displaystyle
Z_{2}^{\prime}-Z_{3}^{\prime}$ $\displaystyle\coloneqq
Z_{2}-Z_{3}-\frac{s_{2}+s_{3}}{2},$ $\displaystyle Z_{3}^{\prime}$
$\displaystyle\coloneqq Z_{3}+\frac{r-s_{3}}{2},$ (4.178)

which implies

$\displaystyle Z_{1}-\frac{r-s_{1}}{2}=Z_{1}^{\prime}-(r-s_{1}-s_{2}-s_{3}).$
(4.179)

We argue as in the proof of (4.65) to see that

$\displaystyle\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\;\operatorname*{ess\,sup}_{X\in\mathbb{R}^{3}}\bigl{|}(\ell_{T,\mathbf{A}}-\widetilde{\ell}_{T,\mathbf{A}})(X,\mathbf{Z},r,\mathbf{s})\bigr{|}\leqslant
J_{T,\mathbf{A}_{h}}(r-s_{1}-s_{2}-s_{3})$

holds with $J_{T,\mathbf{A}}$ in (4.173). When insert this bound into (4.177)
and use

$\displaystyle\bigl{\|}V\alpha_{*}\;\bigl{(}V\alpha_{*}*V\alpha_{*}*V\alpha_{*}*J_{T,\mathbf{A}}\bigr{)}\bigr{\|}_{1}$
$\displaystyle\leqslant
C\;\|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\;\|J_{T,\mathbf{A}}\|_{1},$

as well as (4.174), this finishes the proof. ∎

#####  4.5.10.2 The operator $N_{T,\mathbf{B}}^{(1)}$.

We define the operator $N_{T,\mathbf{B}}^{(1)}$ by

$\displaystyle N_{T,\mathbf{B}}^{(1)}(\alpha)(X,r)$
$\displaystyle\coloneqq\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;\ell_{T}(\mathbf{Z},r,\mathbf{s})\;\mathcal{A}_{\mathbf{B}}(X,\mathbf{Z},\mathbf{s}),$
(4.180)

where

$\displaystyle\ell_{T}(\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq\ell_{T,0}(0,\mathbf{Z},r,\mathbf{s}),$ (4.181)

with $\ell_{T,0}$ in (4.166) and

$\displaystyle\mathcal{A}_{\mathbf{B}}(X,\mathbf{Z},\mathbf{s})$
$\displaystyle\coloneqq\mathrm{e}^{\mathrm{i}Z_{1}\cdot\Pi}\alpha(X,s_{1})\;\overline{\mathrm{e}^{\mathrm{i}Z_{2}\cdot\Pi}\alpha(X,s_{2})}\;\mathrm{e}^{\mathrm{i}Z_{3}\cdot\Pi}\alpha(X,s_{3}).$
(4.182)

The following bound allows us to replace
$\langle\Delta,\widetilde{N}_{T,\mathbf{A}}(\Delta)\rangle$ by
$\langle\Delta,N_{T,\mathbf{B}}^{(1)}(\Delta)\rangle$ in our computation of
the energy.

######  Proposition 4.30.

Let $|\cdot|V\alpha_{*}\in L^{\nicefrac{{4}}{{3}}}(\mathbb{R}^{3})$ for
$k\in\\{0,1\\}$, let $A\in L^{\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be
periodic, assume $\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$, and denote
$\Delta\equiv\Delta_{\Psi}$ as in (3.1). For any $T_{0}>0$ there is $h_{0}>0$
such that for any $T\geqslant T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle|\langle\Delta,\widetilde{N}_{T,\mathbf{A}}(\Delta)-N_{T,\mathbf{B}}^{(1)}(\Delta)\rangle|$
$\displaystyle\leqslant C\;h^{5}\;\max_{k=0,1}\|\ |\cdot|^{k}\
V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}.$

Before we give the proof of Proposition 4.30 we define the functions

$\displaystyle J_{T}^{(1)}$
$\displaystyle\coloneqq\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\,|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{-\mathrm{i}\omega_{n}}|$
$\displaystyle\hskip
100.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
$\displaystyle\hskip
100.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}$
(4.183)

and

$\displaystyle J_{T}^{(2)}$
$\displaystyle\coloneqq\smash{\frac{2}{\beta}\sum_{n\in\mathbb{Z}}}\,\bigl{(}|\cdot|\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{-\mathrm{i}\omega_{n}}|*|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|+|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|$
$\displaystyle\hskip
30.0pt+|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{\mathrm{i}\omega_{n}}|\bigr{)}*|g_{0}^{-\mathrm{i}\omega_{n}}|+|g_{0}^{\mathrm{i}\omega_{n}}|*|g_{0}^{-\mathrm{i}\omega_{n}}|*|g_{0}^{\mathrm{i}\omega_{n}}|*\bigl{(}|\cdot|\,|g_{0}^{-\mathrm{i}\omega_{n}}|\bigr{)}.$
(4.184)

Using Lemma 4.4 and (4.56), we show that for any $T_{0}>0$ there is a constant
$C>0$ such that for $T\geqslant T_{0}$ we have

$\displaystyle\|J_{T}^{(1)}\|_{1}+\|J_{T}^{(2)}\|_{1}\leqslant C.$ (4.185)

###### Proof of Proposition 4.30.

We recall the definition of the phase $\Upsilon_{\mathbf{A}}$ in (4.168). A
tedious but straightforward computation shows that

$\displaystyle\Upsilon_{\mathbf{A}_{\mathbf{B}}}(X,\mathbf{Z},r,\mathbf{s})$
$\displaystyle=Z_{1}\cdot(\mathbf{B}\wedge X)-Z_{2}\cdot(\mathbf{B}\wedge
X)+Z_{3}\cdot(\mathbf{B}\wedge X)+\frac{\mathbf{B}}{2}\cdot
I(\mathbf{Z},r,\mathbf{s}),$

where

$\displaystyle I(\mathbf{Z},r,\mathbf{s})$
$\displaystyle\coloneqq\frac{r}{2}\wedge\bigl{(}Z_{1}-\frac{r-s_{1}}{2}\bigr{)}+\frac{r}{2}\wedge\bigl{(}Z_{3}+\frac{r-s_{3}}{2}\bigr{)}$
$\displaystyle+\bigl{(}Z_{2}-Z_{3}-\frac{s_{2}+s_{3}}{2}\bigr{)}\wedge\bigl{(}Z_{1}-Z_{2}-\frac{s_{1}+s_{2}}{2}\bigr{)}$
$\displaystyle+\bigl{(}Z_{3}+\frac{r-s_{3}}{2}\bigr{)}\wedge\bigl{(}Z_{1}-Z_{2}-\frac{s_{1}+s_{2}}{2}\bigr{)}+\bigl{(}s_{2}+s_{3}-\frac{r}{2}\bigr{)}\wedge\bigl{(}Z_{1}-Z_{2}-\frac{s_{1}+s_{2}}{2}\bigr{)}$
$\displaystyle+\bigl{(}Z_{3}+\frac{r-s_{3}}{2}\bigr{)}\wedge\bigl{(}Z_{3}-Z_{2}+\frac{s_{2}+s_{3}}{2}\bigr{)}+\bigl{(}s_{3}-\frac{r}{2}\bigr{)}\wedge\bigl{(}Z_{3}-Z_{2}+\frac{s_{2}+s_{3}}{2}\bigr{)}.$
(4.186)

By (4.90), the operator $\widetilde{N}_{T,\mathbf{A}}$ can therefore be
rewritten as

$\displaystyle\widetilde{N}_{T,\mathbf{A}}(\alpha)$
$\displaystyle=\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\iiint_{\mathbb{R}^{9}}\mathrm{d}s\;\ell_{T}(\mathbf{Z},r,\mathbf{s})\,\mathrm{e}^{\mathrm{i}\Upsilon_{A_{h}}(X,\mathbf{Z},r,\mathbf{s})}\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot
I(\mathbf{Z},r,\mathbf{s})}\;\mathcal{A}_{\mathbf{B}}(X,\mathbf{Z},\mathbf{s}).$

This formula and the estimate in (4.176) imply the bound

$\displaystyle|\langle\Delta,\widetilde{N}_{T,\mathbf{A}}(\Delta)-N_{T,\mathbf{B}}^{(1)}(\Delta)\rangle|$
$\displaystyle\leqslant
C\;h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}\int_{\mathbb{R}^{3}}\mathrm{d}r\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;|V\alpha_{*}(r)|\;|V\alpha_{*}(s_{1})|\;|V\alpha_{*}(s_{2})|\;|V\alpha_{*}(s_{3})|$
$\displaystyle\times\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\;|\ell_{T}^{n}(\mathbf{Z},r,\mathbf{s})|\;\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\Upsilon_{A_{h}}(X,\mathbf{Z},r,\mathbf{s})}\,\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot
I(\mathbf{Z},r,\mathbf{s})}-1\bigr{|}$ (4.187)

with $\Upsilon_{A}$ in (4.168) and $I$ in (4.186). In terms of the coordinates
in (4.178) and with the help of (4.179), the phase function $I$ can be written
as

$\displaystyle I(\mathbf{Z},r,\mathbf{s})$
$\displaystyle=(Z_{2}^{\prime}-Z_{3}^{\prime})\wedge(Z_{1}^{\prime}-Z_{2}^{\prime})+Z_{3}^{\prime}\wedge(Z_{1}^{\prime}-Z_{2}^{\prime})+Z_{3}^{\prime}\wedge(Z_{3}^{\prime}-Z_{2}^{\prime})$
$\displaystyle\hskip
50.0pt+\frac{r}{2}\wedge\bigl{(}Z_{1}^{\prime}-(r-s_{1}-s_{2}-s_{3})\bigr{)}+\bigl{(}s_{2}+s_{3}-\frac{r}{2}\bigr{)}\wedge(Z_{1}^{\prime}-Z_{2}^{\prime})$
$\displaystyle\hskip
50.0pt+\bigl{(}s_{3}-\frac{r}{2}\bigr{)}\wedge(Z_{3}^{\prime}-Z_{2}^{\prime})+\frac{r}{2}\wedge
Z_{3}^{\prime}.$ (4.188)

Moreover, using the definition of $\Phi_{\mathbf{A}}$ in (4.2), the definition
of $\Upsilon_{A}(X,\mathbf{Z},r,\mathbf{s})$ in (4.168), (4.178), and (4.179),
we obtain the bound

$\displaystyle|\Upsilon_{A}(X,\mathbf{Z},r,\mathbf{s})|$
$\displaystyle\leqslant\|A\|_{\infty}\bigl{(}(Z_{1}^{\prime}-(r-s_{1}-s_{2}-s_{3}))+(Z_{1}^{\prime}-Z_{2}^{\prime})+(Z_{2}^{\prime}-Z_{3}^{\prime})+Z_{3}^{\prime}\bigr{)}.$

In combination with (4.187), (4.188), and an argument that is similar to the
one used to obtain (4.114), we find

$\displaystyle\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\;|\ell_{T,0}^{n}(\mathbf{Z},r,\mathbf{s})|\;\bigl{|}\mathrm{e}^{\mathrm{i}\Upsilon_{A_{h}}(X,\mathbf{Z},r,\mathbf{s})}\,\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}\cdot
I(\mathbf{Z},r,\mathbf{s})}-1\bigr{|}$ $\displaystyle\hskip 20.0pt\leqslant
Ch\;\bigl{[}J_{T}^{(1)}(r-s_{1}-s_{2}-s_{3})+J_{T}^{(2)}(r-s_{1}-s_{2}-s_{3})\;\bigl{(}1+|r|+|s_{1}|+|s_{2}|+|s_{3}|\bigr{)}\bigr{]}$

with the functions $F_{T}^{(1)}$ and $F_{T}^{(2)}$ in (4.183) and (4.184),
respectively. Accordingly, an application of Young’s inequality shows that

$\displaystyle|\langle\Delta,\widetilde{N}_{T,\mathbf{A}}(\Delta)-N_{T,\mathbf{B}}^{(1)}(\Delta)\rangle|$
$\displaystyle\leqslant
C\;h^{5}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}\;\bigl{(}\|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}+\|\,|\cdot|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\bigr{)}\bigl{(}\|J_{T}^{(1)}\|_{1}+\|J_{T}^{(2)}\|_{1}\bigr{)}.$

The claim of the proposition follows when we apply (4.185) on the right side
of the above equation. ∎

#####  4.5.10.3 The operator $N_{T}^{(2)}$.

We define the operator $N_{T}^{(2)}$ by

$\displaystyle N_{T}^{(2)}(\alpha)(X,r)$
$\displaystyle\coloneqq\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{Z}\iiint_{\mathbb{R}^{9}}\mathrm{d}\mathbf{s}\;\ell_{T}(\mathbf{Z},r,\mathbf{s})\,\prod_{i=1}^{3}\alpha(X,s_{i})$
(4.189)

with $\ell_{T}$ in (4.181).

In the computation of the BCS energy we can replace
$\langle\Delta,N_{T,\mathbf{B}}^{(1)}(\Delta)\rangle$ by
$\langle\Delta,N_{T}^{(2)}(\Delta)\rangle$ with the help of the following
error bound. Its proof can be found in [16, Proposition 4.20]. We highlight
that the $H_{\mathrm{mag}}^{2}(Q_{h})$-norm of $\Psi$ is needed once more.

######  Proposition 4.31.

Assume that $|\cdot|^{k}V\alpha_{*}\in
L^{\nicefrac{{4}}{{3}}}(\mathbb{R}^{3})$ for $k\in\\{0,1,2\\}$, let $\Psi\in
H_{\mathrm{mag}}^{2}(Q_{h})$, and $\Delta\equiv\Delta_{\Psi}$ as in (3.1). For
any $T\geqslant T_{0}>0$ there is $h_{0}>0$ such that for $0<h\leqslant h_{0}$
we have

$\displaystyle|\langle\Delta,N_{T,\mathbf{B}}^{(1)}(\Delta)-N_{T}^{(2)}(\Delta)\rangle|$
$\displaystyle\leqslant C\;h^{6}\;\max_{k=0,1,2}\|\ |\cdot|^{k}\
V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{3}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}.$

####  4.5.11 Calculation of the quartic term in the Ginzburg–Landau functional

The quartic term in the Ginzburg–Landau functional in (1.18) is contained in
$\langle\Delta,N_{T}^{(2)}(\Delta)\rangle$. It can be extracted with the
following proposition, whose proof can be found in [16, Proposition 4.21].

######  Proposition 4.32.

Assume $V\alpha_{*}\in L^{\nicefrac{{4}}{{3}}}(\mathbb{R}^{3})$ and let
$\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$ as well as $\Delta\equiv\Delta_{\Psi}$
as in (3.1). For any $h>0$, we have

$\displaystyle\langle\Delta,N_{{T_{\mathrm{c}}}}^{(2)}(\Delta)\rangle=8\;\Lambda_{3}\;\|\Psi\|_{4}^{4}$

with $\Lambda_{3}$ in (3.23). Moreover, for any $T\geqslant T_{0}>0$, we have

$\displaystyle|\langle\Delta,N_{T}^{(2)}(\Delta)-N_{{T_{\mathrm{c}}}}^{(2)}(\Delta)\rangle|$
$\displaystyle\leqslant
C\;h^{4}\;|T-{T_{\mathrm{c}}}|\;\|V\alpha_{*}\|_{\nicefrac{{4}}{{3}}}^{4}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{4}.$

####  4.5.12 Summary: The quartic term and proof of Theorem 3.6

Let the assumptions of Theorem 3.6 hold. We use the resolvent identity in
(4.36) to decompose the operator $N_{T,\mathbf{A},W}$ in (3.12) as

$\displaystyle
N_{T,\mathbf{A},W}=N_{T,\mathbf{A}}+\mathcal{R}_{T,\mathbf{A},W}^{(3)}$
(4.190)

with $N_{T,\mathbf{A}}$ in (4.164) and

$\displaystyle\mathcal{R}_{T,\mathbf{A},W}^{(3)}(\Delta)$
$\displaystyle\coloneqq\frac{2}{\beta}\sum_{n\in\mathbb{Z}}\Bigl{[}\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,W_{h}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}\,\overline{\Delta}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}$
$\displaystyle\hskip
10.0pt-\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A}}}}\,W_{h}\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}\,\overline{\Delta}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}$
$\displaystyle\hskip
10.0pt+\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A}}}}\,\overline{\Delta}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,W_{h}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}$
$\displaystyle\hskip
10.0pt-\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A}}}}\,\overline{\Delta}\,\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A}}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A}}}}\,W_{h}\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}\Bigr{]}.$
(4.191)

We claim that the operator $\mathcal{R}_{T,\mathbf{A},W}^{(3)}$ satisfies the
bound

$\displaystyle\|\mathcal{R}_{T,\mathbf{A},W}^{(3)}(\Delta)\|_{{L^{2}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}}$
$\displaystyle\leqslant
C\,T^{-5}\,h^{5}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{3}.$

This is a direct consequence of Hölder’s inequality in (2.1) for the trace per
unit volume, which implies that the Hilbert-Schmidt norm per unit volume of
the terms in the sum in (4.191) are bounded by
$C\,|\omega_{n}|^{-5}\,\|W_{h}\|_{\infty}\,\|\Delta\|_{6}^{3}$. Moreover, an
application of Lemma 4.1 and (2.9) show that this expression is bounded by
$C\,|2n+1|^{-5}\,T^{-5}\,h^{5}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{3}$,
which proves our claim.

When we combine this bound, Lemma 4.27, and Propositions 4.29-4.32, we find

$\displaystyle\frac{1}{8}\langle\Delta,N_{T,\mathbf{A},W}(\Delta)\rangle=\;\Lambda_{3}\;\|\Psi\|_{4}^{4}+R_{4}(h),$
(4.192)

where the remainder $R(h)$ satisfies the bound

$\displaystyle|R_{4}(h)|\leqslant
C\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{3}\,\bigl{(}h^{5}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}+h^{6}\;\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}\bigr{)}.$

The statement in Theorem 3.6 is a direct consequence of (4.192) and (4.157).

###  4.6 Proof of Proposition 3.2

We assume that the assumptions of Proposition 3.2 hold and recall the
definition of $\Gamma_{\Delta}$ in (3.4). Using the resolvent equation in
(4.36) and (4.37), we write $\alpha_{\Delta}=[\Gamma_{\Delta}]_{12}$ as

$\displaystyle\alpha_{\Delta}$
$\displaystyle=[\mathcal{O}]_{12}+\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta).$

Here,
$\mathcal{O}=\frac{1}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{\mathrm{i}\omega_{n}-H_{0}}\delta\frac{1}{\mathrm{i}\omega_{n}-H_{0}}$,
see (4.38), with $\delta$ in (3.1) and

$\displaystyle\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)$
$\displaystyle\coloneqq\frac{1}{\beta}\sum_{n\in\mathbb{Z}}\Bigl{[}\frac{1}{\mathrm{i}\omega_{n}-H_{0}}\delta\frac{1}{\mathrm{i}\omega_{n}-H_{0}}\delta\frac{1}{\mathrm{i}\omega_{n}-H_{\Delta}}\delta\frac{1}{\mathrm{i}\omega_{n}-H_{0}}\Bigr{]}_{12}.$
(4.193)

Moreover, we have $[\mathcal{O}]_{12}=-\frac{1}{2}L_{T,\mathbf{A},W}\Delta$
with $L_{T,\mathbf{A},W}$ in (3.11). Using the decomposition of
$L_{T,\mathbf{A},W}$ in (4.42), we define

$\displaystyle\eta_{0}(\Delta)$
$\displaystyle\coloneqq\frac{1}{2}\bigl{(}L_{T,\mathbf{A},W}\Delta-
L_{T,\mathbf{A}}\Delta\bigr{)}+\frac{1}{2}\bigl{(}L_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}}\Delta\bigr{)}+\frac{1}{2}\bigl{(}M_{T}^{(1)}\Delta-
M_{{T_{\mathrm{c}}}}^{(1)}\Delta\bigr{)}$ $\displaystyle\hskip
60.0pt+\frac{1}{2}\bigl{(}M_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta\bigr{)}+\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta),$
$\displaystyle\eta_{\perp}(\Delta)$
$\displaystyle\coloneqq\frac{1}{2}\bigl{(}M_{T,\mathbf{A}_{e_{3}}}\Delta-
M_{T}^{(1)}\Delta\bigr{)},$ (4.194)

with $M_{T,\mathbf{A}}$ in (4.108), $L_{T,\mathbf{A}}^{W}$ in (4.43), and
$M_{T}^{(1)}$ in (4.120). From Proposition 4.17 we know that
$-\frac{1}{2}M_{{T_{\mathrm{c}}}}^{(1)}\Delta=\Psi\alpha_{*}$, which allows us
to write $\alpha_{\Delta}$ as in (3.6). The operator
$M_{T,\mathbf{A}_{e_{3}}}$ equals $M_{T,\mathbf{A}}$ in (4.108) with
$\mathbf{A}$ replaced by $\mathbf{A}_{e_{3}}$. The contribution from this
operator needs to be carefully isolated for the orthogonality property in
(3.9) to hold. This should be compared to part (c) of [16, Proposition 3.2].
In the following, we will establish the properties of $\eta_{0}$ and
$\eta_{\perp}$ that are stated in Proposition 3.2.

We will first prove (3.7), and start by noting that

$\displaystyle\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)$
$\displaystyle=\frac{1}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}\,\overline{\Delta}\,\Bigl{[}\frac{1}{\mathrm{i}\omega_{n}-H_{\Delta}}\Bigr{]}_{11}\,\Delta\,\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}.$

An application of Hölder’s inequality shows
$\|\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)\|_{2}\leqslant
C\beta^{3}\|\Delta\|_{6}^{3}$. With the operator
$\pi=-\mathrm{i}\nabla+\mathbf{A}_{\mathbf{B}}$ understood to act on the
$x$-coordinate of the integral kernel of
$\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)$ we also have

$\|\pi\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)\|_{2}\leqslant\frac{1}{\beta}\sum_{n\in\mathbb{Z}}\Bigl{\|}\pi\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\Bigr{\|}_{\infty}\Bigl{\|}\frac{1}{\mathrm{i}\omega_{n}+\overline{\mathfrak{h}_{\mathbf{A},W}}}\Bigr{\|}_{\infty}^{2}\Bigl{\|}\Bigl{[}\frac{1}{\mathrm{i}\omega_{n}-H_{\Delta}}\Bigr{]}_{11}\Bigr{\|}_{\infty}\|\Delta\|_{6}^{3}.$

An application of Cauchy–Schwarz shows

$(-\mathrm{i}\nabla+\mathbf{A}_{\mathbf{B}}+A)^{2}+W_{h}\geqslant\frac{1}{2}(-\mathrm{i}\nabla+\mathbf{A}_{\mathbf{B}})^{2}-Ch^{2}.$
(4.195)

Accordingly, we have

$\displaystyle\Bigl{\|}\pi\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\Bigr{\|}_{\infty}$
$\displaystyle=\Bigl{\|}\frac{1}{\mathrm{i}\omega_{n}+\mathfrak{h}_{\mathbf{A},W}}\pi^{2}\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\Bigr{\|}_{\infty}^{\nicefrac{{1}}{{2}}}$
$\displaystyle\leqslant\Bigl{\|}\frac{1}{\mathrm{i}\omega_{n}+\mathfrak{h}_{\mathbf{A},W}}\left(\mathfrak{h}_{\mathbf{A},W}+\mu+Ch^{2}\right)\frac{1}{\mathrm{i}\omega_{n}-\mathfrak{h}_{\mathbf{A},W}}\Bigr{\|}_{\infty}^{\nicefrac{{1}}{{2}}}\leqslant
C\,|\omega_{n}|^{-\nicefrac{{1}}{{2}}}.$

It follows that

$\displaystyle\|\pi\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)\|_{2}\leqslant
C\,\|\Delta\|_{6}^{3}.$ (4.196)

The same argument with obvious adjustments also shows that
$\|\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)\pi\|_{2}$ is bounded by the
right side of (4.196), too. Finally, (2.13), an application of Lemma 4.1, and
(2.9) allow us to conclude that

$\displaystyle\|\mathcal{R}_{T,\mathbf{A},W}^{(4)}(\Delta)\|_{{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}\leqslant
C\;h^{6}\;\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{6}$ (4.197)

holds.

To control $M_{T,\mathbf{A}}\Delta-M_{T,\mathbf{A}_{e_{3}}}\Delta$, we need
the following proposition.

######  Proposition 4.33.

Let $|\cdot|V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$ for $k\in\\{0,1\\}$, let
$A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be periodic, assume
$\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$
as in (3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any
$T\geqslant T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle\|M_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta\|_{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant C\,h^{5}\,\max_{k=0,1}\|\ |\cdot|^{k}\
V\alpha_{*}\|_{2}^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$ (4.198)

###### Proof.

Let us define the operator

$\displaystyle\mathcal{Q}_{T,\mathbf{B},A}\alpha(X,r)\coloneqq\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\;\mathrm{e}^{2\mathrm{i}A_{h}(X)\cdot
Z}\;(\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\alpha)(X,s),$

where $k_{T}(Z,r)\coloneqq k_{T,0}(0,Z,r,0)$ with $k_{T,0}$ in (4.45) and
$\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$ is understood to act on the
center-of-mass coordinate $X$ of $\alpha$. We start our analysis by writing

$\displaystyle M_{T,\mathbf{A}}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta=\bigl{(}M_{T,\mathbf{A}}\Delta-\mathcal{Q}_{T,\mathbf{B},A}\Delta\bigr{)}+\bigl{(}Q_{T,\mathbf{B},A}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta\bigr{)}.$ (4.199)

In the following we derive bounds on the
${H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}$-norms of the two terms on
the right side of (4.198). When we use that the integrand in the definition of
$M_{T,\mathbf{A}}$ is symmetric with respect to the transformation
$Z\mapsto-Z$ and apply Lemma 4.13, we see that

$\displaystyle(M_{T,\mathbf{A}}\alpha-\mathcal{Q}_{T,\mathbf{B},A}\alpha)(X,r)$
$\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\,\bigl{[}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}-\mathrm{e}^{2\mathrm{i}A_{h}(X)\cdot
Z}\bigr{]}\,(\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\alpha)(X,s)$

holds. Let us also recall that $\Phi_{\mathbf{A}}$ is defined in (4.2).

We have

$\displaystyle\Phi_{2A}(X,X+Z)-2A(X)\cdot
Z=2\int_{0}^{1}\mathrm{d}t\;\bigl{[}A(X+(1-t)Z)-A(X)\bigr{]}\cdot Z,$

and hence

$\displaystyle\bigl{|}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)}-\mathrm{e}^{2\mathrm{i}A(X)\cdot
Z}\bigr{|}\leqslant\|DA\|_{\infty}\;|Z|^{2}.$ (4.200)

When we apply this bound and
$|Z|^{2}\leqslant|Z+\frac{r}{2}|^{2}+|Z-\frac{r}{2}|^{2}$, it follows that

$\displaystyle\|M_{T,\mathbf{A}}\Delta-\mathcal{Q}_{T,\mathbf{B},A}\Delta\|_{2}^{2}$
$\displaystyle\leqslant
C\,\|\Psi\|_{2}^{2}\,\|DA_{h}\|_{\infty}^{2}\,\|F_{T}^{2}\|_{1}\,\|V\alpha_{*}\|_{2}^{2}$

with $F_{T}^{2}$ in (4.93). Using the $L^{1}$-norm bound for $F_{T}^{2}$ in
(4.95), we conclude the claimed estimate for this term.

To obtain a bound for the first gradient term, we start by noting that

$\displaystyle\|\Pi(M_{T,A}\Delta-\mathcal{Q}_{T,\mathbf{B},A}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,\|\Psi\|_{2}^{2}\int_{\mathbb{R}^{3}}\mathrm{d}r\;\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|$
$\displaystyle\hskip
28.45274pt\times\sup_{X\in\mathbb{R}^{3}}\bigl{|}\nabla_{X}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}-\nabla_{X}\mathrm{e}^{2\mathrm{i}A_{h}(X)\cdot
Z}\bigr{|}\,\bigr{|}^{2}$
$\displaystyle+C\,\|\Pi\mathrm{e}^{\mathrm{i}Z\cdot\Pi}\Psi\|_{2}\,\int_{\mathbb{R}^{3}}\mathrm{d}r\;\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|$
$\displaystyle\hskip
28.45274pt\times\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}-\mathrm{e}^{2\mathrm{i}A_{h}(X)\cdot
Z}\bigr{|}\,\bigr{|}^{2},$

where $\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$ is understood to act on
the center-of-mass coordinate. When we additionally use

$\displaystyle\bigl{|}\nabla_{X}\mathrm{e}^{\mathrm{i}\Phi_{2A}(X,X+Z)}-\nabla_{X}\mathrm{e}^{2\mathrm{i}A(X)\cdot
Z}\bigr{|}$
$\displaystyle\leqslant\bigl{|}\nabla_{X}\Phi_{2A}(X,X+Z)-2\nabla_{X}A(X)\cdot
Z\bigr{|}$ $\displaystyle+\bigl{|}\Phi_{2A}(X,X+Z)-2A(X)\cdot
Z\bigr{|}\,|\nabla_{X}A(X)\cdot Z|$
$\displaystyle\leqslant\bigl{[}\|D^{2}A\|_{\infty}+\|DA\|_{\infty}^{2}\bigr{]}\bigl{[}|Z|^{2}+|Z|^{3}\bigr{]},$

as well as

$|Z|^{a}\leqslant\left|Z+\frac{r}{2}\right|^{a}+\left|Z-\frac{r}{2}\right|^{a},\quad
a\geqslant 0$ (4.201)

for the choices $a=2,3$, $\|D^{2}A_{h}\|_{\infty}\leqslant Ch^{3}$,
$\|DA_{h}\|_{\infty}^{2}\leqslant Ch^{4}$, (4.117), and (4.200), we see that

$\displaystyle\|\Pi(M_{T,\mathbf{A}}\Delta-\mathcal{Q}_{T,\mathbf{B},A}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant
C\,h^{5}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\,\|V\alpha_{*}\|_{2}^{2}\,\left(\|F_{T}^{2}\|_{1}^{2}+\|F_{T}^{3}\|_{1}^{2}\right)$

holds with $F_{T}^{a}$ in (4.95). An application of (4.95) proves the claimed
bound for this term.

In the last step we consider

$\displaystyle\|\widetilde{\pi}(M_{T,\mathbf{A}}\Delta-\mathcal{Q}_{T,\mathbf{B},A}\Delta)\|_{2}^{2}$
$\displaystyle\leqslant C\,\|\Psi\|_{2}^{2}$
$\displaystyle\times\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|\widetilde{\pi}k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|\,\sup_{X\in\mathbb{R}^{3}}\bigl{|}\mathrm{e}^{\mathrm{i}\Phi_{2A_{h}}(X,X+Z)}-\mathrm{e}^{2\mathrm{i}A_{h}(X)\cdot
Z}\bigr{|}\Bigr{|}^{2}.$

The estimate in (4.201) and $\frac{1}{4}|\mathbf{B}\wedge
r|\leqslant\frac{1}{4}|\mathbf{B}|(|r-s|+|s|)$ allow us to prove the bound

$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}Z\;|\widetilde{\pi}k_{T}(Z,r-s)|\,|Z|^{2}\leqslant
F_{T}^{3}(r-s)\,(1+|s|)+G_{T}^{2}(r-s)$ (4.202)

with $F_{T}^{a}$ in (4.93) and $G_{T}^{a}$ in (4.94). In combination with
(4.200), this proves the claimed bound for this term. It also ends the proof
of the claimed bound for the first term on the right side of (4.199). It
remains to consider the second term.

A short computation that uses $k_{T}(-Z,r-s)=k_{T}(Z,r-s)$ and
$\cos(x)-1=-2\sin^{2}(\frac{x}{2})$ shows

$\displaystyle(\mathcal{Q}_{T,\mathbf{B},A}\alpha-
M_{T,\mathbf{A}_{e_{3}}}\alpha)(X,r)$
$\displaystyle=-2\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\,\sin^{2}(A_{h}(X)\cdot
Z)(\cos(Z\cdot\Pi)\alpha)(X,s)$
$\displaystyle-\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;k_{T}(Z,r-s)\,\sin(2A_{h}(X)\cdot
Z)\,(\sin(Z\cdot\Pi)\alpha)(X,s).$

From this, we check that

$\displaystyle\|\mathcal{Q}_{T,\mathbf{B},A}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta\|_{2}^{2}$ $\displaystyle\leqslant
C\,\bigl{[}\|\Psi\|_{2}^{2}\,\|A_{h}\|_{\infty}^{4}+\|\Pi\Psi\|_{2}^{2}\|A_{h}\|_{\infty}^{2}\bigr{]}\,\|F_{T}^{2}\|_{1}^{2}\,\|V\alpha_{*}\|_{2}^{2}$

holds with $F_{T}^{a}$ in (4.93). When we use (4.200) to obtain a bound for
the $L^{1}$-norm of $F_{T}^{2}$, this proves the claimed bound for this term.

Next, we note that

$\displaystyle\|\Pi(\mathcal{Q}_{T,\mathbf{B},A}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta)\|_{2}^{2}\leqslant
C\,\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|$
$\displaystyle\times\Bigl{[}\sup_{X\in\mathbb{R}^{3}}|\nabla_{X}\sin^{2}(A_{h}(X)\cdot
Z)|\,\|\cos(Z\cdot\Pi)\Psi\|_{2}+\sup_{X\in\mathbb{R}^{3}}|\sin^{2}(A_{h}(X)\cdot
Z)|\,\|\Pi\cos(Z\cdot\Pi)\Psi\|_{2}$
$\displaystyle+\sup_{X\in\mathbb{R}^{3}}|\sin(2A_{h}(X)\cdot
Z)|\,\|\sin(Z\cdot\Pi)\Psi\|_{2}+\sup_{X\in\mathbb{R}^{3}}|\sin(2A_{h}(X)\cdot
Z)|\,\|\Pi\sin(Z\cdot\Pi)\Psi\|_{2}\Bigr{]}\Bigr{|}^{2}.$ (4.203)

We have

$\displaystyle\|\sin(Z\cdot\Pi)\Psi\|_{2}\leqslant
C\,h^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}\,|Z|.$

Furthermore, from a straight forward computation or from [16, Lemma 5.12], we
know that

$\displaystyle\Pi_{X}\,\sin(Z\cdot\Pi_{X})=\sin(Z\cdot\Pi_{X})\;\Pi_{X}+2\mathrm{i}\,\cos(Z\cdot\Pi_{X})\;\mathbf{B}\wedge
Z,$

and hence

$\displaystyle\|\Pi\sin(Z\cdot\Pi)\Psi\|_{2}$ $\displaystyle\leqslant
C\,h^{2}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}\,(1+|Z|).$

To obtain the bound we used that $|\mathbf{B}|=h^{2}$. For
$\Pi\cos(Z\cdot\Pi)\Psi$ a similar estimate was obtained in (4.116). Putting
these bounds together, we find that the term on the left side of (4.203) is
bounded by a constant times
$h^{6}\|V\alpha_{*}\|_{2}^{2}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$. It
remains to consider the term proportional to $\widetilde{\pi}$.

A straightforward computation shows that

$\displaystyle\|\widetilde{\pi}(\mathcal{Q}_{T,\mathbf{B},A}\Delta-
M_{T,\mathbf{A}_{e_{3}}}\Delta)\|_{2}^{2}$ $\displaystyle\leqslant
C\bigl{[}\|A_{h}\|_{\infty}^{4}\|\Psi\|_{2}^{2}+\|A_{h}\|_{\infty}^{2}\|\Pi\Psi\|_{2}^{2}\bigr{]}$
$\displaystyle\hskip
20.0pt\times\int_{\mathbb{R}^{3}}\mathrm{d}r\,\Bigl{|}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathrm{d}Z\mathrm{d}s\;|\widetilde{\pi}k_{T}(Z,r-s)|\,|V\alpha_{*}(s)|\,|Z|^{2}\Bigr{|}^{2}.$

We use (4.202) to see that the term on the left side is bounded by a constant
times $h^{6}\max_{k=0,1}\|\ |\cdot|^{k}\
V\alpha_{*}\|_{2}^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$. This
proves Proposition 4.33. ∎

The next lemma provides us with a bound for the term in (4.194) that is
proportional to $L_{T,\mathbf{A},W}-L_{T,\mathbf{A}}$.

######  Lemma 4.34.

Let $V\alpha_{*}\in L^{2}(\mathbb{R}^{3})$, let $W\in
L^{\infty}(\mathbb{R}^{3})$ and $A\in
L^{\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ be periodic, assume $\Psi\in
H_{\mathrm{mag}}^{1}(Q_{h})$, and denote $\Delta\equiv\Delta_{\Psi}$ as in
(3.1). For any $T_{0}>0$ there is $h_{0}>0$ such that for any $T\geqslant
T_{0}$ and any $0<h\leqslant h_{0}$ we have

$\displaystyle\|L_{T,\mathbf{A},W}\Delta-
L_{T,\mathbf{A}}\Delta\|_{{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}}^{2}$
$\displaystyle\leqslant
C\,h^{6}\,\|V\alpha_{*}\|_{2}^{2}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$

###### Proof.

To prove the lemma, we write $L_{T,\mathbf{A},W}\Delta-L_{T,\mathbf{A}}\Delta$
as in (4.160) and use the representation of the $H^{1}$-norm in (2.13). The
details are a straight forward application of arguments that have been used
already several times above, and are therefore left to the reader. ∎

When we combine Propositions 4.9, 4.11, 4.15, 4.17 as well as (4.197) and
Lemma 4.34, we obtain the claimed bound for $\eta_{0}(\Delta)$ in (3.7), that
is, part (a) of Proposition 3.2. The proofs of parts (b) and (c) can be found
in [16], see the proofs of Proposition 3.2 (b), (c). This ends our proof of
Proposition 3.2.

###  4.7 Proof of Proposition 3.7

Let the assumptions of Proposition 3.7 hold. In the following we prove that
there are constants $D_{0}>0$ and $h_{0}>0$ such that for $0<h\leqslant h_{0}$
and

$\displaystyle 0<T_{0}\leqslant T<{T_{\mathrm{c}}}(1-D_{0}h^{2})$

there is a function $\Psi\in H_{\mathrm{mag}}^{2}(Q_{h})$, such that the
energy of the Gibbs state $\Gamma_{\Delta}$ in (3.4) with gap function
$\Delta(X,r)=-2V\alpha_{*}(r)\Psi(X)$ satisfies (3.24).

Let $\psi\in H_{\mathrm{mag}}^{2}(Q_{1})$ with
$\|\psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}=1$ and define $\Psi(X)=h\psi(hX)$. The
function $\Psi$ satisfies $\|\Psi\|_{H_{\mathrm{mag}}^{2}(Q_{h})}=1$. When we
apply Propositions 3.2, 3.5, 4.26, as well as (2.9) and (4.192), we find

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{\Delta})-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})$
$\displaystyle<h^{2}\,\bigl{(}-cD_{0}\,\|\psi\|_{2}^{2}+C\bigr{)}$

for $h$ small enough. The proof of Proposition 3.7 is completed when we choose
$D_{0}=\frac{C}{c\|\psi\|_{2}^{2}}$.

##  5 The Structure of Low-Energy States

In this section we prove a priori bounds for low-energy states of the BCS
functional in the sense of (5.1) below. The goal is to show that their Cooper
pair wave function has a structure similar to that of the trial state we use
in the proof of the upper bound in Section 3. These bounds and the trial state
analysis in Section 3 are the main technical ingredients for the proof of the
lower bound in Section 6. To prove the a priori bounds, we show that the
periodic external potentials $W_{h}$ and $A_{h}$ can be treated as a
perturbation, which reduces the problem to proving a priori bounds for the
case of a constant magnetic field. The solution of this problem has been the
main novelty in [16, Theorem 5.1] and we apply it here. In case of a magnetic
field with zero flux through the unit cell such bounds have been proved for
the first time in [21]. The idea to reduce the problem to the case of a
constant magnetic field is inspired by a similar perturbative analysis in
[21].

We recall the definition of the generalized one-particle density matrix
$\Gamma$ in (1.5), its Cooper pair wave function $\alpha=\Gamma_{12}$, as well
as the normal state $\Gamma_{0}$ in (1.12).

######  Theorem 5.1 (Structure of low-energy states).

Let Assumptions 1.1 and 1.2 hold. For given $D_{0},D_{1}\geqslant 0$, there is
a constant $h_{0}>0$ such that for all $0<h\leqslant h_{0}$ the following
holds: If $T>0$ obeys $T-{T_{\mathrm{c}}}\geqslant-D_{0}h^{2}$ and if $\Gamma$
is a gauge-periodic state with low energy, that is,

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})\leqslant
D_{1}h^{4},$ (5.1)

then there are $\Psi\in H_{\mathrm{mag}}^{1}(Q_{h})$ and
$\xi\in{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}$ such that

$\displaystyle\alpha(X,r)=\alpha_{*}(r)\Psi(X)+\xi(X,r),$ (5.2)

where

$\displaystyle\sup_{0<h\leqslant
h_{0}}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$ $\displaystyle\leqslant C,$
$\displaystyle\|\xi\|_{H^{1}(Q_{h}\times\mathbb{R}_{\mathrm{s}}^{3})}^{2}$
$\displaystyle\leqslant
Ch^{4}\bigl{(}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}+D_{1}\bigr{)}.$ (5.3)

######  Remarks.

  1. (a)

Equation (5.3) shows that $\Psi$ is a macroscopic quantity in the sense that
its $H_{\mathrm{mag}}^{1}(Q_{h})$-norm scales as that of the function in
(1.20). It is important to note that the $H_{\mathrm{mag}}^{1}(Q_{h})$-norm is
scaled with $h$, see (2.7). The unscaled $L_{\mathrm{mag}}^{2}(Q_{h})$-norm of
$\Psi$ is of the order $h$, and therefore much larger than that of $\xi$, see
(5.3).

  2. (b)

Theorem 5.1 has been proven in [16, Theorem 5.1] for the case of a constant
external magnetic field, where $A_{h}=0$ and $W_{h}=0$. Our proof of Theorem
5.1 for general external fields reduces the problem to that case.

Although Theorem 5.1 contains the natural a priori bounds for low-energy
states, we need a slightly different version of it in our proof of the lower
bound for the BCS free energy in Section 6. The main reason is that we intend
to use the function $\Psi$ from the decomposition of the Cooper pair wave
function of a low-energy state in (5.2) to construct a Gibbs state
$\Gamma_{\Delta}$ as in (3.4). In order to be able to justify the relevant
computations with this state, we need $\Psi\in H_{\mathrm{mag}}^{2}(Q_{h})$,
which is not guaranteed by Theorem 5.1 above, see also Remark 3.3. The
following corollary provides us with a decomposition of $\alpha$, where the
center-of-mass wave function $\Psi_{\leqslant}$ has the required
$H_{\mathrm{mag}}^{2}(Q_{h})$-regularity. A decomposition with a cut-off
function of the form in the corollary has also been used in [21, 22, 19, 16].

######  Corollary 5.2.

Let the assumptions of Theorem 5.1 hold and let
$\varepsilon\in[h^{2},h_{0}^{2}]$. Let $\Psi$ be as in (5.2) and define

$\displaystyle\Psi_{\leqslant}$ $\displaystyle\coloneqq\mathbbs
1_{[0,\varepsilon]}(\Pi^{2})\Psi,$ $\displaystyle\Psi_{>}$
$\displaystyle\coloneqq\mathbbs 1_{(\varepsilon,\infty)}(\Pi^{2})\Psi.$ (5.4)

Then, we have

$\displaystyle\|\Psi_{\leqslant}\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$
$\displaystyle\leqslant\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2},$
$\displaystyle\|\Psi_{\leqslant}\|_{H_{\mathrm{mag}}^{k}(Q_{h})}^{2}$
$\displaystyle\leqslant C\,(\varepsilon
h^{-2})^{k-1}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2},\qquad k\geqslant
2,$ (5.5)

as well as

$\displaystyle\|\Psi_{>}\|_{2}^{2}$ $\displaystyle\leqslant
C\varepsilon^{-1}h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2},$
$\displaystyle\|\Pi\Psi_{>}\|_{2}^{2}$ $\displaystyle\leqslant
Ch^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}.$ (5.6)

Furthermore,

$\displaystyle\sigma_{0}(X,r)\coloneqq\alpha_{*}(r)\Psi_{>}(X)$ (5.7)

satisfies

$\displaystyle\|\sigma_{0}\|_{H^{1}_{\mathrm{symm}}(Q_{h}\times\mathbb{R}^{3})}^{2}$
$\displaystyle\leqslant
C\varepsilon^{-1}h^{4}\,\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$ (5.8)

and, with $\xi$ in (5.2), the function

$\displaystyle\sigma\coloneqq\xi+\sigma_{0}$ (5.9)

obeys

$\displaystyle\|\sigma\|_{H^{1}_{\mathrm{symm}}(Q_{h}\times\mathbb{R}^{3})}^{2}\leqslant
Ch^{4}\bigl{(}\varepsilon^{-1}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}+D_{1}\bigr{)}.$
(5.10)

In terms of these functions, the Cooper pair wave function $\alpha$ of the
low-energy state $\Gamma$ in (5.1) admits the decomposition

$\displaystyle\alpha(X,r)=\alpha_{*}(r)\Psi_{\leqslant}(X)+\sigma(X,r).$
(5.11)

For a proof of the corollary we refer to the proof of Corollary 5.2 in [16].

###  5.1 A lower bound for the BCS functional

We start the proof of Theorem 5.1 with the following lower bound on the BCS
functional, whose proof is literally the same as that of the comparable
statement in [21].

######  Lemma 5.3.

Let $\Gamma_{0}$ be the normal state in (1.12). We have the lower bound

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})\geqslant\operatorname{Tr}\bigl{[}(K_{T,\mathbf{A},W}-V)\alpha\alpha^{*}\bigr{]}+\frac{4T}{5}\operatorname{Tr}\bigl{[}(\alpha^{*}\alpha)^{2}\bigr{]},$
(5.12)

where

$\displaystyle
K_{T,\mathbf{A},W}=\frac{(-\mathrm{i}\nabla+\mathbf{A}_{h})^{2}+W_{h}-\mu}{\tanh(\frac{(-\mathrm{i}\nabla+\mathbf{A}_{h})^{2}+W_{h}-\mu}{2T})}$
(5.13)

and $V\alpha(x,y)=V(x-y)\alpha(x,y)$.

In Proposition A.1 in Appendix A we show that the external electric and
magnetic fields can lower the lowest eigenvalue zero of
$K_{{T_{\mathrm{c}}}}-V$ at most by a constant times $h^{2}$. We use this in
the next lemma to show that $K_{T,\mathbf{A},W}-V$ is bounded from below by a
nonnegative operator, up to a correction of the size $Ch^{2}$.

######  Lemma 5.4.

Let Assumptions 1.1 and 1.2 be true. For any $D_{0}\geqslant 0$, there are
constants $h_{0}>0$ and $T_{0}>0$ such that for $0<h\leqslant h_{0}$ and $T>0$
with $T-{T_{\mathrm{c}}}\geqslant-D_{0}h^{2}$, the estimate

$\displaystyle K_{T,\mathbf{A},W}-V$ $\displaystyle\geqslant
c\;(1-P)(1+\pi^{2})(1-P)+c\,\min\\{T_{0},(T-{T_{\mathrm{c}}})_{+}\\}-Ch^{2}$
(5.14)

holds. Here, $P=|\alpha_{*}\rangle\langle\alpha_{*}|$ is the orthogonal
projection onto the ground state $\alpha_{*}$ of $K_{{T_{\mathrm{c}}}}-V$ and
$\pi=-\mathrm{i}\nabla+\mathbf{A}_{\mathbf{B}}$.

###### Proof.

Since $W\in L^{\infty}(\mathbb{R}^{3})$ we can use Lemma A.7 to show that
$K_{T,\mathbf{A},W}\geqslant K_{T,\mathbf{A},0}-Ch^{2}$ holds. The rest of the
proof goes along the same lines as that of [16, Lemma 5.4] with the obvious
replacements. In particular, [16, Proposition A.1] needs to be replaced by
Proposition A.1. We omit the details. ∎

We deduce two corollaries from (5.12) and Lemma 5.4. The first statement is an
a priori bound that will be used in the proof of Theorem 2 (b). Its proof goes
along the same lines as that of [16, Corollary 5.5].

######  Corollary 5.5.

Let Assumptions 1.1 and 1.2 be true. Then, there are constants $h_{0}>0$ and
$C>0$ such that for all $0<h\leqslant h_{0}$ and all temperatures
$T\geqslant{T_{\mathrm{c}}}(1+Ch^{2})$, we have
$\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})>0$
unless $\Gamma=\Gamma_{0}$.

The second corollary provides us with an inequality for Cooper pair wave
functions of low-energy BCS states in the sense of (5.1). The left side of
(5.16) appears as a lower bound for the full BCS functional. Despite of its
apparent simplicity, it still contains all the information needed for a proof
of Theorem 5.1. Before we state the corollary, let us define the operator

$\displaystyle U$
$\displaystyle\coloneqq\mathrm{e}^{-\mathrm{i}\frac{r}{2}\Pi_{X}},$ (5.15)

with $\Pi_{X}$ in (2.4), which acts on the relative coordinate $r=x-y$ as well
as on the center-of-mass coordinate $X=\frac{x+y}{2}$ of a function
$\alpha(X,r)$.

######  Corollary 5.6.

Let Assumptions 1.1 and 1.2 be true. For any $D_{0},D_{1}\geqslant 0$, there
is a constant $h_{0}>0$ such that if $\Gamma$ is a low-energy state in the
sense that it satisfies (5.1), if $0<h\leqslant h_{0}$, and if $T$ is such
that $T-{T_{\mathrm{c}}}\geqslant-D_{0}h^{2}$, then $\alpha=\Gamma_{12}$ obeys

$\displaystyle\langle\alpha,[U(1-P)(1+\pi^{2})(1-P)U^{*}+U^{*}(1-P)(1+\pi^{2})(1-P)U]\alpha\rangle$
$\displaystyle\hskip
200.0pt+\operatorname{Tr}\bigl{[}(\alpha^{*}\alpha)^{2}\bigr{]}\leqslant
Ch^{2}\|\alpha\|_{2}^{2}+D_{1}h^{4},$ (5.16)

where $P=|\alpha_{*}\rangle\langle\alpha_{*}|$ and
$\pi=-\mathrm{i}\nabla_{r}+\mathbf{A}_{\mathbf{B}}(r)$ both act on the
relative coordinate $r$ of $\alpha(X,r)$.

In the statement of the corollary and in the following, we refrain from
equipping the operator $\pi$ and the projection
$P=|\alpha_{*}\rangle\langle\alpha_{*}|$ with an index $r$ although it acts on
the relative coordinate. This does not lead to confusion and keeps the
formulas readable. The proof of the corollary is inspired by the proof of [19,
Proposition 23].

###### Proof of Corollary 5.6.

In the following we use the notation
$\pi_{\mathbf{A}}^{x}=-\mathrm{i}\nabla_{x}+\mathbf{A}(x)$. We claim that

$\displaystyle\pi_{\mathbf{A}_{h}}^{x}$
$\displaystyle=U\pi_{\mathbf{A}^{+}_{h}}^{r}U^{*},$
$\displaystyle-\pi_{\mathbf{A}_{h}}^{y}$
$\displaystyle=U^{*}\pi_{\mathbf{A}^{-}_{h}}^{r}U,$ (5.17)

where $\pi_{\mathbf{A}^{\pm}}^{r}=-\mathrm{i}\nabla_{r}+\mathbf{A}^{\pm}(r)$
with

$\displaystyle\mathbf{A}^{\pm}(r)$
$\displaystyle\coloneqq\mathbf{A}_{e_{3}}(r)\pm A(X\pm r).$

To obtain (5.17), we denote $P_{X}=-\mathrm{i}\nabla_{X}$ and note that
$[r\cdot P_{X},r\cdot(\mathbf{B}\wedge X)]=0$ implies
$U=\mathrm{e}^{\mathrm{i}\frac{\mathbf{B}}{2}(X\wedge
r)}\mathrm{e}^{-\mathrm{i}\frac{r}{2}P_{X}}$. Using this identity we conclude
that

$\displaystyle U\;(-\mathrm{i}\nabla_{r}+\mathbf{A}_{\mathbf{B}}(r))\,U^{*}$
$\displaystyle=-\mathrm{i}\nabla_{r}+\frac{1}{2}\mathbf{A}_{\mathbf{B}}(r)-\frac{1}{2}\Pi_{X},$
$\displaystyle U^{*}(-\mathrm{i}\nabla_{r}+\mathbf{A}_{\mathbf{B}}(r))\,U\;$
$\displaystyle=-\mathrm{i}\nabla_{r}+\frac{1}{2}\mathbf{A}_{\mathbf{B}}(r)+\frac{1}{2}\Pi_{X},$

where $\Pi=-\mathrm{i}\nabla+2\mathbf{A}_{\mathbf{B}}$. Eq. (5.17) is a direct
consequence of these two identities.

We also have

$\displaystyle W(x)$ $\displaystyle=U\,W^{+}(r)\,U^{*},$ $\displaystyle W(y)$
$\displaystyle=U^{*}\,W^{-}(r)\,U,$ $\displaystyle W^{\pm}(r)$
$\displaystyle\coloneqq W(X\pm r).$

Consequently, if $K_{T,\mathbf{A},W}^{x}$ and $K_{T,\mathbf{A},W}^{y}$ denote
the operators $K_{T,\mathbf{A},W}$ acting on the $x$ and $y$ coordinate,
respectively, we infer

$\displaystyle K_{T,\mathbf{A},W}^{x}-V(r)$
$\displaystyle=U^{*}(K_{T,\mathbf{A}^{+},W^{+}}^{r}-V(r))\,U,$ $\displaystyle
K_{T,\mathbf{A},W}^{y}-V(r)$
$\displaystyle=U\;(K_{T,\mathbf{A}^{-},W^{-}}^{r}-V(r))\,U^{*}.$ (5.18)

We highlight that $\mathbf{A}^{\pm}$ and $W^{\pm}$ depend on $X$.

The operator $V$ in (5.12) acts by multiplication with the function $V(x-y)$.
We use the symmetry $\alpha(x,y)=\alpha(y,x)$ to deduce

$\displaystyle\operatorname{Tr}\bigl{[}(K_{T,\mathbf{A},W}-V)\alpha\alpha^{*}\bigr{]}$
$\displaystyle=\frac{1}{2}\fint_{Q_{h}}\mathrm{d}x\int_{\mathbb{R}^{3}}\mathrm{d}y\;\overline{\alpha(x,y)}\bigl{[}(K_{T,\mathbf{A},W}^{x}-V)+(K_{T,\mathbf{A},W}^{y}-V)\bigr{]}\alpha(x,y).$
(5.19)

In combination, (5.1), (5.12), (5.18), and (5.19) therefore prove the bound

$\displaystyle\frac{1}{2}\langle\alpha,[U^{*}(K_{T,\mathbf{A}^{+},W^{+}}^{r}-V(r))\,U+U\;(K_{T,\mathbf{A}^{-},W^{-}}^{r}-V(r))\,U^{*}]\alpha\rangle+c\operatorname{Tr}\bigl{[}(\alpha^{*}\alpha)^{2}\bigr{]}\leqslant
D_{1}h^{4}.$

An application of Lemma 5.4 on the left side establishes (5.16). ∎

###  5.2 Proof of Theorem 5.1

The statement in Theorem 5.1 is a direct consequence of Corollary 5.6 above
and the results in [16]. More precisely, we need to combine Corollary 5.6,
[16, Proposition 5.7], [16, Lemma 5.14], and the arguments in [16, Section
5.4].

##  6 The Lower Bound on (1.22) and Proof of Theorem 2 (b)

###  6.1 The BCS energy of low-energy states

In this section, we complete the proofs of Theorems 1 and 2, which amounts to
providing the lower bound on (1.22), the bound in (1.27), and the proof of
Theorem 2 (b). Since these proofs mostly go along the same lines as those in
[16], we only mention the differences and keep the presentation to a minimal
length. Once the a priori estimates in [16, Theorem 5.1] are proved, the
proofs of the lower bound for the free energy, the decomposition of the Cooper
pair wave function of an approximate minimizer, and the upper bound for the
critical temperature shift in [16] follow the same strategy as the related
proofs in [21, 22]. In the following we will, however, only refer to [16]
because our presentation is closer to the analysis in this reference than to
those in [21, 22].

Let $D_{1}\geqslant 0$ and $D\in\mathbb{R}$ be given, choose
$T={T_{\mathrm{c}}}(1-Dh^{2})$, and assume that $\Gamma$ is a gauge-periodic
state that satisfies (5.1). Corollary 5.2 guarantees a decomposition of the
Cooper pair wave function $\alpha=[\Gamma]_{12}$ in terms of
$\Psi_{\leqslant}$ in (5.4) and $\sigma$ in (5.9). The function
$\Psi_{\leqslant}$ satisfies the bounds

$\displaystyle\|\Psi_{\leqslant}\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$
$\displaystyle\leqslant\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}\leqslant C,$
$\displaystyle\|\Psi_{\leqslant}\|_{H_{\mathrm{mag}}^{2}(Q_{h})}^{2}$
$\displaystyle\leqslant C\varepsilon
h^{-2}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2},$ (6.1)

with $\Psi$ in (5.2). Let us define the state $\Gamma_{\Delta}$ as in (3.4)
with $\Delta(X,r)=-2V\alpha_{*}(r)\Psi_{\leqslant}(X)$. We apply Proposition
3.5 and Theorem 3.6 to obtain the following lower bound for the BCS energy of
$\Gamma$:

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})$
$\displaystyle\geqslant
h^{4}\;\mathcal{E}^{\mathrm{GL}}_{D}(\Psi_{\leqslant})-C\left(h^{5}+\varepsilon
h^{4}\right)\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}$ $\displaystyle\hskip
20.0pt+\frac{T}{2}\mathcal{H}_{0}(\Gamma,\Gamma_{\Delta})-\fint_{Q_{h}}\mathrm{d}X\int_{\mathbb{R}^{3}}\mathrm{d}r\;V(r)\,|\sigma(X,r)|^{2}.$
(6.2)

In the next section we prove a lower bound for the terms in the second line of
(6.2).

###  6.2 Estimate on the relative entropy

The arguments in [16, Eqs. (6.1)-(6.14)] apply in literally the same way here,
too. We obtain the correct bounds when we replace $B$ by $h^{2}$ in all
formulas. This, in particular, applies to the statement of [16, Lemma 6.2].
The only difference is that [16, Eq. (6.10)] is now given by

$|\langle\eta_{0},K_{T_{\mathrm{c}},\mathbf{A},W}\sigma\rangle|\leqslant\
C\varepsilon^{-\nicefrac{{1}}{{2}}}h^{\nicefrac{{9}}{{2}}}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}\bigl{(}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}+D_{1}\bigr{)}^{\nicefrac{{1}}{{2}}},$

which is due to the reason that the bound for the $L^{2}$-norm of $\eta_{0}$
in Proposition 3.2 is worse than the comparable bound we obtained in [16,
Proposition 3.2]. This, however, does not change the size of the remainder in
the final bound because other error terms come with a worse rate.

With the choice $\varepsilon=h^{\nicefrac{{1}}{{3}}}$, we therefore obtain the
bound

$\displaystyle\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma)-\mathcal{F}^{\mathrm{BCS}}_{h,T}(\Gamma_{0})$
$\displaystyle\hskip 56.9055pt\geqslant
h^{4}\,\bigl{(}\mathcal{E}^{\mathrm{GL}}_{D}(\Psi_{\leqslant})-C\,h^{\nicefrac{{1}}{{6}}}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}\;\bigl{(}\|\Psi\|_{H_{\mathrm{mag}}^{1}(Q_{h})}^{2}+D_{1}\bigr{)}^{\nicefrac{{1}}{{2}}}\bigr{)},$
(6.3)

which is the equivalent of [16, Eq (6.14)].

###  6.3 Conclusion

The arguments in [16, Section 6.3] apply in the same way also here and we
obtain the correct formulas when we replace $B^{\nicefrac{{1}}{{2}}}$ by $h$.
This concludes the proof of Theorem 1 and Theorem 2.

###  6.4 Proof of the equivalent of [16, Lemma 6.2] in our setting

To obtain a proof of the equivalent of [16, Lemma 6.2] in our setting, we
follow the proof strategy in [16]. The additional terms coming from the