2.1 Background Lattice

Copper oxide superconductors have a relatively complex three-dimensional lattice structure. However, they always contain parallel two-dimensional layers of copper ( $\mathrm {Cu}^{++}$ ) and oxygen ( $\mathrm {O}^{--}$ ) ions. These $\mathrm {CuO}_{2}$ layers are essential to understanding low-temperature superconducting properties because the (superconducting) charge transport takes place within the layers. This is explained in [Reference Saxena12, Reference Plakida13, Reference Wesche14]. Considering a weak inter-layer interaction might also help to increase prediction accuracy, but charge transport between each $\mathrm {CuO}_{2}$ layer or, more generally, in the direction orthogonal to each layer remains negligible.Footnote ⁴ Each $\mathrm {CuO}_{2}$ layer generally has the symmetries of the square. In other words, it is invariant under the group $\{0,\pi /2,\pi ,3\pi /2\}$ generated by $90^{\circ }$ -degree rotations. See, for example, (Reference Tsuei, Kirtley, Bennemann and Ketterson31, Section 9.1.2), [Reference Saxena12, Section 2.3] and (Reference Wesche14, Section 6.3.1). This is an important symmetry property that we keep in mind throughout our study.

Having in mind these physical observations on cuprates, we consider here quantum particles on lattices $\mathbb {Z}^{2}$ . It means in particular that (disregarding internal degrees of freedom of the quantum particles, like their spin) the (separable) Hilbert space $\ell ^{2}(\mathbb {Z}^{2})$ is the ‘one-particle space’ associated with the physical system we are interested in. Its canonical orthonormal basis is $\{ \mathfrak {e}_{x}\}_{x\in \mathbb {Z}^{2}}$ :

(5) $$ \begin{align} \mathfrak{e}_{x}\left( y\right) \doteq \delta _{x,y},\qquad x,y\in \mathbb{ Z}^{2}, \end{align} $$

where $\delta _{\mathfrak {i},\mathfrak {j}}$ is the Kronecker delta.

2.2 Composite of two fermions and one boson

We consider a system of two fermions (electrons or holes in cuprates) with opposite spins interacting via the exchange of one boson in a two-dimensional square lattice. Physically, the boson that we have in mind in cuprate superconductors is a spinless bipolaron, since the very strong Jahn-Teller (JT) effect associated with copper ions is an important property of such cuprates [Reference Müller and Bednorz16, Reference Köppel, Yarkony and Barentzen15]. See Section A.1 for more details. However, the exchanged spinless boson could be of any type, like a phonon or a spin wave, depending on the physical system and mechanism one has in mind.

Hilbert Spaces. All quantum particles possess an intrinsic form of angular momentum known as spin, which is characterized by a quantum number $\mathfrak {s}\in \mathbb {N}/2$ and a finite spin setFootnote ⁵ $ \mathrm {S}\doteq \{-\mathfrak {s},-\mathfrak {s}+1,\ldots \mathfrak {s}-1, \mathfrak {s\}}\subseteq \mathbb {N}$ . If $\mathfrak {s}\notin \mathbb {N}$ is half-integer, then the corresponding particles are named fermions while $\mathfrak {s}\in \mathbb {N}$ means by definition that we have bosons. For example, photons or spinless bipolarons ( $\mathfrak {s}=0$ ) are bosons, while electrons ( $\mathfrak {s}=1/2$ ) are fermions. In the latter case, $\mathrm {S}\doteq \{-1/2,1/2\}$ , and in physics, the spin set is always written as $\mathrm {S}\equiv \{\uparrow ,\downarrow \}$ , and we thus use this completely standard notation. By the celebrated spin-statistics theorem, fermionic wave functions are antisymmetric with respect to permutations of particles, whereas the bosonic ones are symmetric.

Therefore, the one-particle Hilbert space for the fermions is $\ell ^{2}( \mathbb {Z}^{2}\times \{\uparrow ,\downarrow \})$ , $\{\uparrow ,\downarrow \}$ being the usual spin set for electrons or holes, and for two fermions, we hence use the Hilbert space

$$ \begin{align*} \mathfrak{h}_{f}\doteq \bigwedge\nolimits^{2}\ell ^{2}(\mathbb{Z}^{2}\times \{\uparrow ,\downarrow \})\subseteq \mathfrak{F}_{-}\equiv \mathfrak{F} \left( \ell ^{2}(\mathbb{Z}^{2}\times \{\uparrow ,\downarrow \})\right) \end{align*} $$

of antisymmetric functions,Footnote ⁶ which is a subspace of the fermionic ( $-$ ) Fock spaceFootnote ⁷ $\mathfrak {F}_{-}$ associated with the one-particle Hilbert space $ \ell ^{2}(\mathbb {Z}^{2}\times \{\uparrow ,\downarrow \})$ . See Equation (A.4) below for the precise definition of $\mathfrak {F}_{-}$ . The one-particle Hilbert space of the spinless boson is $\ell ^{2}(\mathbb {Z} ^{2})$ , which can also be seen as a subspace of the bosonic ( $+$ ) Fock space

$$ \begin{align*} \mathfrak{F}_{+}\equiv \mathfrak{F}\left( \ell ^{2}(\mathbb{Z}^{2})\right) \end{align*} $$

associated with $\ell ^{2}(\mathbb {Z}^{2})$ . See Equation (A.5) below for the precise definition of $\mathfrak {F}_{+}$ . For a concise review of bosonic and fermionic Fock spaces, as well as the corresponding annihilation and creation operators, see Section A.2.

We study here the effect of processes of annihilation of two fermions of opposite spins to create a boson, which can conversely be annihilated to create two new fermions. The Hilbert space associated with this composite system, made of two fermions and one boson, is the direct sum $\mathfrak {h} _{f}\oplus \ell ^{2}(\mathbb {Z}^{2})$ , and not the tensor product $\mathfrak { h}_{f}\otimes \ell ^{2}(\mathbb {Z}^{2})$ . Note indeed that $\mathfrak {h} _{f}\oplus \ell ^{2}(\mathbb {Z}^{2})$ can naturally be identifiedFootnote ⁸ with a subspace of $\mathfrak {F} _{-}\otimes \mathfrak {F}_{+}$ . This fact already unveils the strong interdependence of the bosonic and fermionic parts. For this reason, from now on, we rather use the term ‘composite of two fermions and one boson instead of ‘three-body system’, in order to avoid any misinterpretation.

Fermionic Hamiltonian. The fermionic part of the (infinite volume) Hamiltonian of the composite is defined to be the restriction $H_{f}\in \mathcal {B}(\mathfrak {h}_{f})$ of the formal expression

(6) $$ \begin{align} -{\frac{\epsilon }{2}}\sum_{s\in \{\uparrow ,\downarrow \},\ x,y\in \mathbb{Z }^{2}:|x-y|=1}a_{x,s}^{\ast }a_{y,s}+2\epsilon \sum\limits_{s\in \{\uparrow ,\downarrow \},\,x\in \mathbb{Z}^{2}}a_{x,s}^{\ast }a_{x,s}+\mathrm{U} \sum_{x\in \mathbb{Z}^{2}}n_{x,\uparrow }n_{x,\downarrow }+\sum_{x,z\in \mathbb{Z}^{2}}\mathrm{u}\left( z\right) n_{x,\uparrow }n_{x+z,\downarrow } \end{align} $$

to the Hilbert space $\mathfrak {h}_{f}$ . Here, $a_{x,s}$ ( $a_{x,s}^{\ast }$ ) denotes the annihilation (creation) operator acting on the fermionic Fock space $\mathfrak {F}_{-}$ of a fermion at lattice position $x\in \mathbb {Z} ^{2}$ , the spin of which is $s\in \{\uparrow ,\downarrow \}$ . As is usual, $ n_{x,s}\doteq a_{x,s}^{\ast }a_{x,s}$ stands for the number operator of fermions at lattice position $x\in \mathbb {Z}^{2}$ and spin $s\in \{\uparrow ,\downarrow \}$ .

The parameter $\epsilon \in \mathbb {R}_{0}^{+}$ quantifies the hopping amplitude of fermions. In high- $T_{c}$ superconductors [Reference Imada, Fujimori and Tokura29, Reference Keimer, Kivelson, Norman, Uchida and Zaanen30], $\epsilon $ is expected to be much smaller than the fermion-fermion interaction energy – more precisely the on-site repulsion strength $\mathrm {U}\in \mathbb {R}_{0}^{+}$ . The function $\mathrm {u}: \mathbb {Z}^{2}\rightarrow \mathbb {R}_{0}^{+}$ , which represents the fermion-fermion repulsion at all distances, is absolutely summable and invariant with respect to $90^{\circ }$ -rotations, that is,

(7) $$ \begin{align} \sum_{z\in \mathbb{Z}^{2}}\left\vert \mathrm{u}\left( z\right) \right\vert <\infty \qquad \text{and}\qquad \mathrm{u}\left( x,y\right) =\mathrm{u} \left( -y,x\right) ,\qquad x,y\in \mathbb{Z}. \end{align} $$

Clearly, one could set $\mathrm {U}=0$ , by redefining the coupling function $ \mathrm {u}:\mathbb {Z}^{2}\rightarrow \mathbb {R}_{0}^{+}$ . It is, however, convenient to have a separate parameter $\mathrm {U}\in \mathbb {R}_{0}^{+}$ for the on-site repulsion because we shall later on consider the ‘hard-core limit’ $\mathrm {U}\rightarrow \infty $ for some fixed coupling function $\mathrm {u}$ .

Extended Hubbard interactions. The above fermion-fermion interactions have been extensively studied in condensed matter physics during the last decade, in particular for two-dimensional systems. For nonzero functions $\mathrm {u}$ , they are named extended Hubbard interactions and they can drastically change the behavior of the system, as compared to the zero-range case (usual Hubbard interaction, $\mathrm {u}=0$ ). As one example, they are used in the context of ultracold atoms, ions and molecules [Reference Dutta, Gajda, Hauke, Lewenstein, Lühmann, Malomed, Sowiński and Zakrzewski11]. Its bosonic version is also experimentally investigated. See, for example, [Reference Lagoin, Bhattacharya, Grass, Chhajlany, Salamon, Baldwin, Pfeiffer, Lewenstein, Holzmann and Dubin32] published in 2022.

In theoretical studies, frequently, only nearest-neighbor interactions added to the on-site (zero-range) Hubbard interactions are considered. Here, we do not need the restriction to one-range (nearest-neighbor) interactions. We only assume that $\mathrm {u}$ is absolutely summable (see (7)), which is physically a very mild restriction, since the effective two-particle repulsive electrostatic potential in crystals is expected to decay exponentially fast in space, because of screening effects.

The rotation invariance in Equation (7) refers to the isotropy of the system under consideration. However, as shown in [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21], the system has low energy states that spontaneously break the isotropy. This refers to unconventional parings, typically of d -wave type, of electrons one experimentally observes in many high- $T_{c}$ superconductors [Reference Tsuei and Kirtley33, Reference Keimer, Kivelson, Norman, Uchida and Zaanen30, Reference Plakida13]. In fact, to derive the existence of d- and p-wave pairings starting from a physically sound microscopic model was the aim of [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21]. Here, instead, we keep a broader perspective and do not study this particular question.

Bosonic Hamiltonian. Similar to the fermionic part, the bosonic part of the (infinite volume) Hamiltonian of the system is defined to be the restriction $H_{b}\in \mathcal {B}(\ell ^{2}(\mathbb {Z}^{2}))$ to the one-boson Hilbert space $\ell ^{2}(\mathbb {Z}^{2})$ of the formal expression

(8) $$ \begin{align} \epsilon \left( -{\frac{h_{b}}{2}\sum\limits_{x,y\in \mathbb{Z} ^{2}\,:\,|x-y|=1}}b_{x}^{\ast }\,b_{y}\,+2h_{b}{\sum\limits_{x\in \mathbb{Z} ^{2}}}b_{x}^{\ast }\,b_{x}\right) \text{ }. \end{align} $$

Here, $b_{x}$ ( $b_{x}^{\ast }$ ) denotes the annihilation (creation) operator acting on the bosonic Fock space $\mathfrak {F}_{+}$ of a boson at lattice position $x\in \mathbb {Z}^{2}$ . Observe that the bosonic part only contains a kinetic term. The parameter $h_{b}\in \mathbb {R}_{0}^{+}$ quantifies the ratio of the effective masses of fermions and bosons: Taking $h_{b}$ smaller than one physically means that the bosons are heavier than the fermions. As experimentally found [Reference Dzhumanov34, Reference Dzhumanov35, Reference Dzhumanov, Karimboev and Djumanov36, Reference Reagor37] for cuprate superconductors, bipolarons should be much more massive than electrons or holes, and, thus, in the physically relevant regime, $h_{b}$ is to be taken very small (or even zero, in an idealized situation). See (Reference Bru, de Siqueira Pedra and de Pasquale21, Section 3.1). In the sequel, we take $h_{b}\in \lbrack 0,1/2]$ , meaning that the boson mass is at least as big as the mass of two fermions, as discussed in Section 3.

Exchange interactions. The term of the Hamiltonian that encodes the decay of a boson into two fermions (i.e., one of the two-electron(hole)-bipolaron-exchange interaction of the Hamiltonian) refers to the bounded operator

(9) $$ \begin{align} W_{\mathrm{b\rightarrow f}}:\ell ^{2}\left( \mathbb{Z}^{2}\right) \rightarrow \mathfrak{h}_{f}, \end{align} $$

which is defined to be the restriction of the formal expression

(10) $$ \begin{align} {2^{-1/2}\sum\limits_{x,y\in \mathbb{Z}^{2}}}\upsilon \left( x-y\right) c_{y}^{\ast }\,b_{x} \end{align} $$

to $\ell ^{2}(\mathbb {Z}^{2})$ , where

(11) $$ \begin{align} c_{y}^{\ast }\doteq {\sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p} _{1}\left( z\right) a_{y+z,\uparrow }^{\ast }\,a_{y,\downarrow }^{\ast }+ \mathfrak{p}_{2}\left( 2z\right) a_{y+z,\uparrow }^{\ast }\,a_{y-z,\downarrow }^{\ast }\right) \end{align} $$

for some fixed functions $\mathfrak {p}_{1},\mathfrak {p}_{2}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}$ that are invariant under $90^{\circ }$ -rotations and exponentially decay in space, that is,

(12) $$ \begin{align} {\sum_{z\in \mathbb{Z}^{2}}}\,\mathrm{e}^{\alpha _{0}\left\vert z\right\vert }\left\vert \mathfrak{p}_{\sharp }\left( z\right) \right\vert <\infty \qquad \text{and}\qquad \mathfrak{p}_{\sharp }\left( x,y\right) =\mathfrak{p} _{\sharp }\left( -y,x\right) ,\qquad x,y\in \mathbb{Z},\ \sharp \in \{1,2\}, \end{align} $$

for some $\alpha _{0}>0$ . In particular, the functions $\mathfrak {p}_{1}, \mathfrak {p}_{2}$ are absolutely summable in space.

By definition, we take $\mathfrak {p}_{2}\left ( z\right ) \doteq 0$ if $z\in \mathbb {Z}^{2}\backslash (2\mathbb {Z})^{2}$ and we also assume that

(13) $$ \begin{align} \mathfrak{p}_{1}+\mathfrak{p}_{2}\neq 0\qquad \mathrm{and}\qquad \mathfrak{p}_{2}\left( x\right) \neq -\mathrm{e}^{i\frac{k}{2}\cdot x}\mathfrak{p}_{1}\left( x\right) ,\qquad x\in \mathbb{Z},\ k\in \lbrack -\pi ,\pi )^{2}. \end{align} $$

The condition $\mathfrak {p}_{1}+\mathfrak {p}_{2}\neq 0$ only ensures the nontriviality of the exchange interaction, while the second condition avoids the singular case of a quasi-momentum $k_{0}\in \lbrack -\pi ,\pi )^{2}$ at which the exchange interaction trivially vanishes; see below (36). This case can easily be analyzed, but it makes the argumentation cumbersome. So, we omit it here, as it is a highly unusual and irrelevant situation. For example, (13) is already satisfied as soon as $\mathfrak {p}_{1}\left ( z\right ) \neq 0$ for some $z\in \mathbb {Z}^{2}\backslash (2\mathbb {Z})^{2}$ , since $\mathfrak {p}_{2}\left ( z\right ) \doteq 0$ for any $z\notin (2\mathbb {Z})^{2}$ . In (Reference Bru, de Siqueira Pedra and de Pasquale22, Eq. (6)), $\mathfrak {p}_{2}=0$ and, given $\kappa>0$ , $\mathfrak {p} _{1}(z)=\mathrm {e}^{-\kappa |z|}$ for $|z|\leq 1$ and $\mathfrak {p}_{1}(z)=0$ otherwise, while in [Reference Bru, de Siqueira Pedra and de Pasquale21, Eq. (4)], $\mathfrak {p}_{2}\left ( 2z\right ) =\mathfrak {p}_{1}\left ( z\right ) =1$ when $|z|\leq 1$ and $ \mathfrak {p}_{1}\left ( z\right ) =\mathfrak {p}_{2}\left ( z\right ) =0$ otherwise. This are the typical examples we have in mind, the point here being the fact two fermions on different lattice sites can interact by exchanging a boson. See also Section A.1.

Physically, $c_{y}^{\ast }$ represents the creation of a fermion pair of zero total spin, the two components of which are slightly spread around the lattice position $y\in \mathbb {Z}^{2}$ . Such pairs have finite size, because of (12). In fact,

(14) $$ \begin{align} r_{\mathfrak{p}}\doteq \frac{1}{2}\left( r_{\mathfrak{p}_{1}}+r_{\mathfrak{p} _{2}}\right) , \end{align} $$

where, for any $\sharp \in \{1,2\}$ , $r_{\mathfrak {p}_{\sharp }}\doteq 0$ if $\mathfrak {p}_{\sharp }=0$ , otherwise it is equal to

(15) $$ \begin{align} r_{\mathfrak{p}_{\sharp }}\doteq \frac{{\sum_{z\in \mathbb{Z}^{2}}} \,\left\vert z\right\vert \left\vert \mathfrak{p}_{\sharp }\left( z\right) \right\vert }{{\sum_{z\in \mathbb{Z}^{2}}}\,\left\vert \mathfrak{p}_{\sharp }\left( z\right) \right\vert }\leq \inf_{\alpha _{0}>0}\alpha _{0}^{-1}\sqrt{ \frac{{\sum_{z\in \mathbb{Z}^{2}}}\,\mathrm{e}^{\alpha _{0}\left\vert z\right\vert }\left\vert \mathfrak{p}_{\sharp }\left( z\right) \right\vert }{ {\sum_{z\in \mathbb{Z}^{2}}}\,\left\vert \mathfrak{p}_{\sharp }\left( z\right) \right\vert }}<\infty , \end{align} $$

is naturally seen as being the actual size of such pairs. Note that the last inequality is a consequence of the Cauchy-Schwarz inequality, along with the bound

$$ \begin{align*} \left\vert z\right\vert ^{2}\mathrm{e}^{-\alpha _{0}\left\vert z\right\vert }\leq \alpha _{0}^{-2},{\qquad }\alpha _{0}>0. \end{align*} $$

Allowing two fermions in different lattice site to interact by exchanging a boson simply means that $r_{\mathfrak {p}}>0$ . For the physical significance of this property for cuprates, see [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21].

Recall that the exchange strength function $\upsilon :\mathbb {Z} ^{2}\rightarrow \mathbb {R}$ is only absolutely summable, not necessarily exponentially decaying as $\mathfrak {p}_{1}$ and $\mathfrak {p}_{2}$ , and invariant under $90^{\circ }$ -rotations, that is,

(16) $$ \begin{align} {\sum_{z\in \mathbb{Z}^{2}}}\,\left\vert \upsilon \left( z\right) \right\vert <\infty \qquad \text{and}\qquad \upsilon \left( x,y\right) =\upsilon \left( -y,x\right) ,\qquad x,y\in \mathbb{Z}. \end{align} $$

Note that the Fourier transforms $\hat {\upsilon }$ , $\mathfrak {\hat {p}}_{1}$ and $\mathfrak {\hat {p}}_{2}$ of $\upsilon $ , $\mathfrak {p}_{1}$ and $ \mathfrak {p}_{2}$ are real-valued continuous functions (on the two-dimensional torus $\mathbb {T}^{2}$ ) that are again invariant under $ 90^{\circ }$ -rotations. Additionally, $\mathfrak {\hat {p}}_{1}$ and $ \mathfrak {\hat {p}}_{2}$ are real analytic, for $\mathfrak {p}_{1}$ and $ \mathfrak {p}_{2}$ are exponentially decaying. The reverse process – that is, the annihilation of two unbound fermions to form a boson – is represented by the adjoint operator

(17) $$ \begin{align} W_{\mathrm{f\rightarrow b}}\doteq W_{\mathrm{b\rightarrow f}}^{\ast }: \mathfrak{h}_{f}\rightarrow \ell ^{2}\left( \mathbb{Z}^{2}\right) . \end{align} $$

Mathematical remarks. The infinite sums (6), (8) and (10) defining formally $H_{f}\in \mathcal {B}\left ( \mathfrak {h}_{f}\right ) $ , $H_{b}\in \mathcal {B}(\ell ^{2}(\mathbb {Z}^{2}))$ and $W_{\mathrm {b\rightarrow f}}\in \mathcal {B}(\ell ^{2}(\mathbb {Z}^{2}),\mathfrak {h}_{f})$ (9) are to be understood as follows: If $\psi \in \mathfrak {h}_{f}$ or $\psi \in \ell ^{2}(\mathbb {Z}^{2})$ is a finitely supported function, then the sum corresponding to $H_{f}\psi $ , $H_{b}\psi $ or $W_{\mathrm {b\rightarrow f} }\psi $ is absolutely convergent. Thus, $H_{f}$ , $H_{b}$ and $W_{\mathrm { b\rightarrow f}}$ are well-defined linear operators acting on the dense subspace of such functions. One checks that $H_{f}$ , $H_{b}$ and $W_{ \mathrm {b\rightarrow f}}$ are all bounded on this subspace, and they thus have a unique bounded linear extension to the whole Hilbert space where they are defined – namely, $\mathfrak {h}_{f}$ for $H_{f}$ , and $\ell ^{2}(\mathbb {Z} ^{2})$ for $H_{b}$ and $W_{\mathrm {b\rightarrow f}}$ . We denote the extensions again by $H_{f}$ , $H_{b}$ and $W_{\mathrm {b\rightarrow f}}$ . Note that, being a bounded operator, $W_{\mathrm {b\rightarrow f}}$ has an adjoint (17), while $H_{f}$ and $H_{b}$ are clearly symmetric and so, self-adjoint, for they are also bounded.

Full model. Finally, the full Hamiltonian for the fermion-boson composite is defined, in matrix notation for the direct sum $ \mathfrak {h}_{f}\oplus \ell ^{2}(\mathbb {Z}^{2})$ , as follows:

(18) $$ \begin{align} \begin{pmatrix} H_{f} & W_{\mathrm{b\rightarrow f}} \\[0.5em] W_{\mathrm{f\rightarrow b}} & H_{b} \end{pmatrix} \in \mathcal{B}\left( \mathfrak{h}_{f}\oplus \ell ^{2}\left( \mathbb{Z} ^{2}\right) \right). \end{align} $$

Observe that this Hamiltonian is invariant under translations, as well as $ 90^{\circ }$ -rotations.

Using the canonical orthonormal basisFootnote ⁹

$$ \begin{align*} \left\{ \mathfrak{e}_{\left( x,s\right) }:x\in \mathbb{Z}^{2},s\in \{\uparrow ,\downarrow \}\right\} \subseteq \ell ^{2}\left( \mathbb{Z} ^{2}\times \{\uparrow ,\downarrow \}\right) \end{align*} $$

to define the closed subspace

(19) $$ \begin{align} \mathfrak{h}_{0}\doteq \text{\ }\overline{\mathrm{span}}\left\{ \mathfrak{e} _{(x,\uparrow )}\wedge \mathfrak{e}_{(y,\downarrow )}\,:\,x,y\in \mathbb{Z} ^{2}\right\} \subseteq \mathfrak{h}_{f}, \end{align} $$

we remark that the zero-spin subspace

(20) $$ \begin{align} \mathfrak{H}\doteq \mathfrak{h}_{0}\oplus \ell ^{2}\left( \mathbb{Z} ^{2}\right) \subseteq \mathfrak{h}_{f}\oplus \ell ^{2}\left( \mathbb{Z} ^{2}\right) \end{align} $$

is invariant under the action of the (full) Hamiltonian (18). We can thus consider its restriction

(21) $$ \begin{align} H\doteq \left. \begin{pmatrix} H_{f} & W_{\mathrm{b\rightarrow f}} \\[0.5em] W_{\mathrm{f\rightarrow b}} & H_{b} \end{pmatrix} \right\vert _{\mathfrak{H}}\in \mathcal{B}\left( \mathfrak{H}\right) \end{align} $$

to this particular subspace $\mathfrak {H}\subseteq \mathfrak {h}_{f}\oplus \ell ^{2}(\mathbb {Z}^{2})$ .

In fact, as the boson is assumed to be spinless, by the conservation of angular momentum, we have that the total spin of the fermion pair resulting from a bosonic decay must be zero. In other words, the physically relevant (vector) states of the fermion-boson compound system always lie in $ \mathfrak {H}$ . Note finally that H inherits the symmetries of the Hamiltonian (18) (i.e., H is invariant under translations and $ 90^{\circ }$ -rotations). Note that this last symmetry (i.e., the rotation invariance) is mainly relevant for the study of unconventional pairings, which is not done here.

2.3 The model in spaces of quasi-momenta

We have a composite of two fermions and one boson whose Hamiltonian is translation invariant. In this case, it is a standard procedure (see, for example, [Reference Reed and Simon38, Chapter XIII.16]) to use the direct integral decomposition of the Hamiltonian in Fourier space in order to study its spectral properties.

For the two-dimensional lattice $\mathbb {Z}^{2}$ , the (Fourier) space of quasi-momenta is nothing else than the torus

$$ \begin{align*} \mathbb{T}^{2}\doteq \lbrack -\pi ,\pi )^{2}\subseteq \mathbb{R}^{2}. \end{align*} $$

This set is endowed with the metric $d_{\mathbb {T}^{2}}$ defined by

(22) $$ \begin{align} d_{\mathbb{T}^{2}}\left( k,p\right) \doteq \min \left\{ \left\vert k-p-q\right\vert :q\in 2\pi \mathbb{Z}^{2}\right\} , \end{align} $$

where $|k-p-q|$ is the Euclidean distance between k and $p+q$ in $\mathbb {R }^{2}$ . This defines a compact metric space $(\mathbb {T}^{2},d_{\mathbb {T} ^{2}})$ . Observe also that the usual group operation in $\mathbb {T}^{2}$ (i.e., the sum in $\mathbb {R}^{2}$ modulo $(2\pi ,2\pi )$ ) is a continuous operation, while any Borel set in $\mathbb {T}^{2}$ is also a Borel set in $ \mathbb {R}^{2}$ (endowed with the Euclidean metric).

We also need the normalized Haar measure $\nu $ on $\mathbb {T}^{2}$ defined for any Borel set $B\subseteq \mathbb {T}^{2}$ by

(23) $$ \begin{align} \nu (B)=(2\pi )^{-2}\boldsymbol{\lambda }(B), \end{align} $$

where $\boldsymbol {\lambda }$ is the Lebesgue measure in $\mathbb {R}^{2}$ . This measure appears in relation with direct integrals of constant Hilbert spaces on the two-dimensional torus $\mathbb {T}^{2}$ , like the Hilbert space

$$ \begin{align*} L^{2}\left( \mathbb{T}^{2}\right) \equiv L^{2}\left( \mathbb{T}^{2},\mathbb{C }\right) \equiv L^{2}\left( \mathbb{T}^{2},\mathbb{C},\nu \right) \doteq { \int_{\mathbb{T}^{2}}^{\oplus }}\mathbb{C}\,\nu (\mathrm{d}k) \end{align*} $$

of square-integrable, complex-valued functions on $\mathbb {T}^{2}$ . Since the Haar measure $\nu $ is used in all our direct integrals on $\mathbb {T} ^{2}$ , for simplicity, we often remove the symbol $\nu $ from the notation of $L^{2}$ -spaces, unless this information is important to recall.

The Fourier transform can be applied in the fermionic and bosonic sectors. In the fermionic one, there is more than one natural way of implementing the transform, as the corresponding functions have two arguments in $\mathbb {Z} ^{2}$ . It turns out that to be very useful to extract the total quasi-momentum of fermionic pairs. In fact, we consider the direct integral

(24) $$ \begin{align} L^{2}\left( \mathbb{T}^{2},\mathcal{H}\right) \equiv L^{2}\left( \mathbb{T} ^{2},\mathcal{H},\nu \right) \doteq {\int_{\mathbb{T}^{2}}^{\oplus }} L^{2}\left( \mathbb{T}^{2},\mathbb{C},\nu \right) \oplus \mathbb{C}\,\nu ( \mathrm{d}k) \end{align} $$

of the (constant fiber) Hilbert space

(25) $$ \begin{align} \mathcal{H}\doteq L^{2}\left( \mathbb{T}^{2}\right) \oplus \mathbb{C}\equiv L^{2}\left( \mathbb{T}^{2},\mathbb{C},\nu \right) \oplus \mathbb{C} \end{align} $$

over the torus $\mathbb {T}^{2}$ , and choose a unitary transformation

$$ \begin{align*} \mathbb{U}:\mathfrak{H}\longrightarrow L^{2}\left( \mathbb{T}^{2},\mathcal{H} \right) \end{align*} $$

in such a way that $k\in \mathbb {T}^{2}$ , the fiber quasi-momentum, is exactly the total quasi-momentum of the fermion pair.

Recall that $\mathfrak {H}$ defined in (20) is the Hilbert space on which H is originally defined. More precisely,

(26) $$ \begin{align} \mathbb{U}\doteq U_{f}\oplus \mathcal{F}, \end{align} $$

where

(27) $$ \begin{align} \mathcal{F}:\ell ^{2}\left( \mathbb{Z}^{2}\right) \rightarrow L^{2}\left( \mathbb{T}^{2}\right) \end{align} $$

is the Fourier transform on $\ell ^{2}(\mathbb {Z}^{2})$ , while the fermionic part

(28) $$ \begin{align} U_{f}\doteq U_{2}U_{1}:\mathfrak{h}_{0}\rightarrow \int_{\mathbb{T} ^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \nu \left( \mathrm{d} k\right) \end{align} $$

is the composition of two unitary (linear) transformations $U_{1}$ and $ U_{2} $ , whose exact definitions are given as follows:

(29) $$ \begin{align} \begin{array}{cccccccc} U_{1}: & \mathfrak{h}_{0} & \rightarrow & \ell ^{2}\left( \mathbb{Z} ^{2}\times \mathbb{Z}^{2}\right) & \rightarrow & \ell ^{2}\left( \mathbb{Z} ^{2}\times \mathbb{Z}^{2}\right) & \rightarrow & \ell ^{2}\left( \mathbb{Z} ^{2}\right) \otimes \ell ^{2}\left( \mathbb{Z}^{2}\right) \\ & \mathfrak{e}_{(x,\uparrow )}\wedge \mathfrak{e}_{(y,\downarrow )} & \mapsto & \mathfrak{e}_{(x,y)} & \mapsto & \mathfrak{e}_{(x,x-y)} & \mapsto & \mathfrak{e}_{x}\otimes \mathfrak{e}_{x-y} \end{array} \end{align} $$

and

(30) $$ \begin{align} \begin{array}{cccccc} U_{2}: & \ell ^{2}\left( \mathbb{Z}^{2}\right) \otimes \ell ^{2}\left( \mathbb{Z}^{2}\right) & \rightarrow & L^{2}\left( \mathbb{T}^{2}\right) \otimes L^{2}\left( \mathbb{T}^{2}\right) & \rightarrow & \int_{\mathbb{T} ^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \nu \left( \mathrm{d} k\right) \\ & \mathfrak{e}_{x}\otimes \mathfrak{e}_{x-y} & \mapsto & \mathfrak{\hat{e}} _{x}\otimes \mathfrak{\hat{e}}_{x-y} & \mapsto & \mathfrak{\hat{e}} _{x}\left( \cdot \right) \mathfrak{\hat{e}}_{x-y} \end{array} . \end{align} $$

Because $\{\mathfrak {e}_{(x,\uparrow )}\wedge \mathfrak {e}_{(y,\downarrow )}\}_{x,y\in \mathbb {Z}^{2}}$ , $\{\mathfrak {e}_{(x,y)}\}_{x,y\in \mathbb {Z} ^{2}}$ and $\{\mathfrak {e}_{x}\otimes \mathfrak {e}_{y}\}_{x,y\in \mathbb {Z} ^{2}}$ are orthonormal bases and $(x,y)\mapsto (x,x-y)$ is a bijection on $ \mathbb {Z}^{2}\times \mathbb {Z}^{2}$ , $U_{1}$ is well-defined as a composition of three unitary linear transformations. Note also that the last unitary linear transformation defining $U_{2}$ is defined as in Proposition A.8, while the first one defining $ U_{2}$ is the tensor product $\mathcal {F}\otimes \mathcal {F}$ of the Fourier transform $\mathcal {F}$ on $\ell ^{2}(\mathbb {Z}^{2})$ , defined for any $ f\in \ell ^{1}(\mathbb {Z}^{2})\subseteq \ell ^{2}(\mathbb {Z}^{2})$ by

(31) $$ \begin{align} \hat{f}\left( k\right) \equiv \mathcal{F}f\left( k\right) ={\sum_{x\in \mathbb{Z}^{2}}}\,\mathrm{e}^{ik\cdot x}f\left( x\right) ,\qquad k\in \mathbb{T}^{2}, \end{align} $$

$k\cdot x$ being the usual scalar product of $k\in \mathbb {T}^{2}$ and $x\in \mathbb {Z}^{2}$ , seen as vectors of $\mathbb {R}^{2}$ . Here, we use the symbol $\widehat {\left ( \cdot \right ) }$ to shorten the notation of the Fourier transform. For instance, for any $x\in \mathbb {Z}^{2}$ , we write above $\hat { \mathfrak {e}}_{x}$ to denote the function $\mathrm {e}^{i(\cdot )\cdot x}$ on the torus $\mathbb {T}^{2}$ . That is, $\{\hat {\mathfrak {e}}_{x}\}_{x\in \mathbb {Z}^{2}}$ is the image under the Fourier transform of the canonical orthonormal basis $\{\mathfrak {e}_{x}\}_{x\in \mathbb {Z}^{2}}$ (5) of $\ell ^{2}(\mathbb {Z}^{2})$ .

For the reader’s convenience and completeness, in Section A.4, we gather key results from the theory of direct integrals with constant fiber Hilbert spaces. In the next subsection, we explain how the properties of the Hamiltonian $H\in \mathcal {B}(\mathfrak {H} ) $ defined by (21) can be studied on the direct integral (24) over total quasi-momenta.

2.4 Fiber decomposition of the Hamiltonian

By explicit computations, exactly like in [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21], we show that the conjugation of the Hamiltonian $H\in \mathcal {B}(\mathfrak {H})$ with the unitary transformation $\mathbb {U}$ of Equation (26) is a decomposable operator on the direct integral $L^{2}(\mathbb {T}^{2},\mathcal {H })$ . To state this result precisely, we need preliminary definitions allowing to define the so-called ‘fiber Hamiltonians’, or ‘fibers’ for short, $A(k)\in \mathcal {B}(\mathcal {H})$ of $ \mathbb {U}H\mathbb {U}^{\ast }$ at total quasi-momenta $k\in \mathbb {T}^{2}$ . In fact, the mapping $k\mapsto A(k)$ defines an element of the von Neumann algebraFootnote ¹⁰

$$ \begin{align*} L^{\infty }\left( \mathbb{T}^{2},\mathcal{B}(\mathcal{H})\right) \equiv L^{\infty }\left( \mathbb{T}^{2},\mathcal{B}(\mathcal{H}),\nu \right) \end{align*} $$

of (equivalence classes of) strongly measurable functions $\mathbb {T} ^{2}\rightarrow \mathcal {B}(\mathcal {H})$ . See Section A.4 for more details.

Given a total quasi-momentum $k\in \mathbb {T}^{2}$ and the parameters $ \epsilon ,h_{b}\in \mathbb {R}_{0}^{+}$ tuning the strengths of the two (fermionic and bosonic) kinetic parts of the model, we define continuous, real-valued functions $\mathfrak {f}(k),\mathfrak {d}(k),\mathfrak {b}\in C\left ( \mathbb {T}^{2}\right ) $ on the torus $\mathbb {T}^{2}$ by

(32) $$ \begin{align} \mathfrak{b}\left( p\right) &\doteq h_{b}\epsilon \left( 2-\cos \left( p\right) \right) , \end{align} $$

(33) $$ \begin{align} \mathfrak{f}\left( k\right) \left( p\right) &\doteq \epsilon \left\{ 4-\cos \left( p+k\right) -\cos \left( p\right) \right\} , \end{align} $$

(34) $$ \begin{align} \mathfrak{d}\left( k\right) \left( p\right) &\doteq \mathfrak{\hat{p}} _{1}\left( k+p\right) +\mathfrak{\hat{p}}_{2}\left( k/2+p\right) , \end{align} $$

for all $p=(p_{1},p_{2})\in \mathbb {T}^{2}$ , where

(35) $$ \begin{align} \cos \left( q\right) \doteq \cos \left( q_{1}\right) +\cos \left( q_{2}\right) ,\qquad q=(q_{1},q_{2})\in \mathbb{R}^{2}. \end{align} $$

Recall that (the $(2\pi ,2\pi )$ -periodic function) $\mathfrak {\hat {p}}_{1}$ and $\mathfrak {\hat {p}}_{2}$ are the Fourier transform of $\mathfrak {p}_{1}$ and $\mathfrak {p}_{2}$ , which are the functions defining the operator $ c_{y}^{\ast }$ in (11), representing the creation of fermion pairs in the model. Note also from (13) that $ \mathfrak {d}(k)\neq 0$ for all $k\in \mathbb {T}^{2}$ . Indeed, using $ \mathfrak {p}_{2}(z)\doteq 0$ for $z\notin 2\mathbb {Z}$ as well as (31) and (34),

(36) $$ \begin{align} \mathfrak{d}\left( k\right) =\mathcal{F}\left[ \mathrm{e}^{ik\cdot x} \mathfrak{p}_{1}\left( x\right) +\mathrm{e}^{i\frac{k}{2}\cdot x}\mathfrak{p} _{2}\left( x\right) \right] , \end{align} $$

where $\mathrm {e}^{ik\cdot x}\mathfrak {p}_{\sharp }(x)$ stands for the function $x\mapsto \mathrm {e}^{ik\cdot x}\mathfrak {p}_{\sharp }(x)$ with $ \sharp \in \{1,2\}$ .

Then, at any quasi-momentum $k\in \mathbb {T}^{2}$ and on-site repulsion strength $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , we define the bounded operators $B_{1,1}\left ( k\right ) $ and $A_{1,1}(\mathrm {U},k)$ acting on the Hilbert space $L^{2}\left ( \mathbb {T}^{2}\right ) $ by

(37) $$ \begin{align} B_{1,1}\left( k\right) &\doteq M_{\mathfrak{f}\left( k\right) }+{ \sum\limits_{x\in \mathbb{Z}^{2}}}\,\mathrm{u}\left( x\right) P_{x},\end{align} $$

(38) $$ \begin{align} A_{1,1}\left( \mathrm{U},k\right) &\doteq B_{1,1}\left( k\right) +\mathrm{U} P_{0}, \end{align} $$

where $M_{\mathfrak {f}(k)}$ stands for the multiplication operator by $ \mathfrak {f}(k)\in C(\mathbb {T}^{2})$ and $P_{x}$ is the orthogonal projection onto the one-dimensional subspace $\mathbb {C}\mathfrak {\hat {e}} _{x}\subseteq L^{2}(\mathbb {T}^{2})$ . Note that the infinite sum defining the bounded operator $B_{1,1}(k)$ is absolutely convergent, for the function $\mathrm {u}:\mathbb {Z}^{2}\rightarrow \mathbb {R}$ is, by assumption, absolutely summable. See (7).

We define next

(39) $$ \begin{align} & \begin{array}{cccl} A_{2,1}\left( k\right) : & L^{2}\left( \mathbb{T}^{2}\right) & \rightarrow & \mathbb{C} \\ & \varphi & \mapsto & \hat{\upsilon}\left( k\right) \left\langle \mathfrak{d} \left( k\right) ,\varphi \right\rangle , \end{array} \end{align} $$

(40) $$ \begin{align} & \begin{array}{@{\ \kern1pt}cccl} A_{1,2}\left( k\right) : & \mathbb{C} & \rightarrow & L^{2}\left( \mathbb{T} ^{2}\right) \\ & z & \mapsto & \hat{\upsilon}\left( k\right) \mathfrak{d}\left( k\right) z \end{array} \end{align} $$

as well as

(41) $$ \begin{align} \begin{array}{cccl} A_{2,2}\left( k\right) : & \mathbb{C} & \rightarrow & \mathbb{C} \\ & z & \mapsto & \mathfrak{b}\left( k\right) z \end{array} \end{align} $$

for any fixed $k\in \mathbb {T}^{2}$ . By compactness of $\mathbb {T}^{2}$ and continuity (in operator norm) of the mappings $k\mapsto A_{i,j}(k)$ for all $ i,j\in \{1,2\}$ , we have

(42) $$ \begin{align} A\left( \cdot \right) \equiv A\left( \mathrm{U},\cdot \right) \doteq \begin{pmatrix} A_{1,1}\left( \mathrm{U},\cdot \right) & A_{1,2}\left( \cdot \right) \\[0.5em] A_{2,1}\left( \cdot \right) & A_{2,2}\left( \cdot \right) \end{pmatrix} \in L^{\infty }\left( \mathbb{T}^{2},\mathcal{B}(\mathcal{H})\right) \end{align} $$

(see Lemma 4.1), which is meanwhile the fiber decomposition of the operator $\mathbb {U}H\mathbb {U}^{\ast }$ :

Proposition 2.1 (Fiber decomposition of the quantum model).

The conjugation of H by $\mathbb {U}$ (26) is decomposable and has $ A(\cdot )$ as its fibers; that is,

$$ \begin{align*} \mathbb{U}H\mathbb{U}^{\ast }={\int_{\mathbb{T}^{2}}^{\oplus }}A(k)\,\nu ( \mathrm{d}k). \end{align*} $$

The fiber decomposition given by Proposition 2.1 is useful because it gives access to spectral properties of H. In fact, for an operator that is decomposable on $L^{2}\left ( \mathbb {T}^{2},\mathcal {H}\right ) $ , that is, an operator unitarily equivalent to an element of the von Neumann algebra $L^{2}\left ( \mathbb {T}^{2},\mathcal {B}(\mathcal {H})\right ) $ , like the Hamiltonian H, the fibers $A(k)$ of which are all self-adjoint, it is known that $\lambda \in \sigma (H)$ if, and only if, for all $\varepsilon>0$ ,

$$ \begin{align*} \nu \left( \left\{ k\in \mathbb{T}^{2}:\sigma \left( A\left( k\right) \right) \cap \left( \lambda -\varepsilon ,\lambda +\varepsilon \right) \neq \emptyset \right\} \right)>0. \end{align*} $$

See Theorem A.3. As is usual, here, $\sigma (X)$ denotes the spectrum of any operator X acting on some Hilbert space.

3.1 Spectral properties

Having in mind Proposition 2.1 and Theorem A.3, we start with the spectral properties of fiber Hamiltonians (42) at any quasi-momentum $k\in \mathbb {T }^{2}$ . This refers to the following theorem:

Remark 3.2. Recall that if, for some natural number $d\geq 1$ ,

$$ \begin{align*} \sum_{x\in \mathbb{Z}^{2}}\left\vert x\right\vert ^{d}\left\vert \upsilon \left( x\right) \right\vert <\infty, \end{align*} $$

then the Fourier transform $\hat {\upsilon }$ of the function $\upsilon : \mathbb {Z}^{2}\rightarrow \mathbb {R}$ , as defined by (31), is of class $C^{d}$ on the whole torus $\mathbb {T}^{2}$ .

Remark 3.3. If $\mathfrak {p}_{1}=\mathfrak {p}_{2}\in \mathbb {C}\mathfrak {e}_{0}$ ,i.e., $ r_{\mathfrak {p}}=0$ , then Theorem 3.1 (iii) remains true, but not uniformly in $\mathrm {U}\in \mathbb {R}_{0}^{+}$ . That is, in this case, one only has

$$ \begin{align*} \min_{k\in \mathbb{T}^{2}}\left\{ \min \sigma _{\mathrm{ess}}\left( A\left( \mathrm{U},k\right) \right) -\mathrm{E}\left( \mathrm{U},k\right) \right\}>0, \end{align*} $$

and there are $C_{\mathrm {U}},\alpha _{\mathrm {U}}\in \mathbb {R}^{+}$ such that, for all $k\in \mathbb {T}^{2}$ ,

$$ \begin{align*} \left\vert \mathcal{F}^{-1}[\hat{\psi}_{\mathrm{U},k}]\left( x\right) \right\vert \leq C_{\mathrm{U}}\mathrm{e}^{-\alpha _{\mathrm{U}}|x|},\qquad x\in \mathbb{Z}^{2}. \end{align*} $$

Assertion (i) of Theorem 3.1 holds true for all $h_{b}\in \mathbb {R}_{0}^{+}$ , but the other assertions need the restriction $h_{b}\in \lbrack 0,1/2]$ to ensure that the eigenvalues are below the essential spectrum, as stated in Assertion (ii). In fact, $h_{b}\in \lbrack 0,1/2]$ iff

(43) $$ \begin{align} \mathfrak{b}\left( k\right) \leq \mathfrak{z}\left( k\right) \doteq 4\epsilon -2\epsilon \cos (k/2)=\min \sigma _{\mathrm{ess}}\left( A\left( \mathrm{U},k\right) \right) \end{align} $$

for all $k\in \mathbb {T}^{2}$ , with equality only at $k=0$ . See Equation (32). Therefore, by Assertion (ii), $\mathrm {E}(\mathrm {U},k)$ belongs to the essential spectrum iff $k=0$ and $\hat {\upsilon }(0)=0$ . Otherwise, we have a uniform spectral gap, as stated in Assertion (iii).

As is explained in [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21], the eigenvalue $\mathrm {E} \left ( \mathrm {U},k\right ) $ given by Theorem 3.1 is associated with the formation of dressed bound fermion pairs with total quasi-momentum $k\in \mathbb {T}^{2}$ . These pairs are generally exponentially localized, thanks to Theorem 3.1 (iii), which basically implies that the two fermions move together confined within some small ball; that is, they are tightly bound in space, provided $\hat {\upsilon }(0)\neq 0$ . When $\mathfrak {p}_{1}(z)\neq 0$ or $\mathfrak {p}_{2}(z)\neq 0$ for some $z\neq 0,$ or, equivalently, $r_{\mathfrak {p}}>0$ , the size of the small does not depend upon the Hubbard coupling constant $\mathrm {U}$ and a very large $\mathrm {U}\gg 1$ only prevents two fermions from occupying the same lattice site. The condition $\mathfrak {p}_{1}(z)\neq 0$ or $\mathfrak {p} _{2}(z)\neq 0$ for some $z\neq 0$ , or equivalently, $r_{\mathfrak {p}}>0$ , is therefore pivotal to get (Cooper) fermion pairs, the natural candidates for superconducting charge carriers, in presence of strong on-site Coulomb repulsions, like in cuprates [Reference Imada, Fujimori and Tokura29, Reference Keimer, Kivelson, Norman, Uchida and Zaanen30].

For the usual (i.e., nonextended) Hubbard interaction ( $\mathrm {u}=0$ ) and one-rangeFootnote ¹² creation operators $c_{y}^{\ast }$ of fermion pairs (in this case (12) holds true for all $\alpha _{0}\in \mathbb {R}_{0}^{+}$ ), note that a weak form of pair localization was previously shown in the ground state. See, for instance, [Reference Bru, de Siqueira Pedra and de Pasquale22, Theorem 3 and Proposition 13]. In this particular case, estimates of $\mathrm {E}\left ( \mathrm {U},k\right ) $ and $\Psi ( \mathrm {U},k)$ are known for large $\mathrm {U}\gg 1$ . See, for example, [Reference Bru, de Siqueira Pedra and de Pasquale22, Theorem 4, Corollary 5, Theorem 16]. Recall that the aim in [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21] was to show the existence of d- and p-wave pairings in the ground state for some physically sound model, and not the systematic study of a general class of models. In [Reference Bru, de Siqueira Pedra and de Pasquale21], we conjecture that such dressed bound fermion pairs represent the charge carriers below the pseudogap temperature in cuprates.

Theorem 3.1 combined with Proposition 2.1 and the theory of direct integrals (cf. Theorem A.3) has direct consequences for the spectrum of the full Hamiltonian $H\in \mathcal {B}(\mathfrak {H})$ , which is defined by Equation (21). Among other things, we obtain the following corollary:

Corollary 3.4 (Spectral properties of H).

Fix $\epsilon ,\mathrm {U}\in \mathbb {R}_{0}^{+}$ and $h_{b}\in \lbrack 0,1/2] $ . Then,

$$ \begin{align*} \sigma \left( H\right) \cap \left( -\infty ,8\epsilon \right] =\left\{ \mathrm{E}\left( \mathrm{U},k\right) :k\in \mathbb{T}^{2}\right\} \cup \left( 0,8\epsilon \right) , \end{align*} $$

where $\sigma \left ( H\right ) $ denotes, as is usual, the spectrum of H, and

$$ \begin{align*} \min \sigma \left( H\right) =E\left( \mathrm{U}\right) \doteq \min_{k\in \mathbb{T}^{2}}\mathrm{E}\left( \mathrm{U},k\right) \leq 0. \end{align*} $$

If additionally $\hat {\upsilon }(0)\neq 0$ and $r_{\mathfrak {p}}>0$ , then

$$ \begin{align*} \sup_{\mathrm{U}\in \mathbb{R}_{0}^{+}}E\left( \mathrm{U}\right) <0. \end{align*} $$

Proof. To prove the first assertion, it suffices to combine Proposition 2.1 and Theorem 3.1 with Theorem A.3. The second one can be proven like in [Reference Bru, de Siqueira Pedra and de Pasquale22] by using Kato’s perturbation theory [Reference Kato40]. In Proposition 4.11, we give an alternative and more direct proof of it. Finally, the last assertion is a consequence of the inequalities

$$ \begin{align*} \min_{k\in \mathbb{T}^{2}}\mathrm{E}\left( \mathrm{U},k\right) \leq \mathrm{E }\left( \mathrm{U},0\right) =\mathrm{E}\left( \mathrm{U},0\right) -\min \sigma _{\mathrm{ess}}\left( A\left( \mathrm{U},0\right) \right) \end{align*} $$

and Theorem 3.1 (iii).

Physically speaking, the spectral values of H represent the energy levels that are available to the composite of two fermions and one boson – in particular, for a fermion pair exchanging a boson. As expected, the minimum energy E, also well-known as the ground state energy, is given by minimizing the eigenvalues $\mathrm {E}(\mathrm {U},k)$ over the torus $ \mathbb {T}^{2}$ .

We now study the model at very large on-site repulsion $\mathrm {U}\gg 1$ . In fact, quoting [Reference Bru, de Siqueira Pedra and de Pasquale21], ‘in all cuprates, there is undeniable experimental evidence of strong on-site Coulomb repulsions, leading to the universally observed Mott transition at zero doping [Reference Imada, Fujimori and Tokura29, Reference Keimer, Kivelson, Norman, Uchida and Zaanen30]. This phase is characterized by a periodic distribution of fermions (electrons or holes) with exactly one particle per lattice site. Doping copper oxides with holes or electrons can prevent this situation. Instead, at sufficiently small temperatures a superconducting phase is achieved, as first discovered in 1986 for the copper oxide perovskite $ \mathrm {La}_{2-x}\mathrm {Ba}_{x}\mathrm {CuO}_{4}$ [Reference Müller and Bednorz16]’. However, instead of the usual s-wave superconductivity, one experimentally observes d-wave superconductivity [Reference Tsuei and Kirtley33, Reference Keimer, Kivelson, Norman, Uchida and Zaanen30, Reference Plakida13]. The fact that only the s-wave pairing is suppressed also advocates for a very local (i.e., on-site) and strong effective repulsion of fermions. For this reason, we consider the limit $\mathrm {U}\rightarrow \infty $ in our model. It corresponds to a hard core limit because it prevents two fermions from being on the same lattice site.

In the limit $\mathrm {U}\rightarrow \infty $ , it is easy to see that the ground state energy $E(\mathrm {U})$ of Corollary 3.4 defines an increasing function of $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , which is bounded from above by $0$ . Hence,

(44) $$ \begin{align} E\left( \infty \right) \doteq \lim\limits_{\mathrm{U}\rightarrow \infty }E\left( \mathrm{U}\right) ={\sup\limits_{\mathrm{U}\in \mathbb{R}_{0}^{+}}} E\left( \mathrm{U}\right) \leq 0. \end{align} $$

For more details, see Lemma 4.14. The limit $\mathrm {U}\rightarrow \infty $ of the eigenvalue and eigenvector of each fiber, given by Theorem 3.1, is less trivial to obtain and is the object of the next theorem:

Theorem 3.5 (Spectral properties of fiber Hamiltonians – Hard-core limit).

Fix $\epsilon ,\mathrm {U}\in \mathbb {R}_{0}^{+}$ and ${h_{b}\in \lbrack 0,1/2] }$ . The following limits exist:

$$ \begin{align*} \mathrm{E}\left( \infty ,k\right) &\doteq {\lim\limits_{\mathrm{U} \rightarrow \infty }}\mathrm{E}\left( \mathrm{U},k\right) ={\sup\limits_{ \mathrm{U}\in \mathbb{R}_{0}^{+}}}\mathrm{E}\left( \mathrm{U},k\right) \leq \mathfrak{b}\left( k\right) ,\qquad k\in \mathbb{T}^{2}. \\ \Psi \left( \infty ,k\right) &\doteq {\lim_{\mathrm{U}\rightarrow \infty }} \,\Psi \left( \mathrm{U},k\right) \in \mathcal{H}\backslash \{0\},\qquad k\in \mathbb{T}^{2}\backslash \{0\}. \end{align*} $$

Assertion (iv) of Theorem 3.1 also holds true for $\mathrm {U} =\infty $ . In addition, when $r_{\mathfrak {p}}>0$ , $\mathrm {E}(\infty ,k)= \mathfrak {b}(k)$ iff $\hat {\upsilon }(k)=0$ . If $r_{\mathfrak {p}}>0$ and $ \hat {\upsilon }(0)\neq 0$ , then $\Psi (\infty ,0)$ exists.

Note that the eigenvalues given by Theorem 3.1 are not explicitly known. The same is of course true in the hard-core limit $\mathrm { U}\rightarrow \infty $ . For applications, it is important to have a sufficiently good control on these objects to be able to compute them, either analytically or numerically. This is done in [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21] for the special case of the usual Hubbard interaction ( $\mathrm {u}=0$ ) and one-range creation operators $c_{y}^{\ast }$ of fermion pairs, by providing estimates for $\mathrm {E}\left ( \mathrm {U},k\right ) $ and $\Psi \left ( \mathrm {U},k\right ) $ at large $\mathrm {U}$ . See [Reference Bru, de Siqueira Pedra and de Pasquale22, Theorem 4, Corollary 5, Theorem 16].

Recall that $\hat {\upsilon }$ is the Fourier transform of $\upsilon $ , which is the function appearing in Equation (10), encoding the (exchange) interaction between fermion pairs and bosons. By Theorem 3.1, if $\hat {\upsilon }(k)=0$ , then $\mathrm {E}(\mathrm {U},k)$ is nothing else than the explicit function $\mathfrak {b}(k)$ (32). Hence, we focus on the physically more relevant case $\hat {\upsilon }(k)\neq 0$ . Using the Birman-Schwinger principle (Theorem A.10), we show in this case that the eigenvalue $\mathrm {E}(\mathrm {U} ,k) $ is the unique solution to a relatively simple equation for real numbers, similar to the characteristic equation used to compute eigenvalues of matrices.

To this end, we define a function $\mathfrak {T}:\mathcal {D}\rightarrow \mathbb {R}$ on the set

$$ \begin{align*} \mathcal{D}\doteq \left\{ \left( \mathrm{U},k,x\right) \in \left[ 0,\infty \right] \times \mathbb{T}^{2}\times \mathbb{R}:x<\mathfrak{z}\left( k\right) \right\} \subseteq \mathbb{R}^{3} \end{align*} $$

by

(45) $$ \begin{align} \mathfrak{T}\left( \mathrm{U},k,x\right) \doteq \left\langle \mathfrak{d} \left( k\right) ,\left( A_{1,1}\left( \mathrm{U},k\right) -x\mathbf{1} \right) ^{-1}\mathfrak{d}\left( k\right) \right\rangle \end{align} $$

for any finite $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , $k\in \mathbb {T}^{2}$ and $x\in \left ( 0,\mathfrak {z}\left ( k\right ) \right ) $ , while for the infinite on-site repulsion, $k\in \mathbb {T}^{2}$ and $x\in \left ( 0,\mathfrak {z} \left ( k\right ) \right ) $ ,

$$ \begin{align*} \mathfrak{T}\left( \infty ,k,x\right) \doteq \lim_{\mathrm{U}\rightarrow \infty }\mathfrak{T}\left( \mathrm{U},k,x\right) , \end{align*} $$

the above limit existing by virtue of Corollary 4.13. Recall that $\mathfrak {z}\left ( k\right ) $ is defined by Equation (75). In fact, for any $k\in \mathbb {T}^{2}$ and $x\in \left ( 0,\mathfrak {z}\left ( k\right ) \right ) $ ,

$$ \begin{align*} \mathfrak{T}\left( \infty ,k,x\right) =R_{\mathfrak{s},\mathfrak{s} }^{-1}\left( {R_{\mathfrak{d},\mathfrak{d}}R_{\mathfrak{s},\mathfrak{s}}-} \left\vert R_{\mathfrak{s},\mathfrak{d}}\right\vert ^{2}\right)>0, \end{align*} $$

where $R_{\mathfrak {s},\mathfrak {s}}$ , $R_{\mathfrak {s},\mathfrak {d}}$ , $R_{ \mathfrak {d},\mathfrak {s}}$ and $R_{\mathfrak {d},\mathfrak {d}}$ are four constants defined by Equations (99)–(102) with $\lambda =x$ . When $\mathrm {u}=0$ , these constants are given by explicit integrals on the torus $\mathbb {T}^{2}$ [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21]. Then, the eigenvalues, the existence of which is stated in Theorem 3.1, as well as their limits (Theorem 3.5), can be studied via the following characteristic equation:

Theorem 3.6 (Characteristic equation for the fiber ground states).

Fix $\epsilon \in \mathbb {R}_{0}^{+}$ , $h_{b}\in \lbrack 0,1/2]$ and $k\in \mathbb {T}^{2}$ such that $\hat {\upsilon }(k)\neq 0$ . Then, for any $\mathrm {U }\in \left [ 0,\infty \right ] $ , $\mathrm {E}(\mathrm {U},k)$ is the unique solution to the equation

$$ \begin{align*} \hat{\upsilon}\left( k\right) ^{2}\mathfrak{T}\left( \mathrm{U},k,x\right) +x-\mathfrak{b}\left( k\right) =0,\qquad x<\mathfrak{z}\left( k\right) . \end{align*} $$

Notice that, more generally, for any fixed $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , the same characteristic equation determines all eigenvalues of the fiber lying in the resolvent set $\rho (A_{1,1}(\mathrm {U},k))$ of the operator $A_{1,1}(\mathrm {U},k)$ . See Theorem 4.5. Also the associated eigenspaces can be explicitly characterized, thanks to Corollary 4.6. In this context, Corollary 4.7 shows that, for any $h_{b}\in \lbrack 0,1/2]$ and total quasi-momentum $k\in \mathbb {T}^{2}$ , there is at most one eigenvalue of $A( \mathrm {U},k)$ in each connected component of $\rho (A_{1,1}(\mathrm {U} ,k))\cap \mathbb {R}$ .

3.2 Dispersion relation of dressed bound fermion pairs

By Theorem 3.1, $\mathrm {E}:\mathbb {T}^{2}\rightarrow \mathbb { R}$ is a continuous family of nondegenerate eigenvalues, generally (at least for $k\neq 0$ ) associated with exponentially localized eigenvectors. Note that the case $k=0$ is particular when $\hat {\upsilon }(0)=0$ , since $ \mathrm {E}(0)$ is not an isolated eigenvalue of $A(\mathrm {U},0)$ . However, the family $(\mathrm {E}(k))_{k\in \mathbb {T}^{2}}$ is still continuous. The peculiar behavior at $k=0$ leads us to only consider total quasi-momenta in the subset

(46) $$ \begin{align} \mathbb{S}^{2}\doteq \left( -\pi ,\pi \right) ^{2}\backslash \left\{ 0\right\} \subseteq \mathbb{T}^{2}, \end{align} $$

as, for instance, in Theorem 3.1 (iv).

Because of Proposition 2.1, the family $(\mathrm {E}(k))_{k\in \mathbb {T}^{2}}$ can thus be seen as the effective dispersion relation of dressed bound fermion pairs. It is expected to determine transport properties of the quantum system at low temperatures. We now define in mathematical terms what a dispersion relation is.

First, a dispersion relation $\varkappa :\mathbb {T}^{2}\rightarrow \mathbb {R} $ should be a functions mapping quasi-momenta $k\in \mathbb {T}^{2}$ on the torus to spectral values of the corresponding fibers. More precisely, $ \varkappa (k)$ should be an isolated eingenvalue of the fiber associated with the total quasi-momentum k. Recall that the dispersion relation of a (nonrelativistic) particle in the d-dimensional continuum (that is, the particle moves in the continuum d-dimensional space $\mathbb {R}^{d}$ ), whose (isotropic) mass is m, is $\ k^{2}/2m$ and velocity $v(k)=k/m$ , $k\in \mathbb {R}^{d}$ . Having this standard example in mind, we would like also to derive from a dispersion relation a group velocity and a mass tensor, at any fixed quasi-momentum $k\in \mathbb {T}^{2}$ , as is usual. These are key objects, for instance, in the study of transport properties. Notice that they require sufficient regularity of the dispersion relation to be defined.

Keeping in mind that all our quantities are parametrized by the on-site repulsion $\mathrm {U}\in \left [ 0,\infty \right ] $ , we define a family of dispersion relations associated with the fiber Hamiltonians $A(\mathrm {U},k)$ as follows:

The first property is a very natural property, having in mind Proposition 2.1 and the theory of direct integrals (Theorem A.3). The second property of Definition 3.7 is needed to define group velocities and mass tensors.

To explain these two concepts, we need the Hessian of functions $f\in C^{2}( \mathbb {S}^{2})$ at fixed k, which is denoted by

(47) $$ \begin{align} \mathrm{Hess}\left( f\right) \left( k\right) \doteq \left( \begin{array}{cc} \partial _{k_{1}}^{2}f & \partial _{k_{1}}\partial _{k_{2}}f \\ \partial _{k_{2}}\partial _{k_{1}}f & \partial _{k_{2}}^{2}f \end{array} \right) \left( k\right) \in \mathsf{M}_{2}\left( \mathbb{R}\right) ,\qquad k\in \mathbb{S}^{2}, \end{align} $$

where $\mathsf {M}_{2}(\mathbb {R})$ is the set of $2\times 2$ matrices with real coefficients. It is a straightforward consequence of the regularity of $ f\in C^{2}(\mathbb {S}^{2})$ that

$$ \begin{align*} \mathrm{Hess}\left( f\right) :\mathbb{S}^{2}\longrightarrow \mathsf{M} _{2}\left( \mathbb{R}\right) \end{align*} $$

is continuous. For any $f\in C^{2}(\mathbb {S}^{2})$ , we consider the set

(48) $$ \begin{align} \mathfrak{M}_{f}\doteq \left\{ k\in \mathbb{S}^{2}\,:\,\mathrm{Hess}\left( f\right) \left( k\right) \in \mathsf{GL}_{2}\left( \mathbb{R}\right) \right\} \subseteq \mathbb{S}^{2} \end{align} $$

with $\mathsf {GL}_{2}\left ( \mathbb {R}\right ) \subseteq \mathsf {M}_{2}\left ( \mathbb {R}\right ) $ being the set of invertible $2\times 2$ matrices with real coefficients. As $\mathsf {GL}_{2}(\mathbb {R})\subseteq \mathsf {M}_{2}( \mathbb {R})$ is an open set (see [Reference Folland50, Theorem 1.4]), it then follows that

$$ \begin{align*} \mathfrak{M}_{f}=\mathrm{Hess}\left( f\right) ^{-1}\left( \mathsf{GL} _{2}\left( \mathbb{R}\right) \right) \end{align*} $$

is also an open set.

We are now in a position to define group velocities and mass tensors of a family of dispersion relations.

Definition 3.8 (Group velocities and mass tensors).

At any $\mathrm {U}\in \left [ 0,\infty \right ] $ , the group velocity $\mathbf { v}_{\varkappa ,\mathrm {U}}:\mathbb {S}^{2}\rightarrow \mathbb {R}$ and the mass tensor $\mathbf {m}_{\varkappa ,\mathrm {U}}:\mathfrak {M}_{\varkappa \left ( \mathrm {U},\cdot \right ) }\rightarrow \mathsf {M}_{2}\left ( \mathbb {R} \right ) $ associated with a family $\varkappa :\left [ 0,\infty \right ] \times \mathbb {T}^{2}\rightarrow \mathbb {R}$ of dispersion relations are respectively defined by

$$ \begin{align*} \mathbf{v}_{\varkappa ,\mathrm{U}}\left( k\right) \doteq \vec{\nabla} _{k}\varkappa \left( \mathrm{U},k\right) \qquad \text{and}\qquad \mathbf{m} _{\varkappa ,\mathrm{U}}\left( k\right) \doteq \mathrm{Hess}\left( \varkappa \left( \mathrm{U},\cdot \right) \right) \left( k\right) ^{-1}. \end{align*} $$

We deduce from Theorem 4.9 that $\mathrm {E}$ is a dispersion relation when the function $\upsilon :\mathbb {Z}^{2}\rightarrow \mathbb {R}$ is at least $2$ times continuously differentiable and, in this case, we can even compute the group velocity via the characteristic equation (Theorem 3.6).

Similar to Remark 3.2, if for some strictly positive constant $\gamma>0$ ,

$$ \begin{align*} \sum_{x\in \mathbb{Z}^{2}}\mathrm{e}^{\gamma \left\vert x\right\vert }\left\vert \upsilon \left( x\right) \right\vert <\infty, \end{align*} $$

then the Fourier transform $\hat {\upsilon }$ of the function $\upsilon : \mathbb {Z}^{2}\rightarrow \mathbb {R}$ , as defined via (31), is real analytic on the whole torus $\mathbb {T}^{2}$ . It is very natural to expect a local interaction between fermion pairs and bosons in (10), meaning here that the function $\upsilon $ should even have finite support. In particular, all conditions of Theorem 3.9, including the ones of the third assertion, should hold true in the application to superconducting cuprates.

In fact, as shown in [Reference Bru, de Siqueira Pedra and de Pasquale21], the dispersion relation of Theorem 3.9 yields the formation of d-wave pairs when one adjusts the parameters of the model (with $\mathrm {u}=0$ ) to fit those of cuprate superconductors – in particular, the ones of the cuprate $\mathrm {La}_{2-x}\mathrm {Sr}_{x}\mathrm {CuO}_{4}$ (LaSr 214) near optimal doping. When considering the usual Hubbard model – that is, the case where there is no other fermionic repulsion than the on-site one (i.e., $\mathrm {u}=0$ ) and no fermion-boson exchange (i.e., $ \upsilon =0$ ) – $\mathrm {E}$ turns out to be the function $\mathfrak {b}: \mathbb {T}^{2}\rightarrow \mathbb {R}$ , defined by (32), which is nothing else than the dispersion relation

$$ \begin{align*} \mathfrak{b}(k)\doteq h_{b}\epsilon \left( 2-\cos \left( k\right) \right) ,\qquad k\in \mathbb{T}^{2}, \end{align*} $$

of free bosons (bipolarons for cuprates).

By turning on the fermion-boson-exchange interaction, the dispersion relation of dressed bound fermion pairs with lowest energy can strongly deviate from $\mathfrak {b}$ , the unperturbed one. Recall, for instance, that $ \mathfrak {b}$ describes bosons with a very large mass as compared to the effective mass of electrons or holes in cuprates. However, as shown in [Reference Bru, de Siqueira Pedra and de Pasquale21], for typical parameters of the cuprate LaSr 214, the effective mass of the bound pair (with dispersion relation $\mathrm {E}$ ) is comparable to the mass of electrons or holes. This is a consequence of the mass of charge carriers calculated in [Reference Chen, Birgeneau, Kastner, Preyer and Thio52], and the fact that a large effective mass of dressed bound fermion pairs and a high fermion-pair depletion,Footnote ¹³ close to $90\%$ as measured in [Reference Božović, He, Wu and Bollinger53], is not compatible with our model. This solves the so-called ‘large mass paradox’ of the microscopic theory of cuprate superconductors, based on some kind mechanism involving bipolarons. For more details, see [Reference Bru, de Siqueira Pedra and de Pasquale21] and references therein.

In fact, the effective mass of dressed bound fermion pairs, or more generally its (effective) mass tensor, depends strongly on the coupling function $\hat {\upsilon }$ near its maximum. Bearing in mind Definitions 3.7 and 3.8, one can therefore provide via Theorems 3.6 and 3.8 not only qualitative but also quantitative information, which is important for describing the physical behavior of fermionic pairs formed in this way by means of a bosonic field. A natural question is then to study its scattering properties and this is precisely what we propose to do in the next section.

3.3 Quantum scattering

Scattering in quantum mechanics constitutes a well-established mathematical theory aiming at analyzing the behavior of quantum systems at large times. To this end, a reference (or free) Hamiltonian Y is chosen and the dynamics $(\mathrm {e}^{itX})_{t\in \mathbb {R}}$ of the quantum system driven by the (full) Hamiltonian X is compared at large (negative and positive) times to $(\mathrm {e}^{itY})_{t\in \mathbb {R}}$ . In fact, scattering theory can be viewed as a kind of perturbation theory for the absolutely continuous spectrum of X. See, for example, [Reference Kato40, Chapter X]. For standard textbooks explaining in detail the scattering theory, we recommend [Reference Reed and Simon41, Reference Newton42, Reference Yafaev43]. Below, for the reader’s convenience, we shortly recall definitions that are relevant here.

Take two bounded self-adjoint operators X and Y acting on two Hilbert spaces $\mathcal {X}$ and $\mathcal {Y}$ , respectively. Let $P_{\mathrm {ac} }(Y) $ be the orthogonal projection onto the absolutely continuous space of $ Y$ , which is defined as follows:

(49) $$ \begin{align} \mathrm{ran}\left( P_{\mathrm{ac}}(Y)\right) &\doteq \big\{ \psi \in \mathcal{Y}:\left\langle \psi ,\chi _{(\cdot )}(Y)\psi \right\rangle _{\mathcal{Y}}\text{ is absolutely continuous} \notag \\ &\qquad \qquad \qquad \qquad \qquad \qquad \text{with respect to the Lebesgue measure} \big\} , \end{align} $$

where $\chi _{\Omega }$ is its characteristic functionFootnote ¹⁴ of any Borel set $\Omega \subseteq \mathbb {R}$ . The so-called wave operators for the pair $(X,Y)$ with identification operator $J\in \mathcal {B}\left ( \mathcal {Y},\mathcal {X} \right ) $ is, by definition, the strong limit

(50) $$ \begin{align} W^{\pm }\left( X,Y;J\right) \doteq s-{\lim\limits_{t\rightarrow \pm \infty }} \mathrm{e}^{itX}J\mathrm{e}^{-itY}P_{\mathrm{ac}}\left( Y\right) , \end{align} $$

when it exists. See, for instance, [Reference Yafaev43, Definition 1.3]. When $ \mathcal {Y}=\mathcal {X}$ and $J=\mathbf {1}$ , like in (Reference Yafaev43, Definition 1.1), we use the shorter notation

(51) $$ \begin{align} W^{\pm }\left( X,Y\right) \equiv W^{\pm }\left( X,Y;\mathbf{1}\right) \doteq s-{\lim\limits_{t\rightarrow \pm \infty }}\mathrm{e}^{itX}\mathrm{e} ^{-itY}P_{\mathrm{ac}}\left( Y\right) . \end{align} $$

In case the above wave operators exist, they are partial isometries [Reference Reed and Simon41, Proposition 1, Sect. XI.3]. They are said to be complete when

$$ \begin{align*} \mathrm{ran}\left( W^{+}\left( X,Y\right) \right) =\mathrm{ran}\left( W^{-}\left( X,Y\right) \right) =\mathrm{ran}\left( P_{\mathrm{ac}}(X)\right) . \end{align*} $$

See [Reference Reed and Simon41, p. 19, Sect. XI.3].

Similarly, in the general case, $W^{\pm }\left ( X,Y;J\right ) $ are said to be complete whenever

$$ \begin{align*} \overline{\mathrm{ran}\left( W^{+}\left( X,Y;J\right) \right) }=\overline{ \mathrm{ran}\left( W^{-}\left( X,Y;J\right) \right) }=\mathrm{ran}\left( P_{ \mathrm{ac}}(X)\right) . \end{align*} $$

See [Reference Reed and Simon41, p. 35, Sect. XI.3]. The corresponding scattering operator is equal to

(52) $$ \begin{align} S\left( X,Y;J\right) \doteq W^{+}\left( X,Y;J\right) ^{\ast }W^{-}\left( X,Y;J\right) \in \mathcal{B}\left( \mathcal{Y}\right) . \end{align} $$

It leads to the scattering matrix (or simply S-matrix) in a representation where Y is diagonal, because the scattering operator commutes with Y. See [Reference Yafaev43, Equation (1.12)].

Remark 3.10. For two bounded self-adjoint operators X and Y acting on two Hilbert spaces $\mathcal {X}$ and $\mathcal {Y}$ , noteFootnote ¹⁵ that

$$ \begin{align*} \mathrm{ran}\left( P_{\mathrm{ac}}(X\oplus Y)\right) =\mathrm{ran}\left( P_{ \mathrm{ac}}(X)\right) \oplus \mathrm{ran}\left( P_{\mathrm{ac}}(Y)\right) . \end{align*} $$

This is an elementary observation used to study the scattering channels in Section 4.6.

In our framework, the Hamiltonian X is the bounded self-adjoint operator

$$ \begin{align*} \mathbb{U}H\mathbb{U}^{\ast }={\int_{\mathbb{T}^{2}}^{\oplus }}A(k)\,\nu ( \mathrm{d}k) \end{align*} $$

of Proposition 2.1, which acts on the Hilbert space $L^{2}(\mathbb {T}^{2},\mathcal {H})$ . Below, two different (reference) Hamiltonians Y are taken into account, corresponding to two scattering channels: the unbound and bound pair channels. For cuprates, the first channel should be associated with the high temperature regime, while the second one is related to sufficiently low temperatures.

3.3.1 Unbound pair scattering channel

Far apart from each other, two fermions only experience a very weak repulsion force due to the extended Hubbard interaction while the probability that they bind together to form a boson is also very small. Thus, in this situation, one expects that the dynamics of such a pair is governed by the fermionic part, and even by the hopping term alone. During intermediate times, they could of course interact, as they may get close to each other, and they could even be bound together via the effective attraction caused by fermion-boson exchange processes. The lifetime of bound fermions should, however, be finite in this situation, and they are expected to be released at some point and behave again as two free fermions that go far apart from each other for large times. See Figure 2. We show below that this heuristics can be put in precise mathematical terms.

Figure 2 Illustration of the unbound pair scattering channel: Two free fermions of (quasi-) momentum $k-p$ and q respectively (i.e., the full momentum of the fermionic pair is k) at time $t=- \infty $ interact in finite time with the composite system – in particular with the bosonic field – to be asymptotically free again at time $t=+ \infty $ , thanks to Theorem 3.11. Here, $S_{k}=S\left ( A\left ( k\right ) ,\left ( M_{ \mathfrak {f}\left ( k\right ) }+R\left ( \mathrm {V},\mathrm {v}\right ) \right ) \oplus A_{2,2}\left ( k\right ) \right ) $ is the scattering operator of this process in each fiber k, which depends explicitly on $ \hat { \upsilon }\left ( k\right ) $ . See Theorem 3.13 and the example given by Equations (63)–(64).

To this end, define the Hilbert space

(53) $$ \begin{align} \mathfrak{H}_{f}\doteq L^{2}\left( \mathbb{T}^{2},L^{2}\left( \mathbb{T} ^{2}\right) ,\nu \right) \doteq \int_{\mathbb{T}^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2},\mathbb{C},\nu \right) \,\nu \left( \mathrm{d}k\right) \end{align} $$

as well as the Hamiltonian

(54) $$ \begin{align} \mathrm{H}_{f}\equiv \mathrm{H}_{f}\left( \mathrm{V},\mathrm{v}\right) \doteq {\int_{\mathbb{T}^{2}}^{\oplus }}\left( M_{\mathfrak{f}\left( k\right) }+R\left( \mathrm{V},\mathrm{v}\right) \right) \,\nu (\mathrm{d} k)\in \mathcal{B}\left( \mathfrak{H}_{f}\right) \end{align} $$

for any $\mathrm {V}\in \mathbb {R}_{0}^{+}$ and absolutely summable function $ \mathrm {v}:\mathbb {Z}^{2}\rightarrow \mathbb {R}_{0}^{+}$ , where

(55) $$ \begin{align} R\left( \mathrm{V},\mathrm{v}\right) \doteq {\sum\limits_{x\in \mathbb{Z} ^{2}}}\,\mathrm{v}\left( x\right) P_{x}+\mathrm{V}P_{0}\in \mathcal{B}\left( L^{2}(\mathbb{T}^{2})\right) , \end{align} $$

$M_{\mathfrak {f}\left ( k\right ) }$ being the fiber Hamiltonian defined as the multiplication operator by $\mathfrak {f}(k)\in C(\mathbb {T}^{2})$ (see ( 33)) while $P_{x}$ is the orthogonal projection onto the one-dimensional subspace $\mathbb {C}\mathfrak {\hat {e}}_{x}\subseteq L^{2}( \mathbb {T}^{2})$ . Observe then that

$$ \begin{align*} \mathbb{U}H\mathbb{U}^{\ast }- \begin{pmatrix} \mathrm{H}_{f} & 0 \\[0.5em] 0 & 0 \end{pmatrix} ={\int_{\mathbb{T}^{2}}^{\oplus }} \begin{pmatrix} {\sum\limits_{x\in \mathbb{Z}^{2}}}\,\left( \mathrm{u}\left( x\right) - \mathrm{v}\left( x\right) \right) P_{x}+\left( \mathrm{U}-\mathrm{V}\right) P_{0} & A_{1,2}\left( k\right) \\[0.5em] A_{2,1}\left( k\right) & A_{2,2}\left( k\right) \end{pmatrix} \,\nu (\mathrm{d}k). \end{align*} $$

Compare indeed (54) with Equations 42–37 and Proposition 2.1. By Lemma 4.21, note that $P_{ \mathrm {ac}}(\mathrm {H}_{f})=\mathbf {1}$ .

Let us consider the identification operator $\mathfrak {U}:\mathfrak {H} _{f}\rightarrow L^{2}\left ( \mathbb {T}^{2},\mathcal {H}\right ) $ defined for any purely fermionic state $\psi \in \mathfrak {H}_{f}$ by

(56) $$ \begin{align} \begin{array}{cccl} \mathfrak{U}\psi : & \mathbb{T}^{2} & \rightarrow & \mathcal{H}\doteq L^{2}\left( \mathbb{T}^{2}\right) \oplus \mathbb{C} \\ & k & \mapsto & \left( \psi \left( k\right) ,0\right) \end{array} . \end{align} $$

See Equation (25). Note that $\mathfrak {U}$ is an isometry (i.e., a norm preserving linear transformation). In fact, it is the canonical fiberwise inclusion of $\mathfrak {H}_{f}$ into $L^{2}\left ( \mathbb {T}^{2}, \mathcal {H}\right ) $ . Recall from Proposition 2.1 that $A\left ( \mathrm {U},\cdot \right ) $ , defined by (37)–(42), is the fiber decomposition of the operator $\mathbb {U}H\mathbb {U}^{\ast }$ . Then, we obtain wave and scattering operators with respect to fermionic parts:

Theorem 3.11 (Unbound pair (scattering) channel).

Let $\mathrm {V}\in \mathbb {R}_{0}^{+}$ and $\mathrm {v}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}_{0}^{+}$ be any absolutely summable function.

1. i.) The wave operators, as defined by (50) for $X=\mathbb {U}H\mathbb {U}^{\ast }$ , $Y=\mathrm {H}_{f}$ and $ J=\mathfrak {U}$ , satisfy

   (57) $$ \begin{align} W^{\pm }\left( \mathbb{U}H\mathbb{U}^{\ast },\mathrm{H}_{f};\mathfrak{U} \right) =\left( {\int_{\mathbb{T}^{2}}^{\oplus }}W^{\pm }\left( A\left( k\right) ,\left( M_{\mathfrak{f}\left( k\right) }+R\left( \mathrm{V},\mathrm{ v}\right) \right) \oplus A_{2,2}\left( k\right) \right) \nu \left( \mathrm{d} k\right) \right) \mathfrak{U} \end{align} $$
   with range equal to
   (58) $$ \begin{align} \mathrm{ran}\left( W^{\pm }\left( \mathbb{U}H\mathbb{U}^{\ast },\mathrm{H} _{f};\mathfrak{U}\right) \right) =\int_{\mathbb{T}^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \oplus \{0\}\,\nu \left( \mathrm{d}k\right) . \end{align} $$

2. ii.) The scattering operator, as defined by (52) for $X=\mathbb {U}H\mathbb {U}^{\ast }$ , $Y=\mathrm {H}_{f}$ and $J= \mathfrak {U}$ , equals

   $$ \begin{align*} S\left( \mathbb{U}H\mathbb{U}^{\ast },\mathrm{H}_{f};\mathfrak{U}\right) = \mathfrak{U}^{\ast }\left( {\int_{\mathbb{T}^{2}}^{\oplus }}S\left( A\left( k\right) ,\left( M_{\mathfrak{f}\left( k\right) }+R\left( \mathrm{V},\mathrm{ v}\right) \right) \oplus A_{2,2}\left( k\right) \right) \nu \left( \mathrm{d} k\right) \right) \mathfrak{U}. \end{align*} $$

Proof. Observe that the operator difference $(\mathbb {U}H\mathbb {U}^{\ast }-\mathrm { H}_{f})$ is not trace-class (it is not even compact) and, thus, the existence of this scattering channel is not a direct consequence of the well-known Kato-Rosenblum theorem [Reference Reed and Simon41, Theorem XI.8]. In fact, one of the main steps of the proof is to show that this difference is the direct integral of a strongly measurable family of trace-class operators. By this means, we are then able to apply the Kato-Rosenblum theorem ‘fiberwise’ to deduce the first assertion. See Section 4.6.1 for more details – in particular, Theorem 4.23. Assertion (ii) is a direct consequence of Assertion (i) together with the theory of direct integrals.

Remark 3.12. If one would like to go back to the original Hilbert space $\mathfrak {h}_{0}$ (19) for fermion pairs with opposite spins – that is, if one wishes to use space coordinates, instead of the quasi-momenta – then one employs Theorem 3.11, along with the observation that

$$ \begin{align*} W^{\pm }\left( H,U_{f}^{\ast }\mathrm{H}_{f}U_{f};\mathbb{U}^{\ast } \mathfrak{U}U_{f}\right) =U^{\ast }W^{\pm }\left( \mathbb{U}H\mathbb{U} ^{\ast },\mathrm{H}_{f};\mathfrak{U}\right) U_{f}, \end{align*} $$

where $U_{f}$ is defined by (28). See also Equation (26) and Proposition 2.1.

Theorem 3.11 refers to the unbound pair (scattering) channel. The subspace $\mathfrak {H}_{f}\subseteq \mathfrak {H}$ corresponds to the ‘incoming’ ( $+$ ) and ‘outcoming’ ( $-$ ) scattering states of the quantum system, in this particular scattering channel. Physically, this theorem shows, among other things, that the bosonic component of e $^{itH}$ vanishes on this channel, as $t\rightarrow \pm \infty $ . This is a direct consequence of Equation (58).

In addition, Equation (57) gives an explicit fiber decomposition of wave operators with respect to the purely fermionic Hamiltonian in terms of $ k$ -dependent wave operators defined naturally from the fiber decomposition of the operator $\mathbb {U}H\mathbb {U}^{\ast }$ . Mutatis mutandis for the scattering operator, thanks to Theorem 3.11 (ii). In other words, the knowledge of scattering properties of each fiber, almost everywhere, entirely determines the scattering properties of the composite system, made of two fermions and one boson. We can now use this property (i.e., Theorem 3.11) to obtain a more computable expression for the wave and scattering operators in each given fiber. This can be done via infinite series (perturbative expansions), thanks to Corollary A.2.

Below, we give an example of such a computation by taking $\mathrm {U}=\mathrm { V}\in \mathbb {R}_{0}^{+}$ and $\mathrm {v}=\mathrm {u}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}_{0}^{+}$ in Theorem 3.11. With this particular choice, the unbound pair channel allows one to isolate the fermion-boson exchange mechanism, which in terms of Hamiltonians refers to the use of off-diagonal operators

(59) $$ \begin{align} B^{(t)}\left( k\right) \doteq \begin{pmatrix} 0 & B_{1,2}^{(t)}\left( k\right) \\[0.5em] B_{2,1}^{(t)}\left( k\right) & 0 \end{pmatrix} \in \mathcal{B}\left( \mathcal{H}\right) ,\qquad t\in \mathbb{R},\ k\in \mathbb{T}^{2}, \end{align} $$

in the fibers, where, for any $t\in \mathbb {R}$ and $k\in \mathbb {T}^{2}$ ,

(60) $$ \begin{align} B_{1,2}^{(t)}\left( k\right) \doteq \mathrm{e}^{itA_{1,1}\left( \mathrm{U} ,k\right) }A_{1,2}\left( k\right) \mathrm{e}^{-itA_{2,2}\left( k\right) }\qquad \text{and}\qquad B_{2,1}^{(t)}\left( k\right) \doteq \mathrm{e} ^{itA_{2,2}\left( k\right) }A_{2,1}\left( k\right) \mathrm{e} ^{-itA_{1,1}\left( \mathrm{U},k\right) }. \end{align} $$

Recall that, for $m,n\in \{1,2\}$ , $A_{m,n}(k)$ is defined by (37)–( 41). Below, $B_{m,n}(k)$ , $m\neq n$ , stands for the norm-continuous family of operators $(B_{m,n}^{(t)}(k))_{t\in \mathbb {R}}$ .

To shorten the notation, for any $s,t\in \mathbb {R}$ , as well as two norm-continuous families $X\equiv (X_{t})_{t\in \mathbb {R}}$ and $Y\equiv (Y_{t})_{t\in \mathbb {R}}$ of bounded operators $X_{t}:\mathcal {X} \rightarrow \mathcal {Y}$ and $Y_{t}:\mathcal {Y}\rightarrow \mathcal {X}$ on two Hilbert spaces $\mathcal {X}$ and $\mathcal {Y}$ , respectively, we define the bounded operators:

(61) $$ \begin{align} \cos _{\succ }\left( XY;s,t\right) &\doteq \mathbf{1}+\sum_{p=1}^{\infty }\left( -1\right) ^{p}\int_{s}^{t}\mathrm{d}\tau _{1}\cdots \int_{s}^{\tau _{2p-1}}\mathrm{d}\tau _{2p}(X_{\tau _{1}}Y_{\tau _{2}})\cdots (X_{\tau _{2p-1}}Y_{\tau _{2p}}),\qquad\qquad\quad \end{align} $$

(62) $$ \begin{align} \sin _{\succ }\left( XY;s,t\right) &\doteq \int_{s}^{t}\mathrm{d}\tau X_{\tau }+\sum_{p=1}^{\infty }\left( -1\right) ^{p}\int_{s}^{t}\mathrm{d} \tau _{1}\cdots \int_{s}^{\tau _{2p}}\mathrm{d}\tau _{2p+1}X_{\tau _{1}}\left( (Y_{\tau _{2}}X_{\tau _{3}})\cdots (Y_{\tau _{2p}}X_{\tau _{2p+1}})\right) . \end{align} $$

The integrals above are Riemann ones, noting that $(X_{t})_{t\in \mathbb {R}}$ and $(Y_{t})_{t\in \mathbb {R}}$ are continuous families in Banach spaces – namely, $\mathcal {B}\left ( \mathcal {X};\mathcal {Y}\right ) $ and $\mathcal {B} \left ( \mathcal {Y};\mathcal {X}\right ) $ , respectively. Note that $\cos _{\succ }\left ( XY;s,t\right ) \in \mathcal {B}\left ( \mathcal {Y}\right ) $ and $\sin _{\succ }\left ( XY;s,t\right ) \in \mathcal {B}\left ( \mathcal {X}, \mathcal {Y}\right ) $ are always absolutely summable series in the operator norm. Then, we obtain the following results:

Theorem 3.13 (Scattering operators as pertubative series).

Let $\varepsilon \in \mathbb {R}^{+}$ and $\mathrm {H}_{f}\equiv \mathrm {H} _{f}\left ( \mathrm {U},\mathrm {u}\right ) $ . Then, for any $\varphi \in \mathfrak {H}_{f}$ , there is $T>0$ such that

$$ \begin{align*} T &<t\implies \left\Vert \left( W^{+}\left( \mathbb{U}H\mathbb{U}^{\ast }, \mathrm{H}_{f};\mathfrak{U}\right) -V_{0,t}\mathfrak{U}\right) \varphi \right\Vert _{\mathcal{X}}\leq \varepsilon , \\ t &<-T\implies \left\Vert \left( W^{-}\left( \mathbb{U}H\mathbb{U}^{\ast }, \mathrm{H}_{f};\mathfrak{U}\right) -V_{0,t}\mathfrak{U}\right) \varphi \right\Vert _{\mathcal{X}}\leq \varepsilon , \end{align*} $$

Moreover, for any $\varphi ,\psi \in \mathfrak {H}_{f}$ , there is $T>0$ such that

$$ \begin{align*} s<-T<T<t\implies \left\langle \psi ,S\left( \mathbb{U}H\mathbb{U}^{\ast }, \mathrm{H}_{f};\mathfrak{U}\right) \varphi \right\rangle _{\mathcal{X} }=\left\langle \mathfrak{U}\psi ,V_{t,s}\mathfrak{U}\varphi \right\rangle _{ \mathcal{X}}+\mathcal{O}\left( \varepsilon \right) , \end{align*} $$

where, for all $s,t\in \mathbb {R}$ ,

$$ \begin{align*} V_{t,s}\doteq {\int_{\mathbb{T}^{2}}^{\oplus }}\left( \begin{array}{cc} \cos _{\succ }\left( B_{1,2}\left( k\right) B_{2,1}\left( k\right) ;s,t\right) & -i\sin _{\succ }\left( B_{1,2}\left( k\right) B_{2,1}\left( k\right) ;s,t\right) \\ -i\sin _{\succ }\left( B_{2,1}\left( k\right) B_{1,2}\left( k\right) ;s,t\right) & \cos _{\succ }\left( B_{2,1}\left( k\right) B_{1,2}\left( k\right) ;s,t\right) \end{array} \right) \nu \left( \mathrm{d}k\right) . \end{align*} $$

Proof. It suffices to combine Lemma 4.24 with Equation (141) and Theorem 3.11, similar to Corollary A.2.

Theorem 3.13 provides a way to approximate the scattering matrix associated with the fermion-boson-exchange interaction. Note for instance from (37)–(41) that the operator $B_{1,2}^{(t)}\left ( k\right ) B_{2,1}^{(s)}\left ( k\right ) $ and $B_{2,1}^{(t)}\left ( k\right ) B_{1,2}^{(s)}\left ( k\right ) $ have a relatively simple form for any $s,t\in \mathbb {R}$ and $k\in \mathbb {T }^{2}$ :

(63) $$ \begin{align} B_{1,2}^{(t)}\left( k\right) B_{2,1}^{(s)}\left( k\right) &=\left( \hat{ \upsilon}\left( k\right) ^{2}\mathrm{e}^{i\left( s-t\right) \mathfrak{b} \left( k\right) }\right) \mathrm{e}^{itA_{1,1}\left( \mathrm{U},k\right) }P_{ \mathfrak{d}\left( k\right) }\mathrm{e}^{-isa:{1,1}\left( \mathrm{U} ,k\right) },\kern2pt \end{align} $$

(64) $$ \begin{align} B_{2,1}^{(t)}\left( k\right) B_{1,2}^{(s)}\left( k\right) &=\left( \hat{ \upsilon}\left( k\right) ^{2}\mathrm{e}^{i\left( t-s\right) \mathfrak{b} \left( k\right) }\right) \left\langle \mathfrak{d}\left( k\right) ,\mathrm{e} ^{i(s-t)A_{1,1}\left( \mathrm{U},k\right) }\mathfrak{d}\left( k\right) \right\rangle ,\quad \end{align} $$

where $P_{\mathfrak {d}\left ( k\right ) }$ is the orthogonal projection onto the one-dimensional subspace $\mathbb {C}\mathfrak {d}\left ( k\right ) \subseteq L^{2}(\mathbb {T}^{2})$ . Similar computations can be done for other choices of $\mathrm {V}\in \mathbb {R}_{0}^{+}$ and $\mathrm {v}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}_{0}^{+}$ in Theorem 3.11, like $\mathrm {V}=0=\mathrm {v}$ (noninteracting fermion systems). See again Corollary A.2.

The understanding of this kind of scattering, regarding free fermion (electron) collisions, is relevant in physics, because it can allow the exchange function $\upsilon $ of a real system to be studied. It is therefore important to have a model from which not only qualitative, but also quantitative, information can be obtained. This is the purpose of this section, in particular of Theorem 3.11, which gives the explicit dependency of scattering in terms of $\upsilon $ .

3.3.2 Bound pair scattering channel

Similarly, we also prove the existence of a scattering channel for dressed bound pairs. As dressed bound pairs are space-localized objects (see, for example, Theorem 3.1 (iii)), the fermions forming the pair efficiently exchange a boson, at a non-negligible rate, via the terms $W_{\mathrm { b\rightarrow f}}$ (9) and $W_{\mathrm {f\rightarrow b}}$ (17) in the Hamiltonian H. In particular, such quantum states must have some non-negligible bosonic component representing the exchanged boson that ‘glues’ the two fermions together. This dressed bound pair is however expected to move like a free (quantum spinless) particle in the real space. See Figure 3. We translate this physical heuristics in precise mathematical terms by considering the effective dispersion relations

$$ \begin{align*} \mathrm{E}:\left[ 0,\infty \right] \times \mathbb{T}^{2}\rightarrow \mathbb{R } \end{align*} $$

given by Theorems 3.1, 3.5 and 3.9.

Figure 3 Illustration of the bound pair scattering channel. Here, k is the full (quasi-)momentum of the (exponentially localized) dressed bound fermion pairs. The oscillating vertical lines between the two fermions (e.g., electrons) before the scattering process and afterwards characterize their bound via a bosonic (e.g., bipolaronic) particle transfer with coupling function $\hat {\upsilon }\left ( k\right ) $ ; see Figure 1. It illustrates the stability of these pairs of fermions in time, as expressed by Theorem 3.14, that is, the pairs cannot decay into an (even only asymptotically) unbound pair of fermions.

For any $\mathrm {U}\in \mathbb {R}_{0}^{+}\cup \{\infty \}$ , we consider the identification operator

$$ \begin{align*} \mathfrak{P}_{\mathrm{U}}:L^{2}\left( \mathbb{T}^{2}\right) \rightarrow L^{2}\left( \mathbb{T}^{2},\mathcal{H}\right) \end{align*} $$

defined for any $\varphi \in L^{2}\left ( \mathbb {T}^{2}\right ) $ byFootnote ¹⁶

(65) $$ \begin{align} \begin{array}{cccl} \mathfrak{P}_{\mathrm{U}}\varphi : & \mathbb{T}^{2}\backslash \{0\} & \rightarrow & \mathcal{H}\doteq L^{2}\left( \mathbb{T}^{2}\right) \oplus \mathbb{C} \\ & k & \mapsto & \varphi \left( k\right) \left\Vert \Psi \left( \mathrm{U} ,k\right) \right\Vert ^{-1}\Psi \left( \mathrm{U},k\right) \end{array} , \end{align} $$

where $\Psi \left ( \mathrm {U},k\right ) $ is the eigenvector associated with the (nondegenerate) eigenvalue $\mathrm {E}\left ( \mathrm {U},k\right ) $ , as given by Theorems 3.1, 3.5 and 3.9. Note from Theorem 3.1 that the mapping

$$ \begin{align*} k\mapsto \left\Vert \Psi \left( \mathrm{U},k\right) \right\Vert ^{-1}\Psi \left( \mathrm{U},k\right) \end{align*} $$

is continuous on $\mathbb {T}^{2}$ for any $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , and its pointwise limit

$$ \begin{align*} k\mapsto \left\Vert \Psi \left( \infty ,k\right) \right\Vert ^{-1}\Psi \left( \infty ,k\right) \end{align*} $$

(cf. Theorem 3.5) is therefore measurable. In particular, the linear transformation $\mathfrak {P}_{\mathrm {U}}$ is well-defined for any $ \mathrm {U}\in \mathbb {R}_{0}^{+}\cup \{\infty \}$ . Moreover, one checks that it is norm-preserving.

Since $\mathrm {E}\left ( \mathrm {U},\cdot \right ) \in C(\mathbb {T}^{2}; \mathbb {R})\subseteq L^{\infty }(\mathbb {T}^{2},\nu )$ (see Theorems 3.1 (iv) and 3.5), we can consider the multiplication operator by $\mathrm {E}\left ( \mathrm {U},\cdot \right ) $ on $ L^{2}(\mathbb {T}^{2})$ , which is denoted by

(66) $$ \begin{align} M_{\mathrm{E}\left( \mathrm{U},\cdot \right) }\doteq {\int_{\mathbb{T} ^{2}}^{\oplus }}\mathrm{E}\left( \mathrm{U},k\right) \,\nu \left( \mathrm{d} k\right) ,\qquad \mathrm{U}\in \mathbb{R}_{0}^{+}\cup \{\infty \}. \end{align} $$

Remark also from Theorems 3.1 (iv) and 3.5 together with Corollary A.5 that $ P_{\mathrm {ac}}(M_{\mathrm {E}\left ( \mathrm {U},\cdot \right ) })=\mathbf {1}$ whenever $\hat {\upsilon }$ is real analytic on $\mathbb {S}^{2}$ . We then study now the (dressed) bound pair scattering channel, which is much simpler than in the unbound pair channel:

Theorem 3.14 (Bound pair (scattering) channel).

Let $h_{b}\in \lbrack 0,1/2]$ . Then the following assertions hold true:

1. i.) Dynamics and wave operators at finite $\mathrm {U}\in \mathbb {R} _{0}^{+}$ :

   $$ \begin{align*} \mathrm{e}^{it\mathbb{U}H\mathbb{U}^{\ast }}\mathfrak{P}_{\mathrm{U}}= \mathfrak{P}_{\mathrm{U}}\mathrm{e}^{itM_{\mathrm{E}\left( \mathrm{U},\cdot \right) }},\qquad t\in \mathbb{R}. \end{align*} $$

2. ii.) Dynamics in the hard-core limit $\mathrm {U}\rightarrow \infty $ :

   $$ \begin{align*} s-{\lim\limits_{\mathrm{U}\rightarrow \infty }}\mathfrak{P}_{\mathrm{U}}= \mathfrak{P}_{\infty }\qquad \text{and}\qquad s-{\lim\limits_{\mathrm{U} \rightarrow \infty }}\,\mathrm{e}^{it\mathbb{U}H\mathbb{U}^{\ast }}\mathfrak{ P}_{\mathrm{U}}=\mathfrak{P}_{\infty }\mathrm{e}^{itM_{\mathrm{E}\left( \mathrm{\infty },\cdot \right) }},\qquad t\in \mathbb{R}. \end{align*} $$

Proof. Assertion (i) is Proposition 4.25. Assertion (ii) results from Proposition 4.26.

From Theorem 3.14, (i) the scattering channel is time independent. For instance, for any $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , one trivially checks that

$$ \begin{align*} W^{\pm }\left( \mathbb{U}H\mathbb{U}^{\ast },M_{\mathrm{E}\left( \mathrm{U} ,\cdot \right) };\mathfrak{P}_{\mathrm{U}}\right) =\mathfrak{P}_{\mathrm{U} }P_{\mathrm{a}\mathrm{c}}\left( M_{\mathrm{E}\left( \mathrm{U},\cdot \right) }\right), \end{align*} $$

and since $\mathfrak {P}_{\mathrm {U}}^{\ast }\mathfrak {P}_{\mathrm {U}}= \mathbf {1}$ and $P_{\mathrm {ac}}$ is a projection, its scattering operator is equal to

$$ \begin{align*} S\left( \mathbb{U}H\mathbb{U}^{\ast },M_{\mathrm{E}\left( \mathrm{U},\cdot \right) };\mathfrak{P}_{\mathrm{U}}\right) =P_{\mathrm{a}\mathrm{c}}\left( M_{\mathrm{E}\left( \mathrm{U},\cdot \right) }\right) . \end{align*} $$

If $\hat {\upsilon }$ is additionally real analytic on $\mathbb {S}^{2}$ , then $ P_{\mathrm {ac}}(M_{\mathrm {E}\left ( \mathrm {U},\cdot \right ) })=\mathbf {1}$ , thanks to Theorem 3.1 and Corollary A.5. In this case, the wave and scattering operators are given by

$$ \begin{align*} W^{\pm }\left( \mathbb{U}H\mathbb{U}^{\ast },M_{\mathrm{E}\left( \mathrm{U} ,\cdot \right) };\mathfrak{P}_{\mathrm{U}}\right) =\mathfrak{P}_{\mathrm{U} }\qquad \text{and}\qquad S\left( \mathbb{U}H\mathbb{U}^{\ast },M_{\mathrm{E} \left( \mathrm{U},\cdot \right) };\mathfrak{P}_{\mathrm{U}}\right) = \mathbf{1} \end{align*} $$

for any $\mathrm {U}\in \mathbb {R}_{0}^{+}$ . Their hard-core limit are then also trivial, thanks to Theorem 3.14 (ii).

This scattering channel is therefore easy to study. In particular, similar to Remark 3.12, we can easily go back to the original Hilbert space $\mathfrak {h}_{0}$ (19), referring to space coordinates instead of the quasi-momenta. With this aim, we first observe that, for any $ \mathrm {U}\in \mathbb {R}_{0}^{+}$ ,

(67) $$ \begin{align} \mathrm{e}^{itH}\mathcal{P}_{\mathrm{U}}=\mathcal{P}_{\mathrm{U}}\mathrm{e} ^{itU_{f}^{\ast }M_{\mathrm{E}\left( \mathrm{U},\cdot \right) }U_{f}},\qquad t\in \mathbb{R}, \end{align} $$

where $\mathcal {P}_{\mathrm {U}}\in \mathcal {B}\left ( \mathfrak {h}_{0}, \mathfrak {H}\right ) $ is the new identification operator

$$ \begin{align*} \mathcal{P}_{\mathrm{U}}\doteq \mathbb{U}^{\ast }\mathfrak{P}_{\mathrm{U} }U_{f},\qquad \mathrm{U}\in \mathbb{R}_{0}^{+}, \end{align*} $$

and $U_{f}\doteq U_{2}U_{1}$ . See Equations (26)–(28) and Proposition 2.1.

On the one hand, Equation (67) together with Theorem 3.9 shows that $ \mathrm {E}\left ( \mathrm {U},\cdot \right ) $ defines (a family of) dispersion relations, in the sense of Definition 3.7. The Fourier transform of $\mathrm {E}\left ( \mathrm {U},\cdot \right ) $ is the (effective) hopping strength for the (spatially localized) dressed bound pairs. On the other hand, the new identification operator $\mathcal {P}_{\mathrm {U}}$ is translation invariant, that is,

$$ \begin{align*} \mathcal{P}_{\mathrm{U}}\theta _{x}=\Theta _{x}\mathcal{P}_{\mathrm{U}},\qquad x\in \mathbb{Z}^{2}, \end{align*} $$

where, for any fixed $x\in \mathbb {Z}^{2}$ , $\theta _{x}\in \mathcal {B}( \mathfrak {h}_{0})$ and $\Theta _{x}\in \mathcal {B}(\mathfrak {H})$ are the unique unitary operators respectively satisfying

$$ \begin{align*} \theta _{x}\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(z,\downarrow )}\right) =\mathfrak{e}_{(y+x,\uparrow )}\wedge \mathfrak{e} _{(z+x,\downarrow )},\qquad y,z\in \mathbb{Z}^{2}, \end{align*} $$

(see Equations (19)) and

$$ \begin{align*} \Theta _{x}\left( \mathfrak{\psi }\oplus \varphi \right) =\left( \theta _{x} \mathfrak{\psi }\right) \oplus \varphi \left( x+\cdot \right) ,\qquad \mathfrak{\psi \in h}_{0},\ \varphi \in \ell ^{2}\left( \mathbb{Z} ^{2}\right) . \end{align*} $$

See Equations (20). In addition, when $\hat {\upsilon }(0)\neq 0$ , Theorem 3.1 (iii) shows that the (dressed) fermion pair in the bound pair channel is exponentially localized in space; that is, the associate fermion-fermion correlation function decays exponentially fast with respect to the distance between the fermions, uniformly in time. Note that it is not required that the range of $\mathcal {P}_{\mathrm {U}}$ has a vanishing bosonic component, because of the expected presence of ‘gluing bosons’ in the dressed bound fermionic pair.

As a consequence, the bound channel describes an effective system of free localized, spinless quasi-particles which minimize the energy at any fixed total quasi-momentum. In particular, such quasi-particles of lowest energy, or dressed fermion pairs, are stable in time; that is, they cannot decay into an (even only asymptotically) unbound pair of fermions. Conversely, we also show in Section 3.3.1 that a pair of fermions that is asymptotically unbound far in the past is not able to bind together to form a stable bound pair in the distant future.

Nevertheless, these quasi-particles should only be stable with respect to external perturbations as soon as their states are related to quasi-momenta k such that $\mathrm {E}\left ( \mathrm {U},k\right ) <0$ . If the (dressed) quasi-particle is in a state whose support contains fibers k such that $\mathrm {E}\left ( \mathrm {U},k\right ) \geq 0$ , it is not in the most energetically favorable state, since

$$ \begin{align*} \min \sigma _{\mathrm{ess}}\left( A\left( \mathrm{U},0\right) \right) =0 \end{align*} $$

(see Theorem 3.1). In fact, if the component corresponding to quasi-momenta k such that $\mathrm {E}\left ( \mathrm {U},k\right ) \geq 0$ has nonvanishing Lebesgue measure, then the quasi-particle should be instable with respect to external perturbations, by possibly creating unbounded fermions with small quasi-momenta to decrease its total energy. This situation clearly appears for quasi-momenta $k\in \mathbb {T}^{2}$ such that $\hat {\upsilon }(k)=0$ or sufficiently small $\left \vert \hat {\upsilon } (k)\right \vert \ll 1$ when $k\neq 0$ , since in these two cases, either $ \mathrm {E}\left ( \mathrm {U},k\right ) =\mathfrak {b}\left ( k\right ) $ ( $\hat { \upsilon }(k)=0$ ) or $\mathrm {E}\left ( \mathrm {U},k\right ) \simeq \mathfrak {b} \left ( k\right ) $ ( $\left \vert \hat {\upsilon }(k)\right \vert \ll 1$ ) with $ \mathfrak {b}\left ( k\right ) \doteq h_{b}\epsilon \left ( 2-\cos \left ( k\right ) \right ) $ (see (32)). If such a decay process really occurs, one should see critical quasi-momenta, like in physical superconductors.

To prevent from this situation, one needs sufficiently strong $\left \vert \hat {\upsilon }\left ( k\right ) \right \vert \gg 1$ to have $\mathrm {E}\left ( \mathrm {U},k\right ) <0$ for all $k\in \mathbb {T}^{2}$ . In the position space, this means that the exchange strength between the two fermions and the boson, represented by the function $\upsilon :\mathbb {Z}^{2}\rightarrow \mathbb {R}$ appearing in (9)–(17), has to be sufficiently strong and localized, like in Remark 3.2, in order to get a sufficiently strong ‘gluing effect’ for dressed pairs, at all quasi-momenta. Recall also that the boson should be heavier than the two fermions (i.e., $h_{b}\in \lbrack 0,1/2]$ ).

Last but not least, all this discussion can be extended to the hard-core limit $\mathrm {U}\rightarrow \infty $ , in view of Theorems 3.9 and 3.14 (ii).

4.1 Notation

The purpose of this section is to fix (or recall) the notation and terminology that is used throughout the rest of the article. Let $\mathcal {X} $ be any complex Hilbert space. We denote its scalar product by $\langle \cdot ,\cdot \rangle _{\mathcal {X}}$ , with the convention that it is antilinear in the first argument and linear in the second one. The norm of $ \mathcal {X}$ is thus

$$ \begin{align*} \left\Vert \varphi \right\Vert _{\mathcal{X}}\doteq \sqrt{\left\langle \varphi ,\varphi \right\rangle _{\mathcal{X}}},\qquad \varphi \in \mathcal{ X}. \end{align*} $$

When there is no danger of confusion, as already said in Remark 1.2, we usually omit the subscript referring to the Hilbert space and write $\Vert \cdot \Vert $ for $\Vert \cdot \Vert _{\mathcal {X}}$ and $ \langle \cdot ,\cdot \rangle $ for $\langle \cdot ,\cdot \rangle _{\mathcal {X }}$ .

Recall that $\mathcal {B}(\mathcal {X})$ denotes the set of bounded (linear) operators on $\mathcal {X}$ . $\mathbf {1}\equiv \mathbf {1}_{\mathcal {X} }\in \mathcal {B}(\mathcal {X})$ is the identity operator. Given $T\in \mathcal {B}(\mathcal {X})$ , $T^{\ast }$ denotes its adjoint operator. The (full) spectrum, essential spectrum and resolvent set of any $T\in \mathcal {B }(\mathcal {X})$ are denoted by $\sigma (T)$ , $\sigma _{\mathrm {ess}}(T)$ and $\rho (T)$ , respectively. The operator norm of $\mathcal {B}(\mathcal {X})$ is

$$ \begin{align*} \Vert T\Vert _{\mathrm{op}}\doteq \sup \left\{ \left\Vert T\varphi \right\Vert _{\mathcal{X}}:\varphi \in \mathcal{X}\ \text{with }\left\Vert \varphi \right\Vert _{\mathcal{X}}=1\right\}. \end{align*} $$

Given $T\in \mathcal {B}(\mathcal {X})$ , $\mathcal {E}_{T}(\lambda )$ stands for the eigenspace associated with an eigenvalue $\lambda \in \sigma (T)$ of T.

In all the Section 4, we study properties of the Hamiltonian $H\in \mathcal {B}(\mathfrak {H})$ defined by (21). As one can see from (18)–(21) combined with (6), (8)–(11) and (17), it depends on several parameters. More precisely, $\epsilon ,\mathrm {U},h_{b}\in \mathbb {R} _{0}^{+}$ and $\alpha _{0}\in \mathbb {R}^{+}$ , while

$$ \begin{align*} \mathrm{u}:\mathbb{Z}^{2}\rightarrow \mathbb{R}_{0}^{+},\quad \mathrm{e} ^{\alpha _{0}\left\vert \cdot \right\vert }\mathfrak{p}_{1}:\mathbb{Z} ^{2}\rightarrow \mathbb{R}\ ,\quad \mathrm{e}^{\alpha _{0}\left\vert \cdot \right\vert }\mathfrak{p}_{2}:\mathbb{Z}^{2}\rightarrow \mathbb{R}\quad \text{and}\quad \upsilon :\mathbb{Z}^{2}\rightarrow \mathbb{R} \end{align*} $$

(with $\mathfrak {p}_{2}(z)\doteq 0$ for $z\notin 2\mathbb {Z}$ ) are all absolutely summable functions that are invariant with respect to $90^{\circ } $ -rotations. See Equations (7), (12) and (16). The parameters of the operator H are always fixed and arbitrary, unless we need to specify them to clarify some particular statement. Recall that the invariance under $90^{\circ }$ -rotation is not that important here. In fact, here, the only important point concerning this symmetry is that it implies that the Fourier transforms $\hat {\upsilon }$ , $ \hat {\mathfrak {p}}_{1}$ and $\hat {\mathfrak {p}}_{2}$ are real-valued, because

$$ \begin{align*} \upsilon (-x)=\upsilon (x)=\overline{\upsilon (x)},{\quad }\mathfrak{p }_{1}(-x)=\mathfrak{p}_{1}(x)=\overline{\mathfrak{p}_{1}(x)}{\quad and\quad }\mathfrak{p}_{2}(-x)=\mathfrak{p}_{2}(x)=\overline{\mathfrak{p} _{2}(x)}, \end{align*} $$

that is, the real valued functions $\upsilon $ , $\mathfrak {p}_{1}$ and $ \mathfrak {p}_{2}$ are reflection invariant, as a consequence of their $ 90^{\circ }$ -rotation invariance. Apart of this technical point, it is mainly relevant for the study of unconventional pairings, which is not done in the present work.

Note additionally that the on-site repulsion $\mathrm {U}\in \mathbb {R} _{0}^{+}$ appears explicitly in all the quantities defined in Sections 2–3. However, in Section 4, this parameter is only important for the Subsections 4.4 and 4.6. Therefore, unless the parameter $\mathrm {U}$ is important for our discussions or statements, we omit it in order to shorten the notation, by writing

$$ \begin{align*} f\left( k\right) \equiv f\left( \mathrm{U},k\right) \end{align*} $$

for any function $f(\mathrm {U},k)$ of the parameters $\mathrm {U}$ and k.

4.2 Computation of the fiber decomposition of the Hamiltonian

For completeness, we first proof in a simple lemma that the fiber Hamiltonians defined by (42) yield an element of $ L^{\infty }\left ( \mathbb {T}^{2},\mathcal {B}(\mathcal {H})\right ) $ . Then, we prove Proposition 2.1.

Proof. Since $\cos :\mathbb {R}^{2}\rightarrow \mathbb {R}$ , as defined by (35), is a nonexpansive mapping with period $2\pi $ , given $ k,k^{\prime },p\in \mathbb {T}^{2}$ , the quantity

$$ \begin{align*} \mathfrak{f}(k^{\prime })\left( p\right) -\mathfrak{f}\left( k\right) \left( p\right) =\epsilon \left\{ \cos \left( p+k\right) -\cos \left( p+k^{\prime }\right) \right\} =\epsilon \left\{ \cos \left( p+k\right) -\cos \left( p+k^{\prime }+2\pi q\right) \right\} \end{align*} $$

(see (33)) is bounded for any $q\in \mathbb {Z}^{2}$ by

$$ \begin{align*} \left\vert \mathfrak{f}(k^{\prime })\left( p\right) -\mathfrak{f}\left( k\right) \left( p\right) \right\vert \leq \epsilon \left\vert \left( p+k\right) -\left( p+k^{\prime }+2\pi q\right) \right\vert =\epsilon \left\vert k-k^{\prime }+2\pi q\right\vert . \end{align*} $$

Hence, taking the minimum over all $q\in \mathbb {Z}^{2}$ and the supremum over all $p\in \mathbb {T}^{2}$ , we obtain from (22) and (38) that

$$ \begin{align*} \left\Vert A_{1,1}(k^{\prime })-A_{1,1}(k)\right\Vert _{\mathrm{o}\mathrm{p} } &=\left\Vert M_{\mathfrak{f}(k^{\prime })}-M_{\mathfrak{f}\left( k\right) }\right\Vert _{\mathrm{o}\mathrm{p}}=\sup_{p\in \mathbb{T}^{2}}\left\vert \mathfrak{f}(k^{\prime })\left( p\right) -\mathfrak{f}\left( k\right) \left( p\right) \right\vert \\ &\leq \epsilon \min_{q\in \mathbb{Z}^{2}}\left\vert k-k^{\prime }+2\pi q\right\vert =\epsilon d_{\mathbb{T}^{2}}(k,k^{\prime }) \end{align*} $$

for all $k,k^{\prime }\in \mathbb {T}^{2}$ . In other words, the mapping

$$ \begin{align*} A_{1,1}\left( \cdot \right) :\mathbb{T}^{2}\rightarrow \mathcal{B}\left( L^{2}\left( \mathbb{T}^{2}\right) \right) \end{align*} $$

is ( $\epsilon $ -Lipschitz) continuous with respect to the metric $d_{\mathbb { T}^{2}}$ . Similarly, we see that $\mathfrak {b}:\mathbb {T}^{2}\rightarrow \mathbb {R}$ , defined by (32), is continuous with respect to $d_{\mathbb { T}^{2}}$ , and hence, so is $A_{2,2}:\mathbb {T}^{2}\rightarrow \mathcal {L}( \mathbb {C})$ (see (41)). In addition, by the triangle and Cauchy-Schwarz inequalities, for any $k,k^{\prime }\in \mathbb {T}^{2}$ and $ \varphi \in L^{2}(\mathbb {T}^{2})$ ,

$$ \begin{align*} \left\vert \hat{\upsilon}(k^{\prime })\left\langle \mathfrak{d}(k^{\prime }),\varphi \right\rangle -\hat{\upsilon}(k)\left\langle \mathfrak{d} (k),\varphi \right\rangle \right\vert \leq \left\vert \hat{\upsilon} (k^{\prime })-\hat{\upsilon}(k)\right\vert \left\Vert \mathfrak{d}(k^{\prime })\right\Vert \left\Vert \varphi \right\Vert +\left\vert \hat{\upsilon} (k)\right\vert \left\Vert \mathfrak{d}(k^{\prime })-\mathfrak{d} (k)\right\Vert \left\Vert \varphi \right\Vert. \end{align*} $$

Because of (12) and (16), $\mathfrak {d}(k),\hat { \upsilon }\in C\left ( \mathbb {T}^{2}\right ) $ . So, since $\mathbb {T}^{2}$ is ( $d_{\mathbb {T}^{2}}$ -)compact and

$$ \begin{align*} \left\Vert A_{2,1}(k^{\prime })-A_{2,1}(k)\right\Vert _{\mathrm{o}\mathrm{p} }=\sup_{\varphi \in L^{2}(\mathbb{T}^{2}),||\varphi ||_{2}=1}\left\vert \hat{ \upsilon}(k^{\prime })\left\langle \mathfrak{d}(k^{\prime }),\varphi \right\rangle -\hat{\upsilon}(k)\left\langle \mathfrak{d}(k),\varphi \right\rangle \right\vert , \end{align*} $$

we deduce from the last inequality and (39) that $A_{2,1}:\mathbb {T} ^{2}\rightarrow L^{2}(\mathbb {T}^{2})^{\ast }$ is continuous. As $ A_{1,2}(k)=A_{2,1}(k)^{\ast }$ for all $k\in \mathbb {T}^{2}$ , we conclude that the mapping $A:\mathbb {T}^{2}\rightarrow \mathcal {B}(\mathcal {H})$ is continuous, and hence bounded on the $d_{\mathbb {T}^{2}}$ -compact set $ \mathbb {T}^{2}$ .

We now compute the following unitary transformation of the Hamiltonian H (see (18)):

(68) $$ \begin{align} \mathbb{U}H\mathbb{U}^{\ast }=\left( \begin{array}{cc} U_{f} & 0 \\ 0 & \mathcal{F} \end{array} \right) \begin{pmatrix} H_{f} & W_{\mathrm{b\rightarrow f}} \\[0.5em] W_{\mathrm{f\rightarrow b}} & H_{b} \end{pmatrix} \left( \begin{array}{cc} U_{f}^{\ast } & 0 \\ 0 & \mathcal{F}^{\ast } \end{array} \right) = \begin{pmatrix} U_{f}H_{f}U_{f}^{\ast } & U_{f}W_{\mathrm{b\rightarrow f}}\mathcal{F}^{\ast } \\[0.5em] \mathcal{F}W_{\mathrm{f\rightarrow b}}U_{f}^{\ast } & \mathcal{F}H_{b} \mathcal{F}^{\ast } \end{pmatrix} \end{align} $$

with $\mathbb {U}$ defined by (26)–(31). In fact, the remaining part of this section is devoted to the computations leading to Proposition 2.1.

To begin with, we observe that, for any lattice site $x\in \mathbb {Z}^{2}$ and spin $s\in \{\uparrow ,\downarrow \}$ , $b_{x}\doteq b(\mathfrak {e}_{x})$ and $a_{x,s}\doteq a(\mathfrak {e}_{(x,s)})$ , where $\{\mathfrak {e}_{x}\doteq \delta _{x,\cdot }\}_{x\in \mathbb {Z}^{2}}$ is the canonical orthonormal basis (5) of $\ell ^{2}(\mathbb {Z}^{2})$ and $a_{x,s}$ ( $ b_{x}^{\ast }$ ) denotes the annihilation operator acting on the fermionic (bosonic) Fock space $\mathfrak {F}_{-}$ ( $\mathfrak {F}_{+}$ ) of a fermion (boson). In both cases, $\Omega $ denotes the vacuum state. We compute each term of the the right-hand side (68) separately:

$\underline{\mathrm{Computation\ of}\ U_{f}H_{f}U_{f}^{\ast}\ \mathrm{in\ relation\ to}\ {A_{1,1}}}.$ We first note from (A.6) that, for any $ x,y,u\in \mathbb {Z}^{2}$ and $s\in \{\uparrow ,\downarrow \}$ ,

$$ \begin{align*} a_{x,s}\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) =\frac{1}{\sqrt{2}}\left( \langle \mathfrak{e}_{(x,s)},\mathfrak{e} _{(y,\uparrow )}\rangle \mathfrak{e}_{(u,\downarrow )}-\langle \mathfrak{e} _{(x,s)},\mathfrak{e}_{(u,\downarrow )}\rangle \mathfrak{e}_{(y,\uparrow )}\right) \end{align*} $$

vanishes whenever $(x,s)\notin \{(y,\uparrow ),(u,\downarrow )\}$ . Using this observation and (A.7), one concludes that, for any $y,u\in \mathbb {Z}^{2}$ ,

(69) $$ \begin{align} &\sum_{s\in \{\uparrow ,\downarrow \},\ x,z\in \mathbb{Z} ^{2}\,:\,|z|=1}a_{x,s}^{\ast }a_{x+z,s}\,\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) \notag \\ &=\sum_{z\in \mathbb{Z}^{2}\,:\,|z|=1}\left( a_{y+z,\uparrow }^{\ast }a_{y,\uparrow }\,\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(u,\downarrow )}\right) +a_{u+z,\downarrow }^{\ast }a_{u,\downarrow }\,\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) \right) \notag \\ &=\frac{1}{\sqrt{2}}\sum_{z\in \mathbb{Z}^{2}\,:\,|z|=1}\left( a_{y+z,\uparrow }^{\ast }\left( \mathfrak{e}_{(u,\downarrow )}\right) -a_{u+z,\downarrow }^{\ast }\left( \mathfrak{e}_{(y,\uparrow )}\right) \right) \notag \\ &=\sum_{z\in \mathbb{Z}^{2}\,:\,|z|=1}\left( \mathfrak{e}_{(y+z,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}+\mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u+z,\downarrow )}\right) . \end{align} $$

Likewise, we see that, for any $y,u\in \mathbb {Z}^{2}$ ,

(70) $$ \begin{align} \sum_{s\in \{\uparrow ,\downarrow \},\ x\in \mathbb{Z}^{2}}a_{x,s}^{\ast }a_{x,s}\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(u,\downarrow )}\right) =2\mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(u,\downarrow )}. \end{align} $$

Moreover, as $\mathrm {u}:\mathbb {Z}^{2}\rightarrow \mathbb {R}$ is absolutely summable and invariant with respect to $180^{\circ }$ -rotations (cf. (7)), we also get that

(71) $$ \begin{align} \sum_{x,z\in \mathbb{Z}^{2}}\mathrm{u}\left( z\right) n_{x,\uparrow }n_{x+z,\downarrow }\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(u,\downarrow )}\right) &=\sum_{z\in \mathbb{Z}^{2}}\mathrm{u}\left( z\right) n_{u-z,\uparrow }\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) =\mathrm{u}\left( u-y\right) \left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) \notag \\ &=\mathrm{u}\left( y-u\right) \left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) , \end{align} $$

for any $y,u\in \mathbb {Z}^{2}$ , which, for $\mathrm {u}\left ( z\right ) =\delta _{z,0}$ , is equal to

(72) $$ \begin{align} \sum_{x\in \mathbb{Z}^{2}}n_{x,\uparrow }n_{x,\downarrow }\left( \mathfrak{e} _{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) =\delta _{y,u}\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) . \end{align} $$

We thus infer from (6) combined with (69)–(72) that

$$ \begin{align*} H_{f}\left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) &=-{\frac{\epsilon }{2}}\sum_{z\in \mathbb{Z}^{2}\,:\,|z|=1} \left( \mathfrak{e}_{(y+z,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}+ \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e}_{(u+z,\downarrow )}\right) \\ &\quad +\left( 4\epsilon +\mathrm{U}\delta _{y,u}+\mathrm{u}\left( y-u\right) \right) \left( \mathfrak{e}_{(y,\uparrow )}\wedge \mathfrak{e} _{(u,\downarrow )}\right) , \end{align*} $$

for any $y,u\in \mathbb {Z}^{2}$ . Then, conjugating $H_{f}$ by the unitary operator $U_{f}$ (28)–(31) gives the equality

$$ \begin{align*} U_{f}H_{f}U_{f}^{\ast }\left( \mathfrak{\hat{e}}_{y}\left( \cdot \right) \mathfrak{\hat{e}}_{y-u}\right) &=U_{f}H_{f}\left( \mathfrak{e} _{(y,\uparrow )}\wedge \mathfrak{e}_{(u,\downarrow )}\right) \\ & =-{\frac{\epsilon }{2}}\sum_{z\in \mathbb{Z}^{2}\,:\,|z|=1}\left( \mathfrak{\hat{e}}_{y+z}\left( \cdot \right) \mathfrak{\hat{e}}_{y+z-u}+ \mathfrak{\hat{e}}_{y}\left( \cdot \right) \mathfrak{\hat{e}} _{y-(u+z)}\right) \\ &\quad +\left( 4\epsilon +\mathrm{U}\delta _{y,u}+\mathrm{u}\left( y-u\right) \right) \mathfrak{\hat{e}}_{y}\left( \cdot \right) \mathfrak{\hat{e}}_{y-u} \end{align*} $$

for any $y,u\in \mathbb {Z}^{2}$ . By first evaluating the above expression at $k\in \mathbb {T}^{2}$ , and then at $p\in \mathbb {T}^{2}$ , and using (33) and (35), we obtain that

$$ \begin{align*} \left( U_{f}H_{f}U_{f}^{\ast }\left( \mathfrak{\hat{e}}_{y}\left( \cdot \right) \mathfrak{\hat{e}}_{y-u}\right) (k)\right) \left( p\right) &= \mathfrak{\hat{e}}_{y}\left( k\right) \mathfrak{\hat{e}}_{y-u}\left( p\right) \left( \mathrm{U}\delta _{y,u}+\mathrm{u}\left( y-u\right) \right) \\ &\quad +\mathfrak{\hat{e}}_{y}\left( k\right) \mathfrak{\hat{e}}_{y-u}\left( p\right) \epsilon \left( 4-{\frac{1}{2}}\sum_{z\in \mathbb{Z} ^{2}\,:\,|z|=1}\left( \mathrm{e}^{i\left( k+p\right) \cdot z}+\mathrm{e} ^{ip\cdot z}\right) \right) \\ &=\mathfrak{\hat{e}}_{y}\left( k\right) \mathfrak{\hat{e}}_{y-u}\left( p\right) \left( \mathrm{U}\delta _{y,u}+\mathrm{u}\left( y-u\right) + \mathfrak{f}\left( k\right) \left( p\right) \right) \\ &=\mathfrak{\hat{e}}_{y}\left( k\right) \left( \mathrm{U}P_{0}+{ \sum\limits_{x\in \mathbb{Z}^{2}}}\,\mathrm{u}\left( x\right) P_{x}+M_{ \mathfrak{f}(k)}\left( p\right) \right) \left( \mathfrak{\hat{e}} _{y-u}\right) \left( p\right) , \end{align*} $$

for any $y,u\in \mathbb {Z}^{2}$ . By (37)–(38), it follows that

(73) $$ \begin{align} U_{f}H_{f}U_{f}^{\ast }\left( \mathfrak{\hat{e}}_{y}\left( \cdot \right) \mathfrak{\hat{e}}_{y-u}\right) =\left( {\int_{\mathbb{T}^{2}}^{\oplus }} A_{1,1}\left( p\right) \,\nu (\mathrm{d}p)\right) \mathfrak{\hat{e}} _{y}\left( \cdot \right) \mathfrak{\hat{e}}_{y-u}. \end{align} $$

As $\{\mathfrak {\hat {e}}_{y}\left ( \cdot \right ) \mathfrak {\hat {e}} _{y-u}\}_{y,u\in \mathbb {Z}^{2}}$ is an orthonormal basis for the Hilbert space

$$ \begin{align*} \int_{\mathbb{T}^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \nu \left( \mathrm{d}k\right) , \end{align*} $$

we deduce from (73) that

$$ \begin{align*} U_{f}H_{f}U_{f}^{\ast }={\int_{\mathbb{T}^{2}}^{\oplus }}A_{1,1}\left( k\right) \,\nu (\mathrm{d}k). \end{align*} $$

$\underline{\mathrm{Computation\ of}\ \mathcal {F}H_{b}\mathcal {F}^{\ast }\ \mathrm{in\ relation\ to}\ A_{2,2}.}$ Using (A.8) and (A.9), we conclude that, for any $y\in \mathbb {Z}^{2}$ ,

$$ \begin{align*} H_{b}\left( \mathfrak{e}_{y}\right) =\epsilon h_{b}\left( 2{ \sum\limits_{x\in \mathbb{Z}^{2}}}\langle \mathfrak{e}_{x},\mathfrak{e} _{y}\rangle b_{x}^{\ast }\Omega -{\frac{1}{2}\sum\limits_{z\in \mathbb{Z} ^{2}\,:\,|z|=1}}\left\langle \mathfrak{e}_{x+z},\mathfrak{e} _{y}\right\rangle b_{x}^{\ast }\Omega \right) =\epsilon h_{b}\left( 2 \mathfrak{e}_{y}-{\frac{1}{2}\sum\limits_{z\in \mathbb{Z}^{2}\,:\,|z|=1}} \mathfrak{e}_{y+z}\right) \end{align*} $$

so that

$$ \begin{align*} \mathcal{F}H_{b}\left( \mathfrak{e}_{y}\right) =\epsilon h_{b}\left( 2 \mathfrak{\hat{e}}_{y}-{\frac{1}{2}\sum\limits_{z\in \mathbb{Z} ^{2}\,:\,|z|=1}}\mathfrak{\hat{e}}_{y+z}\right) ,\qquad y\in \mathbb{Z} ^{2}. \end{align*} $$

Therefore, using that $\mathfrak {\hat {e}}_{y}\equiv \mathcal {F(\mathfrak {e}} _{y}\mathcal {)=}\,\mathrm {e}^{ik\cdot y}$ (see (31)) as well as (32), (35) and (41), we arrive at the result

$$ \begin{align*} \mathcal{F}H_{b}\mathcal{F}^{\ast }(\mathfrak{\hat{e}}_{y})\left( k\right) &=\epsilon h_{b}\mathrm{e}^{ik\cdot y}\left( 2-{\frac{1}{2} \sum\limits_{z\in \mathbb{Z}^{2}\,:\,|z|=1}}\mathrm{e}^{ik\cdot z}\right) =\epsilon h_{b}\left( 2-\cos \left( k\right) \right) \mathrm{e}^{ik\cdot y} \\ &=\mathfrak{b}\left( k\right) \mathrm{e}^{ik\cdot y}=A_{2,2}\left( k\right) \mathfrak{\hat{e}}_{y}\left( k\right) \end{align*} $$

for all $y\in \mathbb {Z}^{2}$ and $k\in \mathbb {T}^{2}$ . As $\{\hat { \mathfrak {e}}_{y}\}_{y\in \mathbb {Z}^{2}}$ is an orthonormal basis for $ L^{2}(\mathbb {T}^{2})$ , it follows that

$$ \begin{align*} \mathcal{F}H_{b}\mathcal{F}^{\ast }={\int_{\mathbb{T}^{2}}^{\oplus }} A_{2,2}(k)\,\nu (\mathrm{d}k). \end{align*} $$

$\underline{\mathrm{Computation\ of}\ \mathcal {F}W_{\mathrm {f\rightarrow b} }U_{f}^{\ast }\ \mathrm{and}\ U_{f}W_{\mathrm {b\rightarrow f}}\mathcal {F}^{\ast }\ \mathrm{in\ relation\ to}\ A_{2,1}\ \mathrm{and}\ A_{1,2}}.$ Observe from (11) and (A.7) that, for all $y\in \mathbb {Z}^{2}$ ,

$$ \begin{align*} c_{y}^{\ast }\Omega =\sqrt{2}{\sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p}_{1}\left( z\right) \mathfrak{e}_{(y+z,\uparrow )}\wedge \mathfrak{e}_{(y,\downarrow )}+\mathfrak{p}_{2}\left( 2z\right) \mathfrak{e} _{(y+z,\uparrow )}\wedge \mathfrak{e}_{(y-z,\downarrow )}\right), \end{align*} $$

and as a consequence, using (9)–(10) as well as ( A.8), we get that, for any $u\in \mathbb {Z}^{2}$ ,

$$ \begin{align*} W_{\mathrm{b}\rightarrow \mathrm{f}}\left( \mathfrak{e}_{u}\right) ={ \sum\limits_{y\in \mathbb{Z}^{2}}}\upsilon \left( u-y\right) { \sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p}_{1}\left( z\right) \mathfrak{e}_{(y+z,\uparrow )}\wedge \mathfrak{e}_{(y,\downarrow )}+ \mathfrak{p}_{2}\left( 2z\right) \mathfrak{e}_{(y+z,\uparrow )}\wedge \mathfrak{e}_{(y-z,\downarrow )}\right) . \end{align*} $$

Therefore, by (28)–(31), for any $u\in \mathbb {Z }^{2}$ ,

$$ \begin{align*} U_{f}W_{\mathrm{b\rightarrow f}}\mathcal{F}^{\ast }\left( \mathfrak{\hat{e}} _{u}\right) ={\sum\limits_{y\in \mathbb{Z}^{2}}}\upsilon \left( u-y\right) { \sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p}_{1}\left( z\right) \mathfrak{\hat{e}}_{y+z}\left( \cdot \right) \mathfrak{\hat{e}}_{z}+ \mathfrak{p}_{2}\left( 2z\right) \mathfrak{\hat{e}}_{y+z}\left( \cdot \right) \mathfrak{\hat{e}}_{2z}\right) . \end{align*} $$

In particular, as $\upsilon ,\mathfrak {p}_{1},\mathfrak {p}_{2}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}$ are absolutely summable and invariant with respect to $180^{\circ }$ -rotations (cf. (12) and (16)), we deduce from the last equality that, for any $u\in \mathbb {Z}^{2}$ and $k,p\in \mathbb {T}^{2}$ ,

$$ \begin{align*} \left( U_{f}W_{\mathrm{b\rightarrow f}}\mathcal{F}^{\ast }\left( \mathfrak{ \hat{e}}_{u}\right) \left( k\right) \right) \left( p\right) &={ \sum\limits_{y\in \mathbb{Z}^{2}}}\upsilon \left( u-y\right) { \sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p}_{1}\left( z\right) \mathrm{e}^{ik\cdot (y+z)}\mathrm{e}^{ip\cdot z}+\mathfrak{p}_{2}\left( 2z\right) \mathrm{e}^{ik\cdot (y+z)}\mathrm{e}^{i2p\cdot z}\right) \\ &={\sum\limits_{z\in \mathbb{Z}^{2}}}\,\left( \mathfrak{p}_{1}\left( z\right) \mathrm{e}^{i\left( k+p\right) \cdot z}+\mathfrak{p}_{2}\left( 2z\right) \mathrm{e}^{i\left( k+2p\right) \cdot z)}\right) \mathrm{e} ^{ik\cdot u}{\sum\limits_{y\in \mathbb{Z}^{2}}}\upsilon \left( y-u\right) \mathrm{e}^{ik\cdot \left( y-u\right) }. \end{align*} $$

Using now that

$$ \begin{align*} {\sum\limits_{z\in \mathbb{Z}^{2}}}\mathfrak{p}_{1}\left( z\right) \mathrm{e} ^{i\left( k+p\right) \cdot z}=\mathfrak{\hat{p}}_{1}\left( k+p\right) \qquad \text{and}\qquad {\sum\limits_{z\in \mathbb{Z}^{2}}}\mathfrak{p}_{2}\left( 2z\right) \mathrm{e}^{i\left( k+2p\right) \cdot z)}=\mathfrak{\hat{p}} _{2}\left( k/2+p\right) \end{align*} $$

( $\mathfrak {p}_{2}\left ( z\right ) \doteq 0$ for $z\notin (2\mathbb {Z})^{2}$ ), we arrive at the equalities

$$ \begin{align*} \left( U_{f}W_{\mathrm{b\rightarrow f}}\mathcal{F}^{\ast }\left( \mathfrak{ \hat{e}}_{u}\right) \left( k\right) \right) \left( p\right) &=\left( \mathfrak{\hat{p}}_{1}\left( k+p\right) +\mathfrak{\hat{p}}_{2}\left( k/2+p\right) \right) \mathrm{e}^{ik\cdot u}{\sum\limits_{z\in \mathbb{Z}^{2}} }\upsilon \left( z\right) \mathrm{e}^{ik\cdot z} \\ &=\hat{\upsilon}\left( k\right) \mathfrak{d}\left( k\right) \left( p\right) \hat{\mathfrak{e}}_{u}(k)=\left[ A_{1,2}\left( k\right) \hat{\mathfrak{e}} _{u}(k)\right] \left( p\right) , \end{align*} $$

with $\mathfrak {d}(k)(p)$ and $A_{1,2}(k)$ defined by (34) and (40), respectively. As $u\in \mathbb {Z}^{2}$ and $k,p\in \mathbb {T}^{2}$ are arbitrary in the above equations, this shows that $U_{f}W_{\mathrm {b} \rightarrow \mathrm {f}}\mathcal {F}^{\ast }$ coincides with the bounded linear transformation

$$ \begin{align*} \begin{array}{cccc} J: & L^{2}\left( \mathbb{T}^{2}\right) & \rightarrow & \int_{\mathbb{T} ^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \nu \left( \mathrm{d} k\right) \\ & & & \\ & \varphi & \mapsto & A_{1,2}\left( \cdot \right) \hat{\mathfrak{e}} _{u}(\cdot ) \end{array} \end{align*} $$

on the orthonormal basis $\{\hat {\mathfrak {e}}_{u}\}_{u\in \mathbb {Z}^{2}}$ and, therefore, $U_{f}W_{\mathrm {b}\rightarrow \mathrm {f}}\mathcal {F}^{\ast }=J $ . By taking adjoints on both sides, we also obtain $\mathcal {F}W_{ \mathrm {f}\rightarrow \mathrm {b}}U_{f}^{\ast }=J^{\ast }$ . Finally, one can easily check from (39) that

$$ \begin{align*} \left( J^{\ast }\psi \right) \left( k\right) =A_{2,1}\left( k\right) \psi \left( k\right) ,\qquad k\in \mathbb{T}^{2},\ \psi \in \int_{\mathbb{T} ^{2}}^{\oplus }L^{2}\left( \mathbb{T}^{2}\right) \nu \left( \mathrm{d} k\right) . \end{align*} $$

This completes the proof of Proposition 2.1.

4.3 Spectrum of the fiber Hamiltonians

We start with the study of the essential spectrum of fiber Hamiltonians (42) at any total quasi-momentum $k\in \mathbb {T}^{2}$ , before considering afterwards the discrete one in Section 4.3.2. Then, in Section 4.3.3, we study the bottom of the spectrum ( $k\in \mathbb {T}^{2}$ being fixed). The whole study leads to important spectral properties of H, as previously explained, via Proposition 2.1 combined with Theorem A.3.

4.3.1 Essential spectrum

The essential spectrum $\sigma _{\mathrm {ess}}(A(k))$ of the fiber Hamiltonian

$$ \begin{align*} A\left( k\right) \equiv A\left( \mathrm{U},k\right) , \end{align*} $$

defined by (42) at fixed $\mathrm {U}\in \mathbb {R} _{0}^{+}$ and total quasi-momentum $k\in \mathbb {T}^{2}$ , is completely determined by the following proposition:

Proposition 4.2 (Essential spectrum of fiber Hamiltonians).

For any $k\in \mathbb {T}^{2}$ and $h_{b},\epsilon ,\mathrm {U}\in \mathbb {R} _{0}^{+}$ , one has

$$ \begin{align*} \sigma _{\mathrm{ess}}\left( A\left( k\right) \right) =\sigma _{\mathrm{ess} }\left( A_{1,1}\left( k\right) \right) =\sigma _{\mathrm{ess}}\left( B_{1,1}\left( k\right) \right) =\sigma \left( M_{\mathfrak{f}\left( k\right) }\right) =2\epsilon \cos \left( k/2\right) \left[ -1,1\right] +4\epsilon , \end{align*} $$

where $M_{\mathfrak {f}\left ( k\right ) }$ stands for the multiplication operator associated with the function $\mathfrak {f}(k)$ (33), while $ B_{1,1}(k)$ and $A_{1,1}(k)\equiv A_{1,1}(\mathrm {U},k)$ are defined by (37) and (38), respectively.

Proof. Recall that $\nu $ is the normalized Haar measure defined by (23) on $\mathbb {T}^{2}$ . Fix $k\in \mathbb {T}^{2}$ . If $\lambda $ is an eigenvalue of $M_{\mathfrak {f}\left ( k\right ) }$ with associated eigenvector $\varphi \in L^{2}(\mathbb {T}^{2})$ , then

$$ \begin{align*} M_{\mathfrak{f}\left( k\right) }\varphi \left( p\right) \doteq \mathfrak{f} \left( k\right) \left( p\right) \varphi \left( p\right) =\lambda \varphi \left( p\right) \end{align*} $$

for almost every $p\in \mathbb {T}^{2}$ . As $\varphi \neq 0$ , there exists $ \Omega \subseteq \mathbb {T}^{2}$ with strictly positive measure $\nu (\Omega )>0$ such that the above equality holds true with $\varphi (p)\neq 0$ for every $p\in \Omega $ . Thus, $\mathfrak {f}(k)(p)=\lambda $ for all $p\in \Omega $ . Because

$$ \begin{align*} \nu ([-\pi ,\pi )^{2}\backslash (-\pi ,\pi )^{2})=0, \end{align*} $$

we can assume without loss of generality that $\Omega \subseteq (-\pi ,\pi )^{2}$ . Since $\mathfrak {f}(k)-\lambda $ is real analytic on the open domain $(-\pi ,\pi )^{2}$ in $\mathbb {R}^{2}$ and the zeros of any nonconstant real analytic function have null Lebesgue measure [Reference Mityagin49], we would have $\nu (\Omega )=0$ , which contradicts our choice of the set $\Omega $ . Recall indeed that $\nu $ is the Lebesgue measure, up to a normalization constant (see (23)). Hence, $M_{\mathfrak {f}\left ( k\right ) } $ has no eigenvalues and, thus,

$$ \begin{align*} \sigma _{\mathrm{ess}}\left( M_{\mathfrak{f}\left( k\right) }\right) =\sigma \left( M_{\mathfrak{f}\left( k\right) }\right) =\,\mathfrak{f}\left( k\right) (\mathbb{T}^{2}). \end{align*} $$

The last equality holds true, for $\mathfrak {f}$ is a continuous function on a compact domain – namely, the torus $\mathbb {T}^{2}$ . Clearly,

$$ \begin{align*} \mathfrak{f}\left( k\right) (\mathbb{T}^{2})=2\epsilon \cos \left( k/2\right) \left[ -1,1\right] +4\epsilon . \end{align*} $$

Observing that $A_{1,2}(k)$ , $A_{2,1}(k)$ , $A_{2,2}(k)$ and $P_{x}$ are all rank-one linear transformations, we can apply the stability of the essential spectrum under compact perturbations (see [Reference Schmüdgen95, Corollary 8.16]) to conclude that

(74) $$ \begin{align} \sigma _{\mathrm{ess}}\left( A\left( k\right) \right) =\sigma _{\mathrm{ess} }\left( M_{\mathfrak{f}\left( k\right) }\right) =\,\mathfrak{f}\left( k\right) (\mathbb{T}^{2}). \end{align} $$

In fact, from the absolute summability of the function $\mathrm {u}:\mathbb {Z} ^{2}\rightarrow \mathbb {R}_{0}^{+}$ (see (7)), along with the closedness of the subspace of compact operators in the Banach space of all bounded operators, the operator defined by the infinite sum

$$ \begin{align*} {\sum\limits_{x\in \mathbb{Z}^{2}}}\,\mathrm{u}\left( x\right) P_{x} \end{align*} $$

is not only bounded, but even compact on the Hilbert space $L^{2}(\mathbb {T} ^{2})$ . Recall that $P_{x}$ denotes the orthogonal projection onto the one-dimensional subspace $\mathbb {C}\hat {\mathfrak {e}}_{x}\subseteq L^{2}( \mathbb {T}^{2})$ for any $x\in \mathbb {Z}^{2}$ . For the same reasons,

$$ \begin{align*} \sigma _{\mathrm{ess}}\left( A_{1,1}\left( k\right) \right) =\sigma _{ \mathrm{ess}}\left( B_{1,1}\left( k\right) \right) =\sigma _{\mathrm{ess} }\left( M_{\mathfrak{f}\left( k\right) }+\mathrm{U}P_{0}\right) =\sigma _{ \mathrm{ess}}\left( M_{\mathfrak{f}\left( k\right) }\right) .\\[-37pt] \end{align*} $$

Corollary 4.3 (Bottom of the spectrum of $A_{1,1}(k)$ and $B_{1,1}(k)$ ).

For any $k\in \mathbb {T}^{2}$ , one has that

$$ \begin{align*} \min \sigma \left( A_{1,1}\left( k\right) \right) =\min \sigma \left( B_{1,1}\left( k\right) \right) =\min \sigma \left( M_{\mathfrak{f}\left( k\right) }\right) =4\epsilon -2\epsilon \cos \left( k/2\right) \doteq \mathfrak{z}\left( k\right) . \end{align*} $$

Proof. Fix $k\in \mathbb {T}^{2}$ . Since $\mathrm {U}\in \mathbb {R}_{0}^{+}$ , one has the operator inequalities

$$ \begin{align*} M_{\mathfrak{f}\left( k\right) }\leq M_{\mathfrak{f}\left( k\right) }+{ \sum\limits_{x\in \mathbb{Z}^{2}}}\,\mathrm{u}\left( x\right) P_{x}\doteq B_{1,1}\left( k\right) \leq B_{1,1}\left( k\right) +\mathrm{U}P_{0}\doteq A_{1,1}(k), \end{align*} $$

for the set of positive operators on a Hilbert space forms a norm-closed convex cone. By combining the last inequalities with Proposition 4.2, one arrives at the assertion.

4.3.2 Discrete spectrum

In the following, it is technically convenient to assume that $\mathfrak {b} (k) $ , which is the kinetic energy of a boson with quasi-momentum k, is below the bottom of the spectrum of $A_{1,1}(k)$ – that is, the minimum energy of the fermion pair for the same total quasi-momentum. In other words, we assume from now on that

(75) $$ \begin{align} \mathfrak{b}(k)\doteq h_{b}\epsilon \left( 2-\cos \left( k\right) \right) \leq \mathfrak{z}(k)\doteq 4\epsilon -2\epsilon \cos (k/2) \end{align} $$

for all $k\in \mathbb {T}^{2}$ , with equality only at $k=0$ . See Equation (32) and Corollary 4.3. By direct computations,Footnote ¹⁷ one verifies that this amounts to take $h_{b}$ in the interval $ [0,1/2]$ . This means that we consider a regime where the boson mass is at least the mass of the two fermions, as physically expected for cuprate superconductors; see [Reference Bru, de Siqueira Pedra and de Pasquale21, Section 3.1].

Proposition 4.4 (Eigenvalues of fiber Hamiltonians – I).

Take any $k\in \mathbb {T}^{2}$ and $h_{b}\in \lbrack 0,1/2]$ .

1. i.) $\lambda \neq \mathfrak {b}(k)$ is an eigenvalue of $A(k)$ iff there is a nonzero vector $\varphi \in L^{2}(\mathbb {T}^{2})$ in the kernel of the bounded operator

   $$ \begin{align*} A_{1,1}\left( k\right) -\lambda \mathbf{1}-\left( \mathfrak{b}\left( k\right) -\lambda \right) ^{-1}A_{1,2}\left( k\right) A_{2,1}\left( k\right) \in \mathcal{B}\left( L^{2}(\mathbb{T}^{2})\right) . \end{align*} $$

   In this case, $\lambda $ is an eigenvalue of $A(k)$ with associated eigenvector

   $$ \begin{align*} \left( \varphi ,-\left( \mathfrak{b}\left( k\right) -\lambda \right) ^{-1}A_{2,1}\left( k\right) \varphi \right) \in \mathcal{H}\doteq L^{2}\left( \mathbb{T}^{2}\right) \oplus \mathbb{C}. \end{align*} $$

2. ii.) $\mathfrak {b}(k)$ is an eigenvalue of $A(k)$ iff $\hat {\upsilon } (k)=0$ .

Proof. Fix $k\in \mathbb {T}^{2}$ and $h_{b}\in \lbrack 0,1/2]$ . We start with the proof of Assertion (i): If $\lambda \neq \mathfrak {b}(k)$ is an eigenvalue of $A(k)$ with associated eigenvector $(\varphi ,z)\in \mathcal {H}\backslash \{0\}$ , then we directly deduce from (42) that

(76) $$ \begin{align} \left( A_{1,1}\left( k\right) -\lambda \mathbf{1}\right) \varphi +A_{1,2}\left( k\right) z &=0, \end{align} $$

(77) $$ \begin{align} \qquad\! A_{2,1}\left( k\right) \varphi +\left( \mathfrak{b}\left( k\right) -\lambda \right) z &= 0. \end{align} $$

By combining these two equations, we obtain

(78) $$ \begin{align} z=-\left( \mathfrak{b}\left( k\right) -\lambda \right) ^{-1}A_{2,1}\left( k\right) \varphi , \end{align} $$

and thus,

$$ \begin{align*} \left[ A_{1,1}\left( k\right) -\lambda \mathbf{1}-\left( \mathfrak{b} \left( k\right) -\lambda \right) ^{-1}A_{1,2}\left( k\right) A_{2,1}\left( k\right) \right] \varphi =0. \end{align*} $$

We have that $\varphi \neq 0$ , for otherwise z would also be zero, by (78), and this would contradict the fact that $(\varphi ,z)$ is a nonzero vector. The converse is obvious and Assertion (i) holds true.

We now prove Assertion (ii): It is easy to check from (42) that $\hat {\upsilon }(k)=0$ implies that

(79) $$ \begin{align} A\left( k\right) \left( 0,1\right) =\mathfrak{b}\left( k\right) \left( 0,1\right) . \end{align} $$

Conversely, suppose that $\mathfrak {b}(k)$ is an eigenvalue of $A(k)$ with associated eigenvector $(\varphi ,z)\in \mathcal {H}\backslash \{0\}$ , but $ \hat {\upsilon }(k)\neq 0$ . Then, by (39) and (42),

(80) $$ \begin{align} \left( A_{1,1}\left( k\right) -\mathfrak{b}\left( k\right) \mathbf{1} \right) \varphi +A_{1,2}\left( k\right) z=0\qquad \text{and}\qquad A_{2,1}\left( k\right) \varphi \doteq \hat{\upsilon}\left( k\right) \left\langle \mathfrak{d}\left( k\right) ,\varphi \right\rangle =0. \end{align} $$

Remark that the second equality says that $\varphi \,\bot \,\mathfrak {d}(k)$ , since we assume $\hat {\upsilon }(k)\neq 0$ . Considering the scalar product of $\varphi $ with both sides of the first equation, we then get that

(81) $$ \begin{align} \left\langle \varphi ,\left( A_{1,1}\left( k\right) -\mathfrak{b}\left( k\right) \mathbf{1}\right) \varphi \right\rangle +z\hat{\upsilon}\left( k\right) \left\langle \varphi ,\mathfrak{d}\left( k\right) \right\rangle =\left\langle \varphi ,\left( A_{1,1}\left( k\right) -\mathfrak{b}\left( k\right) \mathbf{1}\right) \varphi \right\rangle =0\text{ }; \end{align} $$

see (40). Because $h_{b}\in \lbrack 0,1/2]$ , if $k\neq 0$ , then (75) holds true with a strict inequality, and therefore,

$$ \begin{align*} \mathfrak{b}\left( k\right) <4\epsilon -2\epsilon \cos \left( k/2\right) =\min \sigma \left( A_{1,1}\left( k\right) \right) , \end{align*} $$

thanks to Corollary 4.3. Hence,

$$ \begin{align*} A_{1,1}\left( k\right) -\mathfrak{b}\left( k\right) \mathbf{1}\geq c \mathbf{1}, \end{align*} $$

for some constant $c>0$ , which, combined with (81), in turn implies that $\varphi =0$ . If now $k=0$ , then $\mathfrak {b}(0)=0$ (see (32)) and we obtain from (81) that

(82) $$ \begin{align} {\int_{\mathbb{T}^{2}}}\left\vert \varphi \left( p\right) \right\vert ^{2} \mathfrak{f}\left( 0\right) \left( p\right) \,\nu \left( \mathrm{d}p\right) =\langle \varphi ,M_{\mathfrak{f}\left( 0\right) }\varphi \rangle \leq \langle \varphi ,A_{1,1}\left( 0\right) \varphi \rangle =0, \end{align} $$

since $M_{\mathfrak {f}\left ( 0\right ) }\leq A_{1,1}(\mathrm {U},0)$ (see (37)–(38)). As

$$ \begin{align*} \mathfrak{f}(0)(p)\doteq \epsilon \{4-2\cos (p)\},\qquad p\in \mathbb{T} ^{2}, \end{align*} $$

(see (33) and (35)) defines a positive and continuous function that vanishes at $p=0$ only, one deduces from (82) that $ \varphi =0$ also when $k=0$ . In any case, $\varphi =0$ and so, (80) combined with (40) yields

$$ \begin{align*} A_{1,2}(k)z\doteq \hat{\upsilon}(k)\mathfrak{d}(k)z=0. \end{align*} $$

Since $\mathfrak {d}(k)\neq 0$ and $\hat {\upsilon }(k)\neq 0$ , we must have that $z=0$ .Thus, we arrive at $(\varphi ,z)=(0,0)$ , which contradicts the fact that $(\varphi ,z)$ is a nonzero vector. Therefore, if $\mathfrak {b} (k) $ is an eigenvalue of $A(k)$ , then we must have $\hat {\upsilon }(k)=0$ .

The Birman-Schwinger principle (Theorem A.10) allows us to transform the eigenvalue problem for the fiber Hamiltonian $ A(k) $ into a nonlinear equation on the resolvent set $\rho (A_{1,1}( \mathrm {U},k))$ of the operator $A_{1,1}(\mathrm {U},k)$ , which is the resolvent set for a fermion pair with total quasi-momentum $k\in \mathbb {T} ^{2}$ :

Theorem 4.5 (Characteristic equation for eigenvalues).

Fix $h_{b}\in \lbrack 0,1/2]$ and $k\in \mathbb {T}^{2}$ . Then, ${\lambda \in \rho (A_{1,1}(k))}$ is an eigenvalue of $A(k)$ iff it is a solution to the equation

(83) $$ \begin{align} \hat{\upsilon}\left( k\right) ^{2}\mathfrak{T}\left( k,z\right) +z-\mathfrak{ b}\left( k\right) =0,\qquad z\in \rho \left( A_{1,1}\left( k\right) \right) , \end{align} $$

where $\mathfrak {T}$ is the function defined by (45); that is,

(84) $$ \begin{align} \mathfrak{T}\left( k,z\right) \equiv \mathfrak{T}\left( \mathrm{U} ,k,z\right) \doteq \left\langle \mathfrak{d}\left( k\right) ,\left( A_{1,1}\left( k\right) -z\mathbf{1}\right) ^{-1}\mathfrak{d}\left( k\right) \right\rangle , \end{align} $$

Proof. Fix $h_{b}\in \lbrack 0,1/2]$ and $k\in \mathbb {T}^{2}$ . We divide the proof in several cases:

Case 1: We first consider the case $\hat {\upsilon } (k)=0 $ and $k\neq 0$ . In that situation, $\mathfrak {b}(k)$ is trivially the only solution to (83). However, we already know from Proposition 4.4 (ii) that $\mathfrak {b}(k)$ is an eigenvalue of $A(k)$ . We must therefore prove that there is no other eigenvalue $\lambda $ of $A(k)$ in $\rho (A_{1,1}(k))$ but $\mathfrak {b}(k)$ . In fact, if such a $\lambda \in \rho (A_{1,1}(k))$ exists, then, by Proposition 4.4 (i) with $\hat {\upsilon }(k)=0$ , $ A_{1,1}(k)-\lambda \mathbf {1}$ would have a nontrivial kernel, which is not possible, for $\lambda $ is in the resolvent set of $A_{1,1}(k)$ .

Case 2: Suppose that $k=0$ and $\hat {\upsilon }(0)=0$ . We observe that (83) has no solution because

$$ \begin{align*} 0\in \lbrack 0,8\epsilon ]=\sigma _{\mathrm{ess}}(A_{1,1}(0)), \end{align*} $$

thanks to Proposition 4.2. In addition, by applying Proposition 4.4 (i) and noting that $ \mathfrak {b}(0)=0$ , we see that $A(0)$ has no eigenvalues in $\rho (A_{1,1}(0))$ .

Case 3: Finally, assume that $\hat {\upsilon }(k)\neq 0$ and take $\lambda \in \rho (A_{1,1}(k))$ . Observe from Proposition 4.4 (ii) that $\mathfrak {b}(k)$ cannot be an eigenvalue of $A(k)$ . Additionally, $\mathfrak {b}(k)$ cannot be a solution to Equation ( 83). This last observation is proven as follows: When $k=0$ , this is clear because $\mathfrak {b}(0)=0$ is not even in the domain of the equation to be solved in (83). For $k\neq 0$ , if $\mathfrak { b}(k)$ is a solution to (83), then $\mathfrak {T}(k, \mathfrak {b}(k))=0$ , but we know from Corollary 4.3 and $h_{b}\in \lbrack 0,1/2]$ that

$$ \begin{align*} A_{1,1}(k)-\mathfrak{b}(k)\mathbf{1}\geq c\mathbf{1} \end{align*} $$

for some constant $c>0$ . Therefore, $\mathfrak {T}(k,\mathfrak {b}(k))=0$ would yield $\mathfrak {d}(k)=0$ , which is obviously wrong, by (34). Therefore, in all cases, $\mathfrak {b}(k)$ cannot be a solution to Equation ( 83), and we can assume that $\lambda \neq \mathfrak {b}(k)$ . Now, the remaining part of the proof is essentially the same as the one of [Reference Bru, de Siqueira Pedra and de Pasquale22, Proposition 10], but we reproduce it for completeness. By ( 39)–(40), the orthogonal projection S onto the subspace $ \mathbb {C}\mathfrak {d}(k)\subseteq L^{2}(\mathbb {T}^{2})$ can be written as

$$ \begin{align*} S\varphi =\Vert \mathfrak{d}(k)\Vert ^{-2}\langle \mathfrak{d}(k),\varphi \rangle \mathfrak{d}(k)=\hat{\upsilon}(k)^{-2}\Vert \mathfrak{d}(k)\Vert ^{-2}A_{1,2}(k)A_{2,1}(k)\varphi ,\qquad \varphi \in L^{2}\left( \mathbb{T} ^{2}\right) . \end{align*} $$

Then, observe from Proposition 4.4 (i) that $\lambda $ is an eigenvalue of $A(k)$ iff $\lambda $ is an eigenvalue of $T-V^{2}$ with

(85) $$ \begin{align} V &\doteq \hat{\upsilon}(k)(\mathfrak{b}(k)-\lambda )^{-1/2}\Vert \mathfrak{ d}(k)\Vert S, \end{align} $$

(86) $$ \begin{align} T &\doteq A_{1,1}(k). \end{align} $$

Thus, by applying Theorem A.10, we deduce that $ \lambda $ is an eigenvalue of $A(k)$ iff $1$ is an eigenvalue of the corresponding Birman-Schwinger operator, which, with the above operators T and V, is equal to

$$ \begin{align*} \mathrm{B}(\lambda )=\hat{\upsilon}(k)^{2}(\mathfrak{b}(k)-\lambda )^{-1}\Vert \mathfrak{d}(k)\Vert ^{2}S(A_{1,1}(k)-\lambda \mathbf{1})^{-1}S . \end{align*} $$

Remark in this case that

(87) $$ \begin{align} \mathcal{E}_{\mathrm{B}(\lambda )}(1)=\mathbb{C}\mathfrak{d}(k)\qquad \text{ and}\qquad \dim \mathcal{E}_{T-V^{2}}(\lambda )=\dim \mathcal{E}_{\mathrm{B} (\lambda )}(1)=1, \end{align} $$

since, obviously,

$$ \begin{align*} \mathrm{B}(\lambda )L^{2}\left( \mathbb{T}^{2}\right) \subseteq SL^{2}\left( \mathbb{T}^{2}\right) =\mathbb{C}\mathfrak{d}(k). \end{align*} $$

We thus conclude that $\lambda $ is an eigenvalue of $A(k)$ iff

$$ \begin{align*} \mathrm{B}(\lambda )\mathfrak{d}(k)=\mathfrak{d}(k)& \Leftrightarrow \langle \mathfrak{d}(k),\mathrm{B}(\lambda )\mathfrak{d}(k)-\mathfrak{d}(k)\rangle =0 \\[1em] & \Leftrightarrow \langle \mathfrak{d}(k),\mathrm{B}(\lambda )\mathfrak{d} (k)\rangle =\Vert \mathfrak{d}(k)\Vert ^{2} \\[1em] & \Leftrightarrow \hat{\upsilon}(k)^{2}(\mathfrak{b}(k)-\lambda )^{-1}\Vert \mathfrak{d}(k)\Vert ^{2}\langle \mathfrak{d}(k),S(A_{1,1}(k)-\lambda \mathbf{1})^{-1}S\mathfrak{d}(k)\rangle =\Vert \mathfrak{d}(k)\Vert ^{2} \\[1em] & \Leftrightarrow \hat{\upsilon}(k)^{2}\mathfrak{T}(k,\lambda )=\mathfrak{b} (k)-\lambda . \end{align*} $$

This completes the proof.

Corollary 4.6 (Eigenspaces of fiber Hamiltonians).

Fix $h_{b}\in \lbrack 0,1/2]$ and $k\in \mathbb {T}^{2}$ . If $\lambda \in \rho (A_{1,1}(k))$ is an eigenvalue of the fiber Hamiltonian $A(k)$ , then the associated eigenspace is

$$ \begin{align*} \mathcal{E}_{A\left( k\right) }\left( \lambda \right) =\mathbb{C}g\left( k,\lambda \right) , \end{align*} $$

where

$$ \begin{align*} g\left( k,\lambda \right) \doteq \left( \hat{\upsilon}\left( k\right) \left( A_{1,1}\left( k\right) -\lambda \mathbf{1}\right) ^{-1}\mathfrak{d}\left( k\right) ,-1\right) \in \mathcal{H}. \end{align*} $$

In particular, $\lambda $ is a nondegenerated eigenvalue of $A(k)$ .

Proof. Assume that $\hat {\upsilon }(k)\neq 0$ . Recall from the proof of Theorem 4.5 that in this case, $\mathfrak { b}(k)$ is not an eigenvalue of $A(k)$ , and so we assume without loss of generality that $\lambda \neq \mathfrak {b}(k)$ . In this case, observe that we have (87) and a close look at the proof of Theorem A.10, in particular Lemma A.9, leads us to

$$ \begin{align*} \mathcal{E}_{T-V^{2}}(\lambda )\doteq \ker (T-V^{2}-\lambda \mathbf{1})= \mathbb{C}\varphi _{0}, \end{align*} $$

where

(88) $$ \begin{align} \varphi _{0}\doteq (T-\lambda \mathbf{1})^{-1}V\mathfrak{d}(k)=\hat{ \upsilon}(k)(\mathfrak{b}(k)-\lambda )^{-1/2}\Vert \mathfrak{d}(k)\Vert (A_{1,1}(k)-\lambda \mathbf{1})^{-1}\mathfrak{d}(k), \end{align} $$

by (85) and (86). From Proposition 4.4 (i) ( $\hat {\upsilon }(k)\neq 0$ ), one then obtains that

$$ \begin{align*} \mathcal{E}_{A(k)}(\lambda ) &=\left\{ (\varphi ,-(\mathfrak{b}(k)-\lambda )^{-1}A_{2,1}(k)\varphi )\in \mathcal{H}\,:\,\varphi \in \ker (T-V^{2}-\lambda \mathbf{1})\right\} \\ &=\mathbb{C}\ (\varphi _{0},-(\mathfrak{b}(k)-\lambda )^{-1}A_{2,1}(k)\varphi _{0}). \end{align*} $$

In view of Equation (39), (88) and Theorem 4.5, the last vector can be rewritten as follows:

$$ \begin{align*} (\varphi _{0},-(\mathfrak{b}(k)-\lambda )^{-1}A_{2,1}(k)\varphi _{0})& =(\varphi _{0},-(\mathfrak{b}(k)-\lambda )^{-1}\hat{\upsilon}(k)\langle \mathfrak{d}(k),\varphi _{0}\rangle ) \\ & =(\varphi _{0},-(\mathfrak{b}(k)-\lambda )^{-3/2}\hat{\upsilon} (k)^{2}\Vert \mathfrak{d}(k)\Vert \mathfrak{T}(k,\lambda )) \\ & =(\varphi _{0},-(\mathfrak{b}(k)-\lambda )^{-1/2}\Vert \mathfrak{d} (k)\Vert ) \\ & =(\mathfrak{b}(k)-\lambda )^{-1/2}\Vert \mathfrak{d}(k)\Vert g(k,\lambda ), \end{align*} $$

whenever $\hat {\upsilon }(k)\neq 0$ . Finally, if $\hat {\upsilon }(k)=0$ and $ \lambda \in \rho (A_{1,1}(k))$ is an eigenvalue of $A(k)$ , then it is straightforward to check that $\lambda =\mathfrak {b}(k)$ with eigenspace generated by the vector $(0,1)\in \mathcal {H}$ ; see, for instance, (76)–(77) and (79).

Corollary 4.7 (Eigenvalues of fiber Hamiltonians – II).

Fix $h_{b}\in \lbrack 0,1/2]$ and $k\in \mathbb {T}^{2}$ . There is at most one eigenvalue of $A(k)$ in each connected component of $\rho (A_{1,1}(k))\cap \mathbb {R}$ .

Proof. In view of Theorem 4.5, it suffices to show that the derivative of the mapping

$$ \begin{align*} \rho (A_{1,1}(k))\cap \mathbb{R}\ni x\longmapsto \hat{\upsilon}(k)^{2} \mathfrak{T}(k,x)+x-\mathfrak{b}(k)\in \mathbb{R} \end{align*} $$

is strictly positive. For any $x_{0}\in \rho (A_{1,1}(k))\cap \mathbb {R}$ , we have that

(89) $$ \begin{align}\left. \partial _{x}\left\{ \hat{\upsilon}(k)^{2}\mathfrak{T}(k,x)+x-\mathfrak{b}(k)\right\} \right\vert _{x=x_{0}}=\hat{\upsilon}(k)^{2}\Vert(A_{1,1}(k)-x_{0}\mathbf{1})^{-1}\mathfrak{d}(k)\Vert ^{2}+1>1\ . \end{align} $$

4.3.3 Bottom of the spectrum

As is well-known, physical properties of quantum systems at very low temperatures are essentially determined by the bottom of the spectrum of the corresponding Hamiltonian. In our case, having in mind the application to superconductivity in cuprates, we would like to study the bottom of the spectrum of the Hamiltonian $H\in \mathcal {B}(\mathfrak {H})$ defined by (21). By Proposition 2.1 and Theorem A.3, we thus study the bottom of the spectrum of the fiber Hamiltonian $A(k)$ (42) at fixed total quasi-momentum $ k\in \mathbb {T}^{2}$ , similar to [Reference Bru, de Siqueira Pedra and de Pasquale22, Reference Bru, de Siqueira Pedra and de Pasquale21].

Theorem 4.8 (Bottom of the spectrum of $A(k)$ ).

Fix $h_{b}\in \lbrack 0,1/2]$ . If $k \neq 0$ , then there is exactly one eigenvalue $\mathrm {E}\left ( k\right ) \equiv \mathrm {E}\left ( \mathrm {U} ,k\right )$ of the Hamiltonian $A(k)$ strictly below $\sigma _{\mathrm {ess} }\left ( A\left ( k\right ) \right )$ . In this case, the eigenvalue is nondegenerated and $\mathrm {E}(k) < \mathfrak {b}(k)$ when $\hat {\upsilon } (k) \neq 0$ , whereas $\mathrm {E}(k) = \mathfrak {b}(k)$ if $\hat {\upsilon }(k) = 0$ . This statement remains valid for $k = 0$ provided $\hat {\upsilon }(0) \neq 0$ .

Proof. Assume that $k\neq 0$ . Recall that $\mathfrak {z}(k)$ is defined in Corollary 4.3. From Corollaries 4.3 and 4.7, the interval $(-\infty ,\mathfrak {z}(k))$ contains at most one eigenvalue of $A(k)$ . By Corollary 4.6, the eigenvalue is nondegenerate, if it exists. If $\hat {\upsilon }(k)=0$ , we know from Proposition 4.4 (ii) that $\mathfrak {b}(k)$ is such an eigenvalue. Recall that for $k\neq 0$ , one has that $\mathfrak {b} (k)<\mathfrak {z}(k)$ , because $h_{b}\in \lbrack 0,1/2]$ . Now, suppose that $ \hat {\upsilon }(k)\neq 0$ . When $\mathfrak {b}(k)\leq x<\mathfrak {z}(k)$ , we have

$$ \begin{align*} \left( A_{1,1}\left( k\right) -x\mathbf{1}\right) ^{-1}\geq c\mathbf{1} \end{align*} $$

for some constant $c>0$ , and hence, $\mathfrak {T}(k,x)>0$ ; see (84). Consequently,

$$ \begin{align*} \hat{\upsilon}(k)^{2}\mathfrak{T}(k,x)+x-\mathfrak{b}(k)>0, \end{align*} $$

which means that $A(k)$ has no eigenvalues in the interval $[\mathfrak {b}(k), \mathfrak {z}(k))$ , by Theorem 4.5. We shall now look for an eigenvalue in the interval $(-\infty , \mathfrak {b}(k))$ . On the one hand, using Corollary 4.3, observe that

$$ \begin{align*} \Vert \mathfrak{d}(k)\Vert ^{-2}|\mathfrak{T}(k,x)|\leq \Vert (A_{1,1}(k)-x \mathbf{1})^{-1}\Vert _{\mathrm{o}\mathrm{p}}=(\mathfrak{z}(k)-x)^{-1}\leq (\mathfrak{b}(k)-x)^{-1}, \end{align*} $$

whenever $x<\mathfrak {b}(k)$ . Taking $x\rightarrow -\infty $ , $\mathfrak {T} (k,x)$ tends to zero, and hence,

$$ \begin{align*} {\lim\limits_{x\rightarrow -\infty }}\left\{ \hat{\upsilon}(k)^{2}\mathfrak{T }(k,x)+x-\mathfrak{b}(k)\right\} =-\infty . \end{align*} $$

On the other hand, the continuity of the mapping

$$ \begin{align*} \mathfrak{T}(k,\cdot ):\rho (A_{1,1}(k))\rightarrow \mathbb{R} \end{align*} $$

on $(-\infty ,\mathfrak {b}(k)]$ gives us

$$ \begin{align*} {\lim\limits_{x\rightarrow \mathfrak{b}(k)}}\left\{ \hat{\upsilon}(k)^{2} \mathfrak{T}(k,x)+x-\mathfrak{b}(k)\right\} =\hat{\upsilon}(k)^{2}\mathfrak{T }(k,\mathfrak{b}(k))>0. \end{align*} $$

By the intermediate value theorem, there is $\mathrm {E}(k)\in (-\infty , \mathfrak {b}(k))$ such that

$$ \begin{align*} \hat{\upsilon}(k)^{2}\mathfrak{T}(k,\mathrm{E}(k))+\mathrm{E}(k)-\mathfrak{b} (k)=0. \end{align*} $$

By Theorem 4.5, $\mathrm {E} (k)$ must be an eigenvalue of $A(k)$ .

The proof for $k=0$ is done in a similar way. Basically, the only difference is that, in this case, $\mathfrak {b}(0)=\mathfrak {z}(0)=0$ and

$$ \begin{align*} {\lim\limits_{x\rightarrow 0^{-}}}\left\{ \hat{\upsilon}(0)^{2}\mathfrak{T} (0,x)+x\right\} \in (0,\infty] \end{align*} $$

occurs due to other reasons. Indeed, from Corollary 4.3, we deduce that $\mathfrak {T}(0,\cdot )$ is strictly positive on the interval $(-\infty ,0)$ . Because of (89), we also have that $\partial _x\mathfrak {T}(0,x)|_{x = x_0} \geq 0$ whenever $x_0 < 0$ . Thus, the limit of $\mathfrak {T}(0,x)$ as $x \to 0^-$ exists, being possibly infinite.

If $\hat {\upsilon }(0)=0$ , then, by Theorem 4.5, the fiber Hamiltonian $A(0)$ has no negative eigenvalues. In this case, we set $\mathrm {E}(0)=0$ , which is obviously an eigenvalue of $ A(0)$ with associated eigenvector $(0,1)$ . (Note that $\sigma _{\mathrm {ess} }(A(0))=[0,8\epsilon ]$ , by Proposition 4.2.) With this definition, observe that, for all $k\in \mathbb {T}^{2}$ , $ \mathrm {E}(k)$ is the minimum spectral value of $A(k)$ :

(90) $$ \begin{align} \mathrm{E}\left( k\right) =\min \sigma \left( A\left( k\right) \right) \leq \mathfrak{z}\left( k\right) =\min \sigma _{\mathrm{ess}}\left( A\left( k\right) \right) . \end{align} $$

The lowest eigenvalue $\mathrm {E}(k)$ of $A(k)$ , when $\mathrm {E}(k)<0$ , is related to the formation of dressed bound fermion pairs of fermions with total quasi-momentum $k\in \mathbb {T}^{2}$ . In Theorem 4.20, we make this claim more precise and prove the spatial localization of such bounded pairs. Before doing that, we study the regularity of the real-valued function

$$ \begin{align*} \mathrm{E}\equiv \mathrm{E}\left( \mathrm{U},\cdot \right) :\mathbb{T} ^{2}\rightarrow \mathbb{R} \end{align*} $$

on the two-dimensional torus.

To this end, we rewrite the characteristic equation given by Theorem 4.5 via the function $\Phi : \mathcal {O}\rightarrow \mathbb {R}$ defined by

(91) $$ \begin{align} \Phi \left( k,x\right) \equiv \Phi \left( \mathrm{U},k,x\right) \doteq \hat{ \upsilon}\left( k\right) ^{2}\mathfrak{T}\left( k,x\right) +x-\mathfrak{b} \left( k\right) , \end{align} $$

where $\mathcal {O}$ is the open set

(92) $$ \begin{align} \mathcal{O}\doteq \left\{ \left( k,x\right) \in \mathbb{S}^{2}\times \mathbb{ R}:x<\mathfrak{z}\left( k\right) \right\} \subseteq \mathbb{R}^{3}. \end{align} $$

Observe from Equation (89) that $\partial _{x}\Phi>0$ over the whole domain of $\Phi $ .

We now study the continuity of the function $\mathrm {E}:\mathbb {T} ^{2}\rightarrow \mathbb {R}$ and give a sufficient condition for $\mathrm {E}$ to be of class $C^{d}$ on $\mathbb {S}^{2}$ for every $d\in \mathbb {N}\cup \{\omega ,a\}$ , where $C^{d}(\Omega )$ , $d\in \mathbb {N}$ , stands for the space of $d$ times continuously differentiable functions on $\Omega $ , while $C^{\omega }(\Omega )$ and $C^{a}(\Omega )$ refer to the space of smooth and real analytic functions on $\Omega $ , respectively.

Theorem 4.9 (Regularity of the function $\mathrm {E}$ ).

Let $h_{b}\in \lbrack 0,1/2]$ .

1. i.) The family $\{\mathrm {E}(\mathrm {U},\cdot )\}_{\mathrm {U}\in \mathbb {R}_{0}^{+}}$ of real-valued functions on $\mathbb {T}^{2}$ is equicontinuous with respect to the metricFootnote ¹⁸ $d_{\mathbb {T}^{2}}$ .

2. ii.) If $\hat {\upsilon }\in C^{d}(\mathbb {S}^{2})$ ( $\mathbb {S} ^{2}\subseteq \mathbb {R}^{2}$ ) for some $d\in \mathbb {N}\cup \{\omega ,a\}$ , then $\mathrm {E}\equiv \mathrm {E}(\mathrm {U},\cdot )\in C^{d}(\mathbb {S} ^{2}) $ for all $\mathrm {U}\in \mathbb {R}_{0}^{+}$ . In this case,

   (93) $$ \begin{align} \partial _{k_{j}}\mathrm{E}\left( k\right) =-\left( \partial _{x}\Phi \left( {k,\mathrm{E}\left( k\right) }\right) \right) ^{-1}\partial _{k_{j}}\Phi \left( {k,\mathrm{E}\left( k\right) }\right) ,\qquad k\in \mathbb{S}^{2},\ j\in \{1,2\}. \end{align} $$

Proof. By the spectral theorem, we deduce from (90) that, for any $ \mathrm {U}\in \mathbb {R}_{0}^{+}$ ,

(94) $$ \begin{align} \mathrm{E}\left( \mathrm{U},k\right) =\min \sigma \left( A\left( \mathrm{U} ,k\right) \right) ={\inf_{\psi \in \mathcal{H},\Vert \psi \Vert =1}} \left\langle \psi ,A\left( \mathrm{U},k\right) \psi \right\rangle ,\qquad k\in \mathbb{T}^{2}. \end{align} $$

Given any $\varepsilon>0$ and $k_{0}\in \mathbb {T}^{2}$ , by the (operator norm) continuity of the mapping $A\left ( 0,\cdot \right ) :\mathbb {T} ^{2}\rightarrow \mathcal {B}(\mathcal {H})$ at the point $k_{0}$ , we can find $ \delta>0$ such that

$$ \begin{align*} \sup_{\mathrm{U}\in \mathbb{R}_{0}^{+}}{\sup_{\psi \in \mathcal{H},\Vert \psi \Vert =1}}\left\vert \left\langle \psi ,A\left( \mathrm{U},k\right) \psi \right\rangle -\left\langle \psi ,A\left( \mathrm{U},k_{0}\right) \psi \right\rangle \right\vert & \leq \sup_{\mathrm{U}\in \mathbb{R} _{0}^{+}}\left\Vert A\left( \mathrm{U},k\right) -A\left( \mathrm{U} ,k_{0}\right) \right\Vert _{\mathrm{op}} \\ & =\sup_{\mathrm{U}\in \mathbb{R}_{0}^{+}}\left\Vert A\left( 0,k\right) -A\left( 0,k_{0}\right) \right\Vert _{\mathrm{op}}<\varepsilon \end{align*} $$

for every $k\in \mathbb {T}^{2}$ with $d_{\mathbb {T}^{2}}(k,k_{0})<\delta $ . Recall that $\mathcal {H}$ stands for the (fiber) Hilbert space $L^{2}( \mathbb {T}^{2})\oplus \mathbb {C}$ (see (25)), while $d_{\mathbb {T} ^{2}}$ is the metric (22) on the torus $\mathbb {T}^{2}$ . Therefore, $\mathrm {E}:\mathbb {T}^{2}\rightarrow \mathbb {R}$ can be expressed as the infimum over the equicontinuous family $\{\langle \psi ,A( \mathrm {U},\cdot )\psi \rangle \}_{\mathrm {U}\in \mathbb {R}_{0}^{+},\psi \in \mathcal {H},\Vert \psi \Vert =1}$ of (continuous) functions and Assertion (i) follows.

Take again some $k_{0}\in \mathbb {S}^{2}$ . Assume that $\hat {\upsilon }$ is of class $C^{d}$ on $\mathbb {S}^{2}\subseteq \mathbb {R}^{2}$ with $d\in \mathbb {N}\cup \{\omega ,a\}$ . Let $\vartheta =(k_{0},\mathrm {E}(k_{0}))\in \mathcal {O}$ . Using Theorems 4.5 and 4.8 as well as Equation (89) and the (operator norm) continuity of the mapping

(95) $$ \begin{align} A_{1,1}\left( \cdot \right) :\mathbb{T}^{2}\rightarrow \mathcal{B}\left( L^{2}(\mathbb{T}^{2})\right) , \end{align} $$

one checks that $\Phi \in C^{d}(\mathcal {O})$ with $d\geq 1$ , $\Phi (\vartheta )=0$ and $\partial _{x}\Phi (\vartheta )\neq 0$ . See, for instance, (89). We can thus apply the implicit function theorem (see, for example, [Reference de Oliveira91] for an ordinary version and [Reference Toland92] for an analytic version) to obtain open subsets $ U\subseteq \mathbb {R}^{2}$ and $J\subseteq \mathbb {R}$ such that $\vartheta \in U\times J\subseteq \mathcal {O}$ and, for each $k\in U$ , there is a unique real number $\xi (k)\in J$ satisfying $\Phi (k,\xi (k))=0$ . Moreover, the mapping $\xi :U\rightarrow J$ defined in this way is of class $C^{d}$ and its partial derivatives are given by

(96) $$ \begin{align} {\partial }_{k_{j}}{\xi \left( {k}\right) }=-\left( \partial _{x}\Phi { \left( {k,\xi \left( k\right) }\right) }\right) ^{-1}{{\partial } _{k_{j}}\Phi \left( {k,\xi \left( k\right) }\right) },\qquad k\in U,\ j\in \{1,2\}. \end{align} $$

As $\xi (k_{0})=\mathrm {E}(k_{0})<\mathfrak {z}(k_{0})$ (see (90)), by continuity, there exists a neighborhood $V\subseteq U$ of $k_{0}$ such that $\xi (k)<\mathfrak {z}(k)$ for every $k\in V$ . It follows that, for all $ k\in V$ , $\xi (k)$ and $\mathrm {E}(k)$ are in the same connected component $ (-\infty ,\mathfrak {z}(k))$ , and from $\partial _{x}\Phi>0$ , we conclude that $\mathrm {E}\upharpoonright V=\xi \upharpoonright V$ . So, $\mathrm {E}$ is of class $C^{d}$ near $k_{0}$ and (96) yields (93) for any $k\in V$ – in particular, for $k=k_{0}$ . As $k_{0}$ is arbitrary, Assertion (ii) follows.

We can now deduce from Theorem 4.9 that $\mathrm {E}$ is a dispersion relation (see Definition 3.7) when the function $\hat {\upsilon }:\mathbb {S}^{2}\rightarrow \mathbb {R}$ is at least $ 2 $ times continuously differentiable, and in this case, we can even compute the group velocity. To see this, recall that, for any $f\in C^{2}(\mathbb {S} ^{2})$ , we define in (48) the subset

$$ \begin{align*} \mathfrak{M}_{f}\doteq \left\{ k\in \mathbb{S}^{2}\,:\,\mathrm{Hess}\left( f\right) \left( k\right) \in \mathsf{GL}_{2}\left( \mathbb{R}\right) \right\} \subseteq \mathbb{S}^{2} \end{align*} $$

with $\mathsf {GL}_{2}\left ( \mathbb {R}\right ) $ being the set of invertible $ 2\times 2$ matrices with real coefficients.

Corollary 4.10 ( $\mathrm {E}$ as a dispersion relation and group velocity).

Let $h_{b}\in \lbrack 0,1/2]$ and $\mathrm {U}\in \mathbb {R}_{0}^{+}$ . Then, $ \mathrm {E}\equiv \mathrm {E}(\mathrm {U},\cdot )\in C(\mathbb {T}^{2})$ and is of class $C^{2}$ on the open set $\mathbb {S}^{2}\subseteq \mathbb {R}^{2}$ whenever $\hat {\upsilon }$ is of class $C^{2}$ on $\mathbb {S}^{2}$ . In this case, the corresponding group velocity is

$$ \begin{align*} \mathbf{v}_{\mathrm{E}}\left( k\right) \doteq \vec{\nabla}_{k}\mathrm{E} \left( k\right) =-\left( \partial _{x}\Phi \left( {k,\mathrm{E}\left( k\right) }\right) \right) ^{-1}\vec{\nabla}_{k}\Phi \left( {k,\mathrm{E} \left( k\right) }\right) ,\qquad k\in \mathbb{S}^{2}. \end{align*} $$

Moreover, if $\hat {\upsilon }$ is real analytic (i.e., of class $C^{a}$ in the above terminology) on $\mathbb {S}^{2}$ , then either $\mathfrak {M}_{ \mathrm {E}}$ has full measure or is empty.

Proof. The first part of the assertion is a direct application of Theorem 4.9. It remains to study the set $\mathfrak {M}_{\mathrm {E}}$ . If $\hat {\upsilon }$ is real analytic then, from Equation (47) and Theorem 4.9, the function $f:\mathbb {S} ^{2}\longrightarrow \mathbb {R}$ defined by

$$ \begin{align*} f(k)\doteq \det \left( \mathrm{Hess}{\left( \mathrm{E}\right) \left( k\right) }\right) ,\qquad k\in \mathbb{S}^{2}, \end{align*} $$

is real analytic and satisfies $f^{-1}(\{0\})=\mathbb {S}^{2}\backslash \mathfrak {M}_{\mathrm {E}}$ . Since the zeros of any nonconstant real analytic function have null Lebesgue measure (see, for example, [Reference Mityagin49]), either $\mathfrak {M}_{\mathrm {E}}$ has full measure or is empty.

It is natural to derive now the ground state energy of the Hamiltonian $H\in \mathcal {B}(\mathfrak {H})$ defined by (21), which is related to the ground state energy of fiber Hamiltonians, thanks to Proposition 2.1 and Theorem A.3. As expected, one has the following equality for the ground state energy

$$ \begin{align*} E{\left( \mathrm{U}\right) }\doteq \min \sigma \left( H\right) =\min \mathrm{ E}\left( \mathbb{T}^{2}\right) . \end{align*} $$

A proof can be done like [Reference Bru, de Siqueira Pedra and de Pasquale22, Lemma 8] by using Kato’s perturbation theory. In the sequel, we provide an alternative way of proving the equality, which is much more direct.

Proposition 4.11 (Bottom of the spectrum of H).

We have that

$$ \begin{align*} E{\left( \mathrm{U}\right) }={\min\limits_{k\in \mathbb{T}^{2}}}\,\min \sigma \left( A\left( k\right) \right) =\min \mathrm{E}\left( \mathbb{T} ^{2}\right) \leq 0. \end{align*} $$

Proof. We first remark that the union

$$ \begin{align*} \mathcal{K}\doteq {\bigcup }\left\{ \sigma \left( A\left( k\right) \right) \,:\,k\in \mathbb{T}^{2}\right\} \subseteq \mathbb{R} \end{align*} $$

of the spectra of all fiber Hamiltonians is closed. To see this, let $ (\lambda _{n})_{n\in \mathbb {N}}$ be a sequence of real numbers converging to $\lambda \in \mathbb {R}$ with $\lambda _{n}\in \sigma (A(k_{n}))$ for some $k_{n}\in \mathbb {T}^{2}$ at $n\in \mathbb {N}$ . By compactness of $ \mathbb {T}^{2}$ , we can assume without loss of generality that $ (k_{n})_{n\in \mathbb {N}}$ converges to some point $k_{0}\in \mathbb {T}^{2}$ . By the (operator norm) continuity of the mapping $A:\mathbb {T} ^{2}\rightarrow \mathcal {B}(\mathcal {H})$ , it follows that

$$ \begin{align*} {\lim\limits_{n\rightarrow \infty }}\,(A(k_{n})-\lambda _{n}\mathfrak{1)} =A(k_{0})-\lambda \mathbf{1} \end{align*} $$

(in operator norm). Hence, $\lambda \in \sigma (A(k_{0}))$ , for otherwise $ A(k_{n})-\lambda _{n}\mathbf {1}$ would be invertibleFootnote ¹⁹ for sufficiently large n. Thus, $\mathcal {K}$ is a closed set, and as a consequence, for any $s\notin \mathcal {K}$ , there is $\varepsilon>0$ such that

$$ \begin{align*} \left( s-\varepsilon ,s+\varepsilon \right) \cap \sigma \left( A\left( k\right) \right) =\emptyset ,\qquad k\in \mathbb{T}^{2}. \end{align*} $$

With this property, we infer from Proposition 2.1 and Theorem A.3 that $\sigma (H)\subseteq \mathcal {K}$ , which yields the inequality

(97) $$ \begin{align} E{\left( \mathrm{U}\right) }\geq {\min\limits_{k\in \mathbb{T}^{2}}}\,\min \sigma \left( A\left( k\right) \right) =\min \mathrm{E}\left( \mathbb{T} ^{2}\right) . \end{align} $$

Note that the last equality results from (90). By Theorem 4.9 (i), $\mathrm {E}:\mathbb {T}^{2}\rightarrow \mathbb {R}$ is continuous. From the compactness of the torus $\mathbb {T}^{2}$ and the Weierstrass extreme value theorem, $\mathrm {E}$ has a minimizer in $\mathbb {T }^{2}$ , say $k_{0}\in \mathbb {T}^{2}$ . The continuity of $\mathrm {E}$ at $ k_{0}$ implies that, for every $\varepsilon>0$ , there is $\delta>0$ such that, for all $k\in \mathbb {T}^{2}$ satisfying $d_{\mathbb {T} ^{2}}(k,k_{0})<\delta $ ,

$$ \begin{align*} \mathrm{E}\left( k\right) \in \left( \mathrm{E}\left( k_{0}\right) -\varepsilon ,\mathrm{E}\left( k_{0}\right) +\varepsilon \right) , \end{align*} $$

and as a consequence,

$$ \begin{align*} \nu \left( \left\{ k\in \mathbb{T}^{2}:\sigma \left( A\left( k\right) \right) \cap \left( \mathrm{E}\left( k_{0}\right) -\varepsilon ,\mathrm{E} \left( k_{0}\right) +\varepsilon \right) \neq \emptyset \right\} \right) \geq \nu \left( \mathbb{T}^{2}\cap B_{\delta }\left( k_{0}\right) \right)>0 , \end{align*} $$

where $B_{\delta }(k_{0})$ is the open ball (for the metric $d_{\mathbb {T} ^{2}}$ ) centered at $k_{0}\in \mathbb {T}^{2}$ of radius $\delta \in \mathbb {R }^{+}$ . By Theorem A.3, this implies that $\mathrm {E}(k_{0})\in \sigma (H)$ , which, combined with (97), yields the equalities

$$ \begin{align*} E{\left( \mathrm{U}\right) }={\min\limits_{k\in \mathbb{T}^{2}}}\,\min \sigma \left( A\left( k\right) \right) =\min \mathrm{E}\left( \mathbb{T} ^{2}\right) . \end{align*} $$

Using Equations (75) and (90), we note that

$$ \begin{align*} E{\left( \mathrm{U}\right) }=\min \mathrm{E}\left( \mathbb{T}^{2}\right) \leq {\min\limits_{k\in \mathbb{T}^{2}}\ }\mathfrak{z}(k)=4\epsilon -2\epsilon \max_{k\in \mathbb{T}^{2}}\cos (k/2)=\mathfrak{z}(0)=0.\\[-45pt] \end{align*} $$