Eigenvalue Asymptotics near a flat band in presence of a slowly decaying potential

We denote by $\mathcal{X}=(\mathcal{V},\mathcal{A})$ the standard graph structure in $\mathbb{Z}^{n}$. That is, the set of vertices $\mathcal{V}$ consists of points $\mu\in\mathbb{Z}^{n}$ and the set of oriented edges $\mathcal{A}$ is composed of pairs $(\mu,\nu)$ such that $\nu=\mu\pm\delta_{j}$, where $\{\delta_{j}\}_{j=1}^{n}$ denotes the canonical basis of $\mathbb{Z}^{n}$. An edge in $\mathcal{A}$ is written $\mathrm{e}=(\mu,\nu)$ and its transpose $\overline{\mathrm{e}}:=(\nu,\mu)$. Let us consider the vector spaces of $0-$cochains $C^{0}(\mathcal{X})$ and $1-$cochains $C^{1}(\mathcal{X})$ given by

$$C^{0}(\mathcal{X}):=\{f:\mathcal{V}\to\mathbb{C}\}\text{ ; }\qquad C^{1}(\mathcal{X}):=\{g:\mathcal{A}\to\mathbb{C}\mid g(\mathrm{e})=-g(\overline{\mathrm{e}})\}.$$

The Hilbert spaces $\ell^{2}(\mathcal{V})$ and $\ell^{2}(\mathcal{A})$ are naturally defined by the inner products of cochains: $\langle f_{1},f_{2}\rangle_{0}=\sum_{\mu\in\mathcal{V}}f_{1}(\mu)\overline{f_{2}(\mu)}$ and $\langle g_{1},g_{2}\rangle_{1}=\frac{1}{2}\sum_{e\in\mathcal{A}}g_{1}(e)\overline{g_{2}(e)}$, respectively.

The coboundary operator $d:\ell^{2}(\mathcal{V})\to\ell^{2}(\mathcal{A})$ is defined by

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$$df(\mathrm{e}):=f(\nu)-f(\mu),\quad\text{ for }\mathrm{e}=(\mu,\nu)\in\mathcal{A}\ .$$

This is the discrete version of the exterior derivative and its adjoint $d^{*}:\ell^{2}(\mathcal{A})\to\ell^{2}(\mathcal{V})$ is given at each edge by the finite sum

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$$d^{*}g(\mu)=\sum_{j=1}^{n}g(\mu,\mu\pm\delta_{j}),\quad\text{ for }\mu\in\mathcal{V}.$$

Let us define the Hilbert space $\ell^{2}(\mathcal{X})=\ell^{2}(\mathcal{V})\oplus\ell^{2}(\mathcal{A})$ and denote by $P_{\mathcal{V}}$ and $P_{\mathcal{A}}$ the corresponding projections. Further, we introduce the involution $\tau$ on $\ell^{2}(\mathcal{X})$ by

$$\tau(f,g)=(f,-g)\ .$$

Then, for a strictly positive constant $m$ let us consider the free Dirac operator

$$H_{0}=d+d^{*}+m\tau=\begin{pmatrix}0&d^{*}\\ d&0\end{pmatrix}+m\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}=\begin{pmatrix}m&d^{*}\\ d&-m\end{pmatrix}$$

where we have slightly abused notation by considering $d$ and $d^{*}$ acting on $\ell^{2}(\mathcal{X})$. Note that $H_{0}$ is a Dirac-type operator in the sense that

$$H_{0}^{2}=\begin{pmatrix}\Delta_{0}+m^{2}&0\\ 0&\Delta_{1}+m^{2}\end{pmatrix}$$

where $\Delta_{0}$ is the Laplacian on vertices and $\Delta_{1}$ is the (1-down) Laplacian on edges.

Let us denote by $\mathcal{H}:=L^{2}(\mathbb{T}^{n},\mathbb{C}^{n+1})$. In this section we construct a unitary operator $\mathscr{U}:l^{2}(\mathcal{X})\to\mathcal{H}$. Consider the action of $\mathbb{Z}^{n}$ on $\mathcal{X}$ given for $\mu\in\mathbb{Z}^{n}$, $x\in\mathcal{V}$, and $\mathrm{e}=(x,y)\in A$ by

$$\mu x=\mu+x\text{ and }\mu\mathrm{e}:=(\mu+x,\mu+y)\ .$$

Then, a natural class of representatives of the orbits of such action is given by $\mathbf{0}\in\mathcal{V}$ together with the edges $\mathrm{e}_{j}=(\mathbf{0},\delta_{j})$ and $\mathrm{e}_{j}^{-}=(\mathbf{0},-\delta_{j})$.

Let us denote $\mathbb{T}^{n}=\mathbb{R}^{n}/[0,1]^{n}$ and set $C_{c}(\mathcal{X})$ to be the set of cochains with compact support, i.e. , $f\in C_{c}(\mathcal{X})$ if and only if it vanishes except for a finite number of vertices and edges. We define $\mathscr{U}:C_{c}(\mathcal{X})\to L^{2}(\mathbb{T}^{n},\mathbb{C}^{n+1})$ by setting, for $f\in C_{c}(\mathcal{X})$ and $\xi\in\mathbb{T}^{n}$,

$$(\mathscr{U}f)(\xi)=\left(\sum_{\mu\in\mathbb{Z}^{n}}e^{-2\pi i\xi\cdot\mu}f(\mu),\sum_{\mu\in\mathbb{Z}^{n}}e^{-2\pi i\xi\cdot\mu}f(\mu\mathrm{e}_{1}),\ldots,\sum_{\mu\in\mathbb{Z}^{n}}e^{-2\pi i\xi\cdot\mu}f(\mu\mathrm{e}_{n})\right).$$

Then $\mathscr{U}$ extends to a unitary operator, still denoted by $\mathscr{U}$, from $\ell^{2}(\mathcal{X})$ to $\mathcal{H}$. Further, we set $\mathcal{H}_{\mathcal{V}}:=\mathscr{U}P_{\mathcal{V}}(\ell^{2}(\mathcal{X}))\cong L^{2}(\mathbb{T}^{n})$ and $\mathcal{H}_{\mathcal{A}}:=\mathscr{U}P_{\mathcal{A}}(\ell^{2}(\mathcal{X}))\cong L^{2}(\mathbb{T}^{n},\mathbb{C}^{n})$.

We draw the reader’s attention to the fact that this definition of $\mathscr{U}$ correspond to the following choice of the Fourier transform in $\mathbb{Z}^{n}$:

$$\mathcal{F}:l^{2}(\mathbb{Z}^{n})\to L^{2}(\mathbb{T}^{n})\quad;\quad(\mathcal{F}f)(\xi):=\sum_{\mu\in\mathbb{Z}^{n}}e^{-2\pi i\xi\cdot\mu}f(\mu)\ .$$

Finally, let us define the functions

$$a_{j}(\xi):=-1+e^{-2\pi i\xi_{j}}\ .$$

The following Section shows that through conjugation by $\mathscr{U}$, the operator $H_{0}$ becomes a multiplication operator, enabling the study of its spectral properties through the examination of characteristics of its band functions.

The band functions of $h_{0}$ have an explicit expression so we are able to compute $\sigma(H_{0})=\bigcup_{j}\lambda_{j}(\mathbb{T}^{n})$. Indeed, from Eq.˜5 one can see that for $\xi\in\mathbb{T}^{n}$ the characteristic polynomial associated to $h_{0}(\xi)$ is given by

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$$p_{z}(\xi)=(-1)^{n}(m+z)^{n-1}\Big{(}m^{2}-z^{2}+\sum_{j=1}^{n}|a_{i}(\xi)|^{2}\Big{)}\ .$$

For convenience we define $r:\mathbb{T}^{n}\to\mathbb{R}^{+}$ and $r_{i}:\mathbb{T}^{n}\to\mathbb{R}^{+}$ for $i\in\{1,\dots n\}$ by

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$$r(\xi):=\sum_{j=1}^{n}|a_{j}(\xi)|^{2}\text{ and }r_{i}(\xi)=r(\xi)-|a_{i}(\xi)|^{2}=\sum_{j\neq i}|a_{j}|^{2}\ .$$

Thus, there are three band functions:

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$$z_{0}(\xi)=-m\ ,\quad z_{\pm}(\xi)=\pm\sqrt{m^{2}+r(\xi)}.$$

From the identities

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$$|a_{j}(\xi)|^{2}=2(1-\cos(2\pi\xi_{j}))=4\sin^{2}(\pi\xi_{j})$$

we easily see that the spectrum of $H_{0}$ satisfies Eq.˜1. Moreover, as is shown in Figure˜1, we can observe that the threshold $-m$ correspond to both the maximum of $z_{-}$ and the constant value of the flat band. Note from Equations˜9 and 8 that $z_{-}$ attains its maximum only at $\mathbf{0}\in\mathbb{T}^{n}$ for every $n$ and hence Figure˜1 is generic.

From Equation˜5 and for $z\notin\sigma(H_{0})$ on can check that

$$(h_{0}-z)^{-1}=\frac{(-1)^{n}}{p_{z}}\times\\ \begin{pmatrix}(z+m)^{n}&a_{1}(z+m)^{n-1}&\cdots&a_{n}(z+m)^{n-1}\\[5.0pt] \overline{a_{1}}(z+m)^{n-1}&(z+m)^{n-2}(z^{2}-m^{2}-r_{1})&\cdots&a_{n}\overline{a_{1}}(z+m)^{n-2}\\[5.0pt] \vdots&\vdots&\ddots&\vdots\\ \overline{a_{n}}(z+m)^{n-1}&a_{1}\overline{a_{n}}(z+m)^{n-2}&\cdots&(z+m)^{n-2}(z^{2}-m^{2}-r_{n})\end{pmatrix}$$

from where one can obtain

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$$(h_{0}-z)^{-1}\hskip-2.0pt=\frac{1}{(m+z)(m^{2}-z^{2}+r)}\begin{pmatrix}(z+m)^{2}&a_{1}(z+m)&\cdots&a_{n}(z+m)\\[5.0pt] \overline{a_{1}}(z+m)&z^{2}-m^{2}-r_{1}&\cdots&a_{n}\overline{a_{1}}\\[5.0pt] \vdots&\vdots&\ddots&\vdots\\ \overline{a_{n}}(z+m)&a_{1}\overline{a_{n}}&\cdots&z^{2}-m^{2}-r_{n}\end{pmatrix}\ .$$

Notice that from the particular form of $h_{0}(\xi)+m$, that can be obtained directly from Equation˜5, we can check that $\operatorname{Ker}(h_{0}+m)\leq\mathcal{H}_{\mathcal{A}}$. Indeed, one can prove directly that $\operatorname{Ker}(H_{0}+m)\leq\{\mathbf{0}\}\times\ell^{2}(A)$ by constructing for each $\mu\in\mathbb{Z}^{n}$ a closed path over which we define a cochain alternating the values $1$ and $-1$, see [MPR23, Sec. 2] for an explicit construction for the $\mathbb{Z}^{2}$ case. We stress that the flat bands of discrete periodic graphs are known to generate finitely supported eigenfunctions [Kuc91].

Given the Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, we denote by $\mathfrak{S}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2})$ the class of compact operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. When $\mathcal{H}_{1}=\mathcal{H}_{2}=\mathcal{H}$ we will just write $\mathfrak{S}_{\infty}(\mathcal{H})$. For $K=K^{*}\in\mathfrak{S}_{\infty}(\mathcal{H})$ and $s>0$ we set

$$n_{\pm}(s;K):=\operatorname{Rank}\mathds{1}_{(s,\infty)}(\pm K)\ .$$

Thus, the functions $n_{\pm}(\cdot;K)$ are respectively the counting functions of the positive and negative eigenvalues of the operator $K$. For $K\in\mathfrak{S}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2})$ we define

$$n_{*}(s;K):=n_{+}(s^{2};K^{*}K),\quad s>0;$$

thus $n_{*}(\cdot;K)$ is the counting function of the singular values of $K$ which, when ordered non–increasingly, we denote by $\{s_{j}(K)\}$. Let $K_{j}$, $j=1,2$ be self-adjoint compact operators. For $s_{1},s_{2}>0$, we have the Weyl inequalities (see e.g. [BS87, Theorem 9.2.9])

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$$n_{\pm}(s_{1}+s_{2};K_{1}+K_{2})\leq n_{\pm}(s_{1};K_{1})+n_{\pm}(s_{2};K_{2})\ .$$

If instead we only have $\{K_{1},K_{2}\}\subset\mathfrak{S}_{\infty}(\mathcal{H}_{1},\mathcal{H}_{2})$ the Ky Fan inequality (see e.g. [BS87, Subsection 11.1.3]) gives

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$$n_{*}(s_{1}+s_{2};K_{1}+K_{2})\leq n_{*}(s_{1};K_{1})+n_{*}(s_{2};K_{2})\ .$$

Further, for $0<p<\infty$ we define the class of compact operators $\mathfrak{S}_{p,w}$ by

$$\mathfrak{S}_{p,w}:=\{K\in\mathfrak{S}_{\infty}:s_{j}(K)=O(j^{-1/p})\}\ ,$$

together with the quasi-norm

$$||K||_{p,w}:=\sup_{j}\{j^{1/p}s_{j}(K)\}=\left(\sup_{s>0}\{s^{p}n_{*}(s;K)\right)^{1/p}$$

that satisfies the "weakened triangle inequality"

$$||K_{1}+K_{2}||_{p,w}\leq 2^{1/p}(||K||_{p,w}+||K||_{p,w})$$

and the "weakened Hölder inequality"

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$$||K_{1}K_{2}||_{r,w}\leq c(p,q)||K_{1}||_{p,w}||K_{2}||_{q,w}\ ,$$

for $r^{-1}=p^{-1}+q^{-1}$ and $c(p,q)=(p/r)^{1/p}(q/r)^{1/q}$ (see [BS87, Chapter 11]).

Finally, consider the set $l_{p,w}$ of functions $v:\mathbb{Z}^{n}\to\mathbb{C}$ such that

$$\#\{\mu\in\mathbb{Z}^{n}:|v(\mu)|>\lambda\}=O(\lambda^{-p})\ .$$

Let us finish this section by considering the following result, which is a particular case of [BKS91, Theorem 4.8(ii)]

In this section we will use the notation

$$\mathcal{N}((a,b);T):=\operatorname{Rank}\mathds{1}_{(a,b)}(T),$$

where $a<b$ and $T$ is a self-adjoint operator without essential spectrum in $(a,b)$. Following the approach coming from the study of magnetic Schrödinger operators, our aim is to study $PVP$ where $P$ stands for the projection on the flat band, i.e.,

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$$P:=\mathds{1}_{\{-m\}}(H_{0})\ .$$

Now, set $P^{\perp}:=I-P$ and for $\kappa>0$ define

$$H_{\kappa}^{\pm}:=H_{0}+P(V\pm\kappa|V|)P+P^{\perp}(V\pm\kappa^{-1}|V|)P^{\perp}$$

Then, by [PR11, Lemma 4.2]

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$$H_{\kappa}^{-}\leq H\leq H_{\kappa}^{+}.$$

Then, arguing as in the proof of [PR11, Theorem 4.1(ii)] we obtain that:

	$\displaystyle\pm\mathcal{N}(\lambda)\leq$	$\displaystyle\pm\mathcal{N}((-m+\lambda,0);-mP+P(V\pm\kappa|V|)P)$
(23)			$\displaystyle\pm\mathcal{N}((-m+\lambda,0);P^{\perp}(H_{0}+(V\pm\kappa^{-1}|V|))P^{\perp})+O(1).$

In the next Lemma we treat the second term on the right of Equation˜23, showing that the perturbation $V$ interacts with the complement of the degenerated eigenspace only at a lesser order.

From Eqs.˜23 and 4.3

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$$\pm\mathcal{N}(\lambda)\leq\pm\mathcal{N}((\lambda,m);P(V\pm\kappa|V|)P)+o(\lambda^{-n/\gamma}),\quad\lambda\downarrow 0\ .$$

Then, we are led to study the distribution of positives eigenvalues of the compact operator $P(V\pm\kappa|V|)P$.

For ease of notation, for any $\kappa>0$ we define $T_{\kappa}^{\pm}$ in $\mathfrak{S}_{\infty}(\mathcal{H})$ by

$$T^{\pm}_{\kappa}:=\mathscr{U}P(V\pm\kappa|V|)P\mathscr{U}^{*}=M\mathscr{U}(V\pm\kappa|V|)\mathscr{U}^{*}M\ .$$