The landscape law for tight binding Hamiltonians

In this paper, we consider the discrete tight-binding Schrödinger operator $H=-\Delta+V$ on ${\mathbb{Z}}^{d}$. The traditional approaches to the estimates for the integrated density of states of its continuous analogue in $\mathbb{R}^{d}$ can roughly be split into two groups, both pertaining to the asymptotic regimes. The first one are the results akin to the Weyl law, estimating the asymptotics of the spectrum as the eigenvalue $\mu\to+\infty$ in terms of the volume in the phase space of the set $\bigl{\{}(\xi,x):\,|\xi|^{2}+V(x)\leq\mu\bigr{\}}$, or in terms of the associated counting function according to the Fefferman-Phong uncertainty principle [Fe]. Informally speaking, letting $\mu\to+\infty$ corresponds to considering length scales which tend to zero, and hence such estimates, by design, are not relevant for a tight-binding model on ${\mathbb{Z}}^{d}$. And indeed, deterministic potentials on ${\mathbb{Z}}^{d}$ were typically treated by more ad hoc approaches specific to their structure: periodicity, symmetries, etc, see, e.g., [DLY, HJ]. The second type of results pertains to the case when the potential is random. Then the integrated density of states exhibits the so-called Lifschitz tails, an exponential asymptotic behavior at the edge of spectrum. This regime is rather well-understood in both ${\mathbb{Z}}^{d}$ and $\mathbb{R}^{d}$, but is in essence probabilistic, restricted to disordered potentials and insensitive to their individual features exhibited, for instance, on finite sets.

The present paper introduces another approach. It takes advantage of the landscape function $u$ from [FM, DFM] to build a box-counting somewhat analogous to the Weyl law and the Fefferman-Phong uncertainty principle, but associated to a different potential, the reciprocal of the landscape $1/u$. The use of the landscape in place of the original potential $V$ allows one to work in a non-asymptotic regime, contrary to aforementioned results, and in some sense to bring the ideas behind the original Weyl law to the lattice without restrictions on the potential or the pertinent eigenvalues, for both deterministic and random scenarios. In practice this approach provides a “black box”, in which the landscape, evaluated directly from the original Hamiltonian, yields an accurate approximation for the integrated density of states without any adjustable parameters, for deterministic and random potentials alike – see the numerical experiments in [FM, ADFJM1, ADFJM3]. The present paper addresses the estimates from above and below and makes the first step towards the mathematically rigorous understanding of the precision of the landscape predictions in the aforementioned works.

In order to describe our main results, we introduce some notations. Let $\Lambda=({\mathbb{Z}}/K{\mathbb{Z}})^{d}\cong\{\bar{1},\cdots,\bar{K}\}^{d}$ be an integer torus, where $K\in\mathbb{N}$, $K\geq 3$, and $\bar{k}$, $1\leq k\leq K,$ is the congruence class, modulo $K$. For simplicity, we will omit the bar from $\bar{k}$ when it is clear. Let $V=\{v_{n}\}_{n\in{\Lambda}}\in\ell^{\infty}(\Lambda)$ be a real-valued, non-constant, non-negative potential. We denote by $V_{\max}=\max_{n\in{\Lambda}}v_{n}$ the amplitude of the potential. The tight binding Hamiltonian $H$ is the linear operator on $\mathcal{H}:=\ell^{2}({\Lambda})\cong\mathbb{R}^{K^{d}}$ defined by

where $|n|_{1}:=\sum_{i=1}^{d}|n_{i}|$ is the $1$-norm on $\Lambda$. We may think of $\varphi$ either as a periodic sequence $\varphi_{n}$ indexed by $n\in{\mathbb{Z}}^{d}$ or as a periodic function $\varphi(n)$ on ${\mathbb{Z}}^{d}$. We are interested in the normalized integrated density of states of $H$, i.e., the eigenvalue counting function per unit volume:

In 2012, a new concept called the localization landscape was introduced in [FM]. Given an operator $H$ as above, the discrete localization landscape function is the unique solution $u=\{u_{n}\}_{n\in\Lambda}\in\mathcal{H}$ to the equation $(Hu)_{n}=1$. When $V=0$ and $u$ vanishes on the boundary, the landscape is simply the torsion function of the Dirichlet Laplacian.

Next, let us define the landscape box counting function $N_{u}$. We start by defining, for any positive integer $s$, a partition ${\mathcal{P}}(s)$ of the set $\{1,\cdots,K\}^{d}$ into subsets which are mostly boxes of side length $s$, as follows. Writing $K=qs+r$ (where the quotient $q$ and the remainder $r$ are non-negative integers and $r<s$), we define a partition ${\mathcal{P}}_{1}(s)$ of the set $\{1,\cdots,K\}$ into $q$ subsets of $s$ consecutive elements, and, if $r>0$, one additional subset of cardinality $r$. The partition ${\mathcal{P}}(s)$ then consists of the boxes defined by the Cartesian products of $d$ subsets from ${\mathcal{P}}_{1}(s)$, see Figure 1.

For given $\mu>0$, we then set $s(\mu)=\left\lceil\mu^{-1/2}\right\rceil$, and define $N_{u}(\mu)$ as the number of boxes on which the minimum of $1/u_{n}$ does not exceed $\mu$, normalized by the size of the set $\Lambda$:

Our goal is to estimate the integrated density of states $N$ in terms of the landscape box counting function $N_{u}$. To this end, we will establish several estimates, stated below as Theorems 1, 2, and 3, which we collectively refer to as the Landscape Law.

The first result, which will be proven in Section 3.1, shows that, after a proper scaling, $N_{u}$ provides an upper bound for $N(\mu)$ over the whole range of $\mu$.

The next theorem, proved in Section 3.2, contains the key estimate for obtaining a lower bound for $N(\mu)$.

In order to get a positive lower bound on $N(\mu)$, we will remove the negative correction terms on the right-hand side of (1.5) and (1.6) through several complementary mechanisms. We roughly divide potentials into two regimes, corresponding to potentials satisfying a certain scaling condition and to certain disordered potentials. Note that these regimes could overlap and do not between them cover all potentials.

In the tight-binding model, the Hilbert space is taken as the space of sequences $\ell^{2}({\mathbb{Z}}^{d})=\left\{\{\phi_{i}\}_{i\in{\mathbb{Z}}^{d}}\,|\,\sum_{i\in{\mathbb{Z}}^{d}}|\phi_{i}|^{2}<\infty\right\}$ where we may think of $\phi=\{\phi_{i}\}_{i\in{\mathbb{Z}}^{d}}$ either as a function $\phi=\phi(i)$ on ${\mathbb{Z}}^{d}$ or as a sequence $\{\phi_{i}\}$ indexed by $i\in{\mathbb{Z}}^{d}$. The ${\mathbb{Z}}^{d}$ lattice is equipped with the $1$-norm:

which reflects the graph structure of ${\mathbb{Z}}^{d}$. We will also frequently need the infinity (maximum) norm

Two vertices $m=(m_{1},\cdots,m_{d}),n=(n_{1},\cdots,n_{d})\in{\mathbb{Z}}^{d}$ are called the nearest neighbors if $|m-n|_{1}=1$. We also say that nearest neighbors $m,n$ are connected by an edge of the discrete graph ${\mathbb{Z}}^{d}$. We denote by $e_{i}=(0,\cdots,0,1,0,\cdots,0),\,i=1,\cdots,d,$ the elements of the canonical basis of ${\mathbb{Z}}^{d}$. For $\phi=\{\phi_{n}\}_{n\in{\mathbb{Z}}^{d}}\in\mathbb{R}^{{\mathbb{Z}}^{d}}$, its $i$-th directional (forward) difference $\nabla_{i}\phi:\,{\mathbb{Z}}^{d}\to\mathbb{R}$ is defined as

and its gradient $\nabla\phi:{\mathbb{Z}}^{d}\to\mathbb{R}^{d}$ is

We also denote the dot product and the induced norm of the resulting vectors by $(\nabla g\cdot\nabla f)(n)=\sum_{i=1}^{d}\nabla_{i}g_{n}\cdot\nabla_{i}f_{n}$, and $|\nabla f|(n):=\sqrt{(\nabla f\cdot\nabla f)(n)}.$ The discrete (graph) Laplacian $\Delta$ on ${\mathbb{Z}}^{d}$ is defined as usual, acting on $\phi=\{\phi_{n}\}_{n\in{\mathbb{Z}}^{d}}$, via

For a real sequence $\{v_{n}\}_{n\in{\mathbb{Z}}^{d}}$ on ${\mathbb{Z}}^{d}$, the potential $V$ is a multiplication operator acting on $\phi\in\ell^{2}({\mathbb{Z}}^{d})$ as $(V\phi)_{n}=v_{n}\phi_{n}$. The operator $-\Delta+V$ is called the discrete Schrödinger operator on ${\mathbb{Z}}^{d}$. If one takes $v_{n}=v_{n}(\omega)$ as independent, identically distributed random variables (in some probability space), the random operator $-\Delta+V(\omega)$ is usually referred to as the Anderson model. We refer readers to [AW, Ki] for more details and a complete introduction to tight-binding Hamiltonians and the Anderson model.

Throughout the rest of the paper, we consider the discrete Schrödinger operator $-\Delta+V$ restricted to a finite domain in ${\mathbb{Z}}^{d}$. Let $\Lambda=({\mathbb{Z}}/K{\mathbb{Z}})^{d}\cong\{\bar{1},\bar{2},\cdots,\bar{K}\}^{d}$, where $K\in\mathbb{N}$ and $\bar{k}$, $1\leq k\leq K,$ is the congruence class, modulo $K$. For simplicity, we often treat $\Lambda$ as a subset of ${\mathbb{Z}}^{d}$. Slightly abusing the notation, we denote by $|\cdot|_{1}$ the induced 1-norm of ${\mathbb{Z}}^{d}$ on the congruence class $\Lambda$, where, for example, we consider two points $(1,n_{2},\cdots,n_{d})$ and $(K,n_{2},\cdots,n_{d})$ to be nearest neighbors and to have distance one from each other in $\Lambda$. From now on, we will concentrate on the finite dimensional subspace $\ell^{2}({\Lambda})$ of $\ell^{2}({\mathbb{Z}}^{d})$. We frequently write $\mathcal{H}:=\ell^{2}({\Lambda})\cong\mathbb{R}^{K^{d}}$ for simplicity. The linear space $\mathcal{H}$ is equipped with the usual inner product on $\mathbb{R}^{K^{d}}$, which is denoted by $\left\langle\cdot,\,\cdot\right\rangle=\left\langle\cdot,\,\cdot\right\rangle_{\mathcal{H}}$. It is easy to check that for $\phi\in\mathcal{H}$,

which specifies the periodicity of $\phi$.

We assume that $v_{n}\geq 0$ is real valued and non-constant, and that $\min_{\Lambda}v_{n}\geq 0$. We set $V_{\max}:=\max_{\Lambda}v_{n}>0$ and let $H=H_{\Lambda}$ be the restriction of $-\Delta+V$ to $\mathcal{H}$:

Similar to the continuous case, the discrete Hamiltonian can be written in its Dirichlet form on the periodic lattice $\Lambda$:

where $\|\nabla\phi_{n}\|^{2}:=\sum_{i=1}^{d}|\nabla_{i}\phi_{n}|^{2}$.

It is easy to check that all eigenvalues of $H$ in (2.6) are contained in $[0,4d+V_{\max}]$ for any finite $K$. For the Anderson model $H_{\infty}=-\Delta+V(\omega)$ acting on the entire space $\ell^{2}({\mathbb{Z}}^{d})$, it is well known that the spectrum $\sigma(H_{\infty})$ is (almost surely) the non-random set $[0,4d]+{\rm supp}\,V\subset[0,4d+V_{\max}]$.

A linear operator acting a finite dimensional space may be viewed as a matrix. For example, $H_{\Lambda}=-\Delta+V$ acting on $\Lambda={\mathbb{Z}}/K{\mathbb{Z}}$ in (2.6) may be identified with the sum of the two $K\times K$ matrices,

It is easy to verify that $H$ is invertible. Moreover, by the maximum principle (see Lemma A.2), all the matrix elements of its inverse are positive, $H^{-1}(i,j)>0$ for all $i,j\in\Lambda$. Therefore, there is a unique positive vector $u\in\ell^{2}(\Lambda)$ solving the equation $(Hu)_{n}=1,n\in\Lambda$. The equation will be referred to as the landscape equation and the solution $u=\{u_{n}\}_{n\in{\Lambda}}$, will be called the landscape function. The function $u$ thus defined is the discrete analogue of the landscape function in [FM] in the continuum setting. The discrete landscape function was first introduced in [LMF], for a one dimensional lattice with zero boundary conditions. It was studied on a higher dimensional lattice with periodic boundary conditions in [WZ]. The following result can be found in [WZ].

As shown in [ADFJM2] for the continuous case and in [WZ] for the discrete case, $1/u:=\{1/u_{n}\}_{n\in\Lambda}$ serves as an effective potential via the following landscape uncertainty principle:

Let us introduce a few more notations. For $a,b\in{\mathbb{Z}},a\leq b$, we denote by $\llbracket a,b\rrbracket=\{a,a+1,\cdots,b\}$ consecutive integers from $a$ to $b$. We will frequently work with cubes in ${\mathbb{Z}}^{d}$, and their images in $\Lambda=({\mathbb{Z}}/K{\mathbb{Z}})^{d}$. For $r\in\mathbb{N},Q=\llbracket 1,r\rrbracket^{d}$, we say that $Q$ is a cube in ${\mathbb{Z}}^{d}$ of side length $\ell(Q)=r$, which is the cardinality of $Q$ projected in each direction. We denote by $|Q|={\rm Card}\left(Q\right)$ the total cardinality of $Q$ and call it the volume of $Q$ when it is clear. For any $a\in{\mathbb{Z}}^{d}$, $a+Q$ is the translation of $Q$ in ${\mathbb{Z}}^{d}$, respectively having the same side length and volume. For a cube $Q$, we denote by $3Q$ the cube concentric with $Q$ of side length $3\ell(Q)$

see Figure 3.

Let $\partial Q$ be the (inner) boundary of $Q$:

and let $\partial^{\circ}Q\subset\partial Q$ be the flat part of the boundary, removing all the corners:

For an integer interval $I=\llbracket a,a+r-1\rrbracket$ of side length $r\geq 3$, we denote by $I/3:=\llbracket a+\left\lceil r/3\right\rceil,a+\left\lceil r/3\right\rceil+\left\lfloor r/3\right\rfloor-1\rrbracket$ the middle third interval of $I$. For a cube $Q=I_{1}\times I_{2}\times\cdots\times I_{d},I_{i}=\llbracket a_{i},a_{i}+r-1\rrbracket,i=1\cdots,d$, we denote by $Q/3$ the middle third cube of $Q$, defined as:

It is easy to verify that $Q/3$ is the “thin” middle third part of $Q$ in the sense that $\ell(Q/3)=\left\lfloor\ell(Q)/3\right\rfloor\leq\ell(Q)/3$ and $3(Q/3)\subseteq Q$. The relation becomes $3(Q/3)=Q$ if $3\mid\ell(Q)$.

In this section, we study Theorems 1, 2, and 3. Let us recall some of the notations first. Let $\Lambda\cong\llbracket 1,K\rrbracket^{d}$ be the periodic domain of side length $K$. Let $N(\mu)$ be the (finite volume) integrated density of states (IDS) of $H$ on $\Lambda$, as defined in (1.2).

Let $u=\{u_{n}\}$ be the landscape function of $H$ defined in the introduction. For $\mu>0$, let $s(\mu)=\left\lceil\mu^{-1/2}\right\rceil$ and let $N_{u}(\mu)$ be the landscape box counting function, defined with respect to the partition ${\mathcal{P}}={\mathcal{P}}(s(\mu);\Lambda)$ as in (1.3), see Figures 4, 5 for examples on ${\mathbb{Z}}^{1},{\mathbb{Z}}^{2}$.